Critical magnetic Reynolds number of the weakly-collisional turbulent dynamo in supersonic plasmas

Introduction

Weakly-collisional plasmas are found in a broad range of astrophysical environments, including the solar wind, the hot intracluster medium (ICM) of galaxy clusters, hot accretion flows, and collisionless shocks. In these systems, the Coulomb mean free path is comparable to the characteristic macroscopic length scales. Therefore, a collisional magnetohydrodynamic (MHD) description does not correctly capture the plasma evolution, and a kinetic treatment is required (Kulsrud 2005; Schekochihin & Cowley 2006; Simionescu et al. 2019; Kunz et al. 2022). The weak collisionality of these plasmas causes departures from local thermodynamic equilibrium and gives rise to kinetic instabilities that regulate the transport of momentum and energy.

Many weakly-collisional astrophysical plasmas are also turbulent. In galaxy clusters, turbulence can be generated by merger activity, and energy injection from active galactic nuclei (Subramanian et al. 2006; Banerjee & Sharma 2014; Mohapatra & Sharma 2019; Zhuravleva et al. 2014, 2019; Gatuzz et al. 2023). The compressibility of these motions varies substantially among different environments. Turbulence can be subsonic in relaxed cluster interiors (Hitomi Collaboration et al. 2016; Zhuravleva et al. 2022a,b, 2023), whereas merging systems and shocked regions can exhibit transonic or supersonic motions (Hoang et al. 2017).

Magnetic fields are dynamically important in many of these systems. For example, magnetic fields of order $\sim\mu\mathrm{G}$ are observed in the hot ICM (Bonafede et al. 2010, 2013; Botteon et al. 2022). These strong fields can grow from much weaker primordial or astrophysical seed fields through turbulent dynamo action, which converts the kinetic energy of turbulent motions into magnetic energy (Kazantsev 1968; Moffatt 1978; Brandenburg & Subramanian 2005). During the kinematic stage of the dynamo, turbulent motions stretch, twist, and fold the weak magnetic field, causing exponential growth of the magnetic energy. The subsequent back-reaction of the magnetic field on the flow eventually leads to the saturated state of the dynamo (Federrath et al. 2011; Seta & Federrath 2020, 2021; Achikanath Chirakkara et al. 2021).

Whether a turbulent flow can amplify magnetic energy is determined by the competition between inductive amplification and magnetic diffusion. This competition is quantified by the magnetic Reynolds number, Rm, defined as

where $v_{\mathrm{turb}}$, $L_{\mathrm{turb}}$, and $\eta$ are the turbulent speed, turbulence driving scale, and Ohmic diffusivity, respectively. Dynamo action is possible only when Rm exceeds a critical magnetic Reynolds number, $\mathrm{Rm}_{\mathrm{crit}}$. For $\mathrm{Rm} < \mathrm{Rm}_{\mathrm{crit}}$, magnetic diffusion dissipates magnetic energy more rapidly than turbulent motions can generate it. The value of $\mathrm{Rm}_{\mathrm{crit}}$ therefore provides an important measure of the conditions required to excite the turbulent dynamo.

Studies of the collisional MHD dynamo have demonstrated that $\mathrm{Rm}_{\mathrm{crit}}$ is not universal. It depends on the magnetic Prandtl number, Pm, as well as on the compressibility of the plasma [20] (Schober et al. 2012; Federrath et al. 2014; Seta et al. 2020). At fixed Pm, numerical simulations find that the critical magnetic Reynolds number increases as the turbulence transitions from the subsonic to the supersonic regime. For Pm = 1, for example, $\mathrm{Rm}_{\mathrm{crit}}$ increases from approximately 35 in subsonic flows to 70 in supersonic flows [20]. This increase is likely associated with the formation of shocks and strong compressive structures, as dynamo amplification is driven primarily by solenoidal motions, which efficiently stretch, twist, and fold magnetic-field lines, whereas compressive motions contribute less efficiently. In highly compressible, supersonic simulations, Federrath et al. (2014) measured $\mathrm{Rm}_{\mathrm{crit}} = 129^{+43}_{-31}$. Together, these results suggest that increasing compressibility reduces the efficiency of magnetic-field amplification and raises the threshold for dynamo action.

