February 1982

227

T. Satomura

An Investigation of Shear Instability in a Shallow Water
Part II. Numerical experiment
By TakehikoSatomura

Geophysical Institute, Tokyo University, Tokyo 113(Manuscript received 27 luly 1981, in revised Jorm 18,September19Bl)

Abstract
Using a channel model in a shallow water, we numerically consider characteristics offinite-amplitude gravity waves which were found to be unstable waves in a constant shear
flow by the linear analysis. Numerical integration is performed at Froude number equals
to five for both inviscid and viscous case. In the inviscid run, it is shown that the first
appearedmode has the growth rate and the structure which are the same as those expectedfrom the linear analysis. Energy budget shows that disturbances extract their energy fromthe additional part of the mean kinetic energy as in the linear anaiysis.
In the viscous run (Re:3000), it is shown that disturbance energy reaches a quasisteady state. Energy budget shows that energy is supplied to the mean kinetic energy,
converted to the eddy kinetic energy, and then dissipated by the viscosity acting on the
disturbances. Momentum budget indicates that these gravity waves can mix the averaged
momentum permanently.
At a later stage of time integration, the disturbanceenergy oscillates. A linear stability
analysis for the quasi-steadystate is examined, and it shows that the oscillation is produced
by unstable sub-harmonics.
Shape of disturbance depth near the boundaries and change of mean depth are also
discussed.

unstable mode. Further comparison among 51
and Blumen et al. or other studies was presented
In one of previous papers (Satomura, i981a: in 51 and Satomura (1981b) where he used ahere after referred to as S1), stability problems piecewise-linear flow unbounded in both sidesof two types of shear flows were studied: One and compared the results with those of Drazin
is a plane Couette flow bounded in both sides and Davey (1977).
(case I) and another is the same flow but unIn the previous paper 51, a curious nature ofbounded in one side to ccnnect with a rest fluid the unstable wave was pointed out, that is, the(case II). He used analytical solutions expressed unstable wave growing in a constant potential
by power series and calculated eigenvalues and vorticity field can not change potential vorticity
eigenfunctions with high accuracy. He con- at all. It means that a change of zona-l mean
cluded that, when Froude number is greater than flow, Aa/ dt, does not occur unless zonal mean
unity, gravity waves are destabilized even by a depth h changes. But, when disturbances disconstant shear flow, and thus they are not related appear for some reason, Z must be constant and,with the existence of inflection point of the basic thus, no change of n will remain. Then, a
shear. He also concluded that excited gravity question arose whether the unstabie waves canwaves can radiate from the shear zone in the permanently change momenta of basic flows. The
analysis of 51, however, was a linear theory, andcase II.
Also Blumen, T)razin, and Billings (1975) had only suggestions for an answer of this question
found unstable waves similar to those of S1, but was given.
their discussion about the characteristics of the
A purpose of this paper is to investigate theunstable waves had been insufficient. One of the effects of nonlinearity and viscosity upon the
purposes of 51 was to ciarify the nature of the unstable waves, and to exarnine whether permaL.

Introduction

728

Journal o{ the Meteorological Society of Japan

Vol. 60, No. 1

nent redistribution of momentum occurs or not. to the walls. The corresponding velocity comA next step of a iinear analysis may be a ponents are u+ and zr3, respectiveiy. The fluid
flnite ampiitude (but weak nonlinear) analysis. depth is h* and assumed to be 11 at rest. By
But, for the channel model we use, the critical introducing a velocity scale Us, dimensionlesswavenumber which corresponds to the fastest variables are written as
growing mode at the critical Froude number(r, y) : (nx, y*) /La*,t : t*U o/Lu*,(=2) is inflnite. This fact makes the finite ampli(u,a):(u*,o*)/Uo,h:h*/H ,(2.1)
tude analysis difficult. Moreover, no experimentnor numerical simulation, which guides us into where variables without asterisks are dimensionan appropriate hypothesis such as shape assump- less.
The two dimensionless parameters relevant totion, has been performed. In this paper, we willthepresent problem are the Froude number Fr
integrate basic equations rrumerically and present
andthe Reynolds number Re:
results of integrations.
In the next.section, we p.resent shallow water
F":Uo/J gE, R":(JoLyx/v,(2.2)
equations as basic equations and describe numeri- where g is the gravity acceleration andr, is thecal channel model. A constant shear flow is coefrcient of kinetic viscosity
and these quantities
used as a basic flow in the channel. Thus, the are assumed to be constants.
Since it is not amodel is the same as the case I of 51. In section subject of this paper to study
whole aspects of3, results of inviscid run are described at first, high speed flows,
we only consider Fr:5
for alland, then, results of viscous run are discussed. cases, and Re:3000(viscous case) or infinite
Some of the results are the same as thcse sug- (inviscid case).
gested by the linear theory 51 but some of themThen, the dimensionless equations of motion
are not. Summary and conciusions are presented
and continuity in a shallow water are
in section 4. A detailed expression of flnitea v a v a v l _ , 1 _difference scheme is given in appendix A, and a+uI h:o ' 12'31a,a*+o ay+ F;
R"derivation of energy equations is given in appendix B. In appendix C, viscous terms in a() 4\
shallow water equations are derived.
2.

Basic eouations and numerical model

where Z is the velocity, F is the viscous termandWe shall consider a plane parallel flow of aa \shallow water. The model described below is'- / a
A
y)'
\
d
r
'
schematicaliy shown in Fig. 1. The fluid is
The channel width, I_],, is unity, and boundary
bounded by two vertical walls, assumed to be
rigid and to be separated by a distance lox as conditions at the walls arein the case I of 51. Cartesian co-ordinates are(2.5a)
D:O at Y: -1/2
deflned, with r* along the walls and y'' normal
in the inviscid case, and
D:O,

'l
ol_

i1,,

x

I

uo
Fig. 1 The model of numerical integration. Dimensionlesswidth is 1 and dimensionlessbasic florr
varies from 0 to 1. Surface elevates normallyto the paper.

u- +1/2

at y: +1/2

(2.5b)

in the viscous case. In the r direction, thechannel length is Lu, and fluid is assumed tocontinue cyclically.
The initial conditions fcr time intesration are
u:Ylau,
(2.6)

1;:0,
h:l+ah,where da and 6h are small random perturbations,and are connected with each other by a following
relation:
?tu

-

A*

q=Y:

0u
A,'

n

.' - - 6(6u), .3) :-1.

1+6h

e'\

February 1982

229

T. Satomura

Above relation is set to bring the numericalmodel close to the analytical model of S1' because, in the linear theory, no potential vorticity
is added by perturbations superposed on aconstant shear flow (see equations (5.3) and (5.4)
oI 5I ).
Finite-difference scheme for the space coordinates of (2.3) and (2'4) follows Arakawa andLamb (1981). It conserves total mass' energy'
and potential enstrophy q2l2h. Some of the reasons why we choose this complicated scheme arethat zero potential vorticity of the disturbanceis an important characteristic, and, thus, there
is a risk that non-physicai change of q by afinite-difference approximation cause a serious
error. The explicit form of the finite-difference
scheme is shown in appendix A.
The scheme of time difference is the forward
difference for the viscous term and the leapfrog scheme for the other terms.
The number of grid points in the r and the y
directions are 64x20 in the inviscid case, and32x10 in the viscous case. Grid intervals are
and 0.1 in the inviscid andd: /n: /y:0.05
viscous case, respectively. Thus, the domain of
the present numerical integration is that Zo=
in the inviscid case, and that
3.25 and Ly:1.0
in the viscous case in
Lr:3.3and ly:1.0
dimensionless unit. These values of Zr's areabout twice as long as the wavelength of thein 51. So
most rapidly growing wave for Fr:5
far as parameters used in this paper, errors by
are not
the finite-difference approximations
serious, and, thus, we suppose that the numericalsolutions indicate characteristics of the true
solutions.

