^{1}, J. C. McWilliams

^{2}and Z. Kizner

^{3,a)}

### Abstract

Modons, i.e., localized steady-state solutions to the nonlinear equations of potential vorticity conservation, represent a paradigm for coherent structures in geophysical flows. A characteristic property of the baroclinic modons suggested until the present time is that, in such solutions, the boundary of the trapped-fluid area, the separatrix, is essentially independent of depth. In this paper, the existence of translating two-layer modons with noncoincident separatrices in the layers is demonstrated. The solutions are constructed by a combination of analytical and numerical methods. The exterior fields (outside the separatrices), governed by linear equations, are given analytically by explicit formulas; implicitly, these formulas also determine the separatrices. The interior fields are then determined using an iterative numerical algorithm for solving nonlinear partial differential equations. Applying this semianalytical approach, a variety of modon solutions are produced. Among them, the smooth modons (i.e., those with continuous vorticity) are of special geophysical significance. The existence of solutions of even higher smoothness, the so-called supersmooth modons marked by continuity of both the vorticity and its derivatives at the separatrices, is shown. The supersmooth modon possesses a number of features typical of the baroclinic vortical pairs observed earlier in numerical simulations.

This research was supported by Binational Israel–U.S. Science Foundation (BSF) Grant No. 2002392. The authors thank G. Reznik for helpful discussions.

I. INTRODUCTION

II. PROBLEM SETUP

A. Basic equations

B. Exterior problem

C. Interior problem

III. NUMERICAL IMPLEMENTATION

A. Determination of the exterior solution and separatrices

B. Interior solution

1. Linearization

2. Initial guess, polynomial approximation, and collocation

3. Accuracy

IV. RESULTS

A. Smooth modon solutions with noncoincident separatrices

B. Testing the stability

V. CONCLUSION

### Key Topics

- Vortex dynamics
- 19.0
- Rotating flows
- 12.0
- Numerical solutions
- 8.0
- Polynomials
- 6.0
- Vortex stability
- 6.0

## Figures

Upper- and lower-layer separatrices in two-layer vortical pairs: (a) -plane distributed modon equilibrium achieved in a numerical simulation (see text for details); (b) typical separatrices of an -plane heton translating in the direction. Solid closed line, upper layer; dashed line, lower layer.

Upper- and lower-layer separatrices in two-layer vortical pairs: (a) -plane distributed modon equilibrium achieved in a numerical simulation (see text for details); (b) typical separatrices of an -plane heton translating in the direction. Solid closed line, upper layer; dashed line, lower layer.

Notations: domains , , , and , and separatrices and . Solid closed line, upper layer; dashed line, lower layer.

Notations: domains , , , and , and separatrices and . Solid closed line, upper layer; dashed line, lower layer.

Domains and , their boundaries and , and schematic of the numerical grid.

Domains and , their boundaries and , and schematic of the numerical grid.

PV jump across the separatrix in the upper layer, , as a function of . -plane modons with , , and ; (a) general view with two roots 1 and 2; (b) magnified graph in the vicinity of root 1; (c) same in the vicinity of root 2.

PV jump across the separatrix in the upper layer, , as a function of . -plane modons with , , and ; (a) general view with two roots 1 and 2; (b) magnified graph in the vicinity of root 1; (c) same in the vicinity of root 2.

The -plane modon given by the parameters , , , and (root 2). (a) and (b) Upper-layer contours of the comoving streamfunction and potential vorticity , respectively; (c) separatrices (solid line) and (dashed line); (d) and (e) cross sections along the axis of and , respectively; (f) vs scatter-graph. In (a) and (b), the contour interval is 10% of the maximum; dotted line is the separatrix ; shaded, regions of positive values and .

The -plane modon given by the parameters , , , and (root 2). (a) and (b) Upper-layer contours of the comoving streamfunction and potential vorticity , respectively; (c) separatrices (solid line) and (dashed line); (d) and (e) cross sections along the axis of and , respectively; (f) vs scatter-graph. In (a) and (b), the contour interval is 10% of the maximum; dotted line is the separatrix ; shaded, regions of positive values and .

The -plane modon given by the parameters , , , and (root 1). Notations as in Fig. 5.

The -plane modon given by the parameters , , , and (root 1). Notations as in Fig. 5.

The supersmooth -plane modon solution given by the parameters , , , and . (a)–(f) As in Fig. 5; inset in (f), magnified view of vs scatter-graph in the vicinity of the point of contact .

The supersmooth -plane modon solution given by the parameters , , , and . (a)–(f) As in Fig. 5; inset in (f), magnified view of vs scatter-graph in the vicinity of the point of contact .

The -plane modon equilibrium achieved in a numerical simulation with time integration from an unsteady initial state [Fig. 1(a)]. Notations as in Fig. 5.

The -plane modon equilibrium achieved in a numerical simulation with time integration from an unsteady initial state [Fig. 1(a)]. Notations as in Fig. 5.

Evolution of the supersmooth modon (Fig. 7) in two simulations. (a)–(c) Under perturbations caused by the numerical procedure only (including the cutting filter); (d)–(f) tilted modon. (a) and (d) in the initial states; (b) and (e) at ; (c) and (f) vs scatter graphs at .

Evolution of the supersmooth modon (Fig. 7) in two simulations. (a)–(c) Under perturbations caused by the numerical procedure only (including the cutting filter); (d)–(f) tilted modon. (a) and (d) in the initial states; (b) and (e) at ; (c) and (f) vs scatter graphs at .

First simulation: modon characteristics at . Notations as in Fig. 5. Changes from to are fairly small.

First simulation: modon characteristics at . Notations as in Fig. 5. Changes from to are fairly small.

Second simulation: transition to a nonoverlapping state. (a) Separation between the upper- and lower-layer PV peaks vs time; (b) -component, , of the modon speed vs time.

Second simulation: transition to a nonoverlapping state. (a) Separation between the upper- and lower-layer PV peaks vs time; (b) -component, , of the modon speed vs time.

Second simulation: modon characteristics at . Notations as in Fig. 5. The overlap of the upper and lower PV chunks is negligible.

Second simulation: modon characteristics at . Notations as in Fig. 5. The overlap of the upper and lower PV chunks is negligible.

The supersmooth -plane modon solution given by the parameters , , , and . Notations as in Fig. 7; black area in (b), zero .

The supersmooth -plane modon solution given by the parameters , , , and . Notations as in Fig. 7; black area in (b), zero .

## Tables

Maximal difference between and , and difference between and , as functions of shift . The external parameters used are , , and .

Maximal difference between and , and difference between and , as functions of shift . The external parameters used are , , and .

Accuracy of the interior solution: root-mean-square (RMS) and maximal (Max) residuals. Two left columns, Eqs. (29) and (30); two right columns, Eqs. (29) and (31).

Accuracy of the interior solution: root-mean-square (RMS) and maximal (Max) residuals. Two left columns, Eqs. (29) and (30); two right columns, Eqs. (29) and (31).

Accuracy estimates for matching of the layer streamfunctions and their derivatives at and : RMS and maximal residuals. Four left columns, conditions (33), two right columns, conditions (34).

Accuracy estimates for matching of the layer streamfunctions and their derivatives at and : RMS and maximal residuals. Four left columns, conditions (33), two right columns, conditions (34).

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