CONSERVATIVE SCHEMES FOR THE SHALLOW-WATER MODEL

Yuri N. Skiba and Denis M. Filatov

E-mails: skiba@servidor.unam.mxdenisfilatov@mail.ru

Universidad Nacional Autónoma de MéxicoCentro de Ciencias de la Atmósfera

BASIC VARIABLES

( ),u v

x

z

( ), ,H x y t

y

( ),Th x y

- velocity field( , )u v

- free surface height( , , ) Th x y t H h= −

- topography( , )Th x y

( , , )H x y t - fluid depth

( ), ,h x y t

b) periodic channel in x

u u u hu v fv gt x y x

∂ ∂ ∂ ∂+ + − = −

∂ ∂ ∂ ∂

v v v hu v fu gt x y y

∂ ∂ ∂ ∂+ + + = −

∂ ∂ ∂ ∂

0H Hu Hvt x y

∂ ∂ ∂+ + =

∂ ∂ ∂

Initial conditions:

Periodic conditions in x:

Lateral boundary conditions :

( ) ( ) { }0, ,0 , , , ,R x y R x y R u v H= =where

( ) ( ) { }0, , , , , , ,R y t R L y t R u v H= =where

( ) ( ),0, , , 0v x t v x M t= =

SHALLOW WATER MODEL IN PERIODIC DOMAINS ON A PLANE

a) periodic domain in both x and y

( ) 0d HAdt

=Column volume conservation:

Relative hightconservation:

Potential vorticityconservation:

LOCAL CONSERVATION LAWS FOR SHALLOW-WATER MODEL

1 divdA uA dt

=r1 divdH u

H dt= −

r

Since

and

{ } 0Tz hddt H

−=

{ } 0 , (rot ) zd f udt H

ζ ζ+= =

r

( ) constD

M t Hdxdy= =∫

( )2 2

2 2( ) const2 2 T

D

u v gE t H h h dxdy⎛ ⎞+

= + − =⎜ ⎟⎝ ⎠∫

Mass conservation:

Total energy conservation:

The classic schemes by Arakawa & Lamb (1981), Sadourny (1975), Takano & Wurtele (1982), Kim (1984) as well as recent schemes by Ringler &

Randall (2002) and Salmon (2004), conserve the energy and/or the potential enstrophy only if the model is still continuous in time,while the discretization in time destroys all these laws.

Potential enstrophyconservation: ( )

2

2

1 const2D

v uJ t f dxdyx y H

⎛ ⎞∂ ∂= − + =⎜ ⎟∂ ∂⎝ ⎠∫

INTEGRAL CONSERVATION LAWS FOR THE SHALLOW WATER MODEL

( ) 0 (0, )

(0)

d A t Tdt

g

ϕ ϕ

ϕ

+ =

=

inr

r

r r, 0Aϕ ϕ =r r 0ϕ∀ ≠

rLet A be antisymmetric:

( )( ) (0) , 0,t g t Tϕ ϕ= = ∈r r r

11

12

(1 ) 0

( , 0 1)

j jj jAϕ ϕ αϕ α ϕ

τα α

++− ⎡ ⎤+ + − =⎣ ⎦

≠ ≤ ≤

r rr r

111 1

2 2

12

0

( )

j jj jAϕ ϕ ϕ ϕ

τα

++− ⎡ ⎤+ + =⎣ ⎦

=

r rr r

1j jϕ ϕ+ ≠r r

1j jϕ ϕ+ =r r

CONSERVATION LAW AND TIME DISCRETIZATION

Discretization effect:

DIVERGENT FORM OF THE MODELUsing the change of variables (Skiba, 1995)

1 12 2

U uU U vU U hu v fV gzt x x y y x

⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂⎡ ⎤+ + + + − = −⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦

1 12 2

V uV V vV V hu v fU gzt x x y y y

⎡ ⎤∂ ∂ ∂ ∂ ∂ ∂⎡ ⎤+ + + + + = −⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦

0H zU zVt x y

∂ ∂ ∂+ + =

∂ ∂ ∂which is essential for constructing conservative difference schemes.

z H= U zu= V zv=

the equations of shallow-water model are transformed to a divergent form

The discrete divergent forms

1 1 1 1 1 1i i i i i i i i i ii

a R a R R R a R a RaR Ra al l l l l

+ + − + + −− − −∂ ∂+ ⋅ ≈ + =

∂ ∂ Δ Δ Δ

Similarly, the discrete divergent forms of advective terms guaranteethe conservation of total energy, since

( )1

1 1 11 0 0 1

0

1 0I

i i i ii I I I

i

a R a RR a R R a R R

l l

−+ + −

− −=

−= − =

Δ Δ∑

for periodic conditions as well as for conditionsat the lateral boundary of channel.

