Pseudomomentum and the Orthogonality of Modes in Shear Flows

Linear modes on shear flows are not orthogonal in the sense of energy; if two modes are present, the eddy energy is not equal to the sum of the eddy energy in the separate modes. However, linear modes are orthogonal in the sense of pseudomomentum (or pseudoenergy). Two applications of this result to planetary waves in horizontal and vertical sheer are discussed. 1) The qualitative character of the evolution of a disturbance to a stable meridional sheer flow, as described by the barotropic vorticity equation, depends critically on whether the disturbance projects primarily onto discrete modes or onto continuum modes that cascade enstrophy to small meridional scales. It is demonstrated that the pseudomomentum and pseudoenergy orthogonality relations provide a natural framework for examining the relative excitation of discrete and continuum modes. 2) Using the quasi-geostrophic potential vorticity equation, it is shown that pseudomomentum orthogonality provides a simple explanation for how quasi-stationary neutral external modes of large amplitude can be excited by a small initial disturbance.