Spectral energy scaling in precessing turbulence

We study precessing turbulence, which appears in several geophysical and astrophysical systems, by direct numerical simulations of homogeneous turbulence where precessional instability is triggered due to the imposed background flow. We show that the time development of kinetic energy K occurs in two main phases associated with different flow topologies: (i) an exponential growth characterizing three-dimensional turbulence dynamics and (ii) nonlinear saturation during which K remains almost time independent, the flow becoming quasi-two-dimensional. The latter stage, wherein the development of K remains insensitive to the initial state, shares an important common feature with other quasi-two-dimensional rotating flows such as rotating Rayleigh-Bénard convection, or the large atmospheric scales: in the plane k_{∥}=0, i.e., the plane associated to an infinite wavelength in the direction parallel to the principal rotation axis, the kinetic energy spectrum scales as k_{⊥}^{-3}. We show that this power law is observed for wave numbers ranging between the Zeman "precessional" and "rotational" scales, k_{S}^{-1} and k_{Ω}^{-1}, respectively, at which the associated background shear or inertial timescales are equal to the eddy turnover time. In addition, an inverse cascade develops for (k_{⊥},k)