IntroductionIn semi-Lagrangian and semi-implicit (SLSI) modelsof the atmosphere governed by the primitive equa-tions, a semi-Lagrangian time discretisation of theadvective processes combined with a semi-implicittime discretisation of the gravity wave forcing leads

to a numerical algorithm that is (nearly)* uncondition-ally stable and accurate. This allows the use of rela-tively large time steps and thereby achieves a signifi-cant reduction in computational time compared to thecorresponding Eulerian models. The pioneering stud-ies of Dr André Robert in the development of SLSImethods and subsequent application of such methods

Aust. Met. Mag. 49 (2000) 293-317

A semi-Lagrangianand semi-implicit scheme on anunstaggered horizontal grid

Sajal K. Kar and Les W. LoganBureau of Meteorology Research Centre, Melbourne, Australia

(Manuscript received December 1999; revised May 2000)

A two time-level, two-dimensional semi-Lagrangian and semi-implicit (SLSI) scheme based on an unstaggered horizontal gridis presented for limited-area grid-point models of the atmos-phere. A Lorenz grid is used for the vertical discretisation. Anisothermal reference atmosphere with surface topography isused for the semi-implicit linearisation of the governing moist,diabatic, hydrostatic primitive equations. The momentumequation is discretised in vector form. Semi-implicit treatmentof the gravity wave terms and linear part of the Coriolis forceterms followed by a vertical decoupling transformation, lead toa set of two-dimensional separable elliptic equations that aresolved by an efficient iterative solver. The main original contri-bution of this paper is in the use of the unstaggered horizontalA grid to develop the 2D SLSI scheme.The scheme is implemented in the regional NWPmodel LAPS

(Limited Area Prediction System) of the Australian Bureau ofMeteorology. The currently operational LAPS is an Eulerianmodel with 19 σσ-levels in the vertical and a uniform horizontalresolution of 0.75°. While the fully explicit Eulerian modelemploys a time step of 90 s for computational stability, the SLSImodel can use a time step up to 30 min. Results from 48 hourforecasts using the two models show that the newly developedSLSI model provides a stable, accurate, and computationallyefficient alternative to the Eulerian model.

Corresponding author address: Dr. Sajal K. Kar, Bureau ofMeteorology Research Centre, GPO Box 1289K, Melbourne, Vic.3001, Australia.E-mail: S.Kar@bom.gov.au

* The ‘(nearly)’ refers to the stability condition still remaining due tothe Coriolis terms in models that treat it explicitly.

293

in designing atmospheric numerical models have beenduly recognised by many authors in a memorial vol-ume (Lin et al. 1997). Since the first application of theSLSI methods in a hydrostatic primitive equationmodel by Robert et al. (1985), many authors haveapplied this method to essentially reformulate theiralready existing Eulerian models. For example,McDonald and Haugen (1992, hereafter MH) devel-oped a two time-level SLSI limited-area grid-pointmodel, Bates et al. (1993, hereafter BMH) developeda two time-level SLSI global grid-point model, andRitchie (1991) developed a three time-level SLSIglobal spectral model of the atmosphere. Not surpris-ingly, many leading operational centres of the worldcurrently use SLSI models for routine forecasts. Forexample, the Irish Meteorological Service uses a twotime-level SLSI limited-area grid-point model devel-oped initially by McDonald (1986), the CanadianMeteorological Centre (CMC) employs a two time-level semi-Lagrangian implicit global finite-elementmodel (Côté et al. 1998), the European Centre forMedium-Range Weather Forecasts (ECMWF)employs a three time-level SLSI global spectral model(Ritchie 1988; Ritchie et al. 1995), and the AustralianBureau of Meteorology (BoM) uses a three time-levelSLSI global spectral model (BoM 1998).

In this paper, we formulate an efficient two time-level SLSI scheme for limited-area grid-point modelsof the atmosphere. Unlike the two time-level SLSIschemes developed by MH and BMH, this particularformulation employs the unstaggered A grid (a nota-tion introduced by Arakawa; Arakawa and Lamb1977) to horizontally discretise the governing primi-tive equations. Although the use of centred-differenceschemes on the A grid leads to the so-called grid-split-ting problem (Mesinger and Arakawa 1976) in whichnoisy checker-board patterns appear in the model-pre-dicted fields, a numerical strategy introduced byMesinger (1973) has been adapted into the BoM’sexisting and currently operational Eulerian modelLAPS (Limited Area Prediction System) to obviatethis problem. The original time-difference scheme ofthe Eulerian model is marginally modified (Kar 2000)so that a perturbation introduced at one height grid-point can readily travel to the nearest height grid-point and thereby eliminate the stationary two-grid-interval waves from the height field (and consequent-ly from the horizontal wind field), which is the causeof the grid-splitting problem (Mesinger and Arakawa1976).

Purser and Leslie (1988) demonstrated that the useof higher-order spatial difference schemes on the Agrid can produce model solutions that are more accu-rate than the traditional second-order spatial-differ-ence schemes based on the staggered C grid.

Influenced by these results, the Eulerian LAPS wasdesigned with the option of higher-order spatial dif-ference schemes on the A grid (Puri et al. 1998).Arguing in favour of the A grid, Leslie and Purser(1991) pointed out that a semi-Lagrangian treatmentof advection on an unstaggered grid required less gridinterpolations compared to the same task on a stag-gered grid. In view of such arguments, we have stayedwith the A grid in our formulation of the SLSI schemefor LAPS. In formulating the SLSI scheme, as in caseof the Eulerian model, we have taken measures as dis-cussed in the next section to deal with the grid-split-ting problem associated with the A grid.

The SLSI scheme that we formulate in this paperemploys the horizontal momentum equations in avector form so that the metric terms are not treated inan explicit manner. The vector discretisation of thevector-momentum equations follows the tangent-plane method of Ritchie (1988) that was originallydeveloped for three time-level SLSI schemes. In addi-tion to Ritchie (1988), there are two other methods ofvector discretisation introduced independently byCôté (1988) and Bates (1988). Staniforth and Côté(1991) have indicated that all three vector-discretisa-tion methods lead to identical algorithms in the caseof two time-level schemes.

The SLSI scheme that we formulate treats only thehorizontal advection in a semi-Lagrangian manner.Thus, it is a 2D SLSI scheme unlike the 3D SLSIschemes developed by MH and BMH. The scheme isconditionally stable, the (stable) time step for which isdictated by the vertical resolution and the typical ver-tical velocities in the model. Since the typical verticalvelocities associated with the synoptic-to-mesoscalemotions are generally small compared to the corre-sponding horizontal velocities, we may be justified inusing a semi-Lagrangian scheme only for the hori-zontal advection processes. However, for mesoscaleapplications of the model with significantly increasedvertical resolutions, the (potential) computationalefficiency of the scheme can be adversely affected, aspointed out by a reviewer.

In our formulation of the SLSI scheme, the surfacetopography is included in the reference state atmos-phere used for semi-implicit linearisation, a featurenot included in the schemes developed by MH andBMH. Inclusion of this particular feature can reducespurious resonance over steep topographic regionswhen the Courant number is grater than one (Ritchieand Tanguay 1996).

In addition to semi-implicit linearisation of theprimitive equations, we ‘linearise’ the Coriolis forceterms by expressing the Coriolis parameter as a con-stant plus a latitude dependent deviation. Then, thegravity oscillation terms as well as the linearised

294 Australian Meteorological Magazine 49:4 December 2000

Coriolis force terms (an option in MH; McPherson1971) are time discretised in an implicit manner. Thisleads to a coupled implicit system of equations,which, after the vertical decoupling transformation,reduces to a set of 2D Helmholtz type separable ellip-tic equations (Temperton 1994), for which an efficientsolver can be readily developed.

