Turbulent parameters in the middle atmosphere: Theoretical estimates deduced from a gravity-wave resolving general circulation model

We present a scaling analysis for the stratified turbulent and small-scale turbulent regimes of atmospheric flow with emphasis on the mesosphere. We distinguish rotating-stratified macroturbulence turbulence (SMT), stratified turbulence (ST), and small-scale isotropic Kolmogorov turbulence (KT), and we specify the length and time scales and the characteristic velocities for these regimes. It is shown that the buoyancy scale (Lb) and the Ozmidov scale (Lo) are the main parameters that describe the transition from SMT to KT. We employ the buoyancy Reynolds number and horizontal Froude number to characterize ST and KT in the mesosphere. This theory is applied to simulation results from a high-resolution general circulation model with a Smagorinsky-type turbulent diffusion scheme for the sub-grid scale parameterization. The model allows us to derive the turbulent root-mean-square (RMS) velocity in the KT regime. It is found that the turbulent RMS velocity has a single maximum in summer and a double maximum in winter months. The secondary maximum in the winter MLT we associate with a secondary gravity wave breaking phenomenon. The turbulent RMS velocity results from the model agree well with Full Correlation Analyses based on MF-radar measurements. A new scaling for the mesoscale horizontal velocity based on the idea of direct energy cascade in masoscales is proposed. The latter findings for mesoscale and small-scale characteristic velocities supports the idea proposed in this research that mesoscale and small-scale dynamics in the mesosphere are governed by SMT, ST, and KT in the statistical average.

Exploiting self-organized criticality in strongly stratified turbulence

A multiscale reduced description of turbulent free shear flows in the presence of strong stabilizing density stratification is derived via asymptotic analysis of the Boussinesq equations in the simultaneous limits of small Froude and large Reynolds numbers. The analysis explicitly recognizes the occurrence of dynamics on disparate spatiotemporal scales, yielding simplified partial differential equations governing the coupled evolution of slow large-scale hydrostatic flows and fast small-scale isotropic instabilities and internal waves. The dynamics captured by the coupled reduced equations is illustrated in the context of two-dimensional strongly stratified Kolmogorov flow. A noteworthy feature of the reduced model is that the fluctuations are constrained to satisfy quasilinear (QL) dynamics about the comparably slowly varying large-scale fields. Crucially, this QL reduction is not invoked as an ad hoc closure approximation, but rather is derived in a physically relevant and mathematically consistent distinguished limit. Further analysis of the resulting slow–fast QL system shows how the amplitude of the fast stratified-shear instabilities is slaved to the slowly evolving mean fields to ensure the marginal stability of the latter. Physically, this marginal stability condition appears to be compatible with recent evidence of self-organized criticality in both observations and simulations of stratified turbulence. Algorithmically, the slaving of the fluctuation fields enables numerical simulations to be time-evolved strictly on the slow time scale of the hydrostatic flow. The reduced equations thus provide a solid mathematical foundation for future studies of three-dimensional strongly stratified turbulence in extreme parameter regimes of geophysical relevance and suggest avenues for new sub-grid-scale parametrizations.

Energy cascades in rapidly rotating and stratified turbulence within elongated domains

We study forced, rapidly rotating and stably stratified turbulence in an elongated domain using an asymptotic expansion at simultaneously low Rossby number $\mathit {Ro}\ll 1$ and large domain height compared with the energy injection scale, $h=H/\ell _{in}\gg 1$ . The resulting equations depend on the parameter $\lambda =(h \mathit {Ro} )^{-1}$ and the Froude number $\mathit {Fr}$ . An extensive set of direct numerical simulations (DNS) is performed to explore the parameter space $(\lambda,\mathit {Fr})$ . We show that a forward energy cascade occurs in one region of this space, and a split energy cascade outside it. At weak stratification (large $\mathit {Fr}$ ), an inverse cascade is observed for sufficiently large $\lambda$ . At strong stratification (small $\mathit {Fr}$ ) the flow becomes approximately hydrostatic and an inverse cascade is always observed. For both weak and strong stratification, we present theoretical arguments supporting the observed energy cascade phenomenology. Our results shed light on an asymptotic region in the phase diagram of rotating and stratified turbulence, which is difficult to attain by brute-force DNS.

