Dynamo theory

The dynamo theory describes the generation and behavior of magnetic fields in electrically conductive matter ( plasma ). It is part of magnetohydrodynamics (MHD).

The " dynamo effect ", ie the generation of magnetic fields, has its cause in electromagnetic induction due to the interaction between convection in electrically conductive matter and its rotation.

The magnetic fields of the earth, the sun and other astronomical objects can be explained by the dynamo theory. For a better differentiation from technical dynamos ( electrical generators ), such dynamos are then also called magnetohydrodynamic or MHD dynamos .

Induction equation

The theoretical basis for describing dynamos is the induction equation:

{\ displaystyle {\ partial \ mathbf {B} \ over \ partial t} = R_ {m} \ nabla \ times (\ mathbf {v} \ times \ mathbf {B}) - \ nabla \ times \ eta \ nabla \ times \ mathbf {B}}

Here mean:

${\ displaystyle \ mathbf {B}}$	Magnetic field , more precisely: magnetic induction
${\ displaystyle \ mathbf {v}}$	Velocity field of matter
${\ displaystyle \ eta = {1 / \ mu \ sigma}}$	magnetic diffusivity, where mean magnetic permeability and electrical conductivity ${\ displaystyle \ mu}$ ${\ displaystyle \ sigma}$
${\ displaystyle R_ {m} = {V / R \ mathrm {H}}}$	magnetic Reynolds number , with typical orders of magnitude for speed , length and diffusivity . ${\ displaystyle V}$ ${\ displaystyle R}$ ${\ displaystyle \ mathrm {H}}$

The induction equation can be derived from Maxwell's equations and Ohm's law .

Antidynamotheorems

Antidynamo theorems make statements about conditions under which no dynamo process can take place. They provide an insight into how dynamos work, as they limit the variety of solutions to the dynamo equation, and thus show which requirements are essential for a functioning dynamo.

Cowling theorem

The cowling theorem states that an axially symmetric magnetic field cannot be sustained by any dynamo process.

Elsasser theorem

The Elsasser theorem (also toroidal theorem) states that a purely toroidal flow cannot sustain a dynamo. However, in a spherical geometry this is only the case under the restrictive condition that the electrical conductivity is not angle-dependent.