Main

Astrophysical flows are considered to be susceptible to two types of dynamo instability. First, a large-scale 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 question1. 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 effects2, and their presence and astrophysical relevance is no longer strongly debated.

The presence of the other type of dynamo instability, namely the small-scale or fluctuation dynamo (SSD), however, remains controversial in solar and stellar physics. In an SSD-active 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 number3. In contrast to the LSD, excitation of an SSD requires markedly stronger turbulence1. Furthermore, it has been theorized that it becomes increasingly more difficult to excite an SSD at very low magnetic Prandtl number PrM (refs. 4,5,6,7,8,9,10), the ratio of kinematic viscosity ν and magnetic diffusivity η. In the Sun, PrM 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 near-surface solar convection typically operate at PrM ≈ 1 (refs. 12,13,14,15,16,17,18) and thus circumvent the issue of low-PrM 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 rotation19,20, interact with the LSD21,22,23,24,25 or contribute to coronal heating via enhanced photospheric Poynting flux26. 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 small-scale magnetic field on the surface of the Sun has contributions from the SSD or is solely due to the tangling of the large-scale magnetic field by the turbulent motions27,28,29,30,31,32. However, these studies show a slight preference of the small-scale fields to be cycle independent. SSDs at small PrM are also important for the interiors of planets and for liquid-metal experiments33.

Various numerical studies have reported increasing difficulties in exciting the SSD when decreasing PrM (refs. 6,10,34), confirming the theoretical predictions. However, current numerical models reach only PrM = 0.03 using explicit physical diffusion or slightly lower (estimated) PrM, relying on artificial hyperdiffusion7,8. To achieve even lower PrM, one needs to increase the grid resolution massively (see also ref. 35). Exciting the SSD requires a magnetic Reynolds number (ReM) typically larger than 100; hence, for example, PrM = 0.01 implies a fluid Reynolds number Re = 104, where \({{{\rm{Re}}}}={u}_{{{{\rm{rms}}}}}\ell /\nu\), with urms being the volume integrated root-mean-squared velocity, ℓ a characteristic scale of the velocity and ReM = PrMRe. In this Article, we take this path and lower PrM substantially using high-resolution simulations.

Results

We include simulations with resolutions of 2563 to 4,6083 grid points and Re = 46 to Re = 33,000. This allows us to explore the parameter space from PrM = 1 to PrM = 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 in-depth exploration of the effect of PrM, we omit large-scale 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 set-up 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 highest-resolution and -Reynolds-number cases. As might be anticipated for low-PrM turbulence, the flow exhibits much finer, fractal-like 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.

Fig. 1: Visualization of flow and SSD solution.
figure 1

Flow speed (left) and magnetic field strength (right) from a high-resolution SSD-active run with Re = 18,200 and PrM = 0.01 on the surface of the simulation box.

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Growth rates and critical magnetic Reynolds numbers

In Fig. 2, we visualize the growth rate λ as function of Re and ReM. We find positive growth rates for all sets of runs with constant PrM if ReM is large enough. λ increases always with increasing ReM 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 ReM values used, this maps roughly to a PrM interval from about 0.1 to 0.04.

Fig. 2: SSD growth rate as function of the fluid and magnetic Reynolds numbers (Re and ReM).
figure 2

The diamonds represent the results of this work and the triangles represent the results of ref. 10. The colour coding indicates the value of the normalized growth rate λτ with τ = 1/urmskf, a rough estimate for the turnover time. The dotted lines indicate constant magnetic Prandtl number PrM. The white circles indicate zero growth rate for certain PrM, obtained from fitting for the critical magnetic Reynolds number, as shown in Fig. 3; fitting errors are signified by yellow-black bars (Supplementary Section 5). The background colours, including the thin black line (zero growth), are assigned via linear interpolation of the simulation data. The green dashed line shows the power-law fit of the critical ReM for PrM ≤ 0.08, with power 0.125 (Fig. 3b).

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The growth rates for PrM = 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 × 103 (see the thin black line). Looking at \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) as a function of magnetic Prandtl number PrM, it first increases with decreasing PrM and then decreases for PrM < 0.05. Hence, an SSD is easier to excite here than for 0.05 < PrM < 0.1. We could even find a nearly marginal, positive growth rate for PrM = 0.003125. The decrease of λ at low PrM is an important result as the SSD was believed to be even harder4,9 or at least equally hard7,8 to excite when PrM was decreased further from previously investigated values. The growth rates agree qualitatively with the earlier work at low PrM (refs. 6,7,8).

