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
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)
Shrink timestep & spatial step to infinitessimal to write as differential equation
Conservation of Momentum: Newton's second law
- , 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:
- i: plane perpendicular to axis (x)
- j: direction of stress
- Stress tensor is symmetric, so and there are really only six distinct stress components
- 3 equations with 6 unknowns
Force balance in z
Force balance in x
Force balance in y
(we won't concern ourselves with the transverse y direction, which also means we can eliminate the y-derivative terms in the z and x equations above)
Newton's First Law: action / reaction
- What drives glacier flow? Gravity is the "action".
- What is the response? Resistance to flow is the "reaction".
Exercise in deriving force balance in the horizontal
Integrate the force balance equation in x over the depth of the ice column to eventually derive an expression relating driving stress, basal drag, and longitudinal stress gradients.
Mathways, start with:
The integral in is easy because it is just the difference of the stress at the surface and the bed -- the two dz's cancel. But we have to use the Leibniz rule to work with the term, because the limits of integration h and b are really h(z) and b(z) - they depend on z, which we're trying to integrate over. Shucks.
Using the Leibniz Rule, .
The entire balance equation is thus