This initial lecture is devoted
to an exposition of the fundamental principles
of open-channel flow. Conservation of mass and
momentum is stated and then exploited to establish
the Bernoulli principle, which is central tohydraulics.
The lecture concludes with a discussion of the
effects of bottom friction and ends with the
Manning formula for equilibrated flow down a
Lecture 2. Shear turbulence.
Friction in a river flow is
caused by turbulence induced by shear along
the bottom. This lecture derives the dynamics
that lead to the quadratic friction law. Along
the way, the concepts of mixing length, von
Karman constant and logarithmic velocity profile
are reviewed. The lecture ends with a novel
way of parameterizing shear turbulence by means
of a non-local equation.
Lecture 3. Types of river flows.
Equations for non-equilibrated
channel flow, either gradually or rapidly varying,
are established and solved. It is shown how
nonlinearities in the equation account for the
existence of multiple solutions. Sub and super-critical
flows and the concept of control are elucidated.
The lecture concludes with the theory for hydraulic
Lecture 4. Internal waves in lakes.
Internal waves in a thermally
stratified lake propagate both vertically and
horizontally, reflecting both sideways against
the sides and vertically against the bottom
and surface. In the absence of friction, multiple
reflections can cause resonance. This lecture
is devoted to exploring the peculiar mathematical
properties of resonant internal waves in thermally
The Saint-Venant system for shallow water. Derivation
from Navier-Stokes and numerical solution.
The shallow water description through
the Saint-Venant system is usual for many applications
(rivers flow, tidal waves, oceans). This is an hyperbolic
system, relatively simple but that contains source
terms describing the bottom topography and friction,
and thus can undergo shock waves (bores, dam break).
Although Saint-Venant derived it from elementary
principle in 1871, one understands only from less
than 10 years how to derive it including the right
viscosity and thus the right entropy conditions
for shock waves.
The numerical solution of such
system, including the shock waves solutions, has
to be adapted to the balance laws that it expresses.
Classical finite volume schemes, very stable, give
a low accuracy because they do not preserve the
steady state of a lake at rest. This question has
been considered by many authors who modify the most
In this course, we address these questions on the
Lecture 1. Derivation from Navier-Stokes system
(correction to the friction term, viscosity terms,
Lecture 2. Granular materials and non flat topography
(avalanches and landslides).
Lecture 3. Hyperbolic systems, finite volumes
method: an Introduction.
Lecture 4. Well balanced schemes (generalities,
kinetic approach, real datas).
Avalanches: models and mathematical results.
Avalanches, landslides and debris
flows are devastatingly powerful natural phenomena
that are far too little understood. These granular
matters are mixtures of solid particles and of an
interstitial fluid (e.g. water or air) and are easily
modelled on the microscopic level by the laws of
mechanics. On mesoscopic and macroscopic levels
the different scales of the influence of the particles,
of the fluid and of their interaction lead to various
models of avalanching flows.
We discuss several models of granular
materials characterized by height only or by height
and momentum, discuss the existence of similarity
solutions, existence of arbitrary solutions and
the phenomenon of particle segregation. The main
part concerns the Savage--Hutter equations modelling
dense flow avalanches on a curved bed. This model
defines a highly nonlinear system of conservation
laws with source term for which the existence of
weak entropy solutions is shown.
Roughness-induced effects on geophysical systems.
We present some mathematical results related to small-scale roughness in
geophysics. Special attention is paid to two classical models: the rotating
fluids system, and the quasi-geostrophic equations. For such models, we
rigorously derive reduced sytems that take into account roughness-induced
effects. We then give some physical insight on such reduced systems.
Mathematical and numerical analysis of the primitive
equations in oceanography.
After some simplifications (using
"the rigid lid" hypothesis and assuming
that the pressure is hydrostatic), the 3D Navier-Stokes
equations lead to the so-called "primitive
equations" in which the effects of the Coriolis
forces can also be considered. These equations constitute
a general mathematical model in the field of geophysical
fluids. In particular, they describe the water circulation
in lakes and oceans.
In this lecture, we will present
recent mathematical results concerning this model,
both from the analytical point of view (existence,
uniqueness, regularity of solutions) and from the
numerical point of view (stability, convergence,
error estimates). Moreover, we will perform an asymptotic
analysis in order to make a rigorous justification
of the model as a limit of the 3D Navier-Stokes
Adjustment of the global thermohaline circulation
to local forcing anomalies.
There has been much recent interest
in the ocean thermohaline circulation, its stability
and its role in climate. It is well established
that the thermohaline circulation plays an important
role in transporting heat northward in the Atlantic,
and that changes in atmospheric forcing over the
North Atlantic can result in significant thermohaline
circulation anomalies in the North Atlantic basin.
Less clear, however, is the effect on the remainder
of the oceans. A theory is developed for the propagation
of thermohaline circulation anomalies, both in an
idealised basin and in the global ocean, as a function
of latitude and frequency. Our key findings are
that the equator acts as a low pass filter to circulation
anomalies, confining variability in thermohaline
circulation on decadal and shorter time-scales to
the hemispheric basin in which it is generated.
Implications for monitoring of the thermohaline
circulation will be discussed.
The future of climate and the impact
of human activities on the climate system became
recently one of the major concerns of our society.
For several years now, the general public has been
aware of the possibility of a global warming due
to human activities, with unpredictable consequences.
In order to predict the future of climate we need
to use complex climate computational models. These
sophisticated climate models are similar to the
ones used for daily weather prediction and are based
on balance equations (pdes) for momentum, temperature
and water components (vapour, liquid and ice) that
are numerically discretised in a 3D grid over the
However, and although rather complex,
these models are still quite inaccurate when representing
clouds, turbulence and the interaction between the
ocean and the atmosphere. The major problem is that
all these processes can occur in a variety of scales,
from the planetary scale to very small scales that
cannot be represented explicitly in any atmospheric
model. A great challenge in climate and atmospheric
modeling, these days, is in how to improve the representation
of these sub-grid scale physical processes: the
Four issues in particular will
be reviewed in this presentation: (i) how to represent
the sub-grid scale turbulent motions; (ii) how to
represent clouds and the physical processes (e.g.
rain, snow) associated with them; (iii) how to integrate
numerically these equations in an efficient way
and (iv) how to represent these sub-grid scale processes
in predictability studies. A substantial part of
this type of research involves a blend of different
areas of applied mathematics bringing ideas from
numerical analysis together with statistical and