Chandran, 2016; Arzamasskiy et al., 2019; Martinović et al., 2020; Cerri et
al., 2021).

Chandran et al. (2010) used test particle simulations of randomly phased
Alfvén and kinetic Alfvén waves to develop an empirical formula for the rate
of ion stochastic heating by turbulent fluctuations with frequencies much
lower than the ion cyclotron frequency $\omega/\Omega_{i}\ll 1$. Following
this, Chandran et al. (2011) developed a refined implementation to predict the
partitioning of turbulent energy among parallel ion heating, perpendicular ion
heating, and electron heating, with a formula for the damping rate by ion
stochastic heating given by

$\gamma_{SH}=c_{1}\left(\frac{\delta
v_{\perp}^{(\rho_{i})}}{v_{ti}}\right)\Omega_{i}e^{-c_{2}v_{ti}/\delta
v_{\perp}^{(\rho_{i})}}$ (29)

where $\delta v_{\perp}^{(\rho_{i})}$ is the root-mean-square amplitude of the
perpendicular velocity fluctuations at the perpendicular scale of the ion
Larmor radius $k_{\perp}\rho_{i}=1$. Here $c_{1}$ and $c_{2}$ are fitting
constants with typical values $c_{1}=0.18$ and $0.15\lesssim c_{2}\lesssim
0.34$ (Chandran et al., 2011). The key dimensionless parameter controlling the
onset of stochastic heating in this formulation is the ratio of the
perpendicular velocity fluctuation to the thermal velocity,
$\epsilon\equiv\delta v_{\perp}^{(\rho_{i})}/v_{ti}$; these studies found a
critical threshold $\epsilon_{crit}\simeq 0.19$ that determines the minimum
fluctuation amplitude required for the onset of stochastic heating (Chandran
et al., 2010).

Using the relation for Alfvén waves to connect the perpendicular fluid
velocity fluctuations to the perpendicular magnetic field fluctuations $\delta
v_{\perp}/v_{A}=\pm\overline{\omega}\delta B_{\perp}/B_{0}$ (Howes et al.,
2008a) and the MHD scaling of the turbulent perpendicular magnetic field
fluctuation amplitude given by (4) with the B06 scaling with $\alpha=1$, we
obtain the expression

$\frac{\delta
v_{\perp}}{v_{ti}}=\overline{\omega}\beta_{i}^{-1/2}\left(\frac{k_{0}\rho_{i}}{k_{\perp}\rho_{i}}\right)^{1/4}$
(30)

At $k_{\perp}\rho_{i}\sim 1$, we obtain the normalized frequency
$\overline{\omega}\sim 1$ and find the key relation for the amplitude in terms
of our fundamental turbulence parameters

$\epsilon\equiv\frac{\delta
v_{\perp}^{(\rho_{i})}}{v_{ti}}=\beta_{i}^{-1/2}\left(k_{0}\rho_{i}\right)^{1/4}$
(31)

To estimate the importance of damping by ion stochastic heating relative to
the turbulent cascade rate, we can calculate
$\gamma_{SH}/\omega_{nl}\sim\gamma_{SH}/\omega$ to obtain the final result

$\frac{\gamma_{SH}}{\omega}=c_{1}\left(k_{0}\rho_{i}\right)^{-1/4}e^{-c_{2}\beta_{i}^{1/2}(k_{0}\rho_{i})^{-1/4}}$
(32)

where we have used the subsitution
$\Omega_{i}/\omega=\beta_{i}^{1/2}/k_{\parallel}\rho_{i}$.

In summary, the proposed turbulent damping mechanism of ion stochastic heating
depends primarily on the fundamental parameters $\beta_{i}$ and
$k_{0}\rho_{i}$, where lower ion plasma beta and larger isotropic driving
wavenumber (which corresponds to a smaller inertial range) both conspire to
yield larger turbulent amplitude at the ion Larmor radius scale
$k_{\perp}\rho_{i}\sim 1$, enhancing stochastic heating of the ions. The
exponential effectively provides a threshold turbulent amplitude at the ion
scales below which stochastic ion heating is ineffective. Note that the
exponential dependence on the constant $c_{2}$ makes its accurate
determination critical to assess the importance of ion stochastic heating in
any particular case; its value is likely to depend on the detailed nature of
the turbulent fluctuations at the perpendicular scale of the ion Larmor radius
(Chandran et al., 2011). This heating mechanism is predicted to lead to
strictly ion energization $Q_{i}/Q=1$ in the perpendicular degrees of freedom
$Q_{\perp,i}/Q_{i}=1$.

Note that, because the amplitude of the turbulent fluctuations is typically
found to decrease monotonically as the turbulent cascade progress to smaller
scales, it is unlikely that electrons will experience significant stochastic
heating, so we do not explore electron stochastic heating here. Further
research will be necessry to to evaluate thoroughly the potential for electron
stochastic heating to damp turbulence at the perpendicular scale of the
electron Larmor radius, $k_{\perp}\rho_{e}\sim 1$.

###  5.7 Magnetic Pumping

Magnetic pumping (Spitzer & Witten, 1951; Berger et al., 1958; Lichko et al.,
2017; Lichko & Egedal, 2020; Montag & Howes, 2022) is a particle energization
mechanism in which variations of magnetic field magnitude at low frequency
$\omega$ lead to an oscillating transfer of energy from parallel to
perpendicular degrees of freedom and back, obeying the double-adiabatic (Chew-
Goldberger-Low, CGL) equations of state (Chew et al., 1956). In the absence of
any collisions, the net change of particle energy over a full oscillation is
zero. But in the presence small but finite collisionality999Note that the
collisionality can be due to Coulomb collisions or to an effective
collisionality due to pitch-angle scattering through collisionless wave-
particle interactions with small-scale electromagnetic fluctuations, such as
those driven by kinetic temperature anisotropy instabilities (Kunz et al.,
2018; Arzamasskiy et al., 2023). there can be a non-zero transfer of energy
from the oscillating electromagnetic fields to the particles over the course
of a full oscillation.

As a proposed mechanism for the damping of turbulence, magnetic pumping
necessarily requires turbulent fluctuations that involve changes of the
magnetic field magnitude from the equilibrium field strength $B_{0}$, or
$\delta B\equiv|\mathbf{B}|-B_{0}\neq 0$. For Alfvénic turbulence, which is
dominantly incompressible101010Note, however, that the magnetic field
magnitude variations arising from strong turbulent driving at the outer scale
with $|\delta\mathbf{B}|\sim B_{0}$ can lead to the “interruption” of linearly
polarized Alfvén waves by kinetic temperature anisotropy instabilities at
sufficient high $\beta_{i}\gg 1$ (Squire et al., 2016, 2017b, 2017a) in the
MHD regime at $k_{\perp}\rho_{i}\ll 1$, significant fluctuations of the
magnetic field magnitude arise only within the kinetic Alfvén wave regime at
$k_{\perp}\rho_{i}\gtrsim 1$ under conditions of ion plasma beta
$\beta_{i}>1$. Alternatively, if the turbulence has a signficant component of
compressible magnetosonic fluctuations with $E_{comp}/E_{inc}>0$, because the
turbulent amplitudes typically decrease monotonically with decreasing scale
lengths, the largest scale compressible fluctuations at the driving scale may
lead to magnetic pumping, directly energizing particles from these large-scale
fluctuations.

We determine first the parameters controlling magnetic pumping in Alfvénic
turbulence, following a similar procedure later to analyze the magnetic
pumping arising from compressible turbulence with $E_{comp}/E_{inc}>0$. The
foundation of magnetic pumping is the double-adiabatic evolution of the weakly
collisional plasma which conserves the particle magnetic moment $\mu_{m}\equiv
mv_{\perp}^{2}/2B$ and the action integral of the parallel bounce motion
$J_{\parallel}\equiv\oint v_{\parallel}dl$ (Lichko et al., 2017; Montag,
2018). Conservation of these adiabatic invariants requires a scale separation
of the perpendicular wavenumber relative to the particle species Larmor radius
$k_{\perp}\rho_{s}\ll 1$ and of the wave parallel phase velocity to the
particle thermal velocity $\omega/k_{\parallel}v_{ts}\ll 1$.

Kinetic Alfvén waves give rise to non-negligible magnetic field magnitude
variations over the perpendicular wavenumber range $1\lesssim
k_{\perp}\rho_{i}\lesssim(\tau\mu)^{1/2}$ for values of $\beta_{i}\gtrsim 1$.
This range of $\delta B$ fluctuations is incompatible with the necessary
condition $k_{\perp}\rho_{i}\ll 1$ for the ions, so we expect that _magnetic
pumping by kinetic Alfvén waves cannot energize the ions_. For the electrons,
on the other hand, the requisite condition $k_{\perp}\rho_{e}\ll 1$ can be
rewritten as $k_{\perp}\rho_{i}\ll(\tau\mu)^{1/2}$, so we expect that kinetic
Alfvén waves may be able to damp turbulent fluctuations and energize the
electrons under ion plasma beta conditions with $\beta_{i}\gtrsim 1$. The
phase velocity condition for electrons can be expressed as
$\overline{\omega}\ll(\beta_{i}\mu/\tau)^{1/2}$. Under the ion plasma beta
condition $\beta_{i}\gtrsim 1$, (25) can be approximated by
$\overline{\omega}\simeq k_{\perp}\rho_{i}/\beta_{i}^{1/2}$, so this
requirement can be converted to a condition on the perpendicular wavenumber
given by $k_{\perp}\rho_{i}\ll\beta_{i}(\mu/\tau)^{1/2}$, which can certainly
be satisfied for the perpendicular wavenumber range of kinetic Alfvén waves
with significant magnetic field magnitude fluctuations. Therefore, the
necessary conditions for the magnetic pumping of electrons to play a role in
the dissipation of Alfvénic plasma turbulence with $\beta_{i}\gtrsim 1$ appear
to be satisfied.

Previous studies have shown that the ratio of the damping rate due to magnetic
pumping $\gamma_{MP}$ to the pump wave frequency $\omega$ is given by (Lichko
et al., 2017; Montag & Howes, 2022)

$\frac{\gamma_{MP}}{\omega}=\frac{\chi_{c,s}}{1+\chi_{c,s}^{2}}\left(\frac{\delta
B}{B_{0}}\right)^{2}$ (33)

where $\chi_{c,s}\equiv 6\nu_{s}/\omega$ is a dimensionless measure of the
collision frequency $\nu_{s}$ for species $s$ and $\delta B$ is the amplitude
of the magnetic field magnitude fluctuations.

To determine the dependence of the magnetic pumping of electrons on the plasma
and turbulence parameters in Tables 2 and 3, we assume isotropic driving
$k_{\parallel 0}/k_{\perp 0}=1$ for simplicity, write the electron-electron
collision frequency as $\nu_{e}=v_{te}/\lambda_{mfp,e}$, and use the general
parallel wavenumber scaling for arbitrary $\alpha$ given by (1) to obtain

$\chi_{c,e}=\frac{6}{\overline{\omega}(k_{\parallel
0}\lambda_{mfp,e})}\left(\frac{\beta_{i}\mu}{\tau}\right)^{1/2}\left(\frac{k_{0}\rho_{i}}{k_{\perp}\rho_{i}}\right)^{2/(3+\alpha)}.$
(34)

For a sufficiently large turbulent inertial range with $k_{0}\rho_{i}\ll 1$,
kinetic Alfvén waves over the perpendicular wavenumber range $1\lesssim
k_{\perp}\rho_{i}\lesssim(\tau\mu)^{1/2}$ will have relatively small turbulent
amplitudes $\delta B_{\perp}/B_{0}\ll 1$, so the magnetic field magnitude
fluctuations can be expressed to lowest order as variations of the parallel
component of the perturbed magnetic field $\delta B\simeq\delta B_{\parallel}$
(Huang et al., 2024). The turbulent magnetic field magnitude variations may
then be expressed as

$\frac{\delta B}{B_{0}}=\left(\frac{\delta B_{\parallel}}{\delta
B_{\perp}}\right)\left(\frac{k_{0}\rho_{i}}{k_{\perp}\rho_{i}}\right)^{1/(3+\alpha)},$
(35)

where the ratio $\delta B_{\parallel}/\delta B_{\perp}$ for kinetic Alfvén
waves over the relevant range from the linear dispersion relation is strictly
a function of $\beta_{i}$, or $\delta B_{\parallel}/\delta
B_{\perp}=f(\beta_{i})$. Therefore, electron magnetic pumping by Alfvénic
turbulent fluctuations at scales $k_{\perp}\rho_{i}\gtrsim 1$ depends on the
parameters $\beta_{i}$, $\tau$, $\mu$, $k_{0}\rho_{i}$, and $k_{\parallel
0}\lambda_{mfp,e}$.

Next we consider that case that the turbulence is driven compressibly with
$E_{comp}/E_{inc}>0$. In this situation, the magnetic field magnitude
fluctuations at the driving scale, which typically have the largest amplitude
of $\delta B/B_{0}$, may contribute to magnetic pumping of both the ions and
electrons. In this case of turbulent damping at the driving scale, it is
necessary to use the three driving parameters $(k_{\perp
0}\rho_{i},k_{\parallel 0}/k_{\perp 0},\chi_{0})$ in Table 2, rather than the
simplified parameterization through the isotropic driving wavenumber
$k_{0}\rho_{i}$, to determine the parameter dependence of the turbulent
damping via magnetic pumping.

First, the normalized fast and slow magnetosonic wave phase velocities can be
expressed by the linear dispersion relation (Klein et al., 2012)

$\widetilde{\omega}^{4}-\widetilde{\omega}^{2}(1+\beta)+\beta\cos^{2}\Theta_{0}=0,$
(36)

which has four solutions given by

$\widetilde{\omega}^{2}=\frac{1}{2}\left[1+\beta\pm\sqrt{(1+\beta)^{2}-4\beta\cos^{2}\Theta_{0}}\right],$
(37)

where the plus (minus) sign corresponds to the two fast (slow) magnetosonic
wave modes. Here we define the normalized wave frequency
$\widetilde{\omega}\equiv\omega/kv_{A}$, the total (MHD) plasma beta
$\beta=\beta_{i}(1+1/\tau)$, and the wavevector components
$(k_{\parallel},k_{\perp})$ are alternatively parameterized by $(k,\Theta)$,
where $k_{\parallel}=k\cos\Theta$, $k_{\perp}=k\sin\Theta$,
$k=(k_{\parallel}^{2}+k_{\perp}^{2})^{1/2}$, and
$\Theta=\tan^{-1}(k_{\perp}/k_{\parallel})$. The dimensionless form of the
solutions in (37) makes clear that the fast and slow magnetosonic wave phase
velocities depend only on two dimensionless parameters,
$\widetilde{\omega}(\beta,\Theta)$.

We may determine the parameter dependence of the normalized ion collision
frequency by using the relation for the ion-ion collision rate in terms of the
electron-electron collision rate $\nu_{i}=\mu^{-1/2}\tau^{-3/2}\nu_{e}$ and
writing the pumping wave frequency at the driving scale as
$\omega=\widetilde{\omega}_{0}k_{0}v_{A}$ to obtain

$\chi_{c,i}=\frac{6}{\widetilde{\omega}_{0}(k_{\parallel
0}\lambda_{mfp,e})\tau^{2}}\beta_{i}^{1/2}\cos\Theta_{0}.$ (38)

Note that the ion collisional mean free path can be expressed in terms of the
electron collisional mean free path by the relation
$\lambda_{mfp,i}=\lambda_{mfp,e}\tau^{2}$, so the denominator is actually
independent of the ion-to-electron temperature ratio $\tau$, depending rather
only on the normalized ion mean free path, $k_{\parallel 0}\lambda_{mfp,i}$.

To evaluate the turbulent magnetic field magnitude variations for ion magnetic
pumping by compressible fluctuations at the driving scale, we first note that
the phenomenon of dynamic alignment, parameterizied by the alignment angle
$\theta_{0}$, will not play a role: dynamic alignment develops through the
nonlinear interactions as the turbulence cascades to smaller scales, but does
not come into play at the driving scale, so we set $\theta_{0}=1$, thus
reducing the nonlinearity parameter to $\chi_{0}=k_{\perp 0}\delta B_{\perp
0}/(k_{\parallel 0}B_{0})$. If the magnetic perturbations at the driving scale
have $|\delta\mathbf{B}|/B_{0}\sim 1$, the linear theory for waves breaks
down, and the nonlinearity must be taken into account in the lowest-order
description of the wave dynamics; in this case, one may directly take the
measured magnetic field magnitude fluctuations $\delta B/B_{0}$ as the
parameter to estimate the damping rate by magnetic pumping in (33). However,
if the magnetic perturbations at the driving scale have
$\epsilon\equiv|\delta\mathbf{B}|/B_{0}<1$, the binomial expansion may be used
to obtain the expression

$\delta B/B_{0}\simeq\delta B_{\parallel}/B_{0}+(\delta
B_{\parallel}/B_{0})^{2}/2+(\delta B_{\perp}/B_{0})^{2}/2,$ (39)

where the first term is of order $\epsilon$ and the next two terms are of
order $\epsilon^{2}$. Thus, to lowest order, we may estimate $\delta
B\simeq\delta B_{\parallel}$, where the error in this estimation scales as
$\epsilon^{2}/2$, yielding a fractional error of 25% for
$|\delta\mathbf{B}|/B_{0}=1/2$ and of 6% for $|\delta\mathbf{B}|/B_{0}=1/4$.

Using this approximation, we can derive an expression the normalized
fluctuations of the magnetic field magnitude by $\delta B/B_{0}\simeq\delta
B_{\parallel}/B_{0}=(\delta B_{\perp}/B_{0})(\delta B_{\parallel}/\delta
B_{\perp})$. If we estimate $\delta B_{\parallel}/\delta
B_{\perp}\sim(E_{comp}/E_{inc})^{1/2}$, we can then obtain the expression for
the normalized amplitude of the magnetic field magnitude fluctuations

$\frac{\delta B}{B_{0}}=\frac{k_{\parallel 0}}{k_{\perp
0}}\chi_{0}\left(\frac{E_{comp}}{E_{inc}}\right)^{1/2}.$ (40)

Combining the dependencies of $\chi_{c,i}$ in (38) and of $\delta B/B_{0}$ in
(40), ion magnetic pumping by compressible turbulent fluctuations at the
driving scale depends on the parameters $\beta_{i}$, $k_{\parallel
0}\lambda_{mfp,i}=(k_{\parallel 0}\lambda_{mfp,e})\tau^{2}$, $k_{\parallel
0}/k_{\perp 0}$, $\chi_{0}$, and $E_{comp}/E_{inc}$.

