3.1.1. Generation of pressure anisotropy and triggering of kinetic instabilities

We demonstrate this quantitatively by considering the linearized Vlasov equation for the ions,

(3.4)\begin{equation} \biggl( \frac{\partial }{\partial t} + \textrm{i} k_{\parallel} v_{\parallel} \biggr) \delta f_i(t,k_{\parallel},v) + \frac{e\varphi(t,k_{\parallel})}{T_{i0}} \textrm{i} k_{\parallel} v_{\parallel}\, f_{{\rm M},i}(v) = 0, \end{equation}

with the potential (3.3) and integrating it forward in time from the initial condition (3.2). The result is

(3.5)\begin{align} \delta f_i(t,k_{\parallel},v) = f_{{\rm M},i}(v) \exp({-\textrm{i} k_{\parallel} v_{\parallel} t}) \left[ \frac{\delta n(0,k_{\parallel})}{n_0} - \textrm{i} k_{\parallel} v_{\parallel} \int^t_0 \textrm{d} t' \exp({\textrm{i} k_{\parallel} v_{\parallel} t'}) \frac{T_e}{T_{i0}} \frac{\delta n(t',k_{\parallel})}{n_0} \right] . \end{align}

The first term represents phase mixing of the initial perturbation due to particles free streaming along the (unperturbed) magnetic field. The second term includes the self-consistent response; it may be manipulated, after integration by parts, to help recast (3.5) as

(3.6)\begin{align} \delta f_i(t,k_{\parallel},v) + f_{{\rm M},i}(v)\frac{T_e}{T_{i0}} \frac{\delta n(t,k_{\parallel})}{n_0} &= f_{{\rm M},i}(v) \exp({-\textrm{i} k_{\parallel} v_{\parallel} t}) \left(1+\frac{T_e}{T_{i0}}\right) \frac{\delta n(0,k_{\parallel})}{n_0} \nonumber\\ &\quad +f_{{\rm M},i}(v) \int^t_0 \textrm{d} t' \exp({-\textrm{i} k_{\parallel} v_{\parallel} (t-t')}) \frac{T_e}{T_{i0}} \frac{\partial }{\partial t'} \frac{\delta n(t',k_{\parallel})}{n_0}. \end{align}

Taking the difference of the $m_i v^2_{\perp} /2$ and $m_i v^2_{\parallel}$ moments of (3.6) eliminates the second term on the left-hand side, leaving the following expression for the ion pressure anisotropy driven by the IAW:

(3.7)\begin{align} \varDelta_\textrm{IAW}(t,k_{\parallel}) &= 2 \left(\frac{k_{\parallel} v_{\textrm{th}i0} t}{2}\right)^2 \exp({-(k_{\parallel} v_{\textrm{th}i0} t/2)^2}) \left(1+\frac{T_e}{T_{i0}}\right) \frac{\delta n(0,k_{\parallel})}{n_0} \nonumber\\ &\quad +2 \int^t_0 \textrm{d} t' \left[\frac{k_{\parallel} v_{\textrm{th}i0} (t-t')}{2}\right]^2 \exp({-[k_{\parallel} v_{\textrm{th}i0} (t-t')/2]^2}) \frac{T_e}{T_{i0}}\frac{\partial }{\partial t'} \frac{\delta n(t',k_{\parallel})}{n_0} . \end{align}

All terms involving the combination $k_{\parallel} v_{\textrm {th}i0}t/2$ describe the damping effect of phase mixing on the moments of the perturbed distribution function due to the production of fine-scale structure along $v_{\parallel}$. The first term on the right-hand side of (3.7) arises because no accompanying fine-scale structure is produced along $v_{\perp}$ – magnetized particles cannot stream freely across the magnetic field – and so the $v^2_{\perp}$ and $v^2_{\parallel}$ moments of $\delta f_i$ phase mix differently. (Equivalently, the parallel flows of perpendicular and parallel heat are different.) The result is a transiently produced positive pressure anisotropy. The last term on the right-hand side of (3.7) captures the pressure anisotropy driven by adiabatic invariance as the density fluctuates in time.

The integral in (3.7) can be straightforwardly performed numerically. The resulting evolution of the (spatial) root mean square pressure anisotropy, $\langle \varDelta ^2_\textrm {IAW}(t,x)\rangle ^{1/2}$, is shown in figure 1(a) at a selection of values of $T_e/T_{i0}$; the maximum value of the pressure anisotropy is plotted as a function of $T_e/T_{i0}$ in figure 1(b).Footnote ² As the IAW begins to oscillate and decay, regions with $\delta n(0,x) > 0$ ($<0$) initially drive $\varDelta _\textrm {IAW}(t,x) > 0$ ($<0$) towards the mirror (firehose) threshold. Once $\varDelta _\textrm {IAW}(t,x) \gtrsim 1/\beta _i$ ($\lesssim -2/\beta _i$), those regions become mirror (firehose) unstable. Noting that the maximum value of $\varDelta _\textrm {IAW}$ is a relatively weak function of $T_e/T_{i0}$, we arrive at an amplitude threshold for an IAW to trigger both instabilities,

(3.8)\begin{equation} \alpha = \left|\frac{\delta n}{n_0}\right| \gtrsim \frac{2}{\beta_i} \quad \textrm{(IAW amplitude threshold)}. \end{equation}

For example, for $T_e/T_{i0}=1$, figure 1(b) indicates that $\alpha = 1.74/\beta _i$ is required for $\textrm {max}|\varDelta _\textrm {IAW}|=2/\beta _i$. Because $\beta _i \gg 1$ in many astrophysical plasmas, even small-amplitude fluctuations (viz., those with $\alpha \ll {1}$ but $\alpha \beta _i \gtrsim 2$) generate enough pressure anisotropy to destabilize the plasma. This is the first key result of this paper.

