Glen's flow law

Remarks

It is an empirical constitutive law, fitted to laboratory creep experiments of Glen and put into tensor form by Nye. With the effective strain rate $\dot\varepsilon_e$ defined by $\dot\varepsilon_e^2 = \tfrac12 \dot\varepsilon_{ij}\dot\varepsilon_{ij}$, the law gives $\dot\varepsilon_e = A \tau_e^n$. Writing $\tau_{ij} = 2\eta\, \dot\varepsilon_{ij}$ gives the effective viscosity $\eta = 1/(2A\tau_e^{\,n-1})$, which decreases as stress increases, so ice is a shear-thinning non-Newtonian fluid of power-law type; $n = 1$ would give a Newtonian fluid. Example: in a slab of ice of thickness $H$ on a plane of slope $\alpha$, the shear stress at height $z$ above the bed is $\tau_{xz} = \rho g (H - z)\sin\alpha$. Integrating $du/dz = 2A\tau_{xz}^n$ shows that the surface speed exceeds the basal speed by $\frac{2A}{n+1}(\rho g \sin\alpha)^n H^{n+1}$. So the surface speed grows like $H^4$ for $n = 3$, which explains why thick glaciers flow much faster than thin ones.

Going deeper

For a student

Glen's law is the constitutive relation used in almost all models of glaciers and ice sheets, combined with the Stokes equations for slow viscous flow. Its nonlinearity is what makes ice-flow equations nonlinear, for example in the shallow ice approximation, where ice flux grows like a high power of thickness and surface slope. The rate factor is usually taken to follow an Arrhenius law in temperature.