**Advection**, in

chemistryChemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....

,

engineeringEngineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

and

earth scienceEarth science is an all-embracing term for the sciences related to the planet Earth. It is arguably a special case in planetary science, the Earth being the only known life-bearing planet. There are both reductionist and holistic approaches to Earth sciences...

s, is a

transportTransport or transportation is the movement of people, cattle, animals and goods from one location to another. Modes of transport include air, rail, road, water, cable, pipeline, and space. The field can be divided into infrastructure, vehicles, and operations...

mechanism of a substance, or a

conservedThe nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...

property, by a

fluidIn physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

, due to the fluid's bulk

motionIn physics, motion is a change in position of an object with respect to time. Change in action is the result of an unbalanced force. Motion is typically described in terms of velocity, acceleration, displacement and time . An object's velocity cannot change unless it is acted upon by a force, as...

in a particular direction. An example of advection is the transport of

pollutantA pollutant is a waste material that pollutes air, water or soil, and is the cause of pollution.Three factors determine the severity of a pollutant: its chemical nature, its concentration and its persistence. Some pollutants are biodegradable and therefore will not persist in the environment in the...

s or

siltSilt is granular material of a size somewhere between sand and clay whose mineral origin is quartz and feldspar. Silt may occur as a soil or as suspended sediment in a surface water body...

in a

riverA river is a natural watercourse, usually freshwater, flowing towards an ocean, a lake, a sea, or another river. In a few cases, a river simply flows into the ground or dries up completely before reaching another body of water. Small rivers may also be called by several other names, including...

. The motion of the water carries these impurities downstream. Another commonly advected property is

energyIn physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

or

enthalpyEnthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...

, and here the fluid may be

waterWater is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...

, air, or any other thermal energy-containing fluid material. Any substance, or conserved property (such as enthalpy) can be advected, in a similar way, in any

fluidIn physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

.

The fluid motion in advection is described

mathematicallyMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

as a

vector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

, and the material transported is typically described as a

scalarIn physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...

concentration of substance, which is contained in the fluid. Advection requires currents in the fluid, and so cannot happen in rigid solids. It does not include transport of substances by simple

diffusionMolecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

. Advection is sometimes confused with the more encompassing process

convectionConvection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

, which encompasses both advective transport

*and* diffusive transport in fluids. Convective transport is the sum of advective transport and diffusive transport.

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

In

meteorologyMeteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...

and

physical oceanographyPhysical oceanography is the study of physical conditions and physical processes within the ocean, especially the motions and physical properties of ocean waters.Physical oceanography is one of several sub-domains into which oceanography is divided...

, advection often refers to the transport of some property of the atmosphere or

oceanAn ocean is a major body of saline water, and a principal component of the hydrosphere. Approximately 71% of the Earth's surface is covered by ocean, a continuous body of water that is customarily divided into several principal oceans and smaller seas.More than half of this area is over 3,000...

, such as

heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

, humidity (see

moistureWater vapor or water vapour , also aqueous vapor, is the gas phase of water. It is one state of water within the hydrosphere. Water vapor can be produced from the evaporation or boiling of liquid water or from the sublimation of ice. Under typical atmospheric conditions, water vapor is continuously...

) or salinity. Meteorological or oceanographic advection follows isobaric surfaces and is therefore predominantly

horizontalIn geometry, physics, astronomy, geography, and related sciences, a plane is said to be horizontal at a given point if it is perpendicular to the gradient of the gravity field at that point— in other words, if apparent gravity makes a plumb bob hang perpendicular to the plane at that point.In...

.

## Distinction between advection and convection

Occasionally, the term

*advection* is used as synonymous with

*convection*Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

. However, many engineers prefer to use the term convection to describe transport by combined molecular and

eddyEddy diffusion, eddy dispersion, or turbulent diffusion is any diffusion process by which substances are mixed in the atmosphere or in any fluid system due to eddy motion...

diffusion, and reserve the usage of the term advection to describe transport with a general (net) flow of the fluid (like in river or pipeline). 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".

