Critical transitions in thin layer turbulence

We investigate a model of thin layer turbulence that follows the evolution of the two-dimensional motions u(2D)(x, y) along the horizontal directions (x,y) coupled to a single Fourier mode along the vertical direction (z) of the form u(q) (x,y,z) /= [v(x) (x,y) sin (qz) , v(y) (x,y) sin (qz), v(z)(x,y) cos(qz)], reducing thus the system to two coupled, two-dimensional equations. The model, despite its simplicity and ad hoc construction, displays a rich behaviour. Its reduced dimensionality allows a thorough investigation of the transition from a forward to an inverse cascade of energy as the thickness of the layer H = pi/q 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 analyse with numerical simulations for a wide range of Reynolds numbers and aspect ratios.