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New intermediate models for rotating shallow water and an investigation of the preference for anticyclones

Published online by Cambridge University Press:  10 September 2009

MARK REMMEL  and
LESLIE SMITH
MARK REMMEL
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA
LESLIE SMITH*
Affiliation:
Departments of Mathematics and Engineering Physics, University of Wisconsin, Madison, WI 53706, USA
*
†Email address for correspondence: [email protected]
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Abstract

New intermediate models for the rotating shallow water (RSW) equations are derived by considering the nonlinear interactions between subsets of the eigenmodes for the linearized equations. It is well-known that the two-dimensional quasi-geostrophic (QG) equation results when the nonlinear interactions are restricted to include only the vortical eigenmodes. Continuing past QG in a non-perturbative manner, the new models result by including subsets of interactions which include inertial-gravity wave (IG) modes. The such simplest model adds nonlinear interactions between one IG mode and two vortical modes. In sharp contrast to QG, the latter model behaves similar to the full RSW equations for decay from balanced initial conditions as well as unbalanced random initial conditions with divergence-free velocity. Quantitative agreement is observed for statistics that measure structure size, intermittency and cyclone/anticyclone asymmetry. In particular, dominance of anticyclones is observed for Rossby numbers Ro in the range 0.1 < Ro < 1 (away from the QG parameter regime Ro → 0). A hierarchy of models is explored to determine the effects of wave-vortical and wave–wave interactions on statistics such as the skewness of vorticity in decaying turbulence. Possible advantages over previously derived intermediate models include (i) the non-perturbative nature of the new models (not restricting them a priori to any particular parameter regime) and (ii) insight into the physical and mathematical consequences of vortical–wave interactions.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 635 , 25 September 2009 , pp. 321 - 359
Copyright
Copyright © Cambridge University Press 2009

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