


Applied Partial Differential Equations
Org: Anne Bourlioux (Montreal), Reinhard Illner (Victoria) and Boualem Khouider (Victoria) [PDF]
 JOE BIELLO, University of California, Department of Mathematics, Kerr
Hall, One Shields Ave, Davis, CA 95616, USA
Rossby wave interaction between the tropics and midlatitudes:
a novel asymptotic theory and solitary waves
[PDF] 
Simplified asymptotic equations are developed for the nonlinear
interaction of long wavelength equatorial Rossby waves and barotropic
Rossby waves with a significant midlatitude projection in the presence
of suitable horizontally and vertically sheared zonal mean flows. The
simplified equations allow for nonlinear energy exchange between the
barotropic Rossby waves and the baroclinic equatorial waves for
nonzero zonal mean vertical shear through wavewave interactions.
Idealized examples in the model demonstrate that midlatitude Rossby
wave trains in a baroclinic mean shear can transfer their energy to
localized equatorially trapped baroclinic Rossby waves through a
nonlinear "westerly wind burst" mechanism. Conversely,
equatorially trapped baroclinic Rossby wave trains in the idealized
model can transfer substantial energy to the midlatitude barotropic
Rossby waves. From the viewpoint of applied mathematics, the
asymptotic equations derived here have several novel features. In
particular, they admit analytic solitary wave solutions which
correspond to interesting localized waves in the equatorial
troposphere.
 ANNE BOURLIOUX, Université de Montréal
Modelling error analysis and estimation for unsteady
flamelets
[PDF] 
Flamelet models are asymptotic solutions of the nonlinear PDEs used to
simulate turbulent flames. They capitalize on the scale separation
between the flame thickness (assumed to be thin) and the other flow
scales by replacing the distorted, unsteady flame front by a
onedimensional, steady object. This simplifies the computations
tremendously. An idealized setup is used to systematically
investigate this type of strategy, using a combination of asymptotic
analysis and numerical simulations.
The class of unsteady flows under consideration consists of a shear
with a timemodulated crossflow. It is known to lead to very
interesting intermittent mixing regimes for nonreactive problems. In
the present reactive case, it has the property that, to leading order,
the flame indeed matches a flamelet structure, but with a highly
nontrivial (local and unsteady) dissipation, which is the key fitting
parameter of the model. Different strategies to predict this
parameter are examined and their regimes of validity are identified.
One goal is to design an automatic procedure that would select the
appropriate model as the computation proceeds. This requires an
efficient modelling error estimation procedure. Preliminary efforts
in that direction will also be discussed.
Joint work with Oleg Volkov.
 ALEXEI CHEVIAKOV, University of British Columbia, Vancouver, BC
Framework for nonlocallyrelated PDE systems and nonlocal
symmetries: Algorithmic approach
[PDF] 
For a given PDE system, one can construct extended hierarchies
("trees") of nonlocallyrelated PDE systems. Each system in an
extended tree is equivalent, in the sense that the solution set for
any system in a tree can be found from the solution set for any other
system in the tree. Due to the equivalence of solution sets, any
coordinateindependent method of analysis (qualitative, numerical,
perturbation, etc.) can be applied to any system within the
tree, and may yield simpler computations and new results that cannot
be obtained when the method is directly applied to the given system.
Nonlocal symmetries and new local and nonlocal conservation laws for a
given PDE system can arise from any system in its extended tree.
The concept of useful conservation laws plays an essential role in the
construction of an extended tree. Useful conservation laws yield
potential variables and equivalent nonlocallyrelated potential
systems and subsystems for any given system.
We construct extended trees for the systems of Planar Gas Dynamics and
Nonlinear Telegraph equations. Using the described framework, we
demonstrate a direct relation between Eulerian and Lagrangian
descriptions of gas dynamics, and find new families of conservation
laws and new nonlocal symmetries.
The presented research was done in collaboration with George Bluman
(UBC).
