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

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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 [FORMULA] in the right hand side of Eq. (5), we can take

[EQUATION]

where

[EQUATION]

Thus [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA] The function [FORMULA] is derived by Rogachevskii and Kleeorin (2000). Note that in a more simplified model of turbulence the function [FORMULA] (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 [FORMULA] and [FORMULA].

Now we insert the [FORMULA]-coefficient given by Eq. (6) into local disc dynamo problem to obtain the following equations:

[EQUATION]

(here [FORMULA]). We can then obtain the steady-state solution of Eqs. (7) and (8). Recognizing that in cylindrical coordinates

[EQUATION]

we obtain for fields of quadrupole symmetry (cf. Kvasz et al., 1992)

[EQUATION]

in a steady state. The corresponding equation in kinematic theory reads

[EQUATION]

Substituting (6) into (10) we obtain,

[EQUATION]

Using Eq. (9) we rewrite Eq. (12) in the form

[EQUATION]

For the [FORMULA] dynamo [FORMULA] This assumption is justified if [FORMULA], i.e. [FORMULA]. Eq. (13) then becomes

[EQUATION]

Note that Eq. (14) differs from Eq. (11), arising from kinematic theory. For the specific choice of helicity profile [FORMULA] and negative dynamo number D, there is an explicit steady solution, if we assume [FORMULA] (remember that also [FORMULA], i.e. super-equipartition), of the form

[EQUATION]

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

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© European Southern Observatory (ESO) 2000

Online publication: September 5, 2000
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