A&A 380, 359-372 (2001)
DOI: 10.1051/0004-6361:20011388
R. Arlt^{1} - A. Brandenburg^{2,3}
1 -
Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany
2 - NORDITA, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark
3 - Department of Mathematics, University of Newcastle upon Tyne, NE1 7RU, UK
Received 2 July 2001 / Accepted 1 October 2001
Abstract
The origin of large scale magnetic fields in accretion discs is
investigated. Using global three-dimensional simulations of accretion disc
turbulence, a recent suggestion of Vishniac & Cho (2001, ApJ 550, 752)
is re-examined, according to which large scale fields in accretion discs
could be understood without explicitly invoking the usual helicity effect.
Particular emphasis is placed on a certain correlation between vorticity and
azimuthal velocity gradient which has been predicted
to drive large scale dynamo action, independent of the presence or absence
of kinetic helicity.
In the global disc simulations two types of behaviours are found: those
which do show this type of velocity correlation and those which do not.
The former ones are typically also the cases where the resistivity is larger.
The latter ones show signs typical of dynamo action based on the usual helicity
effect. In the idealized simulations without rotation and just shear
the above correlation is found to be
particularly strong. In both cases there is, as expected, a systematic flux of
magnetic helicity through the midplane. However, very little magnetic helicity
leaves the domain through the top and bottom boundaries.
The idealized simulations reveal that much of this systematic flux
comes from the rotational component of the helicity flux and does not
contribute to its divergence.
Key words: MHD - turbulence
A number of different simulations have now established the possibility to generating strong large scale magnetic fields from turbulent motions (Glatzmaier & Roberts 1995; Brandenburg et al. 1995; Ziegler & Rüdiger 2000; Brandenburg 2001, hereafter referred to as B2001). Although most of these simulations were devised to explain magnetism in real astrophysical bodies, it remains debatable whether or not the mechanisms that are at work in those simulations are also those that are responsible for the generation of large scale fields in astrophysical bodies. Up to modestly high magnetic Reynolds numbers, currently accessible to simulations, the by far strongest large scale dynamo effect is based on kinetic helicity of the flow. This effect is described in standard text books (e.g. Moffatt 1978; Krause & Rädler 1980), and is closely related to the inverse cascade of magnetic helicity (Frisch et al. 1975; Pouquet et al. 1976). The reason why one may suspect problems with such mechanisms in astrophysical settings is that they tend to produce large scale magnetic fields that are helical and that, owing to magnetic helicity conservation, such helical fields can only be built up slowly on a resistive time scale (B2001). Of course, shear (or differential rotation) contributes strongly to the dynamo and enhances the growth rate and final field strength, but this reduces the resistively limited saturation time of the dynamo only by a factor of 10-100 for the Sun (Brandenburg et al. 2001, hereafter referred to as BBS2001), or perhaps somewhat more for accretion discs. Typical growth times will still be of the order of . Open boundaries also tend to reduce the time scale, but this is typically at the expense of lowering the final field strength (Brandenburg & Dobler 2001, hereafter referred to as BD2001).
The "helicity problem'' was originally identified in attempts to understand the "catastrophic'' magnetic feedback on turbulent transport coefficients such as turbulent diffusivity (Cattaneo & Vainshtein 1991) and the alpha-effect (Vainshtein & Cattaneo 1992; Cattaneo & Hughes 1996). Although the original arguments did not invoke magnetic helicity, subsequent work by Bhattacharjee & Yuan (1995) and Gruzinov & Diamond (1995) related the quenching to helicity conservation. Furthermore, models using the proposed quenching formulae reproduce the field evolution in the simulations and those predicted by magnetic helicity conservation extremely well (B2001, Fig. 21). Blackman & Field (2000) pointed out that catastrophic quenching of the -effect is peculiar to flows in periodic domains where there is no loss of magnetic helicity. It is important to emphasise that the helicity problem applies to the non-kinematic stage of dynamo activity, without explicitly invoking the concept of -effect dynamos.