## Meteorology

In

meteorologyMeteorology is the interdisciplinary scientific study of the atmosphere. Studies in the field stretch back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw breakthroughs occur after observing networks developed across several countries...

and

physical oceanographyPhysical oceanography is the study of physical conditions and physical processes within the ocean, especially the motions and physical properties of ocean waters.Physical oceanography is one of several sub-domains into which oceanography is divided...

, advection often refers to the transport of some property of the atmosphere or

oceanAn ocean is a major body of saline water, and a principal component of the hydrosphere. Approximately 71% of the Earth's surface is covered by ocean, a continuous body of water that is customarily divided into several principal oceans and smaller seas.More than half of this area is over 3,000...

, such as

heatIn physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

, 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

The advection equation also applies if the quantity being advected is represented by a

probability density functionIn probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

at each point, although accounting for diffusion is more difficult.

## Mathematics of advection

The

**advection equation** is the

partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

that governs the motion of a conserved

scalarIn physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...

as it is advected by a known

velocity fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

. It is derived using the scalar's

conservation lawIn physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves....

, together with Gauss's theorem, and taking the

infinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

limit.

Perhaps the best image to have in mind is the transport of salt dumped in a river. If the river is originally fresh water and is flowing quickly, the predominant form of transport of the salt in the water will be advective, as the water flow itself would transport the salt. If the river were not flowing, the salt would simply disperse outwards from its source in a

diffusiveMolecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...

manner, which is not advection.

In Cartesian coordinates the advection operator is

.

where the velocity vector

has components u, v and w in the x, y and z directions respectively.

The advection equation for a

scalarIn linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....

is expressed mathematically as:

where

is the

divergenceIn vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

operator and

is the velocity vector field. Frequently, it is assumed that the flow is

incompressibleIn fluid mechanics or more generally continuum mechanics, incompressible flow refers to flow in which the material density is constant within an infinitesimal volume that moves with the velocity of the fluid...

, that is, the velocity field satisfies

(it is said to be solenoidal) If this is so, the above equation reduces to

For a vector

, such as magnetic field or velocity, in a solenoidal field it is defined as:

In particular, if the flow is steady,

which shows that

is constant along a

streamlineFluid flow is characterized by a velocity vector field in three-dimensional space, within the framework of continuum mechanics. Streamlines, streaklines and pathlines are field lines resulting from this vector field description of the flow...

.

The advection equation is not simple to solve

numericallyNumerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

: the system is a

hyperbolic partial differential equationIn mathematics, a hyperbolic partial differential equation of order n is a partial differential equation that, roughly speaking, has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along...

, and interest typically centers on

discontinuousIn mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

"shock" solutions (which are notoriously difficult for numerical schemes to handle).

Even in one space dimension and constant velocity, the system remains difficult to simulate. The equation becomes

where

is the scalar being advected and

the x component of the vector

.

According to, numerical simulation can be aided by considering the

skew symmetricIn mathematics, and in particular linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...

form for the advection operator.

where

is a vector with components

and the notation

has been used.

Since skew symmetry implies only

imaginaryAn imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...

eigenvalues, this form reduces the "blow up" and "spectral blocking" often experienced in numerical solutions with sharp discontinuities (see Boyd )

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

This form also makes visible that the

skew symmetricIn mathematics, and in particular linear algebra, a skew-symmetric matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation If the entry in the and is aij, i.e...

operator introduces error when the velocity field diverges.

## See also

- Continuity equation
A continuity equation in physics is a differential equation that describes the transport of a conserved quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described...

- Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

- Courant number
- Péclet number
The Péclet number is a dimensionless number relevant in the study of transport phenomena in fluid flows. It is named after the French physicist Jean Claude Eugène Péclet. It is defined to be the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion of the same...

- Overshoot (signal)
In signal processing, control theory, electronics, and mathematics, overshoot is when a signal or function exceeds its target. It arises especially in the step response of bandlimited systems such as low-pass filters...

- Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

- Earth's atmosphere
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...