148

Journal of the Meteorological Society of Japan

An Investigation of Shear Instability
By TakehikoVol. 59, No. I

in a Shallow Water

Satomura

Geophysical Institute, University ol Tokyo, Tokyo(Manuscript received 15 August 1980, in revised lorm 6 November 1980)

Abstract
The stabiiity of parallel shear flows of a shallow fluid is investigatedby linear analysis.
Following Biumen (1970), a necessarycondition for instability and the Howard semi-circie
theorem are rederived for a shallow water, and energy equations are aiso derived. Then we
examine the stabiiity of two types of basic flows: plane Couette flow bounded in both sides(case I), and the same flow but unbounded in one side to connect with a rest fluid (caseII).
By obtaining soiutions expressed as power series, eigenvaiues and eigenfunctions are
determined to high accuracy, In the case I, it is shown that normal modes of gravity waves
in a channel are modified by the basic shear flow to become unstable in some discrete
ranges of the wavenumber, if Froude number, Fr, is greater than 2. In the case II, it
is shown that two types of unstable waves are found for Fr)1:
One is similar to theunstable waves in the case I but modified by the presence of the rest fluid. The wave
is trapped near the boundary. The other is a wave destabilized by the shear too, butradiates its energy to the unbounded rest fluid. The unstable'regions of the waves of thistype are continuous in wave number space. The structure of all unstable waves found in
this paper are similar to that of gravity waves.
It is shown that the unstable waves extract their energy not from the "ordinary" mean
kinetic energy, but from an additional term of the mean kinetic energy arising from thecorrelation between perturbation zonal velocity and perturbation depth, and thus reduce the"depth-weighted" mean
kinetic energy. Variation of the basic flow with time and somepossibilities of redistribution of momentum by these unstable gravity waves are discussed.
Relation between our results and Blumen et aI. (1975) is also commented.

shear flow of a non-rotating fluid having a hyperbolic tangent proflle, and also found reduction
The purposeof this investigationis to examine of growth rate by the divergenceeffect for Machthe effect of horizontal divergence upon the number M<1. LIe also deriveda necessary
constability of a parallel shear flow and to present dition for instability: M (or Froude number
a possible mechanism of excitation of gravity Fr)>I, or U/U"10 in some region of fluid.
waves.
The former condition relates with the effect of
To investigate the effect of the divergence strong divergenceand the latter condition relates
upon the barotropic instability, Lipps (1963) with the existenceof an inflection point (barotropic instability). Since he discussedonly unused a quasi-geostrophicflow with a jet (U:sechzy)profile in a two-layered incompressible stable waves for M{1, Blumen, Drazin andfluid. He obtained complex eigenvalues by Billings (1975) extendedthe analysisto unstableanaiytical extensionfrom an exaet neutral solu- wavesfor M>l in the samebasicflow as Blumen
tion, and found that both the p-effect and the (1970). An interesting finding of their work isdivergence effect stabilize the flow. Philander that there exists another type of unstable waves(i976) studied a model similar to Lipps. His which are thought to be acousticwaves generated
result was that both the B-effect and the diver- in the shear zone and radiate outward. They
gence effect stabilize westerly jets, but both called this type of unstable waves as second
these effects destabilize easterly jets. Blumen mode. Although they obtained such unstable(1970) discussedthe stability of a compressible wavesfor M>1, and showedthat they have non1. Introduction

February 1981

I.

Satomurazeto phase speed, they did not discuss theenergetics or structure ot the waves at all so thatthe nature of the waves remain unclear. Further,
the coexistence of the inflection point instability
interpretationdifficult.
makes the physicalWhether the second unstable mode is linked withthe point of inflection or not was not explored.
One of the motivations of the present paper is
to examine unstable travelling waves (gravity
waves in a shallow water or acoustic waves in acon.rpressible fluid) occuring in shear flow morecomprehensively and to make their characteristics
ciear.

119

structures and energetics should be discussed in
more detail. In this paper, two cases will be
examined: case I is a Couette flow between twoparallel walls, ard case II is a Couette flowbounded only on one side and joined to a semiinflnite rest fluid. We consider oniy inviscid
fluid. Froude number is greater than I for theboth cases because these flows are stable forFr(las shown in later section. Case I is treated
for the purpose of finding a new unstable mode
which has no relation with barotropic instability.
In the non-divergence limit (Fr-+0), Case (1960)
showed that the perturbation equation for plane
A number of studies on mechanisms of ex- Couette flow has no normal modes (discretecitation of gravity waves by a hlrrizontal shear, spectrum solutions) but only has continuous"weak"solutionswhich is another motivation of this paper, were spectrum solutions, which are
performed in context of the rain hand formation having a singularity at a critical point. He also
in hurricanes. Among theoretical studies, Kuri- showed that the basic flow is stabie even if suchhara (197 6) made a linear stability analysis a continuous solutions are used for expressing
using an approximation that both Z (azimuthal initial disturbances. Thus, if there exists anvelocity) and dV/dr of the basic state are con- unstable discrete-spectrum solution in case I, lt
stants. Although he found unstable out-going can not be connected with barotropic instabiiity
gravity waves, his model may be too much at all. but it should owe its existence to diversimplified to reach a definite conclusion of this gence (gravity wave). In fact, such unstableinstability problem. Willoughby (1977, 1978) modes exist as will be shown in later section. On
treated a similar problem considering the varia- the other hand, case II is treated to examrnetion of basic flow more strictly, and founcl that radiative characteristics of the unstable gravity
neutral forced waves grow spatially in outward waves. We will find unstable out-going gravity
propagation but their growth rates (spatial waves and unstable trapped waves in the case II.
amplification rates) are not large enough to comThe present u,ork has some interesting techpensate geometrical spreading of cylindrical co- nicalities. Complex eigenvalues will be calcuordinates. Recently Broadbent and Moore (1979) lated down to ten decimal places by utilizing
studied the eigenvalue problem for a vortex in analytical solutions expressed by power senes.
a homoentropic and compressible fluid for a wide The method used to obtain eigenvalues andrange of M in the context of the noise generation eigenfunctions enables us to save computer refrom supersonic flows, and they found that sources and to evaluate them with anv accuracy
acoustic waves which propagate to infinity are easiiy. The method is superior in that we neverweakly unstable. Their model is similar to ours miss eigenvalues while we are lree from theexpect for the coordinates. But they did not tedious work of omitting computational modes
discuss structure or energetics sufficiently.
which are often mixed with the true eigenvalues
These studies cited above suggest that hori- if we use a finite difference method.
zontal shear flow may be unstable even if there
2. Basic equations
is no inflection point, and that gravity waves mav
be excited. In connection with excitation of
We shall consider th: stability of a plane
gravity waves, Blumen (197i) investigated the parailel flow, U*(y*), of a shallow water wherecharacteristics of disturbances in a two-dimen- the basic flow U" is in the r* direction, andsionally varying basic flow U:(a*z)tanhy.
He varies in the transverse direction, y*. An indiscussed unstable modes, but his discussion was finitesimal disturbance having the velocity (il*,confined to solutions which are valid onlv for 'u*) in the (n*, y") directions and a surface disFr<1. Thus, the unstable modes which he found placement ftx is superposed on this basic state.
are essentially barotropic instability.
It is convenient to non-dimensionalize the
It is rational to consider that unstable modes. quantities by introducing a velocity scaie U6,
related not with the point of inflection but with and a length scale I, both of which are straightstrong divergence, should be sought and their forwardly derived from the basic flow, andar50

Journal of the Nleteoroiogical Society of Japan

Vo1. 59, No. 1

depth scale 11 which is the basic depth oi the the region of the rest fluid is
fluid. The dimensionless coordinatcs, time, veloc[ : e x p l + / r{ 1- F , ] C ' ] , / , y J .
(2.8' )
it1,, and surface displacement are written as
We do not confine ourselves to 'subsonic' disturb( . n , y ) - ( r + , y * ' ) /L' ,t:t*LroiL,
ances, that is, dimensional phase speed C,x can( u , u ) : ( u * , u x ) 1 L ,o
U:U*/Uo ,be greater than the velocit;r of thc gravity waves^/ llH, so that Fr2C,2 can be greater than 1. Thus,h -_h*,/H(2. 1)The perturbation equations for disturbances u-r the radicant in the right-hand side of (2.8) or(2.8') can be either positive or negative, andan inviscid shallow water (rvithout Coriolis force)accordingly, the solution is exponential or sinuthen become
soidal. We select a solution which decays away
6u -.6u
dU
| 6h
() )\
- : ,
_
from the shear zone or a solution which expresses" d yJ __:tlot'" drF,-6r
a wave radiating from the shear zone'r'. Such
A
u
6
h
|
solution
is appropriate to be used as a boundary
,
,
6
u
_
_ I
: t ,
(2.3)
oI
oJ
condition for a solution in the shear zonc.
o"tfr'

Ah L [ ' 6h

6 u : - : oo u

At

ot

-

"6r

dy

(2.4)

3.

