On the stability of rotating stratified shear flow: An analytical study
Résumé
We study the stability problem of unbounded shear flow, with velocity Ui=Sx3δi1, subjected to a uniform vertical density stratification, with Brunt-Väisälä frequency N, and system rotation of rate Ω about an axis aligned with the spanwise (x2) 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 σφ2=R(R+1)sin2 φ+Ri, where R=(2Ω/S) is the rotation number, φ is the angle between the horizontal wave vector and the streamwise axis, and Ri=N2/S2 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 Ri>1/4, or if 00, or if R(R+1)<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 Ri and R and an arbitrary orientation of the wave vector. Crucial contributions to spectral energies are shown to come from the k1=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.
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