Editing Advection

{{Short description|Transport of a substance by bulk motion}}
{{More citations needed|date=May 2022}}
In the fields of [[physics]], [[engineering]], and [[earth science]]s, '''advection''' is the [[transport]] of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is also a fluid. The properties that are carried with the advected substance are [[Conservation of energy|conserved]] properties such as [[energy]]. An example of advection is the transport of [[pollutant]]s 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 [[Intensive and extensive properties|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 [[Mathematics|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 [[water vapor|moisture]]) or [[salinity]].  
<!-- This is sentence is patently false, e.g. cloud formation is vertical advection of water vapor driven by density gradients! See, for example, the following sentence:
Meteorological or oceanographic advective transport is perpendicular to isobaric surfaces and is therefore predominantly [[Horizontal plane|horizontal]].  
-->
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 field|velocity vector field]].{{sfn | LeVeque | 2002 | p=1}} 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 [[Diffusion|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]] <math>\psi(t,x,y,z)</math> is expressed by a [[Continuity_equation#Differential_form|continuity equation]]:
<math display="block"> \frac{\partial\psi}{\partial t} +\nabla\cdot\left( \psi{\mathbf u}\right) =0, </math>
where [[vector field]] <math>\mathbf{u} = (u_x, u_y, u_z)</math> is the [[flow velocity]] and <math>\nabla</math> is the [[del]] operator.<ref group=note>The subscripts denote the coordinates of the vector field; not to be confused with the [[Notation_for_differentiation#Partial_derivatives|notation for partial derivatives]].</ref>  If the flow is assumed to be [[incompressible flow|incompressible]] then <math>\mathbf{u}</math> is [[solenoidal]], that is, the [[Del#Divergence|divergence]] is zero:<math display="block">\nabla\cdot{\mathbf u} = 0,</math>and (by using a [[Del#Divergence|product rule associated with the divergence]]) the above equation reduces to

<math display="block"> \frac{\partial\psi}{\partial t} +{\mathbf u}\cdot\nabla\psi =0.</math>

In particular, if the flow is [[steady state|steady]], then{{sfn|LeVeque|2002|p=391}}<math display="block">{\mathbf u}\cdot\nabla \psi = 0,</math>which shows that <math>\psi</math> is constant (because <math display="inline">\nabla \psi = 0</math> for any vector <math display="inline">\mathbf u</math>) along a [[Streamlines, streaklines and pathlines|streamline]]. 

If a vector quantity <math>\mathbf{a}</math> (such as a [[magnetic field]]) is being advected by the [[solenoidal]] [[velocity field]] <math>\mathbf{u}</math>, then the advection equation above becomes:

<math display="block"> \frac{\partial{\mathbf a}}{\partial t} + \left( {\mathbf u} \cdot \nabla \right) {\mathbf a} =0. </math>

Here, <math>\mathbf{a}</math> is a [[vector field]] instead of the [[scalar field]] <math>\psi</math>.

===Solution===
{{see also|Method of characteristics}}

[[File:GaussianUpwind2D.gif|thumb|A simulation of the advection equation where {{math|1='''u''' = (sin ''t'', cos ''t'')}} is solenoidal.]]
Solutions to the advection equation can be approximated using [[Numerical_methods_for_partial_differential_equations|numerical methods]], where interest typically centers on [[Continuous function|discontinuous]] "shock" solutions and necessary conditions for convergence (e.g. the [[Courant–Friedrichs–Lewy_condition|CFL condition]]).{{sfn | LeVeque | 2002 | pp=4-6,68-69}} 

Numerical simulation can be aided by considering the [[Skew-symmetric matrix|skew-symmetric]] form of advection
<math display="block"> \tfrac12 {\mathbf u} \cdot \nabla {\mathbf u} + \tfrac12 \nabla ({\mathbf u} {\mathbf u}),</math>
where
<math display="block"> \nabla ({\mathbf u} {\mathbf u}) = \nabla \cdot[{\mathbf u} u_x,{\mathbf u} u_y,{\mathbf u} u_z].</math>

Since skew symmetry implies only [[Imaginary number|imaginary]] [[eigenvalues]], this form reduces the "blow up" and "spectral blocking" often experienced in numerical solutions with sharp discontinuities.{{sfn | Boyd | 2001 | p=213}}

==Distinction between advection and convection==
[[File:Heat-transmittance-means2.jpg|thumb|250px|The four fundamental modes of heat transfer illustrated with a campfire]]
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, although it might seem confusing, 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 cloud]]s (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|reason=Previous reference does not exist|date=September 2024}}

== See also ==
{{div col|colwidth=20em}}
* [[Advection-diffusion equation]]
* [[Atmosphere of Earth]]
* [[Conservation law|Conservation equation]]
* [[Courant–Friedrichs–Lewy condition]]
* [[Kinematic wave]]
* [[Overshoot (signal)]]
* [[Péclet number]]
* [[Radiation]]
{{Div col end}}

==Notes==
{{reflist}}
{{reflist|group=note}}

==References==
* {{cite book | last=Boyd | first=John P. | title=Chebyshev and Fourier Spectral Methods | publisher=Courier Corporation | publication-place=Mineola, NY | url=https://depts.washington.edu/ph506/Boyd.pdf|date=2001 | isbn=0-486-41183-4}}
* {{cite book | last=LeVeque | first=Randall J. | title=Finite Volume Methods for Hyperbolic Problems | publisher=Cambridge University Press | date=2002 | isbn=978-0-521-81087-6 | doi=10.1017/cbo9780511791253}}

{{Meteorological variables}}

[[Category:Vector calculus]]
[[Category:Atmospheric dynamics]]
[[Category:Conservation equations]]
[[Category:Equations of fluid dynamics]]
[[Category:Oceanography]]
[[Category:Convection]]
[[Category:Heat transfer]]
[[Category:Transport phenomena]]