A remarkable change in the turbulent dynamo action occurs if the turbulence is helical. This can be clearly seen for example in the simulations by Brandenburg [19], where a large scale field, on the scale of the box develops when a helical forcing is employed, even though the forcing itself is on a scale about 1/5th the size of the box. The large scale field however in these closed box simulations develops only on the long resistive timescales. It is important to understand how such a field develops and how one can generate a large scale field on a faster timescale. The possible importance of helical turbulence for large-scale field generation was proposed by Parker [20], and is in fact discussed in text books [21]. We summarize briefly below the theory of the mean-field dynamo (MFD) as applied to magnetic field generation in disk galaxies, turn to several potential problems that have been recently highlighted and their possible resolution.
Suppose the velocity field is split into the sum of a mean, large-scale velocity and a turbulent, stochastic velocity u. The induction equation becomes a stochastic partial differential equation. Let us split the magnetic field also as B = + b, into a mean field = <B> and a fluctuating component b. Here the average <>, is defined either as a spatial average over scales larger than the turbulent eddy scales (but smaller than the system size) or as an ensemble average. Taking the average of the induction equation 2.1, one gets the mean-field dynamo equation for ,
(4) |
This averaged equation now has a new term, the mean electromotive force (emf) = , which crucially depends on the statistical properties of the small-scale velocity and magnetic fields. A central closure problem in MFD theories is to compute the mean emf and express it in terms of the mean field itself. In the two-scale approach one assumes that the mean field is spatially smooth over scales bigger than the turbulence coherence scale l, and expresses the mean emf mean magnetic field and its first derivative [21]. For isotropic, homogeneous, helical 'turbulence' in the approximation that the correlation time is short (ideally u / l << 1, where u is the typical turbulent velocity) one employs what is known as the First order smoothing approximation (FOSA) to write
(5) |
Here _{K} = -(/3) is the dynamo -effect, proportional to the kinetic helicity and _{t} = (/3) /3 is the turbulent magnetic diffusivity proportional to the kinetic energy of the turbulence [21].
In the context of disk galaxies, the mean velocity is that of differential rotation. This leads to the -effect, that of shearing radial fields to produce toroidal fields, while the -effect is crucial for regeneration of poloidal from toroidal fields. A physical picture of the -effect is as follows: The interstellar medium is assumed to become turbulent, due to for example the effect of supernovae randomly going off in different regions. In a rotating, stratified (in density and pressure) medium like a disk galaxy, such turbulence becomes helical. Helical turbulent motions of the gas perpendicular to the disk draws out the toroidal field into a loop which looks like a twisted . Such a twisted loop is connected to a current which has a component parallel to the original toroidal field. If the motions have a non-zero net helicity, this parallel component of the current adds up coherently. A toroidal current then results from the toroidal field. Hence, poloidal fields can be generated from toroidal ones. (Of course microscopic diffusion is essential to make permanent changes in the field). This closes the toroidal-poloidal cycle and leads to exponential growth of the mean field. The turbulent diffusion turns out to be also essential for allowing changes in the mean field flux. The kinematic MFD equation 5.1, gives a mathematical foundation to the above picture. One finds exponentially growing solutions, of the MFD equations provided a dimensionless dynamo number has magnitude D = |_{0} G h^{3} _{t}^{-2}| > D_{crit} ~ 6 [11, 22], a condition which can be satisfied in disk galaxies. (Here h is the disk scale height and G the galactic shear, _{0} typical value of , and we have defined D to be positive). The mean field grows typically on time-scales a few times the rotation time scales, of order 3-10 × 10^{8} yr.