-----

2ko
_--

--

-

3k0

higher nodes
nkO for n>5(n-1/21k0 for n>l

9
E
o

o

5o

too

150

,*

Fig. 2 Time variation of potential energy for each
Fourier component in the inviscid case. Thicksolid line indicates a srowth rate o:0.059

From this figure, it is found that the growthrate of the first appeared mode is 0.059, and isin good agreement with the theoretical value
0.0585. Of course, the wavenumber of the first
appeared mode, ftg:3.8,
is the same as thatindicated by the theory.
Since the Froude number is greater lhan 2,
3. Results
the time when nonlinear effects become dominantIn this section, results and discussions of is also interesting. Fig. 2 indicates that higher
inviscid nrn are described in the first subsection, modes /c:2k0, 3k0,. . ., are excited by the nonand those of viscous run are presented \n 3.2. linearity and their amplitudes can be seen only
In both runs, the Froude number is fixed to five, for the later stage of the run (/>150). Thus, theand only the Reynolds numbers are different be- linear stability analysis retains its usefulness evenfor Fr:5.
When we numerically integrate thetween them.
(2.3) and (2.4) to a little further beyond
In the following discussions, t I denotes equations
t:19O, the result has no longer a physical meanz o n a l l y a v e r a g e dq u a n t i t y a n d ( ) ' : ()-fIing due to the errors which increase as theis the deviation from it.
amplitudes of the higher harmonics of the order
3 .1 Inviscid case (Re.-> a )of grid sizes become large.
When Re tends to be infinity, growth rates
The structure of the disturbance at t:1.43.165
and structures of the numerical solutions should is shown in Fig. 3. It js a typicat structure inbe the same, at least its initial stage, as those an early stage of the development of the disturbof the linear theory. Fig. 2 shows time variation ance, and it also agrees well with theoreticallyof potential energy of each Fourier component. composed one: phases of troughs and ridges tiit

234

Journal of the Meteorological Society of Japan

Vol. 60, No. I

0,tr - AxIs

Fig. 3 Structure of disturbance at t:1.43.165 in the inviscid case. Arrows denote velocity(ur,ut) and solid iines are contours of h:1*/z/.
Contour interval is 0.02.

to the positive r direction with increasingy, and, propagating to the negative r direction relative
the maxima of divergenceand convergenceare to the flow. Near the wall at t:-l/Z,
on theat the left side (the smaller r side) of the troughs other hand, waves are supposed to be propagating
and of the ridges respectivelynear the wall at,,:1 lt / 1
')

f

.-.1

uttu

,,;^.

r'arsa near

the

wall

at

y:

-I/2.

The total pattern is similar to the gravity
waves in a channel.
Fig. 4 shows a typical r-variation of the
disturbed depth, H, at y:9.47t for three stages.
In the early stage (t: I43.165), h' varies
sinusoidally and symmetrically with r. But, as
the amplitude increases,asymmetry appears in
an intermediatestage (1:165.165), and a shapesimilar to a shock wave is formed in a strongnonlinear stage (1:187.165). At this time,
perturbation Froude number F"o, which is defined as F,p:Uo/J gFl by introducing characteristic velocity of wave pafi Uo, is nearly
equal to unity. Inspecting the asymmetry of h'in Fig. 4 closely, we notice that lAh/ 0rL islarger in the left side (the smaller a side) ofpeaks than the right side (the larger r side). It
suggestsus that, on the analogy of formation ofshock waves, waves near the wall at y:l/Z
are

(a)

x 10-32-0.4,,t:143.165

-0.2

0.2

.ah

dnv

0.4

O"v

dI

Oh'v'
Oy

(d)
Fig 4 Disturbance depth ht at y:Q.475 for threevalues of / in the inviscid case.

Fig.5 (z) Zonal mean depth ft for t:165.165 andt:187.165 and (b) values of terms in the equation of i for t:159.66 in the inviscid case.

T. Satomura

February 1982

to the positive r directior.r. What are mentioned
above confirm the discussion of 51 about Figs.
3(a) and 4(a) of that paper (p. 157-158), andare consistent with the conclusion of S1 thatthe unstable waves are destabilized gravity waves

L)

-

l

(rru+iiT)-'-(h'u')-----(l'"1

in a ehannel.
Fig. 5(a) depicts zonal mean depth h at t:
165.165 and r: 187.165. We see that zonal
mean mass concentrates on the center of thechannel more and more with time. To find theterm which produces this concentration, let us
examine the equation of ft. It is written from
(2.4) as
^ 7

on

o

=_

*ot+ i - noyo +

o -

ioy
-h'a':0,(3.1)

At the initiai time ir:O and Z:1, thus thefirst term in the left-hand side of (3.i) should
be balanced with the last term. The last term
can be evaluated by the eigenfunctions presented
in 51, and is positive near the center of thechannel and negative near the walls. Thus, the Fig. 6 Time variation of the half momentum MHA
linear theory predicts that h should becomein the inviscid case.
shallow with time near the center and deep near
the walls. But, what happens in the numerical.
a_integration is opposite. An example of vaiues/? ?)| &a+h'u'):2lhua1,.o.
otof terms of (3.1) are shown in Fig. 5(b) for the(Small scale Thus
numerical solution at t:159.66.
MHA is changed by the momentum flux
disturbances may be hardly related with physical from the ' upper' half zone (0<y<I/2) to thephenomena because the truncation error is up 'lower' half zone (-t/2<y<0).
From Fig. 6, theto 10-4.) We find that the last term of (3.1) 'upper' momentum decreases with time, i.e.,
is the same sign as the linear analysis, but the momentum is transportedto the 'lower' half, andsecond term, neglected in the linear analysis('F), redistributed. It is also found that most pafi of.l-hiscontrols the time ihange of ll.
nature does
the decreaseof MHA is t77). These properties
not change through the numerical integrationof momentum change are consistent with those
except at the eariiest time.
in 51 and they also give a hope for a permanentFig. 6 shows time variation of momentum
change of momentum by the unstable waves
averaged in the half area (abbreviated to MHA),
under viscouseltects.
which is defined as

The energy balance at t:165.t65<ni>:<ha+T7):2\ hidy.

is shown(3.2) in Fig. 7. A detailed derivation of the energy

equations and definitions of symbols are given
From the equation (2.3), we obtain a following in appendix B. This four box diagram indicatesthat the disturbance kinetic energy 1(' receives
equation for (Ti) as
energy through the tKKt transformation. The(*) Jf we assume
disturbance potential energy F receives energy
olu nh( u' '!'throughIKP'I and returns only a little energy+ r'o^!-\ .
oI
\
dx
0y/
to the mean kinetic energy K. It is also foundthe second term of (3.1) can be evaluated and from Fig. 7 that the total disturbance energy
is consistent with the numerical results shown K + P':33 x 10-5 and equals to the absolutein figure 5(b). We cannot evaluate, however,value of the additional part of the mean kinetic
ha directly, but only afri/at. tnd, dhu/Ot wtll
energyKo. The relation,affects only the higher order trend a4/ AP.
(3.4)
K'+P'+K':O,
Thus, it seems appropriate to neglect the second
was derived in 51 under the condition of theterm of (3.1) in the linear theory.

