Instability of very long waves in a zonal flow

The eigenvalue problem governing baroclinic disturbances, relative to a zonal flow, of wavelengths comparable with the circumference of the Earth is formulated on the basis of: (a) adiabatic, frictionless motion of a perfect gas in a uniform gravitational field over a spherical Earth and (b) the restriction Ro < Ri−1/2 <1, where Ro and Ri are appropriate Rossby and Richardson numbers. Attention is focused on unstable disturbances, and several theorems governing the existence of such disturbances are deduced from an earlier and more general study of baroclinic stability. Explicit results for the complex wave speed of an unstable disturbance are given in the limits β → 0 and β → ∞, where β = Ro Ricotφ (φ = latitude). It is established that one and only one unstable mode exists for a wide class of monotonically increasing wind profiles.