Figure 1.

(a) Solution of (3.7) for the time-dependent root mean square pressure anisotropy of a linear IAW of wavenumber $k_{\parallel}$ and dimensionless amplitude $\alpha$ for various $T_e/T_{i0}$. (b) Maximum pressure anisotropy (divided by $\alpha$) versus $T_e/T_{i0}$; its values at $T_e/T_{i0}=1/3$, $1$ and $3$ are indicated.

3.1.2. Evolution of IAW-driven kinetic instabilities and regulation of pressure anisotropy

For initial density perturbations satisfying (3.8), the times at which the plasma becomes firehose and mirror unstable – $t_{f}$ and $t_{m}$, respectively – may be estimated by asking when the corresponding (approximate) instability parameters, $\varLambda _{f} \doteq -(\varDelta + 2/\beta _i)$ and $\varLambda _{m} \doteq \varDelta - 1/\beta _i$, pass through zero to become positive. Assuming that the combination $\alpha \beta _i$ is sufficiently large so that these times are ${\ll }(|k_{\parallel} | v_{\textrm {th}i0})^{-1}$, (3.7) may be used to approximate the time evolution of the pressure anisotropy as $|\varDelta (t)| \sim \alpha (k_{\parallel} v_{\textrm {th}i0} t)^2$. Then, neglecting factors of order unity,

(3.9)\begin{equation} t_{f}, t_{m} \sim \left( |k_{\parallel}| v_{\textrm{th}i0} \sqrt{\alpha\beta_i} \right)^{-1} . \end{equation}

For $\alpha \beta _i \gtrsim 2$, this is a fraction of the sound-crossing time ${\sim }2{\rm \pi} /|k_{\parallel} |v_{\textrm {th}i0}$.

After the plasma is driven firehose/mirror unstable (i.e. for $t>t_{f},t_{m}$), the pressure anisotropy will continue to grow in amplitude following (3.7) until the instabilities are able to deplete the pressure anisotropy faster than it is being supplied. To determine the times at which this occurs, we first note that the maximum growth rates of the oblique firehose and mirror instabilities at high $\beta _i$, viz. $\gamma _{f} \sim \varOmega _i \varLambda ^{1/2}_{f}$ and $\gamma _{m} \sim \varOmega _i \varLambda ^2_{m}$, respectively (e.g. Yoon et al. Reference Yoon, Wu and de Assis1993; Hellinger Reference Hellinger2007), depend on time through $\varDelta = \varDelta _\textrm {IAW}(t)$.Footnote ³ For times such that $(t/t_{f}-1), (t/t_{m}-1) \ll {1}$ and that the instability parameters are both ${\ll }1$, we can obtain simple analytical expressions for these time-dependent growth rates by Taylor expanding the instability parameters about $t_{f}$ and $t_{m}$ and using $|\varDelta (t)|\sim \alpha (k_{\parallel}v _{\textrm {th}i0}t)^2$. This gives

(3.10a)\begin{gather} \gamma_{f}(t) \sim \varOmega_i \left(\frac{\alpha}{\beta_i}\right)^{1/4} (|k_{\parallel}| v_{\textrm{th}i0})^{1/2} (t-t_{f})^{1/2}, \end{gather}

(3.10b)\begin{gather}\gamma_{m}(t) \sim \varOmega_i \left(\frac{\alpha}{\beta_i}\right) (k_{\parallel} v_{\textrm{th}i0})^2 (t-t_{m})^2. \end{gather}

(Recall that $k_{\parallel}$ refers to the parallel wavenumber of the IAW, not of the kinetic instabilities.) The time-dependent amplification factors of the energies of the microscale magnetic fluctuations that arise due to the firehose and mirror instabilities are, therefore,

(3.11a)\begin{gather} \frac{\delta B^2_{f}(t)}{\delta B^2_{f}(t_{f})} \sim \exp\left[ 2 \int^t_{t_{f}} \textrm{d} t' \gamma_{f}(t') \right] \sim \exp\left[\frac{4\varOmega_i}{3} \left(\frac{\alpha}{\beta_i}\right)^{1/4} (|k_{\parallel}| v_{\textrm{th}i0})^{1/2}(t-t_{f})^{3/2} \right], \end{gather}

(3.11b)\begin{gather}\frac{\delta B^2_{m}(t)}{\delta B^2_{m}(t_{m})} \sim \exp\left[ 2 \int^t_{t_{m}} \textrm{d} t' \gamma_{m}(t') \right] \sim \exp\left[ \frac{2\varOmega_i}{3} \left(\frac{\alpha}{\beta_i}\right) (k_{\parallel} v_{\textrm{th}i0})^2 (t-t_{m})^3 \right], \end{gather}

respectively. This growth in magnetic energy returns the pressure anisotropy to values bounded approximately by the instability thresholds. We argue that, due to the super-exponential growth of the instabilities, this regulation will occur very rapidly once the exponents in (3.11) become order unity (this neglects order-unity prefactors and logarithmic corrections). Therefore, the times at which the IAW-driven pressure anisotropy starts to be regulated by the firehose ($t_{f,\textrm {reg}}$) and mirror ($t_{m,\textrm {reg}}$) instabilities should satisfy

(3.12a)\begin{gather} |k_{\parallel}| v_{\textrm{th}i0} (t_{f,\textrm{reg}} - t_{f}) \sim (|k_{\parallel}| \rho_{i0})^{2/3} \left(\frac{\beta_i}{\alpha}\right)^{1/6}, \end{gather}

(3.12b)\begin{gather}|k_{\parallel}| v_{\textrm{th}i0} (t_{m,\textrm{reg}} - t_{m}) \sim (|k_{\parallel}| \rho_{i0})^{1/3} \left(\frac{\beta_i}{\alpha}\right)^{1/3}, \end{gather}

where we have normalized these times using the IAW wave frequency ${\sim }|k_{\parallel} |v_{\textrm {th}i0}$.

