Gravitoturbulent dynamos in astrophysical discs
Abstract
The origin of largescale and coherent magnetic fields in astrophysical discs is an important and long standing problem. Researchers commonly appeal to a turbulent dynamo, sustained by the magnetorotational instability (MRI), to supply the largescale field. But research over the last decade in particular has demonstrated that various nonideal MHD effects can impede or extinguish the MRI, especially in protoplanetary disks. In this paper we propose a new scenario, by which the magnetic field is generated and sustained via the gravitational instability (GI). We use 3D stratified shearing box simulations to characterise the dynamo and find that it works at low magnetic Reynolds number (from unity to ) for a wide range of cooling times and boundary conditions. The process is kinematic, with a relatively fast growth rate (), and it shares some properties of mean field dynamos. The magnetic field is generated via the combination of differential rotation and spiral density waves, the latter providing compressible horizontal motions and largescale vertical rolls. At greater magnetic Reynolds numbers the build up of largescale field is diminished and instead smallscale structures emerge from the breakdown of twisted flux ropes. We propose that GI may be key to the dynamo engine not only in young protoplanetary discs but also in some AGN and galaxies.
keywords:
accretion discs — turbulence — dynamo — instabilities — protoplanetary discs — galaxies: nuclei — galaxies: magnetic fields1 Introduction
Magnetic fields appear in almost all astrophysical discs, from those surrounding newly born stars, active galactic nuclei (AGN), dwarf novae and lowmass Xray binaries, to spiral galaxies. They are thought to guide the evolution of these objects via several processes, such as accretion, outbursts, winds, and turbulence (Balbus, 2003; Wardle, 2007; Armitage, 2015). In several protoplanetary (PP) discs, indirect observations, based on polarized synchrotron or dust emission, have detected magnetic fields with various strengths and morphologies (CarrascoGonzález et al., 2010; Stephens et al., 2014; Goddi et al., 2017). In fact, in the bulk of the FUOri disc, fields have been observed directly, using the highresolution spectropolarimeter ESPaDOnS (Donati et al., 2005). On the other hand, magnetic fields appear in many galactic discs, with a horizontal bisymmetricspiral component (e.g. M51, Fletcher et al., 2011) and in our own galaxy, with a vertical structure recently determined by the Planck satellite using the cosmic background polarization. Fields in AGN and masstransferring binaries have not been detected directly, but their existence can be inferred from the magnetised jets launched from their associated disks.
A fundamental goal is to determine the origin of these magnetic fields and their sustenance over the disc’s life. In the case of PP discs (the main focus of our paper), it is possible that the fields issue from the central protostar or are inherited from the primordial nebulae. If the latter, then the transport and amplification of magnetic field through the disk are key, and these processes remain the focus of current research (Guilet & Ogilvie, 2013; Zhu & Stone, 2018). Another possibility is that the field is produced insitu by a turbulent dynamo. This idea, originally suggested by Pudritz (1981), was bolstered by the observations of Donati et al. (2005) which revealed a complex magnetic topology, with both filamentary and toroidal structures, possibly generated by motions internal to the disc. Such an insitu process is appealing because of its generality: PP disks need not rely on the particularities of any external magnetic source; it might also generalise to AGN and galactic disks.
For more than two decades, the magnetorotational instability (MRI) has been proposed as a potential source of turbulence and magnetic fields in these objects (Balbus & Hawley, 1991; Hawley et al., 1995; Brandenburg et al., 1995; Balbus & Henri, 2008), via a subcritical, nonlinear, and cyclic dynamo. That said, in PP discs, this process has been shown to work reliably only in their very inner radii ( AU), where ionization sources are strong enough to permit the MRI. Further out, nonideal MHD effects (Ohmic and ambipolar diffusion plus the Hall effect) tend to weaken or quench the MRI (Wardle, 1999; Sano et al., 2000; Fleming et al., 2000; Kunz & Balbus, 2004; Wardle & Salmeron, 2012; Bai, 2013; Lesur et al., 2014; Bai, 2015). So too in AGN; close to the supermassive black hole the disc is well coupled to the magnetic field, but outward of 0.01 pc the state of the gas is far less clear, as it depends on uncertain nonthermal ionisation sources and the degree of disk warping (e.g. Menou & Quataert, 2001). Putting ionisation aside, several numerical studies have suggested that the MRI dynamo is suppressed in many of these settings because of their low magnetic Prandtl number (ratio between the microscopic viscosity and resistivity; see Fromang et al., 2007; Balbus & Henri, 2008; Käpylä & Korpi, 2011; Meheut et al., 2015; Riols et al., 2017b, for more details).
An alternative to the MRI is growth of magnetic fields via other disc instabilities. Perhaps the most efficient and powerful, after the MRI, is the gravitational instability (GI; for the PP disk context see reviews by Durisen et al., 2007; Kratter & Lodato, 2016). Discs are usually susceptible to the GI in their outer regions, provided that
(1) 
where is the Toomre parameter (Toomre, 1964), is the sound speed, the orbital frequency, and the background surface density. It is believed that 50% of class 0 and 1020% of class I PP discs might be unstable to GI (Tobin et al., 2013; Mann et al., 2015). In particular, the detection of spiral arms and streamers, recently imaged at NIR wavelengths in Ophiuchi and FU Ori systems, supports the idea that their discs are gravitationally unstable (Christiaens et al., 2014; Liu et al., 2016; Dong et al., 2016). Because they are thin, AGN are also susceptible to GI in the regions beyond 0.01 pc and might build up magnetic fields through this process (Menou & Quataert, 2001; Goodman, 2003; Lodato, 2007). The instability, of course, features also in galactic discs (perhaps its most famous venue), shaping the largescale spiral and clumpy structures where stars form (Goldreich & LyndenBell, 1965; Wang & Silk, 1994; Kim & Ostriker, 2001).
The ability of GI to amplify seed magnetic fields has been pointed out in the context of star formation (i.e during the collapse of selfgravitating dense cloud, see Pudritz & Silk, 1989; Federrath et al., 2011). However, the viability of a GIreliant dynamo process is relatively unexplored in the accretion disc context. The recent shearing box simulations of Riols & Latter (2018a, hereafter RL18a) showed that such a dynamo does exist and can sustain magnetic fields to near equipartition values, especially in the regime of efficient cooling; in addition, GI impedes the (zeronet flux) MRI in this regime. However, most of the simulations were run in the “ideal limit" (i.e. without explicit diffusion), omitting all the relevant nonideal effects, such as resistivity, the Hall effect, and ambipolar diffusion, each potentially important in the regions of PP discs where GI is active (see Simon et al., 2015, for instance). Moreover, no theory was proposed that could account for the properties of, and physical mechanism animating, the GI dynamo. This paper is a first in a series that aims to address these issues and lay the foundations of a general theory. Here, we only focus on ohmic resistivity, so as to more easily characterize the dynamo’s fundamental nature: is it kinematic or nonlinear? Fast or slow? Small or large scale?. We stress that the present study is a preliminary and necessary step before including other nonideal physics. However, we point out that both ambipolar diffusion and the Hall effect are necessarily absent in the kinematic phase of any dynamo, and so during this initial phase our results are broadly applicable.
To that end, we perform 3D MHD stratified shearing box simulations with selfgravity using a modified version of the PLUTO code. Simulations maintain a quasisteady energy balance via inclusion of a simple linear cooling law, characterised by a uniform and constant timescale , and enforce a zeronet vertical magnetic flux. We explore a range of magnetic Reynolds numbers, from 1 to about 500, defined with respect to the sound speed and disc scaleheight. It is important to note that the MRI is excluded in this regime (Simon et al., 2012), and so any magnetic field generation must issue from GIturbulent motions. Given the size of the box necessary to capture GI, it is not yet possible to probe the regime of Rm larger than a thousand, typically found in the inner (0.1 AU) and external regions ( 20 AU) of older PP disc, and in other environments like AGN.
