
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
20th
ANNIVERSARY
YEAR

Workshop
on Submesoscale Ocean Processes
June 1114, 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 
Onsite Registration 
9:30  9:45 
Welcome and Introduction 
9:45  10:25 
Cédric Chavanne, l'Université
du Québec à Rimouski
Straindriven submesoscale frontogenesis :
what can surface currents tell us about what is happening below? 
10:2510:45 
Coffee
Break 
10:4511:25 
Michael Waite, University
of Waterloo
Potential enstrophy in stratified turbulence 
11:3012:10 
Vladimir Zeitlin, Ecole
Normale Supérieure
Instabilities of coupled density fronts and their
nonlinear evolution in the twolayer rotating shallow water model. Influence
of the lower layer and of the topography 
12:1014:20 
Lunch Break 
14:2015:00 
Francis Poulin, University
of Waterloo
Spectral characteristics of a turbulent, homogeneous
winddriven gyre flow 
15:1015:50 
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 
9:009:40 
Pascale Lelong, NorthWest
Research Associates
A numerical study of lateral dispersion in a field
of oceanic internal waves 
9:4510:25 
Michael Dunphy, University
of Waterloo
Focussing and normal mode scattering of the first
mode internal tide by mesoscale eddy interaction 
10:2510:45 
Coffee Break 
10:4511:25 
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

11:3012:10 
Peter Bartello, McGill University
From Quasigeostrophic to stratified turbulence

12:1014:20 
Lunch Break 
14:2015:00 
Daniel Kirschbaum, McGill
University
Invigoration of cumulus cloud fields by mesoscale
ascent 
Thursday,
June 13
Stewart
Library, Fields Institute 
9:009:40 
Rob Scott, Université de
Bretagne Occidentale, and CNRS
Eddymodulated, superinertial turbulence

9:4510:25 
MaryLouise Timmermans,
Yale University
Scales of horizontal density structure in the
surface layer of the Arctic Ocean 
10:2510:45 
Coffee Break 
10:4511:25 
David Straub, McGill University
Influence of forced nearinertial motion on nearly
geostrophic flow in a recirculating zonal channel 
11:3012:10 
Eric Skyllingstad, Oregon
State University
Simulations of coherent structures in ocean
frontal zones and effects on dye dispersion 
12:1014:20 
Lunch Break 
14:2015:00 
Gualtiero Badin, University
of Hamburg
Toward outofbalance surface dynamics in the ocean

Friday,
June 14
Stewart
Library, Fields Institute 
9:009:40 
Pascale Lelong, NorthWest
Research Associates
Nearinertial waves within an anticyclonic eddy and
turbulence in the Mediterranean sea during BOUM experiment 
9:4510:25 
Susan Allen, University
of British Columbia
Extending the Validity of a OneDimensional Coupled
Biophysical Model by Parametrization 
10:2510:45 
Coffee Break 
10:4511:25 
Jacques Vanneste, University
of Edinburgh
A surfaceaware projection basis for quasigeostrophic
flow 
11:3012:10 
Alexandre Stegner, Ecole
Polytechnique
Inertialcentrifugal 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 OneDimensional Coupled Biophysical
Model by Parametrization
Processes, such as those at the submesoscale, 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 semienclosed, 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 mixinglayer 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 onedimensional 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 prototype
for parametrization of submesoscale processes into coarse resolution
models.

Gualtiero Badin
University of Hamburg 
Toward outofbalance surface dynamics in the ocean
The surface quasigeostrophic approximation is rewritten
in an oceanic context using the twodimensional semigeostrophic approximation.
The new formulation allows to take into account the presence of outofbalance
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 largescale 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 smallscale
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 smallscale 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

Straindriven 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 highfrequency radars and satellite
radiometers. The surface currents measured by the radars displayed
anticorrelated dipoles of vorticity and divergence across the front,
indicative of an ageostrophic crossfrontal circulation maintaining
alongfront 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 semigeostrophic 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 nonzero potential vorticity (i.e. the equivalent of the surface
quasigeostrophic model in the semigeostrophic approximation) predicts
only half the observed coefficient of proportionality. This suggests
that the front was confined to the surface mixedlayer 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
tidetopography 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 modeone internal tide with a barotropic
and a baroclinic modeone 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, largeeddy
simulations of tradewind 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 tradewind inversion, the simulated island
clouds are more numerous, vigorous, and liquidrich 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 partlycloudy 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 20fold relative to that
in the upstream flow. The island cloud broadening is favored by the
presence of broad watervapour 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 ONRsponsored
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 GarrettMunk
(GM) spectrum, reproduce the observed effective isopycnal diffusivity.
Furthermore, they suggest that shear dispersion by lowfrequency waves
is likely not an efficient stirring mechanism, nor are submesoscale
vortices (vortical motions) created through geostrophic adjustment
of threedimensional 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. Nearinertial waves within an anticyclonic eddy and turbulence in
the Mediterranean sea during BOUM experiment (Author: Pascale BouruetAurbertot;
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 finescale 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 nearinertial wave generation and
turbulence. Indeed observations reveal nearinertial 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 106W.kg1 and 108 W.kg1
respectively. Finescale 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.
Coauthors 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 largescale circulation gives rise to intermediatescale
eddies. The largescale flow and the resultant eddies are both, however,
in approximate geostrophic balancea 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 surfaceintensified
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 winddriven
gyre flow
For over half a century the scientific community has worked
in developing models that idealize the dynamics of winddriven 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 selfconsistent 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 twodimensional 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 QuasiGeostrophic (QG) model is limited in its regime
of applicability it has often been used to study winddriven 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 winddriven 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 highresolution numerical simulations
of a homogeneous single winddriven gyre using both the rigidlid
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 winddriven 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

Eddymodulated, superinertial turbulence
The horizontal velocity vector of linear, internal gravity waves
rotates anticyclonically. Thus rotary spectra allow the decomposition
of superinertial currents into motions consistent and not consistent
with internal waves. We explore the importance of the nonwave component,
denoting this as "superinertial turbulence". A striking
contrast was found between the Northern Hemisphere and the Southern
Hemisphere; in the NH the internal waves dominated with superinertial
turbulence accounting for about 10% to 20% of the superinertial variability.
In contrast, in the Southern Hemisphere, superinertial turbulence
accounted for close to 50% of the superinertial variability. The
monthly internal wave energy was found, unsurprisingly, to be uncorrelated
with the monthly mean currents. In contrast, the monthlymean superinertial
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.
Largeeddy 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 1020 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 
Inertialcentrifugal 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 centrifugalinertial 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 nondissipative
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 centrifugalinertial
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 nearinertial
motion on nearly geostrophic flow in a recirculating zonal channel
We consider a winddriven 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 quasizonal
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 nearinertial
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 nearinertial
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 "balancedtounbalanced"
energy transfer to some of our previous work on the threedimensionalization
of turbulent twodimensional flows.

MaryLouise 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 sheardriven 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 icefree conditions. Based on the evidence
here, it is likely that submesoscale processes play an important role
in setting surfacelayer properties and lateral density variability
in the Arctic Ocean.

Jacques Vanneste
University of Edinburgh
Lecture Notes

A surfaceaware 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 surfacetrapped modes, with
an exponentiallydecaying 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 surfacetrapped 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 submesoscale 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 twolayer 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 twolayer rotating
shallow water model on the fplane 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 phaselocking and resonances of frontal, Rossby, Poincaré
and topographic waves. A swap in the leading longwave 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 newgeneration wellbalanced finitevolume code for multilayer
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 
BouruetAubertot, 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, MariePascale 
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 
UQARISMER 
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, MaryLouise 
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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