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MHD instabilities as an initial boundary-value problem

The gross MHD instabilities of straight cylindrical plasmas with elongated cross-section are investigated by solving the linearized MHD equations as an initial boundary-value problem on the computer. The linearized equations are Fourier-analysed along the ignorable co-ordinate of the equilibrium in order to reduce the computation to two dimensions. The method is applied to find the fixed-boundary instabilities of an equilibrium with rectangular walls. Starting with an arbitrary initial perturbation and following it for many Alfvén transit times, we find that the dominant instability overwhelms all stable oscillations after several e-folding times. We determine the growth rates of the fastest growing instabilities as a function of the equilibrium parameters. Then we examine the spatial structure of the physical variables (v⃗1, B⃗1, p1). We find that the cross-section of the velocity field displays a distinctive convection pattern. This structure becomes spatially concentrated around the point of maximum rotational transform as the equilibrium current is decreased to the marginal point, and concentrates near the wall as the current is increased. Given the equilibrium p′(ψ) = Jzc ψ/ψc, ψ (wall) = 0, we find that the marginal current density Jzc for each mode increases as the cross-section is elongated. But the growth rates of the higher azimuthal m-modes increase with elongation and their intervals of instability overlap with the lower m-modes.