Advection

Advection, in chemistry, engineering and earth sciences, is a transport mechanism of a substance, or a conserved property, by a fluid, due to the fluid's bulk motion in a particular direction. An example of advection is the transport of pollutants or silt in a river. The motion of the water carries these impurities downstream. Another commonly advected property is energy or enthalpy, and here the fluid may be water, 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 fluid.

The fluid motion in advection is described mathematically as a vector field, and the material transported is typically described as a scalar concentration of substance, which is contained in the fluid. Advection requires currents in the fluid, and so cannot happen in solids. It does not include transport of substances by simple diffusion. Advection is sometimes confused with the more encompassing process convection, 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 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. Meteorological or oceanographic advection follows isobaric surfaces and is therefore predominantly horizontal.

Distinction between advection and convection

Occasionally, the term advection is used as synonymous with convection. However, many engineers prefer to use the term convection to describe transport by combined molecular and eddy 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).^[1][2] 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".^[3][4]

Meteorology

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

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

The advection equation is the partial differential equation that governs the motion of a conserved scalar as it is advected by a known velocity field. It is derived using the scalar's conservation law, together with Gauss's theorem, and taking the infinitesimal 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 diffusive manner, which is not advection.

In Cartesian coordinates the advection operator is

$\mathbf{u} \cdot \nabla = u \frac{\partial}{\partial x} + v \frac{\partial}{\partial y} + w \frac{\partial}{\partial z}$.

where the velocity vector $\bold u$ has components u, v and w in the x, y and z directions respectively.

The advection equation for a scalar ψ is expressed mathematically as:

$\frac{\partial\psi}{\partial t} +\nabla\cdot\left( \psi{\bold u}\right) =0$

where $\nabla\cdot$ is the divergence operator and $\bold u$ is the velocity vector field. Frequently, it is assumed that the flow is incompressible, that is, the velocity field satisfies $\nabla\cdot{\bold u}=0$ (it is said to be solenoidal) If this is so, the above equation reduces to

$\frac{\partial\psi}{\partial t} +{\bold u}\cdot\nabla\psi=0.$

For a vector $\bold a$, such as magnetic field or velocity, in a solenoidal field it is defined as:

$\frac{\partial{\bold a}}{\partial t} + \left( {\bold u} \cdot \nabla \right) {\bold a} =0.$

In particular, if the flow is steady, ${\bold u}\cdot\nabla\psi=0$ which shows that ψ is constant along a streamline.

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

$\frac{\partial\psi}{\partial t}+u\frac{\partial\psi}{\partial x}=0$

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

According to,^[5] numerical simulation can be aided by considering the skew symmetric form for the advection operator.

$\frac{1}{2} {\bold u} \cdot \nabla {\bold u} + \frac{1}{2} \nabla ({\bold u} {\bold u})$

where $\nabla ({\bold u} {\bold u})$ is a vector with components $[\nabla ({\bold u} u_x),\nabla ({\bold u} u_y),\nabla ({\bold u} u_z)]$ and the notation ${\bold u} = [u_x,u_y,u_z]$ has been used.

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 ^[6])

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

$\mathbf{u} \cdot \nabla \mathbf{u} = \nabla \left( \frac{\|\mathbf{u}\|^2}{2} \right) + \left( \nabla \times \mathbf{u} \right) \times \mathbf{u}$

$\frac{1}{2} \mathbf{u} \cdot \nabla \mathbf{u} + \frac{1}{2} \nabla (\mathbf{u} \mathbf{u}) = \nabla \left( \frac{\|\mathbf{u}\|^2}{2} \right) + \left( \nabla \times \mathbf{u} \right) \times \mathbf{u} + \frac{1}{2} \mathbf{u} (\nabla \cdot \mathbf{u})$

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

See also

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

References

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. ^ David A. Randall, "General circulation model development", Academic Press, 2000. (Google books)
4. ^ Scott Ryan, "Earth Science (CliffsQuickReview)", Wiley Publishing Inc., 2006. (Google books)
5. ^ 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. 
6. ^ Boyd, John P. (2000). Chebyshev and Fourier Spectral Methods 2nd edition. Dover. pp. 213. http://www-personal.engin.umich.edu/~jpboyd/BOOK_Spectral2000.html.