The classical concept of hydrodynamic stability is reviewed in the light of recent developments in semigeostrophic theory. A certain Jacobian matrix is required to be positive-definite if solution of the semigeostrophic equations is to be well posed. formally, this gives three conditions which must be satisfied, one of which corresponds to the positivity of the semigeostrophic potential vorticity defined as the determinant of the matrix. The full implications of these stability criteria are sought here, together with some simple expression of the positive-definite requirement. The existence of a minimum energy principle, with respect to virtual parcel displacements which conserve absolute momentum and potential temperature is shown to play a key role, as in the equivalent circular vortex problem studied by Fjörtoft. It also appears to provide the basis for extending semigeostrophic theory to the general geophysical situation in which the gravitation vector and axis of rotation are not colinear.
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Shutts et al. (1987) studied this question.