Integrability and regularity of 3D Euler and equations for uniformly rotating fluids

We consider 3D Euler and Navier-Stokes equations describing dynamics of uniformly rotating fluids. Periodic boundary conditions are imposed, and the ratio of domain periods is assumed to be generic (nonresonant). We show that solutions of 3D Euler/Navier-Stokes equations can be decomposed as U(t, x₁, x₂, x₃) = Ũ (t, x₁, x₂) + V(t, x₁, x₂, x₃) + r where Ũ is a solution of the 2D Euler/Navier-Stokes system with vertically averaged initial data (axis of rotation is taken along the vertical e₃). Here r is a remainder of order Ro^1/2_α where Ro_α = (H₀U₀)/(Ω₀L²₀) is the anisotropic Rossby number (H₀ - height, L₀ - horizontal length scale, Ω₀ - angular velocity of background rotation, U₀ - horizontal velocity scale); Ro_α = (H₀/L₀)Ro where H₀/L₀ is the aspect ratio and Ro = U₀/(Ω₀L₀) is a Rossby number based on the horizontal length scale L₀. The vector field V(t, x₁, x₂, x₃) which is exactly solved in terms of 2D dynamics of vertically averaged fields is phase-locked to the phases 2Ω₀t, τ₁(t), and τ₂(t). The last two are defined in terms of passively advected scalars by 2D turbulence. The phases τ₁(t) and τ₂(t) are associated with vertically averaged vertical vorticity curl Ū(t) · e₃ and velocity Ū³(t); the last is weighted (in Fourier space) by a classical nonlocal wave operator. We show that 3D rotating turbulence decouples into phase turbulence for V(t, x₁, x₂, x₃) and 2D turbulence for vertically averaged fields Ū(t, x₁, x₂) if the anisotropic Rossby number Ro_α is small. The mathematically rigorous control of the error r is used to prove existence on a long time interval T* of regular solutions to 3D Euler equations (T* → +∞, as Roα → 0); and global existence of regular solutions for 3D Navier-Stokes equations in the small anisotropic Rossby number case.