We find that the GIdynamo is inherently kinematic and linear, with growth rates depending strongly on Rm. The dynamo process persists even in a very resistive plasma with typical , and appears to be robust even when GI is weak. At lowish Rm (), a very efficient dynamo, supported by the combined action of shear (the “omega effect”) and spiral wave motions, amplifies and organises the magnetic field into largescale spiral patterns. Essential ingredients in the dynamo are the vertical rolls associated with spirals waves (Riols & Latter, 2018b, hereafter RL18b) which allow the regeneration of the mean poloidal field from toroidal field, through a very unusual “alpha" effect. At larger Rm (), the dynamo persists but saturates at a lower amplitude. In this regime the largescale field ropes are twisted and break down into smallscale filamentary structures with a concomitant alteration of the meanfield dynamo effect.
The structure of the paper is as follows: first, in Section 2, we present the basic equations of the problem and the numerical methods. In Section 3 we study the dynamo growth rate and saturated state as a function of Rm. In Section 4, we characterise the essence of the dynamo process and its typical scales. In Section 5 we investigate in more detail the role of spiral waves in the generation of largescale and also smallscale fields. We discuss in Section 6 the dependence of the dynamo on numerical details (boundary conditions, numerical resolution) and its applications to protoplanetary discs and galaxies. Finally, we state our conclusions in Section 7.
2 Numerical model
2.1 Governing equations
We use the local Cartesian model of an accretion disc (the shearing sheet; Goldreich & LyndenBell, 1965; Latter & Papaloizou, 2017) where the differential rotation is approximated locally by a linear shear flow and a uniform rotation rate , with for a Keplerian equilibrium. We denote as the radial, azimuthal and vertical directions respectively, and refer to the projections of vector fields as their poloidal components and to the component as their toroidal one. We assume that the gas orbiting around the central object is ideal, its pressure and density related by , where is the sound speed and the ratio of specific heats. In this paper, we neglect molecular viscosity but will consider nonzero uniform magnetic diffusivity .A uniform greatly simplifies the problem, although it is not especially realistic. In PP discs, outside of 1 AU, the ionisation fraction (and consequently ) depends strongly on , due to irradiation by cosmic, FUV, and X rays. We assume for the moment that this variation does not significantly influence the dynamo process. Future simulations will explore a height dependent , using realistic profiles such as in Simon et al. (2015).
The evolution of density , total velocity field , magnetic field and total energy density follows:
(2) 
(3) 
(4) 
(5) 
where is the electric field. The total velocity field can be decomposed into a mean shear and a perturbation :
(6) 
is the sum of the tidal gravitational potential induced by the central object in the local frame and the gravitational potential induced by the disc itself. The latter obeys the Poisson equation:
(7) 
Cooling in the total energy equation (5) is a linear function of with a typical timescale referred to as the ‘cooling time’. This prescription is not especially realistic but allows us to simplify the problem as much as possible. We also neglect thermal conductivity.
Finally, defines our unit of time and our unit of length. is the standard hydrostatic disc scale height defined as the ratio where is the sound speed in the midplane of a hydrostatic disc in the limit . To characterize the importance of ohmic dissipation in the induction equation, we introduce the magnetic Reynolds number:
(8) 
2.2 Numerical methods
Our computational approach is identical to that used in RL18a. Simulations are performed with the Godunovbased PLUTO code, adapted to highly compressible flow (Mignone et al., 2007), in the shearing box framework. The box has a finite domain of size , discretized on a mesh of grid points. The numerical scheme uses a conservative finitevolume method that solves the approximate Riemann problem at each intercell boundary. It conserves quantities like mass, momentum, and total energy across discontinuities. The Riemann problem is handled by the HLLD solver, suitable for MHD. An orbital advection algorithm is used to increase the computational speed and reduce numerical dissipation. Finally, the divergence of is forced to 0 by the constrainedtransport algorithm of PLUTO.
To calculate the 3D selfgravitating potential, we employ the numerical methods detailed in Riols et al. (2017a) and RL18a. We tested the stratified disc equilibria, as well as the linear stability of these equilibria, to ensure that our implementation is correct (see appendices in Riols et al. (2017a)).
The boundary conditions are periodic in and shearperiodic in . In the vertical direction, we use a standard outflow condition for the velocity field and assume a hydrostatic balance in the ghost cells for pressure, taking into account the largescale vertical component of selfgravity (averaged in and ). For the magnetic field, we usually impose and , as in several previous MRI simulations (Brandenburg et al., 1995; Gressel, 2010; Oishi & Mac Low, 2011; Käpylä & Korpi, 2011). They allow the mean horizontal magnetic field (or total flux) to vary, even when the field has initially zero net or , and thus a potential concern is that a meanfield dynamo might be artificially sustained. That being said, previous studies indicate that the largescale magnetic field behaves similarly when open or closed boundary conditions are employed. In any case, in Section 3.4, we simulate the dynamo in a limited set of runs with periodic boundaries, where this is no issue. (See also the discussion in Section 4.2.) Finally, the boundary conditions for the selfgravity potential, in Fourier space, are:
(9) 
where is the horizontal Fourier component of the potential and , are the radial, azimuthal wavenumbers and . This condition is an approximation of the Poisson equation in the limit of low density.
Lastly, we enforce a density floor of which prevents the timesteps getting too small due to evacuated regions near the vertical boundaries. Mass is replenished near the midplane so that the total mass in the box is maintained constant. We checked that the mass injected at each orbital period is negligible compared to the total mass (less than 1% per orbit).
2.3 Simulation setup and parameters
The large scale waves excited by GI have radial lengthscales . In order to capture these waves, while affording acceptable resolution, we use a box of intermediate horizontal size . The vertical domain of the box spans to . For runs with Rm , we use a resolution of and . This resolution is enough to capture the smallest scales at which magnetic energy is (physically) dissipated. For simulations without explicit resistivity or , we double the resolution in and ( and ).
Throughout we use a fixed heat capacity ratio and consider a zeronet vertical magnetic flux. To run pure hydrodynamic GI simulations, we start from a polytropic vertical density equilibrium, computed with a Toomre Q of order 1. The calculation of such an equilibrium is detailed in the appendix of Riols et al. (2017a). Nonaxisymmetric density and velocity perturbations of finite amplitude are injected to trigger the turbulent state. For the MHD runs, the initialization of will be detailed in the corresponding sections.
2.4 Diagnostics
2.4.1 Averages
To analyse the statistical behaviour of the turbulent flow, we define the standard and weighted box averages
(10) 
respectively, where is the volume of the box. We also define the horizontally averaged vertical profile of a dependent variable:
(11) 
An important quantity is the average 2D Toomre parameter defined by
(12) 
where is the average surface density of the disc. Another quantity that characterises the turbulent dynamics is the coefficient which measures the angular momentum transport efficiency. This quantity is the sum of the total stress (which includes gravitational , Reynolds and Maxwell stresses ) divided by the box average pressure:
(13) 
where
Finally, we denote by and the box averaged kinetic and magnetic energies.