For magnetic pumping of electrons by compressible turbulent fluctuations at
the driving scale, we can follow an analogous procedure to obtain the
normalized electron collision frequency

$\chi_{c,e}=\frac{6}{\widetilde{\omega}_{0}(k_{\parallel
0}\lambda_{mfp,e})}\left(\frac{\beta_{i}\mu}{\tau}\right)^{1/2}\cos\Theta_{0}.$
(41)

and the same expression in (40) for the the normalized fluctuations of the
magnetic field magnitude by $\delta B/B_{0}$, which is independent of the
species properties. Thus, electron magnetic pumping by compressible turbulent
fluctuations at the driving scale depends on the parameters $\beta_{i}$,
$\tau$, $\mu$, $k_{\parallel 0}\lambda_{mfp,e}$, $k_{\parallel 0}/k_{\perp
0}$, $\chi_{0}$, and $E_{comp}/E_{inc}$.

In summary, Alfvénic turbulent fluctuations can damping turbulence and
energize electrons through turbulent fluctuations at perpendicular scales
$1\lesssim k_{\perp}\rho_{i}\lesssim(\tau\mu)^{1/2}$ for plasma conditions
with $\beta_{i}\gtrsim 1$, but ions are not expected to be energized by
Alfvénic fluctuations. The resulting magnetic pumping of electrons by Alfvénic
turbulent fluctuations depends on the parameters $\beta_{i}$, $\tau$, $\mu$,
$k_{0}\rho_{i}$, and $k_{\parallel 0}\lambda_{mfp,e}$. If the turbulence is
driven compressibly with $E_{comp}/E_{inc}>0$, we expect that the turbulent
fluctuations at the driving scales, which have the largest amplitude of
$\delta B/B_{0}$, may lead to damping of the turbulence through the magnetic
pumping of both the ions and electrons. The resulting magnetic pumping of ions
by compressible turbulence at the driving scales depends on the parameters
$\beta_{i}$, $k_{\parallel 0}\lambda_{mfp,i}=(k_{\parallel
0}\lambda_{mfp,e})\tau^{2}$, $k_{\parallel 0}/k_{\perp 0}$, $\chi_{0}$, and
$E_{comp}/E_{inc}$. Analogously, the magnetic pumping of electrons by
compressible turbulence depends on the parameters $\beta_{i}$, $\tau$, $\mu$,
$k_{\parallel 0}\lambda_{mfp,e}$, $k_{\parallel 0}/k_{\perp 0}$, $\chi_{0}$,
and $E_{comp}/E_{inc}$. These parameter dependencies for magnetic pumping of
ions and electrons are summarized in Table 4.

###  5.8 Kinetic Viscous Heating

As discussed above for the case of magnetic pumping, in weakly collisional
plasmas, low-frequency turbulent fluctuations that generate variations in the
magnetic field magnitude will lead to a double-adiabatic (CGL) (Chew et al.,
1956) evolution of the plasma that preserves to lowest order the particle
magnetic moment $\mu_{m}$ and the action integral of the parallel bounce
motion $J_{\parallel}$ (Lichko et al., 2017; Montag, 2018; Arzamasskiy et al.,
2023). This dynamics leads to the development of anisotropies in the velocity
distribution of the plasma particles exhibiting temperature and pressure
anisotropies with $p_{\parallel}\neq p_{\perp}$. These deviations from local
thermodynamic equilibrium can trigger rapidly growing kinetic instabilities
that can disrupt the linear wave dynamics (Squire et al., 2016, 2017b, 2017a)
and mediate nonlocal transfer of turbulent energy directly from the large
driving scales to kinetic length scales (Kunz et al., 2018; Arzamasskiy et
al., 2023), as illustrated by the red arrow in Fig. 1(b). The effect of these
kinetic instabilities on the turbulent dynamics has recently been modeled as
an effective viscosity (Arzamasskiy et al., 2023).

To properly describe the kinetic physics of turbulence in temperature
anisotropic plasmas, it is necessary to employ the full set of plasma and
turbulence parameters in Table 2. The relevant kinetic instabilities driven by
anisotropies in the ion velocity distribution are the four ion temperature
anisotropy instabilities: (i) the parallel (or whistler) firehose instability
(Kennel, 1966; Gary et al., 1976) and (ii) the Alfvén (or oblique) firehose
instability (Hellinger & Matsumoto, 2000), which both are relevant to plasmas
with an ion temperature anisotropy sufficiently less than unity
$A_{i}=T_{\perp,i}/T_{\parallel,i}<1$ and with a parallel ion plasma beta
$\beta_{i}>1$; and (iii) the mirror instability (Vedenov & Sagdeev, 1958;
Tajiri, 1967; Southwood & Kivelson, 1993) and (iv) the ion cyclotron
instability (Gary et al., 1976), which both are relevant to plasmas with an
ion temperature anisotropy sufficiently greater than unity
$A_{i}=T_{\perp,i}/T_{\parallel,i}>1$ for any value of the parallel ion plasma
beta $\beta_{i}$. When the plasma exceeds a threshold value of the proton
temperature anisotropy (Boris:1977; Matteini:2006; Hellinger:2008;
Klein:2015), these instabilities can tap the free energy associated with the
anisotropic proton temperature, driving electromagnetic fluctuations and
ultimately reducing the temperature anisotropy, thereby moving the plasma back
toward a state of marginal stability.

Recent theoretical and numerical studies have highlighted the fact that, at
sufficiently high parallel ion plasma beta $\beta_{i}\gg 1$, the marginal
stability boundary for these kinetic ion temperature anisotropy instabilities
asymptotes towards isotropy, so the double-adiabatic evolution of turbulent
fluctuations can easily exceed these boundaries and trigger these rapidly
growing instabilities. Large-scale Alfvén waves in the MHD limit at
$k_{\perp}\rho_{i}\ll 1$ yield only perpendicular magnetic field perturbations
$\delta B_{\perp}\neq 0$ with $\delta B_{\parallel}=0$, so the variation in
the magnetic field magnitude $\delta B$ is negligible in the limit of small
amplitude waves $\delta B_{\perp}/B_{0}\ll 1$. However, at sufficiently large
amplitudes $\delta B_{\perp}/B_{0}\rightarrow 1$, these fluctuations can lead
to significant variation in the magnetic field magnitude, as made clear by
(39). Thus, at sufficiently high values of parallel ion plasma beta
$\beta_{i}\gg 1$, the physics of linearly polarized111111Note that circularly
polarized Alfvénic fluctuations, which have no magnetic field magnitude
variation, do not suffer this kinetic-instability-driven interruption (Squire
et al., 2016). Alfvén waves can be “interrupted” by these kinetic ion
temperature anisotropy instabilities (Squire et al., 2016, 2017b, 2017a).

Recent theoretical and numerical work has extended this concept to consider
the effect of kinetic instabilities on the turbulent cascade under conditions
of high parallel ion plasma beta $\beta_{i}\gg 1$, finding that these
instabilities can nonlocally transfer energy directly to kinetic length scales
(Arzamasskiy et al., 2023), as illustrated by the red arrow in Fig. 1(b). This
new channel of energy transfer can compete with the traditional local transfer
of energy to smaller scales of the turbulent cascade that is mediated by
nonlinear interactions, as illustrated by the black arrows in Fig. 1(a). The
effect of these kinetic instabilities on the large-scale turbulent dynamics
has been quantified as an effective viscosity by Arzamasskiy et al. (2023).
Here we denote this kinetic-instability-mediated viscosity as the _kinetic
viscosity_ 121212This _kinetic viscosity_ terminology for kinetic-instability-
mediated viscosity should not be confused with the _kinematic viscosity_
defined in hydrodynamic fluids. to distinguish it from the usual definition of
viscosity that arises through Coulomb collisions among the plasma particles.

These ion kinetic instabilities act to limit the ion pressure anisotropy in
the plasma to values corresponding to marginal stability, given by $\Delta
p_{i}/p_{i}\lesssim 1/\beta_{i}$, where $\Delta p_{i}\equiv p_{\perp
i}-p_{\parallel i}$ and $p_{i}\equiv(2p_{\perp i}+p_{\parallel i})/3$ (Kunz et
al., 2014; Melville et al., 2016; Arzamasskiy et al., 2023). Using this
marginal stability scaling with $\beta_{i}$, one can estimate an effective
collision frequency $\nu_{\mbox{eff}}\sim\beta_{i}\omega(\delta
B_{\perp}/B_{0})^{2}$ (Arzamasskiy et al., 2023), where $\omega$ is the linear
wave frequency at the scale where the dynamics generate that largest pressure
anisotropy, and thus have the maximum growth rate. Since the turbulent
fluctuation amplitudes generally decrease monotonically with scale, the
kinetic temperature anisotropy instabilities will be driven most strongly by
turbulent fluctuations at the driving scale. As with the case for magnetic
pumping by compressible fluctuations that is dominated by the driving scale
dyanmics, here we must use the three driving parameters $(k_{\perp
0}\rho_{i},k_{\parallel 0}/k_{\perp 0},\chi_{0})$ in Table 2, rather than
$k_{0}\rho_{i}$, to estimate the turbulent damping rate by the kinetic
viscosity.

Using the effective collision frequency $\nu_{\mbox{eff}}$ above, we can
estimate the ratio of the damping rate due to kinetic viscous heating to the
wave frequency at the driving scales, obtaining

$\frac{\gamma_{VH}}{\omega}\sim\beta_{i}\left(\chi_{0}\frac{k_{\parallel
0}}{k_{\perp 0}}\right)^{2},$ (42)

where the definition of the nonlinearity parameter $\chi_{0}=(k_{\perp
0}/k_{\parallel 0})(\delta B_{\perp 0}/B_{0})\theta_{0}$ with $\theta_{0}=1$
at the driving scale is used to replace the amplitude of the driving-scale
turbulent fluctuations $\delta B_{\perp 0}/B_{0}$.

In summary, the kinetic viscous heating due to ion temperature anisotropy
instabilities arising from the nonthermal velocity distributions driven by the
large-scale turbulent motions occurs in weakly collisional turbulent plasmas
with parallel ion plasma beta $\beta_{i}\gg 1$. The kinetic viscous damping
rate depends on the parallel ion plasma beta $\beta_{i}$, the nonlinearity
parameter $\chi_{0}$, and the anisotropy of the turbulent driving $k_{\perp
0}/k_{\parallel 0}$. Note that the scaling used to derive the form of the
effective collision frequency $\nu_{\mbox{eff}}$ depends on the scaling of the
marginal stability boundaries that are generally calculated in quiescient
plasmas in the absence of turbulence. The scaling of marginal stability
boundaries in the presence of strong plasma turbulence may differ, although
evidence from spacecraft observations (Hellinger et al., 2006; Bale et al.,
2009; Maruca et al., 2011) and from hybrid kinetic ion and fluid electron
simulations of temperature anisotropic plasma turbulence (Kunz et al., 2014;
Bott et al., 2021; Arzamasskiy et al., 2023) appear to suggest that any
modifications of these boundaries due to the presence of turbulence are
relatively small.

###  5.9 Collisionless Magnetic Reconnection

Numerical simulations of plasma turbulence demonstrate the ubiquitous
development of current sheets at small scales (Matthaeus & Montgomery, 1980;
Meneguzzi et al., 1981; Biskamp & Welter, 1989; Spangler, 1998, 1999; Biskamp
& Müller, 2000; Maron & Goldreich, 2001; Merrifield et al., 2005; Greco et
al., 2008), and it has been found that the dissipation of the turbulence is
largely concentrated in the vicinity of these current sheets (Uritsky et al.,
2010; Wan et al., 2012; Karimabadi et al., 2013; TenBarge & Howes, 2013; Wu et
al., 2013; Zhdankin et al., 2013), giving rise to the proposal that magnetic
reconnection may play a role in the damping of plasma turbulence. Motivated by
these findings, statistical analyses of observations of turbulence in the
solar wind have sought evidence for such spatially localized heating (Osman et
al., 2011; Borovsky & Denton, 2011; Osman et al., 2012b, a; Perri et al.,
2012; Wang et al., 2013; Wu et al., 2013; Osman et al., 2014b). Yet the
physical mechanisms that lead to the development of these current sheets in
plasma turbulence is yet to be fully understood.

Under the _B06_ theory for the scaling of MHD turbulence, the phenomenon of
dynamic alignment (Boldyrev, 2006) is predicted to lead to the development of
current sheets with a width $w\sim 1/k_{i}$ to thickness $a\sim 1/k_{\perp}$
ratio in the plane perpendicular to the magnetic field given by
$w/a=k_{\perp}/k_{i}=[(k_{\perp}\rho_{i})/(k_{0}\rho_{i})]^{1/4}$, according
to the scaling in (2) for $\alpha=1$. The geometry of the current sheets
expected to develop in plasma turbulence is illustrated in Fig. 8, where for
sufficiently small perpendicular scales relative to the driving scale,
$k_{\perp}\rho_{i}\gg k_{0}\rho_{i}$, the current sheets have a parallel
extent along the mean magnetic field $\mathbf{B}_{0}$, given by $l\sim
1/k_{\parallel}$, and a scale separation of thickness to width to length
obeying $a\ll w\ll l$; the corresponding three components of the wavenumber
have the complementary scaling $k_{\parallel}\ll k_{i}\ll k_{\perp}$. In Fig.
8, the wavelength of a mode that is unstable to the collisionless tearing
instability is illustrated by $\lambda\sim 1/k$ (red). Note that the minimum
value of this tearing unstable wavenumber $k$ is equal to the intermediate
wavenumber $k_{i}$ corresponding to the width of the current sheet, $w\sim
1/k_{i}$. The maximum value of $k$ assumed in calculations of the linear
collisionless tearing instability (Loureiro & Boldyrev, 2017a; Mallet et al.,
2017b) must be much smaller than the perpendicular wavenumber $k_{\perp}$
characterizing the current sheet thickness ($a\sim 1/k_{\perp}$), thereby
yielding $k/k_{\perp}=ka\ll 1$. Thus, the range of $k$ is given by

$k_{i}\rho_{i}=(k_{0}\rho_{i})^{1/4}(k_{\perp}\rho_{i})^{3/4}\lesssim
k\rho_{i}\ll k_{\perp}\rho_{i}\lesssim 1,$ (43)

where the final constraint $k_{\perp}\rho_{i}\lesssim 1$ is needed for the
tearing instability to arise for a current sheet thickness within the MHD
regime, otherwise different scalings for electron-only reconnection (Phan et
al., 2018) in the regime $k_{\perp}\rho_{i}\gtrsim 1$ would need to be
incorporated.

Figure 8: Diagram of the current sheet geometry generated self-consistently in
plasma turbulence in the MHD regime at $k_{\perp}\rho_{i}\lesssim 1$, with the
scalngs of $(k_{\perp},k_{i},k_{\parallel})$ given by the _B06_ scaling. The
current sheet has thickness $a\sim k_{\perp}$ and width $w\sim 1/k_{i}$ in the
perpendicular plane and length $l\sim 1/k_{\parallel}$ along the equilibirium
magnetic field $\mathbf{B}_{0}$. The wavelength of the mode unstable to the
collisionless tearing instability is given by $\lambda\sim 1/k$ (red), and
must fall within the bounds determined by the current sheet scaling, yielding
an ordering $k_{i}\lesssim k\ll k_{\perp}$.

Note that, in turbulence obeying the _B06_ scaling, the self-consistent
generation of current sheets arises only at sufficiently small perpendicular
scales, $k_{\perp}\rho_{i}\gg k_{0}\rho_{i}$. Therefore, the ratio of the
perpendicular magnetic field perturbation to the mean magnetic field will be
small, $\delta B_{\perp}/B_{0}=(k_{0}\rho_{i}/k_{\perp}\rho_{i})^{1/4}\ll 1$,
so the reconnection that arises in plasma turbulence necessarily falls in the
limit of strong-guide-field magnetic reconnection, as illustrated in Fig. 8
(green). An implication of this fact is that, if numerical simulations are
employed to assess the importance of magnetic reconnection in turbulence, it
is critical that those simulations be modeled in three spatial dimensions
(3D). Previous studies of 2D vs. 3D plasma turbulence have shown that 3D
simulations exhibit a significantly faster turbulent cascade rate (Li et al.,
2016), so enabling full 3D evolution is likely to be essential to determine
the competition between the turbulent cascade and magnetic reconnection.

The tearing instability growth rate typically increases with an increasing
perpendicular aspect ratio $w/a$ ratio of the current sheets. If this ratio
reaches a sufficiently large value, it was recently recognized that the growth
rate of the tearing instability could exceed the turbulent cascade rate at
that scale, $\gamma/\omega_{nl}\gtrsim 1$, triggering a reconnection flow that
would disrupt the turbulent dynamics, deviating from the scaling relations
summarized in §3 and potentially affecting the damping of turbulent cascade
(Boldyrev & Loureiro, 2017; Loureiro & Boldyrev, 2017b, a; Mallet et al.,
2017b, a; Walker et al., 2018).

It is worth noting here that an alternative explanation for the development of
current sheets in plasma turbulence due to the nonlinear interactions between
counterpropagating Alfvén waves—a phenomenon often denoted _Alfvén wave
collisions_ (Howes & Nielson, 2013)—was proposed by Howes (2016), but no
scaling for the resulting aspect ratio of the current sheets was provided. If
the scaling of this alternative mechanism differs from that of the _B06_
theory, than the scaling predictions for the onset of reconnection would also
change.

Here we adopt the _B06_ scaling for the development of current sheets to
determine the dependence of magnetic reconnection as a turbulent damping
mechanism on the fundamental plasma and turbulence parameters for the
isotropic temperature case in Table 3. This calculation follows previous
investigations that assess the conditions under which collisionless magnetic
reconnection may disrupt the turbulent cascade and thereby impact how the
turbulent energy is channeled into particle energy (Loureiro & Boldyrev,
2017a; Mallet et al., 2017b). Our goal here is to express the results from
these previous studies in the dimensionless parameters for the isotropic
temperature case proposed in Table 3.

