2.1 Rate of energy transfer between fields and particles in a collisionless plasma

Dissipation in weakly collisional plasmas consists of a two-step process (Howes Reference Howes2017). First, energy is transferred from the electromagnetic fields to plasma particles, appearing as non-thermal energy in the particle velocity distributions. This collisionless process is reversible. Then, the non-thermal energy is cascaded in phase space, generating fine structures in the velocity space that are eventually smoothed out as thermal energy in the particle distributions via the action of collisions, and hence realizing irreversible heating of the plasma (Howes Reference Howes2008; Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Plunk, Quataert and Tatsuno2008, Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009). The energy transfer directly measured by the field–particle correlation technique is the net energy transferred between the electromagnetic fields and plasma particles in the first step.

The theory describing the energy transfer between fields and plasma particles, which is directly measured in field–particle correlations, was first derived for a 1D–1V Vlasov–Poisson plasma (Klein & Howes Reference Klein and Howes2016; Howes et al. Reference Howes, Klein and Li2017), and subsequently for a 3D–3V electromagnetic plasma (Klein et al. Reference Klein, Howes and TenBarge2017). Here we briefly summarize the key points of the theory in 3D–3V, using insights from the 1D–1V model. Consider the Boltzmann’s equation for species $s(=i,e)$ in a proton–electron plasma

(2.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}t}+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}f_{s}+\frac{q_{s}}{m_{s}}\left(\boldsymbol{E}+\frac{\boldsymbol{v}\times \boldsymbol{B}}{c}\right)\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\boldsymbol{v}}=\left(\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}t}\right)_{c}.\end{eqnarray}$$

Focusing on the net collisionless energy transfer in the first step of dissipation, we neglect the collision term, $(\unicode[STIX]{x2202}f_{s}/\unicode[STIX]{x2202}t)_{c}$ , on the right. We then multiply the whole equation by $m_{s}v^{2}/2$ to obtain an equation for the rate of change of phase-space energy density, $w_{s}\equiv (m_{s}v^{2}/2)f_{s}$ ,

(2.2) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}w_{s}}{\unicode[STIX]{x2202}t}=-\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}w_{s}-q_{s}\frac{v^{2}}{2}\boldsymbol{E}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\boldsymbol{v}}-\frac{q_{s}}{c}\frac{v^{2}}{2}(\boldsymbol{v}\times \boldsymbol{B})\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\boldsymbol{v}}.\end{eqnarray}$$

When integrating over all space and velocity, $w_{s}$ gives the microscopic kinetic energy $W_{s}$

(2.3) $$\begin{eqnarray}W_{s}\equiv \int \text{d}^{3}\boldsymbol{x}\int \text{d}^{3}\boldsymbol{v}\frac{1}{2}m_{s}v^{2}f_{s}.\end{eqnarray}$$

When integrating (2.2) over all space, the first term on the right vanishes for periodic or infinite spatial boundaries. When integrating over velocity space, the magnetic force term involves $\boldsymbol{v}\boldsymbol{\cdot }(\boldsymbol{v}\times \boldsymbol{B})=0$ and therefore does not contribute to the change of the microscopic kinetic energy $W_{s}$ , as expected. The change of the microscopic kinetic energy, or equivalently, the energy transfer to species $s$ , comes from the electric field term. We can further separate the $E_{\Vert }$ and $\boldsymbol{E}_{\bot }$ contributions to the rate of change of the microscopic kinetic energy as

(2.4) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}W_{s}}{\unicode[STIX]{x2202}t}=-\frac{q_{s}}{2}\int \text{d}^{3}\boldsymbol{x}\int \,\text{d}^{3}\boldsymbol{v}\,v^{2}\left(E_{\Vert }\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}v_{\Vert }}+\boldsymbol{E}_{\bot }\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}\boldsymbol{v}_{\bot }}\right),\end{eqnarray}$$

where ‘ $\Vert$ ’ is with respect to the local magnetic field. The first and second terms on the right represent energy transfer mediated by $E_{\Vert }$ and $\boldsymbol{E}_{\bot }$ , respectively. Note that this rate of energy density transfer, when integrated over velocity, simply yields the rate the work done by the electric field, $\boldsymbol{J}_{s}\boldsymbol{\cdot }\boldsymbol{E}=J_{s\Vert }\boldsymbol{\cdot }E_{\Vert }+\boldsymbol{J}_{s\bot }\boldsymbol{\cdot }\boldsymbol{E}_{\bot }$ (Klein & Howes Reference Klein and Howes2016; Howes et al. Reference Howes, Klein and Li2017; Klein et al. Reference Klein, Howes and TenBarge2017).

2.2 Field–particle correlations and gyrokinetics

We can now construct the field–particle correlations based on (2.4). Without integrating over space or velocity, the transfer rate of phase-space energy density can be directly computed by correlating $E_{\Vert }$ or $\boldsymbol{E}_{\bot }$ with the corresponding velocity derivative of $f_{s}$ at a point $\boldsymbol{x}$ and time $t$ over a correlation interval $\unicode[STIX]{x1D70F}$ as

(2.5) $$\begin{eqnarray}C_{E_{\Vert }}(\boldsymbol{x},\boldsymbol{v},t,\unicode[STIX]{x1D70F})=C\left(-q_{s}\frac{v_{\Vert }^{2}}{2}\frac{\unicode[STIX]{x2202}f_{s}(\boldsymbol{x},\boldsymbol{v},t)}{\unicode[STIX]{x2202}v_{\Vert }},E_{\Vert }(\boldsymbol{x},t)\right)\end{eqnarray}$$

(2.6) $$\begin{eqnarray}C_{E_{\bot }}(\boldsymbol{x},\boldsymbol{v},t,\unicode[STIX]{x1D70F})=C\left(-q_{s}\frac{v_{\bot 1}^{2}}{2}\frac{\unicode[STIX]{x2202}f_{s}(\boldsymbol{x},\boldsymbol{v},t)}{\unicode[STIX]{x2202}v_{\bot 1}},E_{\bot 1}(\boldsymbol{x},t)\right)+C\left(\!-q_{s}\frac{v_{\bot 2}^{2}}{2}\frac{\unicode[STIX]{x2202}f_{s}(\boldsymbol{x},\boldsymbol{v},t)}{\unicode[STIX]{x2202}v_{\bot 2}},E_{\bot 2}(\boldsymbol{x},t)\!\right),\end{eqnarray}$$

where the unnormalized correlation on the right is defined as

(2.7) $$\begin{eqnarray}C(A,B)=\frac{1}{N}\mathop{\sum }_{j=i+1}^{i+N}A_{j}B_{j}\end{eqnarray}$$

