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Steadily translating anticyclones on the beta plane
Dynamics o f Atmospheres and Oceans, 16 (1992) 473-498 Elsevier Science Publishers B.V., A m s t e r d a m
473
Steadily translating anticyclones on the beta plane J. Nycander a and G.G. Sutyrin b a Department of Technology, Uppsala Unit'ersity, Box 534, S-751 21 Uppsala, Sweden b p.p. Shirshoc bzstitute of Oceanology, Academy of Sciences o]"the USSR, 23 Krasiko~a St.. 117218 Moscow, Russia (Received 28 February 1991; revised 18 June 1991; accepted 9 November 1991 )
ABSTRACT Nycander, J. and Sutyrin, G.G., 1992. Steadily translating anticyclones on the beta plane. Dyn. Atmos. Oceans, 16: 473-498. Approximately stationary anticyclones in shallow water on a rotating planet are studied both analytically and numerically. They consist of a monopolar part with large amplitude and a dipolar part with small amplitude, proportional to the beta parameter. An explicit solution for the dipolar part is obtained with an arbitrary radial profile of the monopolar part. Numerical experiments show that this dipolar part accelerates or decelerates an initially circular vortex depending on the vortex size. The propagation speed is determined by a general integral relation. It is found that for the vortices to be stationary, this speed should not be close to the phase velocity of linear Rossby waves. This requires a strongly elevated surface (large amplitude). The vortices contain a core region with trapped fluid, unlike one-dimensional KdV solitons. Applications of the solution to tong-lived geophysical eddies are discussed.
1. I N T R O D U C T I O N
Among long-lived geophysical eddies, monopolar anticyclones seem to be the most persistent. The largest of these vortices is the Great Red Spot in the atmosphere of Jupiter. This vortex, which is significantly larger than the Earth, has been coherent for at least 300 years. Other Jovian long-lived vortices, the Large Ovals, have been known for several decades (Smith et al., 1979). Similar large anticyclones, such as Big Bertha, the Brown Spot, Anne's Spot, etc. have been detected in the atmosphere of Saturn (Smith et al., 1982). In the oceans of the Earth the recently discovered compact intrathermocline anticyclones can exist for several years (McWilliams, 1985; Kamenkovich et al., 1986). Observations show that these vortices have trapped fluid which they transport over thousands of kilometers. Because
C o r r e s p o n d e n c e to: J. Nycander, D e p a r t m e n t of A p p l i e d M a t h e m a t i c s and Theoretical Physics, University of Cambridge, Silver Street, C a m b r i d g e CB3 9EW, UK. 0377-0265/92/$05.00 © 1992 - E l s c v i c r Science Publishers B.V. All rights reserved
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of their high intensity, long lifetimes and transport properties, the role of isolated vortices in geophysical flows seems to be very important. Since the horizontal scale of the long-lived geophysical eddies is much larger than the depth of the atmosphere or the ocean, the shallow water approximation is generally used when modelling them. The validity of this approach has been discussed in detail by Williams (1985). During the last decade the problem of the resistance of such localized structures to Rossby wave dispersion has been widely studied (Flierl, 1987). Large-scale eddies are typically near geostrophic balance, thus the perturbed surface in cyclones is lower than the equilibrium level, while anticyclones have an elevated surface. To lowest order in the perturbation of the surface (as in the quasi-geostrophic approximation), the behavior of cyclones and anticyclones is similar. Taking into account higher order terms in the the amplitude breaks this symmetry, and allows non-linear steepening to compensate the dispersion in anticyclones. Models of this kind, with weak non-linearity and a unique analytic relation between the potential vorticity and the Bernoulli function, were proposed by several authors (Flierl, 1979; Petviashvili, 1980; Mikhailova and Shapiro, 1980; Charney and Flierl, 1981). In their solutions the perturbation of the fluid depth is small, and there is a unique relation between the amplitude and the vortex diameter. The size is assumed to be between the deformation radius and the radius of the planet, i.e. in the so-called intermediate geostrophic (IG) scale. The difference between the evolution of cyclones and anticyclones at this scale was emphasized by Yamagata (1982). However, it has been shown by a thorough analysis that the equation governing the dynamics in the IG scaling region has no stationary localized solution with analytic relation between the potential vorticity and the Bernoulli function (Romanova and Tseitlin, 1984; Nycander, 1989). The reason is that the meridional variation of the Rossby wave propagation speed is essential in the IG dynamics, leading to meridional twisting. Thus, the theoretical solutions mentioned above are inconsistent. Laboratory experiments in shallow water in paraboloidal rotating vessels (Antipov et al., 1982, 1988; Nezlin, 1986; Nezlin et al., 1990) and numerical simulations on the beta plane (Matsuura and Yamagata, 1982; Sutyrin and Yushina, 1986) show indeed that anticyclones can exist much longer than cyclones. They propagate westward faster than linear Rossby waves, transporting fluid in the core. However, the experimentally observed long-lived anticyclones have a large amplitude, with the perturbation of the fluid depth comparable to the equilibrium depth. A close examination also revealed that they are significantly smaller than vortices obtained from a generalization to large amplitude of the above-mentioned models with unique analytic form (Sutyrin, 1985).
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These experimental and numerical results have stimulated the development of a broader concept of solitary vortices, incorporating the features of finite amplitude and trapped fluid (Sutyrin, 1985; Nezlin, 1986; Sutyrin and Yushina, 1988; Nezlin and Sutyrin, 1989; Nycander, 1990). The trapping of fluid rotating around the axis of the vortex gives it qualitatively new properties. One is the existence of a 'memory' of the initial disturbance inside the region of closed streamlines, or, in other words, the absence of a unique relation between the amplitude and the diameter of the vortex. Another is that collisions between solitary vortices are inelastic, unlike collisions between one-dimensional KdV solitons. Essentially, the existence of stationary, localized vortices can be understood from only two elements. The first one is the dispersion relation for linear Rossby waves. The phase velocity of these waves has an upper limit, and in Section 4.2 it is shown that any stationary, localized vortex must propagate faster than all linear waves. This complementarity between the phase velocity of linear waves and the velocity of solitary, stationary structures is a general feature in many non-linear wave equations, for instance the KdV equation, and it is easy to explain. If the velocity of a localized structure coincides with the phase velocity of linear waves, it will radiate energy, similar to Cerenkov radiation. It can, therefore, not be stationary. If, on the other hand, the velocity is outside the region of linear phase velocities, the structure can not radiate energy by coupling to linear waves. Instead the field amplitude decreases exponentially outward from the localized structure, analogously to the evanescent wave in total reflection in optics. (This can be described by an imaginary wavenumber.) The second element needed to understand the problem is a general integral relation that determines the velocity of the center of mass of any localized perturbation, and thereby the translational velocity of monopolar vortices. (Again, similar integral relations exist for many other non-linear wave equations, including the KdV equation.) This relation shows that if the amplitude is positive (as in geostrophic anticyclones) and large enough, the center of mass propagates faster than the linear waves. Thus, a vortex with such an amplitude can be stationary and localized. In the present article this conclusion is verified both by analytic calculations and numerical simulations. Explicit stationary solutions are found by perturbation analysis, using a circular vortex with arbitrary radial profile as the zeroth-order solution. The beta effect and the propagation velocity are included in the first-order solution. Numerical simulations then show that this solution is indeed stationary to a very good approximation. The article is organized as follows. In Section 2 we present the basic equations (the shallow water equations) and the relevant integral relations.
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.I. N Y C A N D E R
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In Section 3 we briefly consider the problem using the conventional quasi-geostrophic scaling. In Section 4 we employ a generalized geostrophic scaling which allows for arbitrary amplitude. A dynamic equation which is valid for this scaling is solved both by perturbation analysis and by numerical simulation. In the concluding section the results are summarized, and the necessary amplitude for the vortices to be stationary is estimated, using the simple physical picture that has been presented here. 2. BASIC EQUATIONS
The dynamics of a thin layer of fluid with a free surface on a rotating planet is described by the equation of motion, 3U - - + v " Vv = - V ( g H o h at
) + f(y)v
X.;
(1)
and the equation of continuity, Oh 1 -- + --V[/-/0(1 at Ho
+ h),,] = 0
(2)
Here g is the acceleration of gravity, f = 2w0sin a (where a is the latitude and w 0 the rotation frequency of the planet) is the Coriolis parameter, and H = H0(1 + h ) is the depth of the fluid. The x-axis points east and the y-axis north. The external inhomogeneity that we will mainly consider here is that of the parameter f ( y ) , i.e. the /3-effect, which is the essential one for planetary flows. This is most simply done by introducing the /3-plane approximation:
f(Y) =fo + f ' Y
(3)
where f ' is considered as a constant. In the laboratory experiments, the variation of the effective gravity g(y) must also be included, and the inhomogeneity of the equilibrium depth Ho(y) can be very important. This case is considered in detail by Nycander (1992). A well-known property of eqns. (1) and (2) is the conservation of the potential vorticity along the fluid trajectories: - - + v "V 3t
where
D_,=2.V x v
1 +h
=0
(4)
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ANTICYCLONES
Equations (1) and (2) also conserve the mass anomaly M,
M=fh dS and the energy E, 1
E= ~ f [(l + h)v2 + gHoh2] dS By analogy with the motion of charged particles in a magnetic field we define the generalized m o m e n t u m (Nycander, 1990): P x = f [ ( 1 + h)vx - hF(y)] dS where F ' ( y ) = f ( y ) ,
and
Py= f[(1 + h)c;, + hxf(y)]
as
Using eqns. (1) and (2) we obtain d
~C=0
(5)
and
~-tPv= f (1 + h ) x L , , . f ' ( y ) - H o g ' ( y ) ~
dS
(6)
In the case of a solution propagating with the constant velocity u and preserving its shape, eqn. (6) reduces to
u f hf
a s = f ( 1 + h)xc, y f ' ( y ) as
where we have assumed that g ' ( y ) = 0. For an almost circular vortex, this expression can be approximated as oc
f, U
--
-
-
2f°
[1 + h(r)]C,o r2 dr "U oc
~ h(r)r dr
where c,o is the azimuthal velocity. This is equivalent to the relation (4.6) in the paper by Nof (1983). If we substitute v0 from eqn. (1), expressing the Coriolis force through the pressure gradient and the centrifugal force, following Killworth (1983), it can also be rewritten as
u = -L, R 1 + gi~oM
(7)
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.1. N Y C A N D E R
AND G.G. SUTYRIN
where v a = f ' g H o / f 2 is the maximum phase velocity of the Rossby waves. Since the mass anomaly M is positive for anticyclones and negative for cyclones, while the energy is always positive, only anticyclones can propagate faster than the linear Rossby waves. We have seen that the general integral relation (6) essentially determines the center-of-mass velocity in the e a s t - w e s t direction. Similarly, it may be shown that eqn. (5) means that any localized solution is confined to a certain latitude. Equations (1) and (2) describe high-frequency gravitational waves, as well as low-frequency geostrophic motion. We are only interested in the low-frequency branch, which is characterized by a small Rossby number: f~
(8) We also assume that length scale L of the motion is much smaller than the scale of the variation of the Coriolis parameter, i.e. that the p a r a m e t e r
Lf' ~
f{,
(9)
is much smaller than unity. The last scaling p a r a m e t e r describes the typical amplitude: eh ~ h
(10)
To simplify the equations it is also often assumed that e h << 1. We finally note that the ratio between two of the scaling parameters gives a typical length scale: E,o
p2
el,
L2
(11)
where p is the Rossby radius, p2 = gHo/f{2. Thus, if the amplitude is larger than the Rossby number, the length scale must be larger than the Rossby radius. 3. THE EQUIVALENT BAROTROPIC QUASI-GEOSTROPHIC EQUATION
Using the conventional quasi-geostrophic scaling, et~ ~ % ~ e,o ~ e
(12)
and expanding eqn. (4) to order e 2, we obtain the well-known equivalent barotropic quasi-geostrophic equation, O 0 t ( h - V 2 h ) = {h, V2h + fly} (13)
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479
where we have transformed to dimensionless variables by f o t --, t, r --* por and /3 = P o f ' / f o , where p02 = gHo/f~ ~. The Jacobian (Poisson bracket) is defined by aa ab (a, b}--- - 3x ay
ab aa ax 3y
All terms in eqn. (13) are of order e 2. Linearizing eqn. (13) and Fourier expanding we obtain the dispersion relation for linear Rossby waves: w kX
-/3 -
-
-
1 +k 2
(14)
Thus, the phase velocity t,ph of linear waves travelling in the east-west direction lies in the interval -/3 < Vph < O. Any localized solution of eqn. (13) can be described by the linearized version of the equation at some distance away, where the amplitude is small. In general, stationary solutions of this linear equation travelling with the velocity u2 are sinusoidal oscillations if u is in the linear interval, -/3 < u < 0. Any localized solution of the full non-linear eqn. (13) travelling with a velocity in this interval, therefore, has an oscillating tail of linear waves. If such a solution is stationary, it must have infinite energy, and is poorly localized. Conversely, if it has finite energy and is initially well localized, it can not be stationary, and radiates energy by coupling to linear waves (similarly to ship waves, or Cerenkov radiation). If the velocity is outside the linear interval, on the other hand, the amplitude in the tail region decreases exponentially outward, corresponding to a negative value of k 2 in eqn. (14). Only in this case can stationary and localized solutions of eqn. (13) with finite energy exist. This conclusion can be confirmed by looking directly for stationary solutions of eqn. (13). Setting a / a t = - u a / a x it can be written {h + uy, h - V2h - / 3 y } = 0
(15)
i.e. the potential vorticity Vh - V2h - / 3 y must be constant along streamlines in the moving frame of reference, defined by h + uy = const. On open streamlines the value of the potential vorticity is d e t e r m i n e d by the upstream condition at infinity, h ~ 0 as r ~ ~. The equation for stationary solutions in the region of open streamlines then becomes
If the coefficient on the right-hand side of eqn. (16) is positive the amplitude decreases exponentially outward, while in the opposite case h
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J N Y C A N D E R A N D G.G. S U T Y R I N
oscillates with a slowly decreasing amplitude. Again, we conclude that localized, stationary solutions are possible only if u < -/3 or u > 0. Note that the dispersion relation (14) can be obtained from eqn. (16) by setting u = w/k~. and V 2= - k 2. We also conclude that all localized, stationary solutions of eqn. (13) have a region of closed streamlines, since eqn. (16) has no non-singular solution in the whole plane when 1 + f i / u is positive. To determine whether stationary, localized solutions of eqn. (13) exist, we must then determine the velocity u in some independent way. Multiplying eqn. (13) by x and integrating we obtain d dt
fxh dx dy f h dx dy
-
/3
(17)
We have here integrated partially, assuming that the solution is localized, and used the fact that the mass anomaly f h d x d y is conserved. This equation can also be obtained directly from eqn. (6) by applying the scaling (12) and expanding in c. Equation (17) implies that, regardless of the amplitude, the velocity of the center of mass is -f12, i.e. on the boundary of the linear interval. Thus, any stationary solution must travel with this velocity, unless the mass vanishes (as is the case for dipole vortices). Inserting u = - f i into eqn. (16), we find that stationary, localized vortex solutions of eqn. (13) with finite energy and non-zero mass do not exist (Larichev, 1984; Flierl, 1987; Nycander, 1988). We conclude that the observations can not be explained within the framework of the equivalent barotropic quasi-geostrophic equation. This is also immediately apparent from the fact that eqn. (13) is invariant under the transformation h ---, - h , y ~ - y . Cyclones and anticyclones, therefore, evolve similarly, which contradicts the observations. 4. GENERAL GEOSTROPHIC SCALING 4.1. General analysis In the previous section we saw that the velocity of the center of mass of localized solutions of eqn. (13) is never in the non-linear region u < -/3 or u > 0, and that it is independent of the amplitude. The reason is that the conventional geostrophic scaling (12) involves an expansion in small amplitude. Equation (13), therefore, can not describe the effect of large amplitude, which must be responsible for the difference between cyclones and anticyclones.