 AMIK ST CYR, National Center for Atmospheric Research, 1850 Table Mesa
Drive, Boulder, CO 80305
Optimized Schwarz methods for highorder spectral elements
[PDF] 
In this presentation, it is shown how a small modification of the RAS,
MS and ASaug preconditioners at the algebraic level, motivated by
optimized Schwarz methods defined at the continuous level, leads to a
significant reduction in the iteration count of the iterative Krylov
solver. Numerical experiments on the modified Helmholtz equation
using a model problem and a next generation spectral element general
circulation model on the sphere, illustrate the effectiveness of this
new approach. Experimentally, it is observed that the best condition
number attainable in 2D (without coarse solver), for a nonoverlapping
decomposition, is ÖN where N is the order of the polynomial
basis employed. The performance of the method on the Blue gene/L
supercomputer is investigated.
Collaborators: Martin J. Gander and Stephen J. Thomas.
 REINHARD ILLNER, University of Victoria
Entropy methods for degenerate driftdiffusion equations
[PDF] 
I will present selected results from recent joint work with Jean
Dolbeault, Philippe Bartier and Michal Kowalczyk. Linear
driftdiffusion equations with degenerate or timedependent
coefficients arise in various applications, for example in traffic
flow models or in flashing ratchet models. In more difficult (yet
practically relevant) cases the coefficients may be coupled to moments
of the dependent variable, producing a nonlinear problem.
Entropyentropy production estimates offer natural ways to describe
the asymptotic behaviour of solutions to such problems, and I will
show some of the relevant estimates.
 NICOLAS KEVLAHAN, McMaster University
Multiscale spacetime adaptive simulation of 2D turbulence
[PDF] 
A spacetime adaptive wavelet collocation method is developed to
efficiently simulate twodimensional incompressible turbulence. This
new DNS technique takes advantage of the spatial and temporal
intermittency of turbulence to approximate the solution in the
spacetime domain using an adaptive collocation wavelet method. Both
spatial and temporal resolution are adapted locally to solve the
vorticity equation to the desired tolerance. Note that the global
time integration error is controlled: this is not possible using
conventional time marching methods. We will present results for the
merging of identical vortices at Re = 1000, and for decaying
twodimensional turbulence. We find that the total number of active
spacetime degrees of freedom is significantly smaller than in a
conventional dynamically adaptive time marching method. We also
expect to present an estimate of the number of spacetime degrees of
freedom for decaying 2D turbulence as a function of Reynolds number.
 BOUALEM KHOUIDER, University of Victoria, 3800 Finnerty Road, Victoria, BC
Multicloud parametrizations for convectively coupled tropical
waves
[PDF] 
The tropical large scale circulation has a significant impact on our
weather and climate through various meteorological disturbances
originating near the equator such as El Niño, the MaddenJulian
Oscillation, and various convectively coupled tropical waves. Such
disturbances are often a result of organized tropical convection over
a large range of scales; from the single clouds of 1 to 10 km to cloud
clusters and superclusters of a few hundred to a few thousand
kilometers. However, today's general circulation models (GCMs)
perform very poorly in predicting/representing these phenomena because
on the one hand they occur at the subgrid scale for the GCMs and on
the other hand the underlying physics are still not completely
understoodimpossible to parametrize accurately.
Recent observational analysis reveals the central role of three
multicloud types, congestus, stratiform, and deep convective cumulus
clouds, in the dynamics of large scale convectively coupled Kelvin
waves, westward propagating twoday waves, and the MaddenJulian
oscillation. We present in this talk a systematic model convective
parametrization highlighting the dynamic role of the three cloud types
through two baroclinic modes of vertical structure: a deep convective
heating mode and a second mode with low level heating and cooling
corresponding respectively to congestus and stratiform clouds.
Joint with A. J. Majda (NYU).
 HORST LANGE, Mathematisches Institut, Universität Köln, Weyertal 8690,
D50931 Köln, Germany
On the controllabilty of nonlinear Schrödinger equations
[PDF] 
We consider the controllabilty of nonlinear Schrödinger equations in
two specific cases, namely the nonlinear Hartree equation (of quantum
chemistry), and the GrossPitaevskii equation (in the theory of
BoseEinstein condensation). We study the mathematical structure of
the sets of reachable and nonreachable states, and show, e.g.,
that the set of nonreachable states for the Hartree equation is
"fat" in the Baire categorical sense, and dense in state space,
whereas the set of reachable states is "meagre". In the
GrossPitaevskii case the nonreachable states form a finite
dimensional manifold.
Joint work with Reinhard Illner, Victoria, and Holger Teismann, Acadia.
