Abstract
Magnetic fields on small scales are ubiquitous in the Universe. Although they can often be observed in detail, their generation mechanisms are not fully understood. One possibility is the socalled smallscale dynamo (SSD). Prevailing numerical evidence, however, appears to indicate that an SSD is unlikely to exist at very low magnetic Prandtl numbers (Pr_{M}) such as those that are present in the Sun and other cool stars. Here we have performed highresolution simulations of isothermal forced turbulence using the lowest Pr_{M} values achieved so far. Contrary to earlier findings, the SSD not only turns out to be possible for Pr_{M} down to 0.0031 but also becomes increasingly easier to excite for Pr_{M} below about 0.05. We relate this behaviour to the known hydrodynamic phenomenon referred to as the bottleneck effect. Extrapolating our results to solar values of Pr_{M} indicates that an SSD would be possible under such conditions.
Main
Astrophysical flows are considered to be susceptible to two types of dynamo instability. First, a largescale dynamo (LSD) is excited by flows exhibiting helicity, or more generally, lacking mirror symmetry, due to rotation, shear and/or stratification. It generates coherent, dynamically relevant magnetic fields on the global scales of the object in question^{1}. The characteristics of LSDs vary depending on the dominating generative effects, such as differential rotation in the case of the Sun. Convective turbulence provides both generative and dissipative effects^{2}, and their presence and astrophysical relevance is no longer strongly debated.
The presence of the other type of dynamo instability, namely the smallscale or fluctuation dynamo (SSD), however, remains controversial in solar and stellar physics. In an SSDactive system, the magnetic field is generated at scales comparable to or smaller than the characteristic scales of the turbulent flow, enabled by chaotic stretching of field lines at high magnetic Reynolds number^{3}. In contrast to the LSD, excitation of an SSD requires markedly stronger turbulence^{1}. Furthermore, it has been theorized that it becomes increasingly more difficult to excite an SSD at very low magnetic Prandtl number Pr_{M} (refs. ^{4,5,6,7,8,9,10}), the ratio of kinematic viscosity ν and magnetic diffusivity η. In the Sun, Pr_{M} can reach values as low as 10^{−6}–10^{−4} (ref. ^{11}), thus seriously repudiating whether an SSD can at all be present. Numerical models of SSDs in nearsurface solar convection typically operate at Pr_{M} ≈ 1 (refs. ^{12,13,14,15,16,17,18}) and thus circumvent the issue of lowPr_{M} dynamos.
A powerful SSD may potentially have a large impact on the dynamical processes in the Sun. It can, for example, influence the angular momentum transport and therefore the generation of differential rotation^{19,20}, interact with the LSD^{21,22,23,24,25} or contribute to coronal heating via enhanced photospheric Poynting flux^{26}. Hence, it is of great importance to clarify whether or not an SSD can exist in the Sun. Observationally, it is still debated whether the smallscale magnetic field on the surface of the Sun has contributions from the SSD or is solely due to the tangling of the largescale magnetic field by the turbulent motions^{27,28,29,30,31,32}. However, these studies show a slight preference of the smallscale fields to be cycle independent. SSDs at small Pr_{M} are also important for the interiors of planets and for liquidmetal experiments^{33}.
Various numerical studies have reported increasing difficulties in exciting the SSD when decreasing Pr_{M} (refs. ^{6,10,34}), confirming the theoretical predictions. However, current numerical models reach only Pr_{M} = 0.03 using explicit physical diffusion or slightly lower (estimated) Pr_{M}, relying on artificial hyperdiffusion^{7,8}. To achieve even lower Pr_{M}, one needs to increase the grid resolution massively (see also ref. ^{35}). Exciting the SSD requires a magnetic Reynolds number (Re_{M}) typically larger than 100; hence, for example, Pr_{M} = 0.01 implies a fluid Reynolds number Re = 10^{4}, where \({{{\rm{Re}}}}={u}_{{{{\rm{rms}}}}}\ell /\nu\), with u_{rms} being the volume integrated rootmeansquared velocity, ℓ a characteristic scale of the velocity and Re_{M} = Pr_{M}Re. In this Article, we take this path and lower Pr_{M} substantially using highresolution simulations.