2.4.2 Fourier modes
To analyse the structure of the magnetic field, we use the 2D Fourier decomposition of these fields in and . The Fourier modes are calculated as in Riols et al. (2017a) (Section 2.5.2). We denote by and the radial and azimuthal Eulerian wavenumbers. In this mathematical representation, fields are a sum of a mean ( and ) mode, axisymmetric modes with (), and nonaxisymmetric modes with (commonly referred as “shearing waves"). A largescale wake or spiral wave can be viewed as the result of the constructive interference of multiple shearing waves of different but the same, fundamental, azimuthal wavenumber (see Ogilvie & Lubow, 2002)
2.4.3 Mean magnetic field and dynamo equation
In Section 4, we investigate the ability of GI to sustain a largescale dynamo throughout the domain. A key quantity is the mean magnetic field , averaged in the and directions, corresponding to the and Fourier component of . The equations governing the horizontallyaveraged radial and toroidal fields are:
(14) 
(15) 
with the horizontallyaveraged electromotive force (EMF). If we denote and , i.e. perturbations in the magnetic and velocity fields, then we have
(16) 
The first “laminar" term is due to the mean flow, while the second “turbulent’ term issues from the correlation between velocity and magnetic perturbations. We recall that the mean radial and toroidal fields in the box, respectively and (equal to the average of and ), are free to vary.
3 GIdynamo threshold and saturation
In the regime of moderate cooling ( the GI breaks down into a disordered state, often referred to as “gravitoturbulence", the properties of which have prompted considerable study (Gammie, 2001; Rice et al., 2003; Paardekooper, 2012; Shi & Chiang, 2014; Hirose & Shi, 2017; Riols et al., 2017a). The aim of this section is to determine, for a given Rm and by using numerical simulations, the ability of gravitoturbulent flows to generate magnetic fields. We calculate, in particular, the growth rates and saturated states of the associated dynamo as a function of Rm. We stress that the large resistivity in our simulations prohibits excitation of the MRI, so any field growth cannot be attributed to it.
Run  Resolution  Time (in )  /  /  /  

SGhydro5  100  5  1.239  0.67  0.063  0.117  0.18  0.2  
SGhydro20  200  20  1.194  0.191  0.0132  0.0345  0.048  0.05  
SGhydro100  400  100  1.29  0.088  0.0032  0.00804  0.011  0.01 
3.1 Hydrodynamic flows
Numerically, the usual way to study dynamos in turbulent flows is to start from the hydrodynamic state of fullydeveloped turbulence, and then introduce a magnetic seed to see whether it grows or decays over time. To prepare the initial states, we ran a set of pure hydrodynamic GI simulations, as in Riols et al. (2017a), with moderate resolution (13 points per in the horizontal directions and 16 points in the vertical). This resolution allows us to resolve magnetic Reynolds number up to 200 (see Section 2.3).
As described in Riols et al. (2017a) the strength and saturation of the gravitoturbulence is set by the cooling time . In particular, the timeaveraged stress to pressure ratio follows the Gammie (2001) relation:
(17) 
Three different have been considered: (strong turbulence, close to the fragmentation threshold), (moderate turbulence) and (weak turbulence).
Table 1 shows some box and timeaveraged quantities for the three different hydrodynamical simulations. While the Toomre parameter varies slightly between simulations, kinetic energy, Reynolds and gravitational stress increase significantly when decreases. We checked also that the Gammie relation (Eq. 17) holds in the three cases.
Figure 1 shows the density and radial velocity of the turbulent state associated with each . These representative snapshots are chosen at a random time and serve as initial conditions for the dynamo simulations in the next sections. For and , the turbulence is supersonic, highly compressible, and characterized by largescale spiral density waves, particularly strong in the case . Smallscale motions, potentially driven by a parametric instability (see Riols et al., 2017a) faintly distort the spiral waves at higher altitude, around . They are also marginally visible in the midplane (Fig .1) but with weaker amplitude. Note that the resolution used here allows us to capture only the largest scales of the parametric instability. In the case , the flow looks very different: the turbulence is subsonic, the waves are fainter, smoother and thinner, and their pitch angle much smaller.
3.2 Kinematic regime and growth rates
Having constructed the hydrodynamic GI turbulent states, we now investigate their amplification of magnetic fields. For a given , we run a series of MHD simulations at different Rm. The initial condition corresponds to the state described in Fig. 1. The magnetic seed introduced at the beginning of the simulations is zeronet flux, toroidal, and has a sinusoidal dependence in . In all cases (except for ), its amplitude is fixed to , which corresponds to an initial . This very small value ensures that the Lorentz force has little effect on the gas dynamics during the first few hundred , at least on large scales, and thus any dynamo action is purely kinematic.
We first analyse the intermediate cooling regime with . Figure 2 shows the time evolution of the boxaveraged magnetic energy for different Rm. For , we found that the magnetic energy falls to 0 with decay rate close to 0.14. For Rm between 5 and 100, evolves quasiexponentially with a welldefined growth rate. This is particularly striking in the case of (green curve) for which a clean exponential amplification is obtained for at least . Saturation of magnetic energy occurs after this point (a regime analysed in Section 3.3).
For the largest Rm shown in the figure (, blue curve), we found that the magnetic field is amplified with growth rate , measured by fitting an exponential curve to the numerical data. Unlike the more resistive cases we explored, its evolution is more complex than a simple exponential growth; the field exhibits intermittent bursts with long periods of about and shorter periods of . We checked that the same behaviour is obtained when with double the resolution. Unsurprisingly perhaps, it resembles the dynamo identified by RL18a in the limit of ideal MHD (without explicit resistivity). We are hence tempted to conclude that the simulations are only marginally resolved at this Rm. What we may be seeing is the grid (rather than explicit resistivity) beginning to control the dynamics.
We collect the growth rates obtained from all simulations with cooling time and plot their dependence on Rm in Figure 3 (blue curve). To make a comparison with the ideal limit, we also show in this diagram the growth rate measured in the simulation SGMRI20 of RL18a which had no explicit resistivity (blue diamond marker). We deduce from this figure that the critical threshold for the dynamo is . Its growth rate is maximal for (with ) and decreases at larger Rm.
In addition, we explore the dependence of the growth rates on , by repeating exactly the same procedure as described above, for and . The corresponding growth rates are superimposed on Fig. 3 to make direct comparison with . In the case of inefficient cooling (cyan curve, ), the dynamo is still active and the dependence on Rm is very similar to . However, the growth rates are lower and reduced by a factor 1.52. For efficient cooling (red curve, ), turbulent motions are stronger and growth rates are consequently larger. The maximum amplification occurs again for , with growth rate . In this regime magnetic fields are efficiently amplified so that they reach a quasisteady configuration within a few orbits.
To summarize, GI can amplify a magnetic field for a wide range of and the dynamo persists at very small . In addition to its kinematic and linear properties, we demonstrate that the dynamo behaves like a “slow" dynamo, with growth rates decreasing with increasing Rm above a critical value. The runs at double resolution with and those without explicit resistivity (grid Rm 660) suggest perhaps that the growth rate decreases more slowly at higher Rm, indicative of a different dynamo regime. Despite this trend, we emphasize that predicting the dynamo behaviour at larger Rm or in the ideal limit remains extremely difficult. Testing the convergence of the dynamo growth rate with Rm would require us to go beyond , which is inaccessible with our current resources. In any case, these results suggest that ohmic diffusion is crucial for the maintenance of a powerful and efficient dynamo.
3.3 Saturation and nonlinear regime
We now explore the nonlinear regime of the dynamo, i.e when the Lorentz force provides feedback on the turbulent flow. What is the dependence of the saturated state on Rm and how strongly does resistivity affect GI turbulence?
