Streamline diffusion

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

Explanation

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

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

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:

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

We may now rewrite the equation on the following form:

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

The term below is called streamline diffusion.

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

Crosswind diffusion

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

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