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Journal of the Meteorological Society of Japan

Vol. 59, No. I

NOTES AND CORRESPONDENCE
Supplementary Note on Shear Instability

in a, Shallow Water

By Takehiko Satomura
Geophysical Institute, University ol Tokyo, Tokyo(Manuscript received 2 September 1980, in retised lorm 6 November I9g0)

in a shalloq water. Then it follows (cf. I) thatthe stabilitr of this basic flow is governed by
In a previous paper (Satomura,1981; here- the equations

1. Introduction

after referred to as I), the stability problem ofd2h ,.,-.,k : { F - - t1 - C r r - 1 } l r : 0 f o r l S y ,two types of shear flows was studied. One typedrz4of shearflow was the plane Couetteflow bounded(2)
in both sides by two rigid walls, and the otherdthLlh,0 [ u - ^C. )- 2^-f t,Fr:(y-C)r-llh:O
was the same flow but is unboundcd in one sidedndf
to connect to a rest fluid which extends tofor0':1<1, (3)infinity. One of the purposesof I was to clarify
d2h
the secondmode instability which was first found(4 )f o r _ r ' < o.
;A r o + k ' [ F , z g ' : 1 l h : oby Blumen et al. (1975). Since they used a
hyperbolic tangent profile of a compressiblefluid The boundary conditions are thatft is finite oras a basic shear flow, the physical mechanismof radiating from the shear zoneto ). -) -t-. The
destabilization for the second mode was not interfacial boundary
conditions are that theclear owing to the existenceof a point of inflec- velocity component normal
to the internaltion. A conclusion of I is that those waves are boundary, a, and thesurface displacement. ft,
gravity waves (or acoustic waves in the case of s h o u l d b econtinuous at y:0
and 1.:1. Herea compressiblefluid) destabilizedby the basic we use dimensionless variables
throughout. andshear flow. The other conclusion is that, in assume that the perturbation
surface displaccessence,the unstable waves do not arise from ment has the form as
ft(y)exp[it(.i]-Cr)l in rerms
the existenceof an inflection point but they are of the positive wavenumber
I and contplex phasedestabilizedby the linear shearunder the presence speed C:C,*iCi.
The Froude number Fr isof divergence. The other purpose of I was to defined as the ratio of the characreristic
velocity
examine possibility of generation of gravity of the basic flow U6 to the phasespeed of gravity
waves which radiate from the shear zone. The
wave "/gH, where H is rhe mean depth of thepossibility was demonstrated.
shallow water.
Sincethe basicflows examinedin I are bounded
As in I, general solurion of (3) is expressedas
in one or two sides, the comparison of theresultsof I with Drazin and Davey (1977),whichh : A Z a " ( y- C ) . - B I a n ( y - C ) " ,(4)n=3 odda=0 iev€n)
is an extensionof Blumen et al. (1975), is notstraightforward. In the present note, we will where I and B arc constants and a's defined byobtain eigenvaluesfor a piecewise-linearshear (4.13) of I. Solutions of (2) and (4) which satisfy
flow unboundedin both sides,which is regarded the boundarv conditions at lyi+ oo are written
as an approximationof a hyperbolictangentflow, asand comparethem with the resultsof Drazin andft:1*si*t for 1<y ,(5)Davey (1977).
h: A-ei I for y (0 ,(5)2. Eigenvalue problem
where ,4 - and A- are constantsand
We consider a stability of a piecewise-linear(7)
P* : - k l l - F r ' Q- C 1 z 1 r t z ,plane parallel flow with a velocity
p - : k { l - F t 2 c 2 l t /.2(8)If .ivv o(y{l,o v<o( 1 )We choose branches of square roots in the righthand sides of (7) and (8) so as to make real partsof p + and ,: - non-positive and non-negative,