232

Iournal of the Meteorological Society of Japan

Vol. 60, No. 1

variations are seen:(a) At first, disturbance grows exponentiallyand its growth rate o=0.036 and wavenumber
t:165.165
fto:3.8.
These values are in good agreement
estimated by the linear theory
withthevalueslfR.K' ll(ot=0.032<*), ko: 3.8, respectively),x 10(b) Higher harmonics rrzith wavenumbers 2k6,
3ko, 4ko, . . ., are excited one after another by
nonlinear effects, and viscosity stops growing ofdisturbances. (240 < t 5300)(c) Energy of disturbances decreases and
1
tK'.P'lreaches a quasi-steady state. (30O<l<500)(d) Modes whose wavenumbers are (n-1/2)kogrowwith oscillation, although disturbances with
Fig. 7 Four-box energy diagram at l:165.165 in
are
w
a
v
e
n
u m b e r sn k 6 , w h e r e n : 0 , - + - 1 ,+ 2 , . . . ,the inviscid case.
still in quasi-steady state. (5003t<700)(e) Amplitudes of (r?-1l2)ft6 modes become
constant shear flow, and it means that K" plays large and osciliations both of the potential energy
an important role in the energy budget. The of nke modes and of the total disturbance energy
fact that (3.a) is satisfied in good accuracy in become evident. (7N<t)numerical integration means that the total nature
The stages (a)-(c) are rather ordinary charof the finite-amplitude numerical solution closely acteristics. But, in the stages (d) and (e), an
resembles the characteristics of the eigenfunction interesting feature of growing oscillation appears.
in 51. A projection of this interesting property Fig. 9 is a close-up of Fig. 8(b) for 469<t<522.
to the momentum budeet is the fact that It shows that the period of oscillating mode
T:9
and the phase of ko/Z mode is earlier thanl.
|<F7>t>>|(.hn>- o.2sthat of 3/2ko mode abont x/2.
It also shows
3.2 Viscous case (Re:3000)
that the amplitude of ft2 or ks/2 mode is about
In this subsection,results of viscous run for 1.5 times as large as that of 3/2kn
mode. On
Re:3000 are presented.Before the resultsare theother hand, no osciliation is looked in nks
shown, it should be took notice that the viscous mode from this flgure. But, froma much larger
term F is rather unusual, and it is expressedas
magnified figure (not shown), it is found that,(3.5) at ieast, the potential e.nergy of ftg mode is
F:(F", Fi ,oscillating in the same crder of the amplitude
where
as that of k0/ 2 mode and is out phase withks/Z mode.
hlor\
dfr/
0yt
1
f
r
/
)
_
l
\dy
Then, why do ks/Z and 3/2ko modes grow?
(3.6)
And, why does their potential energy oscillatea o \ l o 1 " , 6 u \ ) with that period?
r :- l, - -6 (l /r l([ A
- -u
ft u
^
l)r-.
lzn ^ ]
ft-drl \oy6r))' ay\-"ayll'
In order to answer above questions. let us(3.7)
examine a linear stability of the quaslsteady
These forms include effects of the free surface state.
on stress tensor. Detailed derivation of (3.6) andAs the first step of the analysis, divide the(3.7) is given in appendix C.
physical quantities into two parts as
Fig. 8(a) shows time variation of total energy
t'r'r The growth rate o1 is estimated by followingof disturbances ,K'*P
and Fig. 8(b) shows time
equationvariation of potential energy of each wavenumber.

l

'1r'

F,:+l!(zh-T"\+*{n(2Y-*:')}l,

Nature of both time variations for t)850 depends
strongiy on both the grid intervals of space difference and the definition of the potential
vorticity at the walls. Thus. we will not discuss
the results for r)850, although numerical integration can be stably extended for a long time.
From these figures, following aspects of time

L2L12
O t : O i - " ; '-t(P,where o,. is the growth rate in the inviscid iinear
theory, fr and / are the wavenumbers in ther and y direction respectively.Here, we assumed
that eigenfunction is the same as that of theinviscid linear theory and that viscous terms are
expressedby the usual Laplacian form.

T. Satomura

February 1982

LJ )

'iii
1[,,

z

z
E
&

600
TIME

400

o)
Fig. 8 Time variation of (a) total perturbation energY and (b) potentiai energy of each
Fourier component for Re:3000. Thick solid line i n 1 b ) i n d i c a t e s a g r o w t h r a t e o :0.036.

(3.8)

y < r >: ( Y t n , <+ ) e L k o/n2+ V t 7 2c,* p x L ko ' / z ) e L ' t

where superscripts (0) and (1) indicate a basic
state and infinitesimal perturbations, respectively.
The basic state is assumed to be composed ofthree parts as* conjugate term'
V (o): VoI YrcLkon| ( V t n , ( - ) e L k o ' / 2+ V z p , a - p 3 1 k o / 2 ) e - 1 ' t

I/:V(o) +v(1) ,h:hQ> +h(r) ,*conjugate terms,(3'10)

h < D: ( h t n , s g L k o r / 2 I h t 1 z , t . + > e 3 1 k o t / 2 ) e 1 ' ta ( h r n , ; p r k o t / 2 I h z 1 r ,r - t 2 3 7 k o t / 2 ) e - 1 u t+ conjugate terms .
whero subscrlpts1/ 2 and 3/ 2 indicate ft6l2 mode
part
zonal
the
and 3kr;/2 mode respectiveiy,and subscripts(t)where subscripts 0 and 1 indicate
indicate signs of frequencies. Thisand (-)and the wavy part, respectivelY.
is appropriate if Fourier expansion
quantities
assumption
perturbation
that
also
assumeWeof perturbation quantities convergesrapidly. It
are written as
hQ>-_6ho*hp'&orlconjugate term,

(3.9)

234

Journalof the MeteorologicalSociety of Japan

Vol. 60, No. 1z:_:_t:ll3
o

7ko
0.5k0
4 9 6 , 0 04 9 8 . 0 0s 0 0 , 0 0s 0 2 . o o5 0 a . o os o G . 0 os o B . o os l o . o 0 s t : . d o s t r . d o s t 6 . o o 5 l 8 . o o 5 2 0 , 0 c
TINE

Fig. 9

1 ,S k O

Close-up of Fig. 8(b) for 496<t=522.

would be also suitable if eigenvalues and eigenfunctions do not change significantly by including higher modes such as -+5ko/ 2. The variationof sub-harmonics, -+k0l2, are examined by removing 1-3ko/ 2 modes instead of including
higher modes, and it is found that thev change
only a little. Thus, the ciosure assumptions (3.10)
and (3.11) is not a poor approximation.
Now, using (3.8) (3.11), perturbation equations
are obtained from (A.1)('") as* iroVs+ qokx tr rr,,*, * q * x f ,,r,r*,/2,(.r)* q s p , r + 1 kx V o * q t p , 1 * ' k x t t+ f ( f o v e , r . ,L
, Vt V tr , , - , +:*{#+ iahztz,l!)+ !

