Linear stability analysis of parallel shear flows for an inviscid generalized two-dimensional fluid system

The linear stability of parallel shear flows for an inviscid generalized two-dimensional (2D) fluid system, the so-called alpha turbulence system, is studied. This system is characterized by the relation q = -(-Delta)(alpha/2)psi between the advected scalar q and the stream function psi. Here, alpha is a real number not exceeding 3 and q is referred to as the generalized vorticity. In this study, a sufficient condition for linear stability of parallel shear flows is derived using the conservation of wave activity. A stability analysis is then performed for a sheet vortex that violates the stability condition. The instability of a sheet vortex in the 2D Euler system (alpha = 2) is referred to as a Kelvin-Helmholtz (KH) instability; such an instability for the generalized 2D fluid system is investigated for 0 < alpha < 3. The sheet vortex is unstable in the sense that a sinusoidal perturbation applied to it grows exponentially with time. The growth rate is finite and depends on the wavenumber of the perturbation as k(3-alpha) for 1 < alpha < 3, where k is the wavenumber of the perturbation. In contrast, for 0 < alpha <= 1, the growth rate is infinite. In other words, a transition of the growth rate of the perturbation occurs at alpha = 1. A physical model for KH instability in the generalized 2D fluid system, which can explain the transition of the growth rate of the perturbation at alpha = 1, is proposed.