4.1. Structure formation

As magnetic energy is injected and transferred to larger scales, the total energy of the system tends to grow secularly (figure 3a,b). In compressible isothermal MHD, the total energy ${\mathcal {E}}^{\textrm {T}}$ is the sum of the kinetic ${\mathcal {E}}^K=\frac {1}{2}\rho |{\boldsymbol {v}}^2|$, magnetic ${\mathcal {E}}^M=\frac {1}{2}|{\boldsymbol {b}}|^2$ and potential energy ${\mathcal {E}}^\rho =\rho c_s^2 \ln (\rho /\rho _0)$, stored as density fluctuations. Since no large-scale energy removal occurs, the asymptotic state of the system would be a condensate of magnetic helicity at the largest scales. This would strongly affect the locality of transfers and the spectral exponents (Alexakis et al. Reference Alexakis, Mininni and Pouquet2006; Linkmann & Dallas Reference Linkmann and Dallas2016). For this reason, in order to avoid the strong modification of the inverse transfer range by finite-size effects, the spectra are taken at an instant when an energetically dominant largest-scale magnetic structure has not yet emerged. The restricted spectral bandwidth of the numerical experiments thus requires caution with regard to the assumed stationarity of the established asymptotic scaling interval at $20 \leq K \leq 30$ or $15 \leq K \leq 25$. Any large-scale damping of magnetic energy or a magnetic large-scale condensate of finite energy would have a finite spectral width of influence reducing the observable scaling region from the large-scale side as does the finite spectral support of the magnetic driving from the small-scale side. The chosen set-up, although certainly not ideal, represents a pragmatic compromise between unquestionable statistical stationarity and a perfectly developed scaling region. Nevertheless, spectral scaling laws consistent with results from previous research in incompressible MHD and natural extensions of the same to compressible MHD have been found. Furthermore, the main results are confirmed through higher resolution runs (§ 5) and the main measurements of spectral exponents are presented as functions in time displaying acceptable saturation to constant levels (cf. figures in §§ 4.2–4.4).

Figure 3.

(a,b) Time evolution of the kinetic, potential, magnetic and total energies for two representative M5s and M5cA runs. (c,d) Time evolution of the magnetic field–velocity alignment (normalised cross-helicity).

For all the runs, the magnetic field–velocity alignment,

(4.1)\begin{equation} \rho_{C}=\frac{\langle {\boldsymbol{v}} \cdot {\boldsymbol{b}} \rangle}{\sqrt{\langle |{\boldsymbol{v}}|^2 \rangle \langle |{\boldsymbol{b}}|^2 \rangle}}, \end{equation}

grows in time but remains very low (under 0.035, see figure 3c,d), so that the average alignment of magnetic and velocity field is small and the dynamics of the system are dominated by the direct cascades of kinetic and magnetic energies and the inverse transfer of magnetic helicity.