169

T. Satomrlra

February 1981

ratc at Fr:o, i.c., non-divergent limit (mode I).
The other appears only for Fr)I
and its phasespeed C. is not 0.5 (mode II). Obviously, theformer type is associatedwith an inflection point(barotropic instability), and the latter can exist
by the effect of divergence and basic shear flow.
Mode I has two different eigenvalues at the samek and Fr for 1.83Fr32.7. For Fr51.8 mode I
is single-valued and for 2.7 3Fr it disappears.
those in L
It is worth noting that propagating neutral modes
3. Results
exist for Fr31 .8, and, from Fig.5, a propagating
Figs. I and 2 display distributions of phase neutral mode exist at Fr-1.8 even for smaller
speed JC, and growth rate kCion the k-Fr plane' wavenumber than the cut off wavenumber ofrespectively, where JC,.: 0.5 - C,l. Figs. 3-7 m o d e I .
show graphs of :JC,. and frCi as a function ofwavenumber k for five typical values of Fr'

lespectively. These branches of square roots alsosatisfy the radiation cotr.lition where lf j-t *.
Then the charactcristic equation is deduced fromthe requirement that all constants,A, B, A.
do not vanish when sclutions (4)-(6)and ,4boundary conditions. Theinterfacialsatisfy
nrethod of solving the characteristic equation andthc accuracy of the eigenvalues are the same asrespectively.
From these figures it is easily found that there
are two different t-vpes of unstable waves; one
has a phase speed C,.:0.5 and the largest growth

2Fig. 1phase speed from 0'5, i.e.,
Multivalued ltnstable region
lCr:\0.5-C,-.
is indicated by shade. A stationary unstableand Ci:0) exists in a region
mode (/C,:0and thebetween a thick solid line IC,:OFr axis and outside the dotted region. The
thin solid lines depict isolines ol 1JC,..

Difference ofuFig. 3

246kGraphs of /C,:19.5-C,l and ftC1 for
Fr-0^5. Solid line depicts lC,. Thismode of lcr+ 0 is neutral. Chainline denotes growth rate kCi tlf
stationary mode (1C,.:0).

cr+0lCrFig. 2

Growth rate AC;. Multi-valued unstable region is indicated by shade.The
thin solid lines depict isolinesof kCi.
Thick solid line depictsC;:0.

2Fig. 44uk

Same as Fig. 3 except for Fr:1.5.
Dashed line depicts growth rate liC;of travelling mode (:lC"*0).

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Journal of the Meteorological Society of Japan

Vol. 59. No.

\\Fig. 7\Same as Fig. 4 exceptfor Fr:3.0.

k4'as Fig. 4 except for F r : 1 . 8 .
2Fig. 8

Real part of eigenfunctionft for stationarymode and travelling mode. Solid line depicts
travelling mode: Fr-2.2, fr:0.7, lC,:0.183, kCi:0.029, ri+: -0.510-0.061i,:0.085-0.788i. Dashed line depicts
19s t a t i o n a r ym o d e : F r : 0 . 5 , k : 0 . 7 5 , i l C , : O ,AC,:6. 130. p * :a0.732-0.024i.

0.5 t rJC,.. Since the solutions in the rest fluids
have a form as (5) and (6), the rvave u,hich has
a faster phase speedC.:0.5*JC,is sinusoidal
f o r y - < 0 a n d e x p o n e n t i a lf o r , 1 ' r : l . F o r t h e w a v eof slower phase speed C',:0.-5 - J(',. its property
is reversed.
When we compare Figs. l-7 with Figs. l-3of Drazin et al. ( 1977t and Fig. I of Blumenet ul. (l 9751'".u e noticc some similaritiesand
dilTerences.Ther arc suntmarized as follows:a ) D e p e n d e n c co f g r o w t h r a t e o f m o d e I o n
I and I'r is sinrilar to that in the case of hyperbolic tangent flow. It is reasonable, becauscnrode I crists only for small k and piecewisek
24l i n c a r f l o u ( l ) i s a g o o d a p p r o x i ; r ' r a t i o no f U :tanh(r') for small k as Esch (1957) showed.
Fig. 6 Same as Fig. 4 except for F r : 2 . 0 .
b) Striking characteristicssuch as mode bifurFig. 8 shows structures of mode I and II. This cation and multivalued nature of C which werefigure indicates that mode I is quickly damped found by Drazin et al. (1977) are also foundin our shear flow.
away from shear zone and mode II propagates near Fr:2from one side of shear zone to infinity. Of course, '1'
Note that cur definitionsof nondimension:rlwavemode Il contains a pair of waves which havenumber and Froude number dilTer fronr those of
the same growth rate but whose phase speeds
Drazin er al. (1977) and Blumen (r al. | 1975)bydiffer from 0.5 by the same value, i.e., C.:a factor two.