* i a V t n , , i : 1 , +q o nx V rn , <r >I e t k x V l t p , a . - .

fil rr , r , , . - _ , )(314t(*]).;' ',,,.,'-,

a,n, +
0,,,,,,*,,:
o,i:, $( 3 . 15 )where a dagger denotes a complex conjugate,* is a vertical unit vector,+ qrl k x frr rr,r*, t q, n, <*>kx to

(-)

(-)

* qi t 12,;1k x t t I qs 12,<*>kx il tl
* F

\YoV

t n , < * '*, V t V I t p , < - ,+ Y J V sn ,1
t. p1 , h tn . , ,\_:, / r a ,at;;r_;_ )/rt0\r)__\Z ) |n,,,.,-_).

to:(ttt')o,(3.16)

t,:@Y),,(3.17)

V r n , < + > : h o Y t p , < + >I h r V l t p , r - 1 i h i Y z p , l * 1* h t n , c + > V*o h t t n , ,- r V , * h s/ 2 ,*( ) y J ,
(*,

(+)

(_)

(3.12)
iko^!tofi112.1*,+7o ^U t t 2 . , . _ , , -- f ? ' t , z . r =: U .

( 3 . 18 )t r rr,r*, : hoYzn,<* >* hrV t n, c, >* hep, 1*',Vx
(-)
(-)
(-J

(3.13)('r) The reason why (A.1) is usedis that we will
use a finite-differencescheme instead of the differentiation tvith y, and that it is easier to makean energy conserving scheme from (A.1) thanfrom (2.3).

lh1p,1y; Vr,(-)(3.19)lfikoAo ' / 2 (, a +ut (S| / 2 ,y / 2 ,+ _ )l:): n lZ)- 1 ( , / i k o u " ' " , - ' - - o u t " ' , - ;\) th" \nt\, 2February i982

T. Satomura

l n r, l.l /- 3
- ; i- k' ao3 2 '

_

Auo,
_
n.
()y

d
oy

)
.- )

are as one order small as the zonal mean pa.rts.

\. --ui.2.

-)

-'I

.-/

By using finite-difference approximation to thedifferentiations with 1', and by using the boundary
conditions for the perturbations

2 1uoko" uJ(3.20)x ( h 1 htIp , r , >+ h r Ih ap , r * r ) 1 1 ,
(-)

(-)

1 ,3iko
Ql,z,.'t-

-

/

J

-

noa z
| (, /iko, \ntl
^
n o l
\ z

235

Vrp,<+>:Yzp,1a1:0

aI Y: +7/2,_.1

a3/2,' -T(3.22)cioyauz/2,'.

(-.
\

Dt,z..-t--Llttz,r-t l
o y
t - /

equations(3.12) (3.15) can be written formallyas(3.23)
Ab:ab,

where D is the eigenvector of A to be determined
and it describes perturbation field, and Z is the()y-)
n0- oy.--lcoefficient matrix of finite-difference equations.
(3.2r)
Then, ro and D can be obtained as a numerica-l
In equations (3.12)-(3.17,
Vo, ,", qo, and ql solution of the eigenvalue problem (3.23), andare directly calculated from the grid data by the results are shown in Table I and Figs. 10
Fourier transformation. In equations (3.20) and a n d 1 1 .
(3.21), square terms of the wavy parts of, theAs seen from Table I, the difference between
basic state are neglected, because the wavy parts the eigenvalue and the frequencies found by the
-

Auo,
^

)

l 1 l t Z ,- r I - -2 Auo,,

I^-111111,2,,.

numerical integration is small for the real part.
On the other hand, the imaginary parts of the
Tabie 1 Frequency of oscillating modes. Frequency frequencies of numerical integration are threeof energy is twice as large as these values times aslarge as that of the eigenvalue, but
disagreement
between them is supposed to beNT,,--ricalStability
Wavenumber,'-.'::'-:
not so serious that we must seek another cause, ' , . r s ,. t i o nanalysisof the growing oscillation.
ko/2
0.36
+0.018to.,r*o.oou, It is worth noting that the real parts of the
3ko/2
0 . 3+
6 0.018t )phase velocities of the | / 2ko mode ( : -+-9. 19

I
t

' r

J , ,

.,
I
:
:
i

i

:

\al

i
I
I
I
t

z

/

l

-\

(b)
I
I

l
I

-x

l
0

r
z

I
I

-n

0

E

Fig. 10 Amplitude (left half) and phase (right ha10 of eigenfunction for (a) exp{i(ku/Zorr)) mode, 1b) exp{i(k6l2 * art)} mode, (c) exp{i(3ks/2- at)} mode, and (d) exp{i(3kn/2*at)} mode. Solid curve is the eigenfunction of 7,cr), dashed curve is thatol u(r), and dotted curve is that of o(1). Phasesindicate the maximurn values of eacheigenfunction.

/30

Journal of the MeteorologicalSociety of Japan

Vol. 60, No. I

from the numerical results and -+0.22 from the direction (see equation (3.11)). For simplicity,
stability analysis) are almost equal to those of the perturbation depth of a wavenumber k rs
the linear inviscid theory, 4-0.225 (see Fig. 3(a) re-written asof S1). This fact suggests that a mechanism
h: hrca>eL(kt-@t)+ ht 1e)e-1'(ku-at)
similar to a resonance is acting, i.e., neutral(3,24)+ ht<o>eL'r'*")l lll rrrrg-L(kt+at),
eigen-mocies are likely destabilized resonantly by
where lz1 and h2 are the amplitudes. Then. theenergy supply from the basic state, especiallyperturbation potential energy P.E. of the wavefrom ft6 mode.
number k becomes
Amplitudes(*) and phases of eigenfunctions
( r/2
r 1/2
the
relation
between10.
The
are shown in Fig.
lhf dye,lP . E . r c\t l l , ' , 2t i h z t z ) d t
location of the maximum amplitude and sign of) -r/2) -t /2
r r/2
the phase velocity of ko/ 2 mode agrees with the(3.25)
ReaI(hthzlezi't)dy,+2\
discussion of 51 (p. 156-157), i.e., the mode
) *r/2
which has a phase velocity faster than the ywhere Real ( ) indicates real part. The second
averaged zonal flow, which is 0.5 in 51 and 0
integration of the right-hand side of (3.25) causesin this paper, has a large amplitude near thethe oscillation with a half period of the originalwall at y:-1\/
2 and vice versa. This agreewaves. Physically, this term comes from the fact
ment supports the suggestion of the resonantthat two waves are in-phase and out-phasedestabilization described iittle earlier.
periodically owing to the propagation in the
For 3/2ko modes, the relation between the
opposite direction with each other. If hr: hz,
amplitude and the phase velocity is reversed.
these two waves form a standing wave. Of
It might be explained that these modes correspondcourse, the sum of the potential energy and theto the destabilized neutral modes which exist
kinetic
energy of the wavenumber should be a
(1.1)
along neutral lines throuehmode and (1.2)
if the waves are linear and neutral.
mode or the line through (1.1) mode and (2.1) constantHence,
theoscillation of P.E. is an apparent one,mode of 51 (see Fig. 3(a) of S1). But, suchin this sense. But. waves .in our case are growing
correspondence to the neutral modes of 51 is
inodes and the perturbation total energy oscillates
worse than ftnl2 modes.
due to the oscillations of a and a, i.e, the oscilFig. 11 depicts time variation of the potential
the Reynolds stress. This oscillationenergy calculated by eigenfunctions. The reason lation of
total enerqv becomes obvious
thedisturbanceof
why the potential energy oscillates is as follows:
for t27N
8.
inFig.
There are two waves in each wavenumber andAs shown in Fig. 1 1, the phase of the potential
one of them propagates to the positive s directionenergyoscillation of 3/Zkg mode lags behind thatand another wave propagates to the opposite
of kx/ 2 mode, and it is the same tendency as theresults of the numerical integration. The valueof the phase lag is, however, about tr / 4 and rs
about a half of the numerical results. The ratioof potential energ-y of 3/2kn mode to that of theko/ 2 mode is also smailer than that of thec.
numerical results. But, we may regard that thesedifferences are not crucial and that these disagreements can be reduced by including the
0.1
higher harmonics both 2ko, 3k0,. . ., in the basic@
state, and 5/2k0, 7/2k0,. . ., in the perturbation
0
5
1
0
.fime
structure.
Fig. 11 Time variation of potential energl' of the
From the above discussion we interpret thateigenfunction. Solid curve is ks/2 mode and the oscillations shown in Figs. 8 and 9 aredashed curve is 3tsl2 mode. Absolute value producedby the oppositely propagating wavesof the potential energy is arbitrarily.
destabilized by both the zonal shear flow and(t') There is a small difference of the maximum the steady disturbance of the wavenumber k6.
is
The structure of the disturbance at t:495
ampiitudes between ftsl2 modes and also between 3ks/2 modes. The cause is an asymmetry shown in Fig. 12. It is the structure in the quasisteady state where the viscosity piays an imof the basic field.