Figure 4(a) shows the total magnetic helicity in the system as a function of time for the M1s2, M11s and M8c runs, taken as representatives. The helicity is normalised by an estimate of its injection rate: ${\epsilon ^{{{\mathcal {H}}^M}}_{inj}}=2{\epsilon ^{M}_{inj}}/(K_Ht_{\mathcal {T}})$ with $K_H=50$ the shell around which the electromotive forcing takes place. Therefore, if all the injected magnetic helicity were conserved, the normalised magnetic helicity in the system should be 1 at $t/t_{\mathcal {T}}=1$. Indeed, at early times, the total magnetic helicity follows this ideal evolution (indicated through the orange dotted line). However, due to numerical non-idealities, magnetic helicity is dissipated already at the forcing scale, and even partially transferred to smaller scales, where it is dissipated more rapidly, giving an effective rate of increase less than unity. The rate of increase is around 40 % to 60 % for these runs, since at $t/ t_{\mathcal {T}} = 1$ approximately half of the injected magnetic helicity is present in the system. These values are consistent with the ones obtained by considering the magnetic helicity flux ${\Pi }^{{{\mathcal {H}}^M}}(K)=\sum _{K_0=1}^K {\mathcal {T}}^{{{\mathcal {H}}^M}}(K_0)$ with ${\mathcal {T}}^{{{\mathcal {H}}^M}}=2{\mathcal {Q}}({\boldsymbol {b}},{\boldsymbol {v}}\times {\boldsymbol {b}})/{\epsilon ^{{{\mathcal {H}}^M}}_{inj}}$ the magnetic helicity transfer function normalised by its injection rate (figure 4b–d). For these curves, an approximate plateau is visible at larger scales, showing that approximately 50 % to 70 % of the injected magnetic helicity experience the inverse transfer, the rest being transferred to smaller scales or dissipated at the forcing scale. This plateau hints at a roughly constant flux of magnetic helicity to larger scales, even though a visible deviation exists that is probably due to non-local transport (Alexakis et al. Reference Alexakis, Mininni and Pouquet2006; Linkmann & Dallas Reference Linkmann and Dallas2016; Teissier Reference Teissier2020) and the finite size of the system. Not all the magnetic helicity transferred to larger scales is conserved, giving, hence, a lower rate of increase, since dissipation still occurs, especially when magnetic helicity has not yet reached the largest scales. This is why at later times the curves are not linear but convex, as can be seen through the tangent of the M1s2 curve, which lies under it. Indeed, as magnetic helicity is transferred to ever larger scales, dissipative effects are smaller so that magnetic helicity is better conserved.

Figure 4.

(a) Evolution of the total magnetic helicity in the system, normalised by its estimated injection rate for some representative runs. The orange dotted line has a slope of 1, indicating the ideal evolution. The black dotted line is the tangent at the instant $t=0.4t_{\mathcal {T}}$ of the M1s2 run, showing that these curves are convex at later times. (b–d) Magnetic helicity fluxes as a function of time for these three runs, five curves regularly spaced in the interval $]t_i,t_f]$ with $t_i$ the instant at which the electromotive driving is switched on and $t_f$ an instant at which ${\mathcal {I}}_{{{\mathcal {H}}^M}} \approx L/6$ – see relation (4.2). The darker the curve, the further forward time has advanced. The dotted line indicates zero on the ordinate.

This inverse transfer of magnetic helicity in supersonic flows is illustrated in figure 5, where the time evolution of the magnetic helicity spectra $H^M$ is shown for the least and most compressible cases considered. Figure 6 shows the time evolution of the magnetic helicity integral scale, defined as

(4.2)\begin{equation} {\mathcal{I}}_{{{\mathcal{H}}^M}}=L\frac{\displaystyle\int_{K} K^{{-}1} H^M_K \,{\rm d} K}{\displaystyle\int_{K} H^M_K \,{\rm d} K}. \end{equation}

Magnetic helicity is not a positive-definite quantity. Even though division by zero is in theory possible, since only magnetic helicity of one sign is injected, positive magnetic helicity dominates the system at all scales so that division by zero does not occur and considering this integral scale remains meaningful. The spectrum of the positive helical magnetic field $H^M_+$ is approximately an order of magnitude larger than its negative part $H^M_-$ at all scales. The time is measured in units of the turbulent turnover time $t_{\mathcal {T}}=L/(2c_s{\mathcal {M}})$ of the hydrodynamic statistically stationary state of the flow, so that ${\mathcal {M}}$ designates the RMS Mach number before the injection of magnetic helicity. The inverse transfer occurs faster with increasing compressibility. While for the solenoidally driven runs, more than $1.5t_{\mathcal {T}}$ are required to reach the integral scale ${\mathcal {I}}_{{{\mathcal {H}}^M}}={L}/{6}$, less than approximately $0.6t_{\mathcal {T}}$ are needed for the compressively driven runs with ${\mathcal {M}} \gtrapprox 5$. For lower densities indeed, the Lorentz force has a greater impact on the velocity field's temporal evolution, since $\partial _t {\boldsymbol {v}} = \cdots + ({1}/{\rho }) (\boldsymbol {\nabla } \times {\boldsymbol {b}}) \times {\boldsymbol {b}}$, which backreacts on the magnetic field. At high compressibility, especially for compressively driven runs, large regions of low density are present (figure 1), which are associated with higher Alfvén velocities ${\boldsymbol {v_A}}={\boldsymbol {b}}/\sqrt {\rho }$ and hence faster dynamical time scales of magnetic field line reconfiguration $\propto l_{rec}/{\boldsymbol {v_A}}$, with $l_{rec}$ characterising the field-parallel extent of the magnetic reconnection region. Magnetic reconnection processes are required to achieve the topological changes of magnetic structures in the course of the inverse transport of magnetic helicity. Those changes are communicated along the affected field lines by Alfvén waves propagating at speed ${\boldsymbol {v_A}}$. This explains why for the M5cB run, whose initial condition presents a peak of the density p.d.f. at lower densities (figure 2d), the growth of the magnetic helicity integral scale is faster than for the M5cA run. This is also manifest in the time evolution of the magnetic helicity spectra: for the M8c run, in contrast to the M01s4, M11s and M1c runs, the magnetic helicity spectrum goes first to significantly larger scales before steepening its spectral slope (figure 5). This is also the reason of the choice of the turbulent turnover time of the mechanically forced velocity field as the unit of time in the plots. This quantity is constant by definition, contrary to the shorter magnetic inverse transfer time scale which evolves dynamically.