Characteristics of the basic equation

Before obtaining solutions of (2.6) for specific^,/ SH is the Froude number, g
where Fr:tlo/
flows, we shall discuss some properties of soluis the gravity acceleration. Equations (2.2) and tionsof (2.6).
(2.3) are the momentum equations, and (2.4) is
3.1 The Howard semi-circle theorem
the mass conservation equation.
Since (2.6) is the same form as (7) in Blumen
We assume a form
(1970), the Howard.semi-circle theorem (Howard,q : 4 (y ) e x p[ ( i k (x - Cr t ] ,( 2 .5 )1961) can hold, as Blumen (1970) has shown.
for each component q ol the perturbation Dividing (2.6)
by (U*C)3: multiplying by hx, thequantities, where ,t is a real wave number andcomplex conjugate of /r, and then integrating,
C:C,iiCi
is the complex phase velocity. The we have the following relationfrom the realcomplex eigenvalue C is determined under the part of the result,condition that perturbation quantities satisfy their.
.
^ lb ap-Ll-.rl
linear equations (2.2), (2.3) and (2.4) and apU --l ) "dy:\ ktF.:htdy
t :U - C , 4odpropriate boundary conditions.
L -,, -L o'n
Upon writing u, u, and ft in the form as (2.5),- l _ I1l |' -L . - \r: r:
we eliminate fi and 0 to obtain a sinsle equation)t\-

IOr /r as( u - C ) h " - 2 u ' h ' \ k , ( U- C )' f F , ' ( U C ) ' ' ,l l. hi : 0)'''

lU^^'-{r'.,"

it,i'

\

'/

r

i l
t )a

Qdy
lu

-L

'. '\
\ J ' ' J)

where boundaries are at y:a and y:D; a and b
may be infinite. U*"* and U,,in denote thewhere prime denotes differentiation with respect
maximum and minimum values of U, respectively,
to .v. Equation (2.6) becomes identical with (7)andin Blumen (1970) for a perturbation in compresq : p z i f i i z ' i, ' ) " r } .
(3.2)
sible fluid with shear when Fr is replaced by, o - .
i " ^Mach number M as he pointed out.
S i n c e k z F r z l , ' 2> 0 . ( 3 . 1 ) i m p l i e s
The boundary condition at a rigid wall is
U^"*tU^in \2 n.,lU^u'-U-i"\:
I c-l f ("'-:=\
wrrtten as
z
I
\
t
t
/'
i,

(2.6)

(2.7)

( 3 .3 )When the fluid is unbounded. we assume that
This is the theoremstatingthat if there is a
U approaches a constant value outside the shear growing
disturbance as eigensolution its eigenzone. Then from (2.6) we have the asymptotic
vaiue C"*iC
should lie in a semi-circle on theform of a solution ascompiex plane, given by (3.3). This theorem isi.
f t- e x P [ - : k U - F t 2 l U - C ) 2 ] ' " y I
lmportant not only from theoretical viewpoint
as lyi--.

(2.8)

If a rest fluid extends in the outside the shear
zone as in the case II, the solution of (2.6) in'F In the case where C has an imaginary part,
thislatter condition becomes the same as the former
one.

l.

F e b r u a r y1 9 8 1

15 t

5alomurlbut in a practical procedure of searching for C
in the complex plane. Thus we will use (3'3)
to restrict the complex C plane in which we mustsearch to find complex eigenvalues'a basic flow is stable if (3.10) or (3.10') holdat least in one o[ the coordinate s1'stemswhich
are moving in the r-direction. Note that sucha consideration leads to the stronger necessary
condition for instability (Fjgrtoft, 1950) concerning the barotropic instabilitv.

3.2 Necessar,t condition t'or instability
We derive a necessary condition for instability
3.3 Energetics
following Blumen (1970). The equation for the
For discussing the energetics of the flow,
potential vorticity c,.,is derived from (2.2)-(2.4)
form is convenient. The total velocdimensionalasity (.ur, oi) and the total depth /z+ can be exAa
- . 6------,t-.:( 3 . 4 ) panded in power series of a small parameter .
-u
u
TU
ot
ofr
(to the second order) as
where
Yl-IJ aeule2u(2) ,y l : 6 . y l e 2 u ( 2 >,(3.11)( 3 s. )oy
0!rhi+H+rh+r'H'",
T h e p e r t u r b a t i o ne n e r g y e q t t a t i o ni s also obtained
where all variables have dimensions and e is aASsmail parameter proportionai to perturbatlon
a l , ' l /h 2 \ ,
amplitude. U is the basic flow, I/ is the basic
t
t
L
t
z
t
.-\ \+{
ldtclYo tJ ) t \fr-,
depth (constant), u, a, and ft are the f,rst orderr:fand tJe), Ve), and I1(2) express
perturbations,(3.6):
1 \ u u L l , d r d y.
variations of the zonal mean quantities at the) J
and inte- second order. So far as we are concerned with
Now (3.4) is multiplied by aU/U"
the energetics, we do not lose validity by conyieldgrated over the whole domain to'sidering only the zonal mean parts of the second
ff
o ll u o'(3 '7\
dxdY '
order variables u(2), v(2), H(2).
; t J t \ \ , 1 , t ;- d t d Y : \ \ a u U
JJL
J.l'Loca1'kinetic energy, E1, ma! be expanded as
of
side
the
right-handUpon rewriting ro on
1
pr::(H,le h+ . r H ( 2 ) ) l ( U + € u l 2 2 f J { z >(3.7) by using (3.5) we integrate by parts to,=Y-!*,'r '1,obtainfr flr ll,":+d \t \S(1 ;t 'g""- 2''ff

u h ,'\ a r a:v\ \ u u U ' d r d v
JJ

(3.8)
Addition of (3.6) and (3.8) givesAI- t t ^1 t u
J ) l - , , , :s _ ; t h t\\
At\\2ltJ"*
rr-

FrtUtt):+t':Il\7-F,:U2)u' ldrdY:9.

)

(3.9)

This implics that thc basic flow is stable if thefollowing relations hold in the whole domain.
1-F,2(J2>o* (euI :2V r'z>121

and *ro.
U(3.10)

IJ..

ur,"*,
t2HUu*h(J2)+;7AUU,",
2'')""+ H (2)(J2+ 2huU * H(u2 l "-')] + O(e3).

By taking the zonal mean ol the above expressron, we have:2')
H U : + ^ l 2 Hl J{ J - r H 2 \ L z,+ 2hutJ- HIT - X)l

P:=

1

z

I

i

-

(

a

- r )

: ) - H r J+z , , . ] 8 , + E( , - + r , ) f . ( 3 .1 2 )
-

\

/ t

where a bar denotes averaging in the :u-direction,andIn the dimensional form, (3.10) is

J s H > t U l, a n d * r o .

