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{{Short description|Mechanism by which a celestial body generates a magnetic field}}
{{About|a proposed theory for the source of a planet's magnetic field|an explanation of the operation of a mechanical dynamo|Dynamo}}
[[File:Dynamo Theory - Outer core convection and magnetic field generation.svg|upright=1.5|thumb|Illustration of the dynamo mechanism that generates the Earth's magnetic field: [[convection]] currents of fluid metal in the Earth's [[Internal structure of Earth|outer core]], driven by heat flow from the inner core, organized into rolls by the [[Coriolis force]], generate circulating electric currents, which supports the magnetic field.<ref>{{cite web |title=How does the Earth's core generate a magnetic field? |website=USGS FAQs |publisher=United States Geological Survey |url=http://www.usgs.gov/faq/?q=categories/9782/2738 |access-date=21 October 2013 |url-status=dead |archive-url=https://web.archive.org/web/20150118213104/http://www.usgs.gov/faq/?q=categories%2F9782%2F2738 |archive-date=18 January 2015 }}</ref>]]

In [[physics]], the '''dynamo theory''' proposes a mechanism 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.  A dynamo is thought to be the source of the [[Earth's magnetic field]] and the magnetic fields of Mercury and the [[Jovian planets]].

== 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 1822, [[André-Marie Ampère]] proposed that internal currents are responsible for Earth's magnetism.<ref>{{cite book |author-last=Ampère |author-first=André-Marie |date=1822  |url=http://catalogue.bnf.fr/ark:/12148/cb37284089w |language=French |location=Paris |publisher=Crochard |title=Recueil d'observations électro-dynamiques : contenant divers mémoires, notices, extraits de lettres ou d'ouvrages périodiques sur les sciences relatifs à l'action mutuelle de deux courants électriques, à celle qui existe entre un courant électrique et un aimant ou le globe terrestre, et à celle de deux aimants l'un sur l'autre}}{{No ISBN}}</ref> 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 alternative explanations. 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| s2cid=143344977 }}</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 | access-date=2007-02-28 }}</ref>

At the dawn of the 21st century, numerical modeling of the Earth's magnetic field is far from precise. 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 indicated that model refinements {{which|date=February 2022}} 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 |author2=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 | 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 = 11 | pages = 741–742 | 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.<ref name="hydromagnetic dynamo">{{cite journal|doi=10.4249/scholarpedia.2309|doi-access=free |title=Hydromagnetic dynamo theory |year=2007 |last1=Brandenburg |first1=Axel |journal=Scholarpedia |volume=2 |issue=3 |page=2309 |bibcode=2007SchpJ...2.2309B }}</ref>

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=https://books.google.com/books?id=qLuVCJtRTV0C&pg=PA316 |isbn=978-1-4419-1683-9 |access-date=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 generation of magnetic field is described by the [[induction equation]]:
<math display="block">\frac{\partial \mathbf{B}}{\partial t} = \eta \nabla^2 \mathbf{B} + \nabla \times (\mathbf{u} \times \mathbf{B}) </math>
where {{math|'''u'''}} is velocity, {{math|'''B'''}} is magnetic field, {{mvar|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 cause friction that heats up their interiors. This is known as tidal heating, and it helps keep the interior in a liquid state. A liquid interior that can conduct electricity is required to produce a dynamo. Saturn's Enceladus and Jupiter's Io have enough tidal heating to liquify their inner cores, but they may not create a dynamo because they cannot conduct electricity.<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 | access-date=August 14, 2012 | author=Steigerwald, Bill | archive-date=March 24, 2015 | archive-url=https://web.archive.org/web/20150324222212/http://www.nasa.gov/mission_pages/cassini/whycassini/cassini20100708-b.html | url-status=dead }}</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 | access-date=August 14, 2012 | author=Cassis, Nikki }}{{dead link|date=September 2017 |bot=InternetArchiveBot |fix-attempted=yes }}</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 | website=MESSENGER | date=March 21, 2012 | access-date=August 14, 2012 | archive-date=January 18, 2015 | archive-url=https://web.archive.org/web/20150118213837/http://carnegiescience.edu/news/mercury%E2%80%99s_surprising_core_and_landscape_curiosities | url-status=dead }}</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 9, 2011 | access-date=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: The model makes no provision for  the flow distorting in response to the magnetic field. This method cannot provide the time variable behaviour of a fully nonlinear chaotic dynamo, but can be used to study 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 a linear eigenvalue equation for magnetic fields ('''{{math|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 the magnetic field dissipates.

