December  4, 2022


Workshop on Sub-mesoscale Ocean Processes
June 11-14, 2013
Location: Stewart Library, Fields Institute
Organizing Committee:
Kevin Lamb (Waterloo), Francis Poulin (Waterloo)

Fields Institute program on the Mathematics of Oceans is to take place in the year 2013 as a part of the initiative for the Mathematics of Planet Earth.

Draft Schedule

Tuesday, June 11
Stewart Library, Fields Institute

9:00 - 9:30 On-site Registration
9:30 - 9:45 Welcome and Introduction
9:45 - 10:25 Cédric Chavanne, l'Université du Québec à Rimouski
Strain-driven submesoscale frontogenesis : what can surface currents tell us about what is happening below?
10:25-10:45 Coffee Break
10:45-11:25 Michael Waite, University of Waterloo
Potential enstrophy in stratified turbulence
11:30-12:10 Vladimir Zeitlin, Ecole Normale Supérieure
Instabilities of coupled density fronts and their nonlinear evolution in the two-layer rotating shallow water model. Influence of the lower layer and of the topography
12:10-14:20 Lunch Break
14:20-15:00 Francis Poulin, University of Waterloo
Spectral characteristics of a turbulent, homogeneous wind-driven gyre flow
15:10-15:50 Balu Nadiga, Los Alamos National Lab
Some experiments on dissipation of balanced energy in the interior
16:00 Reception
Wednesday, June 12
Stewart Library, Fields Institute
9:00-9:40 Pascale Lelong, NorthWest Research Associates
A numerical study of lateral dispersion in a field of oceanic internal waves
9:45-10:25 Michael Dunphy, University of Waterloo
Focussing and normal mode scattering of the first mode internal tide by mesoscale eddy interaction
10:25-10:45 Coffee Break
10:45-11:25 Xavier Carton, Université de Bretagne Occidentale
The influence of mesoscale, surface intensified eddies of the Arabian Sea and adjacent gulfs, on the RSW and PGW outflows
11:30-12:10 Peter Bartello, McGill University
From Quasigeostrophic to stratified turbulence
12:10-14:20 Lunch Break
14:20-15:00 Daniel Kirschbaum, McGill University
Invigoration of cumulus cloud fields by mesoscale ascent
Thursday, June 13
Stewart Library, Fields Institute
9:00-9:40 Rob Scott, Université de Bretagne Occidentale, and CNRS
Eddy-modulated, super­inertial turbulence
9:45-10:25 Mary-Louise Timmermans, Yale University
Scales of horizontal density structure in the surface layer of the Arctic Ocean
10:25-10:45 Coffee Break
10:45-11:25 David Straub, McGill University
Influence of forced near-inertial motion on nearly geostrophic flow in a recirculating zonal channel
11:30-12:10 Eric Skyllingstad, Oregon State University
Simulations of coherent structures in ocean frontal zones and effects on dye dispersion
12:10-14:20 Lunch Break
14:20-15:00 Gualtiero Badin, University of Hamburg
Toward out-of-balance surface dynamics in the ocean
Friday, June 14
Stewart Library, Fields Institute
9:00-9:40 Pascale Lelong, NorthWest Research Associates
Near-inertial waves within an anticyclonic eddy and turbulence in the Mediterranean sea during BOUM experiment
9:45-10:25 Susan Allen, University of British Columbia
Extending the Validity of a One-Dimensional Coupled Bio-physical Model by Parametrization
10:25-10:45 Coffee Break
10:45-11:25 Jacques Vanneste, University of Edinburgh
A surface-aware projection basis for quasigeostrophic flow
11:30-12:10 Alexandre Stegner, Ecole Polytechnique
Inertial-centrifugal instability of intense anticyclonic vortices : linear stability analysis, laboratory experiments and oceanic observations

Speaker & Affiliation
Title and Abstract

Susan Allen
University of British Columbia

Lecture Notes

Extending the Validity of a One-Dimensional Coupled Bio-physical Model by Parametrization