_l(3.10')

1-

E,-Lg<z>92+HLlu(!)-thuU,(3.13a)Inequalities (3.10) or (3.10') are the same asI
frBlumen (1970)'s if Fr is replaced by M.
( 3 .1 3 b )E o : ^ H 1 u :- u ' ) r i h : .
z/.
If we choose a coordinate system moving in
the r-direction, the inequalities (3.10) and (3.10') The first term of the right-hand side of (3.12)
may change. But the physical properties must is the basic kinetic energy, and Eo is the totai
not change. From this consideration we see that energy of the perturbation per unit area. As

152

Journal of the Meteorological Societv of jaoanVol. 59, No. lshown soon later,,E" is the kinetic energy of displacement.
Then, equatron (3.21) is divided
Cepth weighted zonal mean flow contained in a as follows:c o l u m n w i r h u n i t a r e a . W e r r c i n t e r e s t e di n t h e
i r

c n e r g yc o n v e r s i obne t w e etn" a n d r r . o r . i n u*u[ tuJ" r r : i ( r u r ? - n r uo*y\ a ,J'special case. among the terms in the right-handsideor (3.13a).
Now, the first-order perturbation equations

1..-du d\).
lnuU--

i

l

d

dU
Ah
I s
a,
a.r:

o u _ _o udhu I U d- $ I g- d; y- : 0 ,ot( 3 .1 5 )h ../Au du,-6; h- U. , 6^-rH I ^ '-l:0.
ot

\ dI

0I

13'14)oyI(3. 16)and the second-order equations are-4,
AIJ2'- L l - ^
^-.
ot

ofr

au

-l'-;--Vdy_. .- rtu : U .
rdy(3'17)

u+-,-y-;+-n a!::o ( 3 .1 8
o

t

o

r

d

y

"

o

y

aH,2' 6hu 6hu
AV 2-o t * o- I' * -. d^ y * H o- y : 0 .

)cu* t J\ E u r d y :-;;-\( nUu-i
J'

ay

),1

- 1,,r9\\dv .
'dy,(3.23b)
The flrst and the second integrals in the righthand sideof (3.23a)expressthe rate of the energy
conversionsto Epp and Ep, respectively.

It is worth noting that, even if ,ordinary, zonal
mean kinetic energy Es does not change, pertur_
bation energy can grow at the expense of theadditionai energy term E6p, which mav become
negative. In fact, the linear shear flow, that we
consider in detail in the next section, falls in thislnterestlng case.
3.4

1 3 i. 9 )(3.23a)v

A f

Au ..6uE-ua*-'dy ' '

Characteristics of solutions near the singularpoint

Equation (2.6) has a singular poiht at):l,c
M u l t i p l y i n g( 3 . 1 4 ) ,( 3 . 1 5 )a n d ( 3 . 1 6 )b y H u , H u where U(y"): C. To examine thecharacteristrcs
and gHh respectively, adding the results, and of a solution near the singuiar point, fiand (lintegrating over the whole domain, we obtainare expanded as follows#\u'or:-\'nfi^, ,(3.20)

ir:z^V nnz,
z=0(3.24)The right-hand side of (3.20) expresses the rate a n dof energy conversion by the Reynolds stress.
@ t n^-.9"<">( 1: l(3.2s)
Multiplying (3.14), (3.16), (3.17), and (3.19) by/l!
n:0
Uh, Utt, Htl , and (J2/2, respectively, integrating
i is the index which should beover the whoie domain, and adding the results, where z:y-I",
determined later and U"(t) is the n-th order
we obtain
derivative of U(y) at y:y".
After substitution( 3 . 2 r ) of Q.2a) and (3.25) into (2.6), and rearrangement, we obtain the following equations for a,s'Iheright-hand side of (3.21) expresses the rate by using the fact that coefficient of each powerof energy conversion by the Reynolds stress, and of z should be zero;
it is the quantity that causes change of the peru"(1))()_3)co:0.
(3.26a)turbation energy as seen in (3.20). Thus, theidentification of ,E" as the (weighted) zonal meanu , . < t t ( ) t r ) Q - 2 ) a- r :y ; Q - 5 ) a o,
kinetic energy is appropriate.
( 3 '2 6 b )Since, as shown in following sections, the lastLrce)(),+2)()-r)az
term in the right-hand side of (3.i3a), huu, plays
an important role, we divide E, into two parts:a,#\",,,-\"nfi

Es:En*Esp,(3.22)

where EB: H(.2)a2/2+HUU(.2) expressesthe
zonal mean kinetic energyarisingfrom the second
order zonal mean quantities, and Eup: huu isinterpreted as additional part of the zonal mean
kinetic energy due to the perturbation surface: -1il )Q-7)-k'uPlao-u{r" f^n\()-4)at,(3 . 2 6 c )u"(r)(.)+3\at,: -lT^t*s)-k'Y!:1)aoFebruary1981

T. Satomura- l f u " 3 ,' ^. .- 1 t ( )- 6 r - k ' U " t I
,,
lat
-

I |" r)]1(;/,+2)Qt -3)ar,(3.26d)-L)iwhere 7 is a constant and d(_v)is the delta function. This delta function is cau,qedby a discontinuous change of U'(y) at y:ya (for example,
1,:0 in the case II).
Since z, u, and h are assumed to have the form
as (2.5), equation (.1.1)is rewritter.t'"r"", ", the indices
From (3.26ar,;. ;.r.r;t.
(.4.2)i k ( . U- C ) a : 7 u d ( y - y a ) .
as ,i : 3 and ,i :0. The former gives a regularsolution while adopting the latter we faii to When U is a linear function of y, (4.2) means
determine ru3as seen from (3.26d) and we have
(4.3)
o:0 ,
to seek for another independentsolution by the
except for !:!a."'method of Frobenius. Thus we have generai
Deviding (.2) by ik(U- C), multiplyingby uU,
solution as,
and integrating over the whole domain, we obtain
fr:Ahr<">*Bhz<z>,G.n)where ,4 and B are constants,and lz1and ft2 aret w o i n d e p e n d e nsto l u t i o n sg i v e n a sh{z):ztf, onz' ,L b n z n,

(4.4)

J J

(3.28)

where o'4: o(ya) and ^y': ; / ik(U6- C).
Equation (4.4) is the same as (3.7) exceptfor
the left-hand side. Thus, simiiar calcuiation as(3.2e)

before gives

n:0hz(z):aht(z)logzL

ff\\auLrd xdy -T/LI aua2n=0

k2u c(2>d- *i,.D.

(3.30)

bn are coelficients determined from the requirements that (3.29) satisfles (2.6); but their explicit
forms may not necessarily be written here forthe reason as explained soon later.
Although.

ingeneral. fr hu.

alogarithmica :aa*l\u o"o. t \\uuu' n t- 7'uaz(Ja -liuJuo{J a ,(4.5)

where ftfr]o6 is a jump of the Reynolds stress,
LaI, at y:yd,
Adding (3.6) and (4.5), we havc
A t''ta 1;-\ l-lr h -' F- , : [ J u ) , - u 'a t) ) 2 L F ' ' -singularity at y:y"as (3.29) shows. ft becomesregular in the whole domain if U"rz>:O, as seenI-t(1 - F,2U2)u2
ldrdy
from (3.30). This occurs either in the case where)
the shear is linear or in the case where we: - Q ' u a 2 * [ u u l r ) U o.
(4.6)
happen to have U"rzt-O for non-linear shear.
Most of the studies so far made deal with the When there is no discontinuity of U' as in thelatter case, because they consider only the so- caseI or Ua:0 as in the caseII, the right-handcalled inflection-point instability. In this paper, side of (4.6) is zero. In these cases,the above
however. we conflne ourselves to the linear shear equation impiies that the basic flow is stable ifflow. because we are interested in destabilizatron an inequality(.4.7)
I-Fr2U2>0,of gravity waves and therefore it is desirabie to
avoid the occurrence of an inflection-point hoids in the whole domain. In the dimensional/r becomes a reguiar form, (4.7) is written as

instabiiity.
function.

In this .ur.,

4.

of linear shear flows

4.1

Stability

JsH>U.

(4.7' )In the subsequent part of the present paper wetreat stability of flows of two different proflles.
Adopting the consideration in the last part ofsubsection 3.2 to (4.7) we obtain a necessary
condition of instability as

Notes lor the necessary condition of a linear
shear flow
When 7':0,
the derivation of the condition(3.10) or (3.10') is incorrect. But, as we will
Fr>t2
for the caseI.
show subsequently, a similar condition can beobtained. The equation for the potential vorticity But, in the case II, that consideration cannot be
rr.rin the linear shear case is written formaliy as'k Thus we excluded singular normal modes,
rrr:
0a_ _6 aassociatedwith continuous eigenvaluesas
A(U-C),
+
u
;
:
7
u
d
(
y
y
a
)
,
( 4 .1 )^,oI
ox,obtained by Case ( 1 9 6 0 ) .