for real quantities $A_{j}$ and $B_{j}$ measured at discrete times $t_{j}=j\unicode[STIX]{x0394}t$ over a correlation interval $\unicode[STIX]{x1D70F}=N\unicode[STIX]{x0394}t$ that starts from an initial time $t_{i+1}$ . The factor $v^{2}=v_{\Vert }^{2}+v_{\bot }^{2}$ in (2.4) is reduced to $v_{\Vert }^{2}$ in (2.5) upon integration $\int \text{d}\boldsymbol{v}_{\bot }$ as it can be separated from $\int \text{d}v_{\Vert }$ . The correlation interval $\unicode[STIX]{x1D70F}$ is an important parameter in the unnormalized correlation. When $\unicode[STIX]{x1D70F}$ is set to zero, the unnormalized correlation measures the instantaneous energy transfer at each time $t$ , which often contains a large oscillatory component due to undamped wave motions To measure the net secular or long-term energy transfer, $\unicode[STIX]{x1D70F}$ is chosen to be sufficiently long such that the oscillatory component in the instantaneous energy transfer, which is often large but does not contribute to the net energy transfer, can be averaged out. Normally, one wave period of the outer-scale mode is sufficient (Howes et al. Reference Howes, Klein and Li2017). The parallel correlation $C_{E_{\Vert }}(\boldsymbol{x},\boldsymbol{v},t,\unicode[STIX]{x1D70F})$ measures the net energy transfer rate mediated by the parallel electric field, and is therefore a suitable measure for Landau damping (Landau Reference Landau1946), or mechanisms like strong-guide-field magnetic reconnection that may be dominated by $E_{\Vert }$ (Dahlin, Drake & Swisdak Reference Dahlin, Drake and Swisdak2016); it, however, does not capture transit time damping (Barnes Reference Barnes1966; Quataert Reference Quataert1998) that arises from the change in the magnetic field strength. Similarly, the perpendicular correlation $C_{E_{\bot }}(\boldsymbol{x},\boldsymbol{v},t,\unicode[STIX]{x1D70F})$ captures the net transfer rate mediated by the perpendicular electric field, and is therefore suitable for determining cyclotron damping (Coleman Reference Coleman1968; Isenberg & Hollweg Reference Isenberg and Hollweg1983) or stochastic ion heating (Chen, Lin & White Reference Chen, Lin and White2001; Chandran et al. Reference Chandran, Li, Rogers, Quataert and Germaschewski2010). In this study using gyrokinetic simulations, we cannot explore perpendicular energization due to the conservation of the magnetic moment in gyrokinetic theory, so we focus here on the net energy transfer rate accomplished by $E_{\Vert }$ .

In gyrokinetics, the system evolves in a three-spatial and two-velocity dimension (3D–2V) phase space, where the two velocity coordinates are $v_{\Vert }$ and $v_{\bot }$ . The distribution function to the first order in gyrokinetics (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006) is given by

(2.8) $$\begin{eqnarray}f_{s}(v_{\Vert },v_{\bot })=\left(1-\frac{q_{s}\unicode[STIX]{x1D711}}{T_{0s}}\right)F_{0s}(v)+h_{s}(v_{\Vert },v_{\bot }),\end{eqnarray}$$

where $F_{0s}=(n_{0s}/\unicode[STIX]{x03C0}^{3/2}v_{ts}^{3})\exp (-v^{2}/v_{ts}^{2})$ is the equilibrium Maxwellian distribution, $q_{s}\unicode[STIX]{x1D711}/T_{0s}$ is the Boltzmann term ( $q_{s}$ the species charge and $\unicode[STIX]{x1D711}$ the electric potential), $h_{s}$ is the first-order gyroaveraged part of the perturbed distribution. When substituting $f_{s}(v_{\Vert },v_{\bot })$ into the first term in (2.4), we see that $\unicode[STIX]{x2202}F_{0s}(v)/\unicode[STIX]{x2202}v_{\Vert }$ is odd in $v_{\Vert }$ , and is multiplied by an even power $v_{\Vert }^{2}$ . Thus, this term vanishes upon integration over all $v_{\Vert }$ (again, $\int \text{d}\boldsymbol{v}_{\bot }$ can be separated from $\int \text{d}v_{\Vert }$ here). Hence, the equilibrium Maxwellian distribution and the Boltzmann term in (2.8) do not contribute to the net energy transfer in the field–particle correlation in (2.5). The perturbed, gyroaveraged contribution $h_{s}$ can be used in place of $f_{s}$ for the purpose of calculating field–particle correlations.

Another convenient transformation used is the complementary distribution function $g_{s}$ (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009)

(2.9) $$\begin{eqnarray}g_{s}(v_{\Vert },v_{\bot })=h_{s}(v_{\Vert },v_{\bot })-\frac{q_{s}F_{0s}}{T_{0s}}\left\langle \unicode[STIX]{x1D711}-\frac{\boldsymbol{v}_{\bot }\boldsymbol{\cdot }\boldsymbol{A}_{\bot }}{c}\right\rangle _{\boldsymbol{R}_{s}},\end{eqnarray}$$

where $\langle \cdots \!\,\rangle$ represents gyroaveraging at constant guiding centre $\boldsymbol{R}_{s}$ , capturing the perturbations to the Maxwellian distribution in the moving frame of Alfvén waves. It retains the parallel perturbations of $\unicode[STIX]{x1D6FF}f_{s}(\equiv f_{s}-F_{0s})$ in (2.8), and is therefore appropriate for calculating the net energy transfer. Note that the term $\langle \unicode[STIX]{x1D711}-\boldsymbol{v}_{\bot }\boldsymbol{\cdot }\boldsymbol{A}_{\bot }/c\rangle _{\boldsymbol{R}_{s}}$ is independent of $v_{\Vert }$ , and therefore yields zero upon integration $\int \text{d}v_{\Vert }$ when substituted into the first term in (2.4). Therefore, this term is the same, whether evaluated with $h_{s}$ or $g_{s}$ . As a result, the correlation $C_{E_{\Vert }}$ in (2.5), which is based on this term, is quantitatively similar using $h_{s}$ or $g_{s}$ , but $g_{s}$ contains less additional terms and can more clearly reveal structures near the resonant velocity in the distribution functions themselves.

Here we will use $g_{s}$ in the calculation of field–particle correlations, so the specific parallel field–particle correlation used to analyse the gyrokinetic turbulence simulations here takes the form

(2.10) $$\begin{eqnarray}C_{E_{\Vert }}(\boldsymbol{x},v_{\Vert },v_{\bot },t,\unicode[STIX]{x1D70F})=C\left(-q_{s}\frac{v_{\Vert }^{2}}{2}\frac{\unicode[STIX]{x2202}g_{s}(\boldsymbol{x},v_{\Vert },v_{\bot },t)}{\unicode[STIX]{x2202}v_{\Vert }},E_{\Vert }(\boldsymbol{x},t)\right),\end{eqnarray}$$

where $E_{\Vert }$ is gyroaveraged. For simplicity, we refer to this form of parallel correlation that depends on gyrotropic velocity space $(v_{\Vert },v_{\bot })$ as $C_{E_{\Vert }}(v_{\Vert },v_{\bot },t)$ . The parallel reduced correlation $C_{E_{\Vert }}(v_{\Vert },t)$ , defined by integrating $C_{E_{\Vert }}(v_{\Vert },v_{\bot },t)$ over all $v_{\bot }$ , is given by

(2.11) $$\begin{eqnarray}C_{E_{\Vert }}(v_{\Vert },t,\unicode[STIX]{x1D70F})=\int v_{\bot }\,\text{d}v_{\bot }C_{E_{\Vert }}(v_{\Vert },v_{\bot },t,\unicode[STIX]{x1D70F}).\end{eqnarray}$$

2.3 Field–particle correlations in Fourier space

To derive the appropriate form of the field–particle correlation in Fourier space, we first express the electric field at a single position $\boldsymbol{x}$ by a Fourier series

(2.12) $$\begin{eqnarray}\boldsymbol{E}(\boldsymbol{x})=\mathop{\sum }_{\boldsymbol{k}}\boldsymbol{E}_{\boldsymbol{k}}\text{e}^{\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}},\end{eqnarray}$$

summed over all Fourier modes $\boldsymbol{k}$ in a 3-D domain of volume $L^{3}$ . The particle velocity distribution function $f_{s}(\boldsymbol{x})$ is expressed similarly in terms of its Fourier coefficients $f_{s\boldsymbol{k}}$ . The product of two real quantities $A(\boldsymbol{x})$ and $B(\boldsymbol{x})$ integrated over a volume $L^{3}$ can be converted to a sum over the product of the Fourier coefficients

