Saturation mechanism of the fluctuation dynamo in supersonic turbulent plasmas

Magnetic fields in several astrophysical objects are amplified and maintained by a dynamo mechanism, which is the conversion of the turbulent kinetic energy to magnetic energy. A dynamo that amplifies magnetic fields at scales less than the driving scale of turbulence is known as the fluctuation dynamo. We study the properties of the fluctuation dynamo in supersonic turbulent plasmas, which is of relevance to the interstellar medium of star-forming galaxies, structure formation in the universe, and laboratory experiments of laser-plasma turbulence. Using numerical simulations of driven turbulence, we explore the global and local properties of the exponentially growing and saturated (statistically steady) state of the fluctuation dynamo for subsonic and supersonic turbulent flows. First, we confirm that the fluctuation dynamo efficiency decreases with compressibility. Then we show that the fluctuation dynamo-generated magnetic fields are spatially intermittent and the intermittency is higher for supersonic turbulence, but in both cases the level of intermittency decreases as the field saturates. We also find a stronger back-reaction of the magnetic field on the velocity for the subsonic case as compared to the supersonic case. Locally we find that the level of alignment between vorticity and velocity, velocity and magnetic field, and current density and magnetic field in the saturated stage is enhanced in comparison to the exponentially growing phase for the subsonic case, but only the current density and magnetic field alignment is enhanced for the supersonic case. Finally, we show that both the magnetic field amplification (mainly due to weaker stretching magnetic field lines) and diffusion decreases when the field saturates, but the diffusion is enhanced relative to amplification. This occurs throughout the volume in the subsonic turbulence, but primarily in the strong-field regions for the supersonic case. This leads to the saturation of the fluctuation dynamo. Overall both the amplification and diffusion of magnetic fields are affected by the exponentially growing magnetic fields and thus a drastic change in either of them is not required for the saturation of the fluctuation dynamo.

Figure 1

Time evolution (in terms of the eddy turnover time, $t_0$, which also varies with the Mach number) of the ratio of magnetic to turbulent kinetic energies, $E_{\mathrm{mag}}/E_{\mathrm{turb},\mathrm{kin}}$, for $\text{Re}=2000$, $\text{Rm}=6000$, and $\mathcal{M}=0.1$ (blue), 2 (cyan), 5 (magenta), and 10 (red). For all the cases, after the initial transient decay phase, the magnetic field grows exponentially (kinematic stage, dashed-dotted gray line) and then saturates (saturated stage, dotted gray line). The growth rate, $\mathrm{\Gamma }\left[t_0^{-1}\right]$ in the exponentially growing or kinematic phase decreases with compressibility till $\mathcal{M}=5$ and then increases for $\mathcal{M}=10$. The saturated value of the ratio $E_{\mathrm{mag}}/E_{\mathrm{turb},\mathrm{kin}}$, $R_{\mathrm{sat}}$, decreases with increasing Mach numbers. Thus, per unit time, a smaller fraction of the turbulent kinetic energy is converted to magnetic energy for supersonic flows as compared to the subsonic flows. See Table 1 for the values of $\mathrm{\Gamma }$ and $R_{\mathrm{sat}}$ in other runs and Fig. 2 for the dependence of $\mathrm{\Gamma }$ and $R_{\mathrm{sat}}$ on $\mathcal{M}$.

Figure 2

The growth rate in the kinematic phase, $\mathrm{\Gamma }\left[t_0^{-1}\right]$, (a) and the ratio of magnetic to turbulent kinetic energies in the saturated stage, $R_{\mathrm{sat}}$, (b) as a function of the Mach number of the turbulent flow, $\mathcal{M}$, for all $\text{Rm}$s. The growth rate as a function of the Mach number first decreases till $\mathcal{M}=5$ and then increases for $\mathcal{M}=10$. The overall trend with $\mathcal{M}$ does not change much with $\text{Rm}$. For $R_{\mathrm{sat}}$, as the Mach number of the turbulent flow increases, the ratio decreases. This confirms that a smaller fraction of the turbulent kinetic energy is converted to magnetic energy, per unit time, for the supersonic turbulence as compared to the subsonic case. The variation of $\mathrm{\Gamma }$ and $R_{\mathrm{sat}}$ with $\mathcal{M}$ is roughly consistent with the empirical model (shown in dashed, black line) of Federrath et al. [see Eq. (3) and Table 1 in [52]).