=== Practical measure of possible dynamos ===
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 experimentally applying a certain velocity field to a small magnetic field, one can observe whether the magnetic field tends to grow (or not) in response 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 “not a dynamo”.

An analogous method called 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.

=== Spontaneous breakdown of a topological supersymmetry ===
Kinematic dynamo can be also viewed as the phenomenon of the spontaneous breakdown of the topological supersymmetry of the associated stochastic differential equation related to the flow of the background matter.<ref>{{cite journal |author1=Ovchinnikov, I.V. |author2=Ensslin, T.A. |title=Kinematic dynamo, supersymmetry breaking, and chaos |journal=Physical Review D |volume=93 |issue=8 |pages=085023 |date=April 2016 |doi=10.1103/PhysRevD.93.085023 |arxiv=1512.01651 |bibcode=2016PhRvD..93h5023O |s2cid=59367815 }}</ref> Within [[Supersymmetric theory of stochastic dynamics|stochastic supersymmetric theory]], this supersymmetry is an intrinsic property of ''all'' [[stochastic differential equation]]s, its interpretation is that the model's phase space preserves continuity via continuous time flows. When the continuity of that flow spontaneously breaks down, the system is in the stochastic state of [[Chaos theory|''deterministic chaos'']].<ref>{{cite journal |author=Ovchinnikov, I.V. |date=March 2016 |title=Introduction to Supersymmetric Theory of Stochastics |journal=Entropy |volume=18 |issue=4 |pages=108 |doi=10.3390/e18040108 |arxiv=1511.03393 |bibcode=2016Entrp..18..108O |s2cid=2388285|doi-access=free }}</ref> In other words, kinematic dynamo arises because of chaotic flow in the underlying background matter.

==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 ''hydromagnetic dynamos''.<ref>{{cite journal|last1=Parker|first1=Eugene N.|title=Hydromagnetic Dynamo Models|journal=The Astrophysical Journal|date=September 1955|volume=122|pages=293&ndash;314|doi=10.1086/146087|bibcode = 1955ApJ...122..293P }}</ref>
Virtually all dynamos in astrophysics and geophysics are hydromagnetic dynamos.

The main idea of the theory is that any small magnetic field existing in the outer core creates currents in the moving fluid there due to Lorentz force. These currents create further magnetic field due to [[Ampere's law]]. With the fluid motion, the currents are carried in a way that the magnetic field gets stronger (as long as <math>\; \mathbf{u} \cdot ( \mathbf{J} \times \mathbf{B} ) \;</math> is negative<ref name=Kono2002>{{cite journal |last1=Kono |first1=Masaru |first2=Paul H. |last2=Roberts |year=2002 |title=Recent geodynamo simulations and observations of the geomagnetic field |journal=[[Reviews of Geophysics]] |volume=40 |issue=4 |pages=1–53 |doi=10.1029/2000RG000102 |bibcode=2002RvGeo..40.1013K |doi-access=free}}</ref>). Thus a "seed" magnetic field can get stronger and stronger until it reaches some value that is related to existing non-magnetic forces.

