Saturation mechanism of the fluctuation dynamo in supersonic turbulent plasmas

Fig. 1 shows the evolution of the ratio of magnetic to turbulent kinetic energies, $E_{\rm mag}/E_{\rm turb,kin}$, as a function of time normalised by the eddy turnover time, $t_{0}$, for various Mach numbers ($0.1,2,5,$ and $10$) with $\text{Re}=2000$ and $\text{Rm}=6000$. After the initial transient phase, the magnetic energy amplifies exponentially (kinematic stage) for all the cases. Then when the magnetic field becomes strong enough to react back on the flow, the exponential increase slows down, and finally the magnetic energy saturates to a statistically steady value (saturated stage). This happens for all the runs and the corresponding growth rate, $\Gamma[t_{0}^{-1}]$, in the exponentially growing or kinematic stage and the saturated level of $E_{\rm mag}/E_{\rm turb,kin}$, $R_{\rm sat}$, in the saturated stage are given in Table 1.

Fig. 2 shows the growth rate, $\Gamma\,[t_{0}^{-1}]$, (Fig. 2 (a)) and the saturated level of $E_{\rm mag}/E_{\rm turb,kin}$, $R_{\rm sat}$, (Fig. 2 (b)) as a function of $\mathcal{M}$. The growth rate decreases till $\mathcal{M}=5$ but then increases for $\mathcal{M}=10$ and the overall trend is consistent with the empirical model in the literature [52]. Fig. 2 (b) shows the dependence of the saturated level, $R_{\rm sat}$, on $\mathcal{M}$. For all values of Rm, $R_{\rm sat}$ decreases with $\mathcal{M}$. This shows that as the compressibility increases, a smaller fraction of turbulent kinetic energy is converted to magnetic energy, per unit time. Thus, the efficiency of the fluctuation dynamo decreases with increasing compressibility. Here too, the trend of $R_{\rm sat}$ with $\mathcal{M}$ roughly agrees with the known empirical model [52]. These empirical models for the dependence of $\Gamma$ and $R_{\rm sat}$ on $\mathcal{M}$ are constructed from numerical simulations in [52], where most of the simulations used numerical viscosity and resistivity, whereas here, all our simulations include explicit (viscous and resistive) diffusion terms (Eq. 2 and Eq. 3). This might be the reason for differences in the results (coloured lines vs. dashed black line in Fig. 2), but the overall trend is roughly consistent with the empirical models. Thus, generally, the growth rates and the saturated levels agree with previous results. We now show and discuss velocity and magnetic structures in the kinematic and saturated stages for $\mathcal{M}=0.1$ and $\mathcal{M}=10$ with a fixed Rm of $6000$.

Fig. 3 and Fig. 4 shows two-dimensional velocity and magnetic field structures in the kinematic and saturated stages for subsonic and supersonic turbulence. Both the velocity and magnetic fields show random distributions with complex structures and without any significant mean trend. This is expected for the turbulent flows and the magnetic fields they amplify. The velocity structures for subsonic flow look larger in size in the saturated stage (Fig. 3 (b)) as compared to the kinematic stage (Fig. 3 (a)). This is due to the back reaction of the strong magnetic field on the turbulent flow (the back reaction is negligible in the kinematic stage). Such a difference is smaller for the supersonic case. The structures in supersonic turbulence (Fig. 3 (b, d)), in both the kinematic and saturated stages, are of a larger size than in the subsonic case (Fig. 3 (a, c)) but the supersonic case can have locally strong velocity because of random shocks. The difference in structure between the kinematic and saturated stages is more pronounced for the magnetic field. For both the Mach numbers, magnetic fields in the saturated stages (Fig. 4 (c, d)) have larger structures than in their corresponding kinematic stages (Fig. 4 (a, b)). Fig. 5 shows the three-dimensional magnetic structures in both the kinematic and saturated stages for both Mach numbers. Overall, for both Mach numbers, the magnetic fields have visually larger structures in the saturated stage than in their respective kinematic stage. We now discuss the spectral properties of velocity and magnetic fields in the kinematic and saturated stages for subsonic and supersonic turbulent flows.

