4.1.1. Turbulent energy

While the results shown in figures 1–4 clearly demonstrate the significant impact of the pressure-anisotropy feedback on the statistics of $B$ and $\Delta p$, it is important to confirm that this does not occur purely as a result of the viscous damping of those motions that would otherwise drive significant $\Delta p$. Because all simulations are driven with the same energy-injection rate $\varepsilon$, the simplest diagnostic of this is the turbulent fluctuation energy, which is shown in figure 5 for the lrCL runs (we plot $E_{K\perp } + E_{M\perp } \doteq \int \textrm {d} \boldsymbol {x}\ (\rho |\boldsymbol {u}_\perp |^2 + |\boldsymbol {B}_\perp |^2/4{\rm \pi} )/2$, which includes only the fluctuating $y$ and $z$ components of $\boldsymbol {u}$ and $\boldsymbol {B}$). A fluctuation energy that decreased with decreasing $\mathcal {I}$ would imply a cascade efficiency that decreased with $\mathcal {I}$, as would be naïvely expected if fluctuations were increasingly strongly damped above the interruption limit (see § 2.2). Instead, we see in figure 5(a) that such a decrease is quite small (compare, e.g. the active and passive cases at $\beta _{0}=10$), with a similar steady-state energy reached even at $\beta _{0}=100$ (the lowest $\mathcal {I}$ that we explore). Thus, we see that there is only modest viscous damping of pressure-anisotropy-generating fluctuations, a feature that we explore in greater depth below. Note that, in contrast, a single linearly polarised shear-Alfvén wave with similar amplitude would be strongly damped under these conditions because its fluctuations are confined to the plane set by the initial conditions and so it cannot rearrange itself to avoid generating large pressure anisotropies (Squire et al. (Reference Squire, Quataert and Schekochihin2016); S$+$19).

Figure 5. Time evolution of the total perpendicular energy (a) and the Alfvén ratio $r_{A}=\sqrt {\varTheta E_{M\perp }/E_{K\perp }}$ (b) for the lrCL set of collisionless simulations with varying interruption number. Note that $r_{A}$ defined in this way (with $\varTheta$ computed as a volume average) accounts for the effect of the mean pressure anisotropy on Alfvénic fluctuations (see (2.11)). The dotted lines in each panel show the ‘expected’ values assuming $\delta u_\perp /v_{A}=\delta B_\perp /B_0=1/2$: $E_{K\perp } + E_{M\perp }=V\rho _0 v_{A}^2/4$ and $r_\textrm {A}=1$. We see that the bulk properties of collisionless turbulence are rather similar to standard MHD, even for $\mathcal {I}\ll 1$.

Figure 5(b) shows the Alfvén ratio, $r_{A} = \sqrt {\varTheta E_{M\perp } /E_{K\perp }}$, for some of the simulations. This should be approximately unity for Alfvénic turbulence. The time-dependent box-averaged mean anisotropy parameter $\varTheta$ is included in $r_{A}$ because an Alfvén wave satisfies $\delta B_{\perp }=\delta u_{\perp }\sqrt {4{\rm \pi} \rho /\varTheta }$ in the presence of a mean pressure anisotropy $\Delta p$ (see § 2.1.2; we set $\varTheta =1$ for the passive simulation). Figure 5 shows that, after accounting for the change in the Alfvén ratio due to variation of the mean $\Delta p$ with $\mathcal {I}$ (see figure 4), the bulk properties of the turbulence are rather similar to MHD, with just a slight excess of magnetic energy (also observed in MHD; see, e.g. Müller & Grappin Reference Müller and Grappin2005; Chen et al. Reference Chen, Bale, Salem and Maruca2013).

4.2.1. Turbulent energy spectra at $\mathcal {I}\lesssim 1$

Figure 6 shows the kinetic (blue lines) and magnetic (red lines) spectra obtained in the CL10, B100 and CL100 simulations, again comparing active-$\Delta p$ simulations (solid lines) with the passive-$\Delta p$ simulations (dashed lines; note that this case is just isothermal MHD for velocity and magnetic fluctuations). The purpose of this comparison is to exhibit the extremely similar spectra in each case, demonstrating that vigorous turbulence survives even when the expected damping from pressure anisotropy is strong across a wide range of scales starting at the outer scale (as is the case for all three simulations shown here since $\mathcal {I}<1$). A careful examination reveals minor differences caused by the pressure-anisotropy feedback, in particular a slight steepening of the kinetic-energy spectrum compared with MHD. In all cases the magnetic spectra exhibit a scaling that is flatter than ${\sim }k_{\perp }^{-5/3}$ and closer to ${\sim }k_{\perp }^{-3/2}$ (shown with the dashed lines), although the exact slopes vary somewhat across the inertial range. We shall study the eddies’ 3-D structure and dynamic alignment in § 4.2.2.

Figure 6. Kinetic (blue) and magnetic (red) energy spectra for simulations CL10, B100 and CL100, as labelled. Solid lines show CGL-LF (active-$\Delta p$) simulations, and dashed lines show isothermal MHD (passive-$\Delta p$) simulations. The dotted black lines indicate slopes of $k_{\perp }^{-3/2}$, $k_{\perp }^{-5/3}$ and $k_{\perp }^{-2}$ for comparison. We see spectra broadly consistent with a $k_{\perp }^{-3/2}$ scaling, although velocity spectra are steepened modestly by the effect of pressure anisotropy in the active-$\Delta p$ runs (however, they are clearly flatter than the ${\propto }k_{\perp }^{-2}$ spectrum observed in the hybrid-kinetic simulations of A$+$22). This demonstrates that vigorous turbulence, broadly resembling MHD, is maintained even when the feedback from the pressure anisotropy is strong ($\mathcal {I}<1$ in all cases). Note that the cutoff of the spectra at relatively larger scales seen in CL100 is simply a result of that run's lower numerical resolution.

