Advection equation

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The advection equation is the partial differential equation that governs the motion of a conserved scalar as it is advected by a known velocity field. It is derived using the scalar's conservation law, together with Gauss's theorem, and taking the infinitesimal limit.

Perhaps the best image to have in mind is the transport of salt dumped in a river. If the river is originally fresh water and is flowing quickly, the predominant form of transport of the salt in the water will be advective, as the water flow itself would transport the salt. If the river was not flowing the salt would simply disperse outwards from its source in a diffusive manner, which is not advection.

The advection equation expressed mathematically is:

$\frac{\partial\psi}{\partial t} +\nabla\cdot\left( \psi{\bold u}\right) =0$

where $\nabla\cdot$ is the divergence operator. Frequently, it is assumed that the velocity field is solenoidal, that is, that $\nabla\cdot{\bold u}=0$ . If this is so, the above equation reduces to

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

In particular, if the flow is steady, ${\bold u}\cdot\nabla\psi=0$ which shows that $ψ$ is constant along a streamline.

The advection equation is not simple to solve numerically: the system is a hyperbolic partial differential equation, and interest typically centers on discontinuous "shock" solutions (which are notoriously difficult for numerical schemes to handle).

Even in one space dimension and constant velocity, the system remains difficult to simulate (it is a standard test for advection schemes known as the pigpen problem). The equation becomes

$\frac{\partial\psi}{\partial t}+u\frac{\partial\psi}{\partial x}=0$

where $ψ = ψ(x, t)$ .

According to Zang [2], numerical simulation can be aided by considering the skew symmetric form for the advection operator.

$\frac{1}{2} {\bold u} \cdot \nabla {\bold u} + \frac{1}{2} \nabla ({\bold u} {\bold u})$

where $\nabla ({\bold u} {\bold u})$ is a vector with components $[\nabla ({\bold u} u_x),\nabla ({\bold u} u_y),\nabla ({\bold u} u_z)]$ and the notation ${\bold u} = [u_x,u_y,u_z]$ has been used.

Since skew symmetry implies only complex eigenvalues, this form reduces the "blow up" and "spectral blocking" often experienced in numerical solutions with sharp discontinuities (see Boyd [1] pp. 213).

1 Meteorology
2 Other quantities
3 References
4 See also

[edit] Meteorology

In meteorology and physical oceanography, advection often refers to the transport of some property of the atmosphere or ocean, such as heat, humidity (see moisture) or salinity. Meteorological or oceanographic advection follows isobaric surfaces and is therefore predominantly horizontal.

[edit] Other quantities

The advection equation also applies if the quantity being advected is represented by a probability density function at each point, although accounting for diffusion is more difficult.