Figure 5.

Time evolution of the magnetic helicity spectra for the least and most compressible runs. The darker the curve, the later it is. The 10 curves are equally spaced in time and correspond to the time intervals $t\in [0.335t_{\mathcal {T}},3.35t_{\mathcal {T}}], [0.22t_{\mathcal {T}},2.2t_{\mathcal {T}}], [0.24t_{\mathcal {T}},2.4t_{\mathcal {T}}]$ and $[0.16t_{\mathcal {T}},1.6t_{\mathcal {T}}]$ for the M01s4, M11s, M1c and M8c runs, respectively.

Figure 6.

Magnetic helicity integral scale time evolution.

4.2. Scaling relations: magnetic quantities

In order to compare the numerical models more easily with one another, the Fourier spectra of the different set-ups are considered at their respective instant in time when a magnetic helicity integral scale ${\mathcal {I}}_{{{\mathcal {H}}^M}}\approx L /6$ is reached. This corresponds to $t/t_{\mathcal {T}}=2.13, 2.04, 1.73, 1.56, 1.38$ for the runs M01s4, M1s2, M5s, M7s, M11s, respectively, expressed in units of their respective large-eddy turbulent turnover time. For the compressively driven M1c, M3c, M5cA, M5cB, M8c runs, the respective instants are $t/t_{\mathcal {T}}=1.71, 0.95, 0.66, 0.42, 0.62$. This scale of $L/6$ is chosen in order to have a sufficiently large spectral distance between the magnetically forced and the largest scales, since a strong modification of the scaling range by the electromotive forcing at small scales and through large-scale condensation of magnetic helicity at the largest scales at late times due to the limited box size are expected. The magnetic-to-kinetic energy ratio ${\mathcal {P}}({\boldsymbol {b}})/{\mathcal {P}}(\sqrt {\rho }{\boldsymbol {v}})$ is of the order $10^{-2}$ at the largest scales $K<3$ due to the dominance of the kinetic forcing and the deliberate avoidance of a large-scale magnetic condensate. The spectral power-law exponent is determined by a logarithmic linear least squares fit (LSF) to the spectrum in a relevant domain, which depends on the run and the considered quantity. This domain is shown through vertical dashed lines in the figures.

