A&A 439, 835-843 (2005)
DOI: 10.1051/0004-6361:20053221
A. Brandenburg^{1} - K. Subramanian^{2}
1 -
NORDITA, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark
2 -
IUCAA, Post Bag 4, Pune University Campus, Ganeshkhind, Pune 411 007, India
Received 10 April 2005 / Accepted 30 May 2005
Abstract
The validity of a closure called
the minimal tau approximation (MTA),
is tested in the context of dynamo theory,
wherein triple correlations are assumed to
provide relaxation of the turbulent electromotive force.
Under MTA, the alpha effect in mean field dynamo theory becomes
proportional to a relaxation time scale multiplied by
the difference between kinetic and current helicities.
It is shown that the value of the relaxation time is positive and,
in units of the turnover time at the forcing wavenumber,
it is of the order of unity.
It is quenched by the magnetic field - roughly independently of the
magnetic Reynolds number.
However, this independence becomes uncertain
at large magnetic Reynolds number.
Kinetic and current helicities are shown to be dominated by
large scale properties of the flow.
Key words: magnetohydrodynamics (MHD) - turbulence
In many branches of astrophysics it is necessary to use turbulent transport coefficients. This allows one to describe the collective effects of turbulence (e.g. diffusive and non-diffusive effects) without the need to resolve the small scale turbulence. Non-diffusive effects include the effect in mean-field dynamo theory (Krause & Rädler 1989) and the effect in the theory of stellar differential rotation (Rüdiger 1989). In order to calculate turbulent transport coefficients one usually makes the assumption that the equations for the fluctuating quantities can be linearized (Moffatt 1978; Krause & Rädler 1980). This implies that one is forced to restrict oneself to the case where the fluctuations are weak compared with the mean field. This is clearly not very useful for astrophysical applications. Nevertheless, mean-field theory is usually applied in the astrophysically interesting parameter regime (e.g., Rüdiger & Kitchatinov 1993; Kitchatinov et al. 1994), which should in principle be well beyond the regime of validity of such a linear approximation. How is it then possible that the results are actually quite reasonable?
The approximation under which higher order terms are neglected is known as the quasilinear, the first order smoothing, or the second order correlation approximation. With the exception on an important additional term, the results derived under this approximation turn out to be similar to those derived under a more general approximation, which is sometimes referred to as the minimal tau approximation (MTA). This approximation has its roots in early papers by Vainshtein & Kitchatinov (1983), Kleeorin et al. (1990, 1996), and its general usefulness has recently been stressed further with the papers by Blackman & Field (2002) and Rädler et al. (2003). A recent review of nonlinear dynamo theory and MTA has been given by Brandenburg & Subramanian (2005a). MTA has also been applied to the case of passive scalar diffusion (Blackman & Field 2003), where its validity has been tested numerically (Brandenburg et al. 2004).
In the hydromagnetic case, the single most important addition to the theory is the attenuation of the effect by a term proportional to the current helicity of the small scale field. The beauty of MTA is that this term emerges in a completely natural fashion. In the first order smoothing approach this term is absent, although even there it can be incorporated in a more phenomenological and hence less convincing fashion (Brandenburg & Subramanian 2005a). The role of the current helicity term was first identified by Pouquet et al. (1976) and later associated with catastrophic quenching by Gruzinov & Diamond (1994) and many others after them. The purpose of the present paper is to test MTA in the context of dynamo theory.
In mean field dynamo theory one splits the magnetic
field
into a mean
magnetic field
and a small scale
field
,
and
derives the mean-field dynamo equation
(2) |
As was first shown by Blackman & Field (2002), the use of MTA leads to
an explicitly time-dependent equation for the electromotive force which,
in the isotropic approximation, is
(5) |
In the passive scalar case a relatively robust method
for estimating
from turbulence simulations
is to impose a mean concentration gradient across
the box, and to treat the deviations from this mean gradient as periodic
in all three directions (Brandenburg et al. 2004).
In the present case this corresponds to imposing a mean magnetic field,
.
The magnetic field is then given by
,
where is the departure from the mean field, .
Since
,
the evolution of
is given by
In the main part of this paper we adopt combined volume and
time averages, so,
.
This means that in Eq. (4) both
and
vanish.
