Fluctuation Dynamo in a Collisionless, Weakly Magnetized Plasma

Figure 1(a) displays the box-averaged kinetic and magnetic energies versus time for the ${\beta }_{{\rm{i}}0}={10}^{6}$ run. The kinetic energy saturates within $t\approx {t}_{\mathrm{corr}}$ and a large-scale smooth flow is established. On average, this flow amplifies the large-scale seed magnetic field, and rapid growth of magnetic energy occurs at $k{\rho }_{{\rm{i}}}\approx 0.5\mbox{--}1$ (${kL}/2\pi \approx 4\mbox{--}8$), adiabatically driving $\langle {{\rm{\Delta }}}_{{\rm{i}}}\rangle \gt 0$ (Figure 1(b); see also Figure 2, $t/{t}_{\mathrm{corr}}=1$). Because ${\beta }_{{\rm{i}}0}\gg 1$, mirror instabilities are readily excited. From the standpoint of these mirror modes, the initial seed field (${kL}/2\pi =1,2$) behaves as a local "mean" field on which they grow with oblique polarization ${k}_{{\boldsymbol{B}}{\boldsymbol{\times }}{\boldsymbol{J}}}\gt {k}_{\parallel }\gt {k}_{{\boldsymbol{B}}{\boldsymbol{\cdot }}{\boldsymbol{J}}}$ (Figure 1(c), $t/{t}_{\mathrm{corr}}\,\lesssim 1.5$), where

$$\begin{eqnarray*}&&{k}_{\parallel }\equiv {\left(\displaystyle \frac{\langle | {\boldsymbol{B}}{\boldsymbol{\cdot }}{\boldsymbol{\nabla }}{\boldsymbol{B}}{| }^{2}\rangle }{\langle {B}^{4}\rangle }\right)}^{1/2},\end{eqnarray*}$$

$$\begin{eqnarray*}&&{k}_{{\boldsymbol{B}}{\boldsymbol{\times }}{\boldsymbol{J}}}\equiv {\left(\displaystyle \frac{\langle | {\boldsymbol{B}}{\boldsymbol{\times }}{\boldsymbol{J}}{| }^{2}\rangle }{\langle {B}^{4}\rangle }\right)}^{1/2},\,{k}_{{\boldsymbol{B}}{\boldsymbol{\cdot }}{\boldsymbol{J}}}\equiv {\left(\displaystyle \frac{\langle | {\boldsymbol{B}}{\boldsymbol{\cdot }}{\boldsymbol{J}}{| }^{2}\rangle }{\langle {B}^{4}\rangle }\right)}^{1/2}\end{eqnarray*}$$

are the characteristic wavenumbers of magnetic-field variation along (${k}_{\parallel }$) and across (${k}_{{\boldsymbol{B}}{\boldsymbol{\times }}{\boldsymbol{J}}},{k}_{{\boldsymbol{B}}{\boldsymbol{\cdot }}{\boldsymbol{J}}}$) itself, and ${\boldsymbol{J}}={\boldsymbol{\nabla }}{\boldsymbol{\times }}{\boldsymbol{B}}$ is the current density (see Section 3.2.1 of S04). Firehose-unstable modes are also triggered on ion-Larmor scales in regions of locally decreasing field and, in concert with mirror-unstable modes, ultimately generate sharp features in the magnetic field that trap and pitch-angle scatter particles. The latter produces an effective collisionality ${\nu }_{\mathrm{eff}}$, which drives ${{\rm{\Delta }}}_{{\rm{i}}}$ toward marginal stability (Figure 2, $t/{t}_{\mathrm{corr}}=2,5$) and ties the pressure anisotropy to the parallel rate of strain (Figure 1(b), $t/{t}_{\mathrm{corr}}\gtrsim 3$). This leads to a Braginskii-like relation, ${{\rm{\Delta }}}_{{\rm{i}}}\,\approx 3\hat{{\boldsymbol{b}}}\hat{{\boldsymbol{b}}}\,{\boldsymbol{:}}\,\,{\boldsymbol{\nabla }}{\boldsymbol{u}}/{\nu }_{\mathrm{eff}}$, in which a balance is obtained between adiabatic production and collisional relaxation, with ${\nu }_{\mathrm{eff}}\lesssim {{\rm{\Omega }}}_{{\rm{i}}}$.

