Freely evolving two-dimensional turbulence is dominated by coherent vortices. The density of these vortices decays in time as p tt with (=0.75. A new scaling theory is proposed which expresses all statistical properties in terms of g. Thus the average circulation of the vortices increases as t~t' and their average radius as t~. The total energy is constant, the enstrophy decreases as t~, and the vorticity kurtosis increases as t~. These results are supported both by numerical simulations of the fluid equations and by solutions of a modified point-vortex model. PACS numbers: 47.25.cBecause of its geophysical and astrophysical importance, the emergence and evolution of coherent vortices in freely evolving two-dimensional turbulence has been a subject of intense study in the last ten years. ' In thisLetter, we formulate a new scaling theory and present evidence for scaling behavior in both two-dimensional turbulence and a simple, punctuated-Hamiltonian, dynamical model of coherent vortices. In anticipation of this work on vortex dynamics, we have previously performed studies of punctuated-Hamiltonian models of aggregation in one-dimensional systems. The fluid-dynamical equations are g, +J(tlt, g) =v~( -1)~+'V t'tll, g-: V ttt, where ttt is the stream function, g is the vorticity, J(a, b) a"bJb, a~-is the Jacobian, V -=a"'+a, ' is the Laplacian, and v~i s the hyperviscosity for p a positive integer (p =2 here). The domain is a square of side 2trL and the boundary conditions are periodic in both x and y. Numerical solutions of Eq. (1) show that well separated, almost axisymmetric, coherent vortices emerge from structureless initial conditions. Between the vortices there is a background sea of small-scale, incoherent vorticity. After the emergence of the vortices the dynamics appears to be dominated by two processes: (1) mutual advection of well separated vortices in which Hamiltonian point-vortex dynamics is a good approximation, and (2) merger of like-sign vortices during close encounters.As a result of the mergers, the vorticity is concentrated in increasingly larger, fewer, and more widely separated vortices as time increases. A "vortex census" shows that, for a broad class of initial conditions, the number of vortices per area p(t) decreases according to p -t~, with g approximately 0.75. The focus of this Letter is on the scaling properties of this "dilute vortex gas. " There is a simple dimensional argument that predicts (=2. The kinetic energy per area 8 is invariant as v~0, ' where ] fO 2 Vy. Vydx 4 212+8-pg, ", a (3) This is easily seen as an approximation to the second expression in (2) for both spatial arguments within the same vortex (the self-energy 4, ). ' It also characterizes the contribution to 8 from arguments in separate vortices (the configuration energy 8"); 8, scales with the number of vortices, rather than the number of pairs, due to cancellations from vortices of opposite sign. We assume the contributions in (2) from arguments outside the vortices are negligible.Inviscid dynamics [vz =0 in Eq. (...
Rotating Rayleigh-Bénard convection exhibits, in the limit of rapid rotation, a turbulent state known as geostrophic turbulence. This state is present for sufficiently large Rayleigh numbers representing the thermal forcing of the system, and is characterized by a leading order balance between the Coriolis force and pressure gradient. This turbulent state is itself unstable to the generation of depth-independent or barotropic vortex structures of ever larger scale through a process known as spectral condensation. This process involves an inverse cascade mechanism with a positive feedback loop whereby large-scale barotropic vortices organize small scale convective eddies. In turn, these eddies provide a dynamically evolving energy source for the large-scale barotropic component. Kinetic energy spectra for the barotropic dynamics are consistent with a k-3 downscale enstrophy cascade and an upscale cascade that steepens to k-3 as the box-scale condensate forms. At the same time the flow maintains a baroclinic convective component with an inertial range consistent with a k-5/3 spectrum. The condensation process resembles a similar process in two dimensions but is fully three-dimensional.
Decaying two-dimensional turbulence is characterized by the emergence of coherent vortices, which'subsequently dominate the evolution. The temporal scaling behavior of the flow is analyzed using a scaling theory, a long-time integration of the fluid equations, and a dissipative, modified point-vortex model that represents the turbulence as a system of interacting coherent structures. Good agreement is found in the behavior of average vortex properties, low-order moments of the flow fields, and the form of self-similar evolution. I. lNTRODUCTlON
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