Results
We include simulations with resolutions of 256^{3} to 4,608^{3} grid points and Re = 46 to Re = 33,000. This allows us to explore the parameter space from Pr_{M} = 1 to Pr_{M} = 0.0025, which is closer to the solar value than has been investigated in previous studies. For each run, we measure the growth rate λ of the magnetic field in its kinematic stage and determine whether or not an SSD is being excited.
To afford an indepth exploration of the effect of Pr_{M}, we omit largescale effects such as stratification, rotation and shear. We avoid the excessive integration times, required to simulate convection, by driving the turbulent flow explicitly under isothermal conditions. Our simulation setup consists of a fully periodic box with a random volume force (see Methods for details); the flow exhibits a Mach number of around 0.08. In Fig. 1, we visualize the velocity and magnetic fields of one of the highestresolution and Reynoldsnumber cases. As might be anticipated for lowPr_{M} turbulence, the flow exhibits much finer, fractallike structures than the magnetic field. Note that all our results refer to the kinematic stage of the SSD, where the magnetic field strength is far too weak to influence the flow but otherwise arbitrary.
Growth rates and critical magnetic Reynolds numbers
In Fig. 2, we visualize the growth rate λ as function of Re and Re_{M}. We find positive growth rates for all sets of runs with constant Pr_{M} if Re_{M} is large enough. λ increases always with increasing Re_{M} as expected. Surprisingly, the growth rates are distinctly lower within the interval from Re = 2,000 to Re = 10,000 than below and above. With the Re_{M} values used, this maps roughly to a Pr_{M} interval from about 0.1 to 0.04.
The growth rates for Pr_{M} = 0.1 match very well the ones from ref. ^{10}, indicated by triangles in Fig. 2. From Fig. 2, we clearly see that the critical magnetic Reynolds number \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\), defined by growth rate λ = 0, first rises as a function of Re and then falls for Re > 3 × 10^{3} (see the thin black line). Looking at \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) as a function of magnetic Prandtl number Pr_{M}, it first increases with decreasing Pr_{M} and then decreases for Pr_{M} < 0.05. Hence, an SSD is easier to excite here than for 0.05 < Pr_{M} < 0.1. We could even find a nearly marginal, positive growth rate for Pr_{M} = 0.003125. The decrease of λ at low Pr_{M} is an important result as the SSD was believed to be even harder^{4,9} or at least equally hard^{7,8} to excite when Pr_{M} was decreased further from previously investigated values. The growth rates agree qualitatively with the earlier work at low Pr_{M} (refs. ^{6,7,8}).
For a more accurate determination of \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\), we next plot the growth rates for fixed Pr_{M} as a function of Re_{M} (Fig. 3a). The data are consistent with \(\lambda \propto \ln ({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}/{{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}})\) as theoretically predicted^{36,37}. Fitting accordingly, we are able to determine \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) as a function of Pr_{M} (Fig. 3b). This plot clearly shows that there are three distinct regions of dynamo excitation. When Pr_{M} decreases in the range 1 ≥ Pr_{M} ≥ 0.1 it becomes much harder to excite the SSD. In the range 0.1 ≥ Pr_{M} ≥ 0.04, excitation is most difficult with little variation of \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\). For Pr_{M} ≤ 0.04, it again becomes easier as Pr_{M} reduces. In refs. ^{7,8}, the authors already found an indication of \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) to leveloff with decreasing Pr_{M}, however, only when using artificial hyperdiffusion. Similarly, with our error bars, a constant \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) cannot be excluded for 0.01 < Pr_{M} < 0.1. However, at Pr_{M} = 0.005, the error bar allows to conclude that \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) is here lower than at Pr_{M} = 0.05. This again confirms our result that \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) is decreasing with Pr_{M} for very low Pr_{M}.