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At the same time that the firehose and mirror instabilities saturate at $k{\rho }_{{\rm{i}}}\lesssim 1$ with $\delta B/{B}_{0}\sim 1$, the magnetic field acquires energy at sub-ion-Larmor scales due to field-line stretching and folding by the large-scale flow (Figure 1(d), $t/{t}_{\mathrm{corr}}\gtrsim 5$). The result is a much flatter angle-integrated magnetic-energy spectrum, $M(k)\equiv \tfrac{1}{2}\int d{{\rm{\Omega }}}_{{\boldsymbol{k}}}\,{k}^{2}\langle | {\boldsymbol{B}}({\boldsymbol{k}}){| }^{2}\rangle$ (Figure 3, $t/{t}_{\mathrm{corr}}=5$), than is seen in corresponding $\mathrm{Pm}\gg 1$ MHD simulations. A change in the dominant magnetic-field topology accompanies this growth, with ${k}_{{\boldsymbol{B}}{\boldsymbol{\times }}{\boldsymbol{J}}}\gt {k}_{{\boldsymbol{B}}{\boldsymbol{\cdot }}{\boldsymbol{J}}}\gt {k}_{\parallel }$ indicating a folded geometry in which the field varies quickly (slowly) across (along) itself (Figure 1(c), $t/{t}_{\mathrm{corr}}\gtrsim 2$).⁴

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Eventually, this period of rapid growth ends. Figure 1(b) indicates that the reduction in growth rate is due to two effects. First, an appreciable fraction of the magnetic energy migrates to resistive scales, and magnetic diffusion becomes important. Second, $\hat{{\boldsymbol{b}}}\hat{{\boldsymbol{b}}}\,{\boldsymbol{:}}\,\,{\boldsymbol{\nabla }}{\boldsymbol{u}}$ is sharply reduced between $t/{t}_{\mathrm{corr}}\approx 3$ and 5, a feature that we attribute to feedback from firehose/mirror fluctuations (e.g., Schekochihin et al. 2008; Rosin et al. 2011; Rincon et al. 2015) and to parallel-viscous damping. Indeed, while $\langle | {\boldsymbol{\nabla }}{\boldsymbol{u}}{| }^{2}\rangle /\langle {u}^{2}\rangle$ increases substantially in that time interval, $\langle | \hat{{\boldsymbol{b}}}\hat{{\boldsymbol{b}}}\,{\boldsymbol{:}}\,\,{\boldsymbol{\nabla }}{\boldsymbol{u}}{| }^{2}\rangle /\langle {u}^{2}\rangle$ remains nearly constant (Figure 1(e)); i.e., the parallel rate of strain is suppressed. That this is concurrent with the development of an angle-integrated kinetic-energy spectrum, $E(k)\equiv \tfrac{1}{2}\int d{{\rm{\Omega }}}_{{\boldsymbol{k}}}\,{k}^{2}\langle | {\boldsymbol{u}}({\boldsymbol{k}}){| }^{2}\rangle$ (Figure 3, $t/{t}_{\mathrm{corr}}=5,18$), that is Kolmogorov (1941; i.e., $\propto {k}^{-5/3}$) suggests that not all fluid motions cascade to the smallest scales; i.e., Re is anisotropic.

