GLOBAL SPLITTING, INTEGRABILITY AND REGULARITY OF 3D EULER AND NAVIER-STOKES EQUATIONS FOR UNIFORMLY ROTATING FLUIDS

We consider 3D Euler and Navier-Stokes equations describing dynamics o f uniformly rotating fluids. Periodic boundary conditions are imposed, the ratio of domain periods is assumed to be generic (nonresonant). W e show that solutions of 3D Euler/Navier-Stokes equations can be decom posed 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 Euler/Navier-Stokes system with vertically averaged initial data (axi s 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(02) is the anisotrop ic Rossby number (H-0- height, L(0-) horizontal length scale, Omega(0- ) angular velocity of background rotation, U-0- horizontal velocity sc ale); Ro(a) = (H-0/L(0)) R(0) where H-0/L(0) is the aspect ratio and R o = 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 solved in terms of 2D dynamics of vertically averaged fields is phase -locked to the phases 2 Omega(0)t, tau(1) (t) and tau(2) (t). The last two are defined in terms of passively advected scalars by 2D turbulen ce. The phases tau(1) (t) and tau(2) (t) are associated with verticall y averaged vertical vorticity curl (U) over bar(t). e(3) and velocity <(3)over bar (3)> (t); the last is weighted (in Fourier space) by a cl assical non-local wave operator. We show that 3D rotating turbulence d ecouples into phase turbulence for V (t, x(1), x(2), x(3)) and 2D turb ulence for vertically averaged fields (U) over bar(t, x(1), x(2)) if t he anisotropic Rossby number Ro(a) is small. The mathematically rigoro us control of the error r is used to prove existence on a long time in terval T of regular solutions to 3D Euler equations (T* --> +infinity , as Ro(a) --> 0); and global existence of regular solutions for 3D Na vier-Stokes equations in the small anisotropic Rossby number case.