## 4. DiscussionFor 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 where 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 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
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 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 © European Southern Observatory (ESO) 1999 Online publication: June 6, 1999 |