Processes, such as those at the sub-mesoscale, that are unresolved by models need to be parametrized. Here we will review the parametrized processes used in a vertical column model of the Strait of Georgia. The Strait of Georgia is a semi-enclosed, temperate, coastal sea with large freshwater sources. The coupled model has been successfully used to model and hindcast the timing of the spring phytoplankton bloom and investigate the major processes impacting pH in the Strait. The physical model is based on a mixing-layer model; the biological model is NPZD with three nutrients, three phytoplankton, one active and one closing zooplankton and three detritus variables tracked, and the chemistry model includes the carbon and oxygen cycles. The Strait of Georgia is biological productive due to higher dimensional processes that are not resolved in the one-dimensional model. However, the Strait is well studied and we parametrize the important processes based on data. I will review the model and our goals and emphasize the parametrization procedures and how they are implemented in the model as, perhaps, a proto-type for parametrization of sub-mesoscale processes into coarse resolution models.
Gualtiero Badin
University of Hamburg

Toward out-of-balance surface dynamics in the ocean

The surface quasi-geostrophic approximation is re-written in an oceanic context using the two-dimensional semi-geostrophic approximation. The new formulation allows to take into account the presence of out-of-balance flow features at scales comparable to or smaller than the Rossby radius of deformation and for small bulk Richardson numbers. Implications for the nonlinear behavior of submesoscale instabilities as well as for lateral mixing in the ocean are discussed.

Peter Bartello
McGill University

Lecture Notes

From Quasigeostrophic to stratified turbulence

Numerical explorations will be described that illustrate the transition from large-scale quasigeostrophic flow, through a scale range exhibiting the breakdown of balance, to stratified turbulence with negligible rotation. The latter has recently been demonstrated to be inherently unbalanced in that linear wave time scales are not fast with respect to the nonlinear variablity of the turbulence. In this setting the small-scale turbulence follows a shallow -5/3 spectrum with respect to the horizontal wavenumber in contrast to the considerably steeper spectra of quasigeostrophic turbulence in the potential enstrophy cascade range. In addition, the emergence of unbalanced small-scale turbulence from balanced initial conditions also manifests itself via a shallow -5/3 range at high wavenumbers in the horizontal energy spectrum. These results will be related to observations of atmospheric and oceanic spectra with the caveat that statistically homogeneous turbulence is simulated here without boundaries. The latter have been demonstrated to facilitate the breakdown of balance.

Xavier Carton
Université de Bretagne Occidentale

Lecture Notes

The influence of mesoscale, surface intensified eddies of the Arabian Sea and adjacent gulfs, on the RSW and PGW outflows

The Arabian Sea is strongly influenced by the atmospheric forcings, in particular the seasonal monsoon winds and the intense heat fluxes.

These forcings produce a complex variation of the thermal structure in the upper ocean, by firstly increasing the stratification in the early summer, then decreasing it by mixing due to strong southwesterly monsoon winds and finally by providing again a heat gain in late summer/early fall.
These strong winds also generate alongshore currents (the East Arabian Current and the Somali Current) which produce large eddies (dipoles along the Omani and Yemeni coasts, and the Great Whirl and Socotra Eddy near the Somali Coast). The summer monsoon winds also generate intense upwellings along the Somali and Omani coasts, which also generate eddies and filaments.

At depth, the Arabian Sea is influenced by the outflows from the marginal seas ; the Red Sea and Persian Gulf produce very salty waters and export them via Bab el Mandeb into the Gulf of Aden and via the Straits of Hormuz into the Sea of Oman. The Red Sea outflow mixes with Indian Central Water and forms a water mass (RSW) with a salinity maximum between 600 and 1000 m depth. The Persian Gulf outflow also mixes and forms a water mass (PGW) with an even higher salinity maximum between 200 and 400 m depth. RSW and PGW flow on average along the Somali and Omani coasts (continental slopes).

Using satellite and in situ data (Argo float profiles and hydrological data from the Physindien experiment), we show that often, the upper ocean eddies strongly perturb these outflows and eject part of PGW or RSW offshore. More specifically, we show that
1) the upper ocean eddies have a deep dynamical signature (profiling floats at 700 or 1000 m depth follow the upper ocean motion)
2) these profiling floats identify RSW or PGW ejected from the coastal currents under the form of fragments, filaments and sometimes small eddies.
3) A seasonal dipolar surface eddy exists near Ras al Hamra (in the Northern Sea of Oman) and induces this process. Deformation maps associated with the dipole motion are calculated and compared with float trajectories and recordings.
4) PGW thus expelled can follow the Iranian - and also the Pakistani - coast and be expelled offshore again by other surface eddies
5) Another dipole located near Ras al Hadd (southern Sea of Oman) in 2011 ejected the PGW current and produced a small lens eddy of PGW offshore.
6) A dipole in the Sea of Oman also ejected filaments of PGW at that period.
Regional primitive equation modeling at very high resolution, and theoretical studies of the surface signature of these small deep fragments, are under way.