As indicated by a reviewer, there is indeed anargument in favour of maintaining the explicit time-difference schemes for LAPS as opposed to the SLSIscheme presented here. The split-explicit schemes(e.g., Klemp and Wilhelmson 1978; Leslie and Purser1991) are generally efficient, relatively easy to for-mulate and code, and well suited for the architectureof some supercomputers such as the NEC SX-machines acquired by the BoM. In fact, our earlierattempts (G. Dietachmayer 1996 personal communi-cation) in developing a Klemp-Wilhelmson type split-explicit scheme in which the horizontal advectionterms were separately updated on a slow time-scale,resulted in a scheme that offered substantial savingsin CPU time; but unfortunately upon further testingproved to be ‘less’ robust, particularly for relativelyhigh resolution runs of LAPS with full physics. Forthis reason, the split-explicit schemes were not pur-sued any further in our research and we began devel-oping the SLSI scheme presented in this paper.

In the next section, we present a formulation of the2D SLSI scheme for limited-area finite-differencegrid-point models of the atmosphere, in particularLAPS. In the section on numerical results, we discuss48 hour time integrations of the operational Eulerianmodel and the SLSI model, and compare the forecastsmade by the two models to the verifying analyses, toshow that the SLSI model provides a computational-ly stable and accurate, but significantly more efficientalternative to the existing Eulerian model. In the finalsection we present the summary and some conclu-sions. A list of all constants and variables used in thispaper is presented in Appendix A. Appendix B definesthe matrices associated with vertical discretisation ofthe hydrostatic equation and the coupled implicit sys-tem of equations mentioned above.

Formulation of the SLSI schemeContinuous equationsThe governing equations of motion in (λ, θ, σ) coor-dinates for a moist diabatic hydrostatic atmosphere(MH) can be written as

where

The symbols used here and elsewhere in this paper areconventional and listed in Appendix A. Here the ther-modynamic energy equation (Eqn 2) appears in anapproximate form. To derive Eqn 2 from the fullequation (see MH, Eqn 3):

where Rv and Cpv denote the gas constant for watervapour and the specific heat of moist air, respectively,we have used the approximation

that is generally acceptable since r<<1.The model atmosphere is bounded in the vertical

by two constant-σ surfaces, σ = σT and σ = 1, with theupper and lower boundary conditions

Thus, there is no mass transport through the earth’ssurface and the top of the model atmosphere.

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ −

−NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ −

−NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−

∂−σ. σ.

σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−

∂−σ.+ −

−

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

Kar and Logan: A semi-Lagrangian and semi-implicit scheme 295

Integrating Eqn 5 vertically from an intermediateσ to the surface (Φ = ΦS), we obtain

Introducing a generalised geopotential G as

where T0 is a constant reference temperature, we canrewrite Eqn 8 as

where

can be viewed as a ‘moisture correction’ geopotential,which when subtracted from the right hand side ofEqn 8 yields the geopotential (at the intermediate σ)for a dry atmosphere. As in MH, the variable Φw isintroduced so that the virtual effects of moisture ongeopotential do not enter into the implicit part of thescheme, as will be described later. We are not awareof any adverse impact of this simplification from ear-lier studies.

We now invoke a semi-implicit linearisation of theequations of motion to separate out the terms respon-sible for the propagation of gravity waves. A dryisothermal hydrostatic atmosphere with topography isused as the reference state characterised by

where a subscript zero identifies the reference statevariables. This particular reference state includingtopography was originally proposed by Ritchie andTanguay (1996) in their three time-level, global spec-tral SLSI model to reduce spurious resonant amplifi-cation of small-scale model solutions over steep topo-graphic regions, for Courant number larger than unity.To proceed with the semi-implicit linearisation, wethen express the variables (T, Φ, Λ, G) as

where the primed variables identify the perturbationsthat are superposed on the reference state. In additionto Eqn 13, we express the Coriolis parameter as

where f0 is a θ-averaged, thus constant, value of f.This ‘linearisation’ of f (McPherson 1976, MH) sim-plifies the solution of the Helmholtz elliptic problemthat will arise, as will be described later.

Substituting Eqns 13 and 14 into Eqns 1-4 and 9,and then using Eqn 12, we obtain the semi-implicitlinearised system of equations

Here NV~, NT, NΛ, and Nr denote the nonlinear termsdefined by

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

296 Australian Meteorological Magazine 49:4 December 2000

where κ = R/Cp. Notice that the horizontal advectionterms are absorbed in the material time derivatives tofacilitate the semi-Lagrangian treatment of horizontaladvection that follows later.

Vertical discretisationThe vertical discretisation of the equations of motionis based on a Lorenz type staggered grid (Fig. 1),where all model variables, except for σ. , are carried atthe full integer levels. The hydrostatic equation (Eqn5) is applied at the half integer levels and discretisedas

where

Summing Eqn 24 over the levels k, k + 1, ..., K, weobtain a vertically discrete form of Eqn 8:

where ak,l are the elements of an upper triangularmatrix defined in Appendix B. By analogy, the ver-tically discrete forms of Eqns 9 and 11 can be writtenas

The vertical advection terms, as they appear inEqns 20, 21 and 23, are discretised as

where Ψ is an arbitrary variable. On the other hand,the vertical divergence term in Eqns 17 and 21, and anadiabatic cooling related term in Eqns 16 and 21 arediscretised as

where (∆σ)k = σk+1/2 - σk-1/2. Here for specified val-ues of σk(1≤k≤K), σk+1/2 is defined by

Then the linearised system of Eqns 15-19 can be dis-cretised at each level k as follows

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ −

−NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ −

−NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

Kar and Logan: A semi-Lagrangian and semi-implicit scheme 297

Fig. 1 Vertical distribution of variables.

where

The explicit forms of the vertically discrete nonlinearterms are omitted for brevity.

Time discretisationThe linearised, vertically discrete system of equations(Eqns 32-36) are discretised in time along the hori-zontal trajectory of an air parcel, using a two time-level SLSI scheme. Notice that each linearised equa-tion is of the form

where F is one of the prognostic variables V~ , T′ -κT0Λ′, Λ′ and r; L and N denote respectively the lin-ear and nonlinear terms. The damping terms, whichinclude linear horizontal diffusion and horizontaldivergence diffusion, are integrated in a time splitmanner using an explicit two time-level scheme, areomitted from the present development of the SLSIscheme. In the following, the superscripts n + 1 andn denote the two time-level indices, with n + 1/2denoting the temporal mid-level index; ∆t is the timestep; ε(0≤ε≤1) is the off-centring parameter. Presenceof a subscript denoted by an asterisk (*) on any vari-able indicates that it is evaluated at the departurepoint, whereas the absence of such a subscript indi-cates that the variable is evaluated at the arrival grid-point. The semi-implicit time discretisation of Linvolves a weighted temporal average of its values atthe two endpoints of the trajectory. The explicit timediscretisation of N, on the other hand, involves aweighted spatial average of its values at the two (geo-graphical) endpoints of the trajectory, but computed atthe temporal mid-level. A forward time discretisationalong the trajectory is used for P. The SLSI schemefor Eqn 37 is then expressed as

with ℵ defined by

where Nn+1/2 is evaluated using a linear extrapolation:

The parameter ε used in Eqn 38 for the off-centredtime averaging of L, varies the implicit scheme froma neutral, second-order, Crank-Nicholson scheme forε = 0 to a strongly damping, first-order, backwardscheme for ε = 1. However, for relatively small valuesof ε, the high-frequency gravity oscillations are some-what selectively damped by the scheme that helps inreducing small-scale noise from the forecasts. Notethat any non-zero value of ε formally reduces thescheme to less than second-order in accuracy. Thus, arelatively small value of ε is generally recommendedand used. A typical value of ε can range from 0.1 to0.2 depending on the resolution of the model.

The off-centred spatial averaging of the nonlinearterm as shown in Eqn 39, has been adopted previous-ly in a two time-level limited-area grid-point modelby MH, and in three time-level global spectral andlimited-area grid-point models by Tanguay et al.(1992) and Ritchie and Tanguay (1996). Using theirthree time-level SLSI models, Ritchie and Tanguay(1996) have demonstrated that this type of spatialaveraging of the nonlinear terms, together with inclu-sion of topography in the reference state, can signifi-cantly reduce spurious resonance over steep topo-graphic regions when the Courant number is greaterthan one.