Therefore we simply have
(9) |
(12) |
(13) |
So far, and throughout most of this paper, we consider the case of an imposed constant mean field, , with zero mean current, . While this is reasonable for measuring the value of , it is of course unrealistic for dynamo calculations, where there must be a mean current proportional to the curl of the mean field. Furthermore, if a mean field is imposed, this mean field can obviously not change by the dynamo action. Nevertheless, it provides a useful method for calculating . This method gives similar results than the more complicated and less accurate methods that can be invoked in dynamo simulations (Brandenburg 2001, see his Figs. 14 and 15). However, in order to show that Eq. (3) has any justification, we insert a discussion (Sect. 3) where we consider the local correlation in a simulation without an imposed field.
Before we present in detail results for the case of an imposed field, we consider first the case of a dynamo-generated magnetic field. In that case the magnetic field shows a marked sinusoidal variation in one of three possible directions. Which of the three directions is chosen depends on chance. In the following we compare with a simulation of homogeneous helically forced turbulence of Brandenburg (2001); in his Run 3 the magnetic field shows a slow variation in the y direction, so we define mean fields by averaging over the x and z directions.
Figure 1: Comparison of the spatial dependence of two components of the mean magnetic field and the triple correlation in Run 3 of B01. The magnetic field is normalized by the equipartition field strength, , and the triple correlation is normalized by , but scaled by a factor of 2.5 to make it have a similar amplitude as the mean field. Note that (solid line) correlates with T_{x} (filled dots) and (dashed line) correlates with T_{z} (open dots). | |
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Figure 1 shows that the x and z components of have a positive correlation with the mean magnetic field. Note that itself has a negative correlation with , for a negative . And since this simulation has positive kinetic helicity, the obtained is indeed negative. Therefore the positive correlation of with is consistent with Eq. (3) with a positive . (The term does not change the sign of , but only reduces its magnitude. This follows from the fact that in a dynamo simulation has to overcome the magnetic diffusion term.) We have used as a proxy for in order to study the correlation of with , because itself is even more noisy than . This is because almost cancels and the fact that consists of derivatives of which contributes further to the noise.
In conclusion, our first assessment of is very encouraging in that its phase relation with the magnetic field is such that it would indeed contribute as a damping term, i.e. that it has the expected sign.
Having shown that
varies locally in the expected
sense, we now concentrate on the case of a uniform imposed field where
it is possible to use averages over the full box.
(In the dynamo case considered in the previous section such averages
would have given zero.)
We solve Eqs. (6)-(8) numerically.
The forcing amplitude is such that
we are in the low Mach number regime, i.e.
,
so for all practical purposes the flows can be
considered nearly incompressible.
We consider runs with different hydrodynamic and hydromagnetic Reynolds
numbers,
Figure 2: dependence of the normalized for B_{0}=0.03 (or ) and B_{0}=0.1 (or ) in cases where the turbulence is driven kinetically with a body force (lower graph) or magnetically with a forcing term in the induction equation (upper graph). The vertical bars on the data points give estimates of the error (see text). | |
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We measure velocity in units of the sound speed , density in units of the average density , magnetic field in units of , and length in units of k_{1}^{-1}, where k_{1} is the smallest possible wavenumber in the box. Whenever possible we present the result in normalized (explicitly nondimensional) form.
We begin by considering the dependence of on ; see Fig. 2. We focus on the results for two different field strengths, B_{0}=0.03, corresponding to , and B_{0}=0.1, corresponding to . Throughout this paper, error bars are obtained by taking the extrema from 5 separate averages, each over 1/5 of the full time series.
In both kinetically and magnetically driven cases is negative when the helicity of the forcing is positive. It turns out that, when the turbulence is kinetically forced, decreases with increasing like where n is between 1/2 and 1. For large field strengths and large values of we expect an asymptotic dependence with n=1 (Cattaneo & Hughes 1996), but when these values are not yet large enough the data are best fitted with values of n less than 1.
In the magnetically driven case is generally, at comparable values of , larger than in the kinetically driven case. Furthermore, in the magnetically driven case does not show the systematic decrease of seen in the kinetically driven case. This can be understood as a result of a modified Keinigs (1983) relation; see Appendix B.
Next, we consider the values of and in both kinetically and magnetically driven cases; see Figs. 3 and 4. The results depend on the values of and . In the kinetically driven case there is a clear tendency for to increase with until it approaches and slightly exceeds the value of . For larger values of , and are constant and close to each other.
Figure 3: Dependence of and on in the kinetically forced case. Vertical bars give error estimates. | |
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In the magnetically driven case, and depend only weakly on . Furthermore, can exceed by even an order of magnitude; see the first panel of Fig. 4.
