Collapse of generalized Euler and surface quasigeostrophic point vortices

Point-vortex models are presented for the generalized Euler equations, which are characterized by a fractional Laplacian relation between the active scalar and the stream function. Special focus is given to the case of the surface quasigeostrophic (SQG) equations, for which the existence of finite-time singularities is still a matter of debate. Point-vortex trajectories are expressed using Nambu dynamics. The formulation is based on a noncanonical bracket and allows for a geometrical interpretation of trajectories as intersections of level sets of the Hamiltonian and Casimir. Within this setting, we focus on the collapse of solutions for the three-point-vortex model. In particular, we show that for SQG the collapse can be either self-similar or non-self-similar. Self-similarity occurs only when the Hamiltonian is zero, while non-self-similarity appears for nonzero values of the same. For both cases, collapse is allowed for any choice of circulations within a permitted interval. These results differ strikingly from the classical point-vortex model, where collapse is self-similar for any value of the Hamiltonian, but the vortex circulations must satisfy a strict relationship. Results may also shed a light on the formation of singularities in the SQG partial differential equations, where the singularity is thought to be reached only in a self-similar way.

Figure 1

Surfaces given by Eqs. (58) (blue) and (59) (yellow) for the choice of values ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_3=1,{\mathrm{\Gamma }}_2=-0.49$ and (a) $H=0$, (b) $H=-0.1$, and (c) $H=0.1$ (bottom).

Figure 2

Intersections of the curves given by Eqs. (59) and (58) for the choice of values ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_3=1,{\mathrm{\Gamma }}_2=-0.49$ and $H$ ranging from $H=-1$ (most outer curves) to $H=0$ (most inner curves). The dashed lines represent the boundary of the admissible cone. The dot-dashed line represents the line of equilateral triangles defined by $r_{12}=r_{23}$.

Figure 3

Black dots: data from numerical integration of the equations of motion Eqs. (69) when ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_3=1,{\mathrm{\Gamma }}_2=-0.49$. Time increases from right to left and the initial conditions were chosen to satisfy Eqs. (73) and (74) with $B=H=0$. Solid line: analytical solutions of Eqs. (73) and (74), with $B=0$, given by $r_{23}=0.751484r_{12}$.

Figure 4

Black dots: data from numerical integration of the equations of motion Eqs. (69) when ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_3=1,{\mathrm{\Gamma }}_2=-0.49$. Time increases from right to left and the initial conditions were chosen to satisfy (73, 74) with $H=-0.1$. Solid curve: analytical solutions of Eqs. (73) and (74) with the specified choice of parameters.

Figure 5

Intersections of the curves given by Eqs. (59) and (58) for the choice of values ${\mathrm{\Gamma }}_1={\mathrm{\Gamma }}_3=1,{\mathrm{\Gamma }}_2=-0.49$, and $H$ ranging from $H=0.01$ (most outer curves) to $H=0.5$ (most inner curves). The dashed lines represent the boundary of the admissible cone. The dot-dashed line represents the line of equilateral triangles defined by $r_{12}=r_{23}$.