Difference between revisions of "Blatter-Pattyn model"

From Interactive System for Ice sheet Simulation
Jump to: navigation, search
Line 26: Line 26:
 
  & 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 \\  
 
  & 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 \\  
 
  & z:\quad \frac{\partial \tau _{zz}}{\partial z}-\frac{\partial P}{\partial z}=\rho g \\  
 +
\end{align}</math>
 +
 +
 +
 +
Unfortunately, the arguments that support this reduction are complex and we can't go into them in detail here. Two papers given in the references below (''Schoof and Hindmarsh'' (in press) and ''Dukowicz et al.'' (submitted)) provide more details on the mathematical background that allows us to state that these equations are "first order accurate" approximations to the Stokes equations.
 +
 +
 +
We continue by noting that the 3rd (vertical) balance equation above can be integrated through the depth of the ice sheet to give an expression for the pressure, ''P'',
 +
 +
 +
<math>P=\rho g\left( s-z \right)+\tau _{zz}(z)</math>
 +
 +
 +
This expression can be substituted into the horizontal pressure gradient terms above to remove pressure from the equations. For example, for the 'x' component of velocity we have
 +
 +
 +
 +
<math>\begin{align}
 +
  & x:\quad \frac{\partial \tau _{xx}}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=\frac{\partial P}{\partial x} \\
 +
& x:\quad \frac{\partial \tau _{xx}}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=\frac{\partial }{\partial x}\left[ \rho g\left( s-z \right)+\tau _{zz}(z) \right] \\
 +
& x:\quad \frac{\partial \tau _{xx}}{\partial x}-\frac{\partial \tau _{zz}}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=\rho g\frac{\partial s}{\partial x} \\
 
\end{align}</math>
 
\end{align}</math>

Revision as of 15:58, 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}


Unfortunately, the arguments that support this reduction are complex and we can't go into them in detail here. Two papers given in the references below (Schoof and Hindmarsh (in press) and Dukowicz et al. (submitted)) provide more details on the mathematical background that allows us to state that these equations are "first order accurate" approximations to the Stokes equations.


We continue by noting that the 3rd (vertical) balance equation above can be integrated through the depth of the ice sheet to give an expression for the pressure, P,


P=\rho g\left( s-z \right)+\tau _{zz}(z)


This expression can be substituted into the horizontal pressure gradient terms above to remove pressure from the equations. For example, for the 'x' component of velocity we have


\begin{align}
  & x:\quad \frac{\partial \tau _{xx}}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=\frac{\partial P}{\partial x} \\ 
 & x:\quad \frac{\partial \tau _{xx}}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=\frac{\partial }{\partial x}\left[ \rho g\left( s-z \right)+\tau _{zz}(z) \right] \\ 
 & x:\quad \frac{\partial \tau _{xx}}{\partial x}-\frac{\partial \tau _{zz}}{\partial x}+\frac{\partial \tau _{xy}}{\partial y}+\frac{\partial \tau _{xz}}{\partial z}=\rho g\frac{\partial s}{\partial x} \\ 
\end{align}