Session GG: Geophysical Fluid Dynamics IV

On the ``arrest'' of inverse energy cascade and the Rhines scale

The idea of the ``arrest'' of the inverse energy cascade in two-dimensional turbulence with a $\beta$-effect has been promulgated in meteorological and oceanographic literature for the last 30 years. The arrest is associated with the interaction between turbulence and Rossby waves. In relation to the arrest, Rhines (1975) has introduced a scale, $L_R = (2U/\beta)^{1/2}$, obtained by equating the r.m.s. velocity $U$ to the phase speed of the Rossby wave. It has been widely accepted that $L_R$ is a scale at which the nonlinear turbulent cascade gives way to propagation of linear Rossby waves. This interpretation, as well as the notion of the cascade arrest have been used in theories of large-scale atmospheric and oceanic circulations. We revise the notion of the cascade arrest in the context of continuously forced flows and demonstrate that the idea of the cascade arrest by a $\beta$-effect, in the naive sense of a transition from a nonlinear to a linear regime is fallacious; the up-scale energy propagation can only be absorbed by friction. Furthermore, $L_R$ cannot be related to the transition between the regimes of nonlinear turbulence and linear wave propagation. Spectral analysis in the frequency domain shows that Rossby waves coexist with turbulence on scales much smaller than $L_R$.   

Zigzag instability in a stratified fluid: a direct transfer of energy

In a strongly stratified fluid, a columnar counter-rotating vortex pair is subject to the zigzag instability which bends the vortices and ultimately produces layers. We have investigated the nonlinear evolution of this linear instability by means of DNS. We show that the instability grows exponentially without nonlinear saturation and therefore produces rapidly intense vertical shear. The instability growth is only stopped when vertical viscous effects become dominant. This occurs when $F_h^2 Re = O(1)$ with $F_h$ the horizontal Froude number and $Re$ the Reynolds number. No secondary zigzag instabilities or shear instabilities have been observed. This means that the zigzag instability is a mechanism capable of directly transferring the energy from large scales to small vertical scales where it is dissipated without any cascade.   

Inertia and Compressibility Effects in the Boussinesq Approximation

The Boussinesq approximation is typically applied to flows where buoyancy is the dominant driving force. To extend its applicability to flows with substantial inertial perturbations, we examine the flow equations using perturbation analysis about the hydrostatic state. The physical effects corresponding to stratification, compressibility, small initial entropy perturbations, and inertia are characterized in terms of nondimensional parameters derived from the analysis. A simple and computationally efficient extension to the traditional Boussinesq approximation is proposed to include the interaction of buoyancy and inertia. The role of {\it fluid compressibility} in stratified low Mach number flows is highlighted and distinguished from the {\it flow compressibility} which is caused by motion. A nondimensional parameter is derived to demarcate compressible and nearly-incompressible hydrostatic states. The significance of the extended Boussinesq approximation is illustrated with numerical solutions to model problems. Application to the problem of aircraft vortex wake-exhaust jet interaction is discussed.   

Anomalous diffusion in rotating stratified turbulence

Diffusion in rotating and stratified fluids is one of the central subjects in geophysical and astrophysical dynamics. In this paper, we report features of the dispersion of Lagrangian fluid particles in rotating stratified flows using the Direct Numerical Simulations (DNS) of the Navier-Stokes equations. And for calculation of particle dispersion, we use the cubic spline interpolation method by Yeung and Pope. Our main concern is the picture different from the Taylor dispersion theory, i.e. $\left \sim t^2$ for $t\ll t_B$ (ballistic mode) and $\left \sim t$ for $t\gg t_B$(Brownian mode), where $t_B$ is the time after which the Lagrangian velocity auto-correlation function drops rapidly. The different features of diffusion from the standard Taylor picture is often called anomalous diffusion. Particle dispersion shows quite large anisotropy. In the vertical direction, particles stop migration after they pass the time interval of the ballistic mode when only stratification is active and rotation enhances this tendency (for decaying turbulence). In the horizontal direction, we observe that $t^2$ behavior is more evident not only for the initial ballistic mode but also at later times, and that there is a transition period between the two $t^2$ regimes. The duration and the starting time of the transition period are a function of the parameters of stratification and rotation. For a fixed value of stratification, the transition period shifts to earlier time with rotation. And with strong stratification, the transition period disappears.   