Substituting Eqn 39 in Eqn 38, the SLSI schemecan be written in a compact form:

where

Let us now apply Eqns 41 and 42 to the prognos-tic Eqns 32-35, not necessarily in that order, to deter-mine the corresponding time discrete forms. To begin,we apply Eqns 41 and 42 to 34 to derive a time dis-crete form of the prognostic equation for Λ′:

where

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ −

−NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ −

−NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ −

−NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ −

−NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ −

−NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ −

−NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ −

−NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

...1

...2

...3

...4

...5

...6

...7

...8

...9

...10

...11

...12

...13

...14

...15

...16

...17

...18

...19

...20

...21

...22

...23

...24

...25

...26

...27

...28

...29

...30

...31

...34

...35

...36

...37

...40

...38

...39

...33

...32

(2.2)

(2.3)

(2.4)

(2.5)

where

dVdt------~ ~ ~ ~

~

~

~

~~

~ ~ ~ ~ ~

~~

~∇Φ RTv Λ∇− fk V×−+ + + PV KV+=

tddT R

Cp----- 1 0.26r_( )T dHΛ

dt--------- σσ---+ −

−PT KT+=

dHΛdt--------- D

σ∂−∂−

σ+ + KΛ=

tddr Pr Kr+=

σln∂−∂Φ RTv

_=

σppS----- Λ, pS f 2Ω θsin= ,ln= =

Tv T 1 0.61 r1 r+-----------+ −

−V u i v j+=,=

σdσdt------ PV Pu i Pv j KV Ku i Kv j+≡−,+≡−,=

D ∇− V⋅−1

a θcos---------------λ∂−

∂−uθ∂−

∂− v θcos( )+= =

∇−i

a θcos---------------λ∂−

∂− ja--- θ∂−

∂−+=

tdd dH

dt----- σσ∂−

∂−+= , dH

dt----- t∂−∂− u

a θcos---------------λ∂−

∂− va--- θ∂−

∂−+ +=

_.

.

.

.

.

.

tddT Rv

Cpv---------T dHΛ

dt--------- σσ---+ −

−_ PT KT+=

RvCpv--------- R

Cp------ 1 0.26r_( )∼

∼

σ 0 at σ σT , 1= =

Φ ΦS R Tv σlndσ

1

∫−+=

G ΦS RT0Λ R T σlndσ

1∫−+ +=

Φ G RT0Λ_ Φw+=

Φw R Tv T_( ) σlnd

σ

1

∫−=

T0 constant=Φ0 λ θ σ, ,( ) ΦS λ θ,( ) RT0 σln_

_

=

Λ0 λ θ,( )ΦS λ θ,( )

RT0---------------------- =

G0 λ θ σ, ,( ) Φ0 λ θ σ, ,( ) RT0Λ0 λ θ,( )+=

T T0 T′ −+Φ0 Φ′−

Λ Λ0 Λ′−

G+=

= +

G0 G′−+

=Φ

=

f f0 f ′−+=

dHVdt---------- ∇−G′−_

_

_

_ f0 k V×− NV PV KV+ + +=∼ ∼ ∼

∼ ∼ ∼

∼ ∼

∼

∼

∼ ∼

dH

dt----- T′ κT0Λ′−( ) κT0σ.σ--- NT PT KT+ + +=

dHΛ′−dt----------- D

σ∂−∂−

σ.+ − −NΛ KΛ+ +=

_

dHrdt-------- Nr Pr Kr+ +=

G′− RT0Λ′− R T′ σlndσ

1∫−+=

NV Nu i Nv j+≡− =

Φw

∇− R Tv T0_( ) Λ∇−+[ ] f f0_( )k V× −.

σσ

∂−∂−V__

NT κ T0 1 0.26r_( )T_ Dσ∂−∂−

σ. σ.σ---_+=

∼

[ ] − −

1cp-----V ΦS∇−⋅−_ σ.

σ∂−∂−T_

NΛ1

RT0----------V ΦS∇−⋅−=

Nr σ.σ∂−

∂−r_=

Φk 1+ Φk_

_

__

_

_∆ σln( )k 1 2⁄−+

---------------------------------- RTv( )k 1+ Tv( )k+

2----------------------------------------= 1 k K 1≤ ≤ −

ΦS ΦK∆ σln( )K 1 2⁄−+

----------------------------------- R Tv( )K= k K=

∆ σln( )k 1 2⁄−+σk 1+

σk-------------- 1 k K 1≤ ≤ −ln=

σK 1 2⁄−+σK

---------------------ln k K=

Φk ΦS R ak l, Tv( )ll k=

K∑−+= 1 k K≤ ≤ −

Gk ΦS RT0Λ R ak l, Tll k=

K∑−+ +=

=Φkw R ak l, Tv T_( )l

l k=

K∑−

σ.σ∂−

∂Ψ − −

k

12--- σ. k 1 2⁄−+

Ψk 1+ Ψk_

∆σ( )k 1 2⁄−+----------------------------- σ. k 1 2⁄−_ Ψk Ψk 1__

∆σ( )k 1 2⁄−_----------------------------+

=

σ∂−∂−

σ. − −

kσ. k 1 2⁄−+ σ. k 1 2⁄−__

∆σ( )k----------------------------------------------=

σ.σ--- −

−k

1σk------σ. k 1 2⁄−+ σ. k 1 2⁄−_+

2----------------------------------------------=

σk 1 2⁄−+12--- σk 1+ σ+ k( )= 1 k K 1_≤ ≤ −−

dHVkdtk

------------ G′−k∇−_ f0k~~ V~k×−_ NV~

PV~KV~

+ +( )k+=

dH

dtk------- T′−k κT0Λ′−_( ) κ T0

σ.σ--- −

−k NT PT KT+ +( )k+=

dHΛ′−dtk

----------- Dσ∂−∂−

σ.+ − −

k_ NΛ K+

Λ( )k+=

dHrkdtk

---------- Nr Pr Kr+ +( )k=

G′−k RT0Λ′− R ak l, T′−ll k=

K

∑−+=

dH

dtk------- t∂−

∂− V~k ∇−⋅−+=

dHFdt--------- L N P K+ + +=

Fn 1+ F*n_

∆t--------------------------- 12--- 1 ε+( )Ln 1+ 1 ε_( )L*

n+[ ]

−

ℵ −P*n + +

=

ℵ−12--- 1 ε+( )Nn 1 2⁄−+ 1 ε_( )N*

n 1 2⁄−++[=

Ψn 1 2⁄−+ 1

2--- 3Ψn

Ψn 1__( )

...41

...42

...43

=

F αL_( )n 1+ C=

C B* αNn 1 2⁄−++≡−

B F βL+( )n

βNn 1 2⁄−+ Pn∆t+ +≡−

α12--- 1 ε+( )∆t β 1

2--- 1 ε_( )∆t≡−;≡−

Λ′− α Dσ∂−

∂−σ.+ −

−

k+

n 1+CΛ( )k=

]

298 Australian Meteorological Magazine 49:4 December 2000

Then, summing (Eqn 43) x (∆σ)l over the levels 1,2,..., K, and using the upper and lower boundary con-ditions from Eqn 7, we obtain Λ′ at time level n + 1 interms of Dn + 1 and known quantities:

Similarly, summing (Eqn 43) x (∆σ)l over the levels1, 2,..., k, and using the upper boundary conditionfrom Eqn 7, we obtain σ. at time level n + 1 in termsof Dn + 1 and known quantities:

To continue, we now apply Eqns 41 and 42 to Eqns33 and 35, and derive time discrete forms of the prog-nostic equations for T′ and r:

where

Thus, Eqns 47 and 48 respectively determine T′ and rat time level n + 1 in terms of Dn + 1 and known quan-tities. Then, we apply Eqn 36 at time level n + 1 andemploy Eqns 45, 46, 47 and 30 in the resulting equa-tion to determine G′ at time level n + 1 in terms ofDn+1 and known quantities:

where

and Mk,l denotes the elements of the matrixdefined in Appendix B.