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In the general geostrophic scaling we allow for arbitrary amplitude, while still expanding in the small parameters E,o (geostrophic approximation) and % (/3-plane approximation). The dispersion relation (14) is then still essentially unchanged. The main difference is that since the typical length scale can now be much larger than the Rossby radius, cf. eqn. (11), the latitudinal dependence of the phase velocity of linear Rossby waves must be taken into account. This will be discussed in detail below. To find out whether we can expect stationary vortices to exist, we should then generalize the integral relation (17), which determines the velocity of the center of mass, to arbitrary amplitude. This is done by expanding the integral relations (5) and (6) in E,, and Eta, while keeping Eh arbitrary. To lowest order in e~ we obtain the geostrophic velocity from eqn. (1),
Vo =
gH° ~. × Vh f
(18)
Inserting this into eqns. (5) and (6) with g' = 0 we obtain d dt
fyh dx dy = 0 fh d x dy
(19)
and
-
dt
fh dx dy
(20)
/3
fh dx dy
We have here transformed to the same dimensionless variables as in eqn. (13). Equation (20) has earlier been obtained by Nycander (1990) and Cushman-Roisin et al. (1990). Comparing it with eqn. (17), we see that it contains a non-linear correction which is proportional to the amplitude. (The same non-linear correction can also be obtained from eqn. (7) by expanding to lowest order in E~o,which means that we keep the potential energy and neglect the kinetic energy.) If the amplitude is positive, as for anticyclones, the velocity of the center of mass is in the non-linear region u < -/3, and localized stationary solutions are possible. If the amplitude is negative, on the other hand, the velocity is in the region of linear Rossby waves, -/3 < u < 0. (Note that since h can not be smaller than - 1 , the velocity of the center of mass always satisfies the inequality u < - / 3 / 2 , as
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J. N Y C A N D E R
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pointed out by Cushman-Roisin et al. (1990).) Cyclones, therefore, quickly lose energy by coupling to linear waves. Thus, the basic features of the observed vortices can be explained from the dispersion relation of linear Rossby waves and the general integral relation (20). This conclusion will be confirmed by finding explicit stationary anticyclonic solutions, and by numerical simulations. Finally, it should be noted that if a variable effective gravity as in the laboratory experiments were included, the non-linear term in eqn. (20) would be cancelled, and we would get the same result as in eqn. (17). Thus, there is still no difference between cyclones and anticyclones, which makes it necessary to calculate center-of-mass velocity to higher order in e,o. This is done by Nycander (1992).
4.2. Approximate dynamic equation Using the general geostrophic scaling we can derive a simplified equation, where the high-frequency gravitational waves contained in eqns. (1) and (2) have been filtered away. We first obtain the velocity by rewriting the equation of motion (1) as --+(f+f~)~×13= 0t
- V gHoh+
and expanding in e~o. The zeroth-order velocity is given by eqn. (18). Including the first-order correction we obtain
v-
f+~
~.×V gh+
2-
-
(21)
7 V g~-
where
Inserting eqn. (21) into eqn. (3) we obtain
ah 0t
H(,.[l+h
1 [ ah)]
+~---~ a ~ V / g--~-
{l+h
H0[~]
--H o l ) + f , gh+ V
2} (23)
Still assuming that et~ is small, we use the/3-plane approximation, and let f be given by eqn. (3). The right-hand side of eqn. (23) is a generalization of the Jacobian in eqn. (13), with the general expression for the potential vorticity used, and the dynamic pressure included in the pressure term.
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To derive the integral relation for the center-of-mass velocity, we multiply eqn. (23) by x and integrate. Expanding the result to lowest order in e~o, and regarding g as constant, we recover eqn. (20). The KdV-term h ah/ax, which is responsible for the non-linear contribution to the center-of-mass velocity in eqn. (20), comes from the Jacobian {h/f, gh} on the right-hand side of eqn. (23). In the experiments in paraboloidal vessels, where g ( y ) is proportional to 1/f(y), this term vanishes (Nycander, 1992). However, to next order in e~ other contributions appear, corresponding to the kinetic energy in eqn. (7). Equation (23) is valid for small Rossby number, but arbitrary amplitude, and was first proposed by Sutyrin and Yushina (1986). (They did not include the factor 1/f in the second term on the left-hand side, but the difference is only significant outside the region of validity of the equation itself.) According to a recent study (Allen et al., 1990) this model is the most accurate among the various intermediate geostrophic models that have been proposed. It conserves the mass and the potential vorticity, while the energy is approximately conserved for small eo~. It should finally be r e m e m b e r e d that the relation (11) means that a structure with h ~ 1 must have a much larger length scale than the Rossby radius, since e~o is assumed to be small. We first calculate the dispersion relation for linear waves. Linearizing eqn. (23) and expanding in /3 we obtain ah _
02h _
(l+2/3Y)aT+Z/3ayat
__V
ah ah at - / 3 a-X-X= 0
2 __
using the same dimensionless units as in eqn. (13). Strictly speaking, this equation can not be Fourier expanded in the y-direction, since one coefficient is explicitly y-dependent. However, since we have assumed that /3y << 1, we can use the local approximation, which is valid if the coefficient varies on a much larger length scale than the perturbation, i.e. if k~, >>/3. We then obtain o~ -/3 k--~ = 1 + 2/3y + k 2 +
2i/3ky
(24)
The phase velocity of linear waves in the x-direction is now confined to the interval /3 < Re 1 + 2/3y
< 0
(25)
Equation (24) is a generalization of eqn. (14). The new effects are the y - d e p e n d e n c e of the phase velocity of Rossby waves, and the linear
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J. N Y C A N D E R
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amplification of waves with northward phase velocity, due to the imaginary term. These effects are the result of taking into account the finite radius of the planet. Actually, this is not done completely consistently in eqn. (23). If, for instance, one more term in the Taylor expansion (3) is kept, the coefficients of the new terms in eqn. (24) have to be modified. However, the main new effect in eqn. (24), that the phase velocity of Rossby waves increases towards the equator, is true also in reality. To improve the description further, one should start from the shallow water equations in fully spherical coordinates. Equation (23) without the factor 1 / f in the second term has been solved numerically by Sutyrin and Yushina (1986, 1988), using a monopole vortex as initial condition. They found that a cyclone quickly loses its energy by exciting linear waves, while an anticyclone with large amplitude can propagate unchanged for a long time, after an initial adjustment period when the shape becomes slightly non-circular. It will here be shown how a stationary solution of this kind can be constructed explicitly. We first look for a stationary solution of eqn. (23) travelling with the velocity u2, setting O/Ot = - u 3 / O x . The equation can then be written {q, 0} = 0
(26)
where the potential vorticity is given by q-
1+~
+/3y+V"
l+/3~--y
and the Bernoulli function by 9-1( 0=h+_
Vh )2 -3t-b/(y l+/3y
+ /3--~+Y2
1 Oh) l + / 3 y 0y
(28)
The time and space variables have here been normalized as in eqn. (13); thus, u and/3 are dimensionless parameters. In the general solution of eqn. (26), 0 is an arbitrary function of q. In the region of open isolines of q, this function can be uniquely determined by the boundary condition h --+ 0 at infinity, giving V"
l+/3y
=(l+h)
[
- 1 -/3y
( l + / 3 y ) 2+ 2 / 3 h + - u u
1+/3y
t2
+ - l + / 3 y 3y
]
(29)
in agreement with the result obtained by Sutyrin and Yushina (1986). To simplify this expression we note that the ratio between % and et, can be
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ANTICYCLONES
given a simple interpretation: % %
/3 l'
(30)
where L, is the fluid velocity given by eqn. (21). Since the propagation velocity u of the vortex is of the same order as /3, this means that we must have % ~ et~ in the separatrix region, where u ~ t,. (In the core region, on the other hand, we may have t, >> u, and et, can be arbitrarily large.) Assuming that % ~ e~ and expanding eqn. (29) to order e h2 we obtain V2h=
( )
1 + /3 h + 2 / 3 y h + t~-
~
-
2
u2
+2/3--+--(Vh) ay 2u
2
(31)
W e stress that this equation is only valid in the far field and in the vicinity of the separatrix. Also note that the dispersion relation (24) can be obtained from eqn. (31) by linearizing and setting u = w / k X and V = ik. The character of eqn. (31) d e p e n d s on the sign of the two first terms on the right-hand side. In the northern half-plane y > (u + / 3 ) / 2 / 3 2 the solution has an exponential profile, and in the southern, complementary half-plane it has an oscillatory character. For a vortex to be well localized, the oscillatory region must be sufficiently far away. This means that the coefficient 1 + / 3 / u must be positive, and large enough that the first term on the right-hand side of eqn. (31) dominates the second one. W h e t h e r this condition can be satisfied d e p e n d s on the velocity u, which is determined by the amplitude in the main part of the vortex, cf. eqn. (20). If the amplitude is very small, e h ~ e~ as for the conventional geostrophic scaling, then u = -/3, and the line separating the oscillatory and exponential regions runs right through the center of the vortex. The energy is, therefore, dispersed by coupling to linear waves. If the amplitude is slightly larger, e h ~ % as in the IG scaling region (Matsuura and Yamagata, 1982; Williams and Yamagata, 1984), the coefficient 1 + / 3 / u does not vanish, but it is still small, of order e h, and the two first terms on the right-hand side of eqn. (31) are of the same magnitude. The bordering line of the oscillatory region is then close to the vortex. Such an anticyclone can live for a rather long time, but it can not be considered as stationary, since it excites linear waves on the southern side and thereby gradually loses its energy. This has been observed by Matsuura and Y a m a g a t a (1982) in a numerical simulation of a dynamic equation valid for the IG scaling. They also found that cyclones quickly disintegrate, which is natural since they are entirely within the oscillatory region. If the amplitude is positive and large, e h ~ 1, we see from eqn. (20) that the vortex moves significantly faster than linear waves, so that 1 + / 3 / u is of
486
J. N Y C A N D E R A N D G . G . S U T Y R I N
order unity. The oscillatory region y < (u +/3)/2/82 is then far away, and the amplitude decreases exponentially until it is reached. The leakage of energy by coupling to linear waves is, therefore, very small, and in practice the vortex can be considered as stationary. This is the kind of vortices that have been seen in the simulations of eqn. (23).
