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: [math]\displaystyle{ \nabla\cdot{\mathbf u}=0 }[/math], and [math]\displaystyle{ ||{\mathbf u}||=1 }[/math]

[math]\displaystyle{ \frac{\partial\psi}{\partial t} +{\mathbf u}\cdot\nabla\psi=0. }[/math]

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

[math]\displaystyle{ D\nabla^2\psi }[/math],

Giving an equation of the form:

[math]\displaystyle{ \frac{\partial\psi}{\partial t} +{\mathbf u}\cdot\nabla\psi +D\nabla^2\psi =0 }[/math]

The equation may now be rewritten in the following form:

[math]\displaystyle{ \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 }[/math]

The term below is called streamline diffusion.

[math]\displaystyle{ {\mathbf u}({\mathbf u}\cdot D\nabla^2\psi) }[/math]

Crosswind diffusion

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

[math]\displaystyle{ (D\nabla^2\psi-{\mathbf u}({\mathbf u}\cdot D\nabla^2\psi)) }[/math]