At this stage we intend to determine Dn + 1, prior towhich (u, v)n + 1 must be determined. To that effect,we apply Eqns 41 and 42 to Eqn 32, and derive a timediscrete form of the vector-momentum equation:

where

Let and respectively denote thehorizontal unit vectors at the arrival and departurepoints of the trajectory. Then, we may rewrite Eqn 52as

Following the tangent-plane algorithm of Ritchie(1988), and subsequent simplification of the samealgorithm by Ritchie and Beaudoin (1994), we canexpress in terms of as follows:

where

Details of the derivations of Eqns 55 and 56 respec-tively can be found in Ritchie (1988, Appendix) andRitchie and Beaudoin (1994, Section 2.c). As indicat-ed in the introduction, there are two other methods ofvector-discretisation of the vector-momentum equa-tion developed independently by Côté (1988) andBates (1988). Staniforth and Côté (1991, Section 4.c)indicated that all three methods lead to identical algo-rithms in case of two time-level schemes.

Then, substituting Eqn 55 in Eqn 54, we obtain incomponent forms:

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

l 1=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

Kar and Logan: A semi-Lagrangian and semi-implicit scheme 299

where

From Eqns 57 and 58, we solve forto obtain

where

Substituting Eqns 60 and 61 into the definition ofD from Eqn 6, we derive a time discrete form of thedivergence equation:

where

Notice that Eqns 50 and 63 form a coupled implicitsystem of equations in (G′)n+1 and Dn+1. Introducingthe K-dimensional column-vectorswhose respective elements are

, Eqns 50 and 63 can bewritten in matrix forms:

To decouple the matrix equations, a traditionalvertical decoupling linear transformation is intro-

duced as follows. Let be the matrix of whosecolumns are the eigenvectors of , with the associat-ed eigenvalues λ1, λ2, ..., λK. Then, multiplying Eqns66 and 67 on the left by we obtain

where

Here δk,l denotes the Kronecker delta and λk δk,l con-stitutes the elements of a diagonal matrix of dimen-sion K x K. Then, reverting to the component forms ofEqns 68 and 69, we obtain

where are the elements of the k-dimensional vectors defined by Eqn 70.

Eliminating from Eqns 71 and 72, we obtain aset of 2D Helmholtz type elliptic equations

where

Note that the linearisation of the Coriolis force termsusing Eqn 14 is essential to produce this ‘separable’form of the elliptic Eqn 73, for which a fast-iterativesolver based on a conventional successive over-relax-ation (SOR) method (e.g., Haltiner and Williams,1980, pp. 157-159) is used.

Horizontal discretisationAs indicated in the Introduction, the unstaggered A

grid with uniform grid intervals ∆λ and ∆θ, respec-tively, in longitude and latitude, is used to horizontal-ly discretise the governing equations. In Shuman(1962) notation, the finite difference and finite aver-age operators used in the λ direction are defined by

The corresponding operators in θ-direction, namelyδθ()and , are defined similarly.

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

...44

...46

...45

CΛ( )k BΛ( )*, k α NΛ( )kn 1 2⁄−++=

BΛ( )k Λ′− β Dσ∂−

∂−σ.+ −

−k

– nβ NΛ( )k

n 1 2⁄−++=

Λ′−( )n 1+ 1

1 σT_---------------= CΛ αDn 1+_( )l ∆σ( )l

=

K

∑−

ασ. k 1 2⁄−+n + 1 =

CΛ αDn 1+_( )l ∆σ( )ll 1=

k

∑− σk 1 2⁄−+ σT_( ) Λ′ −( )n 1+_

T′−k κT0 Λ′− ασ. kσk------+

− − −_

n 1+CT( )k=

rkn 1+ Cr( )k=

...47

...48

...49

...50

...51

CT( )k BT( )*,k α NT( )kn 1 2⁄−++=

Cr( )k Br( )*,k α Nr( )kn 1 2⁄−++=

BT( )k T′−k κT0 Λ′− βσ. k σk⁄−_( )_[ ]n

β NT( )kn 1 2⁄−++=

PT( )+ kn

∆t

Br( )k rkn

β Nr( )kn 1 2⁄−+ Pr( )

nk ∆t+ +=

G′−k( )n 1+

α Mk l, Dln 1+

l 1=

K

∑−+ Hk 1 k K≤ ≤ − =

Hk R ak l, CT( )ll k=

K

∑− Mk l, CΛ( )ll 1=

K

∑−+=

V~ αLV~_( )

kn 1+ CV~

( )k= ...52

...53

...54

...55

...57

...56

...58

...59

LV~G′− f0k~ V~×−_∇−_=

CV~( )k YV~

( )* k,α NV~

( )kn 1 2⁄−++=

YV~( )k V~ βLV~

+( )kn

β NV~( )k

n 1 2⁄−+ PV~( )k

n∆t + +=

u αLu_( )n 1+

αNun 1 2⁄−+_[ k i~

v αLv_( )n 1+

αNvn 1 2⁄−+_[ ]k j~+

Yu( )* k,i~* Yv( )* k,

j~*+ =

]

i~Y i~* X j~*+

ℜ−-----------------------= j~

Y j~* X_~*

ℜ−--------------------=,

X ∆ta-----u* θ*sin= Y, θ*cos ∆t

a-----v* θ*sin_=

ℜ− X2 Y2+( )1 2⁄−=

i

u α1

a θcos---------------λ∂−

∂−G′− f0v_ − −+

k

n 1+Au( )k=

v α1a--- θ∂−

∂−G′− f0u+ − −+

k

n 1+Av( )k=

Au( )k α Nu( )kn 1 2⁄−+ X Yv( )* k,

Y Yu( )* k,+ℜ−

---------------------------------------------------+=

Av( )k α Nv( )kn 1 2⁄−+ Y Yv( )* k,

X Yu( )* k,_

ℜ−------------------------------------------------+=

Yu u β1

a θcos---------------λ∂−

∂−G′− f0v_ − −_ n

βNun 1 2⁄−+ Pu

n t∆+ +

=

Yv v β1a--- θ∂−

∂−G′− f0u+ − −_ n

βNvn 1 2⁄−+ Pv

n t∆+ +

=

u αb

a θcos---------------λ∂−

∂−G′−ea--- θ∂−

∂−G′−+ −...60

...61

−+k

n 1+Fu( )k=

v αba--- θ∂−

∂−G′−e

a θcos---------------λ∂−

∂−G′−_ − −+

k

n 1+Fv( )k=

Fu( )k b Au( )k e Av( )k+=Fv( )k b Av( )k e_ Au( )k=

F+ f0α b, 1 F+2+( )

1_ e, bF+= = = ...62

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

300 Australian Meteorological Magazine 49:4 December 2000

The horizontal discretisation operators introducedabove are directly applied to the σ and t discreteforms of the equations of motion derived earlier, toconstruct second-order accurate centred schemes. Forsimplicity, we have not considered higher-orderschemes that can be readily designed on the A grid, asindicated earlier.

However, some special attention has been paid tothe discretisation of the pressure gradient vector, ∇Φ+ RTν∇Λ , which sums two terms that can becomelarge and of opposite sign near steep topography,thereby introducing a large truncation error into thecomputed pressure gradient force and the impliedgeostrophic wind. This is a well known difficultyassociated with the σ coordinate system. Here wehave adopted the approach of Corby et al. (1972) ‘bydefining an atmospheric structure which has zero gra-dient of Φ in p coordinates and then to find space dis-cretisations of the two components in σ coordinatesso that when combined they do not lead to a largeapparent or spurious gradient’. Following thisapproach, we have derived a horizontally discreteform of ∇Φ + RTν∇Λ:

For details of the derivation, the reader is referred toCorby et al. (1972). Here Eqn 76 provides a second-order accurate scheme, but higher-order accurateschemes also can be readily derived. In fact, both sec-ond-order and fourth-order schemes for ∇Φ + RTν∇Λhave been implemented into the Eulerian version ofthis model. In our SLSI formulation, we haveattempted to maintain the horizontal-discretisation of∇Φ + RTν∇Λ. Thus, the nonlinear part of the pressuregradient vector in Eqn 20 has also been discretisedfollowing (Eqn 76).