First, we plot in Fig. 4 the boxaveraged pressure, kinetic and magnetic energy obtained in the saturated state (when the magnetic field stops growing) as a function of Rm and for . Since the field concentrates within the bulk of the disc (see Section 4.1), we weighted quantities by the gas density during the averaging procedure. We see that the saturated magnetic energy in the nonlinear regime depends on the parameters in the same way as the kinematic growth rates (see Fig. 3 for comparison). is maximal for and decreases quite abruptly towards . At its maximum, the dynamo sustains strong magnetic fields with a plasma beta and energy typically 56 times larger than the turbulent kinetic energy in the bulk of the disc. This corresponds to a regime of superequipartition fields, for which we expect important feedback of the Lorentz force on the gas flow. For Rm between 10 and 100, the kinetic energy is reduced by a factor almost 3 compared to the ideal limit and by a factor 6 compared to the unmagnetized gravitoturbulence. We checked that the gravitational stress is also reduced in this regime.
In RL18a, we showed that idealMHD gravitoturbulence exhibits a moderate drop in kinetic energy in comparison to the purely hydrodynamical case, a property attributed to the nonlinear suppression of largescale structures, such as oscillatory epicyclic modes. In the low Rm regime, the drop in kinetic energy is more dramatic because of the stronger magnetic field. Additional effects, associated with magnetic pressure and thermodynamics, may also contribute; for instance, since the kinetic to magnetic pressure ratio is of order unity for Rm between 10 and 100, the linear stability and growth rate of spiral GI modes will be altered. Indeed, the effective Toomre parameter, including magnetic pressure, is
(18) 
(Kim & Ostriker, 2001) which is greater than the hydrodynamic in the low Rm states. Moreover, magnetic fields provide an additional reservoir of energy that can be directed into ohmic heating and, consequently, an enhanced .
To study this effect, we plot in Fig. 5 the evolution of the boxaverage Toomre parameter as a function of time for . In the initial state and during the kinematic phase, the magnetic field is too small to influence the flow and settles around its hydrodynamical value (). In the final state, with , the average Toomre parameter is . Therefore there is an increase of almost 15% (and equivalently of the average sound speed in the midplane). Note however that the elevation of disc temperature remains moderate, despite the strong magnetization. In particular, it is relatively small compared to that measured in a 2D disk (Riols & Latter, 2016). Nonetheless, by combining the effect of magnetic pressure and ohmic dissipation, the effective in the low Rm regime can be 40% higher than in pure hydrodynamics, leading to fainter spiral waves and a depression of GI activity.
Interestingly, exhibits notable oscillations when the magnetization becomes important (at ). These oscillations must reflect the backreaction of the field on the flow. They probably originate from a thermal cycle similar to that described in RL18a (Section 5.4), in which (a) GI amplifies , (b) magnetic energy is dissipated into heat and enhances , (c) GI or spiral activity dies because of the high and strong magnetic tension, and finally (d) in the absence of vigorous GI, and return to their original value. This scenario is consistent with the fact that the variations of and in Fig. 5 are correlated.
The dependence of saturated state on cooling times was explored in RL18a in the ideal limit. In the case of finite resistivity, we find that magnetic energy also peaks around for . However for , simulations have not been run for sufficiently long time to be statistically meaningful. The reason is that most stalled after a few tens of an orbit because of transient fragments that considerably reduce the time step.
3.4 Dependence on boundary conditions
In the solar context, boundary conditions strongly influence the excitation and saturation of convective dynamos (e. g. Choudhuri, 1984; Jouve et al., 2008; Käpylä et al., 2010). We now check if the GI dynamo exhibits a similar sensitivity. We compared three different vertical boundaries: vertical field (open boundary), perfect conductor (closed boundary), and periodic. Note that the perfect conductor has , and . In theory, it prevents any EMF from being generated at the boundary or outside the domain, and conserves magnetic helicity in the box. However, in practice, given that the EMF is reconstructed at the cell interfaces while boundaries are given at cells centres, the code still generates a nonvanishing EMF at the boundary that can lead to a mean and . When using such condition, we found similar dynamo growth rates as in the open boundaries case, but ended up with higher magnetization in the saturated state.
Periodic boundary conditions are interesting since they ensure that no EMF is produced at the boundaries and thus provide the best setup to test dynamo action as formally defined. We superimpose in Fig. 3 the growth rates obtained for three different Rm (120, 200, 400) and also without explicit diffusion (diamond markers). Although a dynamo appears, the critical magnetic Reynolds number is shifted to higher Rm, with . Note that this effect appears to be independent of vertical box size because we obtain a similar result with and . As in the open boundary case, the field grows quasiexponentially, but growth rates and saturated magnetic energy are lower. The fact that the dynamo depends strongly on the nature of the boundary is actually not surprising. We will show in Section 4.2 that periodic boundaries prevent flux from leaving the domain and even to be recycled from the upper surface to the lower surface (and vice versa), which is somewhat unrealistic. One quantifiable effect is that the largescale magnetic field is forced to be antisymmetric about the midplane, which enhances ohmic diffusion.
4 Dynamo properties
In this section we turn our attention from the dynamo’s broadbrush features (growth rates, saturated amplitudes) and concentrate on some of its specifics so as to better understand how it works. We focus primarily on the kinematic phase of its evolution.
4.1 Large or small scale?
To understand whether the GIdynamo preferentially amplifies large or smallscale fields, we show in Fig. 6 (top panels) the evolution of magnetic energy associated with different Fourier modes when . The calculation is restricted to the midplane as it contains most of the magnetic energy. For both and , the magnetic field is clearly dominated by the mean component (, ), and especially its toroidal projection . The dynamo is thus predominantly large scale, in the sense that magnetic fields grow on lengths much larger than .
The second important component is the fundamental nonaxisymmetric mode (in green) which corresponds to a large scale spiral magnetic structure (see section 5). Nonaxisymmetric modes are essential to the dynamo process according to the Cowling’s antidynamo theorem, so we expect them to feature. In particular, the interaction of their magnetic and velocity components produce a mean electromotive force that allows to be constantly regenerated so as to compensate ohmic losses. Note that axisymmetric modes with and seem negligible in the magnetic energy budget.
Cases and are distinguished by their ratio of large to small scale magnetic fields. In terms of energy, there is only a factor 4 difference between the mean and the nonaxisymmetric harmonic for , while this ratio exceeds 100 when . This we attribute to a drop in the mean field , the saturation level of the nonaxisymmetric mode being unchanged when Rm is increased. So it would appear that the largescale dynamo is weakened at large Rm, and supplanted by a smallerscale dynamics. The origin of the smallscale magnetic fields is unclear at this stage. They might originate in a turbulent cascade of magnetic energy, possibly from local magnetic instabilites, or from a smallscale dynamo of Zel’dovich type that stretches and folds the field lines in a random way. The mechanism behind the development of the smallscale field is discussed in Section 5.3. We stress that the MRI is not (or marginally) active in the very diffusive regime considered in this paper, and is anyway likely to be strongly impeded by GI (see RL18a, ).
4.2 Vertical profile of mean magnetic field
In this subsection, we look in more detail at the properties of the largescale dynamo during its kinematic phase. The lower panels of Fig. 6 show the timeevolution of the vertical profiles of the mean magnetic field (center) and (bottom) for and open boundary conditions. For both and , the largescale field is mainly toroidal and remains confined within the midplane region of the disc. This is reminiscent of simulations without explicit resistivity (RL18a). Unlike the latter, however, the dynamo in resistive flow does not exhibit timereversals of , or possibly these reversals occur on timescales longer than . Figure 6 shows that both poloidal and toroidal components have the same polarity in the midplane, but changes its sign at . For , the diagram also reveals bursts of poloidal field confined to the midplane. These bursts are correlated with the activity of vigorous density spiral waves. We note finally that for large Rm, both the radial and toroidal mean field almost vanish in the disc atmosphere, above one disc scaleheight.
