If one cross-differentiates (2.10) and (2.11) with respect to y and x and subtracts to eliminate , the result is the vorticity equation (linear form)

In general both the Poincare gravity modes and the vortex
modes have vorticity; however the vortex modes are particularly
rich in vorticity as well as slow in temporal evolution.
Following Rossby we approximate the vorticity
using
the geostrophic approximation (4.1) for , with *f* fixed,
however the variability of *f* is accounted for in
. Moreover
is evaluated using (2.12) rather than the geostrophic approximation.
The resulting quasi-geostrophic vorticity equation is

where

and

the Jacobian of A and B.

Note that (4.3) is only first order in *t* and hence can admit
only one root for ; the gravity waves have essentially
been ``filtered out'' by use of the quasi-geostrophic relation.
The term is the *restoring term*. In order to
gain some appreciation for this and other terms, we may note
that for an inviscid but stratified fluid the potential
absolute vorticity

is conserved following a fluid column of effective thickness
*h*. This holds for each of the vertical modes if *h* is taken
as ; for example for the barotropic mode this
gives . Furthermore, using the geostrophic
approximation for the relative vorticity
for
mode *n* gives for the potential absolute vorticity of
mode *n*:

which in the absence of any disturbance reduces to
for the basic state.
Eq. (4.3) is simply the linearized version of
following the fluid at the geostrophic
velocity (4.1). The restoring term
arises if the flow crosses contours of
(i.e., when is not a function of ).
For example, if the flow is northwards, then *f* increases
following the fluid; in order that remain constant,
either must decrease or must increase,
or both.
The quantity when
multiplied by is the linearized version of the
anomaly of from . If one does allow for variations
of *D* and hence of the with *x*, *y* then a more general
form of the restoring term in (4.3) is

where the vector is defined by

which is a normalized gradient of potential absolute
vorticity for the base state for mode *n*. For the barotropic
mode this reduces to

If *D* and the for are constant then
all have the common value
whose direction is northwards and whose magnitude is
, being the geocentric
distance to the water parcel (about 6370 km). Following
almost universal convention we will denote
as the Rossby parameter.

For barotropic modes on a sloping shelf, the effect of
variable *D* may dominate over that of *f* in determining the
magnitude and direction of the effective parameter
. This can be important in studies of
subinertial quasi-geostrophic shelf waves (Lamb, 1932,
Art. 212; R. O. Reid, *J. Mar. Res.*, 1958; LeBlond and
Mysak, Ch. 10, Vol. 6, *The Sea*, 1977).

Sun May 19 00:59:05 CDT 1996