3.1 Evolution of turbulent and non-thermal energies

In figure 2, we plot the evolution of these three different contributions to the total fluctuating energy normalized to the total initial fluctuating energy $\unicode[STIX]{x1D6FF}W_{0}\equiv \unicode[STIX]{x1D6FF}W(t=0)$ . In figure 2(a), we plot the total fluctuating energy $\unicode[STIX]{x1D6FF}W/\unicode[STIX]{x1D6FF}W_{0}$ (black), the turbulent energy $E^{(\text{turb})}/\unicode[STIX]{x1D6FF}W_{0}$ (purple), the ion non-thermal energy $E_{i}^{(nt)}/\unicode[STIX]{x1D6FF}W_{0}$ (red) and the electron non-thermal energy $E_{e}^{(nt)}/\unicode[STIX]{x1D6FF}W_{0}$ (blue). Note that collisions in AstroGK, as well as in real plasma systems, convert non-thermal to thermal energy, representing irreversible plasma heating with an associated increase of entropy. The energy lost from $\unicode[STIX]{x1D6FF}W$ by collisions is tracked by AstroGK and represents thermal heating of the plasma species, but this energy is not fed back into the code to evolve the equilibrium thermal temperature, $T_{0s}$ (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006; Numata et al. Reference Numata, Howes, Tatsuno, Barnes and Dorland2010; Li et al. Reference Li, Howes, Klein and TenBarge2016). The evolution in figure 2(a) makes clear that, over 7.5 periods of the initial Alfvén waves, more than 60 % of the initial fluctuating energy in the simulation is lost to collisional heating.

Figure 2. (a) Evolution of the normalized energy $E/\unicode[STIX]{x1D6FF}W_{0}$ as a function of time $t/T_{0}$ for the total fluctuating energy $\unicode[STIX]{x1D6FF}W$ (black), the turbulent energy $E^{(\text{turb})}$ (purple), the ion non-thermal energy $E_{i}^{(nt)}$ (red) and the electron non-thermal energy $E_{e}^{(nt)}$ (blue). (b) Evolution of the different components of the turbulent energy $E^{(\text{turb})}$ (purple), dominated by the perpendicular magnetic field energy $E_{B_{\bot }}$ (green dashed) and the perpendicular ion bulk flow kinetic energy $E_{i,u_{\bot }}$ (red dashed), with successively smaller contributions by the perpendicular electron bulk kinetic energy $E_{e,u_{\bot }}$ (blue dashed), the parallel magnetic field energy $E_{B_{\Vert }}$ (green dotted), the parallel ion bulk flow kinetic energy $E_{i,u_{\Vert }}$ (red dotted) and the parallel electron bulk flow kinetic energy $E_{e,u_{\Vert }}$ (blue dotted).

In figure 2(b), we plot the different components that contribute to the turbulent energy $E^{(\text{turb})}$ . In order of decreasing magnitude, these contributions are the perpendicular magnetic energy $E_{B_{\bot }}$ (green dashed), perpendicular ion kinetic energy $E_{u_{i,\bot }}$ (red dashed), perpendicular electron kinetic energy $E_{u_{e,\bot }}$ (blue dashed), parallel magnetic energy $E_{B_{\Vert }}$ (green dotted), parallel ion kinetic energy $E_{u_{i,\Vert }}$ (red dotted) and parallel electron kinetic energy $E_{u_{e,\Vert }}$ (blue dotted). The turbulent energy is dominated by the perpendicular magnetic energy and perpendicular ion kinetic energy. This is expected for Alfvénic fluctuations at $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ : transverse motion of the plasma dominated by ion kinetic energy is first arrested by magnetic tension, followed by the acceleration of the plasma back toward the equilibrium point by magnetic tension, thereby leading to the oscillatory transfer of energy back and forth between perpendicular magnetic energy and perpendicular ion kinetic energy, as evident in figure 2(b). Note that this energy is integrated over the entire simulation domain, so neither of these energies is expected to drop to zero, as would occur for the energy density at a single point in space as an Alfvén wave passes through that point. In the MHD limit $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ , Alfvénic fluctuations also have very little parallel motion, $u_{\Vert }\ll u_{\bot }$ and a very small parallel magnetic field fluctuation, $\unicode[STIX]{x1D6FF}B_{\Vert }\ll \unicode[STIX]{x1D6FF}B_{\bot }$ . Furthermore, the electron kinetic energies are down from the respective ion kinetic energies approximately by a factor of the mass ratio, $m_{e}/m_{i}=1/36$ , so electrons make a subdominant contribution to the turbulent energy. Finally, note that although the volume-integrated energy of each component of $E^{(\text{turb})}$ shows oscillations with the period $T_{0}$ , their sum varies smoothly in time, suggesting that this definition of turbulent energy is physically well motivated.

3.2 Evolution of collisional heating

In figure 3, we present the evolution of the collisional heating rate per unit volume of ions $Q_{i}$ (red) and electrons $Q_{e}$ (blue) as well as the total collisional heating rate $Q_{\text{tot}}=Q_{i}+Q_{e}$ (black) for this nonlinear Alfvén wave collision simulation (thick lines). The heating rates are normalized by a characteristic heating rate per unit volume, $Q_{0}=(n_{0i}T_{0i}v_{ti}/L_{\Vert })(\unicode[STIX]{x03C0}/8)(L_{\bot }/L_{\Vert })^{2}$ . The total fluctuating energy $\unicode[STIX]{x1D6FF}W$ in figure 2(a) diminishes in time due to thermalization by collisions. This collisional energy loss from $\unicode[STIX]{x1D6FF}W$ is tracked in AstroGK by this collisional heating rate, enabling energy conservation to be measured in the simulation.

Figure 3. Ion collisional heating rate $Q_{i}/Q_{0}$ (red), electron collisional heating rate $Q_{e}/Q_{0}$ (blue) and total collisional heating rate $Q_{\text{tot}}=Q_{i}+Q_{e}$ (black) and as a function of time $t/T_{0}$ for the nonlinear simulation (thick lines). Also plotted (thin lines) is the linear evolution from the same initial conditions.