BASIC PROPERTY OF THE DIVERGENT FORM

0 0Ia a= =

1 1i i i ia R a RaRl l

+ + −∂≈

∂ Δ

( )1

1 10 0

0

1 0I

i i i iI I

i

a R a Ra R a R

l l

−+ +

=

−= − =

Δ Δ∑

and

guarantee the conservation of mass, since

( ) ( )cos1 1 1 coscos 2 2

tan ,cos

uU vUU U Uu vt a

u gz hf Va a

ϕϕ

ϕ λ λ ϕ ϕ

ϕϕ λ

⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂ ∂+ + + + −⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

∂⎛ ⎞− + = −⎜ ⎟ ∂⎝ ⎠

( ) ( )cos1 1 1 coscos 2 2

tan ,

uV vVV V Vu vt a

u gz hf Ua a

ϕϕ

ϕ λ λ ϕ ϕ

ϕϕ

⎡ ⎤∂ ∂⎛ ⎞ ⎛ ⎞∂ ∂ ∂+ + + + +⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

∂⎛ ⎞+ + = −⎜ ⎟ ∂⎝ ⎠

( ) ( )cos1 0cos

zU zVht a

ϕϕ λ ϕ

∂ ∂⎡ ⎤∂+ + =⎢ ⎥∂ ∂ ∂⎣ ⎦

SHALLOW WATER MODEL ON A SPHERE

SPLITTING METHOD

0U fVt

∂− =

∂0V fU

t∂

+ =∂

Splitting step 3 (sphere rotation):

Splitting steps 1 (in x) and 2 (in y):