Analytical studies based on the Kazantsev model also predict that the threshold for dynamo action depends strongly on the nature of the turbulence. In particular, the predicted $\mathrm{Rm}_{\mathrm{crit}}$ is considerably larger for highly compressible, Burgers-like turbulence than for incompressible, Kolmogorov-like turbulence (Schober et al. 2012; Bovino et al. 2013). Both analytical and numerical studies indicate that compressibility can inhibit the onset of the turbulent dynamo. These results have been obtained for collisional MHD plasmas, and it remains unclear whether the same behaviour applies to weakly-collisional plasmas.

The turbulent dynamo in weakly-collisional plasmas has only recently been studied using kinetic and hybrid-kinetic numerical methods (Rincon et al. 2016; St-Onge & Kunz 2018; St-Onge et al. 2020; Sironi et al. 2023; Zhou et al. 2024; Achikanath Chirakkara et al. 2024a,b, 2025). In the hybrid particle-in-cell (HPIC) approach, ions are evolved kinetically as charged particles, while electrons are represented as a fluid. In collisional MHD simulations, the viscosity and Ohmic diffusivity can be set independently, allowing the kinetic Reynolds number, Re, and the magnetic Prandtl number, Pm = Rm/Re, to be controlled separately. In a weakly-collisional HPIC plasma, however, the effective viscosity arises self-consistently from wave–particle interactions and may evolve as the magnetic field and plasma magnetisation change (Schekochihin & Cowley 2006). The corresponding Re and Pm are therefore emergent properties rather than fixed input parameters. This complicates direct comparisons between HPIC and MHD dynamos.

Using HPIC simulations, Achikanath Chirakkara et al. (2024a) measured the critical magnetic Reynolds number of the weakly-collisional turbulent dynamo in the subsonic regime. However, the critical magnetic Reynolds number of the weakly-collisional dynamo has not been determined in the supersonic regime. In this work, we investigate the onset of weakly-collisional turbulent dynamo action in a supersonic plasma.

The rest of this paper is organised as follows. We describe the hybrid-kinetic equations, plasma cooling, turbulent driving, and initial conditions in § 2. In § 3, we examine the density, velocity, and magnetic-field structures, follow the time evolution of the magnetic energy, and determine the critical magnetic Reynolds number. We compare our results with previous studies of collisional MHD and subsonic weakly-collisional plasmas in § 4. Finally, we present our conclusions in § 5.

Methods

We perform numerical experiments of the weakly-collisional dynamo using the HPIC code AHKASH (Achikanath Chirakkara et al. 2024b). In this section, we describe the hybrid-kinetic equations solved by AHKASH, the cooling prescription used to maintain an isothermal plasma, and the external forcing used to drive turbulence. We then discuss the plasma parameters and the initial conditions of our simulations.

Hybrid-kinetic equations

Within the hybrid-kinetic framework, ions are evolved kinetically as particles, while the electrons are represented as a massless fluid. The evolution of the $p^{\mathrm{th}}$ ion is governed by the equations of motion,

where $\boldsymbol{r}_p$ and $\boldsymbol{v}_p$ denote the particle position and velocity, respectively, with $p = 1,\ldots,P$ and $P$ the total number of particles. Here, $q_i$ and $m_i$ are the ion charge and mass. The particle trajectories are determined by the Lorentz force together with the turbulence forcing term, $\boldsymbol{f}$, which injects turbulent kinetic energy into the plasma (Section 2.2). The evolving particle distribution is used to evaluate the ion moments required to solve for the electromagnetic fields, $\boldsymbol{E}$ and $\boldsymbol{B}$. The electric field is obtained from Ohm’s law (Rosin et al. 2011),

where $\boldsymbol{J} = (\nabla\times\boldsymbol{B})/\mu_0$ is the total current density and $\mu_0$ is the magnetic permeability of free space. The ion charge density, $\rho_i$, and ion current density, $\boldsymbol{J}_i$, are sampled directly from the particle distribution. We assume an isothermal electron fluid, with the pressure given by the ideal gas equation of state,