The growth rate of the collisionless tearing instabilty, which is the
instability that initiates the process of magnetic reconnection under the
weakly collisional conditions typical of space and astrophysical plasmas, has
been derived in the limit of very low electron plasma beta
$\beta_{e}\sim\mu^{-1}$ (Zocco & Schekochihin, 2011), where we convert
$\beta_{e}=\beta_{i}/\tau$. The physics of the collisionless tearing
instabilty depends on three length scales: (i) the thickness of the current
sheet $a\sim 1/k_{\perp}$, assumed here to occur on MHD scales with
$k_{\perp}\rho_{i}\ll 1$; (ii) the ion sound Larmor radius
$\rho_{s}\equiv\rho_{i}/(2\tau)^{1/2}$ where two fluid effects begin to lead
to differences between the ion and electron dynamics; and (iii) the inner
boundary layer scale $\delta_{in}$ where the magnetic flux can be unfrozen
from the electron fluid flow. We will restrict our analysis to
$\beta_{e}\gg\mu^{-1}$, such that the flux unfreezing arises from electron
inertia (Zocco & Schekochihin, 2011; Loureiro & Boldyrev, 2017a; Mallet et
al., 2017b), occuring approximately at the electron inertial length scale
$d_{e}\equiv c/\omega_{pe}=\rho_{i}/(\beta_{i}\mu)^{1/2}$. These scales are
ordered by $\delta_{in}\sim d_{e}\ll\rho_{s}\ll a$. Together, these
restrictions limit us to moderately small plasma beta,

$\mu^{-1}\ll\beta_{i}/\tau\ll 1.$ (44)

The tearing instabilty depends on the tearing mode instability parameter
$\Delta^{\prime}$, determined by the outer-region solution (at MHD scales
$k_{\perp}\rho_{i}\ll 1$) (Zocco & Schekochihin, 2011). This parameter can be
expressed over two limiting regimes in a dimensionless form (Loureiro &
Boldyrev, 2017a) as

$\Delta^{\prime}a=(ka)^{-n}\quad\quad\mbox{for}\quad\quad 1<n\leq 2,$ (45)

where $k$ is the unstable wavenumber along the width of the current layer
governing the reconnection flow, as illustrated in Fig. 8 by the red
sinusoidal curve. Here $n=1$ corresponds to a Harris current sheet (a
hyperbolic tangent profile of the reconnecting magnetic field across the
current sheet thickness) (Harris, 1962), while $n=2$ corresponds to a
sinusoidal variation of the reconnection magnetic field (Loureiro & Boldyrev,
2017a), which may be the more likely case for the onset of reconnection at
current sheets that self-consistently arise in a turbulent plasma. The tearing
instability growth rates are solved in two limits of $\Delta^{\prime}$ (Zocco
& Schekochihin, 2011; Loureiro & Boldyrev, 2017a; Mallet et al., 2017b): (i)
$\Delta^{\prime}\delta_{in}\ll 1$,

$\gamma\sim
kv_{A\perp}\frac{d_{e}\rho_{s}\Delta^{\prime}}{a}\quad\quad\mbox{with}\quad\quad\delta_{in}\sim
d_{e}\rho_{s}^{1/2}\Delta^{\prime 1/2},$ (46)

and (ii) $\Delta^{\prime}\delta_{in}\gtrsim 1$,

$\gamma\sim
kv_{A\perp}\frac{d_{e}^{1/3}\rho_{s}^{2/3}}{a}\quad\quad\mbox{with}\quad\quad\delta_{in}\sim
d_{e}^{2/3}\rho_{s}^{1/3},$ (47)

where $v_{A\perp}\equiv\delta B_{\perp}/(4\pi n_{i}m_{i})^{1/2}$ is the Alfvén
speed based on the magnitude of _only_ the reconnecting component of the
magnetic field $\delta B_{\perp}$. Since the growth rates above are calculated
in the limit $k_{\perp}\rho_{i}\ll 1$, we will focus our assessment on whether
magnetic reconnection will disrupt the turbulent cascade at MHD scales
$k_{\perp}\rho_{i}\lesssim 1$, which is the same approach as taken in previous
studies (Loureiro & Boldyrev, 2017a; Mallet et al., 2017b).

Using (45) and either (46) or (47) for $\delta_{in}$, the normalized tearing
mode instability parameter $\Delta^{\prime}\delta_{in}$ can be expressed in
terms of the dimensionless quantities in Table 3 by

$\Delta^{\prime}\delta_{in}=\frac{k_{\perp}\rho_{i}}{(2\tau\beta_{i}^{2}\mu^{2})^{1/6}}\left(\frac{k_{\perp}\rho_{i}}{k\rho_{i}}\right)^{n}.$
(48)

This equation shows that $\Delta^{\prime}\delta_{in}$ increases as the
perpendicular wavenumber $k_{\perp}\rho_{i}$ increases, corresponding to a
thinner current sheet, or as the unstable wavenumber $k\rho_{i}$ decreases,
requiring to a wider current sheet. Since the width $w\sim 1/k_{i}$ of the
current sheets in _B06_ scaling theory governed by (2) bounds the unstable
wavenumber $k$ from below by the intermediate wavenumber, $k>k_{i}$, the
maximum value of instability parameter $\Delta^{\prime}\delta_{in}$ increases
with the perpendicular aspect ratio of the current sheets
$w/a=k_{\perp}/k_{i}\gg 1$. Note also that
$\Delta^{\prime}\delta_{in}\propto\beta_{i}^{-1/3}$, so that lower values of
ion plasma beta lead to a larger instability parameter.

Note that the value of the unstable wavenumber $k\rho_{i}$ at the transition
between the limiting regimes of the tearing mode instability parameter, where
$\Delta^{\prime}\delta_{in}=1$, is given by

$k\rho_{i}=k_{t}\rho_{i}\equiv\frac{(k_{\perp}\rho_{i})^{(n+1)/n}}{(2\tau\beta_{i}^{2}\mu^{2})^{1/6n}},$
(49)

which provides a bound on the value of $k\rho_{i}$ for the validity of the
tearing growth rate formulas in (46) and (47).

To assess whether the collisionless tearing instabilty grows fast enough to
enable the onset of magnetic reconnection to disrupt the turbulent cascade, we
will assess $\gamma_{RXN}/\omega_{nl}\sim\gamma_{RXN}/\omega$ assuming
critical balance of linear and nonlinear timescales $\omega_{nl}\sim\omega$
(Goldreich & Sridhar, 1995; Mallet et al., 2015). Taking the linear wave
frequency for MHD Alfvén waves
$\omega=k_{\parallel}v_{A}=(k_{\parallel}\rho_{i})^{1/2}(k_{\perp}\rho_{i})^{1/2}v_{A}/\rho_{i}$
using (1) with $\alpha=1$ for the _B06_ scaling of $k_{\parallel}$, and
substituting $v_{A\perp}=v_{A}(\delta
B_{\perp}/B_{0})=v_{A}[(k_{0}\rho_{i})/(k_{\perp}\rho_{i})]^{1/4}$ using (4),
we find that the growth rate in the $\Delta^{\prime}\delta_{in}\ll 1$ limit,
which corresponds to the large unstable wavenumber $k$ limit by inspection of
(48), can be expressed as

$\frac{\gamma_{RXN}}{\omega}\sim\frac{(k_{\perp}\rho_{i})^{(5+4n)/4}}{(k\rho_{i})^{n-1}(k_{0}\rho_{i})^{1/4}(2\tau\beta_{i}\mu)^{1/2}},$
(50)

where this expression is formally valid in the MHD limit with
$k_{\perp}\rho_{i}\ll 1$. Combining the limitations on the possible unstable
wavenumber from (43) and the requirement $\Delta^{\prime}\delta_{in}\lesssim
1$ using (49), the formula in (50) is valid over the range of normalized
unstable wavenumbers

$\max\left[(k_{0}\rho_{i})^{1/4}(k_{\perp}\rho_{i})^{3/4},\frac{(k_{\perp}\rho_{i})^{(n+1)/n}}{(2\tau\beta_{i}^{2}\mu^{2})^{1/6n}}\right]\lesssim
k\rho_{i}\ll k_{\perp}\rho_{i}\lesssim 1.$ (51)

Note that this valid unstable wavenumber range can be used to show that no
range with $\Delta^{\prime}\delta_{in}\lesssim 1$ exists if
$\beta_{i}\lesssim(2\tau\mu^{2})^{-1/2}$. For typical values of $\tau\sim 1$,
however, the very low values of $\beta_{i}$ required to violate this condition
would already have violated the assumed moderate beta limit given by (44), so
this limitation is of no practical concern.

The tearing growth rate in (50) demonstrates that, in the large $k$ limit, the
normalized growth rate is independent of $k$ with $\gamma_{RXN}/\omega\propto
k^{0}$ for Harris-type current sheets with $n=1$, and is inversely
proportional to $k$ with $\gamma_{RXN}/\omega\propto k^{-1}$ for the $n=2$
case of sinusoidal variations of the perpendicular magnetic field component
that may be expected to occur in plasma turbulence.

The maximum growth rate $\gamma_{RXN}/\omega$ from (50) increases with
$k_{\perp}\rho_{i}$, but also requires the limit $k_{\perp}\rho_{i}\ll 1$ for
the validity of this linear tearing growth rate calculation. Therefore, to
estimate quantitatively the maximum normalized tearing growth rate, we take
$k_{\perp}\rho_{i}\rightarrow\mathcal{F}$, where $\mathcal{F}\ll 1$ is the
maximum value of the normalized perpendicular wavenumber for which (50)
remains valid; numerical simulations of collisionless magnetic reconnection
will be invaluable to provide estimates of the maximum value of $\mathcal{F}$.
The maximum growth rate $\gamma_{RXN}/\omega$ also occurs for the minimum
unstable wavenumber $k\rho_{i}$ allowed by the range in (51). Thus, taking
these two limits and requiring a collisionless tearing instability growth
sufficient to disrupt the turbulent cascade, $\gamma_{RXN}/\omega\gtrsim 1$,
yields the constraint

$k_{0}\rho_{i}\lesssim\frac{\mathcal{F}^{(4+5n)/n}}{(2\tau)^{(4n+2)/3n}(\beta_{i}\mu)^{(2n+4)/3n}}$
(52)

if the minimum $k\rho_{i}$ is constrained by the condition
$\Delta^{\prime}\delta_{in}\lesssim 1$. Alternatively, if the minimum
$k\rho_{i}$ is constrained by the width of the current sheet given by (43),
then the condition $\gamma_{RXN}/\omega\gtrsim 1$ yields the constraint

$k_{0}\rho_{i}\lesssim\frac{\mathcal{F}^{(5+4n)/n}}{(2\tau\beta_{i}\mu)^{2/n}}.$
(53)

Figure 9: (a) Scaling the normalized collisionless tearing instability growth
rate $\gamma_{RXN}/\omega$ for a turblent plasma with $\beta_{i}=0.01$,
$\tau=1$, $\mu=1836$, and $k_{0}\rho_{i}=10^{-4}$ as a function of the
normalized unstable wavenumber $k\rho_{i}$. Colors indicate the solutions in
the low $k$ limit (corresponding to $\Delta^{\prime}\delta_{in}\gtrsim 1$)
(red) and in the high $k$ limit (corresponding to
$\Delta^{\prime}\delta_{in}\ll 1$) (blue). Line thickness indicates the either
a sinusoidal current profile with $n=2$ (thick) or a Harris-like hyperbolic
tangent current profile with $n=1$ (thin). Line styles indicate different
values of maximum perpendicular wavenumber $\mathcal{F}=1.0$ (solid), $0.3$
(dashed), and $0.1$ (dotted). (b) The corresponding normalized tearing mode
instability parameter $\Delta^{\prime}\delta_{in}$ vs. $k\rho_{i}$ for each
case.

The normalized growth rates in the $\Delta^{\prime}\delta_{in}\lesssim 1$
limit vs. normalized unstable wavenumber $k\rho_{i}$ are illustrated in Fig.
9(a) for $n=1$ (thin blue) and $n=2$ (thick blue) for turbulent parameters
$\beta_{i}=0.01$, $\tau=1$, $\mu=1836$, and $k_{0}\rho_{i}=10^{-4}$ and for
three values of the maximum of $k_{\perp}\rho_{i}$ given by $\mathcal{F}=1.0$
(solid), $0.3$ (dashed), and $0.1$ (dotted). The corresponding values of
$\Delta^{\prime}\delta_{in}$ vs. $k\rho_{i}$ are presented in Fig. 9(b) for
each of the $n$ and $k_{\perp}\rho_{i}$ cases. For the turbulence parameters
specified in this example, we find $\gamma_{RXN}/\omega\gtrsim 1$ only for the
$k_{\perp}\rho_{i}=1.0$ case over the unstable wavenumber range $0.3\lesssim
k\rho_{i}\lesssim 1$ for the $n=1$ case (thin solid blue) and over
$0.6\lesssim k\rho_{i}\lesssim 1$ for the $n=2$ case (thick solid blue).

Next we assess the initiation of collisionless tearing in the
$\Delta^{\prime}\delta_{in}\gtrsim 1$ limit with the growth rate given by
(47). Using similar substitutions for $\omega$ and $v_{A\perp}$ (as detailed
above for the $\Delta^{\prime}\delta_{in}\ll 1$ limit), we find that the
growth rate in the $\Delta^{\prime}\delta_{in}\gtrsim 1$ limit, which is the
small unstable wavenumber $k$ limit as shown by (48), can be expressed as

$\frac{\gamma_{RXN}}{\omega}\sim\frac{(k\rho_{i})(k_{\perp}\rho_{i})^{1/4}}{(k_{0}\rho_{i})^{1/4}(4\tau^{2}\beta_{i}\mu)^{1/6}},$
(54)

where this expression is formally valid in the MHD limit with
$k_{\perp}\rho_{i}\ll 1$. Combining the limitations on the possible unstable
wavenumber from (43) and the requirement $\Delta^{\prime}\delta_{in}\gtrsim 1$
using (49), the formula in (54) is valid over the range of normalized unstable
wavenumbers

$(k_{0}\rho_{i})^{1/4}(k_{\perp}\rho_{i})^{3/4}\lesssim
k\rho_{i}\ll\max\left[k_{\perp}\rho_{i},\frac{(k_{\perp}\rho_{i})^{(n+1)/n}}{(2\tau\beta_{i}^{2}\mu^{2})^{1/6n}}\right]\lesssim
1.$ (55)

This valid unstable wavenumber range can be used to show that no range with
$\Delta^{\prime}\delta_{in}\gtrsim 1$ exists if

$k_{\perp}\rho_{i}\lesssim(k_{0}\rho_{i})^{n/(n+4)}(2\tau\beta_{i}^{2}\mu^{2})^{2/3(n+4)}.$
(56)

This finding is consistent with the fact the that value of
$\Delta^{\prime}\delta_{in}$ decreases with decreasing $k_{\perp}\rho_{i}$ in
(48), so by decreasing $k_{\perp}\rho_{i}$ enough, one will always reach a
point where there is no range of $k\rho_{i}$ with
$\Delta^{\prime}\delta_{in}\gtrsim 1$.

The tearing growth rate in (54) shows that, in the small $k$ limit, the
normalized growth rate increases linearly with the unstable wavenumber $k$ and
increases weakly with the perpendicular wavenumber $k_{\perp}\rho_{i}$,
independent of whether the turbulently generated current sheets are Harris-
like or sinusoidal (thus, independent of $n$).

Once again the maximum growth rate $\gamma_{RXN}/\omega$ from (54) increases
with $k_{\perp}\rho_{i}$ (althoughly weakly with the $1/4$ power), so once
again we take the limit $k_{\perp}\rho_{i}\rightarrow\mathcal{F}$, where
$\mathcal{F}\ll 1$ is the maximum value of the normalized perpendicular
wavenumber for which (54) remains valid. The maximum growth rate will also
occur for the maximum unstable wavenumber $k\rho_{i}$ given by (55). Thus,
taking these two limits and requiring a collisionless tearing instability
growth rate sufficient to distrupt the turbulent cascade,
$\gamma_{RXN}/\omega\gtrsim 1$, yields the constraint

$k_{0}\rho_{i}\lesssim\frac{\mathcal{F}^{(4+5n)/n}}{(2\tau)^{(4n+2)/3n}(\beta_{i}\mu)^{(2n+4)/3n}}.$
(57)

if the maximum $k\rho_{i}$ is constrained by the condition
$\Delta^{\prime}\delta_{in}\gtrsim 1$. Note that this condition is the same as
(52), as it must be since this maximum growth rate occurs at
$\Delta^{\prime}\delta_{in}=1$, where the growth rates for the
$\Delta^{\prime}\delta_{in}\lesssim 1$ and $\Delta^{\prime}\delta_{in}\gtrsim
1$ limits cross. Alternatively, if the maximum $k\rho_{i}$ is constrained by
the maximum $k_{\perp}\rho_{i}\rightarrow\mathcal{F}$, then the condition
$\gamma_{RXN}/\omega\gtrsim 1$ yields the constraint

$k_{0}\rho_{i}\lesssim\frac{\mathcal{F}^{(5+4n)/n}}{(4\tau^{2}\beta_{i}\mu)^{2/3}}$
(58)

The normalized growth rates in the $\Delta^{\prime}\delta_{in}\gtrsim 1$ limit
vs. normalized unstable wavenumber $k\rho_{i}$ are illustrated in Fig. 9(a)
(thick red) for turbulent parameters $\beta_{i}=0.01$, $\tau=1$, $\mu=1836$,
and $k_{0}\rho_{i}=10^{-4}$ and for three values of the maximum of
$k_{\perp}\rho_{i}$ given by $\mathcal{F}=1.0$ (solid), $0.3$ (dashed), and
$0.1$ (dotted). The corresponding values of $\Delta^{\prime}\delta_{in}$ vs.
$k\rho_{i}$ are presented in Fig. 9(b) for each of the $n$ and
$k_{\perp}\rho_{i}$ cases. For the turbulence parameters specified in this
example, we find $\gamma_{RXN}/\omega\gtrsim 1$ only for the
$k_{\perp}\rho_{i}=1.0$ case over the unstable wavenumber range $0.2\lesssim
k\rho_{i}\lesssim 0.3$ for the $n=1$ case (thick solid red) and over
$0.2\lesssim k\rho_{i}\lesssim 0.6$ for the $n=2$ case (thick solid red).

It is necessary to highlight here a few important caveats in using the linear
collisionless tearing mode growth rates and MHD turbulence scaling theory to
predict when magnetic reconnection may disrupt the dynamics of the turbulent
cascade. First, the linear tearing mode growth rates used here are rigorously
derived in asymptotic limits, but we employ these same rates up to the
boundary of those limits. For example, the linear growth rates formally
require $k_{\perp}\rho_{i}\ll 1$; also, the $\Delta^{\prime}\delta_{in}\ll 1$
limit is used to estimate the tearing mode growth rates as
$\Delta^{\prime}\delta_{in}\rightarrow 1$. But such rates derived in
asymptotic limits may not remain quantitatively accurate as we extend beyond
their regime of formal validity. We take the approach that the growth rates
from the limits are at least a reasonable lowest-order estimate of the rate
beyond their formally limited range. Furhermore, the growth rates for the
collisionless tearing mode are determined under smooth conditions, whereas to
determine whether reconnection arises in the current sheets that naturally
arise in plasma turbulence one needs to determine the collisionless tearing
growth rates in the presence of turbulence; once again, the estimated tearing
mode growth rates used here we take simply as the lowest-order estimation.
Computing the tearing mode growth rates in a turbulent plasma, and determining
the competition of the tearing mode with the nonlinear cascade of energy due
to turbulence, are both necessary for more accurate calculations, which likely
will require detailed kinetic numerical simulations. Finally, alternative
proposed mechanisms for the development of current sheets in plasma
turbulence—such as the nonlinear interaction of counterpropagating Alfvén wave
collisions (Howes, 2016), as opposed to dynamic alignment (Boldyrev, 2006)—may
yield a different scaling for the width and thickness of current sheets; the
resulting predictions for the onset of magnetic reconnection in a turbulent
plasma would thereby also likely change.

