# Difference between revisions of "Notes/vanderVeen Aug5"

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<math> \sigma_{zz}(z + \Delta z) \Delta x \Delta y + \sigma_{xz}(z+\Delta z) \Delta z \Delta y + \sigma_{yz}(z + \Delta z) \Delta x \Delta z = \rho g \Delta x \Delta y \Delta z </math> | <math> \sigma_{zz}(z + \Delta z) \Delta x \Delta y + \sigma_{xz}(z+\Delta z) \Delta z \Delta y + \sigma_{yz}(z + \Delta z) \Delta x \Delta z = \rho g \Delta x \Delta y \Delta z </math> | ||

− | <math> \frac{\partial \sigma_{xz}}{\partial x} + \frac{\partial | + | <math> \frac{\partial \sigma_{xz}}{\partial x} + \frac{\partial \sigma_{yz}}{\partial y} + \frac{\partial \sigma_{zz}}{\partial z} = -\rho g </math> |

'''Force balance in x''' | '''Force balance in x''' |

## Revision as of 10:09, 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**

**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 z \Delta y + \sigma_{yz}(z + \Delta z) \Delta x \Delta z = \rho g \Delta x \Delta y \Delta z **

**Force balance in x**

**Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \sigma_{xy}}\{\partial y} + \frac{\partial \sigma_{xz}}{\partial z} = 0 **

**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".