A possible way out of the helicity problem is to produce large scale fields without invoking kinetic helicity of the flow. That helicity is not crucial for large scale dynamo action was already known since the work of Gilbert et al. (1988), who found that flows that only lack parity invariance are already capable of producing large scale dynamo action via an -effect. The problem has been investigated further in a recent paper by Zheligovsky et al. (2001), who found that even parity invariant flows are capable of large-scale dynamo action. In that case the dynamo works not via an -effect, but through a negative turbulent magnetic diffusivity effect. A related issue was brought up by Vishniac & Cho (2001, hereafter referred to as VC2001) in an attempt to produce large scale non-helical dynamo action that would survive in the large magnetic Reynolds number limit. Their mechanism requires the presence of a certain correlation between the azimuthal component of the vorticity and the azimuthal gradient of the vertical velocity. If that is the case they predict the presence of a strong vertical magnetic helicity current upwards. This could then drive a field-aligned electromotive force that is proportional to the divergence of this magnetic helicity current. This form of the electromotive force would conserve magnetic helicity and was first proposed by Bhattacharjee & Hameiri (1986).
The purpose of the present paper is to assess the viability and properties of the mechanism proposed by VC2001 using numerical simulations. The required correlation between the azimuthal derivative of the vertical velocity and the azimuthal component of the vorticity is expected to occur in the presence of shear. It should thus be especially important in accretion discs. We therefore begin by determining the presence of such a correlation using global simulations of accretion discs (Sect. 2).
However, in order to isolate the proposed effect from the ordinary helicity effect, which is always present because of rotation, we have also carried out some idealized simulations of forced turbulence with shear, but no rotation. The latter is expected to promote the correlation anticipated by VC2001, but would not lead to kinetic helicity in the flow. Of course, real systems do rotate and have therefore also kinetic helicity. However, at small and modestly large magnetic Reynolds numbers the dynamo effect based on kinetic helicity is so much more powerful than other mechanisms that it is necessary to suppress it artificially if one wants to study it in isolation. This will be done in Sect. 3.
For both the global disc simulations as well as the idealized model we also determine the resulting helicity fluxes, which turn out to be small and fluctuating about zero, however. We conclude in Sect. 4 with summarizing remarks and speculations concerning the viability of conventional helicity-driven dynamos.
(1) |
(3) |
In most of the models we apply closed boundary conditions for the flow on the z-boundaries, with the magnetic field penetrating the boundaries at right angles. One of the models (Model IX; see below) uses open boundaries in the z-direction. The inner radial boundary is open in all models; the mass flux is monitored on the inner boundary and fed back into the domain on the outer radial boundary with either a homogeneous or a Gaussian infall pattern for the density. The maximum infall velocity is constant in time and over zand is either 10^{-2} or 10^{-3} of the sound speed. The -direction has periodic boundaries.
The initial configuration contains a relaxed disk with a slow outflow on the inner boundary due to numerical viscosity. The density scale height varies between and 0.66 between the radii r=4 and 6. In the absence of magnetic fields the system is hydrodynamically stable; this was verified numerically for up to 30 orbits without producing any visible changes at the end of the simulation. The magnetic field imposed to this configuration is merely a vertical B_{z} field with zero net flux through the vertical surfaces.
Table 1 summarizes the global runs used for this analysis. The same numbering of the models as in Arlt & Rüdiger (2001) is used. Two of the simulations are new on this list: Model Va is a repetition of the configuration of Model V, but with an initial magnetic field of mixed parity. The parity of the B_{z} field was zero at the beginning as was the parity of the emerging field. All other models start with an initial field parity of -1 (antisymmetry). Model IX is similar to Model V, but uses outflow boundary conditions for the vertical direction instead of closed boundaries. In Table 1 the magnetic diffusivity is also given for all runs. Since the considerations presented here were made after the actual production runs had been performed, only a limited number of fully three-dimensional snapshots were available, which is the reason for a relatively coarse sampling.
