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In the field of physics, engineering, and earth sciences, advection is the transport of a substance by bulk motion. The properties of that substance are carried with it. Generally the majority of the advected substance is a fluid. The properties that are carried with the advected substance are conserved properties such as energy. 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.

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 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

The term advection often serves as a synonym for convection, and this correspondence of terms is used in the literature. More technically, convection applies to the movement of a fluid (often due to density gradients created by thermal gradients), whereas advection is the movement of some material by the velocity of the fluid. Thus, somewhat confusingly, it is technically correct to think of momentum being advected by the velocity field in the Navier-Stokes equations, although the resulting motion would be considered to be convection. Because of the specific use of the term convection to indicate transport in association with thermal gradients, it is probably safer to use the term advection if one is uncertain about which terminology best describes their particular system.

Meteorology

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

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.^{[citation needed]}

Mathematics of advection

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

In Cartesian coordinates the advection operator is

u⋅∇=ux∂∂x+uy∂∂y+uz∂∂zdisplaystyle mathbf u cdot nabla =u_xfrac partial partial x+u_yfrac partial partial y+u_zfrac partial partial z.

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

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

∂ψ∂t+∇⋅(ψu)=0displaystyle frac partial psi partial t+nabla cdot left(psi mathbf u right)=0

where ∇⋅displaystyle nabla cdot is the divergence operator and again udisplaystyle mathbf u is the velocity vector field. Frequently, it is assumed that the flow is incompressible, that is, the velocity field satisfies

∇⋅u=0displaystyle nabla cdot mathbf u=0

and udisplaystyle mathbf u is said to be solenoidal. If this is so, the above equation can be rewritten as

∂ψ∂t+u⋅∇ψ=0.displaystyle frac partial psi partial t+mathbf u cdot nabla psi =0.

In particular, if the flow is steady, then

u⋅∇ψ=0displaystyle mathbf ucdot nabla psi =0

which shows that ψdisplaystyle psi is constant along a streamline.
Hence, ∂ψ/∂t=0,displaystyle partial psi /partial t=0, so ψdisplaystyle psi doesn't vary in time.

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

∂a∂t+(u⋅∇)a=0.displaystyle frac partial mathbf apartial t+left(mathbf ucdot nabla right)mathbf a=0.

Here, adisplaystyle mathbf a is a vector field instead of the scalar field ψdisplaystyle psi .

Solving the equation

A simulation of the advection equation where u = (sin t, cos t) is solenoidal.

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

∂ψ∂t+ux∂ψ∂x=0displaystyle frac partial psi partial t+u_xfrac partial psi partial x=0

where ψ=ψ(x,t)displaystyle psi =psi (x,t) is the scalar field being advected
and uxdisplaystyle u_x is the xdisplaystyle x component of the vector u=(ux,0,0)displaystyle mathbf u =(u_x,0,0).

Treatment of the advection operator in the incompressible Navier Stokes equations

According to Zang,^[1] numerical simulation can be aided by considering the skew symmetric form for the advection operator.

12u⋅∇u+12∇(uu)displaystyle frac 12mathbf ucdot nabla mathbf u+frac 12nabla (mathbf umathbf u)

where

∇(uu)=[∇(uux),∇(uuy),∇(uuz)]displaystyle nabla (mathbf umathbf u)=[nabla (mathbf uu_x),nabla (mathbf uu_y),nabla (mathbf uu_z)]

and udisplaystyle mathbf 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^[2]).

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

u⋅∇u=∇(‖u‖22)+(∇×u)×udisplaystyle mathbf u cdot nabla mathbf u =nabla left(frac mathbf u 2right)+left(nabla times mathbf u right)times mathbf u

12u⋅∇u+12∇(uu)=∇(‖u‖22)+(∇×u)×u+12u(∇⋅u)displaystyle frac 12mathbf u cdot nabla mathbf u +frac 12nabla (mathbf u mathbf u )=nabla left(frac mathbf u 2right)+left(nabla times mathbf u right)times mathbf u +frac 12mathbf u (nabla cdot mathbf u )

This form also makes visible that the skew symmetric operator introduces error when the velocity field diverges. Solving the advection equation by numerical methods is very challenging and there is a large scientific literature about this.

See also

• Atmosphere of Earth


• Conservation equation


• Convection


• Courant–Friedrichs–Lewy condition


• Del


• Diffusion


• Overshoot (signal)


• Péclet number


• Radiation

References