Free boundary effects on baroclinic instability

Abstract

The effects of a free boundary on the stability of a baroclinic parallel flow are investigated using a reduced-gravity model. The basic state has uniform density stratification and a parallel flow with uniform vertical shear in thermal-wind balance with the horizontal buoyancy gradient. A finite value of the velocity at the free (lower) boundary requires the interface to have a uniform slope in the direction transversal to that of the flow. Normal-mode perturbations with arbitrary vertical structure are studied in the limit of small Rossby number. This solution is restricted to neither a horizontal lower boundary nor a weak stratification in the basic state.

In the limit of a very weak stratification and bottom slope there is a large separation between the first two deformation radii and hence short or long perturbations may be identified:

(a) The short-perturbation limit corresponds to the well-known Eady problem in which case the layer bottom is effectively rigid and its slope in the basic state is immaterial.

(b) In the long-perturbation limit the bottom is free to deform and the unstable wave solutions, which appear for any value of the Richardson number Ri, are sensible to its slope in the basic state. In fact, a sloped bottom is found to stabilize the basic flow.

At stronger stratifications there is no distinction between short and long perturbations, and the bottom always behaves as a free boundary. Unstable wave solutions are found for Ri→∞ (unlike the case of long perturbations). The increase in stratification is found to stabilize the basic flow. At the maximum stratification compatible with static stability, the perturbation has a vanishing growth rate at all wavenumbers.

Results in the long-perturbation limit corroborate those predicted by an approximate layer model that restricts the buoyancy perturbations to have a linear vertical structure. The approximate model is less successful in the short-perturbation limit since the constraint to a linear density profile does not allow the correct representation of the exponential trapping of the exact eigensolutions. With strong stratification, only the growth rate of long enough perturbations superimposed on basic states with gently sloped lower boundaries behaves similarly to that of the exact model. However, the stabilizing tendency on the basic flow as the stratification reaches its maximum is also found in the approximate model. Its partial success in this case is also attributed to the limited vertical structure allowed by the model.