### Topics in Rotating Stratified Turbulence Workshop August 2-5, 2010

### Lectures

**Moisture, Rotation and Stratification in the Atmospheric Mesoscales**

Peter Bartello , McGill University

Recent efforts in these topics will be presented. First, is the simplest extension of the Boussinesq equations that includes water in two phases. It was formulated and implemented numerically in order to investigate the influence of clouds (at least at this level of simplification) on predictability and small-scale turbulence statistics. It was found that moisture does not make the flow inherently more unpredictable, when compared to dry flows with the same energy. Moisture also had a weak but systematic tendency to disrupt small-scale intermittency. Another topic is the breakdown of large-scale geostrophy in proceeding down scale at integral-scale Rossby numbers of order unity. Numerical Boussinesq simulations in triply-periodic geometry show a transition to a more shallow small-scale energy spectrum that is consistent with the Nastrom-Gage data and with stratified turbulence in the small scales.

**Transition to Turbulence via Transient Growth of Perturbation for Internal Gravity Waves and Shear Flows **

Jean-Marc Chomaz, LadHyX

In collaboration with Cristobal Arratia, LadHyX

This work investigates the three-dimensional stability of a horizontal flow sheared horizontally, the hyperbolic tangent velocity profile, in a stably stratified fluid. In an homogeneous fluid, the Squire theorem states that the most unstable perturbation is two-dimensional. When the flow is stably stratified, this theorem does not apply and we have performed a numerical study to investigate the three-dimensional stability characteristics of the flow. When the Froude Number, Fh, is varied from ƒ to 0.05, the most unstable mode remains two-dimensional. However, the range of unstable vertical wavenumbers widens proportionally to the inverse of the Froude Number for Fh<<1. This means that the stronger the stratification, the smaller the vertical scales that can be destabilized. This loss of selectivity of the two-dimensional mode in horizontal shear flows stratified vertically may explain the layering observed numerically and experimentally.

We study the transient growth on this inviscid horizontal shear layer with vertical stratification. We compute the optimal perturbations that maximize the energy growth up to a time horizon T as a function of the streamwise and spanwise wavenumbers.

For strong stratification, the self-similarity found by Billant and Chomaz (2001) holds. The gravity wave components of the perturbations are obtained by means of a Craya-Herring decomposition which, in the absence of shear, corresponds to an exact separation between gravity waves and vortical modes for the linear dynamics. Intense excitation of gravity waves due to transient growth of perturbations is found in a broad region of the wavevector plane, gravity waves being eventually emitted away from the shear layer.

**Using Simple Models and Rigorous Mathematics to Improve Operational Atmosphere and Ocean Modelling**

Mike Cullen , Senior Research Fellow, Met Office, Exeter Devon, United Kingdom

In recent years the mathematical community have done a lot of rigorous analysis of the equations governing the phenomena in the atmosphere (and ocean) that we are trying to simulate. I summarise some key results and implications. Specifically I will show how analysis of the shallow water equations in different regimes supports the computational evidence that the limiting behaviour can either be purely zonal flow or coherent eddies. This is related to the changes of behaviour in the atmospheric circulation between blocked or progressive patterns. I then discuss the mathematical properties of models based on potential vorticity, and identify key properties that allow them to be solved. I illustrate with the semi-geostrophic model with variable rotation. I then show how simple model solutions can be used to validate operational models, by showing that they respect the predicted asymptotic behaviour. I illustrate this for the singular semi-geostrophic limit of the Eady frontogenesis problem.

**What do Experiments Tell Us about Rapidly-rotating Turbulence?**

Peter Davidson, University of Cambridge, DAMTP, UK

The most fundamental question we can ask about rapidly-rotating turbulence is the following: why are the large scales dominated by columnar eddies aligned with the rotation axis? Recent experiments on such turbulence have shed new light on the formation mechanism for these columnar structures.

Contrary to received wisdom, it seems that (quasi-) linear inertial wave propagation may play a crucial, possibly even dominant, role. This new experimental evidence is reviewed, and the question asked: have we been following the wrong paradigm?

