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.
We first summarize briefly several properties concerning the dynamics of two-dimensional (2D) turbulence, with an emphasis on the inverse cascade of energy to the largest accessible scale of the system. In order to study a similar phenomenon in three-dimensional (3D) turbulence undergoing strong solid-body rotation, we test a previously developed Large Eddy Simulation (LES) model against a high-resolution direct numerical simulation of rotating turbulence on a grid of 3072 3 points. We then describe new numerical results on the inverse energy cascade in rotating flows using this LES model and contrast the case of 2D versus 3D forcing, as well as non-helical forcing (i.e., with weak overall alignment between velocity and vorticity) versus the fully helical Beltrami case, both for deterministic and random forcing. The different scaling of the inverse energy cascade can be attributed to the dimensionality of the forcing, with, in general, either a k −3 ⊥ or a k −5/3 ⊥ energy spectrum of slow modes at large scales, perpendicular referring to the direction of rotation. We finally invoke the role of shear in the case of a strongly anisotropic deterministic forcing, using the so-called ABC flow.
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