2.1. Field–particle correlation for transit-time damping

To determine the appropriate form of the field–particle correlation to diagnose TTD via the magnetic mirror force, we start with the Vlasov equation for a species $s$ being acted upon by a general force $\boldsymbol {F}_s$

(2.1)\begin{equation} \frac{\partial f_s}{\partial t} + \boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla} f_s + \frac{\boldsymbol{F}_s}{m_s} \boldsymbol{\cdot} \frac{\partial f_s}{\partial \boldsymbol{v}} =0, \end{equation}

where $f_s$ denotes the distribution function, and $m_s$ is the mass of the particle. Multiplying the Vlasov equation by $m_s v^2/2$, we obtain an expression for the rate of change of the phase-space energy density, $w_s(\boldsymbol {r},\boldsymbol {v},t) \equiv m_s v^2 f_s(\boldsymbol {r},\boldsymbol {v},t)/2$,

(2.2)\begin{equation} \frac{\partial w_s(\boldsymbol{r},\boldsymbol{v},t)}{\partial t} ={-} \boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{\nabla} w_s - \frac{v^2}{2} \boldsymbol{F}_s \boldsymbol{\cdot} \frac{\partial f_s}{\partial \boldsymbol{v}}. \end{equation}

Previous analysis of this equation (Klein & Howes Reference Klein and Howes2016; Howes et al. Reference Howes, Klein and Li2017) has shown that, if integrated over space (with appropriate infinite or periodic boundary conditions), the change in the total kinetic energy of the particles $\mathfrak {W}_s (t)= \int {\rm d}^3\boldsymbol {r} \int {\rm d}^3\boldsymbol {v} w_s(\boldsymbol {r},\boldsymbol {v},t)$ is due to work done on the particle species by the force $\boldsymbol {F}_s$. Therefore, the field–particle correlation due to a general force $\boldsymbol {F}_s$ at spatial position $\boldsymbol {r}_0$ is defined as a time average over a correlation interval $\tau$ of the last term on the right-hand side of (2.2)

(2.3)\begin{equation} C_{\boldsymbol{F}_s}(\boldsymbol{r}_0,\boldsymbol{v},t;\tau) \equiv \frac{1}{\tau}\int^{t+\tau/2}_{t-\tau/2} \frac{-v^2}{2} \boldsymbol{F}_s(\boldsymbol{r}_0,\boldsymbol{v},t') \boldsymbol{\cdot} \frac{\partial f_s(\boldsymbol{r}_0,\boldsymbol{v},t')}{\partial \boldsymbol{v}} {\rm d} t'. \end{equation}

Note here that the correlation interval $\tau$ is a parameter of the field–particle correlation analysis, and so is included as a secondary argument, separated by a semicolon from the primary arguments that define the dimensions of the 3D-3 V phase space in position, velocity and time.

If we consider the force due to an electric field $\boldsymbol {F}_s=q_s \boldsymbol {E}$, we obtain the established field–particle correlation due to the electric field (Klein & Howes Reference Klein and Howes2016; Howes et al. Reference Howes, Klein and Li2017; Klein et al. Reference Klein, Howes and TenBarge2017)

(2.4)\begin{equation} C_{\boldsymbol{E}, s}(\boldsymbol{r}_0,\boldsymbol{v},t;\tau) = \frac{1}{\tau}\int^{t+\tau/2}_{t-\tau/2} \frac{-q_s v^2}{2} \boldsymbol{E}(\boldsymbol{r}_0,t') \boldsymbol{\cdot} \frac{\partial f_s(\boldsymbol{r}_0,\boldsymbol{v},t')}{\partial \boldsymbol{v}} {\rm d} t'. \end{equation}

The collisionless transfer of energy between electromagnetic waves and particles in TTD is mediated by the magnetic mirror force, $\boldsymbol {F}_s = - \mu _s \hat {\boldsymbol {b}} \boldsymbol {\cdot }\boldsymbol {\nabla } \boldsymbol {B}$, where the magnetic moment for a particle of species $s$ is given by $\mu _s=m_s v_\perp ^2/(2 B)$, the unit vector in the direction of the magnetic field is given by $\hat {\boldsymbol {b}} \equiv \boldsymbol {B}/B$ and the magnitude of the magnetic field is $B=|\boldsymbol {B}|$. Substituting the magnetic mirror force into (2.3), we obtain

(2.5)\begin{equation} C_{\boldsymbol{B}, s}(\boldsymbol{r}_0,\boldsymbol{v},t;\tau) = \frac{1}{\tau}\int^{t+\tau/2}_{t-\tau/2} \frac{m_s v^2v_\perp^2}{4 B} (\hat{\boldsymbol{b}} \boldsymbol{\cdot}\boldsymbol{\nabla}) \boldsymbol{B}(\boldsymbol{r}_0,t') \boldsymbol{\cdot} \frac{\partial f_s(\boldsymbol{r}_0,\boldsymbol{v},t')}{\partial \boldsymbol{v}} {\rm d} t'. \end{equation}

A few modifications of the form of the field–particle correlation for TTD given in (2.5) are helpful for its application to the gyrokinetic simulations presented here. First, we exploit two important characteristics of TTD and turbulence: (i) TTD is most effective in damping the dominant Alfvénic fluctuationsFootnote ¹ in turbulence with perpendicular wavelengths at the ion scales, $k_\perp \rho _i \sim k \rho _i \sim 1$, where $k_\parallel /k_\perp \ll 1$; and (ii) for most turbulent space and astrophysical plasmas of interest, the amplitude of the magnetic fluctuations $\delta \boldsymbol {B}$ at ion scales $k \rho _i \sim 1$ is much smaller than the magnitude of the mean magnetic field $\boldsymbol {B}_0$. Therefore, if we separate the magnetic field into its mean plus the fluctuations, $\boldsymbol {B}=\boldsymbol {B}_0+\delta \boldsymbol {B}$, where $|\delta \boldsymbol {B}| \ll |\boldsymbol {B}_0|$, the change in the magnetic field magnitude $\delta | \boldsymbol {B}|$ (which is the key ingredient for the magnetic mirror force) can be expressed as $\delta B_\parallel$ by recognizing

(2.6)\begin{equation} \delta | \boldsymbol{B}| = |\boldsymbol{B}|-|\boldsymbol{B}_0| = \sqrt{(\boldsymbol{B}_0 + \delta \boldsymbol{B})^2} - B_0 = \sqrt{B_0^2 + 2 \delta \boldsymbol{B} \boldsymbol{\cdot}\boldsymbol{B}_0 + |\delta \boldsymbol{B}|^2 } -B_0 \simeq \delta B_\parallel, \end{equation}

where we use a binomial expansion to eliminate the square root, neglect the small $|\delta \boldsymbol {B}|^2$ term, and write $\delta B_\parallel = \delta \boldsymbol {B} \boldsymbol {\cdot } (\boldsymbol {B}_0/B_0)$ as the variation in the component of the perturbed magnetic field parallel to the mean magnetic field. Furthermore, separating term $v^2=v_\perp ^2+v_\parallel ^2$ in the correlation (2.5), it is easy to show that the $v_\perp ^2$ contribution yields a perfect differential in $f_s$ when integrated over all parallel velocity (Howes et al. Reference Howes, Klein and Li2017), so we choose to omit this term since it leads to zero net change in the particle energy. Finally, we write the gradient along the magnetic field direction by $\boldsymbol {\nabla }_\parallel \equiv \hat {\boldsymbol {b}} \boldsymbol {\cdot }\boldsymbol {\nabla }$, leading to our preferred form of the field–particle correlation for TTD

(2.7)\begin{equation} C_{\delta B_\parallel, s}(\boldsymbol{r}_0,\boldsymbol{v},t;\tau) = \frac{1}{\tau}\int^{t+\tau/2}_{t-\tau/2}\frac{m_s v_\parallel^2v_\perp^2}{4 B} \boldsymbol{\nabla}_\parallel \delta B_\parallel \frac{\partial f_s(\boldsymbol{r}_0,\boldsymbol{v},t')}{\partial v_\parallel} {\rm d} t'. \end{equation}

In the gyrokinetic system of equations (Antonsen & Lane Reference Antonsen and Lane1980; Frieman & Chen Reference Frieman and Chen1982; Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006; Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009), the gyroaveraged effect of parallel magnetic field gradients leads to the magnetic mirror force, which can accelerate particles in the direction parallel to the magnetic field via the Landau resonance, and therefore can lead to collisionless TTD of electromagnetic fluctuations. A rigorous derivation of the gyrokinetic equation in Appendix A shows explicitly the two collisionless wave–particle interactions via the Landau resonance – specifically, LD and TTD. The natural form of the field–particle correlation arising from the gyrokinetic version of the generalized energy density equation is slightly different from the field–particle correlation for TTD given in (2.7), but the gyrokinetic form requires the gyroaveraged distribution function, which is not accessible through single-point spacecraft measurements and is not a natural quantity that can easily be derived from other kinetic simulation approaches, such as particle-in-cell or Vlasov simulations. Therefore, we choose here to use the perturbed distribution functions and electromagnetic fields generated by our gyrokinetic simulations, but we analyse them using (2.7), which is more directly applicable to spacecraft measurements or alternative kinetic simulation approaches, such as particle-in-cell codes.