In order for the regulation times given by (3.12) to be valid, they must satisfy our working assumption that $(t/t_{f}-1)$, $(t/t_{m}-1) \ll 1$. This requires $|k_{\parallel} |\rho _{i0} \ll \beta ^{-1/2}_i (\alpha \beta _i)^{-1/2}$ for the firehose and $|k_{\parallel} |\rho _{i0} \ll \beta ^{-2}_i (\alpha \beta _i)^{-1/2}$ for the mirror. All of the numerical simulations presented in § 3.2 easily satisfy the first of these criteria (for the firehose) but not the second (for the mirror). As a result, the onset of the mirror instability occurs late enough that $\varDelta (t)$ grows to be ${\gg }1/\beta _i$ before being regulated. In this situation, the Taylor expansion used to obtain (3.11b) for the mirror instability, and thus (3.12b), is no longer valid. A better approximation is then $\varLambda _{m}(t) \approx \varDelta (t) \sim \alpha (k_{\parallel} v_{\textrm {th}i0}t)^2$, and the argument of the exponential in the amplification factor (3.11b) instead reads $(2\varOmega _i/5) \alpha ^2 (k_{\parallel}v _{\textrm {th}i0})^4 (t-t_{m})^5$. The corresponding ‘mirror regulation’ time should then satisfy

(3.12c)\begin{equation} |k_{\parallel}|v_{\textrm{th}i0} (t_{m,\textrm{reg}}-t_{m}) \sim (|k_{\parallel}|\rho_{i0})^{1/5} \alpha^{-2/5} . \end{equation}

This prediction should hold so long as $t_{m,\textrm {reg}}$ is less than half of an IAW period, otherwise the oscillating wave would drive the pressure anisotropy negative before the mirror is able to regulate it. Therefore, we require $|k_{\parallel} |\rho _{i0}\ll \alpha ^2$ in order for the prediction (3.12c) to be valid. The scalings (3.12a) and (3.12c) are tested in § 3.2.4.

Once the times given by (3.12) are reached, the evolution of the pressure anisotropy will no longer follow (3.7). Instead, it will be regulated to values near the instability thresholds as its unstable portion, viz. $\varLambda _{f}(t_{f,\textrm {reg}})$ for firehose and $\varLambda _{m}(t_{m,\textrm {reg}})$ for mirror, is transferred to (mostly transverse) magnetic-field fluctuations, in the case of firehose, and (approximately pressure-balanced) compressive fluctuations, in the case of mirror. Thereafter, these fluctuations will continue to grow in order to maintain locally a marginally unstable plasma, $|\varDelta (t)|\sim 2/\beta _i$, as the pressure anisotropy is persistently driven by the oscillating IAW. The result is a secular-in-time amplification of the microscale fluctuations (Schekochihin et al. Reference Schekochihin, Cowley, Kulsrud, Rosin and Heinemann2008; Rosin et al. Reference Rosin, Schekochihin, Rincon and Cowley2011; Rincon et al. Reference Rincon, Schekochihin and Cowley2015). Accompanying this growth is an anomalous ‘effective collisionality’ $\nu _\textrm {eff}$, caused by the particles scattering off of these fluctuations. This scattering contributes to maintaining a marginally unstable pressure anisotropy by directly isotropizing the plasma. Once $\nu _\textrm {eff}$ attains values comparable to $S\beta _i$, where $S \sim |\textrm {d}\varDelta _\textrm {IAW}/\textrm {d} t|$ is the rate of pressure-anisotropy production by the oscillating IAW, the enhanced collisionality obviates the need for the microscale fluctuations to continue growing, and the instabilities saturate (Kunz et al. Reference Kunz, Schekochihin and Stone2014a).

This evolution may be formalized somewhat as follows. In the case of the firehose, we adapt the arguments of Schekochihin et al. (Reference Schekochihin, Cowley, Kulsrud, Rosin and Heinemann2008) to write the marginally unstable balance between the production of negative pressure anisotropy directly by the IAW, the adiabatic production of positive pressure anisotropy by the growing firehose fluctuations $\delta B_{f}$, and the isotropization of the plasma by the effective collisionality $\nu _\textrm {eff}$, heuristically as

(3.13)\begin{equation} \frac{\textrm{d} \varDelta}{\textrm{d} t} \approx \frac{\textrm{d} \varDelta_\textrm{IAW}}{\textrm{d} t} + \frac{3}{2} \frac{\textrm{d} }{\textrm{d} t} \frac{\delta B^2_{f}}{B^2_0} - 3\nu_\textrm{eff}\varDelta \approx 0,\quad \textrm{with}\ \varDelta \approx -\frac{2}{\beta_i}. \end{equation}

The collisionless secular phase of growth, in which $\nu _\textrm {eff}\ll S\beta _i$, is achieved by balancing the first two terms on the right-hand side of (3.13), yielding

(3.14)\begin{equation} \frac{3}{2} \frac{\delta B^2_{f}(t)}{B^2_0} \approx \frac{3}{2} \frac{\delta B^2_{f}(t_{f,\textrm{reg}})}{B^2_0} + |\varDelta_\textrm{IAW}(t)| - \frac{2}{\beta_i} . \end{equation}

This phase of growth is, however, short-lived when $S\beta _i\ll \varOmega _i$ (Kunz et al. Reference Kunz, Schekochihin and Stone2014a; Melville et al. Reference Melville, Schekochihin and Kunz2016). The isotropizing effect of particles’ scattering off of the growing firehose fluctuations becomes increasingly important, and ultimately plays the dominant role in regulating the pressure anisotropy:

(3.15)\begin{equation} \frac{1}{3\nu_\textrm{eff}}\frac{\textrm{d} \varDelta_\textrm{IAW}}{\textrm{d} t} \approx - \frac{2}{\beta_i}, \end{equation}

a sort of Braginskii-like weakly collisional closure.

