Difference between revisions of "Blatter-Pattyn model"

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(New page: The Blatter-Pattyn model is given by where ''u'' and ''v'' are the depth-independent ''x'' and ''y'' components of velocity, <math>\bar{\eta }</math> is the depth-averaged effective vi...)
 
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The Blatter-Pattyn model is given by
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The starting point for the Blatter-Pattyn model is the full Stokes equations
  
  
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<math>\begin{align}
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  & x:\quad \frac{\partial \tau _{xx}}{\partial x}-\frac{\partial P}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=0 \\
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& y:\quad \frac{\partial \tau _{yy}}{\partial y}-\frac{\partial P}{\partial y}+\frac{\partial \tau _{xy}}{\partial x}+\frac{\partial \tau _{yz}}{\partial z}=0 \\
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& z:\quad \frac{\partial \tau _{zz}}{\partial z}-\frac{\partial P}{\partial z}+\frac{\partial \tau _{zy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial x}=\rho g \\
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\end{align}</math>,
  
  
where ''u'' and ''v'' are the depth-independent ''x'' and ''y'' components of velocity, <math>\bar{\eta }</math> is the depth-averaged effective viscosity, ''H'' is the ice thickness, &rho; is the ice density, ''g'' is the acceleration due to gravity, and ''s=s(x,y)'' is the ice surface elevation.
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where ''P'' is the pressure and <big>&tau;</big> is the deviatoric stress tensor. The latter is given by
  
Notice the symmetry in the equations. This means that, computationally, many of the same subroutines can be used for discretization.
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<big><math>\tau _{ij}=\sigma _{ij}+P\delta _{ij}</math></big>,
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where <big>&sigma;</big> is the full stress tensor.
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There are a number of ways to argue that because of the "shallowness" of ice sheets - that is because the ratio of <big>''H''/''L''</big>, where ''H'' is the thickness and ''L'' is a relevant horizontal length scale, is small - the equations above can be reduced to the following "first-order approximation"
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<math>\begin{align}
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  & x:\quad \frac{\partial \tau _{xx}}{\partial x}-\frac{\partial P}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=0 \\
 +
& y:\quad \frac{\partial \tau _{yy}}{\partial y}-\frac{\partial P}{\partial y}+\frac{\partial \tau _{xy}}{\partial x}+\frac{\partial \tau _{yz}}{\partial z}=0 \\
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& z:\quad \frac{\partial \tau _{zz}}{\partial z}-\frac{\partial P}{\partial z}=\rho g \\
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\end{align}</math>

Revision as of 16:40, 30 July 2009

The starting point for the Blatter-Pattyn model is the full Stokes equations


\begin{align}
  & x:\quad \frac{\partial \tau _{xx}}{\partial x}-\frac{\partial P}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=0 \\ 
 & y:\quad \frac{\partial \tau _{yy}}{\partial y}-\frac{\partial P}{\partial y}+\frac{\partial \tau _{xy}}{\partial x}+\frac{\partial \tau _{yz}}{\partial z}=0 \\ 
 & z:\quad \frac{\partial \tau _{zz}}{\partial z}-\frac{\partial P}{\partial z}+\frac{\partial \tau _{zy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial x}=\rho g \\ 
\end{align},


where P is the pressure and τ is the deviatoric stress tensor. The latter is given by


\tau _{ij}=\sigma _{ij}+P\delta _{ij},


where σ is the full stress tensor.


There are a number of ways to argue that because of the "shallowness" of ice sheets - that is because the ratio of H/L, where H is the thickness and L is a relevant horizontal length scale, is small - the equations above can be reduced to the following "first-order approximation"


\begin{align}
  & x:\quad \frac{\partial \tau _{xx}}{\partial x}-\frac{\partial P}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=0 \\ 
 & y:\quad \frac{\partial \tau _{yy}}{\partial y}-\frac{\partial P}{\partial y}+\frac{\partial \tau _{xy}}{\partial x}+\frac{\partial \tau _{yz}}{\partial z}=0 \\ 
 & z:\quad \frac{\partial \tau _{zz}}{\partial z}-\frac{\partial P}{\partial z}=\rho g \\ 
\end{align}