Rates, pathways, and end states of nonlinear evolution in decaying two‐dimensional turbulence: Scaling theory versus selective decay

Polar vortex crystals: Emergence and structure

Vortex crystals, geometric arrays of like-signed vortices, are observed in natural systems with vastly different space and time scales: at the poles of Jupiter (∼10,000-km radius and lifetime of at least 5 y) and in laboratory experiments with pure-electron plasma (∼3.5-cm radius, lifetime of about 1.7 s). We follow the adage “less is more” and show that minimal physics is required for polar vortex crystals formation and persistence. Crystals, resembling those of Jupiter, form from the free evolution of an unstratified and rapidly rotating fluid in an axisymmetric geometry. An essential ingredient in this minimal model is the decrease of the vertical component of the Coriolis force with distance from the pole. Once formed, the crystal seems to survive indefinitely.

Vortex crystals are quasiregular arrays of like-signed vortices in solid-body rotation embedded within a uniform background of weaker vorticity. Vortex crystals are observed at the poles of Jupiter and in laboratory experiments with magnetized electron plasmas in axisymmetric geometries. We show that vortex crystals form from the free evolution of randomly excited two-dimensional turbulence on an idealized polar cap. Once formed, the crystals are long lived and survive until the end of the simulations (300 crystal-rotation periods). We identify a fundamental length scale, Lγ=(U/γ)1/3, characterizing the size of the crystal in terms of the mean-square velocity U of the fluid and the polar parameter γ=fp/ap2, with f_p the Coriolis parameter at the pole and a_p the polar radius of the planet.

Violent relaxation in two-dimensional flows with varying interaction range

Understanding the relaxation of a system towards equilibrium is a long-standing problem in statistical mechanics. Here we address the role of long-range interactions in this process by considering a class of two-dimensional flows where the interaction between fluid particles varies with the distance as ∼r(α-2) for α>0. We find that changing α with a prescribed initial state leads to different flow patterns: for small α, a coarsening process leads to the formation of a sharp interface between two regions of homogenized α-vorticity; for large α, the flow is attracted to a stable dipolar structure through a filamentation process. Assuming that the energy E and the enstrophy Z are injected at a typical scale smaller than the domain scale L, we argue that convergence towards the equilibrium state is expected when the parameter (2π/L)(α)E/Z tends to one, while convergence towards a dipolar state occurs systematically when this parameter tends to zero. This suggests that weak long-range interacting systems are more prone to relax towards an equilibrium state than strong long-range interacting systems.

Effects of friction on forced two-dimensional Navier-Stokes turbulence

Large-scale dissipation mechanisms have been routinely employed in numerical simulations of two-dimensional turbulence to absorb energy at large scales, presumably mimicking the quasisteady picture of Kraichnan in an unbounded fluid. Here, "side effects" of such a mechanism--mechanical friction--on the small-scale dynamics of forced two-dimensional Navier-Stokes turbulence are elaborated by both theoretical and numerical analysis. Given a positive friction coefficient α, viscous dissipation of enstrophy has been known to vanish in the inviscid limit ν→0. This effectively renders the scale-neutral friction the only mechanism responsible for enstrophy dissipation in that limit. The resulting dynamical picture is that the classical enstrophy inertial range becomes a dissipation range in which the dissipation of enstrophy by friction mainly occurs. For each α>0, there exists a critical viscosity ν(c), which depends on physical parameters, separating the regimes of predominant viscous and frictional dissipation of enstrophy. It is found that ν(c)=[η'(1/3)/(Ck(f)(2))]exp[-η'(1/3)/(Cα)], where η' is half the enstrophy injection rate, k(f) is the forcing wave number, and C is a nondimensional constant (the Kraichnan-Batchelor constant). The present results have important theoretical and practical implications. Apparently, mechanical friction is a poor choice in numerical attempts to address fundamental issues concerning the direct enstrophy transfer in two-dimensional Navier-Stokes turbulence. Furthermore, as relatively strong friction naturally occurs on the surfaces and at lateral boundaries of experimental fluids as well as at the interfaces of shallow layers in geophysical fluid models, the frictional effects discussed in this study are crucial in understanding the dynamics of these systems.

Similarity decay of enstrophy in an electron fluid

A similarity decay law is proposed for enstrophy of a one-signed-vorticity fluid in a circular free-slip domain. It excludes the metastable equilibrium enstrophy which cannot drive turbulence, and approaches Batchelor's t(-2) law for strong turbulence. Measurements of the decay of a turbulent electron fluid agree well with the predictions of the decay law for a variety of initial conditions.

