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
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).