In the case of the mirror, the fluctuations grow secularly while trapping an increasing fraction of particles $\xi _\textrm {tr} \approx |\delta B_{m}/B_0|^{1/2}$ in the deepening magnetic troughs (Schekochihin et al. Reference Schekochihin, Cowley, Kulsrud, Rosin and Heinemann2008; Rincon et al. Reference Rincon, Schekochihin and Cowley2015). This results in a balance between the production of positive pressure anisotropy by the IAW, and what amounts to betatron cooling of the trapped ion population:

(3.16)\begin{equation} \frac{\textrm{d} \varDelta}{\textrm{d} t} \approx \frac{\textrm{d} \varDelta_\textrm{IAW}}{\textrm{d} t} - 3 \frac{\textrm{d} }{\textrm{d} t} \left|\frac{\delta B_{m}}{B_0}\right|^{3/2} \approx 0,\quad \textrm{with}\ \varDelta \approx \frac{1}{\beta_i}. \end{equation}

In principle, this phase would last all the way until $\delta B_{m} / B_0 \approx 0.3$ at $t \gtrsim S^{-1}$, when the ions begin to scatter appreciably off the sharp ends of the mirrors (Kunz et al. Reference Kunz, Schekochihin and Stone2014a; Riquelme et al. Reference Riquelme, Quataert and Verscharen2015; Melville et al. Reference Melville, Schekochihin and Kunz2016). However, because the large-scale oscillation causes $\varDelta _\textrm {IAW}$ to change sign halfway through the IAW period, we expect that the mirrors will not have enough time to complete their secular growth to large amplitudes and so the associated strong scattering will not be realized.

3.1.3. Effective collisionality and interruption of Landau damping

In either situation – firehose or mirror – these pressure-anisotropy instabilities will ultimately interfere with phase mixing and collisionless damping by impeding the maintenance of Landau resonances through particle scattering and/or trapping. Accordingly, the rate of damping of the IAW should decrease.

To gauge just how effective these instabilities are at reducing the Landau damping, let us consider (3.15) for the balance between pressure-anisotropy production by the IAW and its depletion by the firehose-induced effective collisionality. During this phase, the collisionality is large enough to cause $\varDelta _\textrm {IAW}$ to depart from the collisionless prediction (3.7). To obtain a rough estimate for the evolution of $\varDelta _\textrm {IAW}$ during this weakly collisional phase, we adopt the double-adiabatic scaling $\textrm {d} \varDelta _\textrm {IAW} / \textrm {d} t \approx - 2 (\textrm {d}/\textrm {d} t) (\delta n/n_0) \sim \alpha |k_{\parallel} |v_{\textrm {th}i0}$. The regulating collisionality must then satisfy

(3.17)\begin{equation} \frac{\nu_\textrm{eff}}{|k_{\parallel}|v_{\textrm{th}i0}} \sim \alpha\beta_i. \end{equation}

With $\alpha \gtrsim 2/\beta _i$ being the requirement for the IAW to trigger the firehose instability in the first place, we have $\nu _\textrm {eff} \gtrsim |k_{\parallel} |v_{\textrm {th}i0}$, and so the effective collisionality induced by the firehose is in principle sufficient to halt Landau damping altogether.

3.2.1. Generation of pressure anisotropy and triggering of kinetic instabilities

Figure 2(a) shows the evolution of $\langle 2(\delta n/n_0)^2\rangle ^{1/2}$ using our fiducial set-up for $\alpha = 0.1$ (black) and $\alpha = 0.01$ (red), compared with their expected exponential decay by linear Landau damping (dashed lines; (2.1) gives $\gamma /|k_{\parallel} |v_{\textrm {th}i0} = 0.60$ for $T_e/T_i=1$). While the $\alpha =0.01$ perturbation is efficiently Landau damped, the damping of the $\alpha =0.1$ ($>\!2/\beta _{i0}$) perturbation slows down and ultimately stalls. An IAW with $\alpha =0.1$ and $\beta _{i0}=1$ (blue) is also shown to be Landau damped to small amplitude.Footnote ⁵ Figure 2(b) displays the spread (i.e. minimum to maximum) in $x$ (yellow region) and mean (black line) of the $y$-averaged pressure anisotropy $\langle \varDelta \rangle _y$ versus time for this stalled-decay perturbation. For $\varOmega _{i0} t \lesssim 300$ (roughly half a wave period), the extrema of $\langle \varDelta \rangle _y$ follow the theoretical prediction (3.7). Thereafter, $|\varDelta |$ is large enough for the growth of firehose and mirror instabilities and the consequent relaxation of the pressure anisotropy to overwhelm the production of $\varDelta$ (with firehose occurring earlier than mirror, as predicted). Figure 2(c,d) show the energy densities of the $x$, $y$ and $z$ components of the magnetic-field fluctuations associated with the firehose and mirror instabilities, defined by the spatial regions in which they primarily occur: $\delta B^2_{f} \doteq \langle \delta B^2_{x,y,z}\rangle _{250 \le x/\rho _{i0} \le 750}$ for firehose and $\delta B^2_{m} \doteq \langle \delta B^2_{x,y,z}\rangle _{0\le x/\rho _{i0} < 250 \cup 750 < x/\rho _{i0}\le 1000}$ for mirror. As we demonstrate below (see figure 4), these fluctuations locally scatter and trap particles, respectively, inhibiting strong Landau resonances and thereby stalling the Landau damping of the IAW.

Figure 2.

