Anisotropy and nonuniversality in scaling laws of the large-scale energy spectrum in rotating turbulence

Abstract

Rapidly rotating turbulent flow is characterized by the emergence of columnar structures that are representative of quasi-two-dimensional behavior of the flow. It is known that when energy is injected into the fluid at an intermediate scale Lf, it cascades towards smaller as well as larger scales. In this paper we analyze the flow in the inverse cascade range at a small but fixed Rossby number, Rof≈0.05. Several numerical simulations with helical and nonhelical forcing functions are considered in periodic boxes with unit aspect ratio. In order to resolve the inverse cascade range with reasonably large Reynolds number, the analysis is based on large eddy simulations which include the effect of helicity on eddy viscosity and eddy noise. Thus, we model the small scales and resolve explicitly the large scales. We show that the large-scale energy spectrum has at least two solutions: one that is consistent with Kolmogorov-Kraichnan-Batchelor-Leith phenomenology for the inverse cascade of energy in two-dimensional (2D) turbulence with a ∼k⊥-5/3 scaling, and the other that corresponds to a steeper ∼k⊥-3 spectrum in which the three-dimensional (3D) modes release a substantial fraction of their energy per unit time to the 2D modes. The spectrum that emerges depends on the anisotropy of the forcing function, the former solution prevailing for forcings in which more energy is injected into the 2D modes while the latter prevails for isotropic forcing. In the case of anisotropic forcing, whence the energy goes from the 2D to the 3D modes at low wave numbers, large-scale shear is created, resulting in a time scale τsh, associated with shear, thereby producing a ∼k-1 spectrum for the total energy with the horizontal energy of the 2D modes still following a ∼k⊥-5/3 scaling.

Large-scale anisotropy in stably stratified rotating flows

“…Moreover, in Π T (k) and Π T (k ⊥ ), a range of wavenumbers can be identified at large scales for which the fluxes remain approximately constant. This is consistent with what was observed in previous studies (see, e.g., [35,46]): in the purely rotating case, energy at large scales goes towards 2D modes (modes with k = 0), modes which then undergo an inverse cascade in the 2D plane towards the largest scales in the system.…”

“…Direct numerical simulations have shown that rotating flows in finite domains can develop inverse cascades of energy (see, e.g., [34,35]), and recent experiments have observed the same phenomena [36]. In the case of stratified flows, the situation is less clear.…”

Rotating turbulence under “precession-like” perturbation

“…In most cases, it is customary to fix the orientation of the rotation axis. The general consensus is that when rotation is strong enough, the forward energy cascade from the large scales to the small scales is inhibited and an inverse cascade develops resulting in a quasi-two-dimensional behavior characterized by columnar structures along the fixed rotation axis [5,6]. The dominance of the inverse energy cascade entails the presence of a large scale energy sink in order to reach stationarity [7].…”

Inverse cascades and resonant triads in rotating and stratified turbulence

“…In purely rotating systems energy can cascade both to the large and to the small scales [58,59], with the inverse cascade being associated with the 2D modes of the system. For this case different scaling laws in the inverse cascade range were also reported depending on how the flow is stirred [41]. Finally, for negligible or no rotation, although an increase of energy and of the integral vertical length scale was reported in simulations [37,38], Waite and Bartello [39,40] conclude against the presence of an inverse cascade using an argument based on statistical mechanics, similar to the one used by Kraichnan to predict the inverse cascade in 2D flows.…”