T. Satomura

February 1982

L ) l

ill'18:495.0

0,t

2.5

1.5

t.0

0.:

3 .t )X.MIS

Fig. 12 S,tructureof disturbance at t:495 for Re:3000. Arrows denote velocity (u',u")Contour interval is 0.02.
:rnd soiid curves are contours of h:h*ltt.

0.51

Fig. 13 Disturbance depth ht at r:0.45
3 0 0 0a t t : 4 9 5 .

t-l
l l

TTMtr _

for Re:, 1O q'?

0.45. It is seen that it differs from a sinusoidal
variation, but, due to the viscosity, the amplitudes of higher wavenumbers are not large. Compared with the inviscid results at t:187 .1'65(see Fig. 4), the shape similar to a shock wave
is disappear and symmetry between right andleft sides of ridges is held better.
is shown
The zonal mean depth h at t:495
in Fig. 14. We see that & has its maximum near
the center of the channel and the minima near
the walls. It is the same as the results of theinviscid run and is also an opposite tendency tothe supposition from the linear theory. The reason of this opposite tendency is the same as thatin the inviscid run. Of course, the mean depth
is also quasi-steady about this time and oscillates
when oscillation amplitudes of disturbances become large (t>750).
Fig. 15 depicts time variation of MHA.
Decrease of MHA is abott 2Va in the quasisteady state and the contributions (hDand(h'u'> arc almost the same. Thus, we conclude_ls-ithat, in a viscous fluid, unstable gravity waves

1
-!.:u

-0.{!

-0.30 -0.2! -0.1C 0.00 J.i0
T-AXI5

0,20 O.:O O,tO r.:-C

Fig. 14 Zonal mean depth h for Re:3000 at
t:495. The ordinate is magnifiedby ten times.

portant role, but it is still similar to the structureof the linear theory S1. The asymmetrv of the
maxima and minima of depth near the walls areemphasized by a variation of h with y, whichis shown in Fig. 14. (Note that a contour indicates a totai depth h+h'),Fig. 13 shows the disturbance depth fr, at v:

T0.005rMHA

forvariation of
Re:3000.

the half

momentumJournal of the Meteoroiogical Society of Japan

238

12> u
:R'F]
- 0.0

l:495

x to-'

Re= 3000

Fig. 16 Four-box ener,ey diagram at t:495
Re:3000.

forcan cause a permanent change of the momentum
in spite of the uniform vorticity of the initial
basic field. For /)750, the oscillation appears in

Vol. 60, No. 1

viscosity. The results are summarized as follows:
In the inviscid case, the disturbance which has
the structure quite similar to that of the linear
unstable wave grows exponentially at flrst. lts
growth rate is the same as that of S 1. The
larger the amplitude became, the more significant
the nonlinear effect beca-me.Finally, the structureof the disturbance formed a shape similar to a
shock wave, and the numerical integration was
broken down. This shock wave shape of the laststage of the numerical run confirms the discussion of 51 about the dispersion diagram (Fig.
3(a) or 4(a)).
The momentum mixing, which was one of
the motivations of the p,resentstudy, was occurredat least temporarily in the inviscid case. Theenergy budget showed that only the mean kinetic
energv ,R is the origin of all other energy, andthe transformations [KK'], [Kf], and lK'P'l are
all positive.

(ha+"i-'l>,but the amplitude of the oscillation
In the viscous run, on the other hand, thesame disturbance as that in the inviscid run grewof (hfi
is much smaller than that ot (TV).
It indicates that the energy conversions among exponentially, and reached a quasi-steady state.
oscillating modes are almost closed within the In this quasi-steady state, the momentum of the
disturbance energy K, P', and the additional half area, MHA, was reduced by 2%. It sugmean kinetic energy Ko. It also indicates that gests us that the unstable gravity waves can mix
the momentum permanently.
the relation (3.4) is satisfied.
The wavenumber at which the energy is
The quasi-steady state was unstable for (nmostly dissipated is estimated (not shown) and 1/2)fto mode. The unstabie modes had a freis fts as expected. This fact confirms the above quency a=0.36 and growth rate o=0.018. There
description about the momentum budget. They were two phase velocities of opposite signs forare consistent with the energy budget described each wavenumber, and the energy of each wavebelow.
number oscillates with a frequency 2a. To exThe energy balance in the quasi-steady state plain these unstable waves and oscillations. a
is shown in Fig. 16. This four box diagram shows linear anaiysis in which the basic state was comthat the directions of the energy transformations posed of two parts, the zonai flow and the ko
are the same as those of the inviscid run. Be- mode, was examined. Obtained eigenvalues andcause this diagram is depicted for the quasi- eigenfunctions agreed with the results of thesteady state, the in-flow and the out-flow of the numerical integration.
each potential energ,v P and P'cancel out, andThe energv balance in the quasi-steady state
the transformation [KiK"] : 5 X 10 -5 is the same showed that the viscous effect near the walls
as the energy dissipation rate of the disturbances. creates the mean kinetic energy K. most part ofFrom this figure, it is also seen that the energy the created energy is transformed to the disturbis supplied to R by the mean viscous force D ance kinetic energy K', and, then, dissipated byand is dissipated by the disturbance viscous force the viscosity acting on the disturbances.
Dt acting on K/. This fact confirms the stateIf the channel is much more longer than thatment that disturbances permanently mixed the used in this paper, the most unstable wave growzonal mean momentum.
ing in the quasi-steady state may not be (n 1/2)ko mode but the side-band mode (n-+a)ko
4, Summary and conclusions
where a is a real number. Indeed, by an stability
In the present study, we have numerically analysis for the side band modes, we find thatinvestigated time development of unstable gravitl. side band modes of +-0.4k0, -(1-0.4)fts are thewaves in a shallow water with and without most unstable modes in the quasi-steadv state.

