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Astron. Astrophys. 361, L33-L36 (2000)

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3. Closure model for an infinite system

Under the assumptions that the velocity is delta-correlated in time and the magnetic field is a gaussian random field S97 derived equations for the longitudinal correlation function [FORMULA] and the correlation function for magnetic helicity density, [FORMULA]. The velocity is represented by a longitudinal correlation function [FORMULA] and a correlation function for the kinetic helicity density, [FORMULA]. We change somewhat the notation of S99 and define the operators

[EQUATION]

so the closure equations can be written as

[EQUATION]

[EQUATION]

where [FORMULA] is the correlation function of the current helicity, [FORMULA] is the effective induction,

[EQUATION]

[EQUATION]

are functions resembling the usual [FORMULA]-effect and the total magnetic diffusivity. Here [FORMULA] and [FORMULA]. Note that at large scales

[EQUATION]

[EQUATION]

where [FORMULA]. Expression (9) is similar to the [FORMULA]-suppression formula first found by Pouquet et al. (1976). Here [FORMULA] and [FORMULA] are scale dependent (increasing until the forcing scale) and, in addition, both are affected by AD.

We construct [FORMULA] and [FORMULA] from an analytic approximation of the kinetic energy and helicity spectra, [FORMULA] and [FORMULA], respectively. Zero velocity at large scales means that [FORMULA] for [FORMULA]. At some wavenumber [FORMULA] the spectrum turns to a [FORMULA] Kolmogorov spectrum, followed by an exponential cutoff, so we take

[EQUATION]

We use parameters representative of the simulations of B2000, so [FORMULA], [FORMULA] and [FORMULA], giving [FORMULA] for [FORMULA]. Like in B2000 we assume the turbulence fully helical, so [FORMULA] (e.g. Moffatt 1978). The correlation functions [FORMULA] and [FORMULA] are then obtained via

[EQUATION]

and [FORMULA], where [FORMULA] and [FORMULA] is the correlation time. (We use [FORMULA], representative of the kinematic stage of Run 3 of B2000.)

We solve Eqs. (5) and (6) using second order finite differences and a third order time step on a uniform mesh in [FORMULA] with up to 10,000 meshpoints and [FORMULA], which is large enough so that the outer boundary does not matter. In the absence of helicity, [FORMULA], and without nonlinearity, [FORMULA], we recover the model of Novikov et al. (1983). The critical magnetic Reynolds number based on the forcing scale is around 60. With helicity this critical Reynolds number decreases, confirming the general result that helicity also promotes small scale dynamo action (cf. Kim & Hughes 1997, S99). In our model with nonlinearity the exponential growth of the magnetic field terminates when its energy becomes comparable to the kinetic energy (for [FORMULA]). After that point the magnetic energy continues however to increase nearly linearly. Unlike the case of the periodic box (Sect. 2) the magnetic field can here extend to larger and larger scales; see Fig. 3. The corresponding magnetic energy spectra,

[EQUATION]

are shown in Fig. 4.

[FIGURE] Fig. 3. Evolution of magnetic correlation functions for different times. The correlation function of the magnetic helicity is shown in the inset. [FORMULA].

[FIGURE] Fig. 4. Evolution of magnetic energy spectra. Note the propagation of magnetic helicity and energy to progressively larger scales. The [FORMULA] slope is given for orientation.

The resulting magnetic field is strongly helical and the magnetic helicity spectra (not shown) satisfy [FORMULA]. The development of a helicity wave travelling towards smaller and smaller k, as seen in Fig. 4, is in agreement with the closure model of Pouquet et al. (1976). In the following we shall address the question of whether or not the growth of this large scale field (which is nearly force-free) depends on the magnetic Reynolds number (as in B2000). We have checked that to a very good approximation the wavenumber of the peak is given by

[EQUATION]

This result is familiar from mean-field dynamo theory (see also S99) and is consistent with simulations (B2000, Sect. 3.5). Note that here [FORMULA] decreases with time because [FORMULA] tends to a finite limit and [FORMULA] increases. (This is not the case in the box calculations where [FORMULA].)

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© European Southern Observatory (ESO) 2000

Online publication: October 2, 2000
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