Simulations of strongly stratified turbulence often exhibit coherent large-scale structures called vertically sheared horizontal flows (VSHFs). VSHFs emerge in both twodimensional (2D) and three-dimensional (3D) stratified turbulence with similar vertical structure. The mechanism responsible for VSHF formation is not fully understood. In this work, the formation and equilibration of VSHFs in a 2D Boussinesq model of stratified turbulence is studied using statistical state dynamics (SSD). In SSD, equations of motion are expressed directly in the statistical variables of the turbulent state. Restriction to 2D turbulence makes available an analytically and computationally attractive implementation of SSD referred to as S3T, in which the SSD is expressed by coupling the equation for the horizontal mean structure with the equation for the ensemble mean perturbation covariance. This second order SSD produces accurate statistics, through second order, when compared with fully nonlinear simulations. In particular, S3T captures the spontaneous emergence of the VSHF and associated density layers seen in simulations of turbulence maintained by homogeneous large-scale stochastic excitation. An advantage of the S3T system is that the VSHF formation mechanism, which is wavemean flow interaction between the emergent VSHF and the stochastically excited largescale gravity waves, is analytically understood in the S3T system. Comparison with fully nonlinear simulations verifies that S3T solutions accurately predict the scale selection, dependence on stochastic excitation strength, and nonlinear equilibrium structure of the VSHF. These results facilitate relating VSHF theory and geophysical examples of turbulent jets such as the ocean's equatorial deep jets.
Two-dimensional laminar roll convection is capable of generating energetic horizontal mean flows via a well-understood process known as the tilting instability. Less wellunderstood is the physical mechanism behind the strong dependence of this effect on the horizontal lengthscale of the convection pattern. Mean flows of this type have been found to form for sufficiently large Rayleigh number in periodic domains with a small aspect ratio of horizontal length to vertical height, but not in large aspect ratio domains. We demonstrate that the elimination of the tilting instability for large aspect ratio is due to a systematic eddy-eddy advection mechanism intervening at linear order in the tilting instability, and that this effect can be captured in a model retaining two nonlinearly interacting horizontal wavenumber components of the convection field. Several commonly used low-order models of convection also exhibit a shutdown of the tilting instability for large aspect ratio, even though these models do not contain the eddy-eddy advection mechanism. Instability suppression in such models is due to a different mechanism involving vertical advection. We show that this vertical advection mechanism is excessively strong in the low-order models due to their low resolution, and that the instability shutdown in such models vanishes when they are appropriately extended.
Vertically banded zonal jets are frequently observed in weakly or non-rotating stratified turbulence, with the quasi-biennial oscillation in the equatorial stratosphere and the ocean's equatorial deep jets being two examples. Explaining the formation of jets in stratified turbulence is a fundamental problem in geophysical fluid dynamics. Statistical state dynamics (SSD) provides powerful methods for analyzing turbulent systems exhibiting emergent organization, such as banded jets. In SSD, dynamical equations are written directly for the evolution of the turbulence statistics, enabling direct analysis of the statistical interactions between the incoherent component of the turbulence and the coherent large-scale structure component that underlie jet formation. A second-order closure of SSD, known as S3T, has previously been applied to show that meridionally banded jets emerge in barotropic β -plane turbulence via a statistical instability referred to as the zonostrophic instability. Two-dimensional Boussinesq turbulence provides a simple model of non-rotating stratified turbulence analogous to the β -plane model of planetary turbulence. Jets known as vertically sheared horizontal flows (VSHFs) often emerge in simulations of Boussinesq turbulence, but their dynamics is not yet clearly understood. In this work S3T analysis of the zonostrophic instability is extended to study VSHF emergence in two-dimensional Boussinesq turbulence using an analytical formulation of S3T amenable to perturbation stability analysis. VSHFs are shown to form via an instability that is analogous in stratified turbulence to the zonostrophic instability in β -plane turbulence. This instability is shown to be strikingly similar to the zonostrophic instability, suggesting that jet emergence in both geostrophic and non-rotating stratified turbulence may be understood as instances of the same generic phenomenon. ABSTRACTTest. ABSTRACTTest. ABSTRACTTest.
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