In summary, we have followed previous investigations (Loureiro & Boldyrev,
2017a; Mallet et al., 2017b) to estimate how the collisionless tearing mode
may arise in the current sheets that develop self-consistently in plasma
turbulence, potentially disrupting the nonlinear cascade of turbulent energy
to small scales. We restrict ourselves to the moderately low electron plasma
beta limit $\mu^{-1}\ll\beta_{e}\ll 1$, which can be expressed in terms of the
turbulence parameters in Table 3 as the limit $1\ll\beta_{i}\mu/\tau\ll\mu$.
In this limit, magnetic flux conservation is broken due to electron inertia at
electron inertial length scales $d_{e}$ (Zocco & Schekochihin, 2011; Loureiro
& Boldyrev, 2017a; Mallet et al., 2017b). We consider only the case when
collisionless magnetic reconnection arises in current sheets with thicknesses
$a\sim 1/k_{\perp}$ in the MHD regime $k_{\perp}\rho_{i}\lesssim 1$. We find
that the growth rate of reconnection relative to the turbulent fluctuation
frequencies $\gamma_{RXN}/\omega$ is a function of the isotropic driving
wavenumber $k_{0}\rho_{i}$, ion plasma beta $\beta_{i}$, and ion-to-electron
temperature ratio $\tau$, as well as the ion-to-electron mass ratio $\mu$.

In general, the normalized collisionless tearing instabilty growth rates
$\gamma_{RXN}/\omega$—given by (50) in the $\Delta^{\prime}\delta_{in}\ll 1$
limit and by (54) in the $\Delta^{\prime}\delta_{in}\gtrsim 1$ limit—increase
as $k_{0}\rho_{i}$ and $\beta_{i}$ decrease. For specified values of the
plasma parameters $\beta_{i}$ and $\tau$, we obtain different conditions on
the maxmimum value of $k_{0}\rho_{i}$ needed for reconnection to arise, given
by (52) and (53) in the $\Delta^{\prime}\delta_{in}\ll 1$ limit and by (57)
and (58) in the $\Delta^{\prime}\delta_{in}\gtrsim 1$ limit. These scalings
mean that reconnection is more likely to arise in turbulence with a very large
driving scale relative to the ion Larmor radius scale $k_{0}\rho_{i}\ll 1$ and
for low ion plasma beta condtions $\beta_{i}\ll 1$.

It is worthwhile emphasizing that these estimates on the importance of
magnetic reconnection in plasma turbulence are calculated only if for current
sheets with a thickness $a\sim 1/k_{\perp}$ in the MHD regime
$k_{\perp}\rho_{i}\ll 1$ so that the MHD scaling predictions in §3 apply. It
is also possible that electron-only magnetic reconnection (Phan et al., 2018)
could occur with the range of scales in the kinetic dissipation range
$k_{\perp}\rho_{i}\ll 1$, but to estimate whether this will occur would
require the use of modified turbulence scalings appropriate for the
dissipation range (Howes et al., 2008a, 2011).

Finally, how the incorporation of magnetic reconnection into the turbulent
cascade impacts the damping of the turbulence and the resulting energization
of the plasma remains an open question. Collisionless magnetic reconnection
effectively converts magnetic energy into other forms, specifically the
kinetic energy of the bulk plasma outflows and possibly nonthermal energy in
the particle velocity distribution functions (Drake et al., 2005; Egedal et
al., 2008, 2012; Drake & Swisdak, 2012; Loureiro et al., 2013; Jiansen et al.,
2018; Pucci et al., 2018; Muñoz & Büchner, 2018; Dahlin, 2020; McCubbin et
al., 2022). But, the linear terms in the equations of evolution for the
turbulent plasma mediate the ongoing oscillation of energy from magnetic
energy to the kinetic energy of the turbulent bulk plasma motions and back
(Howes, 2015). For example, the physics of undamped Alfvén waves involve the
transformation of the bulk kinetic energy of the perpendicular wave motions to
magnetic energy as the frozen-in magnetic field is stretched; magnetic tension
serves as the restoring force for the wave, decelerating the perpendicular
wave motions and ultimately re-accelerating them back towards the state where
the magnetic field is not bent, thereby converting magnetic energy back to
kinetic energy. Thus, collisionless magnetic reconnection does not necessarily
lead directly to damping of the turbulent motions, but rather is simply a
channel, in additon to magnetic tension, for converting magnetic to kinetic
energy. The triggering of magnetic reconnection, however, will impact the
scaling of the three-dimensional wave vector of the turbulent motions that
result from the reconnection flow (Boldyrev & Loureiro, 2017; Loureiro &
Boldyrev, 2017b, a; Mallet et al., 2017b, a; Walker et al., 2018), and these
different scalings may channel energy into different kinetic physical
mechanisms that transfer turbulent energy into particle energy.

##  6 Phase Diagrams for Plasmas Turbulence

The scalings of the different proposed kinetic dissipation mechanisms listed
in §5.1 enable us to map the importance of the different mechanisms for the
damping of turbulence as a function of the fundamental plasma parameters by
creating _phase diagrams for plasma turbulence_ , analogous to a similar
effort to create a phase diagram for magnetic reconnection (Ji & Daughton,
2011). Here we present two such phase diagrams based on the plasma and
turbulence parameters estimated for different space and astrophysical plasmas.

###  6.1 Collisionality of the Dissipation Range of Plasma Turbulence

Figure 10: Phase Diagram of the extent of the turbulent inertial range from
driving scales at $k_{\parallel 0}\lambda_{mfp,e}$ to the transition to the
dissipation range at $k_{\parallel T}\lambda_{mfp,e}$ vs. the ion plasma beta
$\beta_{i}$. The plot indicates that, at the small scale end of the inertial
range where damping mechanisms can remove energy from the turbulence, for many
plasma systems of interest, the governing dynamics are weakly collisional with
t $k_{\parallel T}\lambda_{mfp,e}\gg 1$, so kinetic plasma theory is necessary
to capture the physics of turbulent dissipation.

We first create a phase diagram indicating that, for many plasmas of interest,
even if the turbulence is driven at fluid scales with
$k_{\parallel,0}\lambda_{mfp,e}\ll 1$, the end of the inertial range generally
occurs at scales that are weakly collisional. Therefore, the kinetic damping
mechanisms analyzed here are responsible for removing energy from the
turbulent cascade, rather than viscosity and resistivity, which are based on
microscopic Coulomb collisions between particles (Braginskii, 1965; Chapman &
Cowling, 1970). Since the mean free path must be compared to the parallel
length scales of the turbulent fluctuations characterized by $k_{\parallel}$,
we must first determine the parallel length scales associated with the small-
scale end of the inertial range at the transition to the dissipation range,
corresponding to a perpendicular scale $k_{\perp}\rho_{i}\sim 1$. Specifying
the _GS95_ scaling with $\alpha=0$ for this example, the parallel wavenumber
at the transiton to the dissipation range $k_{\parallel T}$, determined by
(1), is given by $k_{\parallel
T}\rho_{i}=(k_{0}\rho_{i})^{1/3}(k_{\perp}\rho_{i})^{2/3}=(k_{0}\rho_{i})^{1/3}$,
here taking $k_{\perp}\rho_{i}\sim 1$ as the end of the inertial range. For
simplicity, we assume here that the turbulence is driven strongly and
isotropically with $\chi_{0}=1$ and $k_{\parallel 0}/k_{\perp 0}=1$.
Normalizing this parallel scale in terms of the electron collisional mean free
path $\lambda_{mfp,e}$, we obtain $k_{\parallel
T}\lambda_{mfp,e}=(k_{0}\rho_{i})^{1/3}(\lambda_{mfp,e}/\rho_{i})$.

The range of plasma and turbulence parameters for a number of plasma systems
of interest are denoted by the colored boxes in Fig. 10. Plotted on the
horizontal axis of Fig. 10 is the range of parallel scales bounding the
turbulent inertial range normalized to the electron collisional mean free path
$\lambda_{mfp,e}$: (i) the normalized parallel driving scale $k_{\parallel
0}\lambda_{mfp,e}$ of the turbulence is represented by the left end of each
box; (ii) the turbulent energy cascades to smaller scales to the right within
each box (wavy red arrow below plot); and (iii) the parallel scale
corresponding to the small-scale end of the inertial range at $k_{\parallel
T}\lambda_{mfp,e}$ is represented by the right end of each box. The vertical
extent of the colored boxes indicates the range of ion plasma beta $\beta_{i}$
for each of the chosen systems. The fluid regime (yellow region) indicates
conditions under which the dynamics of the turbulent fluctuations occur on
parallel scales that are collisional with $k_{\parallel 0}\lambda_{mfp,e}\ll
1$; the kinetic regime (light blue region) indicates conditions under which
the dynamics of the turbulent fluctuations occur on parallel scales that are
weakly collisional with $k_{\parallel 0}\lambda_{mfp,e}\gg 1$.

In Fig. 10 are shown these parameter ranges for the near Earth solar wind
(red), the Earth’s magnetosheath and the solar corona (green), the inner
helisophere at heliocentric radii $R\ll 1$ AU and the outer helisophere at
$R\gg 1$ AU (blue), the Galactic Center and intracluster medium (black), and
terrestrial laboratory experiments including the Large Plasma Device (LAPD)at
UCLA (Gekelman et al., 1991, 2016) and the Joint European Torus (JET) Fusion
experiment (JET Team, 1992). The main message of Fig. 10 is that, for most
turbulent plasma systems of interest, the parallel length scales associated
with the end of the inertial range correspond to weakly collisional plasma
conditions with $k_{\parallel T}\lambda_{mfp,e}\gg 1$. Therefore, to explore
the physical damping mechanisms responsible for removing energy from the
turbulent fluctuations and consequently energizing the plasma particles,
plasma kinetic theory is essential.

###  6.2 Turbulent Damping Mechanisms on the $(\beta_{i},k_{0}\rho_{i})$ Plane

Figure 11: Phase diagram for the kinetic damping mechanisms of weakly
collisional plasmas turbulence as a function of isotropic driving wavenumber
$k_{0}\rho_{i}$ and ion plasma beta $\beta_{i}$, showing regions of this
parameter space where different mechanisms are likely to contribute to the
heating of the plasma species: ion Landau damping (iLD) and ion transit time
damping (iTTD) (blue); electron Landau damping (eLD) and electron transit time
damping (eTTD) (blue); ion cycltron damping (iCD) (black); ion stochastic
heating (iSH) (green); ion kinetic viscous heating (iVH) (magenta); and
collisionless magnetic reconnection (RXN) (red) for intermittent current
sheets ($n=1$, red dashed) and sinusoidal current sheets ($n=2$, red solid).
The extent of the turbulent near-Earth solar wind on the
$(\beta_{i},k_{0}\rho_{i})$ plane is indicated (gray shading).

A single phase diagram that demonstrates the power of using the dimensionless
plasma and turbulence parameters for the isotropic temperature case presented
in Table 3 is a plot of the predicted regions on the
$(\beta_{i},k_{0}\rho_{i})$ plane where different kinetic mechanisms are
expeected to play a role the damping of weakly collisional plasma turbulence,
presented here in Fig. 11.

For this phase diagram, we specify the ion-to-electron temperature ratio
$\tau=1$ and the ion-to-electron mass ratio $\mu=1836$. For the isotropic
temperature case, the species temperature anisotropies $A_{i}=A_{e}=1$, and we
take the turbulence to be driven strongly ($\chi_{0}=0$) and isotropically
($k_{\parallel 0}/k_{\perp 0}=1$) so that the turbulent driving can be
characterized by the isotropic driving wavenumber $k_{0}\rho_{i}$.
Furthermore, we assume the turbulence driving to be balanced
($Z_{0}^{+}/Z_{0}^{-}=1$) and incompressible ($E_{comp}/E_{inc}=0$). In
addition, the turbulence is assumed to be weakly collisional with
$k_{\parallel 0}\lambda_{mfp,e}\ll 1$, eliminating magnetic pumping as a
potential turbulent damping mechanism. Finally, we specifically choose to use
the _B06_ scaling for MHD turbulence with $\alpha=1$ to assess the
contributions of the different proposed damping mechanisms.

We use the dependencies discussed in §5 of the proposed turbulent damping
mechanisms on the dimensionless parameters in Table 3 to assess the regions of
the two-dimensional $(\beta_{i},k_{0}\rho_{i})$ parameter space over which
each mechanism is expected to contribute.

For the Landau-resonant ($n=0$) collisionless damping mechanisms—Landau
damping and transit-time damping—the primary parameter dependencies summarized
in §5.4 are $\beta_{i}$ for ion Landau damping (iLD) and ion transit-time
damping (iTTD) and $(\beta_{i},\tau,\mu)$ for electron Landau damping (eLD)
and electron transit-time damping (eTTD). Since the sum of LD and TTD for each
species is always leads to damping for isotropic equilibrium temperatures, but
seperately the net energy transfer by either mechanism to that species over a
full wave period can be positive or negative—as discussed in §5.4 and
illustrated in Fig. 7— we do not separate the contributions LD and TTD for a
single species. Furthermore, since any turbulent cascade energy not deposited
onto the ions by iLD and iTTD at the ion scales $k_{\perp}\rho_{i}\sim 1$ will
cascade to $k_{\perp}\rho_{i}\gg 1$ and ultimately be transferred to the
electrons, the partitioning of energy between ions and electrons by these
Landau-resonant mechanisms is simply a function of $\beta_{i}$. Solutions of
the linear Vlasov-Maxwell dispersion relation for the normalized ion damping
rate by iLD and iTTD, $(\gamma_{iLD}+\gamma_{iLD})/\omega$ indicate that ion
damping is non-neglible for $\beta_{i}\gtrsim 0.5$, so iLD and iTTD will
contribute in this region (blue line with right arrow), independent of the
drivign scale $k_{0}\rho_{i}$. For $\beta_{i}\lesssim 0.5$, eLD and eTTD will
dominate the Landau-resonant damping (blue line with left arrow), but eLD and
eTTD will still contribute to terminate the remaining turbulent cascade energy
not removed by ions (blue right arrow).

As discussed in §5.5, ion cyclotron damping (iCD) will contribute to the
damping of turbulence if the turbulent frequencies approach the ion cyclotron
frequency $\omega/\Omega_{i}\rightarrow 1$, with the scaling of this ratio
given by (28). We can solve this equation for $k_{0}\rho_{i}$ in terms of
$\beta_{i}$ with $\alpha=1$ for the _B06_ scaling, yielding

$k_{0}\rho_{i}=\left(\frac{\omega}{\Omega_{i}}\frac{1}{\overline{\omega}}\right)^{2}\beta_{i}.$
(59)

Taking the factor $(\omega/\Omega_{i})/\overline{\omega}=1$ gives an estimate
for when $\omega/\Omega_{i}=1$ (black dotted line), but iCD often becomes
significant at lower values of the frequency, $\omega/\Omega_{i}\lesssim 1$.
Thus, we estimate the onset of iCD by taking
$(\omega/\Omega_{i})/\overline{\omega}=1/2\pi$ (black solid line and upward
arrow).

The impact of ion stochastic heating (iSH) on turbulent fluctuations for the
_B06_ scaling is given by (32) in §5.6. The exponential dependence in this
equation for $\gamma_{SH}/\omega$ yields a strong inhibition of ion stochastic
heating unless $c_{2}\beta_{i}^{1/2}(k_{0}\rho_{i})^{-1/4}\lesssim 1$. Thus,
we obtain a condition for $k_{0}\rho_{i}$ in terms of $\beta_{i}$ for the
onset of iSH, given by

$k_{0}\rho_{i}\gtrsim c_{2}^{4}\beta_{i}^{2}.$ (60)

Note that the dependence of this condition on $c_{2}^{4}$ means that an
accurate determination of this constant is critical to assess the role of iSH
on the damping of plasma turbulence. Taking the value of $c_{2}=0.34$ from
Chandran et al. (2010), we plot (60) for the onset of ion stochastic heating
(green line and upward arrow) on Fig. 11.

Kinetic viscous heating of ions (iVH) mediated by temperature anisotropy
instabilities a high values of ion plasma beta $\beta_{i}\gg 1$ depends on the
$\beta_{i}$ and the amplitude of the turbulent fluctuations at the driving
scale, as shown by (42) in §5.8. Because this mechanism only contributes when
$\beta_{i}$ is sufficiently high that the temperature anisotropy exceeds the
threshold values for the proton temperature anisotropy instabilities, we take
this kinetic damping mechanism as dependent only on the value of the ion
plasma beta, with an estimated onset at $\beta_{i}\gtrsim 10$ (magenta line
and right arrow).

Finally, if the driving scale $k_{0}\rho_{i}\ll 1$ and the ion plasma beta
$\beta_{i}\ll 1$ of the turbulence are sufficiently low, current sheets with a
large width-to-thickness ratio in the plane perpendicular to the equilibrium
magnetic field can be unstable to a rapid collisionless tearing instability in
the large-guide-field limit, as detailed in §5.9. If growth rate of the
tearing instability is faster than the rate of the nonlinear cascade of energy
to small scales by the turbulence $\gamma_{RXN}/\omega{nl}\gtrsim 1$,
collisionless magnetic reconnection can disrupt the turbulent cascade. The
condition that the normalized collisionless tearing growth rates satisfy
$\gamma_{RXN}/\omega\gtrsim 1$ occurs for the maximum growth rate of the
tearing instability is given by either (52) or (57) (both expressions yield
that same maximum value at the transitional value of
$\Delta^{\prime}\delta_{in}=1$). In these expressions, $n=1$ corresponds to a
Harris-like current sheet with a hyperbolic tangent profile, and $n=2$
corresponds to a sinusoidal variation of the reconnection magnetic field. The
determination of the tearing growth rate demands $k_{\perp}\rho_{i}\ll 1$ for
the validity of this linear tearing growth rate calculation, so we take
$k_{\perp}\rho_{i}\rightarrow\mathcal{F}$, where $\mathcal{F}\ll 1$ is the
maximum value of the normalized perpendicular wavenumber for which the growth
rate derviations remain valid. Here, we specify a value $\mathcal{F}=0.3$ for
our estimation of the onset of collisionless magnetic reconnection, plotting
the boundaries for reconnection in intermittent current sheets ($n=1$) (dashed
red line and downward arrow) and for reconnection in sinusoidal current sheets
($n=2$) (solid red line and downward arrow). Note that the dependence of
$k_{0}\rho_{i}$ on $\mathcal{F}$ in both (52) and (57) is very strong, scaling
as $\mathcal{F}^{9}$ for $n=1$ and as $\mathcal{F}^{7}$ for $n=2$, so the
determination of the maximal value of $k_{\perp}\rho_{i}\equiv\mathcal{F}$ for
the tearing instability growth has a strong influence on whether reconnection
will arise.