Figure 4: Dependence of and on in the magnetically forced case. Vertical bars give error estimates. | |
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It is not easy to present the data of all runs in a meaningful way in a single graph. This is why we have summarized the results of all runs in tabular form; see Table 1, allowing alternative representations and interpretations to be made.
After these preparations we can now return to the main question addressed
in this paper: what is the relevant relaxation time in MTA and how does
it depend on .
Assuming a steady state, we can use Eq. (4) and calculate
(15) |
Figure 5: Strouhal numbers as a function of for different combinations of B_{0} and . Here, is assumed. The horizontal lines are drawn to indicate the range over which the Strouhal numbers are approximately constant. In the second panel therefore, the slope of the lower graph is -0.68. | |
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Figure 6: Magnetic and kinetic Strouhal numbers as a function of for different values of B_{0} and . Here, kinetically and magnetically forced runs have been used to calculate separately . The horizontal lines are drawn to indicate the range over which the Strouhal numbers are approximately constant. | |
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As we have discussed above, at large ,
can even slightly exceeds the value of
.
Since
itself does not change sign, this would imply that
would become negative for large .
An obvious solution to this problem is that
and
should be preceded by some additional
quenching functions,
and ,
respectively.
Both functions are expected to be of the order of unity, but they may
not be exactly equal; see Kleeorin et al. (2002).
In the following we combine the information from the kinetically and
magnetically driven runs to compute separately
and
,
and assume
and
.
We thus have an equation with two unknowns,
Table 1: Summary of the measured rms velocity and normalized s for kinetically and magnetically forced runs, together with the resulting Strouhal numbers and their corresponding errors obtained by taking the extrema from 5 separate averages over 1/5 of the full time series. Values in parenthesis show departures from the trend and should be regarded as uncertain. For the resolution varies between 64^{3} and 512^{3}meshpoints for and , respectively, while for the resolution varies between 32^{3} and 256^{3}meshpoints for and , respectively.
In order to obtain a second equation, we can consider the simulations for
kinetically and magnetically driven turbulence as two independent measurement.
This leads to two separate measurements of ,
,
and
,
distinguished by superscripts (k) and (m), respectively.
In this way, Eq. (16) becomes a matrix equation
(17) |
(18) |
(19) |
(20) |
Figure 7: Magnetic and kinetic Strouhal numbers as a function of for and . Kinetically and magnetically forced runs have been used to calculate separately . | |
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When the field strength is increased, both and are quenched uniformly, i.e. approximately independently of . This becomes clear when comparing the first two panels of Fig. 6, or by inspecting the data of Table 1. For a fixed value of (around 50) the quenching with increasing field strength is shown in Fig. 7. This quenching, which sets in once is of the order of unity, is rather strong (proportional to B_{0}^{-3}). Similarly strong quenching has previously been found for if the contribution from is ignored (Moffatt 1972; Rüdiger 1974; Rüdiger & Kitchatinov 1993).
In Figs. 8 and 9 we present, both for the kinetically and magnetically driven cases, shell-integrated spectra of magnetic and kinetic energies, M(k) and E(k), respectively, as well as current and kinetic helicities, C(k) and F(k), respectively. They are normalized such that , , , and . The two helicity spectra are subject to a realizability condition, and (Moffatt 1978). If the bound is nearly saturated, we say that the field is fully helical. It turns out that for we have and , so magnetic and velocity fields are in fact not fully helical at small scales.
Figure 8: Compensated shell-integrated power spectra of magnetic energy M(k)and kinetic energy E(k) (the upper solid and dotted lines in the upper panel), and current helicity C(k) and kinetic helicity F(k)(the lower solid and dotted red or gray lines in the upper panel). The energy spectra spectra a made dimensionless by scaling with , where is the total energy dissipation rate. The two helicity spectra are scaled such that, if the fields were nearly fully helical at all scales, they would be close to the corresponding energy spectra, which is not the case. Instead, both energy and helicity spectra show a short subrange where they are best fitted with a k^{-5/3} power law. The helicity spectra, compensated by k^{5/3}, are shown in the lower panel. The turbulence is kinetically forced, with imposed field B_{0}=0.1, using 512^{3} meshpoints. | |
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Since both C(k) and F(k) seem to scale like k^{-5/3} in the inertial range, both helicities are governed by contributions from large scales. This implies in particular that the small scale current helicity, , and hence also , should not depend on the length of the inertial range and hence should be independent of . If this were not the case, i.e. if the magnetic field were fully helical with a k^{-2/3} current helicity spectrum, would be asymptotically proportional to (Blackman & Brandenburg 2002). Our results suggest that this is not the case here. This seems to be in conflict with earlier results for helical dynamos (Brandenburg 2001), but those results were at 4 times smaller resolution (120^{3} instead of 512^{3} meshpoints) and the forcing was at 3 times smaller scale, so the effective length of the inertial range was 12 times smaller and hence completely absent. What remains true, however, is that the field is nearly fully helical at the energy carrying scale, i.e. at wavenumber .