For Pr_{M} ≤ 0.05, the decrease of \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) with Pr_{M} can be well represented by the power law \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\propto {\Pr }_{{{{\rm{M}}}}}^{0.125}\). Extrapolating this to the Sun and solarlike stars would lead to \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\approx 40\) at Pr_{M} = 10^{−6}, which means that we could expect an SSD to be present. For increasing Re, by decreasing ν, it would be reasonable to assert that the statistical properties of the flow and hence \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) become independent of Pr_{M}. However, episodes of nonmonotonic behaviour of \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) when approaching this limit cannot be ruled out.
The welldetermined \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) dependency on Pr_{M} together with its error bars and the powerlaw fit have been added to Fig. 2, and agree very well with the thin black line for λ = 0 interpolated from the growth rates.
Regions of dynamo excitation
Next we seek answers to the obvious question arising: why is the SSD harder to excite in a certain intermediate range of Pr_{M} and easier at lower and higher values? For this, we investigate the kinetic and magnetic energy spectra of a representative subset of the runs (Supplementary Table 2). We show in Fig. 4 the spectra of two exemplary cases: run F005, with Pr_{M} = 0.05, probes the Pr_{M} interval of impeded dynamo action, while run H0005, with Pr_{M} = 0.005, is clearly outside it (see Supplementary Figs. 1 and 2 for spectra of other cases).
In all cases, the kinetic energy as a function of wavenumber k clearly follows a Kolmogorov cascade with E_{kin} ∝ k^{−5/3} in the inertial range. When compensating with k^{5/3}, we find the wellknown bottleneck effect^{38,39}: a local increase in spectral energy, deviating from the power law, as found both in fluid experiments^{40,41,42} and numerical studies^{43,44}. It has been postulated to be detrimental to SSD growth^{4,10}. For the magnetic spectrum, however, yet clearly visible for only Pr_{M} ≤ 0.005, we find a power law following E_{mag} ∝ k^{−3}. A 3/2 slope at low wavenumbers as predicted by ref. ^{45} is seen only in the runs with Pr_{M} close to one, while for the intermediate and lowPr_{M} runs, the positiveslope part of the spectrum shrinks to cover only the lowest k values, and the steep negative slopes at high k values become prominent. A steep negative slope in the magnetic power spectra was also seen by ref. ^{7} for Pr_{M} slightly below unity. However, the authors propose a tentative power of −1 given that the −3 slope is not yet clearly visible for their Pr_{M} values.
Analysing our simulations, we adopt the following strategy. For each spectrum, we determine the wavenumber of the bottleneck, k_{b}, as the location of its maximum in the (smoothed) compensated spectrum, along with its starting point k_{bs} < k_{b} at the location with 75% of the maximum (Fig. 4, middle). We additionally calculate a characteristic magnetic wavenumber, defined as k_{M} = ∫_{k}E_{mag}(k)kdk/∫_{k}E_{mag}(k)dk, which is often connected with the energycarrying scale. Furthermore, we calculate the viscous dissipation wavenumber \({k}_{\nu }={({\epsilon }_{{{{\rm{K}}}}}/{\nu }^{3})}^{1/4}\) following Kolmogorov theory, where ϵ_{K} is the viscous dissipation rate 2νS^{2} with the traceless rateofstrain tensor of the flow, S. From the relations between these four wavenumbers (listed in Supplementary Table 2), we draw insights about the observed behaviour of \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) with respect to Pr_{M}.
