Steady two-layer vortices on the beta-plane

\

ELSEVIER

Dynamics of Atmospheres and Oceans 25 (1996) 67-86

Steady two-layer vortices on the beta-plane Alireza Pakyari, J.

Nycander *

Department of Technology, Uppsala Unioersity, Box 534, 751 21 Uppsala, Sweden

Received 5 April 1995; revised 1 November 1995; accepted 2 January 1996

Abstract

Monopolar vortices are studied in the framework of the quasigeostrophic two-layer equations. Steady, eastward-travelling vortex solutions with a baroclinic vertical structure and closed streamlines in both layers are obtained by perturbation analysis. The radial profile is essentially arbitrary in the region of closed streamlines. Westward-travelling vortices with a similar structure radiate barotropic waves, and are therefore unsteady. The propagation velocity of the vortices is essentially determined by the 'hetonic' mechanism, i.e. by the horizontal shift between the upperand lower-layer vortices. Their dynamics are studied qualitatively, and it is concluded that eastward-travelling vortices are stable, whereas those travelling westward are unstable.

I. Introduction

Many long-lived vortices of different kinds (e.g. intrathermocline anticyclones, meddies and Gulf Stream Rings) have been observed in the oceans, with lifetimes ranging from several months to several years. As this is much longer than the dispersion time in linear theory, much effort has been devoted to finding nonlinear steady solutions that describe such vortices. In many of these studies a reduced-gravity one-layer model (also called the equivalent barotropic model) is used. This means that the barotropic mode (in which the motion is essentially independent of the depth) is neglected. The motivation is that the ratio H u / H L between the depth of the upper and lower layers is small in the ocean, and that the motion is essentially confined to the upper layer. The existence condition for steady and localized vortices in such one-layer models is that they should propagate either westward faster than the Rossby waves (i.e. the first baroclinic mode), or eastward (Flied, 1987; Nycander and Sutyrin, 1992; Nycander, 1994). Otherwise, i.e. if they

* Corresponding author. 0377-0265/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved. PII S0377-0265(96)00475-7

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A. Pakyarl. J. Nvcander / Dynam~'.~ ~/ Atmo~phere.~ and Ocean.s 25 t 1996) 67-86

propagate westward with a velocity within the range of phase velocities of the Rossby waves, their energy will gradually be depleted by wave radiation. Within the framework of the quasigeostrophic approximation, this existence condition can only be satisfied by dipole vortices, and by monopole vortices with zero total mass anomaly, in which the azimuthal flow changes direction at some radius. However, the latter are strongly unstable (Swenson, 1987). The situation is changed if one uses a more general model that allows for a large displacement of the thermocline, for instance the shallow water equations, or one of the many intermediate models. The centre-of-mass velocity then acquires an extra nonlinear contribution, so that anticyclones propagate westward faster than the fastest baroclinic waves (Nof, 1983; Cushman-Roisin et al., 1990). The flow field is then evanescent outside the separatrix, i.e. it decreases exponentially away from the vortex. Such anticyclones can be steady (Nycander and Sutyrin, 1992). Cyclones, on the other hand, propagate slower than the fastest baroclinic waves, and therefore radiate such waves. The most serious weakness of these theories is that they neglect the lower layer, and hence the barotropic waves. The validity of this approximation has been examined by Chassignet and Cushman-Roisin (1991). For the case that the vortex radius is of the same magnitude as the baroclinic Rossby radius, which is typical, they found that the depth ratio H v / H L must be as low as 0.02 tbr the influence of the lower layer to be neglected. With H t J H I . ~ 0.2~ which is realistic for oceanic conditions, the influence of the lower layer was very considerable. In the present paper we will therefore use the quasigeostrophic two-layer equations. This means that the barotropic Rossby waves are included in the model. Using the rigid-lid approximation their maximum westward phase velocity is infinite, and one therefore expects all westward-travelling vortices to radiate barotropic waves, and that only eastward-travelling ones can be steady and localized. Some exact vortex solutions of the quasigeostrophic two-layer equations on the /3-plane can be found by assuming that the potential vorticity depends linearly on the streamfunction. In this way Flierl et al. (1980) found steady dipoles, and dipoles with monopolar riders. Some of them propagate eastward, but, somewhat unexpectedly, westward-propagating dipolar solutions (without riders) also exist. Outside the separatrix they have a purely baroclinic field, which is evanescent, as the vortices travel faster than the baroclinic waves. The barotropic mode is absent in the outer region, as it must be, because it would be oscillatory if present, implying radiative loss of energy. In this paper we investigate the existence of steadily propagating monopoles, rather than dipoles, and without the restriction to a linear vorticity-streamfunction relation. We employ a perturbation method where the zeroth-order solution is a circular vortex with essentially arbitrary radial profile, and treat /3 (the latitudinal derivative of the Coriolis parameter) and the propagation velocity c as small parameters. In this way we find solutions of baroclinic nature, with closed streamlines in both layers, and the upper- and lower-layer vortices rotating in opposite directions. As expected, we only find solutions that propagate eastward. When studying the existence of steadily propagating vortex solutions, the second crucial step, besides determining the range of linear wave propagation, is to find out what determines the propagation velocity of the vortices. In many cases, for instance the

