The Earth’s Magnetic Field - Ultra and Extremely Low Frequency Electromagnetic Fields - page 10

Fig. 1.2 A simple model of magnetic field amplification due to stretch . a ! b / , twisting . b ! c /
and doubling . c ! d / of the magnetic loops

of the random pulsation averaging while in the case of laminar flow the azimuth and
time averaging is considered. It is interesting to note also that in both of these cases
the hydromagnetic dynamo is mainly due to the same characteristics of the flow,
which are the mean helicity and nonuniform/differential rotation.
Some understanding of the hydromagnetic dynamo can be achieved by consider-
ing the following idealized descriptive model (Davies 1958 ; Vaynshtein et al. 1980 ).
Imagine first a closed tube in perfectly conducting fluid as shown in Fig. 1.2 a. There
is a toroidal magnetic field inside the tube. In the framework of ideal MHD approach
the magnetic field is frozen into the conducting fluid, so that the frozen-in field lines
move with the fluid. Suppose that the fluid flow results in tension of the tube so
that the tube length is doubled whereas its cross area is decreased roughly two times
(Fig. 1.2 b). Owing to the local flow rotation or for some other reason, the tube takes a
shape of figure-of-eight (Fig. 1.2 c) and then one loop of the figure-of-eight combines
with another loop (Fig. 1.2 d). The magnetic field is therefore doubled. Each iteration
of this procedure results in duplication of the magnetic field. If this procedure is
repeated n times, the magnetic field becomes 2 n as great as the initial magnetic field.
This means that the magnetic field and magnetic flux increase exponentially. This
mechanism of field amplification can arise in a turbulent flow. Since in the model
the rate of field increase does not depend on the magnetic diffusion, this mechanism
is referred to as fast dynamo.

1.1.5 Turbulent Diffusion and Mean Helicity

Random fluctuations of the conducting fluid flow give rise to fluctuations of currents
and magnetic field. In what follows we study the flow pattern and conditions, which
are required in order to produce a nonzero mean magnetic field in the conducting
media.
Let us consider first the flow pattern where the eddies with certain sign, either
right-handed or left-screw, are predominant. This means that a pronounced direction