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 various 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 unstaggcrcd 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<lkav.:a and Lamb (1977) analy?;c:d inertial gravity
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 vorticitydivc:rgc;ncc sha Ilow 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.k en 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
-
Dh
Iv + .qax
at
Dv
-
ah
-
Dt
Dt +
+ gH
.f-u.
-
Dh
+ .qau
ciu + Dv)
Dy
D:r
0,
(2.1)
0,
(2.2)
= 0,
(2.3)
where u and v are the velocit_y perturbatious, his the height perturbation and II the equivalent 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 combining, can be wri tLen
() (
Dt
an
() l
f( +
+ f n + v8
u,/"J
+g
2
()2
(
---
h
() :i: 2
(2. ·1)
= 0,
()2
h.)
+ --f) y2
0,
') _,)
(-·O
i}h.
Hl
where/]
(2.6)
0,
+HD
df /dy,
av
·"
~
Du
-
/);i;
Dy
Du
av
and
-+
i}a;
au·
D =
To isol<ltc the H.ossby mock rnorc c<lsily \Ve apply the qnasi-gcostrophic <1pproxirnation
to the sd (2.1)-(2.6), (for r;xarnplc, sec Cha.pkr 3 in H<lltinr;r and \Vil Iiams ·1980):
ac
~
Ol
.
1
+ f oD + rJO
__ ro,~.·
JI -.,
ah ,
(JJ
[)2 h [).2 h)
+ .lJ ( a
+ -() l/2
X2
fJ h
Dt
(')
-·· '"'')
{
g ~ do = 0,
+ II D
=
(')-··'s·)
0,
0,
(2.9)
whr;rc fo m1d {J0 , arr; cvahrntcd <lt <ln appropri<ltc ccntr<ll latitude.
The plane 'vave expression for the Rossby wave freq1.tency is obtained by inserting the
wave forrns
(2."lO)
into (2.7) - (2.9) which gives
(2.n)
where >. = (gH) 1 l 2 / fu is the Ross by radii.ts of deformation. The hvo components of the
group velocity are given by
a~.)
fJµ
=
:r [112
t
I_)
(µ2
- (k2
+ >.-2)]
+ k'2 + ).-'.l)2
'
(2."12)
and
GT'Y
*X =
11.Ll.1~
and
~·
= kily
( f.1·2
+ k2 + ),-2)2 .
(2.U)
Scheme
Opc;rator i\ n<11ytic:
Isosceles Tri<1nglc;s
( 3 + cos x + 2 cos
O'
l
()
1-l
Vort icity-Divergence Form
I'iniLe Elements
1-frdan gn Iar
~ cos y) *
(2 + cos x) ( 2 + cos y)
6
2 (sin X + sin ·~ cos )/)
')
6
f.1--
._,
k2
:3
3
sin X (2 +ms Y)
--
;L~:i::
:~
6 :1::
.
2
sm
(~~r
(~xr
3
,. '.!rp- +
2
COS ,
(~1/ )
:3
(:~+ co::; X - 4 cos
"l cos Y)
26y'.!
Slll
2
(2 + cos X) (2 +cos Y)
:3
3
S111
x
--
(2 +cos V)
:~
. 2 x -sm
2 (o + cos y· )
(~~r
\')
.Vlitdwll
• '.! y (Slll 2 ,)
(~y)2
6
+
COS .
\')
6
(:S+cos.X) (5 +ms Y)
a'
Table I: The Operators for the V<1rious
~~q11ation s in Vorticity Divc;rgence Forrn
~\',
D.:i:
x
2 (2 +cos ·v)
sin2 K2
Sbniforth
()
~-inite ~~lc;nwnt
.j
Schemes for the Shallmv W<1ter
6
Scheme
Operator i\ nalytic:
Vorticity-Divergence Form
Finite Differences
Sccond Ordcr
F'o11rt h Ordcr
l
(t
()
l
sinX
D.x
1-l
sin 2
f,1.-
~x
(
sin
k2
c.
1f
c:os 2Y - -1 6 cos y + 1.5
6D.y2
2
~'inite
l)ifforc:nc:c Schcrnc:s for the: Shallmv Water
Primitive Form
Finite Differences
/\
H
c
cos 2 -cos 2
2
1
(}
f-1·
y
sm x
--cosD.:i::
2
--
c
k2
D.a::
')
(
D,;i;2
") y
v
---
D.y~
Table: :3: The Opcrators for the: Various
~~q11ations in Prirnitivc Form
x
.
sm- x2 1 +cos :y
2
~a·)2
sin 2 X
sin 2
sm
sin:.. 2 1 +cos .X
---2
2
~!!)