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69

one-layer models described above, there exists an integral relation that determines the centre-of-mass velocity of localized solutions (Nycander, 1994). However, in a two-layer model with a rigid lid this integral relation degenerates to the 'no net angular momentum' theorem derived by Flierl et al. (1983). In the terminology used here, this theorem says that the integral of the barotropic component of any steady localized solution must vanish. The propagation velocity of vortices that satisfy this constraint is determined by their geometric structure. This is well illustrated by the 'hetons', which consist of one point vortex in each layer (Hogg and Stommei, 1985). The vortices me displaced horizontally relative to one another, and have opposite signs. Each point vortex is pushed forward by the other one, and the whole structure thereby acquires a velocity, much as a dipole vortex in a one-layer model. However, the point vortex model is only valid if/3 = 0, i.e. in the isotropic case where there are no Rossby waves. The same is true of the finite-area analogues of hetons that were found numerically by Polvani (1991). They consist of two vortex patches with opposite signs and situated in different layers. The steady vortices on the /3-plane found here of course satisfy the 'no net angular momentum' theorem. Their propagation velocity is essentially determined by the 'hetonic' mechanism, i.e. by the horizontal displacement of the upper- and lower-layer vortices relative to one another. A general integral relation that clearly displays this mechanism will also be derived. A related study is the one by Sutyrin and Dewar (1992). Applying a similar perturbation method to the present one to the two-layer shallow water equations, they obtained a steady monopole propagating westward faster than the baroclinic waves. Because of the small depth ratio assumed (of the same order as the Rossby number), the radiation of barotropic waves was in their case a weak higher-order effect. It was therefore neglected, which is why their solution could be steady. We will also qualitatively study the dynamics of baroclinic vortices. In simulations by Mied and Lindemann (1982) it was seen that eastward-propagating monopoles easily formed from rather general initial conditions, whereas westward-travelling ones either quickly disintegrated, or turned around and started propagating eastward. This is not unlike the behaviour of dipoles in a one-layer model (Hesthaven et al., 1993). We explain it as a consequence of the conservation of potential vorticity in each layer, and of excitation of linear waves. The paper is organized as follows. In Section 2 we present the basic equations and general conditions for steady solutions. In Section 3 we derive explicit solutions by perturbation analysis, and compare them with simulations and oceanic observations made in previous studies. The dynamics of baroclinic monopoles will be studied in Section 4, and the results are summarized in Section 5.

2. B a s i c e q u a t i o n s

We will use the quasigeostrophic two-layer model with a rigid lid: 0qu --

Ot

+ J ( ~bu,qu ) = 0

(la)

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A. Pakyari. J. Nycander./ Dynamic.s ql Atmo.vdteres and Oceans 25 (1996) 67 86

aql. - - +J(tPL'qL) = 0 at

(lb)

Here 6L and 6L are the streamfunctions in the upper and lower layer, respectively, and the Jacobian is defined by J ( f , g ) - O f a g - a j ~ g. The potential vorticity in the respective layers is defined by

q,: = v 2 ~ , + ~ y + ,,(,( 4 , , . - 4,~ )

(2a)

qL = v 6 L + ~ Y + K [ ( ' / J U - + L )

(2b)

where KI~r

g'Ht:

# g'Hi. f0 is the Coriolis parameter, H U and H L are the unperturbed depths in the respective layers, and g' is the reduced gravity: g ' = g - -P
~/T =

K2ffJU + KL2~ffJL m u ~ U + HLdt,. K2 = H

(3a)

~c = ~bu- 61.

(3b)

where H is the total ocean depth, and 9

K ~ is commonly called the baroclinic Rossby radius, or the deformation radius. Substituting Eqs. (3a) and (3b) into Eqs. (la) and (lb) we obtain

3

OLb"T

J<~i K~.

~tt Ve0.r +/3 3x + J(0T'~72¢pT) + - - ~ - J ( o c ' v 2 0 c ) = 0

(4a)

7, (v2*" -

(48)

~

+

a., + ~ J (Kt p ( : ' ~ 7 2 t p c )

+ J(q'~..v2% -) + ~-'Y( ~,..q,T ) = 0

+J(~OT'V2~bc)

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71

The dispersion relations for barotropic and baroclinic waves are obtained by linearizing Eqs. (4a) and (4b): /3k~ toT = k2 (Sa) /3k, K2 + k2

toc =

(5b)

where toT and toc are the barotropic and baroclinic wave frequencies, respectively, and k is the modulus of the wave vector. It should be noted that a barotropic wave can travel with any phase velocity to the west, whereas the phase velocity of a baroclinic wave propagating in the east-west direction lies in the interval -/3K -2 < toc/k, < 0. Moreover, the group velocity of the barotropic waves, which is the velocity with which they transport energy, can be infinite. This is related to the fact that the Green's function used to obtain 0T at each time step in Eq. (4a) is the logarithm, whereas the corresponding Green's function for the baroclinic mode 0c in Eq. (4b) is the modified Bessel function K 0, which decreases exponentially beyond the baroclinic Rossby radius. Hence, we may assume that the baroclinic field decreases exponentially as r ~ zc, whereas the barotropic field typically decreases only slowly (algebraically) at infinity, even if it was well localized initially. This must be born in mind when performing partial integrations to derive various integral relations. We first multiply Eq. (4a) by x and integrate over the xy-plane. If we assume that ~OT is sufficiently localized that all boundary contributions from the partial integrations vanish, we obtain

fqJTdS =

0

(6)

As pointed out above, one cannot usually expect this assumption to be true. However, we will later be looking for steady solutions that are sufficiently localized for Eq. (6) to be valid, hence this equation is a necessary condition for the existence of such solutions. When one performs the same operations as in the derivation of Eq. (6) on the two-layer equations with a free surface, i.e. with a finite barotropic Rossby radius R v, one obtains an integral relation for the centre-of-mass velocity (Nycander, 1994). In the limit R T ~ ,c (the rigid lid approximation), this velocity tends to infinity, and the integral relation degenerates to Eq. (6), which is a special case of the 'no net angular momentum' theorem of Flierl et al. (1983). There is, however, another integral expression that determines the velocity of localized vortices in the rigid-lid model. We first integrate Eq. (4b) over the xy-plane, and obtain d -gtfV,cdS = 0 (7) As ~c is proportional to the vertical displacement of the interface between the layers, this relation simply expresses mass conservation in both layers. We then multiply Eq. (4b) by x and integrate. Defining the x-component of the 'baroclinic centre of mass' by

