TY - JOUR

T1 - The motion of a fluid ellipsoid in a general linear background flow

AU - McKiver, William J

AU - Dritschel, David Gerard

PY - 2003/1/10

Y1 - 2003/1/10

N2 - The study of the motion of a fluid ellipsoid has a long and fascinating history stretching back originally to Laplace in the late 18th century. Recently, this subject has been revived in the context of geophysical fluid dynamics, where it has been shown that an ellipsoid of uniform potential vorticity remains an ellipsoid in a background flow consisting of horizontal strain, vertical shear, and uniform rotation. The object of the present work is to present a simple, appealing, and practical way of investigating the motion of an ellipsoid not just in geophysical fluid dynamics but in general. The main result is that the motion of an ellipsoid may be reduced to the evolution of a symmetric, 3 x 3 matrix, under the action of an arbitrary 3 x 3 'flow' matrix. The latter involves both the background flow, which must be linear in the Cartesian coordinates at the surface of the ellipsoid, and the self-induced flow, which was given by Laplace.The resulting simple dynamical system lends itself ideally to both numerical and analytical study. We illustrate a few examples and then present a theory for the evolution of a vortex within a slowly varying background flow. We show that a vortex may evolve quasi-adiabatically, that is, it stays close to an equilibrium form associated with the instantaneous background flow. The departure from equilibrium, on the other hand, is proportional to the rate of change of the background flow.

AB - The study of the motion of a fluid ellipsoid has a long and fascinating history stretching back originally to Laplace in the late 18th century. Recently, this subject has been revived in the context of geophysical fluid dynamics, where it has been shown that an ellipsoid of uniform potential vorticity remains an ellipsoid in a background flow consisting of horizontal strain, vertical shear, and uniform rotation. The object of the present work is to present a simple, appealing, and practical way of investigating the motion of an ellipsoid not just in geophysical fluid dynamics but in general. The main result is that the motion of an ellipsoid may be reduced to the evolution of a symmetric, 3 x 3 matrix, under the action of an arbitrary 3 x 3 'flow' matrix. The latter involves both the background flow, which must be linear in the Cartesian coordinates at the surface of the ellipsoid, and the self-induced flow, which was given by Laplace.The resulting simple dynamical system lends itself ideally to both numerical and analytical study. We illustrate a few examples and then present a theory for the evolution of a vortex within a slowly varying background flow. We show that a vortex may evolve quasi-adiabatically, that is, it stays close to an equilibrium form associated with the instantaneous background flow. The departure from equilibrium, on the other hand, is proportional to the rate of change of the background flow.

KW - 2-DIMENSIONAL VORTEX DYNAMICS

KW - VORTICES

KW - SHEAR

KW - TURBULENCE

KW - MERGER

KW - MODEL

UR - http://www.scopus.com/inward/record.url?scp=0037428328&partnerID=8YFLogxK

U2 - 10.1017/S0022112002002859

DO - 10.1017/S0022112002002859

M3 - Article

SN - 0022-1120

VL - 474

SP - 147

EP - 173

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

ER -