Cargese GEOMIX Summer School

19th August - 1st September 2001

Lecture Programme

Week 1

Week 2

Lecture Details

Wiggins Mathematics of chaotic advection (5h)

Lecture 1 (1.5h)
Background: the dynamical systems point of view of transport; brief case studies from theory, experiment, and computation; introduction of dynamical systems terminology; practical issues-dynamical systems defined as data sets, finite versus infinite time.

Lecture 2 (1h)
hyperbolic trajectories; stable and unstable manifolds; lobe dynamics.

Lecture 3 (1h)
KAM theorems and chaos-implications for transport

Lecture 4 (1.5h)
Three dimensionality; Lagrangian versus Eulerian transport.

Plumb Atmospheric observations, modelling and theory (5h)

Lecture 1 (1.5h)
Atmospheric structure and dynamics (overview): Convection, Rossby waves, Baroclinic waves.

Lecture 2 (1h)
Stratospheric dynamics (details; theory and obs): Rossby wave breaking/surf zone/barriers, Gravity waves, Dynamics of mean circulation, QBO, Tropopause. Stratospheric transport, Chaotic advection in the surf zone, Barrier permeability, Formation and fate of filaments.

Lecture 3 (1.5h)
Theory of atmospheric transport: Constraints on transport, Equilibrium tracer structures, Age. Theory for rapid isentropic mixing: Shear dispersion, Equilibrium slopes, Tracer-tracer relationships; joint PDFs, Uses and limitations of tracer-tracer relationships.

Lecture 4 (1h)
Tropospheric transport: Surface zone, Upper troposphere; tropopause folds, Hadley circulation, Transport time scales. Atmospheric transport modelling: Lagrangian modelling (e.g. parcels, CA, RDF), Stratospheric and tropospheric CTMs.

Falkovich Particles in turbulent flows (5h)

Lecture 1 (1h)
Modelling transport and mixing in turbulent flows - introduction and motivation.

Lecture 2 (1.5h)
1- and 2-Particle Dispersion.

Lecture 3 (1h)
Multiparticle dynamics and statistical conservation laws.

Lecture 4 (1.5h)
Inertial particles and cloud physics.

Ricard/Schmalzl Geological data, geophysics and modelling of the mantle (6.5h)

Lecture 1 (1.5h) (Schmalzl)
Geophysical Observation: Earth's mantle, seismic tomography, post-glacial rebound, plate tectonics; Earth's core, magnetic field observations, seismic results.

Lecture 2 (1.5h) (Ricard)
Geochemical observations: composition of the mantle, heat sources, isotope systems, behavior of elements during melting, rare gasses, ridge and hotspot, mantle reservoirs

Lecture 3 (1h) (Schmalzl)
Physics of convection: linear stability analysis, stationary convection pattern, the influence of inertia, Lagrangian structure, time-dependent flows, influence of rotation.

Lecture 4 (1h) (Schmalzl)
Modelling of the mantle: experimental results, numerical approaches, investigation of mixing properties.

Lecture 5 (1.5h) (Ricard)
The connection of mantle convection and geochemical observations: box-models and convection simulations, conservation of primitive mantle: blob-models and lava-lamp, thermochemical convection.

Hua Physical oceanic processes (5h)

Lecture 1 (1h)
Introduction. Transport and mixing in the ocean in stably stratifed cases (i) slow manifold: ventilation/subduction; baroclinic instability and parametrization of geostrophic eddies; geostrophic turbulence.

Lecture 2 (1h)
Introduction (continued) Transport and mixing in the ocean in stably stratifed cases (ii) fast manifold: near-inertial internal waves; internal tides; boundary mixing. Transport and mixing in unstable stratification: upright and slanted convection.

Lecture 3 (1h)
Filaments: alignment dynamics of tracer gradients. Stratified stirring.

Lecture 4 (1h)
Lagrangian approaches based on strain. Hyperbolic trajectories.

Lecture 5 (1h)
Fronts: secondary circulation; cross-frontal exchanges; compensated thermohaline fronts (observations; nonlinear diffusion; geostrophic turbulence). Jets: midlatitude zonal hets; equatorial jets.

Martin Oceanic biological observations and modelling (5h)

Lecture 1 (1.5 hours)
An introduction to marine plankton ecology: what are plankton? what types are there (phyto, zoo, bacteria etc)? what size are they? what habitats do they inhabit? processes of life: growth, grazing etc, nutrient requirements, seasonal cycles, big picture.

Lecture 2 (1 hour)
A primer for plankton modelling: individual versus population modelling, fundamental processes and how they are modelled, limiting effects and their representation, what the models gloss over: ignorance (of important processes), complexity (notably behaviour).

Lecture 3 (1.5 hours)
Plankton patchiness: (focus on mesoscale and smaller) observations: historical, current and future, theories of patchiness: biology versus physics, what do we mean by patchy? ways of characterizing spatial structure.

Lecture 4 (1 hour)
Into the big green yonder - Challenges: behaviour, "all-scale" modelling, from individual to collective descriptions.