Interrupted collisionless damping of an IAW of wavelength $\lambda _{\parallel} = 1000\rho _{i0}$, initial frequency $\omega = 1.45|k_{\parallel} |v_{\textrm {th}i0}$ and initial amplitude $\alpha =0.1$. (a) Amplitude of density perturbation; dashed line shows theoretical Landau damping at the predicted linear rate $\gamma /|k_{\parallel} |v_{\textrm {th}i0} = 0.60$ (see (2.1)). For comparison, identical IAWs but with $\alpha =0.01$ (red line) or with $\beta _{i0}=1$ (blue line) are shown to be damped at the Landau rate. (b) Mean (black line) and spread in $x$ (yellow region) of the $y$-averaged pressure anisotropy, $\langle \varDelta \doteq p_{\perp} /p_{\parallel} - 1\rangle _y$, versus time. Mirror (${\simeq }1/\beta _i$) and firehose (${\simeq }-\!2/\beta _i$) instability thresholds are denoted by the dashed lines. The energy densities of the $x$, $y$ and $z$ components of the magnetic-field fluctuations associated with the firehose ($\delta B^2_{f}/2B^2_0$) and mirror ($\delta B^2_{m}/2B^2_0$) instabilities are shown in panels (c) and (d), respectively (see text for their definitions).

The production of unstable magnetic-field fluctuations and consequent regulation of the pressure anisotropy is also illustrated in figure 3, which shows $\langle \varDelta \rangle _y$ and $\delta B_{x,y,z}$ as functions of $x$ at various times. At $\varOmega _{i0} t = 170$ (blue line), the pressure anisotropy attains its global minimum (${\simeq }\!-\!0.09$), after which the $\langle \varDelta \rangle _y < 0$ region (within $250 \lesssim x/\rho _{i0} \lesssim 750$) is quickly (by $\varOmega _{i0} t = 300$) regulated to the firehose threshold (red line). The magnetic-field fluctuations in this region are oriented primarily in the $z$ direction and are oblique with respect to the mean magnetic field. By $\varOmega _{i0} t = 530$ (yellow line), the regions with $\langle \varDelta \rangle _y > 0$ ($0\lesssim x/\rho _{i0}\lesssim 250$ and $750 \lesssim x/\rho _{i0} \lesssim 1000$) are well regulated to the mirror threshold. Indeed, the magnetic-field fluctuations in these regions are mirror-mode polarized, with $\delta B_{\perp} / \delta B_{\parallel} \simeq 0.36 \simeq \textrm {max}(\varDelta - 1/\beta _i)^{1/2}$ (Hellinger Reference Hellinger2007). At late times (e.g. $\varOmega _{i0} t = 3000$, purple line), the plasma is nearly pressure-isotropic. While the mirror modes appear to be long-lived, pressure-balanced structures, the firehose modes become stably propagating Alfvén waves with parallel wavenumbers comparable to $\rho ^{-1}_i$ (as also seen in the simulations of Alfvén-wave interruption by Squire et al. (Reference Squire, Kunz, Quataert and Schekochihin2017a)).

Figure 3.

(a) Spatial profile of $y$-averaged pressure anisotropy at $\varOmega _{i0} t = 170$ (when $\langle \varDelta \rangle _y$ is most negative), $300$ (at firehose regulation), $530$ (at mirror regulation) and $3000$ (when $\langle \nu _\textrm {eff}\rangle \lesssim 10^{-3}\varOmega _{i0}$ – see figure 4). (b) Spatial profiles of magnetic-field fluctuations (normalized to $B_0$) at $\varOmega _{i0} t = 300$, $530$ and $3000$. (The fluctuations shown in the $\varOmega _{i0}t = 3000$ panels are multiplied by $2$ to enhance their visibility.) All plots correspond to the $(\alpha ,\beta _{i0})=(0.1,100)$ IAW featured in figure 2.

3.2.2. Effective collisionality: particle scattering and trapping

Figure 4 provides the link between these magnetic-field fluctuations and the evolution shown in figure 2. The top two panels illustrate the effective collisionality of the plasma $\nu _\textrm {eff}$, computed following the method used in Kunz et al. (Reference Kunz, Schekochihin and Stone2014a), Melville et al. (Reference Melville, Schekochihin and Kunz2016) and Squire et al. (Reference Squire, Kunz, Quataert and Schekochihin2017a).Footnote ⁶ The box-averaged collisionality $\langle \nu _\textrm {eff}\rangle$ (black line) increases dramatically once the firehose modes are triggered, peaking approximately when the IAW stops Landau damping at a value $\nu _\textrm {eff} \sim |k_{\parallel} |v_{\textrm {th}i0}$. Thereafter, its steady exponential decrease follows the decaying magnetic energy of the firehose fluctuations (red line), viz., $\langle \nu _\textrm {eff}\rangle /\varOmega _{i0} \approx \delta B^2_{f}/2B^2_0$, as it should if this scattering is due to pitch-angle scattering of particles off ion-Larmor-scale magnetic fluctuations (e.g. Kulsrud & Pearce Reference Kulsrud and Pearce1969; Schlickeiser Reference Schlickeiser1989). What sets the exponential decay rate of the firehose fluctuations, and thus the anomalous collisionality, is not yet fully understood, but a scan across $L_x/\rho _{i0}=1000\text {--}16\,000$ returned the empirical result that their decay rate is ${\approx }0.1 |k_{\parallel} |v_{\textrm {th}i0}$ with $|k_{\parallel} |=2{\rm \pi} /L_x$ (see figure 7b). If this trend holds at even larger scale separations, it would imply that the observed interruption of Landau damping by scattering off firehose fluctuations is an asymptotically valid effect. Importantly, if IAWs were to be continuously replenished in a magnetized, high-$\beta$ plasma on a time scale comparable to or smaller than their oscillation period, the resulting dynamics would never be collisionless. We revisit this idea in § 4.

Figure 4.

