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Angular momentum dynamics and the intrinsic drift of monopolar vortices on a rotating sphere

### Abstract

On the basis of the angular momentumequation for a fluid shell on a rotating planet, we analyze the intrinsic drift of a monopolar vortex in the shell. Central is the development of a general angular momentumequation for Eulerian fluid mechanics based on coordinate-free, general tensorial representations of the underlying fluid dynamics on the one hand, and an appropriate representation of the Lie algebra so(3) of rotations on the other hand. We show that angular momentumfluid dynamics concisely describes the motion of vortices along the sphere and explains why both geostrophic cyclones and anticyclones drift in retrograde direction (westward), why anticyclones do so faster than cyclones, and why this difference is enhanced by a cyclostrophic correction. Technically, the analysis is based on a tensorial representation of the integral angular momentumequation for the fluid shell as a whole, and, derived from this, a coordinate representation with respect to coordinates which may move with the vortex along the surface of the planet. Depicting the angular momentum balance of cyclones and anticyclones in terms of vector diagrams, we present an overview of the results achieved.

© 2010 American Institute of Physics

Received 03 July 2009
Accepted 27 May 2010
Published online 30 August 2010

Acknowledgments:
This work has been carried out as part of the Nonlinear Ocean Dynamics Program of the Department of Physical Oceanography of the Netherlands Institute for Sea Research.

Article outline:

I. INTRODUCTION
II. COVARIANT FORMULATION OF MOMENTUM BALANCE OF FLUID ELEMENTS
III. COVARIANT FORMULATION OF ANGULAR MOMENTUM BALANCE
A. Rotation and angular momentum
B. The inner product of the momentum balance with an arbitrary vector field
C. Vector fields that generate rotations
D. Focusing on generators of rotations
IV. APPLICATION TO A REDUCED GRAVITY, SINGLE-LAYER MODEL ON A SPHERE
A. Independence of the radial coordinate
B. Integral angular momentum of the outer shell
C. Integral angular momentumequation for a reduced gravity, shallow-water model on a sphere
V. DISTINCTION BETWEEN PHENOMENON AND BACKGROUND MASS CURRENTS
A. Decomposition of mass currents
B. The angular momentumequation for a phenomenon
VI. CONCRETE REPRESENTATION SUITABLE FOR LOCALIZED PHENOMENA
A. Sign of dimensionless vortex strength
VII. NUMERICAL SOLUTIONS
A. Closing the angular momentumequations: The Gaussian monopole
B. Numerical solutions
VIII. ANALYSIS
A. Decoupled form of the angular momentumequations
B. Intrinsic drift and the stability thereof
C. Angular momentumequations and large drift velocity
1. An overview of all modes
2. Limit of low drift speed
D. The geostrophic balance: Covariant and conventional form
E. Effect of centrifugal correction on drift velocity
IX. INTERPRETATION IN TERMS OF AN ANGULAR MOMENTUM DIAGRAM
X. REMARKS ON THE DYNAMICS OF THE LOCAL VORTEX STRUCTURE AND THE RELATION OF MONOPOLES TO THE ROSSBY WAVE FIELD
XI. CONCLUSION

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2010-08-30

2016-04-30

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