Session M1: Geophysical Flows: General I

On the dynamics of homogeneous turbulence near a surface

It is becoming increasing clear that stably-stratified flows can support a stratified turbulence $k^{-5/3}$ inertial range, different from Kolmogorov's. Stratification inhibits vertical motions, but the large-scale quasi-horizontal motions produce strong vertical shearing and small-scale instabilities. The result is a $k^{-5/3}$ horizontal spectrum for the horizontal velocities at scales larger than the Ozmidov scale, the largest scale that can overturn. For smaller scales, the classical Kolmogorov $k^{-5/3}$ applies. Inspired by data taken near the water surface in a tidal river,\footnote{Chickadel et al. (2011) to appear in {\em IEEE Geosci. Remote Sens. Lett.}} we here explore to what extent the dynamics of the nonlinear spectral energy transfer of near-surface turbulence with no mean shear (i.e., horizontally isotropic turbulence bounded by free-slip and no-slip surfaces) is analogous to stably stratified turbulence. To that end, we perform DNS of decaying isotropic turbulence with $Re_\lambda\sim 100$, but bounded by a non-slip surface and a free slip surface. The behavior of the flow near the free-slip surface is similar to stratified turbulence, with a tentative $k^{-5/3}$ range, but the same is not true for the no-slip surface at the present Reynolds numbers. This research was supported by ARO and NSF.   

The energetics of stably stratified turbulence in the Boussinesq approximation

There has been a recent resurgence of interest in determining the consistency of the Boussinesq approximation to describe the coupling of the dynamics and thermodynamics of turbulent stratified flows. In particular, there is some debate over how energy is converted from internal to mechanical energy in this approximation. To gain some insight into these issues, we derive the evolution equations of the different forms of energy for Boussinesq stratified flows from the Lagrangian point of view. This analysis allows us better physical insight into these issues and allows us to show explicitly how energy is converted between internal and mechanical energy. The physical significance of these results will be discussed.   

Dynamics and structure of stably stratified turbulence

The structural features of stratified turbulence and its relationship to the flow dynamics has been the subject of many recent investigations. In strongly stratified turbulent flows, the formation of large-scale quasi-horizontal vortices in layers with strong vertical variability has been observed in laboratory experiments. In this study, direct numerical simulations (DNS) of stably stratified turbulence are used to investigate the evolution of flows in terms of overturns and their relationship to mixing. Isosurfaces of enstrophy of strongly stable flows indicate the emergence of randomly distributed ``pancake-like'' structures with near horizontal orientation at later times. The vertical dynamics of such strongly stratified flows are dominated by linear internal waves and can be described using rapid-distortion theory (RDT) while their horizontal dynamics are dominated by nonlinear effects that cannot be described by RDT. This suggests a decoupling of the vertical and horizontal dynamics of the flow. The ``pancake'' enstrophy structures appear to be associated with shear layers between layered vortex modes. It has been suggested by others that these shear layers could become locally unstable leading to additional source of turbulence in these strongly stable flows.   

Energy transfer for stably stratified turbulence

To understand the mechanism of producing the power-law transition in the energy spectra for stably stratified turbulence, we investigate the energy transfer using direct numerical simulations (DNS) at a resolution of $1024^3$. The calculation is done by solving the 3D Navier-Stokes equations with horizontal forcing under the Boussinesq approximation pseudo-spectrally. Using toroidal-poloidal decomposition (Craya-Herring decomposition), the velocity field is decomposed into vortex and wave modes. We have observed that the vortex and wave spectra are consistent with a Kolgomorov--like $k^{-5/3}$ range at sufficiently large k, and that at large scales, the wave spectrum is a steeper $k_{\perp}^{-2}$, while that for the vortex component is consistent with $k_{\perp}^{-3}$ for sufficiently strong stratification. Here $k_{\perp}$ is the horizontally gathered wave numbers. Therefore there is a power-law transition in the spectra of the both modes. By looking at the energy budget of the wave and vortex modes, we study the energy transfer functions. We will demonstrate that the energy transfer shows a difference from the conventional Kolmogorov picture for isotropic turbulence, in particular at the large scale behavior in which the energy transfer is balanced with enhanced dissipation presumably because of the rough layer edges suggested by Herring \& M\'etais (1989).\\ Herring, J.R. \& M\'etais, O. : Numerical experiments in forced stably stratified turbulence {\it J. Fluid Mech. }{\bf 202} 97--115 (1989).   

