Cosmicraydriven dynamo in galactic disks
A parameter study
M. Hanasz^{1}  K. OtmianowskaMazur^{2}  G. Kowal^{2,3}  H. Lesch^{4}
1  Torun Centre for Astronomy, Nicolaus Copernicus University,
87148 Torun/Piwnice, Poland
2  Astronomical Observatory, Jagiellonian University,
ul. Orla 171, 30244 Kraków, Poland
3  Department of Physics and Astronomy, McMaster University,
1280 Main St. W., Hamilton, ON L8S 4M1, Canada
4  Astronomical Observatory, Munich University,
Scheinerstr. 1, 81679 Munich, Germany
Received 28 May 2008 / Accepted 19 December 2008
Abstract
Aims. We present a parameter study of the magnetohydrodynamicaldynamo driven by cosmic rays in the interstellar medium (ISM), focusing on the efficiency of magneticfield amplification and the issue of energy equipartition between magnetic, kinetic, and cosmicray (CR) energies.
Methods. We perform numerical CRMHD simulations of the ISM using an extended version of ZEUS3D code in the shearingbox approximation and taking into account the presence of Ohmic resistivity, tidal forces, and vertical disk gravity. CRs are supplied in randomlydistributed supernova (SN) remnants and are described by the diffusionadvection equation, which incorporates an anisotropic diffusion tensor.
Results. The azimuthal magnetic flux and total magnetic energy are amplified in the majority of models depending on a particular choice of model parameters. We find that the most favorable conditions for magneticfield amplification correspond to magnetic diffusivity of the order of
,
SN rates close to those observed in the Milky Way, periodic SN activity corresponding to spiral arms, and highly anisotropic and fieldaligned CR diffusion. The rate of magneticfield amplification is relatively insensitive to the magnitude of SN rates spanning a range of 10% to 100% of realistic values. The timescale of magneticfield amplification in the most favorable conditions is 150 Myr, at a galactocentric radius equal to 5 kpc, which is close to the timescale of galactic rotation. The final magneticfield energies reached in the efficient amplification cases fluctuate near equipartition with the gas kinetic energy. In all models CR energy exceeds the equipartition values by a least an order of magnitude, in contrast to the commonly expected equipartition. We suggest that the excess of cosmic rays in numerical models can be attributed to the fact that the shearing box does not permit cosmic rays to leave the system along the horizontal magnetic field, as may be the case for true galaxies.
Key words: galaxies: ISM  galaxies: magnetic fields  magnetohydrodynamics (MHD)  ISM: cosmic rays  ISM: kinematics and dynamics  ISM: magnetic fields
1 Introduction
An attractive idea of fast galactic dynamo was proposed by Parker (1992). The idea relies on two ingredients: (1) cosmic rays (CR) continuously supplied to the disk by supernova (SN) remnants and (2) fast magnetic reconnection, which operates in current sheets and enables the dissipation and relaxation of the random magneticfield components at the limit of vanishing resistivity. In the past decade, we have investigated the different elements, physical properties, and consequences of Parker's idea and scenario by means of analytical calculations and numerical simulations (Hanasz & Lesch 2000; Hanasz et al. 2002; Hanasz & Lesch 2003b,2001,1998,1997; Hanasz et al. 2004; Hanasz & Lesch 1993; Kowal et al. 2003; OtmianowskaMazur et al. 2007; Kowal et al. 2006; Lesch & Hanasz 2003; Hanasz & Lesch 2003a; Hanasz et al. 2006; OtmianowskaMazur 2003)
The first complete 3D numerical model of the CRdriven dynamo was developed by Hanasz et al. (2006,2004). In this paper, we perform a parameter study of the CRdriven dynamo model by examining the dependence of magneticfield amplification on magnetic diffusivity, supernova rate determining the CR injection rate, temporal modulation of SN activity, grid resolution, and CR diffusion coefficients.
The principle of action of the CRdriven dynamo is based on the cosmicray energy supplied continuously by SN remnants. Due to the anisotropic diffusion of cosmic rays and the horizontal magneticfield configuration, cosmic rays tend to accumulate within the disc volume. However, the configuration stratified by vertical gravity is unstable with respect to the Parker instability. Buoyancy effects induce vertical and horizontal motions of the fluid and formation of undulated patterns, such as magnetic loops in frozenin, predominantlyhorizontal magneticfields. The presence of rotation in galactic disks implies a coherent twisting of the loops by means of the Coriolis force, which leads to the generation of smallscale, radial magneticfield components. The next phase is merging of smallscale loops by the magneticreconnection process to form largescale, radial magneticfields. Finally, the differential rotation stretches the radial magnetic field to amplify the largescale azimuthal magneticfield component. The coupling of amplification processes of radial and azimuthal magneticfield components results in an exponential growth of the largescale magnetic field. The timescale of magneticfield amplification, resulting from the action of the CRdriven dynamo, was found (Hanasz et al. 2006,2004) to be equal to 140250 Myr, depending on the galactocentric radius, which is comparable to the galactic rotation period.
The CRdynamo experiments reported in the aforementioned papers relied on the energy of CRs accelerated in SN remnants. Gressel et al. (2008a,b) reported a series of nonideal MHD simulations demonstrating dynamo action resulting from the SNdriven turbulence, in the absence of CRs. Using a similar set of galactic disk parameters, with angular velocity a factor of 4 higher than the value of , typical of the galactocentric radius of Sun, these authors found amplification of the largescale magnetic fields on a timescale of . This indicates some similarity between the CRdriven dynamo and the dynamo driven by thermal energy output from supernovae. The similarity is presumably related to the buoyancy effect, which can be commonly attributed to the excess of both thermal and cosmic ray energies in the disk volume.
The magnitudes of galactic magnetic fields are usually estimated from measurements of the radio synchrotronemission produced by cosmic ray electrons in the magnetic field. To interpret the radio emission spectrum, it is usually assumed that the energy density in the magnetic field is of the same order of magnitude as the energy density in cosmicray protons (which are assumed to outnumber the electrons by 100 to 1, as they do in our Galaxy). There is however no compelling evidence of energy equipartition. Since the equipartition or minimumenergy assumption is one of the few ways of calculating radio source parameters, it is important to determine suitability of the approach. Strong et al. (2007) and Snodin et al. (2006) raised again the question about the applicability of the equipartition argument.
From the observational point of view (Fitt & Alexander 1993; Vallee 1995), the equipartition assumption seems to hold. In particular, Vallee's comparison of three different methods determining galactic magneticfield strengths (Faraday rotation method, equipartition method, and cosmicray equipartition) indicates that the equipartition fields are in a good agreement. On the other hand, Beck & Krause (2005) considered in detail a problem raised by Chi & Wolfendale (1993), which was that the commonly used classical equipartition or minimumenergy estimate of total magneticfield strengths from radiosynchrotron intensities is of limited practical use because it is based on the hardly known ratio K of the total energies of cosmicray protons and electrons and also has inherent problems. They present a revised formula using the numberdensity ratio K for which they provided estimates. For particle acceleration in strong shocks, K is about 40 and increases with decreasing shock strength. Their revised estimate for the field strength inferred higher values than the classical estimate for flat radio spectra with spectral indices of about 0.50.6, but smaller values for steep spectra and total fields stronger than about 10 . In young supernova remnants, for example, the classical estimate may be too high by up to 10. On the other hand, if energy losses of cosmicray electrons are important, K increases with particle energy and the equipartition field may be underestimated significantly.
From a more global pointofview the assumption of equipartition appears to be natural. A thermodynamical system will always distribute the free energy to all the degrees of freedom available, if the system has time to do so. In the case of accelerated particles diffusing in the largescale and turbulent magnetic fields, one can expect that at least the turbulent magnetic field (since it represents three degrees of freedom) is virialized with respect to any other pressure term, such as cosmicray pressure. This may not be true for the ordered magnetic fields, which are supposed to be amplified by the combined action of differentiallyrotating shearflows in the disk and some helical upward and downward motion driven either by cosmicray pressure or any activity in the disk. In any case, the connection between starformation activity accompanied by enhanced flux of cosmic rays and the amplification of largescale magnetic fields, inherently raises the expectation that magnetic fields should not exhibit higher pressures than cosmic rays. One would expect to find instead magnetic fields whose pressure is lower than the pressure of the cosmic rays, if the cosmic rays represent a source of the galactic dynamo.
The paper is organized as follows: in Sect. 2, we describe the CRdriven dynamomodel and its numerical implementation; in Sect. 3, we present our simulation setup and describe parameters used in numerical simulations; and in Sect. 4, we describe results, focusing on the effect of each parameter on magneticfield amplificationrate. We discuss the final saturated states of models in terms of equipartition between kinetic, magnetic, and CR energies. Finally, in Sect. 5 we present our conclusions.
2 Description of the model
As in papers by Hanasz et al. (2006,2004), we take into account the following elements of the CRdriven dynamo:
 (1)
 the cosmicray component, a relativistic gas, which is described
by the diffusionadvection transport equation (see Hanasz & Lesch 2003b, for
details of the numerical algorithm). The typical values of the diffusion coefficient