4.3. Explicit solution We will now find explicit stationary solutions of the kind described above. The plane is first divided into two regions by a circle with the radius r o. We solve the equations separately in both regions, and finally match the solutions at the boundary. In the inner region, containing the main part of the vortex, we allow the amplitude to be of order unity, e h ~ 1, and the vortex, therefore, must be much larger than the Rossby radius in order to keep e,o small. In the outer region, containing the separatrix, we assume that % ~ e,o. The length scale there (i.e. the width of the separatrix region) is, therefore, much smaller than that of the whole vortex, and of the same order as the Rossby radius. The problem is solved by perturbation analysis, setting h = h 0 ( r ) + hi(r, 0). We assume that the zeroth-order solution is circular, and treat /3 and u as small first-order parameters. A similar expansion was used by Larichev and Reznik (1976) and Nycander (1988) to study stationary vortex solutions with zero mass in the traditional quasi-geostrophic approximation. We first consider the inner region, where all streamlines are closed. Expanding eqns. (27) and (28) to first order we obtain q-
[
1 l+h(~ l + V 2 h o + V Z h l
hi
1+h¢~ (1 +
Oho]
V2ho)+/3y-/3yV2ho-/3~-y~y
(32) and O=ho+~(Vho)
2+h~+vh o-Vhl-/Sy(Vho)'+u
y+
(33)
Integrating eqn. (26) along 0 we obtain dOo dqo q' d r -O~ d~-
(34)
We then transform eqns. (32) and (33) to polar coordinates, setting y = r sin 0 and h I = " q l ( r ) + E l ( r ) sin 0
(35)
STEADILYTRANSLATING ANTICYCLONES
487
Inserting the result into eqn. (34) and separating the variables we obtain a homogeneous equation for r/l(r) and an inhomogeneous equation for s~l(r). For our purposes we only need the equation for ~:~:
1 [ Vzs~1 ~1 r2 1 +h 0
~1
(1 + VZho) +/3r(1 - V2h,,) -/3hl,
]
1 +h 0
d[ 1 ,2 × d r h° + 2 ( h ° ) d
=[~'+hi'~i-/3r(h;')2+u(r+h~))]d- r
1 + V2ho
l+ho
(36)
One of the two homogeneous solutions (i.e. solutions for /3 = u = 0) is st1 -- hi1, corresponding to a uniform displacement of the vortex. By making the ansatz
~l(r) = z(r)hi,(r )
(37)
eqn. (36) can, therefore, be reduced to a first-order differential equation in z'(r):
a(r)z"(r) + b ( r ) z ' ( r ) = c ( r )
(38)
where
a(r) = (h;,)2(1 + h ; ) b(r)=
2ho+ hi~ ( l + h 0 ) h l , r
( l + h 0 ) ( h 0 ) -~r
6(1
c(r) = [u(r + h;i ) - /3r(h;i)2](1 + ho)~-r
1 +h 0
+ V2ho ) 1 +h o
(39)
+/3h;,[h'o-r(1- V2h0)](1 +h~) The integrating factor a(r) of eqn. (38) is a ( r ) = exp
( r tas : ,
r(h°'21+h 0
exp{ rl+h0 d[
l + h 0 ds s ( l + h 0 )
as)
Solving eqn. (38) and requiring regularity of ~:1 at the origin we obtain 1
/.rc(s)
, ,
z'(r) - a(r) Jo ~(-~ats) ds
(40)
We then solve eqn. (31) in the outer (separatrix) region, again assuming the zeroth-order solution to be circular. To lowest order we only include
488
J. N Y C A N D E R
AND
O.O. SUTYRIN
the first term on the right-hand side of the equation, since it must be d o m i n a n t for a well-localized solution:
with the solution h0(r ) =
CKo(Kr)
(42)
where C is a constant and K2 = 1 + /3/U. Using the conventional geostrophic scaling all the other terms of eqn. (31) are of equal magnitude, and they are, therefore, included in the first-order equation. Making the ansatz (35) and separating the variables we again get one equation for ~¿(r) and one for r/l(r). In this case, however, both equations are inhomogeneous, with the inhomogeneous terms in the equation for ~ coming from the quadratic terms in eqn. (31) (the third and fifth terms). The equations for ~1 in the inner and outer regions can be regarded as one second-order i n h o m o g e n e o u s differential equation. Requiring regularity at the origin and rll ---' 0 at infinity we know that this equation has a unique solution. This solution is only interesting for numerical simulations, and will not be discussed further. T h e equation for sol in the outer region is
d2~:, last1 dr 2 + r dr
( 1 ) ~ + K2 ~ = 2/3(rh° +
h°)
(43)
The boundary conditions at r = r 0 are that ~l(r0) and sOl(r0) should be continuous. F u r t h e r m o r e , since the h o m o g e n e o u s solution ~ = h'0 simply corresponds to a uniform displacement of the vortex, we can choose sOl(r0) = 0
(44)
This condition means that we have d e t e r m i n e d the position of the vortex in the y-direction. (In the numerical calculations below a different choice is made.) Equation (43) can easily be integrated using the Green's function. Differentiating the result we obtain
~;("o) -
1 £~rKl(Kro)Zfi(rh o + h;,) dr roK,(Kro )
(45)
u
A n o t h e r expression for ~:i(r0) can be obtained from the solution (40) in the inner region, using eqns. (37) and (44):
hi,(ro)
.r,c(r)a(r)
~;(r°)- a(ro~ ~ol ~ - ~
dr
(46)
STEADILY
TRANSLATING
489
ANTICYCLONES
Requiring eqns. (45) and (46) to be equal, we obtain a relation that determines the velocity u of the vortex. In general, however, the problem is quite complicated, since the value of u is needed even for the zeroth-order solution ho(r), as can be seen from eqn. (41). To simplify it we make two approximations. First, only terms of the lowest non-vanishing order in e o 1/2 in the inner region. We then assume are kept, observing that d / d r ~ E~o that the amplitude of the vortex is large, so that h 0 ~ 1 in the inner region, while h 0 is small in the outer (separatrix) region. This means that the contribution from eqn. (45) can be neglected altogether, while the upper limit of the integral in eqn. (46) can be extended to infinity. In effect, we require the solution in the inner region to approach zero smoothly as r ~ ~. With these approximations we obtain
u ~ hor d r = - / 3 ~
hi,+
dr
(47)
This is a special case of the general relation (20) determining the velocity of the center of mass. Thus, the velocity is mainly determined by the profile in the inner region, while details of the separatrix region only have a small effect. From eqn. (41) this velocity determines how quickly the amplitude decreases in the separatrix region. The first-order correction to the propagation velocity can then be determined from eqns. (45) and (46). Geometrically, the effect of the dipolar part of the solution is that the still approximately circular isobars (isolines of h) are no longer concentric. However, for a particular class of radial profiles the dipolar part exactly vanishes, and the isobars are still concentric circles. These profiles can be found by solving eqn. (39) for c = 0. Expanding this equation to lowest order in the Rossby number, and integrating once with h 0 ---, 0, r ~ oc, we obtain V2h0 = --(1 + h 0 ) In(1 + h 0 ) + h o u If the amplitude is small this equation can be further expanded, giving V2h0 =
1+--/3 h 0 + - - u 2 u
(48)
Equation (48) has a localized solution, describing an anticyclonic vortex with a unique relation between amplitude and size. The coefficient 1/2 of the last term agrees with the weakly non-linear model of Mikhailova and Shapiro (1980), but not with the other weakly non-linear models (Flierl, 1979; Petviashvili, 1980; Charney and Flierl, 1981).
490
J. N Y C A N D E R AND G.G. SUTYR1N
4. 4. N u m e r i c a l s i m u l a t i o n
In the calculations a finite-difference non-dimensional version of eqn. (23) for 64 × 64 grid points was used with the same boundary conditions as were explored by Sutyrin and Yushina (1988): l~. = 0 at all boundaries and shifting the region along the x-axis together with the center of the anticyclone. This assured that an undisturbed state was maintained ahead of the vortex and permitted the wave to pass behind the vortex, at the eastern boundary. Wave disturbances did not propagate ahead of the vortex in this problem, since its velocity exceeded the maximum phase velocity of Rossby waves.
The zeroth-order solution was chosen as ho=A
+ (1 - A )
h~, = C K o ( K r )
e x p ( - s r 2) - B s r 2
at r < r o
(49)
at r > r o
(50)
The coefficients A, B and C were defined by matching h0, hi~ and hi'~ at r = r 0 to avoid discontinuities in the vorticity field. In all simulations the value sr o = 3 was chosen, which was large enough that the propagation velocity calculated from eqn. (47) was approximately the same as for a purely Gaussian vortex with the amplitude h 0 ( 0 ) = 1: u = -1.25/3. This gives K = 0.45, cf. eqn. (42). The first-order solution was calculated numerically from eqns. (37) and (40) in the region r < r 0 with z ( 0 ) = 0. This means that we choose the center of the solution to be situated at the origin. In the outer region the h o m o g e n e o u s solution z ( r o ) h ' o ( r ) was chosen for simplicity, since the contribution from eqn. (45) can be neglected if the amplitude of the vortex is large. In our case h0(0) = 1. In the/3 plane approximation in eqn. (23) we put /3 = 0.007. The only p a r a m e t e r which was varied in the different simulations was s, which determines the diameter of the vortex. In each case the evolution of an initially circular vortex was compared with that of one where the first-order solution was included. We do not show any direct contour plots of h ( r ) , since almost no change is detectable by eye in any of the simulations. Instead, we show plots of the translational velocity of the center of the vortex (defined as the point of maximum amplitude h) as a function of time, and contour plots of the first-order stationary solution, which has a dipole character. (1) The value s = 0.02 corresponds to the case M1 (very large vortex) in the paper by Williams and Wilson (1988). In Fig. 1 it can be seen that the initially circular vortex (dashed line) at first propagates faster than the stationary solution, and that it is displaced poleward during the adaptation stage. Significant velocity oscillations with a period approximately equal to
STEADILY TRANSLATING ANTICYCLONES
491
•. 6 0 ] ',,,,,,,
1.4 01
"\,
i .20
""
£) 2
""
~
'
""
,. /
~j
,£-
soo
l o'oo
ls'oo
2ooo
Fig. 1. Evolution of the westward speed u X (upper) and meridional speed uy of the vortcx center. Here s = 0.02, which is also equal to the characteristic angular velocity l / / L . The time unit is f~] ~ (approximately 4 h). The initially circular case is shown by the dashed line, and the construction with the first-order solution by the solid line.
that of the swirling motion are also seen. They seem to be caused by the different angular velocities in different parts of the vortex. If the first-order solution is included the vortex is almost exactly stationary (solid line). The first-order part (Fig. 2) is a dipole with a cyclone towards the pole and an anticyclone towards the equator, and it reduces the westward drift of the
32
i
i6 1
32 f L 3P
16
0
16
32
Fig. 2. The first-order solution divided by /3 at s = 0.02. The length unit is P0, and thc contour interval is 0.4.
C U a! l ~ " ~ * ' u !
auu.luv,,a
dq,.L
I t U = a
L
ludaA,=,
C
*a!~l ~
dtu~a
~'H,L
0 I
t
17 o . ~
"+ : q
i
i
T
T
~] L
--IDa-
"l~lllJJ!J AII~!*!u! '~LI~ 'J~') ~!LI'I Ul tUOt)[) i ~ j~ u!~J.IUi~l-Ul:;U.ll.[~ll ) ALl, J ~ u ~ u oq}, U! L asea o l s p u o d s a ~ o a (az F mP.~apouJ) [ [ ' 0 = ~ ~ n l e A oqJ~ (E)
XOl.IOA 0141 alO apF U.IOql..IOU aql UO .Iadaals a[tJo.Id aql pue '3!..tluo3uo3 lou uaql aJ-e (74 jo sou!los!) s.reqos! aq,L "paads ale!.Ido~dde aql ol xal.IOA aioq~
I [ U = a OOg
lu,~JA,.,
049 v
I
OOg
o!d
~,-....,tur,~ oq<.L
OOU
C~O L
t
°!zl
:.:
L
,i
#i i* / * t
II
"~"
\\\ \
STEADILY
TRANSLATING
493
ANTICYCLONES
1.4-
15s%
~-
~
I Jx
1 2
0 1 t
D
~
J ~/f
(} 1 -!. . . . 7~ F i g . 5. T h e
•
s
i
? 0 i} same
I 4 (. } i."~
I ,i~<2 /
as Fig. 1 except
T
I ~z [. J :.
• i~! , ,
'
s = 0.05.
vortex moves slower than the stationary solution (Fig. 3), and the center is displaced toward the equator during the adaptation period. The construction with the first-order solution shows quite stationary behavior (solid line). From Fig. 4 it is seen that the dipolar first-order part now has the cyclone toward the equator and the anticyclone toward the pole; thus, it increases the westward velocity of the whole vortex. In this case the profile is steeper on the southern side of the vortex core. (3) The value s = 0.05 corresponds to an intermediate vortex size. The first-order solution now has a small amplitude, and does not significantly influence the drift velocity of the vortex (Fig. 5). It also has a more complicated structure (Fig. 6), while the isobars are almost concentric. Apparently, this case is close to the special class of radial profiles given by c = 0 in eqn. (39). These examples demonstrate the different structure of stationary propagating vortices depending on the vortex size. In all cases the propagation velocity agrees well with eqn. (47), as was also found in the recent paper by Cushman-Roisin et al. (1990). For lens-like eddies (steadily propagating anticyclones with finite Rossby n u m b e r and zero depth at the vortex boundary) with linear azimuthal velocity profile, the dipolar correction was calculated by Killworth (1983) and Flierl (1984). This corresponds to the limit of infinite amplitude and infinite size compared with the Rossby radius (or H 0-~ 0) in our case. T h e n the problem of localization and radiation of baroclinic Rossby waves does not arise. In both these papers the dipolar correction gives a steeper profile on the northern side of the vortex core, just as in case (1) above. (To see this from the solution of Flierl, one should subtract the h o m o g e n e o u s
494
J. N Y C A N D E R A N D G . G . S U T Y R I N
iG
_±_~.~
i
-IG
i
t
0
i
i
i
i
16
Fig. 6. The same as Fig. 2 except s = 0.05. The contour interval is 0.06. first-order solution, which gives a rigid shift, so that the solution has an extremum in the center.) 5. CONCLUSIONS In Section 4 we derived an expression describing the shape of stationary anticyclonic vortices by perturbation analysis. It was found that such vortices propagate westwards faster than the linear Rossby waves, and that the velocity can not be too close to the maximum phase velocity u R of these waves. The vortex amplitude must be large, i.e. the velocity of the swirling motion must be much larger than the translational velocity of the vortex. Thus, the vortex carries trapped fluid. Since the vortex is in geostrophic balance (the Rossby n u m b e r is assumed to be small), the requirement of large amplitude also means that its diameter should be much larger than the Rossby radius, cf. eqn. (11). The correctness of this analytic solution was then confirmed by numerical simulations. These also showed that the main role of the first-order solution (the dipole part) is to adjust the propagation velocity to the value appropriate for stationary solutions. Although in principle straightforward, the perturbation analysis may appear rather complicated, since different scalings must be used in the exterior and interior parts of the vortex. This should, however, not obscure the fact that the underlying physics is simple. Essentially, it can be understood from only two elements.
STEADILY TRANSLATING ANTICYCLONES
495
The first element is the dispersion relation (24) for linear Rossby waves, which is directly connected to the general equation (eqn. (31)) for the outer region, where the isolines of potential vorticity are open. These equations show that stationary, localized vortices must propagate significantly faster than all linear waves. The other element is the general integral relation (20), which determines the translational velocity of the vortex. An important feature which was included in the present analysis is that the maximum phase velocity L'R of the linear Rossby waves is weakly y-dependent, cf. eqn. (25). It is, therefore, not sufficient for a stationary vortex to propagate just slightly faster than the local value of t, a, since there would still be a region nearby (towards the equator) where Rossby waves with long wavelength travel faster than the vortex. In this region the stationary solution is oscillatory, as pointed out in Section 4.2. Dynamically such a vortex would radiate energy by coupling to linear waves, as was observed in numerical simulations by Matsuura and Yamagata (1982). On the basis of this simple physical picture we can estimate the necessary amplitude for anticyclones to be stationary. The criteria are that the distance Los c to the oscillatory region should be much larger than the 'damping length' L d = K- 1 in the outer region, cf. eqn. (42), and also larger than the radius of the vortex (the oscillatory region must be outside the separatrix). The maximum phase velocity of Rossby waves depends on the latitude as c u = Ccosa/sin2a, where C is a constant. The relevant quantity, however, is the angular velocity around the planet, since the vortex travels on a fixed latitude. Waves propagating closer to the equator travel a longer distance, and must, therefore, travel faster in order to keep up with the vortex. Correcting for this geometric effect, we find that the maximum angular velocity era of the linear waves depends on the latitude as eR~ = C/sin2c~, where C is a r e n a m e d constant. The angular velocity of the vortex can be estimated from eqn. (20) as u~ = -V~a(1 + h / 2 ) , where h is a typical amplitude of the vortex. (For a Gaussian vortex h is half the maximum amplitude.) The distance to the region where u a and the local t'aa are equal is then approximately U Ra
h
tan c~
Lo~c- 2 OVR~ -- 2 R ~
-
~y or
Losc = --R 4
(51)
496
J. N Y C A N D E R A N D G . G . S U T Y R I N
at mid-latitudes. Here R is the radius of the planet. The damping length in the outer region can be estimated from the first term on the right-hand side of eqn. (31) as L d - - ( 2 / h ) 1/2, or in dimensional units L d -- po(2/h) ~/2, where P0 is the Rossby radius. The requirement Iosc >> L d can then be written
~ - >> P0 or approximately
>>
--
(52)
The requirement that the oscillatory region should be outside the separatrix can be written 2D
> -k-
(53)
where D is the diameter of the vortex. These conditions are not satisfied in the atmosphere of the Earth, since Losc is only one tenth of the diameter of a typical anticyclone. For oceanic intrathermocline anticyclones, on the other hand, we typically have h ~ 1 and D < 100 km. Hence the inequalities (52) and (53) are satisfied by a wide margin, and the present analysis a.ppears to be fully applicable. For the Great Red Spot of Jupiter we have D ~- R / 6 and h ~ 1 (Dowling and Ingersoll, 1989), while the Rossby radius P0 is much smaller than R, so that the conditions (52) and (53) are satisfied. Thus, the present analysis can help to understand its persistence. However, it is also clear that the shear in the background flow is very important, as is seen in the experiments of Antipov et al. (1985) and Sommeria et al. (1988). In the vortex experiments in paraboloidal rotating vessels (Antipov et al., 1982, 1988; Nezlin et al., 1990) the effective gravitational acceleration g in eqn. (1) is not constant, but a function of y. This does not change the dispersion relation for linear Rossby waves; however, as was realized during the preparation of this paper, it exactly cancels the non-linear term in eqn. (20). Thus, to lowest order in the Rossby number, the center-of-mass velocity is independent of the amplitude, and coincides with the phase velocity of long-wavelength Rossby waves. The expansion should, therefore, be carried out to higher order. This case is considered in detail by Nycander (1992).
S'I'I~'ADILY T R A N S L A T I N G A N T I C Y C L O N E S
497
ACKNOWLEDGMENT
One of the authors (G. Sutyrin) acknowledges stimulating discussions of vortex solutions with Mikhail Nezlin, Glenn Flierl and Vitaly Larichev. REFERENCES Allen, J.S., Barth, J.A. and Newberger, P.A., 1990. On intermediate models for barotropic continental shelf and slope flow fields: Part 1. Formulation and comparison of exact solutions. J. Phys. Oceanogr., 20: 1017-1042. Antipov, S.V., Nezlin, M.V., Snezhkin, E.N. and Trubnikov, A.S., 1982. Rossby soliton in the laboratory. Sov. Phys. JETP, 55: 85-95. Antipov, S.V., Nezlin, M.V., Snezhkin, E.N. and Trubnikov, A.S., 1985. Rossby autosoliton and laboratory model of Jupiter's Great Red Spot. Sov. Phys. JETP, 62: 1097-11(17. Antipov, S.V., Nezlin, M.V., Rodionov, V.K., Rylov, A.Y., Snezhkin, E.N., Trubnikov, A.S. and Khutoretskii, A.V., 1988. Properties of drift solitons in plasma, following from laboratory experiments with rapidly rotating shallow water. Fiz. Plazmy, 14:1104-1121. Charney, J.G. and Flierl, G.R., 1981. Oceanic analogues of large-scale atmospheric motions. In: B.A. Warren and C. Wunch (Editors), Evolution of Physical Oceanography. The Massachusetts Institute of Technology, Cambridge, MA, pp. 504-552. Cushman-Roisin, B., Chassignet, E.P. and Tang, B., 1990. Westward motion of mesoscalc eddies. J. Phys. Oceanogr., 20: 758-768. Dowling, T.E. and Ingersoll, A.P., 1989. Jupiter's great red spot as a shallow water system. J. Atmos. Sci., 46: 3256-3278. Flierl, G.R.. 1979. Planetary solitary waves. Polymode News, 62: 1, 7-14. Flierl, G.R., 1984. Rossby wave radiation from a strongly nonlinear warm eddy. J. Phys. Oceanogr., 14: 47-58. Flierl, G.R., 1987. Isolated eddy models in geophysics. Annu. Rev. Fluid Mech., 19: 493-530. Kamenkovich, V.M., Koshlyalov, M.N. and Monin, A.S., 1986. Synoptic Eddies in the Ocean. Reidel, Dordrecht, 433 pp. Killworth, P.D., 1983. On the motion of isolated lenses on a beta-plane. J. Phys. Oceanogr., 13: 368-376. Larichev, V.D., 1984. Integral characteristics of localized vortices on the beta-plane. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana, 20: 733-740. Larichev, V.D. and Reznik, G.M., 1976. Strongly nonlinear, two-dimensional isolated Rossby waves. Okeanologiya, 16: 961-967. Matsuura, T. and Yamagata, T., 1982. On the evolution of nonlinear planetary eddies larger than the radius of deformation. J. Phys. Oceanogr., 12: 440-456. McWilliams, J.C., 1985. Submesoscale, coherent vortices in the ocean. Rev. Geophys., 23: 165-182. Mikhailova, E.I. and Shapiro, N.B., 1980. Two-dimensional model of synoptic disturbances evolution in the ocean. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana, 16: 823-833. Nezlin, M.V., 1986. Rossby solitons. Soy. Phys. Usp., 29: 807-842. Nezlin, M.V. and Sutyrin, G.G., 1989. Long-lived solitary anticyclones in the planetary atmospheres and oceans, in laboratory experiments and in theory. In: J.C.J. Nihoul and B.M. Jamart (Editors), Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence. Elsevier Science, Amsterdam, pp. 701-720.
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AND G.(L SUTYI?,IN
Nezlin, M.V., Rylov, A.Yu., Trubnikov, A.S. and Khutoretskii, A.V., 1990. Cyclonic-anticyclonic asymmetry and a new soliton concept for Rossby vortices in the laboratory, oceans and the atmospheres of giant planets. Geophys. Astrophys. Fluid Dyn., 52: 211-247. Nor, D., 1983. On the migration of isolated eddies with application to Gulf Stream Rings. J. Mar. Res., 41: 399-425. Nycander, J,, 1988. New stationary vortex solutions of the Hasegawa-Mima equation. J. Plasma Phys., 39: 413-430. Nycander, J., 1989. The existence of stationary vortex solutions of the equations for nonlinear drift waves in plasmas and nonlinear Rossby waves. Phys. Fluids, BI: 17881796. Nycander, J., 1990. Existence criteria and velocity of stationary vortices in plasmas and in shallow water on a rotating planet. In: Nonlinear World, IV Int. Workshop on Non-linear and Turbulent Processes in Physics, Kiev, 1989. World Scientific, Singapore, pp. 933 -954. Nycander, J., 1992. Difference between monopole vortices in planetary flows and experiments with paraboloidal vessels. J. Fluid Mech., in press. Petviashvili, V.I., 1980. Red spot of Jupiter and the drift soliton in a plasma. JETP Lett., 32: 632-635. Romanova, N.N. and Tseitlin, V.Yu., 1984. On quasigeostrophic motions in barotropic and baroclinic fluids. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana, 20: 85-91. Smith, B.A., Soderblom, L.A., Johnson, T.V. et al., 1979. The Jupiter system through the eyes of Voyager-1. Science, 204: 951-972. Smith, B.A., Soderblom, L.A., Batson, R. et al., 1982. A new look at the Saturn system: the Voyager-2 images. Science, 215: 504-537. Sommeria, J., Mcyers, S.D. and Swinney, H.L., 1988. Laboratory simulation of Jupiter's great red spot. Nature, 331: 689-693. Sutyrin, G.G., 1985. On the theory of solitary anticyclones in a rotating fluid. Dokl. Akad. Nauk SSSR, 280:1101-1105 (transl. Earth Sci., pp. 38-41). Sutyrin, G.G. and Yushina, I.G., 1986. On the evolution of isolated eddies in a rotating fluid. Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza No. 4, 52-59 (transl. Fluid Dyn. (USSR), pp. 550-556). Sutyrin, G.G. and Yushina, I.G., 1988. Formation of a vortical soliton. Dokl. Akad. Nauk SSSR, 299:580-584 (transl. Sov. Phys. Dokl., 33(3): 179-181). Williams, G.P., 1985. Jovian and comparative atmospheric modeling. Adv. Geophys., 28A: 381-429. Williams, G.P. and Yamagata, T., 1984. Geostrophic regimes, intermediate solitary vortices and Jovian eddies. J. Atmos. Sci., 41: 453-478. Williams, G.P. and Wilson, R.J., 1988. The stability and genesis of Rossby vortices. J. Atmos. Sci., 45: 207-241. Yamagata, T., 1982. On nonlinear planetary waves: A class of solutions missed by the quasi-geostrophic approximation. J. Oceanogr. Soc. Jpn., 38: 236-244.