Kinetic Energy Dynamics in Forced, Horizontally Homogeneous, Stably Stratified Turbulence

Recent numerical simulations and scaling arguments have established that in stratified turbulence there is net downscale transfer of kinetic energy. The nature of the transfer is less clear, particularly in how it compares to energy transfer in the inertial range of isotropic turbulence, and the literature suggests that the dynamics depend on the buoyancy Reynolds number, $\mathrm{Re}_b$. To gain further insight, three-dimensional direct numerical simulations with 170 billion grid points are considered. The flows are horizontally homogeneous and vertically stratified with Froude number between 0.125 and 1 and $\mathrm{Re}_b$ between 9 and 219. The complete balances of the horizontal and vertical contributions to kinetic energy are presented in terms of two-dimensional spectra. In this format, the extent in wave number space of the inertial range where the dissipation rate is small compared with other terms in the balance is clear. The spectra also show regions of wave space in which energy transfers downscale in the vertical and then upscale in the horizontal, particularly at low $\mathrm{Re}_b$. They also explain why some published results reporting one-dimensional spectra appear to show high dissipation rate at large length scales.   

Transition to turbulence in homogeneous and stratified Ekman boundary layers - an experimental study

A dynamical interplay between the pressure gradient, the Coriolis force and the viscous forces describes the laminar regime of the (Ekman) boundary layers in the ocean and the atmosphere. Early experiments on the homogeneous Ekman layer identified two branches of instabilities that result in roll patterns with distinct wavelengths and orientation. In this talk, we present results based on quantitative velocity field measurements in experiments performed on the 13m diameter Coriolis rotating platform at Grenoble, France. Geostrophic flow outside the boundary layer is set up in a well-controlled manner by changing the tank rotation appropriately. In the homogeneous case, each of the two branches of instabilities is captured in isolation by abruptly increasing the Reynolds number from 0 to a finite value ranging from 50 to 360. The corresponding wavelengths and growth rates as a function of the Reynolds number are deduced from our measurements. Emission of inertial waves is detected for certain parameters. Finally, the influence of a background stratification on the roll patterns of instability is investigated in some exploratory experiments.   

Analysis of cascades in space and in scale for rotating and stratified Boussinesq flows

We use high-resolution simulations of Boussinesq flows, forced in the large-scales, with fixed rotation and stable stratification along the vertical axis, to study the downscale cascades of energy and potential enstrophy in three different regimes of stratification and rotation. (1) For strongly stratified flows with moderate rotation, we observe anisotropic fluxes of energy and potential enstrophy into Fourier modes with large vertical component $k_z$, predominantly due to a highly non-local transfer from the large-scales directly to the smallest scales. The energy cascade is predominantly due to three vortical-mode interactions. (2) For strongly rotating flow with moderate stratification, there are anisotropic fluxes to modes with large $k_h$, due to a ``diffusely'' local transfer much like in isotropic Navier-Stokes turbulence. The energy cascade is primarily due to three vortical-mode interactions, as in the strongly stratified case, although wave-vortical-wave and vortical-wave-vortical interactions also make a noticeable contribution. (3) In the third case of equally strong rotation and stratification, there are only slightly anisotropic fluxes, mostly to modes with large $k_h$, due to an ultra-local transfer in which the energy gained by an inertial scale comes almost exclusively from the adjacent larger scales. We confirm that the cascades in the third regime are primarily due to wave-vortical-wave interactions, in agreement with previous work.   

3D Structure and Internal Circulation of Pancake Vortices in Rotating Stratified Flows

Jovian vortices, Atlantic meddies, and vortices of the protoplanetrary disks are examples of weakly-forced or unforced long-lived vortices in rotating stratified flows. Knowing the 3D structure and internal circulation of these vortices is essential in understanding their physics, which is not well-understood. For example, the aspect ratio of these vortices has been long thought to be $f/N$ where $f$ is the Coriolis parameter and $N$ is the Brunt-Vaisala frequency. However, our recent theoretical and experimental study has shown that the aspect ratio in fact depends not only on $f$ and $N$ but also on the Rossby number and density mixing inside the vortex. The new scaling law also agrees with the available measurements of the meddies and Jupiter's Great Red Spot. High resolution 3D numerical simulations of the Navier-Stokes equation are carried out to confirm this new scaling law for a slowly (viscously) decaying anticyclonic vortex in which the Rossby number and stratification inside the vortex evolve in time. For a wide range of parameters and different distributions of density anomaly, the secondary circulations within the vortices are studied. The effect of a non-uniform background stratification is investigated, and the small cyclonic vortices that form above and below the anticyclone are studied.   

New equations for the slow dynamics in the presence of strong rotation and weak stratification

We explore the strong rotation limit of the rotating and stratified Boussinesq equations with periodic boundary conditions when the stratification is weak. This regime corresponds to the limit where (Rossby number) $Ro = \epsilon$, (Froude number) $Fr = O(1)$, as $\epsilon \rightarrow 0$. We show that the slow dynamics decouples from the fast and derive new equations for the slow dynamics. The new equations for the slow dynamics describe the dynamics of Taylor-Proudman flows and their conservation laws including a new conserved quantity for vertical kinetic energy and potential energy. The leading order potential enstrophy is slow while the leading order total energy retains both fast and slow dynamics. We also perform forced numerical simulations of the rotating Boussinesq equations to demonstrate support for the theory.