found by modeling CR data (see e.g. Strong et al. 2007)
are
at energies
,
and even higher values of
are possible (Jokipii 1999), although, we use
lower values in a majority of our simulations;
 (2)
 following Giacalone & Jokipii (1999) and Jokipii (1999), we presume that
cosmic rays diffuse anisotropically along magneticfield lines. The ratio of the
perpendicular to parallel CR diffusion coefficients propose by these authors
was 5%;
 (3)
 localized sources of cosmic rays: supernova remnants exploding randomly in
the disk volume (see Hanasz et al. 2004). We assume that each SN remnant
supplies cosmic rays almost instantaneously, i.e. the comicray input for a single
SN remnant, distributed over several subsequent timesteps,
equals 10% of the canonical SN kineticenergy output (
);
 (4)
 resistivity of the ISM
(see Hanasz et al. 2002; Tanuma et al. 2003; Hanasz & Lesch 2003a)
responsible for the onset of fast magnetic reconnection and
topological evolution of magneticfield lines. In this paper, we apply the
uniform resistivity and neglect the Ohmic heating of gas by resistive
dissipation of magnetic fields;
 (5)
 shearing boundary conditions and tidal forces following the prescription by
Hawley et al. (1995) aimed at modeling differentially rotating disks
in the local, shearingbox approximation;
 (6)
 realistic vertical disk gravity following the model of ISM in the Milky Way by Ferriere (1998).
(5) 
where is the shearing parameter, R is the distance to the galactic centre, is the resistivity, is the adiabatic index of thermal gas, the gradient of cosmic ray pressure is included in the equation of motion (see e.g. Berezinskii et al. 1990), and other symbols have their usual meaning. The uniform resistivity is included only in the induction equation (see Hanasz et al. 2002). The thermal gas component is currently treated as an adiabatic medium.
The transport of the cosmicray component is described by the
diffusionadvection equation (see e.g. Schlickeiser & Lerche 1985; Berezinskii et al. 1990)
where represents the source term for the cosmicray energydensity, and the rate of production of cosmic rays injected locally in the SN remnants is given by
(7) 
The adiabatic index of the cosmicray gas and the formula for the diffusion tensor of
are adopted by following Ryu et al. (2003).
3 Numerical simulations
3.1 Simulation setup
In this paper we present a series of numerical simulations, whose aim was to search for the most favorable conditions for magneticfield amplification by means of the CRdriven dynamo. The presented numerical simulations were performed with the aid of a Zeus3D MHD code (Stone & Norman 1992a,b) extended with additions to the standard algorithm, corresponding to items (1)(6) of Sect. 2, i.e. the cosmicray component, treated as a fluid and described by the diffusionadvection equation, including anisotropic CR diffusion tensor and cosmicray sources  supernova remnants exploding randomly in the disk volume, resistivity of the ISM leading to magnetic reconnection, shearingperiodic boundary conditions, rotational pseudoforces, and a realistic vertical disk gravity.
All simulations were performed in a Cartesian domain of size 0.5 kpc 1 kpc 2 kpc in x, y and z coordinates, corresponding to the radial, azimuthal, and vertical directions, respectively. The basic resolution of the numerical grid was grid cells in x, y, and z directions, respectively, and for a smaller sample of simulations performed with higher values of CR diffusion coefficients the grid resolution was grid cells. The boundary conditions are shearedperiodic in coordinate x, periodic in coordinate y, and outflow on outer zboundaries, with at the domain boundaries. The positions of SN are chosen randomly, with a uniform distribution in xy coordinates and Gaussian distribution in z coordinate.
The initial density distribution results from integration of the hydrostatic equilibrium equation for the vertical gravity model of Ferriere (1998) and the assumption of constant gas temperature across the disk, equal to approximately 6000 K, which corresponds to a sound speed of about 7 km s^{1}. The integration procedure identifies hydrostatic equilibrium for a given gas column density treated as an input parameter.
The magneticfield strength, incorporated in the initial hydrostatic equilibrium of gas, is defined by means of the parameter by denoting the ratio of initial magnetic to gas pressures. The initial cosmicray pressure equals the initial gas pressure.
The CR energy supplied to the system in SN remnants, randomly distributed
around the disk midplane, implies that the CR pressuregradient force
accelerates a vertical wind of thermal gas. To prevent significant mass
losses from the computational domain, due to the vertical wind, we compensate
the massloss
after each timestep. The compensation mass
is supplied as a mass source term, which is proportional to
the initial mass distribution
where is the total gas mass in the computational domain, and is the initial density distribution, produced by the integration of the hydrostatic equilibrium equation.
3.2 Simulation parameters
Table 1: Parameters of simulations presented in this paper.
The basic input parameters, resulting from the assumed model are the vertical gravity profile, local value of the galactic rotation and shear, gas column density, and supernova rate. We adopt these parameters from the global model of ISM in the Milky Way (Ferriere 1998) for the galactocentric radius , where angular velocity is , gas column density , and the realistic vertical gravity given by Eq. (36) in (Ferriere 1998). In our simulations, the values of gas column density correspond to the total density of all gas components in Ferriere (1998), while SNrate is the rate of type II supernovae. We assume for simplicity that all SN explosions appear as single supernovae, and that the vertical distribution of SN explosions is Gaussian of a fixed halfwidth equal to 100 pc.
In addition to the aforementioned wellestablished local diskparameters, there is a group of less known quantities, such as effective magnetic diffusivity, CR diffusion coefficients, and efficiency of conversion of SN kinetic energy into cosmic ray energy. We assume the standard 10% value of kinetictoCR energyconversion efficiency, and that the magnetic diffusivity and CR diffusion coefficients vary across a wide range.
In this paper, we present the results of five simulation series AE. The summary of all variable simulations parameters for the entire set of simulations is presented in Table 1.
In the simulation series A (runs A1A5), we examine the effects of magneticdiffusivity variations on magneticfield amplification by applying in the range corresponding to in CGS units. We define the magnetic Reynolds number Rm as in Gressel et al. (2008a,b), where is the domain size in the y direction. We assume a continuous and timeinvariable supply of CRs in SN remnants. The simulation runs A1, A2 and A3 are the same as the runs B, C, and D, respectively, discussed by OtmianowskaMazur et al. (2007). For comparison, we note, that the commonly adopted value of turbulent diffusivity in the ISM is ( ) for and . We note that the adopted values of magnetic diffusivity exceed the value corresponding to the Spitzer resistivity ( ; see Parker 1992) by 1518 orders of magnitude. The relative smallness of the Spitzer resistivity implies that an anomalous resistivity, considered as a subscale phenomenon, must be invoked to explain dissipation of the smallscale magnetic fluctuations in the ISM. Following Parker (1992), we assume that reconnection rates in the ISM are comparable to those predicted by the Petscheck's fast reconnection model, i.e. the magnetic cuttingspeeds are of the order of rather than typical for the slow ParkerSweet reconnection model, where Rm is the Lundquist number or the magnetic Reynolds number.
In the simulation series B (runs B1B5), we apply the same range of magnetic diffusivity values, but the CR supply is modulated in a way that mimics passages of subsequent spiral arms, regulating the star formation rate and subsequent SNrate. The effect of spiral arms is modelled (see Hanasz et al. 2006) by supplying cosmic rays in intermittent periods of 25 Myr, for SN rate equal to a factor of 4 higher than the reference , and followed by periods of 75 Myr without any SN activity. The timeaveraged supernova rate in this case equals to the reference .
In the simulation series C (runs C1C5), we apply a constant magnetic diffusivity and vary the surface frequency of SN explosions in the range of , assuming modulated CR supply as in the simulation series B. In the simulation series D (runs D1D2), we repeat the simulations A4 and B4 with a grid resolution that is a factor of two lower in each spatial direction.
Due to the CFL timestep limitation of the currently used explicit algorithm of CR diffusion, the applied values of CR diffusion coefficients are lower than realistic values. The timestep limitation, which ensures stability of explicit numerical schemes applied to the diffusion equation, is , where K is the diffusion coefficient. The timestep becomes prohibitively short when the diffusion coefficient is large or the spatial step is too small. For this reason, the CR diffusion coefficient was reduced in simulation series AD, by about one order of magnitude, with respect to the aforementioned realistic values . The fiducial values of the parallel and perpendicular diffusion coefficients applied in simulation series AD are, respectively, , and .
Finally, in the simulation series E (runs E1E6) we increase the parallel and perpendicular CR diffusion coefficients by factors 3 and 10, with respect to the fiducial values, to examine the magneticfield amplification for more realistic magnitudes of these quantities. In this way, we apply realistic CR diffusion coefficients in a few single simulation runs, even though the maximum CR diffusioncoefficients used do not reach the upper range of realistic values, of the order of , mentioned in the literature.
Figure 1: Exemplary plots illustrating the state of the system at for simulation A4. In the first two panels we present slices through the computational volume in the yzplane for x =0. Panel a) shows cosmic ray energy density with vectors of magnetic field, panel b) shows gas density with velocity vectors. In panel c) we plot horizontally averaged x and y components of magnetic field, and in panel d) horizontally averaged vertical velocity component and its fluctuations. 

