Resonant interactions in rotating homogeneous three-dimensional turbulence

Abstract

Direct numerical simulations of three-dimensional (3D) homogeneous turbulence under rapid rigid rotation are conducted to examine the predictions of resonant wave theory for both small Rossby number and large Reynolds number. The theory predicts that "slow modes" of the velocity, with zero wavenumber parallel to the rotation axis (k z = 0), will decouple from the remaining "fast modes" and solve an autonomous system of twodimensional (2D) Navier-Stokes equations for the horizontal velocity components, normal to the rotation axis, and a 2D 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 parallel simulations for the 2D Navier-Stokes equation and 2D passive scalar equation to compare them with the slowmode dynamics of the 3D rotating turbulence. The simulation results reveal that there is a clear inverse energy cascade to the large scales, as predicted by 2D Navier-Stokes equations for resonant interactions of slow modes. As the rotation rate increases, the vertically-averaged horizontal velocity field from 3D Navier-Stokes converges to the velocity field from 2D Navier-Stokes, as measured by the energy in their difference field. Likewise, the vertically-averaged vertical velocity from 3D Navier-Stokes converges to a solution of the 2D passive scalar equation. The slow-mode energy spectrum approaches k, where k h is the horizontal wavenumber, and energy flux becomes closer to constant, as in 2D, the greater the rotation rate. Furthermore, the energy flux directly into small wave numbers 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.

Low Rossby limiting dynamics for stably stratified flow with finite Froude number

“…In the horizontal, the slow equations are the two-dimensional Navier-Stokes equations along with two conservation laws, the horizontal kinetic energy and the vertical vorticity. In the case where the flow is not stratified this is consistent with other work Chen et al (2005). The vertically averaged vertical momentum equation is an advection-diffusion like equation that couples to the buoyancy through its vertical average.…”

Dimensional transition in rotating turbulence

“…Numerical simulations of rotating flows have also indicated a general trend towards two-dimensionalization [8,17] and some studies [9] have reported a split of the energy cascade into a downscale and an upscale process. Other studies [18] have found that, as the rotation rate increases, the flows exhibit a dynamics very similar to what is found in a 2D system, with the vertical * Corresponding author: deusebio@mech.kth.se velocity behaving as a passive scalar [19]. Bourouiba and Bartello [20] reported a strong transfer of energy from 3D modes to 2D modes at intermediate rotation rates, whereas at large rotations a decoupling of the dynamics was observed.…”

“…Even though such interactions can move energy towards the 2D plane [17], triadic resonance cannot transfer energy into/out of the k z = 0 plane, which, in fact, constitutes a closed resonant set. If only interactions among resonant triads are allowed, 2D modes should thus be dynamically decoupled from 3D modes [18,19]. Sometimes, it is advocated that nearly resonant interactions [17], for which the resonance condition is satisfied only to a certain degree, and higher-order resonance interactions [23] can explain the transfer of energy into the k z = 0 plane.…”

Kolmogorov’s hypotheses and global energy spectrum of turbulence

“…[10][11][12] HWD is also of interest in fluid dynamics, as each helical wave represents a peculiar flow with a certain chirality and thus it is an inherent useful tool in studying the role of helicity in turbulence dynamics. [13][14][15][16][17][18] Nevertheless, most of such studies consider the entire space, wherein the helical waves become helical Fourier modes. 19,20 Recently, Yang et al 21,22 have used HWD to analyze DNS data of turbulent channel flows, wherein the helical waves are different from the helical Fourier modes in the wall-normal direction.…”