Transport in rotating fluids

We consider uniformly rotating incompressible Euler and Navier-Stokesequations. We study the suppression of vertical gradients of Lagrangiandisplacement ("vertical" refers to the direction of the rotation axis). Weemploy a formalism that relates the total vorticity to the gradient of theback-to-labels map (the inverse Lagrangian map, for inviscid flows, a diffusiveanalogue for viscous flows). The results include a nonlinear version of theTaylor-Proudman theorem: in a steady solution of the rotating Euler equations,two fluid material points which were initially on a vertical vortex line, willperpetually maintain their vertical separation unchanged. For more generalsituations, including unsteady flows, we obtain bounds for the verticalgradients of the Lagrangian displacement that vanish linearly with the maximallocal Rossby number.