Dynamo theory

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

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 Earth's liquid iron, 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 the Earth's magnetic field and is misaligned along the rotation axis by approximately 11 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.

In the case of the Earth, the magnetic field is believed to be caused by the convection of molten iron and nickel, within the outer planetary core, along with Coriolis effect caused by the overall planetary rotation. When conducting fluid flows across an existing magnetic field, electric currents are induced, creating 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 contain 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 of kinematic dynamo theory, which is a more simplistic version of the former. It involves the vector velocity field, V, which is prescribed. 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 (J is the current density), which is assumed to be a constant in order to further simplify the investigation.

$\mathbf{E} + \mathbf{V} \times \mathbf{B} = \mathbf{J}$

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:

$\frac {\partial \mathbf{B}} {\partial t} - \nabla \times \left( \mathbf{V} \times \mathbf{B} \right)= \frac{n} {\mu_0}\nabla^2\mathbf{B}$

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’s flow 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 Induction Equation

By dropping the displacement current term in Ampere's law, Maxwell's Equations are used to build an understanding of the induction equation:

\nabla \cdot \mathbf{E} = \frac {\rho}{\mathbf{e}}

\nabla \cdot \mathbf{B} = \mathbf{0}

\nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}

\nabla \times \mathbf{B} = \mu_0 \mathbf{J}

So, by assuming that the process does not consider radiation, it can be deduced that the process' time scale is longer than the time it takes for light to travel across the system. Now, the next equation that will be necessary for further understanding this process is Ohm's law when applied to a conductor that is crossing the magnetic field and electric field when moving at velocity (v), becoming:

$\mathbf{J} = \sigma \cdot \left( \mathbf{E} + \mathbf{v}\times\mathbf{B} \right)$

Then, by utilizing the equation $\nabla \times \nabla \times \mathbf{B} = \nabla \left( \nabla \cdot \mathbf{B} \right) - \nabla^2 \mathbf{B} = -\nabla^2 \mathbf{B}$ , the fundamental equation of kinematic dynamo theory (also known as the induction equation) can be derived.

$\frac {\partial \mathbf{B}}{\partial t} = \nabla \times \left( \mathbf{u} \times \mathbf{B} \right) + \mathbf{n} \nabla^2 \mathbf{B}$

Further Exploration

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.

References

Demorest, Paul. "Dynamo Theory and Earth's magnetic Field." 21 May 2001. ^*

Fitzpatrick, Richard. "MHD Dynamo Theory." 18 May 2002. ^*

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Formal Definition

Kinematic Dynamo Theory

The Induction Equation

Further Exploration

References

See also