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The small-scale dynamo: Breaking universality at high Mach numbers

(Abridged) The small-scale dynamo may play a substantial role in magnetizing the Universe under a large range of conditions, including subsonic turbulence at low Mach numbers, highly supersonic turbulence at high Mach numbers and a large range of magnetic Prandtl numbers Pm, i.e. the ratio of kinetic viscosity to magnetic resistivity. Low Mach numbers may in particular lead to the well-known, incompressible Kolmogorov turbulence, while for high Mach numbers, we are in the highly compressible regime, thus close to Burgers turbulence. In this study, we explore whether in this large range of conditions, a universal behavior can be expected. Our starting point are previous investigations in the kinematic regime. Here, analytic studies based on the Kazantsev model have shown that the behavior of the dynamo depends significantly on Pm and the type of turbulence, and numerical simulations indicate a strong dependence of the growth rate on the Mach number of the flow. Once the magnetic field saturates on the current amplification scale, backreactions occur and the growth is shifted to the next-larger scale. We employ a Fokker-Planck model to calculate the magnetic field amplification during the non-linear regime, and find a resulting power-law growth that depends on the type of turbulence invoked. For Kolmogorov turbulence, we confirm previous results suggesting a linear growth of magnetic energy. For more general turbulent spectra, where the turbulent velocity v_t scales with the characteristic length scale as u_\ell\propto \ell^{\vartheta}, we find that the magnetic energy grows as (t/T_{ed})^{2\vartheta/(1-\vartheta)}, with t the time-coordinate and T_{ed} the eddy-turnover time on the forcing scale of turbulence. For Burgers turbulence, \vartheta=1/2, a quadratic rather than linear growth may thus be expected, and a larger timescale until saturation is reached.


I. INTRODUCTION
The small-scale dynamo has been suggested to operate under a large range of different conditions, including the lence [18], assuming a scaling relation u ℓ ∝ ℓ 1/3 between turbulent velocity u ℓ and length scale ℓ. For Kolmogorov turbulence, it was previously concluded that the magnetic energy grows exponentially in the kinematic regime [e.g. [19][20][21][22] and linearly once the backreactions from the magnetic field become important (e.g. [23][24][25]). The latter was interpreted by Beresnyak [25] as evidence for universality of the small-scale dynamo, suggesting that a fixed fraction of the global turbulence dissipation rate is converted into magnetic energy.
However, observations of turbulence in molecular clouds [e.g. 26,27] and numerical simulations of supersonic turbulence [e.g. [28][29][30][31] often reveal steeper turbulent spectra, typically inbetween the incompressible Kolmogorov turbulence and the highly compressible Burgers turbulence [32]. So far, only a small amount of studies have investigated the turbulent dynamo in this regime. For instance, Haugen et al. [33] provided the first study exploring the dependence of the dynamo on the Mach number in simulations of driven turbulence, and Balsara et al. [34,35], Balsara and Kim [36] explored the amplification of magnetic fields in turbulence produced from strong supernova shocks. The first systematic study covering turbulent Mach numbers from 0.02 to 20 and different types of turbulence driving has been pursued by Federrath et al. [37], while the effect of a large range of different turbulence spectra has been explored by Schober et al. [22] based on the Kazantsev model [19].
We note that the small-scale dynamo has also been studied in the context of so-called shell models [38][39][40][41]. The latter originate from shell models of hydrodynamical turbulence, which originally considered turbulence in 2D [42][43][44], but were extended to 3D once a description of kinetic helicity was obtained [45]. The first 2D MHD shell model has been derived by Frik [46], while 3D models have been developed by Brandenburg et al. [47], Basu et al. [48], Frick and Sokoloff [49]. More sophisticated processes such as non-local interactions [50,51], anisotropies [52] and the Hall effect [53] have been incorporated in more recent studies. These approaches allow to study both the evolution of the power spectrum as well as the saturated regime, and are highly complementary to the methods presented here.