Cédric Chavanne
l'Université du Québec à Rimouski

Lecture Notes

Strain-driven submesoscale frontogenesis : what can surface currents tell us about what is happening below?

Detailed observations of a submesoscale front west of Oahu, Hawaii, were obtained in October 2002 from high-frequency radars and satellite radiometers. The surface currents measured by the radars displayed anti-correlated dipoles of vorticity and divergence across the front, indicative of an ageostrophic cross-frontal circulation maintaining along-front thermal wind balance in the presence of background strain induced by a pair of vortices. The coefficient of proportionality between the observed surface vorticity and divergence quantitatively agrees with a semi-geostrophic model of a front confined between two rigid lids in a surface layer of zero potential vorticity. Removing the bottom lid to model an infinitely deep ocean and assuming a constant but non-zero potential vorticity (i.e. the equivalent of the surface quasi-geostrophic model in the semi-geostrophic approximation) predicts only half the observed coefficient of proportionality. This suggests that the front was confined to the surface mixed-layer and decoupled from the ocean interior by a strong pycnocline.

Michael Dunphy
University of Waterloo

Lecture Notes

Focussing and normal mode scattering of the first mode internal tide by mesoscale eddy interaction

The generation of the internal tide (via, for example, barotropic tide-topography interaction) has been studied by many authors, however, the fate of the internal tide (the propagation, interaction with other processes and ultimately its dissipation) is still under investigation. Here I will report on numerical experiments performed using the MITgcm to investigate the interaction of a mode-one internal tide with a barotropic and a baroclinic mode-one mesoscale eddy.

A suite of experiments are conducted varying the eddy size, velocity, and Coriolis parameter. The barotropic cases show hot and cold beams of energy flux, and the baroclinic cases yield the generation of higher mode internal tide beams. An energy budget analysis is performed to measure the scattering of energy between modes, and conversion efficiencies reach 13 percent for the parameters regime considered here.

Daniel Kirshbaum
McGill University

Lecture Notes

Invigoration of cumulus cloud fields by mesoscale ascent

Forced ascent of atmospheric flow by mesoscale features (e.g., mountains, gravity waves, frontal boundaries, etc.) is a common and widely accepted mechanism for the initiation of cumulus convection. Its impact on the subsequent dynamics of developing cumuli, however, has largely been neglected. This is exemplified by entraining/detraining cloud models that form the basis of most convection parameterization schemes, which treat the background flow as static throughout the cloud life cycle. This theoretical framework breaks down when the background flow itself is undergoing rapid modification. In this study, large-eddy simulations of trade-wind cumuli impinging on a island ridge are conducted to investigate the impact of rapid mesoscale ascent on the morphology, dynamics, and microphysics of a mature cumulus field. Despite being trapped beneath a sinking trade-wind inversion, the simulated island clouds are more numerous, vigorous, and liquid-rich than those over the open ocean. This results from two principal mechanisms: (i) the different lapse rates of dry and saturated air parcels, which enhance the horizontal buoyancy gradients of partly-cloudy air when lifted in bulk and (ii) a sharp increase in horizontal cloud size, which reduces the dilution of the buoyant convective cores by the entrainment of environmental air. The increased coverage and precipitation efficiency of the island clouds increases the mean precipitation rate 20-fold relative to that in the upstream flow. The island cloud broadening is favored by the presence of broad water-vapour anomalies within the impinging airstream that are forcibly lifted to saturation, along with turbulent constraints that support wider, less dilute clouds in areas of rapid ascent.

Pascale Lelong
NorthWest Research Associates Seattle Washington USA

Lecture 1 Notes

Lecture 2 Notes

1. A numerical study of lateral dispersion in a field of oceanic internal waves

An extensive field campaign was conducted as part of the ONR-sponsored Scalable Lateral Mixing and Coherent Turbulence Dedicated Research Initiative (aka LatMix) in June 2011 in the Sargasso Sea. One of the campaign objectives was to better understand the processes that govern submesoscale lateral dispersion in the stratified interior in
regions characterized by low ambient background shear and strain.

