Dynamo theory

The Dynamo theory proposes a mechanism by which a celestial body such as the Earth generates a magnetic field.

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History of theory

In 1905, shortly after composing his Special Relativity paper, Albert Einstein described the origin of the earth’s magnetic field as being one of the great unsolved problems facing modern physicists. Since then, there have been many studies of the geodynamo problem based on historical measurements of the earth’s field and these works often descend into the domain of the theoretical mathematician.

Formal definition

Dynamo theory describes the process through which motion of a conductive body in the presence of a magnetic field acts to regenerate that magnetic field. This theory is used to explain the presence of anomalously long-lived magnetic fields in astrophysical bodies. In such bodies, dynamo action depends on the presence of highly conducting fluids such as the liquid iron of the Earth's outer core or the ionized gas of the sun. Dynamo theory of astrophysical bodies uses magnetohydrodynamic equations to investigate how the flow of the conducting materials in the interior of an object can continuously regenerate the magnetic fields of planetary and stellar bodies. It was actually once believed that the dipole, which comprises much of the Earth's magnetic field and is misaligned along the rotation axis by 11.3 degrees, was caused by permanent magnetization of the materials in the earth. This means that dynamo theory was originally used to explain the sun's magnetic field in its relationship with that of the Earth. However, this theory, which was initially proposed by Joseph Larmor in 1919, has been modified due to extensive studies of magnetic secular variation, paleomagnetism (including polarity reversals), seismology, and the solar system's abundance of elements. Also, the application of the theories of Carl Friedrich Gauss to magnetic observations proved that Earth's magnetic field had an internal, rather than external, origin.

In the case of the Earth, the magnetic field is believed to be caused by the convection of molten iron, within the outer liquid core, along with a Coriolis effect caused by the overall planetary rotation that tends to organize currents in rolls aligned along the north-south polar axis. When conducting fluid flows across an existing magnetic field, electric currents are induced, which in turn creates another magnetic field. When this magnetic field reinforces the original magnetic field, a dynamo is created which sustains itself. Similar magnetic fields are present in many celestial bodies including most stars such as the Sun (which contains conducting plasma) and active galactic nuclei.

Kinematic dynamo theory

Dynamo theory is a very complex concept to study. Often, college courses and research focuses mainly on kinematic dynamo theory, which is a more simplistic version of the former. It involves the vector velocity field, V, which is prescribed, instead of also varying according to the forces exerted upon the fluid. To examine this sector of the theory, an assumption that must be made is that the magnetic field has to be sufficiently small so that it cannot affect the velocity field. Because of this, the approach cannot divulge anything about the long-term behavior of a dynamo system. This analysis begins with the magnetohydrodynamic theory version of Ohm's law once it has been modified to include resistivity (resistivity is the reciprocal of conductivity σ, J is the current density), which is often assumed to be a constant in order to further simplify the investigation.

Using Maxwell’s Equations simultaneously with the curl of the aforementioned equation, one can derive what is basically the 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 it decays.

The most functional feature of kinematic dynamo theory is that it can be used to determine what fields or systems are or are not dynamos. By applying a certain velocity field to a small magnetic field, it can be determined through observation whether the magnetic field tends to grow or not in reaction 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 non-dynamo.

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.

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.

References

• Demorest, Paul. "Dynamo Theory and Earth's magnetic Field." 21 May 2001. [1]
• Fitzpatrick, Richard. "MHD Dynamo Theory." 18 May 2002. [2]

See also

• Dynamo
• Earth's magnetic field
• Maxwell's Equations
• Rotating magnetic field
• Magnetohydrodynamics
• Rapid-decay theory