1_s4

Journal of the Nleteorological Society of Japan

Vol. 59, No. 1

adopted, because (4.7) or (4.7') is derived only (even) means that the respective summation isfor a coordinate system which gives U,1:0. Thus carried out only for odd or even n's. The co(a.7) is written as
efficients a's are determined as
F >1

for the caseII.

ao:7,

ar:0,

az:-k2,12,

dB:1,

4.2

Case I: Plane Couette flow bounded in
k:l.an-zl Fian-)(.4.1.3)
for n)>4.
both sidesn(n-3)First we consider a plane Couette flolv, 1.e., The radius of convergence,p, of the power senes
the fluid is bounded by rigid walls at ,v:0 and ts examined bv use of the relation.
y:1, and the basic flow is a linear function of y:
f /r'r'\ -.'(4.8a)

an-z,.
P-llrn-( 4 .1 4 )

Qn

n+6

Then boundary conditrons are

and turns out to be inflnity, so that (4.12) canbe applied everywhere in the domain.
This situation is schematically shown in Fig. f i.
Now, an eigenvalue problem is formulated as
It is convenient to transform independent foilows: The boundary condition is (,1.10).Then,variable y into z:U-C:yC . T h e n ( 2 . 6 ) b e - substituting (21.12)into (4.10), we obtain,comes
A I na"(- CL-t + B l, na"(- C)"-t -O
u: h/:0

at y: Q and y: I .

dri ^ di, * -2|+k)z(F,22' -1)h:0,dz"
az( 4 .8 b )(4.9)

and boundary condition (4.8b) is
u:

dh
E:0

at z: -Cand z: i -C.

To find solution of (4.9), /z is expanded into
power series ol z as
h : z ) Z a n z ".

(4. 15)AZ(4.10)(4.11)

n:0

n - Jlodd(4.12)

n- 0 cvenwhere ,4 and B are constants, and (odd) orllna"(I-C|-t*BZn:3(odd)n a " ( l- C ) " - t - 0 .
n=2(even)(4. 16)
Requiring tbat A and B do not vanish, we havea characteristic equation for C as,ll,r-f)
1 . , , -"1' ^ \I
l:0. t4.17)'f r k . F , : C r : l ' . 'l/r(t-C) l.tt-Cil

Then from the preceding discussion (subsection where
3.4) we see that both of the two roots of thel.tztindicial equation, ,l:0
and ,1.:3, are valid andthe general solution of (a.9) can be written as
h:AZanzn*BZa,z"n:2(eYn)r= 3(odd)\-nn^zn-I( 4 .1 8 )z= 3(odd)

f , ( z ' l : L n a n z n - r,

(4.19),= 2(even)

When ft and .Fr are given, (4.17) determines the
eigenvalue C. Thus, to obtain an eigenvalue isequivalent to finding a zero point of the complex
function l(C). We seek eigenvalues by successive
approximation as described below.
The basic idea of search is to use a theorem
in complex function theory

N_P:*f "n,,#o'(4.20)

where $ dC means an integral along a closed
L'lcontour L on the complex plane, and N and P
are numbers of zero points and poies of thecompiex function l(C) located in the domain
enclosed by I, respectivelv. In the present case'.A.ll fisrrres in this naner are drawn in the norr the two independent solutions of (4.9) are regulardimensional variables and ampiitudes of perturba- everywhere, so that their first derivatives 11 andEquation (,1.20)
tion variables are normalized by the maximum l2 are regular and hence P:0.
amplitude of I for each parameter.
is transformed into a more convenient form
The model of the case I. Non-dimensional width is 1 and non-dimensional
basic flow varies from 0 to 1.

T . Satomura

F e b r u a r y1 9 8 1

N: +

f ,o

155

vaiue of C's at each corner
the rectangles, c1
to1r :j;b,stt')-are(lo)1, to Ce (see Fig. 2(b)). Thenofaroundthe corner

which gave the smaliest absolute value of l, say
Cs, make new M rectangles, each of which rs
where Arg(l) is the argument of the function l,
I / M of the rectangles of the first iteration, andand fo and fu are the values of I at the flxed
re-evaluate I at new nine corners Cs-Cfi. We
point on L after and before the line integration,
repeat the procedure dividing the rectangle into
respectively.
M smaller rectangles and evaluating I using the
We search for the eigenvalues as follows:
vaiues of C's at corners until the smallest absolute(a) The parameters k and Fr are specified.
smaller than a specifled value, say(b) I' and fb are evaiuated numericaliy at a value of I is
10-10. The approximate eigenvalue C is what
point on a rectangular contour I which includes
makes l(C) the smallest value.
the Howard semi-circle.
Using a quadruple precision program, weparameter(c) If N:0,
then another set of
we divide the obtain eigenvalues to more than ten places ofvalues are given. But if N:0
decimals for ail sets of parameter values. In
rectangle into small sub-domains and f" and loorder to get the above mentroned accuracy, it
are re-evaluated at a point on the contour enwas required to calculate the power series up to
we
find
N
subcircling every sub-domain. If
n:255 in (4.15) and (4.16) for Fr:1
and k:
point,
werepeat
the
zero
a
have
domains which
10, which was the greatest n and the number ofprocedure dividing the domain into smaller subthe iterations is about 20-40.
domains until the domain in which C exists rs
Tn nrincinle- it is oossible to locate z, zero(d)so small that the iteration as described in
pointof I to any accuracy by repeating the
(see
Fig.
2(a)).
easiiyconverges(d) Divide the small sub-domain into M procedure (c) without using (d). There is anotherrectangies, each of which has the same area, and way to evaluate directly the vaiue of C by anevaluate the letl-hand side of (4.I7), f , using the integral(4.20')

c,+0.54 -----:--:-----=i,--,:*f ,#fl,ac"(4.l1)

which cai-r be applied to any (not necessariiy
small) closed contour Z encircling the zero point.
' , .
L ,
( a)
However, it turned out that both of the methodsI
are less convenient compared with the simplest
trial and error method described in (d). Efficiencies of the methods may depend on typesof problems and also on details of numericalcalculations.
Results of the eigenvalue calculations arepresented in Figs. 3 and 4. Figs. 3(a) and 3(b)show the phase speeds C, and grcwth rates ftC;
for Fr:5,
respectively. The meanings of theindices (1,i). (1,2), etc. are explained later. Figs.
.1(a) and 4(b) are the same as Figs. 3(a) and 3(b)respectively. except for Fr:7 . Both Figs. 3(a)and 4(a) are symmetric about the line C':0.5,
as expected from the symmetry of the model.
Inspecting Figs. 3(a) and 4(a), we notice that
Searching procedure for eigenvalues. (a) the k-C, relation (dispersion relation) has partiDividing procedure. Line integral is evalu- cular features. The5r might be interpreted asated along rectangles in the directions of follows.
If we assume that the basic flow isarrows and shaded rectangular includes a constant, equation (4.9) has solutions of simple
zero point. (b) Iteration procedure for M:2.
gravity waves and their dispersion relation is
At first l's are evaluated at nine points
given as
Cr-Ce, then re-evaluated at nine points CqCrr. True eigenvalue locates at the point
which is indicated by a cross, X.

t

I

r--c

c:uor-7 | 1+;,

(4.22)

156

Journai of the NleteorologicalSociety of Japan

Vol. 59, No. I

These relations are shown schematically in Fig.
5(a).
At the next step, we consider distortion of thephase speeds by the basic flow which varies withy. For the sake of simplicitv U is assumed tochange slowly. fhen (a.i) is approximated as
drh: - k 2 { F t : t y- C \ ' - 1 ) h .
(4.24)
ary. '

a
Fio

2

3

4

5

6

7 8

k

ia)
3(a) Curves of non-dimensional phasevelocity for Fr:5 and case I . S o l i dlines are neutral modes and dottedlines are growth modes.

Now, for the family of the faster phase speedsC-tn) (>l/2), the vaiue in the bracket of (4.24)
has greater positive values near the wall at y:0
than near the wall at y:1:
aird a region ofnegative value of the bracket, where the soiutions
cannot be sinusoidal but becomes exponential,
appears (for Fr>2) near the wall at v:1. Thus,
the solutions are more wavy near y:0 than y: 1
and the area where the solutions are sinusoidal

o

r

2

3

4 5 66 7 7 8 8(a)Fig. 4(a) Same as Fig. 3(a), except
Fr:7.
o

1

2

3

4

5

6

7 8

(b)
Fig. 3(b) Curves of non-dimensionai growtL
rate for Fr:5.
where us is the velocity of the constant basic
flow and the factor nr comes from the boundary
conditions at two rigid walls. $/e may assume,
to a first approximation, that the waves are advected by the mean speed of the basic flow. Thus,
we take uo:7/2 in this case and have approxrmate dispersion relations for the two families as
I
I
1
tn-)i
C_ "-,:
1_;:,
2t' F, \'(4.23a)(4.23b)

o

|

2

3

4

5

6

{b)

Eie. a(b) Same as Fig. 3(b), e.rceprfor
Fr:7.