The top panels of Fig. 6 show the shell-integrated turbulent kinetic ($E_{\rm turb,kin}$, Fig. 6 (a)) and magnetic energy ($E_{\rm mag}$, Fig. 6 (b)) spectra, which were averaged over a few eddy turnover times ($7\,t_{0}$ for the kinematic stage and $10\,t_{0}$ for the saturated stage) for the subsonic ($\mathcal{M}0.1\text{Rm}6000$) and supersonic ($\mathcal{M}10\text{Rm}6000$) cases. The turbulent kinetic energy spectra, over a range of wavenumbers ($3\leq k\leq 20$), are seen to be consistent with the Kolmogorov $k^{-5/3}$ spectrum [63] for the subsonic turbulence and Burgers $k^{-2}$ spectrum [64] for the supersonic turbulence (agreement is slightly better for the supersonic case). At smaller scales, the turbulent kinetic energy is higher for the supersonic case as compared to the subsonic case due to strong shocks. For the subsonic case, as the magnetic field saturates, the turbulent kinetic energy spectrum steepens and it shows the effect of the back-reaction due to the Lorentz force on the turbulent flow. However, this effect on the turbulent kinetic energy spectra for the supersonic case is not that significant. The magnetic energy spectra vary between the kinematic and saturated stages for both subsonic and supersonic flows (Fig. 6 (b)). At lower wavenumbers, the magnetic spectra in the kinematic stage seem to follow $k^{3/2}$ spectra as expected from Kazantsev’s analytical work [22]. The agreement seems to be better for the subsonic flow than the supersonic case and this is because Kazantsev’s theory is derived assuming incompressible flows. However, from extensions to Kazantsev’s theory [40], the slope of the magnetic power spectrum for the supersonic case in the kinematic stage might still be $3/2$ [28]. For both cases, the spectra flatten as the magnetic field saturates. On smaller scales, the magnetic energy is higher for the supersonic case in comparison to the subsonic one. This is primarily due to strong shocks and higher turbulent kinetic energy at smaller scales for supersonic turbulence.

We use the turbulent kinetic ($E_{\rm turb,kin}(k)$) and magnetic ($E_{\rm mag}(k)$) energy spectra to compute the correlation (approximately average) length of the velocity ($\ell_{u}$) and magnetic ($\ell_{b}$) fields as

The bottom panels of Fig. 6 show the velocity ($\ell_{u}/L$, Fig. 6 (c)) and magnetic field ($\ell_{b}/L$, Fig. 6 (d)) correlation scale as a function of the normalised time, $t/t_{0}$. After the initial transient phase, $\ell_{u}$ fluctuates around $L/2$ immediately for the supersonic case, but is initially lower ($\ell_{u}/L\approx 0.4$) for the subsonic case in its kinematic stage. For subsonic flows, as the magnetic field grows, $\ell_{u}$ increases and reaches a value of $L/2$ in the saturated stage. Thus, the back reaction of the growing magnetic field on the velocity field enhances its correlation length. This is seen only for the subsonic case (can also be concluded from the steepening of the turbulent kinetic energy spectra in Fig. 6 (a)). For the magnetic field, after the initial transient phase, the correlation length remains roughly constant in the kinematic phase ($\ell_{b}/L\approx 0.05$ and $0.09$ for the subsonic and supersonic case, respectively) and then increases as the magnetic fields become strong enough to react back on the turbulent flow. In the saturated stage, $\ell_{b}$ reaches a statistically steady value ($\ell_{b}/L\approx 0.14$, similar for both Mach numbers). Overall, the magnetic field correlation length increases due to the growing magnetic field for both the subsonic and supersonic cases (this also agrees with the larger size of magnetic structures seen in Fig. 4 and Fig. 5 for the saturated stage in comparison to the kinematic stage for both cases). We now quantify the non-Gaussianity of these magnetic structures.

