The Earth’s Magnetic Field - Ultra and Extremely Low Frequency Electromagnetic Fields - page 14

the possibility of magnetic field generation due to nonzero helicity. Following
Vaynshtein et al. ( 1980 ) we now consider a simple one-dimensional case in order
to derive a straightforward analytical solution of Eq. ( 1.26 ). Suppose first that the
mean magnetic field h B i is a function of only z and t so that the derivatives with
respect to x and y are equal to zero. It follows from Eq. ( 1.3 ) that @ z h B z iD 0 and

hence h B z iD 0. The components h B x i and ǝ B y Ǜ are considered to vary as exp .t/,
where is the increment of growth. In this case Eq. ( 1.26 ) is reduced to
h B x iD Ǜ d ǝ B y Ǜ

d z C d 2

h B x i
d z 2

; (1.27)

d z C d 2 ǝ B y Ǜ

ǝ B y Ǜ D Ǜ d h B x i

: (1.28)

d z 2

In the case of infinite medium the set of Eqs. ( 1.27 )-( 1.28 ) has a particular solution
(Vaynshtein et al. 1980 )

h B x iD B 0 exp .t/ sin .k z /; (1.29)

ǝ B y Ǜ D B 0 exp .t/ cos .k z /;

(1.30)

where B 0 is a constant, and the increment of growth is related to the constant k by

D Ǜk k 2 :

(1.31)

As is seen from Eq. ( 1.31 ) the magnetic field will increase exponentially in time
under the requirement that the helicity is positive and 0<k<Ǜ=. The magnetic
field thus can enhance if its characteristic scale D k 1 >=Ǜ. The increment
of growth reaches a peak value max D Ǜ 2 =.4/, which corresponds to the spatial
scale D 2=Ǜ. The rate of field enhancement is a maximum at this spatial scale.
In contrast to the left-handed spiral field .Ǜ>0/, the right-handed field .Ǜ<0/
exponentially decreases with time at positive k since <0.
The lack of a reflectional symmetry of the fluid flow means that the number of
right-handed eddies in the flow is not equal to that of left-handed eddies. Notice
that a single eddy has not the reflectional symmetry. Indeed, the reflection of eddy
off a mirror plane, which is parallel to the spin axis, results in transformation of the
right-screw eddy to the left-screw one. However, if the numbers of the right-screw
eddies and left-screw ones are equal to each other, the flow has a mirror symmetry
on average. Only if the eddies of certain sign are predominant in the turbulent flow, it
can produce nonzero mean helicity in a way that the flow loses the mirror symmetry.
In such a case the small-scale turbulence is able to produce a large-scale magnetic
field. The credit for the discovery of this fact is given to Parker ( 1955 ) and Steenbeck
et al. ( 1966 ).