Advection
Advection represents directed transport through space, for example movement with a fluid flow, wind, current or persistent migration velocity.
Advection equation
For constant velocity \(v\),
\[\frac{\partial u}{\partial t}+v\frac{\partial u}{\partial x}=0.\]This transports the spatial profile without the smoothing effect produced by diffusion.
Advection and diffusion together
\[\frac{\partial u}{\partial t}=D\frac{\partial^2u}{\partial x^2}-v\frac{\partial u}{\partial x}.\]The diffusion term spreads the population, while the advection term shifts it directionally.
Key idea. Diffusion represents random spreading; advection represents systematic directional transport.