Generalized reduced magnetohydrodynamic equations

Abstract

A new derivation of reduced magnetohydrodynamic ͑MHD͒ equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with ͑1͒ MHD equilibrium, ͑2͒ fluctuations whose wave vector is aligned perpendicular to the magnetic field, and ͑3͒ those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general toroidal configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector ͑shear-Alfven wave time scale͒, which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson ͓Phys. Fluids 18, 875 ͑1975͔͒.

Energy conservation and numerical stability for the reduced MHD models of the non-linear JOREK code

“…The derivation of reduced MHD models is not a new research topic. We can find some derivations of models with parallel velocity for small curvature in the Tokamak in [23,8]. These derivations are based on an asymptotic analysis with the small parameter ε which corresponds to the curvature of the geometry.…”

“…In our case we use an algebraic derivation. Using the same assumptions for the magnetic field and the velocity field as in [23,8]. The same method and the same type of calculation can be found in the works of R. Sart and B. Després in [2,4,21].…”

“…For the full MHD model the total energy is conserved in the ideal case and dissipated in the resistive case. To validate the derivation of the model, to validate the choice of the projection operators and to obtain the stability results, which are important for the numerical methods we prove that the reduced MHD model satisfies an energy balance equation compatible with the energy balance associated with full MHD model [4]- [8]. Before the energy estimate we introduce the natural Dirirchlet and Neumann boundary conditions [2] given by ψ = u = T = ρ = 0 on ∂Ω and ∂u ∂n = ∂ψ ∂n = 0 (36) with n the outgoing normal to the domain.…”

Effect of sheared flows on classical and neoclassical tearing modes

“…For this we have adopted the simplified approach of [13] and dropped the stress tensor contributions as well as the ∇ · q term in the model. In the absence of flow, we generate an equilibrium profile by numerically solving the Grad-Shafranov equation with the help of an equilibrium code called TOQ (http://fusion.gat.com/toq).…”

“…The above equations have been programmed into an initial value code, called NEAR, which is a derivative of an older code called FAR. An early benchmarking of this code was carried out in [13], where terms proportional to Π e , Π and ∇ •q were dropped and the tests were restricted to the linear growth regime of classical tearing modes. Our emphasis, in this work, is to explore the effects of shear flow in the nonlinear regime and in particular to examine its influence on the evolution of NTMs.…”