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We investigate a model of thin layer turbulence that follows the evolution
of the two-dimensional motions along the horizontal directions coupled to a single Fourier mode along the vertical direction of the form, reducing thus the system to two coupled, two-dimensional equations. The reduced dimensionality of the model allows a thorough investigation of the transition from a forward to an inverse cascade of energy as the thickness of the layer H is varied. Starting from a thick layer and reducing its thickness it is shown that two critical heights are met (i) one for which the forward unidirectional cascade (similar to three-dimensional turbulence) transitions to a bidirectional cascade transferring energy to both small and large scales and (ii) one for which the bidirectional cascade transitions to a unidirectional inverse cascade when the layer becomes very thin (similar to two-dimensional turbulence). The two critical heights are shown to have different properties close to criticality that we are able to analyze with numerical simulations for a wide range of Reynolds numbers and aspect
ratios.
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