Thereafter, $\langle {B}^{2}\rangle$ grows exponentially (Figure 1(a), $t/{t}_{\mathrm{corr}}\gtrsim 5$), much as in the kinematic-diffusive stage of the large-$\mathrm{Pm}$ MHD dynamo (e.g., Schekochihin et al. 2002b), with a growth rate $\gamma \doteq d\mathrm{ln}\langle {B}^{2}{\rangle }^{1/2}/{dt}=0.0093{{\rm{\Omega }}}_{{\rm{i}}0}=0.15{t}_{\mathrm{corr}}^{-1}\approx {u}_{\mathrm{rms}}/L$ that becomes comparable at all scales (Figure 1(d), $t/{t}_{\mathrm{corr}}\gtrsim 5$). The folded magnetic-field geometry previously established persists (Figure 1(c)), and M(k) develops a Kazantsev (1968) ${k}^{3/2}$ scaling with a peak near the resistive scale (Figure 3, $t/{t}_{\mathrm{corr}}=18$). Such a folded structure, accompanied by ion-Larmor-scale irregularities driven by firehose/mirror, is evident in the rightmost panels of Figure 3, which display pseudo-color images of B/${B}_{\mathrm{rms}}$ and u/${u}_{\mathrm{rms}}$ in a representative 2D slice. Suppression of parallel velocity variation is also apparent; while the turbulent velocity field is primarily large-scale, filamentary structures of near-constant u develop along magnetic lines of force. Thus, there is a dynamical feedback of the magnetic field on the large-scale flow, even in the absence of a dynamically important Lorentz force, belying the "kinematic" moniker.

Because of the continuous energy injection and consequent magnetic-field amplification, along with insufficient scale separation between L and ${\rho }_{{\rm{i}}}$, exact marginal firehose/mirror stability cannot be maintained and a residual $\langle {{\rm{\Delta }}}_{{\rm{i}}}\rangle \,\approx (0.02\mbox{--}0.03)\gg 1/{\beta }_{{\rm{i}}}$ persists for $t/{t}_{\mathrm{corr}}\gtrsim 5$ (Figure 1(b)), with the bulk of the plasma approximately following the mirror threshold as ${\beta }_{{\rm{i}}}$ decreases (Figure 2, $t/{t}_{\mathrm{corr}}=18$). The regulation of ${{\rm{\Delta }}}_{{\rm{i}}}$ is imperfect; in order for saturated firehose/mirror instabilities to tightly regulate the pressure anisotropy near marginal stability, ${\nu }_{\mathrm{eff}}\sim S{\beta }_{{\rm{i}}}$ (Kunz et al. 2014a; Melville et al. 2016), where S is the parallel rate of strain at the viscous scale (where it is largest). However, at $t/{t}_{\mathrm{corr}}=5$, $S/{{\rm{\Omega }}}_{{\rm{i}}}\sim {10}^{-2}$ and ${\beta }_{{\rm{i}}}\sim {10}^{5}$, thus requiring ${\nu }_{\mathrm{eff}}\sim {10}^{3}\,{{\rm{\Omega }}}_{{\rm{i}}}$! Instead, ${\nu }_{\mathrm{eff}}\ll {{\rm{\Omega }}}_{{\rm{i}}}$ in the kinematic phase in both simulations, a point that we have confirmed both indirectly, by comparing $\hat{{\boldsymbol{b}}}\hat{{\boldsymbol{b}}}\,{\boldsymbol{:}}\,\,{\boldsymbol{\nabla }}{\boldsymbol{u}}$ and ${{\rm{\Delta }}}_{{\rm{i}}}$ to infer ${\nu }_{\mathrm{eff}}\approx 3\langle {\boldsymbol{B}}{\boldsymbol{B}}\,{\boldsymbol{:}}\,\,{\boldsymbol{\nabla }}{\boldsymbol{u}}\rangle /\langle {B}^{2}{{\rm{\Delta }}}_{{\rm{i}}}\rangle$, and directly, by calculating the mean time over which μ changes for individually tracked particles (using the method described in Kunz et al. 2014a and Squire et al. 2017). The result is shown in Figure 4(d) for ${\beta }_{{\rm{i}}0}={10}^{4};$ qualitatively identical behavior is observed for ${\beta }_{{\rm{i}}0}={10}^{6}$.