For a more accurate determination of \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\), we next plot the growth rates for fixed PrM as a function of ReM (Fig. 3a). The data are consistent with \(\lambda \propto \ln ({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}/{{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}})\) as theoretically predicted36,37. Fitting accordingly, we are able to determine \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) as a function of PrM (Fig. 3b). This plot clearly shows that there are three distinct regions of dynamo excitation. When PrM decreases in the range 1 ≥ PrM ≥ 0.1 it becomes much harder to excite the SSD. In the range 0.1 ≥ PrM ≥ 0.04, excitation is most difficult with little variation of \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\). For PrM ≤ 0.04, it again becomes easier as PrM reduces. In refs. 7,8, the authors already found an indication of \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) to level-off with decreasing PrM, 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 < PrM < 0.1. However, at PrM = 0.005, the error bar allows to conclude that \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) is here lower than at PrM = 0.05. This again confirms our result that \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) is decreasing with PrM for very low PrM.

Fig. 3: Growth rate and critical Reynolds number.
figure 3

a, Normalized growth rate λτ as function of magnetic Reynolds number ReM for simulation sets with fixed magnetic Prandtl number PrM, indicated by different colours. Logarithmic functions \(\lambda \tau \propto \ln ({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}/{{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}})\) according to refs. 36,37 were fitted separately to the individual sets, as indicated by the coloured lines (see the dashed-dotted line for the mean slope). b, Critical magnetic Reynolds number \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) as function of PrM obtained from the fits in a. The error bars show the fitting error (Supplementary Section 5). The diamond indicates a run with growth rate λ ≈ 0; hence, its ReM represents \(\sim {{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) for the used PrM = 0.003125. The red dashed line is a power-law fit \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\propto {\Pr }_{{{{\rm{M}}}}}^{0.125}\), valid for PrM ≲ 0.08. The grey shaded area indicates the PrM interval where the dynamo is hardest to excite (\({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\gtrsim 150\)).

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For PrM ≤ 0.05, the decrease of \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) with PrM 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 solar-like stars would lead to \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\approx 40\) at PrM = 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 PrM. However, episodes of non-monotonic behaviour of \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) when approaching this limit cannot be ruled out.

The well-determined \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) dependency on PrM together with its error bars and the power-law 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 PrM 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 PrM = 0.05, probes the PrM interval of impeded dynamo action, while run H0005, with PrM = 0.005, is clearly outside it (see Supplementary Figs. 1 and 2 for spectra of other cases).

Fig. 4: Energy spectra.
figure 4

Kinetic (top) and magnetic (bottom) energy spectra for two exemplary runs with Re = 7,958 and PrM = 0.05 (left), and Re = 32,930 and PrM = 0.005 (right). In the middle row, the kinetic spectra are compensated by k5/3. Vertical lines indicate the forcing wavenumber kf (green solid), the wavenumber of the bottleneck’s peak kb (red solid) and its starting point kbs (red dotted), the viscous dissipation wavenumber kν (orange), the ohmic dissipation wavenumber \({k}_{\eta }={k}_{\nu }{\Pr }_{{{{\rm{M}}}}}^{3/4}\) (dark blue) and the characteristic magnetic wavenumber kM (light blue). All spectra are averaged over the kinematic phase whereupon each individual magnetic spectrum was normalized by its maximum, thus taking out the exponential growth.

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In all cases, the kinetic energy as a function of wavenumber k clearly follows a Kolmogorov cascade with Ekin ∝ k−5/3 in the inertial range. When compensating with k5/3, we find the well-known bottleneck effect38,39: a local increase in spectral energy, deviating from the power law, as found both in fluid experiments40,41,42 and numerical studies43,44. It has been postulated to be detrimental to SSD growth4,10. For the magnetic spectrum, however, yet clearly visible for only PrM ≤ 0.005, we find a power law following Emag ∝ k−3. A 3/2 slope at low wavenumbers as predicted by ref. 45 is seen only in the runs with PrM close to one, while for the intermediate and low-PrM runs, the positive-slope 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 PrM slightly below unity. However, the authors propose a tentative power of −1 given that the −3 slope is not yet clearly visible for their PrM values.

Analysing our simulations, we adopt the following strategy. For each spectrum, we determine the wavenumber of the bottleneck, kb, as the location of its maximum in the (smoothed) compensated spectrum, along with its starting point kbs < kb at the location with 75% of the maximum (Fig. 4, middle). We additionally calculate a characteristic magnetic wavenumber, defined as kM = ∫kEmag(k)kdk/∫kEmag(k)dk, which is often connected with the energy-carrying scale. Furthermore, we calculate the viscous dissipation wavenumber \({k}_{\nu }={({\epsilon }_{{{{\rm{K}}}}}/{\nu }^{3})}^{1/4}\) following Kolmogorov theory, where ϵK is the viscous dissipation rate 2νS2 with the traceless rate-of-strain 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 PrM.