R.,.

f4,c xdS

f~OcdS

(8)

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72

we obtain d R, - -

f.O( ,/q~, ~/q, )d S -

/3K

2 +

(9)

dt

/t/J(-d S

where we have used the relation J(~/,c,0.r ) = J(0t',qJL) in the numerator. It should be noted that Eq. (9) is valid even if the barotropic field decreases slowly at infinity, unlike Eq. (6). A similar expression for the velocity of a baroclinic monopole was derived by Sutyrin and Dewar (1992). As their scaling allowed for a larger perturbation of the thermocline than the quasigeostrophic approximation used here, their expression also contained a nonlinear term which is proportional to the perturbation of the thermocline. This term is absent in Eq. (9). We then assume that the solution is a perturbed circular vortex: Ou = 0 t , 0 ( r ) + ~ u ( r ) s i n 0

(10a)

0t. = 0 L 0 ( r ) + ~:L( r ) s i n 0

(10b)

This is the form we will assume for the steady solutions in the next section. Geometrically, the effect of the 0-dependent first-order terms in Eqs. (10a) and (10b) is that the circular contour lines of qJ are displaced in the y-direction. Denoting this displacement r/u(r) and "qL(r) in the respective layers, we have rh:

s~u , 4'u0

(lla)

tiE--

0'

(lib)

L0

Using Eqs. (10a), (10b), ( l l a ) and (1 lb) and averaging over 0, Eq. (9) can now be written 1

dR~ dt

-2 fvt!vL rldS /3K -2 +

fqtcdS

(12)

where rt(r) = rl U - rtL is the relative displacement in the y-direction of the upper- and t lower-layer vortices relative to one another, and v u = - g ' ~ 0 and v L = - 0 e 0 are the azimuthal velocities in the respective layers. This relation clearly displays the hetonic propagation mechanism.Let us assume, for example, that q'uo > 0 and q'nO < 0. (Remember that the signs must be different to satisfy Eq. (6).) Then the second term of Eq. (12) is positive if r t < 0, i.e. if the upper, anticyclonic vortex is situated south of the lower, cyclonic one. The second, 'hetonic' term in Eq. (12) is proportional to the displacement rl, which is a first-order quantity. Hence, the propagation velocity is not determined by the zerothorder radial profiles, as in the one-layer model. The velocity of such a two-layer vortex is therefore likely to be much more sensitive and variable than the velocity of a monopole vortex in a one-layer model. We will come back to this question in Section 4.

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73

We finally estimate the second term in Eq. (12). Assuming that v U is of the same order as v L or larger, we find that the magnitude of this term is approximately VL~q/ro, where r 0 is the vortex radius. To have closed streamlines in the lower layer, the propagation velocity c of the vortex must be smaller than v L. Hence, the 'hetonic' contribution to c is significant even if the horizontal displacement r1 is smaller than the vortex radius. In the perturbation analysis carried out in the next section, ~l/r o and C//UL are both small, first-order quantities.

3. Steady solutions Steadily propagating monopole vortex solutions of Eqs. (la) and (lb) have two different regions, one with open and another with closed streamlines. In the outer region with open streamlines exact solutions can be found, whereas this is not generally possible in the inner region. There we will solve the equations by a perturbation method, treating /3 and the propagation velocity c as small parameters. We assume the zeroth-order solution to be a circularly symmetric vortex, and use Eqs. (lOa) and (10b) as an ansatz. The first-order solutions must be matched on the boundary between the outer and inner regions. It will be seen that this is only possible if Eq. (6) is satisfied by the zeroth-order profiles. Thus, this general integral condition for localized solutions appears as a solvability condition in our perturbation analysis. If it is satisfied, the number of free first-order parameters is just enough to satisfy all the boundary conditions for an eastward-going vortex, as will be seen below. For a westward-going steady vortex, on the other hand, there is the additional condition that the barotropic mode vanish in the outer region, and the number of free parameters is therefore insufficient. We first transform Eqs. (la) and (lb) to a system moving with the velocity c~. Denoting the streamfunction in this system by X/= q'~ + cy (i = U,L), stationary solutions are obtained from J(Xu,V:Xo + f l y + KuZXL)=0 2

J(XL,V:XL + / 3 y + K L X u ) = 0 The general solution Of Eqs. (13a) and (13b) is V2Xu + / 3 y + K2 XL = f v ( X u )

(13a) (13b)

(14a) (lab)

V2XL"}-/3Y+ KL2 Xu = f L ( X L ) where fo and fL are arbitrary functions. We will now solve Eqs. (14a) and (14b) in the inner and outer regions, and finally match the solutions by requiring Xi and VXi to be continuous everywhere, and V2Xi to be continuous along the streamlines. 3.1. Outer region On open streamlines the functions fu and fL can be determined uniquely by requiring that both ~bu and ~bL vanish at infinity, which implies that f~ is linear:

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74

Inserting this result into Eqs. (14a) and (14b) and diagonalizing, we obtain

V 2tkT = 3' 2qrr

(16a)

V2~c = A2~bc

(168)

where y2 = fl/c and A2 = K 2 + [3/C, and ~T and ~bc are defined in Eqs. (3a) and (3b). Both eigenvalues A and y are real when c > 0, implying that eastward-propagating vortices are exponentially localized and non-radiating. The barotropic eigenvalue 7, on the other hand, is always imaginary when the velocity is westward. The barotropic mode is then oscillatory, and the vortex loses energy. A westward-propagating vortex can therefore only be steady if the barotropic mode is absent in the outer region. Using Eq. (3a) this implies 'Pu -

~ qJl,

(17)