**The Combined Lagrangian Advection Method (CLAM): A New Approach to Modelling Complex Active and Passive Tracer Advection**

David Dritschel , University of St Andrews

In collaboration with Jerome Fontane, ISAE, Toulouse

This talk presents a new hybrid Lagrangian-Eulerian numerical method for accurately modelling complex tracer advection in geophysical flows. The new method radically extends the Contour-Advective Semi-Lagrangian (CASL) method (Dritschel and Ambaum, 1997) by fusing three computational elements:

(1) an Eulerian pseudo-spectral or grid-point method for large scales,

(2) Lagrangian contour advection for intermediate to small scales, and

(3) Lagrangian particles to account for general forcing and dissipation.

The pseudo-spectral method is both efficient and highly accurate at large scales, while contour advection is efficient and accurate at small scales, allowing one to follow extremely fine-scale structure well below the basic grid scale used to represent the velocity field. The particles allow one to efficiently model non-conservative effects. The method is illustrated in a two-layer shallow-water flow in spherical geometry.

**The Turbulent Equilibration of an Unstable Baroclinic Jet**

J. Gavin Esler, Department of Mathematics, University College London

The evolution of an unstable baroclinic jet, subject to a small perturbation, is examined numerically in a quasi-geostrophic two-layer channel model. After a period of initial wave growth, wave breaking leads to two-dimensional turbulence within each layer, and to the eventual equilibration of the flow. The equilibrated flow must satisfy certain constraints; its total momentum is conserved, its total energy is bounded and the flow must be realizable via some area preserving (diffusive) rearrangement of the potential vorticity field of the initial flow. A theory is introduced that predicts the equilibrated flow in terms of the initial flow parameters. The idea is that the final state minimizes available potential energy, subject to constraints on the total momentum and total energy, and the further constraint that the potential vorticity changes through a process of complete homogenization within well-delineated regions in each layer. Within a large region of parameter space, the theory is found to accurately predict the cross-channel structure and strength of the equilibrated jet, the regions where potential vorticity mixing takes place and total eddy mass (temperature) fluxes.

**Locality, Stability, and Anomalous Sinks in Steady Two-dimensional Turbulence**

Eleftherios Gkioulekas , University of Texas-Pan American

I will discuss a new theoretical framework for understanding the robustness (or lack thereof) of the cascades of two-dimensional Navier-Stokes turbulence. The mathematical framework underlying our analysis is the infinite system of balance equations that govern the generalized unfused structure functions, first introduced by L'vov and Procaccia. As a point of departure we use a revised version of the system of hypotheses that was proposed by Frisch for three-dimensional turbulence. We show that both the enstrophy cascade and the inverse energy cascade are local in the sense of non-perturbative statistical locality. We also investigated the stability conditions for both cascades. We have shown that statistical stability with respect to forcing applies unconditionally for the inverse energy cascade. For the enstrophy cascade, statistical stability requires large-scale dissipation and a vanishing downscale energy dissipation. Finally, we have shown that the anomalous sink hypothesis follows as a consequence of our hypotheses.

**Statistical Theories Applied to Stratified Turbulence**

Jack Herring, NCAR

Statistical theories furnish a language and predictions for understanding certain features of turbulent flows. Such predictions are simplest for isotropic flows, and become more complicated as flows become non isotropic and inhomogeneous. Stably stratified turbulence is perhaps the simplest extension from conditions of isotropy which represents significant meteorological interest. We shall discuss the utility of several statistical theories and compare their predictions to recent direct numerical simulations of stably stratified homogeneous turbulence.

**Layering and Stratified Turbulence Surrounding an Anticyclonic Eddy**

Bach Lien Hua, IFREMER, Laboratoire de Physique des Oceans, Brest

Recent geoseismic data in the North-East Atlantic Ocean has revealed ubiquituous pancake-like density anomalies, ranging in thickness from a few metres to 80 m, surrounding anticyclonic eddy structures (Meddies) within the Mediterranean Water outflow. The depths of pancake-like density layering unambiguously coincide with potential energy spectra in k_{h}^{-5/3}, where k_{h} is the horizontal wavenumber, for horizontal spatial scales comprised between 4km and 100m, reminiscent of non-rotating stratified turbulence results.

