# Difference between revisions of "Notes/vanderVeen Aug5"

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* What drives glacier flow? Gravity is the "action". | * What drives glacier flow? Gravity is the "action". | ||

* What is the response? Resistance to flow is the "reaction". | * What is the response? Resistance to flow is the "reaction". | ||

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+ | == 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: | ||

+ | |||

+ | <math> \int_b^h \frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \sigma_{xy}}{\partial y} + \frac{\partial \sigma_{xz}}{\partial z} dz = \int_b^h 0 dz </math> | ||

+ | |||

+ | The integral in <math> \sigma_{xz} </math> 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 <math> \sigma_{xx} </math> 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, <math> \int_b^h \frac{\partial \sigma_{xx}}{\partial x} dz = \frac{\partial}{\partial x} \int_b^h \sigma_{xx} dz - \sigma_{xx}(s)\frac{\partial s}{\partial x} + \sigma_{xx}(b)\frac{\partial b}{\partial x} </math>. | ||

+ | |||

+ | The entire balance equation is thus | ||

+ | |||

+ | \frac{\partial}{\partial x} \int_b^h \sigma_{xx} dz - \sigma_{xx}(s)\frac{\partial s}{\partial x} + \sigma_{xx}(b)\frac{\partial b}{\partial x} + \sigma_{xz}(s) - \sigma_{xz}(b) = 0 </math>. |

## Revision as of 11:39, 5 August 2009

## 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)

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

\frac{\partial}{\partial x} \int_b^h \sigma_{xx} dz - \sigma_{xx}(s)\frac{\partial s}{\partial x} + \sigma_{xx}(b)\frac{\partial b}{\partial x} + \sigma_{xz}(s) - \sigma_{xz}(b) = 0 </math>.