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Register a new userGeneration of turbulence through frontogenesis in sheared stratified flows
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The large-scale structures in the ocean and the atmosphere are in geostrophic balance, and a conduit must be found to channel the energy to the small scales where it can be dissipated. In turbulence this takes the form of an energy cascade, whereas one possible mechanism in a balanced flow at large scales is through the formation of fronts, a common occurrence in geophysical dynamics. We show in this paper that an iconic configuration in laboratory and numerical experiments for the study of turbulence, that of the Taylor-Green or von Karman swirling flow, can be suitably adapted to the case of fluids with large aspect ratios, leading to the creation of an imposed large-scale vertical shear. To this effect we use direct numerical simulations of the Boussinesq equations without net rotation and with no small-scale modeling, and with this idealized Taylor-Green set-up. Various grid spacings are used, up to $2048^2times 256$ spatial points. The grids are always isotropic, with box aspect ratios of either $1:4$ or $1:8$. We find that when shear and stratification are comparable, the imposed shear layer resulting from the forcing leads to the formation of multiple fronts and filaments which destabilize and further evolve into a turbulent flow in the bulk, with a sizable amount of dissipation and mixing, and with a cycle of front creation, instability, and development of turbulence. The results depend on the vertical length scales for shear and for stratification, with stronger large-scale gradients being generated when the two length scales are comparable.
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Simulations of strongly stratified turbulence often exhibit coherent large-scale structures called vertically sheared horizontal flows (VSHFs). VSHFs emerge in both two-dimensional (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 wave-mean flow interaction between the emergent VSHF and the stochastically excited large-scale 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 oceans equatorial deep jets.
Recent numerical simulations showed that the mean flow is generated in inhomogeneous turbulence of an incompressible fluid accompanied with helicity and system rotation. In order to investigate the mechanism of this phenomenon, we carry out a numerical simulation of inhomogeneous turbulence in a rotating system. In the simulation, an external force is applied to inject inhomogeneous turbulent helicity and the rotation axis is taken to be perpendicular to the inhomogeneous direction. No mean velocity is set in the initial condition of the simulation. The simulation results show that only in the case with both the helical forcing and the system rotation, the mean flow directed to the rotation axis is generated and sustained. We investigate the physical origin of this flow-generation phenomenon by considering the budget of the Reynolds-stress transport equation. It is found that the pressure diffusion term has a large contribution in the Reynolds stress equation and supports the generated mean flow. It is shown that a model expression for the pressure diffusion can be expressed by the turbulent helicity gradient coupled with the angular velocity of the system rotation. This implies that inhomogeneous helicity can play a significant role for the generation of the large-scale velocity distribution in incompressible turbulent flows.
Numerical simulations are made for forced turbulence at a sequence of increasing values of Reynolds number, R, keeping fixed a strongly stable, volume-mean density stratification. At smaller values of R, the turbulent velocity is mainly horizontal, and the momentum balance is approximately cyclostrophic and hydrostatic. This is a regime dominated by so-called pancake vortices, with only a weak excitation of internal gravity waves and large values of the local Richardson number, Ri, everywhere. At higher values of R there are successive transitions to (a) overturning motions with local reversals in the density stratification and small or negative values of Ri; (b) growth of a horizontally uniform vertical shear flow component; and (c) growth of a large-scale vertical flow component. Throughout these transitions, pancake vortices continue to dominate the large-scale part of the turbulence, and the gravity wave component remains weak except at small scales.
Kraichnan seminal ideas on inverse cascades yielded new tools to study common phenomena in geophysical turbulent flows. In the atmosphere and the oceans, rotation and stratification result in a flow that can be approximated as two-dimensional at very large scales, but which requires considering three-dimensional effects to fully describe turbulent transport processes and non-linear phenomena. Motions can thus be classified into two classes: fast modes consisting of inertia-gravity waves, and slow quasi-geostrophic modes for which the Coriolis force and horizontal pressure gradients are close to balance. In this paper we review previous results on the strength of the inverse cascade in rotating and stratified flows, and then present new results on the effect of varying the strength of rotation and stratification (measured by the ratio $N/f$ of the Brunt-Vaisala frequency to the Coriolis frequency) on the amplitude of the waves and on the flow quasi-geostrophic behavior. We show that the inverse cascade is more efficient in the range of $N/f$ for which resonant triads do not exist, $1/2 le N/f le 2$. We then use the spatio-temporal spectrum, and characterization of the flow temporal and spatial scales, to show that in this range slow modes dominate the dynamics, while the strength of the waves (and their relevance in the flow dynamics) is weaker.
To investigate the formation mechanism of energy spectra of internal waves in the oceans, direct numerical simulations are performed. The simulations are based on the reduced dynamical equations of rotating stratified turbulence. In the reduced dynamical equations only wave modes are retained, and vortices and horizontally uniform vertical shears are excluded. Despite the simplifications, our simulations reproduce some key features of oceanic internal-wave spectra: accumulation of energy at near-inertial waves and realistic frequency and horizontal wavenumber dependencies. Furthermore, we provide evidence that formation of the energy spectra in the inertial subrange is dominated by scale-separated interactions with the near-inertial waves. These findings support oceanographers intuition that spectral energy density of internal waves is the result of predominantly wave-wave interactions.
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