(2.13) $$\begin{eqnarray}\int \text{d}^{3}\boldsymbol{x}\,A(\boldsymbol{x})B(\boldsymbol{x})=L^{3}\mathop{\sum }_{\boldsymbol{k}}A_{\boldsymbol{k}}^{\ast }B_{\boldsymbol{ k}}=L^{3}\mathop{\sum }_{\boldsymbol{k}}A_{\boldsymbol{k}}B_{\boldsymbol{k}}^{\ast },\end{eqnarray}$$

where the complex Fourier coefficients $A_{\boldsymbol{k}}$ and $B_{\boldsymbol{k}}$ both satisfy the reality condition $A_{\boldsymbol{k}}=A_{-\boldsymbol{k}}^{\ast }$ . Substituting these Fourier series into (2.4) and using this relation, we obtain the rate of change of microscopic kinetic energy in terms of the Fourier coefficients of the electric field and particle distribution function,

(2.14) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}W_{s}}{\unicode[STIX]{x2202}t}=\mathop{\sum }_{\boldsymbol{k}}\left(-\frac{q_{s}}{2}\right)L^{3}\int \text{d}^{3}\boldsymbol{v}\,v^{2}\left(E_{\Vert \boldsymbol{k}}\frac{\unicode[STIX]{x2202}f_{s\boldsymbol{k}}^{\ast }}{\unicode[STIX]{x2202}v_{\Vert }}+\boldsymbol{E}_{\bot \boldsymbol{k}}\boldsymbol{\cdot }\frac{\unicode[STIX]{x2202}f_{s\boldsymbol{k}}^{\ast }}{\unicode[STIX]{x2202}\boldsymbol{v}_{\bot }}\right).\end{eqnarray}$$

Without summing over all Fourier modes, the contribution to the change of particle energy from each Fourier mode $\boldsymbol{k}$ is given by the summand, leading to the following forms of the field–particle correlations in Fourier space corresponding to (2.5) and (2.6):

(2.15) $$\begin{eqnarray}\displaystyle & \displaystyle C_{E_{\Vert }}(\boldsymbol{k},\boldsymbol{v},t,\unicode[STIX]{x1D70F})=C\left(-q_{s}\frac{v_{\Vert }^{2}}{2}\frac{\unicode[STIX]{x2202}f_{s\boldsymbol{k}}^{\ast }(\boldsymbol{v},t)}{\unicode[STIX]{x2202}v_{\Vert }},E_{\Vert \boldsymbol{k}}(t)\right) & \displaystyle\end{eqnarray}$$

(2.16) $$\begin{eqnarray}\displaystyle & \displaystyle C_{E_{\bot }}(\boldsymbol{k},\boldsymbol{v},t,\unicode[STIX]{x1D70F})=C\left(-q_{s}\frac{v_{\bot 1}^{2}}{2}\frac{\unicode[STIX]{x2202}f_{s\boldsymbol{k}}^{\ast }(\boldsymbol{v},t)}{\unicode[STIX]{x2202}v_{\bot 1}},E_{\bot 1\boldsymbol{k}}(t)\right)+C\left(-q_{s}\frac{v_{\bot 2}^{2}}{2}\frac{\unicode[STIX]{x2202}f_{s\boldsymbol{k}}^{\ast }(\boldsymbol{v},t)}{\unicode[STIX]{x2202}v_{\bot 2}},E_{\bot 2\boldsymbol{k}}(t)\right). & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The discussion on using $g_{s}$ in gyrokinetics in place of $f_{s}$ , which is based on velocity integrals, applies also to the correlations in Fourier space (with the gyroaveraging operation $\langle \cdots \!\,\rangle$ in (2.9) reduced to multiplications by Bessel functions (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006; Numata et al. Reference Numata, Howes, Tatsuno, Barnes and Dorland2010)). The parallel correlation in (2.10) then becomes

(2.17) $$\begin{eqnarray}C_{E_{\Vert }}(\boldsymbol{k},v_{\Vert },v_{\bot },t,\unicode[STIX]{x1D70F})=C\left(-q_{s}\frac{v_{\Vert }^{2}}{2}\frac{\unicode[STIX]{x2202}g_{s\boldsymbol{k}}^{\ast }(v_{\Vert },v_{\bot },t)}{\unicode[STIX]{x2202}v_{\Vert }},E_{\Vert \boldsymbol{k}}(t)\right).\end{eqnarray}$$

This represents the net energy transfer rate in Fourier and gyrotropic velocity space. Together with the parallel reduced correlation in (2.11), it is the form of field–particle correlations we use in this work.

2.4 Diagnosing energy transfer in Fourier space

Previous studies have used the field–particle correlation technique at a single point in physical space to evaluate the energy transfer between fields and particles (Klein & Howes Reference Klein and Howes2016; Howes Reference Howes2017; Howes et al. Reference Howes, Klein and Li2017, Reference Howes, McCubbin and Klein2018; Klein Reference Klein2017; Klein et al. Reference Klein, Howes and TenBarge2017; Chen et al. Reference Chen, Klein and Howes2019). Here we diagnose the particle energization in Fourier space, where each Fourier mode $\boldsymbol{k}$ is specified by a normalized wavevector $(k_{x}\unicode[STIX]{x1D70C}_{i},k_{y}\unicode[STIX]{x1D70C}_{i},k_{z}L_{z}/2\unicode[STIX]{x03C0})$ , where $L_{z}$ is the system size in $z$ . There are a number of advantages when applying field–particle correlations in Fourier space compared to physical space. First, by utilizing full spatial information of the whole domain as opposed to a single point, we can determine how the length scale of fluctuations, within the broadband turbulent spectrum, influences the collisionless field–particle energy transfer. Second, in the gyrokinetic limit $k_{\Vert }\ll k_{\bot }$ , well justified for the anisotropic turbulence observed in the solar wind at kinetic scales (Sahraoui et al. Reference Sahraoui, Goldstein, Belmont, Canu and Rezeau2010; Narita et al. Reference Narita, Gary, Saito, Glassmeier and Motschmann2011; Roberts, Li & Li Reference Roberts, Li and Li2013; Roberts, Li & Jeska Reference Roberts, Li and Jeska2015), the parallel phase velocity $v_{p\Vert }=\unicode[STIX]{x1D714}/k_{\Vert }$ of the linear wave modes depends only on the perpendicular wavenumber and the plasma parameters, $\overline{\unicode[STIX]{x1D714}}\equiv \unicode[STIX]{x1D714}/(k_{\Vert }v_{A})=v_{p\Vert }/v_{A}=\overline{\unicode[STIX]{x1D714}}(k_{\bot }\unicode[STIX]{x1D70C}_{i},\unicode[STIX]{x1D6FD}_{i},T_{i}/T_{e})$ (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006). Therefore, the parallel phase velocity that governs resonant collisionless interactions is well defined for each mode $\boldsymbol{k}$ in Fourier space. In contrast, at a single point in physical space, the dispersive nature of kinetic Alfvén waves (see figure 1) would lead to a range of resonant velocities, broadening the energy transfer signal and potentially smearing out any resonant energy transfer signatures in velocity space. Third, the broadband turbulent fluctuations have decreasing amplitudes with increasing perpendicular wavenumber $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , but in Fourier space one can isolate the fluctuations at higher $k_{\bot }$ from the rest of the fluctuations containing much larger amplitude fluctuations at lower $k_{\bot }$ .

Figure 1. The linear gyrokinetic dispersion relation for a $\unicode[STIX]{x1D6FD}_{i}=1$ and $T_{i}/T_{e}=1$ plasma, showing (a) the normalized real frequency or phase velocity $\unicode[STIX]{x1D714}/k_{\Vert }v_{A}$ (dashed black) and (b) damping rates $\unicode[STIX]{x1D6FE}/k_{\Vert }v_{A}$ for the electrons (green), ions (cyan) and the total damping rate (blue). The total shaded region represents the resolved dynamic range $0.25\leqslant k_{\bot }\unicode[STIX]{x1D70C}_{i}\leqslant 10.5$ . The inner yellow shaded region indicates the range of the sampled $k_{\bot }$ spectrum given in table 1, $1.4\leqslant k_{\bot }\unicode[STIX]{x1D70C}_{i}\leqslant 9.9$ .