where $n_e$ is the electron number density, $T_e$ is the constant electron temperature, and $k_{\mathrm{B}}$ is Boltzmann’s constant. Quasi-neutrality is enforced by proton density, $\rho_e = -\rho_i$. In (4), the coefficient $\eta$ is the Ohmic diffusivity, while hyper resistivity term, controlled by $\eta_{\mathrm{hyper}}$, is included to damp numerical fluctuations at the grid scale. The magnetic field evolves according to Faraday’s law,

which is solved using the constrained transport algorithm to maintain the divergence-free condition for magnetic fields, $\nabla\cdot\mathbf{B} = 0$. Interpolation between particles and the computational grid is performed using the cloud-in-cell method. The ion moments are smoothed to suppress particle noise associated with the finite sampling of the ion distribution function. Additional details of the numerical implementation, together with validation tests of the code, are presented in Achikanath Chirakkara et al. (2024b).

Cooling for hybrid particle-in-cell simulations

Turbulent dissipation and shocks in the supersonic regime continuously heat the plasma. To maintain a statistically steady, isothermal state, this heating must be balanced by an appropriate cooling mechanism. While the electron fluid is kept isothermal using an isothermal equation of state (§ 2.1), no corresponding cooling mechanism exists for the kinetic ions. Therefore, an explicit cooling procedure is required to regulate the ion temperature and maintain thermal equilibrium between the ions and electrons.

To achieve this, we use the local ion-cooling scheme introduced by Achikanath Chirakkara et al. (2024b, see their sec. 2.11), which rescales the ion thermal speed, $v_{\mathrm{th}}$. Cooling is applied on a timescale of $(\Delta t)_{\mathrm{cool}} = 0.01\,t_{\mathrm{cool}}$, where $t_{\mathrm{cool}} = t_{\mathrm{th}}/\mathcal{M}$ and $t_{\mathrm{th}} = \Delta l/v_{\mathrm{th}}$ is the thermal crossing time, defined as the time required for a sound wave to travel across a grid cell of size $\Delta l$. The sonic Mach number,

where $v_{\mathrm{turb}}$ and $v_{\mathrm{th}}$ are the turbulent and thermal speeds, respectively. This cooling prescription maintains an approximately constant plasma temperature in space and time throughout the simulation.

Turbulent driving

The external turbulent forcing, $\mathbf{f}$, is generated using the TurbGen method (Federrath et al. 2010, 2022), which uses an Ornstein–Uhlenbeck stochastic process. We use purely solenoidal forcing, $\nabla\cdot\mathbf{f} = 0$ here and leave an investigation of fully compressive forcing to future work. Energy is injected only on large spatial scales, corresponding to wavenumbers in the range $k_{\mathrm{turb}}L_{\mathrm{box}}/(2\pi) \in[1, 3]$, where $L_{\mathrm{box}}$ is the side length of the computational box. The forcing spectrum follows a parabolic function that has its maximum at $k_{\mathrm{turb}}L_{\mathrm{box}}/(2\pi) = 2$ and decreases smoothly to zero at the boundaries of the driving range, $k_{\mathrm{turb}}L_{\mathrm{box}}/(2\pi) = 1$ and $3$. This is standard procedure in turbulence simulations (Federrath & Offner 2026). The turbulent crossing time is defined as

where $L_{\mathrm{turb}} = L_{\mathrm{box}}/2$ is the turbulent driving scale.

Plasma parameters and initial conditions

This section describes the physical parameters and initial conditions used in the HPIC runs. All simulations are performed in a three-dimensional periodic domain with a grid resolution of $N_{\mathrm{grid}} = 128^3$ cells and $N_{\mathrm{ppc}} = 100$ particles per cell. The convergence tests for the grid and particle resolutions are provided in our previous studies (Achikanath Chirakkara et al. 2024b, 2025).

Plasma parameters

The properties of the turbulent dynamo depend on the following plasma parameters. The compressibility of the plasma is characterised by the Mach number. In this study, we focus on the supersonic regime, with $\mathcal{M} = 2$. We consider six magnetic Reynolds numbers, $\mathrm{Rm} = 31.3$, $62.5$, $125$, $250$, $500$, and $1000$, where $\mathrm{Rm}$ is controlled by varying the Ohmic diffusivity $\eta$. To suppress numerical fluctuations at the grid scale, we include a hyper-resistive term (cf. Eq. (4)). The corresponding hyper-resistive Reynolds number is defined as

We use $\mathrm{Rm}_{\mathrm{hyper}} = N_{\mathrm{grid}}^{10/3}$, where $N_{\mathrm{grid}}$ is the grid resolution (see sec. 2.5.2 in Achikanath Chirakkara et al. 2024a). In weakly-collisional plasmas, the effective viscosity arises self-consistently through wave–particle interactions. As a result, both the kinetic Reynolds number, $\mathrm{Re}$, and the magnetic Prandtl number, $\mathrm{Pm} = \mathrm{Rm}/\mathrm{Re}$, are emergent properties of the plasma rather than prescribed parameters in our simulations.