239

T. Satomura

February 1982

Their growth rate is seven times as large as the
sub-harmonics. But, adding to the fact that the
wavelengths are similar to those of the subharmonics, the reai part of the eigenvalue isalmost the same as that of the sub-harmonics.
Thus, results of a long channel model will notchange quantitatively.
As mentioned in the previous paper 51, unstable gravity waves possibly play a certain rolein actual geophysical fluids, because Fr is able
to exceed unity easiiy for internal waves instratified fluids. Aithough wave energy is concentrated near the walls in a channel modei,
Blumen, Drazin, and Billings (1975) and Satomura (1981b) showed that gravity waves cangrow without wails and radiate to infinity. In
many cases, growth rates of the present instability
may be smailer than those of barotropic or baroclinic instability, but the present mechanism hasa point of advantage that a zonal flow becomes
unstable with no inflection point of the flow,
no rotation of planets, and no temperature
variation of the basic field. Thus, if an excitationof disturbances is found by observations in abarotropicaily and baroclinically stable area, thepresent mechanism will be a strong candidate forthe excitation.
Present mechanism is also one of the candidates for mechanisms to excite gravity waves
whose wavelength is hundreds of kilometers andwhich are observed in the middle atmosphere(see e.9,., Heath et al., 1.974, or Muller, 1974),
for gravity waves excited by the present mechanism have long wavelenglhs of the order of thewidth of the zonal shear flow. To discuss it indetail, however, it is necessary to solve a problem
of vertical propagation of excited waves. It isleft for later studies.
Acknowledgements

puter Center of Tokyo UniversitY.
Appendixdifference scheme

FiniteA:

We will present finite difference scheme, which
was developed by Arakawa & Lamb (1981). At
flrst. equations (2.3) and (2.4) are transformed
to following forms:

"

dtt

n

?-qit+

0 t - ? f idat1Kr@):0,d4-qu*i-(K-o)
0

t

-

0

Ah

Au a0

0t

o&

- I - I - : t l

=a,

(4.1)

Y

oy

is the mass flux, O'
where (1, D:(hu,ha)h/ Frz, K: (*'t az)/ 2, and viscous terms are
dropped for simplicity.
Then, required f,nite-difference scheme whichconserves total mass, energy, and potential
enstrophy is given by=

a U i , ,j + t 1 z - t x i ' j + 1 / ' z a^i + l / 2 ,

-

-,)r.
-di.

j-t,20' I/2' j r)-T

i , j + t , 2 t ) t - tt z . j

i41pti411z, i*etat7z,-.i. 1/2,

i+r

i + 1 / 2 i t i - 1 ,i + r n * 7 fi+tpiliar,

ia1,2

d

x l(K I A)i,* r tz, i + t p - (K * @)i- r tz, i * t nf:O,

(4.2)

a

! i + t 1 2 ,1 * 7 t + r , j + r / 2 i t i +1 ' j + L/ 2U;

! d i , 1 1 1 p t u ,i = r / 2 + o t i , j ' 1 / 2 r t i , i - \ / 2iFi*t,i-tnitt*1, j r/2+ 9i+ttz,i'rPA't+rn, i+t- g r r t t r ,i - r P i t a t 1 2i,- t I 1/ dx l ( K * Q ) i * t 1 2i ,+ t p - ( K I @ ) i , *t t z ,i - r p f : 0 ,

(A.3)
a

--

h ' t * r t z ,i + t f l - l

1
Tlittrt,i+tn-fti'

irttz

The author would like to express his thanks(A.4)* A t r 1 1 2 ,j + t - O i . + t t z , j l : O ,to Prof. T. Matsuno of Geophysical Institute ofTokyo University for his encouragements and where d: y'r: ly is the grid intervai,
discussions. He is also grateful to Prof. K.
1 ,,^(ht+rtz, i+tpI
hi-112, i'1p)ut, i+1/2,i,iti, i+ttz:
Gambo and all the staffs of the Meteorological
2
Laboratory in the Geophysical Institute of Tokyo(A.5)
University for their support of this study anduseful discussions. His thanks are extended to( h i * t t " , i + Lf i h i + r 2 , i - t n ) Q i + t t z ,i ,Ai + t t z , i :rDr. Yoshizaki in the Ocean Research Institute(,{.6)of Tokyo Universitv for his discussions andt criticisms.
ei+ttz, j+1t2:
i+tl qt, i+1- qi. i
24Lqt+1,
The computations were performed by the useof the HITAC M-200FI computer at the Com- qi.+t,
(A.7)
il ,

^

r ,.

240

Journalof the Meteoro,logical1

9i'+1/2,
i+t/2:

Societyof Japan

1 , ,i't+ qi.,i+r
24L- Qi+r,

At z! h t - t 1 2 B\ n i . r t z .B = 1. !r/21 qr.B
di1+qi, j-qi+r, jl ,) a 1
-7l12IOi+tt(A.8)

B x r n * i t t , B x t n ) ( Q i +nr ,- q i ' s )Iai, i*rtz:iol2Qi+r, i=1*qr, i*r*2qt, i* qi1r, 11,*(fii,

to'n',
Fi, i*rtz:24lqi,

i*tl2qi-t,

n4p*iti-1,* (Ai+ttz, ert * 0i-t

i+r+ qi-1, 112q", 11,

-\l

1t':.

Bltt

2l(uttttz.

(,{.10)x (qirr,I
f t, i*ttz:^f2qi,

i+r+ qi-1, 1*1!2qi-t, i * qi, il ,(A.11)
I
6t' i-t,r:2

t Q i - t . 1 - t * 2 Q , .j - j L q t . i l 2 q t - t ,il,(A.12)
1 7lut,qi, i:

i-ttz-ui,

i + r / 2 + ' u i + r / 2 ,i - ' U i - r 1 2 , 1 l

1 | ( h t * r t z j,+ r p * h t - t 1 2 ,i a r 1 2
4 |I h i - r t z ,i - r n t h * r n ,Vol. 60, No. 1

-l

itn))
(A.13)

B!1/2)(qi, B-qi-r,

B)

1 2 ,e r t ) ( q i r e r t -qi, n)j[tt1t, e-_ttzLitj.zlt:)

B x t - e i , n ) * ( 1 t t a t 1 2 a, y r I- itr.,Btn)(qtn,,r,, - q,, r)]l'!fu'*r,'f u211t7z,
i+tl'u2't+t12, i) .