As expected from the presence of $\boldsymbol {\nabla }\boldsymbol {\cdot } {\boldsymbol {v}}$ terms in the nonlinear evolution equations of magnetic helicity and magnetic energy (cf. (1.1)), the respective spectra display systematic changes when augmenting the Mach number of the flow. Those are more pronounced for compressively driven turbulence than for solenoidally driven flows. The low Mach number solenoidally driven M01s4 and M1s2 runs exhibit an approximate magnetic helicity power-law scaling $H^M \sim K^m$ with $m \approx -3.3$, gained from an LSF in the region $20 \leq K \leq 30$ (figure 7). This exponent is consistent with the observations in the incompressible case (Malapaka Reference Malapaka2009; Müller et al. Reference Müller, Malapaka and Busse2012). For higher RMS Mach numbers, increasing deviations are observed, namely $m\approx -3.2, -3.1$ and $-3.0$ for the M5s, M7s and M11s runs, respectively. Although significant, these deviations are clearly smaller than those obtained in the compressively forced runs. There, the spectra are clearly flatter at lower Mach numbers, see figure 7. For the M1c, M3c, M5cA and M8c runs, the exponents computed through an LSF in the region $15 \leq K \leq 25$ are $m \approx -3.2, -2.6, -2.2$ and $-2.3$, respectively. For supersonic compressively driven flows, the scaling exponent is very sensitive to the initial conditions: for example, the M5cB run, for which the magnetic helicity injection starts at a different instant during the same hydrodynamic statistically stationary state as the M5cA run, presents a significantly flatter spectrum with $m \approx -1.5$. This is linked with a peak of the initial condition's mass density p.d.f. at lower densities (figure 2d), so that faster time scales are expected for the M5cB run. This illustrates that the single realisations considered here could still be in a transient state. One would either need a significantly larger scale separation between the electromotive driving scale and the largest scales (which implies a very large resolution of the computational domain), or perform ensemble averaging over many realisations in order to obtain a less approximate dependency of the magnetic helicity exponent as a function of ${\mathcal {M}}$ when using a compressive forcing. These studies would imply significantly higher computational costs, which are beyond the resources available to the present investigation. Nevertheless, independent of possible transients, a clear tendency towards flatter exponents with increasing compressibility is observable, which can be largely compensated by an appropriate choice of variables, even with very different initial conditions, as shown below.

Figure 7.

(a,b) Time evolution of the magnetic helicity power-law exponents. (c,d) Magnetic helicity spectra at an instant when ${\mathcal {I}}_{{{\mathcal {H}}^M}}\approx {L}/{6}$ (shown through circles in panels (a,b)), compensated by $K^{3.3}$, since $H^M \sim K^{-3.3}$ in the incompressible case. The exponents measured in panels (a,b) are gained through an LSF in the region delimited with vertical dashed lines in panels (c,d) (here $20 \leq K \leq 30$ and $15 \leq K \leq 25$ for the solenoidally driven and compressively driven runs, respectively).

The spectra of the positive helical magnetic field $H^M_+$ are close to those of the total magnetic helicity $H^M=H^M_+-H^M_-$, which is expected since $H^M_+$ is approximately an order of magnitude greater than $H^M_-$ at all scales. After a time of the order of $t_{\mathcal {T}}$, the exponents measured are close to $-3.2, -3.2, -3.1, -3.0$ and $-2.9$ for the M01s4, M1s2, M5s, M7s and M11s runs, respectively, and close to $-3.0, -2.6, -2.1, -1.5$ and $-2.3$ for the M1c, M3c, M5cA, M5cB and M8c runs, respectively. The negative helical magnetic field $H^M_-$ scales as $-2.7, -2.5, -2.1, -2.2, -2.0$ (M01s4, M1s2, M5s, M7s, M11s runs) and $-2.4, -2.0, -1.7, -1.1, -1.7$ (M1c, M3c, M5cA, M5cB, M8c runs).

The same tendency is observed for the magnetic energy spectra: the scaling exponents are close to the $-2.1$ observed in the incompressible case (Malapaka Reference Malapaka2009; Müller et al. Reference Müller, Malapaka and Busse2012) for the low Mach number M01s4, M1s2 and M1c runs, flatter with increasing compressibility for the solenoidally driven runs and significantly flatter for the compressively driven runs already at lower initial RMS Mach numbers (curves not shown here).