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4 Results
In Fig. 1, we show the distribution of cosmicray gas and magneticfield vectors (panel (a)), and thermal gas density and gas velocity vectors (panel (b)) in the yzslice taken for x=0 at t=1000 Myr.
We note in panel (a) that the dominating, horizontal, magnetic field component in the disk volume undulates in a way resembling the effects of Parker instability. The cosmicray energydensity is smoothed well by the diffusive transport in the computational volume. The vertical gradient of the cosmicray energydensity is maintained by the supply of cosmic rays around the equatorial plane of the disk in the presence of vertical gravity. The cosmicray energydensity is expressed in units in which the thermal, gas energydensity corresponds to and the isothermal sound speed is equal to 1. The velocity field and the gas density are shown in panel (b). It is apparent that the distribution of gas is significantly less smooth that the distribution of cosmic rays.
To examine the structure of the largescale field, we show in panel (c) the horizontally averaged magnetic field components and and . A striking property of the mean magneticfield configuration is the almost exact coincidence of peaks between the oppositelydirected radial and azimuthal field components. This feature resembles the standard picture of an dynamo: the azimuthal, mean magneticcomponent is generated from the radial one and vice versa (see Lesch & Hanasz 2003, for a correspondingly simple analytical model).
In panel (d) we show the horizontallyaveraged vertical velocitycomponent and its fluctuations . It is apparent that the bulk speeds of the wind, driven by the vertical gradient of CR pressure, reach at . Vertical systematic winds of bulk speeds, comparable to rotational galactic velocities, influence largescale structures of galactic magnetic fields and are observed in external starburst galaxies such as NGC 253 (see Heesen et al. 2007,2009).
By varying the parameters discussed in the previous section, we intend to determine the regions of parameter space in which the magneticfield amplification is the most efficient. The amplification of the regular magneticfield is identified by the amplification of the total magnetic energy in the computational domain, associated with the amplification of the azimuthal magneticflux. The magnetic flux shown in subsequent plots represents an azimuthal flux averaged over all xz slices (across y direction) throughout the discretized computationaldomain. In the subsequent subsections, we present the parameter study of the CRdriven dynamo, focusing on the efficiency of magneticfield amplification and the issue of equipartition between magnetic, kinetic, and CR energies.
4.1 Dependence of magnetic field amplification on magnetic diffusivity
As a first step in our parameter study of the cosmicraydriven dynamo, we examine, in simulation series A, the effect of magnetic diffusivity on the efficiency of magneticfield amplification. Time evolution of magnetic energy and magnetic flux are shown in Fig. 2. Magnetic flux plotted in the left panel of Fig. 2 is scaled in the following way. The initial magneticfield induction is defined by the parameter , shown in the second column of Table. 1, where we apply Parker's convention to assign the inverse of plasma beta as . The adopted values of are 10^{4} and 10^{2} in different simulations, while implies that magnetic pressure equals the thermal gas pressure. We scale magnetic flux so that corresponds to the azimuthal magneticflux . The total magnetic energy plotted in the right panel of Fig. 2 is scaled with respect to the timeaveraged total kineticenergy in the computational domain. The latter quantity appears to fluctuate about a mean value, which is practically timeinvariant for all simulation runs, thus this type of scaling is convenient. The scaling described above is applied to all subsequent plots of magnetic flux and magnetic energy.
Figure 2: Time evolution of azimuthal magnetic flux and total magnetic energy for different values of magnetic diffusivity in simulation series A. The curves represent respectively cases of (A1), (A2), (A3), (A4) and (A5) in units . 

