# Stokes equations

From Interactive System for Ice sheet Simulation

### 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,

,

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

,

where *σ* is the full stress tensor and 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

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