This picture of the galactic dynamo faces several potential problems. Firstly, while the mean field dynamo operates to generate the large-scale field, the fluctuation dynamo is producing small-scale fields at a much faster rate. Also the correlation time of the turbulence measured by u / l is not likely to be small in turbulent flows. So the validity of FOSA is questionable. Indeed, based on specific imposed (kinematic) flow patterns it has been suggested that there is no simple relation between and helicity of the flow; see [23]. In order to clarify the existence of -effect and turbulent diffusion as outlined above, and its R_{m} dependence, even in the kinematic limit, we have recently measured directly in numerical simulations of isotropic, homogeneous, helical turbulence [24]. These simulations reach up to a modest Rm ~ 220. We find, somewhat surprisingly, that for isotropic homogeneous turbulence the high conductivity results obtained under FOSA are reasonably accurate up to the moderate values of R_{m} that we have tested. A possible reason for this might be that the predictions of FOSA are very similar to a closure called the minimal approximation (MTA) [27, 2] where the approximation of neglecting nonlinear terms made in FOSA is not done, but replaced by a closure hypothesis. But MTA is also not well justified, although numerical simulations (for R_{m} 300) support some aspects of this closure [28]. Interestingly, this agreement of and _{t} directly measured from the simulation, with that expected under FOSA, is obtained even in the presence of a small-scale dynamo, where b is growing exponentially. This suggests that the exponentially growing part of the small-scale field does not make a contribution to the mean emf , correlated with the mean field . It is essential to extend these results to even higher values of R_{m}, but these preliminary results are quite encouraging.
Another potential problem with the mean field dynamo paradigm is that magnetic helicity conservation puts severe restrictions on the strength of the -effect [2]. The magnetic helicity associated with a field B = × A is defined as H_{t} = A.B dV, where A is the vector potential [21]. Note that this definition of helicity is only gauge invariant (and hence meaningful) if the domain of integration is periodic, infinite or has a boundary where the normal component of the field vanishes. H_{t} measures the linkages and twists in the magnetic field. From the induction equation one can easily derive the helicity conservation equation, dH_{t} / dt = -2 B ⋅ ( × B) dV. So in ideal MHD with = 0, magnetic helicity is strictly conserved, but this does not guarantee conservation of H_{t} in the limit 0. For example magnetic energy, whose Ohmic dissipation is governed by (dE_{B} / dt)_{Joule} = - (4 / c^{2}) J^{2} dV, can be dissipated at finite rates even in the limit 0, because small enough scales develop in the field (current sheets) where the current density increases with decreasing as ^{-1/2} as 0. Nevertheless, since helicity dissipation rate has a milder dependence on the current density (its rate being proportional to J ⋅ B and not J^{2}), in many astrophysical conditions where R_{m} is large ( small), the magnetic helicity H_{t}, is almost independent of time, even when the magnetic energy is dissipated at finite rates.
The operation of any mean-field dynamo automatically leads to the growth of linkages between the toroidal and poloidal mean fields and hence a mean field helicity. In order to satisfy total helicity conservation this implies that there must be equal and oppositely signed helicity being generated in the fluctuating field. What leads to this helicity transfer between scales? To understand this, we need to split the helicity conservation equation into evolution equations of the sub-helicities associated with the mean field, say _{t} = ⋅ dV and the fluctuating field h_{t} = a ⋅ b dV . The evolution equations for _{t} and h_{t} are [2]
Here, we have assumed that the surface terms can be taken to vanish (we will return to this issue below). We see that the turbulent emf transfers helicity between large and small scales; it puts equal and opposite amounts of helicity into the mean field and the small-scale field, conserving the total helicity H_{t} = _{t} + h_{t}. Note that in the limit when the fluctuating field has reached a stationary state one has dh_{t} / dt 0 and ⋅ dV = -2 (4 / c) j ⋅ b dV which tends to zero as 0 for any reasonable spectrum of small scale current helicity. Therefore, the component of the turbulent electromotive force along the mean field is catastrophically quenched in the sense that it tends to zero as 0, or for large R_{m}, a feature also borne out in periodic box simulations [19, 25].
To make the above integral constraint into a local constraint requires one to be able to define a gauge invariant helicity density for at least the random small-scale field. Such a definition has indeed been given, using the Gauss linking formula for helicity [26]. In physical terms, the magnetic helicity density h of a random small scale field (in contrast to the total helicity h_{t}), is the density of correlated links of the field. This notion can be made precise and a local conservation law can be derived (see [26] for details) for the helicity density h,
(5) |
where gives a flux density of helicity. (For a weakly inhomogeneous system, h is approximately in the Coulomb gauge.) In the presence of helicity fluxes, we have in the stationary limit, ⋅ = -(1/2) ⋅ - , and therefore ⋅ need not be catastrophically quenched. Large scale dynamos then seem to need helicity fluxes to work efficiently [29, 30, 31].