It should be noted that the second, 'hetonic' term of Eq. (12) does not give any contribution to the vortex velocity if q't: and g'L are proportional, as then "O= 0. We now choose the circle r = r 0 as the boundary between the inner and outer regions. Inside the circle the functions f, in Eqs. (14a) and (14b) are arbitrary. The separatrix is situated outside of the circle, and in this region we solve Eqs. (16a) and (16b). (This means that the linear relation between the streamfunction and the potential vorticity holds also in a small region inside the separatrix, which is perfectly possible. If the separatrix were situated inside the circle, on the other hand, this relation would be nonlinear in some region outside the separatrix, which is impossible. It should be noted that the separatrix cannot coincide with the circle, as it has a cusp.) We further assume that this circle is a streamline in the upper layer, which gives the boundary condition xu(r0,0)

= kt

(18)

where k U is a constant. This condition fixes the position of the vortex along the y-axis. Together with the condition 17 (i.e. gt1, = 0), it uniquely determines the solution in the outer region for a westward-travelling vortex:

Ko( Ar ) xu=kUKo(Ar~ ~

[ K t( Ar ) roKl(Aro)

KL2 K o ( A r ) Xt, = - k u K21 Ko(Ar0 ) +

] r

csin 0

K2 K , ( A r ) r° K~1

Kl(Aro)

(19a)

+ r]csinO

(19b)

In a steady eastward-going eddy the barotropic mode is allowed in the outer region, as it is evanescent. To determine its amplitude we need to specify the value of both Xu and XL at r = r 0. The streamlines in the inner region are all approximately circular, and the first-order solution gives the displacement of these circles. If the streamline with radius r n in the lower layer is displaced by ,50 in the y-direction, we have

xL( ro,o

) = ~, - 80 x b ) ( r0) sin 0

(20)

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75

where k L =--XL0(r0) is some constant, and XLo(r) the zeroth-order radial profile. (The difference between q, and X is of first order, so by definition XL0------q'L0)" The displacement 6 0 of the streamline and the displacement '/~L of contourlines of I~tL defined in Eq. (11 a) are related by cr o

60 = r/L(ro)

(21)

X[0(r0)

The solution in the outer region satisfying the boundary conditions 18 and 20 is

[ [

Xu = AToKo(Yr) + Aco--~Ko( Ar) + AT,KI( Tr) + Ac,"~K,( Ar) + cr]sinO (22a)

XL=AToKo(Tr) -Aco-~Ko( Ar) + AT,K,(Tr )-Act--~Kt( Ar ) + cr]sinO (22b) where

K,2ku + K~kL

1

K2

Ko(yr0)

AT0 =

kU- kL

Ac0

K0(Ar0)

.[K2K26°XL°(r°) +cr° ]lK,(yro)

AT' = -I AcI =

6o X[o(ro) Kl(Aro)

The factor X'LO(rO)in the first-order coefficients AT~ and Acl can here be calculated from the zeroth-order, 0-independent term in Eq. (22b). Thus, the whole solution is uniquely determined by the amplitudes k U and k L of the zeroth-order profiles at r -- r 0 and the first-order parameters c and 60. It should be noted that in the westward-travelling solution, Eqs. (19a) and (19b), the displacement 80 of the lower-layer streamline at r = r 0 is fixed by the requirement that q'r = 0 for r > r o, which gives r/(r 0) = 0 and 8o = cro[1/X'uo(ro)- 1/X'Lo(ro)]. We will see that the additional free parameter 80 makes it generally possible to find steady eastward-travelling solutions, as long as the integral condition Eq. (6) is satisfied by the zeroth-order profiles.

3.2. Inner region We saw above that the functions fu and fu are necessarily linear on open streamlines, and exact solutions of Eqs. (14a) and (14b) could therefore easily be found in the outer region. One way to solve Eqs. (14a) and (14b) in the inner region is to

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assume that these functions are linear here as well. In this way, exact dipolar solutions valid in the whole domain were found by Flierl et al. (1980). Here we will allow fu and fL to be nonlinear, and find approximate solutions by perturbation analysis, treating c and /3 as small parameters of order E. Keeping terms up to first order we have X, = X~o( r) + X,~( r.O) = Xio( r) + ~,( r)sinO

(23)

where i = U,L. The first-order term in this expansion is proportional to the inhomogeneous term /3y in Eqs. (14a) and (14b). We also expand f,( Xi): .~ ( Xi ) = £ ( X,o + X,, ) =.f)( X,~, ) +./i'( X,o ) X,,

(24)

Then the zeroth- and first-order equations are V2Xuo = f t z ( X , : o ) - K~X, 0

(25)

and V2xu = f ' u ( X u o ) X u ,

V2XI. =fI~(XL0)XL,

K~TX,.,

/3rsin 0

.... K~ yt t - /3rsinO

(26)

The derivative of f, can be found by differentiating Eq. (25). Inserting the result into Eq. (26) we obtain V2~'U =

V _)(U0 + KU---7"-- f l u - KC:;'I.- /3r XUO Xuo

(27)

v2 " = ( One of the homogeneous solutions of Eq. (27) is ~', = X~0, which corresponds to a rigid displacement of the circular vortex. We therefore make the substitution ~'i = - X~0¢5,: "- , l ) = K C~X' 6{'~+a u( 2Xu-----~°+ ~ - - - ~ ( a U - - 6 L ) + - - - ; /3 --r Xuo r X~:o Xu0

(28a)

611+6[ 2 XL°, + - XLO r

(28b)

= K L - - - - 7 ( 6 L - 6 t , ), + - - 7 - - r XLO XLO

Here at(r) is the displacement of the circular streamlines X~ = constant in the y-direction. It is related to the displacement rli(r) of the contourlines of g,~ by cr

6 , ( r ) = 7/,(r)

X;0( r )

(29)

3.3. Matching conditions Both X~ and VX~ must be continuous at r = r 0. In the lower layer this circle is not a streamline, and the streamlines therefore intersect this circle. As the vorticity must be

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77

continuous along the streamlines we must therefore also require continuity of V2XL at r = r 0.