In the incompressible case, the Alfvén velocity ${\boldsymbol {v_A}}={\boldsymbol {b}}/\sqrt {\rho }$ (with $\rho$ a constant) and the magnetic field are in essence the same quantity. However, in the compressible case, low density regions are associated with higher Alfvén velocities, so that the dynamical time scales are shorter there. For this reason, it is tempting to consider the power spectrum of the Alfvén velocity $E^A={\mathcal {P}}({\boldsymbol {v_A}})$ as well as the cospectrum of the Alfvén velocity and its curl $H^A={\mathcal {Q}}({\boldsymbol {v_A}},\boldsymbol {\nabla } \times {\boldsymbol {v_A}})$, which is the helicity of the Alfvén velocity, a quantity hereafter called ‘Alfvénic helicity’. In the incompressible case, the Alfvénic helicity is ${\boldsymbol {b}} \cdot {\boldsymbol {j}}/\rho$ and corresponds thus to the current helicity $H^J = (({2{\rm \pi} }/{L})K)^2 H^M$, which exhibits an approximate scaling law $H^J \sim K^{-1.3}$ (Malapaka Reference Malapaka2009; Müller et al. Reference Müller, Malapaka and Busse2012).

The Alfvénic helicity and Alfvén velocity power spectra display only a weak dependence on the compressible character of the forcing and the flow. While the magnetic energy and helicity spectra become flatter with increasing compressibility, for the Alfvénic helicity the incompressible scaling exponent $-1.3$ is approximately observed for all simulations with ${\mathcal {M}} \gtrsim 3$ as well (see figure 8). This is in particular the case for the M5cA and M5cB runs, which present very different $H^M$ scaling exponents. The scaling exponents of $E^A$, which approach asymptotically $-1.8$ for the most compressible runs (see figure 9), are also closer to the $-2.1$ scaling for the magnetic energy in the incompressible case (Malapaka Reference Malapaka2009; Müller et al. Reference Müller, Malapaka and Busse2012) and present a weaker dependence on the flow's compressibility as compared with $E^M$.

Figure 8.

Same as figure 7, but for the Alfvénic helicity $H^A$ which exhibits a $K^{-1.3}$ scaling in the incompressible case.

Figure 9.

Time evolution of $H^A$ and $E^A$ for the M11s and M8c runs, compensated by $K^m$ with $-m$ the scaling exponent in the incompressible case. The spectra are snapshots taken at different instants (the darker, the later), from $t=t_f/10$ to $t_f=2.2t_{\mathcal {T}}$ and $t_f=1.6t_{\mathcal {T}}$ for the M11s and M8c runs, respectively.

This universality of the exponents over a wide range of compressibility suggests a systematic scale-dependent correlation between the density and the magnetic fields, which is the object of further investigations. The M1c run may present different dynamics since its velocity compressive ratio ${\mathcal {P}}^{comp}({\boldsymbol {v}})/{\mathcal {P}}({\boldsymbol {v}})$ remains large in the inverse transfer region (see figure 10). This is reflected by the lack of low density regions in this simulation.

Figure 10.

(a,b) Time-averaged velocity compressive ratio during the hydrodynamic statistically stationary state, exhibiting values close to 1/3 for the solenoidally driven supersonic runs and 1/2 for the supersonic compressively driven ones in a reasonable wavenumber range, consistently with Federrath et al. (Reference Federrath, Roman-Duval, Klessen, Schmidt and Mac Low2010). (c,d) Snapshots of the compressive ratio at an instant when ${\mathcal {I}}_{{{\mathcal {H}}^M}}\approx L /6$: all supersonic runs exhibit a compressive ratio close to 0.2, independently of the mechanical forcing type. The M1c run continues to exhibit a large compressive ratio.

