Vortices in shear

Dynamics of Atmospheres and Oceans, 14 (1990) 333-386 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

333

VORTICES IN SHEAR STEPHEN P. MEACHAM and GLENN R. FLIERL Center for Meteorology and Physical Oceanography, Massachusetts Institute of Technology, Cambridge, MA 02139 (U.S.A.)

UWE SEND Scripps Institute of Oceanography, La Jolla, CA 92093 (U.S.A.) (Received August 19, 1988; revised May 24, 1989; accepted June 14, 1989)

ABSTRACT Meacham, S.P., Flied, G.R. and Send, U., 1990. Vortices in shear. Dyn. Atmos. Oceans, 14: 333-386. The nature and stability of Kida's exact, time-dependent solutions for uniform, elliptical vortices in an ambient strain field are investigated. In addition to the classical Love mode of instability, we discover a class of resonant instabilities. Our linearized analysis is supplemented by numerical simulations of vortices in shear which follow the non-linear development of some of these instabilities.

1. INTRODUCTION In recent years, observers have steadily accumulated evidence for the existence of a variety of long-lived oceanic vortices. These appear to be both diverse in nature and abundant in occurrence. They include, as larger examples, the recirculating eddies generated at vorticity fronts associated with western b o u n d a r y currents, and small lens-like patches of highly saline water, thought to originate at the Mediterranean outflow. These highly non-hnear features are often distinguished b y their longevity; Gulf Stream rings have been tracked for periods exceeding two years. In addition, they can travel significant distances: 'Meddies' have been observed in the western Atlantic, roughly 1500 miles from their assumed point of origin. These properties suggest a considerable degree of robustness. It is this stability that we wish to explore in this paper. Our focus is the stability of a vortex e m b e d d e d in a background shear flow. We are particularly interested in the stability of time-dependent solutions. In view of the potential complexity of such a problem, we make 0377-0265/90/$03.50

© 1990 Elsevier Science Publishers B.V.

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S.P. M E A C H A M E T AL.

some rather drastic simplifying assumptions. The problem that we address is strictly two-dimensional and inviscid. We consider a situation containing a single vortex which takes the form of a patch of uniform vorticity. The background flow field in which this is embedded is one having uniform vorticity and strain rate. In the nineteenth century, Kirchhoff made the remarkable discovery that an isolated elliptical patch of uniform vorticity, having aspect ratio X and vorticity ~, will rotate, without change of form, at a steady rate given by f~=

(1 + X) 2

Love (1893) presented a linearized analysis of the evolution of small perturbations to Kirchhoff's solution. He obtained a dispersion relation giving the azimuthal velocity of azimuthal normal modes and noted that, if the ellipse were sufficiently elongated (X < 1/3), some of these normal modes were exponentially unstable. Moore and Saffman (1975) demonstrated the existence of exact, steady solutions to the Euler equations that took the form of a non-rotating ellipse of uniform vorticity superposed on a background flow of uniform vorticity and strain rate. These solutions could be found provided that the three free parameters, the vorticities of the ellipse and of the background flow together with the strain rate, satisfied certain constraints. Moore and Saffman also gave a stability analysis of some of these steady solutions using Love's technique. Kida (1981) extended Moore and Saffman's work by obtaining exact, time-dependent solutions that again consisted of uniform elliptic vortices embedded in a uniform strain and shear field. In these solutions, the aspect ratio and rotation rate of the ellipse are functions of time. Numerical calculations by Send (1986) suggested that vortices could persist in shear and seemed quite stable. Marcus (1988) has used a many vortex code and discovered that vortices of like sign to the vorticity in the shear flow persisted whereas opposite vortices were torn apart and destroyed. (His calculations were complicated by the presence of a mean flow with curvature sufficient to cancel the beta effect as well as shear.) In the present work we first present a reworked version of the problem considered by Kida. We show that the evolution of the elliptical vortex is exactly that predicted by the moment model of Melander et al. (1986). We then describe in more detail the different phase plane portraits that the system can have. We subsequently use a contour dynamical model to study the evolution of the vortex numerically and compare this with the theory. The more novel part of our work is an examination of the linear stability of the periodic solutions to non-elliptical perturbations. This is a generaliza-

335

VORTICES IN SHEAR

tion of the work by Love for the Kirchhoff ellipse. We obtain a Floquet problem that we solve numerically. This enables us to explore the way in which the parameter space of the problem is divided into unstable and neutral regions. From this we can say something about when ambient shear fields are likely to destabilize a two-dimensional vortex. This analysis is complemented by some studies of the response of the periodic solutions to finite-amplitude perturbations. Although this study is, perhaps, helpful in understanding the initial behavior of vortices in a shear layer, it does not include three-dimensional effects that may become important shortly after 2-D rolls develop in a non-geophysical flow; the work of Pierrehumbert (1986) and Bayly (1986) suggests that the instabilities leading to three-dimensionality are rapid when there is no other mechanism to restrain them. However, for geophysical flows, the two-dimensional assumption is much more pertinent because of the geostrophic constraints. There is a connection between some of Kida's results and the work of Farrel (1982) and others on the non-modal amplification of initial perturbations to shear flows. This is dealt with more fully in the discussion below. 2. T H E M O M E N T M O D E L

We consider the flow field as having two parts: ~k, the steady background shear_/strain field and ~k', the field associated with the vorticity anomaly. The ~k field is assumed to be a quadratic function of x, y, having uniform vorticity ~ and uniform strain e. Without loss of generality, we can choose the axes such that the xy term is absent. Thus 07= l ( ~ + e)x2 + l ( ~ - e) y2 - uy + vx is the background field. The vortex has additional vorticity oa (total vorticity + o0) and occupies a patch D. We assume that the area of the anomalous vortex patch is rr; this sets the length scale for the flow. The anomalous vorticity evolves according to atv 2~p' + J(07 + ~', vz~p ') = 0 (1) where ~72~' = ¢0 x_~D 0 x~D In section 3, we will track the boundary of D with a contour dynamical model; here we discuss the evolution of physical space moments of the anomaly. Following Melander et al. (1986), we define the moments of the anomaly J("'")-

1 ff

607/"

(x-x0)

,. (y-yo)"V2~k ' = - l f f,, ( X - X o ) 7r

m

(y

__

Yo) n

336

S.P. M E A C H A M

ET AL.

where Xo, Yo are the centers of mass defined by the conditions 0

j o , o ) = j(o,~) =

1

==~ XO =

1

-~ffD

X>

YO=

-Sfz qr

The moments evolve according to

j(m,n) + m.~oj(m-l,n) + n.~oj(m,n-,) _

,

<~;ffv~
_

,ffo mx ,m-, y ,.,_t , , + q ,

=--

97

#

x

~72~ ')

rj jr x , m y , . j ~ + +~T ,,

,) - n x ,. y ,.1,~t , ~ + q / )

#

--x-x

o,

Y =-Y-Yo

Using the definition of 5, we find j(m,~)= _rn 2 o

-- m

~-

2

2 e

Yo

u

n

.Vo +

j(m-l,n+l)+ n T j ( m + l , n - 1

)

~ + e

2

x°

v

x '~' y '~-, ~y + _"SSo x 'my '~ '--' ~x msso ~r F r o m this, it is clear that the area of the vortex, ~rJ (°'°), is conserved and that requiring the first moments to be zero implies that the center of mass of the vortex evolves according to Xo :

~-e 2

Y°=

~ +2e

1 /. #, Yo -- ~r SSo+' Y+u

Xo+

1 i f Dq/x+v

F r o m these we derive

y(m,n)

=

~-e

__mj(m-l,n+l)

2

~+e

.q_ ~ r / j ( m + l , n - 1 )

2

m,, SSo) 7JJ,~ x ,m,,.,_, y {*,¢', nSL"Y' '('SW)

337

VORTICES IN SHEAR

The system is closed by approximating the shape of the vortex anomaly by an ellipse with aspect ratio ~ --- b/a = minor axis/major axis oriented at an angle q~. We will later show that this approximation is, in fact, exact. This implies that the ~k' field is symmetric so that its contribution to the first moments vanishes and therefore the center of mass of the vortex anomaly satisfies the equations

2°= u - ½(~ - e) Y° Vo= O+ ½(~ + e)Xo

(2a)

Thus the center of mass moves as a passive particle along the background streamlines. To find the changes in second moments, we can take advantage of the fact that in the absence of shear/strain (~ = e = 0) the ellipse would rotate counterclockwise with frequency ~=60

(1 +

to express the integrals in terms of the moments. Alternatively we can substitute into the integrals the solution for the streamfunction of an ellipse with vorticity o~

q/=

o~ ) k X 2 +

2

1+~

~o ~-(~-~0)

y2

4 ~-1 4 +----1 ?~ e-2(~-~°) c o s 2 0

~>~0

where X = ~(1 - 72)/~ cosh ~ cos 8 and Y = ((1 - ~2)/), sinh ~ sin 8 are coordinates oriented along the major and minor axes respectively, (~, 0) are elliptic coordinates with 8 measured from the major axis, and tanh ~0 -- X. (The relations between moments and the aspect ratio and orientation of the ellipse given below must also be used.) The resulting equations j(2,0) = _ ( 5 -- e ) J (1'1) - 2f~J O'1) j(1,1) =

~-e

_ _

~+e

j(0,2) + ~ j ( 2 , 0 )

2

+ ~ [ j(2,0) _ j(0,2)]

2

j(0,2) = ( 5 + e ) J 0'1) + 2 ~ J (1'1)

can be reduced to two equations by using the conservation of area

3,[ j(2,o)j(o,2)_ j(,a) 2] = 0 The simplest form for the two equations is in terms of the variables X and

338

s.P. MEACHAM ET AL.

