Interacting SQG vortices and passive scalar transport

Interacting SQG Vortices and Passive Scalar Transport Stefan G. Llewellyn Smith and C. K. Taylor Mechanical & Aerospace Engineering, University of California San Diego sgls@ucsd.edu Abstract The surface quasi-geostrophic (SQG) equations are a model for low-Rossby number geo- physical flows in which the dynamics are governed by potential temperature dynamics on the boundary. The model can be used to explore the effect on transport of the tran- sition from two-dimensional to three-dimensional mesoscale geophysical flows. On an f -plane with linear stratification, we examine the dynamics of SQG vortices and the re- sulting three-dimensional flow including contributions at first order in Rossby number going beyond the usual QG equation to compute the velocity. While it is simple to obtain the vertical velocity, finding O(Ro) horizontal flow is more involved. The Finite Time Braiding Entropy (FTBE) of Thiffeault & Budisic is used to quantify the chaotic mixing induced by three point vortices. We also consider the exact SQG vortex solution devel- oped by Dritschel (2011) from the limit of a QG ellipsoid of constant potential vorticity. We examine the interaction of two such vortices and the resulting transport. Introduction At large scales, the ocean can be modeled as a thin body of fluid spread over a rotat- ing sphere. The vertical velocity is weak compared to the horizontal velocity, of order Rossby number Ro = U/f L. While the leading-order dynamics give horizontal velocities, transport could be significantly altered by the weak three-dimensional flow. This work examines transport due to idealized vortex solutions in the SQG model. Trans- port was studied by Aref and Pomphrey (1980) under the name “chaotic motion” in the simplified case of point vortices in two-dimensional flow. Aref & Pomphrey (1980) found that three interacting vortices would follow regular trajectories, and four vortices could follow chaotic trajectories. Thus, three point vortices that follow periodic paths can pro- duce chaotic flow and transport in the surrounding fluid. In the ocean, mesoscale vortical structures are observed. Here three SQG point vortex solutions are considered and their resulting chaotic mixing is examined. In order to analyze the quantitative effect of O(Ro) corrections, it is necessary to use an appropriate measure. In chaotic advection, a relevant property is the complexity of the flow, and one of the most convenient measures of global complexity seems to be topo- logical entropy. Topological entropy represents the exponential growth of the number of distinguishable orbits under the repeated iteration of the flow map: the higher the topological entropy, the more chaotic mixing is present. However, this value is difficult to compute given only a velocity field (Newhouse and Pignataro, 1993; Bowen, 1973). Thiffeault & Budiˇsi´c have recently developed a tool called Braidlab that, among other functions, calculates the Finite Time Braiding Exponent (FTBE), which approximates topological entropy using particle trajectories (Thiffeault and Budiˇsi´c, 2014; Budiˇsi´c and Thiffeault, 2015). The benefit of FTBE over the more commonly used Finite Time Lya- punov Exponent (FTLE) (Peacock and Dabiri, 2010) is that FTBE provides a global VIII th Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016