Model of convective Taylor columns in rotating Rayleigh-Bénard convection

Observations, and laboratory and numerical studies, of fluid flows with strong rotation and thermal forcing often show long-lived convective Taylor columns (CTCs) which carry a large portion of the vertical heat and mass fluxes. However, owing to experimental and numerical challenges, these structures remain poorly understood. Here we present a nonlinear, analytical multiscale model of CTCs in the context of rotating Rayleigh-Bénard convection that successfully matches numerical simulations and provides a new multiscale interpretation of the Taylor-Proudman constraint.

Effect of the deformation radius on the evolution of vortex properties in geostrophic turbulence

Coherent vortices in geostrophic turbulence grow in size and become fewer as they merge. It is shown that the deformation radius L(D) has no effect on the growth of the average vortex radius a as a grows from a L(D) to a>> L(D) . Its growth is algebraic, given by a proportional to t(xi/4) , where xi=0.72 . However, the deformation radius does have an effect on the decay of the number of vortices or vortex density rho, given by rho proportional to t(-xi) for a <<L(D) and rho t(-xi/2) for a>> L(D) . Thus the decay of rho becomes slower once a has grown to a size comparable to that of L(D) . One scaling theory for the entire range from a<< L(D) to a>> L(D) is presented and verified by numerical experiments. A special method for quadruplicating the numerical domain when the vortices become too few is proposed, which keeps the computation inexpensive. This work generalizes and agrees with previous work, in which the two special cases a<<L(D) and a>> L(D) are independently investigated.

Self-organization phenomena and decaying self-similar state in two-dimensional incompressible viscous fluids

The final self-similar state of decaying two-dimensional (2D) turbulence in 2D incompressible viscous flow is analytically and numerically investigated for the case with periodic boundaries. It is proved by theoretical analysis and simulations that the sinh-Poisson state comega=-sinh (betapsi) is not realized in the dynamical system of interest. It is shown by an eigenfunction spectrum analysis that a sufficient explanation for the self-organization to the decaying self-similar state is the faster energy decay of higher eigenmodes and the energy accumulation to the lowest eigenmode for given boundary conditions due to simultaneous normal and inverse cascading by nonlinear mode couplings. The theoretical prediction is demonstrated to be correct by simulations leading to the lowest eigenmode of {(1,0) + (0,1)} of the dissipative operator for the periodic boundaries. It is also clarified that an important process during nonlinear self-organization is an interchange between the dominant operators, which leads to the final decaying self-similar state.

Relaxation towards localized vorticity states in drift plasma and geostrophic flows

The drift of ions in a magnetized plasma or the height fluctuations of a rotating fluid layer are described by the conservation equation of a potential vorticity. This potential vorticity contains an intrinsic length scale, the hybrid Larmor radius in plasma, and the Rossby length in the quasigeostrophic flow. The influence of this scale in the evolution of a random initial vorticity field is investigated using a thermodynamic approach. In contrast to the perfect fluid case, where the vorticity tends to a well defined stationary state, complete relaxation towards an equilibrium state is not observed in numerical simulations of quasigeostrophic decaying turbulence. The absence of global thermodynamic equilibrium is explained by the relaxation towards states of local equilibrium where the vorticity is concentrated. The interaction between these separated regions is extremely weak. Explicit, axisymmetric, localized solutions of the mean field integrodifferential equation of extremal entropy states are obtained using asymptotic methods. A comparison of the computed solutions with the observed coherent structures shows that they effectively correspond to states in local thermodynamic equilibrium.

Vortex statistics for turbulence in a container with rigid boundaries

The evolution of vortex statistics for decaying two-dimensional turbulence in a square container with rigid no-slip walls is compared with a few available experimental results and with the scaling theory of two-dimensional turbulent decay as proposed by Carnevale et al. Power-law exponents, computed from an ensemble average of several numerical runs, coincide with some experimentally obtained values, but not with data obtained from numerical simulations of decaying two-dimensional turbulence with periodic boundary conditions.

Anisotropic geophysical vortices

A survey is made of many types of coherent vortices in the Earth's ocean and atmosphere. These vortices often occur with strong, environmentally induced anisotropy in their velocity and vorticity fields. We propose a definition of the essential characteristics of coherent vortices and formulate hypotheses concerning their dynamical role in complex, anisotropic fluid motions. Finally, we analyze numerical solutions both for uniformly rotating, stably stratified three-dimensional flow and for two-dimensional flow for the phenomena of enstrophy cascade and dissipation, intermittency, isotropy in the appropriate coordinate frame, coherent vortex emergence, vortex population dynamics, and approach to a nonturbulent end state.