Commonwealth of AustraliaCOMMONWEALTE SCIENTIFIC AND INDUSTR,IAL R,ESEARCE ORGANIZATION

J, Flwtd Mech, (1976), ool,, 78, part 2, 1ryt, 193-213

Printed dn Great Btdtai,n

The stability of planetary waves on a sphere

By P. G. BAINESCSfRO, Division of Atmosphoric Physics, Aspendalo, Yictoria 3195, Australia

(Reoeived 27 August 1974 and in revisod form 10 Ma,rch 1976)

The stability of individual inviscid barotropic planetary waves and zonal flowon a sphere to small disturbances is examined by me&ns of numerical solutionofthe algebraic eigenvalue problem arising from the spectral forrn ofthe govern-ing equations. ft is shown that waves with total wavenumber ro (the lowerind.ex of the Legendre function Pff whicb describes tho waves' meridionalstructure) less than 3 are stable for all amplitudes, whereas those with n 2 3are unstable if their amplitudes are sufficiently large. n'or travolling waves(m * 0) wilh n: 3 and 4 and with disturbances comprised of 30 modes, theamplitudes required for instability are approximated by those obtained fromtriad interaotions, and are smaller than those given by Hoskins (1973). X'or thezonal-flow modes (m:0) the critical amplitudes are smaller than those pre-dicted by triad interactions, and are close to those obtained. from Bayleigh'sclassical oriterion.

1. Introduction

The stability of Rossby w&ves on a p-plane has been disoussed by Lotenz(197 2), Hoskins & Hollingsworth ( 1 973 ) and Gill (157 4) and that of R ossby waveson a sphere by Hoskins (1973). Although the result that Rossby waves may bounstable is perhaps surprising from a meteorological viewpoint, it is less remark-able when viewed in the light of vrave interaction theory (as was done for theoase of surface w&ves by Hasselmann L967 a). It has been suggested by Lorenz(1972) and supported by Lilly (1972, 1973) that this instability is primarilyresponsible for tho loss of predictability observed in numerical atmospheriomodels, i.e. the divorgenae observed between the properties of two time integra-tions with slightly different initial cond.itions.

Gill (1974) has shown that plane Rossby waves of both small and large ampli-tude are unstable, and that for small amplitud.es the unstable distwbance formsa resonant triad with the primary wa,ve. (The latter result may also be deducedfrom Hasselmann's (lg67b) criterion.)McEwan & Robinson (1975)have demon-strated experimentally and theoretically that internal gravity wa,ves in a,stratified fluid are unstable and that at marginal stability this is equivalent totesonant interaction, and suggested that, this process is a controlling factor foroceanic microstructure. Their analysis is based on the Mathieu equation, andthe instability is 'par'ametric' in character. A similar type of analysis may beseen to apply to other kinds of line&r wave fields where the governing equations

r 5 F'-ttt 73

t94 P. G. Baines

are essentially nonlinear, so that they sustain resonant interactions, such as

inertial waves in a rotating fluid. It is shown in the appendix that the stabilityof Rossby waves on an infinite B-plane is governed. by a third.-order equat'ionakin to Mathieu's equation, which helps to elucidate its relationship With reso-

nant interactions (Longuet-Higgins & Gilt 1967) and, by analogy with the rnuch-

studied classical Mathieu equation, to obtain an overall view of the stability

characteristics.This paper is mainly concerned with Rossby waves on a rotating sphere,

sometimes known as Rossby-Hautwitz warres, which have a stream function

or vorticity of the form Pff(p) cosm(S - anf ), where 1z : sin (latitude), / is the

longitude and, P\@)the Legendre function withm,the longitudinal wavenumber

and. n the total wavenumber. Compared with waves on a p-plane the stability

of these waves is affected by two factors: the finite size of the sphere and the

disorete na,ture of the spectrum. As shown below, both of these tend to inhibit

instability: wa,ves with a sufficiently low wavenumber n ( E 2) are always stablo

and very few planetary w&ves are unstable for all amplitudes.Hoskins (19?3) has considered the stability of planetary wa,ves to a restricted

class of disturbances, namely a single planetary wave and a zonal flow. Supported

by some numerical integrations, he concluded that' waves with zonal wave-

numbers of 5 and less are stable whilst those with zonal wavenumbers greater

than 5 may be unstable. However, his numerical procedure only'permits w&ves

with zonal wavenumbers which are multiples of 4, thus eliminating many de-

stabilizing disturbances. fn fact, as shown below, all waves with wavenumber

n 2 3 arcunstable if their amplitudes (expressed in terms of the angular velocity

of the basic rotation) are sufficiently large. X'urther, the critical amplitude re-

quired for instability decreases rapidly with increasing ra (rather t'han m), and

for waves vnth lml > 1 the distwbance does not (necessarily) contain a zonal

flow.one may approach the question of the stability of waves on a sphere in a

ma,nner similar to that used in the appendix for a B-plane, arriving at a form

of Mathieu equation for an arbitrary disturbance. However this is very complex

because of the spherical geometry and instead the spectral form of the baro-

tropic vorticity equation has been used, followingPlat'zman (1962) and Hoskins

(1973). The problem for the stability of any given planetary wave, including a

zonal fl.ow, may then be expressed as an algebraic eigenvalue problem, which

must be truncated. to a finite number of equations for numerioal solution. The

relevant equations a,re given in $ 2. In $ 3 a criterion of n'jortoft is used to show

that waves wilir. n < 3 are stable regardless of their amplitude, and the eigen-

value problems for the stability of zonal flow and planetary waves are treated

in $$a and 5 respectively. n'or zonal flow, the amplitudes required for instability

lie only slightly above those required by Rayleigh's classical criterion, namely

the vanishing of the vorticity gradient somewhere in the flow, and the unstable

eigenvectors show that more energy flows to lower wavenumbers than to higher

ones in every case examined (3 ( z ( 9). X'or planetary w&ves Pff wrlh m * O

and. n: 3 or 4 and, P! the critical ar4plitudes are generally similar to those

given by triad interactions alone.

