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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Dynamo theory | 2/5 | https://en.wikipedia.org/wiki/Dynamo_theory | reference | science, encyclopedia | 2026-05-05T11:05:18.320341+00:00 | kb-cron |
=== 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. 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. 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. 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 (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 holes 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. Within stochastic supersymmetric theory, this supersymmetry is an intrinsic property of all stochastic differential equations, 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 deterministic chaos. 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. 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
u
⋅
(
J
×
B
)
{\displaystyle \;\mathbf {u} \cdot (\mathbf {J} \times \mathbf {B} )\;}
is negative). 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:
∇
⋅
B
=
0
∇
×
B
=
μ
0
J
{\displaystyle {\begin{aligned}&\nabla \cdot \mathbf {B} =0\\[1ex]&\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} \end{aligned}}}
The continuity equation for conservation of mass, for which the Boussinesq approximation is often used:
∇
⋅
u
=
0
,
{\displaystyle \nabla \cdot \mathbf {u} =0,}