Vortex merger in surface quasi-geostrophy

Abstract

The merger of two identical surface temperature vortices is studied in the surface quasigeostrophic model. The motivation for this study is the observation of the merger of submesoscale vortices in the ocean. Firstly, the interaction between two point vortices, in the absence or in the presence of an external deformation field, is investigated. The rotation rate of the vortices, their stationary positions and the stability of these positions are determined. Then, a numerical model provides the steady states of two finite-area, constant-temperature, vortices. Such states are less deformed than their counterparts in two-dimensional incompressible flows. Finally, numerical simulations of the nonlinear surface quasi-geostrophic equations are used to investigate the finite-time evolution of initially identical and symmetric, constant temperature vortices. The critical merger distance is obtained and the deformation of the vortices before or after merger is determined. The addition of external deformation is shown to favor or to oppose merger depending on the orientation of the vortex pair with respect to the strain axes. An explanation for this observation is proposed. Conclusions are drawn towards an application of this study to oceanic vortices.2

Surface Quasi-Geostrophy

“…This last study also examined analytically and numerically the merger of these vortices in the presence or absence of an external deformation field. It was found that the distance of vortex merging was generally smaller than the two-dimensional barotropic case [24]. This can be explained by the rapid decrease of the velocity as a function of distance (see Section 2.5), which prevents the vortices from exerting an influence on each other at a large distance.…”

“…ψ( k, z) is the two-dimensional Fourier transform of the streamfunction ψ at the altitude z, and θ( k) is the Fourier transform of the surface buoyancy. Equation (24) shows that, for each Fourier component, the streamfunction decreases exponentially with z. Observe in particular that the decrease is even faster when small horizontal scales are considered (large K).…”

“…For continuous buoyancy profiles, the analytical solution of such steadily-translating vortex dipoles was described by Muraki and Snyder [23]. Carton et al [24] presented the solution of two co-rotating vortices with uniform buoyancy in the presence or absence of a large-scale strain field. This last study also examined analytically and numerically the merger of these vortices in the presence or absence of an external deformation field.…”

“…where u g is the horizontal velocity field at some depth z that can be obtained through Equation (24). At the surface z = 0, the velocity field is related to a turbulent regime with a well-defined inertial range (either in the direct cascade of generalized enstrophy or the inverse cascade of generalized energy).…”

Oceanic vortex mergers are not isolated but influenced by the β-effect and surrounding eddies

“…Since the 80's, an important effort has been undertaken to understand the physics of oceanic vortex merger, including in situ 12 , laboratory 13,14 , and numerical 15 observations, associated with intense theoretical debates [16][17][18] . Most fundamental studies addressed the 'isolated vortex merger' problem, omitting the influence of neighbouring eddies, large scale currents, or boundary, topographic, and planetary effects [19][20][21][22][23][24][25] . Efforts to include more physical effects in studies of vortex merger [26][27][28][29] were often impaired by the lack of general observations in the global ocean, or by the complexity of the resulting dynamical system.…”