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There are two processes that contribute to ${\nu }_{\mathrm{eff}}$, depending upon whether the majority of the particles' gyroradii are above or below the reversal scale of the magnetic field. In the former case, those particles sample several field-reversing folds during their gyromotion and thus undergo Bohm-like diffusion with ${\nu }_{\mathrm{eff}}\sim {{\rm{\Omega }}}_{{\rm{i}}}$. On the other hand, if the majority of particles have gyroradii below the field-reversal scale and remain well magnetized, then ${\nu }_{\mathrm{eff}}$ is determined mainly by pitch-angle scattering off of firehose fluctuations, which populate regions of weak magnetic field where ${{\rm{\Delta }}}_{{\rm{i}}}\lt 0$.⁵ As these regions occur primarily at the bends of the folded fields, we expect ${\nu }_{\mathrm{eff}}\sim {k}_{\parallel }{v}_{\mathrm{thi}}$, the inverse timescale for a thermal particle to traverse the length of a fold. Both of these contributions may be important, depending upon the structure of the magnetic field and the local magnetization of the plasma. In our runs, we witness only a brief moment in the evolution with ${\nu }_{\mathrm{eff}}\sim {{\rm{\Omega }}}_{{\rm{i}}}$, giving way to ${\nu }_{\mathrm{eff}}\sim {k}_{\parallel }{v}_{\mathrm{thi}}\ll {{\rm{\Omega }}}_{{\rm{i}}}$ in the kinematic phase. It is only once ${k}_{\parallel }{v}_{\mathrm{thi}}\sim S{\beta }_{{\rm{i}}}$ that efficient regulation of ${{\rm{\Delta }}}_{{\rm{i}}}$ is possible (Section 3.3).

One consequence of ${\nu }_{\mathrm{eff}}\ll {{\rm{\Omega }}}_{{\rm{i}}}$ is an anisotropic viscosity, with Reynolds numbers ${\rm{Re}}\,\equiv \,{u}_{\mathrm{rms}}/({k}_{F}{\eta }_{\mathrm{visc}})$ differing in the parallel and perpendicular directions: ${{\rm{Re}}}_{\parallel }\,\ll \,{{\rm{Re}}}_{\perp }$ (Braginskii 1965). While the magnetic-field growth is controlled by ${{\rm{Re}}}_{\parallel }$ (as $d\mathrm{ln}B/{dt}\simeq \hat{{\boldsymbol{b}}}\hat{{\boldsymbol{b}}}\,{\boldsymbol{:}}\,\,{\boldsymbol{\nabla }}{\boldsymbol{u}}\sim ({u}_{\mathrm{rms}}/L){{\rm{Re}}}_{\,\parallel }^{1/2}$), the viscous cutoff ${{\ell }}_{\mathrm{visc}}$ seen in Figure 3 is determined by ${{\rm{Re}}}_{\perp }$ through the Kolmogorov relation ${{\ell }}_{\mathrm{visc}}\sim L\,{{\rm{Re}}}_{\perp }^{-3/4}$. Using classical transport theory to estimate the effective perpendicular ion viscosity ${\eta }_{\mathrm{visc},\perp }\sim 0.1{\rho }_{{\rm{i}}}^{2}{\nu }_{\mathrm{eff}}$, we find $L/{{\ell }}_{\mathrm{visc}}\sim {({ML}{{\rm{\Omega }}}_{{\rm{i}}}/{\rho }_{{\rm{i}}}{\nu }_{\mathrm{eff}})}^{3/4}$. Taking M, ${{\rm{\Omega }}}_{{\rm{i}}}$, ${\rho }_{{\rm{i}}}$, and ${\nu }_{\mathrm{eff}}$ from the run, we calculate a minimum value of $L/{{\ell }}_{\mathrm{visc}}\sim 10$ at $t/{t}_{\mathrm{corr}}\approx 5$, which grows exponentially to $L/{{\ell }}_{\mathrm{visc}}\sim 100$ at $t/{t}_{\mathrm{corr}}\approx 18$. This roughly agrees with the evolution shown in Figure 3. Likewise, ${{\rm{Re}}}_{\parallel }$ can be calculated using the parallel viscosity for a magnetized plasma, ${\eta }_{\mathrm{visc},\parallel }\,\sim {v}_{\mathrm{thi}}^{2}/{\nu }_{\mathrm{eff}}$. Once ${\nu }_{\mathrm{eff}}\sim {k}_{\parallel }{v}_{\mathrm{thi}}$, ${{\rm{Re}}}_{\parallel }\,\sim \,M({k}_{\parallel }/{k}_{F})\sim 1$, suggestive of a $\mathrm{Pm}\gg 1$ dynamo and consistent with the large drop in E(k) at ${kL}/2\pi \approx 2$. Preliminary Braginskii-MHD dynamo simulations with $1\sim {{\rm{Re}}}_{\parallel }\,\ll \,{{\rm{Re}}}_{\perp }\,\sim \,\mathrm{Rm}$ and $-2/{\beta }_{{\rm{i}}}\leqslant {{\rm{\Delta }}}_{{\rm{i}}}\leqslant 1/{\beta }_{{\rm{i}}}$ enforced (e.g., following Sharma et al. 2006 and Kunz et al. 2012) exhibit similar spectra and field-anisotropic flow to those presented here.

