Streamline diffusion

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

Explanation

In an advection equation, for simplicity assume: $\mathrm{\nabla }\cdot \mathbf{𝐮}=0$, and $||\mathbf{𝐮}||=1$

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

Adding a diffusion term, again for simplicity, assume the diffusion to be constant over the entire field.

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

Giving an equation of the form:

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

The equation may now be rewritten in 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:

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