# Using Amp`ere’s law (36), the j × B term in the MHD equation of motion (71) may be written

j×B =

1

(∇ × B) × B = −∇

B^{2 }

µ_{0 }

2µ_{0 }

+

1 µ_{0 }

(B · ∇) B.

(73)

The last term on the left hand side is called the magnetic tension, being related to the curvature of magnetic field lines and in some respects similar to the ordinary tension in a string, while the first term is the gradient of a quantity known as the magnetic pressure,

B^{2 }p_{B }= 2µ_{0 }

.

(74)

The MHD equation of motion (71) thus can be written as

ρ_{m }^{dv }= −∇(p + p_{B}) + dt

1 µ_{0 }

(B · ∇) B.

(75)

This justifies the name magnetic pressure for the term p_{B }

= B^{2}/2µ_{0}, equal to the magnetic energy

density.

# 4.9 Dynamos

Magnetic fields are present almost everywhere in the universe, so there must be some dynamo pro- cesses generating them. Such processes are described by the MHD equation of motion (73) together with equation (73), the Faraday-Henry law (5), and the freezing-in condition (59):

ρ_{m }

dv

= j×B =

1

dt

µ_{0 }

# (∇ × B) × B

(76)

∂B ∂t

= −∇ × E = ∇ × (v × B).

(77)

Together, these two vector equations form a set of six equations for six unknowns (v and B). To- gether with appropriate boundary and initial conditions, this defines the evolution of v and B in time and space. In general, other terms would have to be included, like the pressure p neglected in equation (71) and dissipation effects due to viscosity and resistivity, but in principle, these equations holds the key to the dynamo problem. For instance, the plasma flow in the sun must generate the solar magnetic field as described by (77); this magnetic field then affects the flow as described by (76). The MHD equations are applicable not only to a plasma but also to other conducting fluids, like planetary interiors, and therefore equations (76) and (77), or some generalizations thereof, also describes the generation of planetary magnetic fields in terms of the flows inside the planetary cores.

In general, nature gives solutions to (76) and (77) with non-zero magnetic fields: all planets thought to have fluid interiors also have magnetic fields, and so does stars and galaxies. The dynamo equations have been the subject of much study, and some general properties of the system are known. For example, it is known that they have no stationary axisymmetric solution. In this context, it is interesting to note that no planetary magnetic field has its dipole axis perfectly aligned to the rotation axis of the planet. Also, it is known that the Earth’s field is non-staionary, having had several pole- reversals with intervals of some tens of thousands of years. For the sun, there is at least one period of much faster change: the sunspot cycle of 22 years.

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