Utilizing the gyroaveraged distribution function and corresponding fields, the quantity $C_{\delta B_\parallel, s}(\boldsymbol {r}_0, \boldsymbol {v}, t; \tau )$ reveals the velocity-space signature of TTD on the gyrotropic velocity space $(v_\perp, v_\parallel )$. To simplify the notation, henceforth, we shall employ $C_{\delta B_\parallel, s} (v_\parallel, v_\perp )$ to signify the gyrotropic correlation, explicitly noting the associated spatial position $\boldsymbol {r}_0$, time $t$, and the correlation interval $\tau$ only when necessary. The resonant structure of the velocity-space signature of TTD is primarily a function of $v_\parallel$, so it is often useful to define the reduced parallel field–particle correlation by integrating $C_{\delta B_\parallel, s} (v_\parallel, v_\perp,t)$ over $v_\perp$, given by $C_{\delta B_\parallel, s}(v_\parallel,t) \equiv 2 {\rm \pi}\int C_{\delta B_\parallel, s}(v_\parallel, v_\perp,t) v_\perp \,{\rm d} v_\perp$, where the extra $2 {\rm \pi}v_\perp$ factor arises from the integration over the gyrophase in 3 V phase space. A timestack plot of the reduced parallel correlation $C_{\delta B_\parallel, s}(v_\parallel,t)$ reveals the persistence in time of any resonant velocity-space signatures in $v_\parallel$. We can also consider the rate of change of the kinetic energy density of species $s$ due to TTD by integrating the gyrotropic correlation over all velocity space: $(\partial W_s/\partial t)_{{\rm TTD}} = \int C_{\delta B_\parallel, s}(v_\parallel, v_\perp ) \,{\rm d}^3 \boldsymbol {v}$.

In closing, note that the gyroaveraging procedure employed in the derivation of the system of gyrokinetics enables the variations in the perpendicular components of the electric field $E_\perp$ to be expressed in terms of changes in the parallel component of the magnetic field $\delta B_\parallel$, as shown by (A7). In a system where the gyroaverage has not been performed, the work done by TTD is actually mediated (at the position of the particle) by the perpendicular component of the electric field $E_\perp$ (Howes et al. Reference Howes, Huang and Felix2024). Therefore, the perpendicular electric field correlation, given by summing the two perpendicular contributions to the electric field correlation, $C_{E_\perp }(\boldsymbol {r}_0,\boldsymbol {v},t;\tau )$ (Klein et al. Reference Klein, Howes, TenBarge and Valentini2020; Afshari et al. Reference Afshari, Howes, Shuster, Klein, McGinnis, Martinovic, Boardsen, Brown, Huang and Kletzing2023), can be used to seek the velocity-space signature of TTD at the parallel resonant phase velocity, as seen recently in hybrid particle-in-cell simulations of plasma turbulence (Cerri et al. Reference Cerri, Arzamasskiy and Kunz2021).

2.2. Prediction of the velocity-space signature of transit-time damping

To predict the velocity-space signature of TTD, we begin with a simple model of a magnetic field with an amplitude variation that varies along the mean field direction $z$, given in cylindrical coordinates $(r,\phi,z)$ by

(2.8)\begin{equation} \boldsymbol{B}(r,\phi,z) ={-} \frac{\delta B_z }{4} k r\sin(k z') \hat{\boldsymbol{r}} + \left\{ B_0 + \frac{\delta B_z}{2} [1 - \cos (k z')] \right \} \hat{\boldsymbol{z}}, \end{equation}

where the wavenumber $k$ of the spatial variation of the magnetic field magnitude is along the mean field direction $z$, and $z'=z-U t$, such that that pattern moves in the $+z$ direction with a phase speed $U \geq 0$. The corresponding electric field variation can be determined by the Lorentz transformFootnote ² from the primed (wave) frame $K'$ in which the magnetic field pattern is stationary (and therefore $\boldsymbol {E}'=0$) to the unprimed (laboratory) frame $K$. This Lorentz transformation in the non-relativistic limit $U/c \ll 1$ is given by $\boldsymbol {E}=\boldsymbol {E}' - \boldsymbol {U} \times \boldsymbol {B}$ and $\boldsymbol {B}=\boldsymbol {B}'$ (Howes, Klein & TenBarge Reference Howes, Klein and TenBarge2014), where the transformation velocity is just $\boldsymbol {U} =U \hat {\boldsymbol {z}}$. The resulting induced electric field in the laboratory frame $K$ is given by

(2.9)\begin{equation} \boldsymbol{E}(r,\phi,z) = U\frac{\delta B_z}{4} kr \sin (k z') \hat{\boldsymbol{\phi}}. \end{equation}

With this simple model, we can illustrate how a single particle is accelerated into different regions of velocity space by the electromagnetic fields. Extending this approach to consider a distribution of particles will enable us to predict the qualitative and quantitative features of the velocity-space signature of TTD.

Consider first the acceleration of a single particle in a stationary mirror field with $U=0$, as shown in figure 2(a), for a ‘wave’ amplitude of $\delta B_z/B_0=0.2$, giving a mirror ratio of $B_\text {max}/B_\text {min} = 1.2$. The particle begins at the minimum in the magnetic field at $z=0$ with an initial perpendicular velocity $v_\perp$ and an initial parallel velocity $v_\parallel <0$, given by the red $+$ in the figure at the tip of the initial velocity vector $\boldsymbol {v}_i$ (blue). As the particle moves into the increasing magnetic field at $z<0$, the mirror force increases $v_\perp$ and decreases $v_\parallel$ such that the particle moves through velocity space (green arrow) on a circle of constant total velocity $v = \sqrt {v_\parallel ^2+v_\perp ^2}$ (black dashed circle). The particle will be reflected by the mirror field if the particle has an initial pitch angle $\alpha = \tan ^{-1}(v_\perp /v_\parallel )$ larger than the loss-cone angle $\alpha _{{\rm loss}} = \sin ^{-1}( \sqrt {B_\text {min}/B_\text {max}})$. For $B_\text {max}/B_\text {min} = 1.2$, the loss-cone angle is $\alpha _{{\rm loss}} = 66^\circ$; the particle depicted in figure 2(a) has an initial pitch angle $\alpha > \alpha _{{\rm loss}}$ and is therefore reflected by the mirror field. The particle follows this circular trajectory in $(v_\parallel,v_\perp )$ velocity space until it returns to its initial axial position $z=0$, ending up with a final velocity $\boldsymbol {v}_f$ (blue) with the same perpendicular component but an equal and opposite parallel component. Thus, the kinetic energy of the particle does not change, consistent with the fact that magnetic fields do no work on particles. The particle has simply been reflected by the mirror, reversing the sign of its parallel velocity.

Figure 2. Diagram of the magnetic mirror reflection and prediction for the velocity-space signature of TTD: (a) $v_\perp$ versus $v_\parallel$ for the single particle motion in a static magnetic mirror field; (b) $v_\perp$ versus $v_\parallel$ for the single particle motion in a moving magnetic mirror field, where the vertical black dashed line denotes the wave phase velocity $U$; (c) the predicted velocity-space signature for a Maxwellian velocity distribution function, where the phase-space energy density decreases at $v_\parallel < U$ (blue) and increases at $v_\parallel > U$ (red); (d) effective $v_\perp$ weighting of correlation $v_\perp ^2f(v_\perp )$, which constrains the velocity-space signature in the $v_\perp$ direction.

Next, we consider the case for a magnetic mirror field moving with velocity $\boldsymbol {U} =U \hat {\boldsymbol {z}}$, where $U>0$. In the wave frame, moving at velocity $\boldsymbol {U} =U \hat {\boldsymbol {z}}$ in which the magnetic field is stationary, the acceleration of the particle by the magnetic field must be the same as the stationary case in figure 2(a). But, in the laboratory frame, depicted in figure 2(b), the particle now moves on a circular trajectory in velocity space centred about the mirror velocity $U$, given by a constant magnitude of velocity in the wave frame $v_0=\sqrt {(v_\parallel -U)^2+v_\perp ^2}$. Here, the particle is initially moving in the same direction as the mirror field but with a slower initial parallel velocity $0 \le v_\parallel \le U$, given by the red $+$ in the figure at the tip of the initial velocity vector $\boldsymbol {v}_i$ (blue). If the pitch angle in the wave frame $\alpha _w = \tan ^{-1}[v_\perp /(v_\parallel -U)]$ is greater than the loss-cone angle, $\alpha _w > \alpha _{{\rm loss}}$, the particle will be reflected by the moving mirror field, leading to a net acceleration in the axial direction, ultimately ending up with a parallel velocity greater than the mirror velocity $v_\parallel > U$, with a final velocity vector $\boldsymbol {v}_f$ (blue). In this case, the induced electric field given by (2.9) has done work on the particle (Howes et al. Reference Howes, Huang and Felix2024), ultimately leading to a net acceleration in the axial direction. This process is the fundamental energy transfer underlying the physics of TTD.

