# Difference between revisions of "Blatter-Pattyn model"

From Interactive System for Ice sheet Simulation

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

+ | <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 \\ | ||

+ | & 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}</math>, | ||

− | where '' | + | where ''P'' is the pressure and <big>τ</big> is the deviatoric stress tensor. The latter is given by |

− | + | ||

+ | <big><math>\tau _{ij}=\sigma _{ij}+P\delta _{ij}</math></big>, | ||

+ | |||

+ | |||

+ | where <big>σ</big> 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 <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} | ||

+ | & 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}</math> |

## Revision as of 16:40, 30 July 2009

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

,

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

,

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"