We plot kb/kν and kbs/kν as functions of PrM in Fig. 5. As is expected, kb/kν, or the ratio of the viscous scale to the scale of the bottleneck, does not depend on PrM, as the bottleneck is a purely hydrodynamic phenomenon. The start of the bottleneck kbs should likewise not depend on PrM, but the low Re values for PrM = 1 to PrM = 0.1 lead to apparent thinner bottlenecks, hence an unsystematic weak dependency. The red shaded area between kb and kbs is the low-wavenumber 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 αb and Supplementary Section 1 for a discussion). We note that αb ≈ −1.3 … −1.5 and can thus deviate markedly from −5/3. Overplotting the kM/kν 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 low-wavenumber part of the bottleneck may indeed be responsible for enhancing \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) in the interval 0.04 ≤ PrM ≤ 0.1. Using this plot, we can now clearly explain the three regions of dynamo excitation. For 0.1 ≤ PrM ≤ 1 the low-wavenumber part of the bottleneck and the characteristic magnetic scale are completely decoupled. This makes the SSD easy to excite (region I). For 0.04 ≤ PrM ≤ 0.1, (grey, region II), the dynamo is hardest to excite because of the shallower slope of the kinetic spectra. In region III, where PrM ≤ 0.04 the low-wavenumber part of the bottleneck and the characteristic magnetic scale are again completely decoupled making the dynamo easier to excite.

Fig. 5: Relation of the characteristic magnetic wavenumber kM to the bottleneck.
figure 5

We show its peak kb and its starting point kbs in red, the characteristic magnetic wavenumber kM in light blue and the ohmic dissipation wavenumber kη in dark blue. The red shaded area between kb and kbs corresponds to the low-wavenumber part of the bottleneck where the turbulent flow is rougher than for a −5/3 power law. The Roman numbers indicate the three distinct regions of dynamo excitation. The region of the weakest growth (II) is over-plotted in grey. The characteristic magnetic wavenumber kM can be fitted by two power laws (black dotted lines): \({k}_{{{{\rm{M}}}}}/{k}_{\nu }\propto {\Pr }_{{{{\rm{M}}}}}^{0.54}\) for PrM ≥ 0.05 and \({k}_{{{{\rm{M}}}}}/{k}_{\nu }\propto {\Pr }_{{{{\rm{M}}}}}^{0.71}\) for PrM ≤ 0.05. All wavenumbers are normalized by the viscous one kν. We find that the dynamo is hardest to excite if kM lies within the low-wavenumber side of the bottleneck. Leaving this region towards lower or higher wavenumbers makes the dynamo easier to excite. The inset shows kM/kη as a function of PrM.

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Further, we find that the dependence of kM/kν on PrM also differs between the regions. In region I, kM/kν depends on PrM 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 kM 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 kM and kη in value and scaling. However, in region III, the scaling of kM comes very close to the 3/4 scaling of kη. This behaviour can be even better seen in the inset of Fig. 5, where the ratio kM/kη is 0.3 for PrM = 1 and tends towards unity for decreasing PrM, 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 solar-like stars: previous numerical studies, albeit at PrM ≈ 1, indicate that this influence concerns, for example, the angular momentum transport19,20 and the LSD21,22,23,24,25. Our kinematic study, however, only shows that a positive growth rate is possible at very low PrM, but not whether an SSD is able to generate dynamically important field strengths. As the ReM of the Sun and solar-like 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 PrM = 1 simulations15. However, numerical simulations with PrM down to 0.01 show a decrease of the saturation strength with decreasing PrM (ref. 46).

The results of our study are well in agreement with previous numerical studies considering partly overlapping PrM ranges6,7,8,10. Those studies found some discrepancies with the Kazantsev theory45 for low PrM, 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 wavenumbers7. We could extend this regime to even lower PrM and therefore study these discrepancies further. For PrM ≤ 0.005, we find that the magnetic spectrum shows a power-law scaling k−3, which is substantially steeper than the tentative k−1 one proposed in ref. 7 for 0.03 ≲ PrM ≲ 0.07 (but only for eighth-order 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 PrM is fundamentally different from that at PrM ≈ 1.

Second, we find that the growth rates near the onset follow an ln(ReM) dependence as predicted by refs. 36,37, and not a \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{1/2}\) one as would result from intertial-range-driven SSDs1,7. We do not observe a tendency of the growth rate to become independent of ReM at the highest PrM either, which could be an indication of an outer-scale driven SSD, as postulated by ref. 7. Furthermore, we find that the pre-factor 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 PrM and seems to be of general validity.

