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Center for Nonlinear Studies |
The equations for magnetohydrodynamic turbulence couple velocity v and magnetic B fields, whereas in the Boussinesq framework, one couples fluctuations of velocity v and temperature θ. In either case, waves are supported by these systems (Alfven, inertia-gravity in the presence of rotation), with anisotropic dispersion relations. What kind of turbulence results from the interactions between nonlinear eddies and waves, and in what parameter domains? Formal analogies can guide our intuition, as for example using statistical mechanics to guess the direction of transfer of the energy, either towards small scales or large scales. However, new results have emerged recently, using accurate direct numerical simulations (DNS), which show that in some cases, such as two-dimensional (2D) and 3D MHD, as well as rotating stratified turbulence (RST), a dual cascade of energy is observed, towards both small and large scales and in both cases with constant fluxes. The scaling of the relative strength of the inverse and direct cascades can be obtained through phenomenological arguments in agreement with weak/wave turbulence ideas. These will be backed by numerical data in the case of forced and/or decaying rotating stratified turbulence. The argument relies on the efficiency of the turbulence cascade to small scales, measured by the effective rate of kinetic energy dissipation compared to its dimensional evaluation: it varies as the Froude number Fr, that is the ratio of the wave period to the eddy turn-over time. This is characteristic of an intermediate flow regime which is found to smoothly bridge the gap, in a simple way, between strong waves and strong eddies. Another characteristic of such flows is their propensity to establish an inertial range in which a quasi-equipartition solution is reached, between kinetic and magnetic energy, or kinetic and potential energy, for a broad range of scales and of values of dimensionless parameters. Such a strong wave turbulence also displays intense intermittency, including non-Gaussian wings for the velocity itself, as documented in the purely stratified case. These strong and localized structures affect small-scale dynamics and overall dissipation, as for reconnection and mixing. Finally, if time permits, I shall sketch the phenomenological framework for RST within which one is led to a scaling for the so-called flux Richardson number and mixing efficiency, which both measure the intensity of the wave term (buoyancy flux) to the dissipation term in the momentum equation: they are found to decay as either Fr-2 for low and intermediate Fr, or for high Reynolds number Re and high Fr, as 1/Fr [~RB -1/2, where RB = Re.Fr2 is the so-called buoyancy Reynolds number], in agreement with numerical data and with observations. As a conclusion, much remains to be done using accurate numerical simulations, developing new numerical techniques, analyzing novel experiments and devising new models.
Host: Daniel Livescu