In the following, we will consider the small-scale dynamo in the kinematic and non-linear regime, and present evidence from existing and new calculations suggesting a strong dependence on the magnetic Prandtl number, as well as the Mach number of the flow. In section 2, we summarize the evidence and indications for a nonuniversal behavior in the kinematic regime, which has been derived in previous studies. In section 3, we present the first exploration concerning different types of turbulence during the non-linear phase of the dynamo, where the backreaction of the magnetic field becomes important. We show that a linear growth is only obtained in the case of Kolmogorov turbulence, while steeper power laws result from turbulent spectra with ϑ > 1/3. We discuss the physical implications in section 4, and sum-marize our main results in section 5.

II. NON-UNIVERSALITY IN THE KINEMATIC REGIME
The small-scale dynamo is well-studied in the kinematic regime, where an exponential growth of the magnetic field is expected on the viscous scale. The growth rate of the magnetic field can be calculated in the framework of the Kazantsev model, assuming homogeneous turbulence that is δ-correlated in time, or with 3-dimensional magneto-hydrodynamical simulations. In this section, we discuss hints and evidence for a nonuniversal behavior in the kinematic regime.
A. Indications for non-universality in the Kazantsev model The amplification of magnetic fields is governed by the induction equation, which is given as We assume in the following B = 0, although we note that scenarios considering B = 0 have been recently explored by Boldyrev et al. [54], Malyshkin and Boldyrev [55,56,57]. In the Kazantsev model, the velocity field and the magnetic field are decomposed into a mean field, denoted with brackets , and a fluctuating component denoted with δ: A central input is the correlation function of the turbulent velocity, which is δ-correlated in time and (in the absence of helicity) can be decomposed as with r = | r 1 − r 2 | and T N , T L are the transverse and longitudinal parts of the correlation function, respectively [58]. The same definitions can be applied to the magnetic field, yielding a two-point correlation function M ij (r, t) with transverse and longitudinal components M N (r, t) and M L (r, t). Unlike the velocity field, the magnetic field is always divergence-free, leading to the additional constraint As the Kazantsev model assumes that the flow is δcorrelated in time, concepts such as viscosity or the magnetic Prandtl number cannot be directly incorporated into the flow, as the turbulent velocity field is destroyed and regenerated at each instant, leaving no time for viscosity to act. However, it can be indirectly included by adopting turbulent velocity spectra that become steeper below a given viscous scale ℓ ν . This is the approach employed here. For a given relation of type u ℓ ∝ ℓ ϑ (5) in the inertial range, the longitudinal correlation function of turbulence can be parametrized as [22] T L (r) = with ℓ ν the viscous scale, L the driving scale of turbulence, V the turbulent velocity on scale L, Re = V L/ν the Reynolds number of the gas and ν the kinetic viscosity. Similarly, we have with θ(ϑ) = (21 − 38ϑ)/5. As we expect an exponential growth of the magnetic energy as a function of time, we make the following ansatz for the kinematic regime: Inserting (8) in the induction equation (1), one obtains the Kazantsev equation, which is of the same form as the quantum-mechanical Schrödinger equation: In this framework, the amplification depends on the effective potential U (r) in Eq. (9), which depends on the properties of turbulence via As recently shown by Schober et al. [22], this form of the potential also accounts for the effect of compressibility by keeping terms related to ∇· v during the derivation. The equation can be solved using the WKB approximation in the limit of Pm → ∞ [19][20][21][22]. For Kolmogorov turbulence, one obtains In a recent study, analytical solutions based on the WKB approximation have been derived in the limit Pm ≪ 1 by Schober et al. [59]. For Kolmogorov, they yield with Rm = V L/η the magnetic Reynolds number, and η the magnetic diffusivity. We thus observe a fundamental difference between the limiting cases Pm ≪ 1 and Pm ≫ 1 in the kinematic regime: For Pm ≫ 1, magnetic field amplification occurs predominantly on the viscous scale, corresponding to the most negative range of the potential. For Pm ≪ 1, on the other hand, the resistive scale becomes larger than the visous scale. Amplification on the viscous scale is thus not possible, and the strongest contribution is close to the resistive scale due to the short eddy-times. Correspondingly, the growth rate of the magnetic field depends on the Reynolds number Re for Pm ≫ 1, and on the magnetic Reynolds number Rm for Pm ≪ 1 [see also 60].