In figure 1, we plot (a) the normalized frequency $\unicode[STIX]{x1D714}/k_{\Vert }v_{A}$ and (b) the normalized damping rate $\unicode[STIX]{x1D6FE}/k_{\Vert }v_{A}$ as a function of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ from the linear gyrokinetic dispersion relation (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006) for a plasma with parameters $\unicode[STIX]{x1D6FD}_{i}\equiv 8\unicode[STIX]{x03C0}n_{i}T_{i}/B^{2}=v_{ti}^{2}/v_{A}^{2}=1$ , $T_{i}/T_{e}=1$ , and $m_{i}/m_{e}=25$ . The total shaded region represents the resolved dynamic range of $k_{\bot }$ in the simulation, $0.25\leqslant k_{\bot }\unicode[STIX]{x1D70C}_{i}\leqslant 10.5$ . Table 1 lists the perpendicular wavevectors $(k_{x}\unicode[STIX]{x1D70C}_{i},k_{y}\unicode[STIX]{x1D70C}_{i})$ of Fourier modes that are sampled in the simulation, along with the corresponding normalized perpendicular wavenumber $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , where $k_{\bot }\equiv \sqrt{k_{x}^{2}+k_{y}^{2}}$ . The range of these sampled perpendicular wavenumbers $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ is indicated in figure 1 by the inner yellow shaded region, covering $1.4\leqslant k_{\bot }\unicode[STIX]{x1D70C}_{i}\leqslant 9.9$ or 0.3 ${\leqslant}k_{\bot }\unicode[STIX]{x1D70C}_{e}\leqslant$ 2.0. For each $(k_{x}\unicode[STIX]{x1D70C}_{i},k_{y}\unicode[STIX]{x1D70C}_{i})$ mode, seven $k_{z}L_{z}/2\unicode[STIX]{x03C0}\in$ [ $-3$ ,3] modes are selected, for a total sampling of 42 modes in $(k_{x}\unicode[STIX]{x1D70C}_{i},k_{y}\unicode[STIX]{x1D70C}_{i},k_{z}L_{z}/2\unicode[STIX]{x03C0})$ Fourier space. Also tabulated in table 1 are the parallel phase velocities normalized to the ion thermal velocity, $v_{p\Vert }/v_{ti}=\unicode[STIX]{x1D714}/k_{\Vert }v_{A}$ (valid for a $\unicode[STIX]{x1D6FD}_{i}=1$ plasma) and $v_{p\Vert }/v_{te}$ . The last column gives the linear wave period $\unicode[STIX]{x1D70F}_{k}$ of a kinetic Alfvén wave specified by $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ .

Table 1. List of diagnosed $(k_{x},k_{y})$ modes in Fourier space.

The dispersion relation plot in figure 1(a) shows the characteristic dispersion of kinetic Alfvén waves, where the parallel phase velocity begins increasing at the transition of $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ . Given the energy transfer from fields to particles due to resonant collisionless wave damping is governed by the parallel phase velocity, one would expect to observe that the region of velocity space in which the particles exchange energy with the fields, known as the velocity-space signature of the particle energization, will shift to higher parallel velocities, tracking the increased parallel phase velocity as the perpendicular wavenumber $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ increases. Our analysis in §§ 4.1 and 4.2 confirms this expectation.

Finally, the collisionless damping rates for ions (cyan) and electrons (green) and the total damping rate (blue) are plotted in figure 1(b). The total damping of the waves is primarily attributed to electron damping for $k_{\bot }\unicode[STIX]{x1D70C}_{i}>$ 2 while both ion and electron damping contribute for $k_{\bot }\unicode[STIX]{x1D70C}_{i}<$ 2. Particularly, ion collisionless damping becomes significant and peaks around $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim$ 1, but drops off very rapidly for $k_{\bot }\unicode[STIX]{x1D70C}_{i}>2$ .

2.5 Simulation set-up

The simulation was performed with the gyrokinetic code AstroGK (Numata et al. Reference Numata, Howes, Tatsuno, Barnes and Dorland2010). It has been extensively used to investigate turbulence in weakly collisional space plasmas (Howes et al. Reference Howes, Dorland, Cowley, Hammett, Quataert, Schekochihin and Tatsuno2008b ; Tatsuno et al. Reference Tatsuno, Schekochihin, Dorland, Plunk, Barnes, Cowley and Howes2009; Howes et al. Reference Howes, TenBarge, Dorland, Quataert, Schekochihin, Numata and Tatsuno2011; TenBarge & Howes Reference TenBarge and Howes2012; Nielson, Howes & Dorland Reference Nielson, Howes and Dorland2013; TenBarge & Howes Reference TenBarge and Howes2013; Kobayashi, Rogers & Numata Reference Kobayashi, Rogers and Numata2014; Howes Reference Howes2016; Li et al. Reference Li, Howes, Klein and TenBarge2016; Howes et al. Reference Howes, McCubbin and Klein2018) as well as collisionless magnetic reconnection in the strong-guide-field limit (Numata et al. Reference Numata, Dorland, Howes, Loureiro, Rogers and Tatsuno2011; TenBarge et al. Reference TenBarge, Daughton, Karimabadi, Howes and Dorland2014; Numata & Loureiro Reference Numata and Loureiro2015). AstroGK is an Eulerian continuum code with triply periodic boundary conditions. It has a slab geometry elongated along the straight, uniform background magnetic field, $\boldsymbol{B}_{0}=B_{0}\hat{\boldsymbol{z}}$ . The code evolves the perturbed gyroaveraged Vlasov–Maxwell equations in five-dimensional phase space (3D–2V) (Frieman & Chen Reference Frieman and Chen1982; Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006). The evolved quantities are the electromagnetic gyroaveraged complementary distribution function $g_{s}(x,y,z,\unicode[STIX]{x1D706},\unicode[STIX]{x1D700})$ for each species $s$ , the scalar potential $\unicode[STIX]{x1D711}$ , parallel vector potential $A_{\Vert }$ and the parallel magnetic field perturbation $\unicode[STIX]{x1D6FF}B_{\Vert }$ , where $\Vert$ is along the total local magnetic field $\boldsymbol{B}=B_{0}\hat{\boldsymbol{z}}+\unicode[STIX]{x1D6FF}\boldsymbol{B}$ . Note that the total and background magnetic fields are the same to first-order accuracy, which is retained for perturbed fields in gyrokinetics. The 2V velocity grid is specified by pitch angle $\unicode[STIX]{x1D706}=v_{\bot }^{2}/v^{2}$ and energy $\unicode[STIX]{x1D700}=v^{2}/2$ . The background distribution functions for both species are stationary uniform Maxwellians. Collisions are incorporated using a fully conservative, linearized gyroaveraged Landau collision operator consisting of energy diffusion and pitch-angle scattering between like particles, electrons and ions and inter-species scattering of electrons off ions (Abel et al. Reference Abel, Barnes, Cowley, Dorland and Schekochihin2008; Barnes et al. Reference Barnes, Abel, Dorland, Ernst, Hammett, Ricci, Rogers, Schekochihin and Tatsuno2009).