Stabi,li,ty of planetarg u,aaes 195

In $ 6 the linear stability results are tested by numericalint'egrations with aninviscid spherical barotropic spectral model. The waves P$ and Pua, each withappropriate small disturbances, are discussed in some detail; in the former casethe growth of the instability does not lead to destruction of the primary wavebecause n, is small, whereas it does in the latter case.

2. Basic equations on a sphereThe equations for a non-divergent single layer of fluid on a sphere of unit

radius in non-rotating spherical polar co-ordinates ma,y be written as

D(lDt: o, ( :Yzt ,

AIh 1 A,/toI : ---!- .t] : -*

A0' -

sin9 0Q'

(2 .1a , b )

(2.1c, d,)

where DlDt denotns the total derivative, (the vort'icity and y'" the stream func-tion, 0 is the coJatitude, / the (east) longitud.e alnd. u and o the fluid velocitiesin the easterly and northerly directions respectively. The dependent variablesmay be expressed in spectral form by expanding in spherical harrnonics, e.g.

@ I L

( : 2n:t nx: -n

(2.2)

where k : cos9 and the ff arc functions of time only, the presence of any par-ticular component (ff also implying the presence of its complex conjugateff.We'here define the Legendre functions PW@) such that

P;*(p): PW@), PT,@) PT,@)dp : 3n .,, (2.3)

following Bourke (1972)t and Hoskins (1973), so that

| 2?!J (n- m)llb (- t1,r.+"( 7 - pzltm 1m+n .P T @ ) : l = , @ + m ) t ) f f i f u t r - P \ " ' ( 2 ' 4 )

These functions have n-m zeros in the range (- 1;1), and their properties aredocumented. in many standard references (perhaps most graphically by Jahnke& Emde 1945, p. 112). X'or nx + O, the associated waves have 2m cells around alongitude circle and n-m* 1 cells on a pole-to-pole semicircle. Throughout thispaper the symbol PW will be used to denote the corresponding planetary wa,ve.

Substituting (2.2)into (2.1a), multiplying by PT@)ec"4 andintegrating overthe whole sphere yields the spectral barotropic vorticity equation (SilbermanL9l4;Platzman L962; Hoskins 1973; but with slightly different notation)

d(dt

: d*lor'n"hh

J:,

f Bourke defines P;*(p) = (-t)* Ptr@).

(2.5)

196 P. G. Bai,nes

where y, p and a denote pafus (m*n ), etc., and

[0 if m, * mo*rmp,

f - l / I I \.yfn- l {__l_ lK-.p" i fl \nu(no+ t) nn(nn*L)l

- ' l

K y po : ['_rrr(* o ru# - *, *ffi) rr

I ,,u,f f i . t : f f i o * *Or )

where (2.7)

Krpn may also bo written as K with a subscript rLrrlprLn and superscriptmnmsmo. I d.enotes summation over all pairs (a,p), but without repetition

q < f

(i.e. no permutations). The coefficients Krpohave been evaluated by Silberman(7954), and are zero unless

ln"-npl < ny < no*%pt nd+np+nr: odd,\(mp,np\ I (-m*ny), (rno,nn) I1-m*n). |

We also have the redundancv relations

K^tfn : - Kyrp : KoVy : Kpyd, (2.e)

where d, and p denote (-rtuo,no) and. 1-mp,np).These coefficients havo beencalculated numeriaally when necessary using routines developed by Dr W.Bourke of the Commonwealth Meteorological Research Centre, based. on expres-sions given by Silberman (1954).

It may be readily shown (e.g. Platzman 1962) that the aomponent containingPl(p) : 191* p contains the total angular momentum about the co-ordinate axis,and is equivalent to a rotation about this axis with angular velocity O given by

o: (8)+#.

(2.8)

(2.10)

(2.1r)

Hence (2.5) may be written as

and the linearized solutions to these equations are the Rossby-Haurwitz wa,ves.These are also exact solutions to (2.1I), as indeed is any sum of componentswith the same degree n (Craig 1945; Neamtan 1946).

We consider the stability of a single wave (o, (hereafter referred to as theprimary wave) to infinitesimal disturbances. ff (o, represents a zonal flow theappropriate linearized form of (2.11) for the disturbances is

# : - u*, (' - ffi) nq * u,, 86,,1,*tut",

9.: -a*,(t- -#")og,*;g", 4 l,po,(p.dt ' \ ny\ry+ L)/ 'u*(,,r, (2.12)

ff ft, represents a planetary wave it is best to consider the equations in a framoof reference rotating with angular velocity

O' : o[1 - 2lnor(nnr+ 7)],

Btabi,Ii,ty of planetary wa,aes

which is tho angular velocity of the wave pattern

2("rP|lrp) aosmfi (nL : rnn ,!L : %or).

