The Earth’s Magnetic Field - Ultra and Extremely Low Frequency Electromagnetic Fields - page 11

Fig. 1.3 A schematic plot of
the helicity origination due to
the convection in stratified
media

ʩ

F C

V r

V q

V q

V q

V q

V r

F C

of the helical curling/motion dominates in the flow. In such a way we say that the
flow has the nonzero mean helicity, which is defined as a value which is proportional
to h V . r V / i , where V is the mass velocity. Formally correct definition of the
helicity Ǜ is (e.g., see monographs by Parker 1979 ; Vaynshtein et al. 1980 , and
references therein)

3 h V . r V / i :

Ǜ D (1.23)

Here the correlation time is l=V 0 , where l is scale size and V 0 is characteristic
velocity.
Such helicity arises in outer space due to the spin of celestial bodies whereas
in laboratory environment the helicity is atypical. The theoretical study has shown
that the mean helicity is due to the violation of reflection symmetry of flow (Parker
1979 ). The reflection asymmetrical random movements are unable to generate a
large-scaled magnetic field.
To understand the mechanism of helicity appearance in a little more detail
we study the convection of matter in a rotating globe with radially stratified
density. Suppose that a small convective fluid element with mass m ascends or is
dropped along the radius. Owing to the density and pressure gradients, the element
undergoes deformation and acquires the additional constituents of the velocity V
and V perpendicular to the radius. These constituents result from dilatation or
compression of the moving element. If the element moves up and the density falls
off with increase of radius, the element volume dilates and whence the perpendicular
velocity is directed so as shown on the right side of Fig. 1.3 . The inverse case, when
the element moves down, is shown on the left side of this figure. In the reference
frame fixed to the globe, that is in the reference frame rotating with constant angular
velocity

, the moving element undergoes the Coriolis inertia force

F C D 2m. V /: (1.24)

where V is given in that frame.