Similar information in a different form is shown in figure 7. We integrate the spectra to obtain the scale-dependent amplitudes

(4.1a,b)\begin{equation} \delta B_{{\perp}}(k_{{\perp}})^{2} = \frac{2}{V} \int_{k_{{\perp}}}^{\infty} {\rm d} k_{{\perp}}' \mathcal{E}_{M\perp}(k_{{\perp}}'),\quad \delta u_{{\perp}}(k_{{\perp}})^{2} = \frac{2}{V{\rho}_{0}} \int_{k_{{\perp}}}^{\infty} {\rm d} k_{{\perp}}'\, \mathcal{E}_{K\perp}(k_{{\perp}}'), \end{equation}

and then use these to compute the scale-dependent interruption number (see (2.19))

(4.2)\begin{equation} \mathcal{I}(k_{{\perp}}) = \left( \beta \frac{\delta B_{{\perp}}}{B_{0}}\frac{\delta u_{{\perp}}}{v_{A}}\right)^{{-}1} \max\left(1, \frac{\nu_{c}}{\omega_{A}}\right)\end{equation}

(we ignore the effect of the mean $\varTheta$ on $v_{A}$ and $\omega _{A}$, as it does not make a significant difference for these qualitative estimates). Here $\omega _{A}$ should be considered a function of scale as per critical balance, so we use $\omega _{A}(k_{\perp }) = k_{\perp } \sqrt {\delta u_{\perp }\delta B_{\perp }/{\rho }_{0}^{1/2}}$. We see that, while the estimated local interruption number is slightly larger for the CGL-LF simulations, because the turbulence amplitudes are modestly smaller, $\mathcal {I}(k_{\perp })$ nonetheless remains ${\lesssim }1$ throughout a wide portion of the turbulent cascade in each case. Since the damping rate of a linearly polarised Alfvénic perturbation is comparable to its frequency for $\mathcal {I}\lesssim 1$ (i.e. for amplitude $\delta B_{\perp }/B_{0}\gtrsim \beta ^{-1/2} \max [1,(\nu _{c}/\omega _{A})^{1/2}]$), but the turbulent fluctuations exceed this amplitude throughout most scales of the simulations, the implication is that the fluctuations must be avoiding those motions that would cause strong damping in order for the cascade to proceed as observed. Also of note is the difference between the collisionless and weakly collisional regimes, as shown by the effectively flat $\mathcal {I}(k_{\perp })$ for the B100 simulation up to $k_{\perp }\approx 100$. This occurs because when $\omega _{A}\ll \nu _{c}$ (the weakly collisional and Braginskii MHD regimes; see § 2.1.3), the increasing frequency of the motions towards smaller scales cancels their decreasing amplitudes, conspiring to keep $\omega _{A}\delta u_{\perp } \delta B_{\perp }$ approximately constant through the inertial range (see § 2.2.3). This breaks down once $\omega _{A}\gtrsim \nu _{c}$, which happens here for $k_{\perp }L_{\perp }\gtrsim 100$, around the expected scale based on the parameters (see table 1). Below this, $\mathcal {I}$ increases at smaller scales since the turbulent eddies are fast enough to be effectively collisionless.

Figure 7. ‘Local’ interruption number defined by (4.2) (see also § 2.2.3). We show the three active-$\Delta p$ simulations whose spectra are given in figure 6, comparing with the passive (MHD) runs with dashed lines. In the B100 simulation ($\beta _{0}=100$, $\nu _{c}=33v_{A}/L_{\perp }$), $\omega _{A}$ depends on scale as per critical balance, $\omega _{A}=k \sqrt {\delta u_{\perp }\delta B_{\perp }/\rho _{0}^{1/2}}$, which causes a slower increase in $\mathcal {I}$ with scale. Clearly, $\mathcal {I}(k_{\perp })\lesssim 1$ across a reasonable range in each simulation, indicating that much larger pressure anisotropies would be created without dynamic feedback from $\Delta p$ (as is the case for the passive simulations).

4.2.2. Three-dimensional anisotropy and alignment

Many previous works have studied the 3-D statistical structure of eddies in MHD turbulence, which has important implications for the cascade rate and intermittency (see Schekochihin (Reference Schekochihin2022), and references therein). Starting with Boldyrev (Reference Boldyrev2006), these have argued that eddies become increasingly ‘dynamically aligned’ towards smaller scales, evolving into elongated sheet-like structures satisfying $\ell _\|\gg \xi \gg \lambda$ (where $\ell _\|$, $\xi$ and $\lambda$ are the eddy's correlation lengths in the field-parallel direction, in the direction of the turbulent fluctuation, and in the mutually perpendicular direction, respectively). Motivated by its utility as a sensitive diagnostic of the Alfvénic fluctuations’ structure, in figure 8 we illustrate the 3-D anisotropy of the CL10 simulations, computed using three-point structure functions (§ 3.3.2). In the left panel, we show the directionally conditioned structure functions of $\boldsymbol {Z}^{\pm }_{\perp } = \boldsymbol {Z}^{\pm } - \hat {\boldsymbol {b}}(\hat {\boldsymbol {b}}\boldsymbol {\cdot }\boldsymbol {Z}^{\pm })$. Here, $\boldsymbol {Z}^{\pm } = \boldsymbol {u}\pm \boldsymbol {v_{A}}$ and $\boldsymbol {v}_{A}$ is the local Alfvén speed $\boldsymbol {v}_{A,\textrm {eff}}=\boldsymbol {B}\varTheta ^{1/2}(4{\rm \pi} \rho )^{-1/2}$, with $\varTheta$ and $\rho$ evaluated using their local three-point average ($\varTheta =1$ for the passive simulation). The colours show different directions of the separation vector $\boldsymbol {\ell }=(\lambda,\xi,\ell _{\|})$, with $\ell _{\|}$ applying to separations within $15^{\circ }$ of the local $\hat {\boldsymbol {b}}$, $\xi$ to separations that are perpendicular (${>}45^{\circ }$) to $\hat {\boldsymbol {b}}$ and are within $15^{\circ }$ of the direction of the local increment of $\boldsymbol {Z}^{\mp }_{\perp }$Footnote ¹³ and $\lambda$ to separations that are perpendicular to both $\hat {\boldsymbol {b}}$ and the $\boldsymbol {Z}^{\mp }_{\perp }$ increment (note that we use $\boldsymbol {Z}^{\mp }$ to define the directions of $\boldsymbol {Z}^{\pm }$ because nonlinear interactions are controlled by the opposite Elsässer variable). We see results that are quite similar to standard MHD turbulence (dashed lines), with $S_{2}$ flatter in the $\lambda$ direction than in the $\xi$ direction, and broadly consistent with (though modestly steeper than) the dynamically aligned MHD-turbulence scalings $S_{2}\sim \lambda ^{1/2}$, $S_{2}\sim \xi ^{2/3}$, $S_{2}\sim \ell _{\|}^{1}$ indicated by the dotted lines (Boldyrev Reference Boldyrev2006; Mallet et al. Reference Mallet, Schekochihin and Chandran2015). The right panel compares the parallel–perpendicular anisotropy of eddies measured from different quantities in the turbulence. As seen in the left panel, the Alfvénic eddies are relatively similar to isothermal MHD, with $\ell _{\|}\sim \ell _{\perp }^{1/2}$ as expected for an aligned cascade. However, while in MHD the compressive quantities $u_{\|}$ and $p = T_{i}\rho$ seem to scale as $\ell _{\|}\sim \ell _{\perp }^{2/3}$, suggesting such fluctuations are unaligned and passively mixed by the critically balanced Alfvénic cascade (Lithwick & Goldreich Reference Lithwick and Goldreich2001; Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009; Chen et al. Reference Chen, Mallet, Schekochihin, Horbury, Wicks and Bale2012), the compressive fields in CGL-LF turbulence scale quite differently, with $p_{\|}$ and $u_{\|}$ having nearly constant anisotropy $\ell _{\|}\simeq 5\ell _{\perp }$ throughout the entire box. This provides the first hint of the differences caused by magneto-immutability, which will be discussed in more detail in the next section (§ 4.3).