Initial conditions and parameter study

We initialise each simulation with a weak magnetic field. The initial magnetic-to-turbulent kinetic energy ratio is defined as

where $B_0$ is the initial magnetic field strength and $\rho_{\mathrm{m}}$ is the ion mass density. The initial plasma beta is denoted by $\beta_0$. Throughout this work, we set $(E_{\mathrm{mag}}/E_{\mathrm{kin}})_0 = 10^{-10}$ to allow for turbulent dynamo amplification in the kinematic regime.

The level of plasma magnetisation is characterised by the Larmor ratio, defined as the ratio of the ion Larmor radius to the size of the computational domain (Achikanath Chirakkara et al. 2024a). The initial Larmor ratio is given by

We consider two initial magnetisation states, corresponding to $r_{\mathrm{Larmor},0}/L_{\mathrm{box}} = 1$ (initially magnetised) and $r_{\mathrm{Larmor},0}/L_{\mathrm{box}} = 100$ (initially unmagnetised). More details about the initial conditions and the plasma parameters are provided in sec. 2.5 of Achikanath Chirakkara et al. 2024a. All simulations are performed using the hybrid-precision method, which substantially reduces the computational cost while maintaining numerical accuracy (Federrath et al. 2021; Achikanath Chirakkara et al. 2024b; Federrath & Offner 2026).

The plasma parameters and measured properties of all simulations are summarised in Table 1, including the grid and particle resolutions, Mach number, initial magnetic to kinetic energy ratio, initial plasma beta, initial magnetisation, magnetic Reynolds number, and the resulting growth/decay rate of the magnetic field.

Ser. No	Model	$N_{\mathrm{grid}}^{3}$	$N_{\mathrm{ppc}}$	$\mathcal{M}$	$(E_{\mathrm{mag}}/E_{\mathrm{kin}})_0$	$\beta_0$	$r_{\mathrm{Larmor},0}/L_{\mathrm{box}}$	Rm	$\Gamma$ ($t_{\mathrm{turb}}^{-1}$)
1	Rm1000rL1	$128^3$	100	$2.16 \pm0.15$	$10^{-10}$	$5 \times10^9$	1	$1080 \pm70$	$0.40 \pm0.03$
2	Rm500rL1	$128^3$	100	$2.13 \pm0.14$	$10^{-10}$	$5 \times10^9$	1	$532 \pm34$	$0.24 \pm0.01$
3	Rm250rL1	$128^3$	100	$2.08 \pm0.12$	$10^{-10}$	$5 \times10^9$	1	$260 \pm15$	$0.11 \pm0.01$
4	Rm125rL1	$128^3$	100	$2.06 \pm0.11$	$10^{-10}$	$5 \times10^9$	1	$129 \pm7$	$-0.14 \pm0.07$
5	Rm63rL1	$128^3$	100	$2.07 \pm0.11$	$10^{-10}$	$5 \times10^9$	1	$64.5 \pm3.3$	$-0.61 \pm0.09$
6	Rm31rL1	$128^3$	100	$2.08 \pm0.11$	$10^{-10}$	$5 \times10^9$	1	$32.4 \pm1.8$	$-1.28 \pm0.13$
7	Rm1000rL100	$128^3$	100	$2.11 \pm0.13$	$10^{-10}$	$5 \times10^9$	$10^2$	$1050 \pm70$	$0.48 \pm0.02$
8	Rm500rL100	$128^3$	100	$2.07 \pm0.12$	$10^{-10}$	$5 \times10^9$	$10^2$	$518 \pm30$	$0.33 \pm0.02$
9	Rm250rL100	$128^3$	100	$2.06 \pm0.11$	$10^{-10}$	$5 \times10^9$	$10^2$	$257 \pm13$	$0.10 \pm0.04$
10	Rm125rL100	$128^3$	100	$2.06 \pm0.10$	$10^{-10}$	$5 \times10^9$	$10^2$	$129 \pm6$	$-0.21 \pm0.10$
11	Rm63rL100	$128^3$	100	$2.06 \pm0.12$	$10^{-10}$	$5 \times10^9$	$10^2$	$64.2 \pm3.8$	$-0.62 \pm0.08$
12	Rm31rL100	$128^3$	100	$2.07 \pm0.11$	$10^{-10}$	$5 \times10^9$	$10^2$	$32.3 \pm1.7$	$-1.14 \pm0.12$

