Lagrangian explanation of same-sign vortex aggregation in 2D fluids

Establish a rigorous Lagrangian dynamical mechanism that explains the aggregation of same-sign vortex structures in two-dimensional incompressible flows governed by the Euler equations, beyond spectral or statistical descriptions, and clarify the conditions under which two vortical structures merge into one in finite time.

Background

Aggregation of like-signed vortical structures is widely observed in simulations and in geophysical flows (e.g., blob merging, absorption of smaller vortices by larger ones), yet a direct Lagrangian mechanism has remained elusive. Spectral explanations exist, and Onsager’s negative-temperature theory describes the statistical character of large-scale end states, but neither provides a dynamical Lagrangian explanation for the relatively rapid aggregation events observed between individual structures.

The work develops one concrete Lagrangian mechanism in a specific regime—showing that a small concentrated vortex blob embedded in a background vorticity gradient drifts along the gradient, effectively biasing aggregation toward regions of higher vorticity. This provides a partial mechanism, but a general, fully Lagrangian explanation covering broader configurations and scales remains unresolved.