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Remarks about the arising of the vorticity-based Rossby number in Geophysical Fluid Dynamics

F.Crisciani, F. Cavallini and R.Mosetti

The investigation deals with two-scale quasi-geostrophic flows in which, due to the presence of a boundary, the advective Rossby number and the vorticity-based Rossby numbers take different orders of magnitude. The latter, far larger than the former, plays the role of ordering parameter in the derivation of the quasi-geostrophic equations governing boundary currents. Two models are considered: an analytical one concerning the formation region of the Gulf Stream and a Munk-like weakly non-linear numerical one. In the first of them the smooth matching between the vorticity-based Rossby number and the advective Rossby number is pointed out; the second applies a criterion for the selection of the sub-domain where the two-scale system holds. Main results:
a) while in the proximity of the boundary the vorticity-based and the advective Rossby numbers are significally different, the former being far larger than the latter, this difference tends to vanish for increasing distances from the boundary itself;
b) the pointwise vorticity-based Rossby number may be used as a marker of a double scale of motion in quasi-geostrophic numerical models.
In a recent paper (Cavallini et al., 2017) the possibility of a third scale of the motion, and hence the existence of a further Rossby number has been excluded by means of a scale analysis of quasi-geostrophic flows.