(
~'inite
_,
x
1
µ
D,;i;
6D.x 2
1
0
6
r
O'
2
1sin2X
-
cos2X -16co::; X + 17)
(~!Ir
Table: 2: The Opcrators for the: Various
~~q11ations in Vorticity Divcrgencc: F'orrn
Scheme
Operator ,'\nalytic
4sinX
;3 D.x
----
- -
')
0
l
2
)/
y
sin X
--cos 2 D.J·
2
sin 2 x2
(~a·r
") y
sin:.. 2
---2
(~!!)
l)ifforc:nc:c Schcnws for the: Shallmv Water
Vorticity-Divergence Form
Finite Elements
Scheme
lkriv<1tive
()0
a1,
i\ n<11ytic:
Isosceles Triangk:s
0
-D..;r---------
l
,
sm ,X + sin(X/ 2) cos Y
Ei
2 cos X
+ coii(X/2) cos Y
;3
2 +cos)/ sm X
-D..;r------'.~
:~
2 + COf:i Y
cosX---3
') s m
2p
0
Staniforth ~\'- IV!itc:hdl
H.c:c:bngn Iar
x
2 + cos )/
:~
:~
r 2 +COSY
;3
cos~\'
')sm X .5 +ms V
~
;~
- D..:c
2 +cos V sin X
-D..:r-------
6
D._;i:
-D..:r f:iin X - 2 siu(X/2) cos Y
2D..y2
-D._:r f:i ill X
fl+ COf:i Y
6
0
2 +ms X sm V
-D..y-------
, ms(,X / 2) sin V
-D..y-----'.~
:~
,
:~
0
, sin(X/ 2) sm Y
-D..y D..:c / 2 -'.~-
sin X sin V
-D..y---, D..J·
:~
0
0
-D..l sin (X/2) siu Y
J (D..:r/2F
3
2k
-2cos(X
6
2 + cos ,X Rm )/
-D..y
:~
-'.~-
Sm X Rm)/
-D..y---.-
Ll:t:
:~
2
. / , sin V
')sinV2+cosX
2)~
- D..y
LllJ
:~
')sm Y .5 +ms X
~ LllJ
6
,, 5 + cosX sin Y
-6
6
Da'
-~y
Table 4: The Derivatives Req1_iired for the Group Velocity
6
Sclwrnc
Derivative Analytic
(Jn
D1-1,
Vorti ci ty-1 )i vc:rgc:ncc: Form
Finite: I)ifforcn cc:s
Second Order
Fourth Order
0
0
0
1
cosX
4
1
- cosX - - cos 2X
3
3
2p
2-6.:r
r:; inX
8 r:; in X - r:;in 2X
0
0
0
()
()
()
0
0
0
0
0
0
·y
srn
8 sin Y - sin 2Y
6.y
36.y
[)()
a1,
-
()6
-
D1-1,
(Js
-
a1
36.:r
1
(Jn
-
Dk
()0
-
Dk
as
-
Hk
[);:-
-
Dk
2k
2
Table 5: The Derivatives Req1_1ired for the Group Velocity
7
Primitive Form
Finite Differences
Scheme
lkriv<ltivc ;\nalytic
Da
[)p
()0
-
[)fl
DS
-
[)p
OE
-
[)fl
ao
-
Hk
[)(}
Hk
aS
-
Dk
as
-
Hk
;\
H
()
()
c
y
')
()
I
211
cos
x
cosy
cos
sin 2X
x
2
2
cos X cos 2
sin X (
- - 1 +cos Y)
6.a:
6.x
cos-
-D.:i: siuX
y
-
2
. __
sin X
')
- 6.;r
y
0
0
2
-6.a: siuX sin 2
2 (6.y / 2)2
0
0
0
-6.y sin Y
0
-6.y-- sin Y
0
- 6 . y - - - .D.:i:
2
()
()
2k
sin .X
D.:i:
-
D.9
sin 2
sm
4-
· ( 6.;r / 2)2
sin Y
D.:y
x
2
sm
sin V
2--
T<lblc; 6: The: lkriv<ltivcs H.c:qnirc:d for the; Gr011p Velocity
8
cos 2 x2
()
2
. )/
~(1 + cos X)
D.y .