Results

We first examine the density, velocity, and magnetic-field structures in § 3.1, before investigating the time evolution of the magnetic energy and its growth rate during the kinematic regime in § 3.2. Finally, in § 3.3 we measure the critical magnetic Reynolds number for turbulent dynamo action in supersonic weakly-collisional plasma.

Density, velocity, and magnetic field structures

Figure 1 shows slice plots of the density, velocity, and magnetic fields in the supersonic HPIC simulations during the kinematic phase at $t = 25\,t_{\mathrm{turb}}$. From top to bottom, the rows correspond to $\mathrm{Rm} = 1000$, $250$, $125$, and $31.3$. The corresponding slice plots for an initially unmagnetised plasma with $r_{\mathrm{Larmor},0}/L_{\mathrm{box}} = 100$ are shown in Figure A1.

The first column shows the normalised density field. Strong density fluctuations, shocks, and large-scale filamentary structures are present in all simulations, as expected for supersonic turbulence. The density structures are qualitatively similar over the range of magnetic Reynolds numbers studied here, showing that the large-scale supersonic flow is largely unaffected by magnetic diffusivity. The second column of Figure 1 shows the normalised velocity magnitude. The flow is dominated by motions on large scales, consistent with the forcing described in § 2.2. Similar to the density field, the velocity structures exhibit little dependence on the magnetic Reynolds number.

This is expected because we only study the kinematic regime of the dynamo, in which the magnetic field is dynamically weak compared to the velocity field and its back-reaction on the flow is negligible. Varying $\mathrm{Rm}$ therefore affects the magnetic-field evolution without changing the density or velocity fields in the kinematic regime. Differences may emerge in the nonlinear and saturation regimes of the dynamo as the magnetic field strength increases and becomes comparable to the velocity field. The filamentary density structures are spatially correlated with the velocity field, consistent with previous studies of supersonic turbulence (Padoan et al. 1997; Kowal et al. 2007; Kritsuk et al. 2007; Federrath et al. 2010).

The third column of Figure 1 shows the normalised magnetic-field strength. Unlike the density and velocity fields, the magnetic-field morphology depends strongly on $\mathrm{Rm}$. At large $\mathrm{Rm}$, the field exhibits intermittent structure over small spatial scales. These features become smoother as $\mathrm{Rm}$ decreases, because higher Ohmic diffusion removes small-scale magnetic structure.

Time evolution

Figure 2 shows the time evolution for the two initial magnetisation values, $r_{\mathrm{Larmor},0}/L_{\mathrm{box}} = 1$ and $r_{\mathrm{Larmor},0}/L_{\mathrm{box}} = 100$, spanning magnetic Reynolds numbers of $\mathrm{Rm} = 31.3$, $62.5$, $125$, $250$, $500$, and $1000$. From top to bottom, the four panels show the Mach number, $\mathcal{M}$, the magnetic energy normalised by its initial value, $E_{\mathrm{m}}/E_{\mathrm{m0}}$, the magnetic-to-kinetic energy ratio, $E_{\mathrm{mag}}/E_{\mathrm{kin}}$, and the Larmor ratio, $r_{\mathrm{Larmor}}/L_{\mathrm{box}}$, respectively.

The turbulent driving and ion cooling maintain an approximately constant Mach number, as shown in the top panels of Figure 2. After an initial transient phase of about $2\,t_{\mathrm{turb}}$, the flow settles into a statistically stationary turbulent state. At sufficiently large $\mathrm{Rm}$, the magnetic energy increases exponentially, marking the kinematic phase of the turbulent dynamo. This phase is the primary focus of this study.