(the orientation of the ellipse relative to the x-axis), related to the J values by 1 j(2,o) = 47t [1 - (1 - X2) sin2g~] 1

j ( u ) = _8__~[ 1 _ hE] sin 2g, 1 [1 -- (1 -- X=) cos2q}] j(0,2)_ 4---XAfter some algebra, we find =

-

eX sin

2~

X

~

el+X

2

6 = ~° (1 + X) 2 + -2 + 2 1 - ~ . 2 cos 2q~

(2b)

Scaling time by 1 / 5 yields ~. = - p h

sin 2q~

h 6=r(l+h)2

1 pl+h 2 +-~ + 2 1 - X 2 cos 2(/)

(2c)

where r = o~/~ is the ratio of the vorticity anomaly to the background vorticity, and p = e / ~ is the ratio of the background rate of strain to the background vorticity. We can find steady solutions to eqns. (2) easily. The restriction 0 < )~ < 1 will be made, leading to the following equilibria (Moore and Saffman, 1971, discuss special cases): h 1 (1 + h)2 = - 2----~, r < - 2

(1)

p=0,

(2)

p~O,h=O,

(3)

X 1 p(l+X2 t p :~ 0, h 4: 0, q) = 0, r (1 + 7~)2 + ~ + ~ 1_---~1 = 0

(4)

pe0,

1

cos2q,=---,p<-lorp>l

P

X,0,

~r h q)=~-,r(l+h)2

1 p/l+h2\ + - ~ - ~ - / 1 - - - S ~ /] = 0

Figure 1 shows the different regimes based on the solution of these equations; depending on the values of r and p, we have 1, 2, or 3 stationary

VORTICESINSHEAR

339 R

~"

........... ~

[0,2,2l

[0o1,01

I1,0,0)

iiiiiiiiiiiiiii..ii12111

.......

0

.~ (0,0,2)

(IFll

"-17 . . . . . .

a

/ 11,2,0l

~ d

[3CI [0,0,21

12,1,01

Fig. 1. Regime diagram showing the distribution of singular points in phase space as a function of p and r. The triples (i, j, k) describe the phase plane structure, i, j, and k are the numbers of critical points at q~= 0, at @= ~r/2, and at 7~= 0, 0 < @< ~r, respectively. solutions in 0 < ~, < 1, 0 < q~ < ¢r as well as 0, 1, or 2 solutions on the ~ = 0 axis. The phase plane trajectories for the unsteady solutions can be found by casting eqns. (2) into Hamiltonian form or, alternatively, noting that the quantity 1-~ 2 1+2, 2 /t =p---~ cos 2 ¢ + ~

)~ 2r l n ( l + X ) 2

(3)

is conserved. In the next section, we present the four basic phase plane patterns and discuss these and the various degenerate cases. 3. CONTOUR DYNAMICS MODEL The contour dynamics procedure (Zabusky et al., 1979), first applied to this problem in the numerical calculations of Send (1986), involves calculating the evolution of the b o u n d a r y of D. If the b o u n d a r y at time t is

340

s.e. MEACHAMET AL.

described by the curve S(x,y,t) =0

(4)

then we have that

DSDt

x oD = o

(5)

The contour dynamics procedure follows the evolution of the vortex by solving this equation numerically in a manner to be described later. However, considerable insight into the evolution of a vortex that is initially almost an ellipse can be gained by a closer analytical examination of eqn. (5). With the results of the m o m e n t model approximation in mind, we consider the vortex to be composed of an ellipse, whose aspect ratio and orientation at time t we take to be those furnished by the m o m e n t model approximation, and a time-dependent perturbation about this. The ellipse is chosen so that its center of mass coincides with the center of mass of the vortex. We begin by moving to a more convenient coordinate system. This transformation is a two-step process. We first move to an X, Y coordinate system that coincides with the principal axes of the ellipse. This involves only a rotation through the angle q~(t) and a translation to the rest frame of the center of mass. (For clarity, we have assumed that the x, y coordinate system was chosen to correspond to a rest frame for the center of mass of the vortex.) We next transform to a coordinate system in which the boundary of the ellipse becomes a circle. This can be achieved by using the conformal mapping z = aw + b / w where z = X + iY, w = r exp(i0) and a = (1 + X)/2v/X-, b = (1 - X ) / 2 v ~

r, 0 are cylindrical coordinates in the complex w plane and the boundary of the ellipse is just r = 1. In these coordinates, the boundary of the full vortex is given by S:r-[l+f(O,t)]=O

the Jacobian of the transformation, h - 2 = a(x, y ) / O ( r , O) is given by h -2 = a2r + b 2 / r 3 - ( 2 a b / r ) cos 20 and the Laplacian becomes

Vr2o"=-~r(r~r) + - - 123i f = 1 r

!/" h - 2

lh-2v2 r

Eqn. (5) becomes

h-2ft +fo('~r -4- h - 2 O )

-4- ~It0 - h - 2 / ~ = 0, x ~ a D

(6)

where 0 = O(x, y, t) and r = R ( x , y, t) are the coordinate functions and

VORTICES IN SHEAR

341

~t' = ~ + ~k'. t appears explicitly in these functions because ~, and hence the coordinate transformation, may be time dependent. We shall use eqn. (6) to establish the remarkable fact that the rotating ellipse of the m o m e n t model approximation is, in fact, an exact solution of the Euler equations. Suppose that, at some particular time, to

f(O, to) = 0 for all 0. Then at that same t o

= ¼(o~ +-~

×

a2r 2 + -~ + 2ab cos 20

a2r 2+

b2

cos 2 0 + 2 a b

e

--~

+¼ e c o s 2 , - o ~ a2r2

--~

sin 20 s i n 2 , - ~ -

inside D. On r = 1

1--~t2 't'° ---

e (l+)t 2

4-----~~ sin 28 - ~ / ~

) cos 2 , sin 20 + sin 2 , cos 20

oa 1 - ) t 2 (1---~) sin20 Also on r = 1, h-2/~ = - ~ a b sin 2 0 - (},/2X) cos 20 and h - 2 ~ = - ~ ~[(1 + Xz)/(4X2)] sin 20. If we substitute the values of ~ and ~ obtained from the m o m e n t model approximation, we find that

h

=%1r=1

and so f will remain zero for t > t 0. Therefore, if the initial shape of the vortex is an exact ellipse, then the patch will remain elliptical at all subsequent times and its aspect ratio and orientation will evolve according to the equations furnished by the m o m e n t model. In other words, the moment model gives a complete and exact description of the evolution of the vortex if it is initially elliptical. Thus we have shown that there exists a large class of exact, time-dependent, inviscid solutions to the Euler equations which have the structure of an elliptical patch of anomalous vorticity embedded in a constant vorticity, straining field. This result comes from the quadratic nature of all the flow fields. It can also be readily derived in the X, Y plane. We have included the transformation to the r, 0 plane because this will facilitate an analysis of the stability of some of the time-dependent solutions described above. We now begin the discussion of the characteristics of our solutions in the different regions indicated on Fig. 1. We adopt the notation, (i, j, k), to

342

S.P. M E A C H A M E T AL.

describe the phase plane structure, where i, j and k are the numbers of critical points at @ = 0, at @ = ~r/2, and at X = 0, 0 < @ < ~r, respectively. Figure 2 indicates the locations of the critical points in the various regimes

(0,i,0)

(0,2,2) ;:.

.'C.

.'c.

(0,0,2)

(0,i,0) (0,0,2)

Iu

(2,1,0)

LEGEND I_

~I

(2,0,2)

A =] points

4---

=0 =0

movement as p, r ~ = ~ vary

Fig. 2. Locations of critical points in phase space for the various different regimes. Arrows indicate the way in which different singular points coalesce as a regime boundary is crossed.

VORTICES IN SHEAR

343

and the types of coalescence occurring as a transition from one regime to another occurs. Characteristic phase plane portraits are shown in Figs. 3-6. Figure 3a shows the phase plane at a point in the (0, 0, 2) regime--2 stationary points on the ~ = 0 axis. Any initial ellipse will eventually shear out into an infinitely elongated and thin eddy oriented at a particular angle 0.5 c o s - l ( - l/p). Figure 3b shows an example of this evolution calculated with a numerical contour dynamics algorithm. We represent ~D by a set of points and compute (u', v') with Green's function integrals around the boundary 60

u ' = - 4---~ l n [ ( x - 2) 2 + ( y - ) 3 ) z] d:~ 09

v ' = - 4----~ l n [ ( x - 2) 2 + ( y - ) 3 ) 2] d)3 The integrals are estimated as centered sums as in Wu et al. (1984). We have also used the code of Send (1986) which handles the principal value problem differently; tests with both codes indicate that the results are essentially identical. (We are grateful to L. Polvani for pointing out this simpler procedure). The total velocity field at point (x, y) is then [ - ( 5 - e ) y / 2 + u'(x, y), (5 + e)x/2 + v'(x, y)]. A second-order Runge-Kutta scheme is used to advance the positions of the points. This procedure gives reasonably accurate solutions when enough points are used, the total run length is not too long, and the ellipse does not become very elongated. On workstations (Sun 3s and Sun 4s) out calculations take times ranging from fractions of an hour to a couple of hours. When p = + 1, the two critical points on the ~ = 0 axis coalesce to form a doublet. As this case corresponds to pure shear flow in the background, it is of particular interest. The (0, 0, 1) type of phase plane (the p = + 1 limit of the (0, 0, 2) regime; Fig. 3c) exists for - ( 3 + 2x/2-) < r < 0. Thus weak 'counterrotating' eddies with vorticity anomaly opposite in sense to the shear will be destroyed by being infinitely extended in the direction of the shear. The next simple case, (1, 0, 0) or (0, 1, 0), has a phase plane with both closed and cyclic periodic solutions (Fig. 4a). The closed orbits correspond to the orientation oscillating about 0 or ~r/2, with the aspect ratio also changing slightly (Fig. 4b). These 'nutating' ellipses can be thought of as being slightly too elongated versions of Moore and Saffman's (1971) steady solutions so that there is a cos 20 perturbation upon the otherwise steady ellipse. This perturbation travels clockwise around the vortex but is sufficiently small that the ellipse along the x-axis is always the dominant factor. In the case of the open orbits (Fig. 4c), it is simplest to think of the ellipse as precessing clockwise owing to its own vorticity, like a Kirchhoff ellipse,

344

S.P. MEACHAM ET AL.

p=1.5 r=1.5 I

1.0

I

I

I

•

I

I

I

I

I

I

I

0.8

0.6

0.4

0.2

0.0

I

90

180

270

360

Phi

1.500000

1.000000

1.500000

0.400000

0.800000

0.

4

(b)

o /---..

/

/

/

//

/

Fig. 3. (a) Sample phase plane from the (0, 0, 2) regime. (b) The evolution of a vortex on one of the trajectories shown in (a). Some streamlines of the background flow are sketched on the first image. The vortex is sheared out as the trajectory approaches the singular point. Time increases from left to fight and down the page. The snapshots are separated by a dimensionless time interval of 0.4. (c) Example of a (0, 0, 1) phase plane, the p = + 1 limit of the (0, 0, 2) regime.