Finally, we consider how this understanding of the single particle motion and acceleration can be combined with a distribution of initial particle velocities to predict the velocity-space signature of TTD. Note that for the sinusoidally oscillating magnetic field magnitude given by (2.8), the long time evolution of the particle in velocity space would oscillate back and forth between $\boldsymbol {v}_i$ and $\boldsymbol {v}_f$ along the green trajectory shown in figure 2(a,b); for example, if the particle started with initial velocity $\boldsymbol {v}_f$, the other side of the mirror field would lead to a reflection in the opposite direction, ultimately resulting in the particle ending up with a final velocity $\boldsymbol {v}_i$. In the case of a moving mirror field, for a Maxwellian velocity distribution there will be more particles with parallel velocities $v_\parallel < U$ than with $v_\parallel > U$, so the net effect is that more particles will gain energy than lose energy, leading to a net energization of the particles and consequent damping of the electromagnetic wave. Only particles with pitch angles in the wave frame larger than the loss-cone angle, $\alpha _w > \alpha _{{\rm loss}}$, will undergo the mirror reflection, so that net effect on the distribution is an acceleration of particles from $v_\parallel < U$ to $v_\parallel > U$. The resulting change in the phase-space energy density leads to the prediction of the velocity-space signature of TTD depicted in figure 2(c): a loss of phase-space energy density (blue) in the region $v_\parallel < U$, and a gain of phase-space energy density (red) in the region $v_\parallel > U$. The extent of this velocity-space signature in $(v_\parallel,v_\perp )$ velocity space is confined by two effects: (i) only particles outside of the loss cone will experience a net acceleration; and (ii) the signature is weighted by the $v_\perp ^2$ weighting in (2.7) for the rate of change of phase-space energy density by TTD (which arises from the magnetic moment $\mu =m v_\perp ^2/(2B)$ dependence of the mirror force) combined with the reduced perpendicular velocity distribution $f(v_\perp )$, where this net weighting of $v_\perp ^2 f(v_\perp )$ is shown in figure 2(d). Thus, the velocity-space signature of TTD in figure 2(c) is restricted to ‘Landau resonant’ particles with parallel velocities near the velocity of the magnetic field pattern $v_\parallel \sim U$ and to a region away from the $v_\perp =0$ axis, unlike the velocity-space signature of LD (Klein & Howes Reference Klein and Howes2016; Howes et al. Reference Howes, Klein and Li2017; Klein et al. Reference Klein, Howes and TenBarge2017) which extends down to $v_\perp =0$.

3.1. Single-wave simulation set-up

We first determine the velocity-space signature of TTD by performing simulations of single KAWs for a fully ionized proton–electron plasma with Maxwellian equilibrium velocity distributions with a temperature ratio $T_i / T_e = 1$ and a realistic ion-to-electron mass ratio of $m_i/m_e = 1836$. We perform three simulations with ion plasma beta $\beta _i=0.3, 1, 3$ and sample the time evolution of the electromagnetic fields and gyrokinetic distribution function at discrete single points throughout the simulation domain with dimensions $L_\perp = 2 {\rm \pi}\rho _i$ and $L_\parallel = 2 {\rm \pi}a_0$, yielding an arbitrary expansion parameter $\epsilon = \rho _i/a_0 \ll 1$. The ion plasma beta is defined by $\beta _i=8 {\rm \pi}n_i T_i/B^2=v_{{\rm ti}}^2/v_A^2$, where the ion thermal velocity is $v_{{\rm ti}}= \sqrt {2 Ti/m_i}$, the Alfvén velocity is $v_A=B/\sqrt {4 {\rm \pi}n_i m_i}$ and the Boltzmann constant is absorbed to give temperature in units of energy. Here, $\rho _i\equiv v_{{\rm ti}}/\varOmega _i$ is the ion Larmor radius, where the angular ion (proton) cyclotron frequency is $\varOmega _i=q_i B_0/m_i$. The dimensions of these single-wave simulations are $(n_x, n_y, n_z, n_{\lambda }, n_E, n_s) = (10, 10, 32, 128, 64, 2)$, where $n_s$ denotes the number of species. Using the solutions to the linear gyrokinetic dispersion relation (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006), a single plane-wave KAW with wavevector $(k_x \rho _i,k_y \rho _i, k_\parallel a_0) = (1,0,1)$ and amplitudeFootnote ³ $\delta B_\perp /B_0=0.125 \epsilon$ is initialized throughout the domain and allowed to evolve linearly for 5 wave periods with enhanced collisionality to eliminate any transients associated with the initialization, yielding a clean, single KAW with $k_\perp \rho _i =1$. The simulation is then restarted with lowered collisionalities $\nu _s/(k_{\parallel 0} v_{{\rm ti}}) = 2 \times 10^{-3}$ and evolved to allow collisionless wave–particle interactions to damp the wave. We have verified that the collisionless damping rates of the initialized KAWs agree with the analytical predictions from Vlasov–Maxwell and gyrokinetic linear dispersion relations.

3.2. The velocity-space signature of transit-time damping

We select the $\beta _i = 1$ single KAW simulation as a fiducial case to determine the velocity-space signature of TTD with (2.7) and compare it with the known velocity-space signature of LD using only the parallel contribution to the dot product in (2.4) (Klein & Howes Reference Klein and Howes2016; Howes et al. Reference Howes, Klein and Li2017; Klein et al. Reference Klein, Howes and TenBarge2017; Howes, McCubbin & Klein Reference Howes, McCubbin and Klein2018; Chen et al. Reference Chen, Klein and Howes2019; Horvath, Howes & McCubbin Reference Horvath, Howes and McCubbin2020; Klein et al. Reference Klein, Howes, TenBarge and Valentini2020; Afshari et al. Reference Afshari, Howes, Kletzing, Hartley and Boardsen2021). The linear Vlasov–Maxwell dispersion relation yields a parallel phase velocity normalized to the Alfvén velocity of $\bar {\omega } \equiv \omega /(k_\parallel v_A) = 1.137$ for a KAW with $k_\perp \rho _i =1$, $\beta _i=1$, and $T_i / T_e = 1$, corresponding to a normalized wave period of $T \omega _A = 5.526$.

A key step in the field–particle correlation analysis is to choose an appropriate correlation interval $\tau$ over which to time average the rate of energization to eliminate a possibly larger amplitude signal of oscillatory energy transfer in order to reveal the often smaller secular rate of energy transfer that corresponds to the collisionless damping of the wave (Klein & Howes Reference Klein and Howes2016; Howes et al. Reference Howes, Klein and Li2017; Klein et al. Reference Klein, Howes and TenBarge2017). In figure 3, we present a test of different correlation intervals over the range $0 \le \tau \omega _A \le 10$ for the $\beta _i=1$ single KAW simulation. In (a), we plot the velocity-space-integrated rate of ion energization due to TTD, $(\partial W_i/\partial t)_{{\rm TTD}} = \int C_{\delta B_\parallel, i}(v_\parallel, v_\perp ) \,{\rm d}^3 \boldsymbol {v}$, vs. time. The unaveraged correlation ($\tau =0$, dark blue) exhibits pronounced oscillations of the net energy transfer vs. time. Setting the correlation interval to one linear wave period $\tau \omega _A = T \omega _A \simeq 5.5$ (black) minimizes the oscillations, providing an optimal choice for $\tau$ for a single wave with a well-defined period, as might be expected on theoretical grounds. We show in (b) the evolution of the reduced parallel correlation $C_{\delta B_\parallel, i}(v_\parallel,t)$ at a parallel velocity $v_\parallel = 1.1 v_\text {ti}$ slightly below the resonant velocity. Here again, setting $\tau \omega _A = 5.5$ (black) effectively minimizes the oscillations, revealing clearly the rate of secular energy transfer at that parallel velocity. We illustrate the impact of choosing an appropriate correlation interval $\tau$ on the timestack plot of the correlation $C_{\delta B_\parallel, i}(v_\parallel,t)$ by comparing (c) the instantaneous ($\tau =0$) field–particle correlation with (d) the correlation using $\tau \omega _A = 5.5$, showing a clear bipolar signature about the normalized parallel phase velocity $\omega /(k_\parallel v_{{\rm ti}})=1.137$ that persists over the course of the simulation.