The vorticity equation in this rotating frame is

D(lDt+2Q'a{tlad : 0,

LS7

which in spoctral form becomes

d 6 - r Q ' u * ' r / 2 \ r , ,- - 2" @Tis,

: -i'mr(o-o') \t- ffi.D) 6*d Pu

rvBn(B(o,f ,q+(0 ,1 ,

e . I4 )

where the notation is the same as above except that the time depondence of thecoeffioients will be differrnt. Tho linearized disturbanco equations then are

d( r : l * . -zo( : - - 1 \ "i lt

-v"v!2'a\n*,gn*3t1- nW)Jsy

/ 1 1 \+i(n>\Wm-W))K,n^(t1 1 1 \

ri(o,Z\;Mffi )\na(r P.16)

where (r, is real in this case and the first summation is over all p with

fnf : lfl'Y-lndf

the second over all p vtth fiLp : rltr*r.tlor.

3. Elf,rtoft's theorem and its irnplications for stabilityConservation ofenergy in the system governed by (2.1) yields

n : f,[ "wlnz d,s : -*l yo,t uu,

integrating over the -X:t;rfi;;:t:;,. L) pft(p) ena,

we obtain

1 @ n (ryf t i @ AZ a' : iAr^?-*'"ff i1: " Arff i): ArEo: constant' (3'2)

where

ni:*f-#*, n*:rffi. (3.3)

A further continuous infinity of invariants may be obtained from the vorticityequation, viz.,

(2 .13)

(3 .1 )

#,1"o cls : o' (3.4)

198 P. G. Bai,nes

where 7 is any real or complex number with Re7 2 1. Equation (3.4) with

u : ! is trivial and equivalent to conservation of angular momentum. T : 2

yields a second. quadratic invariant commonly known as the enstrophy:

therefore

l P 6 m

E : ; l g a s : n za J s n - L m : - n

@

I : n > A ' z " = c o n s t a n t .n : L

@ F^ rlU2+-6,:p*, Enlur*. pfafift_ ry,

(3.5)

(3.6)

X'rom equations (equivalent to) (3.2) and (3.6) n'jortoft (1953) deduced that any

energy exchange must take place between components with (at least) three

different values of n, and, if one of three components represents & source or sink

for both the other two, its nvalue must lie between those of the latter. He also

deduced some inequalities concerning the spectral distribution of energy, the

most important of which (x'jortoft's equation 44) is, in the present notation,

n : N * l

IIE -2 (3.7)( I r+ 1) (N +2)-2 ' ,

for specified E and.-F and any positive integer.ly'. Now the component Pl leple-

sents rigid rotation about an axis perpendicular to the co-ordinate axis, and

hence like Pl, its magnitude (taken a8 zeto here) must remain constant by con-

servation of angular momentum. This permits the strengthening of X'jortoft's

above inequality by using the same argument as his but omitting the modes

voilhn: 1, giving(3.8)

where N > 2 and. Er* and. E * are the energy and enstrophy omitting modes

w]rtlnn: 1. If all the energy is initially in a mode or modes wit'hn: ,L this may

be written as(3.e)

ff we consider the stability of individual modes for infinitesimal disturbances,

Pl and Pl must be absolutely stable by conservation of angular momentum, a,nd.

(S.O) shows that if all energy is initially in mod.es wilbln n:2 rt must remain

there. The interaction rules fequation (2.9)] also show that enelgy cannot flow

between P! , PL, and. Pf dfuectly, so that these modes must all be stable, regardless

of their amplitudes. Equation (3.9) indicates that modes wit'h n ) 3 may be

unstable, utta it is shown below that this is in fact the case if their amplitudes

are sufficiently large, although the amount of energy which may flow to higher

wavenumbers n, is limited.

4. The stability of zonal flow

We consider the stability of zonal flows whose vorticity (or stream function)

is represented by a single Legendre polynomial ?$, where z is an integer. n'rom

the preceding section we know that Pl and P! are completely stable, and here

Stabi,Ii,ty of planetary waues 199

we consider values of n ranging from 3 to g. The spectral equations governinginfinitesimal disturbances are given by (2.L2), an infinite set of linear equationswith constant coefficients. lVe look for the conditions under which this systemhas unstable normal modes. Assuming exponential time d.ependence ei,t eqaa-tions (2.12) may be transformed into an infinite set of algebraic equations, viz.,

f,(',,.,€.,- u rl*o(t - "efu) n.,]) (B : o.

The condition for non-trivial solutions is

e p y : € o r r y p o r - B p y m B ( t - f f i ) ,

(4 .1 )

(4.2)det (A-. l l ) : 0,

where 'det' denotes 'determinant', ).: alQ and the matrix A has elements

(4.3)

where E,r: (,rlQ. Since ra,ris zero, (2.6) requiresmpandmrtobe equalfor non-zero intetaction ooefficients. This implies that the set of equations for (p withlmp : 7 is decoupled from that tvtt'h mp : 2, and so on, so that these groups ofequations may be considered separately. fn other words, the equations governingdisturbances ofgiven zonal wavenumber form an independent set.

The eigenvalue problem represented. 1ay $.2), for values of ra, ranging from7 io mn - 1, was solved numerically for truncated systems with given values of6n . €o, was increased from zero to the point where complex eigenvalues ,1. firstappeared, yielding the critical amplitude for instability of mode a, with thegiven truncation. The truncation number -l[, (i.e. the rank of the matrix A, orthe number of equations) was varied up to 20 in the important cases, and thevalues obtained for the critical amplitudes appeared to converge as -lf, increased,with an apparent approximation to within at least two significant figures to thelimiting value where N, :20. fn most cases the critical amplitude decreasedmonotonically with increasing truncation number (by an amount which de-pended on the wavenumber but was typically of order f, overall), so that thevalue for Nr :20 was taken as the critical amplitude.