Figure 4(a) shows the evolution of kinetic and magnetic energies for the ${\beta }_{{\rm{i}}0}={10}^{4}$ run. After evolving through the rapid-growth phase and a brief exponential kinematic phase, the field growth slows to become secular between $t/{t}_{\mathrm{corr}}\approx 12\mbox{--}24$ as the Lorentz force begins to affect the parallel-viscous-scale eddies (Figure 3, $t/{t}_{\mathrm{corr}}=12;$ cf. Maron et al. 2004; Cho et al. 2009). Saturation is ultimately reached with $\langle {B}^{2}/4\pi \rangle \approx \langle {m}_{{\rm{i}}}{{nu}}^{2}\rangle$ via a reduction of $\hat{{\boldsymbol{b}}}\hat{{\boldsymbol{b}}}\,{\boldsymbol{:}}\,\,{\boldsymbol{\nabla }}{\boldsymbol{u}}$ (Figure 4(b), $t/{t}_{\mathrm{corr}}\gtrsim 25;$ S04).⁶ The ordering ${k}_{{\boldsymbol{B}}{\boldsymbol{\times }}{\boldsymbol{J}}}\gt {k}_{{\boldsymbol{B}}{\boldsymbol{\cdot }}{\boldsymbol{J}}}\gt {k}_{\parallel }$ established in the kinematic phase is preserved (Figure 4(c)), but the two perpendicular scales become closer to one another in saturation; i.e., the folded sheets evolve towards a ribbon-like structure, as seen in the $\mathrm{Pm}\gg 1$ MHD dynamo (S04).

Despite the box-averaged equipartition between kinetic and magnetic energies, this balance is not scale-by-scale (Figure 3, $t/{t}_{\mathrm{corr}}=57$). Rather, there is an excess of the former at the forcing scales (as $E(k)\propto {k}^{-5/3}$) and an excess of the latter at smaller scales (as $M(k)\propto {k}^{3/2}$), although its peak has shifted toward smaller wavenumbers (${kL}/2\pi \approx 5\mbox{--}10$) where the resistivity is negligible. It is because the folds exhibit spatial coherence at the flow scale that allows them to exert a back-reaction on the flow via the Lorentz force. Whether the shrinking gap between the parallel-viscous scale and the peak in M(k) persists in higher-resolution simulations is of interest in the context of the intracluster magnetic field, whose spectrum is inferred to peak at scales ($\sim 1\,\mathrm{kpc}$) far larger than the resistive scales (e.g., Vogt & Enßlin 2005).

