Galvanic cell

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In [[geophysics]], the '''dynamo theory''' proposes a mechanism{{which|date=January 2014}} by which a celestial body such as [[Earth]] or a [[star]] generates a [[magnetic field]]. The dynamo theory describes the process through which a rotating, [[convection|convecting]], and [[electric]]ally conducting fluid can maintain a magnetic field over [[astronomical]] time scales.

==History of theory==
When [[William Gilbert (astronomer)|William Gilbert]] published ''[[de Magnete]]'' in 1600, he concluded that the Earth is magnetic and proposed the first hypothesis for the origin of this magnetism: permanent magnetism such as that found in [[lodestone]]. In 1919, [[Joseph Larmor]] proposed that a dynamo might be generating the field.<ref name=Larmor1919>{{cite journal |first=J. |last=Larmor |year=1919 |title=How could a rotating body such as the Sun become a magnet? |journal=Reports of the British Association |volume=87 |pages= 159–160}}</ref><ref>{{cite journal |first=J. |last=Larmor |year=1919 |title=Possible rotational origin of magnetic fields of sun and earth |journal=Electrical Review |volume=85 |pages= 412ff }} Reprinted in ''Engineering'', vol. 108, pages 461ff (3 October 1919).</ref> However, even after he advanced his hypothesis, some prominent scientists advanced alternate explanations. Einstein believed that there might be an asymmetry between the charges of the [[electron]] and [[proton]] so that the [[Earth's magnetic field]] would be produced by the entire Earth. The [[Nobel Prize]] winner [[Patrick Blackett]] did a series of experiments looking for a fundamental relation between [[angular momentum]] and [[magnetic moment]], but found none.<ref>{{cite journal|last=Nye|first=Mary Jo|title=Temptations of theory, strategies of evidence: P. M. S. Blackett and the earth's magnetism, 1947–52|journal=The British Journal for the History of Science|date=1 March 1999|volume=32|issue=1|pages=69–92|doi=10.1017/S0007087498003495}}</ref><ref>{{harvnb|Merrill|McElhinny|McFadden|1996|loc=page 17}} claim that in 1905, shortly after composing his [[special relativity]] paper, [[Albert Einstein]] described the origin of the [[Earth's magnetic field]] as being one of the great unsolved problems facing modern [[physicist]]s. However, they do not provide details on where he made this statement.</ref>

[[Walter M. Elsasser]], considered a "father" of the presently accepted dynamo theory as an explanation of the Earth's magnetism, proposed that this magnetic field resulted from electric currents induced in the fluid outer core of the Earth. He revealed the history of the Earth's magnetic field through pioneering the study of the magnetic orientation of minerals in rocks.

In order to maintain the magnetic field against [[ohm]]ic decay (which would occur for the dipole field in 20,000 years), the outer core must be convecting. The [[convection]] is likely some combination of thermal and compositional convection. The mantle controls the rate at which heat is extracted from the core. Heat sources include gravitational energy released by the compression of the core, gravitational energy released by the rejection of light elements (probably [[sulfur]], [[oxygen]], or [[silicon]]) at the inner core boundary as it grows, latent heat of crystallization at the inner core boundary, and radioactivity of [[potassium]], [[uranium]] and [[thorium]].<ref>{{cite news | first=Robert | last=Sanders | title=Radioactive potassium may be major heat source in Earth's core | publisher=UC Berkeley News | date=2003-12-10 | url=http://www.berkeley.edu/news/media/releases/2003/12/10_heat.shtml | accessdate=2007-02-28 }}</ref>

