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C. JAUPART (Inst. de Physique du Globe de Paris, France)
Introduction of energy equation. Boussinesq approximation.
Plumes and thermals. Use these to derive scaling relations for velocity as a function of power input and to discuss dynamical regimes (laminar vs. turbulent). Use plumes to illustrate Prandtl number, diffusion of heat and momentum and boundary layer structure. (Application of simple convection code)
Rayleigh-Bénard convection (no internal heating). Heuristic argument to introduce Rayleigh number. Dimensional analysis. Rayleigh number, Prandtl number, Reynolds number for convective flow.
Examples of convective systems: mantle, magma reservoirs. Marginal stability: Influence of boundary conditions (Free boundaries, rigid boundaries, fixed temperature, fixed heat flux).
Equation for the horizontally averaged temperature and the convective heat flux Thermal structure of fluid layer: boundary layers and well-mixed interior.
Free boundaries versus rigid boundaries. Large Prandtl number fluids. Convective loop model. Scaling relations for velocity and heat flux (no internal heating).
Scaling for internally heated fluids. Thermal structure of fluid layer
Temperature-dependent viscosity. Scaling for velocity and heat flux.
Non-Newtonian rheology. Non-Newtonian with temperature-dependent rheology. Scaling for velocity, heat flux, stress and strain-rate.
Two-layer mantle: convection regimes, thermal structure.
C. JAUPART, P. MOLNAR (Inst. de Physique du Globe de Paris, France, University of Colorado at Boulder, USA)
C. JAUPART, S. ZHONG (Inst. de Physique du Globe de Paris, France, Univ. of Colorado at Boulder, USA)
G. HOUSEMAN (University of Leeds, UK)
Stress and strain, and basic isostasy (statics).
Diagnostics of thin sheet deformation: matching gradients in strain-rates to gradients in GPE, and faulting response. Case studies: Tibet, Aegean and Indian Ocean.
Scaling of areal extent of deformation to dimensions of boundaries and to n (England, Houseman, and Sonder, 1985), plus dependence on Ar. Exercises on simple boundary-driven deformation problems using basic solutions. Dependence on the Argand number, Ar, and the exponent n.
Rayleigh-Taylor instability, effects of non-Newtonian viscosity. Derivation of the dependence of growth rate on n.
Examples using sybil to examine the time dependence of growth: exponential for n = 1, and super-exponential with n > 1 Scaling of growth rate to n.
G. HOUSEMAN, P. MOLNAR (Univ. of Leeds, UK, Univ. of Colorado at Boulder, USA)
J. NIEMALA (ICTP, Trieste)
M. MANGA (Univ. of California, Berkeley, USA)
Stokes equation, equation of continuity, and constitutive laws (both Newtonian, and non-Newtonian with underlying physics).
Examples of a cooling lithosphere: both cooling plate and cooling half-space, comparison of different boundary conditions (fixed temperature and fixed heat flux).
M. MANGA, P. MOLNAR (Univ. of California, USA , Univ. of Colorado at Boulder, USA)
P. MOLNAR (Univ. of Colorado at Boulder, USA)
Deflection of the surface above a sinking sphere and associated gravity anomalies and geoid. [Morgan, 1965]
P. MOLNAR, G. HOUSEMAN (Univ. of Colorado at Boulder, USA Univ. of Leeds, UK)
Some simple problems to relate stress and strain and illustrate constitutive laws, such as enhanced localization with non-Newtonian viscosity, with channel flow as an example. Exercises illustrating channel flow: Couette flow: problems, non-Newtonian material, exponential viscosity
R. KATZ (Univ. of Oxford, UK)
MAGMATIC SYSTEMS: MECHANICS. Derivation of conservation of mass & momentum equations. Scaling and the compaction length. Magma wave solutions.
Latent heat of crystallization and melting. Stefan number. Diffusive cooling: pure substance. Mushy layers: observations (lava lakes). Mushy layers: models.
S. ZHONG (Univ. of Colorado at Boulder, USA)
Directors: A. Aoudia, P. Molnar, G. Houseman, C. Jaupart, M. Manga