Difference between revisions of "Blatter-Pattyn model"

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

Revision as of 15:42, 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}
  & \frac{\partial \tau _{xx}}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=\frac{\partial P}{\partial x} \\ 
 & \frac{\partial \tau _{yy}}{\partial y}+\frac{\partial \tau _{xy}}{\partial x}+\frac{\partial \tau _{yz}}{\partial z}=\frac{\partial P}{\partial y} \\ 
 & \frac{\partial \tau _{zz}}{\partial z}=\rho g+\frac{\partial P}{\partial z} \\ 
\end{align}