Stability for flows in a generalized two-dimensional fluid

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Stability of parallel shear flows for a generalized two-dimensional (2D) fluid system, which is a unified form of some geophysical 2D fluid systems, is investigated. First, the conservation of the wave activity is derived. Second, it is shown that the governing equation for the generalized 2D fluid system can be written in the form of non canonical Hamiltonian form. Using them, a necessary condition for a linear stability of parallel shear flows is derived as: "if the transverse derivative of the generalized vorticity for the basic state is positive or negative definite, the flow is stable". Moreover, the Green's function for the generalized 2D fluid system is derived. Using the Green's function, physically realizable systems for the generalized 2D fluid system exist only for \alpha less than or equal 3. Here, \alpha is a real parameter describing the scale separation between the generalized vorticity and the velocity. In addition, the transition of the small scale behavior of the generalized vorticity at alpha=2, a well known property of forced and dissipated turbulence for the generalized 2D fluid, is explained in terms of the Green's function.