VORTICES IN SHEAR

345

p=l.0 r=-l.0 1.0

I

I

[

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

(c)

0.8

0~6 E 0.4

0.2

0.0 0

90

180

270

360

Phi

Fig. 3 (continued). but being perturbed by the background fields. Initially it may be elongated by the shear as it rotates down towards the horizontal. The rotational rate decreases as it elongates but it is still sufficient to bring it past horizontal. Now the shear recompacts the eddy into its original ellipse and the process repeats. This behavior will be labelled 'tumbling'. In the (2, 0, 2) regime (for p = + 1, a (2, 0, 1) regime), reached only when the vorticity anomaly is opposite in sign to the background vorticity and is sufficiently strong, we find the appearance of a saddle point (Fig. 5a) in addition to a center. Vortices which are compact enough and which have an orientation that is close enough to 0 (or ¢r/2 in a (0, 2, 2) regime) will survive, with slight nutations. Such vortices lie in phase space sufficiently close to the center that they are on closed orbits. A vortex not satisfying these conditions will be sheared out and destroyed. Depending on whether the separatrix emanating from the saddle point closes back on itself or not, we may or may not have the periodic 'tumbling' solutions (compare Fig. 5b, which has tumbling solutions, with Fig. 5a, which does not). Finally, in the (2, 1, 0) or (1, 2, 0) regime (Fig. 6a), we have stable centers in the phase plane diagram at either 0 or ~'/2 and one saddle point for elongated vortices. There are nutating and tumbling solutions, but all are periodic (some cases are shown in Fig. 6b). (The example shown in Fig. 6b3, is actually unstable and the growth of an unstable disturbance, triggered by numerical noise, is noticeable in the last frame.) The behavior of some of the trajectories in the presence of strain affords an example of the non-modal amplification described by Farrell (1982 et

346

S.P. M E A C H A M ET AL.

p=-0.5 I

1,0

I

I

I

1

r=-l.0 I

I

I

I

I

I

I

I

i

I

(a)

0.8

A

o.3 E 0.4

0.2

-__...

0.0

i

I

I

90

!

I

180

360

270

Phi

1.000000

-0.500000

-i.000000

1.000000

0.250000

O.

4

(b)

CS>

O

O

Fig. 4. (a) Example of a phase plane in the (1, 0, 0) regime. (b) Evolution of a 'nutating' ellipse described by one of the d o s e d trajectories in (a). The interval between frames is t = 1.0. (c) Evolution of a 'tumbling' ellipse described by one of the open trajectories in (a). The interval between frames is t = 1.0.

347

VORTICES IN SHEAR

1.000000

-0.500000

-i.000000

1.000000

0.150000

0.

4

(c)

Fig. 4 (continued).

seq.). Farrell revived interest in the behavior of transient motions in a shear flow by calculating the potential amplification of properly aligned initial disturbances, demonstrating that it can be quite substantial, and then extending these ideas to baroclinic instability as well. In some cases, the amplification of a small, but optimally oriented, perturbation over an O(1) time period can far exceed the growth of the most unstable linear eigenmode from the same initial amplitude in the same period. Other geophysical fluid dynamicists (Boyd, 1983; Tung, 1983; Shepherd, 1985) have continued along these lines, including beta effects and a fuller spectrum of initial waves. But these studies are linear and therefore cannot represent the evolution properly if the amplitude grows large. Thus we anticipate that the linear model may describe the amplification stage properly but cannot correctly predict the behavior for all times. Calculations with two like-signed point vortices embedded in a shear flow indicated that the ensuing motion was periodic, with non-linearity being significant even when the vortices are widely separated and being sheared apart. The more complete calculations reported here indicate some of the behavior which might be expected in the non-linear problem, although we deal only with very simple initial conditions.

348

s.P. MEACHAM ET AL.

p=-1.4 r=2 1.0

I

I

I

1

I

I

I

I

I

I

I

I

1

I

I

I

I

I

I

I

(o)

0.8

0.6

0.4

0.2

0.0

I

0

90

180

270

360

270

360

Phi

1.0

0.8

0.6

_J 0.4

0.2

0.0 0

90

180 Phi

Fig. 5. (a) Example of a phase plane in the (2, 0, 2) regime (no tumbling trajectories). (b) Example of a phase plane in the (2, 0, 2) regime (with tumbling trajectories).

For a given external strain field and a given initial aspect ratio, the subsequent evolution of the vortex depends on the initial orientation of the vortex. As we vary the initial orientation we change the trajectory on which the vortex lies. When its orientation and aspect ratio correspond to a minimum point on a periodic trajectory the vortex is in a sense an optimal 'perturbation' on the external flow. As its evolution progresses, the aspect

VORTICES IN SHEAR

349

)=0.2 r=-3.5 1.0

I

I

I

I

I

i

I

I

I

I

I

(a)

0.8

0.6

0.4

0.2

0.0

i

i

9O

i

I

180

i

i

i

i

i

270

360

Phi

1.000000

0.200000

-3.500000

1.000000

0.600000

O.

4

(bl)

Fig. 6. (a) Example of a phase plane in the (2, 1, 0) regime. (b) Several examples of periodic solutions described by trajectories in (a).

1.000000

0.200000

-3.500000

i. 000000

0.400000

O.

4

(b2)

G

1.000000

0.200000

Fig. 6 (continued).

-3.500000

1.600000

2.000000

0

O.

4

VORTICES IN SHEAR

351

ratio increases, the vortex becomes more compact and the kinetic energy associated with the 'perturbation' increases. Subsequently this amplification ends and the vortex elongates before going through the cycle once more. 4. STABILITY

The work presented above essentially confirms Moore and Saffman's (1971) and Kida's (1981) results and demonstrates the existence of periodic solutions near the fixed points which are centers in the phase p l a n e s - - a result also anticipated by Moore and Saffman's linear stability analysis. In addition there are solutions with large-amplitude nutations, solutions where the vortex 'tumbles' or rotates through 360 ° while stretching and contracting, and solutions which asymptotically stretch out and flatten. To use these periodic solutions as models of vortex motion, we must also explore the stability of time-dependent solutions. We begin with a linear stability analysis based on eqn. (6). We showed that the patch with b o u n d a r y r = 1 (i.e. f = 0) was an exact solution with a streamfunction given by if'. We now consider small perturbations to this given by r = 1 + of(O, t) and an associated streamfunction xI, + c~', where c << 1. Linearizing eqn. (6) we find that

O,(f/hZ)+3,[(h2~r+(9)f/h2]+~b'8=O

on

r=l

We will denote f / h 2 by ~'. Using the k n o w n form of q" and the details of the coordinate transformation, we find that h2'I% + 0 = F ( X , ~) =

o~

e~ cos 2q~

(1 + X)2

1 - X2

Thus ~, + ~o(F(X, 0 ) ~ ) = - q ~

(7)

An important feature of eqn. (7) is that F is i n d e p e n d e n t of 8. This allows us to separate the 6 and t dependence and so greatly simplifies the solution of the linear stability problem. This simplicity again stems from the quadratic nature of the flow field, xI,. The physical significance of the quantity ~"= f / h 2 lies in the fact that, if the b o u n d a r y of the vortex is displaced a small a m o u n t from r = 1 to r = 1 + of, the area increment in the sector lying between 8 and 8 + d8 is given by cf/h2d6 + O(c2). F r o m eqn. (7) it can be seen that the linearized perturbation does conserve the area of the patch at O(c).

352

s.P. MEACHAM ET AL.

Continuity of the total streamfunction and its gradient across the b o u n d a r y of the vortex yields the j u m p conditions

[¢;] =o

[+'r]

(8)

where [(-)] denotes the j u m p in (.) across r = 1. We note that d z / d w = a - b/w z, indicating that the transformation is non-analytic at w=(a/b)a/2; this dictates the appropriate form of the radial basis functions to use. We look for a solution of the form ~ ' = A ( t ) cos n O + B ( t ) sin nO ~k' = G ( r " +

r-"

-

cos n O + H r " -

a

-

r-"

sin nO, r < l

a

n

~'

= Gr-"(1 +

cos nO+Hr -~ 1 -

-

a

-

a

sin nO, r > l

Eqns. (7) and (8) relate G and H to A and B and yield a simple Floquet problem for A and B

At+ n F -

~- 1 -

B=0 (9)

Bt -

nF-~

1+

a

A=0

This result can also be obtained by using a transformation to elliptic coordinates of the sort employed by Love (1893) and Moore and Saffman (1975). Before using eqns. (9) to examine the stability of some of the periodic solutions discussed in section 2, we point out that the n = 1 and n - - 2 perturbations have a simple interpretation. As noted by Love (1893), the n = 1 perturbation corresponds simply to a shift in the center of mass of the vortex whereas the n = 2 case corresponds to a change in both the aspect ratio and the orientation of the ellipse. For n > 2, the location of the center of mass of the vortex is not disturbed. For n = 1, the new center of mass is located at X = c A / v r ~ , Y=cvr~B (assuming that (X, Y ) - - ( 0 , 0) in the unperturbed state). For the n = 2 case, the orientation is increased by an amount c2XB/(1 - X2) whereas the new aspect ratio is ~(1 - c2A). (O(c 2) effects have been neglected.) The Floquet problem for the n = 1 case can then be solved simply by considering the equations of motion of the center of mass: x -- ½(e - ~ ) ~ , ~ -- ½(e + ~ ) ~ from which it can be seen that, in

353

VORTICES IN SHEAR

the unstable case (~,~)=a

)

(1

1,~--o-o(e+3 ) e x p ( o t ) + f l 1 , - ~ o ( e + ~

)

) exp(-ot)

1 2 - 32)1/2 . Then where 0 = 7(e ( 1 ) A = ~/-~ cos ~ + ~ ( e + 3) s i n , exp( ot ) ( + fl~/~- cos , B=-~-

a(

1 ) -~0 ( e + ~) s i n , e x p ( - o t )

sin~-

e+3) cos~

1

fl( ~- sin,+~(e-3)

cos,

)

exp(ot)

) exp(-ot)

The n = 1 mode is exponentially unstable when e 2 > 32. However, we do not consider this a true instability because the elliptical vortex is not appreciably changed; it merely drifts away from its initial position as a result of the velocity shear of the background flow. The background flow is dominated by the strain field and the background streamlines are open, hyperbolic curves. When e 2 < 32, the streamlines of the background flow are closed and the center of mass of the vortex merely orbits around its undisturbed location. When I e I = 131, i.e. when the background flow is a simple shear flow, the perturbed vortex drifts rectilinearly at a uniform rate. For example, when e = ~

x=x(t=O),

y=~(t=O)+½(e+~)tx(t=O)