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The evolution of magnetic energy and magnetic flux in the models of simulation series A, represented by different curves in Fig. 2, illustrates that magneticfield amplification depends strongly on magnetic diffusivity. In the case of vanishing explicitresistivity, magnetic energy increases by about 2.5 orders of magnitude during the first 500 Myr, but magnetic flux is amplified by only a factor of 3 during this same period and decreases later on. This effect can be attributed to a predominant growth in the smallscale turbulent magneticfield component with a little contribution of largescale magnetic field amplification (see OtmianowskaMazur et al. 2007, for a more extended analysis of the simulations presented in this section). We note, however, that numerical resistivity, always present in numerical MHD simulations, may influence the behavior of the simulation run A1, corresponding to . The amount of numerical magnetic diffusivity was quantified, for the present grid resolution, according to the Parker instability simulations by Kowal et al. (2003).
When magnetic diffusivity increases up to , the efficiency of the magneticfield amplification increases. For (run A4), the growth in magnetic flux persists until and saturates in value thereafter. For smaller values of , the growth rate is lower and the maximum values of magnetic flux are lower than those attained for . For negligible or small explicit diffusivity (runs A1 and A2), magnetic energy increases initially more rapidly than at higher resistivity (runs A3 and A4). This behavior means that low resistivity enables an initially more rapid growth of the random magneticfield component, while for higher resistivity, random magnetic fields are quickly dissipated. The increase of the total magnetic energy follows closely, in the latter case, the increase in the mean magnetic flux. It is also apparent that the the amplification of the magnetic flux and magnetic energy for (run A5) is significantly lowered with respect to the other runs.
Figure 3: Time evolution of the ratio of energies of vertical to horizontal magneticfield components for different values of magnetic diffusivity in the simulation series A. Line assignments are the same as in Fig. 2. 