Linear Non-Modal Growth of Planar Perturbations in a Layered Couette Flow

Layered flows that are commonly observed in stratified turbulence are susceptible to the Taylor–Caulfield Instability. While the modal stability properties of layered shear flows have been examined, the non-modal growth of perturbations has not been investigated. In this work, the tools of Generalized Stability Theory are utilized to study linear transient growth within a finite time interval of two-dimensional perturbations in an inviscid, three-layer constant shear flow under the Boussinesq approximation. It is found that, for low optimization times, small-scale perturbations utilize the Orr mechanism and achieve growth equal to that in the case of an unstratified flow. For larger optimization times, transient growth is much larger compared to growth for an unstratified flow as the Kelvin–Orr waves comprising the continuous spectrum of the dynamical operator and the gravity edge-waves comprising the discrete spectrum interact synergistically. Maximum growth is obtained for perturbations with scales within the region of instability, but significant growth is maintained for modally stable perturbations as well. For perturbations with scales within the unstable region, the unstable normal modes are excited at high amplitude by their bi-orthogonals. For perturbations with modally stable scales, the Orr mechanism is utilized to excite at high amplitude neutral propagating waves resembling the neutral Taylor–Caulfield modes.

Thermohaline staircase formation in the diffusive convection regime: a theory based upon stratified turbulence asymptotics

We describe a mechanism that leads to the spontaneous formation of a thermohaline staircase in the high-latitude oceans. Our analysis of this mechanism is based upon a model in which uniform gradients of temperature and salinity are assumed and is applied to a simplified mean-field model of stratified turbulence. Detailed analysis employs a parametrization of turbulent diapycnal diffusivities (Bouffard & Boegman, Dyn. Atmos. Oceans, vol. 61, 2013, pp. 14–34). This parametrization is apparently unique in that it distinguishes between the diapycnal diffusivities for heat and salt on the basis of their Prandtl (Schmidt) numbers. Our model predicts that the temperature and salinity profiles will be susceptible to linear instability if the buoyancy Reynolds number lies in the range 0.18–91, and a nonlinear mean-field model simulation demonstrates that it evolves into a well-defined thermohaline staircase that matches the characteristics of those found in the high-latitude oceans. The criterion for initial instability is furthermore shown to be consistent with the observed regional variability of staircase occurrence in the Arctic Ocean as determined by the most recent observational datasets.

Insights into the mixing efficiency of submesoscale Centrifugal-Symmetric instabilities

Submesoscale processes provide a pathway for energy to transfer from the balanced circulation to turbulent dissipation. One class of submesoscale phenomena that has been shown to be particularly effective at removing energy from the balanced flow are centrifugal-symmetric instabilities (CSIs), which grow via geostrophic shear production. CSIs have been observed to generate significant mixing in both the surface boundary layer and bottom boundary layer flows along bathymetry, where they have been implicated in the mixing and watermass transformation of Antarctic Bottom Water. However, the mixing efficiency (i.e. the fraction of the energy extracted from the flow used to irreversibly mix the fluid) of these instabilities remains uncertain, making estimates of mixing and energy dissipation due to CSI difficult.In this work we use large-eddy simulations to investigate the mixing efficiency of CSIs in the submesoscale range. We find that centrifugally-dominated CSIs (i.e. CSI mostly driven by horizontal shear production) tend to have a higher mixing efficiency than symmetrically-dominated ones (i.e. driven by vertical shear production). The mixing efficiency associated with CSIs can therefore alternately be significantly higher or significantly lower than the canonical value used by most studies. These results can be understood in light of recent work on stratified turbulence, whereby CSIs control the background state of the flow in which smaller-scale secondary overturning instabilities develop, thus actively modifying the characteristics of mixing by Kelvin-Helmholtz instabilities. Our results also suggest that it may be possible to predict the mixing efficiency with more readily measurable parameters (namely the Richardson and Rossby numbers), which would allow for parameterization of this effect.