Note that the rapid initial rise in the collisional damping rate for the electrons $Q_{e}$ at $t/T_{0}\lesssim 0.5$ in figure 3 is due to the fact that the linear initialization uses higher collision coefficients, $\unicode[STIX]{x1D708}_{s}=0.01k_{\Vert }v_{A}$ , than the subsequent nonlinear evolution, $\unicode[STIX]{x1D708}_{s}=6\times 10^{-4}k_{\Vert }v_{A}$ . When the collisional coefficients are reduced, smaller velocity scale structures in the velocity distribution must develop (through the kinetic evolution) before the collisional heating is able to effectively thermalize the non-thermal energy contained in those fluctuations.

Also plotted in figure 3 is the evolution of the collisional heating rates in a linear simulation (thin lines), where the simulation is started from the same initial conditions but the nonlinear terms are turned off. In this linear simulation, there is no nonlinear transfer of energy to other Fourier modes – meaning that there is no nonlinear turbulent cascade of energy to small scales – so the evolution of the energy is solely due to linear Landau damping of the initial Alfvén waves and the subsequent collisional thermalization of the fluctuations in the velocity distribution functions that were generated by this linear Landau damping. It is important to note that the nonlinear evolution eventually leads to a higher collisional heating rate, presumably through the nonlinear transfer of energy to smaller-scale fluctuations that have higher collisionless damping rates than the initial Alfvén waves, although we do not directly analyse that nonlinear cascade of energy in this study.

3.3 Model of energy flow

A physical interpretation of the two-step energy flow in this strong Alfvén wave collision simulation is illustrated by the diagram in figure 4. The energy of turbulent fluctuations $E^{(\text{turb})}$ , consisting of the sum of the electromagnetic field fluctuations and the kinetic energy of the bulk flows (first velocity moment) of each plasma species (Howes Reference Howes2015, Reference Howes2017), can be removed by collisionless interactions ${\dot{E}}_{s}^{(fp)}$ between the electromagnetic fields and the plasma particles. This energy is converted to non-thermal energy of the ions and electrons, $E_{s}^{(nt)}$ . This non-thermal energy is represented by fluctuations in the particle velocity distribution functions that have no associated bulk flow (first moment), and therefore do not contribute to the turbulent motions. A key property of this collisionless energy transfer ${\dot{E}}_{s}^{(fp)}$ is that it is reversible (two-headed arrows in figure 4), representing the electromagnetic work done on the particles by the fields, which can be positive or negative. Note that this diagram of the energy flow applies to any collisionless damping process: resonant processes, such as Landau damping (Landau Reference Landau1946; Mouhot & Villani Reference Mouhot and Villani2011), transit-time damping (Barnes Reference Barnes1966) or cyclotron damping (Coleman Reference Coleman1968; Isenberg & Hollweg Reference Isenberg and Hollweg1983); non-resonant processes, such as stochastic ion heating (Chen, Lin & White Reference Chen, Lin and White2001; Chandran et al. Reference Chandran, Li, Rogers, Quataert and Germaschewski2010); or particle energization associated with collisionless magnetic reconnection (Birn et al. Reference Birn, Drake, Shay, Rogers, Denton, Hesse, Kuznetsova, Ma, Bhattacharjee and Otto2001; Treumann & Baumjohann Reference Treumann and Baumjohann2015).

Figure 4. Diagram of the energy flow in weakly collisional turbulent plasmas, showing that interactions between the electromagnetic fields and plasma particles ${\dot{E}}_{s}^{(fp)}$ can reversibly transfer energy between the turbulent energy $E^{(\text{turb})}$ and the non-thermal energy in the velocity distribution function of each species $E_{s}^{(nt)}$ . Collisional heating $Q_{s}$ then can irreversibly convert this non-thermal energy, represented by fluctuations in velocity space of each species, into heat of each plasma species $s$ . This is the two-step process of reversible particle energization and subsequent irreversible thermalization of that particle energy.

The non-thermal energy $E_{s}^{(nt)}$ is contained in fluctuations in velocity space of the particle velocity distribution functions for each species, $\unicode[STIX]{x1D6FF}f_{s}(\boldsymbol{v})$ . If these fluctuations reach sufficiently small scales in velocity space, arbitrarily weak collisions can smooth out those fluctuations, thermalizing their energy and thereby realizing irreversible plasma heating, $Q_{s}$ . The kinetic equation for each species governs two mechanisms that facilitate the transfer of energy to ever smaller scales in velocity space: linear phase mixing and nonlinear phase mixing.

The first mechanism is linear phase mixing governed by the ballistic term in the kinetic equation, which couples spatial variations with velocity-space fluctuations and can lead to the transfer of energy to small scales in velocity space.Footnote ³ In linear Landau damping, for example, the energy of a damped wave is first transferred collisionlessly into non-thermal velocity-space fluctuations, which subsequently phase mix linearly to small enough scales in velocity space that weak collisions can irreversibly convert the non-thermal energy into plasma heat. Boltzmann’s $H$ theorem proves that the entropy increase associated with irreversible plasma heating is ultimately collisional (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006).

In addition to this linear phase-mixing process, at perpendicular spatial scales comparable to the particle thermal Larmor radii, $k_{\bot }\unicode[STIX]{x1D70C}_{s}\gtrsim 1$ , a nonlinear phase-mixing process (Dorland & Hammett Reference Dorland and Hammett1993), also known as the entropy cascade (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009; Tatsuno et al. Reference Tatsuno, Schekochihin, Dorland, Plunk, Barnes, Cowley and Howes2009; Plunk et al. Reference Plunk, Cowley, Schekochihin and Tatsuno2010; Plunk & Tatsuno Reference Plunk and Tatsuno2011; Kawamori Reference Kawamori2013), can be very effective at transferring energy to ever smaller scales in velocity space. Ultimately, when the non-thermal particle energy in the velocity distribution functions $\unicode[STIX]{x1D6FF}f_{s}(\boldsymbol{v})$ has reached sufficiently small scales in velocity, due to some combination of linear and nonlinear phase mixing, collisions may thermalize that particle energy, completing the final step in the conversion of turbulent energy into plasma heat. In AstroGK, this collisional heating removes energy from fluctuating energy in the plasma, $\unicode[STIX]{x1D6FF}W$ .

