Astron. Astrophys. 361, L5-L8 (2000)

3. The equilibrium magnetic field configuration

We 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 D, there is an explicit steady solution, if we assume (remember that also , i.e. super-equipartition), of the form

where we have restored the dimensional factor . (Note that for this approximate solution.) This solution is remarkably close to the results from the naive Ansatz or , or the model of Moss et al. (1999). For example, the pitch angle of the magnetic field lines is for and .

© European Southern Observatory (ESO) 2000

Online publication: September 5, 2000
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