7.5 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Dynamo theory | 4/5 | https://en.wikipedia.org/wiki/Dynamo_theory | reference | science, encyclopedia | 2026-05-05T11:05:18.320341+00:00 | kb-cron |
=== 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
J
×
B
{\displaystyle \;\mathbf {J} \times \mathbf {B} \;}
on the outer core matter, whose velocity is
u
{\displaystyle \mathbf {u} }
. This work is the result of non-magnetic forces acting on the fluid. Of those, the gravitational force and the centrifugal force are 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 ×10−2 pascal-second) due to its liquidity. Thus the main time-averaged contribution to the work is from Coriolis force, whose size is
−
2
ρ
Ω
×
u
,
{\displaystyle \;-2\rho \,\mathbf {\Omega } \times \mathbf {u} \;,}
though this quantity and
J
×
B
{\displaystyle \mathbf {J} \times \mathbf {B} }
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 J is itself the result of the magnetic field according to 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,
J
B
∼
ρ
Ω
u
{\displaystyle \;J\,B\sim \rho \,\Omega \,u\;}
and
J
∼
σ
u
B
{\displaystyle \;J\sim \sigma uB\;}
, giving
σ
u
B
2
∼
ρ
Ω
u
,
{\displaystyle \;\sigma \,u\,B^{2}\sim \rho \,\Omega \,u\;,}
or:
B
∼
ρ
Ω
σ
{\displaystyle B\sim {\sqrt {{\frac {\,\rho \,\Omega \,}{\sigma }}\;}}}
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 reversed, though in general it lies on a similar axis to that of
Ω
{\displaystyle \mathbf {\Omega } }
. For earth outer core, ρ is approximately 104 kg/m3, Ω = 2π/day = 7.3×10−5/second and σ is approximately 107Ω−1m−1 . This gives 2.7×10−4 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 (Router core⁄REarth )3 = (2890⁄6370)3 = 0.093 , giving 2.5×10−5 tesla, not far from the measured value of 3×10−5 tesla at the equator.
== Numerical models ==
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 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.
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 and one in the United States. 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. Following this breakthrough, there was a large swell in development of reasonable, three dimensional dynamo models. 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. 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 polarity reversals have also been successfully recreated.
=== Observations ===
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 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. Simplified geodynamo models have shown relationships between the dynamo number (determined by 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. In many models, it appears that magnetic fields have somewhat random magnitudes that follow a normal trend that average to zero. 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.