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To explain the physical mechanism controlling magneticfield amplification by means of the magnitude of magnetic diffusivity, we plot the ratio of total (volumeintegrated) energies of vertical to azimuthal magneticfield components in Fig. 3. It is apparent that the energy in the vertical magneticfield is higher than for all lower magnetic diffusivity runs A1A3 ( , and ), and is comparable to the energy in the azimuthal magneticfield of run A4 ( ), providing the strongest magneticfield amplification. Among all simulations in series A, only simulation A4 achieves energetic equipartition between the magneticfield and gas kinetic energy as a result of the amplification process.
In the case of the high magnetic diffusivity (run A5, ), energy in the vertical magneticfield component, however, remains far lower than the energy in the azimuthal magnetic field, and magneticfield amplification does not occur. This can be interpreted in terms of resistive damping of the undulatory mode of the Parker instability in favor of the interchange mode, which does not contribute to the dynamo action. The above finding indicates that the most favorable conditions for magneticfield amplification correspond to approximately equal energies in the vertical and azimuthal magnetic fields in the case of buoyancydriven dynamo.
To demonstrate the effect of resistivity on kinetic and magnetic turbulent spectra, we compute the Fourier transforms of the kinetic and magneticenergy densities, as in the paper by OtmianowskaMazur et al. (2007). The results are shown in Fig. 4. The highly anisotropic nature of turbulence is reflected in the different lines representing Fourier transforms in the x, y and zdirections. We find that the kinetic spectra, which generally emulate the Kolmogorov spectrum , depend weakly on magnetic diffusivity. The high values of magnetic diffusivity lead only to the damping in the shortwavelength components of the Fourier spectrum computed in the ydirection.
The magnetic spectra appear to be far more sensitive to variations in magnetic diffusivity. The plots obtained for runs A1 and A2 exhibit practically identical spectra in all directions. This means that the diffusivity of does not change the results for negligibly low explicit resistivity, or in other words the numerical resistivity of the code corresponds to the resistivity of Run A2. The effect of resistivity is evident in the smaller amplitude of the highk modes for (Run A3) and an apparent cutoff the in magnetic spectra around for . A further increase in magnetic diffusivity up to leads to a significant reduction in amplitudes of all modes for the Fourier transforms performed in zdirection, steepening the entire spectrum in the zdirection and having a surprising effect on the flattening of the spectrum in the xdirection. The latter effect may indicate a qualitative change in the physical nature of the modes, which can plausibly attributed to the mentioned enhancement of the exchange mode of Parker instability.
We briefly note that we neglect any smallscale dynamics of the helical MHD turbulence. This assumption allows the uniformity of our diffusion coefficients, such as the magnetic diffusivity. Detailed investigations of the influence of smallscale helical MHD turbulence on galactic dynamos are presented by Maron & Blackman (2002) and Maron et al. (2004).
We can interpret these results further in terms of the topological evolution of magnetic field, which is controlled by resistivity. The topology of magnetic field lines determines the paths of anisotropic cosmicray transport. For low values of resistivity, the buoyancy of the cosmic rays leads to an opening of magneticfield lines toward the upper and lowerz boundaries. This implies that the diffusive escape of cosmic rays, along the open magneticfield lines, dominates over the buoyancy and limits the effect of the Coriolis force, which is responsible for dynamo action.
Figure 4: Kinetic ( left column) and magnetic ( right column) spectra computed for (runs A1A4) and (Run A5), separately in x, y and zdirections (full, dotted and dashed thin lines, respectively). Lines representing the k^{5/3} slope (thick full lines) are shown for comparison. 