It is worthwhile to contrast this two-step mechanism in weakly collisional plasmas – collisionless particle energization followed by collisional thermalization – with the more familiar picture of turbulent dissipation in the fluid (strongly collisional) limit. A dimensionless measure of the collisionality is the ratio of the thermal collision rate to the frequency of typical fluctuations in the plasma, $\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714}$ . In the strongly collisional limit, $\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714}\gg 1$ , collisions can directly remove energy from both the bulk plasma flows through viscosity and the plasma currents through resistivity. Because both viscosity and resistivity are collisional, entropy increases through these mechanisms, and the energy from the turbulent electromagnetic field and plasma flow fluctuations is immediately thermalized to plasma heat. Thus, the dissipation of turbulence in the strongly collisional, fluid limit is a single-step process. Consider the example of resistive MHD, where Ohm’s law gives the electric field in terms of the plasma fluid velocity, magnetic field and current density, $\boldsymbol{E}+\boldsymbol{U}/c\times \boldsymbol{B}=\unicode[STIX]{x1D702}\boldsymbol{j}$ (Spitzer Reference Spitzer1962; Kulsrud Reference Kulsrud, Galeev and Sudan1983). The work done by the electric field is $\boldsymbol{j}\boldsymbol{\cdot }\boldsymbol{E}=-\boldsymbol{j}\boldsymbol{\cdot }(\boldsymbol{U}/c\times \boldsymbol{B})+\unicode[STIX]{x1D702}\boldsymbol{j}^{2}$ , where the second term is the non-negative ohmic heating due to resistive dissipation of the current, showing that the resistivity leads directly to plasma heating.

The strong Alfvén wave collision simulation presented here has $\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714}\sim 6\times 10^{-4}\ll 1$ , firmly in the weakly collisional limit. Unlike in the MHD Ohm’s law above, where the current density $\boldsymbol{j}$ and electric field $\boldsymbol{E}$ due to the resistive term are in phase, and thereby yield a zero or positive change in energy, in the weakly collisional case the current density $\boldsymbol{j}$ and electric field $\boldsymbol{E}$ need not be in phase, enabling the work done by collisionless interactions between the fields and particles to give energy to or take energy from the particles. In fact, if the current and electric field are exactly 90 degrees out of phase, there is zero net energy transfer between fields and particles over one complete oscillation, corresponding to undamped wave motion. The bottom line, a point that cannot be overstated, is that in a weakly collisional plasma, the electromagnetic work $\boldsymbol{j}\boldsymbol{\cdot }\boldsymbol{E}$ does not correspond to irreversible plasma heating, but rather to reversible work done on the particles by the fields, or vice versa.

Developing a detailed understanding of particle energization and plasma heating in heliospheric plasmas is grand challenge problem in heliophysics, and this simple model of the energy flow provides important constraints to focus efforts in that endeavour. Note that the final step of the process in figure 4, the thermalization of the particle energy, is fundamentally collisional, independent of what mechanism (which we have not specified here) removed energy from the turbulent fluctuations initially. The key question in understanding particle energization and plasma heating in heliospheric plasmas is therefore to understand the first step: what collisionless and reversible mechanism is responsible for the removal of energy from the turbulent fluctuations and conversion of that energy into non-thermal energy of the plasma species?

3.4 Rate of energy transfer

Now we use the strong Alfvén wave collision simulation presented here to analyse the channels of energy transfer shown in figure 4. For each species, the rate of change of non-thermal energy is given by

where the irreversible collisional heating $Q_{s}\geqslant 0$ but the reversible collisionless field–particle energy transfer ${\dot{E}}_{s}^{(fp)}$ can be either positive or negative. In addition, the rate of change of turbulent energy must be the sum of the collisionless field–particle energy transfer for each species,

Note that we have not specified the physical mechanism governing the field–particle energy transfer, but we are simply showing that the transfers of energy indeed follow the diagram in figure 4.

Numerically, the collisional heating $Q_{s}$ is evaluated for each species in AstroGK by multiplying the linearized Landau collision operator (Abel et al. Reference Abel, Barnes, Cowley, Dorland and Schekochihin2008; Barnes et al. Reference Barnes, Abel, Dorland, Ernst, Hammett, Ricci, Rogers, Schekochihin and Tatsuno2009) by $m_{s}v^{2}/2$ and then integrating over velocity space and over the simulation domain. The rate of change of non-thermal energy ${\dot{E}}_{s}^{(nt)}$ is determined by numerically evaluating $E_{s}^{(nt)}$ as a function of time using (3.3) and then differencing to obtain the time derivative. Similarly, the rate of change of turbulent energy ${\dot{E}}^{(\text{turb})}$ is determined by numerically evaluating $E^{(\text{turb})}$ as a function of time using (3.4) and then taking the time derivative. The rate of energy transfer by collisionless field–particle interactions for each species ${\dot{E}}_{s}^{(fp)}$ is then computed using (3.5).

Figure 5. The rate of energy transfer by field–particle interactions ${\dot{E}}_{s}^{(fp)}$ (solid), the rate of change of non-thermal energy ${\dot{E}}_{s}^{(nt)}$ (dotted) and the collisional heating rate $Q_{s}$ (dashed) for (a) ions (red) and (b) electrons (blue). (c) The energy balance between the loss of turbulent energy $-{\dot{E}}^{(\text{turb})}$ (purple solid) and the summed transfer of energy to both ions and electrons, ${\dot{E}}_{i}^{(fp)}+{\dot{E}}_{e}^{(fp)}$ (black dashed).

In figure 5, we present the terms of these energy transfer relations for the (a) ions and (b) electrons, as well as (c) the balance between the loss of turbulent energy and the field–particle energy transfer to each species. A few very interesting aspects of figure 5 are worth highlighting. First, although the change of turbulent energy $E^{(\text{turb})}$ and non-thermal energies $E_{s}^{(nt)}$ in figure 2 appears to be smooth, the time derivative, which gives the rate of change, indeed varies rapidly, including a significant fluctuation with period $T_{0}/2$ .Footnote ⁴

Second, in figure 5(b), the energy transferred into electron non-thermal energy at the rate ${\dot{E}}_{e}^{(fp)}$ (solid) is very quickly thermalized by collisions into electron heat (dashed); the time lag between these two curves is $\unicode[STIX]{x0394}t=0.6T_{0}$ (not shown), suggesting that non-thermal energy transferred into the electron velocity distribution is rapidly transferred by phase mixing to sufficiently small velocity-space scales to be thermalized by the weak collisions. For the ions in figure 5(a), on the other hand, the time lag between the energy transferred into non-thermal ion energy ${\dot{E}}_{i}^{(fp)}$ and the thermalization of that ion energy is approximately $\unicode[STIX]{x0394}t=3.6T_{0}$ , a factor of $\sqrt{m_{i}/m_{e}}=6$ longer, suggesting that the phase mixing occurs more slowly for ions by the ratio of the electron-to-ion thermal velocity. Note also that the collisionless field–particle energy transfer to ions indeed becomes negative at a few points in time, as allowed for a reversible process.