The phase diagram for the turbulent damping mechanisms on the
$(\beta_{i},k_{0}\rho_{i})$ plane in Fig. 11 can be used in the following
manner to predict which damping mechanisms will contribute to the dissipation
of turbulence in a specific space or astrophysical plasma system. In the near-
Earth solar wind, the observed ion plasma beta values span the range
$0.05\lesssim\beta_{i}\lesssim 10$ (Wilson et al., 2018), taking a 5% and 95%
limit on the distribution of $\beta_{i}$ values. Furthermore, the isotropic
driving wavenumber in the near-Earth solar wind spans the typical values
$10^{-4}\lesssim k_{0}\rho_{i}\lesssim 10^{-3}$ (Tu & Marsch, 1990, 1995;
Matthaeus et al., 2005; Howes et al., 2008a; Kiyani et al., 2015). This range
of parameters for the solar wind at 1 AU is plotted as the gray shaded region
on Fig. 11. Reading the phase diagram, the prediction for the damping
mechanisms in the near-Earth solar wind is that Landau-resonant damping for
ions (iLD and iTTD) and electrons (eLD and eTTD) will contribute to the
damping over the full range of solar wind turbulence parameters. At lower
values of $\beta_{i}\lesssim 0.3$ and higher values of $k_{0}\rho_{i}$,
stochastic heating is predicted to play a role. Indeed, in the Earth’s
magnetosheath (which has a similar footprint on the
$(\beta_{i},k_{0}\rho_{i})$ plane as the near-Earth solar wind), there is
direct evidence from _Magnetospheric MultiScale_ spacecraft observations of
electron Landau damping playing a significant role in the damping of the
turbulent fluctuations (Chen et al., 2019; Afshari et al., 2021). In addition,
in the solar wind in the inner heliosphere, where the values of $\beta_{i}$
tend to be lower than in the near-Earth solar wind, analysis of observations
from the _Helios_ spacecraft (Martinović et al., 2019) and _Parker Solar
Probe_ spacecraft (Martinović et al., 2020) find evidence for ion stochastic
heating in the turbulent plasma. Thus, phase diagrams like Fig. 11 can play a
valuable role in predicting the mechanisms that will contribute to the damping
of weakly collisional plasma turbulence in space and astrophysical plasmas.

##  7 Conclusions

Turbulence is a ubiquitous phenomenon that arises in space and astrophysical
plasmas and mediates the transfer of the energy of large-scale plasma flows
and electromagnetic fields to sufficiently small scales that dissipation
mechanisms can convert that energy into the microscopic energy of the plasma
particles. A grand challenge problem in heliophysics and astrophysics is to
predict the heating or acceleration of the different plasma species by the
turbulence in terms of the dimensionless parameters that characterize the
plasma and the nature of the turbulence. In particular, a long-term goal is to
develop predictive models of the turbulent plasma heating that determine the
partitioning of energy between the ion and electron species, $Q_{i}/Q_{e}$,
and between the perpendicular and parallel degrees of freedom for each
species, $Q_{\perp,i}/Q_{\parallel,i}$ and $Q_{\perp,e}/Q_{\parallel,e}$. An
essential step in the development of such predictive turbulent heating models
is to identify the microphysical kinetic processes that govern the transfer of
the turbulent energy to the particles as a function of the dimensionless
parameters, information that can be summarized in a phase diagram for the
dissipation of plasma turbulence, as shown in Fig. 11.

The first step in this long term effort is to specify a set of the key
dimensionless parameters upon which the turbulent energy cascade and its
dissipation depend. In this paper, we propose a specific set of ten
dimensionless parameters that characterize the state of the plasma and the
nature of the turbulence,

$(\beta_{i},\tau,A_{i},A_{e},k_{\parallel 0}\lambda_{mfp,e};k_{\perp
0}\rho_{i},k_{\parallel 0}/k_{\perp
0},\chi_{0},Z_{0}^{+}/Z_{0}^{-},E_{comp}/E_{inc}),$

summarized in Table 2. Although this set of ten parameters is sufficient to
characterize a wide variety of turbulent space and astrophysical plasmas, the
development of completely general turbulent heating models on a ten-
dimensional parameter space is unlikely to be successful. Fortunately, we can
capture many of the dominant dependencies of the turbulence and its damping
mechanisms by adopting of reduced set of parameters in the limits that (i) the
equilibrium velocity distribution for each species is an isotropic Maxwellian
velocity and (ii) the turbulence is driven at sufficiently large scale or
sufficiently strongly that the turbulent fluctuations at the small-scale end
of the inertial range $k_{\perp}\rho_{i}\sim 1$ satisfy a state of strong
turbulence, $\chi\sim 1$. With these idealizations to the _isotropic
temperature case_ , there remain just three dimensionless parameters

$(\beta_{i},\tau;k_{0}\rho_{i}),$

suumarized in Table 3.

A number of kinetic mechanisms have been proposed to govern the damping of
weakly collisional plasma turbulence, as enumerated in §5.1. The critical
advance presented in this paper is to express the dependence of each of these
mechanisms on the _same set of fundamental plasma and turbulence parameters_.
The results of this analysis of the dependencies for each of the turbulent
damping mechanisms is summarized in Table 4.

Finally, the scalings of each of the turbulent damping mechanisms with the
isotropic driving wavenumber $k_{0}\rho_{i}$ and the ion plasma beta
$\beta_{i}$ can be synthesized to generate the first phase diagram for
turbulent damping mechanisms as a function of $(\beta_{i},k_{0}\rho_{i})$,
presented in Fig. 11. For a chosen space or astrophysical plasma system, such
as the near-Earth solar wind, the range of dimensionless plasma and turbulence
parameters can be identified on plotted on the phase diagram, as illustrated
by the gray shaded region in the figure. One may then read off which mechnisms
are predicted to contribute to the damping of the turbulence in that system.
Kinetic numerical simulations and spacecraft observations of weakly
collisional turbulence in space or astrophysical plasmas can be used to update
and refine this phase diagram, or to extend its applicability as additional
dimensionless parameters are varied, such as the ion-to-electron temperature
ratio $\tau=T_{i}/T_{e}$.

Ulimately, the goal is to produce more accurate and complete predictive
turbulent heating models than those reviewed in §4, facilitating improved
global modeling of important turbulent space and astrophysical plasmas, such
as the mesoscale and macroscale energy transport through the heliosphere or
the interpretation of Event Horizon Telescope observations of accretion disks
around supermassive black holes.

## Acknowledgements

More than fifteen years of turbulent modeling efforts are synthesized here,
having benefitted from interactions over many years with Steve Cowley, Bill
Dorland, Eliot Quataert, Alex Schekochihin, Greg Hammett, Tomo Tatsuno,
Ryusuke Numata, Ben Chandran, Stuart Bale, Kevin Nielson, Jason TenBarge, Jan
Drake, Fred Skiff, Craig Kletzing, Steve Vincena, Troy Carter, Chadi Salem,
Kristopher Klein, Chris Chen, Jim Schroeder, Daniel Told, Frank Jenko, Foaud
Sahraoui, Tak Chu Li, Suranga Ruhunusiri, Jasper Halekas, Sofiane Bourouaine,
Jaye Verniero, Matt Kunz, Andrew McCubbin, Francesco Valentini, Robert Wicks,
Daniel Verscharen, Lloyd Woodham, Sarah Conley, Arya Afshari, Peter Montag,
Daniel McGinnis, Rui Huang, and Alberto Felix.

## Funding

Significant funding supporting this long-term was provided by NSF grants
PHY-10033446, CAREER AGS-1054061 and AGS-1842561, DOE grant DE-SC0014599, and
NASA grants NNX10AC91G, 80NSSC18K0643, 80NSSC18K1217, and 80NSSC18K1371.