Fchrrrur

1'tI

T. Satomura

l9tl Ic ) I n t h e c a s e o f ( 1 ) , n o n - s i n g u l a r n e u t r a l the effect of shear (or the effect of abrupt change
nlrrdcs are continuous to unstable mode and they of shear at interfaces) is important even whencrist for, at least, finite range of k-C, plane. wavenumber becomes very large. In this case,\\'hen Fr exceeds l, larger wavenumber part be- neutral normal-mode solutions have a possibility'high-wave rlumber cutoff' to be Ci * 0 and in fact they are unstable forcomes unstable. Thus
Physically, it is rather obvious. The scaledoes not appear for mode II. In the case of Fr)1.
hyperbolic tangent flow, singular neutral mode of the wave in y direction, say Lo, might be small( s e e M i l e s , 1 9 6 1 ) , w h i c h i s c o n t i n u o u st o u n s t a b l e if the scale in r direction is small. Thus, smoothnon-singular mode, is not continuous to any ly varying basic flow will vary only slightly in'wavelength'
by .L, and waves mav not beother singular neutral modes as Yih (1974) sug- onegested. Instead of the singular neutral mode, affected by the basic shear remarkably. But inonly the continuous-spectrum solution may exist the case of piecewise-linearflow, cven if Lo beoutside the marginally stable cur'.'e in k-Fr (or comes very small, waves may feel abrupt changek-M) plane as commented by Drazin et al. (1977). of the basic shear and may be destabilized.
By evaluating (4.2O') of I. we confirmed the fact
Acknowledgement
that, in the case of hyperbolic tangent flow, noThe author wishes to thank Dr. Matsuno fornormal-mode eigen-solution exists outside theuseful discussions and careful readings of therrnstable domain in k-Fr plane (not shown).
The reason why the behaviour of eigen-solution manuscripts. He also wishes to thank Mrs. Kudo
dilTers from that of Drazin et oi. (197'l) in a for typing.
region of large wavenumber of k-Fr plane isReferences
understood as follows: The equation (7) of
Shear layer instability of an1970:
Blumen, W.,
Blumen (1970) is transformed to a normal form
inviscid compressiblefluid. "/. Fluid Mech., 40,
d2f
769-781.
(n - c ) , ' , : - [ ( u - c l u " - 2 u ' :P. G. Drazin and D. F. Billings, 1975:ay"
Shear layer instabilityof an inviscidcompressible- a 2 ( t 1- c ) 21 1- M ' ( u c ) ' l | l(e)fluid. Part 2. l. Fluid Mech., 71, 305-316.
where .l: i I (u-c) and other symbols are the Drazin, P. G. and A. Davey, 1977: Shear layersame as Blumen (1970). In the case of i:
instability of an inviscid compressiblefluid. Part
3. l. Fluid Mech., 82, 255-260.
tanh(l'), the first and the second terms of theright-hand side of (9) can be neglectedcomparing Esch. R. E., 1957: The instability of a shear layer
between two parallel streams. J. Fluid Mech., 2'
the last term when wavenumber a is very large289-303.
lnd r7:c. Then a simple WKB solution of (9)Miles,J. W., 1961: On the stability of heterogeneous.horvs that the effect of the shear only changes
shear flows. l. Fluid Mech., I0, 496-508.
'rravenumber'thc
in y direction and neutral norSatomura, T., 1981: An investigation of shear
nral nr,-rderemains neutral, if it exists. In theinstabilityin a shallow water, in J. Meteor. Suc.
case rri the piecewise-linear flow, however, [i"Japan,59, (in this issue).
beconrcs infinitely large (delta function) at y:9Yih, C.-S., 1974: Instability of stratified flows as aand r': | . Thus, the first term in the right-handresult of resonance. Phys. Fluids, 1?, 14831488.
side oi (9) cannot be neglected. lt means that)fr7ka>f -TEE{:FEf 6ffil€
EI{WEit' H, t-9 4*" tttt' l w.+'rEi r' t l';* tH1,R ii,,\F-rE J-nt,)cr^1))+J,4,4:l: