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Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
On the baroclinic vortex instability: the scattering problem
M. A. Sokolovskiy
1
, J. Verron
2
, V. M. Gryanik
3
1
Water Problems Institute of RAS, Laboratory of Fluid Dynamics,
3 Gubkina Str., GSP-1, 119991, Moscow, Russia;
e-mail: sokol@aqua.laser.ru
2
Laboratoire des Ecoulements G?ophysiques et Industriels (LEGI), CNRS,
UMR 5519 CNRS, BP 53 X, Grenoble Cedex, 38041 France
3
Alfred-Wegener-Institute for Polar and Marine Research, Division of Climate Sciences,
Bussestrasse 24, D-27570, Bremerhaven, Germany
Abtract :
The stability of an isolated vortex in a two-layer stably stratified fluid on a rotating plane (f-plane
approximation) is studied. In the case of axially symmetric vortex structure consistent of a set of vortex
patches, the problem of linear instability with respect to small disturbances of the patches? boundaries
may be solved analytically. Dispersion relationships allow studying the stability properties of the vortex,
depending on the respective thickness of the layers, radii of the vortex patches, the medium stratification
and the value of potential vorticity.
Contour Dynamics Method (CDM) allows investigation of the non-linear stage in the evolution of
unstable vortex structures, and, in particular, the time law of scattering of newly formed two-layer
vortices of smaller size. When unstable vortices with zero total vorticity (heton) decays, the scattering is
shown to follow the ballistic law. The obtained results can be used in the problems of deep convection for
describing the process of spreading of heat and density anomalies.
The use of CDM gives also possibility to examine the stability and peculiarities of the further evolution on
non-axially symmetric vortex structures ? the hetons with ?tilted axis? and two-layer quasi-elliptical
vortices.
Key-words : heton, CDM, instability
1 Introduction
We investigate the stability of two-layer vortices with piecewise constant distribution of
potential vorticity in the layers, and in particular, the non-linear stage of evolution of unstable
vortex patches.
If the summary potential vorticity of the two-layer vortex system is equal to zero, we deal
with the so called hetons (Hogg and Stommel 1985). In the conditions of a stable stratification
and a hydrostatic equilibrium such vortices possess abnormal heat content. After the unstable
heton decay there form two-layer vortices of smaller scale which scatter from the core of the
initial vortex, and in that way they are the carriers of heat anomalies. Legg and Marshall (1993)
proposed a heton analogy for explaining the mechanism of deep convection formation in the
ocean that has become widely known (Marshall and Schott 1999). In the present work a special
attention is paid to the numerical investigation of the thermal anomaly front motion, which is
associated with evolution of the external boundary of the vortex structure.
2 On linear stability analysis of axially symmetric heton
Let us examine an undisturbed axisymmetric heton, which consists of two-layer cylindrical
structure of unit radius circular patches located strictly one under another. Such state is a
stationary solution of the equations of potential vortex conservation
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Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
0=
?
td
d
jj
,
where
( )
jjjjj
ppFp ?+?=?
?3
2
, . 2,1=j
Here
y
v
x
u
dt
d
jj
j
?
?
+
?
?
+?
?
)()(
)(
)(
and
2
2
2
2
2
)()(
)(
yx ?
?
+
?
?
?? - are two-dimentional
operators: a total derivative with respect to time and the Laplacian one; a dot above a variable
means a partial derivative with respect to time ; , - are the components of the velocity
vector of liquid particles in a j layer of thickness along the axis of the orthogonal coordinate
system and correspondingly; , - is the mean value
of the density,
j
u
j
v
j
h
x y ?? ?+?= )(/4
21
22*
hhghDF
jj
*
?
12
??? ?=? ; - is a characteristic linear scale, D g - the gravitational
acceleration, and - are anomalies (with respect to the equilibrium hydrostatic state) of
the hydrodynamic pressure in the layers, connected with the velocity components by
geostrophic relationships
p
1
p
2
x
p
v
y
p
u
j
j
j
j
?
?
?
?
=?= , .
The question about the stability of the corresponding axisymmetric solution is lawful. Let the
liquid lines, coinciding with external boundaries of the vortices, be described with parametric
relations
,2,1),;0,(),;,( === jftfr
jj
?????
where the parameter ? characterizes a radial Lagrangian coordinate of the points belonging to
the contours, and ? - is a polar angle. Let?s write the function f
j
in the form
1,2,1,1)],(exp[)1(1)1;,( ?=<
m
mI ? should be fulfilled. The algorithm of the
stability examination is given in (Kozlov et al 1986). In essence, the problem reduces to the
analysis of the system of linear algebraic equations with respect to
j
? . The stability analysis
shows that at 2/1
21
== hh (the discussed below parameters correspond to this particular case)
the modes with and 2?m 7.1/
3
>=
? jj
hF? may be unstable.