Numerical simulations of passive dye dispersion in flow conditions consisting of randomly phased internal waves, as described by a Garrett-Munk (GM) spectrum, reproduce the observed effective isopycnal diffusivity. Furthermore, they suggest that shear dispersion by low-frequency waves is likely not an efficient stirring mechanism, nor are submesoscale vortices (vortical motions) created through geostrophic adjustment of three-dimensional turbulence patches. These results point to wave/wave interactions as likely candidates for explaining the observed isopycnal dispersion during LatMix. A parameterization for an eddy isopycnal diffusivity based on background stratification, latitude and GM spectral levels is proposed.

2. Near-inertial waves within an anticyclonic eddy and turbulence in the Mediterranean sea during BOUM experiment (Author: Pascale Bouruet-Aurbertot; Presenter: Pascale Lelong)
One main purpose of BOUM experiment was to give evidence of the possible impact of submesoscale dynamics on biogeochemical cycles. To this aim physical as well as biogeochemical data were collected along a zonal transect through the western and eastern basins. Along this transect 3 day fixed point stations were performed within anticyclonic eddies during which both fine-scale CTD/LADCP profiles and microstructure measurements were collected over the first 500m and the first 100m respectively.

We first focus on the analysis of Cyprus eddy which provides a case study for the characterization of near-inertial wave generation and turbulence. Indeed observations reveal near-inertial oscillations over the whole profile, in the mixed layer, within the eddy and at greater depths. Two mechanisms of generation are discussed: inertial pumping at the base of the mixed layer after a wind event and adjustment of the eddy with possible trapping at the base of the eddy.

The analysis of microstructure measurements revealed a high level of turbulence in the seasonal pycnocline and a moderate level below with energy dissipation mean values of the order of and 10-8 respectively. Fine-scale parameterizations developed to mimic energy dissipation produced by internal wavebreaking were then tested against these direct measurements. Once validated a parameterization has been applied to infer energy dissipation and mixing over the whole data set, thus providing an overview over a latitudinal section of the Mediterranean sea. The results evidence a significant increase of dissipation at the top and base of eddies associated with strong near inertial waves. Vertical turbulent diffusivity is increased both in these regions and in the weakly stratified eddy core.

Co-authors Y. Cuypers, M.P. Lelong, L. Prieur, C. Marec and J.L. Fuda.
Balu Nadiga
Los Alamos National Lab

Some experiments on dissipation of balanced energy in the interior

Ocean circulation is forced at the large scales and the instability of the resulting large-scale circulation gives rise to intermediate-scale eddies. The large-scale flow and the resultant eddies are both, however, in approximate geostrophic balance---a balance between pressure gradients and rotational effects. An important aspect of turbulence in the context of such balanced dynamics is an inverse cascade of energy leading to a trapping of energy in the large and mesoscales; in this setting, viscosity is ineffective in dissipating energy. While dissipation can still occur through interactions of large and mesoscale circulation features with bottom topography through turbulent bottom boundary layer dissipation, its effectiveness is reduced by the baroclinic nature of large scale circulation. Thus a fundamental conundrum of turbulent dynamics in the ocean is as to how the system equilibrates in the presence of continuous large scale forc- ing and an inverse cascade at the intermediate scales.

We consider the role of submesoscales (including inertia gravity wave (IGW) processes) in providing bridging forward cascade pathways that can lead to dissipation of balanced energy through viscosity. A number of recent studies have established the importance of surface-intensified frontogenetic processes in leading to submesoscales and ultimately to dissipation. In this idealized study we consider the role of submesoscales in the interior ocean and the interaction of balanced circulation with ambient imbalance in the interior in providing the forward cascade bridge.
Francis Poulin
University of Waterloo

Spectral characteristics of a turbulent, homogeneous wind-driven gyre flow

For over half a century the scientific community has worked in developing models that idealize the dynamics of wind-driven gyres in the world's oceans. The pioneering works explained that Western Boundary Currents (WBCs) are generated because of a balance between the vorticity induced by the winds and subsequently removed by dissipation. This dissipation of the large scale dynamics is intimately connected to the turbulent processes at smaller scales that are essential to obtain a WBC but at present cannot be described in any self-consistent theory. Moreover, unstable WBCs generate eddies that inject energy into the basin and is therefore analogous to the intermediate forcing scale used in simulations of two-dimensional turbulence. Therefore, this model helps to bridge the gap because classical studies of turbulence and the turbulence that actually occurs in the world's oceans.