Run | Grid (z, r, ) | Radial boundary condition | r-range | Init. parity | |||
II | homogeneous accretion | 4.0-6.0 | 0.159 | 14.7 | antisym. | 0.001 | |
III | homogeneous accretion | 4.0-6.0 | 0.159 | 9.7 | antisym. | 0.001 | |
VIII | homogeneous accretion | 3.0-7.0 | 0.103 | 22.4 | antisym. | 0.001 | |
V | Gaussian accretion | 4.0-6.0 | 0.159 | 18.4 | antisym. | 0.01 | |
Va | Gaussian accretion | 4.0-6.0 | 0.159 | 12.4 | mixed | 0.01 | |
VI | Gaussian accretion | 4.0-6.0 | 0.159 | 16.1 | antisym. | 0.01 | |
IX | Gaussian accretion , z open | 4.0-6.0 | 0.159 | 16.9 | antisym. | 0.01 |
In the theory of VC2001 large scale magnetic field generation is possible
when there is a finite value of the correlation
(4) |
A scatter plot of versus is shown in Fig. 2 for the data of a snapshot from Model V, which has the largest (negative) correlation. The plot contains points in the plane at r=5which is in the middle of the computational domain. The plot looks rather noisy, but one sees nevertheless a slight negative correlation.
Figure 1: Correlation of versus in global disc simulation; here of Models II and III (left), as well as Models V and Va (right). | |
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Figure 2: Scatter plot of with for a snapshot of Model V at . The straight line gives the least square fit, which has a correlation coefficient of in this example. | |
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We first discuss the overall field structure. Horizontal slices of the field at z=-0.39are shown for Models VIII (less resistive) and V (more resistive) in Figs. 3 and 4, respectively. The former figure exhibits a spiral pattern whilst the latter is rather dominated by intermediate scale structures or eddies, which is probably directly a consequence of the larger magnetic diffusivity in that case.
Figure 3: Projection of magnetic field vectors in a horizontal slice at z=-0.39of Model VIII after 19.9 orbits. | |
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Figure 4: Projection of magnetic field vectors in a horizontal slice at z=-0.39of Model V after 17.6 orbits. | |
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Next we derive a number of averaged quantities from the simulations. Throughout this section we denote azimuthal averages by an overbar, e.g. . In Fig. 5 are shown the energies contained in the large scale field, , and the energies of the remaining fluctuations, , where . Like in Fig. 1, the temporal evolution behaviours separate into the same two groups: the low- (less resistive) runs which show significant energies in the large scale field, and models with larger that are more resistive, but better able to generate fluctuation energies of at least 50% of the large scale energy. We note that there is one model (Model IX, not shown) where at the end the energy of magnetic fluctuations exceeds the large scale magnetic energy. The fact that the energy of the mean field is typically larger than that of the fluctuating field is somewhat surprising. A possible reason could be that the memory of the initial mean field has not yet been lost. It is also possible, however, that it is because of the global geometry and the shear that a strong large scale field is more easily established when the magnetic Reynolds number is large.
Figure 5: Comparison of energies in the large scale (-averaged) magnetic field (solid lines) and the small-scale magnetic field (dotted lines) for Models II and III (left), as well as Models V and Va (right). Note that the fluctuations are larger in the cases where is larger. | |
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Another distinction between the two groups of runs is given by the magnetic Taylor microscale, , which we define here via . In Fig. 6 we show the value of for the same four models as in Fig. 1. The quantity characterized the typical thickness of flux structures. Its significance is that in runs with dynamo action tends to increase with time until it reaches saturation (e.g. Brandenburg et al. 1996). Conversely, when the field is amplified just by field compression the value of decreases somewhat with time. This is what happened in the more resistive runs (). These were actually the runs that did show evidence for a finite Vishniac-Cho correlation. In contrast, the less resistive runs ( ) do not show any such trend.
A useful quantity for assessing the importance of helicity in the
large scale field is to look at the nondimensional quantity
(5) |
Figure 6: Comparison of the magnetic Taylor microscale for Models II and III (left), as well as V and Va (right). | |
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Figure 7: Evolution of the nondimensional large-scale current helicity parameter (upper disc plane) for Models II and III (left), as well as Models V and Va (right). | |
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We now discuss the sign of . In BD2001 the sign of the kinetic helicity was positive, and so was the sign of the small scale current helicity. The sign of the large scale current helicity is typically opposite, i.e. negative in that case. In the present case, where we consider the upper disc plane, the kinetic helicity is negative, so we would expect a positive value of , which is not what we find (except for one of the more resistive cases, Model Va). However, the unusual sign of is primarily related to an unusual sign of the effective found in the present simulations. This is because in the steady state the energy-generating effect, , must balance turbulent diffusion, . Therefore the sign of must coincide with the sign of . In discs, however, the sign of is negative (in the upper disc plane), so must also be negative, and hence is negative, as observed. The perhaps most convincing explanation for the negative is that intense parts of a flux tube contract (to maintain pressure balance along field lines), but are also most buoyant. If this contraction is stronger than the expansion associated with the rise into a less dense medium, will be negative (Brandenburg 1998, see also Rüdiger & Pipin 2000).
