Instability in stratified accretion flows under primary and secondary perturbations

We consider horizontal linear shear flow (shear rate denoted by Λ) under vertical uniform rotation (ambient rotation rate denoted by Ω(0)) and vertical stratification (buoyancy frequency denoted by N) in unbounded domain. We show that, under a primary vertical velocity perturbation and a radial density perturbation consisting of a one-dimensional standing wave with frequency N and amplitude proportional to w(0)sin(ɛNx/w(0))≈ɛNx(≪1), where x denotes the radial coordinate and ɛ a small parameter, a parametric instability can develop in the flow, provided N(2)>8Ω(0)(2Ω(0)-Λ). For astrophysical accretion flows and under the shearing sheet approximation, this implies N(2)>8Ω(0)(2)(2-q), where q=Λ/Ω(0) is the local shear gradient. In the case of a stratified constant angular momentum disk, q=2, there is a parametric instability with the maximal growth rate (σ(m)/ɛ)=3√[3]/16 for any positive value of the buoyancy frequency N. In contrast, for a stratified Keplerian disk, q=1.5, the parametric instability appears only for N>2Ω(0) with a maximal growth rate that depends on the ratio Ω(0)/N and approaches (3√[3]/16)ɛ for large values of N.