The eigenvalues and eigenvectors were obtained by employing library sub-routines provided by the CSIR,O Division of Computing Research. RoutineEIGENP, based on an algorithm by Grad & Brebner (1968) using Householder'smethod (Wilkinson 1965, p. 290), was mostly used, with routine EIGEN, basedon a method due to Eberlein (t962), used as a check on certain cases, notablythose in which the eigenvalues did not, d.ecrease monotonically with increasingJ/r. fn every ca,se so checked the eigenvalues obtained by the two methodsagreed to at least six significant figures.

The critical amplitudes of the vorticity of the zonal modes, expressed in termsofthe angular velocity ofthe rotation about the co-ordinate axistogether withtrhe zonal wavenumber of the most unstable disturbance, are given in table 1.With the earth's rotation and radius the associated equatorial wind speeds are89.3 ('trade winds') and 29.5 m/s for +P$ respectively, 25.7 and 5.44mls fot+P!, and 9.97 and 1.8m/s for +P$. The mean barotropic zonal flow of the

Wavo

T4era 1. Criticatr amplitud,es for the vorticity and enorgy of tho zonal-flow modes,

togother w'ith tho zonal wavonumber of the most unstable disturbanco.

atmosphere, ropresented approximately by *P$, is accordingly very steble bythis criterion. The corresponding energy amplitudes are plotted in figuro 1,

which also shows the necessary amplitudes for instability obtained from Ray-

leigh's criterion for instability of a shear flow. On a sphere this has the form

d((p)ld,p+zO : 0 somewhere, (4.4)

whore ((p) isthe vorticity of the zonal flow. If ((p) : (*P"(p) this becomes

+P_P

p0

+Pg-Pg

DOf 6

+P9-P2

po

+P3_Po

Yorticityamplitude

A

0.7990.2640.1610.3620.07770'05640.19570.03530.02s20.1220.0204

Zonalwavenumber

M

2I131I311I1

EnorgyE = Az\N(N+tl

6.32 x 10-26.81 x 10-31.30 x 10-34.90 x 1042.01 x 1047.67 x 10-56.84 x 1042.23 x 10-61.18 x 10-61'66 x 10-44.62 x L0-8

(4.5)

It may be seen that in every case the actual amplitude required for instability

lies only slightly above the necessary value obtained. from (4.5).The stability critoria separate the modes into two distinct groups, each with

its own approximate power-law dependence on n, (see figure 1): modes ry,4,P1,eto., with p osi,ti,ae coefficients are much more sta,ble than the others. fn addition,

all the modes of the second gloup (even modes and negative odd modes) are

most unstable to disturbances of zona,l wa,venumber one, which is not the case

for the first group.An inspection of the eigenvectors ((fr,(Tr, '..) representing the most unstablo

disturbances to each zonal-flow harmonic P$ indicates that they have no uni-

form structure other than a general tendency for the magnitude of the corn-

ponents (Toto deuease (not always monotonically) as n4 increases. If z is odd,

all the components voith na odd are zero (i.e. the most unstable disturbance has

nn evetr). fntegral constraints on the unstable eigenvectors ma,y be obtained as

follows. The equation for an infinitesimal disturbance tfrrto an initial zonal flow

{\witln total wavenumber L in a rotating system is

*ro, f , * ft 6 $!r rro,,lr ro -,li roY',!r r,) + z ch/ * : o,

f i f tnwr2:o '

(4.6)

200 P. G. Bai,nes

Stabi,li,tg of planetary wcwes

l0-2

S 1o-3Ftth€

>rsqr-f

10 -4

10-s

3 4 5

m

X'reunn 1. Critica,l amplitudes in torms of energ'y (tnits nazdlz,'whero a is the ea,rth'srad.ius) for tho zonal-flow modes P!,. The continuoug liroes drawn havo the slope indicateda,nd the dottod lines connoct the oorresponding points obtainod from Rayloigh's criterion.

where the suffixes 0 and. / denote derivatives. Multiplying by tr, integratingover tho sphere and invoking Green's theorem yields

*["+w,t ,lzd,B : I'-,#fi' {r,Yztfr2rit\dp: w,

A 1 f

Ti) u\"l")2d8 : L(L + 1) w,

and eliminating zu between (4.7) and (4.8) gives (Karunin 1970)

(4.7)

wlr-ere ar is the rate of working by 'Reynolds stresses'. Multiplying (a.6) byVzl, yietds

(4.8)

1\\

a

\\

\

tt\

\

\It\

\\

\

\

\

\SIopeN n - 6 ' 7

Un+l^t: L(L+DAELIaL (4.e)

202 P. G. Bai,nes

where EL and. Fr are the energy and enstrophy of the disturbance, which may beexpressed as

nt : I Eh* r, : *, na(na*r)Ek, (4.10)i : 2 nd : z

rvhere ideally ft is infinite and the sum does not include tui,: L or tr since theseform no part of an unstable disturbance. n'or a growing mod.e, each Enumay bewritten as Enc: frn'iatt'

where i. has a positive real part and the frnoure constants. Substituting in (4.9)and normalizing the Eroyields

2 frnu: L, l , no(na*r)En.: L(L+I) .M : 2 n : 2

n'rom these relations we ma,y deduce from n'jortoft's argument

o a t r , + 3 ) ( L - 2 )E*: *4*rr-. ifiaffij,

(4.rr)

(4.r2)

which corresponds exactly to (3.9) except that E"is necessarily zero.Equation (4.12) refers to the relative rate of flow of energy to modes with

wavenumber greater than .l[, and if -l[ : L, t'he upper bound on E, is less than

$ only for L:3. It has been pointed out by Merilees & Warn (1975) thatX'jortoft's argument for triad interactions that more energy must flow to lowerwavenumbers than to higher ones when energyis lost by a single mode is spurious,although this happens in a majority of cases. However, for the truncated eigen-vectors obtained numerically with,lfr varying between 12 and 20, ELwas alwaysless than 0.5, implying a greater flux of energy to the lower wavenumbers. Also,the corresponding ratios of the flux of mea,n-squa,re vorticity to higher w&ve-numbers to the flux to lower wavenumbers are close to unity for all L < g

except L:3, where they have values of 2'1 (positive vorticity coefficient) and.1.65 (negative vorticity coefficient).