We shall now return to the question of whether there is any evidence for the presence of a dynamo effect as envisaged by VC2001. We therefore need to look at the possibility of magnetic helicity fluxes through the domain.
Run | slope | ||||||||
North | South | North | South | ||||||
II | 3.7 | +0.00006 | -0.0024 | -0.012 | |||||
III | 7.5 | +0.00007 | -0.0005 | +0.003 | |||||
VIII | 7.6 | +0.00029 | -0.0018 | -0.005 | |||||
V | 3.8 | -0.00013 | -0.0086 | -0.050 | |||||
Va | 5.3 | -0.00038 | -0.0096 | -0.086 | |||||
VI | 4.0 | -0.00157 | -0.060 | ||||||
IX | 2.9 | +0.00007 | -0.0053 | -0.047 |
In VC2001 it was argued that, although the overall magnetic helicity is small, there could still be a significant (spatially constant) flux of magnetic helicity vertically through the domain. The numerical procedures for evaluating gauge invariant magnetic helicity and magnetic helicity flux in cylindrical geometry with open boundaries in the r and zdirections are not yet available. However, for the present purpose most important is the contribution from the large scales. If we adopt horizontal averages (over r and ), the mean fields are one-dimensional and the mean magnetic vector potential can be obtained simply by integration. The corresponding magnetic helicity and magnetic helicity fluxes of the mean field can then be calculated quite easily (see the appendix of BD2001). In Fig. 8 we plot, for the four models, the magnetic helicity flux, , out of the domain through the two boundaries at z=z_{1} and z_{2}. Here, and denote the upward helicity fluxes at z=z_{1}and z_{2}, respectively; see Appendix A.
Figure 8: Estimates of the magnetic helicity flux out through the vertical boundaries. The dotted lines give the zero value. | |
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The mean outward flux, expressed in "dynamical'' units, , is small ( ). A small outward flux was also found in the case of helical turbulence (BD2001). Furthermore, the signs tend to be different in the cases where is small ( for ) and where it is larger ( for ). It is doubtful that this result indicates any significant departure from zero, because the magnetic helicity on the two sides about the midplane of the disc are expected to be different. Thus, equal losses or gains on the two surfaces (z=z_{1} and z_{2}) should result in zero net magnetic helicity flux. It is therefore now necessary to determine the mean upward fluxes of magnetic helicity on the two sides, and . Its average is denoted by . If there is indeed a systematic upwards flux through the two boundaries, this quantity should be finite and positive. As expected, this quantity turns out to be small. In order to check whether this is the result of some cancellation, we need to consider the magnetic helicity fluxes in smaller sub-volumes.
A difficulty associated with calculating magnetic helicity and magnetic helicity fluxes separately in two sub-volumes (e.g., above and below the midplane) is that we want to make sure that the sum of the two is equal to the total magnetic helicity calculated earlier. This will be the case provided the magnetic helicity in each sub-domain is calculated using the same gauge that also made the helicity of the full domain gauge invariant. This then also allows one to calculate the integrated magnetic helicity fluxes out of each sub-domain. The corresponding formulae are given in Appendix B.
It turns out that the helicity fluxes out of each sub-volume are actually large but of opposite sign. This means that there is actually a large magnetic helicity flux through the midplane, but not through the upper and lower boundaries. Having fixed the gauge such that is equal to the helicity of BD2001 for the full domain, we can also calculate the local magnetic helicity fluxes. We denote these by (if evaluated at the midplane), or by (if calculated for all values of z). In Fig. 9 we plot and compare with the averaged boundary fluxes . It turns out that is indeed mostly positive, as predicted by VC2001, but this flux is not sustained all the way to the boundaries: is virtually zero by comparison. An exception is Run Va, where shows large variations about zero and begins to deviate systematically from zero. (We recall that this is the run where the initial field had mixed parity.)
