For some purposes it is advantageous to write momentum advection in eq :eq:`horiz-mtm` and :eq:`vert-mtm` in the (so-called) ‘vector invariant’ form:
\frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}
+\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla
\left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]
This permits alternative numerical treatments of the non-linear terms based on their representation as a vorticity flux. Because gradients of coordinate vectors no longer appear on the rhs of :eq:`vi-identity`, explicit representation of the metric terms in :eq:`gu-spherical`, :eq:`gv-spherical` and :eq:`gw-spherical`, can be avoided: information about the geometry is contained in the areas and lengths of the volumes used to discretize the model.