Steady SSA CPO field

If the velocity field \({\bf u}({\bf{x}},t)\) and CPO field \({\bf s}({\bf{x}},t)\) can be assumed steady, CPO evolution reduces to a high-dimensional boundary value problem in \({\bf s}({\bf{x}},t)\) that can easily be solved using e.g. the finite element method.

When considering the flow glacier ice, it is common to simplify the problem by invoking the Shallow Shelf Approximation (SSA), a depth-integrated version of the full Stokes equations. This approximation conveniently reduces the problem to a two-dimensional horizontal, membrane-like flow with negligible vertical shear.

Performing a similar calculation, the depth-average equation for steady CPO evolution can be shown to take the form (Rathmann et al., 2025)

\[ ({\bf u} \cdot \nabla) {\bf \bar s} = ({\bf M}_{\mathrm{LROT}}+{\bf M}_{\mathrm{DDRX}}+{\bf M}_{\mathrm{CDRX}}) \cdot {\bf \bar s} + \frac{a_{\mathrm{sfc}}}{H}( {\bf s}_{\mathrm{sfc}} - {\bf \bar s} ) + \frac{a_{\mathrm{sub}}}{H}( {\bf s}_{\mathrm{sub}} - {\bf \bar s} ) , \]

where \({\bf \bar s}(x,y)\) is the depth-average CPO state vector field, \({\bf{u}}(x,y)=[u_x(x,y),u_y(x,y)]\) is the horizontal surface velocity field, and \(H\) is the ice thickness, The first term represents CPO advection along stream lines, and the second term represents the depth-average effect of crystal processes. The third and fourth terms are state-space attractors, causing \({\bf \bar s}\) to tend towards the characteristic CPO states of ice that accumulates on the surface \({\bf s}_{\mathrm{sfc}}\) or subglacially \({\bf s}_{\mathrm{sub}}\) (likely isotropic), depending on the positively-defined accumulation rates \(a_{\mathrm{sfc}}\) and \(a_{\mathrm{sub}}\).

Closure

If only lattice rotation is relevant (typical for cold ice), the problem is closed by specifying the horizontal surface velocity field (e.g., satellite-derived velocities), together with accumulation rates and the characteristic CPO state of accumulated ice (typically isotropic).

If DDRX is non-negligible (typical for warm ice), the temperature and stress field must additionally be prescribed. One solution is to presume some temperature field and assert that the stress and strain-rate tensors are coaxial so that \({\boldsymbol\tau} \propto \dot{\boldsymbol\epsilon}\) (Rathmann et al., 2025).

Regularization

Noise in the surface velocity products used, in addition to uncertainties due to model assumptions, can render the steady SSA CPO problem ill-posed. Adding Laplacian regularization of the form \(\gamma \nabla^2 {\bf \bar s}\) to the right-hand side of the problem solves this, at the expense of limiting how large spatial gradients are permitted in the CPO field. The strength of regularization \(\gamma\) is therefore a free model parameter which must be carefully selected, especially in very dynamic regions where the CPO field might change rapidly with distance.

Example for Ross ice shelf

Documentation in prep. Will be updated once new paper is released.