The fluctuation dynamo generated magnetic fields are non-Gaussian or spatially intermittent and we aim to quantify the intermittency using the statistical measure kurtosis, $\mathcal{K}$, which for any random function $f$ with mean zero is defined as

where $\langle\rangle$ denote the average over the entire domain. The kurtosis of a Gaussian distribution is three.

First, in Fig. 7 (a), we show the probability distribution function (PDF) of a single component (here, $x$) of the random velocity field, $u_{x}/u_{\rm rms}$, obtained in subsonic and supersonic turbulence in the kinematic and saturated stages of the fluctuation dynamo. The distribution, for all four cases, is very close to a Gaussian (or normal) distribution (the computed kurtosis is very close to that of a Gaussian, three). Thus, random velocity in these driven turbulence simulations always follows an approximately Gaussian distribution [also see 65].

Even though the turbulent velocity is normally distributed, the magnetic field it generates is non-Gaussian or spatially intermittent, as shown by Fig. 7 (b), which shows the PDF of a single component of the random magnetic field, $b_{x}/b_{\rm rms}$, in their kinematic and saturated stages for subsonic and supersonic turbulent flows. The magnetic field is always more intermittent for the supersonic turbulence than the subsonic one. Moreover, for both cases, the magnetic intermittency decreases as the dynamo saturates. These conclusions can also be confirmed via the computed kurtosis values (Table 1 and Fig. 8). Fig. 8 (a) shows that the kurtosis, in the Rm range $2000~{}\text{--}~{}6000$, does not depend much on Rm but increases with $\mathcal{M}$ for both the kinematic and saturated stages. The difference in kurtosis of the magnetic fields between the kinematic and saturated stages increases with the Mach number of the turbulent flow, $\mathcal{M}$ (Fig. 8 (b)). Thus, the non-Gaussian or intermittent nature of the dynamo generated magnetic fields is enhanced with compressibility.

Fig. 9 shows the PDF of $\ln(b/b_{\rm rms})$ for the subsonic and supersonic cases in their respective kinematic (Fig. 9 (a)) and saturated (Fig. 9 (b)) stages. We fit the PDF of $\ln(b/b_{\rm rms})$ with a normal distribution of form,

where $w0$ (mean) and $\sigma_{w}$ (standard deviation) are parameters of the fit. The black, dashed lines in Fig. 9 show the fitted distributions, and parameters of the fit are given in the legend. For the kinematic phase (Fig. 9 (a)), the lognormal distribution for $b/b_{\rm rms}$ is a better fit to the supersonic case as compared to the subsonic one. Also, the fit worsens for the saturated stage. Overall, the lognormal distribution captures high probability regions quite well in the kinematic phase and performs better for supersonic turbulence. The parameters $|w0|$ and $\sigma_{w}$ are also higher in the kinematic stage as compared to the saturated stage and are always higher for the supersonic flows. This further confirms that the degree of magnetic intermittency decreases as the field saturates and increases with the compressibility of the turbulent flow.

After characterising the global properties of velocity and magnetic fields in the fluctuation dynamo, in the next section, we study the local interaction of velocity and magnetic fields to explore the saturation mechanism of the fluctuation dynamo.

After confirming the change in the local structure of the velocity and magnetic field on saturation, here, we study the local interaction between those two vectors. First, the evolution of vorticity, obtained by taking the curl of the Navier-Stokes equation (the baroclinic term is zero for an isothermal equation of state), is [66, 67]

where the first term on the right-hand side represents growth of vorticity, the second term represents vorticity diffusion, the third term is due to density gradients, and the last term is due to the Lorentz force, $\bm{j}\times\bm{b}$ ($c$ is the speed of light). Since, we start with a zero velocity (and vorticity) and very weak magnetic fields, the vorticity is most likely generated by the third term when the density gradients develop and is then amplified by the first term [see Fig. 10 and in 67, 52]. The amplification of the vorticity field, via the first term, depends on the angle between $\bm{u}$ and $\bm{\omega}$.