The results can be generalized further for different types of turbulence. In the limit Pm ≫ 1, one obtains [22] In the regime Pm ≪ 1, one finds a similar relation [59], with the prefactor α defined through the quantities A numerical evaluation shows that these coefficients are smaller by about two orders of magnitude in the limit Pm ≪ 1, assuming the same type of turbulence. This can be expected, as the amplification then occurs on larger scales, with larger eddy-turnover times. Similarly, also the type of turbulence reflected in the parameter ϑ may change the amplification rate by about an order of magnitude, in case of the same value of Rm. The most efficient amplification rate occurs for Kolmogorov turbulence, ϑ = 1/3, while it is less efficient for highly compressible Burgers turbulence, ϑ = 1/2, for which the turbulent velocities decrease more rapidly with length scale.
The Kazantsev model thus indicates that the behavior of the dynamo depends both on the type of turbulence and the magnetic Prandtl number. A potential restriction of the underlying model is the assumption of δ-correlated turbulence, although the characteristic timescales are certainly small compared to the dynamical time. To investigate the resulting uncertainties, we refer the reader to Schekochihin and Kulsrud [61]. The main results from these considerations are thus the following: • The behavior of the small-scale dynamo depends sensitively on the value of Pm, and in particular whether Pm ≪ 1 or Pm ≫ 1. We note that there is a continuous transition at Pm ∼ 1, as detailed by Bovino et al. [62].
• The adopted type of turbulence has a significant influence on the efficiency of magnetic field amplification, as turbulent spectra with ϑ > 1/3 correspond to larger eddy-turnover times and smaller amplification rates.

B. Results from numerical simulations
Due to the numerical viscosity and resistivity, it is difficult to perform magneto-hydrodynamical simulations with Pm significantly different from 1. However, a limited range of Pm has nevertheless been explored. For instance, Haugen et al. [63] investigated magnetic Prandtl numbers between 0.1 and 30. For Pm < 1, they report that the miminum magnetic Reynolds number required for dynamo action, Rm c , scales as We note that the factor π in the above is due to their definition of the magnetic Reynolds number. They further report differences in the obtained power spectra, indicating a steeper decrease on small scales for small values of Pm.
Schekochihin et al. [64] and Iskakov et al. [65] report numerical simulations exploring the small-scale dynamo from Pm∼ 0.017 up to Pm = 1. In this regime, they find that even for constant values of Rm, the growth rate decreases with decreasing Pm. In particular, for Pm ∼ 1 and Rm ∼ 830, they report a normalized growth rate of 1.8, which decreases to 0.9 for Pm ∼ 0.2 and the same magnetic Reynolds number.
The simulations further indicate that the value of Rm c settles to a constant limit for Pm ≪ 1, Re ≫ 1 and Rm ≫ 1, even though this case is hard to numerically explore. The scaling of the growth rate on Rm, on the other hand, has not been conclusively explored.
All in all, simulations thus show that the growth rate depends on the magnetic Prandtl number even in the range Pm 1. Another quantity which was shown to influence the dynamo is the Mach number of the gas. Haugen et al. [33] explored Mach numbers in the range of 0.1 − 2.1 and reported a clear dependence of the critical magnetic Reynolds number for dynamo action on the Mach number M. For Pm ∼ 5, they report Rm c ∼ 25π for M < 1 and a rapid increase to Rm c ∼ 45π for M > 1. A similar behavior was found for Pm ∼ 1, with critical values of ∼ 40π and ∼ 80π, respectively.
