Mean flow generation via non-resonant interactions in two-dimensional forced stratified turbulence

We perform numerical simulations of two-dimensional strongly stratified flows in a square periodic domain $(y,z)$ forced by a steady mode with vorticity of the form $\sin (k_{\textit{y f}}y)\sin (k_{\textit{z f}}z)$ , where $(k_{\textit{y f}},k_{\textit{z f}})$ are fixed wavenumbers. It is shown that such deterministic forcing can lead to a transition to turbulence and the emergence of horizontal layers (so-called vertically sheared horizontal flows, VSHFs) similarly as for random stochastic forcing. The flow characteristics are studied depending on the Froude and Reynolds numbers. Furthermore, the mechanisms of layers formation are disentangled. Triadic instabilities first lead to the growth of pairs of wavevectors that resonate with each of the four forced wavevectors. Quadratic interactions between these resonant modes and the forcing also drive the growth of several non-resonant modes at the same growth rate. Since the forcing comprises the wavevectors $\pm (k_{\textit{y f}},k_{\textit{z f}})$ and their mirror symmetric with respect to the horizontal $\pm (k_{\textit{y f}},-k_{\textit{z f}})$ , there exist enslaved/bound modes with the same horizontal wavenumber and different vertical wavenumbers. Hence, the quadratic interactions between the latter modes force a second generation of modes among which some are VSHFs. Their growth rate is twice the growth rate of the primary resonant modes. Such a mechanism is similar to resonant quartets (Newell, J. Fluid Mech. , 1969, vol. 35, no 2, pp. 255–271; Smith & Waleffe, Phys. Fluids , 1999, vol. 11, no 6, pp. 1608–1622). When the forcing is restricted to only the two wavevectors $\pm (k_{\textit{y f}},k_{\textit{z f}})$ , the second generation of enslaved/bound modes all have a non-zero horizontal wavenumber. However, further quadratic interactions can force VSHF. Thus, horizontal layers also emerge, but with a growth rate equal to the number of quadratic interactions times the growth rate of the primary instability.