Title: Linear stability analysis of parallel shear flows for an inviscid generalized two-dimensional fluid system
Abstract: The linear stability of parallel shear flows for an inviscid generalized twodimensional (2D) fluid system, the so-called α turbulence system, is studied.This system is characterized by the relation q = -(-∆) α/2 ψ between the advected scalar q and the stream function ψ.Here, α is a real number not to exceed 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 (α = 2) is referred to as a Kelvin-Helmholtz (KH) instability; such an instability for the generalized 2D fluid system is investigated for 0 < α < 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-α for 1 < α < 3, where k is the wavenumber of the perturbation.In contrast, for 0 < α ≤ 1, the growth rate is infinite.In other words, a transition of the growth rate of the perturbation occurs at α = 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 α = 1, is proposed.