# 1.3.8. Vector invariant form¶

For some purposes it is advantageous to write momentum advection in eq (1.1) and (1.2) in the (so-called) ‘vector invariant’ form:

(1.44)$\frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t} +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right]$

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 (1.44), explicit representation of the metric terms in (1.29), (1.30) and (1.31), can be avoided: information about the geometry is contained in the areas and lengths of the volumes used to discretize the model.