^{1}, Eva Kanso

^{1}and Paul K. Newton

^{1,a)}

### Abstract

We consider streamline patterns associated with single and double von Kármán point vortex streets on the surface of a nonrotating sphere, with and without pole vortices. The full family of streamline patterns are identified and the topological bifurcations from one pattern to another are depicted as a function of latitude and pole strength. The process involves first finding appropriate vortex strengths so that the configuration forms a relative equilibrium, then calculating the angular rotation of the configuration about the center-of-vorticity vector. We move in a rotating frame of reference so that the configuration is fixed, identify the separatrices in the flowfield, and plot the global streamline patterns as a function of the pole strengths and latitudinal positions of the rings. We carry the procedure out for single and double von Kármán vortex streets, with and without pole vortices. The single von Kármán street configurations are comprised of evenly spaced vortices on each of two rings that symmetrically straddle the equator and are skewed with respect to each other by half a wavelength, while the double von Kármán ring configurations are made up of four rings of evenly spaced vortices symmetrically straddling the equator.

I. INTRODUCTION

A. Preliminaries

B. Equations of motion

C. SVD primer

II. SINGLE VON KÁRMÁN STREETS

A. No pole vortices

B. With pole vortices

C. Angular frequency formulas

D. Streamline topologies

III. DOUBLE VON KÁRMÁN STREETS

A. The nullspace structure

B. Streamline topologies

IV. DISCUSSION

### Key Topics

- Rotating flows
- 80.0
- Singular values
- 14.0
- Vortex streets
- 12.0
- Bifurcations
- 11.0
- Topology
- 11.0

## Figures

Schematic of a single VKS on the sphere with an illustration of the vorticity vector where the upper ring is placed at colatitude . is the total number of vortices, is the number of vortices per ring, hence . When pole vortices are included, we have .

Schematic of a single VKS on the sphere with an illustration of the vorticity vector where the upper ring is placed at colatitude . is the total number of vortices, is the number of vortices per ring, hence . When pole vortices are included, we have .

Singular values for a single von Kármán vortex street with parameters and . (a) Without pole vortices, the nullspace dimension is one; (b) with pole vortices, the nullspace dimension is three.

Singular values for a single von Kármán vortex street with parameters and . (a) Without pole vortices, the nullspace dimension is one; (b) with pole vortices, the nullspace dimension is three.

(a) Angular velocity as a function of for a single VKS with . (b) Additional angular velocity due to pole vortices with . As approaches , .

(a) Angular velocity as a function of for a single VKS with . (b) Additional angular velocity due to pole vortices with . As approaches , .

Streamline topologies for a single VKS. The fixed parameters are and . The different topology types are obtained by varying . In (a), we begin with the degenerate case , and in (d) we end with the degenerate case . A single topology type exists in the range , as illustrated in (b) and (c). We call these Type I topologies.

Streamline topologies for a single VKS. The fixed parameters are and . The different topology types are obtained by varying . In (a), we begin with the degenerate case , and in (d) we end with the degenerate case . A single topology type exists in the range , as illustrated in (b) and (c). We call these Type I topologies.

Streamline topologies for a single VKS with vortices at the poles. The fixed parameters are , , and . The different topology types are attained by varying from 10 to −0.02. See Fig. 6 for a north pole view of the streamline topology bifurcations in the vicinity of . (a) Type I; (b) ; (c) Type II; (d) ; (e) Type III; (f) ; (g) Type IV; (h) ; (i) Type II.

Streamline topologies for a single VKS with vortices at the poles. The fixed parameters are , , and . The different topology types are attained by varying from 10 to −0.02. See Fig. 6 for a north pole view of the streamline topology bifurcations in the vicinity of . (a) Type I; (b) ; (c) Type II; (d) ; (e) Type III; (f) ; (g) Type IV; (h) ; (i) Type II.

North pole view of the streamline topologies for a single VKS with vortices at the poles. The fixed parameters are , , and . Here, we illustrate the streamline topology bifurcations in the vicinity of . In the range as shown in (c), a flower-shaped contour consisting of five elliptic points and five saddle points appears about the pole. Figures 6(a)–6(e) correspond to Figs. 5(e)–5(i), respectively. (a) Type III; (b) ; (c) Type IV; (d) ; (e) Type II.

North pole view of the streamline topologies for a single VKS with vortices at the poles. The fixed parameters are , , and . Here, we illustrate the streamline topology bifurcations in the vicinity of . In the range as shown in (c), a flower-shaped contour consisting of five elliptic points and five saddle points appears about the pole. Figures 6(a)–6(e) correspond to Figs. 5(e)–5(i), respectively. (a) Type III; (b) ; (c) Type IV; (d) ; (e) Type II.