Numerical models are used to simulate fully nonlinear dynamos.  The following equations are used:

*The induction equation, presented above.
*Maxwell's equations for negligible electric field: <math display="block">\begin{align}
 &\nabla \cdot \mathbf{B} = 0 \\[1ex]
 &\nabla \times \mathbf{B} = \mu_0 \mathbf{J}
\end{align}</math>
*The [[continuity equation]] for [[conservation of mass]], for which the [[Boussinesq approximation (buoyancy)|Boussinesq approximation]] is often used: <math display="block"> \nabla \cdot \mathbf{u} = 0, </math>
*The [[Navier–Stokes equations|Navier-Stokes equation]] for conservation of [[momentum]], again in the same approximation, with the magnetic force and gravitation force as the external forces: <math display="block"> \frac{D\mathbf{u}}{Dt} = -\frac{1}{\rho_0}\nabla p + \nu \nabla^2 \mathbf{u} + \rho' \mathbf{g} + 2 \boldsymbol{\Omega} \times \mathbf{u} + \boldsymbol{\Omega} \times \boldsymbol{\Omega} \times \mathbf{R} + \frac{1}{\rho_0}\mathbf{J} \times \mathbf{B} ~, </math> where <math>\,\nu\,</math> is the kinematic [[viscosity]], <math>\,\rho_0\,</math>is the mean density and <math>\rho'</math> is the relative density perturbation that provides buoyancy (for thermal convection <math>\;\rho' = \alpha \Delta T\;</math> where <math>\,\alpha\,</math> is [[coefficient of thermal expansion]]), <math>\,\Omega\,</math> is the [[Earth's rotation|rotation rate of the Earth]], and <math>\,\mathbf{J}\,</math> is the electric current density.
*A transport equation, usually of heat (sometimes of light element concentration): <math display="block"> \frac{\,\partial T\,}{\partial t} = \kappa \nabla^2 T + \varepsilon </math> where {{mvar|T}} is temperature, <math>\;\kappa = k / \rho c_p \;</math> is the thermal diffusivity with {{mvar|k}} thermal conductivity, <math>\,c_p\,</math> heat capacity, and <math>\rho</math> density, and <math>\varepsilon</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 display="block"> R_\mathsf{a} = \frac{\, g \alpha T D^3\,}{\nu \kappa} \;,\quad E = \frac{\nu}{\,\Omega D^2\,} \;,\quad P_\mathsf{r} = \frac{\,\nu\,}{\kappa} \;,\quad P_\mathsf{m} = \frac{\,\nu\,}{\eta} </math>
where {{mvar|R}}{{sub|a}} is the [[Rayleigh number]], {{mvar|E}} the [[Ekman number]], {{mvar|P}}{{sub|r}} and {{mvar|P}}{{sub|m}} 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>

===Energy conversion between magnetic and kinematic energy===
The scalar product of the above form of Navier-Stokes equation with <math>\;\rho_0 \mathbf{u}\;</math> gives the rate of increase of kinetic energy density, <math>\; \tfrac{1}{2} \rho_0 u^2 c \;</math>, on the left-hand side. The last term on the right-hand side is then <math>\; \mathbf{u} \cdot ( \mathbf{J} \times \mathbf{B} ) \;</math>, the local contribution to the kinetic energy due to [[Lorentz force#Continuous charge distribution|Lorentz force]].

The scalar product of the induction equation with <math display="inline">\tfrac{1}{\mu_0}\mathbf{B}</math> gives the rate of increase of the magnetic energy density, <math>\;\tfrac{1}{2} \mu_0 B^2\;</math>, on the left-hand side. The last term on the right-hand side is then <math display="inline">\tfrac{1}{\mu_0} \mathbf{B} \cdot \left( \nabla \times \left( \mathbf{u} \times \mathbf{B} \right) \right) \;.</math> Since the equation is volume-integrated, this term is [[integration by parts|equivalent up to a boundary term]] (and with the double use of the [[Triple product#Scalar triple product|scalar triple product]] identity) to <math display="inline">\; -\mathbf{u} \cdot \left( \tfrac{1}{\mu_0} \left( \nabla \times \mathbf{B} \right) \times \mathbf{B} \right) = -\mathbf{u} \cdot \left( \mathbf{J} \times \mathbf{B} \right) ~</math> (where one of Maxwell's equations was used). This is the local contribution to the magnetic energy due to fluid motion.