Instability of the n = 2 mode cannot be considered a true instability of the elliptical vortex because it too cannot appreciably change the vortex. An n = 2 perturbation corresponds to starting with an elliptical vortex lying on a phase plane trajectory close to that of the unperturbed vortex. Its subsequent evolution can then be obtained by following the new trajectory. Only when the unperturbed vortex lies at a saddle point is it appropriate to regard the n = 2 perturbation as a linear instability, because then the perturbed state will depart dramatically from the unperturbed states. This latter case was examined by Moore and Saffman (1971). The structure of the system (9) is that of a parametrically forced oscillator. Perhaps the most familiar example of such an oscillator is the Mathieu equation (Mathieu, 1968) 5i + (8 + ~ cos 2f~t)x = 0

354

s.P. MEACHAMETAL.

which arises as an approximation to a variety of physical systems, including that of a p e n d u l u m suspended from an oscillating support. One of the properties of the Mathieu equation is the existence of two types of growing solution. The first occurs even w h e n e = 0 in which limit it corresponds to the case where the natural frequency of the oscillator (81/2 ) is complex; the second occurs when the natural frequency of the oscillator is almost resonant with a multiple of the half-frequency of the parametric forcing. Thus, when [c [ is small, the parameter axis - oo < 8 < co is divided into alternating bands of neutral and growing solutions. The unstable regions are - oo < 8 < 80, where 80 = O(c 2) (note that the u p p e r b o u n d a r y of the single region of instability present in the c = 0 case is no longer exactly 0 when c 4: 0), and bands around 8 = n~2, n = 1, 2, .... The widths of the resonance bands become very narrow as n ~ co. We find that the structural similarity between (9) and the Mathieu equation is a good predictor of the qualitative properties of (9). As we shall see, the growing solutions of (9) do occur in bands in parameter space and do appear to be recognizable as the two types of instability exhibited by the Mathieu equation. We shall consider the behavior of perturbations to the two types of periodic states, the tumbling and the nutating ellipses, separately, beginning with the tumbling vortices. The formal Floquet solution (e.g. Whitham, 1974) of (9) has the f o r m (As, Bs)=(as(t),

b s ( t ) ) e is'

where a fi t) = a fi t + To), bs( t ) = bs( t + To) and T O is the period of the basic state (A(t) = A(t + To), ¢ ( t ) = ¢ ( t + To)). The solution is either growing or neutral according to whether s is complex or real. s can be found as follows. Let A l(t ), A2(t ) be two independent solutions of (9) (Aj = (A j, Bj)) with

AI(0) = -

(~),A2(0 )_- (01 )" W h e n -

the basic state is periodic with period To,

A j ( t + To) are also solutions of (9), thus A l ( t + To) = a11A_1(t ) + a21A2(t ) A 2 ( t + T o ) = ax2Al(/) ÷ a22A2(t) Now, A f i t ) = ?h(t)Al(t) + XE(t)A2(t ) for some hi, ~k2, so h~Al(0) + X2A2(0) = As(0) = A s ( T o ) e -'st° = e-isTo[(~kaall + ~k2a12)A1(0 ) ÷ (~kla21 ÷ ~k2a22)A2(0)]

Thus a(xX;) = •'isr°¢x'' ~x21 where

)=IAI(TO) a, A(TO)1

VORTICES IN SHEAR

355

and the eigenvalue # = e ~sro is given by the quadratic (~ - Ax(T0))(/~ - B2 (T0)) - A 2 ( T o ) B I ( T o ) = 0 AI(T0), A2(T0) , BI(To) ~-ld B2(T0) call be found by numerically integrating (9) for one period of the basic state, i.e. along a periodic trajectory in the phase plane of the system (2b). Before discussing numerically obtained values of s for the tumbling ellipses, we wish to develop an analytical approximation to s in the limit of small strain. We begin by noting that when e = 0, the tumbling mode trajectories have constant ~ = ~0, say, and the bracketed quantities in (9) are constant. The Floquet problem shows that A ( t ) = A ( O ) e m and yields the dispersion relation s2

=T

(1+~0) 2

1

~1 - - ~ 0

(10)

The modes are exponentially unstable, s 2 < 0, if the aspect ratio is less than a certain threshold value that decreases with increasing mode number, n. This is just Love's result. There is also a family of curves in the neutral part of parameter space at which the frequency of the perturbation is an integer multiple of half the rotation frequency of the ellipse. These are given by S 2 = m 2 ~ 2, i.e.

(1 + X0) 2

1

/ 1 + X0

= m2

(1 + X0) 2 +--w

(11)

and are shown in Fig. 7, together with the stability boundary. Note that we have drawn the curves as continuous functions of n although in reality the azimuthal mode number is quantized. Now, taking 0 < l e l < < 1, we can develop a small e-expansion of the Floquet solution using multiple timescales. In doing so, we determine the location of 8o and the locations of the boundaries of the unstable b~knds around ~ = n~. The details of this procedure can be found in Appendix A and an example of a resonance band in a phase plane diagram is shown in Fig. 8. The results of both the weak strain theory and our numerical results for stronger strain fields are summarized in the sketch shown in Fig. 9. For a given choice of azimuthal mode number and zero strain, the boundary of the Love modes is located at X = A 0 whereas s = rnf~ at the points ~ = Am, m = 1, 2, ..., N. There are only a finite number of resonances, N say, between X = A o and ~ = 1. This number increases with the azimuthal mode number. As l e[ is increased, bands of instability open out from A,,,, I < m < N, while the boundary of the Love modes curves in slightly to smaller aspect ratios.

356

s.p. M E A C H A M ET AL.

RESONRNCE CURVES 2O

t8

t6

t4 m=

i0

12 9 8

z 10

7 6

8

5

6

4

4

2

3

1 -

2

Love mode

boundary 0

0

.I

.2

.3

.4 LRMBORO

.5

Fig. 7. The Love mode stability boundary, m = 1 . . . . . 10, when to = - 1 , z~ = 1 .

.6

.7

s = 0,

.8

and the resonance curves,

s = mfl,

The predictions of weak strain theory for the tumbling modes are compared with direct numerical solutions of the Floquet problem (9) in Table 1. The values in Table 1 were obtained for to = - 1 . 6 , ~ = 1 and n = 5. The agreement with the asymptotic results is good at small values of e. The instability of the tumbling modes follows the pattern suggested by the weak strain theory even at relatively strong values of the strain. Figure 10 shows points in (n, A) space at which the tumbling modes were found to be unstable for the case to = - 1 , ~ - 1, e = - 0 . 5 . The abscissa in Fig. 10 is ?~0, as defined in Appendix A, the mean aspect ratio. These results were obtained by solving (9) numerically at values of A that were integral multiples of 0.005. The resonance curves and Love mode boundary obtained by setting e = 0 have been superposed on the figure. Bands of instability in the neighborhood of the resonance curves can be clearly seen. There is a slight twist in the case of tumbling mode trajectories close to A = 1, the circular case. When 1 - h 0 = O ( e ) the series solution form of the periodic tumbling mode is no longer (A1). Instead of being proportional to a simple cosine, A(1) is now a more general elliptic function of time. One can obtain a Mathieu-like expansion of the Floquet problem that is similar to

VORTICES I N SHEAR

357

PHRSE PLRNE 1.0

I

,

i

,

I

I

'

I

,

I

l

I

'

I

,

I

'

I

'

I

'

]

i

I

'

I

l

I

,9

.8

.7

.6

F ~

.q

INZKD

.3 .2

.1 =

O0

I

.2

i

I

i

.4

I

,6

t

I

,a

I

I

,

I

I

I

I

[

I

[

I

I

IZ.I 2 i

1.0 t.2 1.4 1.6 1.8 Z.O

.I

=

~

I

I

,

I

6 2.8 3.0

PHI

Fig. 8. An example of a phase plane showing the region of unstable Love mode trajectories and the first resonance band.

E

LBVE MBOE5

M=O

fl=l

H=2

LflHBOflO

Fig. 9. Sketch showing the width and location of the bands of unstable trajectories as a function of the strain, e, for small strain.

358

s.P. MEACHAM ET AL.

TABLE 1 The locations of the edges of the domain of Love modes (0) and of the first (1 + and 1 - ) and second (2 + and 2 - ) resonant instability bands, as determined by direct numerical integration of the Floquet problem, when ~ = - 1 . 6 , U = 1, and e = 0.02. The differences between these values and the e = 0 values of h0 are shown, and compared with the predictions of weak strain theory e

0

1-

1+

2-

2+

0.02 0.0 Difference Theory

0.1612545 0.1613593 - 0.0001047 - 0.0001056

0.1945189 0.1970888 - 0.0025681 - 0.0025667

0.1996431 0.1970888 0.0025561 0.0025667

0.2526351 0.2527462 - 0.0001121 - 0.0001115

0.2530207 0.2527462 0.0002735 0.0002746

(A2) but with the cosine functions replaced by elliptic functions. Again it should be possible to obtain small e-expansions of the Floquet modes. The second type of periodic trajectory, the nutating solutions, also seem to have banded stability properties. Examples of this are shown in Fig. 11

RESONANCE CURVES 20

18

16

14

12

z I0

8

6

4

2

0

O

.05

.IO

.15

.20

.25 .30 LRI'IBORO

.35

.40

.'~5

.50

Fig. 10. Location of unstable tumbling modes in (n, ~) space. The points correspond to trajectories that were determined to be unstable by direct numerical integration of the Floquet problem. The solid curves denote the resonance curves and the boundary of the Love modes. The unstable trajectories are located in bands around the resonance curves.

359

V O R T I C E S IN S H E A R

PHASE PLANE 1.0

.9

.8

.7

.6

.5

.~t

.3

.2

.1 -1.0

-.8

-.6

-.4

-.2

0 PHI

.2

.4

.6

.8

l.O

Fig. 11. B a n d s o f u n s t a b l e n u t a t i n g t r a j e c t o r i e s for t h e c a s e to = - 1 ,

~ = 1, e = - 0 . 5 .

which shows bands of unstable trajectories on a phase plane and Fig. 12 which shows the instability bands in (X, n) space. Unlike the tumbling trajectories, the nutating trajectories do not have a useful mean aspect ratio. In Fig. 12 and below, the X0 that we associate with a particular nutating trajectory is the maximum value of X along the trajectory; it is thus not comparable with the X0 defined for the tumbling ellipses above. The stability of the nutating trajectories was determined by solving the Floquet problem numerically. There are situations when all of the nutating trajectories seem stable to linear perturbations with n > 3, e.g. when to = - 9 . 7 5 , ~ = 1, e = 2. The fact that when nutating trajectories do exhibit instability they do so in bands suggests that the instability can again be understood in terms of a resonance in the Floquet problem. The behavior of the frequencies of the azimuthal modes as X0 is increased supports this. As X0 is increased, both the rotational frequency of the vortex and the frequency of a given Floquet mode change. Instability seems to occur when the frequencies are near resonance. We offer Fig. 13 as a demonstration of this. When the Floquet exponent, s, is real, the phase of the associated Floquet mode, relative to the principal axes of the ellipse, increases by an amount s T o during a rotation period T0. In Fig. 13, we have plotted this phase change, modulo 2~r, as a

360

s.P. M E A C H A M ET AL.

