INTEGRABILITY AND REGULARITY OF 3D EULER AND EQUATIONS FOR UNIFORMLY ROTATING FLUIDS

We consider 3D Euler and Navier-Strokes equations describing dynamics of uniformly rotating fluids. Periodic boundary conditions are imposed , and the ratio of domain periods is assumed to be generic (nonresonen t). We show that solutions of 3D Euler/Navier-Strokes equations can be decomposed as U(t,x(1),x(2),x(3)) = (U) over tilde(t,x(1),x(2)) + V(t ,x(1),x(2),x(3)) + r where (U) over tilde is a solution of the 2D Eule r/Navier-Strokes 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) = (H0U0)/(Omega(0)L(0)(2)) is the anisotr opic Rossby number (H-0-height, L(0)-horizontal length scale, Omega(0) -angular velocity of back-ground rotation, U-0-horizontal velocity sca le); Ro(a) = (H-0/L(0))Ro where H-0/L(0) is the aspect ratio and Ro = U-0/(Omega(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 exactly solv ed in terms of 2D dynamics of vertically averaged fields is phase-lock ed to the phases 2 Omega(0)t, tau(1)(t), and tau(2)(t). The last two a re defined in terms of passively advected scalars by 2D turbulence. Th e phases tau(1)(t) and tau(2)(t) are associated with vertically averag ed vertical vorticity curl (U) over bar(t). e(3) and velocity (U) over bar(3)(t); the last is weighted (in Fourier space) by a classical non local wave operator. We show that 3D rotating turbulence decouples int o phase turbulence for V(t,x(1),x(2),x(3)) and 2D turbulence for verti cally averaged fields (U) over bar(t,x(1),x(2)) if the anisotropic Ros sby number Ro(a) is small. The mathematically rigorous control of the error r is used to prove existence on a long time interval T of regul ar solutions to 3D Euler equations (T --> +infinity, as Ro(a) --> 0); and global existence of regular solutions for 3D Navier-Stokes equati ons in the small anisotropic Rossby number case.