Figure 7 compares the vertical profile of produced from simulations with open and with periodic boundary conditions, for . Each profile is calculated at a given time of the simulation during the kinematic phase. In the case of periodic boundaries, the field is antisymmetric about the midplane, with a different polarity in the upper and lower parts of the disc. This configuration ensures that the mean and vanish in the box, but not over each side of the disc. Therefore, the largescale field in a periodic box corresponds to the mode in (instead of for open boundaries) which possesses a strong gradient at . The vertical scale associated with this gradient can be measured from Fig. 7 and is . In comparison the vertical scale of in open boundaries simulations is 5 to 6 times larger. As a consequence, ohmic diffusion on is enhanced by a factor of a few tens, which should enhance the critical magnetic Reynolds number of the dynamo. In addition, horizontally averaged fields and did not necessarily dominate the magnetic energy budget and flipped regularly in time, with a short period of 56 orbits (see Fig. 7).
4.3 Electromagnetic forces
We now investigate how the mean field is regenerated. We focus particularly on the simulation with open boundaries, and . In Fig. 8, we plot the vertical profiles of the different terms in Eq. (14) and Eq. (15) governing the evolution of . This includes the laminar and turbulent part of the mean electromotive force , as well as the shear stretching (the “omega effect”) and ohmic diffusion . All these terms are averaged in time, during the kinematic phase (between and ), and over the horizontal plane. Since their evolution is exponential during the amplification phase, we normalize them by the midplane during the averaging procedure.
The top panel shows that the toroidal component is mainly amplified via the radial laminar EMF (green dashed curve) and the omega effect (cyan), since both have a positive gradient in . In contrast, the radial turbulent EMF (blue) has a negative gradient in , like ohmic diffusion (red), and therefore acts as a turbulent diffusion on the mean field. Note that the radial laminar EMF is associated with compression of the disc due to a mean inflow . This inflow remains relatively weak compared to the r.m.s vertical velocity and is confined to the midplane regions . Further away in the disc corona, is now positive and expels magnetic flux outward. Although the EMF generated by the inflow near is comparable to the stretching by the shear (omega effect), it is insufficient on its own to sustain the toroidal field against ohmic and turbulent diffusion. The omega effect is essential, and so we conclude that a poloidal field is required for the largescale dynamo to work.
The bottom panel shows the EMFs and ohmic diffusion profiles associated with the radial field . Here the laminar term (dashed green curve) is almost negligible compared to the turbulent EMF (blue). Moreover, the profiles are more complicated in the bulk of the disk . We point out that the toroidal and turbulent EMF has two local extrema of opposite signs, one around and one around . These extrema correspond to a change in the sign of the EMF gradient, and then to a change of polarity (see Fig. 6 for comparison). Since both are correlated and keep the same sign, the turbulent EMF is directly involved in the maintenance of the mean radial field against ohmic diffusion.
4.4 Nature of turbulent motions involved in
To proceed further in the analysis we need to understand the turbulent EMF. In Fig. 9 we plot the part of the mean EMF that issues from radial motions (red curve) and the part that issues from vertical motions (green). In the midplane () where , the EMF is entirely due to vertical motions (such as upwellings and rolls). In particular, this means that the strong horizontal motions associated with spiral waves do not play a direct role in the regeneration of the mean poloidal field (though they play an indirect role, see Section 5). Higher up, in the disc corona, both radial and vertical motions contribute to the EMF, with being slightly stronger and leading to the change of polarity.
To determine whether the mean field is generated by largescale motions (like spiral waves) or smallscale fluctuations, we calculate the contribution of largescale modes with to the EMF. In practice, this is done by computing the direct FFT of , , and , filtering out all the Fourier modes that have , and then going back to real space to calculate the products and , where refers to a filtered component. This procedure removes the correlation associated with smallscale nonaxisymmetric turbulent structures, such as helical waves generated by parametric instability, for instance (Riols et al., 2017a). We superimpose in Fig. 9 the filtered EMF (dashed lines) and show that they represent approximately 80% of the total EMF. This is strong evidence that the mean field dynamo is supplied by motions associated with large scale spiral waves (with and radial size ) rather than GI smallscale turbulence.
In summary, we have demonstrated that the dynamo functions via (a) the creation of from by the omega effect (mainly) and (b) generation of by relatively largescale, but turbulent, velocity fluctuations issuing from the spiral density waves.
4.5 Mean field theory and its limitations
Figure 10 shows the evolution of the mean turbulent EMF as a function of the mean toroidal field , during the course of the simulation with , and open boundaries. Although the EMF is highly fluctuating in time, on the average it scales linearly with :
(19) 
with of order 0.002 for and 0.005 for . It is hence tempting to describe the dynamo process in the framework of classical mean field theory (Moffatt, 1977; Parker, 1979; Krause & Raedler, 1980; Brandenburg & Subramanian, 2005; Brandenburg, 2018), according to which the mean EMF is assumed to be a linear combination of the mean and its vertical gradient. The coefficients of the linear system encapsulate the properties of the turbulence and can be derived using the socalled “test field method". However, there are several caveats. First, this theory only applies when or when there is a clear separation of scales between the turbulent motions and the largescale field. In our simulations, these assumptions are far from satisfied, since the fundamental nonaxisymmetric mode associated with spiral arms represents 80% of the mean EMF (see 4.4). Second, we showed that as the EMF changes its sign with altitude, so does . Mean field theory predicts that is proportional to the mean helicity of the flow , but according to our simulations, the helicity keeps a constant sign on each side of the disc and is therefore not correlated with . One could probably find a more sophisticated model than Eq. (19) , involving nondiagonal terms and dependences on the mean field gradient, but this is not the object of the paper.
5 Field amplification by density spiral waves: physical picture
We showed in the previous section that the GI efficiently generates and sustains a largescale magnetic field, especially in the regime of low Rm . This dynamo survives at larger Rm, independently of the vertical boundary condition, but produces weaker mean fields and seems to favour smaller scales. Our aim in this section is to propose a physical picture of the dynamo action, drawing together the different results of the previous sections and providing additional numerical evidence to illustrate the picture.
5.1 Model of the large scale dynamo loop
Our interpretation of the dynamo loop is very similar to the “" mechanism proposed for the solar dynamo cycle (Parker, 1955). As we showed earlier in Section 4.2, the toroidal field is regenerated from poloidal field, through an omega effect. This is straightforward and uncontroversial. The poloidal field, however, issues from an “alpha" effect supported by largescale GI spiral motions that stretch and fold the toroidal field. Our task is then to elucidate how the “stretching" and “folding" motions occur, how they amplify , and how these motions connect to the vertical EMF profiles of Section 4.4.
Although the dynamo is a “statistical” effect comprising the work of multiple spiral modes, it is better to start our investigation with a single spiral wave and ask what it does to the magnetic field. For that purpose, we analyse the motion around a simple spiral density wave. Figure 11 sketches out its main dynamical ingredients. The wave is composed of 2D horizontal compressible motions in the midplane (blue arrows) and incompressible vertical roll motions in the poloidal plane (black arrows). These motions are drawn in the frame perpendicular to the wave front, but are actually invariant along the direction of the front. The pattern of four counterrotating largescale rolls has been studied in detail by RL18b. These features have a typical size and appear as a fundamental feature of linear and nonlinear density waves in stratified discs. They can emerge from a baroclinic effect associated with the thermal stratification of the disc and are generally locally supersonic if the wave is shocked. Such a circulation might also arise from hydraulic jumps in very unstable flows (Boley & Durisen, 2006). Lastly, there is the rotational shear flow coming out of the page, though not perpendicularly. Our pareddown sketch of the spiral wave flow is reminiscent of wellstudied and efficient kinematic dynamos such as those of Roberts and Ponamarenko type. It is then perhaps no surprise that we also find dynamo action associated with the density waves.