We plot k_{b}/k_{ν} and k_{bs}/k_{ν} as functions of Pr_{M} in Fig. 5. As is expected, k_{b}/k_{ν}, or the ratio of the viscous scale to the scale of the bottleneck, does not depend on Pr_{M}, as the bottleneck is a purely hydrodynamic phenomenon. The start of the bottleneck k_{bs} should likewise not depend on Pr_{M}, but the low Re values for Pr_{M} = 1 to Pr_{M} = 0.1 lead to apparent thinner bottlenecks, hence an unsystematic weak dependency. The red shaded area between k_{b} and k_{bs} is the lowwavenumber part of the bottleneck where the slope of the spectrum is larger (less negative) than −5/3 (see Supplementary Table 2 for values of the modified slope α_{b} and Supplementary Section 1 for a discussion). We note that α_{b} ≈ −1.3 … −1.5 and can thus deviate markedly from −5/3. Overplotting the k_{M}/k_{ν} curve reveals that it intersects with the red shaded area exactly where the dynamo is hardest to excite (region II). This lets us conclude that the shallower slope of the lowwavenumber part of the bottleneck may indeed be responsible for enhancing \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) in the interval 0.04 ≤ Pr_{M} ≤ 0.1. Using this plot, we can now clearly explain the three regions of dynamo excitation. For 0.1 ≤ Pr_{M} ≤ 1 the lowwavenumber part of the bottleneck and the characteristic magnetic scale are completely decoupled. This makes the SSD easy to excite (region I). For 0.04 ≤ Pr_{M} ≤ 0.1, (grey, region II), the dynamo is hardest to excite because of the shallower slope of the kinetic spectra. In region III, where Pr_{M} ≤ 0.04 the lowwavenumber part of the bottleneck and the characteristic magnetic scale are again completely decoupled making the dynamo easier to excite.
Further, we find that the dependence of k_{M}/k_{ν} on Pr_{M} also differs between the regions. In region I, k_{M}/k_{ν} depends on Pr_{M} via \({k}_{{{{\rm{M}}}}}/{k}_{\nu }\propto {\Pr }_{{{{\rm{M}}}}}^{0.54}\) and in region II and III via \({k}_{{{{\rm{M}}}}}/{k}_{\nu }\propto {\Pr }_{{{{\rm{M}}}}}^{0.71}\). This becomes particularly interesting when comparing the characteristic magnetic wavenumber k_{M} with the ohmic dissipation wavenumber which is defined as \({k}_{\eta }={k}_{\nu }{\Pr }_{{{{\rm{M}}}}}^{3/4}\). In region I, we find a notable difference of k_{M} and k_{η} in value and scaling. However, in region III, the scaling of k_{M} comes very close to the 3/4 scaling of k_{η}. This behaviour can be even better seen in the inset of Fig. 5, where the ratio k_{M}/k_{η} is 0.3 for Pr_{M} = 1 and tends towards unity for decreasing Pr_{M}, but is likely to saturate below 0.75.
Discussion
In conclusion, we find that the SSD is progressively easier to excite for magnetic Prandtl numbers below 0.04, in contrast to earlier findings, and thus is very likely to exist in the Sun and other cool stars. Provided saturation at sufficiently high levels, the SSD has been proposed to strongly influence the dynamics of solarlike stars: previous numerical studies, albeit at Pr_{M} ≈ 1, indicate that this influence concerns, for example, the angular momentum transport^{19,20} and the LSD^{21,22,23,24,25}. Our kinematic study, however, only shows that a positive growth rate is possible at very low Pr_{M}, but not whether an SSD is able to generate dynamically important field strengths. As the Re_{M} of the Sun and solarlike stars is several orders of magnitude higher than the extrapolated \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) value of 40, we yet expect dynamically important SSDs as indicated by Pr_{M} = 1 simulations^{15}. However, numerical simulations with Pr_{M} down to 0.01 show a decrease of the saturation strength with decreasing Pr_{M} (ref. ^{46}).
The results of our study are well in agreement with previous numerical studies considering partly overlapping Pr_{M} ranges^{6,7,8,10}. Those studies found some discrepancies with the Kazantsev theory^{45} for low Pr_{M}, for example, the narrowing down of the positive Kazantsev spectrum at low and intermediate wavenumbers, and the emergence of a negative slope instead at large wavenumbers^{7}. We could extend this regime to even lower Pr_{M} and therefore study these discrepancies further. For Pr_{M} ≤ 0.005, we find that the magnetic spectrum shows a powerlaw scaling k^{−3}, which is substantially steeper than the tentative k^{−1} one proposed in ref. ^{7} for 0.03 ≲ Pr_{M} ≲ 0.07 (but only for eighthorder hyperdiffusivity). This finding of such a steep power law in the magnetic spectrum challenges the current theoretical predictions and might indicate that the SSD operating at low Pr_{M} is fundamentally different from that at Pr_{M} ≈ 1.
