The Earth’s Magnetic Field - Ultra and Extremely Low Frequency Electromagnetic Fields - page 9

core of the Earth. Substituting the mean conductivity of the core g D 7 10 5 S=m
and the scale sizes l D 10 3 km (Stacey 1969 ) into Eq. ( 1.17 ) gives the magnetic
viscosity/diffusion coefficient m 1:1 m 2 =s and the damping time of the Earth's
magnetic field d 3 10 4 years. When the turbulent magnetic viscosity of about
4 m 2 =s is taken into account, the damping time decreases up to 10 4 years (Parker
1979 ). If the characteristic time of the geomagnetic and mass velocity variations
inside the core is smaller than 10 4 years, the concept of frozen-in field can be applied
to the outer core.

1.1.4 A Simple Model of Hydromagnetic Dynamo

The geomagnetic field is mainly generated due to currents in the electrically
conducting core of the Earth. The currents are in turn driven by the convection of
molten matter in the core. This process is often called a dynamo in analogy to a
motor-driven electric generator, which is capable of producing the electric current
without wire and winding. Pioneering investigations of the dynamo mechanism
were provided by Larmor ( 1919 ) who treated the nature of terrestrial and solar
magnetism. The generally accepted theory states that the hydromagnetic dynamo
is a principal origin of the Earth's magnetic field and that this mechanism is capable
to sustain generation and amplification of the magnetic field due to hydrodynamical
flow in the earth interior. It should be noted that the hydromagnetic dynamo is a
self-excited generator, which can operate without maintaining the external supplies.
The effect of the conducting fluid flow on the magnetic field generation is
controlled by the magnetic Reynolds number Re m D lV= m . This dimensionless
parameter is on the order of ratio of the magnetic energy rate to the Joule dissipation
of energy. As Re m 1 then the magnetic field generation prevails over the energy
dissipation due to the medium heating. Assuming the velocity of western drift
of the matter in the Earth core V D 0:3 mm=s and substituting the numerical
parameters alluded to above we obtain that inside the core Re m 10 3 (Parker
1979 ). Nevertheless, the condition Re m 1 is insufficient for excitation of
the hydromagnetic dynamo. In some sense, we need a topological complexity of
the flow pattern as well. The turbulent flow is tangled enough so that the above
requirement of topological complexity holds automatically.
A laminar flow must be nontrivial to produce a dynamo effect. For example,
it was proved that the high symmetric flows of conducting media such as axially
symmetric, centrally symmetric, and two-dimensional flows are unable to generate
magnetic fields (Cowling 1934 , 1953 , 1976 ; Zeldovich 1956 ; Braginsky 1964 ;
Zeldovich et al. 1983 ).
Surprisingly, however, the turbulent dynamo is simpler and more evident than
the laminar dynamo. In the extreme case of large magnetic Reynolds number the
analytic dynamo solutions to the equation for mean field and correlation function
have been found (Moffat 1968 ;Parker 1979 ; Krause and Rädler 1980 ; Molchanov
et al. 1985 ). In the case of turbulent flow the mean magnetic field is treated as a result