RESONRNCE CURVES I

I

I

I

I

I

I

18

|

16

I |

14

| |

12

i |

z I0

• I

II

8 i

6 liB-

4

2

0

I

I

I

I

I

I

I

.1

.2

.3

.4

.5 LRMBDRO

.6

.7

I

I

.8

I

I

.9

I

1.0

Fig. 12. Locations of some of the unstable trajectories of Fig. 11 in (n, ~) space. The lower swath gives rise to the bands shown in Fig. 11. The upper swath gives rise to bands that are not shown in Fig. 11. These latter bands are thin and correspond to instabilities to higher modes (n > 10).

function of the aspect ratio of the basic ellipse, for azimuthal modes 5-10. When this phase change becomes zero, the corresponding m o d e is in resonance with the nutating ellipse and we see an exponential instability (compare the position of the bands of zero phase change in Fig. 13 with the location of growing Floquet modes in Fig. 12). Unfortunately, for the unperturbed, nutating vortex, ~ and ~, though periodic, are not simple functions of time as they were for the low-strain, tumbling modes. It is therefore difficult to approach this case analytically. 5. FINITE AMPLITUDE INSTABILITY The above discussion focused on the linearized stability properties of the periodic vortex solutions of (2). We have used a numerical implementation of the contour dynamics algorithm to examine the non-linear evolution of vortices that are perturbed forms of the solutions to (2). In doing so, we have several objectives, namely:

VORTICES IN SHEAR

361

PHRSE CURVES .5

-.5

-1.0

-1.5

-2.0

-2.5

-3.0

.60

.65

.70

.75 .80 LRIIBORO

.g5

.90

.95

Fig. 13. The phase of several Floquet modes, relative to the ellipse, at the end of a nu" period as a function of )% for o~= -1, ~ =1, e = -0.05. The numbers adjacent to the denote values of n. (1) to verify some of the predictions of linear theory, in pr existence of alternating bands of neutrality and instability, by initial perturbations to neutral and unstable ellipses. (2) to examine the form taken by the unstable modes larger amplitudes. We are interested in observing both w~ occurs and whether the growing perturbation disrupt, causes it to eject a small amount of vorticity. (3) to discover whether some of the 'neutral' stay plitude initial perturbations. To answer the above questions we have perfc vital statistics are catalogued in Table 2. Our initial conditions are given by X=~

1(

cos0 1+

A cos nO

(cos20 + (1/~) sin2e A cos nO

Y = )~ cos 0 (1 +

(2~ cos20 + sin2r

362

S,P. M E A C H A M ET AL.

TABLE 2 Parameter values used in numerical simulations Case

o~

e

a b c d e f g h i j k 1 m n o p q r s t u v

-

-

1.2 1.2 1.2 1.2 1.2 1.2 1.2 1 1 1 1 1 1 1 9.75 1 1 1 1 1 - 9.75 - 9.75

1/9 1/9 1/9 1/9 1//9 1//9 1//9 0.5 0.5 0.5 0.5 0.5 0.5 0.5 2 0 0 0 0 0 2 2

~

~'0

A

n

Figure

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 1 1

0.15 0.15 0.15 0.25 0.36 0.42 0.45 0.755 0.775 0.805 0.845 0.900 0.945 0.975 0.9091 0.2 0.16667 0.1 0.06667 0.1 0.3349 0.3349

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.005 0.005 0.005 0.005 0.005 0.005 0.005 0 0.02 0.02 0.01 0.00625 * 0.07 0.1

3 4 6 3 3 3 3 8 8 7 7 6 6 5 2 2 2 2 * 3 4

17 14 16a 15 16b 18 19e 19a 19b 20a 20b

*: for explanation, see text.

T h i s reduces to a m o d e n p e r t u r b a t i o n of the sort d i s c u s s e d i n section 4 in t h e l i n e a r l i m i t o f s m a l l A . S u c h a c h o i c e is u n s u i t a b l e f o r l a r g e - a m p l i t u d e i n i t i a l p e r t u r b a t i o n s as it r e s u l t s i n a m u l t i p l y c o n n e c t e d c u r v e w h e n A is larger t h a n a t h r e s h o l d w h i c h d e p e n d s o n ~ a n d n. C a s e s (d), (e), (f) a n d (g) a d d r e s s p o i n t (1) f o r a set o f t u m b l i n g s o l u t i o n s . E a c h of the four r u n s has the s a m e vortex s t r e n g t h a n d b a c k g r o u n d

field,

a n d e a c h is s t a r t e d w i t h a s m a l l a m p l i t u d e n = 3 p e r t u r b a t i o n s u p e r p o s e d o n t h e p e r i o d i c s o l u t i o n . T h e s t a b i l i t y d i a g r a m f o r t h e s e v a l u e s o f o~, ~ a n d e is s h o w n i n F i g . 8. T h e s h a d e d a r e a s o f F i g . 8 i n d i c a t e t h o s e t r a j e c t o r i e s t h a t a r e u n s t a b l e t o m o d e 3 p e r t u r b a t i o n s a c c o r d i n g t o l i n e a r t h e o r y . C a s e (d) ( F i g . 14) c o r r e s p o n d s t o a L o v e m o d e a n d is s t r o n g l y u n s t a b l e . T h e p e r t u r b a t i o n grows r a p i d l y a n d the vortex develops a tail n e a r o n e of the regions of high curvature. A l t h o u g h this tail c o n t a i n s a n o n - n e g l i g i b l e f r a c t i o n of the area of the vortex, the b u l k of the v o r t e x r e m a i n s b e h i n d as a c o h e r e n t b l o b . A s t h e e v o l u t i o n p r o g r e s s e s , t h e t a i l is s h e a r e d o u t b y t h e differential rotation between the vortex and the background field and begins t o w r a p a r o u n d t h e m a i n v o r t e x ( f o r t h e s e v a l u e s o f o~, ~ a n d e, t h e

VORTICES IN SHEAR

1.000000

-0.iiii11

363 -1.200000

3.000000

0.250000

l.O0000e-02

Fig. 14. Evolution of a perturbed vortex: Table 2, case (d). The later stages of the evolution of the tail are not accurately represented by this low resolution model. The interval between frames is t = 3.0.

streamlines of the background field are closed). This process cannot be followed very far with our current CD model; nevertheless, we believe that it can be thought of as an adjustment by which the initially unstable vortex loses a fraction of its vorticity, leaving behind a stable, more compact vortex surrounded at a distance by a band of vorticity produced by the wrapping around of the shed filament. We presume that a prerequisite for such a process to occur is the existence of a stable end-state of the sort described. We have not investigated this question further. The significant point here is that no major disruption of the initial vortex was observed. Case (IF) (Fig. 15) is an example of a mode 3 instability in the first resonance band. The growth rate predicted by linear theory for this mode is close to the maximum for this band. The evolution resembles that of case (d); the instability grows without equilibrating and a tail forms near one of the regions of high curvature. The main differences between (f) and (d) are

3

364 1.000000

s.P.M E A C H A M El"AL. -0.iiiiii

-1.200000

6.000000

0.420000

i. 0 0 0 0 0 e - 0 2

Fig. 15. As Fig. 14, but for case (f). The interval between frames is t = 6.0. that the growth of the instability is m u c h slower, in agreement with the prediction of linear theory that the resonant instabilities are weaker than the Love mode instability, and that the tail which develops contains a smaller fraction of the area than in case (d). This m a y be a consequence of the small width of the resonance band; only a small adjustment is required to take the vortex from an unstable trajectory to a stable trajectory. Cases (e) and (g) (Fig. 16) are examples of small m o d e 3 perturbations on solutions that should be neutral to such perturbations according to linear theory. There is no apparent growth of the instability over the relatively long (t = 200) duration of these runs. This goes some way towards confirming the banded grouping of unstable trajectories. Note that the unperturbed trajectory in case (e) lies between the Love modes and the first resonance band. Cases (h)-(n) demonstrate the b a n d e d arrangement of unstable bands of nutating trajectories when to = - 1, ~ = 1, e = - 0.5. The stability of various trajectories as determined by linear theory can be seen in Fig. 11. In this figure we have identified four unstable bands, in each of which trajectories are unstable to a different azimuthal mode. (Actually, each of these bands

3

VORTICES IN SHEAR

365

contains very thin sub-bands in which higher (n > 10) modes are also unstable. The growth rates of the high modes are very small and we will not consider them further here. Each of the trajectories used as the basic state in (h)-(n) is either neutral to all wavenumbers n < 16 or unstable to only one mode with n < 16 and then that mode is one of n = 5, 6, 7, 8.) The results (not shown) are similar to (d)-(g). For neutral trajectories and the specified wavenumbers we do not see any development of tails. In the case of the unstable trajectories we see the growth of the unstable perturbation and the eventual development of either one or two tails. As the perturbation grows, we can discern each peak of the growing unstable wave. However, not all of the peaks develop into tails. As the wave grows it propagates around the b o u n d a r y of the vortex and the crests pass successively through the regions of high curvature at the ends of the vortex. At the same time the vortex itself is nutating and the strength of the curvature at the ends of the vortex varies periodically with time. The generation of a tail seems to be a non-linear

1.000000

-0.iiiiii

-1.200000

8.000000

0.360000

l.O0000e-02

(a)

Fig. 16. As Fig. 14, but for (a) case (e), and (b) case (g). The interval between frames is t = 8.0 in both cases.

366 1.000000

s.P. M E A C H A M ET AL.