Another two angles of interest are the angle between the velocity and magnetic field, which controls the magnetic field induction term (first term in the right-hand side of Eq. 3) and the angle between the current density and magnetic field, which controls the Lorentz force (or the back reaction of the velocity field on the flow). Thus, we compute the cosine of the following three relevant angles,

$\cos(\theta)=0$ implies an angle of $90\degree$ (or $270\degree$) between the two vectors and thus maximum effect of the physically relevant term (for example, angle of $90\degree$ between $\bm{u}$ and $\bm{b}$ implies maximum induction). On the other hand, $\cos(\theta)=1(\text{or}~{}-1)$ implies alignment (or anti-alignment) between them and no effect of the term. Fig. 12 shows the total and conditional (for strong magnetic field regions, $b^{2}/b_{\rm rms}^{2}>1$) PDFs of these three angles in the kinematic and saturated stages for the subsonic and supersonic turbulence. We only show the absolute value of these angles as these distributions are symmetric around $\cos(\theta)=0$ for isotropic random velocity and magnetic fields. For all three angles, as expected from isotropic random fields, all possible values of $|\cos(\theta)|$ have a non-zero probability. However, all angles are not equiprobable and there are clear differences in the distributions with the Mach number of the turbulent flow and the dynamo stage. We discuss these differences and their implications below.

For the subsonic turbulence, the vorticity is slightly more aligned with the velocity in the saturated stage as compared to the kinematic stage (the difference is not much in Fig. 12 (a) but see Fig. 12 (b)). This shows that as the magnetic field grows, the vorticity amplification term is probably suppressed, especially in the strong-field regions. This is a direct consequence of the back reaction of the magnetic field on the flow. Such a difference is not statistically significant for the supersonic turbulence. This is consistent with our previous results (Fig. 6 (c) and Fig. 11 (a)) that the effect of the back reaction on the velocity in the supersonic case is negligible.

The angle between $\bm{u}$ and $\bm{b}$ controls the magnetic induction or amplification term and the alignment between $\bm{u}$ and $\bm{b}$ is statistically higher in the saturated stage as compared to the kinematic stage for subsonic turbulence (Fig. 12 (c)) and the difference is not that significant for the supersonic turbulence (even for the strong-field regions, Fig. 12 (d)). This mostly implies a reduction in induction due to enhanced alignment between $\bm{u}$ and $\bm{b}$ for the subsonic case but not so much for the supersonic case (because of locally strong shocks). Thus, for the subsonic turbulence, the magnetic field evolves in such a way as to enhance the level of alignment between the velocity and magnetic field.

Finally, the back reaction of the magnetic field on the velocity field via the Lorentz force is controlled by the angle between $\bm{j}$ and $\bm{b}$, for which the alignment is enhanced in the saturated stage as compared to the kinematic stage (Fig. 12 (e)). However, here, the difference is higher for the supersonic case as compared to the subsonic one. The effect is enhanced in the strong-field regions, i.e., the field is more aligned with the current density where the magnetic energy is higher than its rms value (Fig. 12 (f)). In the saturated stage of the subsonic turbulence, the enhanced level of alignment between $\bm{j}$ and $\bm{b}$ would diminish the Lorentz force and the field advection by velocity would become dominant. This would give rise to a higher level of alignment between the velocity and magnetic field, which we see in Fig. 12 (c). Such a difference is not seen in the case of supersonic turbulence because of the presence of strong shocks.

Besides the correlation length of the magnetic field (Fig. 6 (d)), the following characteristic magnetic length scales (or equivalently wavenumbers) can be used to further study the local magnetic field structure [36, 42],