Figure 3

Two-dimensional slices of the velocity field at $z=L/2$ with color showing the square of velocity normalized to its root mean square (rms) value, $u^2/u_{\mathrm{rms}}^2$, for subsonic ($\mathcal{M}0.1\text{Rm}6000$) and supersonic ($\mathcal{M}10.0\text{Rm}6000$) cases in their kinematic ($t/t_0=10$, a and $t/t_0=30$, b) and saturated ($t/t_0=75$, c, and $t/t_0=90$, d) stages. Visually, the velocity structures are larger in the saturated stage for $\mathcal{M}0.1\text{Rm}6000$ than in the kinematic stage (effect of the back-reaction of the strong magnetic field on the turbulent flow). The corresponding difference for $\mathcal{M}10.0\text{Rm}6000$ is smaller. As compared to the subsonic case, the velocity field in the supersonic case can be locally high in strength. This is due to random strong shocks for the $\mathcal{M}=10$ case.

Figure 4

Same as Fig. 3 but for $b^2/b_{\mathrm{rms}}^2$. In comparison to the kinematic stage (a, b), the magnetic structures in the saturated stages (c, d) are seen to be larger in size for both the subsonic and supersonic turbulence. Visually, in the kinematic stage and on an average, the structures are of a larger size for the supersonic turbulence in comparison to the subsonic case. This difference is less for the saturated stage.

Figure 5

Same as Fig. 4 but now showing three-dimensional magnetic structures. Visually, the magnetic structures for the supersonic turbulence are of a larger size and this is more evident in the kinematic stage. The magnetic fields, for both subsonic and supersonic turbulent flows, seen to have larger structures in their respective saturated stage as compared to the kinematic stage.

Figure 6

Top panels: The shell-integrated power spectrum of the turbulent kinetic (a) and magnetic (b) energies normalized by their total energies for $\mathcal{M}0.1\text{Rm}6000$ (subsonic, blue) and $\mathcal{M}10\text{Rm}6000$ (supersonic, red) cases. In each panel, the dashed and solid colored line shows the spectra averaged over the kinematic ($7\le t/t_0\le 14$ for the subsonic case and $28\le t/t_0\le 35$ for the supersonic case) and saturated ($70\le t/t_0\le 80$ for the subsonic case and $85\le t/t_0\le 95$ for the supersonic case) stages, respectively, with the shaded region indicating their $1\sigma$ variations. The turbulent kinetic energy spectra, in the range $3\le k\le 20$, for the subsonic turbulence in the kinematic stage seem to be consistent with the Kolmogorov $k^{-5/3}$ (shown in a black, dashed-dotted line) spectrum [63] and for the supersonic turbulence with the Burgers $k^{-2}$ (shown in a black, dotted line) spectrum [64]. The supersonic case has larger power on smaller scales compared to the subsonic one and this is because of strong shocks for $\mathcal{M}=10$ run (see Fig. 3 for locally strong structures in velocity fields). As the magnetic field saturates, the turbulent kinetic energy spectrum for the subsonic case steepens (due to the effect of back-reaction of the strong magnetic fields on the flow) and such an effect is not that significant for the supersonic case. The magnetic energy spectra in the kinematic stage, at lower wave numbers, for the subsonic case agree better with the Kazantsev $k^{3/2}$ (shown with a black, dashed-dotted line) spectrum [22] than the supersonic case. As the dynamo saturates, spectra for both the cases become flat on larger scales. However, the magnetic energy for the supersonic case is higher at smaller scales in comparison to the subsonic case. This is also because of strong shocks and higher turbulent kinetic energy at smaller scales for supersonic flows. Bottom panels: The time evolution of the velocity [${\ell }_u$, (c)] and magnetic [${\ell }_b$, (d)] field correlation scales in units of the numerical domain length, $L$, for subsonic ($\mathcal{M}0.1\text{Rm}6000$, blue) and supersonic ($\mathcal{M}10\text{Rm}6000$, red) cases computed using Eq. (5). The dashed and dotted black lines shows the average value of each length scale in the kinematic and saturated stages, respectively, and the corresponding values with $1\sigma$ fluctuations are given in the legend. For the subsonic case, after the initial transient decay, ${\ell }_u/L$ remains statistically steady at a value of $\approx 0.4$ in the kinematic stage ($7\le t/t_0\le 14$), but then increases as the field saturates and fluctuates around $\approx 0.5$. For the supersonic case, ${\ell }_u$ always fluctuates around $L/2$ after the initial transient phase. Thus, the back-reaction of the growing magnetic fields on the velocity is evident here for the low Mach number run. The magnetic field correlation scale, ${\ell }_b$, decreases in the initial transient phase and then remains roughly constant in the kinematic stage for both subsonic (${\ell }_b/L\approx 0.05$) and supersonic (${\ell }_b/L\approx 0.09$) cases. In fact, in the kinematic stage, ${\ell }_b$ is higher for the supersonic case (as can also be seen from Fig. 4 and Fig. 5). Then, ${\ell }_b$ for both Mach numbers increases as the magnetic fields become strong enough to react back on the flow (see Fig. 1 for corresponding times). Finally, in the saturated stage, ${\ell }_b$ saturates to a statistically steady value, which is roughly the same for both Mach numbers (${\ell }_b/L\approx 0.14$).