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4.2 The effect of spiral arms
Figure 5: Time evolution of azimuthal magnetic flux and total magnetic energy for different values of magnetic diffusivity, in presence of temporal modulations of SNrate mimicking the presence of spiral arms in the simulation series B. The curves represent respectively cases of (B1), (B2), (B3), (B4) and (B5) in units . 

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In this section, we describe the simulation series B performed for the same set of simulation parameters as for series A with the exception that the cosmicray energyinput is currently modulated in time by a step function. Motivation for this type of CR supply is the presence of spiral arms in disk galaxies. We assume that SNe explode in arms at a rate that is proportional to the star formation rate. We assume that arms pass through the volume of our local computational domain once every , and that the arm passage takes , i.e. starting at t=0, we supply CRs for the first of the and cease the CR supply for the remaining . We enhance the SN rate in the spiral arms by a factor of 4, so that the average SN rate during the entire period of density wave remains the same as in the simulation series A. The evolution in the mean magnetic flux and energy is presented in Fig. 5.
We find that in the present set of simulations, both magnetic flux and magnetic energy increase more rapidly than in the case of simulations without CR modulation. Magneticfield amplification is now apparent even in simulations with . The only exception is the simulation, which does not exhibit a noticeable amplification in the magnetic field, even in the presence of CR modulation. This indicates that the temporal modulation of SN rate acts in the same way as increasing magnetic diffusivity in the range of .
To interpret the above results, we suggest the following scenario: CRs supplied to the system trigger Parker instability and leave the disk volume by means of combined buoyant and diffusive transport. Vertical magnetic loops form efficiently during the period of enhanced SN activity, but later on, in the absence of CR perturbations, the magnetic field tends to relax before the next spiralarm passage. In the absence of CR forcing in the interarm regions, even a small resistivity is sufficient to relax magneticfield structure to a horizontal more regular configuration, which suppresses excessive losses of CRs by diffusive transport. Thus, the efficiency of the magneticfield amplification is enhanced.
4.3 Dependence of magnetic field amplification on SNrate
Figure 6: Time evolution of azimuthal magnetic flux and total magnetic energy for different values of SN rate applied in simulation series C, together with run B4, in the presence of temporal modulations of SNrate. Line assignments are respectively: (C1), (C2), (C3), (B4), (C4), and (C5) supernova explosions per squared kpc per Myr. 

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In all runs of the simulation series A and B, we adopted the fiducial value of , derived from the global model of Milky Way by Ferriere (1998) for a galactocentric radius . In this section, we describe the simulation series C performed for different SN rates of , and , together with simulation B4 ( ), to examine the effect of SN rate on magneticfield amplification. The results are shown in Fig. 6. In all cases, the SN input is modulated as described in Sect. 4.2.
We note that the magneticfield amplification rates and the final saturation levels of both magnetic flux and magnetic energy increase with as long as the SN rate is lower or equal to the fiducial, realistic value of at a galactocentric radius , when all other disk parameters are fixed. The efolding times of magnetic flux deduced from the left panel of Fig. 6 are 150 Myr for (Run B4) and 190 Myr for (Run C1). We therefore note that the magneticfield amplification rate is relatively insensitive to the magnitude of the SN rate for SN rates spanning one decade below the fiducial value. We also note that magneticfield amplification saturates at the level of equipartition of magnetic and kinetic energies in the case of those simulation runs of series C for which the amplification holds.
When the SN rate is doubled, with respect to the fiducial value, then only a short period of magneticfield amplification is observed, until , and if the SN rate is doubled once again then the magnetic field decays. The above results illustrate that magneticfield amplification occurs for a wide range of SN rates, and that realistic values of SN rates correspond to those which are optimal for the galactic dynamo process. As in Sect. 4.1, we show the ratio of energies in the vertical to the horizontal magneticfield components in Fig. 7. We find that for SN rates up to , the efficient magneticfield amplification is associated with the ratio of vertical to horizontal magneticfield energies fluctuating about one, and, in the case of excessive CR supply ( and more), the vertical magneticfield energy dominates and the magnetic field ceases increasing in strength.
Figure 7: Time evolution of the ratio of energies of vertical to horizontal magnetic field components for different (modulated) SN rates (C1), (B4) and (C4) supernova explosions per squared kpc per Myr. 

