Representations for Energy of Waves on Steady Flows of a Compressible Non-isentropic Fluid

Stability and bifurcation of fluid flows have broad applications. According to Krein's theory of Hamiltonian spectra, the signature of wave energy constitutes a key ingredient for the stability criterion; coexistence of two modes with opposite signed energy or of zero-energy modes is necessary for triggering instability. Arnold's theorem for the hydrodynamics states that a steady Euler flow of an incompressible fluid is the extremal of the kinetic energy with respect to the isovortical perturbations. In this virtue, the energy of perturbations, second order in amplitude, is expressible in terms solely of the linear disturbance field, which brings a compact formula for the wave energy (Arnold 1966). In this investigation, we extend the energy formula to compressible non-isentropic flows. We start with the Howard-Gupta (HG) equation, an evolution equation of the Lagrangian displacement field. The HG equation is a second-order, in-time, differential equation driven by the force. Self-adjointness of the force operator guarantees to represent the wave energy in a simple integral. Starting from this representation, we manipulate the energy formula to include the compressibility and baroclinic effect. The resulting formula is compared with other formulas.