SQG Point Vortex Dynamics with Order Rossby Corrections

TL;DR

This paper explores the dynamics of point vortices in surface quasi-geostrophic flow with ageostrophic corrections, revealing new explicit solutions, equilibrium configurations, and complex tracer trajectories influenced by ageostrophic effects.

Contribution

It provides the first explicit solutions for point vortex dynamics in SQG+ and characterizes equilibrium configurations, extending understanding beyond classical 2D Euler vortex models.

Findings

Explicit two-vortex solutions in SQG+

Exact three-vortex equilibria including polygons and collinear arrangements

Numerical analysis of tracer trajectories showing vertical excursions and local divergence

Abstract

Quasi-geostrophic flow is an asymptotic theory for flows in rotating systems that are in geostrophic balance to leading order. It is characterized by the conservation of (quasi-geostrophic) potential vorticity and weak vertical flows. Surface quasigeostrophy (SQG) is the special case when the flow is driven by temperature anomalies at a horizontal boundary. The next-order correction to QG, QG+, takes into account ageostrophic effects. We investigate point vortx dynamics in SQG+, building on the work of Weiss. The conservation laws for SQG point vortices that parallel the 2D Euler case no longer exist when ageostrophic effects are included. The trajectories of point vortices are obtained explicitly for the general two-vortex case in SQG and SQG+. For the three-vortex case, exact solutions are found for rigidly rotating and stationary equilibria consisting of regular polygons and collinear configurations. As in the 2D case, only certain collinear vortex configurations are rigid equilibria. Trajectories of passive tracers advected by point vortex systems are studied numerically, in particular their vertical excursions, which are non-zero because of ageostrophic effects. Surface trajectories can manifest local divergence even though the underlying fluid equations are incompressible.