Editing Dynamo theory (section)

===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]].