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Abstract
We consider the motion of a single ellipsoidal vortex with uniform potential vorticity in a rotating stratified fluid at finite Rossby number . Building on previous solutions obtained under the quasigeostrophic approximation (at first order in ), we obtain analytical solutions for the balanced part of the flow at . These solutions capture important ageostrophic effects giving rise to an asymmetry in the evolution of cyclonic and anticyclonic vortices. Previous work has shown that, if the velocity field induced by an ellipsoidal vortex only depends linearly on spatial coordinates inside the vortex, i.e. , then the dynamics reduces markedly to a simple matrix equation. The instantaneous vortex shape and orientation are encapsulated in a symmetric matrix , which is acted upon by the flow matrix to provide the vortex evolution. Under the quasigeostrophic approximation, the flow matrix is determined by inverting the potential vorticity to obtain the streamfunction via Poisson's equation, which has a known analytical solution depending on elliptic integrals. Here we show that higherorder balanced solutions, up to second order in the Rossby number, can also be calculated analytically. However, in this case there is a vector potential that requires the solution of three Poisson equations for each of its components. The source terms for these equations are independent of spatial coordinates within the ellipsoid, depending only on the elliptic integrals solved at the leading, quasigeostrophic order. Unlike the quasigeostrophic case, these source terms do not in general vanish outside the ellipsoid and have an inordinately complicated dependence on spatial coordinates. In the special case of an ellipsoid whose axes are aligned with the coordinate axes, we are able to derive these source terms and obtain the full analytical solution to the three Poisson equations. However, if one considers the homogeneous case, whereby the outer source terms are neglected, one can obtain an approximate solution having a compact matrix form analogous to the leadingorder quasigeostrophic case. This approximate solution proves to be highly accurate for the general case of an arbitrarily oriented ellipsoid, as verified through comparisons of the solutions with solutions obtained from numerical simulations of an ellipsoid using an accurate nonlinear balance model, even at moderate Rossby numbers.
Original language  English 

Pages (fromto)  333358 
Number of pages  26 
Journal  Journal of Fluid Mechanics 
Volume  802 
Early online date  3 Aug 2016 
DOIs  
Publication status  Published  Sept 2016 
Keywords
 Geophysical and geological flows
 Geostrophic turbulence
 Vortex dynamics
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 1 Finished

Geophysical Vortices: The Structure, stability and interaction of geophysical vortices
Reinaud, J. N., Dritschel, D. G. & Scott, R. K.
5/01/10 → 1/11/13
Project: Standard