## References

  * Adhikari et al. (2020) Adhikari, L., Zank, G. P., Zhao, L. L., Kasper, J. C., Korreck, K. E., Stevens, M., Case, A. W., Whittlesey, P., Larson, D., Livi, R. & Klein, K. G. 2020 Turbulence Transport Modeling and First Orbit Parker Solar Probe (PSP) Observations. Astrophys. J. Supp. 246 (2), 38.
  * Afshari et al. (2021) Afshari, A. S., Howes, G. G., Kletzing, C. A., Hartley, D. P. & Boardsen, S. A. 2021 The Importance of Electron Landau Damping for the Dissipation of Turbulent Energy in Terrestrial Magnetosheath Plasma. J. Geophys. Res. 126 (12), e29578.
  * Afshari et al. (2023) Afshari, A. S., Howes, G. G., Shuster, J. R., Klein, K. G., McGinnis, D., Martinovic, M. M., Boardsen, S. A. Brown, C. R., Huang, R., Kletzing, C. A. & Hartley, D. P. 2023 Direct observation of ion cyclotron damping of turbulence in Earth’s magnetosheath plasma. Nature Comm. Submitted.
  * Alexandrova et al. (2008) Alexandrova, O., Carbone, V., Veltri, P. & Sorriso-Valvo, L. 2008 Small-Scale Energy Cascade of the Solar Wind Turbulence. Astrophys. J. 674, 1153–1157.
  * Alexandrova et al. (2009) Alexandrova, O., Saur, J., Lacombe, C., Mangeney, A., Mitchell, J., Schwartz, S. J. & Robert, P. 2009 Universality of Solar-Wind Turbulent Spectrum from MHD to Electron Scales. Phys. Rev. Lett. 103 (16), 165003.
  * Ambrosiano et al. (1988) Ambrosiano, J., Matthaeus, W. H., Goldstein, M. L. & Plante, D. 1988 Test particle acceleration in turbulent reconnecting magnetic fields. J. Geophys. Res. 93, 14383–14400.
  * Antonsen & Lane (1980) Antonsen, Jr., T. M. & Lane, B. 1980 Kinetic equations for low frequency instabilities in inhomogeneous plasmas. Phys. Fluids 23, 1205–1214.
  * Arzamasskiy et al. (2019) Arzamasskiy, Lev, Kunz, Matthew W., Chand ran, Benjamin D. G. & Quataert, Eliot 2019 Hybrid-kinetic Simulations of Ion Heating in Alfvénic Turbulence. Astrophys. J. 879 (1), 53.
  * Arzamasskiy et al. (2023) Arzamasskiy, Lev, Kunz, Matthew W., Squire, Jonathan, Quataert, Eliot & Schekochihin, Alexander A. 2023 Kinetic Turbulence in Collisionless High-$\beta$ Plasmas. Phys. Rev. X 13 (2), 021014.
  * Balbus & Hawley (1991) Balbus, S. A. & Hawley, J. F. 1991 A powerful local shear instability in weakly magnetized disks 1. linear analysis. Astrophys. J. 376, 214–222.
  * Balbus & Hawley (1998) Balbus, S. A. & Hawley, J. F. 1998 Instability, turbulence, and enhanced transport in accretion disks. Reviews of Modern Physics 70, 1–53.
  * Bale et al. (2009) Bale, S. D., Kasper, J. C., Howes, G. G., Quataert, E., Salem, C. & Sundkvist, D. 2009 Magnetic Fluctuation Power Near Proton Temperature Anisotropy Instability Thresholds in the Solar Wind. Phys. Rev. Lett. 103, 211101.
  * Barbhuiya & Cassak (2022) Barbhuiya, M. Hasan & Cassak, Paul A. 2022 Pressure-strain interaction. III. Particle-in-cell simulations of magnetic reconnection. Phys. Plasmas 29 (12), 122308.
  * Barenblatt (1996) Barenblatt, G. I. 1996 Scaling, Self-similarity, and Intermediate Asymptotics. Cambridge University Press.
  * Barnes (1966) Barnes, A. 1966 Collisionless Damping of Hydromagnetic Waves. Phys. Fluids 9, 1483–1495.
  * Belcher & Davis (1971) Belcher, J. W. & Davis, L. 1971 Large-Amplitude Alfvén Waves in the Interplanetary Medium, 2. J. Geophys. Res. 76, 3534–3563.
  * Berger et al. (1958) Berger, J. M., Newcomb, W. A., Dawson, J. M., Frieman, E. A., Kulsrud, R. M. & Lenard, A. 1958 Heating of a Confined Plasma by Oscillating Electromagnetic Fields. Phys. Fluids 1 (4), 301–307.
  * Biskamp & Müller (2000) Biskamp, D. & Müller, W.-C. 2000 Scaling properties of three-dimensional isotropic magnetohydrodynamic turbulence. Phys. Plasmas 7, 4889–4900.
  * Biskamp & Welter (1989) Biskamp, D. & Welter, H. 1989 Dynamics of decaying two-dimensional magnetohydrodynamic turbulence. Phys. Fluids B 1, 1964–1979.
  * Biskmap et al. (1999) Biskmap, D., Schwarz, E., Zeiler, A., Celani, A. & Drake, J. F. 1999 Electron magnetohydrodynamic turbulence. Phys. Plasmas 6, 751–758.
  * Boldyrev (2006) Boldyrev, S. 2006 Spectrum of Magnetohydrodynamic Turbulence. Phys. Rev. Lett. 96 (11), 115002.
  * Boldyrev & Loureiro (2017) Boldyrev, S. & Loureiro, N. F. 2017 Magnetohydrodynamic Turbulence Mediated by Reconnection. Astrophys. J. 844, 125.
  * Boldyrev et al. (2009) Boldyrev, S., Mason, J. & Cattaneo, F. 2009 Dynamic Alignment and Exact Scaling Laws in Magnetohydrodynamic Turbulence. Astrophys. J. Lett. 699, L39–L42.
  * Boldyrev & Perez (2009) Boldyrev, S. & Perez, J. C. 2009 Spectrum of Weak Magnetohydrodynamic Turbulence. Phys. Rev. Lett. 103 (22), 225001\.
  * Borovsky (2008) Borovsky, J. E. 2008 Flux tube texture of the solar wind: Strands of the magnetic carpet at 1 AU? J. Geophys. Res. 113, A08110.
  * Borovsky & Denton (2011) Borovsky, J. E. & Denton, M. H. 2011 No Evidence for Heating of the Solar Wind at Strong Current Sheets. Astrophys. J. Lett. 739, L61.
  * Bott et al. (2021) Bott, A. F. A., Arzamasskiy, L., Kunz, M. W., Quataert, E. & Squire, J. 2021 Adaptive Critical Balance and Firehose Instability in an Expanding, Turbulent, Collisionless Plasma. Astrophys. J. Lett. 922 (2), L35.
  * Bourouaine & Chandran (2013) Bourouaine, S. & Chandran, B. D. G. 2013 Observational Test of Stochastic Heating in Low-$\beta$ Fast-solar-wind Streams. Astrophys. J. 774, 96.
  * Bourouaine et al. (2008) Bourouaine, S., Marsch, E. & Vocks, C. 2008 On the Efficiency of Nonresonant Ion Heating by Coronal Alfvén Waves. Astrophys. J. Lett. 684, L119–L122.
  * Braginskii (1965) Braginskii, S. I. 1965 Transport Processes in a Plasma. Rev. Plasma Phys. 1, 205–+.
  * Brizard (1992) Brizard, A. 1992 Nonlinear gyrofluid description of turbulent magnetized plasmas. Phys. Fluids B 4, 1213–1228.
  * Brizard & Hahm (2007) Brizard, A. J. & Hahm, T. S. 2007 Foundations of nonlinear gyrokinetic theory. Rev. Mod. Phys. 79, 421–468.
  * Brown et al. (2023) Brown, C.R., Juno, J., Howes, G.G., Haggerty, C.C. & Constantinou, S. 2023 Isolation and phase-space energization analysis of the instabilities in collisionless shocks. J. Plasma Phys. 89 (3), 905890308.
  * Bruno & Carbone (2013) Bruno, R. & Carbone, V. 2013 The Solar Wind as a Turbulence Laboratory. Living Reviews in Solar Physics 10, 2.
  * Buckingham (1914) Buckingham, E. 1914 On Physically Similar Systems; Illustrations of the Use of Dimensional Equations. Phys. Rev. 4, 345–376.
  * Cassak & Barbhuiya (2022) Cassak, Paul A. & Barbhuiya, M. Hasan 2022 Pressure-strain interaction. I. On compression, deformation, and implications for Pi-D. Phys. Plasmas 29 (12), 122306.
  * Cassak et al. (2023) Cassak, Paul A., Barbhuiya, M. Hasan, Liang, Haoming & Argall, Matthew R. 2023 Quantifying Energy Conversion in Higher-Order Phase Space Density Moments in Plasmas. Phys. Rev. Lett. 130 (8), 085201.
  * Cassak et al. (2022) Cassak, Paul A., Barbhuiya, M. Hasan & Weldon, H. Arthur 2022 Pressure-strain interaction. II. Decomposition in magnetic field-aligned coordinates. Phys. Plasmas 29 (12), 122307.
  * Catto (1978) Catto, P. J. 1978 Linearized gyro-kinetics. Plasma Phys. 20, 719–722.
  * Catto et al. (1981) Catto, P. J., Tang, W. M. & Baldwin, D. E. 1981 Generalized gyrokinetics. Plasma Phys. 23, 639–650.
  * Cerri et al. (2021) Cerri, S. S., Arzamasskiy, L. & Kunz, M. W. 2021 On Stochastic Heating and Its Phase-space Signatures in Low-beta Kinetic Turbulence. Astrophys. J. 916 (2), 120.
  * Chael et al. (2019) Chael, Andrew, Narayan, Ramesh & Johnson, Michael D. 2019 Two-temperature, Magnetically Arrested Disc simulations of the jet from the supermassive black hole in M87. Mon. Not. Roy. Astron. Soc. 486 (2), 2873–2895.
  * Chandran (2010) Chandran, B. D. G. 2010 Alfvén-wave Turbulence and Perpendicular Ion Temperatures in Coronal Holes. Astrophys. J. 720, 548–554.
  * Chandran et al. (2011) Chandran, B. D. G., Dennis, T. J., Quataert, E. & Bale, S. D. 2011 Incorporating Kinetic Physics into a Two-fluid Solar-wind Model with Temperature Anisotropy and Low-frequency Alfvén-wave Turbulence. Astrophys. J. 743, 197.
  * Chandran et al. (2010) Chandran, B. D. G., Li, B., Rogers, B. N., Quataert, E. & Germaschewski, K. 2010 Perpendicular Ion Heating by Low-frequency Alfvén-wave Turbulence in the Solar Wind. Astrophys. J. 720, 503–515.
  * Chandran et al. (2013) Chandran, B. D. G., Verscharen, D., Quataert, E., Kasper, J. C., Isenberg, P. A. & Bourouaine, S. 2013 Stochastic Heating, Differential Flow, and the Alpha-to-proton Temperature Ratio in the Solar Wind. Astrophys. J. 776 (1), 45.
  * Chapman & Cowling (1970) Chapman, S. & Cowling, T. G. 1970 Cambridge: University Press.
  * Chen et al. (2010) Chen, C. H. K., Horbury, T. S., Schekochihin, A. A., Wicks, R. T., Alexandrova, O. & Mitchell, J. 2010 Anisotropy of Solar Wind Turbulence between Ion and Electron Scales. Phys. Rev. Lett. 104 (25), 255002.
  * Chen et al. (2013) Chen, C. H. K., Howes, G. G., Bonnell, J. W., Mozer, F. S., Klein, K. G. & Bale, S. D. 2013 Kinetic scale density fluctuations in the solar wind. In American Institute of Physics Conference Series (ed. G. P. Zank, J. Borovsky, R. Bruno, J. Cirtain, S. Cranmer, H. Elliott, J. Giacalone, W. Gonzalez, G. Li, E. Marsch, E. Moebius, N. Pogorelov, J. Spann & O. Verkhoglyadova), American Institute of Physics Conference Series, vol. 1539, pp. 143–146.
  * Chen et al. (2019) Chen, C. H. K., Klein, K. G. & Howes, G. G. 2019 Evidence for electron Landau damping in space plasma turbulence. Nature Comm. 10 (1), 740.
  * Chen et al. (2001) Chen, L., Lin, Z. & White, R. 2001 On resonant heating below the cyclotron frequency. Phys. Plasmas 8, 4713–4716.
  * Chew et al. (1956) Chew, G. L., Goldberger, M. L. & Low, F. E. 1956 The Boltzmann Equation and the One-Fluid Hydromagnetic Equations in the Absence of Particle Collisions. ”Proc. R. Soc. London A” 236, 112–118.
  * Cho & Lazarian (2003) Cho, J. & Lazarian, A. 2003 Compressible magnetohydrodynamic turbulence: mode coupling, scaling relations, anisotropy, viscosity-damped regime and astrophysical implications. Mon. Not. Roy. Astron. Soc. 345, 325–339.
  * Cho & Lazarian (2004) Cho, J. & Lazarian, A. 2004 The Anisotropy of Electron Magnetohydrodynamic Turbulence. Astrophys. J. Lett. 615, L41–L44.
  * Cho et al. (2002) Cho, J., Lazarian, A. & Vishniac, E. T. 2002 Simulations of Magnetohydrodynamic Turbulence in a Strongly Magnetized Medium. Astrophys. J. 564, 291–301.
  * Cho & Vishniac (2000) Cho, J. & Vishniac, E. T. 2000 The Anisotropy of Magnetohydrodynamic Alfvénic Turbulence. Astrophys. J. 539, 273–282.
  * Coburn et al. (2015) Coburn, J. T., Forman, M. A., Smith, C. W., Vasquez, B. J. & Stawarz, J. E. 2015 Third-moment descriptions of the interplanetary turbulent cascade, intermittency and back transfer. Phil. Trans. Roy. Soc. A 373 (2041), 20140150–20140150.
  * Coburn et al. (2014) Coburn, Jesse T., Smith, Charles W., Vasquez, Bernard J., Forman, Miriam A. & Stawarz, Julia E. 2014 Variable Cascade Dynamics and Intermittency in the Solar Wind at 1 AU. Astrophys. J. 786 (1), 52.
  * Coleman (1968) Coleman, Jr., P. J. 1968 Turbulence, Viscosity, and Dissipation in the Solar-Wind Plasma. Astrophys. J. 153, 371–388.
  * Dahlin (2020) Dahlin, J. T. 2020 Prospectus on electron acceleration via magnetic reconnection. Phys. Plasmas 27 (10).
  * Dalena et al. (2014) Dalena, S., Rappazzo, A. F., Dmitruk, P., Greco, A. & Matthaeus, W. H. 2014 Test-particle Acceleration in a Hierarchical Three-dimensional Turbulence Model. Astrophys. J. 783, 143.
  * Denskat et al. (1983) Denskat, K. U., Beinroth, H. J. & Neubauer, F. M. 1983 Interplanetary magnetic field power spectra with frequencies from 2.4 X 10 to the -5th HZ to 470 HZ from HELIOS-observations during solar minimum conditions. J. Geophys. Zeit. Geophys. 54, 60–67.
  * Di Mare & Howes (2024) Di Mare, F. & Howes, G. G. 2024 New regime of alfvén wave turbulence in auroral ionosphere. Phys. Rev. Lett. Submitted.
  * Dmitruk et al. (2004) Dmitruk, P., Matthaeus, W. H. & Seenu, N. 2004 Test Particle Energization by Current Sheets and Nonuniform Fields in Magnetohydrodynamic Turbulence. Astrophys. J. 617, 667–679.
  * Dobrowolny et al. (1980) Dobrowolny, M., Mangeney, A. & Veltri, P. 1980 Fully Developed Anisotropic Hydromagnetic Turbulence in Interplanetary Space. Phys. Rev. Lett. 45, 144–147.
  * Dorland & Hammett (1993) Dorland, W. & Hammett, G. W. 1993 Gyrofluid turbulence models with kinetic effects. Phys. Fluids B 5, 812–835.
  * Drake et al. (2013) Drake, D. J., Schroeder, J. W. R., Howes, G. G., Kletzing, C. A., Skiff, F., Carter, T. A. & Auerbach, D. W. 2013 Alfvén wave collisions, the fundamental building block of plasma turbulence. IV. Laboratory experiment. Phys. Plasmas 20 (7), 072901.
  * Drake et al. (2005) Drake, J. F., Shay, M. A., Thongthai, W. & Swisdak, M. 2005 Production of Energetic Electrons during Magnetic Reconnection. Phys. Rev. Lett. 94 (9), 095001.
  * Drake & Swisdak (2012) Drake, J. F. & Swisdak, M. 2012 Ion Heating and Acceleration During Magnetic Reconnection Relevant to the Corona. Space Sci. Rev. 172, 227–240.
  * Dubin et al. (1983) Dubin, D. H. E., Krommes, J. A., Oberman, C. & Lee, W. W. 1983 Nonlinear gyrokinetic equations. Phys. Fluids 26, 3524–3535.
  * Egedal et al. (2012) Egedal, J., Daughton, W. & Le, A. 2012 Large-scale electron acceleration by parallel electric fields during magnetic reconnection. Nature Physics 8, 321–324.
  * Egedal et al. (2008) Egedal, J., Fox, W., Katz, N., Porkolab, M., ØIeroset, M., Lin, R. P., Daughton, W. & Drake, J. F. 2008 Evidence and theory for trapped electrons in guide field magnetotail reconnection. J. Geophys. Res. 113, 12207.
  * Elsasser (1950) Elsasser, W. M. 1950 The Hydromagnetic Equations. Physical Review 79, 183.
  * Event Horizon Telescope Collaboration et al. (2022) Event Horizon Telescope Collaboration, Akiyama, Kazunori, Alberdi, Antxon, Alef, Walter, Algaba, Juan Carlos, Anantua, Richard, Asada, Keiichi, Azulay, Rebecca, Bach, Uwe, Baczko, Anne-Kathrin, Ball, David, Baloković, Mislav, Barrett, John, Bauböck, Michi, Benson, Bradford A., Bintley, Dan, Blackburn, Lindy, Blundell, Raymond, Bouman, Katherine L., Bower, Geoffrey C., Boyce, Hope, Bremer, Michael, Brinkerink, Christiaan D., Brissenden, Roger, Britzen, Silke, Broderick, Avery E., Broguiere, Dominique, Bronzwaer, Thomas, Bustamante, Sandra, Byun, Do-Young, Carlstrom, John E., Ceccobello, Chiara, Chael, Andrew, Chan, Chi-kwan, Chatterjee, Koushik, Chatterjee, Shami, Chen, Ming-Tang, Chen, Yongjun, Cheng, Xiaopeng, Cho, Ilje, Christian, Pierre, Conroy, Nicholas S., Conway, John E., Cordes, James M., Crawford, Thomas M., Crew, Geoffrey B., Cruz-Osorio, Alejandro, Cui, Yuzhu, Davelaar, Jordy, De Laurentis, Mariafelicia, Deane, Roger, Dempsey, Jessica, Desvignes, Gregory, Dexter, Jason, Dhruv, Vedant, Doeleman, Sheperd S., Dougal, Sean, Dzib, Sergio A., Eatough, Ralph P., Emami, Razieh, Falcke, Heino, Farah, Joseph, Fish, Vincent L., Fomalont, Ed, Ford, H. Alyson, Fraga-Encinas, Raquel, Freeman, William T., Friberg, Per, Fromm, Christian M., Fuentes, Antonio, Galison, Peter, Gammie, Charles F., García, Roberto, Gentaz, Olivier, Georgiev, Boris, Goddi, Ciriaco, Gold, Roman, Gómez-Ruiz, Arturo I., Gómez, José L., Gu, Minfeng, Gurwell, Mark, Hada, Kazuhiro, Haggard, Daryl, Haworth, Kari, Hecht, Michael H., Hesper, Ronald, Heumann, Dirk, Ho, Luis C., Ho, Paul, Honma, Mareki, Huang, Chih-Wei L., Huang, Lei, Hughes, David H., Ikeda, Shiro, Violette Impellizzeri, C. M., Inoue, Makoto, Issaoun, Sara, James, David J., Jannuzi, Buell T., Janssen, Michael, Jeter, Britton, Jiang, Wu, Jiménez-Rosales, Alejandra, Johnson, Michael D., Jorstad, Svetlana, Joshi, Abhishek V., Jung, Taehyun, Karami, Mansour, Karuppusamy, Ramesh, Kawashima, Tomohisa, Keating, Garrett K., Kettenis, Mark, Kim, Dong-Jin, Kim, Jae-Young, Kim, Jongsoo, Kim, Junhan, Kino, Motoki, Koay, Jun Yi, Kocherlakota, Prashant, Kofuji, Yutaro, Koch, Patrick M., Koyama, Shoko, Kramer, Carsten, Kramer, Michael, Krichbaum, Thomas P., Kuo, Cheng-Yu, Bella, Noemi La, Lauer, Tod R., Lee, Daeyoung, Lee, Sang-Sung, Leung, Po Kin, Levis, Aviad, Li, Zhiyuan, Lico, Rocco, Lindahl, Greg, Lindqvist, Michael, Lisakov, Mikhail, Liu, Jun, Liu, Kuo, Liuzzo, Elisabetta, Lo, Wen-Ping, Lobanov, Andrei P., Loinard, Laurent, Lonsdale, Colin J., Lu, Ru-Sen, Mao, Jirong, Marchili, Nicola, Markoff, Sera, Marrone, Daniel P., Marscher, Alan P., Martí-Vidal, Iván, Matsushita, Satoki, Matthews, Lynn D., Medeiros, Lia, Menten, Karl M., Michalik, Daniel, Mizuno, Izumi, Mizuno, Yosuke, Moran, James M., Moriyama, Kotaro, Moscibrodzka, Monika, Müller, Cornelia, Mus, Alejandro, Musoke, Gibwa, Myserlis, Ioannis, Nadolski, Andrew, Nagai, Hiroshi, Nagar, Neil M., Nakamura, Masanori, Narayan, Ramesh, Narayanan, Gopal, Natarajan, Iniyan, Nathanail, Antonios, Navarro Fuentes, Santiago, Neilsen, Joey, Neri, Roberto, Ni, Chunchong, Noutsos, Aristeidis, Nowak, Michael A., Oh, Junghwan, Okino, Hiroki, Olivares, Héctor, Ortiz-León, Gisela N., Oyama, Tomoaki, Özel, Feryal, Palumbo, Daniel C. M., Filippos Paraschos, Georgios, Park, Jongho, Parsons, Harriet, Patel, Nimesh, Pen, Ue-Li, Pesce, Dominic W., Piétu, Vincent, Plambeck, Richard, PopStefanija, Aleksandar, Porth, Oliver, Pötzl, Felix M., Prather, Ben, Preciado-López, Jorge A., Psaltis, Dimitrios, Pu, Hung-Yi, Ramakrishnan, Venkatessh, Rao, Ramprasad, Rawlings, Mark G., Raymond, Alexander W., Rezzolla, Luciano, Ricarte, Angelo, Ripperda, Bart, Roelofs, Freek, Rogers, Alan, Ros, Eduardo, Romero-Cañizales, Cristina, Roshanineshat, Arash, Rottmann, Helge, Roy, Alan L., Ruiz, Ignacio, Ruszczyk, Chet, Rygl, Kazi L. J., Sánchez, Salvador, Sánchez-Argüelles, David, Sánchez-Portal, Miguel, Sasada, Mahito, Satapathy, Kaushik, Savolainen, Tuomas, Schloerb, F. Peter, Schonfeld, Jonathan, Schuster, Karl-Friedrich, Shao, Lijing, Shen, Zhiqiang, Small, Des, Sohn, Bong Won, SooHoo, Jason, Souccar, Kamal, Sun, He, Tazaki, Fumie, Tetarenko, Alexandra J., Tiede, Paul, Tilanus, Remo P. J., Titus, Michael, Torne, Pablo, Traianou, Efthalia, Trent, Tyler, Trippe, Sascha, Turk, Matthew, van Bemmel, Ilse, van Langevelde, Huib Jan, van Rossum, Daniel R., Vos, Jesse, Wagner, Jan, Ward-Thompson, Derek, Wardle, John, Weintroub, Jonathan, Wex, Norbert, Wharton, Robert, Wielgus, Maciek, Wiik, Kaj, Witzel, Gunther, Wondrak, Michael F., Wong, George N., Wu, Qingwen, Yamaguchi, Paul, Yoon, Doosoo, Young, André, Young, Ken, Younsi, Ziri, Yuan, Feng, Yuan, Ye-Fei, Zensus, J. Anton, Zhang, Shuo, Zhao, Guang-Yao, Zhao, Shan-Shan, Chan, Tin Lok, Qiu, Richard, Ressler, Sean & White, Chris 2022 First Sagittarius A* Event Horizon Telescope Results. V. Testing Astrophysical Models of the Galactic Center Black Hole. Astrophys. J. Lett. 930 (2), L16.
