Structure dominated two-dimensional turbulence: formation, dynamics and interactions of dipole vortices

Abstract

Two dimensional turbulence in geophysical fluids and plasma physics tends to be spotty, intermittent and rich in large scale structures such as coherent vortices or zonal flows, due to various mechanisms of self organization. Nonlinear solutions that rely on the vanishing of nonlinearity, especially the dipole vortex solution, stand out as key aspects of this structure dominated turbulence state. Using numerical simulations, it is demonstrated that an initial condition with a small number of high intensity turbulent patches, evolves towards a state dominated by coherent structures, and in particular dipole vortices, as each patch is organized into a finite number of dipole vortices that are ejected from this initially active region. In order to study the details of this process, an initial condition of two Gaussian peaks of the stream function is considered, and it was shown to result in a Chaplygin–Lamb dipole if the peaks have the same amplitude, or a Flierl–Stern–Whitehead dipole that rotates in the direction implied by the excess of vorticity if they do not. Analytical estimates for the velocity, the radius and the radius of curvature of the resulting dipole vortex is given in terms of the peaks and widths of the initial conditions. These are then verified by a detailed comparison of the analytical form of the vorticity of the dipole vortex and its numerical realization. It is argued that since these coherent structures are spared from the strong shear forces normally exerted by the nonlinearities, and can coexist with other localized solutions, or large scale flow patterns, they provide the backbone of the structure dominated or ‘sporadic’ turbulent state in two dimensions, on top of which other structures, waves and instabilities can develop. In order to elucidate these, a number of collision scenarios are considered. It is also shown that a simple two point vortex approximation to a dipole vortex seems to be appropriate for describing their evolution far from each-other, or for computing head on collisions between two or more dipole vortices, but not in the case of close or grazing collisions or their interaction with a nontrivial large scale flow.