Figure 8. Structure functions in the $\beta _{0}=10$, $\nu _{c}=0$ simulations. The left panel shows (3-point) second-order structure functions of $\boldsymbol {Z}_{\perp }^{\pm }$ (see § 3.3.2) with increments taken along the field (blue; $\ell _{\|}$), parallel to the local fluctuation in the opposite Elsässer variable $\boldsymbol {Z}_{\perp }^{\mp }$ ($\xi$; red), and perpendicular to the local fluctuation in $\boldsymbol {Z}_{\perp }^{\mp }$ ($\lambda$; black). Solid and dashed lines show the active-$\Delta p$ and passive-$\Delta p$ cases, respectively, with all length scales normalised to $L_\perp$. The scalings are very similar in both active and passive runs, and are broadly consistent with the expected $\lambda ^{1/2}$, $\xi ^{2/3}$ and $\ell _{\|}^{1}$ scalings of dynamic alignment (shown with dotted black lines). The right-hand panel shows the measured $\ell _{\|}(\ell _{\perp })$ for the same simulation, which is computed from the second-order structure functions of a variety of different quantities as labelled (this provides a measure of the average shape of an eddy for different variables). While $\boldsymbol {Z}_{\perp }^{\pm }$ and $p_{\perp }$ show the expected scale-dependent anisotropy $\ell _{\|}\sim \ell _{\perp }^{1/2}$ of a dynamically aligned cascade, $p_{\|}$ and $u_{\|}$ have nearly constant anisotropy $\ell _{\|}\sim 5\ell _{\perp }$ throughout the inertial range. This differs from MHD (dashed lines), where $u_{\|}$ and $p$ seem to follow $\ell _{\|}\sim \ell _{\perp }^{2/3}$, as expected for non-aligned eddies in a critically balanced cascade.

4.3.1. Pressure statistics

Let us first examine the spectra of pressure fluctuations, which are shown in figure 9, again comparing active- and passive-$\Delta p$ for the CL10, B100 and CL100 simulations. Here, unlike for the kinetic- and magnetic-energy spectra, we see substantial differences due to the feedback of $\Delta p$ on the flow. The most obvious feature is the significant reduction and steepening of the pressure spectra in all active-$\Delta p$ runs. While the passive-$\Delta p$ simulations have quite flat spectra – approximately ${\propto }k_{\perp }^{-1}$ in B100 and yet flatter in the collisionless simulations – all three active cases exhibit pressure spectra ${\propto }k_{\perp }^{-5/3}$ across a reasonable range near the outer scales. Notably, the latter is in agreement with the hybrid-kinetic simulations of A$+$22, providing evidence that similar physics operates in their more realistic simulations. The steepening of pressure spectra due to the pressure-anisotropy feedback is clear evidence of the modified flow structure in the active-$\Delta p$ cases, despite their similar velocity spectra.

Figure 9. Spectra of $\Delta p$ (purple lines), $p_{\perp }$ (green lines), $p_{\|}$ (yellow lines) and the magnetic pressure $B^{2}/8{\rm \pi}$ (grey lines) for the same simulations as in figure 6, and again with the passive-$\Delta p$ simulations shown with dashed lines. The dotted lines indicate the spectral scalings $\mathcal {E}\propto k_{\perp }^{-1}$ and $\mathcal {E}\propto k_{\perp }^{-5/3}$ (see § 4.3.3 for discussion). We see a significant difference between active- and passive-$\Delta p$ cases, with $\mathcal {E}_{\Delta p}\sim \mathcal {E}_{p_{\|}}\gg \mathcal {E}_{p_{\perp }}$ observed most clearly in the active-$\Delta p$ simulations (and, to a lesser degree, collisionless cases). The active-$\Delta p$ spectra bear encouraging resemblance to those seen in the hybrid-kinetic simulations of A$+$22.

Another interesting feature of the active-$\Delta p$ pressure spectra is that $\mathcal {E}_{\Delta p}\sim \mathcal {E}_{p_{\|}}\gg \mathcal {E}_{p_{\perp }}$, while in the passive runs the difference between $\mathcal {E}_{p_{\|}}$ and $\mathcal {E}_{p_{\perp }}$ is far less pronounced (particularly for B100, where $\mathcal {E}_{p_{\|}}\sim \mathcal {E}_{p_{\perp }}$, as might be naïvely expected).Footnote ¹⁴ This feature was also clearly observed in the hybrid-kinetic simulations of A$+$22. As discussed in more detail in § 4.3.3 and in A$+$22, it likely results from perpendicular pressure balance between $\boldsymbol {\nabla } p_{\perp }$ and the magnetic pressure $\boldsymbol {\nabla } B^{2}/8{\rm \pi}$, while $p_{\|}$ is driven more strongly by $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$, thus dominating $\Delta p$. The basic feature seems generic to anisotropic collisionless (CGL) dynamics, occurring because $\delta p_{\perp }$ remains highly constrained by perpendicular pressure balance, while $\delta p_{\|}$ is not, affecting the plasma only through the parallel force (see Appendix A.1.2 for a simple linear calculation to illustrate the effect). We provide evidence for the scenario in figure 9 by plotting the spectra of the magnetic pressure $\mathcal {E}_{B^{2}/8{\rm \pi} }$ (grey lines), which matches $\mathcal {E}_{p_{\perp }}$ very well below the forcing scales. In the CL10 simulation, where the feedback of the pressure anisotropy is only important at larger scales because $\mathcal {I}(k_{\perp })$ increases to ${\gg }1$ at small scales (see figure 7), the feature reverses to $\mathcal {E}_{p_{\|}}\simeq \mathcal {E}_{p_{\perp }}> \mathcal {E}_{\Delta p}$ for $k_{\perp }\gtrsim 200$ (there is no similar reversal for the passive-$\Delta p$ simulation). This is presumably the signature pressure-anisotropy feedback becoming subdominant (i.e. magneto-immutability no longer being important) once the driving of $\Delta p$ via $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$ becomes subdominant to standard MHD effects (the perpendicular pressure balance). We see hints that similar behaviour would occur in B100 and CL100, but at smaller scales than in CL10 (as expected), which unfortunately are unresolved in these simulations. Overall, the general similarity of the active-$\Delta p$ spectra, but not the passive-$\Delta p$ spectra, to the large-scale hybrid-kinetic results of A$+$22 is encouraging for the applicability of the CGL-LF model.