A larger series of simulations has been reported by Federrath et al. [37], exploring Mach numbers from 0.02 up to 20, with compressive and solenoidal forcing, respectively. They performed a fit to the growth rate and saturation levels as a function of Mach number, using the function The fit coefficients for the different cases are given in Table I, and the normalized growth rates are given in Fig. 1. In the subsonic regime, their results indicate that the growth rate (normalized by the eddy-turnover time T ed on the forcing scale) strongly decreases with decreasing Mach number for compressive driving, while it is almost constant at solenoidal driving. At M > 1, there is an initial drop due to the appearance of shocks, but increases as M 1/3 at larger values. A similar dependence is reported on the saturation level, which is particularly high for solenoidal driving, and decreases in the regime of large, supersonic Mach numbers. We thus summarize the results from numerical simulations as follows: • The critical magnetic Reynolds number for dynamo action as well as the resulting spectra for the magnetic field depend on the magnetic Prandtl number.
• Both the growth rates and the saturation levels of the dynamo depend significantly on the turbulent Mach number and the type of forcing that is employed.

III. NON-UNIVERSALITY IN THE NON-LINEAR REGIME
The exponential growth phase will come to an end when the tension force of the magnetic field, B · ∇ B, becomes comparable to the inertial term of the flow, u · ∇ u. At this point, magnetic field amplification will stop on the scales that fulfill this condition, and continue to proceed on larger scales. As discussed by Schekochihin et al. [23], this condition translates to where ℓ a denotes the smallest scale where amplification still occurs. In this regime, a linear growth of the magnetic energy has been reported in previous studies, based on the assumption of Kolmogorov turbulence [e.g. [23][24][25].
In the following, we will generalize these investigations by employing a simplified toy model as well as a more sophisticated Fokker-Planck model previously suggested by Schekochihin et al. [23]. As a result, we will show that different types of power-law growth can be expected depending on the adopted type of turbulence.
We further point out that in the non-linear regime, we expect the magnetic Prandtl number to play a less critical role, as the amplification scale of the magnetic field is now expected to be larger than both the viscous and the resistive scale, such that no strong dependence on Re or Rm can be expected.
We note that the models considered in this section have previously been motivated in the context of the incompressible induction equation, given as However, they can be naturally extended into the compressible regime with the replacement Inserting this replacement as well as the continuity equation,ρ it is straightforward to show that one obtains the compressible form of the induction equation, equivalent to Eq. (1). As long as the mean density ρ in the box is constant, a significant growth of the quantity B/ρ nevertheless implies a corresponding growth of the magnetic energy, assuming that the density distribution function will not change significantly over time.
In the case of well-developed driven turbulence, one indeed expects a characteristic log-normal density probability distribution function, which naturally complies with these requirements [66][67][68]. Strictly speaking, the following considerations apply to the quantityB = B/ρ andW = W/ρ 2 , with W the magnetic energy. In the following, the˜is however dropped for simplicity.

A. First considerations based on a toy model
In the toy model previously proposed by Schekochihin et al. [23], the dominant fraction of the magnetic energy resides on the scale ℓ a , the smallest scale where magnetic field amplification still occurs (thus yielding the shortest amplification timescale). On that scale, the magnetic energy is expected to be already close to saturation. The magnetic energy W (t) can thus be related to the amplification scale ℓ a by the approximate relation The magnetic energy is evaluated here at the mean density ρ of the turbulent box, as we are interested only in the magnetic field amplification by shear. Adopting the eddy-turnover rate on the scale ℓ a as the growth rate for the magnetic field, i.e.
the magnetic energy evolves as with and Now, we have Γ(t)W (t) ∼ ρ u 3 ℓa(t) /ℓ a (t) =: ǫ(t). Inserting in Eq. (28) yields where χ is a constant of order unity. For Kolmogorov turbulence, the quantity ǫ(t) = ρ u 3 ℓa(t) /ℓ a (t) is a constant [18]. In this case, and as long as magnetic energy dissipation is negligible, dW/dt = const, implying a phase of linear growth. In this limit, we obtain the result of Beresnyak [25], where a constant fraction of the turbulence dissipation rate is converted into magnetic energy.