3 Numerical modeling of the nonlinear stage of the heton evolution
A convenient instrument for the numerical study of the evolution of unstable vortex
structures is the Method of Contour Dynamics (CDM) which allows calculating the
configurations of the boundaries of the vortex patches at any time moment. The two-layer
version of CDM was for the first time given by Kozlov et al (1986); then it was used by
Sokolovskiy and Verron (2000), Gryanik et al 2006 and in a series of other works.
The configurations of the vortex patches of the upper (solid line) and bottom (dashed line)
layers are shown in the below figures for the indicated moments of the non-dimension time. A
half of a rotational period of the undisturbed (circular) two-layer vortex is chosen to be a time
unit.
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18
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Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
F
Figure 1 shows the calculation result for the case linear
analysis of the stability
time moment the perturbat 2,1,02.0 == j
were applied here.
This experi d cascade
instability:
- at the first stage in the upper and lowe
with respect to
- nine two-la s and
scatter in the radial directions;
- at the periphery
IG. 1 ? Time evolution of the circular unstable heton: cascade instability
14=? , when, according to the
, the mode with 9=m occurs to be the maximum unstable. In the initial
ions, corresponding to this mode, with amplitude
j
?
ment gives an example of the process realization for the so calle
r layers, there form 9-rayed figures, displaced one
another;
yer pairs with tilted axes break away from the extremities of these ray
of the residuary core, there form 9 new rays;
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18
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Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
- the extremity parts of the rays, shifted one with respect to other, irradiate a new series of
9 small-scale hetons, departing behind the first echelon;
- at the boundary of the central core, new (now irregular) vortex structures form, and they
also move out from the center.
The motion law of the departing vortices is of obvious interest. The corresponding
calculation results are shown in Figure 2 which demonstrates the behavior of radial coordinates
of fore (first, second and third) and rear vortex fronts. The fore front of the vorticity we assume
to be some fictitious material particle whose coordinates are calculated as an arithmetic average
of and coordinates of 20 contour markers that are at the maximum distance of the
center of the initial vortex structure. For designating the rear front we took liquid particles
which are at the minimum distance from the center.
?x ?y
FIG. 2 ? Time evolution of potential vorticity fronts
We see that the motion laws of the fore fronts reach asymptotically the ballistic (linear in
time) law. Therefore we have reason to believe that the front of the heat anomaly also has to
propagate with a constant velocity.
The example given above corresponds to the case when the initial boundary of the heton
has the shape close to the circular one. In the case of a quasi-elliptical two-layer vortex, the
linear analysis of the stability is much more complicated, but the calculations we carried out
showed that the regimes of the cascade instability are also possible for the hetons of elliptical
shape. Figure 3 gives the example of the decay of the initially compact vortex structure.
4 Conclusions
In the models of the general ocean circulation there is assumed that the heat transport has a
diffusion nature with the law of the propagation of the temperature anomaly
2/1
~ tR . At the
same time it is known that the thermic component of the existing models is the least
satisfactory. The heton theory, based on the results of the numerous numerical experiments
(and, in particular, given here) on the modeling the non-linear evolution stage of unstable two-
layered vortices certify in the favor of the mesoscale heat transport which more effective, i.e.
tR ~ . It seems that this parameterization has to provide more realistic results.
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Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
Acknowledgments Investigation was conducted within the frames of the GDRE ?Regular and
chaotic hydrodynamics? (Project 07-05-92210).
Fig. 3 ? Time evolution of the elliptical unstable heton
References
Gryanik, V.M., Sokolovskiy, M.A. & Verron, J. 2006 Dynamics of heton-like vortices. Regular
& Chaotic Dyn. 11, 417-438.
Hogg, N.G. & Stommel, H.M. 1985 The heton, an elementary interaction between discrete
baroclinic geostrophic vortices, and its implications concerning eddy heat-flow. Proc. R. Soc.
Lond. A397, 1-20.
Kozlov, V.F., Makarov, V.G. & Sokolovskiy, M.A. 1986 A numerical model of baroclinic
instability of axially symmetric vortices in a two-layer ocean. Izvestiya, Atmospheric Oceanic
Physics. 22, 868-874.
Legg, S. & Marshall, J. 1993 A heton model of the spreading phase of open-ocean deep
convection. J. Phys. Oceanogr. 23, 1040-1056.
Marshall, J. & Schott, F. 1999 Open-ocean convection: observation, theory, and models. Rev.
Geophys. 37, 1-64.
Sokolovskiy, M.A. & Verron, J. 2000 Finite-core hetons: Stability and interactions. J. Fluid
Mech. 423, 127-154.
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