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Register a new userWeakly nonlinear stability of magnetohydrodynamic systems with a center of symmetry to perturbations, involving large scales
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Added by
Vladislav Zheligovsky A.
Publication date
2006
fields
Physics
and research's language is
English
Authors
V.Zheligovsky
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I consider the problem of weakly nonlinear stability of three-dimensional parity-invariant magnetohydrodynamic systems to perturbations, involving large scales. I assume that the MHD state, the stability of which I investigate, does not involve large spatio-temporal scales, and it is stable to perturbations involving the same small spatial scales, as the perturbed MHD state. Mean-field equations, which I derive for the perturbation using asymptotic techniques for multiscale systems, are a generalization of the standard equations of magnetohydrodynamics (the Navier-Stokes equation with the Lorentz force and the magnetic induction equation). In them, the operator of combined eddy diffusivity emerges, which is in general anisotropic and not necessarily negatively defined, and new quadratic terms, analogous to the ones describing advection. A method for efficient computation of coefficients of the eddy diffusivity tensor and eddy advection terms in the mean-field equations is proposed.
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I consider the problem of weakly nonlinear stability of three-dimensional convective magnetohydrodynamic systems, where there is no alpha-effect or it is insignificant, to perturbations involving large scales. I assume that the convective MHD state (steady or evolutionary), the stability of which I investigate, does not involve large spatio-temporal scales, and it is stable to perturbations involving the same small spatial scales, as the perturbed state. Mean-field equations, which I derive for the perturbation using asymptotic techniques for multiscale systems, are a generalization of the equations of magnetohydrodynamics (the Navier-Stokes and magnetic induction equations). The operator of combined eddy diffusivity emerges, which is in general anisotropic and not necessarily negatively defined, as well as new quadratic terms analogous to the ones describing advection.
I construct a complete asymptotic expansion of solutions to the problem of linear stability of three-dimensional steady space-periodic magnetohydrodynamic states to perturbations involving large periods. Eddy diffusivity tensor is derived for parity-invariant steady states. I present numerical evidences that if perturbations of the flow are permitted, then the effect of negative eddy diffusivity emerges at much larger magnetic molecular diffusivities than in the kinematic dynamo problem (where no perturbations of the flow are assumed).
Weakly non-linear stability of regimes of free hydromagnetic thermal convection in a rotating horizontal layer with free electrically conducting boundaries is considered in the Boussinesq approximation. Perturbations are supposed to involve large spatial and temporal scales. Applying methods for homogenisation of parabolic equations, we derive the system of amplitude equations governing the evolution of perturbations under the assumption that the alpha-effect is insignificant in the leading order. The amplitude equations involve the operators of anisotropic combined eddy diffusivity correction and advection. The system is qualitatively different from the system of mean-field equations for large-scale perturbations of forced convective hydromagnetic regimes. It is mixed: equations for the mean magnetic perturbation are evolutionary, all the rest involve neither time derivatives, nor the molecular diffusivity operator.
A shear-improved Smagorinsky model is introduced based on recent results concerning shear effects in wall-bounded turbulence by Toschi et al. (2000). The Smagorinsky eddy-viscosity is modified subtracting the magnitude of the mean shear from the magnitude of the instantaneous resolved strain-rate tensor. This subgrid-scale model is tested in large-eddy simulations of plane-channel flows at two different Reynolds numbers. First comparisons with the dynamic Smagorinsky model and direct numerical simulations, including mean velocity, turbulent kinetic energy and Reynolds stress profiles, are shown to be extremely satisfactory. The proposed model, in addition of being physically sound, has a low computational cost and possesses a high potentiality of generalization to more complex non-homogeneous turbulent flows.
A single-wavenumber representation of nonlinear energy spectrum, i.e., stretching energy spectrum is found in elastic-wave turbulence governed by the Foppl-von Karman (FvK) equation. The representation enables energy decomposition analysis in the wavenumber space, and analytical expressions of detailed energy budget in the nonlinear interactions are obtained for the first time in wave turbulence systems. We numerically solved the FvK equation and observed the following facts. Kinetic and bending energies are comparable with each other at large wavenumbers as the weak turbulence theory suggests. On the other hand, the stretching energy is larger than the bending energy at small wavenumbers, i.e., the nonlinearity is relatively strong. The strong correlation between a mode $a_{bm{k}}$ and its companion mode $a_{-bm{k}}$ is observed at the small wavenumbers. Energy transfer shows that the energy is input into the wave field through stretching-energy transfer at the small wavenumbers, and dissipated through the quartic part of kinetic-energy transfer at the large wavenumbers. A total-energy flux consistent with the energy conservation is calculated directly by using the analytical expression of the total-energy transfer, and the forward energy cascade is observed clearly.
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