Linear instability of broad baroclinic zones

Abstract

Baroclinic energy conversions play a central role in mid-latitude synoptic-scale dynamics, drawing energy from the meridional temperature gradient and feeding it into the eddy dynamics. The physical mechanisms underlying these conversions can be investigated through the theory of small-amplitude instabilities growing on a steady basic state. In a recent study the author analysed the instability associated with a boundary temperature transition concentrated in a narrow front. This paper looks at the complementary problem of a broad baroclinic zone. The solution is applicable both to the Eady model and to the Phillips two-level quasi-geostrophic model. The reduction in the growth rate relative to an infinitely broad, homogeneous baroclinic mne is inversely proportional to the width of the baroclinic zone, as found in previous studies. The constant of proportionality is expressed in a simplified and more general form, related to the derivative with respect to zonal wave number of the complex phase speed of the homogeneous problem. If the flow is purely baroclinic the maximum growth rate is reduced compared with that of a homogeneous, infinitely broad frontal zone by a factor 1 - (2kmaxr)-1. where kmax is the wave number of the most unstable mode of the homogeneous problem and r is the width of the baroclonic zone. The width, r, is defined in terms of the second meridional derivative of the boundary temperature gradient evaluated at the maximum of the latter. Results in the literature give consistent, though less general, expressions for the growth rate, but various predictions for the structure of the waves. The numerical solutions show that as the jet width is reduced a new type of mode appears in which critical lines at the lid (tropopause) play an important role.