Figure 3. Analysis of correlation interval selection for $\beta _i = 1$ AstroGK single KAW simulation. (a,b) Time evolution of (a) the rate of change of ion kinetic energy density due to TTD, denoted as $\partial W_i /\partial t$, and (b) the reduced correlation $C_{\delta B_{\parallel, i}} (v_\parallel, t)$ at $v_\parallel = 1.1 v_{{\rm ti}}$. Both quantities are presented over a range of $\tau \omega _A$ values from 0 to 10. The selected $\tau \omega _A$ value of 5.5 is marked with a black line. (c,d) Timestack plots of the reduced correlation $C_{\delta B_{\parallel, i}} (v_\parallel, t)$ for (c) $\tau \omega _A = 0$ and (d) $\tau \omega _A = 5.5$, with the vertical dashed line at $v_\parallel /v_\text {ti} = 1.137$ labelling the normalized parallel phase velocity $\omega /(k_\parallel v_\text {ti})$.

Specifying the correlation interval to be approximately one wave period $\tau \omega _A = 5.5$,Footnote ⁴ we now present in figure 4 the velocity-space signatures of (a) TTD and (b) LD from the $\beta _i = 1$ single KAW simulation. Each panel presents three sub-plots, explained here for the TTD case in (a): (i) the main plot presents the gyrotropic velocity-space signature $C_{\delta B_\parallel, i}(v_\parallel,v_\perp )$ at time $t \omega _A = 63.57$, with the parallel phase velocity indicated (vertical dotted line); (ii) the lower plot shows the time-integrated parallel velocity-space signature $C_{\delta B_\parallel, i}(v_\parallel )$ to highlight the variation of the net energy transferred as a function of $v_\parallel$, showing a clear bipolar signature at the parallel phase velocity; and (iii) the left plot shows the velocity-space-integrated net energy density transfer rate due to TTD $(\partial W_i(t)/\partial t)_{{\rm TTD}}$ vs. time, with the centred time of the correlation interval shown in the gyrotropic signature indicated (horizontal solid line). Panel (b) presents the corresponding plots for the LD case.

Figure 4. Velocity-space signatures of (a) TTD and (b) LD, from the AstroGK simulation of a single KAW with $k_\perp \rho _i =1$, $\beta _i=1$ and $T_i / T_e = 1$, each showing the gyrotropic signatures in the main panel, the time-integrated reduced parallel signatures in the lower panel and the net rate of ion energization vs. time for each mechanism in the left panel. The correlation interval is chosen as $\tau \omega _A = 5.5$. The normalized parallel phase velocity is labelled by the two vertical dashed lines at $v_\parallel /v_\text {ti} = \pm 1.137$.

A key result of this paper is the velocity-space signature of TTD plotted in gyrotropic velocity space $C_{\delta B_\parallel, i}(v_\parallel,v_\perp )$ in figure 4(a). From the same simulation, the gyrotropic velocity-space signature of LD $C_{E_\parallel, i}(v_\parallel,v_\perp )$, given by the parallel contribution to the dot product in (2.4), is presented in (b) for comparison. The TTD signature agrees qualitatively with our prediction presented in figure 2(c), showing the key features: (i) the bipolar signature of the rate of loss (blue) and gain (red) of phase-space energy density is centred about the parallel wave phase velocity $v_\parallel \sim \omega /k_\parallel$ (vertical dotted black line); (ii) the gyrokinetic velocity-space signature does not extend down to $v_\perp =0$, due to a combination of the $v_\perp ^2$ weighting arising from the magnetic moment $\mu =m v^2_\perp /(2B)$ in the mirror force and from the loss-cone angle of the mirror force, as explained in § 2.2. The LD signature in (b) likewise yields a bipolar signature near the parallel wave phase velocity. Besides the fact that $E_\parallel$ governs energization through LD and $E_\perp$ governs energization through TTD (Howes et al. Reference Howes, Huang and Felix2024), a key way to distinguish these two mechanisms is that in gyrotropic velocity space the TTD signature does not extend down to $v_\perp =0$, whereas the LD signature extends right down to $v_\perp =0$.

Careful inspection of these single KAW velocity-space signatures reveals a ‘twist’ seen in both panels of figure 4, where the bipolar pattern of energy loss (blue) to energy gain (red) across the parallel phase velocity reverses sign as $v_\perp$ changes. This feature arises from the linear combination of three terms contributing to the linear response for perturbed distribution function $f_1$, which involves the components of the electric field perturbation and the zeroth-order Bessel function and its derivative, as seen in (4.180) from Swanson (Reference Swanson2003). Although a detailed decomposition of these contributions to understand this twist for a single KAW could be performed, for the case of TTD or LD in a turbulent plasma, this twist is obscured when damping a broadband spectrum of wave modes, as shown in § 4, so we do not pursue this line of investigation further.

3.3. Variation of signature with ion plasma beta $\beta _i$

Resonant damping of electromagnetic fluctuations through TTD and LD depends strongly on the plasma beta, typically with LD dominant at $\beta _i \ll 1$ and TTD dominant at $\beta _i \gg 1$ (Quataert Reference Quataert1998), so we vary the value of $\beta _i$ here to determine its impact on the characteristics of the velocity-space signature of the Landau-resonant damping mechanisms. Using the same parameters $T_i/T_e=1$, $m_i/m_e=1836$ and $k_\perp \rho _i =1$, we perform additional single KAW simulations with $\beta _i = 0.3$ and $\beta _i = 3$. For $\beta _i = 0.3$, the normalized parallel phase velocity is given by $\bar {\omega } \equiv \omega /(k_\parallel v_A) = 1.267$, yielding a normalized wave period of $T \omega _A = 4.959$; for $\beta _i = 3$, we obtain $\bar {\omega }= 1.009$ and $T \omega _A = 6.227$.

In figure 5, we plot the velocity-space signatures for (a) TTD and (b) LD for the $\beta _i = 0.3$ case and for (c) TTD and (d) LD for the $\beta _i = 3$ case, where each panel has three subplots in the same format as in figure 4. These bipolar velocity-space signatures look qualitatively similar to the $\beta _i = 1$ case in figure 4, with two important quantitative differences.

Figure 5. Velocity-space signatures of TTD (a,c) and LD (b,d) in AstroGK single KAW simulations with $k_\perp \rho _i =1$, $T_i / T_e = 1$ and $\beta _i = 0.3$ (a,b) and $\beta _i = 3$ (c,d). The correlation intervals are set to the corresponding linear wave periods, with $\tau \omega _A = 5.0$ for the $\beta _i = 0.3$ case and $\tau \omega _A = 6.2$ for the $\beta _i = 3$ case. The normalized parallel phase velocity is labelled by the two vertical dashed lines at $v_\parallel /v_\text {ti} = \pm 2.313$ for $\beta _i = 0.3$ and $v_\parallel /v_\text {ti} = \pm 0.583$ for $\beta _i = 3$. Each panel follows the layout format of figure 4.

First, the position of the bipolar signature in $v_\parallel /v_{{\rm ti}}$ changes as $\beta _i$ is varied, consistent with the variation of the parallel phase velocity normalized to the ion thermal velocity as $\beta _i$ varies, given by $\omega /(k_\parallel v_{{\rm ti}})= \bar {\omega }/ \beta _i^{1/2}$. For $\beta _i = 0.3$, we obtain $\omega /(k_\parallel v_{{\rm ti}})= 2.313$, indicated by the vertical black dashed line in figure 5(a,b); for $\beta _i = 3$, we obtain $\omega /(k_\parallel v_{{\rm ti}})= 0.583$, as shown in (c,d). In both cases, the bipolar signature remains closely associated in $v_\parallel$ with the parallel phase velocity $\omega /k_\parallel$, as expected for a Landau-resonant energy transfer between the fields and the ions.

Second, looking at the vertical subplots on the left for each panel, which shows the net rate of change of ion energy density $W_i$ mediated by each mechanism averaged over a correlation interval equal to one wave period $\tau =T$, we find the surprising result for $\beta _i = 0.3$ that, although LD leads to a net gain of energy by the ions (as expected for collisionless damping of a wave), TTD leads to a net loss of energy from the ions. This finding suggests that the ions are losing energy through the magnetic mirror force while gaining energy through acceleration by the parallel electric field. The rate of ion energy gain by LD is approximately ten times larger than the rate of loss by TTD, so the summed effect of these two mechanisms is energization of the ions by collisionless damping of the wave, as expected. For the $\beta _i = 3$ case, we find the equally surprising result that, although TTD leads to a net gain of energy by the ions, LD is leading to a net loss of energy from the ions; the rate of ion energy gain by TTD is larger than the rate of ion energy loss by LD, so the summed contributions yield a net gain of ion energy as expected for collisionless damping.