As $\mathrm{Rm}$ is reduced by increasing the Ohmic diffusivity, the dynamo becomes less efficient, leading to lower magnetic energy growth rates. Eventually, below a critical magnetic Reynolds number, $\mathrm{Rm}_{\mathrm{crit}}$, the magnetic field is no longer amplified and instead decays with time. To quantify the dynamo efficiency, we determine the magnetic energy growth rate, $\Gamma$, by fitting the magnetic energy in the kinematic phase with

We exclude the initial transition phase during the first $5\,t_{\mathrm{turb}}$ from the fit. The resulting growth and decay curves are over-plotted as black dashed lines in the second panel of Figure 2. Uncertainties in $\Gamma$ are estimated following the method described in appendix C of Achikanath Chirakkara et al. (2024a), and the fitted values are listed in Table 1.

The third panel of Figure 2 shows the evolution of the magnetic-to-kinetic energy ratio, $E_{\mathrm{mag}}/E_{\mathrm{kin}}$. As the magnetic field is amplified by the turbulent dynamo, $E_{\mathrm{mag}}/E_{\mathrm{kin}}$ increases until the magnetic energy becomes comparable to the kinetic energy of the turbulent eddies at the viscous scale. At this stage, the exponential amplification transitions to a linear growth phase before eventually reaching saturation (Seta & Federrath 2020). We note that the present simulations have not been evolved into the saturation regime, which would require substantially more computational resources and is not required for the measurements of $\mathrm{Rm}_{\mathrm{crit}}$.

The fourth panel of Figure 2 shows the evolution of the Larmor ratio. As the magnetic field is amplified, the average ion Larmor radius decreases, resulting in stronger plasma magnetisation, whereas magnetic field decay results in an increase in the Larmor radius.

Measuring the critical magnetic Reynolds number

Figure 3 shows the energy growth/decay rate $\Gamma$ as a function of $\mathrm{Rm}$ for both initial magnetisation values. For each initial Larmor ratio, we model the dependence of $\Gamma$ on $\mathrm{Rm}$ as

where $\mathrm{Rm}_{\mathrm{crit}}$, $\Gamma_{\mathrm{sat}}$, and $\alpha$ are fit parameters (Achikanath Chirakkara et al. (2024a)). In the above equation, $\mathrm{Rm}_{\mathrm{crit}}$ is the critical magnetic Reynolds number below which Ohmic diffusivity suppresses dynamo amplification, resulting in magnetic field decay. $\Gamma_{\mathrm{sat}}$ represents the asymptotic magnetic energy growth rate in the high-Rm limit, and $\alpha$ is a power-law exponent that determines how rapidly $\Gamma$ rises above $\mathrm{Rm}_{\mathrm{crit}}$ and approaches this asymptotic value. The best-fit curves obtained using Eq. (13) are shown in Figure 3 as solid lines.

The vertical dashed lines show the fitted values of $\mathrm{Rm}_{\mathrm{crit}}$, while the shaded regions represent their respective uncertainties. The best-fit values of $\mathrm{Rm}_{\mathrm{crit}}$, $\Gamma_{\mathrm{sat}}$, and $\alpha$ are listed in Table 2. For $\mathrm{Rm} > \mathrm{Rm}_{\mathrm{crit}}$, the positive growth rate ($\Gamma> 0$) indicates magnetic-field amplification by the weakly collisional dynamo, with $\mathrm{Rm}_{\mathrm{crit}} = 190^{+21}_{-17}$ for initially magnetised plasma and $\mathrm{Rm}_{\mathrm{crit}} = 198^{+16}_{-18}$ for initially unmagnetised plasma.