Figure 7

Probability distribution function (PDF) of the $x$ component of the normalized velocity, $u_x/u_{\mathrm{rms}}$ (a), and $x$ component of the normalized magnetic field, $b_x/b_{\mathrm{rms}}$ (b), for the subsonic (blue) and supersonic (red) turbulence in their respective kinematic (kin, dashed) and saturated (sat, solid) stages of the fluctuation dynamo. The lines show the mean obtained after averaging over 10 eddy turnover times (in the range $7\le t/t_0\le 17$ and $70\le t/t_0\le 80$ for the subsonic case and $25\le t/t_0\le 35$ and $85\le t/t_0\le 95$ for the supersonic case) and the shaded regions of the same color show $1\sigma$ deviations. The computed kurtosis (${\mathcal{K}}_{u_x}$ and ${\mathcal{K}}_{b_x}$) with its corresponding error (estimated from the time averaging) for each case is shown in the legend. For both the subsonic and supersonic flows, in the kinematic and saturated stages, the velocity PDF is close to a Gaussian distribution ($\mathcal{N}$, the dashed-dotted, black line). Also, the computed kurtosis is always very close to that of a Gaussian distribution (${\mathcal{K}}_{\mathcal{N}}=3$). Thus, the random velocity field in solenoidally driven turbulence follows a Gaussian distribution, immaterial of the Mach number and the dynamo stage. The magnetic field is always non-Gaussian (the dashed-dotted, black line shows the Gaussian distribution, $\mathcal{N}$). This is also confirmed from their computed kurtosis values, they are significantly higher than three. The magnetic field is always more non-Gaussian (or spatially intermittent) in the supersonic turbulence than in the subsonic case. Furthermore, for both Mach numbers, the magnetic field in the kinematic stage is more intermittent than their respective saturated stages.

