Relations between Adiabatic and Incompressible (Non-Adiabatic) Systems and Their Stability

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Abstract
We examine in general whether the results valid for an incompressible, non-adiabatic system can be deduced from the results valid for an adiabatic (or a more general) system. A simple rule will be established by which an energy principle for incompressible, non-adiabatic perturbations is obtained from the energy principle for adiabatic perturbations. Application yields in particular the energy principles for magnetodynamic, respectively gravitational, respectively gravitational and magnetodynamic stability for incompressible, non-adiabatic perturbations which are the analogues of the energy principles of Bernstein et al., respectively Chandrasekhar, respectively Krüger and Callebaut for adiabatic perturbations. It is proved that an equilibrium state is more stable or at least equally stable for incompressible, non-adiabatic perturbations than for adiabatic ones. The conditions under which the adiabatic regime and the incompressible one are both stable or both unstable are studied. More detailed comparison theorems are enunciated for the case of magnetodynamic stability and all cases where the energy integral for γ=0 is independent of ξ∥ the component of % parallel to B. If div ξ can be chosen arbitrarily when ξ⊥ is given then the adiabatic and the incompressible regimes are both stable or both unstable. A detailed examination whether div ξ can be chosen arbitrarily or not due to the presence of closed field lines leads to a classification of the perturbations in two cases. We compare the stability between these two cases for the adiabatic regime and for the incompressible one and for each case, the stability between both regimes. A similar analysis is given for restrictive conditions on div ξ due to the presence of closed pressure shells. In general only one case of the complete classification has to be considered to decide on the stability. More-over the adiabatic and incompressible regimes are both stable or both unstable for most infinitely long tubes. The whole treatment is illustrated by the example of the linear pinch.

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