4.3.2. Rate-of-strain spectra

Given that the pressure anisotropy is driven by the plasma motions, the pressure-anisotropy feedback must be driving important changes in the flow and magnetic-field structure, despite having only a small influence on the kinetic-energy spectrum (figure 6). We diagnose this in figure 10 via the spectra of the local rates of strain, as defined in § 3.3.1. These reveal the expected substantial suppression of $\boldsymbol {\nabla }_{\|}u_{\|}=\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$ by the pressure-anisotropy feedback, as needed to suppress the turbulently driven fluctuations in $\Delta p$ and $B$. Of some interest is the comparison of $\boldsymbol {\nabla }_{\|}u_{\|}$ with $\boldsymbol {\nabla }_{\perp }u_{\|}$ and $\boldsymbol {\nabla }_{\|}\boldsymbol {u}_{\perp }$, with neither of the latter two significantly suppressed. This demonstrates that the system does not reduce $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$ purely via a reduction in field-parallel flows, or via the reduction of field-parallel gradients, but through the combination of the two. We see broadly similar features in the collisionless (CL10) and weakly collisional (B100) regimes, with the approximate spectral scalings $\mathcal {E}_{\boldsymbol {\nabla }_{\perp }u_{\perp }}\propto k_{\perp }^{1/3}$, $\mathcal {E}_{\boldsymbol {\nabla }_{\perp }u_{\|}}\propto k_{\perp }^{1/3}$, $\mathcal {E}_{\boldsymbol {\nabla }_{\|}u_{\perp }}\propto k_{\perp }^{-1/3}$ and $\mathcal {E}_{\boldsymbol {\nabla }_{\|}u_{\|}}\propto k_{\perp }^{-1/3}$ for both the active- and passive-$\Delta p$ simulations. As discussed below (§ 4.3.3), the relatively flat $\mathcal {E}_{\boldsymbol {\nabla }_{\|}u_{\|}}$ scaling may be a signature of some form of cascade of compressive fluctuations, since a $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$ spectrum that is dominated by linearly polarised Alfvénic fluctuations would be naïvely expected to scale as ${\sim }k_{\perp }^{-1}$. The most interesting difference between the collisionless and weakly collisional spectra is the closer-to-constant offset between the active and passive $\boldsymbol {\nabla }_{\|}u_{\|}$ spectra in the weakly collisional case (i.e. the modestly steeper $\boldsymbol {\nabla }_{\|}u_{\|}$ spectrum in B100), which may be a result of the pressure-anisotropy feedback being approximately constant across scales when $\omega _{A}<\nu _{c}$, rather than being dominated by the outer scale (see figure 7).

Figure 10. Rate-of-strain and pressure-gradient spectra in the CL10 and B100 simulations (left and right panels, respectively). Dashed lines show the equivalent passive-$\Delta p$ runs, while the colours indicate the specific strain direction considered. Details and definitions are described in § 3.3.1 (see (3.5) and (3.6)); the dotted black lines indicate the ${\sim }k_{\perp }^{1/3}$, ${\sim }k_{\perp }^{-1/3}$ and $k_{\perp }^{-1}$ scalings (see text). While $\boldsymbol {\nabla }_{\perp }u_{\perp }$, $\boldsymbol {\nabla }_{\perp }u_{\|}$ and $\boldsymbol {\nabla }_{\|}u_{\perp }$ are each relatively similar in active- and passive-$\Delta p$ runs, $\boldsymbol {\nabla }_{\|}u_{\|}=\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$, which is responsible for the creation of $\Delta p$, is markedly reduced when $\Delta p$ is active. Note that neither $\boldsymbol {\nabla }_{\|}u_{\perp }$ nor $\boldsymbol {\nabla }_{\perp }u_{\|}$ is similarly reduced, showing that the effect is not just a reduction in $u_{\|}$ or in $\boldsymbol {\nabla }_{\|}$. The bottom panels show that $\boldsymbol {\nabla }_{\|}\Delta p$ is suppressed more than $\boldsymbol {\nabla }_{\perp }\Delta p$ in active-$\Delta p$ turbulence, which is a signal that the pressure-anisotropy stress is reduced beyond just the reduction in the variance of $\Delta p$, viz., $\boldsymbol {\nabla }\boldsymbol {\cdot }(\hat {\boldsymbol {b}} \hat {\boldsymbol {b}} \Delta p)/\Delta p_\textrm {rms}$ is smaller in active-$\Delta p$ than in passive-$\Delta p$ turbulence.

The lower panels in figure 10 show spectra of gradients of the pressure anisotropy. As is clear from the $\boldsymbol {u}$ evolution (2.2), it is not the pressure anisotropy itself that feeds back on the flow, but only its divergence

(4.3)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot}(\hat{\boldsymbol{b}} \hat{\boldsymbol{b}} \Delta p) = \hat{\boldsymbol{b}} \boldsymbol{\nabla}_{\|}\Delta p - \hat{\boldsymbol{b}} \Delta p \boldsymbol{\nabla}_{\|}\ln B + \Delta p \hat{\boldsymbol{b}} \boldsymbol{\cdot}\boldsymbol{\nabla}\hat{\boldsymbol{b}} .\end{equation}

Computing each of these terms separately, one finds that $\boldsymbol {\nabla }_{\|}\Delta p$ is larger than the other terms, and there are no significant correlations between the different terms. Thus, if an active-$\Delta p$ run exhibits a lower ratio of $\boldsymbol {\nabla }_{\|}\Delta p$ to $\boldsymbol {\nabla }_{\perp }\Delta p$ than an equivalent passive-$\Delta p$ run, this signifies that the effect of pressure anisotropy on the flow, $\boldsymbol {\nabla }\boldsymbol {\cdot }(\hat {\boldsymbol {b}} \hat {\boldsymbol {b}} \Delta p)$, is reduced beyond what would be assumed by just looking at the variance of $\Delta p$ or $\boldsymbol {\nabla }_{\|}u_{\|}$. We see from figure 10 that this is indeed the case, viz. there is a secondary mechanism – the turbulence reduces $\boldsymbol {\nabla }\boldsymbol {\cdot }(\hat {\boldsymbol {b}} \hat {\boldsymbol {b}} \Delta p)$ beyond just $\Delta p$ (or $\boldsymbol {\nabla }\Delta p$) – which will further suppress the damping of plasma motions by pressure anisotropy.