Figure 8

(a) The kurtosis of the $b_x/b_{\mathrm{rms}}$ distribution [see Fig. 7] as a function of $\text{Rm}$ in the kinematic (kin, dashed) and saturated (sat, solid) stages of the fluctuation dynamo for $\mathcal{M}=0.1$ (blue), 2 (cyan), 5 (magenta), and 10 (red). The points show the mean obtained after averaging over 10 eddy turnover times and the error bars of the same color show $1\sigma$ deviations (values are also provided in Table 1). The kurtosis for all runs in both the kinematic and saturated stages are higher than the kurtosis of a Gaussian distribution (the dashed-dotted, black line). This confirms that all the distributions are non-Gaussian. The estimated kurtosis increases with $\mathcal{M}$ and this shows that the magnetic intermittency increases with the compressibility of the turbulent flow. Within this range of $\text{Rm}$, the kurtosis for each stage does not seem to vary much with the $\text{Rm}$. The kurtosis is always lower for the saturated stage in comparison to the kinematic stage and this difference [shown in (b)] increases with the Mach number of the turbulent flow.

Figure 9

PDF of $ln(b/b_{\mathrm{rms}})$ for subsonic (blue) and supersonic (red) cases in their respective kinematic (kin, a) and saturated (sat, b) stages with black, dashed line showing the fit to a normal distribution [Eq. (7)]. The normal distribution captures high probability regions better than the tails. In the kinematic stage, the lognormal fit to $b/b_{\mathrm{rms}}$ performs better for the supersonic case. The fit worsens, for both Mach numbers, when the magnetic fields saturate (b). Thus, PDF of $b/b_{\mathrm{rms}}$ in the kinematic stage can be represented by a near lognormal distribution. Furthermore, higher values of the fit parameters ($\left|w0\right|$ and ${\sigma }_w$) for the kinematic stage confirm that the magnetic intermittency decreases as the field saturates. Also, these parameters are always higher for the supersonic case as compared to the subsonic one. Thus, confirming that the degree of magnetic intermittency increases with the Mach number of the turbulent flow.

Figure 10

Temporal evolution of the (normalized) rms vorticity (${\omega }_{\mathrm{rms}}/{\omega }_{\mathrm{rms}}(t/t_0\approx 2)$, dashed) and current density ($j_{\mathrm{rms}}/j_{\mathrm{rms}}(t/t_0=0)$, solid) for subsonic ($\mathcal{M}0.1\text{Rm}6000$, blue) and supersonic ($\mathcal{M}10\text{Rm}6000$, red) turbulence. The vorticity remains roughly the same once the turbulence is established and is slightly higher for the subsonic case as compared to the supersonic (the mean value, ${\left({\omega }_{\mathrm{rms}}\right)}_{\mathrm{sat}}$, shown in the legend). The overall trend of the current density evolution is the same as the magnetic field, exponential amplification after the initial transient phase and then saturation. The growth rate, ${\mathrm{\Gamma }}_{j_{\mathrm{rms}}}$, is higher for the subsonic case but the saturated value, ${\left(j_{\mathrm{rms}}\right)}_{\mathrm{sat}}$, is higher for the supersonic case. The growth rate of the current density is roughly half of the magnetic field growth rate (see Fig. 1) for both the subsonic and supersonic flows.

Figure 11

PDF of the normalized vorticity, ${\omega }_x/{\omega }_{\mathrm{rms}}$ (a), and current density, $j_x/j_{\mathrm{rms}}$ (b), for subsonic (blue) and supersonic (red) flows in the kinematic (kin, dashed) and saturated (sat, solid) stages of the fluctuation dynamo (the lines show the mean over 10 eddy turnover times and the shaded region show the $1\sigma$ fluctuations). Both the vorticity and current density distributions are non-Gaussian or intermittent (the dashed-dotted, black line shows the Gaussian distribution) and the estimated kurtosis (shown in the legend for each case) is also higher than three. The vorticity distribution for the subsonic case is less intermittent in the saturated stage as compared to the kinematic stage and this is due to the back-reaction of the amplified magnetic field on the velocity flow. Such a statistical difference in the kurtosis of the vorticity between the kinematic and saturated stages is not seen for the supersonic turbulence. The current density distribution, for both the subsonic and supersonic turbulence, is less intermittent in the saturated stage as compared to their respective kinematic stage but is always higher for the supersonic case.