These wavenumbers (or length scales) can be related to the physical effects and structure of magnetic fields. $k_{\parallel}$ is a probe of variation of the magnetic field along itself and is related to the length scale associated with greatest stretching of magnetic field lines (which maximises magnetic field amplification). $k_{\bm{b}\times\bm{j}}$ is a probe of the magnetic field across itself and is related to resistive dissipation. $k_{\bm{b}\cdot\bm{j}}$ is a probe of magnetic field variations along a direction perpendicular to both magnetic field ($\bm{b}$) and Lorentz force ($\bm{j}\times\bm{b}$) and is related to highest compressive motions (maximising the effect of compression on magnetic field). $k_{\mathrm{rms}}$ measures the overall variation of magnetic fields. For a subsonic fluctuation dynamo with $\text{Pm}\gg 1$, it is shown that the magnetic field organises itself into folded structures [42, 68]. Furthermore, the structures are folded sheets in the kinematic stage (identified by the condition, $k_{\parallel}\lesssim k_{\bm{b}\cdot\bm{j}}\ll k_{\bm{b}\times\bm{j}}\sim k_{\mathrm{rms}}$) and folded ribbons in the saturated stage (identified by the condition, $k_{\parallel}\ll k_{\bm{b}\cdot\bm{j}}\lesssim k_{\bm{b}\times\bm{j}}\sim k_{\mathrm{rms}}$). Since our simulations are with $\text{Pm}\gtrsim 1$ (highest Pm is $3$, for $\text{Rm}=6000$ and $\text{Re}=2000$ runs), we do not necessarily expect to see such folded structures. However, we compute these characteristic scales (Eq. 13 – Eq. 16) to further study the dynamics of growing and saturated magnetic fields in subsonic and supersonic turbulence.

Fig. 13 shows the temporal evolution of wavenumbers, $k_{\parallel}$ (a), $k_{\bm{b}\cdot\bm{j}}$ (b), $k_{\bm{b}\times\bm{j}}$ (c), and $k_{\mathrm{rms}}$ (d), for subsonic ($\mathcal{M}0.1\text{Rm}6000$) and supersonic ($\mathcal{M}10\text{Rm}6000$) turbulent flows (both at $\text{Pm}=3$). Furthermore, the average values of these wavenumbers with one-sigma fluctuations in the kinematic and saturated stages are provided in the legend. For subsonic turbulence, in the kinematic stage, $k_{\parallel}<k_{\bm{b}\cdot\bm{j}}\lesssim k_{\bm{b}\times\bm{j}}\lesssim k_{\mathrm{rms}}$, and thus magnetic structures can probably be characterised as folded ribbons, but structures in the saturated stage are neither folded ribbons nor folded sheets. All four wavenumbers decrease as the magnetic field saturates for the subsonic case.

For the supersonic turbulence, based on these wavenumbers, the structures are never folded sheets or folded ribbons. All wavenumbers except $k_{\bm{b}\cdot\bm{j}}$ decrease as the field grows and saturates. This shows that the dissipation takes place at a smaller length scale in the saturated stage as compared to the kinematic stage for the supersonic turbulence. However, the overall magnetic wavenumber, $k_{\mathrm{rms}}$, still increases. The length scales associated with both the greatest stretching and compression also grows as the magnetic field saturates. On comparing all four wavenumbers between subsonic and supersonic turbulence, $k_{\parallel}$ and $k_{\bm{b}\times\bm{j}}$ are always smaller for the subsonic flow, but $k_{\bm{b}\cdot\bm{j}}$ and $k_{\mathrm{rms}}$ are comparable in the kinematic stage and are higher for the supersonic case in the saturated stage. This shows that the length scale associated with greatest stretching and compression grow as the field saturates for both the subsonic and supersonic cases. However, as the magnetic field grows and saturates, the resistive dissipation scale increases for the subsonic case, but decreases for the supersonic case. The overall magnetic field scale grows as the magnetic field saturates for both Mach numbers (same as the magnetic correlation length in Fig. 6 (d)).