Even though the Quasi-Geostrophic (QG) model is limited in its regime of applicability it has often been used to study wind-driven gyres and, to its credit, with great success. QG is an asymptotic limit of the more general Shallow Water (SW) model that is more adept in describing a wider range of length scales. In this work we focus on studying the dynamics of wind-driven gyres in a homogeneous SW model where the small scales have order one Rossby number and therefore ageostrophic dynamics are expected to arise.

We present the results of a series of high-resolution numerical simulations of a homogeneous single wind-driven gyre using both the rigid-lid QG and full gravity SW models. The diagnostics we use to help quantify the evolution of the gyre and the differences between the two models includes the energy spectra and spectral transfers. Fourier analysis is ideal in studies of homogeneous turbulence but it is not evidence that it is the best metric to study wind-driven gyres because of the inherent inhomogeneities associated with the WBC. It is for this reason that we take a novel approach and compute wavelet spectra that allow us to see how the spectra vary with space between the turbulent and laminar regions. For the range of scales that we are able to resolve we find that there is very little difference between the two models, however by looking at Probability Density Functions of the vorticity we see clear distinctions are present.

Rob Scott
Université de Bretagne Occidentale, and CNRS

Lecture Notes

Eddy-modulated, super­inertial turbulence

The horizontal velocity vector of linear, internal gravity waves rotates anticyclonically. Thus rotary spectra allow the decomposition of super-inertial currents into motions consistent and not consistent with internal waves. We explore the importance of the non-wave component, denoting this as "super-inertial turbulence". A striking contrast was found between the Northern Hemisphere and the Southern Hemisphere; in the NH the internal waves dominated with super-inertial turbulence accounting for about 10% to 20% of the super-inertial variability. In contrast, in the Southern Hemisphere, super-inertial turbulence accounted for close to 50% of the super-inertial variability. The monthly internal wave energy was found, unsurprisingly, to be uncorrelated with the monthly mean currents. In contrast, the monthly-mean super-inertial turbulence was significantly correlated with the monthly mean currents.

Eric Skyllingstad
Oregon State University

Simulations of coherent structures in ocean frontal zones and effects on dye dispersion.

Large-eddy simulation cases are presented focusing on the role of fronts in generating coherent structures in the surface mixed layer. This work is motivated by field observations of dye releases made during the LATMIX experiment showing dye patches organized in bands with horizontal scales 10-20 times the mixed layer depth. Processes that could be responsible for these bands include Langmuir circulation, shear generated roll vortices or frontal generated instabilitiies (e.g. symmetric instability). Model results suggest that frontal instabilities can produce strong roll structures that are many times larger than typical Langmuir cells, and that these structures actively subduct dye into the pycnocline. Comparison between cases with and without a frontal system demonstrate that dye patch size increases in the frontal case. Analysis of the frontal cases indicate that conditions are sufficient for symmetric instability, however the coherent structures have along front variations that suggest more complex processes.
Alexandre Stegner
Ecole Polytechnique

Inertial-centrifugal instability of intense anticyclonic vortices : linear stability analysis, laboratory experiments and oceanic observations

We investigated the stability of various meso and submesoscale circular vortices to three dimensional centrifugal-inertial perturbations. The main purpose of this work was to build a stability diagram taking into account the stratification and the dissipation of realistic oceanic eddies.

By means of asymptotic expansion, we first derive for the Rankine vortex a generalized stability limit equation which depends only on three dimensionless parameters: the vortex Rossby number, the Burger number and the Ekman number. This stability equation is more relevant to oceanic vortices than the generalized Rayleigh criterion which is valid only for non-dissipative and non stratified eddies. Indeed, our stability analysis has shown that a strong stratification enhances the impact of dissipation, making the Ekman number a most crucial parameter for the centrifugal-inertial instability. This analysis was extended to other vortices having a parabolic, a conical or a gaussian vorticity profile. We have shown that when using the vortex Rossby number to quantify the vortex intensity instead of the normalized core vorticity (often used for this purpose) the marginal stability curves of the various vortices collapse to a single one. We then build a stability diagram in the Rossby, Burger and Ekman parameter space which is probably valid for a wide range of eddies.

