Calhoun: The NPS Institutional Archive

Faculty and Researcher Publications Faculty and Researcher Publications

1989

Rossby Wave Frequencies and Group

Velocities for Finite Element and Finite

Difference Approximations to the

Vorticity-Divergence and the Primitive

Forms of the Shallow Water Equations

Neta, Beny

http://hdl.handle.net/10945/39480

Rossby Wave Frequencies and Group Velocities for Finite Element and Finite Difference

Approximations to the Vorticity-Divergence and the Primitive Forms of the Shallow Water

Equations

Deny ~et.a IL T. \\Tilliains

Naval Postgraduate School l\:Iontcrcy~ CA 93943~ U. S. A.

Abstract

In this paper l{ossby wave frequenciPs and group velocitiPs are analyzed for vari-

ous finitP elPment and finite diffPrence approximaJiorrn to the vorticity-divergence form

of the shallow ·water eaquations. Also included a.re finite difference solutions for the primitive equations for the staggered grids B and C from \Vajsowicz and for the unstag-gcrcd grid A. The result;,; arc pre;scntcd for three ratio:,< bct>vccn 1.hc grid 8izc and the l{ossby radius of deformation. The vorticity-divergence equation schemes givP supPrior

solutions to those based on thP primitive equations. The lwst results come from thP

finite element schemes that use linear basis functions on isosceles triangles and bilinear functions on rectangles. All of the primitive equation finite difference schemes have problem;,; for at least. one Ro;,;;,;by deformation-grid size ratio.

1 Introduction

The hydrostatic primitive equa.tion munerica.l models that are used for atmospheric a.ncl oceanographic prediction permit inertial gravity 'vaves, Rossby vrnves, a.ncl a.clvective effects. The influence of a. numerical scheme on :each of these types of motion is most easily analy.zed by separating the linearized prediction equatious into vertical modes with an equi va.lent depth analysis (for example, see Gill 1982). In this ca.se the equations for each vertical mode a.re just the linearized shallow equations with the appropriate equi va.lent depth. In fact~ oue must also consider the vertical dilTerencing in deriving the shallow water system, but we will not treat these dfrds in this paper. /\ r

wave rnotions for fom finite diifrrc;ncc grids that they labc:k:d .1, H, C, and /)_ They found that the geostrophic adj 1.istment for the unstaggered grid A. a.nd grid D is poor and that the adj1.1stment for grids B a.ncl C is good. Schoenstadt (1980) studied geostrophic adjustment for finite elements v;ith piece>vise linear basis f1rnctions with the noda.l points located a.t the finite difference grid points. He determined tha.t the unstaggrred finite element scheme (grid A) gives poor a.dj ustment for small sea.le motions, but the schemes B and C are excellent. \Villi a.nm ( 1981) examined geor::;trophic adjustment in the vorticity-divergence form of the shallow water equatious with finite di.ITerence and finite element schemes. Ile r::;howed that the nonstaggered vorticity-divergence schemes give as good geostrophic a.djur::;tment a.r::; the best staggered shallow water r::;chemer::;. Since finite element models with staggered ba.r::;ir::; functiom a.re much more complicated, especially in two dimeusious, the best finite element sc:hcrnc;s for gcostrophic: adjnstrnc;nt use: the: vorticity-divc:rgc;nccforrm1lation. Some cxarnpk:s

of atrnosphcric prcdidion rnodcls of this type arc given by Staniforth and IV!itc:hcll (1977, 1978), Staniforth and Dalc;y Cl979), and C11llc;n and Hall (1979).

The: objc:divc: of this study is to investigate the trc:atrnc;nt of J{ossby v,·avc;s in vorticity-

divc:rgc;ncc sha I low v.:afrr forn111 lations v.:it h vari01rn fin ik clcrnc;nt and finite diifrrc;ncc sc:hcrnc;s_

~·or cornparison the finik difforcncc: primitive: cq1rntion solutions for grids /,, H, and Carr; also included. The finite difference solutions for grids B a.nd C are ta.ken from a recent a.nd very complete study by \Vajsmvicz (1986). An earlieL one-dimensiona.l study on these grids was ca.rried out by }fesinger (1979).

2 Basic equations

The linearized shallmv 'va.ter eq1.1ations on a beta plane can be written

Du Iv

Dh (2.1) - + q- - 0, at . ax

Dv .f-u.