In the general case with u ℓa ∝ ℓ ϑ a , ǫ(t) is however not constant, but varies as ℓ 3ϑ−1 a . In the case of Burgers turbulence, we thus obtain ǫ ∝ ℓ 0.5 a . In this case, the growth of the magnetic energy is no longer linear, as the turbulent energy dissipation rate is not independent of scale! For comparison, we note that the quantityǫ = ρe 3/2 SGS /ℓ, with e SGS the specific energy density of subgridscale turbulence, is practically independent of ℓ. It however has a weak dependence on the Mach number, and a strong dependence on the type of forcing [69]. As the density fluctuations will however not contribute to the shearing, we will adopt ǫ as the quantity of interest here.
To quantify the expected behavior, we need to solve Eq. (26) for ℓ a . For this purpose, we recall that u ℓa is related to the turbulence driving scale L and the velocity V on that scale via From (26), we thus obtain We can now evaluate (27) and (28), yielding (34) For Kolmogorov turbulence (ϑ = 1/3), we confirm that dW/dt = const, while in the more general case, this quantity will increase with increasing W . This can be intuitively understood, as the steep spectra for ϑ > 1/3 imply a more modest increase of the eddy-timescale with length scale, suggesting that the amplification rate remains larger when increasing the scale. We re-assess these results with the Fokker-Planck model below and explore the physical implications in more detail.

B. Implications of the Fokker-Planck model for universality
The starting point for our investigations is the Fokker-Planck model of Schekochihin et al. [23]. Here, the timeevolution of the magnetic-energy spectrum is given as with the diffusion coefficient D(k) = Γ(t)k 2 /5 and the drift velocity in k-space V (k) = 4Γ(t)k/5. We recall that the magnetic-energy spectrum M is related to the magnetic energy W via To describe the evolution in the nonlinear regime, Schekochihin et al. [23] postulated the following expressions: The constants c 1 and c 2 are of order unity, E(k) is the hydrodynamic energy spectrum neglecting the influence of the magnetic field, and the wave vector k s (t) is defined via Eq. (38). It corresponds to the smallest scale where amplification efficiently occurs. As input for the Fokker-Planck model, we require an energy spectrum of the turbulence. As before, we assume that the velocity in the inertial range scales as The hydrodynamic energy spectrum is then approximately given as with C t a constant which depends on the type of turbulence, k f and k ν the wave vectors describing the injection scale of turbulence and the viscous scale, respectively. The value of k ν is set to enforce the condition ǫ = 2ν ∞ 0 dkk 2 E(k). Unlikely in (6) and (7), we do not explicitly model the turbulent spectra in the viscous regime, as these no longer contribute during the nonlinear stage. With these input data, Eq. (38) can be evaluated as We further introduce the quantities We note that in the above expressions, the integral ∞ 0 dk corresponds to an integration from k f to k ν , as the turbulent energy is non-zero only in this regime (see 40).
Using these definitions, the wave vectors k ν , k s and k f can be expressed as Integrating Eq. (37) now yields the following: Substituting Eqs. (44)-(46) into (47) yields the expression Considering turbulence models between Kolmogorov and Burgers, we have 1/3 ≤ ϑ ≤ 1/2. We further assume that W (t) ≪ W 0 , implying that the magnetic field is far from saturation on the current amplification scale. In this case, we can neglect the second term in the square brackets. As we focus here on the non-linear regime, we can further neglect W ν compared to W (t), and obtain the expression (50) As in our toy model, the growth of the magnetic energy thus scales as For Kolmogorov turbulence, the growth is thus linear, while it grows faster than linear for ϑ > 1/3. Integrating Eq. (51), we obtain with (53) From this expression, we already see that the energy grows linearly in t for Kolmogorov, while it grows as t 2 for Burgers turbulence. For a physical interpretation, the normalization in terms of the eddy-turnover time T ed on the forcing scale is still required, which we perform below.

