Collapse of generalized Euler and surface quasigeostrophic point vortices.

@article{Badin2018CollapseOG,
  title={Collapse of generalized Euler and surface quasigeostrophic point vortices.},
  author={Gualtiero Badin and Anna M. Barry},
  journal={Physical review. E},
  year={2018},
  volume={98 2-1},
  pages={
          023110
        }
}
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… 

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References

SHOWING 1-10 OF 195 REFERENCES
Motion of three vortices near collapse
A system of three point vortices in an unbounded plane has a special family of self-similarly contracting or expanding solutions: during the motion, the vortex triangle remains similar to the
Self-similar motion of three point vortices
One of the counter-intuitive results in the three-vortex problem is that the vortices can converge on and meet at a point in a finite time for certain sets of vortex circulations and for certain
Vortex merger in surface quasi-geostrophy
The merger of two identical surface temperature vortices is studied in the surface quasi-geostrophic model. The motivation for this study is the observation of the merger of submesoscale vortices in
Dynamics and transport properties of three surface quasigeostrophic point vortices.
TLDR
Compared with point vortices in two-dimensional flow, the SQG vortsices are found to produce flows with higher FTBE, indicating more mixing, and results are presented for analyzing mixing for arbitrary vortex strengths.
Asymptotic scale-dependent stability of surface quasi-geostrophic vortices: semi-analytic results
ABSTRACT The scale-dependent stability of surface quasi-geostrophic (SQG) vortices is studied both analytically and numerically. In particular, we study the sensitivity of the stability of SQG
Growth of solutions for QG and 2D Euler equations
The work of Constantin-Majda-Tabak [1] developed an analogy between the Quasi-geostrophic and 3D Euler equations. Constantin, Majda and Tabak proposed a candidate for a singularity for the
Asymptotics and numerics of a family of two-dimensional generalized surface quasi-geostrophic equations
We study the generalised 2D surface quasi-geostrophic (SQG) equation, where the active scalar is given by a fractional power α of Laplacian applied to the stream function. This includes the 2D SQG
A geometric application of Nambu mechanics: the motion of three point vortices in the plane
In this paper the dynamics of three point vortices in the plane is analysed in terms of Nambu mechanics and compared to the classical Hamiltonian dynamics. Two new aspects are introduced: (i) the
Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow
The growth of the gradient of a scalar temperature in a quasigeostrophic flow is studied numerically in detail. We use a flow evolving from a simple initial condition which was regarded by Constantin
Chaotic advection near a three-vortex collapse.
TLDR
The anomalous properties of tracer statistics are the result of the complex structure of the advection phase space, in particular, of strong stickiness on the boundaries between the regions of chaotic and regular motion.
...
...