The same simulation set-up as Li et al. (Reference Li, Howes, Klein and TenBarge2016) is used, a 3-D generalization of the classic 2-D Orszag–Tang Vortex (OTV) problem (Orszag & Tang Reference Orszag and Tang1979). The 2-D problem was widely used in fluid and MHD turbulence simulations; various 3-D generalizations have also been used for studying turbulence (Dahlburg & Picone Reference Dahlburg and Picone1989; Politano, Pouquet & Sulem Reference Politano, Pouquet and Sulem1989; Picone & Dahlburg Reference Picone and Dahlburg1991; Politano, Pouquet & Sulem Reference Politano, Pouquet and Sulem1995; Grauer & Marliani Reference Grauer and Marliani2000; Mininni, Pouquet & Montgomery Reference Mininni, Pouquet and Montgomery2006; Parashar et al. Reference Parashar, Shay, Cassak and Matthaeus2009; Parashar, Vasquez & Markovskii Reference Parashar, Vasquez and Markovskii2014). This 3-D OTV set-up consists of counterpropagating Alfvén waves along $\boldsymbol{B}=B_{0}\hat{\boldsymbol{z}}$ such that on the $z=0$ plane, its initial condition reduces to that of the 2-D OTV problem. An initial amplitude $z_{0}$ of Elsässer variables in the OTV set-up (Li et al. Reference Li, Howes, Klein and TenBarge2016) is chosen to yield a nonlinearity parameter $\unicode[STIX]{x1D712}=k_{\bot }z_{0}/(k_{\Vert }v_{A})=1$ (where $v_{A}=B_{0}/\sqrt{4\unicode[STIX]{x03C0}m_{i}n_{0}}$ is a characteristic Alfvén speed), corresponding to critical balance (Goldreich & Sridhar Reference Goldreich and Sridhar1995), or a state of strong turbulence. Note that previous studies using AstroGK have shown consistency with the prediction of a critically balanced cascade in the dissipation range (TenBarge & Howes Reference TenBarge and Howes2012; TenBarge et al. Reference TenBarge, Howes and Dorland2013).

To resolve the turbulent cascade from the inertial range to the dissipation range for both ions and electrons, a reduced mass ratio of $m_{i}/m_{e}=25$ is used. The simulation domain is $L_{\bot }=8\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{i}$ and dimensions are $(n_{x},n_{y},n_{z},n_{\unicode[STIX]{x1D706}},n_{\unicode[STIX]{x1D700}},n_{s})=(128,128,32,64,32,2)$ , which resolves a dynamic range of $0.25\leqslant k_{\bot }\unicode[STIX]{x1D70C}_{i}\leqslant 10.5$ , or $0.05\leqslant k_{\bot }\unicode[STIX]{x1D70C}_{e}\leqslant 2.1$ , covering both the inertial and dissipation ranges of the system. Typical conditions of solar wind turbulence are considered with ion plasma beta $\unicode[STIX]{x1D6FD}_{i}=8\unicode[STIX]{x03C0}n_{i}T_{0i}/B_{0}^{2}=1$ , where $T_{0i}$ is the constant background ion temperature, and equal temperatures are used for ions and electrons ( $T_{0i}/T_{0e}=1$ ).

To realize collisional dissipation of turbulent energy in particle velocity space, weak finite collision frequencies are required (Howes Reference Howes2008; Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Plunk, Quataert and Tatsuno2008, Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009). Low collision frequencies $\unicode[STIX]{x1D708}_{s}/\unicode[STIX]{x1D714}_{A0}\ll 1$ are chosen to avoid altering the weakly collisional dynamics, where collision frequencies of $\unicode[STIX]{x1D708}_{i}=3\times 10^{-3}\unicode[STIX]{x1D714}_{A0}$ and $\unicode[STIX]{x1D708}_{e}=0.06\unicode[STIX]{x1D714}_{A0}$ (where $\unicode[STIX]{x1D714}_{A0}\equiv k_{\Vert }v_{A}$ is a characteristic Alfvén wave frequency) are used. We also employ a constant ion hypercollision frequency of $\unicode[STIX]{x1D708}_{Hi}=6\times 10^{-3}\unicode[STIX]{x1D714}_{A0}$ and adaptive electron hypercollision frequency of $\unicode[STIX]{x1D708}_{He}=0.12\unicode[STIX]{x1D714}_{A0}$ to ensure fluctuations in velocity space remain well resolved (Howes Reference Howes2008; Howes et al. Reference Howes, TenBarge, Dorland, Quataert, Schekochihin, Numata and Tatsuno2011). The electron hypercollisional coefficient is adjusted based on nonlinear estimations of the total collisional damping rate (including hypercollisions) $\unicode[STIX]{x1D6FE}_{nl}$ and of the energy transfer frequency $\unicode[STIX]{x1D714}_{nl}$ (that is given by the cascade model (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2008a ) and dependent on the magnitude of the magnetic field fluctuations) such that a value of $\unicode[STIX]{x1D6FE}_{nl}/\unicode[STIX]{x1D714}_{nl}\simeq 1/2\unicode[STIX]{x03C0}$ is achieved within some tolerance. The hypercollisionality chosen has the form of a pitch-angle-scattering operator with a wavenumber-dependent collision rate $\unicode[STIX]{x1D708}_{Hs}(k_{\bot }/k_{\bot \max })^{p_{s}}$ , where $k_{\bot \max }$ is a grid-scale wavenumber with $p_{i}=4$ and $p_{e}=8$ . It produces positive–definite heating close to the grid scale and is sufficient to terminate the turbulent cascade at the smallest resolved scales.

4.1 Field–particle correlations in gyrotropic velocity space

To identify the regions of gyrotropic velocity space $(v_{\Vert },v_{\bot })$ where particles play a role in the collisionless transfer of energy density from the parallel electric field to the plasma ions and electrons, we employ the field–particle correlation technique to compute $C_{E_{\Vert }}(\boldsymbol{k},v_{\Vert },v_{\bot },t,\unicode[STIX]{x1D70F})$ , given by (2.17).

Figure 5. Electron gyrotropic correlation $C_{E_{\Vert }}(v_{\Vert },v_{\bot },t)$ for the $(2,2,-1)$ Fourier mode using (a) $\unicode[STIX]{x1D70F}=0$ and (c) $\unicode[STIX]{x1D70F}=2\unicode[STIX]{x1D70F}_{k}$ , plotted at $t_{c}/\unicode[STIX]{x1D70F}_{0}=0.86$ , corresponding to $t/\unicode[STIX]{x1D70F}_{k}=0.81$ for $\unicode[STIX]{x1D70F}=2\unicode[STIX]{x1D70F}_{k}$ . Also plotted are $C_{E_{\Vert }}(v_{\Vert },v_{\bot },t)$ for the $(6,6,+3)$ Fourier mode using (b) $\unicode[STIX]{x1D70F}=0$ and (d) $\unicode[STIX]{x1D70F}=2\unicode[STIX]{x1D70F}_{k}$ at $t_{c}/\unicode[STIX]{x1D70F}_{0}=0.34$ , corresponding to $t/\unicode[STIX]{x1D70F}_{k}=0.42$ for $\unicode[STIX]{x1D70F}=2\unicode[STIX]{x1D70F}_{k}$ . Dashed lines denote Landau resonant velocities: $v_{p\Vert }/v_{te}=\pm 0.42$ for the $(2,2,-1)$ Fourier mode and $v_{p\Vert }/v_{te}=\pm 0.84$ for the $(6,6,+3)$ Fourier mode. Arbitrary units are used in the colour bars while the relative amplitudes between the $\unicode[STIX]{x1D70F}=0$ and $2\unicode[STIX]{x1D70F}_{k}$ correlations are preserved for each Fourier mode.

4.1.1 Electron gyrotropic correlations

Here we explore the net energy transfer rate to the electrons by the parallel electric field as a function of particle velocity in gyrotropic velocity space $(v_{\Vert },v_{\bot })$ for particular Fourier modes, each denoted by their normalized wavevector $(k_{x}\unicode[STIX]{x1D70C}_{i},k_{y}\unicode[STIX]{x1D70C}_{i},k_{z}L_{z}/2\unicode[STIX]{x03C0})$ . On the top row of figure 5, we plot the instantaneous correlation $C_{E_{\Vert }}(\boldsymbol{k},v_{\Vert },v_{\bot },t,\unicode[STIX]{x1D70F}=0)$ for modes (a) $(2,2,-1)$ and (b) $(6,6,+3)$ . The instantaneous energy transfer is spread over a wide range of $v_{\Vert }$ relative to $v_{p\Vert }$ of each Fourier mode of the kinetic Alfvén wave, given by (a) $\unicode[STIX]{x1D714}/(k_{\Vert }v_{te})=\pm 0.42$ and (b) $\unicode[STIX]{x1D714}/(k_{\Vert }v_{te})=\pm 0.84$ , indicated by the vertical dashed black lines. This shows that particles over a relatively broad range of $v_{\Vert }$ participate in energy exchange with the parallel electric field.