-0.Iiiiii

-1.200000

8.000000

0.450000

1.00000e-02

(b}

Fig. 16 (continued).

interaction between a single crest of the perturbation wave and the region of high curvature at the end of the vortex. There seem to be at least two requirements for tail formation: the perturbation peak must have reached a sufficient amplitude and it must approach the end of the vortex at a time when the curvature is sufficiently high, i.e. at an approriate phase of the nutation of the fundamental vortex. The interaction of a large-amplitude perturbation peak and the high curvature region of the nutating vortex produces a localized region of very high curvature. This can be expected to accelerate the rate at which the perturbation will break. When the crest is small, or the curvature at the end of the vortex is small, the peak propagates through the region of high curvature before the accelerated non-linear steepening has had much of an effect. When the two are sufficiently large, the crest steepens and breaks, forming a tail, before it has a chance to escape the high curvature region. Polvani (1988) has studied this effect for perturbations to Kirchhoff ellipses by locating the stagnation points of the flow in a reference frame rotating at the same rate as the ellipse. When the ellipse is

3

VORTICES IN SHEAR

367

acute, these lie close to the ends of the ellipse. The crests of unstable waves are distorted by the flow in the neighborhood of the stagnation points as the former propagate around the end of the ellipse. Eventually the crest wraps around the stagnation point and a tail is formed. When the mode number is even, each time a crest arrives at one end of the vortex a similar crest arrives at the other end of the vortex and so tail formation occurs simultaneously at either end. W h e n the mode n u m b e r is odd one tail forms at one end of the vortex. The formation of a tail alters the rate at which the remaining peaks propagate around the vortex; it is possible that, if we continued our numerical integrations further, we would see a second tail forming at the second end of the vortex in the case of moderately large, odd wavenumber, unstable perturbations (this does not seem to happen for wavenumber 3). Thus far we have seen some confirmation of the predictions of linear theory and we have gained some qualitative insight into the non-linear evolution of some of the unstable modes. A better understanding of the process of tail formation (including some tests of our rather speculative remarks) would be desirable. One of the points that should be pursued in more detail is the relation of the tail formation processes described here and the fairly pernicious instability observed by Dritschel (1988) in which a classically stable rotating vortex (for example a stable Kirchhoff vortex) throws off very small filaments of vorticity. We will not attempt to develop these points further in this paper. In the results described so far we have observed neither the equilibration of an unstable wave at moderate amplitude, nor the radical breakup of an isolated vortex. N o n e of our subsequent results appear the demonstrate equilibration. To investigate further the question of breakup, we examined a Love mode on an ellipse with a larger aspect ratio ~0 = 0.15. We perturbed this with modes 3, 4 and 6 (cases (a), (b) and (c)), all of which are linearly unstable. Increasing the mode n u m b e r ((b) and (c)) reduces the area of the tails. Comparing (a) (Fig. 17) with (d) (Fig. 14), we see that decreasing the aspect ratio of the unstable vortex increases the a m o u n t of vorticity which ends up in the tail; nevertheless, a substantial a m o u n t of vorticity remains behind as a readily identifiable vortex. One circumstance in which the vortex is certainly disrupted occurs when the vortex lies on a trajectory that ends in an attractor. The vortex rapidly becomes sheared out as a result of the background field rather than any instability (see case (o), Fig. 18). Cases (q-t) (q and t are shown in Fig. 19a,b) differ from the results so far in that they exhibit a dramatic breakup of the original vortex. Cases (q-s) are all examples of Kirchhoff vortices (e = 0 = 5 ) initially perturbed with an m = 2 mode. The linear m = 2 mode is neutral in the sense discussed

368 1.000000

S.P.M E A C H A M

-0.iiiiii

-1.200000

2.000000

0.150000

ET AL.

1 .0 0 0 0 0 e - 0 2

Fig. 17. As Fig. 14, but for case (a). The interval between frames is t = 2.0.

previously. However, all of the higher, even, azimuthal modes are included when one makes a finite amplitude m = 2 perturbation. T h e y are not, however, free modes, their phases being linked to that of the m = 2 m o d e by non-linearity. In these examples the vortex breaks up into two similar fragments that contain the bulk of the original vorticity. The remaining vorticity is stretched out and wrapped around the principal fragments in thin filaments. Very elongated initial vortices can break up into m o r e than two fragments. This type of behavior can be characterized as vortex fission. We do not fully understand the underlying mechanism but believe it to be a non-linear phenomenon. Perturbations with single, unstable, azimuthal modes, instead of the m = 2 mode lead to the usual tail production, instead of fission. Case (q), an ellipse with A = g1 is particularly illustrative of this. The only two linearly unstable azimuthal modes are n = 3 and n = 4. Our initial perturbation (m = 2) is symmetric and, from the symmetry of the subsequent evolution, we conclude that no n = 3 contribution of any significant size is

3

369

VORTICES IN SHEAR

1.000000

2.000000

-9.750000

1)o(I

J

0.300000

i.i00000

O.

O

3

O

j

Fig. 18. As Fig. 14, but for case (o). The interval between frames is t = 0.3.

present during the period shown in Fig. 19a. A n ellipse of similar aspect ratio perturbed with the only other linearly unstable mode, n = 4, shows a quite different evolution: a tail forms at either end of the ellipse. We deduce that, in the observed evolution resulting from the m = 2 perturbation, non-linear interactions between a n u m b e r of even azimuthal modes m u s t be playing a significant role. We note that w h e n e = 0, the m = 2 m o d e has zero phase velocity in a frame rotating with the ellipse (see eqn. 10). We have not attempted to develop a weakly non-linear theory for the evolution of such disturbances, though this would probably shed some light on the problem. It appears that vortex fission can only occur for elongated ellipses; case (p) (Fig. 19c), in which a X = 0.2 Kirchhoff ellipse is perturbed by an m = 2 mode, appears to show some narrowing at the middle of the ellipse b u t this does not advance towards fission. Even small-amplitude noise will kick off the fission process for an elongated ellipse. Case (t) demonstrates this. The initial conditions consist of a X = 0.1 Kirchhoff ellipse without any formal perturbation. In practice, small-amplitude noise will be introduced by b o t h the discretization error of

370

s.P. MEACHAM ET AL.

the numerical model and time-stepping errors as the model evolves. This noise is presumably of a fairly broad spectrum. We have experimented with a variety of small, finite-amplitude perturbations containing several, low wavenumber (2-8) azimuthal modes with

.oooooo.

.?oo o.

,.ooooo_o,

(a)

1. 0 0 0 0 0 e - 0 5

0.

1.000000

5. 0 0 0 0 0 0

0. :1.00000

0.

2

(b)

/

/

t

\

Fig. 19. As Fig. 14, but for (a) case (q), (b) ease (t), and (c) case (p). The interval between frames is t = 5.0. The model has ceased to be accurate in the last few frames of some of these plots when the contour extension has become very large.

371

VORTICES IN SHEAR

i. O0000e-05

O.

1.000000

5.000000

0.200000

2.00000e-02

2

(¢)

Q_

_3

Fig. 19 (continued).

randomly distributed amplitudes and phases. The subsequent evolution of the perturbed Kirchhoff ellipses can be quite complicated. When the initial ellipse is sufficiently elongated (roughly, h < 0.2), it may break into several blobs, throw off filaments or cast off one or two small blobs, connected by filaments to a less-elongated central core. For initial ellipses that are more compact, only tail formation seems to occur. The onset of the more dramatic behavior seems to require that the initial ellipse be somewhat more elongated than the linear stability threshold of h = 0.33. Our last set of numerical examples are cases (u)-(v), (Fig. 20) in which we have perturbed nutating trajectories that lie close to the boundary between nutating and non-periodic (shearing) solutions. The basic state nutating trajectory is stable according to linear theory. The purpose of this is to see if any finite-amplitude instability is evident. One might expect such an instability, given the proximity, in phase space, of the rather different types of trajectories.

S.P. MEACHAM ET AL.

372 1.000000 (o)

2.000000

-9.750000

1.700000

0.334900

6. O0000e-02

II 1.000000 (b)

2.000000

-9.750000

0.200000

0.334900

0.100000

4

Fig. 20. As Fig. 14, but for cases (u)-(v). T h e intervals between frames are t = 1.7 a n d t = 0.2 respectively.

In neither of our attempted cases does the addition of a perturbation change the character of the evolution by causing the ellipse to be sheared out. In both of these cases, the angular frequencies of none of the n > 3 azimuthal modes are particularly close to zero. However, one finite-amphtude effect is observed after a sufficient amount of time: tails form at the end of the vortex (1 for the n = 3 m o d e and 2 for the n = 4 mode). These

3

VORTICES IN SHEAR

373

tails contain very little area, rather less than those tails produced by unstable modes. This phenomenon seems consistent with the mechanism of tail production described above. Repeated non-linear steepening of a crest of the perturbation wave as it passes the end of the ellipse forms the tail but as the amplitude of the neutral perturbation is not being fed by linear instability, the amount of area in the tail produced is very small. 6. CONCLUDING REMARKS Developing the work of Moore and Saffman (1975) and Kida (1981), we have analysed in some detail the possible evolution of an initially elliptical vortex in a simple background flow. The existence of both non-periodic and neutral, periodic solutions to the governing equations demonstrates that vortices may survive a background shear for a significant length of time or be torn apart quite quickly according to the values of the vorticity of the vortex and the parameters of the background shear. Such background flows may occur when a vortex approaches a strong oceanic jet or another mesoscale eddy, or may represent the slow circulation of a gyre interior. Estimates of vorticity and strain parameters for an oceanic example are given in Appendix C together with some justification for the neglect of dissipation and Rossby wave radiation. An important extension of this is the study of the stability of the periodic trajectories. The acute trajectories are unstable to Love modes as one might expect from the properties of Kirchhoff vortices in the absence of any external flow. However the existence of bands of trajectories that are unstable to resonant instabilities represents a new mechanism for the evolution of the structure of a vortex under the influence of a background shear flow. Given the results of the numerical simulations, it seems likely that resonant instabilities will cause the vortex to shed small amounts of vorticity as filaments and relax to a more stable state. If the background field is slowly varying (e.g. when the vortex slowly drifts through a background flow varying on a large length scale) then repeated filamentation may occur, causing the vortex to decay slowly. We do not yet fully understand the non-linear aspects of the process of tail formation. A more detailed investigation of this would probably be worthwhile. Vortex fission, although possible under some circumstances, appears to require a fairly special set of initial conditions. Tail formation appears to be the preferred end result of vortex instability, even for the Love modes. We must emphasize, however, that we have not explored this point thoroughly; our conclusion is based more on our not observing much vortex fission (when excited with single mode, rn > 3, perturbations) rather than a clear understanding of when vortex fission can occur.

374

S.P. M E A C H A M

ET AL.