The continuity of Xi implies that 8,. must be continuous, and ~i(r0) is then obtained from Eqs. (19a) and (19b) or Eqs. (22a) and (22b). In particular, 8w(r0) = 0. Using this result and differentiating the relation sri = -X~o t~i(r), we also see that the continuity of VXi and X~o implies that ~] must be continuous. Moreover, 8i should be regular at the origin. Thus, them are six conditions, but four is enough to determine the solution of EClS. (28a) and (28b) uniquely. We will see that the remaining two conditions can be satisfied by using the free parameter 8 o (i.e. the displacement of the streamline with radius r0), and choosing a zeroth-order profile that satisfies the solvability condition Eq. (6). We multiply Eqs. (28a) and (28b) by the integrating factors KL2r(X~O) 2 and 2 p Kur(XL o) 2 , respectively, add the equations and integrate the result from zero to r o. After integrating by parts and extending the upper limit to infinity, we obtain + ,`2 XLo)rdr=

--~

r° [,`2r

2~

~

,

Lt Xuo(ro)]

2 ,

8u(ro)-F,`2[XLO(rO)]2~'L(rO)}

+ r~ [,`L2 Xuo(ro) + K~ XLo(ro)] + ];o(,`L2 XUO + 2 ~

,`~XLo)rdr (30)

In the first and second terms on the right-hand side of Eq. (30) we should use the streamfuctions for the inner region, and in the integral (the last term) we use the streamfunction for the outer region. (It should be noted that this integral is zero in a steady westward-travelling eddy). Writing Eq. (30) in terms of the barotropic and baroclinic modes, by using Eqs. (3a) and (3b) and defining ~'v and ~c similarly as in Eqs. (3a) and (3b), we obtain r0

.~

tt

,'

r0

io,~To,'dr= ~'-~(~'T0~'T -- $~0~'T),=,o- + 2~

2

2

K L K U

,,4 (~0~'¢ -- '~0~";)'='o-

+ - ~ ~b~ro(ro_ ) + f~o~bTordr (31) where r 0_ and ro+ correspond to the inner and outer solution, respectively. We have here used ¢C0,T0= XC0.T0- Assuming Cuo and eL0 to be continuous up to the second derivative, and ~'u and ~'L to be continuous at r = r 0, we have from Eqs. (22a), (22b) and (23) ~b~o( ro- )

AT0

--

y[ ~ ( r o + ) - c]

(32a)

ATt Aco

~b~o( r o- ) = - ~ A ~ r ~ ( ro. ) Acl

(32b)

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Substituting Eqs. (32a) and (32b) into Eq. (31) we finally obtain f°0T°rdr=

2/3

0To(ro)[(T(ro-)--(~-(ro

+ K~K~T ,

)]

ro )1

(33)

This equation is valid for steady vortices that travel both eastward and westward, if we note that 0~r0(r0)= 0 in the latter case. We also remark that Eqs. (32a) and (32b) is modified if ~0u0 is only continuous up to the first derivative, but the final result, Eq. (33), is still valid without any modification. Continuity of 6IT and 6'L in Eqs. (28a) and (28b) is equivalent to ("r and (~. being continuous. We see directly from Eq. (33) that it is necessary that the zeroth-order profile satisfies the integral condition Eq. (6) for ("r and ((', to be continuous. On the other hand, satisfying this integral condition is not sufficient for continuity of both (~and (~'.. In the westward case, for example, the integral condition only ensures the continuity of (('.. For the derivative of both components of the first-order solution to be continuous we need one more adjustable parameter. Such a first-order parameter exists in the eastward case, where the exterior solution is not only proportional to c but also to the horizontal displacement 6 o of the vortices in the two layers relative to one another. For a westward-going eddy, however, there are not enough free parameters to satisfy all boundary conditions, as 6 o is fixed by the condition that ~0T = 0 in the outer region. We have tried to find a solution by treating c as a free parameter, and iterating the solution to satisfy the boundary condition for ~'2. In the starting approximation the velocity was westward, but the iterations converged to a solution with a very small eastward ~,elocity. We also tried to find a solution by introducing one free parameter in the zeroth-order stream function, and keeping c fixed. We added a term C6r6 tO the upper-layer zeroth-order profile and applied the same method as we used for an eastward-travelling eddy, adjusting c 6 instead of 60. However, we were unable to find any solution in this way. We now show two numerical examples of steady, eastward-propagating vortices. In the first one we use parameters similar to those of McWilliams and Flierl (1979): Hvo = 700m; HL0 = 4300m; r 0 = 70km; g' = 2.7 × 10 -2 m s - 2 ; f0 = 0.9 × 10 -4 s - I ; /3= 1.7× 10 l ~ m - ~ s - ~ ; /3K 2 = 3 . 4 × 10-2 m s - ~ ; K -~ = 4 5 k m . The zeroth-order solution in the inner region is chosen as polynomials with six free parameters: Xuo = Co "}- c2 r2 + c4 r4

XLO = do + d2 r2 + d4 r4

(34)

The solution in the outer region is given by Eqs. (22a) and (22b), where k u and k L must be chosen so that the angular velocity at r = r 0 in both layers is greater than the propagation velocity of the vortex. This is necessary so that the separatrix be situated in the outer region. We assume that the vortex propagates eastward with the velocity

A. Pakyari, J. Nycander / Dynamics of Atmospheres and Oceans 25 (1996) 67-86

79

y (North)

(a) 4

2

0

x (East)

'!

-2

-4

'-'4

....

.'2'

0 ....

~ ....

~'

y (North)

(6)

x (EasO

-2

-4

'-.4

. . . .

-)'

'

0

'

'~

. . . .