5. The stability of planetary waves

The stability of planetary wa,ves on a sphere has been discussed by Hoskins(1973), who considered perturbations consisting of a single planetary wavetogether with a zonal flow. This form of disturbance w&s chosen because of theresults of numerical integrations, which were, however, restricted. to zonalwavenumbers which are multiples of 4, so that other firyes of disturbance wereprohibited.

We consider first of all the stability of planetary waves to perturbations con-sisting of (o) a single planetary wave and azonal flow as per Hoskins (1973)and (b) two planetary waves neither of which is a zonal flow. The criteria forinstability due to disturbances of type (a) may be obtained from equation (4.t7)of Hoskins' pa,per (to which the reader is referred for the relevant analysis),and these are presented in table 2 for all planetary wa,ves up to a wavenumber of

Stabi,li,ty of planetury watses

\ M 1 2_lI \

3 2 '04 0.77345 0'39726 0'23597 0.1535 1.19748 0.1065 0.36019 0'07737 0.2148

2.0871.551 0.82480.8680 0.6316 0.51610.5101 0.4540 0'4155 0.36620.3278 0'3245 0.3218 0'30620.2268 0.2368 0.2472 0.24830'1638 0. t774 0.1914 0.1994

0.27750.2390 0'21940.2001 0.1933 0 '1787

Teer.e 2. Critical amplitudes for tho vorticity l$l0l ot planetary waves Pff subject todisturbances consisting only of a singlo planetary wave and a zonal flow, as in the analysisoflfoskins (1973).

9.t X'or an initial unstable wave Pt the most destabilizing disturbance consistsof the ware Ph4plus the interacting zonal-flow harmonics, which act inunison;for an initial wave Pff with m 2 2 the disturbance is Pff*1plus zonal flow.

X'or disturba,nces of type (b) we consider (2.15) with only two disturbancecomponents (, and (, (and their complex conjugates), with wavenumbers'nlp 7L1t m, and. nz, where nl1,nt2 > 0. Consideration of the selection rules (2.6)and (2.8) shows that the only mutually interacting pairs are those which havernL+ rnz: nlo, the zonal wavenumber of the primary w&ve. It transpires that,for every wave with ffio) 7, the members of the triad with the lowest criticalamplitude satisfy the condition nxt+ nxz : nlo, a"rrd nob nxL- rnz : *o; if mn : 1the critical amplitudes for all cases nxL-nlz: ffio are greater than those given intable 2. The equations for disturbances with rnL+rnz: fiLntake the form

d ! ] : - o * . l c , L - ! - - | \ r . . " ' l 1 1 \d,t - yn,1n"+r1- ^pT4y)h+i("\Wt,- nW))'*("

(5.ra)d5 , ' : i , * , "zeL- ] - 1 \v '? l r 1 \d,t - lnolnoi!

- nr@r+ r)) 6- ;t"\mr.l -

nfu"+ L)) Ki&(5.1b)

where c : Tnt-frlzrn, rr : nr?LzrLa2 T : ffiz-ffitmn antd p :nlznrno, togethetwith their complex conjugates. Eliminating (, gives

ffi *m,Lft +b,(,: s,where

b,:z{t( f f i* f f i - f f i ) ," | 1 1 \ 1 1 1 \b' :

\n,1n, a'1 -

#" + D) \nlry + r) -

nW)) (# K' + + o' m'*,)'(5 .3b)

K : KX: -KT.

t The critical values for tho waves P! and P! agreo with the lowest curve of figuro 5 (b)of Hoskins ((* apparently representinC lqD, but the values for PH*r, M ( 6, do not.

(5.2)

(5.3a)

204 P. G. Bai,nes

, .yMI Y \

3

4

o

6

d

8

9

2 9 4 6 6

0.4536 0.2036lDl pl Dl DzL z t l ! z z r l 4

0.0693 0.2102 0.4684Pr"Pl P!,P8 P1',Pz0.031 0.1735 0.0402 0.0870Pl, Pr" Pl, P? Pl, Pl Prn, P3

0.00927 0.00768 0.0132 0.0346 0.0375P;,P* Prn,4 PE,PA PL,PI PB,P3

0.0578 0.00728 0.0578 0.0 0.0521Pl,P!, Pl,P!, Pf,,Plo Pg,P8 P?,Pn

0.0191 0.0201 0.0309 0.00613 0.023Ptr,Pl" Pi, P!,n PL, Pl, Pt, Pto 4, P?,0.0028 0'0 0.00599 0.00266 0'0233Pl, Plu Pl, P?n P?, P?u .P1,4, Pl, Plu

0.0497P3,PT0.0331 0.0658pt Ps p2 p6" z r - t o " 6 r - g

0.0120 0.00696 0.0P!,Pl, PE,PI, P8,PL

Tesr,E 3. Critioal amplitudes for tho vortioity l(fr/Ol of planotary waves Pff subject todisturbancoe consisting only of two planota,rSr waves neither of which is a zonal flow.The two wavos forrning tho most unstablo digturbanoe fot eaah Pfr aro indicated.

