Streamline diffusion
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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]
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