We have examined a wide variety of correlations between the different terms that drive the pressure-anisotropy evolution (2.12). For example, one might imagine that a positive correlation between $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$ and $\boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {u}$ would aid in reducing $\Delta p$ generation, particularly at lower $\beta$ (see (2.13)); or similarly for correlations between the heat fluxes and other terms in the $\Delta p$ evolution (2.12). However, directly comparing such measures between the active- and passive-$\Delta p$ simulations has not yielded any further effects that are worthy of note, so we do not discuss these here.

4.3.3. Possible theoretical interpretation

In this section we speculate on possible phenomenologies that could explain the observed pressure and rate-of-strain spectra in the inertial range below the forcing scales. We examine two qualitatively different possibilities: the first is that the pressures and $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$ are dominated by the residual magnetic-field variation that arises from a collection of linearly polarised Alfvénic fluctuations; the second is that they are dominated by a cascade of compressive fluctuations, which is set up around the forcing scales as the system rearranges itself. In the first scenario, the $\Delta p$ and $B^{2}$ spectra are the residual local ‘left-overs’ that cannot quite be eliminated by magneto-immutability; in the second, they are diagnosing another type of cascade or mixing of larger-scale fluctuations, similar to the passive slow-mode cascade in reduced MHD or gyrokinetics (Schekochihin et al. (Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009), Kunz et al. (Reference Kunz, Schekochihin, Chen, Abel and Cowley2015) and Meyrand et al. (Reference Meyrand, Kanekar, Dorland and Schekochihin2019); however, unlike these gyrokinetic cases, the compressive and Alfvénic cascades could lack individually conserved invariants, meaning the distinction between them may be less precise than implied above).Footnote ¹⁵ Motivated by the observation in figure 9 that $\mathcal {E}_{p_\|}\gg \mathcal {E}_{p_\perp }$, we will imagine the compressive cascade to consist of oblique slow-magnetosonic-like modes,Footnote ¹⁶ which, because they maintain perpendicular pressure balance, satisfy $\delta p_{\perp }\ll \delta p_{\|}$. This property causes the mode's frequency to scale as $k_{\|}v_\textrm {th}$, rather than $k_{\|}v_{A}$ (as for the MHD slow mode), as well as leading to significant damping by heat fluxes and/or collisions. Further details are given in Appendix A.1.2 and Majeski et al. (Reference Majeski, Kunz and Squire2023).

Let us examine predictions for the rate-of-strain and pressure spectra for each possibility in turn, starting with the idea that finite-amplitude Alfvénic fluctuations dominate. In this case, a simple estimate for the spectrum of $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$ comes from (2.13), using $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u} \approx (1/2) B_{0}^{-2} \,\textrm {d}(\delta B_{\perp }^{2})/\textrm {d} t \sim \omega _{A} \delta B_{\perp }^{2}/B_{0}^{2}$. For a critically balanced Goldreich–Sridhar cascade ($\delta B_{\perp }\propto k_{\perp }^{-1/3}$, $\omega _{A}\propto k_{\perp }^{2/3}$), this suggests that $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u} \sim k_{\perp }^{0}$, or a $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$ spectrum $\mathcal {E}_{\boldsymbol {\nabla }_{\|}u_{\|}}\propto k_{\perp }^{-1}$, which is clearly significantly steeper than observed (cf. the steepest dotted line in figure 10). However, this prediction is less rigorous than it seems because ignores the influence of magneto-immutability: larger $\delta B_{\perp }$, with larger local $\mathcal {I}(k_{\perp })$, will be more affected by $\Delta p$ forces, thus reducing $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$ below $\omega _{A} \delta B_{\perp }^{2}/B_{0}^{2}$ and flattening its spectrum, perhaps to what is observed. However, the most straightforward estimate for this effect – that magneto-immutability would lead to $\Delta p$ fluctuations of approximately constant dynamical importance across scale (see § 2.2.3) – would suggest $\delta \Delta p\propto B_{0}^{2}$ or $\mathcal {E}_{\Delta p}\sim \mathcal {E}_{p_\|}\sim k_{\perp }^{-1}$, which also does not agree with what is observed (figure 9). So, there is no obvious phenomenological explanation for the measured spectral slopes within this framework. As described above, the observation that $\delta p_{\perp }\ll \delta p_{\|}$ can be naturally explained by postulating that the finite-amplitude Alfvénic fluctuations are in perpendicular pressure balance with the magnetic-field-strength fluctuations, $\delta p_{\perp }\sim \delta (B^{2})$ (as observed in figure 9). Because $\delta p_{\|}$ is instead driven by $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$ to $\delta p_{\|} \sim \beta \max (1,\omega _{A}/\nu _{c})\delta (B^{2})\gg \delta (B^{2})$, this suggests that $\delta \Delta p\sim \delta p_{\|}\gg \delta p_{\perp }$.