In order to gain more insight into the nature of the turbulence in kinetically and magnetically forced cases we show visualizations of the field-aligned components of velocity and magnetic field, u_{z} and B_{z}, respectively. In the kinetically driven case we see a large scale pattern in velocity, consistent with power at the expected energy-carrying scale corresponding to the forcing wavenumber ; see Fig. 10. The magnetic field, on the other hand, is dominated by smaller scale structures. The strength of the imposed field is still less than the equipartition value ( ) and there is no obvious anisotropy. Although the velocity field is helical, there is no obvious large-scale pattern in the magnetic field. There are two possible reasons for this: lack of scale separation (see Sect. V of Haugen et al. 2004) and suppression of large scale dynamo action by the large scale field (Montgomery et al. 2002; Brandenburg & Matthaeus 2004).
The magnetically driven case is quite different.
There seems to be a large scale pattern in the magnetic field (Fig. 11).
Although there is no scale separation, it is possible to drive a larger
scale field in this case.
The main difference is that in the kinetically driven case,
dynamo-generated large scale and small scale fields have opposite sign
of current helicity (Brandenburg 2001).
In the absence of sufficient scale separation this leads to cancelation.
In the magnetically driven case the current helicity has the same sign
at all scales, so there is no cancellation and therefore the build-up
of a large scale field is possible.
However, when the field is too strong (B_{0}=0.1, corresponding to
)
no large scale pattern in the magnetic field
develops; see Fig. 12.
Figure 9: Same as Fig. 8, but for the magnetically forced case with B_{0}=0.1, 512^{3} meshpoints. | |
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Although a closure hypothesis of the form has a plausible physical interpretation as a relaxation term, (Blackman & Field 2002), it is nevertheless quite crude and lacks a rigorous physical basis. However, on empirical grounds this closure assumption is quite appealing. First of all, unlike the first order smoothing approximation, MTA provides a natural and convincing explanation for the correction term to the effect in Eq. (10). This is indeed a key ingredient to the dynamical quenching model (Kleeorin & Ruzmaikin 1982), where the evolution of is obtained by solving the magnetic helicity equation for the small scale field. The dynamical evolution of the term provides a good model for the resistively limited saturation (Field & Blackman 2002; Blackman & Brandenburg 2002; Subramanian 2002), which is seen in helically forced simulations in periodic boxes (Brandenburg 2001; Mininni et al. 2005). Another appealing property of MTA is that the additional time derivative, in Eq. (4), restores causality and prevents infinitely fast signal propagation. In the case of turbulent passive scalar diffusion, one can easily see that this term turns the heat equation into a damped wave equation, where the maximum signal speed is the rms velocity of the turbulence (Brandenburg et al. 2004).
One of the striking features of the dynamical quenching model is that the resistively limited saturation and the resistive quenching of the effective can already be explained without invoking any explicit dependence of . Instead, the dependence of the steady state values of emerges solely as a result of magnetic helicity conservation and might hence be alleviated if there is a helicity flux that offsets magnetic helicity conservation (Blackman & Field 2000; Kleeorin et al. 2000, 2002; Vishniac & Cho 2001; Subramanian & Brandenburg 2004; Brandenburg & Sandin 2004; Brandenburg 2005; Brandenburg & Subramanian 2005b). However, there is no compulsory reason that there might not be an explicit dependence of after all. Our present simulations begin to shed some light on this question. However, the results are still not as clear cut as one would like them to be. Looking at Fig. 6, it is evident that for , St is independent of and depends only on ; cf. the uniform decrease of St between the first and second panels of Fig. 6. However, for , St may actually decrease with , although the error bars also increase. If this decrease is indeed real, it might indicate that the closure hypothesis adopted here is too simplistic.