$r_{\mathrm{Larmor},0}/L_{\mathrm{box}}$	$\mathrm{Rm}_{\mathrm{crit}}$	$\Gamma_{\mathrm{sat}}$	$\alpha$
1	$190^{+21}_{-17}$	$0.55^{+0.14}_{-0.07}$	$-0.68^{+0.13}_{-0.13}$
100	$198^{+16}_{-18}$	$0.95^{+0.05}_{-0.17}$	$-0.43^{+0.03}_{-0.03}$

The critical magnetic Reynolds numbers for the initially magnetised and initially unmagnetised plasmas are the same within their uncertainties, indicating that, in the supersonic regime, $\mathrm{Rm}_{\mathrm{crit}}$ is not affected by the initial plasma magnetisation. This contrasts with our previous study of the weakly-collisional turbulent dynamo in the subsonic regime of turbulence ((Achikanath Chirakkara et al. 2024a)), in which we found that the critical magnetic Reynolds number was marginally larger for initially magnetised plasmas than for initially unmagnetised plasmas.

Comparison with the collisional MHD turbulent dynamo

In this section, we compare the critical magnetic Reynolds number measured in this work with the $\mathrm{Rm}_{\mathrm{crit}}$ values obtained in collisional MHD studies. Haugen et al. [20] performed direct numerical simulations of compressible, isothermal MHD turbulence to investigate the dependence of $\mathrm{Rm}_{\mathrm{crit}}$ on the Mach number. They estimated $\mathrm{Rm}_{\mathrm{crit}}$ by interpolating between simulations in which the magnetic field grew and those in which magnetic field decayed. They found that $\mathrm{Rm}_{\mathrm{crit}}$ increases sharply around $\mathcal{M} \sim1$, with subsonic and supersonic values of $\mathrm{Rm}_{\mathrm{crit}} \sim35$ and 70, respectively, for $\mathrm{Pm} = 1$, and $\mathrm{Rm}_{\mathrm{crit}} \sim25$ and 50 for $\mathrm{Pm} = 5$. These are characteristic values for the two Mach-number regimes rather than measurements at specific Mach numbers. These results show that increasing compressibility raises the threshold for turbulent dynamo action and that $\mathrm{Rm}_{\mathrm{crit}}$ depends on the magnetic Prandtl number. Using simulations of highly compressible MHD turbulence with Mach numbers in the range $4 \lesssim\mathcal{M} \lesssim11$, Federrath et al. (2014) studied turbulent dynamo action over the range $0.1 \leq\mathrm{Pm} \leq10$ and found a critical magnetic Reynolds number. However, neither Re nor Pm is a fixed parameter in HPIC simulations, making this comparison difficult. Both Re and Pm are self-consistently determined by wave–particle interactions in a weakly-collisional plasma and can evolve as the magnetic field strength changes due to dynamo action.

Our results further reveal that the value of $\mathrm{Rm}_{\mathrm{crit}}$ for the weakly-collisional turbulent dynamo is higher in the supersonic regime than in the subsonic regime ((Achikanath Chirakkara et al. 2024a)). This is consistent with the trend seen in MHD simulations. For the subsonic weakly-collisional turbulent dynamo, Achikanath Chirakkara et al. (2024a) found that an initially magnetised plasma had a marginally higher value of $\mathrm{Rm}_{\mathrm{crit}} \approx124 \pm8$ than an initially unmagnetised plasma, for which $\mathrm{Rm}_{\mathrm{crit}} \approx107 \pm4$. We find that this is not the case in the supersonic regime.

Using a Wentzel–Kramers–Brillouin (WKB) approximation to the Kazantsev (1968) equation, Schober et al. (2012) found $\mathrm{Rm}_{\mathrm{crit}} \approx110$ for Kolmogorov turbulence and $\mathrm{Rm}_{\mathrm{crit}} \approx2700$ for Burgers turbulence, showing that increasing compressibility makes the dynamo instability more difficult to excite. Bovino et al. (2013) investigated the onset of the turbulent dynamo by numerically solving the Kazantsev equation in the regimes $\mathrm{Pm} \ll1$ and $\mathrm{Pm} \gg1$, for both incompressible Kolmogorov and highly-compressible Burgers turbulence. They predicted that the critical magnetic Reynolds number depends strongly on the nature of the turbulence, increasing from $\mathrm{Rm}_{\mathrm{crit}} \approx320$ for Kolmogorov turbulence to $\mathrm{Rm}_{\mathrm{crit}} \approx3.2 \times10^{4}$ for Burgers turbulence, again implying that highly compressible turbulence substantially suppresses turbulent dynamo action. We note that the approach used in these theoretical studies differs from the 3-D HPIC simulations used here. As in simulations of the supersonic MHD turbulent dynamo, the weakly-collisional dynamo in the supersonic regime exhibits critical magnetic Reynolds numbers that are substantially lower than those predicted by these semi-analytical studies, which deserves further investigation into the theoretical modelling of turbulent dynamo amplification in both collisional and weakly-collisional plasmas.

