CIESM Congress 2004, Barcelona, article 0134

Rapp. Comm. int. Mer Médit., 37,2004

134

A SIMPLE BAROCLINIC MODEL FOR DIFFUSIVE VORTICES

J. L. Pelegrí

1 *

, M. Auladell

1, 2

, E. García-Ladona

, J. Font

, and P. Sangrà

Institut de Cienciès del Mar, CMIMA-CSIC, Barcelona, Spain

Facultad de Ciencias del Mar, ULPGC, Gran Canaria, Spain

Abstract

A simple layered diffusive model describes the temporal evolution of the angular velocity and vertical structure of oceanic vortices. An

initial state specifies the core angular velocity of each layer and the density changes between adjacent layers. The temporal evolution of

the angular velocity for each layer is explicitly obtained solving the radial diffusive equation, with the diffusion coefficient determined

from a stability analysis, and the shape of the interfaces follows from the assumption of gradient ?ow.

Keywords: oceanic vortices, radial diffusion, inertial stability, layered model.

Transient mesoscalar vortices are a common feature in the

Mediterranean. A simple model for vortex evolution consists on radial

diffusion of angular velocity

(r,t),

(1)

where K(t)is a horizontal diffusion coefficient, for an initially rotating

cylinder

(2)

where aand

are the vortex’s initial radius and angular velocity

(positive/negative for cyclonic/anticyclonic). The solution, for

constant K, is [1]:

(3)

Figure 1a displays this solution at several times for an initial

cyclonic vortex with a= 25 km, |

| = 3

-5

-1

(period 2.5 days),

and K= 90 m

-1

. The size of the vortex slowly increases with time

while a central core remains in near solid body rotation. This core

remains discernable in time, although after some 30 days its rotation

rate decreases.

The above model may be generalized to a layered ocean (Fig. 2),

with a time-dependent K(t) for each layer, provided that

. The pressure pis hydrostatic and the interfaces’

shape may be determined under the assumptions that (a) the deep ?ow

is motionless and (b) the ?ow in each layer is in gradient balance:

)

(

)

(

)

(

erf

)

(

(4)

where fis the planetary vorticity. For a 1.5 layer model this equation

gives the slope of the active only interface,

(5)

where

and

are the upper and lower

layer densities. The actual interface depth is obtained through

integration from a distant unperturbed radial coordinate, .

A first approximation has the vortex in solid body rotation

for , the interface h(r,t) becoming a function of

(6)

Figure 1b illustrates the interface evolution for a cyclonic vortex at

52ºN, the sloping interface resembling observations of tilted

isopycnals in vortices.

The model requires a knowledge of K(t) for all layers. Within each

layer the angular velocity is approximately constant at the vortex core

so any diffusion there simply redistributes particles with similar

angular velocities. Hence, for each layer the effective diffusion

coefficient depends on the stability at the vortex edge, which we

assess using the theory of radial stability in barotropic structures [2].

At lowest order the radial velocity is unstable when

, where

and

Figures 3a,b show the radial distribution of

for a cyclonic

vortex, at different times, using two different diffusion coefficients.

For cyclonic vortices edge stability leads to estimates of K(t): a

negative minimum at the vortex edge is interpreted as a too small

coefficient (Figures 3a,b). For anticyclonic vortices the eddy’s edge is

rather stable leading to a small effective diffusion [1].

References

[1] Sangrà P., Pelegrí J.L., Arregui I., Martín J.M., Hernández-Guerra A.,

Marrero-Díaz A., Martínez A., and Rodríguez-Santana A., 2004. Life story

of an anticyclonic eddy. J. Geophys. Res., submitted.

[2] Hopfinger E.J., and Van Heijst G.J.F., 1993. Vortices in rotating ?uids.

Ann. Rev. Fluid Mech., 25: 241-289.

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