(A.15)

I

where B: bl and b2 for the plus sign and theminus sign, respectively. This scheme (A.15) rs
used to calculate ei,61 and 7i,az in the inviscidcase.
The scheme (A.15) is stable in a sense thatsquare quantities such as the potential enstrophyand energy are finite and conserved. A little
calculation shows, however, that (A.15) can betransformed as
1 "K i + t / 2i,+ 1 / 2 : !

rtiat, z4 7z

7i+rplu2t, ia112:(,A..14)'6q''B\ n t t t t z . B = t t z - l l l t , 1/ 2 , B - r / 2 )-Qi, a0| ,,*,(hi,+rtz,a-nxrni_hi-rtz> Bxrft)

In order to prove the conservation char(4.16)
lG(q, it, A) ,acteristics of this scheme, an attention shouldq, fi, and 0. If
be given to grid points near the boundaries. To where G is a function ofasuffice the requirements of the conservation of(A.17)
Batp*hi-t12, n:rn)10 ,|(hiatlz,
OT
totai mass and total energy, it is adequate to set
o-points at the rigid walls as shown Fig. A1, and (,4.16) seems unstable. In the inviscid case ofa:0
at the walls. On the other hand, it is easy this paper, there is no problem because q:-l
to derive that following forms for qi,61 and qi,62, initially everywhere in the domain and, thus,
which are potential vorticity on the walls, satisfy the right-hand side of (A.15) is always zero
the conservation law of potential enstrophy:
within the truncation and round-off errors.
On the other hand, however, q will varyphysically with time in the viscous case, and theright-hand side of (A.15) is no longer zero dueto the viscosity. Because (A.15) has the ambiguityin the stabiiity property as seen in (.4..16) andbecause it is hard to find out an appropriate
definition of the viscous terms which should beo q
added to the right-hand side of (A.15), we define. hQ;61 and Qt.tz aS_

q t . b1"u-(ut,ett1z-Ltr,sazp]_'l)i+1/2,8lt-'l)i-tn,ext)fd(ht*rn,nttp*ht-ttz,Bxtn)/2(4.18)(li x V)i,n1r
Qt.B: .
\h)i,n=,.12
Fig. A1model.

(A.19)

Instead of (A.15), we use above scheme in thea/1

T. Satomura

February 1982

the disturbance potential energy P' are definedviscous run.
Although scheme (A.18) destroys the con- b y
_
1
1fr/2
servation law of the total potential enstrophy,(8.6)
P: -.;\\n-t)zay,
the effect of the destruction on numerical soiurr- L J -t/2
tions is found to be small not only in the inviscid|
1 fr/z _(8.6)
h''dy.
case but aiso in the viscous case. Thus, weP'::;.+\tr"Z )_1/2
regard numericai solutions as physical solutions.
As a preparation of derivation of energy
equations,we derive the equationsof time change(2.3) atd (2.4)
Because energy equations are rather strange of momentum. Zonal averageof
B:

Appendix

Energy equations

form, we present detaiied derivation of them.
The total kinetic energy is
1 ft/2I(:+ \za a - /,t \Lu' . lv'
^ V+D*-vt(u'^dt

Ei'+a5dy

<

:+

f r'

{hlur+rr)+2(h'u'il+h'!'a)}dyz )I , r / z+;
L(h+h')(u'z+a'2)dy.

\* i-hDi-"
0t
0y

(8.1)

a | / e1 lt/z _Instead of the zonal average, we can use a'V : ::(B.3).
@in'11u'a11a1a,weighted mean velocity /defned as

oy

h t1

'

I

(B.e)

o.o' ,

9!*!o,r ,+ - hu,+ h,r j)
0t

0n
n

-

a -(htu/ -l ha' + h'u - h'a'):O .

h '

(8.8)

- ( o! -tl-\v' - ),r t'
- r'Y
0y
oy/
fr"
\ dtr

(B.2)
Z ) -1/2

(8.7)

^ F,

Iae

Yo t * ( \ , ' dLf * u 'd!y /\ r ' * \(o !o J+ , ' ! \d Yv/ '

)-l/2

K':+\

0y/

-

Subtracting (8.7) and (B.8) from (2.3) and (2.4)
respectively,we obtain)_t/2| |"{h(it,+u,)+2(h'u'aah'a'D)}dy,
\
L
I

dn

\

1

d k a . _ a - ^: v

e1/,

The mean kinetic energy K and the disturbance energy I( are defined by

,K:;

_

* *vh:

F l / t

1

oy

I

J-1/2

(8.10)0y

and a deviation from it,

Multiplying (B.7) and (B.8) by h and / respectively, and adding the results, we obtain

I':I/-7,
to construct energy equations (ex. Eliassen and
Kleinschmidt, 1951). The energy equations ex-a_:0ht v +a __^- l h Y Dh+v ' u ' ).
oy:v'r*trt-rT-#tr* E.

pressed by fr and V have two advantages: Ko is
implicitly included in the 'mean' energy, andtransformation terms are little simpler. Those(B.1i)
equations are, however, essentially equivalent to
the present equations, and the strange trans- This is the equation of zonaily averagedmomenformation (B.17) does not disappeared even in tum.
Subtracting(8.11) from the equationof total
those equations. Moreover, the importance ofK" would be overlooked. Thus, we use the momentum, which is written as
ordinarv zonal average.
aad; hv+;-(huV)*. (haV)
The totai available potential energy is
ot
ofi
oy
1

1 fI/2rr-z ) _ r / 2\

P: :,.+\
1| gtrz ',-.+ z |
rr-

/

f1/2

(h-\

(h-r)' dy.

\2h a ya
)y

J_r/2

(B.4)

)_1/2

The

h

h

.F r"

Ile

+;,vh:+F,/

mean avaiiable potential energy F

andand averaging zonally, we obtain;,_a _u' + h'Y'a+ Vh'a');d .t h'V' I -yh'Yt
dy'(B.12)

1/1

Journalof the Meteorological^ _|: - v'r '1hv'1- V+Td - +.h'fn'
oY
fr'
1 -

9(Jt * , : - [ K " K '] + [ K . K " ] - I K P ' l -D , .
(8.23)(B.13)I ^ h'F'.
Ae

where

This is the equation of
momentum.

averaged disturbanceThen, multiplying rB.7) and (B.ttt ay hV'Z
andV/Z respectively,multiplying (B.7) and (B.13)
by ilTand V respectively, adding the resulis,
and integrating with :r, we obtain the equationof the mean kinetic energy as

a - K: -IKK' ]-[KP]-[KP' ]-D,
dt
where the transformationterms tff'],
f<P'l are defined as(B.14)
Ifpl,[K"K,]: -\"-,',,rwr, {ay.
f I/2

-/

-h'v'.\,r'
D.:*

)-trz

\

d?'

D,:+\

(8.15)[ K P ] : - ; r r ' ) -I t t z h -0(yh D + h ' u ' ) d y ,(8.16)I (t/z , aflD .
U ( P ' l : -l rl' . 1 - s / z h ' "d' y. " d y .

8.17)

_

I

lr/2

D: - \
Keir/2

r \ e ) . 1/ 2

dY/

(nVf-t'V'.F+h'F'.Vtdy.

i,6

a*-a'-

r)v']at:.

(8.26)(8.27)

rTF?.F+T'F.tt)dy,(8.28)

The time rate of change of the disturbancekinetic energy K' is obtained by subtracting(B.14) from the equation of the totai kinetic
energy, and is written as

4cJt x ' : l i x ' l - l K ' P ' J _ D ' .

(8.2e)

where

)-Ln

1 ft/z _(B18)(B.25)

FO,,\",,,1,'
1

-/

,
^
;
i T ' . ( u 'l - + u ' I \ v ' a v.

^v

lt/2 taV
lK,K'l:-J_,,luT" 7;

).ttz

f1 r,

( ..