IV. PHYSICAL IMPLICATIONS
To explore the physical implications of the abovementioned results, we now perform a normalization in terms of the eddy-turnover time T ed on the forcing scale k −1 f . For this purpose, we note that the expression within the central brackets of Eq. (53) is identical to W ν k ν , and it is straightforward to show that Model and reference ϑ W ∝ ℓa ∝ Kolmogorov [18] 1/3 t 1 t 3/2 Intermittency of Kolmogorov turbulence [70] 0.35 t 1.077 t 1. 54 Driven supersonic MHD turbulence [28] 0.37 t 1.17 t 1.59 Observation in molecular clouds [26] 0.38 t 1.23 t 1.61 Solenoidal forcing of turbulence [31] 0.43 t 1.51 t 1.75 Compressive forcing of turbulence [31] 0.47 t 1.77 t 1.89 Observation in molecular clouds [27] 0.47 t 1.77 t 1.89 Burgers turbulence [32] 1/2 t 2 t 2 If we normalize Eq. (52) in terms of T ed ∼ (k f √ W 0 ) −1 , we thus obtain Adopting a system of units with W 0 = 1 and k f = 1, it is evident that E f ∼ 1, v(k f ) ∼ 1 and thus T ed ∼ 1. From Eq. (40), we also expect ǫ ∼ 1. In these units, our evolution equations simplifies as We illustrate the behavior for the different types of turbulence in Fig. 2 for Re = 10 4 , and summarize the power-law behavior in Table II. The solution suggests that turbulence spectra closer to Kolmogorov saturate earlier (in terms of the eddy-turnover time on the forcing scale k f ), and initially start at a higher value. The latter is fully consistent with our expectations for the kinematic regime, where the growth rates are higher for Kolmogorov turbulence, and a larger amount of magnetic energy may build up before the non-linear regime is reached (due to the increased amount of turbulent energy that is available on the same scale). We note that in the final stage close to saturation, the evolution may start to deviate from the power-law behavior reported here, providing a transition to the regime where W (t) = const. From the relation derived above, we further calculate the characteristic scaling of the current amplification scale l s as a function of time t. Adopting Eq. (26), we have For Kolmogorov turbulence, the characteristic length scale of the magnetic field thus grows as t 3/2 , while it grows as t 2 for Burgers turbulence. The results are summarized for all types of turbulence in Table (II). The power-laws derived here depend on the type of turbulence due to the different eddy-turnover timescales as a function of scale, as we sketch in Fig. (3). We summarize the main ingredients based on the toy model developed in section III A: Considering a driving scale L with a turbulence velocity V on that scale, the ratio of the eddy-turnover times on scale l ≪ L for Kolmogorov and Burgers turbulence is given as During the growth of the magnetic energy, the relevant length scale however shifts to larger scales. According to Eq. (58), the ratio of the eddy timescales approaches unity for ℓ → L. For Burgers turbulence, the magnetic field amplification is thus initially delayed with respect to Kolmogorov, and catches up later, resulting into the non-linear behavior and the power-law growth described here. Due to these results, it is clear that the growth rate of the dynamo is not a fixed fraction of the global turbulence dissipation rate, as previously proposed by Beresnyak [25]. Due to the dependence on the turbulent spectrum, such a consideration may only hold locally, i.e. on a given scale, where the growth rate of the field is indeed related to the local eddy timescale. From a more global perspective, however, the turbulence dissipation rate changes as a function of scale for models different from Kolmogorov, such that the previously postulated universal behavior cannot be expected. From Eq. (56), it is further evident that the evolution depends on the Reynolds number of the gas, and that larger Reynolds numbers imply stronger magnetic fields at earlier times. While the turbulent energy is considerably smaller for Burgers spectra (ϑ = 1/2) on small scales, it approaches the values for Kolmogorov turbulence (ϑ = 1/3) on larger scales. As a result, the magnetic energy grows faster than linear for Burgers turbulence, as the growth rates gradually approach the Kolmogorov values at later times.

