Notes/vanderVeen Aug5

From Interactive System for Ice sheet Simulation
Revision as of 09:56, 5 August 2009 by Kpoinar (Talk | contribs)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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)
  •  M \Delta x + H(x)U(x) - H(x + \Delta x) U(x + \Delta x) = \frac{\Delta H}{\Delta t} \Delta x
  •  \frac{\Delta H}{\Delta t} = -\frac{H(x + \Delta x)U(x+\Delta x) - H(x)U(x)}{\Delta x} + M
  • Shrink timestep & spatial step to infinitessimal to write as differential equation
  •  \frac{\partial H}{\partial t} = -\frac{partial}{\partial x} HU + M

Conservation of Momentum: Newton's second law

  •  F = ma , 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: Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma_{ij} <\math> * i: plane perpendicular to axis (x) * j: direction of stress * Stress tensor is symmetric, so <math> \sigma_{ij} = \sigma_{ji}
and there are really only six distinct stress components
  • 3 equations with 6 unknowns

Force balance in z

Failed to parse (Missing texvc executable; please see math/README to configure.): 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