Note that pure 2D horizontal motions, like those associated with wave compression, are not able to sustain a mean field dynamo on their own. It is true that locally they efficiently produce radial fields, but the mean induced by these motions (averaged over and ) necessarily vanishes. Thus vertical flow structures, such as the counterrotating rolls, are essential: this can be seen immediately from the mean toroidal EMF, a large fraction of which is supplied by either vertical motions or vertical magnetic field (see section 4.4).
Putting these ideas together, our interpretation of the dynamo loop, sketched in Fig. 12, is then the following:

Initially, the disc is threaded by a pure toroidal field with .

A spiral wave, tilted with respect to the azimuthal direction, passes through the disc. The field is then compressed horizontally by the spiral wave, along its pitch angle. On average, this process does not generate any mean radial field: negative component is created within the spiral arms, while a positive component is produced outside (in the interarm region). See Section 5.2.1 for numerical evidence.

Where the field line crosses the centre of the spiral wave, it is pinched and lifted by the vertical velocity associated with the largescale rolls. This process vertically redistributes the radial field and creates a mean of opposite polarity in the midplane and in the corona (through the EMF term ). See Section 5.2.2 for numerical evidence. A key point here is that the field in the wave isn’t perfectly aligned with the spiral front. Because of this nonalignment, the field lines can be pulled apart by the two different rolls on each side of the wave.

As the rolls “turn over" above the disk, they horizontally stretch and fold the field, producing a net with opposite sign to in the corona (through the EMF term ). See Section 5.2.2 for more details.

Finally, the mean radial field is stretched by the shear (omega effect) which produces a net toroidal field (not shown in Fig. 12)
Naturally, this theoretical picture remains to be demonstrated or at least validated by numerical simulations. The next sections will focus especially on steps (b), (c) and (d).
5.2 Numerical evidence of the spiral wave dynamo loop
To check that our dynamo model can be applied to GI turbulence, we analyse the magnetic field topology around spiral waves in our GI simulations. All the following plots correspond to times when magnetic energy grows linearly (i.e. in the kinematic phase).
5.2.1 Horizontal compression and magnetic spiral patterns: step (b)
We first study the effect of spiral waves’ horizontal motion on magnetic field in simulations with . We compare in Fig. 13 the density structures in the midplane (top panels) with the structures (center panels), for both and . In both cases, the radial magnetic field takes the shape of the density spiral waves and concentrates within these structures. In particular, is always negative (as opposed to ) within the density maxima, and positive outside the spirals. To complete the analysis, we superimpose the horizontal projection of magnetic fields lines over the maps. When , the magnetic field is relatively strong and almost aligned with the direction of the wave front inside the arms. Each field line, however, crosses the spiral wave since it has to connect with the interarm field at some point (therefore, it is not perfectly aligned with the front).
In the interarm regions, the field is weaker and has a component perpendicular to the fronts. This configuration is similar to that illustrated in Fig. 12 b) and results from the squeezing of initially toroidal field by compressible motions within the waves. Because the wave has a positive pitch angle ^{1}^{1}1In turbulent shearing box simulations, the pitch angle is generally around and corresponds to the angle for which most of the power from the shear is transferred to the nonaxisymmetric waves. Its relatively small value remains however unclear and might depend on the excitation process, wave phase velocity and gas diffusivity., the field inside is tilted in the same direction which leads to a negative (assuming an initially positive toroidal field). In the simulation with , which has been run at double resolution, the field appears fragmented and more filamentary. However it is again oriented along the spiral front with negative inside the density maximum.
5.2.2 Vertical rolls and step (c) and (d)
As suggested by our model (step c), the vertical rolls surrounding each density wave stretch the field lines vertically. To show this effect numerically, we plot in Fig. 14 (top) the vertical structure of the gravitoturbulent flow in the poloidal plane, at a random time. Around the most prominent density waves, marked by a pink rectangle, we clearly identify a pair of counterrotating rolls, similar to those depicted in Fig. 11.
The left panels of Fig. 15 show (top) and (bottom) around one of the rolls. For , as the field passes through the spiral wave (from “left" to “right"), it rises upward and produces a positive component (shown in red). Once it reaches the centre of the spiral, field lines move back toward the midplane and finally connect with the interarm field, leading to a negative (shown in blue) on the other side of the wavefront. Counterintuitively, the magnetic field is not wrapped around the centre of each roll, but is simply pinched at the centre of the wave, where the vertical velocity of the roll is maximum, exactly as depicted in Fig. 12 (c).
Ultimately, the field that rises up to is
stretched horizontally by the counterrotating motions of each roll and produces a
radial field component in the corona (step d). The stretching leads to
a “mushroomlike" structure in the contours of , reflected on either side of the midplane. These contours are plotted in white in the bottom left panel of Fig. 15.
In order to illustrate this process as cleanly as possible we simulate a simple laminar nonlinear shearing density wave initially embedded in a uniform toroidal field with . The setup is the same as in RL18b: the wave is forced not by the GI but by a potential of the form where is the amplitude of the forcing and are the radial and azimuthal wavenumbers. We fix the amplitude in order to obtain a nonlinear shock wave. The initial pressure profile is polytopic with index (to mimic the entropy gradient found in our GI simulations, see RL18b). The spiral wave structure and its magnetic topology are shown in the right panels of Fig.15. The flow exhibits a roll structure, and this generates a cleaner version of the magnetic topology seen in the turbulent case (compare with the left panels). Again the stretching of the field gives rise to a mushroom structure, clearly visible in the bottom right panel of Fig.15.
5.2.3 Spiral EMFs and generation of mean
To further develop the discussion in Section 4.4 and Fig. 9, we examine in detail the connection between the velocities and the EMF profiles. First, the bottom panels of Fig. 13 show the derivative of the EMF
in , projected onto the midplane. For
, this quantity is concentrated within the density
spiral waves and has a positive feedback on the radial field.
Averaged over the box, this leads to a mean
, consistent with Fig. 6.
In Section 4.4, we demonstrated that such an EMF is dominated by the component near the midplane.
Fig. 15 shows that at the location of the density maxima, where , the vertical velocity of the rolls is always directed outward (positive if and negative
if ). In the interarm regions, this turns to be the opposite configuration. In summary, there is a
clear correlation between and which directly generates a positive gradient of . and then a positive near the midplane region (). Physically, this can be
interpreted as a vertical redistribution of radial magnetic flux. The
rolls expel the negative flux associated with the spiral outward, but
compress the positive flux present in the interarm regions. This
redistribution happening globally around each spiral wave, favours
the configuration depicted in Fig. 6, with a
segregation between the midplane region (that builds up ) and the rest of the disc (that builds up ).
This vertical redistribution is however incomplete, as we showed that at higher altitudes, most of the radial field is produced through horizontal motions, via the term (see Section 4.4). Again there is strong evidence that the rolls are involved in this process. In fact, the maximum of (Fig. 9), located at corresponds exactly to where the radial velocity associated with the rolls is maximum (this also corresponds roughly to their vertical extent). Moreover, Fig. 15 shows that for , the component is correlated with the rolls’ horizontal motions. In regions where is negative, the magnetic field points outward, and viceversa. Such correlations lead to a negative EMF at , resulting in a negative gradient for the EMF and therefore a negative , as expected from Fig. 6 and 9.
5.3 Emergence of smallscale braided structures in the large Rm regime
We showed in Sections 3 and 4.1 that the mean field (or largescale) dynamo is severely curtailed when Rm increases. Instead, smallscale structures seem to emerge within the spiral waves while the mean field is diminished. We now explore how the flow develops the smallscale magnetic structures and why the meanfield dynamo is optimal at large resistivity.
