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.