In order to eliminate the often large contribution to the instantaneous energy transfer given by the oscillating energy transfer associated with undamped wave motions (which is measured in a kinetic Alfvén wave (Gershman et al. Reference Gershman, F-Viñas, Dorelli, Boardsen, Avanov, Bellan, Schwartz, Lavraud, Coffey and Chandler2017)), the field–particle correlation technique performs a time average of the correlation product (in the unnormalized correlation) over a suitably long time period (Klein & Howes Reference Klein and Howes2016; Howes et al. Reference Howes, Klein and Li2017). This effectively removes the oscillating component of the energy transfer, isolating the often smaller net secular energy transfer from fields to particles. In both of these cases, we choose a correlation interval $\unicode[STIX]{x1D70F}=2\unicode[STIX]{x1D70F}_{k}$ where $\unicode[STIX]{x1D70F}_{k}$ is the linear wave period of a kinetic Alfvén wave specified by the normalized wavevector, given by the final column in table 1. The correlation interval is chosen such that the time evolution of the parallel reduced correlation qualitatively converges, which corresponds to $\unicode[STIX]{x1D70F}\geqslant \unicode[STIX]{x1D70F}_{k}$ (see Appendix). The resulting time-averaged field–particle correlations are plotted in figure 5 for Fourier modes (c) $(2,2,-1)$ and (d) $(6,6,+3)$ .

For the time-averaged correlations, time $t$ is defined at the beginning of the correlation interval $\unicode[STIX]{x1D70F}$ . Another convenient way to specify time is at the centre of the correlation interval, giving a centred time of $t_{c}=t+\unicode[STIX]{x1D70F}/2$ . Both $t$ and $t_{c}$ are given in figure 5 for reference.

Several key features of the net energy transfer over the correlation interval $\unicode[STIX]{x1D70F}$ are observed in figure 5(c,d). First, the region in velocity space of largest net transfer of energy between $E_{\Vert }$ and the electrons is closely connected to $v_{p\Vert }$ . This contrasts with the broader spread over $v_{\Vert }$ of instantaneous energy transfer when taking $\unicode[STIX]{x1D70F}=0$ in panels (a,b). The resulting velocity-space characteristic of the energy transfer is consistent with the velocity-space signature of Landau damping of the turbulent fluctuations, as found in previous studies (Klein & Howes Reference Klein and Howes2016; Klein et al. Reference Klein, Howes and TenBarge2017; Howes et al. Reference Howes, Klein and Li2017, Reference Howes, McCubbin and Klein2018; Chen et al. Reference Chen, Klein and Howes2019). The contrast in the energy transfer signals between the instantaneous and long-time-averaged correlations shows that while electrons over a broad range of parallel velocities participate in instantaneous energy transfer, the net secular energy transfer is largely contributed by ‘near-resonance’ electrons that are closely connected to $v_{p\Vert }$ . This reflects the resonant nature of secular net energy transfer in Landau damping of the turbulent fluctuations. Second, the variation in the energy transfer is largely a function of $v_{\Vert }$ , with weak $v_{\bot }$ dependence other than the monotonic drop off as $v_{\bot }$ increases. For the $(6,6,+3)$ mode in (d) which has $k_{\bot }\unicode[STIX]{x1D70C}_{e}\approx 1.7>1$ , however, there is weak $v_{\bot }$ dependence near $v_{\Vert }/v_{te}\sim 1$ , which lasts very briefly, for ${\sim}0.2\unicode[STIX]{x1D70F}_{k}$ . This $v_{\bot }$ dependence in the energy transfer reveals structuring of the distribution in $v_{\bot }$ as $E_{\Vert }$ does not depend on velocity. This qualitative feature is consistent with the expectation of the electron 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; Schoeffler et al. Reference Schoeffler, Loureiro, Fonseca and Silva2014), which predicts some structuring of the distribution in $v_{\bot }$ due to nonlinear phase mixing for $k_{\bot }\unicode[STIX]{x1D70C}_{e}>1$ .

Figure 6. Ion gyrotropic correlation $C_{E_{\Vert }}(v_{\Vert },v_{\bot },t)$ for the $(1,1,+3)$ Fourier mode using (a) $\unicode[STIX]{x1D70F}=0$ and (c) $\unicode[STIX]{x1D70F}=2\unicode[STIX]{x1D70F}_{k}$ , plotted at $t_{c}/\unicode[STIX]{x1D70F}_{0}=1.2$ , corresponding to $t/\unicode[STIX]{x1D70F}_{k}=0.48$ for $\unicode[STIX]{x1D70F}=2\unicode[STIX]{x1D70F}_{k}$ . (c) Shows observable weak $v_{\bot }$ variations at $v_{\Vert }<0$ in addition to more prominent $v_{\Vert }$ dependence. Also plotted are $C_{E_{\Vert }}(v_{\Vert },v_{\bot },t)$ for the $(2,2,-1)$ Fourier mode using (b)  $\unicode[STIX]{x1D70F}=0$ and (d) $\unicode[STIX]{x1D70F}=2\unicode[STIX]{x1D70F}_{k}$ at $t_{c}/\unicode[STIX]{x1D70F}_{0}=0.90$ , corresponding to $t/\unicode[STIX]{x1D70F}_{k}=0.90$ for $\unicode[STIX]{x1D70F}=2\unicode[STIX]{x1D70F}_{k}$ . Dashed lines denote Landau resonant velocities: $v_{p\Vert }/v_{ti}=\pm 1.3$ and $\pm 2.1$ for the $(1,1,+3)$ and $(2,2,-1)$ Fourier modes, respectively. The same format is used as figure 5.

4.2 Parallel reduced correlations

Because the variation of the rate of energy density transfer, shown in figures 5 and 6, is largely a function of $v_{\Vert }$ with little variation in $v_{\bot }$ , we may integrate over $v_{\bot }$ to obtain reduced parallel correlations $C_{E_{\Vert }}(v_{\Vert },t,\unicode[STIX]{x1D70F})$ , given by (2.11). These reduced correlations can conveniently be plotted, for a given correlation interval $\unicode[STIX]{x1D70F}$ , as timestack plots of $C_{E_{\Vert }}(v_{\Vert },t)$ as a function of $v_{\Vert }$ and time $t$ to illustrate the evolution of the field–particle energy transfer due to a single Fourier mode over the course of the entire simulation.

4.2.1 Electron parallel reduced correlations

Figure 7. Electron parallel reduced correlations $C_{E_{\Vert }}(v_{\Vert },t)$ for seven $k_{z}$ modes and summed $C_{E_{\Vert }}(v_{\Vert },t)$ for the $(2,2)$ Fourier mode. Correlation interval of $\unicode[STIX]{x1D70F}=2\unicode[STIX]{x1D70F}_{k}$ is chosen. Time $t$ is defined at the beginning of the correlation interval. Vertical black lines indicate Landau resonant velocities: $v_{p\Vert }/v_{te}=\pm 0.42$ . An arbitrary unit is used. Line plots on the right of each correlation are the $v_{\Vert }$ -integrated correlation as a function of time (red curve) and accumulated over time (green curve). The $\times 10^{n}$ factor above each colour bar applies also to the $x$ -axis of the line plots. Also plotted on the right of panel (h) is an axis for the normalized centred time $t_{c}/\unicode[STIX]{x1D70F}_{0}$ .