How do we reconcile these surprising results with the general expectation of Landau-resonant collisionless damping of waves for a Maxwellian equilibrium velocity distribution? As it turns out, this behaviour is exactly what is predicted by the linear Vlasov–Maxwell dispersion relation. To be specific, we take the complex eigenfrequency from the linear dispersion relation to be given by $\omega + {\rm i}\gamma$, so that time evolution of a plane-wave mode is given by $\exp (-{\rm i}\omega t) \exp (\gamma t)$: positive imaginary components $\gamma >0$ correspond to growth of the wave, and negative imaginary components $\gamma <0$ correspond to damping of the wave. In figure 6, we plot the normalized absolute value of the imaginary component of the wave frequency $|\gamma |/\omega$ vs. $k_\perp \rho _i$ for KAWs using the PLUME solver (Klein & Howes Reference Klein and Howes2015) for a fully ionized proton–electron plasma with isotropic Maxwellian velocity distributions with $T_i/T_e=1$, $m_i/m_e = 1836$, $v_{{\rm ti}}/c=10^{-4}$, $k_\parallel \rho _i=10^{-3}$ over the range $10^{-3} \le k_\perp \rho _i \le 10^2$ for the ion plasma beta values (a) $\beta _i=0.3$, (b) $\beta _i=1$ and (c) $\beta _i=3$. We plot separately the total collisionless damping rate (black solid) due to both ions and electrons, the total ion damping rate (green solid) and the separate contributions to the ion damping rate from TTD (red) and LD (blue). For the separated TTD and LD contributions, we plot negative imaginary components (which correspond to collisionless damping of the wave) using dashed lines, and positive imaginary components (which correspond to collisionless growth of the wave) using dotted lines.

Figure 6. Linear dispersion relations for KAWs from PLUME calculations with the realistic mass ratio $m_i/m_e = 1836$, showing the absolute value of the normalized wave growth rate $|\gamma |/\omega$ as a function of the dimensionless perpendicular wave vector $k_\perp \rho _i$ for (a) $\beta _i = 0.3$, (b) $\beta _i = 1$ and (c) $\beta _i = 3$. The vertical black dashed line at $k_\perp \rho _i = 1$ indicates the values used in the single KAW AstroGK simulations. We plot $\gamma$ (total damping rate, black), $\gamma _i$ (total ion damping rate, green), $\gamma _{i{\rm TTD}}$ (ion growth or damping rate via the magnetic mirror force, red) and $\gamma _{i{\rm LD}}$ (ion growth or damping rate via the electrostatic force, blue). Line styles – solid, dashed and dotted – represent the total damping rates, damping rates separated by mechanism and growth rates separated by mechanism, respectively.

Although all of the KAW dispersion relations plotted in figure 6 yield a net effect of collisionless damping by the ions, over some ranges in $k_\perp \rho _i$ (dotted lines) either TTD or LD individually may lead to a net transfer of energy from the ions to the waves over the course of a single-wave period. For example, for the $\beta _i=3$ case in figure 6(c), the imaginary component due to LD is positive over $1.4 \times 10^{-3} \lesssim k_\perp \rho _i \lesssim 2.1$, corresponding to a transfer of energy from the ions to the wave, and is negative outside of that range, corresponding to a transfer of energy from the wave to the ions. A transfer from ions to the wave would lead to a growth of the wave, but the sum of both of the LD and TTD contributions is always negative for these cases with a Maxwellian velocity distribution, leading to a net collisionless damping of the wave.

The perpendicular wavenumber $k_\perp \rho _i=1$ of our simulated single waves is indicated in figure 6 by the vertical black dashed line, and with these plots we can understand the results presented in figure 6. For the $\beta _i=0.3$ case, at $k_\perp \rho _i=1$ we have $\gamma _{i{\rm LD}} <0$ (blue dashed) and $\gamma _{i{\rm TTD}} >0$ (red dotted), suggesting that ions gain energy due to $E_\parallel$ but lose energy due to the mirror force when averaged over the full wave period. This finding agrees with the net gain of ion energy by LD in figure 5(b) and the net loss of ion energy by TTD in (a). Similarly, for the $\beta _i=3$ case, at $k_\perp \rho _i=1$ we have $\gamma _{i{\rm TTD}} <0$ (red dashed) and $\gamma _{i{\rm LD}} >0$ (blue dotted), suggesting that ions gain energy due to the mirror force but lose energy due to $E_\parallel$ when averaged over the full wave period. Again, this finding from the linear Vlasov–Maxwell dispersion relation agrees with the net gain of ion energy by TTD in figure 5(c) and the net loss of ion energy by LD in (d).

To further understand the physical meaning of $\gamma _{i{\rm TTD}} >0$ in the $\beta _i=0.3$ case, we first point out that the only mechanism that can change the particle energy is work done by the electric field, and the rate of change of the ion energy density $W_i$ is given by $\boldsymbol {j}_i \boldsymbol {\cdot } \boldsymbol {E}$. For TTD, this energization arises from the component of the electric field that is perpendicular to the magnetic field (Howes et al. Reference Howes, Huang and Felix2024) (whereas LD energizes particles through the parallel component of the electric field), so here we consider the perpendicular contribution to the energization ${j}_{\perp i} {E}_\perp$. For a single plane wave, the net transfer of energy to or from the ions by ${j}_{\perp i} {E}_\perp$ over a single wave period depends on the phase of the perpendicular electric field fluctuation ${E}_\perp$ relative to the phase of the self-consistent perpendicular component of the ion current density associated with the wave, ${j}_{\perp i}$. If the phase difference $\delta \phi$ is such that there is an in-phase component $0 < \delta \phi < {\rm \pi}/2$, there will be a net energization of the ions; if there is an out-of-phase component ${\rm \pi} /2 < \delta \phi < {\rm \pi}$, the ions will lose energy. The eigenfunctions arising from solutions of the linear Vlasov–Maxwell dispersion relation dictate the phases of the components of the electric field and current density. For the $\beta _i=0.3$ case, this eigenfunction dictates that ${j}_{\parallel i}$ and ${E}_\parallel$ are in phase, leading to ion energization and wave damping by ${E}_\parallel$ (yielding LD), but ${j}_{\perp i}$ and ${E}_\perp$ are out of phase, so the magnetic mirror force partly counteracts the damping of the wave.

4.1. Turbulence simulation set-up

We perform kinetic simulations of strong plasma turbulence using the astrophysical gyrokinetics code AstroGK (Numata et al. Reference Numata, Howes, Tatsuno, Barnes and Dorland2010) for three values of the ion plasma beta $\beta _i=0.3,1,3$. Each simulation has numerical resolution $(n_x, n_y, n_z, n_{\lambda }, n_E, n_s) = (96, 96, 32, 64, 32, 2)$ within a simulation domain $L_\perp ^2 \times L_\parallel = (8 {\rm \pi}\rho _i)^2 \times (2{\rm \pi} a_0)$, where the elongation of the domain along the equilibrium magnetic field $\boldsymbol {B}_0=B_0 \hat {\boldsymbol {z}}$ is characterized by the arbitrary gyrokinetic expansion parameter $\epsilon \sim \rho _i/a_0 \ll 1$. The proton-to-electron temperature ratio of the Maxwellian equilibrium is $T_i/T_e=1$, and we choose a reduced mass ratio $m_i/m_e = 36$ to ensure that we fully resolve the kinetic damping mechanisms needed to achieve a steady-state turbulent cascade in a driven simulation, as discussed in Howes et al. (Reference Howes, McCubbin and Klein2018). These parameters lead to a fully resolved range of perpendicular wavenumbers $0.25 \le k_\perp \rho _i \le 7.75$, or $0.042\le k_\perp \rho _e \le 1.29$. For the $\beta _i = 0.3$ and $\beta _i = 1$ simulations, the proton and electron collisionalities are set to $\nu _s/(k_{\parallel 0} v_{{\rm ti}})=0.1$, and for the $\beta _i = 3$ simulation, $\nu _s/(k_{\parallel 0} v_{{\rm ti}}) = 0.05$. These collisionalities ensure weakly collisional plasma conditions, yet prevent the small-scale variations that develop in velocity space from becoming unresolved on the velocity grid.