We now study the local stretching and compression of magnetic field lines by the turbulent velocity using the eigenvalues and eigenvectors of the strain tensor. Such an analysis is done for isotropic subsonic turbulence [36, 35, 68], with a different motivation for slightly supersonic turbulence [$\mathcal{M}\sim 2$ in 69], and for different setups such as magnetic fields in rotating convection simulations [70] and decaying magnetic field simulations [71]. Here, we aim to understand the effect of growing magnetic fields and compressibility on the local stretching and compression of magnetic field lines by analysing the strain tensor. First, we compute the strain tensor, $S_{ij}=(1/2)\,(u_{i,j}+u_{j,i})$, at each point in the domain using a sixth-order finite difference scheme and then calculate its eigenvalues ($\lambda_{i}$) and eigenvectors ($\bm{e}_{i}$). We then arrange the eigenvalues in the order $\lambda_{1}>\lambda_{2}>\lambda_{3}$ and let the corresponding eigenvectors be $\bm{e}_{1},\bm{e}_{2},$ and $\bm{e}_{3}$. For an incompressible flow, the sum of eigenvalues, $\lambda_{1}+\lambda_{2}+\lambda_{3}=0$ (approximately applicable for our subsonic case) and $\lambda_{1}>0$ and $\lambda_{3}<0$ [72, 73]. This need not be the case for our supersonic runs. In general, $\bm{e}_{1}$ (positive $\lambda_{1}$) corresponds to the direction of local magnetic field line stretching, $\bm{e}_{3}$ (negative $\lambda_{3}$) corresponds to the direction of local magnetic field line compression, and $\bm{e}_{2}$ (also referred to as the ‘null’ direction) can be either, depending on the sign of $\lambda_{2}$ [36, 42] (especially, see Fig. 10 in [42]). The magnitude of $\lambda_{1}$ (when it is positive) and $\lambda_{3}$ (when it is negative) can be considered as the strength of local magnetic field line stretching and compression, respectively. For clarity, throughout the rest of the paper, we refer $\lambda_{1},\lambda_{2},$ and $\lambda_{3}$ as ${\lambda}_{\mathrm{stre}},{\lambda}_{\mathrm{null}}$, and ${\lambda}_{\mathrm{comp}}$ and the corresponding eigenvectors as ${\bm{e}}_{\mathrm{stre}},{\bm{e}}_{\mathrm{null}}$, and ${\bm{e}}_{\mathrm{comp}}$.

Fig. 14 show the PDF of all three eigenvalues normalised to the rms velocity for subsonic ($\mathcal{M}0.1\text{Rm}6000$, a) and supersonic ($\mathcal{M}10.0\text{Rm}6000$, b) turbulence in kinematic and saturated stages. As expected, for the subsonic case, it can be seen that ${\lambda}_{\mathrm{comp}}$ is always negative and ${\lambda}_{\mathrm{stre}}$ is always positive. This is not the case for the supersonic run, i.e., ${\lambda}_{\mathrm{comp}}$ can be positive for $\mathcal{M}10\text{Rm}6000$ (though the probability of this happening is low, see Fig. 14 (b)). When ${\lambda}_{\mathrm{null}}\approx 0$, the magnetic field is unaffected along the ${\bm{e}}_{\mathrm{null}}$ direction. The magnitude of ${\lambda}_{\mathrm{null}}$ is always smaller than the other two eigenvalues. On average, ${\lambda}_{\mathrm{stre}}\sim-{\lambda}_{\mathrm{comp}}\sim(3~{}\text{--}~{}5){\lambda}_{\mathrm{null}}$ and this is consistent with previous results for isotropic hydrodynamic turbulence [73]. The absolute value of all three eigenvalues is smaller for the supersonic case (compare $x$-axis in Fig. 14 (a) and Fig. 14 (b)). Thus the efficiency of local stretching and compression of magnetic field lines, which leads to magnetic field amplification, is lower for supersonic turbulence. This is also probably the reason for the smaller ratio of saturated magnetic to kinetic energy for compressible runs (see Fig. 2 (b)). As the magnetic field saturates, the absolute value of all three eigenvalues statistically decreases for the subsonic case, but increases for the supersonic case (as shown by the mean value in the legend of Fig. 14). Thus, the local field line stretching and compression decreases for the subsonic case in the saturated stage, but increases in the supersonic case. This is probably to counter high enhancement in diffusion compared to amplification for supersonic turbulence (see Fig. 12 and Table 2).