F e b r u a r y1 9 8 1

157

Satomura

11*t'Lts7/2. Then, (4.23) mav be rewritten asIl" t--P(1.25a)cL(o-u-@)*",
V1+';'
6-1nt-u*t")lalFig. 5(a) Dispersion relations for C 1 ( n) 3nd
C-(n). Upper dashed line is C":
0.5*Fr-1 and lower one is C,:
0.5- Fr- 1.

-

I
j 7; \ J

L\ + ( unco\1[ 'uk2(J.25b)Above relations are shown schematically in Fig.
5(b). Fig. 5(b) is very similar to Figs. 3(a) or
4(a). In general, if two neutral :nodes cross inthe k-C, piane, i.e., if a degeneracy occurs, it isexpected that phase speed C becomes complexvalued near the crossing point (see, e.9., Gustavsson et al. 1980), and it really occurs as shownin Figs. 3(a) and 4(a).
From the above discussion we interpret thateigen-solutions shown in Figs. 3 and 4 areessentially gravity waves modified and destabilized
by the basic shear flow.
Now, the meanings of the indices (1,1), (1,3),
etc. in Figs. 3 and 4 are clear. The first index
in brackets indicates the number, ,?, of the faster
phase speed mode expressed by @.25a) and thesecond index indicates that of (4.25b). The twomodes are mixed to produce this unstable mode.
Thus, for example, (1,2) means the unstable mode
whose parents are the C+irr *oda and the C-(2)mode.
Next. we shall look into the structure of unstable waves. Fies. 6(a), 6(b), 6(c) and 6(d) show
amplitude of h for the (1,1), (2,2), (2,1) and (3,2)
respectivelv. As understoodmodes for Fr:1,
from the previous discussion, a mode (nr nz) has
zr1maxima near y:0
and n2 maxima near ):1.
Fig. 7 shows the amplitudes of the divergenceWeand the vorticity for (1,1) mode of Fr:1.
aFig. 5(b) Dispersion relations forc , rn) and c,(n).

modifiedis larger near y:0
than y:1x.
Then we may
considerthat the family of C -tnt has more energyonear y:Q than y:lo
and that they are advected
not by the mean speed l,rs: L/2 but by the basic
flow at a point cioser to 1t:O, i.e., by a speed Fig. 6 Ampiitude of non-dimensional surface disu-tntll/2.
placement h lor Fr:1 and case I. (a) t:
For the famiiv of the slower ohase
speeds C- (o,. the eflect of the basic shear is
2.75. Cr:0.5 and kCi:0.076 (mode (1,1))'(b) k:6.7, Cr:0.5 and kCi:0.033 (modereversed and they may be advected by a speed(2,2)). (c) k:4.45, Cr:0.6 and ftCi:0.054'r'This supposition will be verified by the structures(mode (2,1)). (d) ft:8.65, Cr:0.55 andof the eigensolutions as shown in Fie. 6.
kCi:0.021 (mode (3,2)).

::.-r.;-J.i..,

158

Journai of the lvleteorological Society of JapanoFig. 7

D.V

Amplitudeofdivergence (D)andVorticity (V) for Fr:1,
k:2.75, Cr:
0.7 and kCi:0.076 (mode (1,i) in case
Solid line is the vorticity
D.
anddashed line is the divergence.

Vol. 59, No. I

is a down-gradient transport of the zonal momentum and hence the disturbance gains energy fromthe mean field as seen from (3.19). It is interesting that this negative az correlation is a resultof the tiit of the wave as mentioned before,
because the velocity vector tends to be perpendicular to an equal phase line (or it is parallel withthe wave number vector), which is a property ofgravity waves. If the disturbances were ofrotation type, the same tilt of equal phase line
wouid be accompanied by a positive rru andhence the wave would decay. Thus we knowthat this unstabie wave owes its existence to thegravity wave mechanisms.
We will call these unstable waves as SGwaves.
4.3

Case II: Plane Couette flow bounded in
one side
Next we consider a fluid wbich is bounded bya rigid wall at ];:1 but extends to l,-+- *. The
basic flow is
Y

.

tI

:

lzztrzzr,rzrrzlzzrrzrrz.rrzzrrrzlrl,,- ou{vl:.

,: :..-ly0<y<1(region I ) ,

i0

ys10

(region lI ) ,U.26)

and boundarv conditions are
Fig. 8Structure of the unstable wave forthe same parameters as Fig. 7. Solidlines are contours of ,4 and arrows
denote velocitv (u, t;). Contour interval
is 0.2.

(4.2:7)u:h':0at ):1,and the perturbation quantities remain finite as
)

--

nr

ror{ioio

fn.- )'+-z.

The interfacial boundary condition is that thevelocity component normal to the internalboundary and the surface displacement ft should
be continuous at y:0,

see that the divergence is greater than the vorticity
near the boundary, where both of them attainmaxima. This fact can be an evidence that these(4.28)
L-t:ur, hr--ht
at y:Q,
unstable waves are of gravity wave origin.
where suffices I and II denote those quantities
Fig. 8 illustrates the structure of the unstablebelonging to the regions I and II, respectively.
wave. This structure is very simiiar to that ofThis situation is schematicaliy shown in Fig. 9.
gravit-y wave in a channel, especially near. theIn region I (shear zone), the posed problem isboundaries (structure of a gravity wave is shown
the same as case I except for the boundary conin Appendix). As shown in Fig. 8, the wave
dition at y:0;
the governing equation in regron
crest has a tilt; the phase at a larger y point
I is the same as (4.9) and its solution is expressedleads that at a small y, possibly owing to difas
ferential advection by the basic shear flow.
Examining the relationship between h and u weh : A t Z a n z " * B t l , u n z o.
(4.2e)
notice that the wave tends to move toward ther=J oddn-2 evenleft side (r(0)
near the y: 1 boundarv whiie Above equation is the same form as (4.12) but
near the y:0 boundary the wave has a structure constants A1 and 81 are determined from theto move to the right, if there is no basic flow. boundary conditions (4.27) and (.4.28).
Since the basic flow has a shear which justIn region II (rest fluid), equation (2.6) is
opposes the above mentioned motions arising
written as
from the wave structure, th.is structure can be,l2i
maintained. Inspecting the veiocity field closely,(4.30)
i*+/.tF,:C, -ttft:0,dy'
we notice that a and .o are negatively correlated
in the middle part of the channel. Thus there and its solution is

T. Satomura

February 1981

u
i
r--

159

Substituting (,1.29) into (4.27), we obtain anotherrelation between ,4r and B1 as
----

----

At Z na"(l-C)"-lBrIIn:3(odd)nan(l-C)-1:0,:2.

(even)(1. 37)r e g r c nl

Requiring that Ar and 87 do not vanish, wehave a characteristic equation for C asIF,
F ( k . F , ' ,C t : l - - 'lr1-F- ,' |I
frl

:0 ,

(4.38l

wnere
F,:

f

(n+pC)a"(-C)"-r,r:3(odd)

"ro

Fz:

-C)-t
Z nan(I,n:2(even't

Fig. 9The model"u." IL Fluid extends
to 1,+-co. Shear zone (region I) is
0<y<1
and rest fluid (region II) isy<0.