Plotted in figure 7 are timestack plots of the parallel reduced correlations $C_{E_{\Vert }}(v_{\Vert },t)$ for the electrons for the $(2,2)$ Fourier mode, separately for each of the seven $k_{z}\in [-3,+3]$ modes in panels (a–g) and the sum of these seven $k_{z}$ modes in panel (h), labelled $(2,2)$ . The correlation interval chosen for all panels is $\unicode[STIX]{x1D70F}$ = $2\unicode[STIX]{x1D70F}_{k}$ . Plotted to the right of each timestack plot is the $v_{\Vert }$ -integrated correlation, giving the net transfer rate of energy density for that Fourier mode $\int C_{E_{\Vert }}(v_{\Vert },t)\,\text{d}v_{\Vert }$ as a function of time (red curve) and the accumulated energy transferred $\int _{0}^{t}\text{d}t\int C_{E_{\Vert }}(v_{\Vert },t)\,\text{d}v_{\Vert }$ over time (green curve). The Landau resonant parallel phase velocity of each Fourier mode is indicated by vertical black lines. We note that, for a given value of $k_{\bot }$ , the normalized phase velocity $\overline{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}/k_{\Vert }v_{A}$ is independent of $k_{\Vert }$ , so the phase velocity ( $\unicode[STIX]{x1D714}/k_{\Vert }$ ) is constant for all $k_{z}$ (note that $k_{\Vert }$ and $k_{z}$ are the same in the linear dispersion relation by which $\overline{\unicode[STIX]{x1D714}}$ is given). This leads to a constant Landau resonant velocity, $v_{p\Vert }/v_{te}$ = $\pm$ 0.42, for all $(2,2,k_{z})$ Fourier modes in figure 7. Also plotted on the right of the $v_{\Vert }$ -integrated correlation for (h) the sum of the seven $k_{z}$ mode is an axis for the normalized centred time $t_{c}/\unicode[STIX]{x1D70F}_{0}$ .

A key characteristic of the parallel reduced correlations for all $(2,2,k_{z})$ Fourier modes is that the energy transfer is largely dominated by particles connected to $v_{p\Vert }$ , with an increase in electron energy (red) just above $v_{p\Vert }$ . This reduced parallel velocity-space signature is consistent with previous results showing Landau damping (Klein & Howes Reference Klein and Howes2016; Klein et al. Reference Klein, Howes and TenBarge2017; Howes et al. Reference Howes, Klein and Li2017, Reference Howes, McCubbin and Klein2018; Chen et al. Reference Chen, Klein and Howes2019). This localization of the energy transfer in parallel velocity indicates that collisionless damping by Landau resonant electrons is the dominant mechanism for transferring energy from the turbulent electromagnetic fluctuations to the electrons, a primary result of this study.

Furthermore, the timestack plots show clearly that this resonant energization of the electrons by the parallel electric field is sustained over the course of the simulation, although the amplitude of this rate of energization decreases in time as the initial fluctuations (the total turbulence energy) damp in time. In fact, the decrease in the rate of electron energization by the electric field over the course of the simulation is qualitatively consistent with the decrease in the spatially integrated rate of electron energization ${\dot{E}}_{e}^{(fp)}$ (solid blue) in figure 3(b). Additionally, the characteristic time scale of the energy transfer in the parallel reduced correlations is approximately $\unicode[STIX]{x1D70F}_{k}$ of this $(2,2)$ Fourier mode. This time scale is consistent with the linear wave period of this Fourier mode, as naturally expected.

The panels (a–g) for the seven different $(2,2,k_{z})$ Fourier modes in figure 7 show that most of the different $k_{z}$ modes yield a net energy gain (green curve) at the end of the run. The (h) sum of these seven $k_{z}$ components of the $(2,2)$ Fourier mode indeed demonstrates a net heating of electrons over the course of the simulation.

Figure 8. Ion parallel reduced correlations $C_{E_{\Vert }}(v_{\Vert },t)$ for seven $k_{z}$ modes and summed $C_{E_{\Vert }}(v_{\Vert },t)$ for the $(1,1)$ Fourier mode. Correlation interval of $\unicode[STIX]{x1D70F}=2\unicode[STIX]{x1D70F}_{k}$ is used. $C_{E_{\Vert }}(v_{\Vert },t)$ for each individual $k_{z}$ mode shows resonant signatures associated with Landau resonances, $v_{p\Vert }/v_{ti}=\pm 1.3$ , indicated by vertical black lines. Highly localized energy transfer signals are observed for all seven Fourier modes. The same format is used as figure 7. Arbitrary unit is used.

4.2.3 Reduced parallel correlations: $k_{\bot }$ dependence and $\unicode[STIX]{x1D6FD}_{i}$ dependence

Previous applications of the field–particle correlation have analysed the energy transfer at a single point in space (Klein & Howes Reference Klein and Howes2016; Klein et al. Reference Klein, Howes and TenBarge2017; Howes et al. Reference Howes, Klein and Li2017, Reference Howes, McCubbin and Klein2018; Chen et al. Reference Chen, Klein and Howes2019). Although this has the advantage of enabling the spatial localization of the energy transfer between fields and particles to be studied, it cannot provide any information about how that energy transfer is mediated by fluctuations at different scales. By performing the field–particle correlation analysis in Fourier space, we can examine how the energy transfer as a function of the scale of the Fourier modes plays a role in the energy transfer. In the gyrokinetic limit $k_{\Vert }\ll k_{\bot }$ , the resonant parallel phase velocity $\unicode[STIX]{x1D714}/k_{\Vert }$ of the kinetic Alfvén wave modes is strictly a function of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ (as shown in figure 1 a), so the reduced parallel correlation for a single perpendicular wavevector $(k_{x},k_{y})$ summed over $k_{z}$ has a single resonant velocity. We may obtain a clean velocity-space signature for each distinct $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ mode.

In figure 9, we present timestack plots of the reduced parallel correlations for the electrons summed over the seven $k_{z}$ modes for each diagnosed $k_{\bot }$ case spanning $1.4\leqslant k_{\bot }\unicode[STIX]{x1D70C}_{i}\leqslant 9.9$ , or $0.3\leqslant k_{\bot }\unicode[STIX]{x1D70C}_{e}\leqslant 2$ . The same format as the summed- $k_{z}$ results in figure 7(h) is used. The results clearly show the distinct advantages of the Fourier implementation of the field–particle correlation technique. First, the energy transfer signal for different $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ modes follows closely the increasing Landau resonant velocity (vertical black lines) as $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ increases from (a) through (f). A net energy energization of the electrons is obtained in all of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ modes, as shown in the accumulated energy transferred (green curve). Clearly, Landau resonant interactions with the electrons must play a key role in the damping of fluctuations across the sampled range of scales, consistent with expectations for a significant rate of collisionless damping by the electrons, $\unicode[STIX]{x1D6FE}_{e}/\unicode[STIX]{x1D714}_{A0}\gtrsim O(0.1)$ , as shown in figure 1(b). Second, the magnitude of energy transfer drops by approximately 3 orders of magnitude from $k_{\bot }\unicode[STIX]{x1D70C}_{i}=1.4$ to $k_{\bot }\unicode[STIX]{x1D70C}_{i}=9.9$ . Of course, the number of perpendicular Fourier modes increases linearly with $k_{\bot }$ , so the decrease in amplitude with $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ is partly compensated by the larger number of Fourier modes at higher $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ . Nonetheless, the first one or two $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ modes that we diagnosed here (specifically $k_{\bot }\unicode[STIX]{x1D70C}_{i}=1.4$ and $k_{\bot }\unicode[STIX]{x1D70C}_{i}=2.8$ ) would dominate the signals in a single-point diagnostic, so one would not be able to identify the energy transfer mediated by higher $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ modes without Fourier decomposition. Finally, the characteristic time scale of the energy transfer is consistent with $\unicode[STIX]{x1D70F}_{k}$ of each $(k_{x},k_{y})$ Fourier mode, even though $\unicode[STIX]{x1D70F}_{k}$ decreases with increasing $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ . As seen in the accumulated energy transferred (green curve), most of the electron energization occurs before a normalized centred time of $t_{c}/\unicode[STIX]{x1D70F}_{0}\sim 1.5$ . This is consistent with the spatially integrated rate of electron energization ${\dot{E}}_{e}^{(fp)}$ (solid blue) in figure 3(b) being the most significant by $1.5\unicode[STIX]{x1D70F}_{0}$ .

