Streamline diffusion

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

[edit] 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)

[edit] 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))
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