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4.4 Dependence of magnetic field amplification on the grid resolution
To check the effect of the grid resolution on the simulation results, we increase the cell size to in the simulations D1 and D2 and apply the same parameters as in simulations A4 and B4, respectively. In Fig. 8, we show the evolution in the total flux of the azimuthal magnetic field component and the total magnetic energy for simulations D1 and D2, and analogous curves for simulations A4 and B4, shown previously in Figs. 2 and 5. The results obtained at both resolutions are evidently similar, although a slightly more rapid increase in magneticfield strength is observed in simulations performed at the lower resolution, which can be explained by the higher numerical resistivity.
4.5 Dependence of magneticfield amplification on CRdiffusion coefficients
The aim of simulation series E is to examine the effect of variations in the CRdiffusion coefficients on magneticfield amplification. All simulations in series E are performed with resolution . This lower grid resolution makes it possible to enlarge the CR diffusion coefficients to realistic values, while preserving acceptable timesteps in the explicit integration algorithm of the CR diffusionadvection equation.
In the present simulation series E, we vary both the parallel and perpendicular diffusion coefficients, choosing different pairs from the set of , and , and , and . The results of these new simulations compared with those of simulation D1 are presented in Fig. 9.
The results of simulation series E can be summarized as follows. We note that the magneticfield growth rate and the saturation values of magnetic flux and energy depend in particular on the choice of and . When is increased by a factor of 3 and 10 with respect to D1, the initial growth in the magnetic field increases slightly, and the saturation level decreases by a factor of 2  3, provided that is not too high. For , the amplification holds for and , but for , the magnetic field decays. Similarly, for , amplification holds for and , but for , we detect an initial growth only until and decay thereafter. The present results indicate that magneticfield amplification is possible only for . Therefore, the anisotropy in the CR diffusion seems to be a crucial condition for magneticfield amplification in the process of the CRdriven dynamo.
In the subsequent Fig. 10, we show the energy ratio of vertical to azimuthal magneticfield components for simulations E4, E5 and E6, corresponding to three different values of and . Comparing Figs. 9 and 10 we find that in the case of two simulation runs E4 and E5 (two lower values of ) the energy ratio of vertical to azimuthal magneticfield components varies in the range , corresponding to an efficient growth in magnetic energy. For the highest value of , the magneticenergy ratio increases occasionally by an order of magnitude, and the magnetic field decays.
4.6 The issue of energy equipartition
Figure 8: Time evolution of the azimuthal magnetic flux and the total magnetic energy for simulations with grid resolutions (runs A4 and B4) and (runs D1 and D2). 

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Figure 9: Time evolution of the azimuthal magnetic flux and the total magnetic energy for different values of the parallel and perpendicular CR diffusion coefficients. Thin lines are used for (runs D1 and E1), mid lines are used for (runs E2 E3 and E4) and thick lines are used for (runs E5 E6 and E7). Full lines denote (runs D1, E2 and E5), dotted lines denote (runs E1, E3 and E6), dashed lines (runs E4 and E7). All diffusion coefficients given in units . 

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Figure 10: Time evolution of the ratio of energies of vertical to horizontal magnetic field components for different values of the perpendicular CR diffusion coefficients and (E4), and (E5) and , (E6). All diffusion coefficients Gaven in units . 

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The results presented so far demonstrate that magnetic fields amplified by the CRdriven dynamo saturate close to the equipartition of magnetic and gas kinetic energies. It is commonly expected that CRs also remain in energetic equipartition with the gas and magnetic field.
To investigate the relationship between CR and other forms of energies, in Fig. 11 we plot the time evolution in the ratio of the CR to the timeaveraged turbulent kinetic energy (after subtracting the kinetic energy of the largescale shear flow) for different values of CR diffusion coefficients (the simulation series E). Depending on the diffusion coefficients, we find that, CR energy is higher than the turbulent kinetic energy by a factor of 1050, while the magneticfield energy, according to the results presented in Fig. 9, remains close to the gas turbulent energy.
For the first few hundred Myr, CRs appear to accumulate quickly in the disk since they are trapped by the horizontal magnetic field. The ratio of CR to kinetic energies saturates as soon as the vertical magneticfield component becomes significant, due to buoyancy, enabling the diffusive transport of CRs out of the disk.
When the cosmicray diffusioncoefficients are higher, one can find that the ratio of CR to kinetic energies is lower. We note that cosmic rays find an easier way of leaving the disk when the parallel and perpendicular diffusion coefficients are higher. Shown in Fig. 11, the results indicate that the ratio of CR to kinetic energies anticorrelates with both: the parallel and perpendicular CR diffusioncoefficients. Due to the aforementioned timestep limitation, in the simulations presented in this paper we could only adopt the values of reaching at most , which are still lower than the value of mentioned by other authors (e.g. Jokipii 1999). Although the lowering the CR energy with the magnitude of the parallel CR diffusion coefficient appears a promising solution, one should not expect values as high as to reduce the problem of the CR energy excess.
Another factor, that may significantly influence the relationship between CR and other form of energies, is the choice of periodic and shearperiodic boundary conditions in the horizontal directions of the computational box. In true galactic disks, CR diffusion is expected to occur along the horizontal magnetic field lines. In the case of periodictype boundary conditions, CRs are trapped inside the disk volume by a predominantly horizontal magneticfield. This type of trapping can be released only in the global galacticdisk simulations.
5 Summary and conclusions
We have described an extensive series of simulations and presented a parameter study of the CRdriven dynamo in a galaxy, characterized by parameters that are typical of the Milky Way galaxy at galactocentric radius of . We considered the magnetic diffusivity, as well as the parallel and perpendicular CR diffusion coefficients to be free parameters, and dedicated simulation series have been performed to investigate their influence on the efficiency of the CRdriven dynamo process. The results of the parameter study can be summarized as follows:
 (1)
 The magnitude of the magnetic diffusivity influences the efficiency
of the magneticfield amplification. The most favorable value of magnetic
diffusivity is
,
value comparable to, although lower than, the value of the turbulent diffusivity of
the ISM deduced from observational data.
 (2)
 The efficiency of the magneticfield amplification is enhanced by the temporal
modulation of the CR supply. An effect of this kind may be associated with the
periodicity of the starformation and supernova activity induced by the spiral arms. The
enhancement is apparent at lower values of magnetic diffusivity and is less
significant at the optimal value of
.
 (3)
 The magneticfield amplification rate is relatively insensitive to the magnitude
of the SNrate for SN rates spanning one decade below the value
,
which are typical for the galactocentric radius
.
We note also that magneticfield amplification saturates at the
level of equipartition in the magnetic and kinetic energies at all supernova rates
for which amplification holds. The magnetic field is no longer amplified, if the SN rate
is enhanced further by factors of 2 and 4 with respect to realistic values,
while other quantities (such as e.g. gas column density) remain fixed.
 (4)
 Magneticfield amplification in the CRdriven dynamo relies on the anisotropic
diffusion of cosmic rays. From the limited set of simulations of series E, one
can deduce that the magneticfield amplification is possible only for
,
and for all
considered values of the parallel diffusioncoefficient
in the range
.
Therefore, the 5% ratio of the
perpendicular to parallel diffusion coefficients postulated by
Giacalone & Jokipii (1999) falls within the amplification range.
Figure 11: Time evolution in the ratio of CR to timeaveraged kinetic energy for different values of the CR diffusion coefficients in simulation series E. Line assignments are the same as in Fig. 9.
Open with DEXTER  (5)
 By varying the parameters of magnetic diffusivity, supernova rate, and the
CR diffusion coefficients, we have found that the favorable conditions for
magneticfield amplification correspond to approximately equal energies of the
vertical and azimuthal magnetic field components in the case of the buoyancydriven
dynamo. An excess or deficit of the vertical magnetic field with respect to the
azimuthal one corresponds to a significantly less efficient amplification or even
a decay in the magnetic field.
 (6)
 We recall problem indicated previously by Snodin et al. (2006), that in all simulations the CR energy in the computational domain exceeds the turbulent kinetic energy and magnetic energy by more than one order of magnitudes. The lowest ratios of CR to kinetic energies also correspond to the highest values of the parallel diffusion coefficient. It seems implausible, however, that an increase in the diffusion coefficients to fully realistic values would reduce the excess of cosmicray energy in the disk. It also seems that the ratio of CR to other forms of energy in the ISM is not yet well constrained on observational grounds (Strong et al. 2007). On the other hand, the currently used shearingbox approximation does not permit CRs to leave the disk by means of diffusion along the predominantly horizontal magnetic field. Therefore, we suggest that future studies of the CRdriven dynamo, designed to solve this problem, should be completed in the framework of global galacticdisk simulations.
Acknowledgements
This work was supported from the Polish Committee for Scientific Research (KBN) through the grants PB 0656/P03D/2004/26 and 2693/H03/2006/31.
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All Tables
Table 1: Parameters of simulations presented in this paper.
All Figures
Figure 1: Exemplary plots illustrating the state of the system at for simulation A4. In the first two panels we present slices through the computational volume in the yzplane for x =0. Panel a) shows cosmic ray energy density with vectors of magnetic field, panel b) shows gas density with velocity vectors. In panel c) we plot horizontally averaged x and y components of magnetic field, and in panel d) horizontally averaged vertical velocity component and its fluctuations. 