Fz: f * Z (n * FC)a"(- C)n- L
z = 3(odd)

F+: Z na"(l-C)"-t.

z:3(odd)h, : A o exptk'/ 1 * F,zCzy\

Equation (4.38) is solved by the same methodthat
as used in case I to the same accuracy.
- t - . 8r e x P (- k J 1 - p - , : 6 : r ,(.+.31),Resultsof the eigenvalue caiculations arewhere A11 and Bs are constants. From thepresented
inFigs. 10, 11 and 12. Figs. 10(a) andboundary condition at y-+ - *, i1 should have10(b) show the phase speeds C' and the growtha form as
respectively. Figs. 11(a)rates ftC; for Fr:3.5,
fio: au"a,G.32) and 11(b) are the same as Figs. 10(a) and 10(b)Figs. 12(a) andwhere ,49 is a constant andrespectively, except for Fr:5.
12(b) are the same as Figs. 10(a) and 10(b) refor Fr:C:)1, 14.33a)^ l-ikJFler=
From these figures,
spectively, except for Fr:1.
P
f
1
*
4
2
6
2
f o r F , 2 C 2 ' - 7 , ( 4 . 3 3 b ) it is easily found that there are two differentt
Of course, the right-hand side of (4.33a) gives types of unstable waves: one is fast to move andthe same value as what (4.33b) gives in the case siow to grow, and the other is slow to move andfast to grow. We will call the latter unstable
where C has an imaginary part.
waves
as SGr-waves and former ones as SGzThe interfacial boundary condition for o rs
equivalent to the continuity of ft'. Thus, using waves.
Comparing Figs. 10, 11, and 12 with Figs. 3(4.29) and (4.32), boundary conditions (4.28) are
4, we also notice that these ft-C' relations
andrewritten as
have similar characteristics with each other. TheyF A e : A , 7 n a " ( - C L - L * U , I n a " ( - C 1 - r . may be interpreted as follows. When the wal1
z- 3(odd)
n:0(even)at y: g is removed, almost all members of the(4.34)
famity of the faster phase speed C r cnr -ut U'.andappear, because they have more energy near thatwall and they may be greatly affected by the
Ap:Ar Z a"(-C)"+Bt Z a"(-C).(4.35)
wall. On the other hand the family of C-('r may
z:3(odd)n= 2(even)
From (4.34) and (4.35), we eliminate Ae to not be modifled remarkably because they havemore energy near the wall at y: 1 which exists
obtainin this case, too. But some parts of them whose
Fr can notphase speeds exceed the value C,:l/Ar Z @+ FC)an(-C)"-'n: l(odd)
remain to be neutrai, because they radiate toy->- co as the boundary condition (4.33a) shows,+ B r { F + | @ + p C ) a " ( - C ) 2 - 1: }0 .
that is momentum is lost to infinity and thusn:3(odd)( 4 . 3 6 ) energy conversion may occur, Thus the phase

-T

160

Journal of the MeteorologicalSociety of Japan

Vol. 59. No. I

u l z 3 4 5 6 7 8 9 l o
(a)

ta)

Fig. 10(a) Curves of non-dimensional
phase velocity for Fr:3.5
in case II. Solid lines areneutral modes and dashedlines are growth modes.

Fig. 11(a) Same as Fig. 10(a),except for Fr:5.

o

2

a

4

5

6

i

.

J

U

(b)

o
o

i

2

3

4

5

6

7

8

q

tn

(h)
Fig. 10(b) Curves of non-dimensional
growth rate for Fr: j.5.
Rather discrete curves andcontinuously extended curves
correspond SGr_waves andSGr-waves, respectively.

Fig. i 1(b) Same as Fig. 10(b),except for Fr:5.

respond to (1, n) modes. where n:7, Z, 3, . . .,in the last subsection, though the reason why
only the (1, n) could survive is not clear. Indeed,
SG1-waves have a second maximum of the surface dispiacement I as a trace of the energy
concentration near l, : 0 as shown in Fig. 15,
speeds must have complex values in that upper which is a characteristicof the familv of C*(').
region of the k-C, plane. On these grounds, SG2_
There exist the most unstable wave near Fr:
waves may be interpreted as the destabiiized part 3.5. WhenFr becomes greater or smaller (notaf the wave famiiy C_(") whiie SG1 ma) cor_ shown) than 3.5, the growthrate is reduced.

161

T. Satomura

FebruarY 1981

0 , VzFig. 13o

I 2 3 4 5 6 7 I
(a)

9 to

Fig. i2(a) Same as Fig. 10(a),except for Fr':7.

of non-dimensional divergence
Amplitudein Case IL Solidand vorticity for Fr:3.5
lines are the vorticity and dashed lines are
the divergence. (a) ft:1.9, Cr:0.155 and(SGr-waves). (b) t:3.1,
Cr:
kCi:0.096(SGe-wave)'
0.319 and tCi:0.005

Theof the perturbation velocity at Fr:3.5.
modes which are shown in Fig. 13(a) and 13(b)correspond to the ground mode of the SGrwaves and SG2-waves, respectively. It is evident
that divergence is dominant near the boundary
for both two waves.
To verify that these unstable waves are destabilized gravity waves, the vorticity equationand the divergence equation are derived from
(2.2)-(2.4). Then we estimate the amplitudes of
the terms in these equations. The vorticity equation is written as:-+ CD:o,=ot: + U j0x)(4.39)and the divergence equation is written as
AD

..aD

^ -6 r d U

|

ot

dx

dtr ayfr"_ L t t _ L /,- "- ,n : \ t|(4.40)

a

?

4

1

5

6

7

8

9

0

(bj

Fig. 12(b) Same as Fig. 10(b),except for Fr:7.

where ( is the perturbation vorticity 6u/ An-dU/dy, D isAui 6y-, g is ttre basic vorticity
the. perturbation divergenceand
.

d.

nL

ofr.

oy-

The results of the calculation are shov''n inWhen Fr increases, all unstable waves shift to Fig. 14. From this figure, following points are
the longer wavelength side and the phase speeds observed:(a) Near the boundary, the second term andof SG1-waves and SGz-waves become closer asthese two the'last term of (4.39) almost cancel each other.
shown in Fig. 11. Finally (Fr:1),
unstable regions of the ground modes are joined It means that vorticity variation due to stretchtogether completely as shown in Fig. 12. Simul- ing of vortex tube by the divergence maintains
taneously, the neutral solution vanishes at the the wave structure against the advection by thebranch point of the soiution, FrC:1,.
This fast basic flow.
vanishing point is denoted by a small circle in(b) Near the boundary, the sum of the first
Fig. 12(a).
and the second terms of (4.40) balances the lastFig. 13 shows the divergence and the vorticity term. This is a characteristics of gravity waves,

i62

Journal of the MeteorologicalSociety of Japan. r,iVol. 59. No. I

r
j,.'

,,,
..........n?Q
1 _ ^
lit

tD

at

"

',

. , . . , , . . , . ..t i.. ;: :f:):

'ar

1_-.
fr

Fig. 14 Absoiute values of each term of the vorticity equation and the divergenceequation.
(a) Absolute values of terms of the vorticity equation for SGr-wave. Parameters
are same as Fig. 13(a). (b) Same as (a) except for SG2-wavesand parameterswhich are same as Fig. 13(b). (c) Same as (a) except for the terms of the divergenceequation. (d) Same as (b) except for the ter-msof the divergenceequation.

Fig. 15(a) and 15(b) depict the structures of
the ground modes of the SGl-waves and the SG2waves for Fr:3.5,
respectively. As alreadymentioned, these figures cleariy show that the
SGe-wave radiates to y-> - cc as gravity wave,
while the SGr-wave quickiy damps away fromthe interface. Near the boundarv at y:1,
thestructure of either wave is similar to that of agravity wave in a channei. The waves also havedesired forms appropriate to match the requirement of maintaining themselves against the basic
shear flow as explained in the case I.

aspect of the disturbance in iinear shear flows.
From (3.20) and (3.21), it is obvious that disturbances extract their energy from the zonai
mean kinetic energy through the conversion term
due to the Reynolds stress. In the case of thelinear shear, however, this conversion term isexpressed as (4.5). From a comparison between(4.5), (3.21), and (3.22) we see that the 'ordinary'
zonal mean kinetic energv .Es must not change,
and that the perturbation quantities can grow at
the expense of the additional energy term -86p,
and thus reduce the 'depth weighted' zonal mean
kinetic energy ^85. This is a difference from usuai
5. Potential vorticity budget, change of zonal
barotropically unstable waves. In order to makemean flow, and energetics
this difference clearer, the tendencv of U with
I n t h i s s e c t i o n , w e w i l l s h o w a n i n t e r e s t i n g time is calculated from that gf the basic potential

February 1981

T. Srtomuraj^ - j - :t6l

On the other hand, the equation of the conservation of the potential vorticity,

- ' o

aa
ot

J--

aQ

aa

15 ?\

lltT--l'*-:t)oyotr.

is derived from the original equations for ui, ui,
S u b s t i t u t i n gt 3 . l I t a n d ( 5 . 1) i n t o ( 5 . 1 ) . a r r a n g ing each power of € and note that first order
quantities to have the form as (2.5), then we
obtain (in the linear shear zone),(s.3)

O ( 1 :) O

F i g l5(a)perturbation
Structure ofthequantities in case II. Parameters
are the same a s F i g . 1 3 ( a ) a n d
p : 1 . 6 5 * 0 . 2 1 i . Solid lines arecontour of h and arrows are velocity (u, r). Contour interval is
0.2.

