Difference between revisions of "Notes/vanderVeen Aug5"

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(Crash Course in Glacier Dynamics)
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== Crash Course in Glacier Dynamics==
 
== Crash Course in Glacier Dynamics==
 
Kees van der Veen, University of Kansas
 
Kees van der Veen, University of Kansas
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Force balance in z
 
  
<math> F_z = 0 </math>
+
'''Force balance in z'''
 +
 
 +
<math> 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 </math>
 +
 
 +
<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'''
 +
 
 +
<math> \frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \sigma_{xy}}\{\partial y} + \frac{\partial \sigma_{xz}}{\partial z} = 0 </math>
 +
 
 +
'''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)
  
<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>
+
'''Newton's First Law: action / reaction'''
 +
* What drives glacier flow?  Gravity is the "action".
 +
* What is the response?  Resistance to flow is the "reaction".

Revision as of 10:07, 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

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


 \frac{\partial \sigma_{xz}}{\partial x} + \frac{\partial |sigma_{yz}}{\partial y} + \frac{\partial \sigma_{zz}}{\partial z} = -\rho g

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".