Using ultra-high three-dimensional resolution nonhydrostatic simulations of a Meddy structure on the Earth Simulator, with a horizontal grid size down to 100m and a vertical grid size of 3m, we have been able to reproduce the pancake-like layering surrounding the eddy. Furthermore, potential and kinetic energy with k_{h}^{-5/3} inertial ranges are simulated. Energy fluxes budgets reveal that a forward energy cascade is occuring at the depths of the layering. A rationale is proposed for the formation mechanism of the layering and we discuss the influence of rotation on stratifed turbulence properties.

**The Nonhydrostatic Balanced-Geostrophic Equations**

Keith Julien, Department of Applied Mathematics, University of Colorado at Boulder

Convection under the influence rotation has been the subject of a great deal of theoretical and experimental research. This problem is relevant to convectively driven fluid flows in the Earth's ocean and interior and also in the Sun and other stars, where the influence of rotation is generally important. In general numerical simulations of rotationally constrained flows are unable to reach realistic parameter values, e.g., Reynolds Re and Richardson Ri numbers. In particular, low values of Rossby number Ro, defining the extent of rotational constraint, compound the already prohibitive temporal and spatial restrictions present for high-Re simulations by engendering high frequency inertial waves and the development of thin (Ekman) boundary layers.

Recent work in the development of reduced partial differential equations (pde's) that filter fast waves and relax the need to resolve boundary layers has been extended to construct a hierarchy of balanced equations that span the stably- and unstably-stratified limits. By varying the aspect ratio for spatial anisotropy characterizing horizontal and vertical scales, rapidly rotating convection and stably-stratified quasi-geostrophic motions can be described within the same framework.

In this talk, the asymptotic pde's relevant for rotating convection are explored. Special classes of fully nonlinear exact solutions are identified and discussed. Direct numerical solutions that correctly capture the regular vortex columnar and irregular geostrophic turbulence regime of recent laboratory experiments are also presented and discussed.

**Structure Formation in Stratified Turbulence**

Yoshi Kimura, Graduate School of Mathematics, Nagoya University

We investigate energy spectra of stably stratified turbulence using direct numerical simulations (DNS) at a resolution of 1024^{3}. The calculation is done by solving the 3D Navier-Stokes equations under the Boussinesq approximation pseudo-spectrally. Using toroidal-polodial decomposition (Craya-Herring decomposition), the velocity field is decomposed into the vortex mode and the wave mode. In general, both the wave and vortex spectra are consistent with a Kolmogorov-like *k ^{-5/3}* range at sufficiently large

*k*. At large scales, and for sufficiently strong stratification the wave spectra is steeper than this (

*k⊥*), while that for the vortex component is consistent with

^{-2}*k⊥*. Here

^{-3}*k*⊥ is the horizontally gathered wave numbers. In contrast to the horizontal wave number spectra, the vertical wave number spectra show very different features. We can observe clear

*k*dependence for small scales while the large scales show rather flat spectra. We model the flat spectra by using a simple layer model for the vertical vorticity. We link these spectra to the 2nd order structure functions of the velocity correlations in the horizontal and vertical directions. Finally we study the inviscid limit in which the highest wave-numbers are progressively thermalized, leaving the smaller wave numbers to adjust to their internal dynamics

_{z}^{-3}*sans*dissipation. In this case, we see -for the non-thermalized components-similar dynamics as that for the finite Reynolds case.

**Stratified Turbulence in the Atmosphere and the Oceans**

Erik Lindborg, KTH Mechanics, Sweden

I will give an overview of my research aimed at understanding the atmospheric kinetic and potential energy spectra from length scales of the order of one kilometre up to several thousands kilometres. Results from theoretical analyses, data analyses and numerical simulations will be presented and discussed. In particular, I will focus on the interpretation of the extended K^{-5/3} mesoscale energy spectrum as a spectrum of stratified turbulence with an associated downscale energy cascade. Open issues in connection with the stratified turbulence interpretation will be critically discussed. I will also point out that a number of observations of ocean dynamics at scales of the order of 10 metres up to 100 kilometres are consistent with the hypothesis that the dynamics at these scales are strongly nonlinear with an associated downscale energy cascade, in other words a stratified turbulence interpretation.

