Linearized potential vorticity mode and its role in transition to baroclinic instability

Abstract

In addition to the effect of stable stratification and uniform shear in turbulent flows, a case which has already been modelled using linear approaches (rapid distortion theory) and direct numerical simulations, we introduce the effect of uniform rotation. Assuming the rotation axis to be vertical, and aligned with the gradients of density and mean velocity, an enslaved horizontal stratification is forced and baroclinic instability can occur. In that case, the linear analysis shows an extension of the unstable domain up to a gradient Richardson number \documentclass[12pt]{minimal}\begin{document}$\textit {Ri}$\end{document}Ri of 1 for kx = 0 modes. First, we extend this asymptotic analysis at t → ∞ to kx ≠ 0 modes. We show that these modes lead asymptotically to oscillating bounded solutions whose frequency depend only on the rotation number \documentclass[12pt]{minimal}\begin{document}$R=1/\textit {Ro}$\end{document}R=1/Ro. The analysis of the transition to baroclinic instability is then completed with a discussion of the role of kx ≈ 0 modes—small streamwise wavenumbers—and of the importance of coupling with the potential vorticity mode \documentclass[12pt]{minimal}\begin{document}$u^{(\Omega _{pot})}$\end{document}u(Ωpot) or Ertel's mode. The latter is shown to be determinant for dramatic transient growth at intermediate times. The structure of the stratified, sheared, rotating flow is predicted using a stochastic linear approach (referred to as kinematic simulation) to complete the rapid distortion theory analysis.