 ADAM MONAHAN, School of Earth and Ocean Sciences, University of Victoria
Covariance Structure of a Fluctuating Midlatitude Jet
[PDF] 
Principal Component Analysis (PCA) is a standard technique used in
ocean/atmosphere physics to look for structure in large multivariate
datasets; mathematically, PCA involves finding the eigenstructure of
the covariance matrix. Individual PCA basis functions are often
assumed to represent distinct physical "modes" of variability. In
this talk, we will develop analytic expressions for the covariance
structure of an idealised midlatitude jet that can vary in strength,
width, and position. Through a systematic perturbation analysis, we
can read off the leading few eigenvectors (PCA modes) of the
covariance matrix.
This analysis demonstrates that even in this idealised system, many of
the assumptions commonly made in interpreting PCA structures are
false. In particular:
(1) the PCA time series are uncorrelated, but not independent,
(2) individual PCA "modes" do not represent individual
physical processes, and
(3) PCA structures arising due to individual processes alone
can be mixed, or "hybridised", when these processes occur
simultaneously.
 ADAM OBERMAN, SFU
Numerical approximation of first and second order nonlinear
elliptic PDEs
[PDF] 
The theory of viscosity solutions gives powerful existence, uniqueness
and stability results for first and second order degenerate elliptic
equations. The approximation theory developed by Barles and Sougandis
in the early nineties gave conditions for the convergence of numerical
schemes.
Building on this work, we develop convergent schemes for nonlinear
second order equations, including: infinity laplacian, motion by mean
curvature, the MongeAmpere equation.
We'll also discuss adaptive schemes on unstructed grids for first
order equations. A motivating example is the highdimensional control
problem of airplane flight.
 VLADISLAV PANFEROV, McMaster University
Global regularity of onedimensional solutions of the
Boltzmann equation
[PDF] 
For the Boltzmann equation the setting of a narrow shock tube implies
that solutions depend only on one spatial coordinate, while having a
threedimensional velocity dependence. We study the propagation of
some regularity estimates, such as supnorms of the macroscopic
density, for the corresponding solutions of the Boltzmann equation.
Using the methods based on the relative entropy control and on a
certain nonlinear functional introduced by Bony and Cercignani, we
establish the global in time existence of regular solutions for some
model cases of particle interactions.
Joint work with A. Biryuk and W. Craig.
 OLIVIER PAULUIS, New York University
Toward the end of cumulus parameterization
[PDF] 
The current generation of General Circulation Models can simulate the
global atmosphere with a resolution of the order of 50 km. Such a
resolution is insufficient to explicitly simulate the deep convective
motions that are responsible for the bulk of the vertical energy
transport in the atmosphere. To compensate for this limitation,
climate models have used various cumulus parameterizations to account
for the effects of convective motions on the atmospheric temperature,
humidity and cloud distribution. However, due to the continuous
increase in computational power, the next generation of global models
might be run at a sufficiently fine resolution as to make the use of
cumulus parameterization unnecessary.
This paper investigates the impacts of horizontal resolution on the
statistical behavior of convection. An idealized radiativeconvective
equilibrium is simulated for model resolutions ranging between 2 and
50 km. The simulations are compared based upon the analysis of the
mean state, of the energy and water vapor transport, and of the
probability distributions functions for various quantities. It is
found that, despite some bias in temperature and humidity, coarse
resolution simulations are able to reproduce reasonably well the
statistical properties of deep convective towers. This is
particularly apparent in the cloud ice and vertical velocity
distributions that exhibit a very robust behavior at resolution up to
16 km.
A theoretical scaling for the vertical velocity as function of the
grid resolution is derived, based upon the behavior of an idealized
air bubbles. The vertical velocity of an ascending bubble is
determined by its aspect ratio, with a wide, flat parcel rising at a
much slower pace than a narrow one. This theoretical scaling law can
explain the behavior of the numerical simulations, and be used to
renormalize the probability distribution functions for vertical
velocity.
 FRANCIS POULIN, University of Waterloo
Turbulent selfdiffusion in isopycnal coordinates
[PDF] 
A recent paper by Dukowicz and Smith (1997), henceforth referred to as
[DS97], extends the classical theory of turbulent transport of a
tracer particle (Morin and Yaglom, 1987) to encompass the problem of
selfdiffusion in stratified mesoscale oceanic turbulence, thereby
shedding light onto the mathematical status and physical meaning of
the recent parametrization of Gent and McWilliams (1990). This is
interesting and important in view of the fact that the theory of
geostrophic turbulence is still in its infancy.