The amplification of magnetic fields due to field line stretching will be maximal when the field line stretching direction (${\bm{e}}_{\mathrm{stre}}$) is aligned with the magnetic field. Similarly, the effect of field line compression is maximal when the field line compression direction (${\bm{e}}_{\mathrm{comp}}$) is perpendicular to the local magnetic field. Fig. 15 show the total (Fig. 15 (a, c, e)) and conditional (based only on the strong-field regions, $b^{2}/b_{\rm rms}^{2}>1$, Fig. 15 (b, d, f)) PDF of the angle between all three eigenvectors (${\bm{e}}_{\mathrm{comp}},{\bm{e}}_{\mathrm{null}},$ and ${\bm{e}}_{\mathrm{stre}}$) and magnetic fields ($\bm{b}$) in the kinematic and saturated stages for subsonic and supersonic turbulent flows. For the supersonic case (see Fig. 14 (b)), we only consider those regions where ${\lambda}_{\mathrm{comp}}<0$ (implying local magnetic field line compression) and ${\lambda}_{\mathrm{stre}}>0$ (implying local magnetic field line stretching). All the possible angles between these three eigenvectors and magnetic fields are not equiprobable.

In the kinematic stage, for the subsonic turbulence, the highest probable angle between the field line compression direction and magnetic field is $90\degree\text{or}~{}270\degree(|\cos(\theta)_{{\bm{e}}_{\mathrm{comp}},\bm{b}}|\approx 0)$ and that between the field line stretching direction and magnetic field is $0\degree\text{or}~{}180\degree\left(|\cos(\theta)_{{\bm{e}}_{\mathrm{stre}},\bm{b}}|\approx 1\right)$. Thus, the magnetic field is more aligned with the field line stretching direction (and also the null direction) and perpendicular to the direction of the field line compression direction. This maximises magnetic field amplification. This also suggest that the magnetic field statistically lies more in the ${\bm{e}}_{\mathrm{stre}}~{}\text{--}~{}{\bm{e}}_{\mathrm{null}}$ plane. As the field saturates in subsonic turbulence, the more probable angles for both ${\bm{e}}_{\mathrm{comp}}$ and ${\bm{e}}_{\mathrm{stre}}$ lie in the range of $|\cos(\theta)|=0.6~{}\text{--}~{}0.7$ (${\bm{e}}_{\mathrm{null}}$ is still more aligned with $\bm{b}$, but the level of alignment has decreased). Thus, the amplification due to field line stretching and the effect of local compression is reduced in the saturated stage as compared to the kinematic stage via changes in these alignments. Moreover, these differences between the kinematic and saturated stages are enhanced in the strong-field regions (Fig. 15 (b, d, f)).

These trends are different for supersonic turbulence. In the kinematic stage, the magnetic field is overall more aligned with the ${\bm{e}}_{\mathrm{stre}}$ and ${\bm{e}}_{\mathrm{null}}$ and also orthogonal to ${\bm{e}}_{\mathrm{comp}}$ as in the subsonic case. Thus, here too, the magnetic field statistically lies in the ${\bm{e}}_{\mathrm{stre}}~{}\text{--}~{}{\bm{e}}_{\mathrm{null}}$ plane. However, as the field saturates, the difference in the distribution of these angles is not as significant as in the subsonic case. In fact, ${\bm{e}}_{\mathrm{comp}}$ is more aligned with $\bm{b}$ in the saturated stage (although not in the strong-field regions; see Fig. 15 (b)), enhancing the effect of local compression regions with $b/b_{\rm rms}\leq 1$. On the other hand, in strong-field regions (Fig. 15 (f)), the orthogonality between ${\bm{e}}_{\mathrm{stre}}$ and $\bm{b}$ is slightly enhanced in the saturated stage, but the overall distribution of $|\cos(\theta)_{{\bm{e}}_{\mathrm{stre}},\bm{b}}|$ (Fig. 15 (e)) is roughly the same for both the kinematic and saturated stages.

