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The two major ingredients of geophysical fluid dynamics, rotation and density stratification, break the standard picture of turbulence. Several inertial ranges may coexist, and the spectral slope cannot be predicted by dimensional arguments because of the introduction of new timescales, corresponding to the propagation of fast inertia-gravity waves, in addition to the slow vortices whose characteristic time is the eddy turnover time. We will discuss the relative role of these two types of motion in the energy cascade of stratified turbulence, with or withour rotation, using high-resolution direct numerical simulations of the Boussinesq equations in a periodic box for a large range of non-dimensional parameters (Rossby and Froude numbers). This discussion will be supplemented by a theoretical argument based on equilibrium statistical mechanics, extending the classical argument of Kraichnan for 2D flows: introducing a probability distribution restricted to the slow manifold predicts an inverse cascade in the case with rotation, but not in the purely stratified case.
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