## 3. The equilibrium magnetic field configurationWe now present some asymptotic expansions for galactic dynamo models with the nonlinearity (5). First of all, we recognize that, because of the large parameter in the right hand side of Eq. (5), we can take where Thus for and for The function is derived by Rogachevskii and Kleeorin (2000). Note that in a more simplified model of turbulence the function (see Field et al. 1999). We stress that the qualitative behaviour of the model does not depend on these uncertainties in estimates for the scaling functions and . Now we insert the -coefficient given by Eq. (6) into local disc dynamo problem to obtain the following equations: (here ). We can then obtain the steady-state solution of Eqs. (7) and (8). Recognizing that in cylindrical coordinates we obtain for fields of quadrupole symmetry (cf. Kvasz et al., 1992) in a steady state. The corresponding equation in kinematic theory reads Substituting (6) into (10) we obtain, Using Eq. (9) we rewrite Eq. (12) in the form For the dynamo This assumption is justified if , i.e. . Eq. (13) then becomes Note that Eq. (14) differs from Eq. (11), arising from
kinematic theory. For the specific choice of helicity profile
and negative dynamo number where we have restored the dimensional factor
. (Note that
for this approximate solution.) This
solution is remarkably close to the results from the naive
© European Southern Observatory (ESO) 2000 Online publication: September 5, 2000 |