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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 $\mathrm{\nabla }\cdot \mathbf{u}=0$, and $\left|\right|\mathbf{u}\left|\right|=1$

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

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

$D{\mathrm{\nabla }}^2\psi$,

Giving us an equation of the form:

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

We may now rewrite the equation on the following form:

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

The term below is called streamline diffusion.

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

Crosswind diffusion

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

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

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