Advection

During advection, a fluid transports some conserved quantity or material via bulk motion. The fluid's motion is described mathematically as a vector field, and the transported material is described by a scalar field showing its distribution over space. Advection requires currents in the fluid, and so cannot happen in rigid solids. It does not include transport of substances by molecular diffusion.

Advection is sometimes confused with the more encompassing process of convection, which is the combination of advective transport and diffusive transport.

In meteorology it is the transfer by the wind of an atmospheric mass. Advection is important for the formation of orographic clouds and the precipitation of water from clouds, as part of the hydrological cycle.

Mathematical description

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

One easily visualized example of advection is the transport of ink dumped into a river. As the river flows, ink will move downstream in a "pulse" via advection, as the water's movement itself transports the ink. If added to a lake without significant bulk water flow, the ink would simply disperse outwards from its source in a diffusive manner, which is not advection. Note that as it moves downstream, the "pulse" of ink will also spread via diffusion. The sum of these processes is called convection.

The advection equation

The advection equation for a conserved quantity described by a scalar field ${\displaystyle \psi (t,x,y,z)}$ is expressed by a continuity equation: $${\displaystyle \frac{\mathrm{\partial }\psi }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \left(\psi \mathbf{u}\right)=0,}$$ where vector field ${\displaystyle \mathbf{u}=(u_x,u_y,u_z)}$ is the flow velocity and ${\displaystyle \mathrm{\nabla }}$ is the del operator. If the flow is assumed to be incompressible then ${\displaystyle \mathbf{u}}$ is solenoidal, that is, the divergence is zero:$${\displaystyle \mathrm{\nabla }\cdot \mathbf{u}=0,}$$and (by using a product rule associated with the divergence) the above equation reduces to

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

In particular, if the flow is steady, then$${\displaystyle \mathbf{u}\cdot \mathrm{\nabla }\psi =0,}$$which shows that ${\displaystyle \psi }$ is constant (because $\mathrm{\nabla }\psi =0$ for any vector $\mathbf{u}$) along a streamline.

If a vector quantity ${\displaystyle \mathbf{a}}$ (such as a magnetic field) is being advected by the solenoidal velocity field ${\displaystyle \mathbf{u}}$, then the advection equation above becomes:

$${\displaystyle \frac{\mathrm{\partial }\mathbf{a}}{\mathrm{\partial }t}+\left(\mathbf{u}\cdot \mathrm{\nabla }\right)\mathbf{a}=0.}$$

Here, ${\displaystyle \mathbf{a}}$ is a vector field instead of the scalar field ${\displaystyle \psi }$.

Solution

Solutions to the advection equation can be approximated using numerical methods, where interest typically centers on discontinuous "shock" solutions and necessary conditions for convergence (e.g. the CFL condition).

Numerical simulation can be aided by considering the skew-symmetric form of advection $${\displaystyle \frac{1}{2}\mathbf{u}\cdot \mathrm{\nabla }\mathbf{u}+\frac{1}{2}\mathrm{\nabla }(\mathbf{u}\mathbf{u}),}$$ where $${\displaystyle \mathrm{\nabla }(\mathbf{u}\mathbf{u})=\mathrm{\nabla }\cdot [\mathbf{u}{u}_x,\mathbf{u}{u}_y,\mathbf{u}{u}_z].}$$

Since skew symmetry implies only imaginary eigenvalues, this form reduces the "blow up" and "spectral blocking" often experienced in numerical solutions with sharp discontinuities.