A quick look at the lower panels of Fig. 13 show that the toroidal EMF for is still contained within the spiral wave but its coherence on longer scales is lost. Parasitic smallscale modes interact with the largescale magnetic structures depicted in Fig. 15 and subsequently reduce the efficiency of the mean field dynamo.
We show in Fig. 16 (bottom) a 3D rendering of the field line topology around a given spiral wave, chosen randomly in our GI turbulent simulation with . To aid interpretation, we show in the top panel the original snapshot containing the full domain with a 2D projection of the field. There is clear a evidence that field lines are rolled and highly twisted within the spiral wave, forming a braided magnetic structure along the wave front. Note that a single structure is formed and settles at the intersection of the four counterrotating rolls, near the density maximum. The twisting probably results from the combined effect of the differential rotation and the helical flow associated with the spiral rolls. Such complex and intricate structures do not appear in resistive runs, probably because they relax rapidly through diffusion.
Twisted magnetic fields are known to be important in in the context of solar flares and coronal heating, because they are subject to kink instabilities when the twist exceeds a critical value (Dungey & Loughhead, 1954; Roberts, 1956; Hood & Priest, 1981; Bennett et al., 1999). The kink instability mainly deforms and breaks magnetic tubes, and thus efficiently produces turbulent fields and ultimately magnetic energy dissipation. If we define to be the pitch of the twisted tube and its radius, the criterion for instability is
(20) 
(Linton et al., 1996). In our simulations this condition is marginally satisfied, which itself revealing. Though this instability criterion has been derived with simple assumptions and welldefined equilibria (which probably do not apply to GI flows), we believe that braided magnetic fields fields within spiral waves are important in the the generation of smallscale field and the breakdown of the mean field dynamo at larger Rm. Stronger resistivity prevents magnetic field from being twisted into a large number of loops and therefore keeps magnetic energy from cascading to small scales.
6 Discussion
6.1 Largescale dynamo
We showed that largescale fields result from an “alpha" effect which involves vertical folding and horizontal stretching by the rolls associated with spiral waves. The connection between the alpha effect and helicity has often been pointed out, and indeed it can be shown that the spiral rolls are helical, since the wave generates prograde motions where the roll is anticlockwise and retrograde motions where it is clockwise. As mentioned earlier, the helical motion bears strong similarities with the Roberts flow (Roberts, 1972). But the dynamo loop described above differs significantly from dynamos supported by smallscale subsonic turbulent flow, such as convection in the Sun. The motions here are supersonic, large scale and highly localized in radius, which leads to a rather patchy and inhomogeneous , different from the initial uniform . Fields are inevitably filamentary and sparse in such GI flows. Finally, we point out that 20% of the mean field might also come from smallscale turbulence, or inertial and helical waves triggered by a parametric instability.
6.2 Boundary conditions
One concern is that the dynamo behaviour differs when using different boundary conditions (see Section 3.4). In particular, the strong and largescale dynamo at low Rm is conditioned by the existence of a net toroidal field throughout the box. However we think that this issue is not critical for several reasons.
First, and perhaps most obviously, the dynamo still works with periodic boundaries. This suggests that magnetic field amplification mechanism is not an artefact of the boundary. In particular, the growth rates obtained with open boundaries increase with decreasing , which is physically expected. The EMF at the boundaries in that case is relatively small compared to the EMF near the midplane (see Fig. 8). In a periodic box, we also checked that a largescale field can be generated, although it exhibits a weaker amplitude and flips regularly, every 5 or 6 orbits. Perhaps, one key question is to understand whether the dynamo in this configuration is supported by the largescale or by a smallscale process of Zeld’ovich type. The answer is not obvious, since the Rm regime allowing the dynamo is marginally resolved by our simulations. In any cases, the stretching and folding by the spiral rolls must be crucial in the dynamo mechanism, whatever kind it is.
Second, boundary conditions are part of the physical problem, and open boundaries are the most realistic of the options available to us. Periodic boundary conditions introduce unnatural communication between the two surfaces of the discs which can alter the nature of the largescale field, as we see. Moreover, periodic boundaries do not allow magnetic buoyancy to evacuate excess magnetic field; its continued buildup may be unphysical. They also force a particular geometry upon the field, which can artificially enhance its dissipation. In real systems, the disc exchanges field with an external “corona”, though the details of how this works is not easily determined.
6.3 Application to protoplanetary discs
The dynamo identified in this work is perhaps most relevant for young and massive protoplanetary discs, subject to GI. Indeed, except for the innermost regions ( 1 AU), the coupling between the gas and magnetic field in these object is believed to be weak, and nonideal MHD effects should be important throughout, and certainly in the regions susceptible to gravitational instability (typically beyond 20 AU). According to Simon et al. (2015), using the standard MMSN disc model, the magnetic Reynolds number in these regions varies between and . However these numbers must be considered gross upper limits for the denser and flatter discs susceptible to GI (see the case Elias 227 for instance, Pérez et al., 2016). The reason is that FUV cannot penetrate very far into the disc when the surface density is high. We expect then that some of the Rm probed by our simulations are relevant.
In any case, ohmic diffusion is not the most dominant nonideal effect at radii larger than 20 AU. Ambipolar diffusion is believed to be the leading effect in these regions with typical Reynolds (or Elsasser) numbers between 1 and 1000 (Simon et al., 2015). Although our numerical setup does not capture ambipolar diffusion, it is not excluded that its effect may be similar to ohmic diffusion in the nonlinear phase of the dynamo, at least if the magnetic field is not highly anisotropic. We point out again that during the dynamo’s kinematic phase, both the Hall effect and ambipolar diffusion are subdominant to ohmic resistivity, though this may change if there is a net magnetic field. Future simulations are necessary to address how ambipolar diffusion and the Hall effect influence the GI dynamo process, but also more generally on GI turbulence. For example, it is possible that zonal flows enhanced by nonideal effects (Lesur et al., 2014; Bai, 2015) significantly perturb the spiral waves motions or become even unstable to nonaxisymmetric perturbations (Vanon & Ogilvie, 2016), leading to different flow behaviour.
It has been postulated that somewhat older disks may periodically undergo GI in their magnetically inactive deadzones (located between roughly 1 and 10 AU) during outbursts of FU Ori or EX Lupi type (Gammie, 1996; Armitage et al., 2001; Zhu et al., 2010). In affected regions the magnetic Reynolds number is low, and a GI dynamo is a distinct possibility in the initial stages of an outburst, perhaps supplying the seed field for the onset of MRI. The interplay of the GI, GI dynamo, the MRI, and the Hall effect are expected to present some intriguing dynamics.
We finally give estimates of the field intensity in PP discs that a GIdynamo might be able to generate. In the range of Reynolds number probed by our simulations, we found that the ratio of magnetic to thermal pressure lies in a range between 0.01 and 0.5 (depending on Rm). We have . We consider a typical disc with surface density
(21) 
and in agreement with observations of Elias 227 (Cieza et al., 2017), a massive disc susceptible to GI. By considering a disc with a central star mass equal to that of the Sun, we then hobtain
(22) 
At AU we estimate the field strength to be around . We remind the reader that such fields are mainly toroidal, if generated through GI spiral waves.
6.4 Application to AGN
The GI dynamo could be also excited in the regions of AGN beyond 0.01 pc, where selfgravity is thought to dominate (Menou & Quataert, 2001; Goodman, 2003; Lodato, 2007). The typical Rm in these objects at these radii is poorly constrained: though the disk is very large, the ionisation fraction could dip to low levels as the gas is too cool to support significant thermal ionisation. One must then appeal to photoionisation, the efficiency of which is uncertain at the midplane, and partly controlled by the morphology of the disk. It is likely, however, that typical Rm values are greater than those probed here. The same claims can be made with respect to the ambipolar Reynolds number (Menou & Quataert, 2001). Thus making a direct comparison with AGN is not straightforward.