sin 2Y
0
D.y
)/
0
ï0.5
A grid
ï1
B grid
C grid
ï1.5
tF
Analytic
Rectangles
ï2
Isosceles
FD 2nd
FD 4th
ï2.5
Staniforth
ï3
ï3.5
0
0.2
0.4
0.6
µ d//
0.8
1
0
ï0.5
A grid
ï1
B grid
C grid
ï1.5
tF
Analytic
Rectangles
ï2
Isosceles
FD 2nd
FD 4th
ï2.5
Staniforth
ï3
ï3.5
0
0.2
0.4
0.6
µ d//
0.8
1
0
ï0.5
A grid
ï1
B grid
C grid
ï1.5
tF
Analytic
Rectangles
ï2
Isosceles
FD 2nd
FD 4th
ï2.5
Staniforth
ï3
ï3.5
0
0.2
0.4
0.6
µ d//
0.8
1
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.x imum 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<llytic: solution is sti 11 dcc.rrnsing at p d/ ri = ·1. The isosc:clcs triangle F~~IVI scheme is <lg<lin
the best. The ~-]) scheme C considera.bly overshoots the <ln<llytic: solution before it drops to
7'ero <lt 11 d / -:r = I v.:hich also oc:cnrs in ~-ig. 2 to a lesser extent. This behavior is cmrncd
by the averaging that is required in sc:hemc C represented by o: in Table I. This is crncial
in this c:ase bec:ause the >.- 2 is the domin<lnt krm in the denominator of (3.1) . .\ok that
sc:hemc C has exc:cssivc group velocity of the v.:rong sign near I' d/ 7r = 1. These three c<lses
with k = 0 all shmv tha.t the isosceles triangles FE~vI scheme in the vorticity-divergence
form gives the best results and the FE~vI scheme ·w ith bilinear ba.sis functions on rectangles
the second best. All of the FD schemes (A., B, a.nd C) for the primitive equa.tions give poor
res1.1lts in a.t lea.st one ca.se.
\\Tith this ba.ckground for the k = 0, we ·will now examine the frequency and group
velocity components as fundious of /-l and/;; for each r:;cheme. The qua.ntitier:; U:F, Gfc, and G}
are computed from (:L4), ('.L5), and (:L 6) respectively, with the relatious given in Tables l - 6.
figures 4 - 9 contain ;;., p, Gfc, and G} for the ca.r:;e J'.l /(4>. 2 ) = 0.1. The analytic solution for
is given in fig. 4d. for this case, FD scheme C (fig. 4c) ir:; dearly bell.er than fD r:;chemer:;
A (fig. 4a) and I3 (fig. 4b) when compared with the analytic solution. In particular, Fig .
.fa shows a rapid ch<lnge in u,'F for scheme .1 nc<lr (p d / -:r = 1.0, k = 0) leading to excessive
vahics of Gp m1d Gj;- as c<ln be seen v.:hen Figs. 6a <lnd Sa arc mmpared \Vith Figs. 6d and
8d . .:\ simi Jar problem ocrnrs for schcnw H nrnr (p d/ ri = ·1.0, A~d/-:r = 1.0) (Fig. -tb) whic:h
is associafrd v.:ith spnri01rnly large values of G'P a.nd G}- in Figs. 6b m1d 8b. The follmving
sc:hemcs tha.t a.re b<lsed on the vortic:ity-divergenc:c form of the eq1rntions: semnd-order ~-])
(Fig. 5e), fourth-order FD (Fig. 5f) a.nd FEl\I on rectangles (Fig. og), are very similaL
and they compare v;ell with the exact solution (Fig. 4d). The FE:\:I scheme is the best of
these three, a.nd the second-order FD is the poorest. The isoscelecs FE~-I (Fig. 5h) has a
generally simila.r behavior, but it is better for sma.11 k a.nd a little poorer near the corner
(11 d/ 7r = 1.0, kd/ri = 1.0). The fl· - k plots for d 2 /(4>. 2 ) = 1.0 will not be given because
the res1_1lts given in Fig. 2 a.r e quite representa.tive.
The frequency ;;., p and the group velocities Gfc and G} are given in figs. 10 - 15 re8pectively, for lP/(4>. 2 ) = 10. The FD scheme C (Fig. lOc) ha.r:; very large gra.dientr:; in ;;., p near
(µ d/7r = l, /;; = 0), and the wrong behavior above the diagonal from this comer. Figure l'.2
shows that the :G-group velocity has the wrong sign and is an order of magnitude too large.
A check of the other schemes in fig. l'.2,U r:;}wws that they all give the wrong group velocit_y
diredion in this region, but the speeds arc an order of n1<lgnitude less th<ln for FD sc:hemc C.
