^{1,a)}, R. Khvoles

^{2}and J. C. McWilliams

^{3}

### Abstract

A family of semianalytical solutions is presented describing multipolar vortical structures with zero total circulation in a variety of two-dimensional models. Analytics are used to determine the form of a multipole edge, or separatrix, and the solution outside this separatrix. The interior is solved using a Newton-Kantorovich (successive linearization) procedure combined with a collocation method. The models considered are the quasigeostrophic - and -planes, with either the rigid-lid or free-surface conditions. A multipole, termed also an -pole, is a vortical system that possesses an -fold symmetry and is comprised of a central core vortex and satellite vortices surrounding the core. Fluid parcels in the core and the satellites revolve oppositely, and the multipole as a whole rotates steadily. The characteristics of the multipoles are examined as functions of and a parameter that incorporates the Rossby deformation radius, -effect, and the vortex’s angular velocity. The analogy between the -plane modons and -plane multipoles is tracked.

This research was supported by Binational Israel–U.S. Science Foundation (BSF) Grant No. 2002392. We thank G. J. F. van Heijst and the two anonymous referees for helpful comments on the manuscript.

I. INTRODUCTION

II. BASIC MODELS

III. RIGID-LID -PLANE MULTIPOLES

A. Formulation and method

B. Smooth multipoles on the rigid-lid -plane

C. Accuracy estimates

IV. EQUIVALENT-BAROTROPIC AND -PLANE MULTIPOLES

A. Unification of models

B. Allowed angular velocity of a -plane multipole: Analogy with -plane modons

C. Exterior solution and the separatrix form

D. Smooth -plane and equivalent-barotropic -plane multipoles

V. CONCLUSION

### Key Topics

- Vortex dynamics
- 52.0
- Rotating flows
- 37.0
- Quadrupoles
- 9.0
- Numerical solutions
- 7.0
- Polynomials
- 7.0

## Figures

Smooth semianalytical multipole solutions on the rigid-lid -plane. (a) Tripole ; (b) quadrupole ; (c) pentapole ; (d) hexapole ; (e) heptapole . Left column, stream-function contours in a corotating frame of reference; central column, cross section of the vorticity fields along the dashed line shown in the left-column panel; right column, scatter graph—the computed vorticity–stream-function relation. All variables are nondimensional; stream-function contours are given at a 0.25 step. Straight lines in the scatter graphs represent the vs relation outside the separatrix; curved lines, same in the interior region.

Smooth semianalytical multipole solutions on the rigid-lid -plane. (a) Tripole ; (b) quadrupole ; (c) pentapole ; (d) hexapole ; (e) heptapole . Left column, stream-function contours in a corotating frame of reference; central column, cross section of the vorticity fields along the dashed line shown in the left-column panel; right column, scatter graph—the computed vorticity–stream-function relation. All variables are nondimensional; stream-function contours are given at a 0.25 step. Straight lines in the scatter graphs represent the vs relation outside the separatrix; curved lines, same in the interior region.

Form characteristics of a smooth two-mode -pole on the rigid-lid -plane as functions of . Black bars, to ratio ( and are the maximal and minimal separatrix radii); white bars, ratio of the core-vortex area to the total area of satellites.

Form characteristics of a smooth two-mode -pole on the rigid-lid -plane as functions of . Black bars, to ratio ( and are the maximal and minimal separatrix radii); white bars, ratio of the core-vortex area to the total area of satellites.

Relation between and on the equivalent-barotropic -plane at a fixed ; , .

Relation between and on the equivalent-barotropic -plane at a fixed ; , .

The minimal separatrix radius, , and the ratio of the maximal radius, , to as functions of and in a unimodal multipole. (a) ; (b) . Indices indicate the values of to which the graphs correspond.

The minimal separatrix radius, , and the ratio of the maximal radius, , to as functions of and in a unimodal multipole. (a) ; (b) . Indices indicate the values of to which the graphs correspond.

The minimal separatrix radius, , and the ratio of the maximal radius, , to as functions of and in a smooth two-mode multipole. (a) ; (b) . Notations as in Fig. 4 .

The minimal separatrix radius, , and the ratio of the maximal radius, , to as functions of and in a smooth two-mode multipole. (a) ; (b) . Notations as in Fig. 4 .

Smooth semianalytical two-mode tripole solutions at different values of in the rigid-lid -plane and equivalent-barotropic -plane models. (a) ; (b) ; (c) . Left column, stream-function contours in the corotating frame of reference; central column, cross section of the potential vorticity field, , along the axis on a rigid-lid -plane ; right column, cross section of the absolute vorticity field, , along the axis on the equivalent-barotropic -plane . Notations as in Fig. 1 .

Smooth semianalytical two-mode tripole solutions at different values of in the rigid-lid -plane and equivalent-barotropic -plane models. (a) ; (b) ; (c) . Left column, stream-function contours in the corotating frame of reference; central column, cross section of the potential vorticity field, , along the axis on a rigid-lid -plane ; right column, cross section of the absolute vorticity field, , along the axis on the equivalent-barotropic -plane . Notations as in Fig. 1 .

Smooth semianalytical multipole solutions on the rigid-lid -plane at . (a) Quadrupole ; (b) pentapole ; (c) hexapole ; (d) heptapole . Notations as in Fig. 1 .

Smooth semianalytical multipole solutions on the rigid-lid -plane at . (a) Quadrupole ; (b) pentapole ; (c) hexapole ; (d) heptapole . Notations as in Fig. 1 .

Smooth semianalytical clockwise-rotating tripoles on the equivalent-barotropic -plane at . (a) , i.e., and ; (b) , i.e., and . Notations as in Fig. 1 .

Smooth semianalytical clockwise-rotating tripoles on the equivalent-barotropic -plane at . (a) , i.e., and ; (b) , i.e., and . Notations as in Fig. 1 .

## Tables

Accuracy estimates for smooth two-mode multipoles on a rigid-lid -plane at the output of the iterative procedure. Root-mean-square and maximal residuals in Eq. (21) and boundary conditions (22) computed over the interior and boundary grid points, respectively. , multipole order; , polynomial power.

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