Third, we find that the measured characteristic magnetic wavenumber kM is always smaller than the estimated kη, and furthermore, kM does not always follow the theory-predicted scaling of \({k}_{\eta }\propto {\Pr }_{{{{\rm{M}}}}}^{3/4}\) with PrM. For region I, where PrM is close to 1, this discrepancy is up to a factor of three and the deviation from the expected PrM scaling is most pronounced here. These discrepancies have been associated with the numerical set-ups injecting energy at a forcing scale far larger than the dissipation scale, that is kf ≪ kη (ref. 1). Furthermore, our runs in region I also have relatively low Re and therefore numerical effects are not dismissible. In region III (low PrM), kM/kη is approaching the constant offset factor 0.75. Hence, the scaling of kM/kη with PrM gets close to the expected one. This result again indicates that the SSD at low PrM is different from that at PrM ≈ 1.

An increase of \({{{{\rm{Re}}}}}_{{{{\rm{M}}}}}^{{{{\rm{crit}}}}}\) with decreasing PrM followed by an asymptotic levelling-off for PrM ≪ 1 was expected in the light of theory and previous numerical studies. Instead, we found non-monotonic behaviour as function of PrM; we could relate it to the hydrodynamical phenomenon of the bottleneck. If the characteristic magnetic wavenumber lies in the positive-gradient 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 ≥ PrM ≥ 0.04). For higher or lower PrM, 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 flow1,10,43, which is expected to harden dynamo excitation based on theoretical predictions4,9 from kinematic Kazantsev theory45. In line with theory, the roughness-increasing part of the bottleneck appears decisive in our results, however, only when kM is used as a criterion. The usage of kη would in contrast suggest that the peak of the bottleneck is decisive10. Such interpretation appears incorrect, as the rough estimate of kη 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 set-up

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.

$$\frac{{{{\rm{D}}}}{{{\bf{u}}}}}{{{{\rm{D}}}}t}=-{c}_{{{{\rm{s}}}}}^{2}{\nabla}\ln \rho +{{{\bf{J}}}}\times {{{\bf{B}}}}/\rho +{\nabla}\cdot (2\rho \nu {\mathsf{S}})/\rho +{{{\bf{f}}}},$$
(1)
$$\frac{\partial {{{\bf{A}}}}}{\partial t}={{{\bf{u}}}}\times {{{\bf{B}}}}+\eta {\nabla}^{2}{{{\bf{A}}}},$$
(2)
$$\frac{{{{\rm{D}}}}\rho }{{{{\rm{D}}}}t}=-{\nabla}\cdot (\rho {{{\bf{u}}}}),$$
(3)

where u is the flow speed, cs 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/μ0 is the current density with magnetic vacuum permeability μ0, while ν and η are constant kinematic viscosity and magnetic diffusivity, respectively. The rate-of-strain tensor Sij = (ui,j + uj,i)/2 − δij∇ ⋅ u/3 is traceless, where δij denotes the Kronecker delta, and the Einstein notation convection applying to their indices i and j. The forcing function f provides random white-in-time non-helical 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 kf = 2k1 with k1 = 2π/L. Its amplitude is chosen such that the Mach number Ma = urms/cs is always around 0.082, where \({u}_{{{{\rm{rms}}}}}=\sqrt{{\langle {{{{\bf{u}}}}}^{2}\rangle }_{V}}\) is the volume and time-averaged root-mean-square 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/(urmskf )≈ 6/(k1cs). 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 kf as

$${{{\rm{Re}}}}={u}_{{{{\rm{rms}}}}}/\nu {k}_{{{{\rm{f}}}}},\quad {{{{\rm{Re}}}}}_{{{{\rm{M}}}}}={u}_{{{{\rm{rms}}}}}/\eta {k}_{{{{\rm{f}}}}}.$$
(4)

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

$${\int}_{k}{E}_{{{{\rm{kin}}}}}(k)\,{\mathrm{d}}k={u}_{{{{\rm{rms}}}}}^{2}\,{\langle \rho \rangle }_{V}/2,$$
(5)
$${\int}_{k}{E}_{{{{\rm{mag}}}}}(k)\,{\mathrm{d}}k={B}_{{{{\rm{rms}}}}}^{2}/2{\mu }_{0},$$
(6)

where \({B}_{{{{\rm{rms}}}}}=\sqrt{{\langle {{{{\boldsymbol{B}}}}}^{2}\rangle }_{V}}\) is the volume-averaged root-mean-square value and 〈ρ〉V is the volume-averaged density.

Our numerical set-up 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 PrM and Ma. Their chosen values best represent the weakly stratified layers within the bulk of the solar convection zone, where PrM ≪ 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 set-up.