1
1
;;.,
1
1
12
~-igme
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
J
.j
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<lllcr than the 1-\ossby radius of deformation [d2 / (L\ 2 ) = o.-l]
grid C is the best of the primitive eqlrntion sc:hcmes bec:a.use grids .1 and H both give
spuriously large gro1_1p velocities when the 1.vave resolution is poor. All of the vorticitydivergence schemes give good results, ·w ith the isosceles triangle FE~vI being the best. The
arrangement of model points in the isosceles triangle FE~vI is favorable for evalua.tion of the
beta term in the vorticity equation, and this effect for advection has a.lso been disntssed
by ~eta and \Vllliams (1986). \Vhen the grid size is of the order of the Rossby radius
[d2 /(4.\ 2 ) = 1.0], all the mnnerical iichemef:i have a reaf:ionable behavior with Lhe isosceles
Lriangle and the rectangle finite elemenL iichemef:i giving Lhe beiit results. for the case where
Lhe grid size is greater Lhan Lhe Hosf:lby radius [d2 /( 4X2 ) = 10] primitive equation grid C haf:l
very· large group velocities when Lhe wave resolution is poor. The rectangular finite element
scheme gives Lhe besL iioluLion.
\Vajsowic.z (1986) pointed out Lhe large group velocities for Lhe grid C finite diIIerence
crn1ld lead to serious errors in western boundary current simulations in bi'iroclinic occa.n
models m1d thi'it sc:hcme H muld <llso ha.ve problems. In i'iddition we have found thi'it finite
diifrrencc scheme /' can also he poor on the brnmdarics.
Our results show that numerical sc:hcmes bi'iscd on the vorticity-divergenc:e form of the
shadow v.:akr equations give hctkr J{osshy wave simulations on the \vhole thi'in schemes
based on the primitive form of the shallmv water eq1.tations. This is not surprising because
Ross by wave dynamics a.re partially or totally controlled by the vorticity eq ua.tion, a.nd the
discrete vorticity equation derived from the discrete equations of motion v;ill normally have
more trunca.tion errror.
These res1.tlts indicate that a finite element vorticity-divergence model would be particularly useful for ocean prediction since theiie models have excellent ad vecLi ve and geosLrophic
<ldjnstrnent properties (sec Neta <lnd \Villiarns 1986 <lnd \VI lliarns 1981 ), and they c:an be 1rned
easily v;i th va.ria.ble element size. Staniforth a.ncl Daley ( 1979) and Cullen and Ha.11 ( 1979)
have demonstra.ted the effectiveness of this type of model for atmospheric prediction.
Acknowledgrnents
The authors vwuld to thank Dr. A. N. Staniforth for his careful reviei;,v of the mamtscript.
This resea.r ch 'vas conchicted for the Office of Na.val Research and was hmded by the Na val
Postgra.clua.te School. The manuscript was carehilly typed by ~vis. P. .] ones and ~vis. J.
}forray, a.ncl the munerica1 calnila.tions were ca.rried out at the \\T. R Church Computer
Center.
REFERENCES
Arakawa, A., and V. R. Lamb, 1977: Computational rlfsign of the basic dynamical process of
tlu UCLA gent.ml circulation model, l\.fethods In Computational Physics.Vol. 17, Aca.clemic
Press, 17:~-'.W."J.
CVte, .J., J-'1. Ilelaud and A. Staniforth, 198:~: Stability of vertical discrtlizalion schemts
foi' sr:mi-implir:d primitive r:q1wtion modds: th wry and applirntion;;, \/]on. \Vrn. H.cv .111,
11891 - 1207.
Cullen, ~vI. .J. P., and C. D. Hall, 1979: Forcasting and ge1nral circulation nsults from finite
elem.ent modfls, Q1_1art . .J. Roy. ~vieteor, Soc 105, 571-592.
Gill, A. E., 1982: i1tmosphert - Ocean Dynamics, Academic Press, 622 pp.
IIahiner, G . .J., and R. T. \\!illiams, 1980: JVumerical lFwther Prulidion and Dynamic
Afrtr:ornlogy, Wiley, ·177 pp
\ksingcr, F., 1979: /Jr;pr:ndwa of vortidty rma!ogur; and thr: Hos;;by wnvr; pha;;r; spcr:d on
tlu choia of horizontal grid, Sciences }Iathema.tiques, 10, 5 - 15.
:\"eta, B., and R. T. \Villiarns, 1986: Stability and phast. spad for various ffr1ite demrnt
formulations of tlu aduection fquation, Comput. Fluids, 14, :39:3 - 410.