Figure 12

PDFs of the absolute value of the cosine of angle between the velocity and vorticity (${|cos\left(\theta \right)}_{\mathit{u},\mathit{\omega }}|$, a, b), velocity and magnetic field (${|cos\left(\theta \right)}_{\mathit{u},\mathit{b}}|$, c, d), and current density and magnetic field (${|cos\left(\theta \right)}_{\mathit{j},\mathit{b}}|$, e, f) for subsonic (blue) and supersonic (red) turbulence in the kinematic (kin, dashed) and saturated (sat, solid) stages of the fluctuation dynamo (the line shows the mean over 10 eddy turnover times and the shaded regions shows the corresponding $1\sigma$ variations). The left panels (a), (c), (e) show the PDFs over the entire domain, and the right panels (b), (d), (f) show the conditional PDFs based only on the strong-field regions ($b^2/b_{\mathrm{rms}}^2>1$). The absolute value of the cosine of the angle is studied, because the distributions are symmetric about the $cos\left(\theta \right)=0$ for isotropic velocity and magnetic fields. $|cos(\theta \left)\right|=0$ implies an angle of ${90}^{\circ }$ (or ${270}^{\circ }$) between the two vectors and $|cos(\theta \left)\right|=1$ implies alignment (or anti-alignment) between them. For the subsonic turbulence, as the magnetic field saturates, velocity and vorticity, velocity and magnetic fields, and current density and magnetic fields are more aligned. Each trend gets more enhanced in the strong-field regions (b, d, f). The corresponding differences between the kinematic and saturated stages for the supersonic case are only significant for the angle between the current density and magnetic field. For the supersonic case, ${|cos\left(\theta \right)}_{\mathit{u},\mathit{\omega }}|$ and ${|cos\left(\theta \right)}_{\mathit{u},\mathit{b}}|$ are statistically the same in the kinematic and saturated stage. In the saturated stage, the current density and magnetic field are more aligned (the level of alignment is enhanced in the strong-field regions) and the difference between the two stages is higher for the supersonic turbulence in comparison to the subsonic case.

Figure 13

The evolution of characteristic magnetic length scales, $k_{\parallel }$ [a, Eq. (13), associated with the length scale for greatest field line stretching], $k_{\mathit{b}·\mathit{j}}$ [b, Eq. (14), associated with resistive dissipation length scale], $k_{\mathit{b}\times \mathit{j}}$ [c, Eq. (15), associated with the length scale for greatest field line compression], and $k_{\mathrm{rms}}$ [d, Eq. (16), associated with overall magnetic field variation], as a function of the normalized time, $t/t_0$, for subsonic (blue, $\mathcal{M}0.1\text{Rm}6000$) and supersonic (red, $\mathcal{M}10\text{Rm}6000$) turbulence. The dashed, gray line shows average for each characteristic wave number in the kinematic and saturated stages (see Fig. 1 for corresponding times for each stage and Mach number) and the average value with $1\sigma$ variation is given in the legend. All length scales, except the resistive dissipation length scale for supersonic turbulence, are enhanced as the magnetic field grows and saturates. $k_{\parallel }$ and $k_{\mathit{b}\times \mathit{j}}$ are always smaller for the subsonic turbulence and $k_{\mathit{b}·\mathit{j}}$ and $k_{\mathrm{rms}}$ are comparable in the kinematic stage but are higher for the supersonic turbulence in the saturated stage.

Figure 14

PDFs of the eigenvalues of the strain tensor, ${\lambda }_{\mathrm{comp}}$ (green), ${\lambda }_{\mathrm{null}}$ (orange), and ${\lambda }_{\mathrm{stre}}$ (magenta) normalized by $u_{\mathrm{rms}}$ for subsonic (a) and supersonic (b) turbulence in the kinematic (kin, dashed) and saturated (sat, solid) stages. The solid lines show the mean over 10 eddy turnover times, and the shaded region of the same color shows the $1\sigma$ variations. The legend shows the mean value (denoted by $\mu$) of the distribution for each case, where the value and the related error are obtained by time averaging. For the subsonic turbulence, ${\lambda }_{\mathrm{comp}}$ is always less than zero (strength of local magnetic field line compression), and ${\lambda }_{\mathrm{stre}}$ is always greater than zero (strength of local magnetic field line stretching). This is not the case for supersonic turbulence. All three eigenvalues decreases (in a statistical sense, as also reflected by the absolute value of their mean) as the magnetic field saturates for the subsonic case but increases for the supersonic turbulence.

