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Geophysical turbulence is observed to self-organize into large-scale flows such as zonal jets and coherent vortices. Previous studies of barotropic $\unicode[STIX]{x1D6FD}$ -plane turbulence have shown that coherent flows emerge from a background of homogeneous turbulence as a bifurcation when the turbulence intensity increases. The emergence of large-scale flows has been attributed to a new type of collective, symmetry-breaking instability of the statistical state dynamics of the turbulent flow. In this work, we extend the analysis to stratified flows and investigate turbulent self-organization in a two-layer fluid without any imposed mean north–south thermal gradient and with turbulence supported by an external random stirring. We use a second-order closure of the statistical state dynamics, that is termed S3T, with an appropriate averaging ansatz that allows the identification of statistical turbulent equilibria and their structural stability. The bifurcation of the statistically homogeneous equilibrium state to inhomogeneous equilibrium states comprising zonal jets and/or large-scale waves when the energy input rate of the excitation passes a critical threshold is analytically studied. Our theory predicts that there is a large bias towards the emergence of barotropic flows. If the scale of excitation is of the order of (or larger than) the deformation radius, the large-scale structures are barotropic. Mixed barotropic–baroclinic states with jets and/or waves arise when the excitation is at scales shorter than the deformation radius with the baroclinic component of the flow being subdominant for low energy input rates and insignificant for higher energy input rates. The predictions of the S3T theory are compared with nonlinear simulations. The theory is found to accurately predict both the critical transition parameters and the scales of the emergent structures but underestimates their amplitude. [ABSTRACT FROM AUTHOR]

Large-scale mean flows often emerge in turbulent fluids. In this work, we formulate a stability theory, the stochastic structural stability theory (SSST), for the emergence of jets under external random excitation. We analytically investigate the structural stability of a two-dimensional homogeneous fluid enclosed in a channel and subjected to homogeneous random forcing. We show that two generic competing mechanisms control the instability that gives rise to the emergence of an infinitesimal jet: advection of the eddy vorticity by the mean flow that is shown to be jet forming and advection of the vorticity gradient of the jet by the eddies that is shown to hinder the formation of the mean flow. We show that stochastic forcing with small streamwise coherence and an amplitude larger than a certain threshold leads to the emergence of jets in the channel through a bifurcation of the non-linear SSST system. [ABSTRACT FROM PUBLISHER]

Non-modal mechanisms underlying transient growth of propagating acoustic waves and non-propagating vorticity perturbations in an unbounded compressible shear flow are investigated, making use of closed form solutions. Propagating acoustic waves amplify mainly due to two mechanisms: growth due to advection of streamwise velocity that is typically termed as the lift-up mechanism leading for large Mach numbers to an almost linear increase in streamwise velocity with time and growth due to the downgradient irrotational component of the Reynolds stress leading to linear growth of acoustic wave energy for large times. Synergy between these mechanisms along with the downgradient solenoidal component of the Reynolds stress produces large and robust energy amplification. On the other hand, non-propagating vorticity perturbations amplify due to kinematic deformation of vorticity by the mean flow. For weakly compressible flows, an initial vorticity perturbation abruptly excites acoustic waves with exponentially small amplitude, and the energy gained by vorticity perturbations is transferred back to the mean flow. For moderate Mach numbers, a strong coupling between vorticity perturbations and acoustic waves is found with the energy gained by vorticity perturbations being transferred to acoustic waves that are abruptly excited by the vortex. Calculation of the optimal perturbations for a viscous flow shows that for low Mach numbers, acoustic wave excitation by vorticity perturbations and the subsequent growth of acoustic waves leads to robust energy growth of the order of Reynolds number, while for large Mach numbers, synergy between the lift-up mechanism and the downgradient solenoidal component of the Reynolds stress dominates the growth and leads to a comparable large amplification of streamwise velocity. [ABSTRACT FROM AUTHOR]