nonlinear: 0,1, or 2

Advection scheme for momentum:

0

: No advection of momentum, i.e. $\mathbf{u}\cdot\nabla\mathbf{u}=0$.

1

: First-order upwind for momentum.

2

: Second-order central for momentum. Note that diffusion must be present for stability. While it is difficult to perform an exact stability analysis for the unstructured-grid equations, the 1-d stability limitation requires that grid Peclet number according to

$$Pe_\Delta<\frac{2}{C} ,$$

where $Pe_\Delta = \frac{\max(\vert U\vert)\max(D_g)}{\min(\nu_H)}$ and $C=\frac{\max(\vert U\vert)\Delta t}{\max(D_g)}$, which translates into the requirement that the horizontal diffusion be restricted to

$$\nu_H>\frac{\max(\vert U\vert)^2\Delta t}{2} .$$

Equivalently, the vertical diffusion is restricted by

$$\nu>\frac{\max(\vert w\vert)^2\Delta t}{2} .$$