At the dawn of the 21st century, numerical modeling of the Earth's magnetic field has not been successfully demonstrated, but appears to be in reach. Initial models are focused on field generation by convection in the planet's fluid outer core. It was possible to show the generation of a strong, Earth-like field when the model assumed a uniform core-surface temperature and exceptionally high viscosities for the core fluid. Computations which incorporated more realistic parameter values yielded magnetic fields that were less Earth-like, but also point the way to model refinements which may ultimately lead to an accurate analytic model. Slight variations in the core-surface temperature, in the range of a few millikelvins, result in significant increases in convective flow and produce more realistic magnetic fields.<ref>{{Cite journal | last = Sakuraba | first = Ataru | coauthors = Paul H. Roberts | title = Generation of a strong magnetic field using uniform heat flux at the surface of the core | journal = Nature Geoscience | volume = 2 | pages = 802–805 | publisher = Nature Publishing Group | date = 4 October 2009 | doi = 10.1038/ngeo643|bibcode = 2009NatGe...2..802S | issue=11}}</ref><ref>{{Cite journal | last = Buffett | first = Bruce | title = Geodynamo: A matter of boundaries | journal = Nature Geoscience | issue = 2 | pages = 741–742 | publisher = Nature Publishing Group | year = 2009 | doi = 10.1038/ngeo673|bibcode = 2009NatGe...2..741B | volume=2}}</ref>

==Formal definition==
Dynamo theory describes the process through which a rotating, convecting, and electrically conducting fluid acts to maintain a magnetic field. This theory is used to explain the presence of anomalously long-lived magnetic fields in astrophysical bodies. The conductive fluid in the geodynamo is liquid iron in the outer core, and in the [[solar dynamo]] is ionized gas at the [[tachocline]]. Dynamo theory of astrophysical bodies uses [[Magnetohydrodynamics|magnetohydrodynamic]] equations to investigate how the fluid can continuously regenerate the magnetic field.

It was once believed that the [[dipole]], which comprises much of the [[Earth's magnetic field]] and is misaligned along the rotation axis by 11.3 degrees, was caused by permanent magnetization of the materials in the earth. This means that dynamo theory was originally used to explain the Sun's magnetic field in its relationship with that of the Earth. However, this hypothesis, which was initially proposed by [[Joseph Larmor]] in 1919, has been modified due to extensive studies of magnetic secular variation, [[paleomagnetism]] (including [[geomagnetic reversal|polarity reversal]]s), seismology, and the solar system's abundance of elements. Also, the application of the theories of [[Carl Friedrich Gauss]] to magnetic observations showed that Earth's magnetic field had an internal, rather than external, origin.

There are three requisites for a dynamo to operate:

*An electrically conductive fluid medium
*Kinetic energy provided by planetary rotation
*An internal energy source to drive convective motions within the fluid.<ref>{{cite book |author=E. Pallé |title=The Earth as a Distant Planet: A Rosetta Stone for the Search of Earth-Like Worlds (Astronomy and Astrophysics Library) |publisher=Springer |location=Berlin |year=2010 |pages=316–317 |url=http://books.google.com/books?id=qLuVCJtRTV0C&pg=PA316&dq=the+dynamo+theory+holds+that#v=onepage&q=the%20dynamo%20theory%20holds%20that&f=false|isbn=1-4419-1683-0 |accessdate=17 july 2010}}</ref>

In the case of the Earth, the magnetic field is induced and constantly maintained by the convection of liquid iron in the outer core. A requirement for the induction of field is a rotating fluid. Rotation in the outer core is supplied by the [[Coriolis effect]] caused by the rotation of the Earth. The coriolis force tends to organize fluid motions and electric currents into columns (also see [[Taylor column]]s) aligned with the rotation axis. Induction or creation of magnetic field is described by the induction equation:

:<math>\frac{\partial \mathbf{B}}{\partial t} = \eta \nabla^2 \mathbf{B} + \nabla \times (\mathbf{u} \times \mathbf{B}) </math>

where '''u''' is velocity, '''B''' is magnetic field, ''t'' is time, and <math>\eta=1/\sigma\mu</math> is the [[magnetic diffusivity]] with <math>\sigma</math> electrical conductivity and <math>\mu</math> [[Permeability (electromagnetism)|permeability]]. The ratio of the second term on the right hand side to the first term gives the [[Magnetic Reynolds number]], a dimensionless ratio of advection of magnetic field to diffusion.