473
Steadily translating anticyclones on the beta plane J. Nycander a and G.G. Sutyrin b a Department of Technology, Uppsala Unit'ersity, Box 534, S-751 21 Uppsala, Sweden b p.p. Shirshoc bzstitute of Oceanology, Academy of Sciences o]"the USSR, 23 Krasiko~a St.. 117218 Moscow, Russia (Received 28 February 1991; revised 18 June 1991; accepted 9 November 1991 )
ABSTRACT Nycander, J. and Sutyrin, G.G., 1992. Steadily translating anticyclones on the beta plane. Dyn. Atmos. Oceans, 16: 473-498. Approximately stationary anticyclones in shallow water on a rotating planet are studied both analytically and numerically. They consist of a monopolar part with large amplitude and a dipolar part with small amplitude, proportional to the beta parameter. An explicit solution for the dipolar part is obtained with an arbitrary radial profile of the monopolar part. Numerical experiments show that this dipolar part accelerates or decelerates an initially circular vortex depending on the vortex size. The propagation speed is determined by a general integral relation. It is found that for the vortices to be stationary, this speed should not be close to the phase velocity of linear Rossby waves. This requires a strongly elevated surface (large amplitude). The vortices contain a core region with trapped fluid, unlike one-dimensional KdV solitons. Applications of the solution to tong-lived geophysical eddies are discussed.
1. I N T R O D U C T I O N
Among long-lived geophysical eddies, monopolar anticyclones seem to be the most persistent. The largest of these vortices is the Great Red Spot in the atmosphere of Jupiter. This vortex, which is significantly larger than the Earth, has been coherent for at least 300 years. Other Jovian long-lived vortices, the Large Ovals, have been known for several decades (Smith et al., 1979). Similar large anticyclones, such as Big Bertha, the Brown Spot, Anne's Spot, etc. have been detected in the atmosphere of Saturn (Smith et al., 1982). In the oceans of the Earth the recently discovered compact intrathermocline anticyclones can exist for several years (McWilliams, 1985; Kamenkovich et al., 1986). Observations show that these vortices have trapped fluid which they transport over thousands of kilometers. Because
C o r r e s p o n d e n c e to: J. Nycander, D e p a r t m e n t of A p p l i e d M a t h e m a t i c s and Theoretical Physics, University of Cambridge, Silver Street, C a m b r i d g e CB3 9EW, UK. 0377-0265/92/$05.00 © 1992 - E l s c v i c r Science Publishers B.V. All rights reserved
474
J. N Y C A N D E R
AND G.G. SUTYRIN
of their high intensity, long lifetimes and transport properties, the role of isolated vortices in geophysical flows seems to be very important. Since the horizontal scale of the long-lived geophysical eddies is much larger than the depth of the atmosphere or the ocean, the shallow water approximation is generally used when modelling them. The validity of this approach has been discussed in detail by Williams (1985). During the last decade the problem of the resistance of such localized structures to Rossby wave dispersion has been widely studied (Flierl, 1987). Large-scale eddies are typically near geostrophic balance, thus the perturbed surface in cyclones is lower than the equilibrium level, while anticyclones have an elevated surface. To lowest order in the perturbation of the surface (as in the quasi-geostrophic approximation), the behavior of cyclones and anticyclones is similar. Taking into account higher order terms in the the amplitude breaks this symmetry, and allows non-linear steepening to compensate the dispersion in anticyclones. Models of this kind, with weak non-linearity and a unique analytic relation between the potential vorticity and the Bernoulli function, were proposed by several authors (Flierl, 1979; Petviashvili, 1980; Mikhailova and Shapiro, 1980; Charney and Flierl, 1981). In their solutions the perturbation of the fluid depth is small, and there is a unique relation between the amplitude and the vortex diameter. The size is assumed to be between the deformation radius and the radius of the planet, i.e. in the so-called intermediate geostrophic (IG) scale. The difference between the evolution of cyclones and anticyclones at this scale was emphasized by Yamagata (1982). However, it has been shown by a thorough analysis that the equation governing the dynamics in the IG scaling region has no stationary localized solution with analytic relation between the potential vorticity and the Bernoulli function (Romanova and Tseitlin, 1984; Nycander, 1989). The reason is that the meridional variation of the Rossby wave propagation speed is essential in the IG dynamics, leading to meridional twisting. Thus, the theoretical solutions mentioned above are inconsistent. Laboratory experiments in shallow water in paraboloidal rotating vessels (Antipov et al., 1982, 1988; Nezlin, 1986; Nezlin et al., 1990) and numerical simulations on the beta plane (Matsuura and Yamagata, 1982; Sutyrin and Yushina, 1986) show indeed that anticyclones can exist much longer than cyclones. They propagate westward faster than linear Rossby waves, transporting fluid in the core. However, the experimentally observed long-lived anticyclones have a large amplitude, with the perturbation of the fluid depth comparable to the equilibrium depth. A close examination also revealed that they are significantly smaller than vortices obtained from a generalization to large amplitude of the above-mentioned models with unique analytic form (Sutyrin, 1985).
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These experimental and numerical results have stimulated the development of a broader concept of solitary vortices, incorporating the features of finite amplitude and trapped fluid (Sutyrin, 1985; Nezlin, 1986; Sutyrin and Yushina, 1988; Nezlin and Sutyrin, 1989; Nycander, 1990). The trapping of fluid rotating around the axis of the vortex gives it qualitatively new properties. One is the existence of a 'memory' of the initial disturbance inside the region of closed streamlines, or, in other words, the absence of a unique relation between the amplitude and the diameter of the vortex. Another is that collisions between solitary vortices are inelastic, unlike collisions between one-dimensional KdV solitons. Essentially, the existence of stationary, localized vortices can be understood from only two elements. The first one is the dispersion relation for linear Rossby waves. The phase velocity of these waves has an upper limit, and in Section 4.2 it is shown that any stationary, localized vortex must propagate faster than all linear waves. This complementarity between the phase velocity of linear waves and the velocity of solitary, stationary structures is a general feature in many non-linear wave equations, for instance the KdV equation, and it is easy to explain. If the velocity of a localized structure coincides with the phase velocity of linear waves, it will radiate energy, similar to Cerenkov radiation. It can, therefore, not be stationary. If, on the other hand, the velocity is outside the region of linear phase velocities, the structure can not radiate energy by coupling to linear waves. Instead the field amplitude decreases exponentially outward from the localized structure, analogously to the evanescent wave in total reflection in optics. (This can be described by an imaginary wavenumber.) The second element needed to understand the problem is a general integral relation that determines the velocity of the center of mass of any localized perturbation, and thereby the translational velocity of monopolar vortices. (Again, similar integral relations exist for many other non-linear wave equations, including the KdV equation.) This relation shows that if the amplitude is positive (as in geostrophic anticyclones) and large enough, the center of mass propagates faster than the linear waves. Thus, a vortex with such an amplitude can be stationary and localized. In the present article this conclusion is verified both by analytic calculations and numerical simulations. Explicit stationary solutions are found by perturbation analysis, using a circular vortex with arbitrary radial profile as the zeroth-order solution. The beta effect and the propagation velocity are included in the first-order solution. Numerical simulations then show that this solution is indeed stationary to a very good approximation. The article is organized as follows. In Section 2 we present the basic equations (the shallow water equations) and the relevant integral relations.
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.I. N Y C A N D E R
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In Section 3 we briefly consider the problem using the conventional quasi-geostrophic scaling. In Section 4 we employ a generalized geostrophic scaling which allows for arbitrary amplitude. A dynamic equation which is valid for this scaling is solved both by perturbation analysis and by numerical simulation. In the concluding section the results are summarized, and the necessary amplitude for the vortices to be stationary is estimated, using the simple physical picture that has been presented here. 2. BASIC EQUATIONS
The dynamics of a thin layer of fluid with a free surface on a rotating planet is described by the equation of motion, 3U - - + v " Vv = - V ( g H o h at
) + f(y)v
X.;
(1)
and the equation of continuity, Oh 1 -- + --V[/-/0(1 at Ho
+ h),,] = 0
(2)
Here g is the acceleration of gravity, f = 2w0sin a (where a is the latitude and w 0 the rotation frequency of the planet) is the Coriolis parameter, and H = H0(1 + h ) is the depth of the fluid. The x-axis points east and the y-axis north. The external inhomogeneity that we will mainly consider here is that of the parameter f ( y ) , i.e. the /3-effect, which is the essential one for planetary flows. This is most simply done by introducing the /3-plane approximation:
f(Y) =fo + f ' Y
(3)
where f ' is considered as a constant. In the laboratory experiments, the variation of the effective gravity g(y) must also be included, and the inhomogeneity of the equilibrium depth Ho(y) can be very important. This case is considered in detail by Nycander (1992). A well-known property of eqns. (1) and (2) is the conservation of the potential vorticity along the fluid trajectories: - - + v "V 3t
where
D_,=2.V x v
1 +h
=0
(4)
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ANTICYCLONES
Equations (1) and (2) also conserve the mass anomaly M,
M=fh dS and the energy E, 1
E= ~ f [(l + h)v2 + gHoh2] dS By analogy with the motion of charged particles in a magnetic field we define the generalized m o m e n t u m (Nycander, 1990): P x = f [ ( 1 + h)vx - hF(y)] dS where F ' ( y ) = f ( y ) ,
and
Py= f[(1 + h)c;, + hxf(y)]
as
Using eqns. (1) and (2) we obtain d
~C=0
(5)
and
~-tPv= f (1 + h ) x L , , . f ' ( y ) - H o g ' ( y ) ~
dS
(6)
In the case of a solution propagating with the constant velocity u and preserving its shape, eqn. (6) reduces to
u f hf
a s = f ( 1 + h)xc, y f ' ( y ) as
where we have assumed that g ' ( y ) = 0. For an almost circular vortex, this expression can be approximated as oc
f, U
--
-
-
2f°
[1 + h(r)]C,o r2 dr "U oc
~ h(r)r dr
where c,o is the azimuthal velocity. This is equivalent to the relation (4.6) in the paper by Nof (1983). If we substitute v0 from eqn. (1), expressing the Coriolis force through the pressure gradient and the centrifugal force, following Killworth (1983), it can also be rewritten as
u = -L, R 1 + gi~oM
(7)
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.1. N Y C A N D E R
AND G.G. SUTYRIN
where v a = f ' g H o / f 2 is the maximum phase velocity of the Rossby waves. Since the mass anomaly M is positive for anticyclones and negative for cyclones, while the energy is always positive, only anticyclones can propagate faster than the linear Rossby waves. We have seen that the general integral relation (6) essentially determines the center-of-mass velocity in the e a s t - w e s t direction. Similarly, it may be shown that eqn. (5) means that any localized solution is confined to a certain latitude. Equations (1) and (2) describe high-frequency gravitational waves, as well as low-frequency geostrophic motion. We are only interested in the low-frequency branch, which is characterized by a small Rossby number: f~
(8) We also assume that length scale L of the motion is much smaller than the scale of the variation of the Coriolis parameter, i.e. that the p a r a m e t e r
Lf' ~
f{,
(9)
is much smaller than unity. The last scaling p a r a m e t e r describes the typical amplitude: eh ~ h
(10)
To simplify the equations it is also often assumed that e h << 1. We finally note that the ratio between two of the scaling parameters gives a typical length scale: E,o
p2
el,
L2
(11)
where p is the Rossby radius, p2 = gHo/f{2. Thus, if the amplitude is larger than the Rossby number, the length scale must be larger than the Rossby radius. 3. THE EQUIVALENT BAROTROPIC QUASI-GEOSTROPHIC EQUATION
Using the conventional quasi-geostrophic scaling, et~ ~ % ~ e,o ~ e
(12)
and expanding eqn. (4) to order e 2, we obtain the well-known equivalent barotropic quasi-geostrophic equation, O 0 t ( h - V 2 h ) = {h, V2h + fly} (13)
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where we have transformed to dimensionless variables by f o t --, t, r --* por and /3 = P o f ' / f o , where p02 = gHo/f~ ~. The Jacobian (Poisson bracket) is defined by aa ab (a, b}--- - 3x ay
ab aa ax 3y
All terms in eqn. (13) are of order e 2. Linearizing eqn. (13) and Fourier expanding we obtain the dispersion relation for linear Rossby waves: w kX
-/3 -
-
-
1 +k 2
(14)
Thus, the phase velocity t,ph of linear waves travelling in the east-west direction lies in the interval -/3 < Vph < O. Any localized solution of eqn. (13) can be described by the linearized version of the equation at some distance away, where the amplitude is small. In general, stationary solutions of this linear equation travelling with the velocity u2 are sinusoidal oscillations if u is in the linear interval, -/3 < u < 0. Any localized solution of the full non-linear eqn. (13) travelling with a velocity in this interval, therefore, has an oscillating tail of linear waves. If such a solution is stationary, it must have infinite energy, and is poorly localized. Conversely, if it has finite energy and is initially well localized, it can not be stationary, and radiates energy by coupling to linear waves (similarly to ship waves, or Cerenkov radiation). If the velocity is outside the linear interval, on the other hand, the amplitude in the tail region decreases exponentially outward, corresponding to a negative value of k 2 in eqn. (14). Only in this case can stationary and localized solutions of eqn. (13) with finite energy exist. This conclusion can be confirmed by looking directly for stationary solutions of eqn. (13). Setting a / a t = - u a / a x it can be written {h + uy, h - V2h - / 3 y } = 0
(15)
i.e. the potential vorticity Vh - V2h - / 3 y must be constant along streamlines in the moving frame of reference, defined by h + uy = const. On open streamlines the value of the potential vorticity is d e t e r m i n e d by the upstream condition at infinity, h ~ 0 as r ~ ~. The equation for stationary solutions in the region of open streamlines then becomes
If the coefficient on the right-hand side of eqn. (16) is positive the amplitude decreases exponentially outward, while in the opposite case h
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J N Y C A N D E R A N D G.G. S U T Y R I N
oscillates with a slowly decreasing amplitude. Again, we conclude that localized, stationary solutions are possible only if u < -/3 or u > 0. Note that the dispersion relation (14) can be obtained from eqn. (16) by setting u = w/k~. and V 2= - k 2. We also conclude that all localized, stationary solutions of eqn. (13) have a region of closed streamlines, since eqn. (16) has no non-singular solution in the whole plane when 1 + f i / u is positive. To determine whether stationary, localized solutions of eqn. (13) exist, we must then determine the velocity u in some independent way. Multiplying eqn. (13) by x and integrating we obtain d dt
fxh dx dy f h dx dy
-
/3
(17)
We have here integrated partially, assuming that the solution is localized, and used the fact that the mass anomaly f h d x d y is conserved. This equation can also be obtained directly from eqn. (6) by applying the scaling (12) and expanding in c. Equation (17) implies that, regardless of the amplitude, the velocity of the center of mass is -f12, i.e. on the boundary of the linear interval. Thus, any stationary solution must travel with this velocity, unless the mass vanishes (as is the case for dipole vortices). Inserting u = - f i into eqn. (16), we find that stationary, localized vortex solutions of eqn. (13) with finite energy and non-zero mass do not exist (Larichev, 1984; Flierl, 1987; Nycander, 1988). We conclude that the observations can not be explained within the framework of the equivalent barotropic quasi-geostrophic equation. This is also immediately apparent from the fact that eqn. (13) is invariant under the transformation h ---, - h , y ~ - y . Cyclones and anticyclones, therefore, evolve similarly, which contradicts the observations. 4. GENERAL GEOSTROPHIC SCALING 4.1. General analysis In the previous section we saw that the velocity of the center of mass of localized solutions of eqn. (13) is never in the non-linear region u < -/3 or u > 0, and that it is independent of the amplitude. The reason is that the conventional geostrophic scaling (12) involves an expansion in small amplitude. Equation (13), therefore, can not describe the effect of large amplitude, which must be responsible for the difference between cyclones and anticyclones.