V. DISCUSSION AND CONCLUSIONS
In this paper, we have explored both the kinematic regime of the small-scale dynamo, where an exponential growth of the magnetic energy is generally observed, and the non-linear regime, where backreactions start occuring on small scales and shift the amplification scale of the magnetic field to larger scales.
In the kinematic regime, analytical studies based on the Kazantsev model suggest a fundamental dependence on the magnetic Prandtl number. In particular, for Pm ≪ 1, the growth rate of the dynamo is a function of the magnetic Reynolds number Rm, while for Pm ≫ 1, it depends on the kinematic Reynolds number Re. In addition, the amplification rates significantly depend on the adopted type of turbulence. For Pm ≫ 1, it scales as Re 1/2 for Kolmogorov turbulence and as Re 1/3 for Burgers turbulence. The same scaling relations, with a different normalization, were found for Pm ≪ 1, with the replacement Re→Rm.
Numerical simulations confirm the dependence on Pm also in the range Pm ∼ 1, and find a strong dependence of the growth rate and the saturation level on the turbulent Mach number M and the type of turbulence forcing. Magnetic field amplification is particularly efficient for solenoidal forcing and low Mach numbers, but also occurs for high Mach numbers and solenoidal / compressive forcing. If the Mach numbers are very small, compressive forcing is hardly able to trigger magnetic field amplification, as the presence of density gradients are required for the production of solenoidal turbulence in this case.
To investigate the non-linear regime of the dynamo, we employed the Fokker-Planck model of Schekochihin et al. [23] and explored the effect of different turbulent spectra on the magnetic field amplification rate. We find that the previously known linear growth only occurs for Kolmogorov turbulence, while in the general case with u ℓ ∝ ℓ ϑ , we expect the magnetic energy to scale as t 2ϑ/(1−ϑ) . The energy growth is thus faster than linear, and may even become quadratic for Burgers turbulence (ϑ = 1/2). However, we note that the growth rate is initially smaller for Burgers turbulence, as the turbulent energy available for amplification is initially much smaller on small scales. While magnetic field amplification is shifted to larger scales, the difference in the turbulent energy decreases, implying the reported power-law behavior as a function of time.
We have further shown that also the scaling of the characteristic length scale ℓ a for magnetic field amplification depends on the turbulent slope. Specifically, we find a scaling as t 1/(1−ϑ) , corresponding to t 3/2 for Kolmogorov and t 2 for Burgers turbulence. The change of length scales proceeds thus in a fashion analogous to the inverse-cascade in case of helicity [e.g. 71,72]. The evolution of this quantity may thus provide another relevant diagnostic for a comparison with numerical simulations.
Due to the above considerations, we point out that the non-linear stage of the small-scale dynamo does not generally correspond to converting a fixed fraction of the turbulence dissipation rate into magnetic energy, as previously suggested by Beresnyak [25]. While their results agree with our model for the case of Kolmogorov turbulence (low Mach numbers), steeper power laws may occur in the highly compressible regime. Universality in the sense of a uniform behavior under all conditions can thus not be expected. Nevertheless, we note that there are still universal laws governing the behavior of the dynamo, which relate the growth of the magnetic energy to the eddy-turnover time on the current amplification scale. This quantity in general does depend on the Mach number and the type of turbulence involved, such that the breaking of universality is a result of the properties of different environments. We propose to explore such effects in further detail with numerical simulations to improve our understanding of such non-universal behavior.