(a) Box-averaged effective collision frequency, $\langle \nu _\textrm {eff}\rangle /\varOmega _{i0}$ (black) and magnetic energy of firehose fluctuations, $\delta B^2_{f}/2B^2_0$ (red), versus time. (b) Space–time diagram of $\langle \nu _\textrm {eff}\rangle _y/\varOmega _{i0}$ (colour), overlaid white lines are trajectories of five representative particles. (c) Magnetic moments $\mu (t)$ (black) and parallel velocities $v_{\parallel} (t)$ (red) of those particles, labelled ${\bigcirc{\kern-8pt 1}}$–${\bigcirc{\kern-8pt 5}}$. All plots correspond to the $(\alpha ,\beta _{i0})=(0.1,100)$ IAW featured in figures 2 and 3.

The space–time evolution of $\nu _\textrm {eff}/\varOmega _{i0}$ is also shown in figure 4 (colour), with tracks of five representative particles overlaid (white lines, labelled $1$–$5$). Those particles’ adiabatic invariants ($\mu$; black lines) and parallel velocities ($v_{\parallel}$; red lines) are plotted in panel (c), correspondingly labelled ${\bigcirc{\kern-8pt 1}}$–${\bigcirc{\kern-8pt 5}}$. All particles begin their lives by streaming freely (i.e. $x(t) = x(0) + v_{\parallel} t$) and thus phase-mixing the distribution function. Between $\varOmega _{i0} t \approx 200\text {--}300$, particles 1, 3 and 4 encounter firehose fluctuations and promptly scatter, changing their $\mu$ by large factors. In contrast, particles 2 and 5 first encounter mirror modes at $\varOmega _{i0} t \approx 400$, become trapped, and bounce, conserving $\mu$ while exhibiting oscillations in $v_{\parallel}$ about $0$. After this initial scattering or trapping, these particles either stay trapped in mirrors (${\bigcirc{\kern-8pt 2}}$), possibly leaking out at late times (${\bigcirc{\kern-8pt 5}}$); or eventually encounter a mirror and become trapped, only subsequently to scatter out of the mirror and once again encounter $\mu$-breaking firehose fluctuations (${\bigcirc{\kern-8pt 3}}$, ${\bigcirc{\kern-8pt 4}}$); or continue scattering while streaming untrapped, but with decreasing changes in $\mu$ as the fluctuations decay (${\bigcirc{\kern-8pt 1}}$). Note that the $\mu$ conservation of particle 3 is broken at $\varOmega _{i0} t \approx 1200$ when the particle encounters a second burst of firehose fluctuations produced by the oscillating IAW.

3.2.3. Reduction of heat conduction by firehose and mirror

An enhanced effective collisionality in the firehose-unstable regions and efficient particle trapping in the mirror-unstable regions suggest that appreciable reductions in the (ion) heat conductivity occur throughout the IAW. In regions where the firehose fluctuations break $\mu$ conservation at the rate $\nu _\textrm {eff}$, we may anticipate the parallel flow of parallel heat

(3.18)\begin{equation} q_{\parallel} \doteq \int\textrm{d}^3\boldsymbol{v}\, m_i (v_{\parallel}-u_{\parallel i})^3 f_i \approx -\frac{3}{2} n \frac{v^2_{\textrm{th}\parallel i}}{\nu_\textrm{eff}} \boldsymbol{\nabla}_{\parallel} T_{\parallel i} \doteq q_{\parallel,\textrm{Brag}}, \end{equation}

following Braginskii (Reference Braginskii1965) and using a pitch-angle-scattering (Lorentz) collision operator with collision frequency $\nu _\textrm {eff}$; $u_{\parallel i} \doteq (1/n)\int \textrm {d}^3\boldsymbol {v}\, v_{\parallel} \,f_i$ is the parallel ion bulk velocity. In regions where the mirror fluctuations are in their late nonlinear stage of evolution and efficiently trap a majority fraction of the nearby particles, $q_{\parallel}$ should be close to $0$. Figure 5 confirms these expectations.

Figure 5.

Spatial profile of $y$-averaged parallel ion heat flux $\langle q_{\parallel} \rangle _y$ at (a) $\varOmega _{i0}t = 500$ (near the time when the firehose-induced collisionality is at its peak value; see figure 4a) and (b) $\varOmega _{i0}t = 1000$ (after mirror-mode saturation). In panel (b), near-complete suppression of $\langle q_{\parallel} \rangle _y$ by particle trapping in mirrors occurs (indicated by the blue arrows), despite the presence of a strong ion temperature gradient (panel c). The heat flux in the central (firehose-unstable) region is accurately described by a Braginskii heat flux (red) using the effective collisionality measured directly from tracked particles (see (3.18)).

Once the firehose fluctuations become efficient scatterers ($\varOmega _{i0}t\gtrsim {500}$; panel a), the Braginskii heat flux $q_{\parallel ,\textrm {Brag}}$ given by (3.18) (red line), with $\nu _\textrm {eff}$ taken from figure 4, is an excellent description of the actual heat flux (black line) in the firehose-infested region. And once the mirrors become efficient particle traps ($\varOmega _{i0}t\gtrsim {1000}$; figure 5b), $q_{\parallel}$ is substantially suppressed in those regions (marked by the blue arrows) – the conductivity there is effectively zero, despite the persistent temperature gradient (figure 5c). A rough quantitative measure of the suppression factor may be obtained by calculating $\langle q_{\parallel} \rangle _y / \langle -\boldsymbol {\nabla }_{\parallel} T_{\parallel i}\rangle _y$ in the mirror-infested regions of the simulation and comparing it to the corresponding collisionless value without mirrors obtained using a linear Landau-fluid model, viz., $(2/\sqrt {{\rm \pi} }) n_0 v_{\textrm {th}i0} / |k_{\parallel} |$ (see (39) of Snyder, Hammett & Dorland (Reference Snyder, Hammett and Dorland1997)). The result gives a suppression factor ${\sim }200$. (A suppression factor of ${\sim }5$ was calculated by Komarov et al. (Reference Komarov, Churazov, Kunz and Schekochihin2016) for the electron parallel heat conductivity in the presence of saturated, ion-Larmor-scale mirrors.)