Second, we find that the growth rates near the onset follow an ln(Re_{M}) dependence as predicted by refs. ^{36,37}, and not a \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{1/2}\) one as would result from intertialrangedriven SSDs^{1,7}. We do not observe a tendency of the growth rate to become independent of Re_{M} at the highest Pr_{M} either, which could be an indication of an outerscale driven SSD, as postulated by ref. ^{7}. Furthermore, we find that the prefactor of \(\gamma \propto \ln ({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}/{{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}})\) is nearly constant with its mean around 0.022, in agreement with 0.023 of ref. ^{10}. A constant value means that the logarithmic scaling is independent of Pr_{M} and seems to be of general validity.
Third, we find that the measured characteristic magnetic wavenumber k_{M} is always smaller than the estimated k_{η}, and furthermore, k_{M} does not always follow the theorypredicted scaling of \({k}_{\eta }\propto {\Pr }_{{{{\rm{M}}}}}^{3/4}\) with Pr_{M}. For region I, where Pr_{M} is close to 1, this discrepancy is up to a factor of three and the deviation from the expected Pr_{M} scaling is most pronounced here. These discrepancies have been associated with the numerical setups injecting energy at a forcing scale far larger than the dissipation scale, that is k_{f} ≪ k_{η} (ref. ^{1}). Furthermore, our runs in region I also have relatively low Re and therefore numerical effects are not dismissible. In region III (low Pr_{M}), k_{M}/k_{η} is approaching the constant offset factor 0.75. Hence, the scaling of k_{M}/k_{η} with Pr_{M} gets close to the expected one. This result again indicates that the SSD at low Pr_{M} is different from that at Pr_{M} ≈ 1.
An increase of \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) with decreasing Pr_{M} followed by an asymptotic levellingoff for Pr_{M} ≪ 1 was expected in the light of theory and previous numerical studies. Instead, we found nonmonotonic behaviour as function of Pr_{M}; we could relate it to the hydrodynamical phenomenon of the bottleneck. If the characteristic magnetic wavenumber lies in the positivegradient part of the compensated spectrum, where the spectral slope is markedly reduced from −5/3 to about −1.4, the dynamo is hardest to excite (0.1 ≥ Pr_{M} ≥ 0.04). For higher or lower Pr_{M}, the dynamo becomes increasingly easier to excite. The local change in slope due to the bottleneck has often been related to an increase of the ‘roughness’ of the flow^{1,10,43}, which is expected to harden dynamo excitation based on theoretical predictions^{4,9} from kinematic Kazantsev theory^{45}. In line with theory, the roughnessincreasing part of the bottleneck appears decisive in our results, however, only when k_{M} is used as a criterion. The usage of k_{η} would in contrast suggest that the peak of the bottleneck is decisive^{10}. Such interpretation appears incorrect, as the rough estimate of k_{η} employed here does not represent the magnetic spectrum adequately and the peak of the bottleneck does not coincide with the maximum of ‘roughness’.
Methods
Numerical setup
For our simulations, we use a cubic Cartesian box with edge length L and solve the isothermal magnetohydrodynamic equations without gravity, similar to refs. ^{5,47}.
