# Difference between revisions of "Notes/vanderVeen Aug5"

From Interactive System for Ice sheet Simulation

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* Assumption: ice is incompressible, so density is constant. Mass conservation ~ volume conservation | * Assumption: ice is incompressible, so density is constant. Mass conservation ~ volume conservation | ||

* Assumption: ignore firn layer (100-150m in Antarctica, less in Greenland) | * Assumption: ignore firn layer (100-150m in Antarctica, less in Greenland) | ||

− | * <math> M \Delta x + H(x)U(x) - H(x + \Delta x) U(x + \Delta x) = \frac{\Delta H}{\Delta t} \Delta x < | + | * <math> M \Delta x + H(x)U(x) - H(x + \Delta x) U(x + \Delta x) = \frac{\Delta H}{\Delta t} \Delta x </math> |

− | * <math> \frac{\Delta H}{\Delta t} = -\frac{H(x + \Delta x)U(x+\Delta x) - H(x)U(x)}{\Delta x} + M < | + | * <math> \frac{\Delta H}{\Delta t} = -\frac{H(x + \Delta x)U(x+\Delta x) - H(x)U(x)}{\Delta x} + M </math> |

* Shrink timestep & spatial step to infinitessimal to write as differential equation | * Shrink timestep & spatial step to infinitessimal to write as differential equation | ||

− | * <math> \frac{\partial H}{\partial t} = -\frac{\partial}{\partial x} HU + M < | + | * <math> \frac{\partial H}{\partial t} = -\frac{\partial}{\partial x} HU + M </math> |

Conservation of Momentum: Newton's second law | Conservation of Momentum: Newton's second law | ||

− | * <math> F = ma < | + | * <math> F = ma </math>, with zero acceleration |

* so the sum of all forces must be zero. | * so the sum of all forces must be zero. | ||

* stresses are easier to work with than forces: stress is force per unit area | * stresses are easier to work with than forces: stress is force per unit area | ||

* Nine stress components: | * Nine stress components: | ||

− | <math> \sigma_{ij} < | + | <math> \sigma_{ij} </math> |

* i: plane perpendicular to axis (x) | * i: plane perpendicular to axis (x) | ||

* j: direction of stress | * j: direction of stress | ||

− | * Stress tensor is symmetric, so <math> \sigma_{ij} = \sigma_{ji} < | + | * Stress tensor is symmetric, so <math> \sigma_{ij} = \sigma_{ji} </math> and there are really only six distinct stress components |

* 3 equations with 6 unknowns | * 3 equations with 6 unknowns | ||

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Force balance in z | Force balance in z | ||

− | <math> F_z = 0 < | + | <math> F_z = 0 </math> |

− | <math> \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 | + | <math> \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 </math> |

## Revision as of 09:58, 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