ENERGY PRINCIPLES FOR LINEAR DISSIPATIVE SYSTEMS WITH APPLICATION TO RESISTIVE MHD STABILITY

A formalism for the construction of energy principles for dissipative systems is presented. It is shown that dissipative systems satisfy a c onservation law for the bilinear Hamiltonian provided the Lagrangian i s time invariant. The energy on the other hand, differs from the Hamil tonian by being quadratic and by having a negative definite time deriv ative (positive power dissipation). The energy is a Lyapunov functiona l whose definiteness yields necessary and sufficient stability criteri a. The stability problem of resistive magnetohydrodynamic (MHD) is add ressed: the energy principle for ideal MHD is generalized and the stab ility criterion by Tasso [Phys. Lett. 147, 28 (1990)] is shown to be n ecessary in addition to sufficient for real growth rates. An energy pr inciple is found for the inner layer equations that yields the resisti ve stability criterion D-R<0 in the incompressible limit, whereas the tearing mode criterion Delta'<0 is shown to result from the conservati on law of the bilinear concomitant in the resistive layer. (C) 1997 Am erican Institute of Physics.