Diagram of a double VKS with and without pole vortices. The configuration consists of one vortex street in the northern hemisphere, and a second in the southern hemisphere, where each street consists of two symmetrically skewed -vortex rings. One ring in each hemisphere has a latitude of from its respective hemisphere’s pole; these are referred to as the -rings. The second ring in each hemisphere has an angle of , and these are referred to as the -rings.

Diagram of a double VKS with and without pole vortices. The configuration consists of one vortex street in the northern hemisphere, and a second in the southern hemisphere, where each street consists of two symmetrically skewed -vortex rings. One ring in each hemisphere has a latitude of from its respective hemisphere’s pole; these are referred to as the -rings. The second ring in each hemisphere has an angle of , and these are referred to as the -rings.

Singular values for a double von Kármán vortex street: (a) without pole vortices and (b) with pole vortices. The fixed parameters are , , and .

Singular values for a double von Kármán vortex street: (a) without pole vortices and (b) with pole vortices. The fixed parameters are , , and .

Curves relating the pole vortex strength vs the angle ratio . The fixed parameters are and . (a) (i.e., ). (b) (i.e., ).

Curves relating the pole vortex strength vs the angle ratio . The fixed parameters are and . (a) (i.e., ). (b) (i.e., ).

Angular velocity as a function of for . The fixed parameter is . In (a), we illustrate the curves when , while the curves in (b) correspond to .

Angular velocity as a function of for . The fixed parameter is . In (a), we illustrate the curves when , while the curves in (b) correspond to .

Streamline topologies for a double VKS with vortices at the poles. The fixed parameters are , , and . The different streamline topologies are attained by varying . We use , and is increased from 0.9 to 0.996. The bifurcation point in (f) corresponds to the point at which . See Fig. 12 for a north pole view of the streamline topology bifurcations in the vicinity of . The topology types above correspond to all those observed in the range (i.e., ). (a) Type I; (b) ; (c) Type II; (d) ; (e) Type III; (f) ; (g) Type II; (h) ; (i) Type I.

Streamline topologies for a double VKS with vortices at the poles. The fixed parameters are , , and . The different streamline topologies are attained by varying . We use , and is increased from 0.9 to 0.996. The bifurcation point in (f) corresponds to the point at which . See Fig. 12 for a north pole view of the streamline topology bifurcations in the vicinity of . The topology types above correspond to all those observed in the range (i.e., ). (a) Type I; (b) ; (c) Type II; (d) ; (e) Type III; (f) ; (g) Type II; (h) ; (i) Type I.

North pole view of the streamline topologies for a double VKS with vortices at the poles in the vicinity of . At this point, the pole vortices switch signs. The fixed parameters are , , and . Figures 12(a)–12(c) correspond to Figs. 13(e)–13(g), respectively. (a) Type III; (b) ; (c) Type III.

North pole view of the streamline topologies for a double VKS with vortices at the poles in the vicinity of . At this point, the pole vortices switch signs. The fixed parameters are , , and . Figures 12(a)–12(c) correspond to Figs. 13(e)–13(g), respectively. (a) Type III; (b) ; (c) Type III.

A continuation of Fig. 11, the figures above are streamline topologies for a double VKS with vortices at the poles. The fixed parameters are again , , and . The different streamline topologies are attained by varying . We use , and is increased from 1.02 to 2. The bifurcation point in (f) corresponds to the point at which . See Fig. 14 for a north pole view of the streamline topology bifurcations in the vicinity of . The topology types above correspond to all those observed in the range (i.e., ). (a) Type I; (b) ; (c) Type II; (d) ; (e) Type IV; (f) ; (g) Type III; (h) ; (i) Type V.

A continuation of Fig. 11, the figures above are streamline topologies for a double VKS with vortices at the poles. The fixed parameters are again , , and . The different streamline topologies are attained by varying . We use , and is increased from 1.02 to 2. The bifurcation point in (f) corresponds to the point at which . See Fig. 14 for a north pole view of the streamline topology bifurcations in the vicinity of . The topology types above correspond to all those observed in the range (i.e., ). (a) Type I; (b) ; (c) Type II; (d) ; (e) Type IV; (f) ; (g) Type III; (h) ; (i) Type V.

North pole view of the streamline topologies for a double VKS with vortices at the poles in the vicinity of . At this point, the pole vortices switch signs. The fixed parameters are , , and . In the range as shown in (c), a flower-shaped contour consisting of five elliptic points and five saddle points appears about the poles. (a) Type II; (b) ; (c) Type IV; (d) ; (e) Type III. Figures 14(a)–14(e) correspond to Figs. 13(c)–13(g), respectively.

North pole view of the streamline topologies for a double VKS with vortices at the poles in the vicinity of . At this point, the pole vortices switch signs. The fixed parameters are , , and . In the range as shown in (c), a flower-shaped contour consisting of five elliptic points and five saddle points appears about the poles. (a) Type II; (b) ; (c) Type IV; (d) ; (e) Type III. Figures 14(a)–14(e) correspond to Figs. 13(c)–13(g), respectively.

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