Mathematical Approaches to Geophysics

I. Fundamental Physics and Materials Technology of Ice.- 1, General Concepts.- 1. Introduction.- 2. Equations of Balance.- 3. Material Response.- (a) General constitutive relations, simple materials.- (b) The rule of material objectivity.- (c) Material symmetry.- (d) Constitutive response for isotropic bodies.- (e) Materials with bounded memory - some constitutive representations.- (f) Incompressibility.- (g) Some representations of isotropic functions.- 4. The Entropy Principle.- (a) The viscous heat-conducting compressible fluid.- (b) The viscous heat-conducting incompressible fluid.- (c) Pressure and extra stress as independent variables.- (d) Thermoelastic solid.- (e) Final remarks.- 5. Phase Changes.- (a) Phase changes for a viscous compressible heat-conducting fluid.- (b) Phase changes for a viscous incompressible heat-conducting fluid.- References.- 2. A Brief Summary of Constitutive Relations for Ice.- 1. Preliminary Remarks.- 2. The Mechanical Properties of Hexagonal Ice.- (a) The crystal structure of ordinary ice.- (b) The elastic behavior of hexagonal ice.- (c) The inelastic behavior of single-crystal ice.- 3. The Mechanical Properties of Polycrystalline Ice.- (a) The elastic behavior of polycrystalline ice.- (b) Linear viscoelastic properties of polycrystalline ice.- (?) General theory.- (?) Experimental results.- (c) Non-linear viscous deformation and creep.- (?) Results of creep tests.- (?) Generalization to a three-dimensional flow law.- (?) Other flow laws.- 4. The Mechanical Properties of Sea Ice.- (a) The phase diagram of standard sea ice and its brine content.- (b) Elastic properties.- (c) Other material properties.- References.- II. The Deformation of an Ice Mass Under Its Own Weight.- 3. A Mathematical Ice-flow Model and its Application to Parallel-sided Ice Slabs.- 1. Motivation and Physical Description.- 2. The Basic Model - Its Field Equations and Boundary Conditions.- (a) The field equations.- (?) Cold ice region.- (?) Temperate ice region.- (b) Boundary conditions.- (?) At the free surface.- (?) Along the ice-water interface.- (?) Along the bedrock surface.- (?) Along the melting surface.- 3. The Response of a Parallel-sided Ice Slab to Steady Conditions.- (a) Dimensionless forms of the field equations.- (b) Parallel-sided ice slab, a first approximation to glacier and ice-shelf flow dynamics.- (?) Velocity and temperature fields.7V-independent.- (?) Extending and compressing flow.- (?) Floating ice shelves.- 4. Concluding Remarks.- References.- 4. Thermo-mechanical Response of Nearly Parallel-sided Ice Slabs Sliding over their Bed.- 1. Motivation.- 2. The Basic Boundary-value Problem and its Reduction to Linear Form.- 3. The Solution of the Boundary-value Problems.- (a) Zeroth-order problem.- (b) First-order problem.- (?) Harmonic perturbation from uniform flow for a zero accumulation rate.- (?) Analytic solution for a Newtonian fluid.- (?) Numerical solution for non-linear rheology.- (?) Effect of a steady accumulation rate.- (?) A historical note on a previous approach.- (?) The first-order temperature problem.- (c) Numerical results for steady state.- (?) Transfer of bottom protuberances to the surface.- (?) Basal stresses.- (?) Surface velocities.- (?) Effect of a steady accumulation rate.- 4. Remarks on Response to a Time-dependent Accumulation Rate.- 5. Surface-wave Stability Analysis.- (a) The eigenvalue problem.- (b) Discussion of results.- 6. Final Remarks.- References.- 5, The Application of the Shallow-ice Approximation.- 1. Background and Previous Work.- 2. Derivation of the Basal Shear-stress Formula by Integrating the Momentum Equations over Ice Thickness.- (a) Derivation.- (b) The use of the basal shear-stress formula in applied glaciology.- 3. Solution of the Ice-flow Problem using the Shallow-ice Approximation.- (a) Governing equations.- (b) Shallow-ice approximation.- (c) Construction of the perturbation solution.- (d) Results.- (e) Temperature field.- 4. Theoretical Steady-state Profiles.- (a) Earlier theories and their limitations.- (b) Surface profiles determined by using the shallow-ice approximation.- 5. An Alternative Scaling - a Proper Analysis of Dynamics of Ice Sheets with Ice Divides.- (a) Finite-bed inclination.- (b) Small-bed inclination.- (c) Illustrations.- References.- 6. The Response of a Glacier or an Ice Sheet to Seasonal and Climatic Changes.- 1. Statement of the Problem.- 2. Development of the Kinematic Wave Theory.- (a) Full non-linear theory.- (b) Perturbation expansion - linear theory.- (c) An estimate for the coefficients C and D.- (d) Boundary and initial conditions.- 3. Theoretical Solutions for a Model Glacier.- (a) Solutions neglecting diffusion.- (b) Theoretical solutions for a diffusive model.- (?) Coefficient functions for the special model.- (?) Solution for a step function.- (?) General solution for uniform accumulation rate.- (?) The inverse problem - calculation of climate from variations of the snout.- 4. General Treatment for an Arbitrary Valley Glacier.- (a) Fourier analysis in time.- (?) Low-frequency response.- (?) High-frequency response.- (?) Use of the results.- (b) Direct integration methods.- 5. Derivation of the Surface-wave Equation from First Principles - Non-linear Theory.- (a) Surface waves in the shallow-ice approximation.- (?) Integration by the methods of characteristics.- (?) An illustrative example.- (?) A remark on linearization.- (?) Effects of diffusion.- (b) Remarks regarding time-dependent surface profiles in ice sheets.- (c) Long waves in an infinite ice slab - Is accounting for diffusion enough?.- (?) Basic equations.- (?) Construction of perturbation solutions.- (?) Numerical results.- 6. Concluding Remarks.- References.- 7. Three-dimensional and Local Flow Effects in Glaciers and Ice Sheets.- 1. Introduction.- 2. Effect of Valley Sides on the Motion of a Glacier.- (a) Solutions in special cases.- (?) Exact solutions for the limiting cases.- (?) Solution for a slightly off-circular channel.- (?) A note on very deep and wide channels.- (b) A useful result for symmetrical channels with no boundary slip.- (c) Numerical solution - discussion of results.- 3. Three-dimensional Flow Effects in Ice Sheets.- (a) Basic equations.- (b) Decoupling of the stress-velocity problem from the problem of surface profile.- (c) The equation describing the surface geometry.- (d) The margin conditions.- 4. Variational Principles.- (a) Fundamental variational theorem.- (b) Variational principle for velocities.- (c) Reciprocal variational theorem.- (d) Maximum and minimum principles.- (e) Adoption of the variational principles to ice problems.- 5. Discussion of Some Finite-element Solutions.- References.- Appendix. Detailed Calculations Pertaining to Higher-order Stresses in the Shallow-ice Approximation.- Author Index.