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In the text 
Figure 2: Time evolution of azimuthal magnetic flux and total magnetic energy for different values of magnetic diffusivity in simulation series A. The curves represent respectively cases of (A1), (A2), (A3), (A4) and (A5) in units . 

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In the text 
Figure 3: Time evolution of the ratio of energies of vertical to horizontal magneticfield components for different values of magnetic diffusivity in the simulation series A. Line assignments are the same as in Fig. 2. 

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In the text 
Figure 4: Kinetic ( left column) and magnetic ( right column) spectra computed for (runs A1A4) and (Run A5), separately in x, y and zdirections (full, dotted and dashed thin lines, respectively). Lines representing the k^{5/3} slope (thick full lines) are shown for comparison. 

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In the text 
Figure 5: Time evolution of azimuthal magnetic flux and total magnetic energy for different values of magnetic diffusivity, in presence of temporal modulations of SNrate mimicking the presence of spiral arms in the simulation series B. The curves represent respectively cases of (B1), (B2), (B3), (B4) and (B5) in units . 

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In the text 
Figure 6: Time evolution of azimuthal magnetic flux and total magnetic energy for different values of SN rate applied in simulation series C, together with run B4, in the presence of temporal modulations of SNrate. Line assignments are respectively: (C1), (C2), (C3), (B4), (C4), and (C5) supernova explosions per squared kpc per Myr. 

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In the text 
Figure 7: Time evolution of the ratio of energies of vertical to horizontal magnetic field components for different (modulated) SN rates (C1), (B4) and (C4) supernova explosions per squared kpc per Myr. 

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In the text 
Figure 8: Time evolution of the azimuthal magnetic flux and the total magnetic energy for simulations with grid resolutions (runs A4 and B4) and (runs D1 and D2). 

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In the text 
Figure 9: Time evolution of the azimuthal magnetic flux and the total magnetic energy for different values of the parallel and perpendicular CR diffusion coefficients. Thin lines are used for (runs D1 and E1), mid lines are used for (runs E2 E3 and E4) and thick lines are used for (runs E5 E6 and E7). Full lines denote (runs D1, E2 and E5), dotted lines denote (runs E1, E3 and E6), dashed lines (runs E4 and E7). All diffusion coefficients given in units . 

Open with DEXTER  
In the text 
Figure 10: Time evolution of the ratio of energies of vertical to horizontal magnetic field components for different values of the perpendicular CR diffusion coefficients and (E4), and (E5) and , (E6). All diffusion coefficients Gaven in units . 

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In the text 
Figure 11: Time evolution in the ratio of CR to timeaveraged kinetic energy for different values of the CR diffusion coefficients in simulation series E. Line assignments are the same as in Fig. 9. 

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In the text 
Copyright ESO 2009