The saturated magnetic energy of the dynamo never exceeds the thermal energy (cf. Fig. 4), but can nonetheless reach significant levels, with . Concurrently, the saturated Toomre Q does attain somewhat larger values (Fig. 5). We might then conclude that a saturated GI dynamo, when up and running, could help prevent a disk from fragmenting when it might otherwise. The thin AGN disk would then extend to larger radii than current hydrodynamical estimates predict, though it is unlikely that the dynamo’s magnetic pressure on it own can completely solve the disk truncation problem (see Goodman, 2003). Adding a vertical magnetic field of sufficient strength may be one possible solution here (RL18a). (Note that a completely analogous discussion can be had regarding giant planet formation in PP disks.) In any case, it is certainly likely that the GI and its interaction with magnetic fields are foremost elements in the mysterious dynamics of the broad line regions of AGN.
6.5 Applications to galaxies
Multiwavelength radio observations suggest that some spiral galaxies (M81, M51 and possibly M33) are dominated by bisymmetricspiral magnetic fields. These magnetic spiral patterns are strongly correlated with the large scale galactic spiral arms.
To explain the correlation, Chiba & Tosa (1990) and Hanasz et al. (1991) proposed a parametric swing excitation due to resonances between weak magnetic wave oscillations and density spiral waves. More sophisticated models of this resonance have been studied (Moss, 1997; Rohde et al., 1999), but most are based on a meanfield theory, in which the interstellar turbulence, driven by supernovae, is parametrized by an alpha effect (Elstner et al., 2000; Rüdiger & Hollerbach, 2004; Gressel et al., 2008). On the whole, the mean field theory provides an acceptable match to the observations, though is not without its challenges (see, for example Nixon et al., 2018; Tabatabaei et al., 2016). Alternative theories, based on the vertical shear of a primordial poloidal field, have been also proposed but remain highly speculative (Nixon et al., 2018). Surprisingly, density spiral waves have rarely been considered as capable of amplifying magnetic fields on their own. Instead density waves have been historically treated as a process that could shape largescale fields already generated. Only recent MHD simulations of galactic discs have suggested some link between the dynamo and the spiral arms (Dobbs et al., 2016; Khoperskov & Khrapov, 2018). Our GIsimulations show that such waves can, in principle, act as a dynamo and produce spiral magnetic patterns, similar to those observed in galaxies.
Although there is a consensus that GI is active in galaxies, there is however some uncertainty regarding the magnetic diffusivity. The microscopic or Spitzer resistivity in galactic disc is about cm/s (Parker, 1972), which yields gigantic magnetic Reynolds numbers. On the other hand, we could invoke an anomalous resistivity due to smallscale turbulence in order to explain dissipation of small scale fluctuations in the ISM (Hanasz et al., 2009). The common value used is cm/s (300 pc Myr), which presents magnetic Reynolds number between 10 and . This indeed overlaps the range of Rm explored by our simulations.
Assuming a surface density of g/cm for kpc, a typical scaleheight of 1000 parsecs, and a rotation period of 200 Myrs, we find that the GIdynamo produces for in a handful of orbits. On the other hand, radiofaint galaxies like M31 and M33 possess fields strength of about 5 , while gasrich galaxies with high starformation rates, like M51, M83 and NGC 6946 possess on average. We stress, however, that direct comparison is difficult and relies on various, somewhat dubious, assumptions. The value of the anomalous resistivity found in the literature is adhoc and estimated from simulations or mean field models. Moreover, our simulations do not describe the rich assortment of physical processes occurring in galactic disc gas, such as supernova driventurbulence or other stellar feedback. Simulations taking into account these effects are necessary to assess their relative importance in comparison to GI. Other than differences in thermodynamics, one of the more glaring discrepancies is our use of a Keplerian rotation law which is, of course, inappropriate for galaxies, though easily rectified in future simulations. Finally, our local model cannot describe the development of the bar instability, which itself may impact significantly (positively or negatively) on a galaxy’s magnetic field generation.
7 Conclusions
In summary, we performed 3D MHD stratified shearing box simulations, with zeronet magnetic flux, in order to assess the ability of gravitoturbulence to sustain a dynamo in the presence of finite and uniform resistivity. In Section 3 and 4, we explored the fundamental properties of the dynamo by probing a wide range of magnetic Reynolds numbers, cooling times, and boundary conditions. We obtained the following results:

The dynamo works for cooling times between (which corresponds roughly to the fragmentation threshold) and . Growth rates and saturated magnetic energy decrease with .

The dynamo is robust to boundary conditions, but its properties depends on which boundary condition is applied vertically (periodic or open).

In the case of open boundaries, the critical magnetic Reynolds number is of order unity ( for ). It is around 200 when periodic boundary conditions are used.

Magnetic energy reaches quasiequipartition with kinetic pressure () for . For larger Rm however, the dynamo growth rate and saturated magnetic energy drop with Rm, which suggests that the dynamo is “slow". However this last statement has to be taken with caution since it is not possible to predict the dynamo behavior in the large magnetic Reynolds number limit.

The dynamo is essentially large scale, kinematic and of mean field type, although smaller magnetic scales emerge at large Rm. It is powered by both differential rotation and largescale spiral waves motion.
In Section 5, we investigated the role of spiral waves in the magnetic field amplification and proposed a model inspired by the solar dynamo cycle and based on the classical  theory of Parker (1955). This model assumes that:

Horizontal motions associated with spiral waves compress an initial toroidal field along the density waves.

Pairs of vertical counterrotating rolls (associated with the density wave) pinch the field, lift, and fold it, generating new radial field.

The disk’s differential rotation shears out the radial field to regenerate toroidal field, and the loop is closed.
We demonstrated that the vertical EMF profiles giving rise to the largescale field are generated by the rolls. At large Rm, we found however that the largescale magnetic ropes inside the spiral waves are braided and broken by smallscale parasitic modes, probably associated with a kink instability. Smallscale fast reconnection processes, like the tearing instability, might also be involved in the destruction of the large scale field at large Rm.
As discussed in Section 6, the GIdynamo could have direct application to young and massive gravitoturbulent stellar discs, since these objects are cold and exhibit large resistivities. In the coming years, ALMA will have the potential to directly infer magnetic field orientation by measuring Zeeman splitting of cyanide lines. This technique has been already utilised to explore largescale massive objects such as molecular clouds, but is now applicable to circumstellar discs (Brauer et al., 2017). This raises the hope of detecting spiral magnetic patterns in PP discs and therefore of characterizing their dynamo behaviour. The highly resistive dead zones of older PP disks undergoing outbursts are another venue in which the GI dynamo phenomena might appear, possibly seeding the field necessary to “kickstart” the MRI. More speculatively, our work might be applied to galactic dynamos, which so far have only been addressed within the framework of mean field theory. Our main result is that spiral waves can directly produce dynamo action in these objects as well as the observed alignment between magnetic and density spiral patterns This result is in particular reminiscent of galactic dynamo simulations by Khoperskov & Khrapov (2018). So too, might the dynamo work in the cooler outer radii of AGN disks.
Finally, we emphasise that characterising the behaviour of the dynamo at Rm larger than a thousand is not yet possible with our simulations, because of resolution issues. This potentially limits application of the theory to more astrophysical settings. In addition, future work will be necessary to better understand the effect not only of low resistivity but also other nonideal effects, ambipolar diffusion especially. Discs are also generally threaded by a net vertical field; the impact of such large scale component remains unclear and will be treated separately.
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