The Laplacian operator defined by Eqn 64 is hori-zontally discretised as:

over a compact stencil of five grid-points, where theneighbouring points are one grid interval apart. Thescheme (Eqn 77) has been employed in the Helmholtztype equation (Eqn 73) to eliminate the spurious two-grid-interval gravity wave solutions that may appearas noisy checkerboard patterns in the model forecasts.Following the approach of Mesinger (1973), a brieftheoretical development behind the choice of Eqn 77is outlined below.

Originally developed for the 2D shallow-watergravity wave propagation on the semi-staggered Egrid, Mesinger’s approach essentially modifies theoriginal time-difference schemes marginally so that aperturbation introduced at one height grid-point canreadily travel to the nearest height grid-points andthereby eliminate the stationary two-grid-intervalwaves from the height field. This in turn keeps thesame waves from appearing in the horizontal wind ordivergence field.

For our SLSI scheme, Eqns 71 and 72 can be iden-tified as the equivalent shallow-water system in asemi-implicit time discrete form for the k-th verticaleigenmode. If second-order centred-differences wereused to discretise the original equations that lead toEqns 71 and 72, we would have arrived at the follow-ing space-discrete form for Eqn 72:

where the horizontally-discrete Laplacian operatoris based on a wide stencil of five grid-points,

where the neighbouring points are two grid intervalsapart. Following the approach of Mesinger (1973),the scheme Eqn 78 can be modified to the form:

where 0≤γ≤1 for consistency. Using normal-modesolutions of the modified scheme Eqns 79 and 71, onecan show that stationary two-grid-interval wave solu-tions are indeed eliminated for values of γ≥1/2. Wehave set γ = 1 in our application of Eqn 79, noting thatthis particular choice of γ replaces the discrete opera-tor by .

For implementation of the 2D semi-Lagrangianadvection procedure, we need (a) an algorithm tocompute the departure points and (b) some horizontalinterpolation schemes to interpolate the field vari-ables and other functions from the grid-points to thedeparture point. The departure points are computedfollowing a space and time-centred iterative proce-dure, similar to the ones used by earlier authors (e.g.,MH, BMH) in their two time-level SLSI models. Theprocedure employs the linear extrapolation (Eqn 40)to determine V~ n+1/2 at the grid-points and a bilinearhorizontal interpolation scheme to compute V~ n+1/2 atthe mid-point of the trajectory. The trajectories arecalculated using an iterative procedure (Robert 1981),that generally requires two iterations for convergence.Aside from this, to interpolate the field variables andother functions from the grid-points to the departurepoint, we have employed the quasi-bicubic interpola-tion scheme (Ritchie et al. 1995), that is relatively lessexpensive (computationally) and nearly as accurate asthe conventional bicubic interpolation scheme.

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

Kar and Logan: A semi-Lagrangian and semi-implicit scheme 301

The limited area domain in (λ,θ) space coveredby LAPS is shown schematically in Fig. 2. TheBoM’s operational Global Assimilation andPrediction (GASP) system (Seaman et al. 1995;Bourke et al. 1995) provides external time historydata every six hour for the lateral boundary condi-tions nesting in terms of the prognostic variablesu, v, T, r, and pS for the total range, typically 48hours, of the LAPS forecast. Clearly, this involvessome spatial and temporal interpolations of theGASP data.

Over a single time step, the Eulerian model firstmakes a prediction of (u, v, T, r, pS ) over the subdo-main LD+B1+B2, where the ‘buffer’ zone B3+B4helps implement a third-order upwind scheme forhorizontal advection in the model. Similarly, theSLSI model first makes a prediction of (u, v, T, r, pS)over the subdomain LD+B1, where the buffer zoneB3+B4 ensures that the departure points computedfor the arrival grid-points in LD+B1+B2, do not falloutside the limited area. The predicted variables arethen blended in with the external data near theboundary over a relaxation zone starting at B4 in Fig.2, using the boundary relaxation technique devel-oped by Davies (1976). The width of the relaxationzone and the profile of the relaxation weights (orblending coefficients) are prescribed externally.Typical eight to 12 grid-points are used for the relax-ation zone and the relaxation weights satisfy a half-cosine profile.

As indicated above, over each time step, the SLSIscheme first predicts (u, v)n+1 over the subdomainLD+B1. A second-order accurate centred-scheme forEqns 60 and 61 would then require that (G′)n+1 bedetermined over the subdomain LD+B1+B2. This inturn requires that we solve the horizontally discreteform of the Helmholtz type Eqn 73 for each verticalmode k over the subdomain LD+B1, with specified(using (G′)n) along B2. As indicated ear-lier, an SOR algorithm has been used to solve for(G′)n+1 over the subdomain LD+B1. The remainingunknowns (Λ′, σ. ,Τ′, r)n+1 are determined over thesame subdomain LD+B1 using Eqns 45, 46, 47, and48, respectively.

At the beginning of each SLSI time step, the aux-iliary variables (Nu, Nv, NT, Nr, NΛ) and (Yu, Yv, BT,Br, BΛ) are computed at each grid-point of the entiredomain. The departure points are then computed foreach grid-point over the subdomain LD+B1+B2. Thevariables (Yu, Yv, BT, Br, BΛ)*,k are computed usingquasi-bicubic horizontal interpolation. Lastly, the

remaining auxiliary variables (CT, Cr, CΛ, Fu, Fv, H,P) are computed at each grid-point of the subdomainLD+B1+B2.Numerical resultsTo determine the overall accuracy of the model solu-tions obtained using the SLSI scheme and the compu-tational economy associated with it, we have per-formed a number of numerical time integrations tocompare the SLSI model with the currently opera-tional Eulerian model. The Eulerian LAPS modelemploys a third order upwind scheme for the hori-zontal advection terms, an explicit quasi-second-order accurate Miller-Pearce scheme (Miller andPearce 1974) for time discretisation and a fourth-order accurate centred scheme designed after Corbyet al. (1972) for the horizontal pressure gradientterms. Following Mesinger (1973), the Miller-Pearcescheme is marginally modified (Kar 2000) to elimi-nate the grid-splitting problem associated with the useof the A grid.

An adiabatic digital filter initialisation (DFI)scheme (Lynch and Huang 1992) is applied to theanalysed fields at the base time, to suppress typicalhigh-frequency noise from imbalance in the initial con-dition. The parametrised physical processes include aconstant flux layer, vertical eddy transport, convection,

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

302 Australian Meteorological Magazine 49:4 December 2000

Fig. 2 A schematic of the horizontal limited area cov-ered by LAPS. The innermost subdomain isdenoted by LD . The dashed-lined rectanglesB1, B2, and B3 respectively represent the first,second, and third lines of grid points sur-rounding LD . The solid-lined rectangle B4 rep-resents the lateral boundary of the limitedarea.

ground hydrology, radiation, and clouds. The para-metrisation schemes used in the model are the same asin the Global Assimilation and Prediction System(GASP). Details of the various parametrisations aregiven in Puri et al. (1998). The parametrised physicalprocesses are computed at every ‘physics time step’,that is defined as an integer multiple of the ‘dynamicstime step’. Optionally, a horizontal ∇2 diffusion isapplied to the model predicted u, v, T, and r. In the caseof the Eulerian model, a horizontal ∇2 divergence-dif-fusion is also applied to the model predicted u and v toreduce small-scale noise. The initialisation, physicalparametrisation and diffusion schemes are the same forboth Eulerian and SLSI models.