where u is the flow speed, c_{s} is the sound speed, ρ is the mass density, B = ∇ × A is the magnetic field with A being the vector potential and ∇ is the gradient vector. J = ∇ × B/μ_{0} is the current density with magnetic vacuum permeability μ_{0}, while ν and η are constant kinematic viscosity and magnetic diffusivity, respectively. The rateofstrain tensor S_{ij} = (u_{i,j} + u_{j,i})/2 − δ_{ij}∇ ⋅ u/3 is traceless, where δ_{ij} denotes the Kronecker delta, and the Einstein notation convection applying to their indices i and j. The forcing function f provides random whiteintime nonhelical transversal plane waves, which are added in each time step to the momentum equation (see ref. ^{5} for details). The wavenumbers of the forcing lie in a narrow band around k_{f} = 2k_{1} with k_{1} = 2π/L. Its amplitude is chosen such that the Mach number Ma = u_{rms}/c_{s} is always around 0.082, where \({u}_{{{{\rm{rms}}}}}=\sqrt{{\langle {{{{\bf{u}}}}}^{2}\rangle }_{V}}\) is the volume and timeaveraged rootmeansquare value. The Ma values of all runs are listed in Supplementary Table 1. To normalize the growth rate λ, we use an estimated turnover time τ = 1/(u_{rms}k_{f} )≈ 6/(k_{1}c_{s}). The boundary conditions are periodic for all quantities and we initialize the magnetic field with weak Gaussian noise.
Diffusion is controlled by the prescribed parameters ν and η. Accordingly, we define the fluid and magnetic Reynolds numbers with the forcing wavenumber k_{f} as
We performed numerical free decay experiments (Supplementary Section 7), from which we confirm that the numerical diffusivities are negligible.
The spectral kinetic and magnetic energy densities are defined via
where \({B}_{{{{\rm{rms}}}}}=\sqrt{{\langle {{{{\boldsymbol{B}}}}}^{2}\rangle }_{V}}\) is the volumeaveraged rootmeansquare value and 〈ρ〉_{V} is the volumeaveraged density.
Our numerical setup employs a markedly simplified model of turbulence compared with the actual one in the Sun. There, turbulence is driven by stratified rotating convection being of course neither isothermal nor isotropic. However, these simplifications were so far necessary when performing a parameter study at such high resolutions as we do. Nevertheless, we can connect our study to solar parameters in terms of Pr_{M} and Ma. Their chosen values best represent the weakly stratified layers within the bulk of the solar convection zone, where Pr_{M} ≪ 1 and Ma ≪ 1. The anisotropy in the flow on small scales is much weaker there than near the surface and therefore close to our simplified setup.
Data availability
Data for reproducing Figs. 2, 3 and 5 are included in the article and its Supplementary Information files. The raw data (time series, spectra, slices and snapshots) are provided through IDA/Fairdata service hosted at CSC, Finland, under https://doi.org/10.23729/206af66907fd4a309968b4ded5003014. From the raw data, Figs. 1 and 4 can be reproduced.
Code availability
We use the Pencil Code^{48} to perform all simulations, with parallelized fastFouriertransforms to calculate the spectra on the fly^{49}. Pencil Code is freely available at https://github.com/pencilcode/.
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Acknowledgements
We acknowledge fruitful discussions with A. Brandenburg, I. Rogachevskii, A. Schekochihin and J. Schober during the Nordita programme on ‘Magnetic field evolution in low density or strongly stratified plasmas’. Computing resources from CSC during the Mahti pilot project and from Max Planck Computing and Data Facility (MPCDF) are gratefully acknowledged. This project, including all authors, has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Project UniSDyn, grant agreement number 818665). This work was done in collaboration with the COFFIES DRIVE Science Center.
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J.W. led and all the authors contributed to the design and performing the numerical simulations. J.W. led the data analysis. M.J.K.L. was in charge of acquiring the computational resources from CSC. All the authors contributed to the interpretation of the results and writing the paper.
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Supplementary Figs. 1–5, Tables 1–3, and discussions in Supplementary Sections 1–6 with references.
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Warnecke, J., KorpiLagg, M.J., Gent, F.A. et al. Numerical evidence for a smallscale dynamo approaching solar magnetic Prandtl numbers. Nat Astron 7, 662–668 (2023). https://doi.org/10.1038/s41550023019751
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DOI: https://doi.org/10.1038/s41550023019751
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