===Tidal heating supporting a dynamo===
Tidal forces between celestial orbiting bodies causes friction that heats up the interiors of these orbiting bodies. This is known as tidal heating, and it helps create the liquid interior criteria, providing that this interior is conductive, that is required to produce a dynamo. For example, Saturn's Enceladus and Jupiter's Io have enough tidal heating to liquify its inner core, even if a moon is not conductive to support a dynamo.
<ref name="Enceladus">
{{cite web | url=http://www.nasa.gov/mission_pages/cassini/whycassini/cassini20100708-b.html | title=Saturn's Icy Moon May Keep Oceans Liquid with Wobble | publisher=NASA | date=October 6, 2010 | accessdate=August 14, 2012 | author=Steigerwald, Bill}}</ref>
<ref name="Io geologic">{{cite web | url=https://clas.asu.edu/node/12161 | title=Geologic map of Jupiter’s moon Io details an otherworldly volcanic surface | publisher=Astrogeology Science Center | date=March 19, 2012 | accessdate=August 14, 2012 | author=Cassis, Nikki}}
</ref> Mercury, despite its small size, has a magnetic field, because it has a conductive liquid core created by its iron composition and friction resulting from its highly elliptical orbit.<ref name="mercury core">{{cite web | url=http://carnegiescience.edu/news/mercury%E2%80%99s_surprising_core_and_landscape_curiosities | title=Mercury’s Surprising Core and Landscape Curiosities | publisher=Carnegie Institution for Science | work=MESSENGER | date=March 21, 2012 | accessdate=August 14, 2012}}</ref> It is theorized that the Moon once had a magnetic field, based on evidence from magnetized lunar rocks, due to its short-lived closer distance to Earth creating tidal heating.
<ref name="lunar dynamo">
{{cite web | url=http://news.ucsc.edu/2011/11/lunar-dynamo.html | title=Ancient lunar dynamo may explain magnetized moon rocks | publisher=University of California | date=November 09, 2011 | accessdate=August 14, 2012 | author=Stevens, Tim}}
</ref> An orbit and rotation of a planet helps provide a liquid core, and supplements kinetic energy that supports a dynamo action.

==Kinematic dynamo theory==
In kinematic dynamo theory the velocity field is prescribed, instead of being a dynamic variable. This method cannot provide the time variable behavior of a fully nonlinear chaotic dynamo but is useful in studying how magnetic field strength varies with the flow structure and speed.

Using [[Maxwell's equations]] simultaneously with the curl of [[Ohm's Law]], one can derive what is basically the linear eigenvalue equation for magnetic fields ('''B''') which can be done when assuming that the magnetic field is independent from the velocity field. One arrives at a critical ''magnetic [[Reynolds number]]'' above which the flow strength is sufficient to amplify the imposed magnetic field, and below which it decays.

The most functional feature of kinematic dynamo theory is that it can be used to test whether a velocity field is or is not capable of dynamo action. By applying a certain velocity field to a small magnetic field, it can be determined through observation whether the magnetic field tends to grow or not in reaction to the applied flow. If the magnetic field does grow, then the system is either capable of dynamo action or is a dynamo, but if the magnetic field does not grow, then it is simply referred to as non-dynamo.

The [[membrane paradigm]] is a way of looking at [[black hole]]s that allows for the material near their surfaces to be expressed in the language of dynamo theory.

==Nonlinear dynamo theory==
The kinematic approximation becomes invalid when the magnetic field becomes strong enough to affect the fluid motions. In that case the velocity field becomes affected by the [[Lorentz force]], and so the induction equation is no longer linear in the magnetic field. In most cases this leads to a quenching of the amplitude of the dynamo. Such dynamos are sometimes also referred to as
[http://www.scholarpedia.org/article/Hydromagnetic_Dynamo_Theory hydromagnetic dynamos].
Virtually all dynamos in astrophysics and geophysics are hydromagnetic dynamos.

