Stokes equations

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Stokes Equations

The conservation equations that describe the balance of momentum for an ice sheet (assuming that accelerations are negligible) are the non-linear Stokes equations. In Cartesian coordinates, these are given by,


\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


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


where σ is the full stress tensor and \delta _{ij} is the Kronecker delta (or identity matrix).

The "nonlinearity" is not obvious here, but becomes clear when we invoke a constitutive law that links stresses and strain rates. A standard Newtonian (constant viscosity) constitutive law for ice looks like


\begin{align}\quad \tau _{ij}=2\eta \dot{\varepsilon }_{ij} \\ 
\end{align}


where η is the viscosity. For glaciers and ice sheets, we are probably more familiar with seeing the constitutive law written in terms of Glen's flow law


\begin{align}\quad \tau _{ij}=B\dot{\varepsilon }_{e}^{\frac{1-n}{n}}\dot{\varepsilon }_{ij},\quad B=B(T) \\\end{align}


where B is the temperature dependent pre-factor. From these two equations, which give the deviatoric stress in terms of the strain rate, we can define an "effective viscosity" according to


\begin{align}\quad \eta \equiv \frac{1}{2}B\dot{\varepsilon }_{e}^{\frac{1-n}{n}} \\\end{align}.


\begin{align}\quad\dot{\varepsilon }_{e}^{\frac{1-n}{n}} \\\end{align} is the "effective strain rate", a norm of the strain-rate tensor. With this definition, as the strain rate at any point increases, the effective viscosity of the ice decreases (i.e. the ice becomes "softer" and deforms more easily). The stress depends on the strain rate, but in this case the coefficient that links the two also depends on the strain rate (hence the nonlinearity in the equations).