3.2.4. Dependence on scale separation

As a last point of comparison with the theory developed in § 3.1, we demonstrate how some of our quantitative results depend on scale separation, $L_x/\rho _{i0}$. First, in figure 6(a), we show that the values of $|k_{\parallel} |v_{\textrm {th}i0}(t_{f,\textrm {reg}}-t_{f})$ measured in IAW simulations with $(\alpha ,\beta _{i0})=(0.1,100)$ across a range of $L_x/\rho _{i0}$ are in remarkable agreement with the predicted scaling (3.12a); the omitted proportionality constant in (3.12a) is found empirically to be ${\approx }8$. For the mirror instability, we find $|k_{\parallel} |v_{\textrm {th}i0}(t_{m,\textrm {reg}}-t_{m}) \propto (L_x/\rho _{i0})^{-1/5}$ once $L_x/\rho _{i0} \ge 4000$, consistent with the corresponding prediction (3.12c), with a proportionality constant ${\approx }2$. This is as it should be, because the predicted asymptotic scaling (3.12b) is not applicable unless $|k_{\parallel} |\rho _i\ll \beta _{i0}^{-2} (\alpha \beta _{i0})^{1/2}$, a condition that is never satisfied even in our very largest numerical simulations. These scalings demonstrate that, in the limit $\rho _{i0}/L_x\rightarrow 0$, the pressure anisotropy will be regulated by the kinetic instabilities almost instantly after the system becomes unstable at $t\sim t_{f}, t_{m}$ (see (3.9)).Footnote ⁷

Figure 6.

(a) Time elapsed between $t_{f}$, when the plasma first becomes firehose unstable, and $t_{f,\textrm {reg}}$, when the negative pressure anisotropy is rapidly regulated by the firehose instability, versus scale separation $L_x/\rho _{i0}$. The dashed line is the theoretical prediction, (3.12a). (b) As in panel (a) but for positive pressure anisotropy and the mirror instability; the dashed line is the theoretical prediction (3.12c). (c) Early evolution of the firehose instability parameter $\varLambda _{f}$ (blue) and (three times) the energy density of the firehose magnetic-field fluctuations (red) for $L_x/\rho _{i0}=16\,000$. The times $t_{f}$ and $t_{f,\textrm {reg}}$ are indicated. (d) As in panel (c) but for the mirror instability. Also for $L_x/\rho _{i0}=16\,000$: (e) logarithmic evolution of the energy density of the firehose magnetic-field fluctuations (red) and of the box-averaged effective collision frequency (orange); and (f) logarithmic evolution of the energy density of the mirror magnetic-field fluctuations (red). These final panels highlight the secular phase of unstable growth, occurring with power laws close to ${\propto }t$ and ${\propto }t^{4/3}$ after the plasma becomes weakly collisional, $\langle \nu _\textrm {eff}\rangle /|k_{\parallel} |v_{\textrm {th}i0}>1$ (vertical dotted line). Before that time, $(3/2)(\delta B_{f}/B_0)^2$ follows well the collisionless prediction (3.14) (blue dot-dashed line). All times are normalized using $|k_{\parallel} |v_{\textrm {th}i0}=2{\rm \pi} \varOmega _{i0}(\rho _{i0}/L_x)$; all data refer to the case $(\alpha ,\beta _{i0})=(0.1,100)$.

In panels (c) and (d), we show the ‘early time’ evolution of the firehose ($\varLambda _{f}$) and mirror ($\varLambda _{m}$) instability parameters (blue), as well as of (three times) the energy densities of the magnetic-field fluctuations associated with the firehose and mirror instabilities (red), all from the IAW simulation with $(\alpha ,\beta _{i0})=(0.1,100)$ and $L_x/\rho _{i0}=16\,000$.Footnote ⁸ The times at which the plasma goes firehose (mirror) unstable, $t_{f}$ ($t_{m}$), and at which the pressure anisotropy is regulated by the growth of magnetic-field fluctuations, $t_{f,\textrm {reg}}$ ($t_{m,\textrm {reg}}$), are indicated by the arrows. The observed behaviour matches that predicted in § 3.1.2: at $t \sim t_\textrm {reg}$, rapid super-exponential growth of the firehose and mirror fluctuations curtails the production of pressure anisotropy by the IAW, after which the fluctuations grow secularly to help maintain a marginally unstable pressure anisotropy. This secular growth is highlighted in panel (e,f) using logarithmic axes; panel (e) also includes the evolution of the box-averaged effective collision frequency (orange line). For times during which $\langle \nu _\textrm {eff}\rangle < |k_{\parallel} |v_{\textrm {th}i0}$, the secular growth of the firehose fluctuations follows well the collisionless prediction (3.14) (blue dot-dashed line). Once $\langle \nu _\textrm {eff}\rangle$ becomes large enough to interfere with the collisionless evolution (i.e. for times to the right of the vertical dotted line at $|k_{\parallel} |v_{\textrm {th}i0} \approx 0.7$), the firehose fluctuations continue to grow secularly but with a shallower slope close to $\delta B^2_{f} \propto t$. The mirror fluctuations satisfy $\delta B^2_{m} \propto t^{4/3}$ during their secular phase. These power laws match those measured by Kunz et al. (Reference Kunz, Stone and Bai2014b) during the secular growth phase of their dedicated simulations of driven firehose and mirror instabilities, and are consistent with (3.14) and (3.16) after the effects of suppressed phase mixing and Landau damping are accounted for in modifying the drive, $\textrm {d} \varDelta _\textrm {IAW}/\textrm {d} t$. The consequent regulation of the pressure anisotropy is aided by an increasing $\langle \nu _\textrm {eff}\rangle$, which attains a maximum value ${\approx }0.003\varOmega _{i0} \approx 8|k_{\parallel} |v_{\textrm {th}i0}$ at $|k_{\parallel} |v_{\textrm {th}i0} \sim 2$, when the oscillating IAW changes the sign of $\textrm {d}\varDelta _\textrm {IAW}/\textrm {d} t$ and the growth of the firehose fluctuations is disabled.