The other possibility – a compressive magnetosonic cascade – more naturally explains some spectral features, but appears to disagree with other key properties. Its most problematic aspect is that oblique CGL magnetosonic slow waves satisfy $\delta p_{\|}/\delta p_{\perp }\approx (5\beta +6)/2$ (see Appendix A.1.2), a far larger ratio than what is observed in all three simulations shown in figure 9.Footnote ¹⁷ Furthermore, although the observed ratio $\delta p_{\|}/\delta p_{\perp }$ increases modestly with $\beta$, the increase is far slower than the predicted linear dependence, which, for example, would yield $\mathcal {E}_{p_{\|}}/\mathcal {E}_{p_{\perp }}\sim 1000$ in the CL100 simulation. This suggests that at least some other component or complication is necessary to decrease $\delta p_{\|}/\delta p_{\perp }$ well below the linear prediction to match the simulations. On the other hand, the observed spectral scalings of the rates of strain and pressure seem to be most naturally explained via a compressive cascade: if one postulates $u_{\|}\propto u_{\perp }\sim k_{\perp }^{-1/3}$ (assuming the compressive and Alfvénic cascades scale similarly), $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$ should scale as ${\sim }k_{\|}u_{\|}\propto k_{\perp }^{1/3}$, giving $\mathcal {E}_{\boldsymbol {\nabla }_{\|}u_{\|}}\propto k_{\perp }^{-1/3}$, which is close to what is seen in figure 10. Predicted scalings of the other rates of strain ($\boldsymbol {\nabla }_{\|}u_{\perp }\propto k_\perp ^{1/3}$, $\boldsymbol {\nabla }_{\perp }u_{\|}\propto k_\perp ^{2/3}$ and $\boldsymbol {\nabla }_{\perp }u_{\perp }\propto k_\perp ^{2/3}$) are also consistent with what is observed, being the same as what is expected for passive slow-mode cascade in isothermal MHD (and indeed, the passive-$\Delta p$ rates of strain exhibit similar scalings in figure 10). In this scenario, pressure spectra would result predominantly from the passive mixing of larger-scale fluctuations, as opposed to local driving by $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$, which also naturally yields the observed scaling $\mathcal {E}_{p_\|}\propto \mathcal {E}_{p_\perp }\sim k_\perp ^{-5/3}$ through the inertial range where $u_{\perp }\sim k_{\perp }^{-1/3}$, as well as explaining the similarity of the collisionless and weakly collisional simulations. Another interesting observation is the long parallel scale of $p_{\|}$ and $u_{\|}$ fluctuations (see $\ell _{\|}(\lambda )$ in figure 8b), which would be expected if for some reason the frequencies of Alfvénic and compressive fluctuations were matched around the outer scale.

Overall, we see a plausible agreement of the Alfvénic hypothesis with most diagnostics, although its predictions remain too qualitative to say much of substance. Its biggest flaw is the similarity of the collisionless and weakly collisional rate-of-strain and pressure spectra, which argues against pressure fluctuations being driven locally in scale by $\hat {\boldsymbol {b}}\hat {\boldsymbol {b}}:\boldsymbol {\nabla } \boldsymbol {u}$. A compressive cascade alone cannot explain the observed spectra because the $\delta p_\|/\delta p_\perp$ ratio characteristic of linear slow-mode fluctuations is too large to fit the data (particularly the dependence on $\beta$); however, we cannot rule out their importance in some form (e.g. if $\delta p_\perp$ was dominated by residual Alfvénic fluctuations), and various other properties, particularly spectral scalings, seem to be more naturally explained via the compressive-cascade hypothesis. Clearly, more work is needed to make further progress.

4.4.1. Energy transfers and fluxes

As discussed in § 3.3.3, the turbulence energetics can be usefully quantified using ‘transfer functions’ $\mathcal {T}^\textrm {AB}_{q\rightarrow k}$, which measure the energy transfer between shells in Fourier space due to different types of nonlinear interactions. Each of these transfers can involve several terms in the compressible system, so, as detailed in Grete et al. (Reference Grete, O'Shea, Beckwith, Schmidt and Christlieb2017), we combine relevant compressive terms into the non-compressive terms in order to reduce the number of quantities under consideration. The compressive terms are all individually very small and do not show interesting features, although they are necessary to include in order to respect energy conservation. Their full forms, as we compute them, are as defined in Grete et al. (Reference Grete, O'Shea, Beckwith, Schmidt and Christlieb2017), but we simply use the placeholder $\mathcal {C}_\textrm {AB}$ to refer to them below. We follow Grete et al. (Reference Grete, O'Shea, Beckwith, Schmidt and Christlieb2017) in our nomenclature; $\textrm {AB}=\textrm {UU}$ refers to the transfer between shells of the kinetic energy through $\boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol {u} + \mathcal {C}_\textrm {UU}$ (see (3.9)), $\textrm {AB}=\textrm {BU}$ refers to the transfer from magnetic to kinetic energy through $\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol {B} + \mathcal {C}_\textrm {BU}$, $\textrm {AB}=\textrm {UB}$ refers to the transfer from kinetic to magnetic energy through $\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol {u}+\mathcal {C}_\textrm {UB}$, $\textrm {AB}=\textrm {BB}$ refers to the transfer between shells of the magnetic energy through $\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol {B}+\mathcal {C}_\textrm {BB}$, $\textrm {AB}={ \Delta p U}$ refers to the transfer from the kinetic to thermal energy via the pressure anisotropy through $\boldsymbol {\nabla }\boldsymbol {\cdot }(\hat {\boldsymbol {b}} \hat {\boldsymbol {b}} \Delta p)$ (see below) and $\textrm {AB}=\textrm {F U}$ refers to the transfer to kinetic energy from the external forcing.Footnote ¹⁸ We compute transfers as a function of the isotropic $k$ as opposed to $k_{\perp }$, but results versus $k_{\perp }$ are similar.

As introduced in A$+$22, the only important new term compared with Grete et al. (Reference Grete, O'Shea, Beckwith, Schmidt and Christlieb2017) is that due to the pressure-anisotropy stress $\mathcal {T}_{q\rightarrow k}^{ \Delta p U}$. There is some freedom in the definition of this; A$+$22 use

(4.4)\begin{equation} \mathcal{T}_{q\rightarrow k}^{\Delta p\mathrm{U}} = \int\mathrm{d}\boldsymbol{x} \left\langle\sqrt{\rho}\boldsymbol{u}\right\rangle_{k} \boldsymbol{\cdot} {\rm sign}({\Delta p}) \frac{\sqrt{|\Delta p|} }{B}\frac{\boldsymbol{B}}{\sqrt{4{\rm \pi} \rho}} \boldsymbol{\cdot} \boldsymbol{\nabla}\langle\sqrt{|\Delta p|}\hat{\boldsymbol{b}}\rangle_{q},\end{equation}

which has the advantage that the square of $\langle \sqrt {|\Delta p|}\hat {\boldsymbol {b}}\rangle _{k}$ can be interpreted as the pressure anisotropy's contribution to the thermal energy (Kunz et al. Reference Kunz, Schekochihin, Chen, Abel and Cowley2015), but the disadvantage of sign discontinuities that arise from $\sqrt {|\Delta p|}$. If we instead interpret the pressure-anisotropy stress as a damping of kinetic energy (as opposed to the transfer between energy reservoirs) a more natural definition is

