In chemistry, engineering and earth sciences, advection is a transport mechanism of a substance or conserved property by a fluid due to the fluid's bulk motion. An example of advection is the transport of pollutants or silt in a river by bulk water flow downstream. Another commonly advected quantity is energy or enthalpy. Here the fluid may be any material that contains thermal energy, such as water or air. In general, any substance or conserved, extensive quantity can be advected by a fluid that can hold or contain the quantity or substance.

In 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 simple diffusion. Advection is sometimes confused with the more encompassing process of convection. In fact, convective transport is the sum of advective transport and diffusive transport.

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. Advection is important for the formation of orographic clouds and the precipitation of water from clouds, as part of the hydrological cycle.

Distinction between advection and convection[link]

The term advection is sometimes used as a synonym for convection. Technically, convection is the sum of transport by diffusion and advection. Advective transport describes the movement of some quantity via the bulk flow of a fluid (as in a river or pipeline).^[1][2]

Meteorology[link]

In meteorology and physical oceanography, advection often refers to the transport of some property of the atmosphere or ocean, such as heat, humidity or salinity. Advection is important for the formation of orographic clouds and the precipitation of water from clouds, as part of the hydrological cycle.

Other quantities[link]

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.

Mathematics of advection[link]

The advection equation is the 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[link]

In Cartesian coordinates the advection operator is

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{u} \cdot \nabla = u_x \frac{\partial}{\partial x} + u_y \frac{\partial}{\partial y} + u_z \frac{\partial}{\partial z}

.

where u = (u_x, u_y, u_z) is the velocity field, and ∇ is the del operator (note that Cartesian coordinates are used here).

The advection equation for a conserved quantity described by a scalar field ψ is expressed mathematically by a continuity equation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial\psi}{\partial t} +\nabla\cdot\left( \psi{\bold u}\right) =0

where ∇∙ is the divergence operator and again u is the velocity vector field. Frequently, it is assumed that the flow is incompressible, that is, the velocity field satisfies

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla\cdot{\bold u}=0

and u is said to be solenoidal. If this is so, the above equation reduces to

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial\psi}{\partial t} +{\bold u}\cdot\nabla\psi=0.

In particular, if the flow is steady, then

Failed to parse (Missing texvc executable; please see math/README to configure.): {\bold u}\cdot\nabla\psi=0

which shows that ψ is constant along a streamline.

If a vector quantity a (such as a magnetic field) is being advected by the solenoidal velocity field u, the advection equation above becomes:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial{\bold a}}{\partial t} + \left( {\bold u} \cdot \nabla \right) {\bold a} =0.

Here, a is a vector field instead of the scalar field Failed to parse (Missing texvc executable; please see math/README to configure.): \psi }.

Solving the equation[link]

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 with one space dimension and a constant velocity field, the system remains difficult to simulate. The equation becomes

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial\psi}{\partial t}+u_x \frac{\partial\psi}{\partial x}=0

where ψ = ψ(x, t) is the scalar field being advected and u_x is the x component of the vector u = (u_x,0,0).

According to Zang^[3], numerical simulation can be aided by considering the skew symmetric form for the advection operator.

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2} {\bold u} \cdot \nabla {\bold u} + \frac{1}{2} \nabla ({\bold u} {\bold u})

where

Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla ({\bold u} {\bold u}) = [\nabla ({\bold u} u_x),\nabla ({\bold u} u_y),\nabla ({\bold u} u_z)]

and u is the same as above.

Since skew symmetry implies only imaginary eigenvalues, this form reduces the "blow up" and "spectral blocking" often experienced in numerical solutions with sharp discontinuities (see Boyd ^[4])

Using vector calculus identities, these operators can also be expressed in other ways, available in more software packages for more coordinate systems.

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{u} \cdot \nabla \mathbf{u} = \nabla \left( \frac{\|\mathbf{u}\|^2}{2} \right) + \left( \nabla \times \mathbf{u} \right) \times \mathbf{u}

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{2} \mathbf{u} \cdot \nabla \mathbf{u} + \frac{1}{2} \nabla (\mathbf{u} \mathbf{u}) = \nabla \left( \frac{\|\mathbf{u}\|^2}{2} \right) + \left( \nabla \times \mathbf{u} \right) \times \mathbf{u} + \frac{1}{2} \mathbf{u} (\nabla \cdot \mathbf{u})

This form also makes visible that the skew symmetric operator introduces error when the velocity field diverges.

See also[link]

• Continuity equation
• Convection
• Courant number
• Péclet number
• Overshoot (signal)
• Partial differential equation
• Del
• Earth's atmosphere
• Diffusion

References[link]

1. ^ Suthan S. Suthersan, "Remediation engineering: design concepts", CRC Press, 1996. (Google books)
2. ^ Jacques Willy Delleur, "The handbook of groundwater engineering", CRC Press, 2006. (Google books)
3. ^ Zang, Thomas (1991). "On the rotation and skew-symmetric forms for incompressible flow simulations". Applied Numerical Mathematics 7: 27–40. DOI:10.1016/0168-9274(91)90102-6. 
4. ^ Boyd, John P. (2000). Chebyshev and Fourier Spectral Methods 2nd edition. Dover. pp. 213. http://www-personal.engin.umich.edu/~jpboyd/BOOK_Spectral2000.html.