Figure 9: Estimates of the integrated magnetic helicity fluxes at the midplane, , (solid line) and on the boundaries, (dotted line). | |
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In order to see whether the magnetic helicity flux at the midplane is typical of the entire interior of the simulation domain, we plot in Fig. 10 the vertical distribution of the magnetic helicity flux, which was derived from horizontal averages of field and flow and then averaged in time (again excluding the first 2 to 4 orbits). Figure 10 shows that a positive (i.e. upwards) flux of magnetic helicity is indeed typical of the interior of the entire domain, and that it vanishes only near the boundaries. Thus the boundaries seem to play an important role, which may of course be unrealistic. We note, however, even Model IX with open boundaries does show a rapid drop of magnetic helicity flux near the z-boundaries. Clearly, an abrupt change of this flux implies generation and destruction of magnetic helicity near the vertical boundaries.
To summarize the global disc simulations, the correlation anticipated by VC2001 is present, provided the magnetic diffusivity is not too small. If the magnetic diffusivity is smaller, the magnetic field tends to be stronger and can become more important and may hence be able to suppress this correlation. Nevertheless, these are also the cases which show most clearly a systematic magnetic helicity flux within the simulation domain, even though it is unable to leave or enter it through the boundaries. It is difficult to say whether the effect of VC2001 was really responsible for the field generation found in the disc simulations. We recall that in the present simulations there is also some evidence for a -effect, although it is based on a rather noisy correlation between the turbulent electromotive force and the mean field; see Arlt & Rüdiger (2001). In the following section we isolate the VC-effect by studying a more idealized model with no net helicity. This model also allows for longer runs and therefore clearer statistics.
In order to study the effect proposed by VC2001 in isolation we need to suppress artificially the ordinary dynamo effect due to kinetic helicity of the flow. Following VC2001 the crucial factor that is necessary for causing the anticipated correlation is shear, but not rotation. To check this we have used a simulation of externally driven turbulence with zero net helicity. This is done by using the forcing function of B2001, but with helicity whose sign changes randomly. In addition, the model is supplemented by the effects of sinusoidal shear (as in BBS2001) and vertical stratification with a conducting halo above a turbulent layer (as in BD2001).
Figure 10: Vertical distribution of the estimates of the upward magnetic helicity flux through planes at various locations z (negative sign means downward). The values at each zare averages over time, excluding the first five snapshots of the simulations. | |
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We solve the isothermal compressible MHD equations for the logarithmic
density ,
the velocity
,
and the magnetic vector potential
,
(6) |
(9) |
We use nondimensional units where . Here, is the isothermal sound speed, k_{1} the smallest wavenumber of the two horizontal directions (so its size is in the horizontal direction). The vertical extent of the domain is . Periodic boundary conditions are adopted in all three directions. The wavenumber of the forcing is . As in BBS2001 the forcing amplitude is f_{0}=0.01 and the nominal shear is . This means that the resulting shear velocity (in the absence of magnetic stresses) is then also of order unity, i.e. close to the speed of sound, and the turbulent rms velocity is about a hundred times smaller. As in BD2001, the gravitational potential varies sinusoidally in z with an amplitude g_{0}=0.5, so the density contrast is . The main parameter that is varied in the models considered below is the magnetic diffusivity , which is in the range .
Run | ||||||
A | 0.01 | 0.0030 | 2.2 | 0.005 | 0.002 | |
B | 0.01 | 0.0030 | 0.7 | 0.018 | 0.002 | |
C | 0.01 | 10^{-4} | 0.0035 | 0.23 | 0.013 | 0.001 |
C2 | 0.005 | 10^{-4} | 0.005 | 0.35 | >0.012 | >0.0004 |
Once the magnetic field becomes strong the shear motion becomes reduced significantly due to magnetic forces. We define the total kinetic energy (per unit surface) as , where and are the boundaries between halo and disc plane. (This expression for includes the energy contained in the shear, in contrast to which does not.) When the field is still weak we have , but once the field is strong this value gets significantly reduced by magnetic stresses; see Table 3. One should keep this in mind when comparing the magnetic energies for the different runs. In the following we use rms values of velocity and magnetic field for normalization purposes; these quantities are defined in terms of and M via and , respectively.
