Reduction of the linear stability problem for general zonal flows with any Rossby and Richardson numbers to a PDE

The stability to linear disturbances of a general laterally and/or vertically sheared zonal flow in thermal wind balance is considered for any Rossby number and any Richardson number. The problem for normal mode perturbations is reduced to a second‐order partial differential equation (PDE) for a single dependent variable, the pressure perturbation, in which the complex phase speed is an eigenvalue. It is shown that the boundary conditions of no normal flow are expressible in terms of the pressure and its first‐order derivatives, and that the PDE can be derived from the linearized equation for conservation of the Ertel potential vorticity. The derivations are performed first for an incompressible Boussinesq fluid in Cartesian coordinates allowing the perturbations to be either hydrostatic or non‐hydrostatic. They are shown to generalize to a perfect gas and to motions on a sphere. A PDE for general (non‐normal mode) linear perturbations can also be derived. The Ertel potential vorticity of the zonal flow is shown to arise in the component of the momentum equation parallel to the zonal flow in a generalized Coriolis acceleration term. An ordinary differential equation (ODE) is also derived for the vertical velocity perturbation to laterally uniform zonal flows which is consistent with previous results. The differences between the singularities of the equations for pressure and vertical velocity perturbations are discussed. Integral constraints on the perturbations are used to prove the standard necessary condition for symmetric instabilities and a generalization of the Charney–Stern–Pedlosky integral constraints.