Difference between revisions of "Notes/vanderVeen Aug5"

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m (Crash Course in Glacier Dynamics)
m (Crash Course in Glacier Dynamics)
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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>
+
* <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>
+
* <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>
+
* <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>, with zero acceleration
+
* <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>
+
<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} <\math> and there are really only six distinct stress components
+
* 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>
+
<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)
  •  M \Delta x + H(x)U(x) - H(x + \Delta x) U(x + \Delta x) = \frac{\Delta H}{\Delta t} \Delta x
  •  \frac{\Delta H}{\Delta t} = -\frac{H(x + \Delta x)U(x+\Delta x) - H(x)U(x)}{\Delta x} + M
  • Shrink timestep & spatial step to infinitessimal to write as differential equation
  •  \frac{\partial H}{\partial t} = -\frac{\partial}{\partial x} HU + M


Conservation of Momentum: Newton's second law

  •  F = ma , 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:

 \sigma_{ij}

  • i: plane perpendicular to axis (x)
  • j: direction of stress
  • Stress tensor is symmetric, so  \sigma_{ij} = \sigma_{ji} and there are really only six distinct stress components
  • 3 equations with 6 unknowns


Force balance in z

 F_z = 0

 \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