Dh (2.2) - + + q- 0, Dt . au

ah gH ciu Dv) 0, (2.3) - + + = Dt D:r Dy

where u and v are the velocit_y perturbatious, his the height perturbation and II the equiv-alent depth, and I is the Coriolis para.meter. The vorticity-divergence equation r::;et, which is obtained by di.ITerentiating (2. l) and ( 2.2) with respect Lo 1' and y respectively and com-bining, can be wri tLen

an () l

() ( Dt + f n + v8 = 0,

(

()2 h ()2 h.) f ( + u,/"J + g --- + ---() :i: 2 f) y2

2

(2. ·1)

0, (') _,) -·O

i}h. Hl +HD 0, (2.6)

where/] df /dy,

·" av Du

~ -/);i; Dy

and

Du av D = -+

i}a; au· To isol. = (gH) 1 l 2 / fu is the Ross by radii.ts of deformation. The hvo components of the group velocity are given by

a~.) = :r [112 - (k2 + >.-2)] fJµ t I_) (µ2 + k'2 + ).-'.l)2 ' (2."12)

and

G'Y T ( f.1·2 + k2 + ),-2)2 . (2.U)

* X = 11.Ll.1~ and ~· = kily

Vort icity-Divergence Form Scheme I'iniLe Elements

Opc;rator i\ n

Vorticity-Divergence Form Scheme Finite Differences

Operator i\ nalytic: Sccond Ordcr F'o11rt h Ordcr

(t l l l

() sinX 4sinX 1sin2X

1-l -- ---- -D.x ;3 D.x 6 D,;i;

0 ') sin 2 1- cos2X -16co::; X + 17)

f,1.-

( ~x r 6D.x2 k2

sin2 f c:os 2Y - -1 6 cos y + 1.5 c.

(~!Ir 6D.y2

Table: 2: The Opcrators for the: Various ~'inite l)ifforc:nc:c Schcrnc:s for the: Shallmv Water ~~q11ations in Vorticity Divcrgencc: F'orrn

Primitive Form Scheme Finite Differences

Operator ,'\nalytic /\ H c _,

2 x 2 )/ O' 1 1 1 cos -cos -

2 2

sm x y sm x sin X 2 y (} f-1· --cos- -- --cos -D.:i:: 2 D.a:: D.J· 2

sin2 X . ') x 1 +cos :y sin 2 x sm-

0 2 2 2 µ D,;i;2 ( ~a·)2 2 ( ~a·r

sin 2 ") y ") y

k2 v sin:.. 2 1 +cos .X sin:.. 2 c --- ---2 ---2 D.y~ ( ~!!) 2 ( ~!!)

Table: :3: The Opcrators for the: Various ~'inite l)ifforc:nc:c Schcnws for the: Shallmv Water ~~q11ations in Prirnitivc Form

Scheme lkriv

Vorti ci ty-1 )i vc:rgc:ncc: Form Sclwrnc Finite: I )ifforcn cc:s

Derivative Analytic Second Order Fourth Order

(Jn 0 0 0

D1-1,

[)() cosX

4 1 - 1 - cosX - - cos 2X a1, 3 3 ()6 r:; inX 8 r:; in X - r:;in 2X - 2p 2--D1-1, 6.:r 36.:r

(Js - 0 0 0 a11 (Jn

() () () -Dk

()0 - 0 0 0 Dk

as - 0 0 0 Hk

[);:- ·y 8 sin Y - sin 2Y - 2k 2

srn

Dk 6.y 36.y

Table 5: The Derivatives Req1_1ired for the Group Velocity

7

Primitive Form Scheme Finite Differences

lkriv

A grid

B grid

C grid

Analytic

Rectangles

Isosceles

FD 2nd

FD 4th

Staniforth

0 0.2 0.4 0.6 0.8 13.5

3

2.5

2

1.5

1

0.5

0

µ d/

F

A grid

B grid

C grid

Analytic

Rectangles

Isosceles

FD 2nd

FD 4th

Staniforth

0 0.2 0.4 0.6 0.8 13.5

3

2.5

2

1.5

1

0.5

0

µ d/

F

A grid

B grid

C grid

Analytic

Rectangles

Isosceles

FD 2nd

FD 4th

Staniforth

0 0.2 0.4 0.6 0.8 13.5

3

2.5

2

1.5

1

0.5

0

µ d/

F

skep slope of the frcq1wncy rnrve. . . . 2 . 2.

