Diffusive draining and growth of eddies

The diffusive effect on barotropic models of mesoscale eddies is addressed, using the Melnikov method from dynamical systems. Simple geometric criteria are obtained, which identify whether a given eddy grows or drains out, under a diffusive (and forcing) perturbation on a potential vorticity conserving flow. Qualitatively, the following are shown to be features conducive to eddy growth (and, thereby, stability in a specific sense): (i) large radius of curvature of the eddy boundary, (ii) potential vorticity contours more tightly packed within the eddy than outside, (iii) acute eddy pinchangle, (iv) small potential vorticity gradient across the eddy boundary, and (v) meridional wind forcing, which increases in the northward direction. The Melnikov approach also suggests how tendrils (filaments) could be formed through the breaking of the eddy boundary, as a diffusion-driven advective process. 1 Eddies and their stability Rings (or eddies) are significant oceanographic features which contribute considerably to fluid transport in the ocean. In particular, mesoscale (of the order of 100 km) rings formed near the Gulf Stream sometimes survive as coherent structures for periods of up to one year (Richardson, 1983). Submesoscale (of the order of 10 km ) eddies may also be long-lived, and we address both mesoscale and submesoscale eddies in the present work. The observational persistence of such eddies has led to theoretical (Flierl, 1988; Helfrich and Send, 1988; Dewar and Gailliard, 1994; Dewar and Killworth, 1995; Paldor, 1999), numerical (Helfrich and Send, 1988; Dewar and Gailliard, 1994; Dewar and Killworth, 1995; Dewar et al., 1999; Paldor, 1999; McWilliams et al., 1986) and experimental (Voropayev et al., 1999) analyses of stability. Since many results indicate that eddies would tend to be unstable, explaining their persistence remains an Correspondence to: S. Balasuriya ([email protected]) active area of research. In this paper, we address a particular aspect of stability of such eddies, which reflects the effect of small diffusivity on the eddy boundary. Though characterised by swirling fluid motions, eddies are often identified experimentally through Eulerian contour plots of temperature, height, salinity, or potential vorticity fields, usually obtained from two-dimensional satellite imaging data (for a review and pictures of contours, see Richardson, 1983), or from numerical schemes. Since fluid motion in the upper ocean tends to remain on surfaces of constant temperature (resp. salinity, potential vorticity, etc), rotational motion results around maxima/minima points of the appropriate scalar field, thereby forming a ‘ring’ (or vortical motion) in the expected sense. Often, tendrils (or filaments) are seen to emanate from these eddies, which appear to wrap around the eddy (see Fig. 4 in the experimental paper by Voropayev et al., 1999, for example). The dynamics governing the behaviour of such eddies is assumed to be close to a two-dimensional incompressible flow in which potential vorticity is conserved (Pedlosky, 1987). Under strict conservation with the dynamics steady in a moving frame, no substantial deformations of eddies are to be expected, since the Lagrangian trajectories are integrable for finite times (Brown and Samelson, 1994). In many of the standard stability analyses, such a system is perturbed through an arbitrary mode, whose growth rate is determined by linearising the potential vorticity conservation equation. In this study, we adopt a different approach, which specifies the physical reason for imposing a perturbation, and also does not rely on a linearisation of the dynamics. Our perturbation shall be the result of small scale turbulence in the ocean, which is frequently modelled by a diffusive term in the governing differential equation (see Haidvogel et al., 1983, for example). The dynamics are then governed by an advection-diffusion equation for the scalar potential vorticity. Such eddy diffusivity has significant consequences in the advection of passive scalars, in general, fluids, and has been addressed in statistical (Poje et al., 1999), numerical (Miller et al., 1997; Poje et al., 1999) and theoretical (Fannjiang 242 S. Balasuriya and C. K. R. T. Jones: Diffusive draining and growth of eddies and Papanicolaou, 1994) senses. Bounds on the eddy diffusivity (Fannjiang and Papanicolaou, 1994; Biferale et al., 1995; Mezić et al., 1996), and descriptions of chaotic motion (Rom-Kedar and Poje, 1999; Klapper, 1992; Jones, 1994), are several features of interest. Even when not modelling flows with diffusivity, numerical methods often introduce a diffusivity in the interest of numerical stability, and, therefore, such numerical models could also be thought of as including eddy diffusivity effects (Rogerson et al., 1999). Unlike in regular advection-diffusion equations, the scalar quantity here is an active (as opposed to passive) scalar, since the potential vorticity possesses a relationship to the fluid velocity field (Pedlosky, 1987). In this study, we shall investigate how the dynamic process of eddy diffusivity affects the geometry of eddies, using a new approach, which uses elements from dynamical systems theory (Balasuriya et al., 1998), and simple geometric arguments. Our first focus in this paper is to obtain a relationship between the growth (or decay) of such eddies, and the characteristics of the scalar potential vorticity field. Would it be possible, for example, to view the field, identify a particular eddy, and predict its chances of survival based on simple geometric properties of the scalar field? In response to this, we are able to develop a collection of (diffusivity-driven) geometric conditions for eddy growth, outlined in Sect. 5. It would be instructive to test our criteria upon available data sets with sufficient resolution. Moreover, in Sect. 7, we also obtain a qualitative condition on (small) wind forcing, which also contributes to eddy growth. ‘Growth,’ as specified in both these cases, will be defined through the enlargement of the eddy boundary; a shrinking boundary will correspond to a ‘draining’ eddy. Growing eddies have the potential of being more visible, and, therefore, are expected to be the longer lasting eddies in the ocean. Draining (shrinking) eddies, on the other hand, will eventually lose their constituent water to the ambient flow, and disappear. Therefore, in a sense, our eddy growth criteria reflect a form of eddy stability in the presence of (small) eddy diffusivity and wind forcing. It must be re-emphasised that this ‘stability’ is not in the traditional sense of linear stability, in which the growth rate of various modes of imposed perturbations is analysed, as in Flierl (1988); Helfrich and Send (1988); Paldor (1999); Dewar and Killworth (1995); Dewar et al. (1999). The analysis we follow in this study is generic, and should be applicable to any system satisfying a similar advectiondiffusion partial differential equation in two dimensions (for example, in tracer mixing in hydrodynamics, or in atmospheric flows). In other words, we are not using a specific model for the flows; rather, we are simply assuming that the flow satisfies the appropriate dynamical equation, and possesses the necessary kinematical properties of an eddy. These statements are made precise in Sect. 2. Section 3 then outlines the Melnikov approach from dynamical systems theory, which leads to the eddy growth criteria in Sects. 5 and 7. A secondary goal of this paper is to give a possible explanation for the tendrils which emanate from eddies. Numerical and experimental studies, even in the laboratory rather than in the oceans, display such filaments (see Voropayev et al., 1999, for example), whose presence is certainly linked to eddy diffusivity (Robinson, 1983). Nevertheless, a geometric description of the process is lacking. Our analysis of the advection-diffusion process, from a dynamical systems viewpoint, affords an immediate and simple reasoning for the appearance of a tendril in a certain type of eddy, as explained in Sect. 6. In this case, too, it is necessary to address the deformation of the eddy boundary, which links the two aspects of this paper.