Figure 7 shows how the characteristics of $\langle \nu _\textrm {eff}\rangle$ depend on $L_x/\rho _{i0}$, using the same parameter scan as in figure 6(a) plus an additional scan with $(\alpha ,\beta _{i0})=(0.1,25)$. Panel (a) confirms that the maximum effective collision frequency in all of our runs is large enough to inhibit Landau damping (i.e. $\textrm {max}\langle \nu _\textrm {eff}\rangle \gtrsim |k_{\parallel} |v_{\textrm {th}i0}$), with the collisionality decreasing with scale separation as $\textrm {max}\langle \nu _\textrm {eff}\rangle /\varOmega _{i0} \propto (L_x/\rho _{i0})^{-1/6}$. (This is a purely empirical scaling for which we have no theory at present, although it is qualitatively consistent with larger scale separations resulting in a pressure anisotropy that overshoots the firehose threshold less before being regulated.) This translates to ${\max }\langle \nu _\textrm {eff}\rangle /|k_{\parallel} |v_{\textrm {th}i0}\propto (L_x/\rho _{i0})^{5/6}$, a scaling that holds until $\textrm {max}\langle \nu _\textrm {eff}\rangle /|k_{\parallel} |v_{\textrm {th}i0} \sim \alpha \beta _{i0}$ at $L_x/\rho _{i0} \gtrsim 32\,000\ (8000)$ for $\beta _{i0}=100\ (25)$. This value is consistent with the asymptotic prediction (3.17) for the approximate effective collisionality required to regulate the IAW-driven pressure anisotropy through pitch-angle scattering alone (dotted line). Indeed, the $(\beta _{i0},L_x/\rho _{i0})=(25,32\,000)$ simulation demonstrates that $\textrm {max}\langle \nu _\textrm {eff}\rangle /|k_{\parallel} |v_{\textrm {th}i0}$ saturates asymptotically at a value ${\approx }2\alpha \beta _{i0}$. The similar scaling of the $\beta _{i0}=25$ and $100$ runs, with the latter shifted to larger scale separations due to the larger value of $\alpha \beta _{i0}$, suggests that a run with $L_x/\rho _{i0} = 128\,000$ would also demonstrate saturation at $\textrm {max}\langle \nu _\textrm {eff}\rangle /|k_{\parallel} |v_{\textrm {th}i0} \approx 2\alpha \beta _{i0}$. Unfortunately, such a simulation is too computationally expensive to be practical at this time. Panel ($b$) demonstrates that, when time is normalized using $|k_{\parallel} |v_{\textrm {th}i0}$ and its origin is offset by the firehose-regulation time $t_{f,\textrm {reg}}$, the effective collisionality in all $\beta _{i0}=100$ runs follows a similar evolution, with a sharp rise followed by a long exponential decay at a rate ${\approx }0.1|k_{\parallel} |v_{\textrm {th}i0}$. (The runs with $L_x/\rho _{i0}\ge 8000$ were too computationally expensive to be continued profitably for longer times.)

Figure 7.

(a) Maximum value of the box-averaged effective collision frequency $\langle \nu _\textrm {eff}\rangle$, normalized using $\alpha \beta _{i0}|k_{\parallel} |v_{\textrm {th}i0}$ with $|k_{\parallel} |=2{\rm \pi} /L_x$. Its value increases with scale separation as $(L_x/\rho _{i0})^{5/6}$ until a value ${\sim }1$ (dotted line; see (3.17)) is achieved. Black (red) circles correspond to $\beta _{i0}=100$ ($\beta _{i0}=25$). (b) Time evolution of $\langle \nu _\textrm {eff}\rangle$ for different $L_x/\rho _{i0}$ at $\beta _{i0}=100$. In all cases with long enough integration times, $\langle \nu _\textrm {eff}\rangle$ is measured to decay exponentially at a rate ${\approx }0.1|k_{\parallel} |v_{\textrm {th}i0}$, independent of scale separation. (c) Root mean square density fluctuation versus time for various combinations of $\beta _{i0}$ and $L_x/\rho _{i0}$. For all points and curves, $\alpha =0.1$.

Finally, figure 7(c) presents the root mean square density fluctuation amplitude versus time for various combinations of $\beta _{i0}$ and $L_x/\rho _{i0}$. Larger scale separations cause a slight decrease in the rate at which the IAW is Landau damped and smaller values of $\alpha \beta _{i0}$ result in a larger initial damping rate, at least for $|k_{\parallel} |v_{\textrm {th}i0}t \lesssim 5$. Both trends are consistent with the scaling of $\textrm {max}\langle \nu _\textrm {eff}\rangle$ shown in panel (a): larger values of $\nu _\textrm {eff}/|k_{\parallel} |v_{\textrm {th}i0}$ at larger scale separation reduce the potency of collisionless damping. After this time, the evolution of the IAW is roughly independent of both scale separation and $\alpha \beta _{i0}$.

In summary, figures 6 and 7 demonstrate that our findings are robust with respect to scale separation: IAWs with amplitudes satisfying (3.8) produce enough pressure anisotropy to trigger firehose and mirror instabilities, which grow super-exponentially and then secularly, with the former generating an anomalous collisionality that is both large and lasting enough to make the plasma behave as though it were weakly collisional.