The Earth’s Magnetic Field - Ultra and Extremely Low Frequency Electromagnetic Fields - page 13

average overall realization of random function B and V in Eq. ( 1.14 ), it is necessary
to calculate the correlation function h V B i . In the course of this text we cannot
come close to the exploration of this extremely complicated mathematical apparatus
used for the study of MHD turbulence. The interested reader is referred to the text
by Moffat ( 1968 ) for a more complete treatise on MHD turbulent processes (see also
the treatise by Krause and Rädler 1980 ;Parker 1979 ; Vaynshtein et al. 1980 ).
Here we only reproduce the main equation describing the generation of mean
large-scaled magnetic field h B i in turbulent conducting media with nonzero mean
helicity

2

@ t h B iD . m C t / r h B iCr Ǜ h B iCr . h V ih B i /: (1.25)

2 =3 denotes the
coefficient of turbulent diffusion where is the correlation time. The last term on the
right-hand side of Eq. ( 1.25 ) describes the large-scaled motion of conducting media
at the mean velocity h V i , including the so-called nonuniform/differential rotation.
In practice, the turbulent diffusion may be so much that t m . Some under-
standing of the turbulent diffusion can be achieved by considering the following
example. As has already been stated, the magnetic field is able to diffuse in a fixed
conducting space in accordance with Eq. ( 1.15 ). A close analogy exists with the
diffusion of impurity molecules in still air. Once the intermixing or turbulence
occurs in the air, the impurity will propagate rapidly as compared to still air.
The magnetic field propagates in moving conducting media analogously to the
impurity in the air because of the “frozen-in” effect. Therefore, the magnetic field
can propagate much more rapidly in turbulent media. Certainly, this analogy is
incomplete since the diffusion of the impurity is described by a scalar equation
whereas the magnetic field diffusion follows a vector equation. Other difference
between diffusion of the magnetic field and of the impurity is that in contrast to the
impurity, the field, if it is not weak, may greatly affect the flow.

Here Ǜ is the mean helicity given by Eq. ( 1.23 ), and t h V i

1.1.6 Magnetic Field Generation

The effects of large-scaled magnetic field generation due to the helicity and
nonuniform rotation of turbulent flow are the underlying physical principles on
which the theory of the turbulent MHD is based. Since the gyrotropic turbulence
due to the helicity is believed to play a major role in MHD dynamo excitation, we
focus our attention on the following equation:

2

@ t h B iD r h B iCr Ǜ h B i ; (1.26)

where D m C t . Here we have dropped the term describing a large-scaled
motion in Eq. ( 1.25 ). This equation has been studied repeatedly to demonstrate