Thus the term <math>\;-\mathbf{u} \cdot ( \mathbf{J} \times \mathbf{B} ) \;</math> is the rate of transformation of kinetic energy to magnetic energy. This has to be non-negative at least in part of the volume, for the dynamo to produce magnetic field.<ref name=Kono2002/>

From the diagram above, it is not clear why this term should be positive. A simple argument can be based on consideration of net effects. To create the magnetic field, the net electric current must wrap around the axis of rotation of the planet. In that case, for the term to be positive, the net flow of conducting matter must be towards the axis of rotation. The diagram only shows a net flow from the poles to the equator. However mass conservation requires an additional flow from the equator toward the poles. If that flow was along the axis of rotation, that implies the circulation would be completed by a flow from the ones shown towards the axis of rotation, producing the desired effect.

===Order of magnitude of the magnetic field created by Earth's dynamo===
The above formula for the rate of conversion of kinetic energy to magnetic energy, is equivalent to a rate of work done by a force of <math>\;\mathbf{J} \times \mathbf{B}\;</math> on the outer core matter, whose velocity is <math>\mathbf{u}</math>. This work is the result of non-magnetic forces acting on the fluid.

Of those, the gravitational force and the [[centrifugal force]] are [[conservative vector field|conservative]] and therefore have no overall contribution to fluid moving in closed loops. Ekman number (defined above), which is the ratio between the two remaining forces, namely the viscosity and Coriolis force, is very low inside Earth's outer core, because its viscosity is low (1.2–1.5 ×10{{sup|&minus;2}} [[pascal-second]]<ref name="Wijs1998"/>) due to its liquidity.

Thus the main time-averaged contribution to the work is from Coriolis force, whose size is <math>\;-2\rho\,\mathbf{\Omega} \times \mathbf{u} \;,</math> though this quantity and <math>\mathbf{J} \times \mathbf{B}</math> are related only indirectly and are not in general equal locally (thus they affect each other but not in the same place and time).

The current density {{mvar|J}} is itself the result of the magnetic field according to [[Ohm's law#Magnetic effects|Ohm's law]]. Again, due to matter motion and current flow, this is not necessarily the field at the same place and time. However these relations can still be used to deduce orders of magnitude of the quantities in question.

In terms of order of magnitude, <math>\; J \, B \sim \rho\, \Omega\, u \;</math> and <math>\; J \sim \sigma u B\;</math>, giving <math>\;\sigma\,u\, B^2 \sim \rho\, \Omega\,u \;,</math> or:
<math display="block">B \sim \sqrt{\frac{\,\rho\,\Omega\,}{\sigma}\;}</math>

The exact ratio between both sides is the square root of [[Elsasser number]].

Note that the magnetic field direction cannot be inferred from this approximation (at least not its sign) as it appears squared, and is, indeed, sometimes [[Earth's magnetic field#Magnetic field reversals|reversed]], though in general it lies on a similar axis to that of <math>\mathbf{\Omega}</math>.

For earth outer core, {{mvar|&rho;}} is approximately 10<sup>4</sup> kg/m<sup>3</sup>,<ref name = "Wijs1998">de Wijs, G. A., Kresse, G., Vočadlo, L., Dobson, D., Alfe, D., Gillan, M. J., & Price, G. D. (1998). [https://web.archive.org/web/20180420202850/https://pdfs.semanticscholar.org/8072/57a2b0ecc863b2f7ae143b9f54ae4b6d10cd.pdf The viscosity of liquid iron at the physical conditions of the Earth's core.] Nature, 392(6678), 805.</ref> &nbsp; {{math|&Omega;}} = 2{{pi}}/day = 7.3×10<sup>−5</sup>/second &nbsp; and &nbsp; {{mvar|&sigma;}} &nbsp; is approximately &nbsp; 10<sup>7</sup>Ω<sup>−1</sup>m<sup>−1</sup>&nbsp;.<ref>Ohta, K., Kuwayama, Y., Hirose, K., Shimizu, K., & Ohishi, Y. (2016). Experimental determination of the electrical resistivity of iron at Earth’s core conditions. Nature, 534(7605), 95. Link to a [http://www.spring8.or.jp/pdf/en/res_fro/16/094_095.pdf summary]</ref>
This gives &nbsp; 2.7×10<sup>−4</sup>&nbsp;[[Tesla (unit)|Tesla]].

