Editing Dynamo theory (section)

==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]].