In order to illustrate more clearly how the energy transfer to the electrons closely tracks that Landau resonant velocity of kinetic Alfvén waves with a given perpendicular wavenumber $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , we zoom into the central part of the $v_{\Vert }$ range in figure 10(a–c). The localized region dominating electron energization moves to higher  $v_{\Vert }$ with higher $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , consistent with the increasing resonant parallel phase velocity of kinetic Alfvén waves as $k_{\bot }$ increases. This analysis strongly suggests that collisionless damping via the Landau resonance with electrons plays a key role in the removal of energy from the turbulence and consequent energization of the electrons.

Figure 9. Parallel reduced correlations summed over seven $k_{z}$ modes $\sum _{k_{z}=-3}^{+3}C_{E_{\Vert }}(v_{\Vert },t)$ for $6(k_{x},k_{y})$ values: $(1,1)$ , $(2,2)$ , $(3,4)$ , $(5,5)$ , $(6,6)$ and $(7,7)$ , representing a total of 42 Fourier modes of electron parallel reduced correlations. Their Landau resonant velocities are: $v_{p\Vert }/v_{te}=\pm (0.25,0.42,0.66,0.79,0.84,0.85)$ respectively. The same format is used as figure 7(h).

Figure 10. (a–c) Zoomed-in plot of summed- $k_{z}$ correlations from figure 9 for 3 $(k_{x},k_{y})$ values – $(1,1)$ , $(3,4)$ and $(7,7)$ with the corresponding Landau resonant velocities being $v_{p\Vert }/v_{te}=\pm (0.25,0.66,0.85)$ , respectively – showing how the field–particle energy transfer rate closely tracks the resonant parallel phase velocities as $(k_{x},k_{y})$ increases. (d–f) Electron parallel reduced correlations $C_{E_{\Vert }}(v_{\Vert },t)$ for the $(2,2,-1)$ Fourier mode from simulations with $\unicode[STIX]{x1D6FD}_{i}=1$ (current run), 0.1 and 0.01 in which the Landau resonant velocity, $v_{p\Vert }/v_{te}=\pm (0.42,1.0,1.4)$ , respectively, increases with decreasing $\unicode[STIX]{x1D6FD}_{i}$ . The energy transfer signals from all three simulations show remarkable agreement with the increasing parallel resonant velocity.

Another plasma parameter that has a strong impact on the resonant parallel phase velocity of kinetic Alfvén waves is the plasma $\unicode[STIX]{x1D6FD}_{i}$ . We performed simulations with an identical set-up as the $\unicode[STIX]{x1D6FD}_{i}=1$ simulation described in § 2.5, but with values $\unicode[STIX]{x1D6FD}_{i}=0.1$ and $\unicode[STIX]{x1D6FD}_{i}=0.01$ . To analyse both of these additional simulations, we take a correlation interval $\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{k}$ for the $(2,2,-1)$ mode analysed here. In figure 10(d–f), we present timestack plots of the reduced parallel correlation $C_{E_{\Vert }}(v_{\Vert },t,\unicode[STIX]{x1D70F})$ for electrons for Fourier mode $(2,2,-1)$ . The resonant parallel phase velocity $v_{p\Vert }$ for the $(2,2)$ Fourier mode, when normalized to $v_{te}$ , increases with values $\unicode[STIX]{x1D714}/(k_{\Vert }v_{te})=\pm 0.42,\pm 1.0,\pm 1.4$ (vertical black lines) as the values of $\unicode[STIX]{x1D6FD}_{i}$ decreases for these three simulations $\unicode[STIX]{x1D6FD}_{i}=1,0.1,0.01$ . We indeed find a remarkably close association between $v_{p\Vert }$ and the region of velocity space where the electrons participate in energy exchange with the parallel electric field $E_{\Vert }$ . This close quantitative agreement with the $\unicode[STIX]{x1D6FD}_{i}$ dependence of phase velocities observed in the simulations again suggests the dominant role that the Landau resonance plays in the field–particle energy transfer for electrons in the turbulence systems.

4.2.4 Accumulated particle energization: $k_{\bot }$ dependence

The collisionless field–particle energy transfer rate is directly measured by the field–particle correlations. With the sampled Fourier spectrum, we can see how the accumulated particle energization varies as a function of the length scales of the fluctuations. In figure 11, we plot this accumulated particle energization at the end of the simulation $\unicode[STIX]{x0394}E_{s}^{(fp)}$ as a function of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ . Each point represents the accumulated energization in each $(k_{x},k_{y})$ Fourier mode summed over $k_{z}$ for ions (blue) and electrons (green). The electron energization $\unicode[STIX]{x0394}E_{e}^{(fp)}$ is then the value of the green curve at the end of the simulation for each $(k_{x},k_{y})$ mode in figure 9. The same normalization is used for both ion and electron energization, $\unicode[STIX]{x0394}E_{i}^{(fp)}$ and $\unicode[STIX]{x0394}E_{e}^{(fp)}$ .

Figure 11. Accumulated particle energization $\unicode[STIX]{x0394}E_{s}^{(fp)}$ at the end of the simulation for the sampled $(k_{x},k_{y})$ modes (summed over $k_{z}$ ) as a function of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ . Each $(k_{x},k_{y})$ mode is represented by a point and all points are connected by lines. The same normalization is used for both ion $\unicode[STIX]{x0394}E_{i}^{(fp)}$ (blue) and electron $\unicode[STIX]{x0394}E_{e}^{(fp)}$ (green) energization.

Several features of $\unicode[STIX]{x0394}E_{s}^{(fp)}$ are notable. First, the particle energization is dominated by the lowest $k_{\bot }$ mode that has the strongest amplitude, as expected. Second, the total ion and electron energization over the sampled spectrum are comparable given that both species are nearly equally energized at this dominant, lowest $k_{\bot }$ mode. This is consistent with the comparable species heating rates due to field–particle interactions ${\dot{E}}_{s}^{(fp)}$ (solid) in figure 3 in the turbulence systemFootnote ² . This is also consistent with the ion and electron collisionless damping rates, $\unicode[STIX]{x1D6FE}_{i}$ and $\unicode[STIX]{x1D6FE}_{e}$ , being of the same order of magnitude at $k_{\bot }\unicode[STIX]{x1D70C}_{i}\sim 1$ , represented by the lowest $k_{\bot }$ mode, in figure 1. Third, when plotted as a function of $k_{\bot }\unicode[STIX]{x1D70C}_{i}$ , $\unicode[STIX]{x0394}E_{s}^{(fp)}$ here can represent the energy transfer spectra; for electrons, $\unicode[STIX]{x0394}E_{e}^{(fp)}$ indicates a spectral rollover at the electron gyroradius scale of $k_{\bot }\unicode[STIX]{x1D70C}_{e}=1$ ( $k_{\bot }\unicode[STIX]{x1D70C}_{i}=5$ ), becoming less steep at $k_{\bot }\unicode[STIX]{x1D70C}_{e}>1$ . Lastly, ion energization shows a dip at $k_{\bot }\unicode[STIX]{x1D70C}_{i}=7$ , rather than a monotonic drop for all $k_{\bot }\unicode[STIX]{x1D70C}_{i}\geqslant 7$ modes as would be expected from the ion collisionless damping rate, suggesting that collisionless damping cannot completely explain ion energization. These interesting features deserve further investigation in the future.