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In the general geostrophic scaling we allow for arbitrary amplitude, while still expanding in the small parameters E,o (geostrophic approximation) and % (/3-plane approximation). The dispersion relation (14) is then still essentially unchanged. The main difference is that since the typical length scale can now be much larger than the Rossby radius, cf. eqn. (11), the latitudinal dependence of the phase velocity of linear Rossby waves must be taken into account. This will be discussed in detail below. To find out whether we can expect stationary vortices to exist, we should then generalize the integral relation (17), which determines the velocity of the center of mass, to arbitrary amplitude. This is done by expanding the integral relations (5) and (6) in E,, and Eta, while keeping Eh arbitrary. To lowest order in e~ we obtain the geostrophic velocity from eqn. (1),
Vo =
gH° ~. × Vh f
(18)
Inserting this into eqns. (5) and (6) with g' = 0 we obtain d dt
fyh dx dy = 0 fh d x dy
(19)
and
-
dt
fh dx dy
(20)
/3
fh dx dy
We have here transformed to the same dimensionless variables as in eqn. (13). Equation (20) has earlier been obtained by Nycander (1990) and Cushman-Roisin et al. (1990). Comparing it with eqn. (17), we see that it contains a non-linear correction which is proportional to the amplitude. (The same non-linear correction can also be obtained from eqn. (7) by expanding to lowest order in E~o,which means that we keep the potential energy and neglect the kinetic energy.) If the amplitude is positive, as for anticyclones, the velocity of the center of mass is in the non-linear region u < -/3, and localized stationary solutions are possible. If the amplitude is negative, on the other hand, the velocity is in the region of linear Rossby waves, -/3 < u < 0. (Note that since h can not be smaller than - 1 , the velocity of the center of mass always satisfies the inequality u < - / 3 / 2 , as
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pointed out by Cushman-Roisin et al. (1990).) Cyclones, therefore, quickly lose energy by coupling to linear waves. Thus, the basic features of the observed vortices can be explained from the dispersion relation of linear Rossby waves and the general integral relation (20). This conclusion will be confirmed by finding explicit stationary anticyclonic solutions, and by numerical simulations. Finally, it should be noted that if a variable effective gravity as in the laboratory experiments were included, the non-linear term in eqn. (20) would be cancelled, and we would get the same result as in eqn. (17). Thus, there is still no difference between cyclones and anticyclones, which makes it necessary to calculate center-of-mass velocity to higher order in e,o. This is done by Nycander (1992).
4.2. Approximate dynamic equation Using the general geostrophic scaling we can derive a simplified equation, where the high-frequency gravitational waves contained in eqns. (1) and (2) have been filtered away. We first obtain the velocity by rewriting the equation of motion (1) as --+(f+f~)~×13= 0t
- V gHoh+
and expanding in e~o. The zeroth-order velocity is given by eqn. (18). Including the first-order correction we obtain
v-
f+~
~.×V gh+
2-
-
(21)
7 V g~-
where
Inserting eqn. (21) into eqn. (3) we obtain
ah 0t
H(,.[l+h
1 [ ah)]
+~---~ a ~ V / g--~-
{l+h
H0[~]
--H o l ) + f , gh+ V
2} (23)
Still assuming that et~ is small, we use the/3-plane approximation, and let f be given by eqn. (3). The right-hand side of eqn. (23) is a generalization of the Jacobian in eqn. (13), with the general expression for the potential vorticity used, and the dynamic pressure included in the pressure term.
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To derive the integral relation for the center-of-mass velocity, we multiply eqn. (23) by x and integrate. Expanding the result to lowest order in e~o, and regarding g as constant, we recover eqn. (20). The KdV-term h ah/ax, which is responsible for the non-linear contribution to the center-of-mass velocity in eqn. (20), comes from the Jacobian {h/f, gh} on the right-hand side of eqn. (23). In the experiments in paraboloidal vessels, where g ( y ) is proportional to 1/f(y), this term vanishes (Nycander, 1992). However, to next order in e~ other contributions appear, corresponding to the kinetic energy in eqn. (7). Equation (23) is valid for small Rossby number, but arbitrary amplitude, and was first proposed by Sutyrin and Yushina (1986). (They did not include the factor 1/f in the second term on the left-hand side, but the difference is only significant outside the region of validity of the equation itself.) According to a recent study (Allen et al., 1990) this model is the most accurate among the various intermediate geostrophic models that have been proposed. It conserves the mass and the potential vorticity, while the energy is approximately conserved for small eo~. It should finally be r e m e m b e r e d that the relation (11) means that a structure with h ~ 1 must have a much larger length scale than the Rossby radius, since e~o is assumed to be small. We first calculate the dispersion relation for linear waves. Linearizing eqn. (23) and expanding in /3 we obtain ah _
02h _
(l+2/3Y)aT+Z/3ayat
__V
ah ah at - / 3 a-X-X= 0
2 __
using the same dimensionless units as in eqn. (13). Strictly speaking, this equation can not be Fourier expanded in the y-direction, since one coefficient is explicitly y-dependent. However, since we have assumed that /3y << 1, we can use the local approximation, which is valid if the coefficient varies on a much larger length scale than the perturbation, i.e. if k~, >>/3. We then obtain o~ -/3 k--~ = 1 + 2/3y + k 2 +
2i/3ky
(24)
The phase velocity of linear waves in the x-direction is now confined to the interval /3 < Re 1 + 2/3y
< 0
(25)
Equation (24) is a generalization of eqn. (14). The new effects are the y - d e p e n d e n c e of the phase velocity of Rossby waves, and the linear
484
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amplification of waves with northward phase velocity, due to the imaginary term. These effects are the result of taking into account the finite radius of the planet. Actually, this is not done completely consistently in eqn. (23). If, for instance, one more term in the Taylor expansion (3) is kept, the coefficients of the new terms in eqn. (24) have to be modified. However, the main new effect in eqn. (24), that the phase velocity of Rossby waves increases towards the equator, is true also in reality. To improve the description further, one should start from the shallow water equations in fully spherical coordinates. Equation (23) without the factor 1 / f in the second term has been solved numerically by Sutyrin and Yushina (1986, 1988), using a monopole vortex as initial condition. They found that a cyclone quickly loses its energy by exciting linear waves, while an anticyclone with large amplitude can propagate unchanged for a long time, after an initial adjustment period when the shape becomes slightly non-circular. It will here be shown how a stationary solution of this kind can be constructed explicitly. We first look for a stationary solution of eqn. (23) travelling with the velocity u2, setting O/Ot = - u 3 / O x . The equation can then be written {q, 0} = 0
(26)
where the potential vorticity is given by q-
1+~
+/3y+V"
l+/3~--y
and the Bernoulli function by 9-1( 0=h+_
Vh )2 -3t-b/(y l+/3y
+ /3--~+Y2
1 Oh) l + / 3 y 0y
(28)
The time and space variables have here been normalized as in eqn. (13); thus, u and/3 are dimensionless parameters. In the general solution of eqn. (26), 0 is an arbitrary function of q. In the region of open isolines of q, this function can be uniquely determined by the boundary condition h --+ 0 at infinity, giving V"
l+/3y
=(l+h)
[
- 1 -/3y
( l + / 3 y ) 2+ 2 / 3 h + - u u
1+/3y
t2
+ - l + / 3 y 3y
]
(29)
in agreement with the result obtained by Sutyrin and Yushina (1986). To simplify this expression we note that the ratio between % and et, can be
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given a simple interpretation: % %
/3 l'
(30)
where L, is the fluid velocity given by eqn. (21). Since the propagation velocity u of the vortex is of the same order as /3, this means that we must have % ~ et~ in the separatrix region, where u ~ t,. (In the core region, on the other hand, we may have t, >> u, and et, can be arbitrarily large.) Assuming that % ~ e~ and expanding eqn. (29) to order e h2 we obtain V2h=
( )
1 + /3 h + 2 / 3 y h + t~-
~
-
2
u2
+2/3--+--(Vh) ay 2u
2
(31)
W e stress that this equation is only valid in the far field and in the vicinity of the separatrix. Also note that the dispersion relation (24) can be obtained from eqn. (31) by linearizing and setting u = w / k X and V = ik. The character of eqn. (31) d e p e n d s on the sign of the two first terms on the right-hand side. In the northern half-plane y > (u + / 3 ) / 2 / 3 2 the solution has an exponential profile, and in the southern, complementary half-plane it has an oscillatory character. For a vortex to be well localized, the oscillatory region must be sufficiently far away. This means that the coefficient 1 + / 3 / u must be positive, and large enough that the first term on the right-hand side of eqn. (31) dominates the second one. W h e t h e r this condition can be satisfied d e p e n d s on the velocity u, which is determined by the amplitude in the main part of the vortex, cf. eqn. (20). If the amplitude is very small, e h ~ e~ as for the conventional geostrophic scaling, then u = -/3, and the line separating the oscillatory and exponential regions runs right through the center of the vortex. The energy is, therefore, dispersed by coupling to linear waves. If the amplitude is slightly larger, e h ~ % as in the IG scaling region (Matsuura and Yamagata, 1982; Williams and Yamagata, 1984), the coefficient 1 + / 3 / u does not vanish, but it is still small, of order e h, and the two first terms on the right-hand side of eqn. (31) are of the same magnitude. The bordering line of the oscillatory region is then close to the vortex. Such an anticyclone can live for a rather long time, but it can not be considered as stationary, since it excites linear waves on the southern side and thereby gradually loses its energy. This has been observed by Matsuura and Y a m a g a t a (1982) in a numerical simulation of a dynamic equation valid for the IG scaling. They also found that cyclones quickly disintegrate, which is natural since they are entirely within the oscillatory region. If the amplitude is positive and large, e h ~ 1, we see from eqn. (20) that the vortex moves significantly faster than linear waves, so that 1 + / 3 / u is of
486
J. N Y C A N D E R A N D G . G . S U T Y R I N
order unity. The oscillatory region y < (u +/3)/2/82 is then far away, and the amplitude decreases exponentially until it is reached. The leakage of energy by coupling to linear waves is, therefore, very small, and in practice the vortex can be considered as stationary. This is the kind of vortices that have been seen in the simulations of eqn. (23).
