School

Peter Constantin (University of Chicago, USA)

Title: Turbulent geophysical flows and transport in rotating fluids. Summary: Lecture 1. Bounds on the total dissipation for rotating fluids: shear forcing. Lecture 2. Bounds for the total dissipation for rotating fluids: penetrative forcing. Lecture 3. The Littlewood-Paley spectrum: Navier-Stokes equations. Lecture 4. The Littlewood-Paley spectrum: quasi-geostrophic equations.

Benoît Cushman-Roisin (Dartmouth College, USA)

Title: Hydrodynamics of rivers and lakes. Summary: Lecture 1. Review of hydraulics. 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 slope. 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 jumps. 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 stratified lakes.

Emmanuel Grenier (ENS-Lyon, France)

Title: Rotating fluids and associated boundary layers. Summary: Lecture 1. Wave propagation -- Rossby waves and acoustic waves, high rotation speed limit, incompressible limit. Lectures 2-3. Boundary layers -- Ekman, Hartman, Munk and Stommel layers; derivation, stability and instability. Lecture 4. Application to geophysical flow -- Derivation of wind driven, quasigeostrophic, planetary equations.

Joseph Pedlosky (Woods Hole Oceanographic Institution, USA)

Title: Elements of geophysical fluid dynamics. Summary: Lecture 1. A review of the homogeneous models of the ocean circulation, western boundary currents and the relation to circulation strength. Lecture 2. The theory of the thermocline. The abyssal circulation. Lecture 3. Elements of instability theory in geophysical fluid dynamics. Lecture 4. New problems in instability theory.

Benoît Perthame (ENS-Paris, France)

Title: The Saint-Venant system for shallow water. Derivation from Navier-Stokes and numerical solution. Summary: 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 classical solvers. In this course, we address these questions on the Saint-Venant system. Lecture 1. Derivation from Navier-Stokes system (correction to the friction term, viscosity terms, Boussinesq coefficient). 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).

Workshop

Reinhard Farwig (TU Darmstadt, Germany)

Title: Avalanches: models and mathematical results. Abstract: 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.

David Gérard-Varet (École Normale Supérieure de Lyon, France)

Title: Roughness-induced effects on geophysical systems. Abstract: 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.

Francisco Guillén-González (Universidad de Sevilha, Spain)

Title: Mathematical and numerical analysis of the primitive equations in oceanography. Abstract: 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 equations. Several open problems will also be pointed out.

David Marshall (University of Reading, UK)

Title: Adjustment of the global thermohaline circulation to local forcing anomalies. Abstract: 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.

João Teixeira (UCAR/NRL, USA)

Title: Turbulence, clouds and climate models. Abstract: 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 entire globe. 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 parameterization problem. 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 stochastic concepts.