(4.5)\begin{equation} \mathcal{T}_{q\rightarrow k}^{\Delta p\mathrm{U}} = \int\mathrm{d}\boldsymbol{x} \left\langle\sqrt{\rho}\boldsymbol{u}\right\rangle_{k} \boldsymbol{\cdot} \frac{\boldsymbol{B}}{\sqrt{4{\rm \pi} \rho}} \boldsymbol{\cdot} \boldsymbol{\nabla}\left\langle \frac{\Delta p}{B^{2}}{\boldsymbol{B}}\right\rangle_{q}.\end{equation}

We have computed both versions, finding qualitatively similar results, but will focus on (4.5), because it fits more naturally with our focus on pressure-anisotropy damping. Note also that the influence of the sign-discontinuity issue of (4.4) was mitigated in A$+$22 because the $\Delta p$ distribution was mostly confined to negative values, but the spread of $\Delta p$ is somewhat broader in our simulations (see figures 3 and 4).

In figure 11, we plot the net energy transfers $\textrm {T}^\textrm {AB}(k)$ (see (3.10)), which measure the net contribution from each term to the rate of change of the kinetic- or magnetic-energy spectrum at each $k$. For a conservative cascade, these are zero, because the transfer into a given shell from larger scales is balanced by the transfer out of the shell to smaller scales. Thus, $\textrm {T}^\textrm {AB}(k)$ provides a direct measure of the damping of kinetic or magnetic energy at each scale due to pressure anisotropy or other terms. To plot the results clearly, we normalise $\textrm {T}^\textrm {AB}(k)$ by the dimensional (Kolmogorov) estimate $\partial _{t}\mathcal {E}\sim k u_{k}\mathcal {E}\sim \varepsilon (k/k_{0})^{-1}$ (where $k_{0}=2{\rm \pi} /L_{\perp }$); with this normalisation, a line that is constant and non-zero with $k$ symbolises a contribution that is of approximately constant importance compared a conservative cascade towards smaller scales. As above, we show the CL10 and B100 simulations, comparing with the results from the corresponding passive-$\Delta p$ simulations (lower subpanels). The thick black lines, which show the sum of all terms (excluding $\textrm {T}^{\Delta p U}$ for the passive runs), are approximately zero throughout the inertial range, as expected, with no clear difference between the active and passive runs. While the contribution from $\textrm {T}^{\Delta p U}$ is slightly negative in both CL10 and B100, indicating a slight damping of inertial-range motions, it is small compared with the cascade rate. Contrast this to the equivalent $\textrm {T}^{\Delta p U}$ from the passive simulations, which gives an indication of what the damping would be in the absence of feedback from $\Delta p$. Although this measurement is counterfactual – the cascade could not have proceeded in the presence of such strong damping – it concisely illustrates how motions driven by normal turbulence would be strongly damped, at a rate comparable to the cascade rate, across all scales. This is not entirely obvious for the collisionless case particularly, because the contribution to the kinetic energy from $\boldsymbol {\nabla }\boldsymbol {\cdot }(\hat {\boldsymbol {b}} \hat {\boldsymbol {b}} \Delta p)$ can be both positive and negative (it is negative definite in the Braginskii-MHD regime, but not otherwise; see (2.10)). The effect of magneto-immutability all but eliminates such damping below the outer scale, leaving only a modest damping of outer-scale motions for the weakly collisional case.

Figure 11. Important terms in the net energy-transfer spectra, as defined in § 3.3.3 and (3.10). The top panels show the simulations with $\beta _{0}=10$, $\nu _{c}=0$ and the bottom panels show the simulations with $\beta _{0}=100$, $\nu _{c}=33$, with the upper and lower subpanels for each showing the active- and passive-$\Delta p$ cases, respectively. Insets in the plots of the two active-$\Delta p$ cases show a zoom of the grey-box region. All transfers are normalised to $\varepsilon /(k/k_{f})$, which approximately represents the local cascade rate (see text). The different colours show different energy-transfer terms as labelled in the top panel: $\mathrm {T}^{\Delta p U}$ is the transfer from $\boldsymbol {u}$ to thermal energy due to $\boldsymbol {\nabla }\boldsymbol {\cdot }(\hat {\boldsymbol {b}} \hat {\boldsymbol {b}} \Delta p)$; $\mathrm {T}^\textrm {UU}$ is momentum transfer via $\boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {\nabla }\boldsymbol {u}$; $\mathrm {T}^\textrm {BU}$ is the contribution to kinetic energy from magnetic tension and pressure; $\mathrm {T}^\textrm {BB}$ is the advection of $\boldsymbol {B}$ in the induction equation; $\mathrm {T}^\textrm {UB}$ is the contribution to magnetic energy from field stretching; and $\mathrm {T}^\textrm {FU}$ is the forcing contribution. In the active-$\Delta p$ simulations below the outer scale of the turbulence, we see little contribution of $\Delta p$ to the energy transfer, which shows directly that the effect of the pressure-anisotropy stress is minimised below the outer scale, even though the local interruption number is still smaller than unity in this range (see figure 7). In contrast, in the passive-$\Delta p$ simulations (bottom panel), the pressure anisotropy that develops is such that, if it did feed back on the flow, it would cause $\mathcal {O}(1)$ dissipation at all scales in the cascade (blue line), which would completely damp the turbulence.

Figure 12 shows similar information in a different form by plotting the contributions to the turbulent energy flux (3.11). We show only the active-$\Delta p$ runs because the comparison with passive simulations is less interesting in this case. Unlike the net transfers (figure 11), individual terms in the fluxes are not easily interpreted because they do not include the diagonal $\mathcal {T}^\textrm {AB}_{k\rightarrow k}$ transfer. While this diagonal contribution necessarily vanishes for the total energy flux, it dominates the transfer between different energy reservoirs, including the damping of the flow by the pressure anisotropy (blue line). Thus the fact that $\varPi ^{\Delta p U}$ is small and positive is not particularly relevant, while the individual lines are of interest only insofar as they indicate a contribution to the total flux (black lines). The most important feature that we observe is that the total energy fluxes are nearly constant for $10\lesssim k\lesssim 200$, consistent with the result of figure 11 that there is little pressure-anisotropy damping through the inertial range (a slight decrease in $\varPi (k)$ is observed for B100, indicating a slight damping). The value of the $\varPi /\varepsilon$ in the inertial range is a direct measure of the cascade efficiency, which, as expected, is less than unity because there is some pressure-anisotropy damping near the forcing scales (the reduction of ${\simeq }25\,\%$ for C100 and ${\simeq }50\,\%$ for B100 is consistent with the total inferred pressure-anisotropy heating, which is discussed below)