In a second step, we performed large scale laboratory experiments on the Coriolis rotating platform to check the stability analysis. In these experiments a linear salt stratification was set in the upper layer on top of a thick barotropic layer, and a cylinder was towed in the upper layer to produce shallow cyclones and anticyclones of similar size and intensity. Towing speed, cylinder size and stratification, were changed in order to cover a large range of the parameter space, staying in a relatively high horizontal Reynolds number (2000- 7000). We identify a new stability test, the so called gamma test, who detect the signature of the unstable growth of inertial perturbations when there is no complete breakdown of anticyclones. On one hand, we found that some anticyclones remain stable even for very intense negative vorticity values, when the Burger number is large enough. On the other hand, unstable vortices were located close to the marginal stability limit we derived. Hence, this latter appear to be an useful tool to check the three dimensional stability of circular anticyclones to inertial perturbations.
Finally we applied our analysis and the gamma test to estimate the stabie or the unstable evolution of an intense anticyclone in the lee of Oahu island in the Hawaii archipelago .
David Straub
McGill University
Influence of forced near-inertial motion on nearly geostrophic flow in a recirculating zonal channel

We consider a wind-driven primitive equation channel flow with stratification and mean wind forcing chosen to correspond to typical Southern Ocean values. This produces a base state flow having features, such as quasi-zonal jets, that are familiar from beta plane turbulence and from models of the Antarctic Circumpolar Current. To this base state, we then apply an additional, high frequency, forcing designed to excite near-inertial motion. Our focus is on how this addition of high frequency energy influences the low frequency (and nearly geostrophic) part of the flow. In the regime studied, we find that the presence of near-inertial motion serves to decrease the low frequency kinetic energy. Moreover, this reduction is found to be mainly due to a removal of the energy from the barotropic mode. An attempt is also made to relate this "balanced-to-unbalanced" energy transfer to some of our previous work on the three-dimensionalization of turbulent two-dimensional flows.

Mary-Louise Timmermans
Yale University

Scales of horizontal density structure in the surface layer of the Arctic Ocean

Arctic Ocean measurements in the surface layer beneath sea ice are shown to exhibit horizontal density structure on scales of hundreds of kilometers to the order 1 km submesoscale. The observed density fronts are dynamically important in that they are associated with restratification of the surface ocean; restratification is prevalent in wintertime and competes with convective mixing upon buoyancy forcing (e.g., ice growth and brine rejection) and shear-driven mixing when the ice moves relative to the ocean. Frontal structure and estimates of the balanced Richardson number point to the likelihood of dynamical restratification by isopycnal tilt and submesoscale baroclinic instability. It is further shown that similar horizontal density structure is observed in the surface Arctic Ocean in ice-free conditions. Based on the evidence here, it is likely that submesoscale processes play an important role in setting surface-layer properties and lateral density variability in the Arctic Ocean.

Jacques Vanneste
University of Edinburgh

Lecture Notes

A surface-aware projection basis for quasigeostrophic flow

A number of recent studies have demonstrated that altimetric observations of the ocean’s mesoscale eddy field reflect the combined influence of both surface buoyancy anomalies and interior potential vorticity anomalies. The former are associated with surface-trapped modes, with an exponentially-decaying vertical structure, and the latter with the standard baroclinic modes, the oscillating eigenfunctions of the quasigeostrophic potential vorticity stretching operator. In order to assess the relative importance of the two contributions to the signal, one would like to project the observed field onto a set of complete modes that separates the influence of each aspect of the dynamics in a natural way. However, because the surface-trapped modes are not orthogonal to the interior baroclinic modes, any projection contains energetic overlaps.

Here we propose a modal decomposition that results from the simulateous diagonalization of two quadratic forms: the energy and a generalization of potential enstrophy that includes contributions from the surface buoyancy variances. These modes provide an orthonormal basis that represents surface and interior components in a natural way. We compute these modes for a given stratification, and demonstrate their use by projecting out the energy of a set of simulations of mesoscale eddies. (Joint work with K S Smith, Courant Institute.)