The stated objective of [DS97] is to develop the stochastic theory of
turbulent diffusion from the standard FokkerPlanck equation (Morin
and Yaglom, 1987; Gardiner, 2004) in such a way that it also applies
to compressible flow. The reason why this is deemed necessary is that
when the classical, incompressible Boussinesq equations are expressed
in isopycnal coordinates the velocity field ceases to be solenoidal.
We will show that the argument presented in [DS97] is incorrect,
although their main result can, fortunately, be salvaged.
 STEVEN RUUTH, Simon Fraser University, Burnaby, BC V5A 1S6
Threshold Dynamics for Willmore Flow
[PDF] 
Many important models of image processing and computer vision involve
curvaturedependent functionals. The minimization of these
functionals can involve the solution of fourth order geometric PDEs.
The numerical solution of such PDEs with standard methods can be very
costly.
Recently, Grzibovskis and Heintz (2005) proposed a threshold dynamics
algorithm that approximates the gradient flow for an important
curvature dependent functional known as the Willmore energy. This
energy consists of the integral of the square of a surface's mean
curvature over that surface. Furthermore, it constitutes an essential
part of certain variational image models for segmentation with depth,
disocclusion, and image inpainting. This talk discusses our recent
work on practical threshold dynamics algorithms for Willmore Flow and
the application of these algorithms to higher order models of image
processing and computer vision.
This is joint work with Selim Esedoglu and Richard Tsai.
 JOHN SCINOCCA, CCCma, MSC, University of Victoria
Wave Forcing, Parameterization, and the Breakdown of Newton's
Third Law
[PDF] 
Current parameterizations of gravitywave drag (GWD) in general
circulation models (GCMs) of the Earth's atmosphere explicitly
conserve wave pseudomomentum flux and, therefore, satisfy Newton's
Third Law. This approach assumes a basicstate flow that is
horizontally uniform and it allows a direct connection between wave
dissipation and waveinduced forcing of the flow. When the horizontal
structure of the basicstate flow is no longer uniform this approach
fails. In this instance the more fundamental principle of wave action
conservation must be invoked. In this more general framework one can
no longer associate all waveinduced forcing with wave dissipation.
Newton's Third Law may be violated and when it is, the basicstate flow
will be subjected to wave induced forces arising from wave dynamics
that are conservative rather than dissipative in nature (Buhler and
McIntyre 2005). In this study we reformulate a current
parameterization of orographic GWD (Scinocca and McFarlane 2000) to
allow horizontally nonuniform flow and to employ wave action flux,
rather than pseudomomentum flux, as its primary conserved variable.
The impact of this new formulation is investigated by offline
calculations and fully interactive GCM simulations.
 HOLGER TEISMANN, Acadia University, Wolfville, NS
Bilinear control of Schrödinger equations with confining
potentials
[PDF] 
We will discuss the control problem for linear and nonlinear
Schrödinger equations with confining potentials, where the controls
are given by applying spatially homogeneous fields. For quadratic
potentials it can be shown that the equation is not controllable; the
manifold of reachable states is finitedimensional. On the other
hand, K. Beauchard (2005) recently proved that the (linear)
Schrödinger equation with an infinite square well (particle in a
box) is controllable in the vicinity of the ground state. We will
discuss some open problems and conjectures deriving from these observations.
 VITALI VOUGALTER, University of Toronto, Department of Mathematics,
40 St. George Street, Toronto, ON M5S 2E4
Eigenvalues of zero energy in the linearized NLS problem
[PDF] 
We study a pair of neutrally stable eigenvalues of zero energy in the
linearized NLS equation. We prove that the pair of isolated
eigenvalues of geometric multiplicity two and algebraic multiplicity
2N is associated with 2P negative eigenvalues of the energy
operator, where P=N/2 if N is even and P=(N1)/2 or P=(N+1)/2
if N is odd. When the potential of the linearized NLS problem is
perturbed with a parameter continuation, we compute the exact number
of unstable eigenvalues that bifurcate from the neutrally stable
eigenvalues of zero energy.

