Resonant interactions in rotating homogeneous three-dimensional turbulence

Direct numerical simulations of three-dimensional homogeneous turbulence under rapid rigid rotation are conducted for a fixed large Reynolds number and a sequence of decreasing Rossby numbers to examine the predictions of resonant wave theory. The theory states that 'slow modes' of the velocity, with zero wavenumber parallel to the rotation axis (k(z) = 0), will decouple at first order from the remaining 'fast modes' and solve an autonomous system of two-dimensional Navier-Stokes equations for the horizontal velocity components, normal to the rotation axis, and a two-dimensional passive scalar equation for the vertical velocity component, parallel to the rotation axis. The Navier-Stokes equation for three-dimensional rotating turbulence is solved in a 128(3) mesh after being diagonalized via 'helical decomposition' into normal modes of the Coriolis term. A force supplies constant energy input at intermediate scales. To verify the theory, we set up a corresponding simulation for the two-dimensional Navier-Stokes equation and two-dimensional passive scalar equation to compare them with the slow-mode dynamics of the three-dimensional rotating turbulence. The simulation results reveal that there is a clear inverse energy cascade to the large scales, as predicted by two-dimensional Navier-Stokes equations for resonant interactions of slow modes. As the rotation rate increases, the vertically averaged horizontal velocity field from three-dimensional Navier-Stokes converges to the velocity field from two-dimensional Navier-Stokes, as measured by the energy in their difference field. Likewise, the vertically averaged vertical velocity from three-dimensional Navier-Stokes converges to a solution of the two-dimensional passive scalar equation. The slow-mode energy spectrum approaches k(h)(-5/3), where k(h) is the horizontal wavenumber, and, as in two dimensions, energy flux becomes closer to constant the greater the rotation rate. Furthermore, the energy flux directly into small wavenumbers in the k(z) = 0 plane from non-resonant interactions decreases, while fast-mode energy concentrates closer to that plane. The simulations are consistent with an increasingly dominant role of resonant triads for more rapid rotation.