Turbulence in the simulations is driven from zero initial conditions using an oscillating Langevin antenna (TenBarge et al. Reference TenBarge, Howes, Dorland and Hammett2014) with characteristic frequency $\omega _0/(k_{\parallel 0} v_A) = 0.9$ and decorrelation rate $\sigma _0/(k_{\parallel 0} v_A) = -0.3$ to drive four Alfvén wave modes at the domain scale with wave vectors $(k_x \rho _i, k_y\rho _i, k_z a_0) = (0.25, 0, \pm 1)$ and $(0, 0.25, \pm 1)$, generating four perpendicularly polarized Alfvén waves propagating in both directions along the equilibrium magnetic field. The amplitude of the driving is chosen to satisfy critical balance with a nonlinearity parameter $\chi \equiv k_\perp \delta B_\perp /(k_\parallel B_0) \simeq 1$ (Goldreich & Sridhar Reference Goldreich and Sridhar1995; Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2008a) at the driving scale $k_\perp \rho _i=0.25$. This driving has been shown to generate effectively a strong plasma turbulent cascade to small scales in previous kinetic simulations through nonlinear interactions between the counterpropagating Alfvén waves (Howes et al. Reference Howes, Dorland, Cowley, Hammett, Quataert, Schekochihin and Tatsuno2008b, Reference Howes, TenBarge, Dorland, Quataert, Schekochihin, Numata and Tatsuno2011; Howes & Nielson Reference Howes and Nielson2013; TenBarge & Howes Reference TenBarge and Howes2013; TenBarge, Howes & Dorland Reference TenBarge, Howes and Dorland2013; Howes et al. Reference Howes, McCubbin and Klein2018; Verniero, Howes & Klein Reference Verniero, Howes and Klein2018; Verniero & Howes Reference Verniero and Howes2018; Horvath et al. Reference Horvath, Howes and McCubbin2020; Conley et al. Reference Conley, Howes and McCubbin2023). To provide the data needed to apply the field–particle correlation analysis, the electromagnetic fluctuations and proton velocity distributions are sampled at a high cadence at twenty-four probe points that are distributed throughout the domain, sixteen in the $xy$-plane at $z=0$, and the remaining eight along the $z$-axis, as illustrated in figure 2 of Horvath et al. (Reference Horvath, Howes and McCubbin2020).

The time scale associated with the outer scale of the turbulent cascade in each simulation is the wave period of the domain-scale Alfvén wave $T$, and the simulations are run for $6.78 T$ for the $\beta _i=0.3$ simulation, $5.51 T$ for the $\beta _i=1$ simulation and $3.61 T$ for the $\beta _i=3$ simulation. The perpendicular magnetic energy spectrum $E_{B_\perp }(k_\perp )$ at the end of each of the simulations is shown in figure 7, showing the spectrum for $\beta _i=0.3$ (red), $\beta _i=1$ (black) and $\beta _i=3$ (blue). These spectra demonstrate that each simulation yields a broadband turbulent spectrum, with the spectral slope for each simulation consistent with the expectation of $-5/3$ for strong plasma turbulence (Goldreich & Sridhar Reference Goldreich and Sridhar1995) in the inertial range at $k_\perp \rho _i<1$, and steepening of the spectrum at the transition to the dissipation range at $k_\perp \rho _i\sim 1$. In the dissipation range at $k_\perp \rho _i> 1$, the spectral slopes begin around $-3.2$, steepening as $k_\perp \rho _e \rightarrow 1$ due to the resolved kinetic dissipation mechanisms that remove energy from the turbulent cascade. Note that these dissipation range slopes are slightly steeper than the values ranging from $-2.7$ to $-3.1$ typically observed in the solar wind (Sahraoui et al. Reference Sahraoui, Huang, Belmont, Goldstein, Rétino, Robert and De Patoul2013), but this is to be expected due to the unphysical mass ratio of $m_i/m_e = 36$, which effectively enhances the damping rate due to electrons relative to the realistic mass ratio case (TenBarge et al. Reference TenBarge, Howes and Dorland2013), leading to slightly steeper dissipation range spectra for stronger damping (Howes, Tenbarge & Dorland Reference Howes, Tenbarge and Dorland2011).

Figure 7. Perpendicular magnetic energy spectra at the end of each of the turbulence simulations, showing $\beta _i=0.3$ (red), $\beta _i=1$ (black) and $\beta _i=3$ (blue). Vertical dotted lines indicate the limit of fully resolved perpendicular wavenumbers in the simulation, $0.25 \le k_\perp \rho _i \le 7.75$ or $0.042\le k_\perp \rho _e \le 1.29$.

In figure 8, we plot the normalized damping rates $|\gamma |/\omega$ from the linear dispersion relation for the simulation parameters with the reduced mass ratio $m_i/m_e=36$, presenting the results for (a) $\beta _i=0.3$, (b) $\beta _i=1$ and (c) $\beta _i=3$, in the same format as presented in figure 6. Here, we plot vertical black dashed lines at the perpendicular wavenumber limits of the simulation at $k_\perp \rho _i=0.25$ and $k_\perp \rho _i=7.75$. Note the salient features that TTD yields a loss of ion energy for $\beta _i=0.3$ in (a), and LD yields a loss of ion energy for $\beta _i=3$ at $k_\perp \rho _i \lesssim 1.5$. These calculations of the linear wave properties and the effective ion energization rates by TTD and LD provide an important theoretical framework for the interpretation of our field–particle analysis results.

Figure 8. Linear dispersion relations for KAWs from PLUME calculations with the reduced mass ratio $m_i/m_e = 36$, showing the absolute value of the normalized wave damping or growth rate $|\gamma |/\omega$ as a function of the dimensionless perpendicular wavenumber $k_\perp \rho _i$ for (a) $\beta _i = 0.3$, (b) $\beta _i = 1$ and (c) $\beta _i = 3$. The two vertical black dashed lines at $k_\perp \rho _i = 0.25$ and $7.75$ label the range consistent with the AstroGK turbulence simulations, and the two vertical green dashed lines mark the range of $1/e$ of the peak value of $\gamma _i$. The horizontal dashed black line at $|\gamma |/\omega = 10^{-1}$ indicates the threshold above which significant damping or growth occurs. Each panel follows the layout format of figure 6.

4.2. Choosing the correlation interval $\tau$

Unlike in the single KAW simulations, where the single-wave period $T$ is the obvious choice for the correlation interval $\tau$ to eliminate the oscillatory contribution to the transfer of energy from fields to particles, choosing $\tau$ for a plasma supporting broadband turbulent fluctuations is less straightforward. The longest wave period for an Alfvén wave at the domain scale in our $\beta _i=1$ simulation is $\tau \omega _A \simeq 6.28$. In figure 9, for a single probe position in the $\beta _i=1$ simulation, we present (a) the total energy transfer rate to ions due to TTD $(\partial W_i/\partial t)_{{\rm TTD}}$ and (b) the energy transfer rate at $v_\parallel /v_{{\rm ti}}=-1.3$, for a correlation interval spanning $0 \le \tau \omega _A \le 15$. The instantaneous values of the local energy transfer rates (with $\tau =0$, blue) exhibit large fluctuations with both positive and negative signs, but for $\tau \omega _A = 6.4$ (black) and longer correlation intervals, those large fluctuations are averaged out, leading to a time-averaged energy transfer rate that is approximately an order of magnitude smaller in amplitude than the peaks of the instantaneous value. In figure 9, we also show timestack plots $C_{\delta B_\parallel, i}(v_\parallel, t; \tau )$ for (c) $\tau =0$ and (d) $\tau \omega _A =6.4$, showing that a relatively persistent bipolar signature of TTD is revealed at $v_\parallel /v_{{\rm ti}}=-1.1$ in the $\tau \omega _A =6.4$ case with peak amplitudes about an order of magnitude smaller than the instantaneous case with $\tau =0$. Thus, we choose a correlation interval $\tau \omega _A = 6.4$ to perform the field–particle correlation analysis of our $\beta _i=1$ turbulence simulation.

Figure 9. Analysis of correlation interval selection for the $\beta _i = 1$ AstroGK turbulence simulation at probe 13 $({\rm \pi} \rho _i, 7 {\rm \pi}\rho _i, 0)$. (a,b) Time evolution of (a) the rate of change of ion kinetic energy density due to TTD, denoted as $\partial W_i /\partial t$, and (b) the reduced correlation $C_{\delta B_{\parallel, i}} (v_\parallel, t)$ at $v_\parallel = -1.3 v_{{\rm ti}}$. Both quantities are presented over a range of $\tau \omega _A$ values from 0 to 15. The selected $\tau \omega _A$ value of 6.4 is marked with a black line. (c,d) Timestack plots of the reduced correlation $C_{\delta B_{\parallel, i}} (v_\parallel, t)$ for (c) $\tau \omega _A = 0$ and (d) $\tau \omega _A = 6.4$, where the range of parallel phase velocities of KAWs that experience significant damping by ions is indicated by vertical dashed lines at $v_\parallel / v_\text {ti} = \pm 1.020$ and $v_\parallel / v_\text {ti} = \pm 1.704$.

