Lack of balance in continuously stratified rotating flows

Abstract

Periodic linear waves in a vertically sheared flow are considered in a continuously stratified layer of rotating fluid between homogeneous layers along a sloping bottom. This generalized Phillips' configuration has cyclonic horizontal shear and supports the Rossby modes related to the thickness variations of the homogeneous layers and inertia–gravity waves (IGW). While long Rossby modes with streamwise wavenumber κ < f/V (f is the Coriolis parameter, V is the maximum velocity) can be approximated by a neutral balanced solution, short waves with κ > f/V are found to have an inertial critical level and unbalanced gravity-wave-like structure beyond this level. Such ageostrophic unstable normal modes are shown explicitly to couple short Rossby waves with Doppler-shifted gravity waves. They exist even for small Froude number, although the growth rate of ageostrophic unstable modes is exponentially small in Froude number as in the Eady model. This lack of balance in continuously stratified flows agrees with the ultraviolet problem for Ripa's sufficient conditions of stability in a multi-layer model when the number of layers tends to infinity.