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,

  & 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 \\ 

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 comes in 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} \\ 

where η is the viscosity. We can write the constitutive law the same way, but define the viscosity instead as an "effective" viscosity such that

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