Mathematical Geoscience. A. C. Fowler

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1 Mathematical Geoscience A. C. Fowler January 10, 2011

2 Contents Preface x 1 Mathematical modelling Conservation laws and constitutive laws Non-dimensionalisation Scaling Approximations Qualitative methods for differential equations Oscillations Relaxation oscillations Hysteresis Resonance Qualitative methods for partial differential equations Waves Burgers equation The Fisher equation Solitons Nonlinear diffusion: similarity solutions The viscous droplet Advance and retreat: waiting times Blow-up Reaction-diffusion equations Notes and references Exercises Climate dynamics Radiation budget Radiative heat transfer Local thermodynamic equilibrium Equation of radiative heat transfer Radiation budget of the Earth The Schuster-Schwarzschild approximation Radiative heat flux Scattering Troposphere and stratosphere The ozone layer i

3 2.2 Convection The wet adiabat Energy balance models Zonally averaged energy balance models Carbon dioxide and global warming The runaway greenhouse effect Ice ages Ice-albedo feedback The Milankovitch theory Nonlinear oscillations Heinrich events Dansgaard Oeschger events The 8,200 year cooling event North Atlantic salt oscillator Snowball Earth The carbon cycle The rôle of the oceans Ocean acidity Notes and references Exercises Oceans and atmospheres Atmospheric and oceanic circulation The geostrophic circulation Eddy viscosity Energy transport Global energy balance Choosing coordinates Non-dimensionalisation Day and night, land and ocean Parameter estimates Basic reference state A reduced model Geostrophic balance The planetary boundary layer Poincaré and Kelvin waves The quasi-geostrophic approximation Boundary conditions The day after tomorrow Rossby waves Baroclinic instability The Eady model Frontogenesis Depressions and hurricanes The mixed layer and the wind-driven oceanic circulation ii

4 3.9 Western boundary currents: the Gulf Stream Effects of basal drag Effects of lateral drag Global thermohaline circulation Tides and tsunamis The tidal equations Ocean tides Seiches Amphidromic points Tsunamis Notes and references Exercises River flow The hydrological cycle Chézy s and Manning s laws The flood hydrograph St. Venant equations Non-dimensionalisation Long wave and short wave approximation The monoclinal flood wave Waves and instability Nonlinear waves Characteristics Roll waves Tidal bores Notes and references Exercises Dunes Patterns in rivers Dunes Sediment transport Bedload Suspended sediment The potential model St. Venant type models A suspended sediment model Eddy viscosity model Orr-Sommerfeld equation Orr-Sommerfeld-Exner model Well-posedness Mixing-length model for aeolian dunes Mixing-length theory Turbulent flow model iii

5 5.7.3 Boundary conditions Eddy viscosity Surface roughness layer Outer solution Determination of p Matching Shear layer Linear stability Separation at the wave crest Formulation of Hilbert problem Calculation of the free boundary Notes and references Exercises Landscape evolution Weathering The erosional cycle River networks Denudation models Sediment transport Non-dimensionalisation The issue of time scale Channel-forming instability Boundary conditions Steady state solution Uplift and denudation Geomorphically concave slopes are unstable WKB approximation at high wavenumber Turning point analysis Rivulet theory: δ Channel formation Channel solutions Bank migration, stability and blow up Channels and hillslope evolution Hillslope evolution Detachment limited erosion Headward erosion Side-branching Notes and references Exercises Groundwater flow Darcy s law Homogenisation Empirical measures iv

6 7.2 Basic groundwater flow Boundary conditions Dupuit approximation Unsaturated soils The Richards equation Non-dimensionalisation Snow melting Similarity solutions Immiscible two-phase flows: the Buckley-Leverett equation Heterogeneous porous media Dual porosity models Contaminant transport Reactive dual porosity models Environmental remediation Reactive groundwater flow Biomass modelling Non-dimensionalisation Three specific remediation problems Precipitation and dissolution Consolidation Compaction Notes and references Exercises Mantle convection Plate tectonics Rayleigh-Bénard convection Boundary conditions Non-dimensionalisation Boussinesq approximation Linear stability Nonlinear stability; planforms Landau equation Plan forms Dislocations and chaos High Rayleigh number convection Boundary layer theory Variable viscosity Rheology of polycrystalline rocks Governing equations Boundary conditions Boundary layer analysis Summary Subduction and the yield stress Near-surface mantle rheology v

7 8.6.2 The plastic lid: failure and subduction Tectonics on Venus Notes and references Exercises Magma transport The magmatic cycle Mechanisms of magma ascent Phase diagrams and geochemistry Phase diagrams Ternary phase diagrams Olivine Summary Melting Continental crust MORB, OIB, CFB Granite Melt transport in the asthenosphere Summary Simplification Boundary conditions Thermodynamic equilibrium Stefan condition Steady state solution, one dimension Outer solution Boundary layer at z = Boundary layer at z = Magmafracturing in the lithosphere Fracture mechanics Magma dynamics Stress intensity factor Non-dimensionalisation and solution Crystallisation in magma chambers The formation of magma chambers Nucleation and crystallisation Double-diffusive convection Layered igneous intrusions Volcanic eruptions Types of eruption Strombolian eruptions Plinian eruptions Notes and references Exercises vi

8 10 Glaciers and ice sheets Dynamic phenomena Waves on glaciers Surges Ice streams Ice shelf instability Tidewater glaciers Jökulhlaups The shallow ice approximation Glaciers Ice sheets Temperature equation A simple non-isothermal ice sheet model Using the equations Ice shelves The grounding line Marine ice sheet instability Sliding and drainage Röthlisberger channels Linked cavities Canals Ice streams Waves, surges and mega-surges Waves on glaciers Surges Sliding and ice streams Heinrich events and the Hudson strait mega-surge Drumlins and eskers Drumlins Eskers Glaciology on Mars Non-dimensionalisation Multiple steady states Trough formation Multiple troughs Notes and references Exercises Jökulhlaups The Nye model Nondimensionalisation Boundary conditions and lake refilling Simplification of the model Periodic oscillations Breaking the seal vii

9 Wide channels and the 1996 eruption Cauldrons and calderas The beam boundary layer Similarity solutions Floods from ice sheets Notes and references Exercises Appendix A The Schwarzschild-Milne integral equation 796 Exercises Appendix B Turbulent flow 801 B.1 The Reynolds equation B.2 Eddy viscosity B.3 Pipe flow B.4 Extension to rivers B.5 Manning s law B.6 Entry length B.7 Sediment deposition Appendix C Asymptotic solution of the Orr Sommerfeld equation 808 Appendix D Melting, dissolution, and phase changes 817 D.1 Thermodynamics of pure substances D.2 The energy equation D.3 Phase change: Clausius-Clapeyron equation D.4 Phase change in multi-component materials D.5 Melting and freezing D.6 Precipitation and dissolution D.7 Evaporation and boiling D.8 Chemical reactions D.9 Surface energy D.10 Pre-melting D.11 Liesegang rings Exercises Appendix E Averaged equations in two phase flow 835 E.1 Discontinuities and jump conditions E.2 Averaging methods E.3 Mass and momentum equations viii

10 E.4 Energy equation E.5 Jump conditions E.6 Nye s energy equation in a subglacial channel Exercises References 847 ix

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