^{1,a)}and Joseph T. F. Zimmerman

^{2,b)}

### 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.

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.

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

### Key Topics

- Rotating flows
- 135.0
- Angular momentum
- 77.0
- Vortex dynamics
- 26.0
- Vector fields
- 24.0
- Cyclones
- 22.0

## Figures

Diagram of relation (38): the coordinate system rotates with respect to the planet about the vector . As a result, the vector , which is steady with respect to the planet, is observed to trace a circular path with respect to the coordinate system.

Diagram of relation (38): the coordinate system rotates with respect to the planet about the vector . As a result, the vector , which is steady with respect to the planet, is observed to trace a circular path with respect to the coordinate system.

Longitudinal velocities as a function of time (days) of, from top to bottom, a cyclogeostrophic cyclone, a geostrophic cyclone, a geostrophic anticyclone, and a cyclogeostrophic anticyclone. All these vortices had a Gaussian profile and obey the full angular momentum equations (48)–(50). Spatially the vortices oscillate about a fixed latitude, in this case about , while drifting westward.

Longitudinal velocities as a function of time (days) of, from top to bottom, a cyclogeostrophic cyclone, a geostrophic cyclone, a geostrophic anticyclone, and a cyclogeostrophic anticyclone. All these vortices had a Gaussian profile and obey the full angular momentum equations (48)–(50). Spatially the vortices oscillate about a fixed latitude, in this case about , while drifting westward.

Trajectories of several geostrophic anticyclones as found by numerical integration. At the vortices are released at several latitudes near . They also have different initial longitudinal velocities. The vortex that starts at 30° has initial velocity as given by Eq. (63). Overall the vortices have initial velocities , where, from bottom to top, .

Trajectories of several geostrophic anticyclones as found by numerical integration. At the vortices are released at several latitudes near . They also have different initial longitudinal velocities. The vortex that starts at 30° has initial velocity as given by Eq. (63). Overall the vortices have initial velocities , where, from bottom to top, .

The dimensionless drift speed [solution of Eq. (77)] of monopoles as a function of geographical latitude for various modes: slow regular monopoles (sr), slow irregular anticyclones (si), fast regular monopoles (fr), and fast irregular anticyclones (fi)

The dimensionless drift speed [solution of Eq. (77)] of monopoles as a function of geographical latitude for various modes: slow regular monopoles (sr), slow irregular anticyclones (si), fast regular monopoles (fr), and fast irregular anticyclones (fi)

Angular momentum diagram for an anticyclonic monopole showing the mechanism of retrograde drift. The encircled cross symbols represent vectors perpendicular to, and into, the plane of the paper.

Angular momentum diagram for an anticyclonic monopole showing the mechanism of retrograde drift. The encircled cross symbols represent vectors perpendicular to, and into, the plane of the paper.

Angular momentum diagram for a cyclonic monopole, showing that cyclones, too, drift in retrograde direction (westward). The encircled dot symbols represent vectors perpendicular to the plane of the paper and in the direction toward the reader.

Angular momentum diagram for a cyclonic monopole, showing that cyclones, too, drift in retrograde direction (westward). The encircled dot symbols represent vectors perpendicular to the plane of the paper and in the direction toward the reader.

## Tables

Local inner products are encountered during the evaluation of Eq. (40).

Local inner products are encountered during the evaluation of Eq. (40).

Turning senses (top views and signs), signs of mass anomalies , and the resulting signs of the ratio of these quantities, for cyclones (C) and regular and irregular anticyclones on the northern and southern hemispheres.

Turning senses (top views and signs), signs of mass anomalies , and the resulting signs of the ratio of these quantities, for cyclones (C) and regular and irregular anticyclones on the northern and southern hemispheres.

Correspondence of branches of Fig. 4 and the parameters and of Eq. (79).

Correspondence of branches of Fig. 4 and the parameters and of Eq. (79).

The nonvanishing components of the metric and nonvanishing Christoffel symbols of spherical coordinate system (E3).

The nonvanishing components of the metric and nonvanishing Christoffel symbols of spherical coordinate system (E3).

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