Ergodicity and spectral cascades in point vortex flows on the sphere

A.C.P. was supported under DOD (MURI) Grant No. N000141110087 ONR. The computations were supported by the CUNY HPCC under NSF Grants No. CNS-0855217 and No. CNS-0958379.We present results for the equilibrium statistics and dynamic evolution of moderately large [n = O (102 - 103)] numbers of interacting point vortices on the sphere under the constraint of zero mean angular momentum. For systems with equal numbers of positive and negative identical circulations, the density of rescaled energies, p(E), converges rapidly with n to a function with a single maximum with maximum entropy. Ensemble-averaged wave-number spectra of the nonsingular velocity field induced by the vortices exhibit the expected k-1 behavior at small scales for all energies. Spectra at the largest scales vary continuously with the inverse temperature of the system. For positive temperatures, spectra peak at finite intermediate wave numbers; for negative temperatures, spectra decrease everywhere. Comparisons of time and ensemble averages, over a large range of energies, strongly support ergodicity in the dynamics even for highly atypical initial vortex configurations. Crucially, rapid relaxation of spectra toward the microcanonical average implies that the direction of any spectral cascade process depends only on the relative difference between the initial spectrum and the ensemble mean spectrum at that energy, not on the energy, or temperature, of the system.Publisher PDFPeer reviewe

Asymmetric vortex merger: mechanism and criterion

The merging of two unequal co-rotating vortices in a viscous fluid is investigated. Two-dimensional numerical simulations of initially equal sized Lamb-Oseen vortices with differing relative strengths are performed. Results show how the disparity in deformation rates between the vortices alters the interaction. Key physical mechanisms associated with vortex merging are identified. A merging criterion is formulated in terms of the relative timing of core detrainment and destruction. A critical strain parameter is defined to characterize the establishment of core detrainment. This parameter is shown to be directly related to the critical aspect ratio in the case of symmetric merger

Discrete exterior calculus (DEC) for the surface Navier-Stokes equation

We consider a numerical approach for the incompressible surface Navier-Stokes equation. The approach is based on the covariant form and uses discrete exterior calculus (DEC) in space and a semi-implicit discretization in time. The discretization is described in detail and related to finite difference schemes on staggered grids in flat space for which we demonstrate second order convergence. We compare computational results with a vorticity-stream function approach for surfaces with genus 0 and demonstrate the interplay between topology, geometry and flow properties. Our discretization also allows to handle harmonic vector fields, which we demonstrate on a torus.Comment: 21 pages, 9 figure

Offsprings of a point vortex

The distribution engendered by successive splitting of one point vortex are considered. The process of splitting a vortex in three using a reverse three-point vortex collapse course is analysed in great details and shown to be dissipative. A simple process of successive splitting is then defined and the resulting vorticity distribution and vortex populations are analysed

Vortex merger near a topographic slope in a homogeneous rotating fluid

This work is a contribution to the PHYSINDIEN research program. It was supported by CNRS-RFBR contract PRC 1069/16-55-150001.The effect of a bottom slope on the merger of two identical Rankine vortices is investigated in a two dimensional, quasi-geostrophic, incompressible fluid. When two cyclones initially lie parallel to the slope, and more than two vortex diameters away from the slope, the critical merger distance is unchanged. When the cyclones are closer to the slope, they can merge at larger distances, but they lose more mass into filaments, thus weakening the efficiency of merger. Several effects account for this: the topographic Rossby wave advects the cyclones, reduces their mutual distance and deforms them. This along shelf wave breaks into filaments and into secondary vortices which shear out the initial cyclones. The global motion of fluid towards the shallow domain and the erosion of the two cyclones are confirmed by the evolution of particles seeded both in the cyclone sand near the topographic slope. The addition of tracer to the flow indicates that diffusion is ballistic at early times. For two anticyclones, merger is also facilitated because one vortex is ejected offshore towards the other, via coupling with a topographic cyclone. Again two anticyclones can merge at large distance but they are eroded in the process. Finally, for taller topographies, the critical merger distance is again increased and the topographic influence can scatter or completely erode one of the two initial cyclones. Conclusions are drawn on possible improvements of the model configuration for an application to the ocean.PostprintPeer reviewe

Generalized Contour Dynamics: A Review

Contour dynamics is a computational technique to solve for the motion of vortices in incompressible inviscid flow. It is a Lagrangian technique in which the motion of contours is followed, and the velocity field moving the contours can be computed as integrals along the contours. Its best-known examples are in two dimensions, for which the vorticity between contours is taken to be constant and the vortices are vortex patches, and in axisymmetric flow for which the vorticity varies linearly with distance from the axis of symmetry. This review discusses generalizations that incorporate additional physics, in particular, buoyancy effects and magnetic fields, that take specific forms inside the vortices and preserve the contour dynamics structure. The extra physics can lead to time-dependent vortex sheets on the boundaries, whose evolution must be computed as part of the problem. The non-Boussinesq case, in which density differences can be important, leads to a coupled system for the evolution of both mean interfacial velocity and vortex sheet strength. Helical geometry is also discussed, in which two quantities are materially conserved and whose evolution governs the flow

The effect of slip length on vortex rebound from a rigid boundary

The problem of a dipole incident normally on a rigid boundary, for moderate to large Reynolds numbers, has recently been treated numerically using a volume penalisation method by Nguyen van yen, Farge, and Schneider [Phys. Rev. Lett.106, 184502 (2011)]. Their results indicate that energy dissipating structures persist in the inviscid limit. They found that the use of penalisation methods intrinsically introduces some slip at the boundary wall, where the slip approaches zero as the Reynolds number goes to infinity, so reducing to the no-slip case in this limit. We study the same problem, for both no-slip and partial slip cases, using compact differences on a Chebyshev grid in the direction normal to the wall and Fourier methods in the direction along the wall. We find that for the no-slip case there is no indication of the persistence of energy dissipating structures in the limit as viscosity approaches zero and that this also holds for any fixed slip length. However, when the slip length is taken to vary inversely with Reynolds number then the results of Nguyen van yen et al. are regained. It therefore appears that the prediction that energy dissipating structures persist in the inviscid limit follows from the two limits of wall slip length going to zero, and viscosity going to zero, not being treated independently in their use of the volume penalisation method.Publisher PDFPeer reviewe

The HyperCASL algorithm

This paper outlines a major new extension to the diabatic Contour-Advective Semi-Lagrangian (CASL) algorithm [5; 8]. The extension, called the 'HyperCASL' (HCASL) algorithm, advects material potential vorticity contours like in CASL, but treats diabatic forcing or damping with a Vortex-In-Cell (VIC) algorithm. As a result, HCASL is fully Lagrangian regarding advection. A conventional underlying grid is used as in CASL for 'inversion', namely for obtaining the advecting velocity from the potential vorticity.</p