-
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