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An 'energy principle' derived previously by Tasso (1978) is applied to the stability of shaped large aspect ratio Tokamaks. Most of the previous results on MHD, resistive and universal-type instabilities can be recovered in a simple way that demonstrates the power of this formation. This allows 3-dimensional tearing modes for general cross-sections to be investigated. An exact proof of the dissipative universal instability is given at least within the assumption of adiabatic motion. The relation of this to current work on collisionless drift is discussed.

An energy principle for 'helical' incompressible perturbations in shaped cross-section plasmas is derived in the Tokamak scaling ( epsilon identical to ka approximately=Bperpendicular to /Bz<

An energy principle for 2-d resistive instabilities has been found. It leads to a necessary and sufficient condition for stability allowing the use of test functions. One simple consequence is that the current density in a plasma with arbitrary cross section should not increase to the outside. Otherwise the plasma would be unstable against resistive instabilities.

An energy principle has been found for all resistive perturbations of a circular plasma cylinder in the Tokamak scaling (kr approximately=Btheta /Bz approximately= epsilon ). It allows stability to be determined independently of the resistive scaling because resistivity is taken to be finite. The inclusion of FLR effects and viscosity would only reduce the growth rates. Using simple test functions near the resonances always leads to instability. Estimates of growth rates can be made, these being only rough if FLR effects dominate.

The linear tearing mode stability is evaluated for 1D equilibria in cylindrical geometry with arbitrary current profiles. This treatment combines the energy method and the Ritz-Galerkin procedure using finite elements. Particularly for hollow currents with two singular q surfaces extremely accurate results showing the coupling influence of both singularities are obtained.

For the stationary one-dimensional Vlasov equation quasi-neutrality is used instead of the Poisson equation. This allows one to obtain explicit solutions for the density as a function of the potential and to prove the positiveness of the trapped particle distribution function in a potential interval. This method has obvious application to oscillatory waves and shock waves.