**3D Dynamics and Turbulence Induced by Mountain and Inertia-Gravity Waves in the Upper Troposphere and Lower Stratosphere (UTLS)**

Alex Mahalov, Center for Environmental Fluid Dynamics, Department of Mechanical and Aerospace Engineering, School of Mathematical and Statistical Sciences, Arizona State University

The generation and physical characteristics of inertia-gravity waves radiated from an unstable forced jet at the tropopause are investigated through high-resolution numerical simulations of the three-dimensional Navier-Stokes anelastic equations. Such waves are induced by Kelvin-Helmholtz instabilities on the flanks of the inhomogeneously stratified jet. From the evolution of the averaged momentum flux above the jet, it is found that gravity waves are continuously radiated after the shear-stratified flow reaches a quasi-equilibrium state. The time-vertical coordinate cross-sections of potential temperature show phase patterns indicating upward energy propagation. The sign of the momentum flux above and below the jet further confirms this, indicating that the group velocity of the generated waves is pointing away from the jet core region. Space-time spectral analysis at the upper flank level of the jet shows a broad spectral band, with different phase speeds. The spectra obtained in the stratosphere above the jet show a shift toward lower frequencies and larger spatial scales compared to the spectra found in the jet region. The three-dimensional character of the generated waves is confirmed by analysis of the co-spectra of the spanwise and vertical velocities. Imposing the background rotation modifies the polarization relation between the horizontal wind components. This out-of-phase relation is evidenced by the hodograph of the horizontal wind vector, further confirming the upward energy propagation. The background rotation also causes the co-spectra of the waves high above the jet core to be asymmetric in the spanwise modes, with contributions from modes with negative wavenumbers dominating the co-spectra. In the second part of the talk, we present high resolution simulations in real atmospheric conditions of mountain waves in the UTLS during the Terrain-induced Rotor Experiment (T-REX). In these simulations, the finest nest of WRF is coupled with microscale nests, within which the three-dimensional fully nonhydrostatic compressible moist atmospheric equations are solved with refined grid in the vertical and improved resolution in the UTLS region. Comparison of simulations with in situ balloon and aircraft measurements obtained during T-REX show favorable agreement.

**References:**

1. A. Mahalov and M. Moustaoui (2009), "Vertically Nested Nonhydrostatic Model for Multi-Scale Resolution of Flows in the Upper Troposphere and Lower Stratosphere", *Journal of Computational Physics*, **vol. 228**, p. 1294-1311.

2. A. Mahalov, M. Moustaoui and B. Nicolaenko (2008), Three-Dimensional Instabilities in Non-Parallel Shear Stratified Flows", *Kinetic and Related Models*, **vol. 2, No. 1**, p. 215-229.

3. A. Mahalov, M. Moustaoui and B. Nicolaenko (2007), "Computational Studies of Inertia-Gravity Waves Radiated from Upper Tropospheric Jets", *Theor. and Comp. Fluid Dynamics*, **vol. 21, No. 6**, p. 399-422.

4. A. Mahalov and M. Moustaoui (2010), "Characterization of Atmospheric Optical Turbulence for Laser Propagation, Laser and Photonics Reviews', **Volume 4 Issue 1**, p. 144-159, (January 2010), Special Issue: 50 Years of Laser.

**Surface Fronts and Filaments: Genesis, Instability, and Arrest**

Jim McWilliams, UCLA Institute of Geophysics and Planetary Physics and Department of Atmospheric and Oceanic Sciences

Geophysical fluid dynamicists have developed a mature perspective on the dynamical influence of Earth's rotation, while most other areas of fluid dynamics can safely disregard rotation. Similarly, geophysical problems usually arise under the influence of stable density stratification at least as importantly as velocity shear. In this talk the dominant turbulence and wave behaviors in the rotating and non-rotating, stratified and non-stratified fluid-dynamical realms are described, and particular attention is given to their borderlands, where rotational and stratified influences are significant but not dominant. Contrary to the inverse energy cascade of geostrophic turbulence toward larger scales, a forward energy cascade develops within the borderlands from the breakdown of diagnostic force balances, frontogenesis, and frontal instabilities, and then it continues further through the small-scale, non-rotating, unstratified (a.k.a. universal) realm until it dissipates at the microscale. In particular, this submesoscale cascade behavior is of interest as a global route to kinetic and available-potential energy dissipations in the oceanic general circulation, as well as an energy source for microscale material mixing across stably-stratified density surfaces. The transition to submesoscale flow is especially effective near topography.

