Analytical Study of Certain Magnetohydrodynamic-alpha Models

Abstract

In this paper we present an analytical study of a subgrid scale turbulence model of the three-dimensional magnetohydrodynamic (MHD) equations, inspired by the Navier-Stokes-alpha (also known as the viscous Camassa-Holm equations or the Lagrangian-averaged Navier-Stokes-alpha model). Specifically, we show the global well-posedness and regularity of solutions of a certain MHD-alpha model (which is a particular case of the Lagrangian averaged magnetohydrodynamic-alpha model without enhancing the dissipation for the magnetic field). We also introduce other subgrid scale turbulence models, inspired by the Leray-alpha and the modified Leray-alpha models of turbulence. Finally, we discuss the relation of the MHD-alpha model to the MHD equations by proving a convergence theorem, that is, as the length scale alpha tends to zero, a subsequence of solutions of the MHD-alpha equations converges to a certain solution (a Leray-Hopf solution) of the three-dimensional MHD equations.

26 pages, no figures, will appear in Journal of Math Physics; corrected typos, updated references

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• Analysis of a General Family of Regularized Navier-Stokes and MHD Models
• On the convergence rate of the Euler-$\alpha$, an inviscid second-grade complex fluid, model to the Euler equations
• Global regularity for a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations
• Magnetic Relaxation of a Voigt-MHD System
• The Navier-Stokes-Voight Model for Image Inpainting
• Onsager's Conjecture for Subgrid Scale $\alpha$-Models of Turbulence
• Global regularity and convergence of a Birkhoff-Rott-alpha approximation of the dynamics of vortex sheets of the 2D Euler equations
• A remark on the two-dimensional magneto-hydrodynamics-alpha system
• A Characteristic Mapping Method with Source Terms: Applications to Ideal Magnetohydrodynamics
• Convolutions on the Sphere: Commutation with Differential Operators