~'

'

Fig. 1. Streamlines of a steady eastward-propagating vortex, with the upper layer shown in (a) and the lower layer in (b). The parameter values are close to the simulation by McWilliams and Flied (1979), with H u / H E = 0.16. Length is measured by the baroelinic Rossby radius K- ~ = 45km. The contour interval in (a) outside r = 1.5K- ~ is four times larger than in (b), and inside this circle it is six times larger.

80

A. Pakyari, J. Nycander / Dynamics o f Atmospheres and Oceans 25 (1996) 6 7 - 8 6

v (North)

(a) 4

-

-

2

x (East)

0

-2

!

-4

~

'

'

:

.

.

.

.

.

;

'

'

'

'4

i (North

(b) 1

\ //

x (East)

1

h

3

0 ¸

;

~

Fig. 2. The s a m e as in Fig. 1, but with p a r a m e t e r v a l u e s close to the v o r t e x o b s e r v e d by S a v e h e n k o et al. (1978), with H v / H L = 1. L e n g t h is m e a s u r e d b y the baroclinic R o s s b y radius K - I = 3 5 k i n . T h e c o n t o u r interval is the s a m e in (a) and (b),

A. Pakyari, J. Nycander / Dynamics of Atmospheres and Oceans 25 (1996) 67-86

81

0.5j~K -2, and set k U = --29.6flk -3 and k L = 5 . 4 •K - 3 . The coefficients in Eq. (34) are then chosen so that Xu0 is continuous up to the first derivative and ~bL0 up to the second derivative at r = r0, and so that the integral condition 6 is satisfied. These six conditions determine the coefficients in Eq. (34) uniquely. The maximum velocity of the resulting vortex flow is 2 m s - ~ in the upper layer and 26 cm s - ~ in the lower layer, and the pressure perturbation at r = 60 km has decreased to about 30% of its peak value, in rough agreement with the simulation by McWilliams and Flied (1979). We then find the first-order solution by integrating Eqs. (28a) and (28b) numerically, starting from r = 0. The continuity of gl and g2 at r = r 0 is ensured by choosing proper values of g 1(0) and g2(0). The displacement parameter g0 can be adjusted iteratively so that g~(r 0) becomes continuous; the integral relation for the zeroth-order solution then guarantees the continuity of g'l(r0), as was also verified numerically. We obtained g0 = 0.078K-i. The streamlines in the moving coordinate system of the resulting solution are shown in Fig. 1. We note that if we had chosen a too large propagation velocity, for instance c = ~[~K-2, while keeping the maximum flow velocity at 2 m s - l , the separatrix in the lower layer would appear inside of r = r 0, indicating that the perturbation analysis breaks down. This is a real effect: the propagation velocity of a vortex with given amplitude can only be increased up to a certain limit by increasing the horizontal displacement. If the displacement is increased beyond this limit the upper- and lowerlayer vortices separate, and the hetonic structure disintegrates. This is apparently what happens in the simulation by McWilliams and Flied. In the second example the parameters are chosen to be close to the vortex observed by Savchenko et al. (1978), which appeared to have a hetonic structure, although its deep structure was not very certain. It was also seen to propagate to the north-northeast. We use the following parameter values: Hu0 = 2000m; HE0 = 2000m; r 0 = 40km; g ' = 1.0× 10-2 ms-2; f0 =0.9 × 10-4s-I; fl= 1.7X 10-1~ m - I s - l ; ~K-2 =2.1 × 10 -2 m s - l ; K- l = 35km. The explicit solution is obtained in the same way as in the previous example, now assuming that the vortex propagates eastward with the velocity c = 0.75ilk -2. The maximum velocity of the resulting vortex flow is in this case 40cm s - i in the upper layer and 29cm s - i in the lower layer, and the horizontal displacement is g0 = 0.25K- i. The streamlines of this solution are shown in Fig. 2. C=

4. Dynamics of baroclinic monopole vortices In this section we qualitatively investigate the stability of monopole vortices to perturbations of the direction of propagation, and also their response to wave radiation. As we have seen above, there exists a large family of steady baroclinic monopoles, with essentially arbitrary radial profile, that propagate to the east. The vortices are propelled by the hetonic mechanism, with the cyclonic vortex situated to the north of the anticyclonic one (regardless of which o n e is in the upper layer and in the lower layer). Because all waves propagate to the west, there is no possibility for wave radiation. Let us suppose now that the angle of propagation is slightly perturbed, say, to the northeast.

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The effect on the vortex can be found by writing Eqs. (2a) and (2b) in terms of the baroclinic and barotropic components: K~ q U + K~3ql, V2///T ~-~

KI2. + K2T

-- f l y

(35)

V2qtc - ( K2 + K2)tbc = q v - q,, In the region of trapped fluid the potential vorticities qu and qL are conserved. Hence the barotropic vorticity component decreases as the vortex latitude y increases, whereas the baroclinic component is essentially unchanged. This means that the cyclone (situated northward), with positive vorticity, becomes weaker, and the anticyclone stronger. Because of this imbalance the trajectory then curves southward, and the vortex starts propagating to the southeast. After a while the decreasing value of y has made the anticyclone weaker and the cyclone stronger, and the process is repeated. The result is an oscillatory motion, as was observed in the numerical simulation by Mied and Lindemann (1982). Although we have not been able to find steady monopoles travelling to the west, we may hypothetically assume that they exist. They must then travel faster than the baroclinic waves, i.e. c < - i l K -z, which requires that the cyclonic part is situated south of the anticyclonic one, according to Eq. (12). Such vortices are unstable to perturbations of the direction of propagation, because of the same mechanism that explains the oscillatory motion of eastward-propagating vortices. The mechanism is exactly the same as in the instability of westward-propagating dipoles in a one-layer model, and the result should also be the same: the vortex either turns around and starts propagating to the east, or it disintegrates (Hesthaven et al., 1993). The reason why we could not find steady westward-propagating vortex solutions was the extra condition imposed by the requirement that the barotropic component vanish in the outer region. Hence, a westward-propagating vortex will normally radiate barotropic waves, and thereby lose energy and enstrophy, the two quadratic invariants of Eqs. (la) and (lb). We will now study the response of the vortex to this loss. The expression for the energy is derived by multiplying Eqs. (l a) and (lb) by ~btj and qJL, respectively, and integrating over the xy-plane. By combining the result suitably and writing it in terms of barotropic and baroclinic components, we obtain d E / d t = 0, where E= P

