Finite-time circulation changes from topological rearrangement of distinguished curves and non-advective fluxes

Abstract

A vorticity budget diagnoses the growth or decay of a vortex from advective transport, or non-advective fluxes such as those due to friction or vortex tilting. However, when a budget calculation is performed with respect to a fixed coordinate, errors may result from time-dependence of the flow, leading to disagreement between the vorticity tendency and the observed vorticity field. An adaptive Lagrangian coordinate resolves this problem, provided that the resulting Lagrangian structure does not become too complicated.

In this study, a numerical simulation of Hurricane Nate (2011), the vorticity tendency is evaluated along distinguished material curves. There can be no net advective flux along a closed material curve, therefore, the total circulation tendency for a material region includes only the non-advective uxes acting along its boundary. A distinguished set of material curves (DMCs) associated with a distinguished hyperbolic trajectory (DHT) form a Lagrangian topology similar to that of a cat’s eye flow or “pouch” at each Eulerian snapshot. The time-dependence of velocities allows additional regions called lobes, which are formed by the intersections of DMCs, to exchange fluid across the vortex boundary by redefining the boundary.

Because the vortex boundary changes, we refer to this redefinition of material boundary as “topological rearrangement”. The method is useful for unsteady cat’s-eye flows and more complex interactions of multiple waves, vortices and background shear. All advective changes of the vortex circulation are identified by exchanges of the lobes, and all non-advective uxes act between the vortex and either the lobes or environmental flow. The Lagrangian topology and combination of advective and non-advective uxes relative to the topology is used to describe the evolution of the circulation of Nate during its time of formation.