Solutions of (5.2) of the form ept will be unstable if

l 2 L \ 2 1 r 1 \ l t 1 \( :o/(" '@r-f f i, 4 ( , * n , , - , * ' , , - , * ' r , \ ' . 6 . 4 \ ,- -

\n*(n,+ t) nr(nr+ L) nz(nz+ I) I ' \ - ' - t '

which requites noto lie between nrand, mz a,s expected. The minimum amplitudesA : Z(,IQ for which inequality (5.a) is satisfied for each primary w&ve, and thetwo components of the corresponding unstable triad, are given in table 3. Exceptfor primary w&ves with a zonal wavenumber of one the critical amplitudes areall considerably smaller than the corresponding ones in table 2, so that the pre-ferred form of disturbance consists of two planetary waves. n'or any initialwave Pff there are a number of unstable triad.s, and those given in table 3 areonly those with the lowest critical amplitudes. n'or three wa,ves, namely Pl,4and P$, the critical amplitudes a,re zero because of the vanishing of the last termin (6.4). This is in fact just the frequency condition for resonant interactionbetween these waves and the other members of their appropriate triads, asdisaussed in $ 1.

fn order to discuss the stability of planetary waves oompletely the eigenvaluoproblem for (2.16) must be solved in the samo manner as in $4, extended tosufficiently largo wavenumbers. The number of oquations in each particulareigenvalue problem may be reduced by use of the selection rules, which show thatinteracting disturbance components may be grouped as shown in figure 2, andeach of these groups treated separately. The eigenvalue problem for the onsetof instability has been solved for all cases for all modes with n: 3 or 4 and for

Stabi,Ii,ty of plametarg waues

X'reuno 2. X'or a given primary wavo Pff the boxos show indepondent groupsof interacting disturbanao wavos Pf.

206

2.00.454

Pun

PlPrs

P3PlP?PlPiPi

0.7890.403

0'204o'0820.0770.2160.2s60.040

m ovonm ovorl,m odd70 evonm-rtu evellrn odd.m-rn oventtu ovoltn odd,rn odd

0.2040.7730.0690.2100.4680.040

IIz' = 30 Txiad

Pf,+zonal modosP,,,PL

Pr,4,P!+zonal modesPI,PIPI, P'UPg,PAPl, Pr,

Tasrn 4. Critical arnplitudes for instability of planetary wavos obtained for N2' - gQ,compared with tho values from tablos 2 and 3 for tho most unsteble triads.

tho mode Pfr, with tho truncation number ffz : 30 for eaoh case.f tr'or this valuoof -lI7 the critical amplitudes seemed to be close to asymptotio values for largoff'. The critical amplitudes .4, obtained for the vorticity A:2(,lQ a,ro givenin table 4, whero they are compared with the corresponding triad values fromtables 2 and 3. X'or eaeh mode, these two figures are generally of tho same ordorof magnitude and in most cases bear a striking similarity; for truncation numbers.il[, Iess than 30 the similarity with the triad values is not as good (for tho twowavos whero the oomparison is loast satisfactory, namoly Prl and P|, the triad

f This onsures the inclusion of mosf, modes for which za ( 8, and for P! and Pf, allmodes foywhich to ( 10.

206 P. G. Bai,nes

Pl+Pj, Pl*zonal flow

/I '

lr' 4'ni' 4

otr//r1

/6

AIA"

//4-P1"lzonalflow

Pfl+fi+zonal flow,//

Z fi+fiazonal flowD3

7 p',

AIAo

X'reunu 3. Grov'th rates i.a (tho growth factor for 1 day is oxp (22).a)) for tho fastest-growing oigenfurrctions for tho unstable modes indicated, as a fi.rnction of their vorticityamplitudes ,4 scaled with their amplitudes A" ofr, bho onset of instability, given in tablo 4.Continuous lines donoto those eigenfunctions which becomo r:nstable whon A = Ao,and for those eigenfrmctions the components of tho most unstable triad are prominent;dashed lines donoto oigonfunctions with the highest growth rate for each primary wavewhen theso aro different from the oigenfunction which frst becomes supercritical; someprincipal components of theso latter eigenfulctions are indicated. Othor unstable eigen-firnctions appearing at supercritical values of A and with lower growth ratos havo beonomitted.

is not a triad. at all but consists of a planetary wave (P| or Pl) and all the inter-a,cting zonal modes; if the most unstable 'pure' triad. is considered the criticalamflitudes a,re closer to the ffr : 30 values). This similarity lead.s to the con-jecture that the critical amplitudes for the most unstable triads for allplanet'ary

l ,

Pt

0.01

0

0.6

0.5

0.3

0.2

0 .1

Ir

Stabi,li,ty of planetarg waues 207

waves (excluding zonal flows) provide at least a good approximation to theircritical amplitudes for the complete infinite system. The reasonableness of thissupposition is supported by the experimental results of McEwan (1971) for theanalogous system of a discrete spectrum ofinteracting internal waves in a finitetank: the critical amplitudes for forced internal waves agreed very well with theirtheoretical values obtained from oalculations based on triads, after allowing fordissipation.