Furthermore, note that the magnitudes of ${\dot{E}}_{i}^{(fp)}$ and ${\dot{E}}_{e}^{(fp)}$ are fairly similar, as expected because the linear damping rates, shown in figure 1, are fairly similar for ions and electrons, $\unicode[STIX]{x1D6FE}_{i}\simeq \unicode[STIX]{x1D6FE}_{e}$ , over the range of spatial scales $k_{\bot }\unicode[STIX]{x1D70C}_{i}<1$ that contain most of the energy in the simulation. Finally, in figure 5(c), we see that the energy lost by the turbulence $-{\dot{E}}^{(\text{turb})}$ (purple solid) is indeed balanced by the sum of the field–particle energy transfer to ions and electrons (black dashed).

3.5 Evolution of the total energy budget

Plots of the total energy budget as a function of time in the simulation nicely summarize the flow of energy in the simulation. First, we account for the energy lost from $\unicode[STIX]{x1D6FF}W$ in the AstroGK simulation to collisional plasma heating by accumulating the thermalized energy in each species over time, $E_{s}^{(\text{coll})}(t)=\int _{0}^{t}\text{d}t^{\prime }Q_{S}(t^{\prime })$ .

Figure 6. (a) The energy budget of the simulation versus time, showing the turbulent energy $E^{(\text{turb})}$ , non-thermal ion energy $E_{i}^{(nt)}$ , non-thermal electron energy $E_{e}^{(nt)}$ , ion heat $E_{i}^{(\text{coll})}$ and electron heat $E_{e}^{(\text{coll})}$ . (b) The same energy budget decomposed according to (3.1), showing the perpendicular magnetic field energy $E_{B_{\bot }}$ , parallel magnetic field energy $E_{B_{\Vert }}$ (cyan, not labelled, appearing between $E_{B_{\bot }}$ and $E_{i}^{(\unicode[STIX]{x1D6FF}f)}$ ), total fluctuating ion kinetic energy $E_{i}^{(\unicode[STIX]{x1D6FF}f)}$ , total fluctuating electron kinetic energy $E_{e}^{(\unicode[STIX]{x1D6FF}f)}$ , ion heat $E_{i}^{(\text{coll})}$ and electron heat $E_{e}^{(\text{coll})}$ . The total fluctuating energy $\unicode[STIX]{x1D6FF}W$ is shown in both panels (thick black line).

In figure 6(a), we plot the evolution of the energy budget over the course of the simulation, showing that turbulent energy $E^{(\text{turb})}$ , which dominates at the beginning of the simulation, is largely converted to ion heat $E_{i}^{(\text{coll})}$ and electron heat $E_{e}^{(\text{coll})}$ by the end of the simulation, with a smaller fraction of the lost turbulent energy persisting as non-thermal ion energy $E_{i}^{(nt)}$ and electron energy $E_{e}^{(nt)}$ . Also indicated in figure 6(a) is the evolution of the total fluctuating energy $\unicode[STIX]{x1D6FF}W$ (thick black line), showing that 60 % of this energy has been lost to plasma heating over 7.5 periods of the initial Alfvén waves. Another interesting point is that, although electrons are heated twice as much as ions, the non-thermal electron energy content of the simulation always remains very small. This point is consistent with the idea, introduced in § 3.4 above, that non-thermal energy transferred into the electron velocity distribution function by collisionless damping of the turbulence is very rapidly thermalized into electron heat. This analysis of the evolution of the total energy budget shows that energy is conserved to within 0.1 % over the course of the simulation.

One can alternatively divide the contributions to the energy budget in terms of (3.1), as shown in figure 6(b), showing the perpendicular magnetic field energy $E_{B_{\bot }}$ (green), the parallel magnetic field energy $E_{B_{\Vert }}$ (cyan), the total fluctuating ion kinetic energy $E_{i}^{(\unicode[STIX]{x1D6FF}f)}$ (red) and the total fluctuating electron kinetic energy $E_{e}^{(\unicode[STIX]{x1D6FF}f)}$ (blue). Note that, as anticipated from the contributions to the turbulent energy in figure 2(b), the turbulent energy in figure 6(a) is largely composed of perpendicular magnetic energy $E_{B_{\bot }}$ and kinetic energy of the perpendicular ion bulk flows $E_{i,u_{\bot }}$ . The wiggly boundary between $E_{B_{\bot }}$ and $E_{i,u_{\bot }}$ is a consequence of the Alfvénic fluctuations, and their nonlinear interactions, in the simulation.

One final point is that, although one may choose to decompose the different contributions to the energy using (3.1) in figure 6(b), by organizing the energies instead according to the turbulent energy $E^{(\text{turb})}=\int \text{d}^{3}\boldsymbol{r}[(|\unicode[STIX]{x1D6FF}\boldsymbol{B}|^{2}+|\unicode[STIX]{x1D6FF}\boldsymbol{E}|^{2})/8\unicode[STIX]{x03C0}+\sum _{s}(1/2)n_{0s}m_{s}|\unicode[STIX]{x1D6FF}\boldsymbol{u}_{\boldsymbol{s}}|^{2}]$ and the species non-thermal energies $E_{s}^{(nt)}$ , the interpretation of the energy flow is much more physically motivated, as illustrated by figure 4. By simply plotting $E_{i}^{(\unicode[STIX]{x1D6FF}f)}$ as a function of time, one does not see the important split between the large fraction of the total fluctuating ion kinetic energy $E_{i}^{(\unicode[STIX]{x1D6FF}f)}$ that is associated with turbulent fluctuations of the bulk ion velocity and the remainder that corresponds to the non-thermal energy that is not associated with those turbulent fluctuations.