Conclusions

Using three-dimensional hybrid particle-in-cell (HPIC) simulations, we determined the critical magnetic Reynolds number of the weakly-collisional turbulent dynamo in the supersonic regime. We drove purely solenoidal turbulence at $\mathcal{M} \approx2$ and measured the magnetic Reynolds number over the range $\mathrm{Rm} \approx31$–1000. We considered initially magnetised and initially unmagnetised plasmas with initial Larmor ratios $r_{\mathrm{Larmor},0}/L_{\mathrm{box}} = 1$ and 100, respectively.

We find that in the kinematic regime of the dynamo, the density and velocity structures show little variation with Rm, whereas the magnetic field becomes progressively smoother as Rm decreases because of increasing Ohmic diffusion. At sufficiently large Rm, the magnetic energy grows exponentially during the kinematic phase of the dynamo, while at sufficiently low Rm, magnetic diffusion dominates and the magnetic energy decays. By fitting the magnetic-energy growth rate as a function of Rm, we obtain $\mathrm{Rm}_{\mathrm{crit}} = 190^{+21}_{-17}$ and $198^{+16}_{-18}$ for the initially magnetised and unmagnetised case, respectively.

The two measurements of $\mathrm{Rm}_{\mathrm{crit}}$ are the same within their uncertainties, indicating that the initial plasma magnetisation does not significantly affect the onset of the weakly-collisional turbulent dynamo in the supersonic regime. This differs from the marginal dependence on initial magnetisation found previously for the weakly-collisional dynamo in the subsonic regime. The values of $\mathrm{Rm}_{\mathrm{crit}}$ in the supersonic regime are also higher than the corresponding subsonic measurements, indicating that increased compressibility makes the weakly-collisional dynamo more difficult to excite. Consistent with collisional MHD simulations, compressibility raises the threshold for dynamo action and our measured values, $\mathrm{Rm}_{\mathrm{crit}} \simeq190$–$198$, are approximately 1.5 times higher than the supersonic MHD value $\mathrm{Rm}_{\mathrm{crit}}=129^{+43}_{-31}$ found by Federrath et al. (2014).

Acknowledgements

C.F. acknowledges funding provided by the Australian Research Council (Discovery Projects DP230102280 and DP250101526), and the Australia-Germany Joint Research Cooperation Scheme (UA-DAAD). This research was enabled in part by support provided by Compute Ontario (computeontario.ca) and the Digital Research Alliance of Canada (alliancecan.ca). Computations were performed on the Trillium supercomputer at the SciNet HPC Consortium. SciNet is funded by Innovation, Science and Economic Development Canada; the Digital Research Alliance of Canada; the Ontario Research Fund: Research Excellence; and the University of Toronto (Loken et al. 2010; Ponce et al. 2019). We further acknowledge high-performance computing resources provided by the Leibniz Rechenzentrum and the Gauss Centre for Supercomputing (grant pr32lo), the Australian National Computational Infrastructure (grant ek9) and the Pawsey Supercomputing Centre (project pawsey0810) in the framework of the National Computational Merit Allocation Scheme and the ANU Merit Allocation Scheme. The simulation software FLASH, which forms the bases for AHKASH, was in part developed by the DOE-supported Flash Center for Computational Science at the University of Chicago and the University of Rochester. All revisions were reviewed by the authors.

Data Availability

The simulation data will be shared upon reasonable request to Radhika Achikanath Chirakkara, noting that the simulations have a total size of 17 TB.

Density, velocity, and magnetic field structures for initially unmagnetised plasma

Figure A1 shows the density, velocity, and magnetic-field structure for the supersonic turbulent dynamo simulations with an initial larmor ratio of $L_{\rm armor,0}/L_{\rm box}=100$. The overall behaviour is qualitatively similar to that discussed in § 3.1 for the initially magnetised runs. The density and velocity fields exhibit little dependence on the magnetic Reynolds number, while the magnetic field becomes progressively smoother as Rm decreases.

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