  We consider uniformly rotating incompressible Euler and Navier-Stokes
equations. We study the suppression of vertical gradients of Lagrangian
displacement ("vertical" refers to the direction of the rotation axis). We
employ a formalism that relates the total vorticity to the gradient of the
back-to-labels map (the inverse Lagrangian map, for inviscid flows, a diffusive
analogue for viscous flows). The results include a nonlinear version of the
Taylor-Proudman theorem: in a steady solution of the rotating Euler equations,
two fluid material points which were initially on a vertical vortex line, will
perpetually maintain their vertical separation unchanged. For more general
situations, including unsteady flows, we obtain bounds for the vertical
gradients of the Lagrangian displacement that vanish linearly with the maximal
local Rossby number.