Numerical models are used to simulate fully nonlinear dynamos. A minimum of 5 equations are needed. They are as follows. The induction equation, see above. Maxwell's equation:

:<math> \nabla \cdot \mathbf{B}=0</math>

The (sometimes) [[Boussinesq approximation|Boussinesq]] conservation of mass:

:<math> \nabla \cdot \mathbf{u} = 0 </math>

The (sometimes) Boussinesq conservation of momentum, also known as the Navier-Stokes equation:

:<math> \frac{D\mathbf{u}}{Dt} = -\nabla p + \nu \nabla^2 \mathbf{u} + \rho'\mathbf{g} + 2\mathbf{\Omega} \times \mathbf{u} + \mathbf{\Omega} \times \mathbf{\Omega} \times \mathbf{R} + \mathbf{J} \times \mathbf{B} </math>

where <math>\nu</math> is the kinematic [[viscosity]], <math>\rho'</math> is the density perturbation that provides buoyancy (for thermal convection <math>\rho'=\alpha\Delta T</math>, <math>\Omega</math> is the [[Earth's rotation|rotation rate of the Earth]], and <math>\mathbf{J}</math> is the electrical current density.

Finally, a transport equation, usually of heat (sometimes of light element concentration):

:<math> \frac{\partial T}{\partial t} = \kappa \nabla^2 T +\epsilon </math>

where ''T'' is temperature, <math>\kappa=k/\rho c_p</math> is the thermal diffusivity with ''k'' thermal conductivity, <math>c_p</math> heat capacity, and <math>\rho</math> density, and <math>\epsilon</math> is an optional heat source. Often the pressure is the dynamic pressure, with the hydrostatic pressure and centripetal potential removed. These equations are then non-dimensionalized, introducing the non-dimensional parameters,

:<math>Ra=\frac{g\alpha T D^3}{\nu \kappa} , E=\frac{\nu}{\Omega D^2} , Pr=\frac{\nu}{\kappa} , Pm=\frac{\nu}{\eta} </math>

where ''Ra'' is the [[Rayleigh number]], ''E'' the [[Ekman number]], ''Pr'' and ''Pm'' the [[Prandtl number|Prandtl]] and magnetic [[Prandtl number]]. Magnetic field scaling is often in Elsasser number units <math>B=(\rho \Omega/\sigma)^{1/2}</math>.

==See also==
{{Wikipedia books|Geomagnetism}}
* [[Antidynamo theorem]]
* [[Dynamo]]
* [[Maxwell's equations]]
* [[Rotating magnetic field]]

==References==
{{reflist|30em}}
{{refbegin}}
* {{cite web |last=Demorest |first=Paul |title=Dynamo Theory and Earth's magnetic Field (term paper) |date=21 May 2001 |url=http://setiathome.berkeley.edu/~pauld/etc/210BPaper.pdf |format=pdf |accessdate=14 October 2011}}
* {{cite web |last=Fitzpatrick |first=Richard |title=MHD Dynamo Theory |date=18 May 2002 |url=http://farside.ph.utexas.edu/teaching/plasma/lectures/node70.html |work=Plasma Physics |publisher=[[University of Texas at Austin]] |accessdate=14 October 2011}}
* {{cite book|last=Merrill|first= Ronald T.|last2=McElhinny|first2=Michael W.|last3=McFadden|first3=Phillip L.|title=The magnetic field of the earth: Paleomagnetism, the core, and the deep mantle|publisher=[[Academic Press]]|year=1996|isbn=978-0-12-491246-5 |ref=harv}}
* {{cite web |title=Chapter 12: The dynamo process |url=http://www.phy6.org/earthmag/dynamos.htm |work=The Great Magnet, the Earth |last=Stern |first=David P. |accessdate=14 October 2011}}
* {{cite web |title=Chapter 13: Dynamo in the Earth's Core |url=http://www.phy6.org/earthmag/dynamos2.htm |work=The Great Magnet, the Earth |last=Stern |first=David P. |accessdate=14 October 2011}}
{{refend}}

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Galvanic cell