In Fig. 11 we plot the resulting correlation between ( ) and for a particular snapshot at the end of Run A. The anticipated effect is relatively well pronounced - much more than in the global disc simulations. As expected (see VC2001), its sign changes where the local shear, , is reversed. This supports the validity of the basic result of VC2001 that such a correlation exists owing to the presence of shear.
It turns out that the Vishniac-Cho correlation is in fact the most significant correlation coefficient that changes sign when shear changes sign. In Appendix C we have calculated, for this flow, all 81 correlation coefficients of . The Vishniac-Cho correlation is given by the sum of and . These two terms are indeed the most important ones. We note that the correlators (for ) are also relatively large, but they do not change sign when the sign of the shear changes.
Next, we need to check whether this flow is capable of dynamo action and whether large scale fields can be generated. If so, then this effect should be associated with a significant vertical transport of magnetic helicity (VC2001). Whether or not such a flux really helps the dynamo needs of course to be seen.
Figure 11: Scatter plot showing the correlation between and for a snapshot taken at a late time of the run with . In this example the correlation coefficients are -0.66 (for , where S<0) and +0.55 (for x=0, where S>0). | |
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In this model we find dynamo action provided the magnetic Reynolds number is large enough. In Table 3 we give various parameters for the models considered. In Run A where (the value used in BBS2001) the dynamo growth rate, , is about . This is much less than in the case with helical forcing where . As in B2001 and BBS2001 the initial field strengths are about 10^{-6}. Thus, in the present case the saturation time is about , compared to about 10^{3} in BBS2001. For (Run C) the growth rate is about five times larger (0.0053) and the resulting saturation time correspondingly shorter. The field is concentrated at small scales and shows some loop-like pattern within the turbulent layer; see Fig. 12. In Fig. 13 we plot the evolution of magnetic energies of the mean and fluctuating field components within the turbulent zone, . Here, mean fields are defined with respect to averaging in the toroidal direction. Large scale magnetic fields are not present in the bulk of the turbulent layer, but can be seen in the halo, especially at later times.
Figure 12: Vertical slice of B_{y} (left) and its y-average (right) for Run C2. The boundaries between halo and disc plane are indicated by white lines. The resolution of this run is meshpoints and t=810, corresponding to . Dark and light shades indicate negative and positive values, respectively. | |
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In Run A where there is significant magnetic energy in the mean magnetic field (upper panel of Fig. 13), but this mean field diminishes and the fluctuating field gains in strength as the magnetic diffusivity is lowered to (Run C).
Figure 13: Normalized magnetic energies in the mean and fluctuating components for Runs A and C. Time is given in units of the inverse growth rate ( for Run A and for Run C). | |
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Next we look at the evolution of magnetic and current helicities;
(10) |
As expected, because there is no net helicity in the forcing of the flow, there are also no net magnetic and current helicities in the fields; see Fig. 14. The fluctuations of magnetic helicity are generally weak, but the contributions from the mean and fluctuating fields are comparable. For the current helicity there is a clear dominance of the small scale fields over the large scale fields. This is explained by the fact that the current helicity has two k-factors more than the magnetic helicity and hence the ratio of small scale to large scale contributions are larger by a factor for current helicity relative to magnetic helicity.
Figure 14: Magnetic and current helicities in the mean and fluctuating components for the run with . H and C are given in units of and , respectively. | |
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Figure 15: Magnetic helicity flux density in Coulomb gauge (upper panel) compared with that in the gauge (lower panel). The local flux density has been averaged in y and in t (between and 42). Run C. | |
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Vishniac & Cho (2001) pointed out that the gauge in Eq. (8) is not suited for analysing local helicity flux densities. We recall that the procedures of Berger & Field (1984) and BD2001 only apply to helicities in a given volume and the corresponding helicity fluxes through its bounding surface. Magnetic helicity densities and the corresponding flux densities are only possible to define in a given gauge.
In the following we denote by the magnetic vector potential in the Coulomb gauge, whilst still refers to the vector potential in the gauge. The conversion between the two is given by , where and . In the following we use the notation .
In the
gauge the magnetic helicity flux density is simply
,
but when converted into the Coulomb gauge it becomes
(see Appendix D)
There may be additional solenoidal contributions also from
.