4.3. Explicit solution We will now find explicit stationary solutions of the kind described above. The plane is first divided into two regions by a circle with the radius r o. We solve the equations separately in both regions, and finally match the solutions at the boundary. In the inner region, containing the main part of the vortex, we allow the amplitude to be of order unity, e h ~ 1, and the vortex, therefore, must be much larger than the Rossby radius in order to keep e,o small. In the outer region, containing the separatrix, we assume that % ~ e,o. The length scale there (i.e. the width of the separatrix region) is, therefore, much smaller than that of the whole vortex, and of the same order as the Rossby radius. The problem is solved by perturbation analysis, setting h = h 0 ( r ) + hi(r, 0). We assume that the zeroth-order solution is circular, and treat /3 and u as small first-order parameters. A similar expansion was used by Larichev and Reznik (1976) and Nycander (1988) to study stationary vortex solutions with zero mass in the traditional quasi-geostrophic approximation. We first consider the inner region, where all streamlines are closed. Expanding eqns. (27) and (28) to first order we obtain q-
[
1 l+h(~ l + V 2 h o + V Z h l
hi
1+h¢~ (1 +
Oho]
V2ho)+/3y-/3yV2ho-/3~-y~y
(32) and O=ho+~(Vho)
2+h~+vh o-Vhl-/Sy(Vho)'+u
y+
(33)
Integrating eqn. (26) along 0 we obtain dOo dqo q' d r -O~ d~-
(34)
We then transform eqns. (32) and (33) to polar coordinates, setting y = r sin 0 and h I = " q l ( r ) + E l ( r ) sin 0
(35)
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487
Inserting the result into eqn. (34) and separating the variables we obtain a homogeneous equation for r/l(r) and an inhomogeneous equation for s~l(r). For our purposes we only need the equation for ~:~:
1 [ Vzs~1 ~1 r2 1 +h 0
~1
(1 + VZho) +/3r(1 - V2h,,) -/3hl,
]
1 +h 0
d[ 1 ,2 × d r h° + 2 ( h ° ) d
=[~'+hi'~i-/3r(h;')2+u(r+h~))]d- r
1 + V2ho
l+ho
(36)
One of the two homogeneous solutions (i.e. solutions for /3 = u = 0) is st1 -- hi1, corresponding to a uniform displacement of the vortex. By making the ansatz
~l(r) = z(r)hi,(r )
(37)
eqn. (36) can, therefore, be reduced to a first-order differential equation in z'(r):
a(r)z"(r) + b ( r ) z ' ( r ) = c ( r )
(38)
where
a(r) = (h;,)2(1 + h ; ) b(r)=
2ho+ hi~ ( l + h 0 ) h l , r
( l + h 0 ) ( h 0 ) -~r
6(1
c(r) = [u(r + h;i ) - /3r(h;i)2](1 + ho)~-r
1 +h 0
+ V2ho ) 1 +h o
(39)
+/3h;,[h'o-r(1- V2h0)](1 +h~) The integrating factor a(r) of eqn. (38) is a ( r ) = exp
( r tas : ,
r(h°'21+h 0
exp{ rl+h0 d[
l + h 0 ds s ( l + h 0 )
as)
Solving eqn. (38) and requiring regularity of ~:1 at the origin we obtain 1
/.rc(s)
, ,
z'(r) - a(r) Jo ~(-~ats) ds
(40)
We then solve eqn. (31) in the outer (separatrix) region, again assuming the zeroth-order solution to be circular. To lowest order we only include
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AND
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the first term on the right-hand side of the equation, since it must be d o m i n a n t for a well-localized solution:
with the solution h0(r ) =
CKo(Kr)
(42)
where C is a constant and K2 = 1 + /3/U. Using the conventional geostrophic scaling all the other terms of eqn. (31) are of equal magnitude, and they are, therefore, included in the first-order equation. Making the ansatz (35) and separating the variables we again get one equation for ~¿(r) and one for r/l(r). In this case, however, both equations are inhomogeneous, with the inhomogeneous terms in the equation for ~ coming from the quadratic terms in eqn. (31) (the third and fifth terms). The equations for ~1 in the inner and outer regions can be regarded as one second-order i n h o m o g e n e o u s differential equation. Requiring regularity at the origin and rll ---' 0 at infinity we know that this equation has a unique solution. This solution is only interesting for numerical simulations, and will not be discussed further. T h e equation for sol in the outer region is
d2~:, last1 dr 2 + r dr
( 1 ) ~ + K2 ~ = 2/3(rh° +
h°)
(43)
The boundary conditions at r = r 0 are that ~l(r0) and sOl(r0) should be continuous. F u r t h e r m o r e , since the h o m o g e n e o u s solution ~ = h'0 simply corresponds to a uniform displacement of the vortex, we can choose sOl(r0) = 0
(44)
This condition means that we have d e t e r m i n e d the position of the vortex in the y-direction. (In the numerical calculations below a different choice is made.) Equation (43) can easily be integrated using the Green's function. Differentiating the result we obtain
~;("o) -
1 £~rKl(Kro)Zfi(rh o + h;,) dr roK,(Kro )
(45)
u
A n o t h e r expression for ~:i(r0) can be obtained from the solution (40) in the inner region, using eqns. (37) and (44):
hi,(ro)
.r,c(r)a(r)
~;(r°)- a(ro~ ~ol ~ - ~
dr
(46)
STEADILY
TRANSLATING
489
ANTICYCLONES
Requiring eqns. (45) and (46) to be equal, we obtain a relation that determines the velocity u of the vortex. In general, however, the problem is quite complicated, since the value of u is needed even for the zeroth-order solution ho(r), as can be seen from eqn. (41). To simplify it we make two approximations. First, only terms of the lowest non-vanishing order in e o 1/2 in the inner region. We then assume are kept, observing that d / d r ~ E~o that the amplitude of the vortex is large, so that h 0 ~ 1 in the inner region, while h 0 is small in the outer (separatrix) region. This means that the contribution from eqn. (45) can be neglected altogether, while the upper limit of the integral in eqn. (46) can be extended to infinity. In effect, we require the solution in the inner region to approach zero smoothly as r ~ ~. With these approximations we obtain
u ~ hor d r = - / 3 ~
hi,+
dr
(47)
This is a special case of the general relation (20) determining the velocity of the center of mass. Thus, the velocity is mainly determined by the profile in the inner region, while details of the separatrix region only have a small effect. From eqn. (41) this velocity determines how quickly the amplitude decreases in the separatrix region. The first-order correction to the propagation velocity can then be determined from eqns. (45) and (46). Geometrically, the effect of the dipolar part of the solution is that the still approximately circular isobars (isolines of h) are no longer concentric. However, for a particular class of radial profiles the dipolar part exactly vanishes, and the isobars are still concentric circles. These profiles can be found by solving eqn. (39) for c = 0. Expanding this equation to lowest order in the Rossby number, and integrating once with h 0 ---, 0, r ~ oc, we obtain V2h0 = --(1 + h 0 ) In(1 + h 0 ) + h o u If the amplitude is small this equation can be further expanded, giving V2h0 =
1+--/3 h 0 + - - u 2 u
(48)
Equation (48) has a localized solution, describing an anticyclonic vortex with a unique relation between amplitude and size. The coefficient 1/2 of the last term agrees with the weakly non-linear model of Mikhailova and Shapiro (1980), but not with the other weakly non-linear models (Flierl, 1979; Petviashvili, 1980; Charney and Flierl, 1981).
490
J. N Y C A N D E R AND G.G. SUTYR1N
4. 4. N u m e r i c a l s i m u l a t i o n
In the calculations a finite-difference non-dimensional version of eqn. (23) for 64 × 64 grid points was used with the same boundary conditions as were explored by Sutyrin and Yushina (1988): l~. = 0 at all boundaries and shifting the region along the x-axis together with the center of the anticyclone. This assured that an undisturbed state was maintained ahead of the vortex and permitted the wave to pass behind the vortex, at the eastern boundary. Wave disturbances did not propagate ahead of the vortex in this problem, since its velocity exceeded the maximum phase velocity of Rossby waves.
The zeroth-order solution was chosen as ho=A
+ (1 - A )
h~, = C K o ( K r )
e x p ( - s r 2) - B s r 2
at r < r o
(49)
at r > r o
(50)
The coefficients A, B and C were defined by matching h0, hi~ and hi'~ at r = r 0 to avoid discontinuities in the vorticity field. In all simulations the value sr o = 3 was chosen, which was large enough that the propagation velocity calculated from eqn. (47) was approximately the same as for a purely Gaussian vortex with the amplitude h 0 ( 0 ) = 1: u = -1.25/3. This gives K = 0.45, cf. eqn. (42). The first-order solution was calculated numerically from eqns. (37) and (40) in the region r < r 0 with z ( 0 ) = 0. This means that we choose the center of the solution to be situated at the origin. In the outer region the h o m o g e n e o u s solution z ( r o ) h ' o ( r ) was chosen for simplicity, since the contribution from eqn. (45) can be neglected if the amplitude of the vortex is large. In our case h0(0) = 1. In the/3 plane approximation in eqn. (23) we put /3 = 0.007. The only p a r a m e t e r which was varied in the different simulations was s, which determines the diameter of the vortex. In each case the evolution of an initially circular vortex was compared with that of one where the first-order solution was included. We do not show any direct contour plots of h ( r ) , since almost no change is detectable by eye in any of the simulations. Instead, we show plots of the translational velocity of the center of the vortex (defined as the point of maximum amplitude h) as a function of time, and contour plots of the first-order stationary solution, which has a dipole character. (1) The value s = 0.02 corresponds to the case M1 (very large vortex) in the paper by Williams and Wilson (1988). In Fig. 1 it can be seen that the initially circular vortex (dashed line) at first propagates faster than the stationary solution, and that it is displaced poleward during the adaptation stage. Significant velocity oscillations with a period approximately equal to
STEADILY TRANSLATING ANTICYCLONES
491
•. 6 0 ] ',,,,,,,
1.4 01
"\,
i .20
""
£) 2
""
~
'
""
,. /
~j
,£-
soo
l o'oo
ls'oo
2ooo
Fig. 1. Evolution of the westward speed u X (upper) and meridional speed uy of the vortcx center. Here s = 0.02, which is also equal to the characteristic angular velocity l / / L . The time unit is f~] ~ (approximately 4 h). The initially circular case is shown by the dashed line, and the construction with the first-order solution by the solid line.
that of the swirling motion are also seen. They seem to be caused by the different angular velocities in different parts of the vortex. If the first-order solution is included the vortex is almost exactly stationary (solid line). The first-order part (Fig. 2) is a dipole with a cyclone towards the pole and an anticyclone towards the equator, and it reduces the westward drift of the
32
i
i6 1
32 f L 3P
16
0
16
32
Fig. 2. The first-order solution divided by /3 at s = 0.02. The length unit is P0, and thc contour interval is 0.4.
C U a! l ~ " ~ * ' u !
auu.luv,,a
dq,.L
I t U = a
L
ludaA,=,
C
*a!~l ~
dtu~a
~'H,L
0 I
t
17 o . ~
"+ : q
i
i
T
T
~] L
--IDa-
"l~lllJJ!J AII~!*!u! '~LI~ 'J~') ~!LI'I Ul tUOt)[) i ~ j~ u!~J.IUi~l-Ul:;U.ll.[~ll ) ALl, J ~ u ~ u oq}, U! L asea o l s p u o d s a ~ o a (az F mP.~apouJ) [ [ ' 0 = ~ ~ n l e A oqJ~ (E)
XOl.IOA 0141 alO apF U.IOql..IOU aql UO .Iadaals a[tJo.Id aql pue '3!..tluo3uo3 lou uaql aJ-e (74 jo sou!los!) s.reqos! aq,L "paads ale!.Ido~dde aql ol xal.IOA aioq~
I [ U = a OOg
lu,~JA,.,
049 v
I
OOg
o!d
~,-....,tur,~ oq<.L
OOU
C~O L
t
°!zl
:.:
L
,i
#i i* / * t
II
"~"
\\\ \
STEADILY
TRANSLATING
493
ANTICYCLONES
1.4-
15s%
~-
~
I Jx
1 2
0 1 t
D
~
J ~/f
(} 1 -!. . . . 7~ F i g . 5. T h e
•
s
i
? 0 i} same
I 4 (. } i."~
I ,i~<2 /
as Fig. 1 except
T
I ~z [. J :.