The model configuration used here employs a verti-cal resolution of 19 sigma levels, as shown in Table 1,and a uniform horizontal resolution of 0.75° coveringthe standard Australian operational domain (65°S-15°N; 65°E-185°E), with 160 and 110 grid-pointsrespectively in longitude and latitude. The horizontalgrid-points cover the domain LD+B1+B2+B3+B4 shownin Fig. 2. For this particular configuration of LAPS, theEulerian model employs a dynamics time step of 90 s,which is dictated by the CFL criterion of linear compu-tational stability of the fastest external/Lamb verticalmode of the model. The physics time step is 12 min.Both ∇2 diffusion and ∇2 divergence-diffusion areapplied at the physics time steps with the coefficients 5x 104 m2 s-1 and 1 x 105 m2 s-1, respectively. Such dif-fusions are primarily needed to reduce small scale noisefrom the model forecasts when physics is included.

For the SLSI model, we have employed the fol-lowing dynamics time steps: 6 min, 12 min, 18 min,24 min, and 30 min, that are 4, 8, 12, 16, and 20 times,respectively, greater than the dynamics time step of90 s employed by the Eulerian model. Thus, for laterconvenience, we have labelled the Eulerian model asE1.5, the SLSI models as L6, L12, L18, L24, andL30. The reference temperature T0 is assigned a con-stant value of 250K. The off-centring parameter ε isset to zero for L6 to L24, but set to 0.15 for L30 tocontrol noise near orographic regions. The coefficientof horizontal ∇2-diffusion is the same as that in E1.5.No horizontal divergence diffusion is used in theSLSI time integrations. Presumably, some damping ofnear grid-scale noise is an implicit feature of anySLSI scheme. The physics time step is 12 min forboth L6 and L12, but is set to 18 min, 24 min, and 30min for L18, L24, and L30 respectively.

A number of different initial and boundary datasetshave been employed to run the Eulerian and SLSImodels for 48 hours and the results compared. Thecase presented here covers the period 2300 UTC 29August 1996 to 2300 UTC 31 August 1996. It showsthe intense development of a low as it crosses the

eastern Australian coast. In this case it is east-coastcyclogenesis, a situation of critical importance pro-ducing flood rains on the eastern seaboard ofAustralia and affecting maritime operations due to theextremely strong winds associated with these sys-tems. Figures 3, 4, and 5 show the LAPS analyses at24-hour intervals for the mean sea-level pressure(MSLP) and 500 hPa height starting at 2300 UTC 29August 1996 (the base time for the forecast). Theanalyses clearly depict the intensification of the lowfrom a central pressure of 1010 hPa over southernQueensland to 995.6 hPa off the NSW coast 48 hourslater. Figures 6 and 7 show the 24 hour and 48 hourforecasts made by the Eulerian model E1.5. The cor-responding forecasts made by the SLSI model areshown in Figs 8 and 9 for L12 and in Figs 10 and 11for L24. A subjective comparison of the MSLP and500 hPa height forecasts made by E1.5 to those madeby L12 and L24 reveal that in terms of overall accu-racy, the SLSI model forecasts are stable and as accu-rate as the Eulerian-model forecasts.

For a more objective comparison of the forecastsmade by the Eulerian and the SLSI model, we havecomputed the root mean square (rms) height differ-ence between the forecast and the verifying analysisat the hour 24 and hour 48, for each of the modelsE1.5, L6, L12, L18, L24, and L30. Note that the rmsheight differences are computed over the entire lim-ited area including the boundary-relaxation zone.Figure 12 shows the rms errors at hour 24 and hour48. The rms errors associated with the SLSI fore-

Kar and Logan: A semi-Lagrangian and semi-implicit scheme 303

Table 1. Specified values of s at the 19 full integer lev-els of the model.

k σk

1 0.0502 0.0703 0.1004 0.1505 0.2006 0.2507 0.3008 0.3509 0.40010 0.50011 0.60012 0.70013 0.75014 0.80015 0.85016 0.90017 0.95018 0.97519 0.991

304 Australian Meteorological Magazine 49:4 December 2000

Fig. 3 Analysis of 2300 UTC 29 August 1996. Charts of (a) the MSLP with a contour interval of 2 hPa and (b) the 500hPa height with a contour interval of 20 m.

(a)

(b)

Kar and Logan: A semi-Lagrangian and semi-implicit scheme 305

Fig. 4 As in Fig. 3, but valid at 2300 UTC 30 August 1996.

(a)

(b)

306 Australian Meteorological Magazine 49:4 December 2000

Fig. 5 As in Fig. 3, but valid at 2300 UTC 31 August 1996.

(a)

(b)

Kar and Logan: A semi-Lagrangian and semi-implicit scheme 307

Fig. 6 As in Fig. 4 but for 24 h forecast using the Eulerian model starting from the analysis of 2300 UTC 29 August 1996.The time step is 90 s.

(a)

(b)

308 Australian Meteorological Magazine 49:4 December 2000

Fig. 7 As in Fig. 5 but for 48 h forecast using the Eulerian model starting from the analysis of 2300 UTC 29 August1996. The time step is 90 s.

(a)

(b)

Kar and Logan: A semi-Lagrangian and semi-implicit scheme 309

Fig. 8 As in Fig. 6 but for the SLSI model with a time step of 12 min.

(a)

(b)

310 Australian Meteorological Magazine 49:4 December 2000

Fig. 9 As in Fig. 7 but for the SLSI model with a time step of 12 min.

(a)

(b)

Kar and Logan: A semi-Lagrangian and semi-implicit scheme 311

Fig. 10 As in Fig. 8 but with a time step of 24 min.

(a)

(b)

312 Australian Meteorological Magazine 49:4 December 2000

Fig. 11 As in Fig. 9 but with a time step of 24 min.

(a)

(b)

casts remain comparable in magnitude to the rmserrors associated with the Eulerian model at all ver-tical levels.

To examine the convergence of the SLSI forecastsas the time step is reduced, we have computed the rmsheight differences between L6 and each of L12, L18,L24, and L30. The results at hour 24 and 48 areshown in Fig. 13. In general, the rms height differ-ences decrease progressively with ∆t suggestingnumerical convergence (in time) of the SLSI fore-

casts. Note that these rms errors are several timessmaller (not surprisingly) than the rms forecast errorsshown in Fig. 12.

The use of larger time steps in the SLSI modeldoes not necessarily guarantee it will always be com-putationally more efficient, in terms of CPU time,compared to the Eulerian model. This is because theSLSI formulations, in general, involve a larger num-ber of calculations for the semi-Lagrangian treatmentof advection (Bartello and Thomas 1996), the semi-

Kar and Logan: A semi-Lagrangian and semi-implicit scheme 313

Fig. 12 (a)Area averaged rms height differencesbetween the analysis of 2300 UTC 30 August1996 and the 24 h forecasts verifying at thesame time produced by the different models.E1.5 refers to the Eulerian LAPS with ∆∆t = 1.5min. The forecasts produced by the SLSImodel with ∆∆t = 6 min, 12 min, etc., arereferred to as L6, L12, ... The results are inmeters and are presented for pressure levels,varying from 100 hPa at the top to 1000 hPa atthe bottom. Area average is over the domain(65°S-15°N; 65°E-185°E). (b) As in (a) butvalid at 2300 UTC 31 August 1996.

(a)

(b)

Fig. 13 Area averaged rms height differences betweenthe forecast produced by L6 and the corre-sponding forecast produced by each of L12,L18, L24 and L30, when physics is included.The results are shown (a) at hour 24 and (b) athour 48. The pressure levels presented and thedomain used for area average are the same asin Fig. 12.

(a)

(b)

implicit treatment of the gravity-inertia waves requir-ing the solution of many elliptic equations at eachtime step. Two more issues must also be considered:(a) the SLSI formulation applies only to the dynamicspart of the model; (b) the physics computations areroughly as expensive computationally as the dynam-ics computations; thus, how frequently physics is

called during the time integrations of a particularmodel can significantly influence the total CPU timeused by that model.