Figure 15

Total and conditional (regions with $b^2/b_{\mathrm{rms}}^2>1$) PDFs of the angle between the three eigenvectors of the rate of strain tensor (${\mathit{e}}_{\mathrm{comp}},{\mathit{e}}_{\mathrm{null}},$ and ${\mathit{e}}_{\mathrm{stre}}$) and magnetic felds ($\mathit{b}$) for subsonic ($\mathcal{M}0.1\text{Rm}6000$, blue) and supersonic ($\mathcal{M}10.0\text{Rm}6000$, red) turbulence in kinematic (kin, dashed) and saturated (sat, solid) stages. The effect of field line compression is enhanced if the field line compression direction is perpendicular to the local magnetic field, i.e., ${|cos\left(\theta \right)}_{{\mathit{e}}_{\mathrm{comp}},\mathit{b}}|\approx 0$. Magnetic field amplification is enhanced when the field line stretching direction is aligned with the local magnetic field, i.e., ${|cos\left(\theta \right)}_{{\mathit{e}}_{\mathrm{stre}},\mathit{b}}|\approx 1$. This is seen in the kinematic stage for the subsonic flows. As the field saturates in the subsonic case, $|cos(\theta \left)\right|$ for both those cases moves to the range $0.6\text{–}0.7$, and thus the field amplification and the effect of local compression is reduced. Such a significant difference between the kinematic and saturated stages is not seen for the supersonic case. For the supersonic case, the direction of local magnetic field line compression is more aligned with the magnetic field in the saturated stage as compared to the kinematic stage (though not in the strong-field regions), whereas, in strong-field regions only, there is a slight decrease in the level of alignment between ${\mathit{e}}_{\mathrm{stre}}$ and $\mathit{b}$ in the saturated stage of the supersonic run.

Figure 16

Total and conditional (regions with $b^2/b_{\mathrm{rms}}^2>1$) PDFs of the first $\left[{\sum }_{i=\mathrm{comp},\mathrm{null},\mathrm{stre}}({\lambda }_i/u_{\mathrm{rms}}){({\mathit{b}}_i/b_{\mathrm{rms}})}^2\right]$, second $(-(1/2){(b/b_{\mathrm{rms}})}^2[\mathbf{\nabla }·(\mathit{u}/u_{\mathrm{rms}})])$, and the combination of the first two terms in the right-hand side of the Eq. (17) for subsonic and supersonic cases ($\mu$ in the legend shows the mean of the distribution for each case). The mean of combination of both terms (e, f) is always positive. This implies that, on an average, the first two terms leads to local growth of the magnetic energy. On saturation, the first term and first two terms combined statistically decreases for the subsonic case (as expected the second term has negligible effect). This shows reduction in amplification, and this reduction is enhanced in strong-field regions. For the supersonic case, the local growth term (e) is not statistically different between the kinematic and saturated stages but is lower in the saturated stage for strong-field regions (f).

Figure 17

The total and conditional (regions with $b^2/b_{\mathrm{rms}}^2>1$) PDF of the dissipation (last) term in Eq. (17) for subsonic (blue) and supersonic (red) turbulence in the kinematic (kin, dashed) and saturated (sat, dashed) stages. The magnetic dissipation is statistically higher for the supersonic turbulence in both the kinematic and saturated stages. For both Mach numbers, the dissipation decreases as the field saturates and the decrease is higher for the supersonic case. The differences are enhanced in the strong-field regions (this can also be seen via the mean of the distribution, $\mu$, in each case)