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{{Short description|Transport of a substance by bulk motion}}
In [[physics]], [[engineering]], and [[earth science]]s, '''advection''' is a [[transport]] mechanism of a substance or [[Conservation of energy|conserved]] property by a [[fluid]] due to the fluid's bulk [[Motion (physics)|motion]]. 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.
{{More citations needed|date=May 2022}}
In the field 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.
 
InDuring 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]].
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==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'' sometimes serves as a synonym for ''[[convection]]'', but technically, ''convection'' covers the sum of transport both by [[diffusion]] and by advection. Advective transport describes the movement of some quantity via the bulk flow of a fluid (as in a river or pipeline).<ref>Suthan S. Suthersan, "Remediation engineering: design concepts", CRC Press, 1996. [http://books.google.ca/books?id=Rd2QIcNKl1UC&pg=PA13&dq=advection+and+convection#v=onepage&q=advection%20and%20convection&f=false (Google books)]
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.
</ref><ref>
Jacques Willy Delleur, "The handbook of groundwater engineering", CRC Press, 2006. [http://books.google.ca/books?id=EiXxaGH3CzcC&pg=PT485&dq=advection+versus+convection#v=onepage&q=advection%20versus%20convection&f=false (Google books)]
</ref>
<!--
An example of convection is flow over a hot plate or below a chilled water surface in a lake. In the ocean and atmospheric sciences, advection is understood as horizontal movement resulting in transport "from place to place", while convection is vertical "mixing". <ref>David A. Randall, "General circulation model development", Academic Press, 2000. [http://books.google.ca/books?id=pRibtFBDNDAC&pg=PA648&dq=advection+and+convection#v=onepage&q=advection%20and%20convection&f=false (Google books)]</ref><ref>Scott Ryan, "Earth Science (CliffsQuickReview)", Wiley Publishing Inc., 2006. [http://books.google.com/books?id=PV_BabxTTkcC&pg=PA99&dq=advection+convection&lr=&as_drrb_is=q&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=&as_brr=0#v=onepage&q=advection%20convection&f=false (Google books)]</ref> Another view is that advection occurs with fluid transport of a point, while convection may be considered as fluid transport of a vector.
-->
 
==Meteorology==
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== 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.{{factcitation needed|reason=Previous reference does not exist|date=AprilSeptember 20132024}}
 
==Mathematical description==
==Mathematics of advection==
The '''advection equation''' is thea 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]].
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===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]]:
In Cartesian coordinates the advection [[Operator (mathematics)|operator]] is
<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. 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>\mathbf{u} \cdot \nabla = u_x \frac{\partial}{\partial x} + u_y \frac{\partial}{\partial y} + u_z \frac{\partial}{\partial z}</math>.
<math display="block">\nabla\cdot{\mathbf u} = 0,</math>
 
and the above equation reduces to
where '''u''' = (''u<sub>x</sub>, u<sub>y</sub>, u<sub>z</sub>'') is the [[velocity field]], and ∇ is the [[del]] operator (note that [[Cartesian coordinate system|Cartesian coordinates]] are used here).
<math display="block"> \frac{\partial\psi}{\partial t} +{\mathbf u}\cdot\nabla\psi =0 </math>
 
The advection equation for a conserved quantity described by a [[scalar field]] ''ψ'' is expressed mathematically by a [[continuity equation]]:
 
{{Equation box 1
|indent =:
|equation = <math> \frac{\partial\psi}{\partial t} +\nabla\cdot\left( \psi{\bold u}\right) =0 </math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}
 
where ∇∙ is the [[divergence]] operator and again '''u''' is the [[velocity field|velocity vector field]]. Frequently, it is assumed that the flow is [[incompressible flow|incompressible]], that is, the [[velocity field]] satisfies
 
:<math>\nabla\cdot{\bold u}=0</math>
 
and '''u''' is said to be [[solenoidal]]. If this is so, the above equation can be rewritten as
 
:{{Equation box 1
|equation=<math> \frac{\partial\psi}{\partial t} +{\bold u}\cdot\nabla\psi=0. </math>
|indent=:
|cellpadding
|border
|border colour = #0073CF
|background colour=#F5FFFA}}
 
In particular, if the flow is steady, then
<math display="block">{\mathbf u}\cdot\nabla \psi = 0</math>
which shows that <math>\psi</math> is constant 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>, the advection equation above becomes:
:<math>{\bold u}\cdot\nabla\psi=0</math>
<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>.
which shows that ''ψ'' is constant along a [[Streamlines, streaklines and pathlines|streamline]]. Hence, <math> \partial\psi/\partial t=0,</math> so ''ψ'' doesn't vary in time.
 