• i~! , ,
'
s = 0.05.
vortex moves slower than the stationary solution (Fig. 3), and the center is displaced toward the equator during the adaptation period. The construction with the first-order solution shows quite stationary behavior (solid line). From Fig. 4 it is seen that the dipolar first-order part now has the cyclone toward the equator and the anticyclone toward the pole; thus, it increases the westward velocity of the whole vortex. In this case the profile is steeper on the southern side of the vortex core. (3) The value s = 0.05 corresponds to an intermediate vortex size. The first-order solution now has a small amplitude, and does not significantly influence the drift velocity of the vortex (Fig. 5). It also has a more complicated structure (Fig. 6), while the isobars are almost concentric. Apparently, this case is close to the special class of radial profiles given by c = 0 in eqn. (39). These examples demonstrate the different structure of stationary propagating vortices depending on the vortex size. In all cases the propagation velocity agrees well with eqn. (47), as was also found in the recent paper by Cushman-Roisin et al. (1990). For lens-like eddies (steadily propagating anticyclones with finite Rossby n u m b e r and zero depth at the vortex boundary) with linear azimuthal velocity profile, the dipolar correction was calculated by Killworth (1983) and Flierl (1984). This corresponds to the limit of infinite amplitude and infinite size compared with the Rossby radius (or H 0-~ 0) in our case. T h e n the problem of localization and radiation of baroclinic Rossby waves does not arise. In both these papers the dipolar correction gives a steeper profile on the northern side of the vortex core, just as in case (1) above. (To see this from the solution of Flierl, one should subtract the h o m o g e n e o u s
494
J. N Y C A N D E R A N D G . G . S U T Y R I N
iG
_±_~.~
i
-IG
i
t
0
i
i
i
i
16
Fig. 6. The same as Fig. 2 except s = 0.05. The contour interval is 0.06. first-order solution, which gives a rigid shift, so that the solution has an extremum in the center.) 5. CONCLUSIONS In Section 4 we derived an expression describing the shape of stationary anticyclonic vortices by perturbation analysis. It was found that such vortices propagate westwards faster than the linear Rossby waves, and that the velocity can not be too close to the maximum phase velocity u R of these waves. The vortex amplitude must be large, i.e. the velocity of the swirling motion must be much larger than the translational velocity of the vortex. Thus, the vortex carries trapped fluid. Since the vortex is in geostrophic balance (the Rossby n u m b e r is assumed to be small), the requirement of large amplitude also means that its diameter should be much larger than the Rossby radius, cf. eqn. (11). The correctness of this analytic solution was then confirmed by numerical simulations. These also showed that the main role of the first-order solution (the dipole part) is to adjust the propagation velocity to the value appropriate for stationary solutions. Although in principle straightforward, the perturbation analysis may appear rather complicated, since different scalings must be used in the exterior and interior parts of the vortex. This should, however, not obscure the fact that the underlying physics is simple. Essentially, it can be understood from only two elements.
STEADILY TRANSLATING ANTICYCLONES
495
The first element is the dispersion relation (24) for linear Rossby waves, which is directly connected to the general equation (eqn. (31)) for the outer region, where the isolines of potential vorticity are open. These equations show that stationary, localized vortices must propagate significantly faster than all linear waves. The other element is the general integral relation (20), which determines the translational velocity of the vortex. An important feature which was included in the present analysis is that the maximum phase velocity L'R of the linear Rossby waves is weakly y-dependent, cf. eqn. (25). It is, therefore, not sufficient for a stationary vortex to propagate just slightly faster than the local value of t, a, since there would still be a region nearby (towards the equator) where Rossby waves with long wavelength travel faster than the vortex. In this region the stationary solution is oscillatory, as pointed out in Section 4.2. Dynamically such a vortex would radiate energy by coupling to linear waves, as was observed in numerical simulations by Matsuura and Yamagata (1982). On the basis of this simple physical picture we can estimate the necessary amplitude for anticyclones to be stationary. The criteria are that the distance Los c to the oscillatory region should be much larger than the 'damping length' L d = K- 1 in the outer region, cf. eqn. (42), and also larger than the radius of the vortex (the oscillatory region must be outside the separatrix). The maximum phase velocity of Rossby waves depends on the latitude as c u = Ccosa/sin2a, where C is a constant. The relevant quantity, however, is the angular velocity around the planet, since the vortex travels on a fixed latitude. Waves propagating closer to the equator travel a longer distance, and must, therefore, travel faster in order to keep up with the vortex. Correcting for this geometric effect, we find that the maximum angular velocity era of the linear waves depends on the latitude as eR~ = C/sin2c~, where C is a r e n a m e d constant. The angular velocity of the vortex can be estimated from eqn. (20) as u~ = -V~a(1 + h / 2 ) , where h is a typical amplitude of the vortex. (For a Gaussian vortex h is half the maximum amplitude.) The distance to the region where u a and the local t'aa are equal is then approximately U Ra
h
tan c~
Lo~c- 2 OVR~ -- 2 R ~
-
~y or
Losc = --R 4
(51)
496
J. N Y C A N D E R A N D G . G . S U T Y R I N
at mid-latitudes. Here R is the radius of the planet. The damping length in the outer region can be estimated from the first term on the right-hand side of eqn. (31) as L d - - ( 2 / h ) 1/2, or in dimensional units L d -- po(2/h) ~/2, where P0 is the Rossby radius. The requirement Iosc >> L d can then be written
~ - >> P0 or approximately
>>
--
(52)
The requirement that the oscillatory region should be outside the separatrix can be written 2D
> -k-
(53)
where D is the diameter of the vortex. These conditions are not satisfied in the atmosphere of the Earth, since Losc is only one tenth of the diameter of a typical anticyclone. For oceanic intrathermocline anticyclones, on the other hand, we typically have h ~ 1 and D < 100 km. Hence the inequalities (52) and (53) are satisfied by a wide margin, and the present analysis a.ppears to be fully applicable. For the Great Red Spot of Jupiter we have D ~- R / 6 and h ~ 1 (Dowling and Ingersoll, 1989), while the Rossby radius P0 is much smaller than R, so that the conditions (52) and (53) are satisfied. Thus, the present analysis can help to understand its persistence. However, it is also clear that the shear in the background flow is very important, as is seen in the experiments of Antipov et al. (1985) and Sommeria et al. (1988). In the vortex experiments in paraboloidal rotating vessels (Antipov et al., 1982, 1988; Nezlin et al., 1990) the effective gravitational acceleration g in eqn. (1) is not constant, but a function of y. This does not change the dispersion relation for linear Rossby waves; however, as was realized during the preparation of this paper, it exactly cancels the non-linear term in eqn. (20). Thus, to lowest order in the Rossby number, the center-of-mass velocity is independent of the amplitude, and coincides with the phase velocity of long-wavelength Rossby waves. The expansion should, therefore, be carried out to higher order. This case is considered in detail by Nycander (1992).
S'I'I~'ADILY T R A N S L A T I N G A N T I C Y C L O N E S
497
ACKNOWLEDGMENT
One of the authors (G. Sutyrin) acknowledges stimulating discussions of vortex solutions with Mikhail Nezlin, Glenn Flierl and Vitaly Larichev. REFERENCES Allen, J.S., Barth, J.A. and Newberger, P.A., 1990. On intermediate models for barotropic continental shelf and slope flow fields: Part 1. Formulation and comparison of exact solutions. J. Phys. Oceanogr., 20: 1017-1042. Antipov, S.V., Nezlin, M.V., Snezhkin, E.N. and Trubnikov, A.S., 1982. Rossby soliton in the laboratory. Sov. Phys. JETP, 55: 85-95. Antipov, S.V., Nezlin, M.V., Snezhkin, E.N. and Trubnikov, A.S., 1985. Rossby autosoliton and laboratory model of Jupiter's Great Red Spot. Sov. Phys. JETP, 62: 1097-11(17. Antipov, S.V., Nezlin, M.V., Rodionov, V.K., Rylov, A.Y., Snezhkin, E.N., Trubnikov, A.S. and Khutoretskii, A.V., 1988. Properties of drift solitons in plasma, following from laboratory experiments with rapidly rotating shallow water. Fiz. Plazmy, 14:1104-1121. Charney, J.G. and Flierl, G.R., 1981. Oceanic analogues of large-scale atmospheric motions. In: B.A. Warren and C. Wunch (Editors), Evolution of Physical Oceanography. The Massachusetts Institute of Technology, Cambridge, MA, pp. 504-552. Cushman-Roisin, B., Chassignet, E.P. and Tang, B., 1990. Westward motion of mesoscalc eddies. J. Phys. Oceanogr., 20: 758-768. Dowling, T.E. and Ingersoll, A.P., 1989. Jupiter's great red spot as a shallow water system. J. Atmos. Sci., 46: 3256-3278. Flierl, G.R.. 1979. Planetary solitary waves. Polymode News, 62: 1, 7-14. Flierl, G.R., 1984. Rossby wave radiation from a strongly nonlinear warm eddy. J. Phys. Oceanogr., 14: 47-58. Flierl, G.R., 1987. Isolated eddy models in geophysics. Annu. Rev. Fluid Mech., 19: 493-530. Kamenkovich, V.M., Koshlyalov, M.N. and Monin, A.S., 1986. Synoptic Eddies in the Ocean. Reidel, Dordrecht, 433 pp. Killworth, P.D., 1983. On the motion of isolated lenses on a beta-plane. J. Phys. Oceanogr., 13: 368-376. Larichev, V.D., 1984. Integral characteristics of localized vortices on the beta-plane. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana, 20: 733-740. Larichev, V.D. and Reznik, G.M., 1976. Strongly nonlinear, two-dimensional isolated Rossby waves. Okeanologiya, 16: 961-967. Matsuura, T. and Yamagata, T., 1982. On the evolution of nonlinear planetary eddies larger than the radius of deformation. J. Phys. Oceanogr., 12: 440-456. McWilliams, J.C., 1985. Submesoscale, coherent vortices in the ocean. Rev. Geophys., 23: 165-182. Mikhailova, E.I. and Shapiro, N.B., 1980. Two-dimensional model of synoptic disturbances evolution in the ocean. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana, 16: 823-833. Nezlin, M.V., 1986. Rossby solitons. Soy. Phys. Usp., 29: 807-842. Nezlin, M.V. and Sutyrin, G.G., 1989. Long-lived solitary anticyclones in the planetary atmospheres and oceans, in laboratory experiments and in theory. In: J.C.J. Nihoul and B.M. Jamart (Editors), Mesoscale/Synoptic Coherent Structures in Geophysical Turbulence. Elsevier Science, Amsterdam, pp. 701-720.
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Nezlin, M.V., Rylov, A.Yu., Trubnikov, A.S. and Khutoretskii, A.V., 1990. Cyclonic-anticyclonic asymmetry and a new soliton concept for Rossby vortices in the laboratory, oceans and the atmospheres of giant planets. Geophys. Astrophys. Fluid Dyn., 52: 211-247. Nor, D., 1983. On the migration of isolated eddies with application to Gulf Stream Rings. J. Mar. Res., 41: 399-425. Nycander, J,, 1988. New stationary vortex solutions of the Hasegawa-Mima equation. J. Plasma Phys., 39: 413-430. Nycander, J., 1989. The existence of stationary vortex solutions of the equations for nonlinear drift waves in plasmas and nonlinear Rossby waves. Phys. Fluids, BI: 17881796. Nycander, J., 1990. Existence criteria and velocity of stationary vortices in plasmas and in shallow water on a rotating planet. In: Nonlinear World, IV Int. Workshop on Non-linear and Turbulent Processes in Physics, Kiev, 1989. World Scientific, Singapore, pp. 933 -954. Nycander, J., 1992. Difference between monopole vortices in planetary flows and experiments with paraboloidal vessels. J. Fluid Mech., in press. Petviashvili, V.I., 1980. Red spot of Jupiter and the drift soliton in a plasma. JETP Lett., 32: 632-635. Romanova, N.N. and Tseitlin, V.Yu., 1984. On quasigeostrophic motions in barotropic and baroclinic fluids. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana, 20: 85-91. Smith, B.A., Soderblom, L.A., Johnson, T.V. et al., 1979. The Jupiter system through the eyes of Voyager-1. Science, 204: 951-972. Smith, B.A., Soderblom, L.A., Batson, R. et al., 1982. A new look at the Saturn system: the Voyager-2 images. Science, 215: 504-537. Sommeria, J., Mcyers, S.D. and Swinney, H.L., 1988. Laboratory simulation of Jupiter's great red spot. Nature, 331: 689-693. Sutyrin, G.G., 1985. On the theory of solitary anticyclones in a rotating fluid. Dokl. Akad. Nauk SSSR, 280:1101-1105 (transl. Earth Sci., pp. 38-41). Sutyrin, G.G. and Yushina, I.G., 1986. On the evolution of isolated eddies in a rotating fluid. Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza No. 4, 52-59 (transl. Fluid Dyn. (USSR), pp. 550-556). Sutyrin, G.G. and Yushina, I.G., 1988. Formation of a vortical soliton. Dokl. Akad. Nauk SSSR, 299:580-584 (transl. Sov. Phys. Dokl., 33(3): 179-181). Williams, G.P., 1985. Jovian and comparative atmospheric modeling. Adv. Geophys., 28A: 381-429. Williams, G.P. and Yamagata, T., 1984. Geostrophic regimes, intermediate solitary vortices and Jovian eddies. J. Atmos. Sci., 41: 453-478. Williams, G.P. and Wilson, R.J., 1988. The stability and genesis of Rossby vortices. J. Atmos. Sci., 45: 207-241. Yamagata, T., 1982. On nonlinear planetary waves: A class of solutions missed by the quasi-geostrophic approximation. J. Oceanogr. Soc. Jpn., 38: 236-244.