12

U uU U hu gzt x x x

∂ ∂ ∂ ∂⎡ ⎤+ + = −⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦

1 02

V uV Vut x x

∂ ∂ ∂⎡ ⎤+ + =⎢ ⎥∂ ∂ ∂⎣ ⎦

0H zUt x

∂ ∂+ =

∂ ∂

1 02

U vU Uvt y y

⎡ ⎤∂ ∂ ∂+ + =⎢ ⎥∂ ∂ ∂⎣ ⎦

12

V vV V hv gzt y y y

⎡ ⎤∂ ∂ ∂ ∂+ + = −⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦

0H zVt y

∂ ∂+ =

∂ ∂

x

yu t⋅Δr

TWO MAPS USED IN THE SPLITTING METHOD

THE SPLITTING ALONG THE LONGITUDINAL DIRECTION λ

11 1 1 1 1 1

1 1

1 1cos 2 2 2

,cos 2

n nk k k k k k k k

kl

k k k

l

U U u U u U U Uua

gz h ha

τ ϕ λ λ

ϕ λ

++ + − − + −

+ −

− − −⎛ ⎞+ + =⎜ ⎟Δ Δ⎝ ⎠−

= −Δ

11 1 1 1 1 11 1 0

cos 2 2 2

n nk k k k k k k k

kl

V V u V u V V Vuaτ ϕ λ λ

++ + − − + −− − −⎛ ⎞+ + =⎜ ⎟Δ Δ⎝ ⎠

11 1 1 11 0

cos 2

n nk k k k k k

l

H H z U z Uaτ ϕ λ

++ + − −− −

+ =Δ

PROBLEM 1: THE SPLIT PROBLEM IN

11 ( )2

n nR R R+= +

λ

THE SPLITTING ALONG THE LATITUDINAL DIRECTION ϕ

11 1 1 1 1 1

1 1

cos cos1 1cos 2 2

cos 0,2

n nl l l l l l l l

l

l ll l

U U v U v Ua

U Uv

ϕ ϕτ ϕ ϕ

ϕϕ

++ + + − − −

+ −

− ⎛ −+ +⎜ Δ⎝

− ⎞+ =⎟Δ ⎠

11 1 1 1 1 1

1 1 1 1

cos cos1 1cos 2 2

cos ,2 2

n nl l l l l l l l

l

l l l l ll l

V V v V v Va

V V gz h hva

ϕ ϕτ ϕ ϕ

ϕϕ ϕ

++ + + − − −

+ − + −

− ⎛ −+ +⎜ Δ⎝

− ⎞ −+ = −⎟Δ Δ⎠

11 1 1 1 1 1cos cos1 0

cos 2

n nl l l l l l l l

l

H H z V z Va

ϕ ϕτ ϕ ϕ

++ + + − − −− −

+ =Δ

PROBLEM 2: THE SPLIT PROBLEM IN

11 ( )2

n nR R R+= +

ϕ

1

tan 0n nkl kl kl

l l klU U uf V

aϕ

τ

+ − ⎛ ⎞− + =⎜ ⎟⎝ ⎠

1

tan 0n n

kl kl kll l kl

V V uf Ua

ϕτ

+ − ⎛ ⎞+ + =⎜ ⎟⎝ ⎠

PROBLEM 3: A SPHERE ROTATION

MASS AND TOTAL ENERGY CONSERVATION LAWS

The mass and total energy conservation laws

( )2 2

21 const

2 2D

u v gE t H h dxdy⎛ ⎞+

= + =⎜ ⎟⎝ ⎠∫

( ) constD

M t Hdxdy= =∫

( )2 2

2 2( ) const2 2 T

D

u v gE t H h h dxdy⎛ ⎞+

= + − =⎜ ⎟⎝ ⎠∫

Evidently, it is sufficient to conserve only the variable part of total energy

CONSERVATION LAWS FOR DISCRETE MODELS

In discrete SW system, the mass and total energy conservation laws

1 2 cosn n nl kl

l k

M M a Hλ ϕ ϕ+ = ≡ Δ Δ ∑ ∑

11 1n nE E+ =

( )2 2 22 11 2cosn n n n

l kl kl kll k

E a U V g hλ ϕ ϕ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤= Δ Δ + +⎣ ⎦ ⎣ ⎦ ⎣ ⎦∑ ∑

where

accept the forms

1n nM M+ = and

EXPERIMENT 1: PLANE. THE HEIGHT h

T=1,3,4,6,7,14 days 4th approximation order

EXPERIMENT 1: PLANE. VELOCITY T=1,3,4,5,6,7 days 4th approximation order

EXPERIMENT 1: PLANE. POTENTIAL VORTICITY

T=1,3,4,5,6,7 days 4th approximation order

INITIALCONDITIONS

T=0

U

h

EXPERIMENT 2: SPHERE.

ROSSBY WAVE

SPHERE

0 50 100 150 200 250 300 350

−80

−60

−40

−20

0

20

40

60

80

λ

ϕ

Velocity at t = 3.5 days

0 50 100 150 200 250 300 350

−80

−60

−40

−20

0

20

40

60

80

λ

ϕ

Velocity at t = 3.5 days

2-order scheme 4-order scheme

0 50 100 150 200 250 300 350

−80

−60

−40

−20

0

20

40

60

80

λ

ϕ

Velocity at t = 7 days

0 50 100 150 200 250 300 350

−80

−60

−40

−20

0

20

40

60

80

λ

ϕ

Velocity at t = 7 days

T=3.5 days

T=7 days

EXPERIMENT 2: SPHERE. VELOCITY

GRID 6º X 6º GRID 3º X 3º

T=7 days

h

T=3.5 days

EXPERIMENT 2: SPHERE. HEIGHT

4-order scheme

0 1 2 3 4 5 6 72.852

2.854

2.856

2.858

2.86

2.862

2.864

2.866

2.868

2.87

2.872x 10

4 Potential Enstrophy

Time (days)

J(t)

0 1 2 3 4 5 6 72.852

2.854

2.856

2.858

2.86

2.862

2.864

2.866

2.868

2.87

2.872x 10

4 Potential Enstrophy

Time (days)

J(t)

0 1 2 3 4 5 6 72.852

2.854

2.856

2.858

2.86

2.862

2.864

2.866

2.868

2.87

2.872x 10

4 Potential Enstrophy

Time (days)

J(t)

0 1 2 3 4 5 6 72.852

2.854

2.856

2.858

2.86

2.862

2.864

2.866

2.868

2.87

2.872x 10

4 Potential Enstrophy

Time (days)J(

t)

EXPERIMENT 2: SPHERE. POTENTIAL ENSTROPHY

1-2 order scheme 3-4 order scheme

4 order scheme2 order scheme

MAXIMUM VARIATION OF POTENTIAL ENSTROPHY

FOR DIFFERENT GRIDSin %

EXPERIMENT 3: SPHERE.TIME BEHAVIOR OF POTENTIAL ENSTROPHY

WITH RANDOM INITIAL PERTURBATIONS

10( ) (0)log

(0)J t J

J−

2nd order scheme 4th order scheme

Yuri N. Skiba and Denis M. Filatov

E-mails: skiba@servidor.unam.mxdenisfilatov@mail.ru

Universidad Nacional Autónoma de MéxicoCentro de Ciencias de la Atmósfera

CONSERVATIVE SCHEMES FOR THE SHALLOW-WATER MODEL