Fast relaxation of a viscous vortex in an external flow

We study the evolution of a concentrated vortex advected by a smooth, divergence-free velocity field in two space dimensions. In the idealized situation where the initial vorticity is a Dirac mass, we compute an approximation of the solution which accurately describes, in the regime of high Reynolds numbers, the motion of the vortex center and the deformation of the streamlines under the shear stress of the external flow. For ill-prepared initial data, corresponding to a sharply peaked Gaussian vortex, we prove relaxation to the previous solution on a time scale that is much shorter than the diffusive time, due to enhanced dissipation inside the vortex core.

💡 Research Summary

The paper investigates the dynamics of a highly concentrated vortex advected by a smooth, divergence‑free external flow in two‑dimensional viscous fluid. Two types of initial data are considered: (i) an idealized point vortex represented by a Dirac mass, and (ii) a sharply peaked Gaussian vortex, which is more realistic for physical applications. The governing equations are the vorticity form of the Navier–Stokes system with an added external velocity field f(x,t):
∂ₜω + (u+f)·∇ω = νΔω, u = BS