4.4. Variations with ion plasma beta $\beta _i$

Next, we explore how the velocity-space signatures of TTD and LD in plasma turbulence vary with changing the ion plasma beta $\beta _i = v_{{\rm ti}}^2/v_A^2$. Because $\beta _i$ is a function of the ratio of the ion thermal velocity to the Alfvén velocity, it directly characterizes where the parallel wave phase velocity falls with the ion velocity distribution, making it the most important parameter controlling resonant wave–particle interactions in a weakly collisional plasma. Specifically, $\omega /(k_\parallel v_{{\rm ti}}) = \bar {\omega } \beta _i^{-1/2}$, where the parallel phase velocity normalized to the Alfvén velocity $\bar {\omega } \equiv \omega /(k_\parallel v_{A})$ typically has a value $\bar {\omega } \sim 1$ for the perpendicular wavelengths $k_\perp \rho _i \sim 1$ at which the ions strongly interact with the waves. Note that, at the perpendicular scale of the domain $k_{\perp 0} \rho _i=0.25$, the Alfvén wave has the $\bar {\omega } \simeq 1$ for all values of $\beta _i$, so we simply choose a correlation interval $\tau \omega _A = 6.4$ for all of the turbulence simulation analysis below.

We plot some typical velocity-space signatures for TTD and LD for the $\beta _i = 0.3$ simulation in figure 11. Note that, for this value of $\beta _i = 0.3$, the contribution of TTD is always to remove energy from the ions, as shown in figure 8(a), so we expect only negative TTD signatures, similar to that shown in figure 5(a). Consequently, we expect LD to dominate the removal of energy from the turbulence. Performing the TTD analysis to determine $C_{\delta B_\parallel, i}(v_\parallel,v_\perp ;\tau )$, we display a typical gyrotropic velocity-space signature in figure 11(a) with the associated timestack plot at the same probe in (b), showing two bipolar signatures with a net negative energy transfer rate and zero crossings at $v_\parallel /v_{{\rm ti}} \simeq \pm 0.7$. These look like typical reversed TTD signatures, but the phase velocity is not within the expected range $1.8 \lesssim |v_\parallel /v_{{\rm ti}}| \lesssim 2.4$ for a KAW with $\beta _i = 0.3$. Only 3 of the 24 probes showed reversed TTD signatures with phase velocities closer to the expected range, as shown in figure 11(c,d) with a bipolar zero crossing around $v_\parallel /v_{{\rm ti}} \simeq -1.5$.

Figure 11. Velocity-space signatures of TTD (a–d) and LD (e–h) sampled from AstroGK turbulence simulation with $0.25 \leq k_\perp \rho _i \leq 7.75$, $T_i / T_e = 1$ and $\beta _i = 0.3$. The correlation interval is set as $\tau \omega _A = 6.4$. (a,c,e,g) Gyrotropic plane $(v_\parallel, v_\perp )$ signatures, and (b,d,f,h) Timestack plots of the $v_\perp$-integrated reduced correlation; both following the layout format of figure 10. The resonant parallel phase velocity ranges are marked by the four vertical dashed lines at $v_\parallel /v_\text {ti} =\pm 1.832$ and $v_\parallel /v_\text {ti} =\pm 2.373$. From top to bottom, data are taken from probe 7 $(5 {\rm \pi}\rho _i, 3 {\rm \pi}\rho _i, 0)$, probe 15 $(5 {\rm \pi}\rho _i, 7 {\rm \pi}\rho _i, 0)$, probe 20 $(4 {\rm \pi}\rho _i, 4 {\rm \pi}\rho _i, -0.125 {\rm \pi}a_0)$ and probe 5 $({\rm \pi} \rho _i, 3 {\rm \pi}\rho _i, 0)$, respectively.

Performing the LD analysis to determine $C_{E_\parallel, i}(v_\parallel,v_\perp ;\tau )$ in the same $\beta _i = 0.3$ turbulence simulation, we find a similar intriguing result that we commonly find bipolar signatures associated with positive energy transfer to ions, but with zero crossings well below the expected range of $1.8 \lesssim |v_\parallel /v_{{\rm ti}}| \lesssim 2.4$. In figure 11(e), we see two bipolar signatures at $v_\parallel /v_{{\rm ti}} \simeq -0.7$ and $v_\parallel /v_{{\rm ti}} \simeq 1.2$. Only 2 of the 24 probes recover LD velocity-space signatures in the expected range, such as that shown in figure 11(g,h).

Turning next to the field–particle correlation analysis of the $\beta _i = 3$ turbulence simulation, the linear dispersion relation plot for KAWs in figure 8(c) shows that LD removes energy from ions for waves with $k_\perp \rho _i \lesssim 1.7$, but TTD positively energizes ions over perpendicular wavenumbers with $k_\perp \rho _i \lesssim 4$. At all probes in the simulation, we find bipolar velocity-space signatures of positive energy transfer by TTD to ions at both positive and negative parallel velocities near the expected range $0.58 \lesssim |v_\parallel /v_{{\rm ti}}| \lesssim 0.94$, with a typical case illustrated in figure 12, showing (a) the gyrotropic velocity-space signature $C_{\delta B_\parallel, i}(v_\parallel,v_\perp ;\tau )$ and (b) the timestack plot $C_{\delta B_\parallel, i}(v_\parallel,t;\tau )$ indicating that these signatures are persistent in time. On the other hand, analysing the signatures of LD using (c) the gyrotropic velocity-space signature $C_{E_\parallel, i}(v_\parallel,v_\perp ;\tau )$ and (d) the timestack plot $C_{E_\parallel, i}(v_\parallel,t;\tau )$, the pattern of energy transfer to ions is widely variable, with ions losing energy more than gaining energy, in agreement with the expectations from the linear dispersion relation. During the majority of the time when clear reversed bipolar patterns are visible for LD, they appear to have zero crossings that are close to the expected range of parallel resonant velocities.

Figure 12. Velocity-space signatures of TTD (a,b) at probe 22 $(4 {\rm \pi}\rho _i, 4 {\rm \pi}\rho _i, 0.375 {\rm \pi}a_0)$ and LD (c,d) at probe 5 $({\rm \pi} \rho _i, 3 {\rm \pi}\rho _i, 0)$ sampled from AstroGK turbulence simulation with $0.25 \leq k_\perp \rho _i \leq 7.75$, $T_i / T_e = 1$ and $\beta _i = 3$. The correlation interval is set as $\tau \omega _A = 6.4$. (a,c) Gyrotropic plane $(v_\parallel, v_\perp )$ signatures, and (b,d) Timestack plots of the $v_\perp$-integrated reduced correlation; both following the layout format of figure 10. The resonant parallel phase velocity ranges are marked by the four vertical dashed lines at $v_\parallel /v_\text {ti} =\pm 0.583$ and $v_\parallel /v_\text {ti} =\pm 0.936$.

Since the field–particle correlation analysis of our single-wave simulations in § 3.3 shows that the velocity-space signatures of TTD and LD appear near the resonant parallel velocities of the waves, as illustrated in figure 5, it raises the question of why the velocity-space signatures in the $\beta _i = 0.3$ simulation in figure 11 often do not appear in the expected range of resonant velocities. There exist at least two possible explanations for this finding: (i) nonlinear transfer from Alfvén waves to other wave modes, and (ii) nonlinear TTD. Before going into the details of each of these possibilities below, it is worthwhile emphasizing that the field–particle correlation technique determines the total energy transfer to particles over the full velocity distribution, meaning that the net energy transfer within a region of velocity space is effectively weighted by the number of particles within that region of the velocity distribution. To illustrate this point, let us focus on the specific example of figure 11(a), where the range of resonant parallel phase velocities for the Alfvén mode (vertical black lines) occurs in the range $1.8 \lesssim v_\parallel /v_{{\rm ti}} \lesssim 2.4$. Even if there exists resonant energy transfer to the particles at these suprathermal parallel velocities, a potentially competing mechanism (such as the two possibilities named above) could dominate the net particle energization if it transfers energy to particles lying within the bulk of the velocity distribution at $v_\parallel /v_{{\rm ti}} \sim 1$, where many more particles can participate in the energy transfer. Thus, the velocity-space signatures arising from the field–particle correlation analysis are generally weighted by the underlying velocity distribution to emphasize energy transfer mechanisms interacting with the main part of the velocity distribution at $v_\parallel /v_{{\rm ti}} \sim 1$.

