Asymptotic behavior of solutions to barotropic vorticity equation on a sphere

The behavior of a viscous incompressible fluid on a rotating sphere is described by the nonlinear barotropic vorticity equation (BVE). Conditions for the existence of a bounded set that attracts all BVE solutions are given. In addition, sufficient conditions are obtained for a BVE solution to be a global attractor. It is shown that, in contrast to the stationary forcing, the dimension of the global BVE attractor under quasiperiodic forcing is not limited from above by the generalized Grashof number.

On the Vorticity Balance over Steep Slopes: Kuroshio Intrusions Northeast of Taiwan

The circulation of the Kuroshio northeast of Taiwan is characterized by a large anticyclonic loop of surface intrusion and strong upwelling at the shelfbreak. To study the mechanisms of Kuroshio intrusions, the vorticity balance is examined using a high-resolution nested numerical model. In the 2D depth-averaged vorticity equation, the advection of geostrophic potential vorticity (APV) term and the joint effect of baroclinicity and relief (JEBAR) term are dominant. On the other hand, in the 2D depth-integrated vorticity equation, the main balance is between nonlinear advection and bottom pressure torque. It is shown that JEBAR and APV tend to compensate, and their difference is comparable to bottom pressure torque. Perhaps most significantly, a general framework is provided for examination of vorticity balance over steep slopes through a full 3D depth-dependent vorticity equation. The 3D analysis reveals a well-defined bottom boundary layer over the shelfbreak, about 40 m deep and capped by the vertical velocity maximum. In the upper frictionless layer from the surface to about 100 m, the primary balance is between nonlinear advection and horizontal divergence. In the lower frictional layer, viscous stress is balanced by nonlinear advection and horizontal divergence. The bottom pressure torque, which corresponds to the depth-integrated viscous effect, is a proxy for viscous stress divergence at the bottom. The importance of nonlinear advection is further demonstrated in a sensitivity experiment by removing advective terms from momentum equations. Without nonlinear advection, the bottom pressure torque becomes trivial, the boundary layer vanishes, and the on-shelf intrusion is considerably weakened.

Regularized vortex approximation for 2D Euler equations with transport noise

We study a mean field approximation for the 2D Euler vorticity equation driven by a transport noise. We prove that the Euler equations can be approximated by interacting point vortices driven by a regularized Biot–Savart kernel and the same common noise. The approximation happens by sending the number of particles [Formula: see text] to infinity and the regularization [Formula: see text] in the Biot–Savart kernel to [Formula: see text], as a suitable function of [Formula: see text].

Regularity of Desingularized Models for Vortex Filaments in Incompressible Viscous Flows: A Geometrical Approach

Summary We establish the regularity of weak solutions for the vorticity equation associated to a family of desingularized models for vortex filament dynamics in 3D incompressible viscous flows. These generalize the classical model ‘of an allowance for the thickness of the vortices’ due to Louis Rosenhead in 1930. Our approach is based on an interplay between the geometry of vorticity and analytic inequalities in Sobolev spaces.