AQo

69r*t

ot

dt

- _ - _ - { l

(5.4)

where
/

Q - 9 , 0 t _ r - ; 1 2 ,_. i 2 ( 9 , , _

h

;,

"

\

)

n.r:**X
-

'j:r

t

--:-:

) : i11151,.-;=

I / 6_ u- t_ /1\drQ < 1 ):

v

6u hdU\
4 '__ l
Ay H dy /n",:#(o*,#-ro#)

1 _
J_':

(5.5)

tbt
Fig. 15(b) Same as Fig. 15(a), except for
r3:0.04-1.55t and the parameterswhich are the same as Figs. 13(b),vorticity and of the depth H. To do this, we use
the expansion (3.11) again and also assume thatthe second order variables U(2), l/tz), vnf, [_ltzt
are independent on ,.
Then, potential vorticity Q may be expandedasPhysically the above results are rather obvious.
Since the basic potential vorticity is uniform, nopotential vorticity can accompany the disturbanceas seen from (5.3). Then the disturbance cannot
cause any change of the zonal mean vorticity
field through the non-linear effect as shown in(5,4). In the case of a non-divergent flow (e.g.
barotropic instability problem), it means that thezonal mean flow and the kinetic energy associatedwith it do not change at all; hence disturbances
do not g.row. In the present problem, we havea somewhat strange situation as explained below.
For convenience, we rewrite (5.4) as
A ALI(2) | dLl aH(2)-:;=-.-

1

6ut
^

o

ot

AuI

x

o

A u r d U ! : - aA: l u\ay

l

ox

r

H +.h-^oy

A { J : I,loy /€' \1t- - n l d U , e / 6 u A u- ,r oh nd U \at,-n\it-ay)
L.:'Lldu

- t

H'L

"\_

_ h t

_

6r

du
_

Ay

I

h_ _d U t

dt

AH(z)
A
H (2)

Ahu

At

ay

(s.6)

/\J{ r/r\(s.7)

Substitute (5.7) into (5.6) and integrate withrespect y to obtain
AU'
IdU,
at:-na,nu+G(t)'

I

H dyl

_ H, ,: .._. d
j _u_ H - . a^u , 2 ) 1
l _ t 0 1 e .r 1
dy
0y _l

--

Hay

Now, upon using (2.5) and the condition that
Z(2) should be zero at the walls or y--) - :c for
unstable waves, (3.19) is rewritten asyhl- _ - - l -

.

oy

(5'8)

where Gu) is a function of time only, and weused the fact that dU/dy is a constantin theJournal of the N{eteorological Society of Japan

16"1

Vol. 59, No.

a _shear zone. For convenience, we consider the_ o ( : )\ o
(5. 12)ot
situation in case I, then, from the symmetry ot
the model, G(r) should be zero. Then from (5.7) With the fact that Qt\) - - (l/ H)(dU / dv).'-O'and (5.8) we have
above inequality implies that the absolute vaiue^Aofthe basic potential vorticity will be reduced.
- . tuht).
(5.9)
J-tHu'-rH'z IJ1:
It means that the basic shear wiil be weakenedoyot
and the redistribution of the momentum canThis implies that the zonal mean momentum
may change, despite the fact that '0(t) does not
From another point of view, this redistributtonchange. F{owever, if the perturbation disappears
is also suggested. The perturbation momentum
at the final stage, U should return to the initial'zonalmean ia is calculated from the eigenfunction and rs
distribution, because 0:const. Then
(g:U'ou*) andfound to be negative near y:lacceleration by waves' does not take place in theis positivenear y:0 (U:U,,in). Then. the basic
p r e s e n tp r o b l e m . H o w e v e r i f w e c o n s i d e r i i a t
shear wiil decrease, if we assume that the waves
a part of the zonal mean momentum, it canwill be dissipated by viscosity but the perturba(cf.3'21
andchange as mentioned previously
tion momentum will remain where it is.
3.22). Redistribution of physical quantities such
Of course, to prove strictly what are discussed
as momentum or mass by the waves may be one
above, a finite amplitude theory with the ellectof the interests of instability problems. Althoughof the viscosity should be required. We do not
the unstable waves found in this paper haveextend our discussion to the finite amplitude
strange characteristics in their energetics, they
theory.
have a possibility to redistribute the momentum
The right-hand side of (5.8) is calculatedof the basic flow. To show this, assume theexplicitly
and it is shown schematically in Fig.
situation that the energy conversion from the
16. From Fig. 16 only, it seems as it thebasic flow to the perturbation is compensated by 'ordinary'mean kinetic energy En will decrease.
the energy loss due to viscosity which is con- The energetics shows, however, Ee must be considered to act only on the perturbation vorticity. stant with time. This apparent contradiction may
but not to affect the surface dispiacement /r. be avoided by considering the variation of the(So far we consider an initial tendency of the mean surface. He). because definition of Eg
effect of the viscosity, these assumptions may inciudes that effect. This is also shown as folhave no great difficulty, because the viscosity lows: The explicit form of the left-hand side
affect the surface displacement ft not directly but of (3.23a) is
A f 1
through the change of velocity field.) From these| 2 H . U U ( 2 ) d y.
j
(5.13)\*tr#ru2
ot J/.
assumptionsand (5.1), the tendency of the second
order potential vorticity may be written as
Substituting (4.8a) and (5.7) into (5.13), andaQ(2) | Aintegrating by parts to rewrite the left-hand side
-H
"""
At
At
-

- - -

h ( ) t . L )t H2'"'

t d u a h ,
Ht dy

1 a -

-Hz

0t'"'

0t

(5.10)

where the constancyof ft with time is used.
is negative,
Since the basic vorticity -dU/dy
the stretchingof the vortex tube by the ft creates
negative vorticity. Thus a correlation between
h and ( may be negative, anda _Fig. 16 The tendency of the variation of the basrc
flow. Solid line is the basic flow, U, anddashed line is the modified basic florv
because dissipation reduces the absolute value
U+I2UQ)' But the value of e is set toof the perturbation vorticity. Substituting(5.11)
be arbitrary large for convenienceto show
the effect of Ue).
into (5.10), we obtain an inequality- ch)o,ot(5.11)

T. Satomura

Februar-v1981

of (3.23a) as; I
f+ \ E B d y- : \ y l h r - H
ot)
J'\

oL 2
o- I /I a v .

t 5 .l 4 t

Equation (5.8) implies that the right-hand sidei5.i4) should be zero. Thus, even if the zonal
mean shear changes as in the case of barotropic
instabitity, the compensating effect by the eleva'ordinarv'
tion of the mean surface will make themean kinetic energy E6 unchange.
6.