4.1 Spatial distribution of parallel electromagnetic work, $j_{\Vert }E_{\Vert }$

As shown in § 3, over the full time of the simulation, $7.5T_{0}$ , 60 % of the fluctuating energy $\unicode[STIX]{x1D6FF}W$ of the initial Alfvén waves is removed from the fluctuations in the plasma. Figure 3 shows that this energy is ultimately irreversibly converted into electron and ion heat through the weak but finite collisionality in the plasma. As the model of energy flow illustrated in figure 4 shows, this energy is initially removed from the turbulent electromagnetic fluctuations (Howes Reference Howes2015, Reference Howes2017) through collisionless interactions between the electromagnetic fields and the individual plasma particles. In a kinetic plasma, the rate of electromagnetic work done on the particles by the fields is given by $\text{d}W/\text{d}t=\int \text{d}^{3}\boldsymbol{r}\boldsymbol{j}\boldsymbol{\cdot }\boldsymbol{E}$ (Howes, Klein & Li Reference Howes, Klein and Li2017; Klein Reference Klein2017). Therefore, plotting the rate of electromagnetic work $\boldsymbol{j}\boldsymbol{\cdot }\boldsymbol{E}$ as a function of position provides useful insights into the particle energization in the plasma.

Figure 8. Plots of $j_{\Vert }E_{\Vert }$ (colour bar) and contours of the parallel vector potential $A_{\Vert }$ (contours, positive solid, negative dashed) at times $t/T_{0}=$  (a) 1.75, (b) 1.86, and (c) 2.03, as well as (d) $\langle j_{\Vert }E_{\Vert }\rangle _{\unicode[STIX]{x1D70F}}$ , the rate of electromagnetic work per unit volume averaged over approximately one full wave period, $\unicode[STIX]{x1D70F}=0.992T_{0}$ , centred at time $t/T_{0}=1.86$ .

As shown in appendix B, in this simulation the dominant electromagnetic work is done by the component of the electric field parallel to the magnetic field, $E_{\Vert }$ , so in figure 8 we plot the instantaneous value of dimensionless rate of work per unit volume $j_{\Vert }E_{\Vert }/Q_{0}$ as a function of position in the plane $z/L_{\Vert }=-0.25$ at three different times during the simulation $t/T_{0}=$  (a) 1.75, (b) 1.86 and (c) 2.03. Note that the value of $j_{\Vert }E_{\Vert }$ is physically interpreted as the rate of transfer of spatial energy density between the parallel electric field $E_{\Vert }$ and the plasma ions and electrons. Since this electromagnetic work is reversible, its value can be positive or negative, where positive means work done on the particles by the field, and negative means work done on the field by the particles.

As emphasized in Howes et al. (Reference Howes, Klein and Li2017), the instantaneous energy transfer between fields and particles has two components: (i) an oscillating energy transfer back and forth between fields and particles that is typical of undamped linear wave motion in a kinetic plasma, and (ii) a secular energy transfer that represents the energy lost from the electromagnetic fluctuations to the plasma particles through collisionless damping. To determine the particle energization, it is the secular energy transfer that is of interest, but the challenge is that the oscillating energy transfer often has a much larger amplitude than the secular energy transfer. However, if a time average is taken over a suitably chosen averaging interval, the oscillating energy transfer will largely cancel out, exposing the smaller secular energy transfer that is sought. In this strong Alfvén wave collision simulation, the linear period $T_{0}$ of the initial Alfvén waves is an appropriate choice for this time averaging, and we plot in figure 8(d) the time average of $\langle j_{\Vert }E_{\Vert }\rangle _{\unicode[STIX]{x1D70F}}$ over an interval $\unicode[STIX]{x1D70F}=0.992T_{0}$ centred at time $t/T_{0}=1.86$ .

The plots shown in figure 8 convey a number of valuable insights into the particle energization in this simulation. First, the plots in figure 8(a–c) show clearly that the instantaneous rate of energy transfer is both spatially and temporally non-uniform, with the energy transfer localized in sheet-like structures reminiscent of the current sheets plotted in figure 7. An example of the temporal variation is illustrated by observing the changes in the instantaneous energy transfer rate at point A marked on each plot. At (a) $t/T_{0}=1.75$ , the plasma is energized by $E_{\Vert }$ , but later at (b) $t/T_{0}=1.86$ the plasma is losing energy to $E_{\Vert }$ and finally at (c) $t/T_{0}=2.03$ there is very little energy transfer either direction. Averaged over one period, figure 8(d) shows that the plasma gains energy from the parallel electric field at point A. Curiously, the instantaneous energy transfer from fields to particles at $t/T_{0}=1.86$ in figure 8(b) is negative at point A, but the single-period average, centred at that same time $t/T_{0}=1.86$ in figure 8(d) shows a positive transfer of energy to the particles at the same position. This plot stresses the importance of appropriate time averaging to properly understand the net particle energization in a turbulent plasma.

Figure 9. Plots of $\langle j_{\Vert }E_{\Vert }\rangle _{\unicode[STIX]{x1D70F}}$ averaged over approximately one full wave period, $\unicode[STIX]{x1D70F}=0.992T_{0}$ , centred at times $t/T_{0}=$  (a) 1.75, (b) 2.03 and (c) 2.85 and (d) 3.84, showing that the qualitative spatial pattern of time-averaged particle energization is remarkably static over the evolution of the simulation.

Second, the net plasma energization over one period in figure 8(d) is also spatially localized, with plasma energization at point A, a net loss of energy at point B and little energy change at point C. It is also worthwhile pointing out that the magnitude of the time-averaged energy transfer is smaller in magnitude than the instantaneous energy transfer, as expected if some fraction of this energy transfer is oscillatory and largely cancels out when averaged over one period $T_{0}$ . Together, the four panels demonstrate the key point that that the particle energization is spatially non-uniform, both instantaneously as well as when averaged over one period $T_{0}$ of the initial Alfvén waves.