4.3. Scaling relations: kinetic quantities

In the incompressible case, the kinetic energy and helicity exhibit approximate scaling laws with exponents close to $-1.2$ and $-0.4$, respectively (Malapaka Reference Malapaka2009; Müller et al. Reference Müller, Malapaka and Busse2012). In the compressible case, several alternative definitions for the incompressible kinetic energy have been considered in the literature, e.g. $E^V={\mathcal {P}}({\boldsymbol {v}})$, i.e. the specific kinetic energy spectrum, $E^K={\mathcal {P}}(\boldsymbol {w})$, with $\boldsymbol {w}=\sqrt {\rho }{\boldsymbol {v}}$, which is dimensionally a kinetic energy spectrum, $E^U={\mathcal {P}}(\boldsymbol {u})$, with $\boldsymbol {u}=\rho ^{1/3}{\boldsymbol {v}}$, a quantity for which the Kolmogorov spectrum with exponent $-\frac {5}{3}$ is recovered in compressible hydrodynamic turbulence with a low $\boldsymbol {\nabla }\boldsymbol {\cdot }{\boldsymbol {v}}$ component (Fleck Reference Fleck1996; Kritsuk et al. Reference Kritsuk, Norman, Padoan and Wagner2007; Federrath et al. Reference Federrath, Roman-Duval, Klessen, Schmidt and Mac Low2010; Federrath Reference Federrath2013), $\frac {1}{2}{\mathcal {Q}}(\rho {\boldsymbol {v}},{\boldsymbol {v}})$, the cospectrum of velocity and momentum (or more generally, any cospectrum $\frac {1}{2}{\mathcal {Q}}(\rho ^{\alpha }{\boldsymbol {v}},\rho ^{1-\alpha }{\boldsymbol {v}})$ with $\alpha \in [0,1]$ Aluie (Reference Aluie2013)), or $E^{V,sol}={\mathcal {P}}^{sol}({\boldsymbol {v}})$, the solenoidal part of the specific kinetic energy spectrum.

From the above-mentioned quantities, the specific kinetic energy spectrum $E^V={\mathcal {P}}({\boldsymbol {v}})$ and its solenoidal part $E^{V,sol}$ (shown in figure 11) present the most universal behaviour, with an exponent close to $-1.2$ for all the runs. The deviations from the $-1.2$ exponent become weaker at later times for the M01s4 and M1s2 runs and the stronger deviations for the M1c run are due to the high velocity-spectrum compressive ratio. The other above-mentioned spectra, as well as that of the density, are displayed in appendix A.

Figure 11.

Same as figure 7, but for the solenoidal part of the specific kinetic energy spectrum $E^{V,sol}$ which exhibits a $K^{-1.2}$ scaling in the incompressible case.

As for the kinetic helicity, the $-0.4$ incompressible exponent appears to be the asymptotic behaviour for all levels of higher compressibility (figure 12). The negative kinetic helicity values at large scales for some runs are not an established feature of the inverse transfer experiment: their sporadic occurrence is a consequence of the random mechanical forcing. The spectral transient region influenced by the small-scale magnetic driving is wider at lower Mach numbers. This is probably linked with the presence of a large-scale mechanical driving in the present experiments, as compared with those of Müller et al. (Reference Müller, Malapaka and Busse2012). In hydrodynamic turbulence, the energy cascade tends to an equipartition of energy in the positive and negative helical parts of the velocity field at small scales, so that the kinetic helicity tends to zero (Chen, Chen & Eyink Reference Chen, Chen and Eyink2003). On the other hand, the injection of magnetic helicity at small scales leads to the generation of kinetic helicity. It is plausible that for runs with pronounced compressible effects and low density regions, the dynamical time scale of the kinetic helicity production is faster than that of the direct energy cascade leading to vanishing kinetic helicity, so that the $-0.4$ exponent observed in the absence of large-scale mechanical driving is recovered, but only for the supersonic flows. As shown in figure 12(c), a low-Mach-number run performed in the absence of a large-scale mechanical driving gives a kinetic helicity scaling compatible with $-0.4$.

Figure 12.

(a,b) Kinetic helicity spectra $H^V$ at an instant when ${\mathcal {I}}_{{{\mathcal {H}}^M}}={L}/{6}$ compensated by $K^{0.4}$. (c) time evolution of $H^V$, compensated by $K^{0.4}$, for a low-Mach-number run without large-scale mechanical driving, with ${\epsilon ^{M}_{inj}}=7\times 10^{-4}$ (for which ${\mathcal {M}} \approx 0.2$). The curves are equally spaced in time from an instant when ${\mathcal {I}}_{{{\mathcal {H}}^M}}\approx L /19$ until ${\mathcal {I}}_{{{\mathcal {H}}^M}} \approx L/3$. The darker the curve, the larger ${\mathcal {I}}_{{{\mathcal {H}}^M}}$.