We should emphasize that, although we have in this work focused on the effect, this is not the only effect that produces large scale dynamo action. An example is the shear-current effect of Rogachevskii & Kleeorin (2003, 2004). Here the electromotive force has a component in the direction of , where is the vorticity of the mean flow. (In the present work this effect is of course absent because there is no mean flow and our mean fields are defined as volume averages, so .) Our work is relevant even in this nonhelical case because, like the effect, the effect produces mean fields that are helical. Depending on the amount of magnetic helicity losses from open boundaries, there will be a tendency to limit the production of net magnetic helicity within the domain, so small scale current helicity of opposite sign must be produced at the same time as large scale field is generated. The presence of small scale current helicity implies that there must be a finite . This is an example where there can be an term even though . Related examples where , but , are the reversed field pinch (e.g. Ho et al. 1989), and the so-called selective decay (e.g. Montgomery et al. 1978), as described in Blackman & Brandenburg (2002) and Yousef et al. (2003). Another example where , but , is the Vishniac & Cho (2001) mechanism whereby dynamo action is accomplished through the current helicity flux that drives ; see Brandenburg & Subramanian (2005b) for model calculations showing that this effect produces saturation field strengths independent of the magnetic Reynolds number when the flux exceeds a certain threshold.
In the present paper we have only discussed the case where the helicity of the turbulence is driven explicitly. Although this is a common assumption that allows progress to be made in that the flow can be taken to be statistically isotropic, it should be noted that in astrophysical applications helicity is usually introduced by the interaction between stratification and rotation (Krause & Rädler 1980). To calculate this effect, MTA has to be used once more (e.g. Rädler et al. 2003), which then renders the effect proportional to . Nevertheless, as is emphasized in Brandenburg & Subramanian (2005a), also in this case the effect is attenuated by the current helicity contribution from the small scale field.
Earlier simulations of helical dynamos (no imposed field) suggested that the field is bihelical, i.e. with one sign of helicity at large scales, corresponding to wavenumber k_{1}, and the opposite sign at small scales, corresponding to wavenumber (Brandenburg 2001). If the field were to continue to be nearly fully helical even at smaller scales, this would have implied that would be attenuated by a factor (Blackman & Brandenburg 2002). Our present results suggest that this is not the case. In agreement with both closure calculations (André & Lesieur 1977) and direct numerical simulations (Borue & Orszag 1997) we find approximate k^{-5/3} scaling for the kinetic helicity (see also Ditlevsen & Giuliani 2001). Again, this implies that the velocity field is not fully helical beyond wavenumber , so should also be independent of . This is indeed the case, except for very small values of .
There are various ways of extending the present studies of turbulent transport to other fields. An obvious possibility is turbulent viscosity and the effect for modeling the Reynolds stress tensor (Käpylä et al. 2005). For example, one would expect the relaxation time to depend on the angular velocity (Kitchatinov & Rüdiger 1993). This dependence could be measured in a similar way as the dependence on the magnetic field studied in the present paper. Another related application is the study of passive scalar transport in the present of rotation. Both aspects are quite essential for many astrophysical applications.
Acknowledgements
We thank Eric G. Blackman for the many discussions we had on the subject, and Günther Rüdiger for comments on the manuscript. We also thank the organizers of the programs on "Astrophysical Discs'' in Aspen (USA), "Magnetohydrodynamics of Stellar Interiors'' at the Isaac Newton Institute in Cambridge (UK), and "Grand Challenge Problems in Computational Astrophysics'' at the Institute for Pure and Applied Mathematics at the University of California in Los Angeles, for providing a stimulating environment. The Danish Center for Scientific Computing is acknowledged for providing time on the Linux cluster in Odense.
For completeness we specify here the forcing function used in the
present paper^{}.
It is defined as
(A.1) |
(A.2) |
The
dependence of
can generally be understood
as a direct consequence of magnetic helicity conservation
(Gruzinov & Diamond 1994).
Dotting Eq. (6) with the vector potential ,
where
,
and adding the corresponding evolution of
,
averaging over a periodic volume, and using
we obtain
(B.1) |
(B.2) |
Figure 10: Visualizations of u_{z} ( left) and b_{z} ( right) in the kinetically forced case. B_{0}=0.03, , , 512^{3} meshpoints. | |
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Figure 11: Visualizations of u_{z} ( left) and b_{z} ( right) in the magnetically forced case. B_{0}=0.03, , , 512^{3} meshpoints. | |
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Figure 12: Same as Fig. 11, but for B_{0}=0.1. There is no clear evidence for a large scale field. | |
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