>$\mathbf{u} = (u,v)$</span>. Hybrid models assume that ice sheet stresses are depth independent, but vertical velocity gradients are included in the non-linear viscosity and treatment of basal stress. This means that unlike models that make the <a href="#Shallow-Ice-Models">shallow-ice</a> or <a href="#Shallow-Shelf-Models">shallow-shelf</a> approximations, they are appropriate to describe both regions of fast and slow ice flow, while being computationally two-dimensional and thus far less expensive than <