Editing Dynamo theory (section)

==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.