Summary and discussionsIb)

sufficiently, simiiarity between the two waves rs
not so obvious. In addition to this vagueness ofthe simiiarity, they found unstable waves ev.dn
Thein the case where Mach number M{I.
reason why subsonic flow is unstable is not soclear. Their model is compressible homoentropic
fluid, and they used the cylindrical coordinates
and assumed that the basic vortex has uniform
vorticity in the core and zero outside the core.
This means that a pressure gradient along theradial direction exists and it corresponds to an
inclination of the undisturbed surlace in ourshallow water model. We only suppose that these
dillerences might break our necessary condition
4.7\.
In view of our results which show that gravity
waves are unstable for Fr) 1 even without apoint of inflection, the second mode of Biumen
et al. (I915) may be interpreted as unstableaccoustic waves which are similar to ours andhave little connection with the point of inflection
essentially. In order to connect our results with
theirs straightforwardly, we examined the stabilityof the basic flow which has a r;rofile as

Using an anaiytical solution of the linearized
forms of the shallow water equations, we havestudied the stabiiity of plane Couette flows.
The characteristics cf unstable '.vavesare summarized:(a) In the case I, there are unstabie gravity
waves (SG-waves). The instability occurs indiscrete intervals of the zonal wave number, andthey are interpreted to be produced as a resultof the mixing of two gravity wave modes which
would propagate opposite directions if the basic
flow did not exist. A strong shear (Fr>2) cancause the mode mixing.
( I 1{v(b) In the case II, there exist two difterent
t
'
(6.1)
types of unstable waves. One is the almost
U{y):l y 0<y<1 .
trapped but radiating waves; part of their energylo Y(1(SG2-waves). The other rs
radiates to y->-:o
The resuits of the calculations of the eigenthe trapped waves (SG1-waves). The SG2-waves
values verify the above interpretation. But, rn
rre interpreted as destabiiized parts of a family
contrast with the results of Blumen et al., theol waves which are neutra.l in the case I. The
high wave number cut off of the sccond unstable
SG1-wavesare interpreted as one of the unstablemode in k-Fr plane is not found in this model.
wave series in the case I modifled bv the unFurther results of this model will be presented
bounded fluid.
elsewhere.
(c) Near the boundaries, the vorticity patternIt
Our instabilities are found only for Fr)l.
of the SG, SG1 and SG2-waves :rre maintained
should be noted that possibility of meeting thisby stretching and shrinking of the vortex tube
condition in actual geophysical fluids is expected
due to the divergence. In the divergence equation,rather iarge, because, in a stably stratified fluid,these waves show the characteristics of gravity
the mean depth of the shallow water, H, may bewaves near the boundaries.
replaced by the verticai scale of the waves, I1",(d) These SG, SGr and SG2-waves obtainand thus Fr:Uo/ NH,, where N is the Brunt'additional"
their energy from themean energy
V?ilslilii frequency. Thus defined Fr may be able'ordinary'huU and nct from themean energy
to exceed 1. In concluding the paper, the author
1/2 H(2)LP+HUUQ).
wishes to mention one interesting appiication.
(e) Perturbation potential vorticitv equals zero,In the upper atmosphere of Venus where theand the mean potential vorticitv does not change
4-day circulation is dominant, a shear in thewith time.
angular velocity is expected to exist (cf. Matsuda,
These instabilities does not change the 'ordi1980), and those unstable waves may exist.
nary' mean kinetic energy ffi€a€ndii,m.
Indeed, if we use a model as
The SGg-waves might be the same, in essence,( rv
0 ( v ( -l . : .
as the unstable waves which are found by Broad(6'2\
U (-Y t : \,<-y" : ! 0
_
r
t
_
y
bent and Moore (1979) and the second mode ofBlumen et a!. (1975). But, since Broadbent e/ a/. with boundary conditions(1979) discussed neither structures nor energetics( 6 .3 )and y: - 1 ,u:0
at y:1

t66

Journal of the MeteorologicalSociety of Japan

Vol. 59, No. I

eigenfunctions will be symmetric about the line where all perturbation quantities are assumed to
1l:0. Thus the pattern in the domain 0<y<1have a form as (2.5). Boundarv conditions areis the same as Fig. 8 and the other haif is a
u:O at y:0(A.3)
and y:L ,mirror image of it with respect the line y:0.
where I is a channel width.
This looks like the Y shape disturbance in the
Then dispersion relation is obtained as
Venus atmosphere (see, for example, Belton etr n-i
al. 1976). Further, the tilting of the equal bright(A.41
c : - y ' g' H t . l - - + .
Vr -,.ness contour of this disturbance is the same direction of the shear of the angular velocity. Thus and u, z', and I are obtained as
this disturbance may not be of a rotational motion
inn cos(nry,tL)
type but a gravity wave type, as pointed in theu:A
elkG-e/,)Lk {I -(c' / gH)}last part of the subsection 4-2. Here we only
point out the similarity between their shapes.
(A.5)
u:Asinfr,n , r*',-rt
L 'Further comparison is left for a study using amore realistic spherical model. If the unstable
icn:t cos(n-viL\.
r rt ':-:----!
h:'A^
eik
waves found in this paper exist, they may be a
LklI-|c' , gH)]mechanism of large horizontal eddy viscosity
where ,4 is a constant.
which is assumed in some modeis of the 4-day
Results of the calculation of the structure by
circulation (e.9. Gierasch, I975; Matsuda, 1980).
(A.5) are shown in Fig. A1.
Aclmowledgements
The author is greatly indebted to Dr. T.
Matsuno for leading the author into this problem,
useful discussions, encouragements, and carefulreadings of the manuscript. He wishes to thankother staffs in Meteorological Laboratory ofTokyo University and Dr. M. Miyata in Laboratory of Physical Oceanography for useful discussions. He is also grateful to Mrs. K. Kudo
who typed the manuscript.
The computations were performed with the useof a HITAC 8700/ 88C0 computer at the Computer Center of Tokyo
and a
University
FACOM M-190 computer at the LICEPP whichbeiongs to Tokyo University.
Appendix:

Gravity

waves in a channel

Here we derive structures of neutral gravity
waves in a channel for reader's convenience forcomparing with the structures of the unstablewaves which is found in this paper.
Shallow water equations for gravity waves ina rest fluid are written as
6,
dt
Au

ar

_ on A h : n
6x

+s

Ah

a"

:0,

Ah .-/6u
6ut- -H { . +=- }:0.
oI
0t /
\0,r
Eliminating a and h fromsingle equation for z- aso,'t --1r'1"'-oHlD.
dy"( A .1 )(A.1), we obtain(A 2)

va11z/,/z//v'//2,/&'///z72272?/4,.//27/?"4/2./z/z/z//z/////1?"2

vtX

Fig. A1

Structure of neutrai gravity wave.
Solid lines are contours of li andarrows are veiocity (2, z'). Contour
interval is 0.2.
References

Belton, M. J. S. and G. R. Smith, 1976: Cloud
patterns, waves and convection in the Venus
atmosphere. J. Atmos. Sci., 33, \394-7417.
Blumen, W., 1970: Shear iayer instabiiity of aninviscid compressible fluid. J. Fluid Mech., 40,
76$-781.
-,1971. Hydrostatic neutral waves in aparallel shear flow of a stratified fluid. "I. Atntos.
Scl., 28, 344*44.
Blumen, W., P. G. Drazin and D. F. Billings, 1975:
Shear layer instability for an inviscid compressible fluid. Pan 2. l. Fluid Mech.. 71. 305-316.
Broadbent, E. G., and D. W. Moore, 1979: Acoustic
destabilization of vortices. Proc. Ro^t. Soc. London, A29O, 353-371.
Case, K. M., 1960: Stability of inviscid plane
Couette flow. Pftys. Fluid., 3, 143-148.
Fjflrtoft, R., 1950: Application of integral theorems
in. deriving criteria of stabiiity for laminar flows
and for the baroclinic circular vortex. Geolys.
publ., t7, 7-52.

T' Satomura

February 1981

theGierasch, P., 19?5: Meridional circulation andl'maintenance oi Venus atmospheric rotation'
Atmos. Sci., 32, 103&1044'
A
1980:
Gustavsson, L. H. and L. S. Hultgren,resonance mechanism in plane Couette flow. J.
Fluid Mech., 98, 149-159.
Howard, L. N., 1961: Note on a paper of John W.
Miles. "/. Fluid Mech., 10, 509-512.
Kurihara, Y., 1976: On the development of spiralband in a tropical cyclone. I. Atmos. Scl., 33,
940-958.
Lipps, F. B., 1963:

Stabilityof jets in a divergent

t61

barotropic fluid. "/. Armos. Sci., 2O, 720-429
Matsuda, Y., 1980: Dynamics of the four-day
circulation in the Venus atmosphere. J. Meteor.
Soc. 1qpt , 58, 443-470.
Philander. S. G. H., ag76: Instabiiities of zonal
equatorial current. J. Geophys. Res., 8L, 3725aa2<

Willoughby, H. E., 1977: Inertia-buoyancy waves rnhnrricanes. J. Atmos. Sci., 34, 1028-1039.
-,
1978: A possible mechanism for theformation of hurricane band. ./. Atmos. Sci., 35,
838-848.

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