For illustration, consider the behaviour of the mean field in a turbulent flow between two planes, . Assume that the mean velocity of the flow is given by . In the induction equation, the electromotive force (27) has to be complemented by the standard term representing turbulent magnetic diffusivity and the advective term, . Then, the induction equation reads
is the scallar turbulent magnetic diffusivity (see, e.g., Krause & Rädler 1980); , and is the molecular magnetic diffusivity. Note that the dissipative term does not appear in the expression (27) because we neglect the magnetic diffusivity from the very beginning for the sake of simplicity.
The evolution of the mean field is determined by the behaviour of its x-component which turns out to be not influenced by and . However, these two components are generated from due to either the stretching effect or the turbulent electromotive force thus and are coupled to . The equation for is
Consider as an example the shear flow with . The solution of Eq. (30) can be represented in the form
where and are the wavevectors in the y- and z-directions, respectively. Then, the equation for f reads
where , , and . The solution of Eq. (32) can easily be obtained by making use of the WKB-approximation. Substituting and assuming , we have
If is vanishing on the surfaces , then the eigenvalues are given by the condition
where n is integer. We can rewrite this equation as
where is some average value of , . Then, the eigenvalues are given by
where , and . The first term on the r.h.s. of Eq. (36) describes oscillations of the magnetic field caused by advection of field lines by the mean flow, the second terms represents the generating effect of the electromotive force , and the third term describes turbulent dissipation. Note that always gives a positive contribution to resulting in a generation of the magnetic field for any wavevectors. The generating effect becomes stronger for a lower magnetic diffusivity (or, in other words, for a higher conductivity). For a relatively large magnetic Reynolds number, we have and the flow becomes unstable to a generation of the mean field. Note that only the field components which depend on the y-coordinate can be unstable.
© European Southern Observatory (ESO) 1999
Online publication: June 6, 1999