4.4. Alfvénic balance

The observations presented so far suggest that the Alfvénic balance found in the incompressible case (relation (3.1)) can be extended to the compressible case when considering, instead of the current helicity $H^J$ and the magnetic energy $E^M$, the Alfvénic helicity $H^A$ and the power spectrum of the Alfvén velocity $E^A$, respectively, and by considering the specific kinetic energy $E^V$ or its solenoidal part. Since the energy associated with $H^A$ and the kinetic helicity $H^V$ is the solenoidal part of $E^A$ and of $E^V$, respectively, the following extension of the Alfvénic balance in the compressible case is proposed:

(4.3)\begin{equation} \left(\frac{E^{V,sol}}{E^{A,sol}}\right)^{\gamma} \propto \frac{H^V}{H^A}. \end{equation}

This choice is a straightforward extension of the balance in the incompressible case, where only solenoidal modes exist. It suggests that the shear, not the compressive waves, balance the twisting in compressible turbulence as well (cf. text around relation (3.1)). To test this relation, the ratio

(4.4)\begin{equation} \varLambda(r_H,r_E,\gamma)=\frac{r_E^\gamma}{r_H}, \end{equation}

with $r_H=r_{H^V_A}=H^V/H^A$ and $r_E=r_{E^{V,sol}_{A,sol}}=E^{V,sol}/E^{A,sol}$ is plotted in figure 13 for the different runs and different $\gamma$. This relation is well followed for the least compressible M01s4, M1s2 and M1c runs with $\gamma =2$, with $\varLambda \approx 1.1 \pm 0.3$ on the domain $20 \leq K \leq 44$, consistent with Müller et al. (Reference Müller, Malapaka and Busse2012). For the most compressible cases, the curves for the M3c, M5cA  M5cB and M8c runs show a similar behaviour for $\gamma =1$, with $\varLambda \approx 2.7 \pm 0.5$ on the $13 \leq K \leq 33$ domain. Regarding the supersonic solenoidally driven runs, they present an intermediate behaviour: for $\gamma =2$, the M5s curve would be included in the ‘M01s4–M1s2–M1c bundle’ on the domain $12 \leq K \leq 40$, whereas for $\gamma =1$, the M11s curve would be included in the ‘M3c–M5c–M8c bundle’ on the domain $16 \leq K \leq 31$.

Figure 13.

Test of the Alfvénic balance (relation (4.3)) by plotting $\varLambda (r_H,r_E,\gamma )=r_E^\gamma /r_H$ (relation (4.4)) with $r_H=r_{H^V_A}=H^V/H^A$ and $r_E=r_{E^{V,sol}_{A,sol}}=E^{V,sol}/E^{A,sol}$ for different runs and different exponents $\gamma$: (a) for the subsonic and transonic runs, $\gamma =2$, compatible with the results in the incompressible case; (b) for the supersonic compressively driven runs, which verify the Alfvénic balance for $\gamma =1$; (c,d) for the supersonic solenoidally driven runs, which present an intermediate behaviour with respect to the $\gamma$ exponent.

Replacing the solenoidal parts $E^{V,sol}$ and $E^{A,sol}$ by the total energies $E^V$ and $E^A$ in $\varLambda$ also gives a good horizontal for the high Mach number compressively driven runs. The ‘M3c–M5c–M8c bundle’ verifies then $\varLambda \approx 3 \pm 0.5$ on the same $13 \leq K \leq 33$ domain. With this choice, however, the M1c curve does not stay in the ‘M01s4–M1s2’ bundle, which is why the choice of relation (4.3) is preferred.

Although other variants of relation (4.3) are partially consistent with the data as well, this choice shows overall the best agreement with the simulations, as well as the least spread between the curves. This also holds at higher numerical resolution, which is not the case for some other possibilities, as shown in § 5.