The magnetic field of a [[magnetic dipole]] has an inverse cubic dependence in distance, so its order of magnitude at the earth surface can be approximated by multiplying the above result with {{nowrap|1= {{big|(}}{{frac|{{mvar|R}}{{sub|outer core}}|{{mvar|R}}{{sub|Earth}} }}{{big|)}}{{sup|3}} = ({{frac|2890|6370}}){{sup|3}} = 0.093 ,}} giving &nbsp; 2.5×10<sup>−5</sup>&nbsp;Tesla, not far from the measured value of 3×10<sup>−5</sup>&nbsp;Tesla at the [[equator]].

==Numerical models==
[[File:Geodynamo Before Reversal.gif|thumb|A visual representation of the Glatzmaier model before dipole reversal]]
Broadly, models of the geodynamo attempt to produce magnetic fields consistent with observed data given certain conditions and equations as mentioned in the sections above. Implementing the [[Magnetohydrodynamics|magnetohydrodynamic]] equations successfully was of particular significance because they pushed dynamo models to self-consistency. Though geodynamo models are especially prevalent, dynamo models are not necessarily restricted to the geodynamo; solar and general dynamo models are also of interest. Studying dynamo models has utility in the field of geophysics as doing so can identify how various mechanisms form magnetic fields like those produced by astrophysical bodies like Earth and how they cause magnetic fields to exhibit certain features, such as pole reversals.

The equations used in numerical models of dynamo are highly complex. For decades, theorists were confined to two dimensional ''kinematic dynamo'' models described above, in which the fluid motion is chosen in advance and the effect on the magnetic field calculated. The progression from linear to nonlinear, three dimensional models of dynamo was largely hindered by the search for solutions to magnetohydrodynamic equations, which eliminate the need for many of the assumptions made in kinematic models and allow self-consistency. 
[[File:Geodynamo In Reversal.gif|thumb|A visual representation of the Glatzmaier model during dipole reversal]]
The first ''self-consistent'' dynamo models, ones that determine both the fluid motions and the magnetic field, were developed by two groups in 1995, one in Japan<ref>{{cite journal| last1=Kageyama|first1=Akira| last2=Sato|first2=Tetsuya| date=1 January 1995|title=Computer simulation of a magnetohydrodynamic dynamo. II|journal=Physics of Plasmas| volume=2| issue=5| pages=1421–1431| bibcode=1995PhPl....2.1421K|doi=10.1063/1.871485}}</ref> and one in the United States.<ref name="selfconsistent">{{cite journal|last1=Glatzmaier|first1=Gary A.|last2=Roberts|first2=Paul H.|year=1995|title=A three-dimensional self-consistent computer simulation of a geomagnetic field reversal| journal=Nature| volume=377| issue=6546| pages=203–209|bibcode=1995Natur.377..203G|doi=10.1038/377203a0|s2cid=4265765}}</ref><ref>{{cite journal| last1=Glatzmaier| first1=G| last2=Roberts|first2=Paul H.|year=1995|title=A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle|journal=Physics of the Earth and Planetary Interiors| volume=91| issue=1–3| pages=63–75| bibcode=1995PEPI...91...63G|doi=10.1016/0031-9201(95)03049-3}}</ref> The latter was made as a model with regards to the geodynamo and received significant attention because it successfully reproduced some of the characteristics of the Earth's field.<ref name="Kono2002" /> Following this breakthrough, there was a large swell in development of reasonable, three dimensional dynamo models.<ref name="Kono2002" />

Though many self-consistent models now exist, there are significant differences among the models, both in the results they produce and the way they were developed.<ref name="Kono2002" /> Given the complexity of developing a geodynamo model, there are many places where discrepancies can occur such as when making assumptions involving the mechanisms that provide energy for the dynamo, when  choosing values for parameters used in equations, or when normalizing equations. In spite of the many differences that may occur, most models have shared features like clear axial dipoles. In many of these models, phenomena like [[secular variation]] and [[Geomagnetic reversal|geomagnetic polarity reversals]] have also been successfully recreated.<ref name="Kono2002" />