Third, something that cannot be appreciated by the single time slice in figure 8(d), is the surprising result that the single-period-averaged plasma energization has very little temporal variation as the centre of the time-average window is advanced over one period. In figure 9, we present $\langle j_{\Vert }E_{\Vert }\rangle _{\unicode[STIX]{x1D70F}}$ time averaged over an interval $\unicode[STIX]{x1D70F}=0.992T_{0}$ centred at times $t/T_{0}=$  (a) 1.75, (b) 2.03 and (c) 2.85 and (d) 3.84. Although the instantaneous spatial distribution of $j_{\Vert }E_{\Vert }$ changes significantly over the wave period, for example shown at $t/T_{0}=$  (a) 1.75, (b) 1.86 and (c) 2.03 in figure 8, the corresponding single-period-averaged plasma energization $\langle j_{\Vert }E_{\Vert }\rangle _{\unicode[STIX]{x1D70F}}$ centred at the same times changes little, shown in figure 9(a) at $t/T_{0}=1.75$ , figure 8(d) at $t/T_{0}=1.86$ and figure 9(b) at $t/T_{0}=2.03$ . In fact, one observes only a very slow evolution of this particle energization pattern over a number of periods, as seen in figure 8(d) at $t/T_{0}=1.86$ , in figure 9(c) at $t/T_{0}=2.85$ and in figure 9(d) at $t/T_{0}=3.84$ . The very small variations are in large part due to the long-term evolution which involves the inevitable accumulated loss of fluctuating energy $\unicode[STIX]{x1D6FF}W$ over the course of the simulation.

A final point is that the plasma energization – the sum of the energy transfer to both ions and electrons – has a net positive value when integrated over the entire simulation domain, as demonstrated by the sum ${\dot{E}}_{i}^{(fp)}+{\dot{E}}_{e}^{(fp)}$ plotted in figure 5(c). Therefore, although there is a loss of plasma energy in some regions of the domain, the net effect is that plasma species gain energy at the expense of the turbulent electromagnetic field and bulk plasma flow fluctuations, as depicted in the energy flow diagram in figure 4.

Figure 10. Comparison of time averaged $\langle j_{\Vert }E_{\Vert }\rangle _{\unicode[STIX]{x1D70F}}$ over an interval $\unicode[STIX]{x1D70F}=0.992T_{0}$ centred at time $t/T_{0}=1.86$ for both (a) a nonlinear run and (b) a linear run, starting from identical initial conditions, showing a much more spatially localized distribution of plasma energization in the nonlinear case.

Further insight into the effect of the nonlinear evolution on the resulting plasma energization can be gained by comparing a linear simulation starting from identical initial conditions. The AstroGK code has the capability of simply turning off the nonlinear term in the gyrokinetic equation (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006; Numata et al. Reference Numata, Howes, Tatsuno, Barnes and Dorland2010), enabling strictly linear evolution from identical conditions with the same simulation code. In the linear case, no energy is transferred to other Fourier modes, and all of the particle energization is due to linear collisionless damping via the Landau resonance. In figure 10, we plot the time averaged $\langle j_{\Vert }E_{\Vert }\rangle _{\unicode[STIX]{x1D70F}}$ over an interval $\unicode[STIX]{x1D70F}=0.992T_{0}$ centred at time $t/T_{0}=1.86$ for both (a) the nonlinear run and (b) the linear run. This figure directly demonstrates the striking fact that the spatial non-uniformity of particle energization arises due to the nonlinear transfer of energy to other Fourier modes. This is fully consistent with the picture of current sheet generation by constructive interference among the initial Alfvén wave modes and the nonlinearly generated fluctuations (Howes Reference Howes2016). In § 5, the field–particle correlation technique will be used to identify the nature of the collisionless energy transfer that yields this spatially non-uniform particle energization.

The rate of plasma energization is the sum of the rates of ion and electron energization, $j_{\Vert }E_{\Vert }=j_{\Vert i}E_{\Vert }+j_{\Vert e}E_{\Vert }$ , and, in appendix B, we plot in figures 18 and 19 the separate ion and electron energization contributing to figure 8. In this simulation, the single-period-averaged particle energization in the plane $z/L_{\Vert }=-0.25$ shown in figure 8 yields approximately twice the energy transfer to electrons relative to the ions at $t/T_{0}=1.86$ .

5.1 Timestack plots of field–particle correlations

Here we present the results of the field–particle technique applied at three points in the simulation domain, labelled A, B and C in figure 8. From figure 8(d), we see that, averaged over one period $\unicode[STIX]{x1D70F}/T_{0}=0.992$ centred at $t/T_{0}=1.86$ , there is a net gain of energy by the plasma at point A, a net loss of energy by the plasma at point B and little net change in the plasma energy at point C. Note that the reduced parallel correlation $C_{E_{\Vert },s}(v_{\Vert },t,\unicode[STIX]{x1D70F})$ in the plots presented in this section is normalized by the energy transfer rate per unit volume per unit velocity, $Q_{0}/v_{ti}$ .

Figure 11. Timestack plots of ion and electron energization at position A for a correlation interval $\unicode[STIX]{x1D70F}/T_{0}=0.992$ . (a) Velocity-space integrated correlation, giving the rate of ion energization $\unicode[STIX]{x2202}w_{i}/\unicode[STIX]{x2202}t$ due to ion interactions with $E_{\Vert }$ . (b) The reduced parallel field–particle correlation for the ions $C_{E_{\Vert },i}(v_{\Vert },t,\unicode[STIX]{x1D70F})$ . (c) Velocity-space integrated correlation, giving the rate of electron energization $\unicode[STIX]{x2202}w_{e}/\unicode[STIX]{x2202}t$ due to electron interactions with $E_{\Vert }$ . (d) The reduced parallel field–particle correlation for the electrons $C_{E_{\Vert },e}(v_{\Vert },t,\unicode[STIX]{x1D70F})$ . Vertical solid black indicate resonant velocities for a parallel phase velocity at the Alfvén speed $\unicode[STIX]{x1D714}/(k_{\Vert }v_{ts})=v_{A}/v_{ts}.$ Vertical dashed lines indicate the highest parallel phase velocities for modes with significant collisionless damping in the simulation.