  * Event Horizon Telescope Collaboration et al. (2019) Event Horizon Telescope Collaboration, Akiyama, Kazunori, Alberdi, Antxon, Alef, Walter, Asada, Keiichi, Azulay, Rebecca, Baczko, Anne-Kathrin, Ball, David, Baloković, Mislav, Barrett, John, Bintley, Dan, Blackburn, Lindy, Boland, Wilfred, Bouman, Katherine L., Bower, Geoffrey C., Bremer, Michael, Brinkerink, Christiaan D., Brissenden, Roger, Britzen, Silke, Broderick, Avery E., Broguiere, Dominique, Bronzwaer, Thomas, Byun, Do-Young, Carlstrom, John E., Chael, Andrew, Chan, Chi-kwan, Chatterjee, Shami, Chatterjee, Koushik, Chen, Ming-Tang, Chen, Yongjun, Cho, Ilje, Christian, Pierre, Conway, John E., Cordes, James M., Crew, Geoffrey B., Cui, Yuzhu, Davelaar, Jordy, De Laurentis, Mariafelicia, Deane, Roger, Dempsey, Jessica, Desvignes, Gregory, Dexter, Jason, Doeleman, Sheperd S., Eatough, Ralph P., Falcke, Heino, Fish, Vincent L., Fomalont, Ed, Fraga-Encinas, Raquel, Friberg, Per, Fromm, Christian M., Gómez, José L., Galison, Peter, Gammie, Charles F., García, Roberto, Gentaz, Olivier, Georgiev, Boris, Goddi, Ciriaco, Gold, Roman, Gu, Minfeng, Gurwell, Mark, Hada, Kazuhiro, Hecht, Michael H., Hesper, Ronald, Ho, Luis C., Ho, Paul, Honma, Mareki, Huang, Chih-Wei L., Huang, Lei, Hughes, David H., Ikeda, Shiro, Inoue, Makoto, Issaoun, Sara, James, David J., Jannuzi, Buell T., Janssen, Michael, Jeter, Britton, Jiang, Wu, Johnson, Michael D., Jorstad, Svetlana, Jung, Taehyun, Karami, Mansour, Karuppusamy, Ramesh, Kawashima, Tomohisa, Keating, Garrett K., Kettenis, Mark, Kim, Jae-Young, Kim, Junhan, Kim, Jongsoo, Kino, Motoki, Koay, Jun Yi, Koch, Patrick M., Koyama, Shoko, Kramer, Michael, Kramer, Carsten, Krichbaum, Thomas P., Kuo, Cheng-Yu, Lauer, Tod R., Lee, Sang-Sung, Li, Yan-Rong, Li, Zhiyuan, Lindqvist, Michael, Liu, Kuo, Liuzzo, Elisabetta, Lo, Wen-Ping, Lobanov, Andrei P., Loinard, Laurent, Lonsdale, Colin, Lu, Ru-Sen, MacDonald, Nicholas R., Mao, Jirong, Markoff, Sera, Marrone, Daniel P., Marscher, Alan P., Martí-Vidal, Iván, Matsushita, Satoki, Matthews, Lynn D., Medeiros, Lia, Menten, Karl M., Mizuno, Yosuke, Mizuno, Izumi, Moran, James M., Moriyama, Kotaro, Moscibrodzka, Monika, Mul$\ddot{}$ler, Cornelia, Nagai, Hiroshi, Nagar, Neil M., Nakamura, Masanori, Narayan, Ramesh, Narayanan, Gopal, Natarajan, Iniyan, Neri, Roberto, Ni, Chunchong, Noutsos, Aristeidis, Okino, Hiroki, Olivares, Héctor, Oyama, Tomoaki, Özel, Feryal, Palumbo, Daniel C. M., Patel, Nimesh, Pen, Ue-Li, Pesce, Dominic W., Piétu, Vincent, Plambeck, Richard, PopStefanija, Aleksandar, Porth, Oliver, Prather, Ben, Preciado-López, Jorge A., Psaltis, Dimitrios, Pu, Hung-Yi, Ramakrishnan, Venkatessh, Rao, Ramprasad, Rawlings, Mark G., Raymond, Alexander W., Rezzolla, Luciano, Ripperda, Bart, Roelofs, Freek, Rogers, Alan, Ros, Eduardo, Rose, Mel, Roshanineshat, Arash, Rottmann, Helge, Roy, Alan L., Ruszczyk, Chet, Ryan, Benjamin R., Rygl, Kazi L. J., Sánchez, Salvador, Sánchez-Arguelles, David, Sasada, Mahito, Savolainen, Tuomas, Schloerb, F. Peter, Schuster, Karl-Friedrich, Shao, Lijing, Shen, Zhiqiang, Small, Des, Sohn, Bong Won, SooHoo, Jason, Tazaki, Fumie, Tiede, Paul, Tilanus, Remo P. J., Titus, Michael, Toma, Kenji, Torne, Pablo, Trent, Tyler, Trippe, Sascha, Tsuda, Shuichiro, van Bemmel, Ilse, van Langevelde, Huib Jan, van Rossum, Daniel R., Wagner, Jan, Wardle, John, Weintroub, Jonathan, Wex, Norbert, Wharton, Robert, Wielgus, Maciek, Wong, George N., Wu, Qingwen, Young, André, Young, Ken, Younsi, Ziri, Yuan, Feng, Yuan, Ye-Fei, Zensus, J. Anton, Zhao, Guangyao, Zhao, Shan-Shan, Zhu, Ziyan, Anczarski, Jadyn, Baganoff, Frederick K., Eckart, Andreas, Farah, Joseph R., Haggard, Daryl, Meyer-Zhao, Zheng, Michalik, Daniel, Nadolski, Andrew, Neilsen, Joseph, Nishioka, Hiroaki, Nowak, Michael A., Pradel, Nicolas, Primiani, Rurik A., Souccar, Kamal, Vertatschitsch, Laura, Yamaguchi, Paul & Zhang, Shuo 2019 First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring. Astrophys. J. Lett. 875 (1), L5.
  * Frieman & Chen (1982) Frieman, E. A. & Chen, L. 1982 Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria. Phys. Fluids 25, 502–508.
  * Furth et al. (1963) Furth, Harold P., Killeen, John & Rosenbluth, Marshall N. 1963 Finite-Resistivity Instabilities of a Sheet Pinch. Phys. Fluids 6 (4), 459–484.
  * Galtier et al. (2000) Galtier, S., Nazarenko, S. V., Newell, A. C. & Pouquet, A. 2000 A weak turbulence theory for incompressible magnetohydrodynamics. J. Plasma Phys. 63, 447–488.
  * Gary (1999) Gary, S. P. 1999 Collisionless dissipation wavenumber: Linear theory. J. Geophys. Res. 104, 6759–6762.
  * Gary et al. (1976) Gary, S. P., Montgomery, M. D., Feldman, W. C. & Forslund, D. W. 1976 Proton temperature anisotropy instabilities in the solar wind. J. Geophys. Res. 81, 1241–1246.
  * Gekelman et al. (1991) Gekelman, W., Pfister, H., Lucky, Z., Bamber, J., Leneman, D. & Maggs, J. 1991 Design, construction, and properties of the large plasma research device - The LAPD at UCLA. Rev. Sci. Instrum. 62, 2875–2883.
  * Gekelman et al. (2016) Gekelman, W., Pribyl, P., Lucky, Z., Drandell, M., Leneman, D., Maggs, J., Vincena, S., Van Compernolle, B., Tripathi, S. K. P., Morales, G., Carter, T. A., Wang, Y. & DeHaas, T. 2016 The upgraded Large Plasma Device, a machine for studying frontier basic plasma physics. Rev. Sci. Instrum. 87 (2), 025105.
  * Glatzmaier (1984) Glatzmaier, G. A. 1984 Numerical simulations of stellar convective dynamos. I. The model andmethod. J. Comp. Phys. 55, 461–484.
  * Glatzmaier (1985) Glatzmaier, G. A. 1985 Numerical simulations of stellar convective dynamos. II - Field propagation in the convection zone. Astrophys. J. 291, 300–307.
  * Goldreich & Sridhar (1995) Goldreich, P. & Sridhar, S. 1995 Toward a Theory of Interstellar Turbulence II. Strong Alfvénic Turbulence. Astrophys. J. 438, 763–775.
  * Goldstein et al. (1994) Goldstein, M. L., Roberts, D. A. & Fitch, C. A. 1994 Properties of the fluctuating magnetic helicity in the inertial and dissipation ranges of solar wind turbulence. J. Geophys. Res. 99, 11519–11538.
  * Grad (1963) Grad, H. 1963 Asymptotic Theory of the Boltzmann Equation. Phys. Fluids 6, 147–181.
  * Greco et al. (2008) Greco, A., Chuychai, P., Matthaeus, W. H., Servidio, S. & Dmitruk, P. 2008 Intermittent MHD structures and classical discontinuities. Geophys. Res. Lett. 35, 19111.
  * Grošelj et al. (2019) Grošelj, Daniel, Chen, Christopher H. K., Mallet, Alfred, Samtaney, Ravi, Schneider, Kai & Jenko, Frank 2019 Kinetic Turbulence in Astrophysical Plasmas: Waves and/or Structures? Phys. Rev. X 9 (3), 031037.
  * Gruzinov (1998) Gruzinov, A. V. 1998 Radiative Efficiency of Collisionless Accretion. Astrophys. J. 501, 787–+.
  * Gurnett & Bhattacharjee (2017) Gurnett, Donald A. & Bhattacharjee, Amitava 2017 Introduction to Plasma Physics: With Space, Laboratory and Astrophysical Applications.
  * Hahm et al. (1988) Hahm, T. S., Lee, W. W. & Brizard, A. 1988 Nonlinear gyrokinetic theory for finite-beta plasmas. Phys. Fluids 31, 1940–1948.
  * Hammett et al. (1992) Hammett, G. W., Dorland, W. & Perkins, F. W. 1992 Fluid models of phase mixing, Landau damping, and nonlinear gyrokinetic dynamics. Phys. Fluids B 4, 2052–2061.
  * Harris (1962) Harris, E. G. 1962 On a plasma sheath separating regions of oppositely directed magnetic field. Il Nuovo Cimento 23 (1), 115–121.
  * Hawley & Balbus (1991) Hawley, J. F. & Balbus, S. A. 1991 A Powerful Local Shear Instability in Weakly Magnetized Disks. II. Nonlinear Evolution. Astrophys. J. 376, 223.
  * Hellinger & Matsumoto (2000) Hellinger, P. & Matsumoto, H. 2000 New kinetic instability: Oblique Alfvén fire hose. J. Geophys. Res. 105, 10519–10526.
  * Hellinger et al. (2006) Hellinger, P., Trávníček, P., Kasper, J. C. & Lazarus, A. J. 2006 Solar wind proton temperature anisotropy: Linear theory and WIND/SWE observations. Geophys. Res. Lett. 33, 9101\.
  * Hnat et al. (2003) Hnat, B., Chapman, S. C. & Rowlands, G. 2003 Intermittency, scaling, and the Fokker-Planck approach to fluctuations of the solar wind bulk plasma parameters as seen by the WIND spacecraft. Phys. Rev. E 67 (5), 056404.
  * Hollweg & Isenberg (2002) Hollweg, J. V. & Isenberg, P. A. 2002 Generation of the fast solar wind: A review with emphasis on the resonant cyclotron interaction. J. Geophys. Res. 107, 1147.
  * Howes (2008) Howes, G. G. 2008 Inertial range turbulence in kinetic plasmas. Phys. Plasmas 15 (5), 055904.
  * Howes (2010) Howes, G. G. 2010 A prescription for the turbulent heating of astrophysical plasmas. Mon. Not. Roy. Astron. Soc. 409, L104–L108.
  * Howes (2015) Howes, G. G. 2015 A dynamical model of plasma turbulence in the solar wind. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 373 (2041), 20140145\.
  * Howes (2015) Howes, G. G. 2015 The inherently three-dimensional nature of magnetized plasma turbulence. J. Plasma Phys. 81, 3203.
  * Howes (2016) Howes, G. G. 2016 The Dynamical Generation of Current Sheets in Astrophysical Plasma Turbulence. Astrophys. J. Lett. 82, L28.
  * Howes (2017) Howes, G. G. 2017 A prospectus on kinetic heliophysics. Phys. Plasmas 24 (5), 055907.
  * Howes et al. (2012a) Howes, G. G., Bale, S. D., Klein, K. G., Chen, C. H. K., Salem, C. S. & TenBarge, J. M. 2012a The Slow-mode Nature of Compressible Wave Power in Solar Wind Turbulence. Astrophys. J. Lett. 753, L19.
  * Howes et al. (2006) Howes, G. G., Cowley, S. C., Dorland, W., Hammett, G. W., Quataert, E. & Schekochihin, A. A. 2006 Astrophysical Gyrokinetics: Basic Equations and Linear Theory. Astrophys. J. 651, 590–614.
  * Howes et al. (2008a) Howes, G. G., Cowley, S. C., Dorland, W., Hammett, G. W., Quataert, E. & Schekochihin, A. A. 2008a A model of turbulence in magnetized plasmas: Implications for the dissipation range in the solar wind. J. Geophys. Res. 113 (A12), A05103.
  * Howes et al. (2008b) Howes, G. G., Dorland, W., Cowley, S. C., Hammett, G. W., Quataert, E., Schekochihin, A. A. & Tatsuno, T. 2008b Kinetic Simulations of Magnetized Turbulence in Astrophysical Plasmas. Phys. Rev. Lett. 100 (6), 065004.
  * Howes et al. (2012b) Howes, G. G., Drake, D. J., Nielson, K. D., Carter, T. A., Kletzing, C. A. & Skiff, F. 2012b Toward Astrophysical Turbulence in the Laboratory. Phys. Rev. Lett. 109 (25), 255001.
  * Howes et al. (2024) Howes, G. G., Huang, R. & Felix, A. A. 2024 The Physics of the Magnetic Mirror Force and Transit-Time Damping: Asymptotic Solution. J. Plasma Phys. In preparation.
  * Howes et al. (2017) Howes, G. G., Klein, K. G. & Li, T. C. 2017 Diagnosing collisionless energy transfer using field-particle correlations: Vlasov-Poisson plasmas. J. Plasma Phys. 83 (1), 705830102.
  * Howes et al. (2014) Howes, G. G., Klein, K. G. & TenBarge, J. M. 2014 Validity of the Taylor Hypothesis for Linear Kinetic Waves in the Weakly Collisional Solar Wind. Astrophys. J. 789, 106.
  * Howes et al. (2018) Howes, G. G., McCubbin, A. J. & Klein, K. G. 2018 Spatially localized particle energization by Landau damping in current sheets produced by strong Alfvén wave collisions. J. Plasma Phys. 84 (1), 905840105\.
  * Howes & Nielson (2013) Howes, G. G. & Nielson, K. D. 2013 Alfvén wave collisions, the fundamental building block of plasma turbulence. I. Asymptotic solution. Phys. Plasmas 20 (7), 072302.
  * Howes et al. (2013) Howes, G. G., Nielson, K. D., Drake, D. J., Schroeder, J. W. R., Skiff, F., Kletzing, C. A. & Carter, T. A. 2013 Alfvén wave collisions, the fundamental building block of plasma turbulence. III. Theory for experimental design. Phys. Plasmas 20 (7), 072304.
  * Howes et al. (2011) Howes, G. G., Tenbarge, J. M. & Dorland, W. 2011 A weakened cascade model for turbulence in astrophysical plasmas. Phys. Plasmas 18, 102305.
  * Howes et al. (2011) Howes, G. G., TenBarge, J. M., Dorland, W., Quataert, E., Schekochihin, A. A., Numata, R. & Tatsuno, T. 2011 Gyrokinetic simulations of solar wind turbulence from ion to electron scales. Phys. Rev. Lett. 107, 035004\.
  * Huang et al. (2024) Huang, Rui, Howes, Gregory G. & McCubbin, Andrew J. 2024 The Velocity-Space Signature of Transit-Time Damping. arXiv e-prints p. arXiv:2401.16697.
  * Hunana et al. (2011) Hunana, P., Laveder, D., Passot, T., Sulem, P. L. & Borgogno, D. 2011 Reduction of Compressibility and Parallel Transfer by Landau Damping in Turbulent Magnetized Plasmas. Astrophys. J. 743, 128\.
  * Isenberg & Hollweg (1983) Isenberg, P. A. & Hollweg, J. V. 1983 On the preferential acceleration and heating of solar wind heavy ions. J. Geophys. Res. 88, 3923–3935.
  * Isenberg et al. (2001) Isenberg, P. A., Lee, M. A. & Hollweg, J. V. 2001 The kinetic shell model of coronal heating and acceleration by ion cyclotron waves: 1. Outward propagating waves. J. Geophys. Res. 106, 5649–5660.
  * Isenberg & Vasquez (2019) Isenberg, Philip A. & Vasquez, Bernard J. 2019 Perpendicular Ion Heating by Cyclotron Resonant Dissipation of Turbulently Generated Kinetic Alfvén Waves in the Solar Wind. Astrophys. J. 887 (1), 63.
  * JET Team (1992) JET Team 1992 Fusion energy production from a deuterium-tritium plasma in the JET tokamak. Nuclear Fusion 32 (2), 187–203.
  * Ji & Daughton (2011) Ji, H. & Daughton, W. 2011 Phase diagram for magnetic reconnection in heliophysical, astrophysical, and laboratory plasmas. Phys. Plasmas 18 (11), 111207–111207.
  * Jiansen et al. (2018) Jiansen, He, Xingyu, Zhu, Yajie, Chen, Chadi, Salem, Michael, Stevens, Hui, Li, Wenzhi, Ruan, Lei, Zhang & Chuanyi, Tu 2018 Plasma Heating and Alfvénic Turbulence Enhancement During Two Steps of Energy Conversion in Magnetic Reconnection Exhaust Region of Solar Wind. The Astrophysical Journal 856 (2), 148.
  * Johnson & Cheng (2001) Johnson, J. R. & Cheng, C. Z. 2001 Stochastic ion heating at the magnetopause due to kinetic Alfvén waves. Geophys. Res. Lett. 28, 4421–4424.
  * Karimabadi et al. (2013) Karimabadi, H., Roytershteyn, V., Wan, M., Matthaeus, W. H., Daughton, W., Wu, P., Shay, M., Loring, B., Borovsky, J., Leonardis, E., Chapman, S. C. & Nakamura, T. K. M. 2013 Coherent structures, intermittent turbulence, and dissipation in high-temperature plasmas. Phys. Plasmas 20 (1), 012303.
  * Kasper et al. (2002) Kasper, J. C., Lazarus, A. J. & Gary, S. P. 2002 Wind/SWE observations of firehose constraint on solar wind proton temperature anisotropy. Geophys. Res. Lett. 29, 20–1.
  * Kawamori (2013) Kawamori, E. 2013 Experimental Verification of Entropy Cascade in Two-Dimensional Electrostatic Turbulence in Magnetized Plasma. Phys. Rev. Lett. 110 (9), 095001.
  * Kawazura (2022) Kawazura, Y. 2022 CALLIOPE: Pseudospectral Shearing Magnetohydrodynamics Code with a Pencil Decomposition. Astrophys. J. 928 (2), 113.
  * Kawazura & Barnes (2018) Kawazura, Y. & Barnes, M. 2018 A hybrid gyrokinetic ion and isothermal electron fluid code for astrophysical plasma. J. Comp. Phys. 360, 57–73.
  * Kawazura et al. (2019) Kawazura, Y., Barnes, M. & Schekochihin, A. A. 2019 Thermal disequilibration of ions and electrons by collisionless plasma turbulence. Proc. Nat. Acad. Sci. 116, 771–776.
  * Kawazura et al. (2022) Kawazura, Y., Schekochihin, A. A., Barnes, M., Dorland, W. & Balbus, S. A. 2022 Energy partition between Alfvénic and compressive fluctuations in magnetorotational turbulence with near-azimuthal mean magnetic field. J. Plasma Phys. 88 (3), 905880311.
  * Kawazura et al. (2020) Kawazura, Y., Schekochihin, A. A., Barnes, M., TenBarge, J. M., Tong, Y., Klein, K. G. & Dorland, W. 2020 Ion versus Electron Heating in Compressively Driven Astrophysical Gyrokinetic Turbulence. Phys. Rev. X 10 (4), 041050.