=== Observations ===
[[File:Geodynamo After Reversal.gif|thumb|A visual representation of the Glatzmaier model after dipole reversal]]
Many observations can be made from dynamo models. Models can be used to estimate how magnetic fields vary with time and can be compared to observed [[Paleomagnetism|paleomagnetic]] data to find similarities between the model and the Earth. Due to the uncertainty of paleomagnetic observations, however, comparisons may not be entirely valid or useful.<ref name="Kono2002" /> Simplified geodynamo models have shown relationships between the dynamo number (determined by [[Differential rotation|variance in rotational rates]] in the outer core and mirror-asymmetric convection (e.g. when convection favors one direction in the north and the other in the south)) and magnetic pole reversals as well as found similarities between the geodynamo and the Sun's dynamo.<ref name="Kono2002" /> In many models, it appears that magnetic fields have somewhat random magnitudes that follow a normal trend that average to zero.<ref name="Kono2002" /> In addition to these observations, general observations about the mechanisms powering the geodynamo can be made based on how accurately the model reflects actual data collected from Earth.

=== Modern modelling ===
The complexity of dynamo modelling is so great that models of the geodynamo are limited by the current power of [[supercomputer]]s, particularly because calculating the [[Ekman number|Ekman]] and [[Rayleigh number|Rayleigh]] number of the outer core is extremely difficult and requires a vast number of computations.

Many improvements have been proposed in dynamo modelling since the self-consistent breakthrough in 1995.  One suggestion in studying the complex magnetic field changes is applying [[spectral method]]s to simplify computations.<ref>{{Cite journal| last1=Avery|first1=Margaret S.| last2=Constable|first2=Catherine G.|author-link2=Catherine Constable| last3=Davies|first3=Christopher J.|last4=Gubbins|first4=David|date=2019-01-01|title=Spectral methods for analyzing energy balances in geodynamo simulations|url=http://eprints.whiterose.ac.uk/140318/1/energy_balance%28freq%29_revised_v2.pdf | journal=Physics of the Earth and Planetary Interiors | volume=286| pages=127–137| bibcode=2019PEPI..286..127A| doi=10.1016/j.pepi.2018.10.002| issn=0031-9201|via=|doi-access=free}}</ref> Ultimately, until considerable improvements in computer power are made, the methods for computing realistic dynamo models will have to be made more efficient, so making improvements in methods for computing the model is of high importance for the advancement of numerical dynamo modelling.

==Notable people==
<!---♦♦♦ Only add a person to this list if they already have their own article on the English Wikipedia ♦♦♦---> 
<!---♦♦♦ Please keep the list in alphabetical order by LAST NAME ♦♦♦--->
*[[Stanislav I. Braginsky]], research geophysicist

==See also==
* [[Antidynamo theorem]]
* [[Rotating magnetic field]]
* [[Secular variation]]

==References==
{{reflist|25em}}
* {{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 |access-date=14 October 2011 |url-status=dead |archive-url=https://web.archive.org/web/20070221094040/http://setiathome.berkeley.edu/~pauld/etc/210BPaper.pdf |archive-date=21 February 2007 }}
* {{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 |website=Plasma Physics |publisher=[[University of Texas at Austin]] |access-date=14 October 2011}}
* {{cite book|last1=Merrill|first1= 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 }}
* {{cite web |title=Chapter 12: The dynamo process |url=http://www.phy6.org/earthmag/dynamos.htm |website=The Great Magnet, the Earth |last=Stern |first=David P. |access-date=14 October 2011}}
* {{cite web |title=Chapter 13: Dynamo in the Earth's Core  |url=http://www.phy6.org/earthmag/dynamos2.htm |website=The Great Magnet, the Earth |last=Stern |first=David P. |access-date=14 October 2011}}

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