This conclusion can be understood more physically as follows: As the large-scale mean field grows the turbulent emf is transferring helicity between the small and large scale fields. The large scale helicity is in the links of the mean poloidal and toroidal fields, while the small scale helicity is in what can be described as "twist" helicity of the small-scale field. Lorentz forces associated with the "twisted" small-scale field would like to untwist the field. This would lead to an effective magnetic -effect which opposes the kinetic produced by the helical turbulence (see below). The cancellation of the total -effect is what leads to the catastrophic quenching of the dynamo. This quenching can be avoided if there is some way of transferring the twists in the small scale field out of the region of dynamo action, that is if there are helicity fluxes out of the system.
Blackman and Field (in [29]) first suggested that the losses of the small-scale magnetic helicity through the boundaries of the dynamo region can be essential for mean-field dynamo action. Such a helicity flux can result from the anisotropy of the turbulence combined with large-scale velocity shear or the non-uniformity of the -effect [29]. Another type of helcity flux is simply advection of the small scale field and its associated helicity out of the system, with = h [30]. This effect naturally arises in spiral galaxies where some of the gas is heated by supernova explosions producing a hot phase that leaves the galactic disc, dragging along the small-scale part of the interstellar magnetic field.
In order to examine the effect of helicity fluxes in more detail, one also needs to fold in a model of how the dynamo co-efficients get altered due to Lorentz forces. Closure models either using the EDQNM closure or the MTA or quasi-linear theory, suggest that the turbulent emf gets re-normalized, with = _{K} + _{m}, where _{m} = (/3) < b. × b> / (4 ), is proportional to the small-scale current helicity and is the magnetic alpha effect mentioned above [2, 27, 32]. The turbulent diffusion _{t} is left unchanged to the lowest order, although to the next order there arises a non-linear hyperdiffusive correction [16]. Some authors have argued against an _{m} contribution, and suggest instead that the -effect can be expressed exclusively in terms of the velocity field, albeit one which is a solution of the full momentum equation including the Lorentz force [33]. To clarify this issue, we have studied the nonlinear -effect in the limit of small R_{m},Re << 1 using FOSA applied to both induction and momentum equations [34]. We show explicitly in this limit that one can express completely in terms of the helical properties of the velocity field as in traditional FOSA, or, alternatively, as the sum of two terms, a so-called kinetic -effect and an oppositely signed term proportional to the helical part of the small scale magnetic field, akin to the above closures.
Adopting = _{K} + _{m} as above, one can now look for a combined solution to the helicity conservation equation (5.3), and the mean-field dynamo equation [2, 35], after relating the current helicity arising in _{m} to the magnetic helicity density h. The effect of the advective flux in resolving the quenching of the dynamo was worked out in detail in Ref [30]. In the absence of an advective flux, the initial growth of magnetic field is catastrophically quenched and the large-scale magnetic field decreases at about the same rate as it grew. The initial growth occurs while the current helicity builds up to cancel the kinetic -effect. However, even a modest advective flux compensates the catastrophic quenching of the dynamo and the mean field stays steady at about 10% of the equipartition value. (Excessive advection, however, hinders the dynamo as it removes the mean field from the dynamo active region). The vertical advection of magnetic helicity by galactic fountain flow can therefore resolve in a straightforward fashion the catastrophic quenching of nonlinear mean-field galactic dynamos. Further, advection of small-scale magnetic fields may help the mean-field dynamo action in other ways. For example, there is a concern that the back-reaction of Lorentz forces due to the rapidly growing small-scale field could quench turbulent transport processes, simply by suppressing the required Lagrangian chaos. In this case their advection out of the galaxy will still allow the dynamo to operate efficiently.
We have emphasized so far the role of helicity for the dynamo generation of large-scale fields. Intriguingly, several recent simulations show that the generation of large-scale fields may arise even in non-helical turbulence in the presence of strong enough shear [36]. It is at present an important open question as the exact cause of such large-scale field generation.