The Effect of Stable Stratification on Fluid Particle Dispersion

The dispersion of fluid particles in statistically stationary stably stratified turbulence is studied by means of direct numerical simulations. Due to anisotropy of the flow, horizontal and vertical dispersion show different behaviour. Single-particle dispersion in horizontal direction is similar to that in isotropic turbulence. In vertical direction, however, three regimes can be identified: a classical $t^{2}$-regime, a plateau which scales as $N^{-2}$ and a diffusion limit $\propto t$, successively. By forcing the flow and performing long-time simulations we were able to observe this last regime, which was predicted but not observed before in purely stratified forced turbulence. A model based on the assumed shape of the velocity autocorrelation function correctly predicts these three regimes. The vertical mean-squared separation of particle pairs shows two plateaus that are not present in isotropic turbulence. They can be linked with characteristics of the flow. Also here the diffusion limit is found.   

Modeling small scale mixing in stably stratified turbulence

We use direct numerical simulations (DNS) to study mixing and dispersion in decaying stably stratified turbulence from a Lagrangian perspective. We track the changes in the density of fluid particles due to small scale mixing to provide insight into the mixing process. These changes are driven by spatially and temporally intermittent events that are strongly suppressed as the stratification increases and overturning motions disappear. The density changes of fluid particles are linked fundamentally to diapycnal mixing. We provide a (simple) general scaling prediction for the diapycnal diffusivity based on a model for these density changes. The scaling highlights the fundamental role of the Ellison overturning scale as an indicator of diapycnal mixing in these flows. We demonstrate the validity of the scaling by comparison with data from other DNS experiments for stratified turbulence, both with and without the presence of shear. The results reported here have implications for the development of improved models for dispersion and mixing in stably stratified turbulence.   

Large Eddy Simulation of a Stratified Shear Layer

Large Eddy Simulation (LES) of a stably stratified turbulent shear layer is performed with several sub-grid models: the simple Smagorinsky model, dynamic Smagorinsky model, and the mixed model. The ability of LES to predict the experimentally observed collapse at a critical value of bulk Richardson number is assessed. {\em A priori} and {\em a posteriori} tests are performed with a Direct Numerical Simulation (DNS) database serving as the benchmark. The budgets of turbulent kinetic energy, turbulent potential energy, Reynolds stress, and mixed budgets are utilized to elucidate the behavior of the sub-grid models. The simple Smagorinsky model, picks up some effects of stratification, but over-predicts the turbulent viscosity, and hence the bulk-behavior of the flow is quantitatively incorrect. The performance of the more advanced subgrid models are contrasted with the Smagorinsky model. The budgets of turbulent kinetic energy, turbulent potential energy, Reynolds stress, and buoyancy flux are utilized to elucidate the behavior of the more advanced sub-grid models.   

The three layer structure of a stratified oceanic bottom boundary layer

A turbulent boundary layer, generated when a uniformly stratified fluid in geostrophic balance flows over an adiabatic wall, is examined through large eddy simulation. Three layers with different dynamics are identified. The lowest layer remains relatively well-mixed and stratification effects are mild. The growth of the mixed region is inhibited when the external stratification is large. Above the mixed layer, a thermocline forms and is associated with elevated mean shear and strong buoyancy effects. The interaction of unsteady turbulence and the thermocline leads to the generation of internal gravity waves. The waves then propagate into the outer region to create a third layer with fluctuation energy but no mean shear. The characteristics of the waves are dependent on the local stratification, and turbulence time and length scales at the generation site.   

Quantifying anisotropy in stratified and rotating turbulent flows

We study freely decaying homogeneous anisotropic turbulent flows, submitted to either rotation or stratification, similar to those encountered in geophysical flows. We solve the three-dimensional Navier-Stokes equations with Boussinesq hypothesis by direct numerical simulation, using a pseudo-spectral method at resolution $512^3$. We propose new diagnostics to characterize and quantify the anisotropy of these flows, which are based on three-dimensional orthogonal vector-valued wavelet decomposition. We thus show the energy distribution in terms of both scale and direction for each component of the velocity vector and quantify the flow anisotropy. We also apply the coherent vortex extraction algorithm, based on the nonlinear filtering of the wavelet coefficients of the vorticity field, to different anisotropic flows, yielding a strong data compression.