In general
we may split
into an irrotational and a
solenoidal part,
(12) |
Figure 16: Time averaged vertical magnetic helicity flux density at x=0 as a function of z in different gauges. refers to the gauge and refers to the Coulomb gauge. refers to the irrotational part and refers to the first part in Eq. (11). Like in Fig. 15, only the contributions from the fluctuating components of and are included. Run C. | |
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In order to assess the possibility of dynamo action we plot in
Fig. 17 vertical profiles of the divergence
of the helicity flux density, in units of
.
This non-dimensional quantity is reminiscent of a local dynamo number,
,
where
(13) |
Figure 17 shows that the non-dimensional measure of the fluctuates around . This value should be compared with the critical value of above which dynamo action is possible. We define the dynamo number as , where . With and we have , and since the critical dynamo number is around 2 (BBS2001) we have 0.01, which agrees with the estimate above (cf. Fig. 17). However, this estimate has been too optimistic in several ways: the actual value of S is smaller than S_{0}and the incoherent effect dynamo will be less efficient. This may explain why the Vishniac-Cho effect does not seem to operate in the present simulations, but it may become more important at higher magnetic Reynolds numbers.
Figure 17: Divergence of the mean helicity flux, normalized to make it similar to a local dynamo number. As in Figs. 15 and 16, only the contributions from the fluctuating components of and are included in the calculation of the y and t averaged magnetic helicity flux. Run C. | |
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In this paper we have re-examined the recent suggestion by VC2001 that large scale dynamo action can result from velocity correlations involving higher derivatives. We found this effect to be present both in global accretion disc simulations as well as in models of forced turbulence with no rotation and just shear. Nevertheless, it does not seem to produce large scale dynamo action in the parameter regime considered here. In particular, we find no signs of any net vertical flux of magnetic helicity through the domain. This was thought to be an important property of the model suggested by VC2001.
Of course, the range of parameters considered in the present work is limited, and the degree of stratification is relatively weak. Nevertheless, the anticipated velocity correlations are strongly present and yet there are no signs of large scale dynamo action. As the magnetic Reynolds number is increased, the anticipated velocity correlations that are necessary to drive non-helical large scale dynamo action become smaller, making this new mechanism an unlikely candidate for explaining the field found in accretion disc simulations. Although we cannot exclude the possibility of different behaviour at larger magnetic Reynolds number or in more realistic representations of accretion discs, it is clear that the anticipated effect will not be easily detectable. This is quite important given the fact that large scale dynamo action owing to the helicity effect is so much stronger than nonhelical dynamo action. Thus, if one is to find the effect anticipated by VC2001 it will be quite important to isolate it from the much stronger helicity-driven dynamo effect.
Some concluding speculations regarding the viability of helicity-driven dynamo action in astrophysical settings are now in order. At large magnetic Reynolds numbers dynamo action will always generate strong magnetic fields within a short period of time. What the helicity constraint does is to prevent the formation of large scale helical patterns in a time less than a certain fraction of the magnetic diffusion time. It does not exclude however the formation of large scale patterns where the magnetic helicity cancels to zero. This would however require an exchange of magnetic helicity between various sub-domains. This does unfortunately not come automatically, as the simulations of BD2001 have shown. However, for the sun the relevant resistive time scales are estimated to be around or less than 10^{6} years. On the one hand this is long enough for large scale dynamo to be well established at the present time. On the other hand, it would suggest that the time scale for the solar cycle must essentially be controlled by non-resistive effects. One idea that deserved further attention is the possibility that the dynamo wave corresponds to actual fluid motions within the solar convection zone, such that the magnetic helicity within a Lagrangian fluid patch remains to be conserved. Since this would correspond to a systematic flux of magnetic helicity, this mechanism would be similar to that of VC2001. Here, however, the magnetic helicity flux would not be self-driven, but driven externally, e.g. by the meridional circulation. A number of recent investigations have shown that meridional circulation would be capable of reversing the sense of the dynamo wave driven by the -dynamo (Durney 1995; Choudhuri et al. 1995; Küker et al. 2001). This is similar to the possibility discussed above where the dynamo wave itself drives the meridional circulation.
