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Astron. Astrophys. 335, 488-499 (1998)

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4. Dynamo parameters

The effect of a turbulent velocity field, [FORMULA], on the mean magnetic field, [FORMULA], is described by the turbulent electromotive force,

[EQUATION]

which is customarily expressed in terms of the alpha-tensor, [FORMULA], and the diffusivity tensor, [FORMULA]:

[EQUATION]

(Moffatt 1978). Under the approximations introduced in Sect. 3, these tensors can be written in the form

[EQUATION]

(Paper I) and

[EQUATION]

(Paper II) in the Galactocentric cylindrical coordinate system [FORMULA]. Briefly, [FORMULA] represents the effective vertical velocity at which the mean magnetic field is advected by turbulent motions; [FORMULA], and [FORMULA] give the effective rotational velocity associated with the alpha-effect when [FORMULA] is, respectively, radial, azimuthal, and vertical; [FORMULA] and [FORMULA] are the horizontal and vertical turbulent magnetic diffusivities; [FORMULA] is the unit tensor and [FORMULA] is the permutation tensor (equal to [FORMULA], or 0, according to whether [FORMULA] forms an even permutation of [FORMULA], an odd permutation of [FORMULA], or neither).

In Papers I and II, we demonstrated that explosions occurring between [FORMULA] and [FORMULA] with a rate per unit area [FORMULA] give rise to the following contributions to the dynamo parameters:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

at a vertical distance [FORMULA] above the explosion site. In the above expressions, [FORMULA] is the horizontal radius of a shell a time [FORMULA] after the explosion; [FORMULA] is its horizontal cross-sectional area; [FORMULA] is the instantaneous power law index for the evolution of the shell radius, [FORMULA] ; [FORMULA] is the time-averaged value of [FORMULA] up to the current time, [FORMULA] ; and [FORMULA] are the values of [FORMULA] at the merge time, [FORMULA]. A detailed physical interpretation of Eqs. (26)-(31) is provided in Papers I and II and a summary is presented in Paper III.

The dynamo parameters in the Galactic disk are obtained by integrating Eqs. (26)-(31) over the observed SN and SB distributions, i.e., over

[EQUATION]

with [FORMULA] and [FORMULA] given by Eqs. (18) and (20), respectively.

The numerical code used here to compute the dynamo parameters is an extension of the code written by Ferrière (1995) for the purpose of calculating the filling factor of hot cavities. It follows the temporal evolution of a large number of shells produced by isolated SNs and by SBs, with [FORMULA] and [FORMULA] spanning the observed range. It determines the merge time, [FORMULA], and the maximum horizontal cross-sectional area, [FORMULA], of each shell as well as the other quantities relevant to dynamo action ([FORMULA] and the integrals appearing in Eqs. (28) and (29)). The contributions from individual shells are then integrated over the SN/SB distribution given by Eq. (32). For a more detailed description of the numerical procedure, the interested reader is referred to Paper III.

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998
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