In figure 11(b), we present a timestack plot of the reduced parallel field–particle correlation for the ions $C_{E_{\Vert },i}(v_{\Vert },t,\unicode[STIX]{x1D70F})$ at position A with a correlation interval $\unicode[STIX]{x1D70F}/T_{0}=0.992$ , showing the distribution of the energy transfer to the ions as a function of the parallel velocity $v_{\Vert }/v_{ti}$ versus normalized time $t/T_{0}$ . Vertical solid and dashed black lines indicate the limits of resonant parallel phase velocities from figure 1, $1.0\lesssim |\unicode[STIX]{x1D714}/k_{\Vert }v_{ti}|\lesssim 1.5$ for ions; there are both positive and negative ranges of parallel phase velocities, corresponding to Alfvén waves travelling up or down the mean magnetic field. Also plotted in figure 11(a) is the velocity-space-integrated correlation, $\unicode[STIX]{x2202}w_{i}/\unicode[STIX]{x2202}t=\int \text{d}v_{\Vert }C_{E_{\Vert }i}(v_{\Vert },t,\unicode[STIX]{x1D70F})$ , equivalent to the parallel ion contribution of the electromagnetic work $j_{\Vert i}E_{\Vert }$ at position A, showing a net energization of the ions over the course of the simulation.

The distribution of the energy transfer as a function of $v_{\Vert }/v_{ti}$ in figure 11(b) is the velocity-space signature of the energy transfer mechanism. The localization of the energy transfer in the marked range of resonant parallel phase velocities for kinetic Alfvén waves clearly indicates that the energy transfer is resonant. The specific distribution of this energy transfer, with a transfer of energy from $E_{\Vert }$ to the ions (red) at $|v_{\Vert }/v_{ti}|>|v_{\text{res}}/v_{ti}|$ and a loss of energy from the ions (blue) at $|v_{\Vert }/v_{ti}|<|v_{\text{res}}/v_{ti}|$ is the characteristic signature of the Landau damping of kinetic Alfvén waves (Howes Reference Howes2017; Klein et al. Reference Klein, Howes and TenBarge2017). The change of sign in the energy transfer occurs at the resonant phase velocity for the collisionlessly damped wave. Here the change of sign occurs at $v_{\Vert }/v_{ti}=\unicode[STIX]{x1D714}/k_{\Vert }v_{ti}\simeq v_{A}/v_{ti}=\pm 1$ , indicating that larger-scale Alfvén waves with $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ , which have a parallel phase velocity $\unicode[STIX]{x1D714}/k_{\Vert }=v_{A}$ , appear to dominate the energy transfer at point A. This is consistent with the fact the energy in the electromagnetic and plasma bulk flow fluctuations in this simulation is dominated by the low $k_{\bot }\unicode[STIX]{x1D70C}_{i}\ll 1$ modes. This novel field–particle correlation analysis shows definitively the key result that ion Landau damping contributes to the energization of ions at position A in this strong Alfvén wave collision simulation.

In figure 11(d), we plot the reduced parallel field–particle correlation for the electrons $C_{E_{\Vert },e}(v_{\Vert },t,\unicode[STIX]{x1D70F})$ at position A with the same correlation interval $\unicode[STIX]{x1D70F}/T_{0}=0.992$ , where vertical solid and dashed black lines indicate the range of resonant parallel velocities from figure 1, $0.17\lesssim \unicode[STIX]{x1D714}/k_{\Vert }v_{te}\lesssim 0.6$ for electrons. Also plotted in figure 11(c) is the velocity-space-integrated correlation, $\unicode[STIX]{x2202}w_{e}/\unicode[STIX]{x2202}t=\int \text{d}v_{\Vert }C_{E_{\Vert }e}(v_{\Vert },t,\unicode[STIX]{x1D70F})$ , equivalent to the parallel electron contribution of the electromagnetic work $j_{\Vert e}E_{\Vert }$ , showing a net energization of the electrons at position A.

The velocity-space signature of the electron energization in figure 11(d) also shows the typical characteristics of electron Landau damping, with a slight difference from the ion case. Because kinetic Alfvén waves are dispersive, with a parallel phase velocity that increases for $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gtrsim 1$ given approximately by $\unicode[STIX]{x1D714}=k_{\Vert }v_{A}\sqrt{1+(k_{\bot }\unicode[STIX]{x1D70C}_{i})^{2}/[\unicode[STIX]{x1D6FD}_{i}+2/(1+T_{e}/T_{i})]}$ (Howes, Klein & TenBarge Reference Howes, Klein and TenBarge2014), the parallel resonant velocity will increase for kinetic Alfvén waves with larger $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ . The velocity-space signature of linear Landau damping typically shows the change of sign of the energy transfer at the resonant velocity. In figure 11(d), that change of sign for $1\leqslant t/T_{0}\leqslant 3$ occurs at a resonant velocity slightly larger than $v_{A}/v_{te}$ (vertical black line), suggesting that the kinetic Alfvén wave involved in the electron Landau damping has a value of $k_{\bot }\unicode[STIX]{x1D70C}_{i}\gtrsim 1$ leading to a higher resonant parallel velocity. Despite this minor detail, the energy transfer still shows that the electron energization is mediated by resonant electrons, with a velocity-space signature typical of electron Landau damping (Howes Reference Howes2017). Therefore, this analysis definitively yields a second key result, that electron Landau damping contributes to the energization of electrons at position A in this strong Alfvén wave collision simulation.

Figure 12. Plots of the same field–particle correlation analysis as figure 11, but taken at point B.

We can also investigate the regions in the simulation domain where the plasma loses energy to the parallel electric field at point B, plotted in figure 12. The (a) velocity-integrated ion energization $\unicode[STIX]{x2202}w_{i}/\unicode[STIX]{x2202}t$ due to $E_{\Vert }$ and (c) velocity-integrated electron energization $\unicode[STIX]{x2202}w_{e}/\unicode[STIX]{x2202}t$ due to $E_{\Vert }$ both show that the net energy transfer to ions and electrons at point B is negative. Here, as in figure 11, we see that the energy transfer for both ions in (b) and electrons in (d) is dominated by particles with velocities that fall within the range of parallel velocities expected to be resonant with the parallel phase velocity of Alfvén waves, demonstrating directly that energy transfer between the particles and the parallel electric field $E_{\Vert }$ is governed by Landau resonant interactions.

Figure 13. Plots of the same field–particle correlation analysis as figure 11, but taken at point C.

