## 3. Closure model for an infinite systemUnder 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 and the correlation function for magnetic helicity density, . The velocity is represented by a longitudinal correlation function and a correlation function for the kinetic helicity density, . We change somewhat the notation of S99 and define the operators so the closure equations can be written as where is the correlation function of the current helicity, is the effective induction, are functions resembling the usual -effect and the total magnetic diffusivity. Here and . Note that at large scales where . Expression (9) is similar to the -suppression formula first found by Pouquet et al. (1976). Here and are scale dependent (increasing until the forcing scale) and, in addition, both are affected by AD. We construct and from an analytic approximation of the kinetic energy and helicity spectra, and , respectively. Zero velocity at large scales means that for . At some wavenumber the spectrum turns to a Kolmogorov spectrum, followed by an exponential cutoff, so we take We use parameters representative of the simulations of B2000, so , and , giving for . Like in B2000 we assume the turbulence fully helical, so (e.g. Moffatt 1978). The correlation functions and are then obtained via and , where and is the correlation time. (We use , 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
with up to 10,000 meshpoints and
, which is large enough so that the
outer boundary does not matter. In the absence of helicity,
, and without nonlinearity,
, we recover the model of Novikov et
al. (1983). The are shown in Fig. 4.
The resulting magnetic field is strongly helical and the magnetic
helicity spectra (not shown) satisfy
. The development of a helicity wave
travelling towards smaller and smaller This result is familiar from mean-field dynamo theory (see also S99) and is consistent with simulations (B2000, Sect. 3.5). Note that here decreases with time because tends to a finite limit and increases. (This is not the case in the box calculations where .) © European Southern Observatory (ESO) 2000 Online publication: October 2, 2000 |