Variable energy flux in turbulence

In three-dimensional hydrodynamic turbulence forced at large length scales, a constant energy flux $ \Pi_u $ flows from large scales to intermediate scales, and then to small scales. It is well known that for multiscale energy injection and dissipation, the energy flux $\Pi_u$ varies with scales. In this review we describe this principle and show how this general framework is useful for describing a variety of turbulent phenomena. Compared to Kolmogorov's spectrum, the energy spectrum steepens in turbulence involving quasi-static magnetofluid, Ekman friction, stable stratification, magnetohydrodynamics, and solution with dilute polymer. However, in turbulent thermal convection, in unstably stratified turbulence such as Rayleigh-Taylor turbulence, and in shear turbulence, the energy spectrum has an opposite behaviour due to an increase of energy flux with wavenumber. In addition, we briefly describe the role of variable energy flux in quantum turbulence, in binary-fluid turbulence including time-dependent Landau-Ginzburg and Cahn-Hillianrd equations, and in Euler turbulence. We also discuss energy transfers in anisotropic turbulence.

Goldilocks mixing in oceanic shear-induced turbulent overturns

We present a new, simple and physically motivated parameterization, based on the ratio of Thorpe and Ozmidov scales, for the irreversible turbulent flux coefficient $\varGamma _{\mathcal {M}}= {\mathcal {M}}/\epsilon$ , i.e. the ratio of the irreversible rate ${\mathcal {M}}$ at which the background potential energy increases in a stratified flow due to macroscopic motions to the dissipation rate of turbulent kinetic energy $\epsilon$ . Our parameterization covers all three key phases (crucially, in time) of a shear-induced stratified turbulence life cycle: the initial, ‘hot’ growing phase, the intermediate energetically forced phase and the final ‘cold’ fossilization decaying phase. Covering all three phases allows us to highlight the importance of the intermediate one, to which we refer as the ‘Goldilocks’ phase due to its apparently optimal (and so neither too hot nor too cold, but just right) balance, in which energy transfer from background shear to the turbulent mixing is most efficient. The value of $\varGamma _{\mathcal {M}}$ is close to 1/3 during this phase, which we demonstrate appears to be related to an adjustment towards a critical or marginal Richardson number for sustained turbulence ${\sim }0.2\text {--}0.25$ . Importantly, although buoyancy effects are still significant at leading order for the turbulent dynamics during this intermediate phase, the marginal balance in the flow ensures that the turbulent mixing of the (density) scalar is nevertheless effectively ‘locked’ to the turbulent mixing of momentum. We present supporting evidence for our parameterization through comparison with six oceanographic datasets that span various turbulence generation regimes and a wide range of geographical location and depth. Using these observations, we highlight the significance of parameterizing an inherently variable flux coefficient for capturing the turbulent flux associated with rare energetic, yet fundamentally shear-driven (and so not strongly stratified) overturns that make a disproportionate contribution to the total mixing. We also highlight the importance of representation of young turbulent patches in the parameterization for connecting the small scale physics to larger scale applications of mixing such as ocean circulation and tracer budgets. Shear-induced turbulence is therefore central to irreversible mixing in the world's oceans, apparently even close to the seafloor, and it is critically important to appreciate the inherent time dependence and evolution of mixing events: history matters to mixing.

Order out of chaos: Shifting paradigm of convective turbulence

Turbulence is ever produced in the low-viscosity/large-scale fluid flows by the velocity shears and, in unstable stratification, by buoyancy forces. It is commonly believed that both mechanisms produce the same type of chaotic motions, namely, the eddies breaking down into smaller ones and producing direct cascade of turbulent kinetic energy and other properties from large to small scales towards viscous dissipation. The conventional theory based on this vision yields a plausible picture of vertical mixing and remains in use since the middle of the 20th century in spite of increasing evidence of the fallacy of almost all other predictions. This paper reveals that in fact buoyancy produces chaotic vertical plumes, merging into larger ones and producing an inverse cascade towards their conversion into the self-organized regular motions. Herein, the velocity shears produce usual eddies spreading in all directions and making the direct cascade. This new paradigm is demonstrated and proved empirically; so, the paper launches a comprehensive revision of the theory of unstably stratified turbulence and its numerous geophysical or astrophysical applications.