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

We present an α-regularization of the Birkhoff-Rott equation, induced by the two-dimensional Euler-α equations, for the vortex sheet dynamics. We show that initially smooth self-avoiding vortex sheet remains smooth for all times under the α-regularized dynamics, provided the initial density of vorticity is an integrable function over the curve with respect to the arc-length measure.

We study the convergence rate of the solutions of the incompressible Euler-α, an inviscid second-grade complex fluid, equations to the corresponding solutions of the Euler equations, as the regularization parameter α approaches zero. First we show the convergence in H s , s > n/2 + 1, in the whole space, and that the smooth Euler-α solutions exist at least as long as the corresponding solution of the Euler equations. Next we estimate the convergence rate for two-dimensional vortex patch with smooth boundaries.

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We present an˛-regularization of the Birkhoff-Rott equation (BR-˛equation), induced by the two-dimensional Euler-˛equations, for the vortex sheet dynamics. We show the convergence of the solutions of Euler-˛equations to a weak solution of the Euler equations for initial vorticity being a finite Radon measure of fixed sign, which includes the vortex sheets case. We also show that, provided the initial density of vorticity is an integrable function over the curve with respect to the arc length measure, (i) an initially Lipschitz chord arc vortex sheet (curve), evolving under the BR-˛equation, remains Lipschitz for all times, (ii) an initially Hölder C 1;ˇ, 0 ġ< 1, chord arc curve remains in C 1;ˇf or all times, and finally, (iii) an initially Hölder C n;ˇ; n 1; 0 <ˇ< 1, closed chord arc curve remains so for all times. In all these cases the weak Euler-˛and the BR-˛descriptions of the vortex sheet motion are equivalent.