Notes/vanderVeen Aug5
From Interactive System for Ice sheet Simulation
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