Notes/vanderVeen Aug5

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Crash Course in Glacier Dynamics

Kees van der Veen, University of Kansas August 5, 2009 Portland Summer Modeling School

What's the objective of an ice sheet model?

  • Understand evolution of ice sheet given some forcing (global warming, etc.)

Fundamental equations: conservation of xxx

  • Mass
  • Energy
  • Momentum

Conservation of Mass: Continuity Equation

  • What comes in (flux, basal freezing if, accumulation if) to some control volume must go out (flux, basal melting if, ablation if).
  • Assumption: ice is incompressible, so density is constant. Mass conservation ~ volume conservation
  • Assumption: ignore firn layer (100-150m in Antarctica, less in Greenland)
  • Failed to parse (Missing texvc executable; please see math/README to configure.): M \Delta x + H(x)U(x) - H(x + \Delta x) U(x + \Delta x) = \frac{\Delta H}{\Delta t} \Delta x <\math> * <math> \frac{\Delta H}{\Delta t} = -\frac{H(x + \Delta x)U(x+\Delta x) - H(x)U(x)}{\Delta x} + M <\math> * Shrink timestep & spatial step to infinitessimal to write as differential equation * <math> \frac{\partial H}{\partial t} = -\frac{\partial}{\partial x} HU + M <\math> Conservation of Momentum: Newton's second law * <math> F = ma <\math>, with zero acceleration * so the sum of all forces must be zero. * stresses are easier to work with than forces: stress is force per unit area * Nine stress components: <math> \sigma_{ij} <\math> * i: plane perpendicular to axis (x) * j: direction of stress * Stress tensor is symmetric, so <math> \sigma_{ij} = \sigma_{ji} <\math> and there are really only six distinct stress components * 3 equations with 6 unknowns Force balance in z <math> F_z = 0 <\math> <math> \sigma_{zz}(z + \Delta z) \Delta x \Delta y + \sigma_{xz}(z+\Delta z) \Delta x \Delta y + \sigma_{yz}(z + \Delta z) \Delta x \Delta y