**Scaling Laws in Helical Rotating Turbulence: Do They Change with Reynolds Number?**

Annick Pouquet, National Center for Atmospheric Research

In collaboration with Julien Baerenzung (Grenoble), Pablo Mininni (Buenos Aires & NCAR)and Duane Rosenberg (NCAR)

In turbulent flows with rotation, the presence of helicity leads to significant differences in the flow dynamics when compared to flows without global velocity-vorticity correlations, as seen using direct numerical simulations (DNS) up to a grid of 1536^{3} points and down to Rossby numbers *Ro*≈ 0.06. Long-lived laminar cyclonic vortices together with turbulent vortices are found to co-exist, somewhat reminiscent of recent tornado observations but in a much simpler and idealized physical context.

The energy undergoes both a large-scale (inverse) and a small-scale (direct) cascade. In the latter case, it is self-similar with spectrum *E(k)~k ^{-e}* and transfer rate ∈

_{E}with no deviations from Gaussianity and dominated by the helicity cascade (with spectrum

*H(k)~k*and transfer rate ∈

^{-h}_{H}). This result points to the plausibility of the existence of a new small parameter in helical turbulence with solid body rotation, namely ∈

_{E}/[L

_{0}∈

_{H}], with L

_{0}a large-scale characteristic length. We also find that the spectral indices obey the scaling law

*e*+

*h*= 4 when taking into account the inertial wave mediation of nonlinear helicity transfer to small scales, with

*e = h = 2*appearing not to be the only solution at least at the intermediate Reynolds numbers at which we compute.

Using an isotropic model based on the Eddy Damped Quasi-Normal Markovian two-point closure (EDQNM), and taking into account the contribution of helicity to eddy viscosity and eddy noise, we show that we can recover the DNS results at substantially lower costs, up to almost four orders of magnitude less: the DNS run on a grid of 1536

^{3}points can be "reliably" modeled using grids of 96

^{3}points. Performing with the model the beginning of a parametric study, we find that, at fixed Reynolds number, strong rotation leads to this new

*e*+

*h*= 4 regime with e ≠ h, whereas one may be recovering the classical Kolmogorov law

*e*=

*h*= 5/3 when increasing the Reynolds number at fixed rotation rate.

**References**

[1] A. Pouquet and P.D. Mininni, "The interplay between helicity and rotation in turbulence: implications for scaling laws and small-scale dynamics."

*Phil. Trans. Roy. Soc.*

**368**, 1635-1662 (2010).

[2] P. Mininni and A. Pouquet, "Rotating helical turbulence. I. Global evolution and spectral behavior;" and "II. Intermittency, scale invariance and structures."

*Phys. Fluids*

**22**, 0351-05 & -06 (2010).

[3] J. Baerenzung, P.D. Mininni, A. Pouquet and D. Rosenberg, "Spectral Modeling of Turbulent Flows and the Role of Helicity in the presence of rotation," submitted to

*J. Atmos. Sci.*, see also arXiv:0912.3414.

**Study of Very Stable Atmospheric Boundary Layers Using Direct Numerical Simulation**

James J. Riley, University of Washington

In collaboration with Oscar Flores, University of Washington

Stable atmospheric boundary layers often occur on clear nights due to radiative cooling. The stable stratification can strongly influence the boundary layer turbulence, causing it to weaken or, in very stable cases, to become intermittent. This can then have profound effects on processes occurring in the boundary layer, e.g., pollutant dispersion and the evolution of the surface temperature.

In the present study direct numerical simulations have been carried out of initially neutrally stable, turbulent Ekman layers subject to cooling at the ground, much as occurs at sunset in the atmosphere. Cases are considered where ground cooling strongly affects the turbulence, in some cases completely extinguishing it. The processes by which the turbulence is affected are studied in detail.

**Evaluation of a Reduced Model for Investigating Hurricane Formation from Turbulence**

David A. Schecter, NorthWest Research Associates, Redmond, WA

Although hurricane formation has been studied for decades, it is not fully understood. Complex, cloud-system resolving models are commonly used to investigate the process, but are computationally expensive and often difficult to interpret. In principle, reduced models can be used to clarify the essential dynamics, and to efficiently discover new phenomena.

