**Cargese GEOMIX Summer School **

**19th August - 1st September 2001 **

**Lecture Programme **

**Week 1 **

Wiggins |
5h |
Mathematics of chaotic advection |

Plumb |
5h |
Atmospheric observations, modelling and theory |

Falkovich |
5h |
Particles in turbulent flows |

Ricard/Schmalzl |
4h |
Geological data, geophysics and modelling of the mantle |

Hua |
5h |
Physical oceanic processes |

Martin |
5h |
Oceanic biological observations and modelling |

**Week 2 **

Vergassola |
3h |
Fields in turbulent flows |

Pierrehumbert |
3h |
Atmospheric observations, modelling and theory |

Garcon |
(cancelled) |
Oceanic biological observations and modelling |

Tel |
3h |
Chaotic advection in open flows |

Hernandez-Garcia |
5h |
implications for reacting systems |

Ricard/Schmalzl |
2.5h |
Geological data, geophysics and modelling of the mantle |

Hernandez-Garcia |
3h |
Spatial structures in reacting systems |

McKenzie |
1.5h |
Geochemical issues |

Cho |
1h |
Tropospheric chemical observations |

Moyer/Ray |
1.5h |
Stratospheric chemical observations |

Tabeling |
1h |
Laboratory experiments on transport and mixing |

Legg |
0h45 |
Turbulent entrainment |

Legras |
0h45 |
Hyperbolic material lines and stirring |

**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 10^{19} 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.