The freq1.1ency curves for k = U a.ncl d /(4,\ ) = 1.0 a.re given in Fig. 2. The general beha.vior is similar to Fig. 1 v;ith certa.in exceptions. All schemes ha.ve larger errors as 1-l d/ri a.pproaches 1 because the analytic sohition is nea.r its ma.ximum va.lue there, a.nd the isosceles FEl\I scheme is the best in this area since it does not drop all the 'vay to zero. ::\ ea.r 1-l d/ri = 1/2, FD scheme C gives the best results, but it then drops off to zero. The poorest schemes are FD scheme I3 and the second-order vorticity-divergence FD scheme. Ther:;e schemes are equivalent whenever/;; = 0. The FD scheme il doer:; not give poor results in this ca.fie became the >.-'.l term in the denominator of ( :L4) is not small, so that the underestimate of 6 ir:; not so important.

The frequenc_y curves for k = 0 and lt2 / ( 4,\ '.l) = 10 a.re given in fig. :L In thir:; ca.r:;e the ;rn

~-igme J .j indicaks that G~. for sc:hcnw C is also an order of magnitmk too large above the diagona.l. The FD schemes A_ (Fig. 14a) and B (Fig. 14b) do not have poor behavior, a.ncl the other schemes are similar in pattern to the other cases. The exception is the isosceles triangle FEl\:I scheme (Fig. 15h) which gives a spurious positive frequency near ri.rl/'rr = 1. This leads to excessivley large va.lues of OF. The behavior in this region is rela.ted to the expression for [) h/ [) :r on the isosceles triangles tha.t leads to a poor representa.tion for small y-scales (see ~eta and \Villiams 1986).

5 Conclusions

In this paper we analyze Rossby wave freq1_1encies and grmtp velocities for va.nous finite element and fini Le diITerence approximaLiorn:; to the vorLici Ly-divergence form of Lhe shallow waLer equatious. Also included are finite diITerence iioluLiom; for Lhe primitive equatious for grids il, fl, and C. The resulLii for the staggered grids fl and C are taken from \Vaji:iowicz ( 1986). The equal.ions are evaluated in Lhree caLegorief:i where Lhe grid f:ii.ze is smaller than, Lhe same order aii, or larger than Lhe Hosf:lby radim of deformation. The TI.of:lsby radim of deformation can be v.:ritkn in krms of the equivalent depth so that Vi'lrious vertic:al modes can be considered.

The results shmv that all sc:hcmes converge in the large scale limit (pd, kd ----+ 0). For the case where the grid si?;e is sm

Staniforth,/\ .. \., and H. L. !Vtitchdl, "1978: A vrwiabfr; rr;so!ution finifo - rlcmr.nt tcdmiqur. for regional forecasting with the prirnitive equations, :Mon. \Vea. Rev. 106, 4:39 - 447.

ScaniforLk A. ~., aud IL \V. Daley, 1979: /1 baroclinic fi.nil t -tltrntnl nwdtl fur regional f01Y0 r.asting with thr. primdivr. uprntion;;, l'vlon. \Vrn. 1-lev. 107, "l07- 121.

\Va.jsowicz, R. C., 1986: Frtt plamlary ·leaves in finit e - d~{ft:rtnct numerical nwdtls, .J. Phys. Oceauogr., 16, 77:~-789.

\Villiams, R.T., 1981: On the formulation of finite - element prediction models, :\Ion. \Vea Rev. 109, 46:3-466.

/"..icnkievvicz, 0. C., 1977: '/hr. Finitr. f;Jr.mrnt Air.t/wd in F~nginr.r.ring Sr.irna, Wiley, 787 pp.

APPENDIX A

Coefficients for Finite Element Schemes

\Ve illustrate the general procedmc by deriving (:3.1) from (2. 7). First express the dependent variables in terms of the basis function (p;(x,y) as follows:

[ '] [(jl n_ = L D_j C!r h J hj To apply· Lhe Ga1erkin procedure we subsLiLuLe (A.l) iuLo ('.2.7), rnulLipl,y by q'J; and integrate over the domain to force the error Lo be orthogonal to the basis fuucLions gi viug

0. (A.2)

The isosceles triangle basis function i::; shown in fig. Hi. The following express10us for in Legra.Lion over Lhe Lriaugles cau be found iu Zienkiewicz ( 1977):

j. - ' I ~ { /' I 6 i = j (p.; OJ t: • 1 = ,,1.1 ') : _.;_ . T I - /, I], ( /,_:3)

(A.4)

(A.5)

I. a(p.; D(;b,; _

--:::;--- --;::;-- d A , J' uy uy

b,: b:i 4A'

a ,: a .i

4A

where T is a. triangular element, A is the area of T and aj and bj are defined by

(A.6)

(A.7)