This study evaluates the adequacy of a reduced (3-layer) model for understanding hurricane formation from turbulent initial conditions. The evaluation is based on a direct comparison to tropical cyclogenesis in a cloud-system resolving model (RAMS-6.0) that uses single-moment warm rain microphysics. The reduced model has three alternative cumulus parameterizations. One parameterization is a variant of the classic convergence-based (CB) scheme of Ooyama 1969. Another regulates cumulus activity by enforcing boundary layer quasi-equilibrium (BLQ). The third resembles the CB parameterization, but provides a selective boost (SB) to convection in regions of exceptionally high instability.

Regardless of the cumulus parameterization, the reduced model produces hurricanes on the same time-scale as the cloud-system resolving model. Generally speaking, the hurricanes emerge from turbulence through the coalescence and convective intensification of cyclonic vorticity. Moreover, in both the reduced and cloud-system resolving models, the onset of rapid intensification coincides with a peak in the local time-series of the eta-variable of Ooyama 1969, which is a combined measure of deep convective instability and middle tropospheric moisture. Eliminating the surface flux of moist entropy or surface friction in either model prevents or severely inhibits hurricane formation; however, hurricanes eventually form without surface friction in the BLQ or SB versions of the reduced model.

An analytical approximation is derived for steady-state hurricane intensity in the context of the reduced model. As in the more realistic but involved theory of Emanuel 1986, the square of the maximum wind speed is roughly proportional to the ratio of entropy to momentum exchange coefficients, times a measure of the ambient thermal disequilibrium between the sea-surface and the upper troposphere (not to be confused with CAPE). The analytical approximation compares favorably to a set of 3-layer numerical simulations that covers a broad range of parameter space. Limitations of the analysis are briefly addressed, and a supergradient wind correction is estimated.

Despite some measure of success, the reduced model has notable deficiencies that are apparent during the intermediate stage of genesis. Compared to the cloud-system resolving model, rotational storms are less sporadic and their winds are less severe. In the small-to-intermediate mesoscale, the horizontal kinetic energy spectrum is relatively steep, and horizontal divergence is relatively weak. Furthermore, the Lagrangian autocorrelation time of vertical vorticity is relatively long. These discrepancies indicate a simplified (quasi two-dimensional) form of rotational convective turbulence. The simplified turbulence has comparatively robust mesoscale cyclones, and tends to produce more hurricanes than its analogue generates in the cloud-system resolving model.

The behavior of the simplified turbulence is interesting in its own right. In some parameter regimes, the time required for tropical cyclogenesis varies dramatically with subtle changes to the initial conditions. Moreover, hurricanes do not always form. In a sufficiently large domain, the turbulence can evolve into a state with several hurricanes amid numerous vorticity filaments. Occasionally, hurricanes develop outer eye walls by entraining nearby filaments, and isolated filaments spawn new hurricanes.

This work is supported by NSF grant ATM-0750660.

**Geostrophic Turbulence Near Rapid Changes in Stratification**

Shafer Smith, Courant Institute of Mathematical Sciences, New York University

When vertical boundaries are present and surface advection of buoyancy is resolved vertically, numerical simulations of quasigeostrophic turbulence driven by baroclinic instability reveal a mix of surface-quasigeostrophic (SQG) and "standard" geostrophic turbulence, the type predicted by Charney (1971). Specifically, the kinetic energy spectrum at the surfaces exhibits a *K ^{-3}* power law in the enstrophy-cascade range, at wavenumbers greater than the first deformation wavenumber, but flattens to

*K*at even larger wavenumbers. In these simulations, the spectral break occurs at a horizontal scale determined by the relative baroclinic forcing of surface and interior modes. The numerically and theoretically predicted spectrum at a rigid upper boundary is relatively consistent with the atmospheric energy spectrum observed near the tropopause by the GASP and MOZAIC programs.