(8.24)[ K . K o ] :- J , , ^\ h ' a ' v . : -)_tY.V't;(hv,)ldy,
)

anO

va-,
I KK ' l :- 1 " ' o i -T '.loY+I

VoI. 60. No. 1

Society of Japanl K ' P ' l : ;t"r "\hv"vh'dy,(8.30)-l_t/zMeanings of transformations will be explained
1 l)/2 _(8.3i)
D': ^ \
hV' .F'dy,
later.
rae )-L/z
Since the last term of the integrand of (B.2)
The rate of time change of F and P' areplays an important role as shown in 51, we
obtainedfrom (B.8) and (B.10) respectively, anddivide the mean kinetic energy K into two parts:
are written as
K:K"+KO,(B.1e)iwhere
1 (r/zK.:+ \ h@z+o'1dv
L J-r/zat(8.20)

A(8.32)

P:lKPl ,_;P':LK' P' l+IKP' ],(B.33)

expresses the mean kinetic energy arising fromonly the zonal averaged quantities, a n d

The transformation [KrK'] represents a conversion
from the mean kinetic energy K into
f 1/2
(h'u'a+h'a'D)dy
Ko: \( B . 2 1 ) the disturbance kinetic energy K' and is a sum
J-t/2
of the conversion by the Reynolds stress andis interpreted as the additional part of the mean the higher order correlatioa. The transformation
kinetic energy due to the disturbance surface
It<pl, tf pl, and [KP'] are the elements of total
displacement. Then, equation (8.14) is divided[KP] transformation which manifests the work
as follows:of the flow to raise the surface against theagravity.

- K.: -lK,K')-LK,K.I-[KP]- D, .

dt

(8.21)The [K,Ko] transformation represents the inner conversion between the constituents K" andT. Satomura

February 1982

Ko of the mean kinetic energy K-. Transformations [K"K'] and [KoKf are the elements of theD, D', D., and Do represents the dissiIKK'|.
rates
of the kinetic energy K. K' K,, andpationKo, respectively.
C:

Appendix

Derivation

h

,f ,

of viscous terms

In cases of viscous fluids with a free surface
as a shallow water, we must include effects ofthe free surface into stress tensor of a viscousterm. It comes from both facts that a volume
eiement on the free surface feels no stress fromthe out side of the fluid and that we treat thefluid only as a set of columns. We will take such
effects on stress tensor into account.
Fig. C1 Schematic ."p..."nfutlo.t of viscous force.
In general cases, stress tensor is written as
At r1:a, depth is h(a) and stress is 2err('c).
/
1\depth is ft(b) and stress is
At xt:6:o:rAr,(C.1)
P , i : - P d t i - 1 2 1 1 ( e i 1 - " e n d i 1l ,2en(b,\
J/
where dl1 is the Kronecker's delta and is definedAS,"r ; : \ 0f l

for i:ifor i+i,| / Au*(c.2)€mn:-:\.
Z\Ofin

Au"\
f

;-J
ofim/

(c.7)

We used the facts that ely nand m, n:1,2.
the right-hand side of (C.1) becomes zero from
(2.4) and that there
6rzi\| / - .Aut+
P;i:-t|(c.3) the equation of continuity
is no momentum flux through the bottom or the
2 \y Ar1
or, /water.
is the rate of strain tensor. In this appendix, surface of the shallow
transformationfrom the integral equaThen,
j:t,
we use subscripts i,
2, and 3 to indicate(C.5) to a differential equation gives thethe r, y, and z components respectively, for tionrequired equation as
simplifying tensor expression.
6a (qh2_./6u* -l orun\)
The equation of momentum is written as^ hu*: ^ i"--orn+ril{ rl}olIn/ )
dtn \ z
Ot
\ ofn
, rif
ff
ff
pudV:\\
Pi;n,dS-\\ putulnldS
4 . ,o- r )\ \) \) v
JJs
JJs
(c.8)
.

7 is the first coefrcient of the viscosity, and.llL pKidv,(c.4)

where I/ is a volume enclosed by a surface S,
ni is a l-th component of the normal vector ofS, and K; is a body force.
Since, in case of a shallow water, hydrostaticbalance is assumed, and both u1 ard u2 ar:
independent of tr3, we can integrate (C.4) for a
fixed column which has a cross section S" anda periphery C to obtain.
o f l
f
f
- \f hundS":Q P^"ds-$ hu*u,ds.
ot))scJc
Jc(c.5)
where P-,P*n:

is defined as1-;gh26*,*2vhe^n,

z-

(c.6)

where r is the coefficient of kinetic viscositv.

^ \llumun)
0fin

Thus, the 11 and :r2 cofilponent of the viscosity
term of (C.8) are the right-hand sides of (3.6)
and (3.7), respectively. Of course, (3.6) and(3.7) are reduced to the ordinary Laplacian
forms if h is a constant.
Now, it is easy to derive them physically. We
will confine ollrselves to considering only one
direction of the stress tensor for simplicity (seeFig. C1). The momentum flux by stress per unit
area is 2a11. Thus, the total momentum flux
through the left-hand side of the column at!x1:a is 2e11,1h61, and the flux through theright-hand side is 2e161h161. Because the force
which acts the column. is the divergence of the
momentum flux. the viscous term should havethe form as
F<"=o>: iimlx-0

h < o * , 1 " .>2 e n c o + 7 r y* h @ 1. 2 e n c o >244Journal of the Meteo,rologicalSociety of Japan-rra:1 ^ 12heiIVol. 60, No. 1

3. l. Fluid Mech., 82,255-260.
Eliassen,
A.. and E. Kleinschmidt, 1957: 'Dynamic
Lofi
_)a=a
Meteorology' in Handbuch der Physik, Springer,
If we extend above discussion to all directions,
Berlin, 48, 1-154.
we will get the same terms as (3.6) and (3.7).
Heath, D. F., E. Hilsenrath, A. J. Krueger, W.
Nordberg, C. Prabhakara and J. S. Theon, 1974:
References
In 'Structure and Dynamics of the Upper Atmosphere', F. Verniani (ed.), Elsevier Scientific publ.
Arakawa, A., and V.R. Lamb, 1981: A potentialCo., 131-198.
enstrophy and energy conserving scheme for theshallow water equations. Mon. Wea. Rev., AOg, Muller, H. G., 1974: ibid., 34?-388.
Satomura, T., 1981a: An investigation of shear
37-51.
instability in a shallo'.v water. J. Meteor. Soc.
Blumen, W., P.G. Drazin and D.F. Billings, 7975:
Japan, 59,148-167.
Shear layer instabiiity for an inviscid compres1981b: Supplementary note on shear
sible fluid. Part 2. J. Fluid Mech.. 71. 306-316.
instabilrty in a shallow water. ibi .., E9, 168-121.
Drazin, P. G., and A. Davey, 7977: Shear layer
instability of an inviscid compressiblefluid. Part(c.e)'&tka>7 -TF.ft.(:Hf 6ffifr,
HII*IJ W .fE T

ffi

E T { f f i 8& i"c, i:Z<eE€)J'tfr.Dffi*\,'Lf{{EBiciE^:/:"
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fh&rt\b bE&, WD' i )v+ - tr+EH)ft Eqr.i+L, EgElrEl(ElrriEAt
9t+ 6 . L t ^LL.
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6o;zJ-tvf
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fs.cafftf!-f
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