At point C in the simulation domain, there is very little net energy transfer from the parallel electric field to the plasma particles. The same field–particle correlation analysis at point C, presented in figure 13, shows that the velocity-integrated (a) ion energization $\unicode[STIX]{x2202}w_{i}/\unicode[STIX]{x2202}t$ and (c) electron energization $\unicode[STIX]{x2202}w_{e}/\unicode[STIX]{x2202}t$ due to $E_{\Vert }$ yield a very small positive transfer of energy to the particles over the first couple of periods $T_{0}$ , with an amplitude approximately an order of magnitude smaller than the energy transfer at points A and B. The reduced parallel field–particle correlation $C_{E_{\Vert },s}(v_{\Vert },t,\unicode[STIX]{x1D70F})$ for (b) the ions and (d) the electrons shows that the majority of this very small amount of energy transfer is still dominated by resonant particles.

But there is a very significant difference between the reduced parallel field–particle correlation $C_{E_{\Vert },s}$ at point C for both ions and electrons compared to the same correlation at points A and B: the pattern of energy transfer at point C is dominantly odd in $v_{\Vert }$ , whereas the patterns at points A and B are dominantly even in $v_{\Vert }$ . When integrated over the parallel velocity to obtain the net change of energy of a species, an odd pattern largely cancels out, whereas an even pattern does not. Therefore, there is little net particle energization at point C, even though individual particles do gain and lose energy through resonant interactions with $E_{\Vert }$ . Particles with $v_{\Vert }>0$ gain nearly the same amount of energy as that lost by particles with $v_{\Vert }<0$ , yielding little net change of particle energy.

The important point that the field–particle correlation analysis here demonstrates is that collisionless interactions of the Landau resonance between $E_{\Vert }$ and the ions and electrons contribute to the spatially non-uniform pattern of time-averaged particle energization, shown in figure 8(d). This result disproves by counterexample the commonly stated belief that Landau damping can only lead to spatially uniform particle energization. Rather, we see clearly here that collisionless damping via the Landau resonance can indeed be responsible for spatially localized particle energization, as previously suggested in the literature (TenBarge & Howes Reference TenBarge and Howes2013; Howes Reference Howes2015, Reference Howes2016). Furthermore, the nonlinear energy transfer by collisionless damping via the Landau resonance is not inhibited by the strong nonlinear interactions that play an important role in this strong Alfvén wave collision simulation. Indeed, nonlinear gyrokinetic simulations of strong, broadband plasma turbulence have indeed shown that the collisionless transfer of energy between fields and ions is dominated by particles approximately in Landau resonance with the parallel phase velocity of Alfvénic fluctuations (Klein et al. Reference Klein, Howes and TenBarge2017).

5.2 Particle energization in gyrotropic velocity space

Finally, we can examine the distribution of particle energization in gyrotropic velocity space $(v_{\Vert },v_{\bot })$ (Howes Reference Howes2017) using the field–particle correlation $C_{E_{\Vert },s}(v_{\Vert },v_{\bot },t,\unicode[STIX]{x1D70F})$ . Although plots of this analysis are limited to the correlation centred at just a single point in time, by not integrating over perpendicular velocity $v_{\bot }$ one obtains complete information about which particles in gyrotropic velocity space $(v_{\Vert },v_{\bot })$ participate in the collisionless transfer of energy. Note that the parallel correlation in gyrotropic velocity space, $C_{E_{\Vert },s}(v_{\Vert },v_{\bot },t,\unicode[STIX]{x1D70F})$ in the plots presented in this section is normalized by the energy transfer rate per unit volume per unit velocity squared, $Q_{0}/v_{ti}^{2}$ .

In figure 14, we plot $C_{E_{\Vert },s}(v_{\Vert },v_{\bot },t,\unicode[STIX]{x1D70F})$ for the same correlation interval $\unicode[STIX]{x1D70F}/T_{0}=0.992$ centred at time $t/T_{0}=2.10$ : (a) ion and (b) electron energization at point A, (c) ion and (d) electron energization at point B and (e) ion and (f) electron energization at point C. As before, vertical solid and dashed black lines denote the range of resonant parallel velocities for Alfvén waves. Three important points can be inferred from these gyrotropic velocity-space plots. First, other than a steady decrease of the amplitude of the signal with increasing $v_{\bot }$ – as expected because the equilibrium Maxwellian distribution drops off exponentially as $\exp (-v^{2}/v_{ts}^{2})$ , so the amplitude of fluctuations $\unicode[STIX]{x1D6FF}f(\boldsymbol{v})$ would be expected to have a similar drop off in amplitude – the energy transfer shows very little variation with $v_{\bot }$ . The variation in the energy transfer is organized almost entirely by $v_{\Vert }$ , as expected for a Landau resonant energy transfer process. Second, this energy transfer is dominated by particles with parallel velocities resonant with Alfvénic fluctuations, $v_{\Vert }\simeq v_{A}$ , demonstrating that the energy transfer is governed by the Landau resonance. Third, the odd or even character in $v_{\Vert }$ at the different points A, B and C seen in the timestack plots is also clearly apparent here in these gyrotropic velocity-space plots.

Figure 14. Plots of the field–particle correlation $C_{E_{\Vert }}(v_{\Vert },v_{\bot },t,\unicode[STIX]{x1D70F})$ on gyrotropic velocity space $(v_{\Vert },v_{\bot })$ for a correlation interval $\unicode[STIX]{x1D70F}/T_{0}=0.992$ centred at time $t/T_{0}=2.10$ : (a) ion and (b) electron energization at point A, (c) ion and (d) electron energization at point B and (e) ion and (f) electron energization at point C. Vertical solid lines denote the resonant parallel velocities for a parallel phase velocity at the Alfvén speed $\unicode[STIX]{x1D714}/(k_{\Vert }v_{ts})=v_{A}/v_{ts}$ .

Summarizing, the gyrotropic velocity space $(v_{\Vert },v_{\bot })$ plots in figure 14 demonstrate how the field–particle correlation technique maximizes the use of the full particle velocity distribution function information, enabling the physical mechanism responsible for the removal of energy from turbulent fluctuations and consequent particle energization to be identified definitively. In this case, the velocity-space signature of the field–particle correlation is unmistakably that of Landau damping of a kinetic Alfvén wave (Howes Reference Howes2017), proving that Landau damping indeed plays a role in the spatially non-uniform removal of the energy of electromagnetic and bulk plasma flow fluctuations, even in the presence of strong nonlinearity.