The vertices of the triangles (a:.i, Y.i) arc m1rnbered c01111krclocbvise. \Vhcn ( ,-\.2) is cvah1atcd for the isosceles triangles 'vc obtain

. ·1 . . (u,u + G [(1,u + (-i.u + (1 ; 2,1 + (-1 ; 2,1 + ( 1;2,-1 + (-1 /2.-1]

+Io { Do.o + l [D1,o + D-1.0 + D1;2,1 + D-1/2,1 + D1;2,-1 + D-1/2,-il} (i\.8) 3o + :3fuf:Lr {2 [hi.o - h-i.o] + h1; 2,1 + h-1 / 2,1 + h1/ '2,-1 + h-1; 2,-d 0,

where each triangle has a base of D.. x and a height of D.. y. The super clot indicates a. partial time derivative and D 1; 2._ 1 is eq1_1al to D(x + D..x /2, y - D..y). The final form of (:3.1) is obtained by introducing Lhe spatial dependence exp [i (p ;r + k:i;)] for each dependent variable. Equatious (;L2) and (;L:~)are obtained in Lhe same manner buL inLegraLion by parLs is required for Lhe Laplacian of h in (2.8).

The equations for the bilinear basis fundious on reel.angles , are obtained in Lhe same manner as with the triangles. The integration formulae corresponding Lo (A.:~) to (A.7) are given by Staniforth and .Vlitdwll ('1977), and the details v.:ill not be reproch1ccd here.

APPENDIX B

Coefficients for Finite Difference Scheme A

The coefficients for the nonstaggerecl finite difference scheme A are derived here. The eq ua.-Lion seL (2.1) - ('.2.;n for this scheme can be wriLLen

16

au -~· 7il - f v + 96.r: h · o ,

[) :::iv + f u. + g6y h Y O , (J t

[)h -x ---y at+ II(6.r u + 6,,v) 0,

S~.h [h(x + D.x/ 2) - h(a; - D.a: / 2)] / D.a: and

hx = h(x + D.x/2) + h(x - D.:r/2)]/2. To obtain the vorticity-divergence fornrnla.tion 'Ne let

.- ---x -- -y ~ = 6); v - c)y u, ,

(B.1)

(B.2)

(n.:n

(fl.4)

Ily subLracLing and adding (Il.l) and (Il.2) and using (Il.4) Lhe vorticity-divergence f:iyf:itern becomef:i

f} .

~ + f JJ + 3v2 !! = o . iJ t ' '

/) h al +HD= 0,

( H . .5)

0, ( H.6)

(B.7)

where f = fo + /Jy and v2Y = [v (y + D. y) + v (y - D. y)]/2 is used to develop this form. The q1_iasi-geostrophic set is obtained by replacing (B.5) and (B.6) v;ith

D( B - - J;2Y ~. + r0 D + --:- lJ6,J1 = 0 , bf }' k. .. . (fl.8)

-.fo( + g (6.; hu + ;s; hyy) = (), (fl.9) which are analogous Lo (2.7) and (2.8). The required coefficients can be obtained by subf:lti-Luting the wave forms into (Il.7), (I3.8), and (I3.9).