X'or every unstable planetary wave considered", the eigenfunction correspond-ing to the unstable eigenvalue at the onset of instability (A : Ac) contains thecomponents of the most unstable triad. as principal components. Growth ratesfor amplitudes larger than critical are shorm in figure 3, where continuous linesrefer to modes which commence growth ab A: A". Other urrstable eigen-functions with different principal components ma,y a,ppear for supercriticalvalues of A, and, where these have growbh rates greater than those of the initialeigenfunctions for I < AlAc < 10 they have been denoted by dashed lines. ft isinteresting to note that, as with the triad interactions, all the components ofeach unstable eigenfunction have the same time dependence (real and imaginary)when viewed relative to the frame rotating with the primary wavo.

6. Numerical integrations for P$ and Pua

Two integrations using the non-divergent form of the barotropic spectralmodel developed by Bourke (1972) were carried. out to examine the behaviourof the system beyond the limits of linear theory, the first, for P$, lasting for 12days with a rhomboidal truncation number J : L0 (i.e. all waves with

l t t [ l . J , 1 < n < l M l + Jincluded) and a f h time step, and the second, for Pf,,lasting for 20 days withJ : I5 and a t h time step. These truncation numbers were chosen on thebasis of avoiding truncation effects as discussed by Puri & Bourke (1974), andeconomical computer time.

We consider first P$, with positive vorticity coeffi.cient (the atmospheric case),but before discussing the spectral-model integration we consider the behaviourof the 'most unstable triad' in isolation for both sub- and supercritical ampli-tudes. P$ is the unstable member of a resonant triad containing waves P/ and,PZ, wilh a critical amplitude from linear theory (for the vorticity) of 1.08 O.Results of considering this triad in isolation are shown in figure 4 for two sets ofinitial amplitudes. ff one writes

(3: C, (E: +Aei" , ( i :+g"oB, 0 : f -e, (6 .1 )

so that A, B and C denote the vorticity amplitudesf of the respective wa,ves, thegoverning equations are

A : O. l543BCsin?, B : 0 '38576AC sin?, C : 0.27003 ABsin0,

0 : -0.46667 Q-0.5588C+(0.38576,4. lB +0.rE4B BIA)C cos?. (O.z)

f The amplitude of tho vorticity or stream function refers to l(ll or ltl\l tor nt, : 0, and2l(yl or 2l1tgl for nx + 0.

P. G. Baines

-2a--i 2=a---Z

sO

0

-20"

-40" 6 6

Days

X'reu;1u 4. Results for two numerical integrations of (6,2) for the intoracting triad

(P3, PZ,P|) with the initia,l amplitudes of P$ on either side of tho critical amplitude

(indic;ted by arrow) grvon by linea,r stability theory. The lowost curve shows the phase

difforenco d botweon (fand' (!.

These equations were integlated using a fourth-order Runge-Kutta method

with a timo step equiva,lent to t h for two amplitudes of P$ on either side of the

critical amplitude (0.938 Q and 1.3 O) a,nd with the amplitudes of Pl and' Pf;

smaller by a factor of 100 in each case. The difference between tho stable and

unstable cases is evident: for the former, all the functions oscillate with only

slight variations in amplitude, whereas for the latGr the disturbance w&ves

increase by two ord.ers of magnitude. The functions are periodic with the expected

elliptic-function (Platzm an L962) appea,ra,nce.The spectral-mod.el integration, whose results are shown in figure 5, wa,s

Stabi,li,ty of planetary waaes

0

-20"

400| ? 3 4 s 6 7 8 9 l 0 r t t 2

DaYs

X'rerrnn 5. Rosults in terms of vorticity amplitudes from integration of a barotropiospectral model (soo toxt) with primary wave +P$ and disturbances as shown. The lowestcuwo shows tho phaso difforence d betweon (! and (!, for comparison with figure 4.

begun with the same initial conditions as for the stable triad case, except thatthe waves P! and P| were added with the same initial amplitude and phase as

fi and P3. AII 'waves which reach amplitud.es above 0.01 Q are shown, exceptfor several whose amplitudes only just exceed this value. The critical amplitudefor this degree of truncation (^'" : 10 in this case) is 0'805 Q (as. 0'799 Q forJ > 20), so that the P$ mode is unstable; the modes PA, Pe, Pl and, Pfi (andPlo and P!r, which are not shown) all grow exponentially as components of theeigenvector comprising the destabilizing disturbance. The modes 1, P? and P$together with P54 appear at a later stage as a result of mutual interactions betweenthe growing waves wilhm: 2 when their amplitudes have become appreciable.The wavcs Prl and P| (which constitute a growing disturbance for ($ negative)

209

P3pn_

Dt)t 9

r 8

dr 5

4

\pli P l

PZltt

r+ eLM 73

3a

x .+

2LO P. G. Bai,nes

1.0'PT, F,

FrerrRo 6. Results in terms of stream-function amplitudes with the earth's radius androtation, from intogration similar to that for figure 4 with primary wavo Pf. With threeoxceptions, only modes with amplitudes greater tha,:n 2.0 at t}r.o end of tho day 7 are shown,thoro being a multitude of curves at lower amplitudes exceeding 0'1.

initially oscillate like the stable triad of figure 3, and a,ppea,r to interact onlyweakly with the other modes shown. In this integration the primary wave P$retains most of its initial energy (at least over t2 days) because, of the energyit loses, a, substantial fraction must fl.ow to waves wit'h n < 3, i.e. n: 2. Theonly such mode interacting strongly with P$ here is P!, and this is clearly coupledto P!, causing a quasi-oscillatory behaviour which limits its growth. In summ&ry,as predicted by linear theory, the expansion of the spectrum has lowered the

.Pl'4

J

4P2

24

Pt

Pl),4z4

DO

l:pip).