+- -

--+

O ]}dS

(36)

To find the enstrophy we multiply Eqs. (la) and (lb) by y and integrate over the xy-plane, which gives a conservation law with some linear terms. To get rid of these we add the Casimir integral

f[

+ 4(q - fl 2y2)] dS

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83

which is also conserved. Writing the result in terms of the barotropic and baroclinic components, we find = 0, where the enstrophy is defined by

dS/dt

S:

V2¢T) 2 + KL2KU 2

¢c

dS

(37)

This can also be written in terms of the upper- and lower-layer potential vorticity:

S=f[ Kz(qu-~y)2+Kz(qL-~y)z ]K2 dS

(38)

Let us consider now two simple ways in which the vortex may respond to radiation losses: by a rigid displacement of whole vortex in the y-direction, and by independent displacements of the upper- and lower-layer vortices. In the case of a rigid displacement we obtain ~V2¢T = - / 3 ~ y and ~V2¢c = 0 from Eq. (35), where ~y has the same value for all trapped fluid elements. Minimizing S in the form of Eq. (37) with respect to y (in effect only the contribution from the first term is minimized), we obtain f V2~0TdS = 0

(39)

where the integral is taken over the region of trapped fluid. Thus, insofar as the vortex responds to the enstrophy loss by changing its latitude, it will drift toward what may be called its 'barotropic latitude of rest', in analogy with the terminology of Larichev (1983). At this latitude its barotropic component is minimized. From a consideration of the energy loss a qualitatively similar conclusion can be reached: the vortex responds by a drift that minimizes the barotropic component. However, the minimization of Eq. (36) is made more complicated by the presence of a nonvanishing boundary term that arises when a partial integration is performed. If independent displacements of the upper- and lower-layer vortices are allowed, then the enstrophy can be decreased by displacing the cyclone (with positive q) northward and the anticyclone southward, as is immediately seen from Eq. (38). In effect, the two vortices try to approach their individual latitudes of rest. For a vortex that initially travels westward, this means that the relative horizontal displacement decreases, so that the hetonic interaction weakens, and the vortex slows down. When the vortices completely overlap, i.e. r / = 0, the eddy moves with velocity -/3,< -2 (see Eq. (12)). At this stage the eddy begins to radiate baroclinic waves, too, and the enstrophy loss becomes stronger. The process continues until the eastward contribution from the hetonic term in Eq. (12) is large enough that the vortex propagates eastward. The wave radiation then ceases, and the vortex becomes steady. This process is clearly seen in the numerical simulations by Mied and Lindemann (1982), in particular their Experiment 42. It is also seen in the simulations by McWilliams and Flierl (1979), but in their case the hetonic structure soon disintegrates because of the small amplitude of the lower-layer vortex. Westward-propagating monopoles have also been studied by Sutyrin and Dewar (1992), using a two-layer shallow water model. Their asymptotic expansion allows for a larger perturbation of the thermocline than here, and a vortex which is anticyclonic in

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the upper layer then travels faster than the baroclinic waves even without the hetonic mechanism (i.e. with vertically aligned upper- and lower-layer vortices). As they also assumed the depth ratio H~/H to be small, the barotropic radiation was weak, and up to the orders considered explicitly the vortex was steady. Nevertheless, the vortex does of course radiate barotropic waves, and if there is trapped fluid in the lower layer the process described here will ultimately occur: retardation, onset of baroclinic radiation, and finally eastward propagation (or alternatively, disintegration). The main effect of the small depth ratio and the large perturbation of the thermocline is probably only that this process is delayed. 5. Conclusion We have investigated the existence of steady monopole vortex solutions of the quasigeostrophic two-layer equations on the fl-plane. A necessary condition for the existence of such a solution is that the integral of the barotropic component over the xy-plane must vanish (see Eq. (6)). This implies that it has a baroclinic structure, with counter-rotating vortices in the two layers. The velocity of the baroclinic centre of mass is determined by the relative horizontal displacement of the upper- and lower-layer vortices (the hetonic mechanism), as seen from Eq. (12). If the relative displacement is zero, the vortex moves to the west with the same speed as the fastest baroclinic waves. At the same time, this displacement determines the magnitude of the barotropic and baroclinic dipolar field components outside the separatrix, as shown in Section 3.1. The general condition for a vortex to be steady is that it must not radiate linear waves, For a vortex propagating to the east this is not a problem, as the phase velocity of all waves is westward. Both the barotropic and the baroclinic field components are therefore evanescent in the outer region. We were able to find a large family of solutions of this kind by perturbation analysis. A circular vortex was taken as the zeroth-order solution, with /3 and the propagation velocity c treated as small parameters. The first-order problem could be solved for a prescribed value of c by choosing the horizontal relative displacement required for this velocity. The zeroth-order radial profiles are arbitrary, except that they must satisfy the integral condition Eq. (6), which arises as a solvability condition in the perturbation analysis. Two numerical example of such steady solutions were also shown, with parameter values chosen to be close to the vortex simulated by McWilliams and Flierl (1979) and to the vortex observed by Savchenko et al. (1978). For a vortex propagating westward faster than the baroclinic waves, the barotropic field is oscillatory, and the vortex can therefore only be steady if this component vanishes in the outer region. This condition fixes the relative displacement of the vortices in the upper and lower layers, so that the first-order problem with a prescribed value of c is overdetermined. We tried to find particular steady solutions by using a zeroth-order profile with some ad hoc extra free parameters, but without success. Even if steady solutions of this kind in fact do exist, our conclusion is that westward-travelling vortices in general radiate barotropic waves, and are therefore unsteady. The dynamics of such vortices was investigated qualitatively. We showed that the vortex responds to the enstrophy loss caused by wave radiation by displacing the

