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O o s t e n d e - B elgium
in s t it u t e
ISSUE 97
APRIL 1993
FEATURING:
D E L E E R SN IJD E R and CA M PIN - On A Peculiarity O f The B-Grid
B E R ESTO V and M A R U S H K E V IC H - Coupled Num erical Model For Altimetric Data
Assimilation
C A S E R T A and SA LU ST I - Bottom Erosion Due To Small-Scale Perturbations O f Density
C urrents
K A N T A R D G I - O n T h e Effective Fetch For W ind W aves On An O cean Current
ED ITO R IA L
23796
n
a
p e c u l ia r it y
o f
t h e
b
-g r ii;
Vlaams Instituut voor de Zae
E . D e le e r s n ijd e r a n d J e a n -M ic h e l C a m p in
To determ ine the stab ility conditions of a nu­
m erical scheme, it is cu sto m ary to exam ine a lin­
earized version in an unb o u n d ed dom ain. T his is usu­
ally referred to as the von N eum an analysis. A lthough
this technique resorts to m any sim plifying assum p­
tions, it is generally capable of providing a satisfac­
to ry ap p ro x im atio n to th e stab ility condition of the
original, non-idealized, num erical alg o rith m . Here, we
present a sim ple geophysical fluid problem where th e
von N eum an analysis provides a stab ility lim it th a t
is com pletely irrelevant an d , hence, useless.
We consider linear, d am p ed inertia-gravity
waves, which obey the following equations:
^
— + fe
X
+ h V -u = 0,
u = -g
y
r? + A H V 2 u ,
(1)
(2)
w here t is tim e; V denotes th e horizontal gradient op­
erato r; r¡ a n d u represent th e sea surface elevation and
th e horizontal velocity; e, h, ƒ, g , and A h are th e ver­
tical u n it vector, the (co n stan t) sea d ep th , th e (con­
s ta n t) Coriolis p aram eter, th e g ra v ita tio n a l accelera­
tion, a n d th e horizontal viscosity, respectively.
T h e dom ain of interest is assum ed to be closed so
th a t, on its boundary, u m ust vanish. Hence, if A h >
0, th e to ta l— p o ten tial -f kinetic— energy o f th e flow
m u st decrease u n til a s ta te o f rest is a tta in e d . If A h is
zero, only th e com ponent o f u n o rm al to th e bou n d ary
is prescribed to b e zero an d th e to ta l energy rem ains
c o n sta n t as th e tim e increases. It is desirable th a t
any num erical solution of ( l) - ( 2 ) exhibit th e sam e
properties.
We define all variables on th e B -grid a n d we ap­
ply th e well known forw ard-backw ard schem e to (1 )(2), i.e., all tim e derivatives are discretized by forw ard
differences, the o th er term s are tak en explicitly, w ith
th e exception o f th e pressure te rm th a t is ev aluated at
in s ta n t i - f A í, in stead o f t, A í being th e tim e step. In
ad d itio n , th e Coriolis te rm is num erically evaluated
by
/ e x u —+/[(1 —<*)e
0 < a < 1,
X
u ( i) -f a e x u ( i + A í)],
(3)
where a is th e “ra te of im plicitness” of th e tim e dis­
c retizatio n of f e X u . O ne m ay believe th a t, when
Flanders Marine Institute
a > 1 /2 , artificial dam ping is ad ded to th e scheme,
which should th e n becom e m ore stable. T his seems to
be confirm ed by the von N eum an analysis, which for
A h = 0, leads to th e following stability conditions:
a = l/2:
o = l:
Aí < — ,
(4)
A í < — — -------- 1
V ^m ax(0,1 —
(5)
w ith c2 = gh a n d A s = m in(A ® , A y)— A x and A y
denoting th e space increm ents in th e direction of the
x — and y — axis, respectively.
As can be seen from (4)—(5), tak in g o = 1 should
im prove th e schem e’s stability and, if A s2f 2/(Ac2) >
1, th e num erical algorithm should be unconditionally
stable. O ne m ay believe th a t introducing a non-zero
viscosity should not render th e schem e less stable.
We have carried o u t a series of num erical ex­
perim ents in a square dom ain of 50 x 50 grid points
bounded by im perm eable walls. Very surprisingly, the
num erical results are in m arked co n strast w ith w hat
could be expected from th e theoretical stability analy­
sis!
For A h = 0 a n d a = 1, th e dom ain of stability
of th e schem e should be delim ited by
f 2* 2 > « ( ^
■-1) •
(6)
T h e num erical sim ulations indicate a com pletely
different stability dom ain (Figure la ) . If som e diffu­
sion is present, th e stab ility dom ain som ew hat ex­
pands b u t rem ains m uch sm aller th a n th a t corre­
sponding to (6)(F igure lb ).
It is striking to no te th a t, for a = 1 /2, the nu­
m erical experim ents (Figure le a n d Id ) lead to a
stability dom ain th a t seems to be approxim ately in
agreem ent w ith (4). A s a m a tte r of fact, it is only for
a = 1 /2 th a t th e scheme presents a large area of sta­
bility. T h u s, for all p ra ctica l purposes, it is clear th a t
one should only consider a = 1 /2.
C an anyone provide a th eoretical explanation for
th a t odd behaviour?
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