We now explore the magnetic energy evolution equation and directly calculate the local growth and dissipation terms. The evolution of magnetic energy, for a fixed resistivity, $\eta$, can be described by the following equation (see Sec. A for the derivation),

The first term is due to stretching and compression of magnetic field lines by the turbulent flow and can increase (e.g., by stretching of magnetic field lines) or decrease (e.g., unstretching of magnetic field lines) the magnetic energy. Statistically, both can happen in a turbulent medium. This term is computed as follows. First, the local magnetic field is projected along the three eigenvectors of the rate of strain tensor (${\bm{e}}_{\mathrm{comp}},{\bm{e}}_{\mathrm{null}},$ and ${\bm{e}}_{\mathrm{stre}}$) and let these projected vectors be $\bm{b}_{\rm comp},\bm{b}_{\rm null},$ and $\bm{b}_{\rm stre}$. The first term is then calculated as the sum $\sum_{i=\rm comp,null,stre}(\lambda_{i}/u_{\rm rms})(\bm{b}_{i}/b_{\rm rms})^{2}$. The second term can also reduce or enhance magnetic energy locally depending on the divergence of the velocity. As expected, this term is negligible for the subsonic case, but does play an important role in the supersonic turbulence. The third term is the dissipative term, which always reduces the magnetic energy.

Fig. 16 shows the total and conditional (only in the strong-field regions) PDFs of the first, second, and first and second terms combined of Eq. 17 for subsonic and supersonic turbulence. For the subsonic case, the second term makes a small difference since $\nabla\cdot(\bm{u}/u_{\rm rms})$ is negligible. Thus, the magnetic field growth term (first term or combination of first two terms in the magnetic energy evolution equation for the subsonic case, Eq. 17) statistically decreases as the dynamo saturates. This implies that the growing magnetic field reduces its own amplification. The differences are further enhanced in strong-field regions (Fig. 16 (f)). For the supersonic case, the combined term is statistically higher in comparison to the subsonic one, but there is not much difference between the corresponding kinematic and saturated stages (Fig. 16 (e), also very similar mean value, $\mu$, for the kinematic and saturated stages). In strong-field regions, the local growth term is slightly reduced on saturation for the supersonic case. Thus, on saturation, the local growth term is reduced throughout the volume for the subsonic case, but mostly in the strong-field regions for the supersonic case. However, the mean of the local growth term ($\mu$ in Fig. 16 (e) and Fig. 16 (f)) always remains positive for all cases. Thus, the magnetic field always grows. Even in the saturated state, the magnetic energy is amplified to counter diffusion.

Fig. 17 show the PDF for the last term on the right-hand side of Eq. 17, $j^{2}$, which probes the dissipation of magnetic energy. The dissipation is also statistically higher for the supersonic case in comparison to the subsonic one. For both cases, the magnetic dissipation statistically decreases as the dynamo saturates and the decrease is more statistically significant for the strong-field regions.

Table 2 shows the ratio of the mean value of local growth term ($\sum_{i=\rm comp,null,stre}(\lambda_{i}/u_{\rm rms})(\bm{b}_{i}/b_{\rm rms})^{2}-(1/2)(b/b_{\rm rms})^{2}(\nabla\cdot(\bm{u}/u_{\rm rms}))$ to the mean value of the local dissipation term ($j^{2}/j_{\rm rms}^{2}$) in the kinematic and saturated stages for subsonic and supersonic turbulence (the magnetic resistivity, $\eta$, is the same for both runs). The ratio is always higher in the kinematic stage as compared to the saturated stage (see (kin-sat) in Table 2), especially in the strong-field regions. Note that, unlike the subsonic case, the difference of the ratio between the kinematic and saturated stages is small for the supersonic case and turns out to be significant only in the strong-field regions. Thus, although both growth and dissipation of magnetic fields decrease as the field saturates, the magnetic dissipation relative to the amplification is enhanced. This leads to the saturation of the fluctuation dynamo.