2. Equations for magnetic helicity
Following Kleeorin and Ruzmaikin (1982), we parameterize the back-reaction of dynamo generated magnetic field in terms of a differential equation for the -coefficient, using arguments from the magnetic helicity conservation law. It is necessary to introduce the large-scale vector potential , small-scale vector potential , and the corresponding representations for the magnetic fields, and . We then write the total magnetic field as , and the total vector potential as , thus decomposing the fields into mean and fluctuating parts. The equation for the vector potential follows from the induction equation for the total magnetic field
where and is the mean fluid velocity field, is the magnetic diffusion due to the electrical conductivity of the fluid, is an arbitrary scalar function. Now we multiply the induction equation for the total magnetic field by and Eq. (1) by , add them and average over the ensemble of turbulent fields. This yields an equation for the magnetic helicity in the form
where is the turbulent magnetic diffusivity, and it is assumed that is the total alpha-effect which at the nonlinear stage includes both the original hydrodynamical, and the magnetic, contributions. Note that the magnetic part of the effect is proportional to the magnetic helicity, i.e. (see, e.g., Kleeorin and Rogachevskii, 1999), where is the density. The simplest form of the magnetic helicity flux for an isotropic turbulence is given by where is the mean fluid velocity, e.g. that of the differential rotation (see Kleeorin and Ruzmaikin, 1982; Kleeorin and Rogachevskii, 1999). Thus, the equation for the magnetic part of the effect in dimensionless form is given by
(see Kleeorin and Ruzmaikin, 1982), where is the scale of turbulent motions. We adopt here the standard dimensionless form of the galactic dynamo equation from Ruzmaikin et al. 1988; in particular, the length is measured in units of the disc thickness h, the time is measured in units of and B is measured in units of the equipartition energy . Here u is the characteristic turbulent velocity in the scale l, and , where and are measured in units of (the maximum value of the hydrodynamic part of the effect). For an axisymmetric dynamo
When and , Eq. (4) yields (see, e.g., Vainshtein and Cattaneo, 1992). However, the latter equation is not valid for galaxies because . In addition, the condition seems not to be valid for galaxies.
Eq. (4) has been later reproduced, e.g. by Gruzinov and Diamond (1995). However, although this equation has never been included into detailed galactic dynamo calculations, nevertheless its qualitative properties are more or less clear. Provided that dissipative losses are taken into account, Eq. (4) leads to the same type of behaviour as that obtained by the ad hoc prescription of the result of Vainshtein and Cattaneo (1992), i.e. the steady state strength of magnetic field is about (see, e.g. Field, 1999). The real advantage of Eq. (4) is the fact that it is derived from first principles rather than prescribed ad hoc . If the dissipative losses in Eq. (4) are neglected, the magnetic field decays for . We stress that Eq. (4) contains a large factor typically.
Kleeorin and Rogachevskii (1999) extended the calculations to include a flux of magnetic helicity. Based on Eq. (13) of that paper, the approximate relation
can be formulated. In Eq. (5), describes the inhomogeneity of the turbulent diffusivity, and we define . The profile depends on details of the galactic structure. Also, is the total effect and , where . Here is the hydrodynamic part of the effect, with its modification due to nonlinear effects. Correspondingly, is the magnetic part of the effect, and is the modification caused by nonlinear effects (see Rogachevskii and Kleeorin, 2000). and the function is defined below. The magnetic part of the effect is proportional to the magnetic helicity, i.e., (see, e.g., Kleeorin and Rogachevskii, 1999). For galaxies the term is very small and can be dropped. The gauge conditions have been used; our results can be shown to be gauge invariant (see Berger and Ruzmaikin, 2000).
The last term in Eq. (5) is related to the turbulent flux of magnetic helicity. This turbulent flux is proportional to the hydrodynamic part of the effect and the turbulent diffusivity (see Kleeorin and Rogachevskii, 1999). The turbulent flux of magnetic helicity serves as an additional nonlinear source in the equation for the magnetic part of the effect and it causes a drastic change in the dynamics of the large-scale magnetic field.
For simplicity we replace the flux divergence in the right hand side of Eq. (5) by a decay term, i.e. we replace by (in principle, there is no problem in treating this point more carefully).
© European Southern Observatory (ESO) 2000
Online publication: September 5, 2000