^{-5/3}Here we consider geostrophic turbulence in the presence of a more general background environment, characterized by a rapid jump in mean stratification rather than by an artificial rigid upper boundary. Model buoyancy frequency profiles are specified analytically as N

*(z)*=

*N*

_{0}+

*N*tanh(z/δ), with both the atmospheric tropopause and upper ocean pycnocline in mind. The rapidity of change is controlled by the length scale δ, and the profile approaches a step function as δ →0. The Green's function for the PV-streamfunction relationship is determined using a WKB approximation, and is used to predict the structure of the spectrum, under various assumptions about the vertical and wavenumber distribution of PV.

_{d}Numerical simulations of freely-evolving and baroclinically forced turbulence verify the predictions, and reveal that the jump in stratification has two effects: it alters the Green's function in the region of the jump, and it produces a peak in PV near the jump (due to the second derivative in

*z*in the PV-streamfunction relation), approaching a delta-function as δ →0. When the Green's function is integrated against this sharp PV distribution, contributions far from the jump (|z| >> δ) are supressed, and the kinetic energy spectrum flattens. This occurs for a range of wavenumbers above the deformation wavenumber associated with the vertical extent

*H*

_{0}of the domain,

*K*/(

_{d}= f*N*

_{0H0), but smaller than a wavenumber associated with the vertical scale of the jump, Kδ = f/(Ndδ). The vertical distribution of the flattened spectrum decays over a distance proportional to δ. Implications of these results for the atmosphere and ocean will be discussed. }

**Modeling Mixing in Stratified Fluids by Statistical Mechanics**

Joel Sommeria, LEGI, Grenoble

In collaboration with Antoine Venaille, Princeton University

A phenomenological model for turbulent mixing in a stratified fluid is presented. It is inspired from a statistical mechanics approach of potential vorticity mixing in geostrophic turbulence. This model describes the evolution of the local probability distribution for the fluid density. Numerical results are presented and compared with a test laboratory experiment of grid turbulence in a stratified fluid.

**Local Generation of Internal Solitary Waves in an Oceanic Thermocline**

Chantal Staquet, LEGI, Grenoble, France

In collaboration with Nicolas Grisouard,
LEGI, Grenoble, France

Oceanic observations have provided evidence of the generation of internal solitary waves due to an internal tidal beam impinging on a seasonal pycnocline from below. The purpose of this talk is to present direct numerical simulations of such a generation process with a fully nonlinear nonhydrostatic model. An academic two-dimensional configuration in a vertical plane is considered. We shall show that, depending on the parameters, different modes of internal solitary waves can be excited in the pycnocline and we shall provide examples of internal solitary waves as first, second and third modes, trapped in the pycnocline. A criterion for the selection of a particular mode will be put forward, in terms of phase speeds. In addition, another simpler geometrical criterion will be presented to explain the selection of modes.

Application to the Bay of Biscay will then be considered. We shall show that, in this more realistic context as well, a beam impinging on a pycnocline initially at rest can induce a displacement of the isopycnals, large enough for internal solitary waves to be generated. These internal solitary waves however differ from those observed in the Bay of Biscay through their amplitude and distance between wave trains. The latter feature is recovered when the background flow around the pycnocline as found in the Bay of Biscay is included in the forcing, thereby yielding a more accurate view of the local generation mechanism in that region.

This abstract is based on the PhD work of Nicolas Grisouard and has been performed in collaboration with Theo Gerkema (NIOZ, The Netherlands).

References :

Grisouard N., C. Staquet & T. Gerkema. Conditions for local generation of solitary waves in a pycnocline. Submitted to *J. Fluid Mechanics*.

Grisouard N. & C. Staquet. Numerical simulations of the local generation of internal solitary waves in the Bay of Biscay. Submitted to *Nonlin. Proc. in Geophys*.

**Are Quasi-2D Turbulent Flows (Somehow) Conformally Symmetric?**

Simon Thalabard, NCAR IMAGe

Bernard & al showed in 2006 that zero vorticity isolines in 2D numerical turbulent flows that exhibit a strong inverse cascade could strongly be believed to be "conformally invariant", and were in a way no different than percolation paths.

Does this property still hold for turbulent flows that are not perfectly bidimensional? For instance, can we see some hints of conformal invariance in (real) experimental 2D flows? And what about 3D flows bidimensionalized by a strong rotation?

