A contribution to the baroclinic instability problem

The problem of the baroclinic instability of zonal flow is formulated under assumptions similar to Green's model. The dynamic equations of the model lead to a single differential equation for the vertical velocity, which is of the confluent hyper-geometric type. The eigen-value problem is solved and numerical methods are only used to find the roots of the characteristic equation.

It is shown that, for large values of the nondimensional shear ? (which is related to Green's parameter γ = 2A^-1), there is a continuous transition from fast growing unstable modes to unstable modes with small growth-rate. The eigen-values of the characteristic equation of the model do not show the continuous transition from the conjugate complex roots to the real roots as appears in the numerical computations of Green (1960) and Hirota (1968). On the other hand, for smaller values of the shear, a continuous transition from the unstable to stable solutions can occur. On the long wave side of such a transition point there are weak unstable waves whose steering level is outside the flow; on the short wave side there is a slight gap in the spectrum of unstable waves.

The structure of the waves is discussed and it is shown that the unstable waves with slow growth rate have the structure of a baroclinically unstable wave in a shallow layer close to the lower boundary.