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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)

displaymath6073

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

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where

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and

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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

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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:

displaymath6117

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

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where the vector tex2html_wrap_inline6157 is defined by

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

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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