**How to Break the Self-consistency of Geostrophic Turbulence**

Joe Tribbia, NCAR CGD

As is well known, a result of geophysical fluid dynamics is the fact that, unlike three-dimensional turbulence in which the eddy turnover time decreases with eddy length scale, in two dimensional and quasi-geostrophic turbulence the eddy turnover time is constant independent of eddy length scale in the enstrophy cascading range. The consequence of this is that the Rossby number is constant through the enstrophy cascade. This further implies that instabilities which depend on ageostrophic processes are restricted because the scaling laws which imply balanced, quasi-geostrophic dynamics are valid at all length scales. However, even assuming that all of the above statements are true and maintained in the dynamics, I will discuss a mechanism through which quasi-geostrophic turbulence becomes inconsistent and develops the seeds of its own destruction at small scales.

**Buoyancy Scale Dynamics in Geophysical Turbulence **

Michael L. Waite, University of Waterloo

Numerical simulations of turbulence in stratified fluids exhibit a direct cascade of energy that has been proposed as a model for the atmospheric mesoscale energy spectrum. This talk will focus on two unresolved issues in the relationship between stratified turbulence and the mesoscale cascade: inconsistencies between idealized turbulence and more realistic atmospheric simulations, and the implications of using relatively coarse horizontal resolution. Both issues are fundamentally connected to the role of motions on the buoyancy scale *L _{b} ≡ U⁄N*, where

*U*is the r.m.s. velocity and

*N*is the Brunt-Väisälä frequency. In the atmosphere, L

_{b}~ 1 km.

Simulations of stratified turbulence require fine vertical grids that resolve the buoyancy scale; with coarser resolutions, the cascade is damped and a very steep spectrum develops. Simulations of the atmospheric mesoscale with global and limited-area models frequently do not use such fine grids, and yet they are able to yield realistic mesoscale spectra. On the other hand, we will show that resolving

*L*in the vertical is not enough to guarantee realistic spectra. We will discuss the mesoscale dynamics of dry baroclinic life cycle simulations with extremely high vertical resolution that nevertheless exhibit significant differences with stratified turbulence phenomenology.

_{b}Although buoyancy scale dynamics play a fundamental role in the stratified turbulence cascade, it is often only possible to resolve them in the vertical. Yes stratified fluids support a rich variety of motions with horizontal scales of

*L*, including shear instability, overturning internal waves, breakdown of coherent vortices, and the subsequent transition to isotropic three-dimensional turbulence. These dynamics are distorted, if not completely suppressed, in simulations with Δ

_{b}*z ‹‹ L*Δ

_{b}‹‹*x*. We will discuss high-resolution simulations of stratified turbulence that clarify the effect of buoyancy scale motions on the downscale cascade. Implications for the atmospheric mesoscale spectrum will be discussed.

**Navier-Stokes Equations on the Beta-plane**

Djoko Wirosoetisno, Department of Mathematical Sciences, University of Durham

We show that, given a sufficiently regular forcing, the solution of the two-dimensional Navier-Stokes equations on the Β-plane will become nearly zonal as *t*→ ∞. A qualitatively improved bound on the dimension of the global attractor is also given.

Joint work with MAH Al-Jaboori.

**Evidence for an Oceanic Forward Kinetic Energy Cascade from High-Frequency Radio Doppler Surface Current Meters**

Edward D. Zaron, Department of Civil and Environmental Engineering, Portland State University

High-frequency radio Doppler current meters were deployed from Dec 2002 to May 2003 to observe ocean surface currents off of Oahu, Hawaii, USA. Here we use these data to compute the third-order longitudinal velocity structure function, from which we infer the scale-dependent energy flux. The sign of the third order structure function is usually negative at small separations, consistent with a downscale transport of kinetic energy (i.e., a forward cascade) at scales ranging from 3km to 20km. Furthermore, the largest scale of the downscale flux is found to be proportional to the rate of energy flux such that Ro > 0.05, approximately, where Ro is a large-scale turbulence Rossby number. These results are consistent with the phenomenology of stratified turbulence, but there are still many questions about how lack of isotropy and homogeneity of the observed flow fields influence the 3rd order statistics.
In collaboration with Cedric Chavanne, School of Environmental Sciences, University of East Anglia, Norwich, United Kingdom.