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The Quasi-Geostrophic Vorticity Equation

If one cross-differentiates (2.10) and (2.11) with respect to y and x and subtracts to eliminate tex2html_wrap_inline5879 , 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 tex2html_wrap_inline6075 using the geostrophic approximation (4.1) for tex2html_wrap_inline5881 , tex2html_wrap_inline5883 with f fixed, however the variability of f is accounted for in tex2html_wrap_inline6085 . Moreover tex2html_wrap_inline6087 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 tex2html_wrap_inline5133 ; the gravity waves have essentially been ``filtered out'' by use of the quasi-geostrophic relation. The term tex2html_wrap_inline6099 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 tex2html_wrap_inline6107 ; for example for the barotropic mode this gives tex2html_wrap_inline6109 . Furthermore, using the geostrophic approximation for the relative vorticity tex2html_wrap_inline6111 for mode n gives for the potential absolute vorticity of mode n:


which in the absence of any disturbance reduces to tex2html_wrap_inline6119 for the basic state. Eq. (4.3) is simply the linearized version of tex2html_wrap_inline6121 following the fluid at the geostrophic velocity (4.1). The restoring term tex2html_wrap_inline6099 arises if the flow crosses contours of tex2html_wrap_inline6125 (i.e., when tex2html_wrap_inline5879 is not a function of tex2html_wrap_inline6125 ). For example, if the flow is northwards, then f increases following the fluid; in order that tex2html_wrap_inline6133 remain constant, either tex2html_wrap_inline5943 must decrease or tex2html_wrap_inline5879 must increase, or both. The quantity tex2html_wrap_inline6139 when multiplied by tex2html_wrap_inline6141 is the linearized version of the anomaly of tex2html_wrap_inline6133 from tex2html_wrap_inline6125 . If one does allow for variations of D and hence of the tex2html_wrap_inline5885 with x, y then a more general form of the restoring term in (4.3) is


where the vector tex2html_wrap_inline6157 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 tex2html_wrap_inline5885 for tex2html_wrap_inline6169 are constant then all tex2html_wrap_inline6157 have the common value tex2html_wrap_inline6173 whose direction is northwards and whose magnitude is tex2html_wrap_inline6175 , tex2html_wrap_inline6177 being the geocentric distance to the water parcel (about 6370 km). Following almost universal convention we will denote tex2html_wrap_inline6179 as tex2html_wrap_inline5157 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 tex2html_wrap_inline5157 parameter tex2html_wrap_inline6189 . 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).

next up previous contents
Next: Rossby Modes for Constant Up: Quasi-Geostrophic Modes (Rossby Waves) Previous: Quasi-Geostrophic Modes (Rossby Waves)

Steve Baum
Sun May 19 00:59:05 CDT 1996