As in the ${\beta }_{{\rm{i}}0}={10}^{6}$ run, the pressure anisotropy becomes Braginskii-like, with $\langle {{\rm{\Delta }}}_{{\rm{i}}}\rangle \propto \langle {\boldsymbol{B}}{\boldsymbol{B}}{\boldsymbol{:}}{\boldsymbol{\nabla }}{\boldsymbol{u}}\rangle /\langle {B}^{2}\rangle \gt 0$ (Figure 4(b), $t/{t}_{\mathrm{corr}}\gtrsim 5$) and ${\nu }_{\mathrm{eff}}\sim {k}_{\parallel }{v}_{\mathrm{thi}}$ (Figure 4(d), $t/{t}_{\mathrm{corr}}\gtrsim 5$). However, once ${\beta }_{{\rm{i}}}$ decreases to $\sim 50$ ($t/{t}_{\mathrm{corr}}\gtrsim 20$), ${\nu }_{\mathrm{eff}}\sim S{\beta }_{{\rm{i}}}$ and ${{\rm{\Delta }}}_{{\rm{i}}}$ is regulated close to the firehose/mirror thresholds (Figure 2,$t/{t}_{\mathrm{corr}}=57$).

Figure 5(a) shows the probability distribution function P(K) of the magnetic curvature $K\equiv | \hat{{\boldsymbol{b}}}\,{\boldsymbol{\cdot }}\,{\boldsymbol{\nabla }}\hat{{\boldsymbol{b}}}|$. In the MHD case, the tail of P(K) relaxes to a ${K}^{-13/7}$ scaling (Schekochihin et al. 2002a) throughout the kinematic and saturated phases, depending only weakly on $\mathrm{Pm}$ (see Figure 25 of S04). While P(K) in the plasma dynamo is peaked at similar values as those found in S04 (${KL}/2\pi \approx 2$), it is generally broader, and is dependent upon whether the host plasma is mirror unstable (blue; 54% by volume), firehose unstable (green; 27%), or stable (purple). Regions that are firehose unstable tend to have the largest curvature, for two reasons. First, ${\rm{\Delta }}\lt 0$ is generically produced in the stretched bends of the field lines, where $d\mathrm{ln}B/{dt}\lt 0$ and K is large. The reduction in effective field-line tension by ${\rm{\Delta }}\lt 0$ reinforces this trend. Second, firehose grows fastest at $k{\rho }_{{\rm{i}}}\sim 1$ and generates sharp kinks in the field lines on these scales. K in mirror-unstable regions is also enhanced by the generation of mirror-shaped field lines. Despite this difference, there remains a strong anti-correlation between B and K in saturation (Figure 5(b)), with $B\propto {K}^{-1/2}$ similar to the MHD case (cf. Figure 17 of S04).

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Finally, Figure 5(c) displays the joint distribution of ${\rho }_{{\rm{i}}}$ and ${k}_{{\boldsymbol{B}}{\boldsymbol{\times }}{\boldsymbol{J}}}$, each computed cell by cell, initially (orange), at the start of the kinematic phase (blue), and in saturation (green). Points rightward (leftward) of the dotted–dashed line exhibit perpendicular magnetic structure on scales $\lesssim {\rho }_{{\rm{i}}}$ ($\gtrsim {\rho }_{{\rm{i}}}$). At early times, this structure is driven by kinetic instabilities and the emergent folded-field geometry, with an appreciable fraction of the plasma having ${\rho }_{{\rm{i}}}$ larger than the field-reversal scale. As B increases, the mode of the distribution crosses into the magnetized region at $t/{t}_{\mathrm{corr}}\approx 5$ and settles when the dynamo saturates ($t/{t}_{\mathrm{corr}}\approx 25$). As this happens, the bulk of the plasma becomes well magnetized on the folding scale.