# Notes/vanderVeen Aug5

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

## 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 [itex] \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> [itex] \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