Crash Course in Glacier Dynamics
Kees van der Veen, University of Kansas
August 5, 2009
Portland Summer Modeling School
notes by Kristin Poinar
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
We can work on the first term, by writing out what is.
We don't know anything about how varies in z, so we won't touch that. But we can integrate the second term:
So the entire force balance equation is now
We can simplify the terms involving now. Those are:
If we differentiate the first term, we get .
We should note that since , it is also true that .
This lets us compare the first and second terms: their sum is just . This is the definition of the driving stress, .
So now the balance equation is
We will apply the zero-stress surface boundary condition, which says that
So we can lose those terms, and the balance equation becomes
And, finally, we'll substitute the basal drag, , for its definition:
to arrive at the final answer,