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cyclonic vortex northward and the anticyclonic one southward. This means that the hetonic term in Eq. (12) gives a more positive (or less negative) contribution to the propagation velocity. This eastward acceleration continues until the vortex propagates to the east, and the radiation ceases. We also identified an instability of westward-travelling vortices that is very similar to the 'tilting' instability of westward-travelling dipoles in a one-layer model (Nycander, 1992). If an asymmetric perturbation causes the vortex to travel slightly northward, the cyclone situated to the south will be weakened, and the anticyclone situated to the north strengthened, because of the conservation of potential vorticity in the trapped fluid. As a result, the trajectory curves away from the x-axis. Thus, if steady westward-propagating vortices exist (we could not find any), small perturbations can trigger two different mechanisms that quickly decrease their westward velocity: wave radiation and tilting instability. There are two possible final states of such a vortex: either it disintegrates into two vortices that propagate independently of one another, or it becomes a steady eastward-travelling vortex. This behaviour is distinct from that of simple monopoles, with only one sign of the relative vorticity. These drift north-westward or south-westward under the influence of the beta-effect, while emitting Rossby waves. In fact, although the vortices studied here (with a counter-rotating flow in the lower layer) look like monopoles, they are dynamically more related to dipoles than to monopoles in a one-layer model. The basic reason for this is that their propagation velocity is mainly determined by the relative positions of the upper- and lower-layer vortices, i.e. by their internal geometry, rather than by external gradients. The best and most abundant observational data from real oceanic vortices are temperature and salinity fields. However, when using these to compute the geostrophic flow there is an unknown integration constant, corresponding to the barotropic component, and this constant determines the direction of the deep flow. Only direct flow measurements can therefore show whether there is a counter-rotating deep flow, but such data are sparse. One conclusion from the present work is that this question is perhaps answered more easily from the dynamic behaviour of the vortex: if it propagates westward with a fairly constant velocity, then it is a simple monopole, i.e. there is no counter-rotating deep vortex with closed streamlines. If it propagates eastward, it has a counter-rotating deep flow. Judging from this criterion, most observed vortices are simple monopoles. However, the cyclonic eddy south of Australia observed by Savchenko et al. (1978) may have had similar properties to those studied here. The data concerning the deep flow were not very good, but hinted at anticyclonic circulation. More importantly, the vortex propagated north-northeastward, which would be difficult to explain if it were a simple monopole.

Acknowledgements This work has been supported by the Swedish Natural Science Research Council (NFR), Grant F - A A / F U / G U 09917-312.

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References Chassignet, E.P. and Cushman-Roisin, B., 1991. On the influence of the lower layer on the propagation of nonlinear oceanic eddies. J. Phys. Oceanogr.. 21:939 957. Cushman-Roisin, B., Chassignet, E.P. and Tang, B., 1990. Westward motion of mesoscale eddies. J. Phys. Oceanogr., 20: 758-768. Flierl, G.R., 1987. Isolated eddy models in geophysics. Annu. Rcv. Fluid Mech., 19: 493-530. Flied, G.R., Larichev, V.D., McWilliams, J.C. and Reznik, G.M., 1980. The dynamics of baroclinic and barotropic solitary eddies. Dyn. Atmos. Oceans, 5: 1-41. Flierl, G.R., Stern, M.E. and Whitehead, Jr.. J.A., 1983. The significance of modons: laboratory experiments and general integral constraints. Dyn. Atmos. Oceans, 7: 233-263. Hesthaven, J.S., Lynov, J.P. and Nycander, J., 1993. Dynamics of nonstationary dipole vortices. Phys. Fluids A, 5: 622-629. Hogg, N. and Stommel, H., t985. The heton, an elementary interaction between discrete baroclinic geostrophic vortices, and its implication concerning eddy heat flow, Proc. R. Soc. London, Ser. A, 397: 1-20. Larichev, V.D., 1983. General properties of nonlinear synoptic dynamics in the simplest model of barotropic ocean. Okeanologiya, 23: 551-558. McWilliams, J.C. and Flierl, G.R., 1979. On the evolution of isolated, nonlinear vortices. J. Phys. Oceanogr., 9: 1155-1182. Mied, R.P. and Lindemann, G.J.. 1982. The birth and evolution of eastward-propagating modons. J. Phys. Oceanogr., 12: 213-230. Nof, D., 1983. On the migration of isolated eddies with application to Gulf Stream Rings. J. Mar. Res., 41: 399-425. Nycander, J., 1992. Refutation of stability proofs for dipole vortices. Phys. Fluids A, 4: 467-476. Nycander, J., 1994. Steady vortices in plasmas and geophysical flows. Chaos, 4: 253-267. Nycander, J. and Sutyrin, G.G., 1992. Steadily translating anticyclones on the beta plane. Dyn. Atmos. Oceans, 16: 473-498. Polvani, L.M., 1991. Two-layer geostrophic vortex dynamics. Part 2. Alignment and two-layer V-states. J. Fluid Mech., 225: 241-270. Savchenko, V.G., Emery, W.J. and Vladimirov, O.A.. 1978. A cyclonic eddy in the Antarctic Circumpolar Current south of Australia: results of Soviet-American observations aboard the R / V Professor Zubov. J. Phys. Oceanogr., 8: 825-837. Sutyrin, G.G. and Dewar, W.K., 1992. Almost symmetric solitary eddies in a two-layer ocean. J. Fluid Mech., 238: 633-659. Swenson, M., 1987. Instability of equivalent-barotropic riders. J. Phys. Oceanogr., 17: 492-506.