Reduced equations for finite beta tearing modes in tokamaks

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Abstract
The equations of resistive magnetohydrodynamics (MHD) are recast in a form that is useful for studying the evolution of those toroidal systems where the fast magnetosonic wave plays no important role. The equations are exact and have ∇ ⋅ B=0 satisfied explicitly. From this set of equations it is a simple matter to derive the equations of reduced MHD to any order in the inverse aspect ratio ε of the torus and for β∼ε or smaller. This is demonstrated by deriving a reduced set of MHD equations that are correct to fifth order in ε. These equations contain the exact equilibrium relation and, as such, can be used to find three‐dimensional stellarator equilibria. In addition, if a subsidiary ordering in η, the resistivity, is made, the equations of Glasser, Greene, and Johnson [Phys. Fluids 8, 875 (1967); 1 9, 567 (1967)] are recovered. This set of reduced equations has been coded by extending the initial value code hilo [Phys. Fluids 2 6, 3066 (1983)]. Results obtained for both ideal and resistive linear stability from the reduced equations are compared with those obtained by solving the full set of MHD equations in a cylinder. Good agreement is shown for both zero and finite‐beta calculations. Comparisons are also made with analytic theory illuminating the present limitations of the latter.