In view of these issues, we have turned off thephysics calls and repeated the 48 hour time integra-tions of E1.5, L6, L12, L18, L24, and L30, and notedthe CPU times used for each run. The single-CPUtimes used by the Eulerian and the SLSI models, withand without physics, on the NEC SX-4 supercomput-er are shown in Table 2. Without physics, L12-L30reduces the single-CPU time of E1.5 by 62-83%. Asanticipated, such percentage reductions of CPU timeare compromised when physics is included. With aphysics time step of 12 min, L12 reduces the single-CPU time of E1.5 by 36%. When the physics is calledless frequently with time steps of 18-30 min, the SLSImodel further reduces the single-CPU time by 52-66% relative to E1.5. This efficiency is not bought atthe expense of accuracy, as we have shown earlier bymaking subjective and objective comparisons of the48 hour forecasts made by E1.5 and L24. Thus, adynamics and physics time step of 18 min for theSLSI model with a percentage reduction of the single-CPU time by 52% is considered satisfactory for thepresent configuration of LAPS.

To examine the convergence of the SLSI forecastswith no physics, we have once again computed therms height differences between L6 and each of L12,L18, L24, and L30 in the no physics case. The resultsat hour 24 and 48 are shown in Fig. 14. At all verticallevels, the rms height differences decrease progres-sively with ∆t indicating numerical convergence (intime) of the no-physics SLSI forecasts.

Summary and conclusionA two time-level, 2D SLSI scheme has been devel-oped for limited-area finite-difference grid-pointmodels of the atmosphere based on the moist, diabat-ic, quasi-static primitive equations in σ-coordinates.A key distinction of the new scheme is that it employsthe unstaggered A grid to horizontally discretise thegoverning equations. One advantage of this choice is

314 Australian Meteorological Magazine 49:4 December 2000

Fig. 14 As in Fig. 13 but physics is not included. Theresults are shown (a) at hour 24 and (b) athour 48.

(a)

(b)

Table 2. The single-CPU time in seconds used by the Eulerian and SLSI models, with and without physics, on the NECSX-4 supercomputer. E1.5 refers to the Eulerian LAPS with a time step of 1.5 min. The SLSI model integrationswith time steps of 6 min, 12 min,...., are respectively denoted by L6, L12,....

E1.5 L6 L12 L18 L24 L30

Full Physics 522 431 336 248 211 179

No Physics 302 207 113 79 62 52

that a semi-Lagrangian treatment of advection on theunstaggered grid requires less grid interpolationscompared to the same task on a staggered grid (Leslieand Purser 1991). The disadvantage of this choice isthe problem of grid splitting associated with the Agrid (Mesinger and Arakawa 1976). Here the problemis eliminated, using the strategy of Mesinger (1973),where a straight forward five-point finite-differenceLaplacian is used to discretise the 2D Helmholtz- typeelliptic equations associated with the semi-implicittreatment of gravity waves.

The other main features of the scheme include (a)surface topography in the reference state isothermalatmosphere; (b) vector discretisation of the momen-tum equations expressed in a vector form; (c) only thehorizontal advective processes are treated in a semi-Lagrangian manner, so the scheme is a 2D semi-Lagrangian scheme; (d) the Coriolis force terms havebeen linearised and combined with the semi-implicitlinearised terms, so that after vertical decouplingtransformation, a set of 2D Helmholtz type equationsare produced that can be solved efficiently using aconventional SOR solver; (e) the nonlinear termshave been treated in a time-explicit, spatially aver-aged manner. Some of these features are commonwith existing two time-level and three time-levelSLSI schemes described already in the literature. Forexample, feature (a) was originally adopted byRitchie and Tanguay (1996); feature (b) was intro-duced independently by Ritchie (1988), Côté (1988),and Bates (1988); feature (d) was adopted byMcDonald and Haugen (1992); feature (e) was adopt-ed by McDonald and Haugen (1992), Tanguay et al.(1992), and Ritchie and Tanguay (1996).

To compare the stability, accuracy, and efficiencyof the newly developed SLSI model to the existingfully explicit Eulerian model, we carried out numer-ous 48 hour forecasts with different initial and bound-ary conditions datasets, but presented one case forbrevity. Both subjective and objective comparison ofthe results indicate that the SLSI model provides fore-casts that are meteorologically equivalent to theEulerian model, numerically stable and convergent,and computationally faster. In particular, we havefound that for the present configuration of the modelwith 0.75° uniform horizontal resolution and 19 σ-levels in the vertical, the SLSI model with a time stepof 18 min, that is 12 times greater than the corre-sponding fully explicit Eulerian model time step of1.5 min, can reduce the single-CPU time on the NECSX-4 supercomputer by 52 per cent.

There are some important issues related to theSLSI scheme that we have not addressed in this paper,but are the subject of ongoing work. Such issues inter

alia include, (a) the semi-Lagrangian treatment ofvertical advection, so that the scheme becomesunconditionally stable for both horizontal and verticaladvection; (b) the semi-implicit treatment of theCoriolis terms leading to an unconditional stability ofthe gravity-inertia waves; and (c) the use of higher-order schemes on the A grid. Our immediate goal,however, is to test the 2D SLSI scheme for a larger setof cases, at various horizontal and vertical resolu-tions, before we implement the scheme operationally.

AcknowledgmentsThe authors are thankful to Dr Kamal Puri, groupleader of the Model Development group at BMRC,for his continued support and encouragement for theresearch reported in this paper. One of the authors(Sajal Kar) is grateful to Dr Michael Naughton fornumerous fruitful discussions regarding the semi-Lagrangian method. The authors are thankful to DrsWilliam Bourke, Michael Naughton and YuqingWang for their insightful review of the original man-uscript as a part of the BMRC internal review process.Finally, the authors wish to thank Prof. Lance Leslieof UNSW and another anonymous reviewer for theircomments, which led to substantial improvements inthe presentation of the material.

Appendix AList of Constants and Variablesa radius of the earth (6.37 x 106 m)Ω angular speed of the earth (7.292 x 10-5 s-1)g acceleration due to gravity (9.806 m s-1)f Coriolis parameter (s-1)R gas constant of dry air (287.0 J kg-1 K-1)cp specific heat of dry air (1004.0 J kg-1 K-1)λ longitudeθ latitudeσ vertical coordinateσT value of σ at the model topσ. σ−vertical velocity (s-1)t time (s)u zonal velocity (m s-1)v meridional velocity (m s-1)r water vapour mixing ratioT temperature (K)Tv virtual temperature (K)p pressure (Pa)ps surface pressure (Pa)Φ geopotential (m2 s-2)ΦS surface geopotential (m2 s-2)D horizontal divergence (s-1)

Kar and Logan: A semi-Lagrangian and semi-implicit scheme 315

PΨ tendency for Ψ due to the physical parametrisa-tion

KΨ tendency for Ψ due to the damping termsAppendix BDefinition of the matrices A and MThe non-zero elements, ak,l, of the upper triangularmatrix of dimension K x K are defined as follows:

The elements, Mk,l , of the matrix of dimensionK x K are defined as follows:

where Jm,l are the elements of the matrix ofdimension K x K, defined by

In (B.3), σ~m is defined as an arithmetic average ofσm+1/2 and σm-1/2.

ReferencesArakawa, A. and Lamb, V.R. 1977. Computational design of the basic

dynamical processes of the UCLA general circulation model.Meth. Comput. Phys., 17, 173-265.

Bartello, P. and Thomas, S.J. 1996. The cost-effectiveness of semi-Lagrangian advection. Mon. Weath. Rev., 124, 2883-97.

Bates, J.R. 1988. Finite-difference semi-Lagrangian techniques for

integrating the shallow water equations on the sphere. Proc.Workshop on Techniques for Horizontal Discretisation inNumerical Weather Prediction Models (Nov. 1987), Reading,ECMWF, 97- 116.