First, let us analyse how nonlinear energy to other wave modes may explain the velocity-space signatures observed in figure 11. Due to the predominantly Alfvénic nature of fluctuations in space plasma turbulence (Tu & Marsch Reference Tu and Marsch1995; Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009; Bruno & Carbone Reference Bruno and Carbone2013), we focus on the collisional damping rates of Alfvénic fluctuations by TTD and LD in this study. However, the plasma turbulence in our gyrokinetic simulations is broadband and may contain slow magnetosonic fluctuations that arise from nonlinear couplings among the turbulent fluctuations. Although slow magnetosonic waves are not generated by nonlinear couplings among the dominantly Alfvénic fluctuations in the magnetohydrodynamics (MHD) limit at $k_\perp \rho _i \ll 1$ (Schekochihin et al. Reference Schekochihin, Cowley, Dorland, Hammett, Howes, Quataert and Tatsuno2009), at the ion kinetic scales $k_\perp \rho _i \sim 1$ it is possible that energy can be nonlinearly transferred into slow magnetosonic fluctuations. These kinetic slow wave fluctuationsFootnote ⁵ may have a different parallel phase velocity from the Alfvénic fluctuations and thereby mediate damping of the turbulent energy through either TTD or LD at phase velocities that fall within the bulk of the velocity distribution at $v_\parallel /v_{{\rm ti}} \sim 1$. The second possibility is that nonlinear TTD may occur, whereby the beat mode fluctuations (which are not natural wave modes of the system) – arising from nonlinear interactions between Alfvénic fluctuations with different frequencies and wave vectors – can have an effective phase velocity that falls in the core of the velocity distribution, leading to efficient net energy transfer to the ions via collisionless wave–particle interactions.

In closing this discussion about the velocity-space signatures observed in figure 11, a final point to emphasize is that the clear bipolar signatures of negative energy transfer from ions by TTD in figure 11(a) and of positive energy transfer to ions by LD in figure 11(e) for $\beta _i=0.3$ suggest that some resonant energy transfer mechanism is governing the energization in these simulations; the details of this turbulent damping mechanism, which may include nonlinear couplings to other linear wave modes or nonlinear beat modes as suggested above, are clearly a ripe avenue for exploration in future work.

4.5. Net turbulent damping as a function of ion plasma beta $\beta _i$

To estimate the net energy density transfer to or from ions mediated by TTD or LD averaged over the volume of the simulation, we integrate the energy density transfer rate due to each mechanism over the duration of each turbulence simulation (after the energy spectra have reached a statistically steady state) and average over all 24 probes for each simulation. This procedure provides a statistical estimate of the total change in ion energy density by each mechanism, denoted by $W_{i,{\rm TTD}}$ and $W_{i, {\rm LD}}$. We compute the ratio of these ion energy density changes by each mechanism to the total ion energy density change $W_i = W_{i,{\rm TTD}} + W_{i, {\rm LD}}$, and plot $W_{i,{\rm TTD}}/W_i$ (green) and $W_{i, {\rm LD}}/W_i$ (blue) versus the ion plasma beta $\beta _i$ in figure 13. Variability in the energy density changes due to TTD and LD is indicated by error bars on each point, computed using the standard deviation of the time-integrated energy density changes at all 24 probes. The red solid line at $1$ presents the sum of $W_{i,{\rm TTD}}/W_i$ and $W_{i, {\rm LD}}/W_i$, and the black dashed line at zero highlights whether a mechanism yields a positive net energy density transfer to ions or a negative net energy density transfer from ions.

Figure 13. Ratio of the change of the ion kinetic energy density due to TTD and LD to the total change of the ion kinetic energy density during the analysis time, both averaged over all 24 probes, plotted against $\beta _i$. The error bars represent the standard deviations calculated across all probes.

For $\beta _i=0.3$, the ion energization is dominated by LD, with TTD yielding a small and slightly negative energy transfer. The ion LD and TTD contributions to the damping rate from the linear dispersion relation in figure 8(a) provide intuition about the relative role of these two collisionless damping mechanisms in a turbulent plasma. Note that the broadband turbulence in the simulation consists of wave modes spanning the range $0.25 \le k_\perp \rho _i \le 7.75$, denoted by the two vertical dashed black lines in figure 8(a), with a monotonically decreasing magnetic energy spectrum, as shown in figure 7. The net ion energization by each mechanism expected in a given simulation can be intuitively estimated by an integration over $k_\perp \rho _i$ of the product of the damping rate for that mechanism with the electromagnetic energy at each scale. For $\beta _i=0.3$, figure 8(a) shows that $\gamma _{i{\rm TTD}}>0$ and $\gamma _{i{\rm LD}}<0$ over the full range of perpendicular wavenumbers in the simulation, so the numerical finding of net energy transfer rates in figure 13 appears to be consistent with the expectation from the linear dispersion relation.

For $\beta _i=1$, both LD and TTD yield net positive energization of the ions, but LD dominates over TTD once more, as seen in figure 13. This result is broadly consistent with the linear damping rates from the linear dispersion relation plotted in figure 8(b), where the LD rate integrated over $k_\perp \rho _i$ would be expected to dominate over the TTD rate, but both mechanisms would be expected to yield net positive ion energization rates.

At $\beta _i=3$, on the other hand, figure 13 shows that TTD mediates a net positive energy transfer to the ions, while LD governs a net negative energy transfer with a slightly smaller magnitude than the positive TTD energy transfer. The linear dispersion relation results in figure 8(c) predict a more complicated behaviour than the $\beta _i=0.3$ and $\beta _i=1$ cases: TTD yields ion energization for all $k_\perp \rho _i \lesssim 5$, but LD leads to ion-to-wave energy transfer for $k_\perp \rho _i \lesssim 1.5$ and wave-to-ion energy transfer for $k_\perp \rho _i \gtrsim 1.5$. The velocity-space-integrated rates of energy transfer by each mechanism at each probe – for example, the vertical left panels of figure 12(a,c) – show variations in time that appear to be consistent with these expectations from linear theory. The TTD rates of ion energy density change are generally positive with moderate amplitudes. On the other hand, the LD rates of ion energy density change are somewhat larger in amplitude but vary with both positive and negative rates, such that the net LD rate averaged over all probes is negative and slightly smaller in amplitude than the net TTD rate. This broader variation is evident in the larger standard deviation of the $\beta _i=3$ values in figure 13 relative to the $\beta _i=0.3$ and $\beta _i=1$ cases.

In conclusion, it is worthwhile emphasizing that the expectations from linear theory of the relative contributions to the energy density transfer rates from each mechanism, including their overall sign, are only expected to hold in statistical sense, averaged over a sufficiently large volume. This is particularly true for plasma parameter values where the expected linear damping rate for a given mechanism changes sign within the range of $k_\perp \rho _i$ where significant damping is expected, such as the case for ion LD in the $\beta _i=3$ simulation. Based on kinetic numerical simulations of turbulence, the rate of change of energy density for species $s$ given by $\boldsymbol {j}_s \boldsymbol {\cdot } \boldsymbol {E}$ is highly variable spatially, with variations that can be much larger than the spatially or temporally averaged net value. An example from a simulation of a strongly nonlinear Alfvén wave collision – the nonlinear interaction between counterpropagating Alfvén waves that is argued to be the fundamental building block of astrophysical plasma turbulence (Howes et al. Reference Howes, Drake, Nielson, Carter, Kletzing and Skiff2012; Drake et al. Reference Drake, Schroeder, Howes, Kletzing, Skiff, Carter and Auerbach2013; Howes & Nielson Reference Howes and Nielson2013; Howes et al. Reference Howes, Nielson, Drake, Schroeder, Skiff, Kletzing and Carter2013; Nielson, Howes & Dorland Reference Nielson, Howes and Dorland2013; Howes Reference Howes2016; Verniero et al. Reference Verniero, Howes and Klein2018; Verniero & Howes Reference Verniero and Howes2018) – is seen in figure 8 of Howes et al. (Reference Howes, McCubbin and Klein2018), where the wave-period-averaged energy density transfer rates are found to be an order of magnitude smaller than the instantaneous rates. Furthermore, even for an overall damping rate that energizes the plasma particles at the expense of the turbulent electromagnetic energy, the net energy transfer rate averaged over time at single position may be negative, even when the spatially averaged energy transfer rate is positive, as seen in figure 8(d) of Howes et al. (Reference Howes, McCubbin and Klein2018).

That the turbulent energy damping rate is highly oscillatory, both in space and time, with a much smaller amplitude net energy damping rate that is generally positive (indicating a damping of turbulence and consequent energization of particles), should not be especially surprising. The nonlinear energy transfer from large to small scales by the turbulence is likewise found to be highly oscillatory, with third-moment calculations of the instantaneous energy transfer rates (Kolmogorov Reference Kolmogorov1941; Politano & Pouquet Reference Politano and Pouquet1998) yielding large values of both signs, with the net averaged value having a much smaller amplitude of positive energy transfer to small scales (Coburn et al. Reference Coburn, Smith, Vasquez, Forman and Stawarz2014, Reference Coburn, Forman, Smith, Vasquez and Stawarz2015). Thus, field–particle correlation calculations of the energy density transfer rate at a single point, even averaged over a long duration, may still lead to a result in a turbulent plasma that appears to conflict with the expectation from linear theory. Only with a sufficient large statistical average, here estimated using the average over all 24 probe positions, should we expect the results to be consistent with the general expectations of linear theory – the results in figure 13 indeed appear to be consistent with the general expectations for the relative contributions of LD and TTD from linear theory.