===Solution===
If a vector quantity '''a''' (such as a [[magnetic field]]) is being advected by the [[solenoidal]] [[velocity field]] '''u''', the advection equation above becomes:
[[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 condtion]]).{{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> \frac{\partial{\bold a}}{\partial t} + \left( {\bold u} \cdot \nabla \right) {\bold a} =0. </math>
<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}}
Here, '''a''' is a [[vector field]] instead of the [[scalar field]] ''ψ''.
 
===Solving the equation===
 
The advection equation is not simple to solve [[numerical analysis|numerically]]: the system is a [[hyperbolic partial differential equation]], and interest typically centers on [[Continuous function|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
 
:<math> \frac{\partial\psi}{\partial t}+u_x \frac{\partial\psi}{\partial x}=0 </math>
 
where ''ψ'' = ''ψ''(''x'', ''t'') is the [[scalar field]] being advected and ''u<sub>x</sub>'' is the ''x'' component of the vector '''u''' = (''u<sub>x</sub>'',0,0).
 
===Treatment of the advection operator in the incompressible Navier Stokes equations===
 
According to Zang,<ref>{{cite journal
| last = Zang
| first = Thomas
| year = 1991
| title = On the rotation and skew-symmetric forms for incompressible flow simulations
| journal = Applied Numerical Mathematics
| volume = 7
| pages = 27–40
| doi = 10.1016/0168-9274(91)90102-6
}}</ref> numerical simulation can be aided by considering the [[Skew-symmetric matrix|skew symmetric]] form for the advection operator.
 
:<math> \frac{1}{2} {\bold u} \cdot \nabla {\bold u} + \frac{1}{2} \nabla ({\bold u} {\bold u}) </math>
 
where
 
:<math> \nabla ({\bold u} {\bold u}) = [\nabla ({\bold u} u_x),\nabla ({\bold u} u_y),\nabla ({\bold u} u_z)]</math>
 
and '''u''' is the same as above.
 
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 (see Boyd<ref>{{cite book |last=Boyd
|first=John P. |title= Chebyshev and Fourier Spectral Methods 2nd edition |url= http://www-personal.engin.umich.edu/~jpboyd/BOOK_Spectral2000.html |year=2000 |publisher=Dover |location= |isbn= |pages=213}}</ref>).
 
Using [[vector calculus identities#Vector dot product|vector calculus identities]], these operators can also be expressed in other ways, available in more software packages for more coordinate systems.
 
:<math>\mathbf{u} \cdot \nabla \mathbf{u} = \nabla \left( \frac{\|\mathbf{u}\|^2}{2} \right) + \left( \nabla \times \mathbf{u} \right) \times \mathbf{u}</math>
 
:<math> \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}) </math>
 
This form also makes visible that the [[Skew-symmetric matrix|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 ==
{{div col|colwidth=20em}}
* [[Advection-diffusion equation]]
* [[Atmosphere of Earth]]
* [[Conservation law|Conservation equation]]
* [[Courant–Friedrichs–Lewy condition]]
* [[Convection]]
* [[CourantKinematic numberwave]]
* [[Péclet number]]
* [[Overshoot (signal)]]
* [[DelPéclet number]]
* [[Earth's atmosphereRadiation]]
{{Div col end}}
* [[Diffusion]]
 
==Notes==
{{reflist}}
 
==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}}
<!--See [[Wikipedia:Footnotes]] for an explanation of how to generate footnotes using the <ref(erences/)> tags-->
* {{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}}
<references/>
 
{{Meteorological variables}}
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