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# Streamline diffusion

Given an advection-diffusion equation, streamline diffusion refers to all diffusion going on along the advection direction.

## Explanation

If we take an advection equation, for simplicity of writing we have assumed $\nabla\cdot\mathbf{F}=0$, 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 on 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: $(D\nabla^2\psi-{\bold u}({\bold u}\cdot D\nabla^2\psi))$