:\"eta, Il., IL T. \Villiarns, aud D. E. IIiusrnan, 1986: Studies in a shallmc waltr .fl,uid modtl
u:ith lopography, Numerical Mathematics aud A pplicaLions, IL ViclmeveLsky and .J. Vignes,
~~ds., ~~I sevi er, 3·17 - ;351.
Schocnstadt, :\. L., 1980: A trnnsfcr fundion annlysi;; of num.r:rir:at ;;dnmcs usr:d to simulntr:
geostrophic adjustment, Mon. \Vea.. Rev. 108, 1248- 1259.
Staniforth, A. :\"., and H. L. }Iitchell, 1977: A .st.mi-implicit findt. flemrnt barotropic modeL
J-fon. \Vea. Rev. 105, 1.14 - 169.
1·1
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
f01Y r.asting with thr. primdivr. uprntion;;, l'vlon. \Vrn. 1-lev. 107, "l07- 121.
0
d~{ft:rtnct
\Va.jsowicz, R. C., 1986: Frtt plamlary ·leaves in finit e Phys. Oceauogr., 16, 77:~-789.
numerical nwdtls, .J.
\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
pp.
F~nginr.r.ring
Sr.irna, Wiley, 787
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:
[n_']
=
h
L
[(jl
D_j
C!r
hj
J
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 . -OJ' I ~
T
(p.;
t: • 1
=
{
/'I.6')
11
,,
I
-
i
=
j
: _.;_ .
/,
( /,_:3)
I],
(A.4)
(A.5)
b,: b:i
(A.6)
4A'
. a(p.;
D(;b,; _
--:::;--- --;::;-- d A
I uy uy
, J'
a ,: a .i
(A.7)
4A
where T is a. triangular element, A is the area of T and aj and bj are defined by
The vertices of the triangles (a:.i, Y.i) arc m1rnbered c01111krclocbvise. \Vhcn ( ,-\.2) is cvah1atcd
for the isosceles triangles 'vc obtain
.
(u,u
·1
.
+ G [( 1,u + (-i.u +
+Io {Do.o +
3o
+ :3fuf:Lr
l
[D1,o
.
(1 ; 2,1
+ (-1 ; 2,1 + ( 1;2,-1 + (-1 / 2.-1]
+ D-1.0 + D1;2,1 + D-1/2,1 + D1;2,-1 + D-1/ 2,-il}
{2 [hi.o - h-i.o]
+ h1; 2,1 + h-1 / 2,1 + h1/ '2,-1 + h-1 ; 2,-d
(i\.8)
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
7il - f v + 9 6.r: h ·
au
-~·
o,
(B.1)
[)
:::iv
+ f u. + g6y h Y
O,
(B.2)
(J
t
[)h
at+
S~.h
[h(x
-x
II(6.r u
+ D.x/ 2)
---y
+ 6,,v)
(n.:n
0,
- 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
6); v -
-- - y
c)y u, ,
(fl.4)
Ily subLracLing and adding (Il.l) and (Il.2) and using (Il.4) Lhe vorticity-divergence f:iyf:itern
becomef:i
f} .
~
iJ t
+ f JJ + '3v2 = o.'
( H ..5)
!!
0,
/) h
al
( H.6)
(B.7)
+HD= 0,
where f = fo + /Jy and v 2 Y = [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(
~.
bf
-.fo(
B
-
- J;2Y
+ }'r0 D + --:k.lJ6,J1
..
+ g (6.; hu + ;s; hyy)
=
0 .,
(fl.8)
= (),
(fl.9)
which are analogous Lo (2.7) and (2.8). The required coefficients can be obtained by subf:ltiLuting the wave forms into (Il.7), (I3.8), and (I3.9).