Vergassola Fields in turbulent flows (3h)

Lecture 1 (1h)
Passive scalar decay

Lecture 2 (1h)
Anomalous scaling for scalar and magnetic fields

Lecture 3 (1h)
Fronts in scalar turbulence

Pierrehumbert Atmospheric observations, modelling and theory (3h)

Lecture 1 (1h)
Applications of advection-diffusion theory to stratospheric tracers. Basic solution balancing strain against diffusion: The concentration PDF for stratospheric mixing, Effect of transport barriers, What we can learn from the Lyapunov exponent PDF. The gradient PDF for idealized and realistic stratospheric mixing: Stretched exponentials, Gradient PDF's for red-noise processes, What can we learn from the gradient PDF regarding dissipation scale? Observed gradient PDFs. PDFs of tracer differences over finite increments. Structure functions and scaling.

Lecture 2 (1h)
Tropospheric problems. Importance of 3-dimensionality in the troposphere. The water vapor problem: Basic observations, The advection-condensation model. Random walk models of water vapor: Maximum excursion probabilities, The reflection principle for brownian motion.

Lecture 3 (1h)
Reactive turbulence: Flame analogies, Kpp flames, and worm diffusion, Condensation as a chemical reaction, Isotope fractionation.

Garcon Oceanic biological observations and modelling (3h)

Lecture 1 (1h)
Biological production in the oceans, Spatio-temporal variability of the biological production, The vertical dimension.

Lecture 2 (1h) Sensitivity to the parametrization of upper ocean turbulence. 3D context : ecosystem model in the North Atlantic ocean. Sensitivity studies: tracer advection sheme and upper ocean turbulence.

Lecture 3 (1h) The role of mesoscale variability on plankton dynamics: Characterization of the physical eddy environment, Biogeochemical oceanic mesoscale variability: observational evidence, Measures of time/space variability, Impact on biota, biogenic elements, and fluxes, Community structure, Biogeochemical oceanic mesoscale variability: modelling studies, Process models, Regional models, Basin scale models, How to quantify the role of mesoscale variability on plankton dynamics?

Tel Chaotic advection in open flows, implications for reacting systems (3h)

Lecture 1 (1h) Open flows; periodic flows: the von Karman vortex street; periodic orbits, stable and unstable manifolds, the skeletons of the advection dynamics.

Lecture 2 (1h) Fractality vs Lyapunov exponent(s) in chaotic flows; evolution of fractal tracer patterns; the effect of diffusion; chaotic adevction in temporally chaotic open flows.

Lecture 3 (1h) Implications for passively advected active tracers: steady state of fractal product distributions, a novel chemical kinetics, coexistence of competing species along unstable manifolds.

Hernandez-Garcia Spatial structures in reacting systems (3h)

Lecture 1 (1h)
Reactions in geophysical flows: the examples of atmospheric ozone chemistry and marine plankton biology. The importance of spatial structure. The impact on reaction rates.

Lecture 2 (1h)
Some sources of patchiness in reacting systems: turbulence, diffusion, Turing instabilities, excitability, chaotic advection. Lagrangian approaches in Analysis and in Numerics. Statistical turbulence approaches: Corrsin, etc.

Lecture 3 (1h)
Structure from chaotic advection. Fractal and multifractal description. Smooth vs. filamental patterns. Excitability and plankton blooms. Excitability under chaotic advection. Some results with individual based models and its relationship with macroscopic modelling.

Cho Tropospheric chemical observations and their interpretation (1h)

Types of data: Platforms, species, resolution, coverage, public availability. Data examples: Tropopause folds, pollution plumes and layers, water vapor filaments. Example diagnostics for transport and mixing: PDFs, Fourier spectra, structure functions, multifractal characterizations. Crucial transport and mixing issues in tropospheric chemistry.

Moyer/Ray Stratospheric chemical observations and their interpretation (1.5h)

Tracer transport from observations. Tracer-tracer relations from balloon and aircraft data.

McKenzie Mixing and stirring in the Earth's mantle: How can we see what is going on? (1.5h)

We believe that the viscosity of the Earth's mantle is everywhere greater than about 10¹⁹ Pa s, and that all advection of momentum can be ignored. It is therefore now possible to carry out three dimensional calculations that properly resolve the smallest scale motions, and to follow the trajectories of particles as they are transported by the circulation. In this respect it is easier to model stirring in the mantle than that in the oceans or atmosphere. But observing what is happening in the mantle is far more difficult, because we can only sample the mantle as small nodules brought up by basalts, or by studying the composition of the basalts themselves. Both types of observations are strongly affected by the two pase flow that both stirs and mixes the material as it is extracted from the mantle and transported to the surface. Some progress has been made in using geochemical observations to separate the influence of source variations from those resulting from transport, by using isotopic variations in the daughter isotopes, such as , ,, , and that are produced by the decay of long lived parent isotopes, such as , , , , and . The values of isotopic ratios of heavy elements at high temperatures are unaffected by melting and melt transport. However observational progress has been slow, and has up until now largely been concerned with attempts to classify sources using the observed isotopic variations. The subject badly needs some guidance from fluid dynamical studies. The mixing and stirring that occurs during melt generation and movement is a more difficult problem than the stirring caused by mantle circulation, partly because we are not yet confident that we know the relevant equations that govern two phase flow, and partly because diffusion is much more important in the liquid than it is in the solid state.

Tabeling Laboratory experiments on transport and mixing(1h)

Transport and mixing from the point of view of an experimentalists. Measurements of statistical properties and comparisons with thoretical predictions.

Legg Turbulent entrainment (0h45)

Modelling of a turbulent plume. Effect of rotation.

Legras Hyperbolic material lines and stirring (0h45)

Mathematical results for finite-time intervals. Diagnostic criteria. Applications.