17

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.76 2.76

2.49 2.21

1.93

1.66

1.38

1.11

0.829

0.553

0.276

a

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.63

2.63

2.37

2.37

2.11

2.11

1.84

1.84

1.58

1.58

1.32

1.32

1.05

1.05

0.79

0.79 0.526

0.526

0.263

0.263

b

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.76

2.49 2.21

1.94 1.66

1.38

1.11 0.829

0.553

0.276

c

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.77 2.49 2.21

1.94 1.66

1.38 1.11

0.83

0.553

0.277

d

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.63 2.37

2.11 1.84

1.58

1.32

1.05

0.79

0.526 0.526

0.263

0.263

e

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.76

2.48 2.2

1.93 1.65

1.38

1.1

0.827 0.551

0.551

0.276

0.276

f

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.72 2.45 2.18 1.9 1.63

1.36

1.09 0.816

0.544

0.272

0.272

g

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.69 2.37 2.05

1.73 1.42

1.1

0.78

0.462

0.144

0.144 0.175

h

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.377

0.754

1.13

1.13

1.51

1.51

1.88

1.88

2.26

2.26

2.64 2.64

2.64 3.02 3.02

3.39 3.39

3.77 3.77

a

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

3.19 2.48

1.77

1.06

0.354

0.354

1.06

1.77

2.48

3.19

b

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.423

0.847

1.27

1.69

2.12

2.54 2.96 3.39

3.81 4.23

c

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.376

0.376

0.753

1.13 1.51

1.88

2.26

2.63

3.01

3.39 3.76

d

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.366

0.732

1.1

1.46 1.83

2.2

2.56

2.93 3.29

3.66

e

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.376

0.752

1.13

1.5 1.88

2.26

2.63

3.01 3.38 3.76

f

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.38

0.761 1.14

1.52

1.9

2.28

2.66 3.04

3.42

3.8

g

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.392

0.784

1.18

1.57

1.96

2.35

2.74

3.14

3.53

3.92

h

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

15.2

11.8 8.42 5.05

1.68 1.68

5.05

8.42

11.8

15.2

a

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

15.2

11.8

8.42

5.05

1.68

1.68 1.68

1.68

5.05

8.42

11.8 15.2

b

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

16.6

14.6 12.6 10.7

8.73

6.77

4.81 2.85

0.888

1.07

c

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

16.6 14.8 12.9 11 9.1

7.22

5.33

3.44

1.56

0.328

d

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

16.6 14.6 12.7

10.7 8.78 6.83

4.88 2.93

0.983

0.967

e

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

16.6 14.7 12.7

10.8 8.86 6.92

4.99

3.05

1.12

0.816

f

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

16.6

14.6 12.7 10.7 8.75

6.79 4.84

2.88

0.925 1.03

g

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

16.6

14.7 12.7

10.8

8.87 6.94 5.01

3.08

1.15

0.783

h

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.865

0.779

0.692 0.606

0.519

0.433

0.346

0.26

0.173

0.0865

a

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.845

0.845

0.76

0.76

0.676

0.676

0.591

0.591

0.507

0.507

0.422

0.422

0.338

0.338

0.253

0.253

0.169

0.169

0.0845

0.0845

b

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.77 2.49 2.21 1.94

1.66

1.38 1.11

0.83

0.553

0.277

c

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.22 2

1.78

1.56

1.33

1.11

0.89

0.667

0.445

0.222

d

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.845

0.845 0.76

0.76

0.676

0.676

0.591

0.591

0.507

0.507

0.422

0.422

0.338

0.338 0.253

0.253

0.169

0.169

0.0845

0.0845

e

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.12 1.01

1.01

0.897

0.897

0.785 0.785

0.673

0.673

0.56

0.56

0.448

0.448

0.336

0.336 0.224

0.224

0.112

0.112

f

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.34 1.2

1.07

1.07

0.937

0.937

0.803

0.803

0.669

0.669 0.535

0.535

0.401

0.401

0.268

0.268

0.134

0.134

g

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.46

1.17

0.886 0.599 0.312

0.312

0.0255 0.0255

0.261

0.548 0.835 1.12

h

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.821

1.64

2.46

3.28

4.11

4.93

5.75 6.57

7.39

8.21

a

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.335

0.261

0.186

0.112

0.0373

0.0373

0.112

0.186

0.261

0.335

b

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 2.62

5.25

7.87

10.5

13.1

15.7

18.4

21

23.6

26.2

c

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.376

0.753

1.13

1.51

1.88 2.26 2.63

3.01

3.39

3.76

d

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0713

0.143

0.214

0.285

0.356 0.428

0.499

0.57

0.641

0.713

e

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.123

0.246 0.369

0.492

0.615

0.737

0.86

0.983 1.11

1.23

f

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.319 0.639

0.958

1.28

1.6 1.92

2.24

2.56 2.88

3.19

g

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.43

4.86

7.29

9.72

12.2

14.6

17

19.4

21.9

24.3

h

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

15.2

11.8 8.42

5.05

1.68

1.68

5.05 8.42

11.8

15.2

a

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

15.2

11.8

8.42

5.05

1.68

1.68

5.05

8.42

11.8

15.2

b

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

19.3

8.33

2.62

2.62

13.6

13.6

24.5

24.5

35.5

35.5 46.4 46.4

57.4

57.4 68.3 68.3

79.3

79.3

c

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

17.6

16.6

15.7 14.8

13.8

12.9

11.9

11

10

9.09

d

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 15.3

12.1

8.89

5.68

2.47

0.741

3.95 7.17

10.4

13.6

e

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

14.4

10.2

6.07

1.92

2.24

6.39

10.5

14.7

18.9

23

f

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

13

7.44

1.89

3.65

9.2

14.7

20.3

25.8

31.4

36.9

g

µ d/

kd/

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

14.3

10

5.8

1.56

2.69

6.93 11.2

15.4

19.7

19.7 23.9

23.9

h

µ d/

kd/

Fignrc 16: The isosceles triangle h;rnis function

:30