Baroclinic instability of three-layer flows Part 1. Linear stability

The stability of quasi-geostrophic three-layer stratified flow in a channel is examined. The mean zonal velocity Ūᵢ is uniform within each layer (i = 1, 2, 3). Thus, as in the two-layer model of Phillips (1954), the only source of energy for growing disturbances is the potential energy stored in the sloping interfaces. Attention is focused upon the case in which ε = Δρ2/Δρ1 [Lt ] 1 (Δρ1, Δρ2 are the changes in density across the upper and the lower interfaces). Two scales of instability are possible: long waves (wavenumber O(1)) associated with the upper interface and short waves (wave-number O(ε−½)) associated with the lower interface. It is found that short waves are unstable only when S (the ratio of the slope of the lower interface to that of the upper interface) is greater than one or less than zero, i.e. when the gradients of potential vorticity in the two lower layers have opposite signs. The short waves have the largest growth rates when S2ε (the ratio of the potential energy stored in the lower interface to that stored in the upper interface) [gsim ] 1. The results of this analysis are used in an accompanying paper to interpret some experiments with three-layer eddies.