  * Kennel (1966) Kennel, C. 1966 Low-Frequency Whistler Mode. Phys. Fluids 9, 2190–2202.
  * Kiyani et al. (2009) Kiyani, K. H., Chapman, S. C., Khotyaintsev, Y. V., Dunlop, M. W. & Sahraoui, F. 2009 Global scale-invariant dissipation in collisionless plasma turbulence. Phys. Rev. Lett. 103, 075006.
  * Kiyani et al. (2015) Kiyani, K. H., Osman, K. T. & Chapman, S. C. 2015 Introduction: Dissipation and heating in solar wind turbulence: from the macro to the micro and back again. Philosophical Transactions of the Royal Society of London Series A 373, 40155.
  * Klein & Chandran (2016) Klein, K. G. & Chandran, B. D. G. 2016 Evolution of The Proton Velocity Distribution due to Stochastic Heating in the Near-Sun Solar Wind. Astrophys. J. 820, 47.
  * Klein & Howes (2015) Klein, K. G. & Howes, G. G. 2015 Predicted impacts of proton temperature anisotropy on solar wind turbulence. Phys. Plasmas 22 (3), 032903.
  * Klein & Howes (2016) Klein, K. G. & Howes, G. G. 2016 Measuring Collisionless Damping in Heliospheric Plasmas using Field-Particle Correlations. Astrophys. J. Lett. 826, L30.
  * Klein et al. (2017) Klein, K. G., Howes, G. G. & TenBarge, J. M. 2017 Diagnosing collisionless enegy transfer using field-particle correlations: gyrokinetic turbulence. J. Plasma Phys. 83 (4), 535830401.
  * Klein et al. (2012) Klein, K. G., Howes, G. G., TenBarge, J. M., Bale, S. D., Chen, C. H. K. & Salem, C. S. 2012 Using Synthetic Spacecraft Data to Interpret Compressible Fluctuations in Solar Wind Turbulence. Astrophys. J. 755, 159.
  * Klein et al. (2020) Klein, Kristopher G., Howes, Gregory G., TenBarge, Jason M. & Valentini, Francesco 2020 Diagnosing collisionless energy transfer using field-particle correlations: Alfvén-ion cyclotron turbulence. J. Plasma Phys. 86 (4), 905860402.
  * Kolmogorov (1941) Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large reynolds numbers. Dokl. Akad. Nauk SSSR 30, 9–, english Translation: Proc. Roy. Soc. London A, 434, 9 (1991).
  * Kraichnan (1965) Kraichnan, R. H. 1965 Inertial range spectrum of hyromagnetic turbulence. Phys. Fluids 8, 1385–1387.
  * Krall & Trivelpiece (1973) Krall, N. A. & Trivelpiece, A. W. 1973 Principles of plasma physics. International Student Edition - International Series in Pure and Applied Physics, Tokyo: McGraw-Hill Kogakusha, 1973.
  * Krishan & Mahajan (2004) Krishan, V. & Mahajan, S. M. 2004 Magnetic fluctuations and Hall magnetohydrodynamic turbulence in the solar wind. J. Geophys. Res. 109 (A18), A11105.
  * Kulsrud (1983) Kulsrud, R. M. 1983 MHD description of plasma. In Basic Plasma Physics I (ed. A. A. Galeev & R. N. Sudan), Handbook of Plasma Physics, vol. 1, chap. 1.4, pp. 115–145. North Holland.
  * Kunz et al. (2018) Kunz, M. W., Abel, I. G., Klein, K. G. & Schekochihin, A. A. 2018 Astrophysical gyrokinetics: turbulence in pressure-anisotropic plasmas at ion scales and beyond. J. Plasma Phys. 84 (2), 715840201.
  * Kunz et al. (2014) Kunz, M. W., Schekochihin, A. A. & Stone, J. M. 2014 Firehose and Mirror Instabilities in a Collisionless Shearing Plasma. Phys. Rev. Lett. 112 (20), 205003.
  * Landau (1946) Landau, L. D. 1946 On the Vibrations of the Electronic Plasma. Journal of Physics 10, 25.
  * Leamon et al. (1998a) Leamon, R. J., Matthaeus, W. H., Smith, C. W. & Wong, H. K. 1998a Contribution of cyclotron-resonant damping to kinetic dissipation of interplanetary turbulence. Astrophys. J. 507, L181–L184.
  * Leamon et al. (2000) Leamon, R. J., Matthaeus, W. H., Smith, C. W., Zank, G. P., Mullan, D. J. & Oughton, S. 2000 MHD-driven Kinetic Dissipation in the Solar Wind and Corona. Astrophys. J. 537, 1054–1062.
  * Leamon et al. (1998b) Leamon, R. J., Smith, C. W., Ness, N. F., Matthaeus, W. H. & Wong, H. K. 1998b Observational constraints on the dynamics of the interplanetary magnetic field dissipation range. J. Geophys. Res. 103, 4775–4787.
  * Leamon et al. (1999) Leamon, R. J., Smith, C. W., Ness, N. F. & Wong, H. K. 1999 Dissipation range dynamics: Kinetic alfvén waves and the importance of $\beta_{e}$. J. Geophys. Res. 104, 22331–22344.
  * Li et al. (2016) Li, T. C., Howes, G. G., Klein, K. G. & TenBarge, J. M. 2016 Energy Dissipation and Landau Damping in Two- and Three-dimensional Plasma Turbulence. Astrophys. J. Lett. 832, L24.
  * Lichko & Egedal (2020) Lichko, E. & Egedal, J. 2020 Magnetic pumping model for energizing superthermal particles applied to observations of the Earth’s bow shock. Nature Comm. 11, 2942.
  * Lichko et al. (2017) Lichko, E., Egedal, J., Daughton, W. & Kasper, J. 2017 Magnetic Pumping as a Source of Particle Heating and Power-law Distributions in the Solar Wind. Astrophys. J. Lett. 850, L28.
  * Lionello et al. (2009) Lionello, Roberto, Linker, Jon A. & Mikić, Zoran 2009 Multispectral Emission of the Sun During the First Whole Sun Month: Magnetohydrodynamic Simulations. Astrophys. J. 690 (1), 902–912.
  * Lithwick & Goldreich (2001) Lithwick, Yoram & Goldreich, Peter 2001 Compressible Magnetohydrodynamic Turbulence in Interstellar Plasmas. Astrophys. J. 562 (1), 279–296.
  * Loureiro & Boldyrev (2017a) Loureiro, N. F. & Boldyrev, S. 2017a Collisionless Reconnection in Magnetohydrodynamic and Kinetic Turbulence. Astrophys. J. 850, 182.
  * Loureiro & Boldyrev (2017b) Loureiro, N. F. & Boldyrev, S. 2017b Role of Magnetic Reconnection in Magnetohydrodynamic Turbulence. Phys. Rev. Lett. 118 (24), 245101.
  * Loureiro et al. (2013) Loureiro, N. F., Schekochihin, A. A. & Zocco, A. 2013 Fast Collisionless Reconnection and Electron Heating in Strongly Magnetized Plasmas. Phys. Rev. Lett. 111 (2), 025002.
  * Mallet et al. (2015) Mallet, A., Schekochihin, A. A. & Chandran, B. D. G. 2015 Refined critical balance in strong Alfvenic turbulence. Mon. Not. Roy. Astron. Soc. 449, L77–L81.
  * Mallet et al. (2017a) Mallet, Alfred, Schekochihin, Alexander A. & Chandran, Benjamin D. G. 2017a Disruption of Alfvénic turbulence by magnetic reconnection in a collisionless plasma. J. Plasma Phys. 83 (6), 905830609.
  * Mallet et al. (2017b) Mallet, A., Schekochihin, A. A. & Chandran, B. D. G. 2017b Disruption of sheet-like structures in Alfvénic turbulence by magnetic reconnection. Mon. Not. Roy. Astron. Soc. 468 (4), 4862–4871.
  * Markovskii & Vasquez (2011) Markovskii, S. A. & Vasquez, B. J. 2011 A Short-timescale Channel of Dissipation of the Strong Solar Wind Turbulence. Astrophys. J. 739, 22.
  * Maron & Goldreich (2001) Maron, J. & Goldreich, P. 2001 Simulations of incompressible magnetohydrodynamic turbulence. Astrophys. J. 554, 1175–1196.
  * Marsch & Tu (1990) Marsch, E. & Tu, C. Y. 1990 On the radial evolution of MHD turbulence in the Inner heliosphere. J. Geophys. Res. 95 (A6), 8211–8229.
  * Martinović et al. (2019) Martinović, Mihailo M., Klein, Kristopher G. & Bourouaine, Sofiane 2019 Radial Evolution of Stochastic Heating in Low-$\beta$ Solar Wind. Astrophys. J. 879 (1), 43\.
  * Martinović et al. (2020) Martinović, Mihailo M., Klein, Kristopher G., Kasper, Justin C., Case, Anthony W., Korreck, Kelly E., Larson, Davin, Livi, Roberto, Stevens, Michael, Whittlesey, Phyllis, Chandran, Benjamin D. G., Alterman, Ben L., Huang, Jia, Chen, Christopher H. K., Bale, Stuart D., Pulupa, Marc, Malaspina, David M., Bonnell, John W., Harvey, Peter R., Goetz, Keith, Dudok de Wit, Thierry & MacDowall, Robert J. 2020 The Enhancement of Proton Stochastic Heating in the Near-Sun Solar Wind. Astrophys. J. Supp. 246 (2), 30.
  * Maruca et al. (2011) Maruca, B. A., Kasper, J. C. & Bale, S. D. 2011 What Are the Relative Roles of Heating and Cooling in Generating Solar Wind Temperature Anisotropies? Phys. Rev. Lett. 107 (20), 201101.
  * Mason et al. (2006) Mason, J., Cattaneo, F. & Boldyrev, S. 2006 Dynamic Alignment in Driven Magnetohydrodynamic Turbulence. Phys. Rev. Lett. 97 (25), 255002.
  * Matthaeus et al. (2005) Matthaeus, W. H., Dasso, S., Weygand, J. M., Milano, L. J., Smith, C. W. & Kivelson, M. G. 2005 Spatial Correlation of Solar-Wind Turbulence from Two-Point Measurements. Phys. Rev. Lett. 95 (23), 231101.
  * Matthaeus et al. (1982) Matthaeus, William H., Goldstein, Melvyn L. & Smith, Charles 1982 Evaluation of magnetic helicity in homogeneous turbulence. Phys. Rev. Lett. 48 (18), 1256–1259.
  * Matthaeus & Montgomery (1980) Matthaeus, W. H. & Montgomery, D. 1980 Selective decay hypothesis at high mechanical and magnetic Reynolds numbers. Annals of the New York Academy of Sciences 357, 203–222.
  * Matthaeus et al. (2014) Matthaeus, W. H., Oughton, S., Osman, K. T., Servidio, S., Wan, M., Gary, S. P., Shay, M. A., Valentini, F., Roytershteyn, V., Karimabadi, H. & Chapman, S. C. 2014 Nonlinear and Linear Timescales near Kinetic Scales in Solar Wind Turbulence. Astrophys. J. 790 (2), 155.
  * Matthaeus & Velli (2011) Matthaeus, W. H. & Velli, M. 2011 Who Needs Turbulence?. A Review of Turbulence Effects in the Heliosphere and on the Fundamental Process of Reconnection. Space Sci. Rev. 160, 145–168.
  * McCubbin et al. (2022) McCubbin, Andrew J., Howes, Gregory G. & TenBarge, Jason M. 2022 Characterizing velocity-space signatures of electron energization in large-guide-field collisionless magnetic reconnection. Phys. Plasmas 29 (5), 052105.
  * Melville et al. (2016) Melville, S., Schekochihin, A. A. & Kunz, M. W. 2016 Pressure-anisotropy-driven microturbulence and magnetic-field evolution in shearing, collisionless plasma. Mon. Not. Roy. Astron. Soc. 459, 2701–2720.
  * Meneguzzi et al. (1981) Meneguzzi, M., Frisch, U. & Pouquet, A. 1981 Helical and nonhelical turbulent dynamos. Phys. Rev. Lett. 47, 1060–1064.
  * Merrifield et al. (2005) Merrifield, J. A., Müller, W.-C., Chapman, S. C. & Dendy, R. O. 2005 The scaling properties of dissipation in incompressible isotropic three-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 12 (2), 022301.
  * Meyrand et al. (2019) Meyrand, R., Kanekar, A., Dorland, W. & Schekochihin, A. A. 2019 Fluidization of collisionless plasma turbulence. Proc. Nat. Acad. Sci. 116, 1185–1194.
  * Meyrand et al. (2021) Meyrand, R., Squire, J., Schekochihin, A. A. & Dorland, W. 2021 On the violation of the zeroth law of turbulence in space plasmas. J. Plasma Phys. 87 (3), 535870301.
  * Mikić et al. (1999) Mikić, Zoran, Linker, Jon A., Schnack, Dalton D., Lionello, Roberto & Tarditi, Alfonso 1999 Magnetohydrodynamic modeling of the global solar corona. Phys. Plasmas 6 (5), 2217–2224.
  * Montag & Howes (2022) Montag, P. & Howes, Gregory G. 2022 A field-particle correlation analysis of magnetic pumping. Phys. Plasmas 29 (3), 032901.
  * Montag (2018) Montag, Peter Katsumi 2018 Modeling the formation of current sheets in symmetric and asymmetric reconnection. PhD thesis, Massachusetts Institute of Technology.
  * Muñoz & Büchner (2018) Muñoz, P. A. & Büchner, J. 2018 Two-stage Electron Acceleration by 3D Collisionless Guide-field Magnetic Reconnection. The Astrophysical Journal 864 (1), 92.
  * Narita et al. (2010) Narita, Y., Glassmeier, K.-H., Sahraoui, F. & Goldstein, M. L. 2010 Wave-Vector Dependence of Magnetic-Turbulence Spectra in the Solar Wind. Physical Review Letters 104 (17), 171101.
  * National Research Council (2013) National Research Council 2013 Solar and Space Physics: A Science for a Technological Society. The National Academies Press, committee on a Decadal Strategy for Solar and Space Physics (Heliophysics); Space Studies Board; Aeronautics and Space Engineering Board; Division of Earth and Physical Sciences; National Research Council.
  * Navarro et al. (2016) Navarro, A. B., Teaca, B., Told, D., Groselj, D., Crandall, P. & Jenko, F. 2016 Structure of Plasma Heating in Gyrokinetic Alfvénic Turbulence. Physical Review Letters 117 (24), 245101.
  * Ng & Bhattacharjee (1996) Ng, C. S. & Bhattacharjee, A. 1996 Interaction of Shear-Alfven Wave Packets: Implication for Weak Magnetohydrodynamic Turbulence in Astrophysical Plasmas. Astrophys. J. 465, 845.
  * Nielson et al. (2013) Nielson, K. D., Howes, G. G. & Dorland, W. 2013 Alfvén wave collisions, the fundamental building block of plasma turbulence. II. Numerical solution. Phys. Plasmas 20 (7), 072303.
  * Opher et al. (2020) Opher, Merav, Loeb, Abraham, Drake, James & Toth, Gabor 2020 A small and round heliosphere suggested by magnetohydrodynamic modelling of pick-up ions. Nature Astronomy 4, 675–683.
  * Osman et al. (2014a) Osman, K. T., Kiyani, K. H., Chapman, S. C. & Hnat, B. 2014a Anisotropic Intermittency of Magnetohydrodynamic Turbulence. Astrophys. J. Lett. 783, L27.
  * Osman et al. (2014b) Osman, K. T., Matthaeus, W. H., Gosling, J. T., Greco, A., Servidio, S., Hnat, B., Chapman, S. C. & Phan, T. D. 2014b Magnetic Reconnection and Intermittent Turbulence in the Solar Wind. Phys. Rev. Lett. 112 (21), 215002.
  * Osman et al. (2011) Osman, K. T., Matthaeus, W. H., Greco, A. & Servidio, S. 2011 Evidence for Inhomogeneous Heating in the Solar Wind. Astrophys. J. Lett. 727, L11.
  * Osman et al. (2012a) Osman, K. T., Matthaeus, W. H., Hnat, B. & Chapman, S. C. 2012a Kinetic Signatures and Intermittent Turbulence in the Solar Wind Plasma. Phys. Rev. Lett. 108 (26), 261103.
  * Osman et al. (2012b) Osman, K. T., Matthaeus, W. H., Wan, M. & Rappazzo, A. F. 2012b Intermittency and Local Heating in the Solar Wind. Phys. Rev. Lett. 108 (26), 261102.
  * Ossendrijver (2003) Ossendrijver, Mathieu 2003 The solar dynamo. Astron. Astrophys. Rev. 11 (4), 287–367.
  * Pao (1965) Pao, Yih-Ho 1965 Structure of Turbulent Velocity and Scalar Fields at Large Wavenumbers. Phys. Fluids 8 (6), 1063–1075.
  * Parker (1955) Parker, E. N. 1955 Hydromagnetic dynamo models. Astrophys. J. 122, 293–314.
  * Parker (1993) Parker, E. N. 1993 A Solar Dynamo Surface Wave at the Interface between Convection and Nonuniform Rotation. Astrophys. J. 408, 707\.
  * Parker et al. (2016) Parker, J. T., Highcock, E. G., Schekochihin, A. A. & Dellar, P. J. 2016 Suppression of phase mixing in drift-kinetic plasma turbulence. Phys. Plasmas 23 (7), 070703.
  * Perez & Boldyrev (2009) Perez, J. C. & Boldyrev, S. 2009 Role of Cross-Helicity in Magnetohydrodynamic Turbulence. Phys. Rev. Lett. 102 (2), 025003\.
  * Perez & Boldyrev (2010) Perez, J. C. & Boldyrev, S. 2010 Strong magnetohydrodynamic turbulence with cross helicity. Phys. Plasmas 17 (5), 055903.
  * Perez et al. (2012) Perez, J. C., Mason, J., Boldyrev, S. & Cattaneo, F. 2012 On the Energy Spectrum of Strong Magnetohydrodynamic Turbulence. Phys. Rev. X 2 (4), 041005.
  * Perri et al. (2012) Perri, S., Goldstein, M. L., Dorelli, J. C. & Sahraoui, F. 2012 Detection of Small-Scale Structures in the Dissipation Regime of Solar-Wind Turbulence. Phys. Rev. Lett. 109 (19), 191101.
  * Phan et al. (2018) Phan, T. D., Eastwood, J. P., Shay, M. A., Drake, J. F., Sonnerup, B. U. Ö., Fujimoto, M., Cassak, P. A., Øieroset, M., Burch, J. L., Torbert, R. B., Rager, A. C., Dorelli, J. C., Gershman, D. J., Pollock, C., Pyakurel, P. S., Haggerty, C. C., Khotyaintsev, Y., Lavraud, B., Saito, Y., Oka, M., Ergun, R. E., Retino, A., Le Contel, O., Argall, M. R., Giles, B. L., Moore, T. E., Wilder, F. D., Strangeway, R. J., Russell, C. T., Lindqvist, P. A. & Magnes, W. 2018 Electron magnetic reconnection without ion coupling in earth’s turbulent magnetosheath. Nature 557 (7704), 202–206.
  * Plunk (2013) Plunk, G. G. 2013 Landau damping in a turbulent setting. Phys. Plasmas 20 (3), 032304.
  * Plunk et al. (2010) Plunk, G. G., Cowley, S. C., Schekochihin, A. A. & Tatsuno, T. 2010 Two-dimensional gyrokinetic turbulence. J. Fluid Mech. 664, 407–435.
  * Plunk & Tatsuno (2011) Plunk, G. G. & Tatsuno, T. 2011 Energy Transfer and Dual Cascade in Kinetic Magnetized Plasma Turbulence. Phys. Rev. Lett. 106 (16), 165003.
  * Podesta et al. (2010) Podesta, J. J., Borovsky, J. E. & Gary, S. P. 2010 A Kinetic Alfvén Wave Cascade Subject to Collisionless Damping Cannot Reach Electron Scales in the Solar Wind at 1 AU. Astrophys. J. 712, 685–691.
  * Pucci et al. (2018) Pucci, F., Usami, S., Ji, H., Guo, X., Horiuchi, R., Okamura, S., Fox, W., Jara-Almonte, J., Yamada, M. & Yoo, J. 2018 Energy transfer and electron energization in collisionless magnetic reconnection for different guide-field intensities. Physics of Plasmas 25 (12).
  * Quataert (1998) Quataert, E. 1998 Particle Heating by Alfvénic Turbulence in Hot Accretion Flows. Astrophys. J. 500, 978–991.
  * Quataert & Gruzinov (1999) Quataert, E. & Gruzinov, A. 1999 Turbulence and Particle Heating in Advection-dominated Accretion Flows. Astrophys. J. 520, 248–255.
  * Ripperda et al. (2021) Ripperda, B., Mahlmann, J. F., Chernoglazov, A., TenBarge, J. M., Most, E. R., Juno, J., Yuan, Y., Philippov, A. A. & Bhattacharjee, A. 2021 Weak Alfvénic turbulence in relativistic plasmas. Part 2. current sheets and dissipation. J. Plasma Phys. 87 (5), 905870512\.
  * Roberts et al. (2015) Roberts, O. W., Li, X. & Jeska, L. 2015 A Statistical Study of the Solar Wind Turbulence at Ion Kinetic Scales Using the k-filtering Technique and Cluster Data. Astrophys. J. 802, 2.