-*rg

+nt4

.4P2

4

Days

Stabi,li,ty of pl,anetary woues 2r1

critical amplitude for P$ below that, predicted for the simple triad, but the growthof the disturbance has been limited to about one order of magnitude, apparentlybecause of the small number of interacting modes with z < B.

Results for the first 7 days of the integration with Pt4 as the unstable wave areshown in figure 6, where the variable employed is the magnitude of the streamfunction in units of km2/s (related to the vorticity via the earth's radius). Theinitial amplitude of n$ (l(/Ol : 0.815) is approximately the same as that usedby Phillips (1959), and subsequently by several others, to test the efficacy ofnumerical models. The waves 4,4, Pl, Pl, Pl, P|and Pf; were also introduced.,all with vorticity coefficients S of t'he same magnitude and phase .but muchsmaller than (ua and 180o out of phase with it, yielding the initial stream-functionamplitudes shown in figure 6.

At this amplitude (a - 20 Ac) Pf has a number of unstable eigenfunctions, themost rapidly growing of which is that which first becomes unstable whenA : A" and has as its principal components P$, Pl, P?, Pf and pl. The growthrate for this mode is z\r:0.0803 (the growth factor for 1 day is exp(2lllc)),yielding an e-folding growth time of 2 days, which is in good agreement with theobserved growth rates of P| and P|. Another unstable eigenfunction, withprincipal components Pl, Pt, P! and -P$, has a growbh rate of 0.0418, giving ane-fold.ing time of 3'8 days, so that the agreement for these modes is less-satis-factory. Continuing the integration beyond day z shows that the amplitude ofPu4 continues to decrease monotonically, so that half its energy has been lost byday 15 and it has completely lost its dominance by day 20, when several othermodes have greater energies, and there is no indication that it would re-emergefrom the large number of other modes with comparable amplitude.

Hence fot P[at the above amplitude the growth rate of sma]l disturbances aspredicted by the linear theory is of a magnitude comparable with that obtained.with a J : I0 spectral model. This growth rato is maintained beyond the limitsof the linear theory, and for disturbances introduced at approximately ro/o ofthe amplitude of Pf, the 'destruction time'for the primary wave is of the orderof 2-3 weeks.

The author is grateful to Dr w. Bourke and Dr K. K. Puri of the common-wealth Meteorological Research Centre, Melbourne, for numerous discussionsconcerning spectral models and carrying out the integrations described in $ 6,and to Dr A. D. McEwan for various discussions on resonant interactions.

Appendix. The infinite B-planeGill (1974) has considered the stability of plane non-divergent Rossby waves

on an infinite p-plane for small and large values of M : UKrlp, where U is thevelocity amplitude of the planetary wave and K is its wavenumber. He foundthem to be unstable in both these limits and, presumably, in between as well.x'or small M the appropriate form of disturbance consists of the two componentsof a resonantly interacting triad. This problem may be approached in a slightlydifferent m&nner as follows.

r4-2

whero

212 P. G. Bui'nes

n'olowing Gill's notation, if the initial or primary wave is given by the streamfunction

t ! : (U lK)sin?, I : fusaly aat,

Kza: f lc, K2: k2+12,

the equation for the disturbanco stream function ry' is

Y\/rt+ p{r,+ (U lK) cos?lk(Y\lr *k't)o-l'(Y\t +k'tlr)' l : 0. (A 3)

If we take K-l and K I p as the units of length and time respectively the non-dimensional form of this equation is

Y\rr+t/ro+ M cos?lfr1Y',lt +t)r-I(Y\r +ty')*l : 0, (A 4)

where 1fr,t1 : @1K,UK). The most simple general solutions to this equationhave tho form

,1, : f (0) ei(Pa+qu+qt).

Substituting this in (A 4) gives the equation for/:

(A 5)

f"' +i,(ar+ Mbrcos0)f" - (ar+ Mbtcos?)f' - i(as+ Mbrcos?)f : 0, (A 6)

where the coefficients aiand b6 are functions of fi, l, p, q and q. Equation (A 6)is a third-order version of the Mathieu equation, to which n'loquet's theoremapplies, so that it has solutions of the form ei^oP(d), where P(d) is periodic

with period. 2a. Ifowever, the exponential factor 4*0 may be regarded as beingabsorbed into the factor di@a+w+rt) so that it is sufficient to look for periodic

solutions of (A 6). If M is small one may expand the solution in a n'ourier series,substitute into (A 6) and equate coefficients of powers of M in a m&nner similarto(forexample)Rhines (1970,$2).Thedetailsarecomparativelystraightforward,and it transpires that if M is very small the equation has solutions with realp, q and.7 except when

ao*nar*nzaz*na: O(M),

where ro is an integer. If n : 1 it may be shown that this implies that two compo-nents of the Fourier series are O(1) whilst the remainder are O(M) and t'hatthese two components form a resonant triad with the primary Rossby wave.If p and q are regarded as real,7 is complex; the growth rate (the imaginarypart of ry) increases linearly wLth M as M increases, &s does the range of valuesof (f + q\t for which unstable solutions exist. Higher values of n correspond tohigher-order wave interactions. The picture obtained is very similar to that oftho classical Mathieu equation, and by analogy one would expect the unstableregions to broaden so as to cover almost all p and q as M increa"ses.

(A 1)

(a 2)

Stability of pl,anetarg waaes 273

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