17
b
a
1
1
0.8
0.8
0.7
0.7
0.6
0.6
kd//
0.9
0.5
ï0.526
ï0.79
ï2.11
ï1.58
ï1.58
0.5
ï1.05
ï1.38
0.4
0.3
0.4
ï1.66
ï1.11
0.2
0.3
ï2.21
ï0.553
ï2.49
0.1
ï0.829
0
0
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
0
0
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
kd//
1
0.5
0.4
ï1.32
ï2.37
ï2.63
0.1
0.2
0.3
0.4
0
0
0.9
1
ï0.553
0.5
ï1.38
ï0.276
ï2.76
0.3
0.1
ï2.21
0.4
ï1.66
ï1.94
0.2
ï1.94
ï2.49
0.2
0.8
0.3
ï1.66
0.1
0.7
ï1.11
ï0.829
ï1.38
0.6
ï0.83
ï1.11
0.1
0.5
µ d/ /
ï0.277
0.4
ï0.553
0.2
ï0.79
ï2.11
d
1
0.3
ï0.263
ï1.05
ï1.84
ï0.263
ï0.526
0.1
ï2.76 ï1.93
ï2.76
0.2
ï1.32
c
kd//
kd//
0.9
ï2.37
ï2.63
ï1.84
ï0.276
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
0
0
0.1
ï2.21
ï2.49
ï2.77
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
e
1
f
ï0.263
1
0.9
0.9
ï0.263
0.8
0.8
ï0.526
ï0.526
0.7
0.7
0.6
ï0.79
kd//
kd//
0.6
0.5
ï0.276
0.5
ï0.276
ï1.1
0.4
0.4
0.3
0.3
ï1.32
0.2
0.1
0
0
ï1.58ï2.37
0.1
0.2
ï1.05
ï2.11 ï1.84
ï0.551
ï0.551
ï0.827
ï1.65
ï1.93
ï1.38
ï2.2
ï2.48
0.1
ï2.76
ï2.63
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
0
0
1
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
h
g
1
1
0.9
0.9
0.8
ï0.144
0.175
0.8
ï0.272
0.7
0.7
ï0.462
ï0.272
0.6
kd//
kd//
0.6
0.5
ï0.816
ï1.09
0.4
ï0.144
0.5
0.4
ï1.36
0.3
0.3
0.2
0.2
0.1
0.1
ï1.42
ï1.73
0
0
ï2.45
ï2.72
0.1
0.2
ï2.18ï1.9 ï1.63
0.3
0.4
0.5
µ d/ /
0.6
ï0.544
0.7
0.8
0.9
1
0
0
ï1.1
ï0.78
ï2.05 ï2.69
ï2.37
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
1
a
2.64
b
2.64
1.13
1
2.26
0.8
ï2.48
1.51
1.88
0.9
0.9
ï3.19
ï1.06
0.8
0.377
0.7
0.7
0.6
0.6
kd//
kd//
ï1.77
0.5
0.4
0.2
3.02
2.26
3.02
3.39
1.77
2.48
0.2
3.77
3.19
0.1
0.754
0
0
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
ï0.354
0.3
1.13
3.39 2.64
1.51
3.77
0.1
0.5
0.4
1.88
0.3
0.354
0
0
1
0.1
0.2
1.06
0.3
0.4
0.5
µ d/ /
1
0.9
0.9
0.8
0.8
0.7
1
0.376
0.6
kd//
kd//
0.9
0.376
0.6
0.5
1.27
3.81
2.54
3.39
4.23
0.2
2.63
3.01
0.2
0.423
2.96
1.88
3.76
3.392.26
0.1
2.12
0.1
1.13
1.51
0.3
1.69
0.1
0.5
0.4
0.3
0
0
0.8
0.7
0.847
0.2
0.7
d
c
1
0.4
0.6
0.753
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
f
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
kd//
kd//
e
1
0.5
0.4
2.63
1.1
1.46
2.93
1.83
0.2
2.2
3.29
0.732
0.2
0.3
0.4
0.5
µ d/ /
3.76 3.01
3.38
0.6
0.7
0.8
0.9
0
0
1
0.376
1.88
1.5
0.1
0.366
0.1
2.26
0.3
2.56
3.66
0.2
0
0
1.13
0.5
0.4
0.3
0.1
0.752
0.1
0.2
0.3
0.4
g
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
0.8
0.9
1
h
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.392
0.6
1.52
0.5
0.4
0.3
kd//
kd//
0.6
1.9
0.4
0.38
2.28
0.3
3.42
0.761
0.5
2.35
1.14
3.92
0.2
0.2
1.57
3.14
1.18
2.66
0.1
3.8 3.04
0.1
2.74
3.53
1.96
0.784
0
0
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
a
b
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
ï1.68
15.2
11.8
kd//
0.5
ï1.68
0.4
1.68
1.68
8.42
5.05
0.5
0.4
5.05
0.3
0.3
ï5.05
0.2
0.2
ï8.42 ï5.05
ï11.8
11.8
ï15.2
15.2
ï11.8
0.1
0.1
8.42
0
0
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
0
0
1
ï15.2 ï8.42
0.1
1.68
0.2
0.3
0.4
c
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
kd//
0.5
0.4
0.2
0.1
0
0
0.6
0.7
0.8
0.9
1
0.9
1
ï1.56
ï0.888
0.3
0.5
µ d/ /
d
1
0.6
kd//
kd//
ï1.68
0.5
0.328
0.4
1.07
ï4.81
0.3
ï2.85
ï7.22
0.1
ï16.6 ï8.73
0.1
ï3.44
0.2
ï6.77
ï10.7
ï12.6
ï14.6
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
0
0
ï14.8
ï12.9
ï11
ï9.1ï5.33
ï16.6
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
f
e
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.967
0.5
ï0.983
0.4
0.3
0.2
0.1
0
0
0.816
0.6
kd//
kd//
0.6
0.5
0.4
ï4.88
0.3
ï2.93
ï8.78
ï10.7
ï6.83
ï8.86ï6.92
ï10.8
0.1
ï14.6
ï12.7
ï16.6
0.1
ï3.05
0.2
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
ï14.7
ï12.7
ï16.6
0
0
1
0.1
ï4.99
ï1.12
0.2
0.3
0.4
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.4
0.6
0.7
0.8
0.9
1
h
1
kd//
kd//
g
0.5
µ d/ /
1.03
ï0.925
0.5
0.4
ï2.88
0.3
ï1.15
ï3.08
0.783
0.3
ï6.79 ï4.84
0.2
ï14.6
0.1
ï12.7
ï10.7
ï8.75
ï16.6
0
0
0.1
0.2
0.2
0.1
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
0
0
ï8.87 ï5.01
ï12.7 ï6.94
ï14.7
ï10.8
ï16.6
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
a
b
1
1
ï0.0865
0.9
ï0.0845
ï0.591
ï0.845
0.9
ï0.26
0.8
0.8
ï0.845
ï0.346
0.7
ï0.519
ï0.173
0.5
ï0.606
ï0.692
0.4
0.3
0.3
ï0.433
ï0.779
0.2
ï0.422
0.5
0.4
ï0.76
ï0.338
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
ï0.507
ï0.338
ï0.169
ï0.253
ï0.76
0.1
ï0.865
0
0
ï0.676
ï0.507
0.2
0.1
0.8
0.9
ï0.676
0
0
1
0.1
0.2
0.3
0.4
0.5
µ d/ /
1
0.9
0.9
0.8
0.8
ï0.277
0.7
kd//
ï1.66
0.4
0.3
0.3
0.2
0.2
ï2.21
ï2.49ï1.94
ï2.77
ï1.38
0.2
0.3
0.4
0.5
µ d/ /
0.9
1
ï1.11
ï1.56
0.5
0.4
0.1
0.8
ï1.78
0.6
ï0.553
0.5
0
0
0.7
0.7
ï0.83
0.1
0.6
ï0.169
d
c
1
0.6
ï0.253
ï0.0845
ï0.422
0.6
kd//
kd//
0.6
kd//
ï0.591
0.7
ï1.11
0.6
0.7
0.8
0.9
1
ï1.33
ï0.222
ï2.22
ï0.667
0.1
0
0
ï2
ï0.445
0.1
0.2
ï0.89
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
e
f
1
1
0.9
0.9
0.8
kd//
0.6
0.8
ï0.676
ï0.591
ï0.169
ï0.507
ï0.845
ï0.0845
ï0.253
0.5
0.6
ï0.338
ï0.338
ï0.422ï0.591
ï0.76
ï0.0845
0.4
ï0.845
ï0.112
ï0.673
ï0.336
ï0.448
ï0.336ï0.56 ï0.785
ï0.897
ï0.507
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
0
0
1
ï0.56
ï0.448
0.1
0.2
0.3
0.4
g
0.9
ï0.937
ï0.669
ï1.07
kd//
kd//
0.9
1
ï0.312
ï0.886
0.548
0.835
1.12
0.261
ï0.599
0.5
0.4
ï0.401
ï0.134
ï1.34
ï1.2
ï0.535 ï0.803
ï0.937
ï0.535
ï1.46
0.3
ï1.17
0.2
ï0.669
0.1
0
0
ï0.0255
0.6
ï1.07
0.2
0.8
ï0.312
0.7
ï0.401
ï0.268
ï0.134
0.6
0.3
0.7
ï0.0255
0.8
0.7
0.4
0.6
h
0.9
0.5
0.5
µ d/ /
1
ï0.803
ï0.268
ï0.224
ï0.112
ï0.673
0.1
1
0.8
ï0.785
0.2
0.1
0
0
ï0.224
ï1.12
0.3
ï0.253
ï0.676
ï0.76
0.5
ï1.01
0.4
ï0.422 ï0.169
0.3
0.2
ï0.897
0.7
kd//
0.7
ï1.01
0.1
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
a
1
b
1.64
1
8.21
0.9
0.0373
0.9
0.8
6.57
5.75
0.7
0.8
4.11
0.261
0.7
7.39
0.112
0.335
kd//
0.6
0.5
4.93
0.4
0.5
0.186
0.4
3.28
0.3
ï0.186 ï0.335
ï0.0373
0.3
2.46
ï0.261
0.2
0.2
ï0.112
0.821
0.1
0
0
0.1
0.2
0.1
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
0
0
1
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
2.62
0.9
1
d
c
1
7.87
3.76
0.9
10.5
18.4
0.8
0.8
23.6
0.7
0.7
0.6
0.6
3.39
2.63
1.88
26.2
kd//
kd//
kd//
0.6
0.5
2.26
3.01
0.5
1.13
0.4
13.1
0.3
0.4
0.3
0.376
21
0.2
1.51
0.753
0.2
15.7
5.25
0.1
0
0
0.1
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
f
e
1
1
0.246
0.214
0.9
0.8
0.356
0.6
0.641
0.0713
0.499
kd//
kd//
0.615
0.7
0.6
0.285
0.4
0.713
0.3
0.3
0.2
0.2
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
0
0
1
0.737
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
h
1
1.28
4.86
0.9
0.9
2.24
3.19
0.8
0.8
0.7
7.29
21.9
0.7
2.56
0.5
1.6
1.92
0.958
0.4
0.3
0.3
17
9.72
14.6
19.4
2.43
0.2
0.639
0.319
24.3
0.5
0.4
0.2
12.2
0.6
2.88
kd//
0.6
kd//
0.492
0.983
g
1
0.1
0
0
0.86
0.1
0.143
0.1
1.23
0.5
1.11
0.57
0.4
0
0
0.123
0.8
0.428
0.7
0.5
0.369
0.9
0.1
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
a
b
1
1
1.68
0.9
0.9
0.8
0.8
0.7
0.7
ï11.8
ï15.2
5.05
11.8
ï5.05
0.6
0.5
5.05
kd//
kd//
0.6
8.42
ï8.42
0.4
15.2
0.3
ï5.05
0.3
11.8
0.2
ï8.42
0.4
ï1.68
ï11.8
0.5
1.68
0.2
ï15.2
0.1
8.42
ï1.68
0.1
0
0
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
0
0
1
15.2
0.1
0.2
0.3
0.4
c
0.6
0.7
0.8
0.9
1
d
1
1
0.9
0.9
2.62
0.8
0.5
µ d/ /
ï10
ï12.9
ï14.8
ï15.7
0.8
ï8.33
ï13.8
0.7
0.7
0.6
24.5
57.4
0.5
ï19.3
0.4
79.3
kd//
kd//
0.6
0.5
ï11.9
0.4
57.4
68.3 68.3
0.3
79.3
0.2
13.6
46.4
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
ï17.6
0.3
0.2
35.5
0.1
0
0
24.5
13.6
2.62
46.4
35.5
1
ï16.6
ï9.09
0.1
0
0
ï11
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
e
1
f
1
ï15.3
0.9
0.9
ï5.68
0.8
ï8.89
kd//
kd//
0.6
0.5
3.95 7.17
0.4
0.5
10.4
0.3
0.2
0.2
0.1
0.1
13.6
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
14.7
0.4
0.3
0.8
0.9
0
0
1
ï1.92
23
ï14.4
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
h
1
3.65
0.9
23.9
19.7
0.9
0.8
15.4
23.9
0.8
9.2
31.4
0.7
2.69
ï10
0.7
25.8
ï5.8
0.6
0.5
ï1.89
36.9
0.4
kd//
0.6
kd//
10.5
ï6.07
g
1
11.2
6.93
0.5
0.4
0.3
ï13
0.3
14.7
0.2
0.2
0.1
0.1
0
0
2.24
0.7
ï2.47
0.6
0
0
ï10.2
18.9
0.8
0.741
ï12.1
0.7
6.39
ï7.44
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
20.3
0.7
0.8
0.9
1
0
0
ï14.3
ï1.56
19.7
0.1
0.2
0.3
0.4
0.5
µ d/ /
0.6
0.7
0.8
0.9
1
Fignrc 16: The isosceles triangle h;rnis function
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