EPS 219 Special topics in Geological Fluid Mechanics I:
Convection in magma chambers

Mark Jellinek

Basaltic magmas, ascending from the upper mantle or lithosphere, intrude the Earth’s crust, where they spread out forming elongate magma chambers that, in turn, cool and solidify. Thermal and compositional buoyancy sources arise as an inevitable consequence of the cooling and crystallization of these magmas and, in turn, drive convection. In this class we will investigate how buoyancy-driven fluid motions influence the internal differentiation of basaltic magma chambers. We will also explore ways to find evidence of these processes in geological data sets.

My role in this class in less a lecturer and more a coach and so we will learn science by doing science. Using a combination of laboratory experiments, critical discussion of the literature led mostly by students, and the occasional lecture from me we will work as a team to explore the physics of thermal, compositional, double diffusive, and particle convection. Quantitative analysis of the convection will stress physical insight gained from dimensional arguments and scaling analyses and be developed along side of our laboratory experiments. We will apply our knowledge of convection to understanding the internal differentiation of basaltic magma bodies emplaced in the deep crust, subaerially, and beneath mid ocean ridges. Specifically, we will investigate how to use geological data to identify the relevant boundary conditions determining the heat and mass transfer in a given basaltic magma body as well as the main buoyancy-driven processes at work during its solidification. We will use an analysis of the dynamics of these processes and their influence on heat and mass transfer to understand in a deep way petrological and geochemical data retrieved from several well-studied basaltic bodies.

Senior or graduate student standing. Basic coursework in physics, chemistry, and/ or engineering and math through ODEs or permission. Experience with fluid mechanics, heat and mass transfer will be useful but not necessary. The study of magma chambers is a strongly interdisciplinary endeavor involving physics, geochemistry, and basic geology. Hence, my goal for enrollment is to have a blend of students with physical, chemical, and trtraditional geological backgrounds. Maximum enrollment is 12.

Tritton, D.J. Physical Fluid Dynamics. 1987. Oxford.
Incropera and DeWitt, Fundamentals of Heat and Mass Transfer.
Novak and Gowens Learning how to Learn

Additional Reading (not including the fluids and geological literature):
Bejan, Convection heat transfer.

Other Useful texts:
Potter and Foss, Fluid Mechanics. (undergraduate level)
Lighthill, An informal introduction to theoretical fluid mechanics. (undergraduate/ beginning graduate level)
Turner, Buoyancy effects in fluids. (advanced undergraduate/ graduate level)
Furbish, Fluid physics in geology. (advanced undergraduate/ graduate level)
White, Viscous fluid flow. (graduate level)
Batchelor, Introduction to fluid dynamics. (graduate level)

Course Outline
I. What are magma chambers and why are they interesting? Overview of the physical problems involved in the solidification and chemical differentiation of magmas emplaced within the Earth’s crust and erupted subaerially. Solving physical problems.

II. Introduction to heat transfer, conservation laws and rate equations. Systems and control volumes. Integral and differential forms of the rate equations for conservation of mass and energy.

III. Conduction and Fourier’s law. 1D steady/unsteady conduction. Conduction through laminated materials. 1D Diffusion mass transfer. The analogy with thermal conduction.

IV. The nucleation and growth of crystals. Diffusive melting and dissolving. Thermal conduction with a phase change.

V. Introduction to convection and the convection equations. Thermal, compositional, and velocity boundary layers. The role of fluid motions in heat and mass transfer problems.

VI. Dimensional analysis and scaling.

VII. Heat and mass transfer by natural convection. Significance of the Rayleigh, Reynolds and Nusselt numbers, and the viscosity ratio.

VIII. Cooling, crystallization and thermal and compositional convection in magma chambers.

IX. A model for the cooling of a komatiite lava flow erupted on the ocean floor.

X. A simple convection/ solidification model of a binary fluid cooled from above. Application of the model to magma chambers and lava lakes.

XI. Further topics: Bubble convection in lava lakes and lava flows; magma chamber replenishment and mixing.