Bates, J.R, Moorthi, S. and Higgins, R.W. 1993. A global multilevelatmospheric model using a vector semi-Lagrangian finite-differ-ence scheme: Part I: Adiabatic formulation. Mon. Weath. Rev.,121, 244-63.

BoM 1998. BoM-NMC Analysis and Prediction Operations BulletinNo. 45. NMOC, Bur. Met., Australia.

Bourke, W., Hart, T., Steinle, P., Seaman, R., Embery, G., Naughton,M. and Rikus, L. 1995. Evolution of the Bureau of Meteorology’sGlobal Assimilation and Prediction system. Part 2: resolutionenhancements and case studies. Aust. Met. Mag., 44, 19-40.

Corby, G.A., Gilchrist, A. and Newson, R.L. 1972. A general circula-tion model of the atmosphere suitable for long period integra-tions. Q. Jl R. Met. Soc., 98, 809-32.

Côté, J. 1988. A Lagrange multiplier approach for the metric terms ofsemi-Lagrangian models on the sphere. Q. Jl R. Met. Soc., 114,1347-52.

Côté, J., Desmarais, J.-G., Gravel, S., Méthot, A., Patoine, A., Roch,M. and Staniforth, A. 1998. The operational CMC-MRB GlobalEnvironmental Multiscale (GEM) model. Part II: Results. Mon.Weath. Rev., 126, 1397-418.

Davies, H.C. 1976. A lateral boundary formulation for multi-levelprediction models. Q. Jl R. Met. Soc., 102, 405-18.

Haltiner, G.J. and Williams, R.T. 1980. Numerical Prediction andDynamic Meteorology. 2nd ed. John Wiley & Sons, 477 pp.

Kar, S.K. 2000. Stable centred-difference schemes, based on anunstaggered A grid, that eliminate two-grid interval noise. Mon.Weath. Rev., 128, 3645-53.

Klemp, J.B., and Wilhelmson, R. 1978. The simulation of three-dimensional convective storm dynamics. J. Atmos. Sci., 35,1070-96.

Leslie, L.M. and Purser, R.J. 1991. High-order numerics in anunstaggered three-dimensional time-split semi-Lagrangian fore-cast model. Mon. Weath. Rev., 119, 1612-23.

Lin, C.A., Laprise, R. and Ritchie, H. 1997. Numerical methods inAtmospheric and Oceanic Modelling: The Andre J. RobertMemorial Volume. NRC Research Press, 581 pp.

Lynch, P. and Huang, X.-Y. 1992. Initialization of the HIRLAMmodel using a digital filter. Mon. Weath. Rev., 120, 1019-34.

McDonald, A. 1986. A semi-Lagrangian and semi-implicit two time-level integration scheme. Mon. Weath. Rev., 114, 824-30.

McDonald, A. and Haugen, J.E. 1992. A two-time-level, three-dimensional semi-Lagrangian, semi-implicit, limited-area grid-point model of the primitive equations. Mon. Weath. Rev., 120,2603-21.

McPherson, R.D. 1971. Note on the semi-implicit integration of afine mesh limited-area prediction model on an offset grid. Mon.Weath. Rev., 99, 242-6.

Mesinger, F. 1973. A method for construction of second-order accu-racy difference schemes permitting no false two-grid-intervalwave in the height field. Tellus, 5, 444-57.

Mesinger, F. and Arakawa, A. 1976. Numerical methods used inatmospheric models. WMO/ ICSU Joint Organizing Committee,GARP Publications Series No. 17, 64 pp.

Miller, M.J. and Pearce, R.P. 1974. A three-dimensional primitiveequation model of cumulonimbus convection. Q. Jl R. Met. Soc.,100, 133-54.

Puri, K., Dietachmayer, G.S., Mills, G.A., Davidson, N.E., Bowen,R.A. and Logan, L.W. 1998. The new BMRC Limited AreaPrediction System, LAPS. Aust. Met. Mag., 47, 203-23.

Purser, J. and Leslie, L.M. 1988. A semi-implicit, semi-Lagrangianfinite-difference scheme using high-order spatial differencing ona nonstaggered grid. Mon. Weath. Rev., 116, 2069-80.

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

where Dk bα∇−2G′−k+( )

n 1+ Pk , 1 k K≤ ≤ −= ...63

...64

...65

...66

...67

...68

...69

...70

...71

...72

...73

...75

...77

...78

...79

...76

...74

∇−2Ψ

1a θcos( )2

-----------------------λ

2

2

∂−

∂ Ψθ

θ∂−∂−

θθ∂−

∂Ψcos − −cos+ =

Pk1

a θcos---------------λ∂−

∂− Fu( )k θ∂−∂− Fv( )k θcos[ ]+

−−−

−−−=

G≈− αM≈−

D≈−+ H≈−

=

D≈−

bα∇−2G≈−+ P

≈−=

G≈− α λkδk l,( )D≈−+ H≈−=

( ) = ( )

D≈−bα∇−

2G≈−+ P≈−=

D≈−

G≈−

P≈−

H≈−, , , E

≈−1_ D

≈−G≈−

P≈−

H≈−, , ,

Gk αλkDk+ Hk=

Dk bα∇−2Gk+ Pk=

b∇−2Gk (µ

2)kGk_ ℑ−k , 1 k K≤ ≤ −=

(µ2)k (α

2λk) 1_

= ; ℑ−k (µ2)k αλkPk Hk_( )=

δλΨ1

∆λ------- Ψ λ

∆λ2-------+ θ, −

−Ψ λ

∆λ2-------

_ θ, − −_=

Ψλ 1

2--- Ψ λ∆λ2-------+ θ, −

−Ψ λ∆λ2-------

_ θ, − −+=

Φ RTv Λ∇−+∇− =1

a θcos--------------- δλΦλ R Tλ

v δλΛλ

+ − −1

a--- δθΦθ R Tθ

v δθΛθ

+ − −,

∇−| 2Ψ

1a θcos( )2

----------------------- δλ δλΨ( ) θ δθ θ δθΨcos( )cos+ ][

=

Dk bα∇−| w2 G+ Pk =

Dk bα 1 γ_( )∇ −| w2 G γ ∇ −| 2 G+[ ]+ Pk =

For

...B1

...B2

...B3

for

k 1 2 …−K 2_, , ,=

a k l,

12--- ∆ σln( )l 1 2⁄−+ l, k=

12--- ∆ σln( )l 1 2⁄−_ ∆ σln( )l 1 2⁄−++[ ] l, k 1 …−K 1_, ,+=12--- ∆ σln( )K 1 2⁄−_ ∆ σln( )K 1 2⁄−++ l, K ;=

−−−−−−−−−

=

k K 1_=

For ; for

...B4

ak l,

12--- ∆ σln( )l 1 2⁄−+ l, K 1_=

12--- ∆ σln( )K 1 2⁄−– ∆ σln( )K 1 2⁄−++ l, K=−

−−−−

=

k 1 2 …−K, , ,= l 1 2 …−K, , ,=

Mk l,

RT0 ∆σ( )l1 σT

_------------------------ 1 κak m,σm

----------- 1 1 σT_( )Jm l,

_ σm σ~ m_( )+

m k=

K

∑−+ ,

=

Jm l,

0 l m<,

12--- l, m=

1 l, m>−−−−−

=

A~~

(D, G, P, H)~~

E~~ E-1~~M~~J~~

~~ ~~ ~~

(Dk, Gk, Pk, Hk)

(D, G, P, H)~~ ~~ ~~ ~~

~ ~(i, j) ~ ~(i*, j*)

un+1,υv+1( )k k

Dn+1,(G')n+1[ ]k k , Pn+1k , Hn+1

k

^

^ ^ ^ ^

Dk^

Gk^

^ ^ ^

( )θ

∇−| 2w

∇−| 2

( )

( )

316 Australian Meteorological Magazine 49:4 December 2000

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