# Streamline diffusion

Streamline diffusion, given an advection-diffusion equation, refers to all diffusion going on along the advection direction.

## Explanation

If we take an advection equation, for simplicity of writing we have assumed $\nabla\cdot{\bold u}=0$, and $||{\bold u}||=1$

$\frac{\partial\psi}{\partial t} +{\bold u}\cdot\nabla\psi=0.$

we may add a diffusion term, again for simplicty, we assume the diffusion to be constant over the entire field.

$D\nabla^2\psi$,

Giving us an equation of the form:

$\frac{\partial\psi}{\partial t} +{\bold u}\cdot\nabla\psi +D\nabla^2\psi =0$

We may now rewrite the equation on the following form:

$\frac{\partial\psi}{\partial t} +{\bold u}\cdot \nabla\psi +{\bold u}({\bold u}\cdot D\nabla^2\psi) +(D\nabla^2\psi-{\bold u}({\bold u}\cdot D\nabla^2\psi)) =0$

The term below is called streamline diffusion.

${\bold u}({\bold u}\cdot D\nabla^2\psi)$

### Crosswind diffusion

Any diffusion orthogonal to the streamline diffusion is called crosswind diffusion, for us this becomes the term:

$(D\nabla^2\psi-{\bold u}({\bold u}\cdot D\nabla^2\psi))$