In a region where the streamlines of the background flow are closed (e.g. a recirculation region near a strong, meandering jet), the filaments of vorticity shed by an unstable vortex do not appear to escape but instead are wrapped around the vortex at some distance from it. When supplemented by weak diffusion this may lead naturally to a vortex structure that consists of concentric bands of vorticity of varying magnitudes and widths. Our study is limited in a number of ways. The analytical results are made possible by the assumption of a quadratic background streamfunction (the property that an initial ellipse remains an ellipse depends u p o n this). When the structure of the background flow is modified to include non-quadratic terms there are two possible cases to be considered: (1) uniform and (2) non-uniform background vorticity. In case (1), qualitatively similar streamfunction patterns may lead to qualitatively similar vortex evolution; however, 'tumbling' and 'nutating' modes are likely to correspond to nearly periodic rather than periodic contours. It remains to be seen how this affects the tendency of the vortices to filament. In case (2) the background vorticity gradient will probably cause a vortex with non-zero, net, angular m o m e n t u m to drift and radiate Rossby waves. It may be possible to treat the problem of a weak background gradient using perturbation techniques provided that the timescales associated with the rotation of a vortex are O(1). We have also excluded baroclinic processes and horizontally continuous vorticity fields. The symmetrization processes noted by Melander et al. (1986) may affect the stability of continuous analogues of the nutating and tumbling discrete vortices that we have found. In summary, the most probable effect of an ambient flow on a uniform elliptical vortex depends on both the ellipticity of the vortex and the ratio of the background vorticity to the background strain rate. If the magnitude of the strain exceeds that of the background vorticity, so that the background streamlines are locally hyperbolic, then it is likely that the vortex will be sheared out, although periodic equilibria are possible when the vorticity anomaly associated with the vortex is sufficiently strong. The stronger the vortex, the larger is the area of the phase plane that contains closed trajectories, and hence the more likely is a periodically vacillating vortex. If the magnitude of the background strain is less than that of the background vorticity, then the streamlines of the background flow are locally ell~tical and, in general, the aspect ratio of the vortex will vacillate. In such a case, if the 'average' aspect ratio of the vortex is sufficiently acute that it is unstable to Love modes, rapid instability will occur and substantial amounts of vorticity may be shed as tails. If the average aspect ratio is less acute so that no Love mode instability occurs, the vortex may lose relatively small amounts of vorticity as a result of the generation of small tails by either resonant instability or non-linear effects. Both of these latter processes are

375

VORTICES IN SHEAR

slow compared with the rotation rate of the vortex and p r o d u c e only relatively small changes in the net vorticity of the vortex. If the average aspect ratio is very acute, a combination of fission and tail production are likely to occur, producing a rapid and substantial change in the vortex; however, a significant (and more stable) vortex core m a y remain. APPENDIX A Taking 0 < ] e [ < < 1, we develop a small e-expansion of the Floquet solution of (9) using multiple timescales. The first step is to develop a series solution form for the basic state. Using (2) one can show that ?t _ ?t(o) + e?to) + e2?t(2) + _ ~(o) + e#)O) + e2(])(2) +

... ...

?t(0) = ?to,

constant

#¢o) = ~t,

~2 = ~o + e2q2,

~0?t0 f~°= (1 + Xo) 2 + 2-

?to ?to) = _ _ cos 212t, 2f~

P~ ~(1) = _ _ sin 2~2t 2f~

12 ?t(2) = _ _ cos 4~2t, 4f~

P2 ~(2) = _ _ sin 4 ~ t 4~

(A1)

The constants Pl, P2, q2, 12 a r e given in Appendix B. Using the above forms for ?t and ~ we write (9) in the form A, + [Go + eG, cos 2f~t + e 2 ( K 2 + G 2 cos 4f~/)] B = 0

(A2)

B , - [ U 0 + ell, cos 2 a t + e2(L2 + H 2 cos 4 a t ) ] A = 0 Gj, Hi, K z, and L 2 are constants, the details of which are included in Appendix B. W e will locate the boundaries of the region of unstable Love modes and of the first two resonance bands. We expand A and B in powers of e A = A (°) + eA 0) + e2A (2) + ... B = B (°) + eB (1) +

e 2 B (2) + . . .

and introduce slow timescales r = et, T = e2t so that 0t ~

0 t -~

eO, + e20r

The O(1) part of (A2) is A~°) + Go B(°) = 0 B, ~°) - Ho A(°) = 0

376

S.P. M E A C H A M E T AL.

with normal modes

A(°)=Ao(,r, T)ei~t + ,

(A3)

is B(O)= _ - - A o ( ~ - ' T)eiSt+ , Go Love modes

The Love m o d e b o u n d a r y is close to s = 0. This condition defines a critical 2~0 (see eqn. 10) which we will denote ~00. This satisfies Ho(Xoo ) = 0. We consider ~o values close to this so that Xo = ~oo + e26 Then no(Xoo + e 2 8 ) = e 2 6 1 _ Go(Xoo + e=8) = Go(Xoo ) + e281+ I _ and I+ are included in A p p e n d i x B. The O(1) part of (A2) becomes 0tA (°) = 0 = A (°) = Ao(~) At O(e), 0t B(1) =AoHl(hOO)cos 2fit so that

AoH1 B (1) = - -

212

sin 2f~t + Bl('r )

while

AoGoHt 2----if-- sin 2f~t

OtAO) = - G ° B 1 - ~ A °

Secularity requires that ~,A o + GoB 1 = 0, thus (I GoH1 A ) = Ao-~-~ - cos 2 ~ t

At the next order 0t B ( 2 ) =

H 1 cos 2~tA (1) + ( 6 I _ + L: + H 2 cos 4f]t)A (°) - O~B(1)

= A o L2+8I_+

+ ~ o A o , , + Ao H 2

~

The secularity condition at this order is

Ao,,+Go

(

L2+8I_+

8f~-------~

t

Ao=0

8a 2

cos4~t

377

VORTICES IN SHEAR

W h e n Go(L 2 + 81_+ Go(H(/8~2)) < 0, we have the possibility of an unstable solution in which A 0 grows on the slow O(e) timescale associated with "r. Thus, if 8 < I L21 - I G0 I(H~/8~2)/I I_ I, the tumbling trajectory with h o = ? % o + e 2 8 is unstable to non-elliptical perturbations. The stability b o u n d a r y for the Love modes is then ho

=

X00(n)+e2[ I L z l - I a 0 l ( n 2 / 8 ~ II_1

l

2)

+ ...

The trajectory of the form (A1) with h ¢°) = ?% as given above forms the boundary of the range of trajectories unstable to the Love instability. A n example can be seen in Fig. 8.

First resonance band We assume that s lies close to f~o and define a parameter o by s = f~o + eo. o is then a measure of how close the parametric forcing is to being resonant with the natural frequency, s, of the system. Both G o and H o are O(1) and the leading-order solution is given by (A3). At O(e) 0tAra) + Go B~I) = - ~ A ~°) - (G 1 cos 2f~t)B (°)

OtB~I)- Ho A°) = - 0 , B (°) + ( H a cos 2 ~ t ) A ~°) whence (~2 + s2)A(1)

°°+s'2°+s'GllIA°ei II 2°+S+ o .ll lI2,sAo 11

+ ~HIG° +

s(2~2G0 + s)G1]A'~e-i2°*+i2qzTe ist ..~ , )

Secularity requires that

2isA°* + ½HAG°+

s)G1]A~e_i2o.+i2q2T= 0

s(2f~ + 2Go

(A4)

while

AO) = Rl[ Aoei~Zn+s)t + *]

1(

B<')=Sl[iAoei
.1 +

2G 0LiA~ ei<2n-s)'+*]

The constants R 1 and $1 can be found in A p p e n d i x B.

)

378

S.l,. M E A C H A M E T AL.

At next order 3tA (2) + Go B(2) = - 0 ~ A O) - (G l cos 2 ~ t ) B (1) - 0TA ~°) B(°)(K2 + G2 cos 4 ~ t )

-

OtB(2) -- Ho A(2) = - 3 , B (a) + ( H 1 cos 2f~t)A (1) - 3 r B (°) + A(°)(L2 + H 2 cos 4fit) which can be reduced to (32 + s2)AO) = -

/[( ei't A°

-( e'"[Ao.

Go

2 S'G1 -

4G----~o + isA°r

+*

)

+ (L2 + ~M1R,)GoAo + iSAoT] + * }

This yields a secularity condition of the form 2isAor + PA o + Ao, ¢ = 0

(A5)

where the constant P is defined in Appendix B. We rewrite (A4) as 2isAo, + QA~ e - 2i°" +2iqzT = 0 and we use this to substitute for Ao¢, in (A5), obtaining 2isAor + ( P + Q Z / 4 s 2 ) A o + Q°A~e-2iO'e2iq2r= 0

(A6)

S

We now combine (A4) with (A6) and return to t as the independent variable: 2is3,A o + e Q + e s

A~e-2~e(-eq2)t + e2 P + ~

A° = 0

We look for a solution of the form Ao

=

Doe-ie(o-eq2) t

where D o satisfies 2is3,Do + 2 s e ( o - eq2)D o + e Q + e

s

Do* + e 2 P + - Do = 0 4s z

(A7)

Expressing D o as D r + iD i, taking real and imaginary parts of (A7), and then eliminating D i between them results in the relation

[[ (°

4s23~ + e 2 2so + e P + 4s 2

2sq2

I;(°°;]) -

Q + e

S

Dr=0

379

VORTICES IN SHEAR

This finally affords a condition for the instability of the near resonant trajectories, namely that

2so+e

P+

4S 2

2sq2

--

Q+e°Qs

<0

W e note that

Q = ~1(H1Go

osGa HoGa) + e

Go

o

+ O( e21 = Q o + e-HoG, + O ( e 2 ) S

After some manipulation we find that when -

~s


-e

~

-

s

2Qo

+e

-q2+--+--

2Q °

2s

8s 3

+q2

2s

+O(e

2)

8s 3

+O(e2)

the tumbling trajectories are unstable. This corresponds to a band of X 0 values of width O ( e ) around the first resonance curve. If we define 3 by X o = Xoo + e3 where Xoo is the location of the resonance at e = 0, then

=o/

Ox

aX + ° ( e )

An example of such a band can be seen in Fig. 8.

Second resonance band We now assume that s lies close to 2f~ 0 and redefine o so that s = 2f~ o + e2o, noting that the second resonance b a n d will have width O ( e 2) rather than the O ( e ) scale appropriate to the first resonance band. It will not be necessary to consider the O ( e - 1 ) timescale, 3~. The leading-order solution is again given b y (A3), while at O ( e )

OtA(1) + GoB(a) = (G 1 cos 2 f l t ) B (°)

- 3 , B (a) - H o A(,) = ( H i cos 2 ~ t ) A (°) so that

(3tZ+s2)A
380

S.P. M E A C H A M E T AL.