As far as discs is concerned, the long resistive time scale is perhaps not a problem, because the possibility of strong outflows always shortens the saturation time scales, albeit at the expense of lowering the final saturation field strengths (see BD2001). The final solution to the problem may require more realistic global simulations with explicit resistivities, combined with suitable analytic approaches to enable one to extrapolate to astrophysical conditions.
Acknowledgements
We thank Wolfgang Dobler, Kandu Subramanian and Ethan Vishniac for interesting discussions and comments on the manuscript. Use of the PPARC supported supercomputers in St Andrews and Leicester (UKAFF) is acknowledged. We thank the John v. Neumann-Institut for Computing at the Forschungszentrum Jülich, Germany, for using the T90 computer. R.A. acknowledges the kind support by the Deutsche Forschungsgemeinschaft and the hospitality of Nordita, where much of this work has been carried out.
The evolution equation of the mean (one-dimensional)
magnetic vector potential is
(A.4) |
(A.5) |
(A.6) |
(A.7) |
(A.8) |
(A.9) |
(A.10) |
Similar to Eq. (A.2), we can now derive an evolution equation for
.
The
flux term is then like in Eq. (A.2), but with
being replaced by
.
The magnetic helicity flux out of an individual sub-domain is then
,
where
(C.1) |
(C.2) |
l=1 | l=2 | l=3 | ||||||
1.00 | 0.04 | -0.09 | 0.04 | 1.00 | -0.14 | -0.09 | -0.14 | 1.00 |
0.02 | -0.01 | -0.01 | -0.26 | -0.03 | 0.03 | -0.03 | 0.01 | 0.03 |
0.01 | 0.05 | -0.23 | 0.10 | 0.10 | -0.02 | -0.54 | -0.01 | 0.04 |
0.02 | -0.26 | -0.03 | -0.01 | -0.03 | 0.01 | -0.01 | 0.03 | 0.03 |
1.00 | -0.00 | -0.03 | -0.00 | 1.00 | -0.01 | -0.03 | -0.01 | 1.00 |
-0.03 | 0.02 | -0.04 | 0.00 | 0.07 | -0.07 | -0.03 | -0.60 | 0.06 |
0.01 | 0.10 | -0.54 | 0.05 | 0.10 | -0.01 | -0.23 | -0.02 | 0.04 |
-0.03 | 0.00 | -0.03 | 0.02 | 0.07 | -0.60 | -0.04 | -0.07 | 0.06 |
1.00 | -0.00 | -0.03 | -0.00 | 1.00 | -0.13 | -0.03 | -0.13 | 1.00 |
l=1 | l=2 | l=3 | ||||||
0.00 | -0.16 | -0.01 | -0.16 | 0.00 | -0.06 | -0.01 | -0.06 | 0.00 |
-0.01 | 0.00 | 0.00 | 0.01 | -0.17 | 0.09 | 0.00 | 0.02 | -0.13 |
-0.15 | -0.02 | 0.00 | -0.03 | -0.04 | 0.21 | 0.01 | 0.28 | -0.07 |
-0.01 | 0.01 | 0.00 | 0.00 | -0.17 | 0.02 | 0.00 | 0.09 | -0.13 |
0.00 | -0.04 | 0.11 | -0.04 | 0.00 | -0.01 | 0.11 | -0.01 | 0.00 |
0.01 | 0.03 | 0.10 | 0.00 | 0.03 | -0.00 | 0.05 | -0.02 | 0.01 |
-0.15 | -0.03 | 0.01 | -0.02 | -0.04 | 0.28 | 0.00 | 0.21 | -0.07 |
0.01 | 0.00 | 0.05 | 0.03 | 0.03 | -0.02 | 0.10 | -0.00 | 0.01 |
0.00 | -0.51 | 0.10 | -0.51 | 0.00 | 0.00 | 0.10 | 0.00 | 0.00 |
In Table C.1 the largest contributions come from C^{(S)}_{xyyx}=-0.26, C^{(S)}_{yzzy}=-0.60, and C^{(S)}_{zxxz}=-0.54. In Table C.2 the largest contributions come from C^{(A)}_{xzzy}=+0.28 and C^{(A)}_{zxzy}=-0.51. These are also the coefficients that are important in the Vishniac-Cho correlation.
(D.2) |
(D.4) |
(D.7) |
(D.8) |