Michael Waite
University of Waterloo

Lecture Notes

Potential enstrophy in stratified turbulence

In geophysical flows with strong rotation and stratification, the Ertel potential vorticity is approximately linear in the flow variables. As a result, the integrated squared potential vorticity, or potential enstrophy, is an approximately quadratic invariant, a fact that has important implications for energy transfers between scales in geostrophic turbulence. However, for flows with Rossby numbers O(1) or larger - as in the oceanic sub-mesoscale and atmospheric mesoscale - the assumption of quadratic potential enstrophy becomes questionable. Some recent results have pointed to quadratic potential enstrophy in this regime, but the universality of these findings has not been established. In this talk, direct numerical simulations of stratified turbulence without rotation will be presented. The potential enstrophy will be shown to be approximately quadratic only when the vertical Froude number is small. However, at large Rossby number, small vertical Froude numbers are only expected for small buoyancy Reynolds numbers, i.e. when the vertical scale of the turbulence is set by viscosity. This regime is common in laboratory experiments and in simulations where viscosity (physical, parameterized, or numerical) damps the buoyancy scale, but not in geophysical turbulence. For larger buoyancy Reynolds numbers, the quadratic, cubic, and quartic contributions to the potential enstrophy are all of the same order. These results raise doubts about the applicability of cascade theories based on quadratic potential enstrophy to stratified turbulence in the atmosphere and ocean.

Vladimir Zeitlin
Ecole Normale Superieure/Université P. et M. Curie

Lecture Notes

Instabilities of coupled density fronts and their nonlinear evolution in the two-layer rotating shallow water model. Influence of the lower layer and of the topography

We undertake a detailed analysis of linear stability of geostrophically balanced double density fronts in the framework of the two-layer rotating shallow water model on the f-plane with topography, the latter being represented by an escarpment beneath the fronts. We use the pseudospectral collocation method to identify and quantify different kinds of instabilities resulting from phase-locking and resonances of frontal, Rossby, Poincaré and topographic waves. A swap in the leading long-wave instability from the classical barotropic one, resulting from the resonance of two frontal waves, to a baroclinic one, resulting from the resonance of Rossby and frontal waves, takes place with decreasing depth of the lower layer. Nonlinear development and saturation of these instabilities, and of an instability of topographic origin, resulting from the resonance of frontal and topographic waves, are studied and compared with the help of a new-generation well-balanced finite-volume code for multi-layer rotating shallow water equations. The results of the saturation for different instabilities are shown to produce very different secondary coherent structures. The influence of the topography on these processes is highlighted.

Participants as of June 7, 2013
* Awaiting confirmation

Full Name University/Affiliation
Albanese, Claudio University of Toronto
Allen, Susan University of British Columbia
Badin, Gualtiero University of Hamburg
Bartello, Peter* McGill University
Bouruet-Aubertot, Pascale* Universit'e Pierre et Marie Curie
Carton, Xavier Université de Bretagne Occidentale
Castaing, M. Richard Ecole Polytechnique
Chavanne, Cédric L'Université du Québec à Rimouski
Connaughton, Colm University of Warwick
Cossu, Remo University of Toronto
Craig, Walter McMaster University
Davarpanah Jazi, Shahrzad University of Toronto
Dunphy, Michael University of Waterloo
Garcia, Carlos McMaster University
Henry, Legena University of the West Indies
Irwin, Rob University of Waterloo
Jackson, Ken University of Toronto
Kirshbaum, Daniel McGill University
Kuksin, Sergei CNRS and Universite de Paris 7
Lamb, Kevin University of Waterloo
Lannes, David* Ecole Normale Superieure - Paris
Lelong, Marie-Pascale Northwest Research Associates
Liu, Guoqiang Bedford Institute of Oceanography
Magcalas, Moriah Redeemer University College
Nadiga, Balasubramanya Los Alamos National Lab
Nazarenko, Sergey University of Warwick
Pelinovsky, Efim Russian Academy of Sciences
Poulin, Francis University of Waterloo
Rakhimov, Shokhrux McMaster University
Restrepo, Juan* University of Arizona
Robitaille, Julien UQAR-ISMER
Salehipour, Hesam University of Toronto
Sartori, Matthew University of Waterloo
Scott, Robert Université de Bretagne Occidentale, and CNRS
Skyllingstad, Eric Oregon State University
Spyksma, Kyle Redeemer University College
Stegner, Alexandre Ecole Polytechnique
Storer, Ben University of Waterloo
Straub, David McGill University
Timmermans, Mary-Louise Yale University
Vanneste, Jacques University of Edinburgh
Waite, Michael University of Waterloo
Yawney, John University of Waterloo
Zeitlin, Vladimir Ecole Normale Supérieure



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