Figure 12. The energy fluxes (see (3.11)), which measure the transfer of energy across a particular $k$, for the two active-$\Delta p$ simulations with $\beta _{0}=10$, $\nu _{c}=0$ (a) and $\beta _{0}=100$, $\nu _{c}=33v_{A}/L_{\perp }$ (b). We normalise to the forcing input $\varepsilon$, which implies that the total flux would be $\varPi /\varepsilon =1$ across the full inertial range for a conservative cascade with no damping. As also seen in figure 11, although there is some energy loss due to $\Delta p$ at the outer (forcing) scale, at smaller scales the flux remains constant, showing that there is little energy damping through the inertial range.

4.4.2. Pressure-anisotropy heating fraction and cascade efficiency

The net energy transfer $\textrm {T}^{\Delta p U} (k)$ also gives a convenient way to evaluate the total heating rate due to pressure anisotropy. Because the pressure-anisotropy (viscous) heating is confined to the outer scales (figure 11), this is also a measure of the cascade efficiency – the energy that is available to cascade in the usual way to heat via small-scale collisionless mechanisms. Its precise value depends on the forcing scheme, since it mostly occurs at the outer scales where the forcing drives the flow, and could thus differ more significantly in systems with more realistic turbulence generation mechanisms (e.g. magneto-rotational turbulence; Kunz, Stone & Quataert Reference Kunz, Stone and Quataert2016; Kempski et al. Reference Kempski, Quataert, Squire and Kunz2019). Nonetheless, its dependence on parameters ($\beta$ and the collisionality) is interesting to consider, as is a comparison with A$+$22.

We define

(4.6)\begin{equation} \mathcal{H}_{\Delta p} = \int_{k_{0}}^{k_{\rm max}} {\rm d} k \,{\rm T}^{\Delta p U}(k),\end{equation}

so $\mathcal {H}_{\Delta p} /\varepsilon$ measures the fraction of the energy input that is converted into heat via pressure anisotropy up to wavenumber $k=k_\textrm {max}$. Because the grid-scale dissipation in our simulations is non-physical – it is supposed to represent energy absorbed by the smaller scales, which would eventually cascade to mostly dissipate around the ion gyro-radius scale (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009) – it is most appropriate to choose $k_\textrm {max}$ to lie near the bottom of the inertial range. We set $k_\textrm {max}=100$ as appropriate for the resolution $400\times 200^{2}$, but the choice hardly affects the results for the active-$\Delta p$ simulations anyway, because almost all of the viscous heating is confined near the outer scales.Footnote ¹⁹

We evaluate $\mathcal {H}_{\Delta p}$ for all of the lrCL, lrB and $\beta 16$ simulations (see table 1), time averaging across the steady state of each. Results are shown in figure 13, plotted against the measured interruption number, with marker colour indicating the collisionality and marker style indicating the simulation set (lrCL, lrB or $\beta 16$). The CGL-LF (active-$\Delta p$) simulations, shown with large symbols, have $\mathcal {H}_{\Delta p}$ between $\mathcal {H}_{\Delta p}\simeq 0.2\varepsilon$ for the collisionless simulations and $\mathcal {H}_{\Delta p}\simeq 0.45\varepsilon$ for the weakly collisional and Braginskii-MHD simulations at $\mathcal {I}\sim 1$, with no significant dependence on $\beta$. In other words, for this choice of forcing, the cascade efficiency is always above ${\simeq }50\,\%$, meaning most of the input energy can participate in an MHD-like turbulent cascade. Contrast this with the small symbols, which show the passive-$\Delta p$ simulations for lrCL and lrB, indicating what the pressure-anisotropy heating rate would be in the absence of magneto-immutability. The numerical values are of course meaningless in this case – the situation is counterfactual and having $\mathcal {H}_{\Delta p}>\varepsilon$ is clearly not possible – but, as above, it demonstrates that the motions involved in standard MHD Alfvénic turbulence would be strongly damped by pressure anisotropy were it not for the pressure-anisotropy feedback modifying their structure. From the $\beta 16$ simulations, we see also that $\mathcal {H}_{\Delta p}(\mathcal {I})$ remains relatively large up to large $\mathcal {I}$, despite the effective driving of pressure-anisotropy fluctuations decreasing towards large $\mathcal {I}$ due to their larger $\nu _{c}$. This shows that as $\mathcal {I}$ is decreased below ${\simeq }10$, the suppression of viscous heating is sufficiently strong that it balances the stronger driving of $\Delta p$ fluctuations that results from lower $\nu _{c}$ (otherwise $\mathcal {H}_{\Delta p}$ would continue increasing below $\mathcal {I}\simeq 10$).

Figure 13. Net pressure-anisotropy heating rate $\mathcal {H}_{\Delta p}$ normalised to the cascade rate $\varepsilon$ from all simulations. The cascade efficiency, defined as the fraction of energy that participates in the cascade to small scales, is $1 - \mathcal {H}_{\Delta p}/\varepsilon$. This is computed using the net transfer functions (figure 11) by summing up the contributions from all $k\leq 100$, viz. $\mathcal {H}_{\Delta p}=\sum _{k\leq 100} \mathrm {T}^{U\Delta p}(k)$, so as to exclude the grid scales. The marker style denotes the simulation set (see § 3.2 and table 1): circles denote the lrB set at constant $\mathcal {I}$, crosses denote the lrCL set with $\nu _{c}=0$, and stars denote the $\beta 16$ set. We see that viscous (pressure-anisotropy) heating damps up to ${\simeq }45\,\%$ of the cascade in collisional regimes ($\nu _{c}/\omega _{A} \gtrsim 1$), and less in collisionless cases. This value is approximately constant for $\mathcal {I}\lesssim 10$ and independent of $\beta$ and $\nu _{c}/\omega _{A}$ (so long as $\nu _{c}/\omega _{A}\gtrsim 1$). The small symbols show the same computation from the passive-$\Delta p$ simulations; in this case the heating rate is larger than unity, because $\Delta p$ does not actually feed back on the flow. Such large values imply that if $\Delta p$ had not been reduced by magneto-immutability, it would have nearly completely damped the cascade.