Global splitting, integrability and regularity of 3D Euler and Navier-Stokes equations for uniformly rotating fluids

Abstract

Periodic boundary conditions are imposed, the ratio of domain periods is assumed to be generic (non-resonant). We show that solutions of 3D Euler/Navier-Stokes equations can be decomposed as U (t, x 1 , x 2 , x 3 ) = U (t, x 1 , x 2 ) + V (t, x 1 , x 2 , x 3 ) + r where U is a solution of the 2D Euler/Navier-Stokes system with vertically averaged initial data (axis of rotation is taken along the vertical e 3 ). Here r is a remainder of order Ro a 1/2 where Ro a = H 0 U 0 /Ω 0 L 0 2 is the anisotropic Rossby number (H 0 - height, L 0 - horizontal length scale, Ω 0 - angular velocity of background rotation, U 0 - horizontal velocity scale); Ro a = (H 0 /L 0 ) R 0 where H 0 /L 0 is the aspect ratio and Ro = U 0 /Ω 0 L 0 is a Rossby number based on the horizontal length scale L 0 . The vector field V (t, x 1 , x 2 , x 3 ) which is exactlty solved in terms of 2D dynamics of vertically averaged fields is phase-locked to the phases 2Ω 0 t, τ 1 (t) and τ 2 (t). The last two are defined in terms of passively advected scalars by 2D turbulence. The phases τ 1 (t) and τ 2 (t) are associated with vertically averaged vertical vorticity curl U(t).e 3 and velocity U 3 (t); the last is weighted (in Fourier space) by a classical non-local wave operator.