Stability of rotating stratified shear flow: An analytical study

Abstract

We study the stability problem of unbounded shear flow, with velocity U(i)=Sx(3)delta(i1), subjected to a uniform vertical density stratification, with Brunt-Väisälä frequency N, and system rotation of rate Omega about an axis aligned with the spanwise (x(2)) direction. The evolution of plane-wave disturbances in this shear flow is governed by a nonhomogeneous second-order differential equation with time-dependent coefficients. An analytical solution is found to be described by Legendre functions in terms of the nondimensional parameter sigma(phi)(2)=R(R+1)sin(2) phi+R(i), where R=(2Omega/S) is the rotation number, phi is the angle between the horizontal wave vector and the streamwise axis, and R(i)=N(2)/S(2) is the Richardson number. The long-time behavior of the solution is analyzed using the asymptotic representations of the Legendre functions. On the one hand, linear stability is analyzed in terms of exponential growth, as in a normal-mode analysis: the rotating stratified shear flow is stable if R(i)>1/4, or if 0<R(i)<1/4 and R(R+1)>0, or if R(R+1)<0<R(R+1)+R(i) and 0<R(i)<1/4. It is unstable if R(i)<0 and R(R+1)+R(i)<0. On the other hand, different behaviors for the "exponentially stable" case can coexist in different wave-space regions: some modes undergo a power-law growth or a power-law decay, while other exhibit damped oscillatory behavior. For geophysical and astrophysical applications, stability diagrams are shown for all values of R(i) and R and an arbitrary orientation of the wave vector. Crucial contributions to spectral energies are shown to come from the k(1)=0 mode, which corresponds to an infinite streamwise wavelength. Accordingly, two-dimensional contributions to both kinetic and potential energies are calculated analytically in this streamwise direction.