Because s = 2f~ + O(e 2) the forcing is non-secular and we obtain A (1)=

(hoRlei(2~+s) t+ , } + (AoR2 eiAT+ , }

O `1, = {iAoSaei(2"+s' t + , } + { iAoS2e `aT + , } where A = o - 2q2. The coefficients Rj and Sj can be found in Appendix B. At next order

OtA(2) + Go B(z) --- - ( ) T A(O) -- (Ga cos 2f~t)B (a) - B(°)(K2 + G2 cos 4fat)

OrB(2) - Ho A(2) = - 3rB (°~ + ( H 1 cos 2f~t)A (a) + A(°)(L2 + H 2 cos 4~2t) which can be combined to yield ( # + s2)A(2) = -2(isAoreiSt+ *)

+ {[ 2 G , S 1 - ½H1GoR, - ( HoKz + GoL2)]Ao era+ *} + ([}SalS2_ + ([lU0a 2

112- I--II) ] A ,~i(2~t+AT) $,a0-.1...2] --a0~ + *} 1/2'- /-4" ]A*oi(4~+s)t

-- $ , , a 0 . . 2 1

~a0~

-'~ , }

+ non-resonant terms Requiring a non-secular .4 (2) furnishes the condition

iAor + MA~e -2jar + NA o = 0 where M and N are constants given in Appendix B. This is similar in form to (A6) and under the transformation

A o = (D r + i D i ) e -iAT becomes

DrTT+[(N+A)2-M2IDr=O. This admits exponentially growing solutions when M2>(N+A)

2

so that the tumbling m o d e trajectories in the band

2q2-N-

[M[ < o < 2 q 2 - N +

[M[

are unstable. If we define 3 by h 0 = )k0o + e23

where ~oo is the location of the resonance at e = O, then -

+ O(e/

VORTICESIN SHEAR

381

When comparing these theoretical results with the numerical results we need to take the higher-order corrections to X and q~ into account. Thus a trajectory that passes through X = X,,, ff = - ~ ' / 2 corresponds to )k°-----~"n- e2{ 4fa 12

X°Pl}2~ 2

The widths of higher resonance bands can be calculated in a similar manner. APPENDIX B Complicated constants occurring in Appendix A. Oaho(1 - ho) Pl

2f](1 + Xo) 3

+

1 1 + X2 2 1 - X2

I+X 2 1 [ 4aX:0 P2=8-~ ( 1 _ X2) 2 + 2f~p I 1 ~ o

4aX2o

1 q2 = - ~

(1 _ X2) 2

2-X o 1-X o ] *°X2(1 + Xo) 4 + 2f~°~/2 (1 + X0) 3

1 +X~o 2aPl 1 ~

~o

2n o (1+~oo) 2

1+

l+h

o/ ]

(.or I ) Ga =

ngl

~o [

2

2nh o

H°=2 -[(l+ho) 2

/41=

OOU1 )

,g, + --~-

[ 1 - ho 1-~l+ho

]

4!

(1 + Xo) ]

/2 = ~-~(1 + 2p,)

Go=

2 2-X°

("0~ 0 . . . .

o

n]

382

S.P. MEACHAM ET AL t.dV2 )

92 = L2=

gl =

rig2

2

(nf2 ou2 ) ,&o(1 -~'o)

~o

2~2(1 + Ao)'

1 - ~2o

1

1 - Ao

g2 =

2~°~lz (1 + ~.o) 3 1 + )~20 - 2 a ~ k ° (1 _ ~2)2

1~

~,o

[2 = -~-~

ap~ 17X2 °

nXo V1 =

1

4~2Pl - -

]

°~k20 (1 + ~k0)4 J

1 + )~2o 2fD~° (1 - ~.2o)2

--~.01 n-1

g(1 + Xo) 2 1 +)~o}

1

( 1 - A o ) "-2

v2 = 4f~2(1 + Ao)41 1 + 2~o

[nX2o( n - ~ - o ) - 2~n12(1 - X2o)]

nX2o(n - Xo) ( 1 - Xo I "-2 u2-- 4~2( 1 + ~'o) 4 / 1 + Xo ] Love modes

I_

2-A o (1 + ~o) 4

(1

+

~kO0)2 1 + 2~00 1 +

1_oo[

I+= (l+Aoo) 2 I+A~

1-

1 + )'00

l+h-------~

First resonance band

1 [~ s(2f~ + s)G~ ] R1 = 4~(f~ +s) 2H1G°+ 2Go

VORTICES IN SHEAR

1 ( sG 1 Sl = ~

383

2~2+s

[

4f~ ( f~ + s ) ½H1G°

2Go

+

s(2f~ + s)G 1

P = GoLz + HoK 2 + ½-(GoRaH1 - sS, G,)

2G0

4Go

Second resonance band 1 R, = 24~2o2(GoH~ + 2HoGa) R2=

1/4, 2Ho

S, =

6aoGo(GoHl-2HoG1)

$2 =

sG1 2G~

1

1

M = s -4 Ho

N=-~

--~

GO + 8

11o

Go

G~

H2o

+ G---~+ ~o

APPENDIX C: DIMENSIONAL ESTIMATES In this appendix, we estimate some of the non-dimensional parameters used in our theoretical treatment from dimensional data for an oceanographic example. We also make crude estimates of the dissipative and radiative spin down timescales in order to demonstrate that these effects should be small. We shall compare these timescales with the rotation time, which also characterizes the timescale for the stronger instabilities. As an example of an oceanic vortex let us consider a Meddy, a lense-like, mid-thermocline vortex of the type observed by McDowell and Rossby (1978) and Armi et al. (1988). This is a three-dimensional feature and so our theory can be only a rough indication of the behavior of such entities.

Estimate of e, w and Though Meddies can vary in size and strength, a representative structure seems to be a core in solid body rotation f~ out to a m a x i m u m azimuthal

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velocity V at a radius R and a decaying azimuthal velocity field beyond this. The eddy is confined vertically in a band of fluid of depth H. We will taken V - 20 cm s -a and R - 30 km. Then f ~ - 2 × 10-5 s-1 and the rotation period of the core - 9 × 105 s - 10 days. The relative vorticity of the core is w = 2f~ - 4 X 10 -5 S - 1 ( - f/10). To estimate the strength of ambient strain and vorticity fields we note that a simple shear contains equal amounts of both strain and vorticity. We will therefore just estimate the ambient strain and vorticity by the magnitude of typical velocity shears. In the recirculation region close to the Gulf Stream, velocity shears may be as high as 10 cm s -1 in 10 km, whereas in the interior of the subtropical gyre they are likely to be smaller than this. We therefore estimate e and ~ by O(10 -5 s -1) - O(~0). Typical values for the non-dimensional parameter r might then be O(1) - O(10), whereas p should lie in the range 0-O(10). In our discussion, we have neglected dissipative effects and losses due to the radiation of Rossby waves associated with the planetary vorticity gradient. We now give crude but conservative estimates of the timescales associated with the latter two effects, in order to demonstrate that they are much longer than the rotation period of a typical eddy.

Dissipation Direct observations of the effects of mixing on a Meddy (Ruddick and Hebert, 1988; Armi et al., 1988) suggest that lateral thermohaline intrusions took approximately one year to penetrate the core of this particular Meddy. This mixing reduced the strength of the vortex but a substantial velocity field remains. Further mixing occurred over the following year through the mechanism of salt-fingering at the base of the eddy. It seems safe to conclude that dissipative effects are small on the timescale of the rotation period of the Meddy (O(10 days)).

Rossby wave radiation The following estimate of the radiative timescale uses little more than dimensional analysis. We assume that Rossby waves are supported by the planetary vorticity gradient (propagation could also be thought of as taking place along the geostrophic contours of the mean basin-wide circulation). The velocity field of the eddy extends beyond the core radius used above. We take 50 km as the characteristic peak-to-trough scale (half-wavelength) and assume an amplitude of 7 / - 30 k m for the particle displacements associated with the Rossby wave source. The meridional wavelength of the

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waves produced from a localized source should be comparable to the horizontal dimension of the source so we use a meridional wavenumber 1 - 2~r/(100 km) - 6 × 10 -5 m -1 for the R o s s b y wave. In the absence of a mean flow, the m a x i m u m frequency and group velocity for such a wave occur when the zonal and meridional wavenumbers are comparable; then the wave frequency v - B / l - -~ × 10 - 6 S - 1 . The associated minimum wave period is T ~ 200 days. This brings out an important point: compared with planetary scale phenomena, the vortex is small and fast. The coupling between the circulation within the vortex and planetary wave processes is very weak. The group velocity - f l / l 2 - 5 x 10 -3 m S-1. We can estimate the kinetic energy density of the wave as E - p(va~) 2 - 10 -1 kg s -2 m -1, i.e. 10-1 j m - 3 close to the vortex (this corresponds to velocities associated with the R o s s b y wave of O(1 cm s-1). An estimate of wave energy flux density is then cgE - 5 X 10 - 4 J s - 1 m - 2 . If we assume that this flux crosses a surface defined b y an arc rr × 100 km long around half of the vortex and 1 km deep (our aim is to overestimate the wave energy radiated), i.e. a total area of ~ r × 1 0 s m 2 then the total radiated energy flux - 1 . 5 × 1 0 s J s -1. We underestimate the total kinetic energy of the eddy b y integrating the kinetic energy density of the core out to the radius of maximum velocity R (neglecting the tail of the velocity field b e y o n d this) and over a conservative estimate of the depth of the eddy, H - 330 m. In doing so, we obtain a figure of 1013 J. Combining this with our estimate of the radiative flux, we obtain a conservative estimate of the spin-down timescale as - 2 yr. Because of the w a y in which we have approximated quantities the true spin-down time associated with R o s s b y wave radiation is likely to be considerably longer than this. We feel that our estimate serves to show that energy losses due to R o s s b y wave radiation are small on time scales comparable to the rotation period of the eddy. A second type of oceanic vortex about which we would like to make inferences using the theory developed in this paper are Gulf Stream rings. These have larger lateral dimensions than Meddies, tend to be less confined in the vertical and possess stronger circulations. Some are known to have weak circulations near the seafloor suggesting that E k m a n dissipation m a y play a role. We estimate that dissipative timescales are likely to be comparable to those observed for Meddies, a point borne out b y the length of time over which some individuals have been observed ( 2 - 3 yr). Because of their larger diameter, they are likely to radiate R o s s b y wave energy more effectively than Meddies; however, they have a much larger reserve of kinetic energy than Meddies and, once again, model calculations (c.f., Flierl, 1977; McWilliams and Flied, 1979) as well as the long, observed lifetimes of those Rings which manage to avoid early reabsorbtion b y the Gulf Stream suggest that radiative processes are negligible on the timescale of the rotation period.

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