3.1 Energy conservation

To derive the expression for the conserved energy of electrostatic fluctuations in an unmagnetized plasma, we begin with the electrostatic analogue of Poynting’s theorem. Beginning with the 1-D electrostatic limit of the Ampere–Maxwell Law, $\unicode[STIX]{x2202}E/\unicode[STIX]{x2202}t=-4\unicode[STIX]{x03C0}j$ , we multiply by $E$ to obtain the rate of change of electrostatic field energy density,

(3.3) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\frac{E^{2}}{8\unicode[STIX]{x03C0}}\right)=-jE. & & \displaystyle\end{eqnarray}$$

Next, we multiply the Vlasov equation (3.1) for species $s$ by $m_{s}v^{2}/2$ and integrate over velocity and position. Exchanging the order of differentiation and integration of the independent variables, we may obtain the form

(3.4) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\int \,\text{d}x\int \,\text{d}v\frac{1}{2}m_{s}v^{2}f_{s}+\int \,\text{d}x\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\left[\int \,\text{d}v\frac{1}{2}m_{s}v^{3}f_{s}\right]\nonumber\\ \displaystyle & & \displaystyle -\,\int \,\text{d}x\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}x}\int \,\text{d}v\left(\frac{q_{s}v^{2}}{2}\right)\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}v}=0.\end{eqnarray}$$

The first term of this equation represents the rate of change of the microscopic kinetic energy of species $s$

(3.5) $$\begin{eqnarray}\displaystyle W_{s}\equiv \int _{-L/2}^{L/2}\,\text{d}x\int _{-\infty }^{\infty }\,\text{d}v\frac{1}{2}m_{s}v^{2}f_{s}. & & \displaystyle\end{eqnarray}$$

The second term, associated with the ballistic behaviour of particles, is a perfect differential in $x$ , yielding zero for either periodic boundary conditions, $f_{s}(x=-L/2,v)=f_{s}(x=L/2,v)$ , or boundary conditions at infinity, $\lim _{L\rightarrow \infty }f_{s}(x=\pm L/2,v)=0$ . Physically, the ballistic term can only transport energy from one position to another, so when integrated over the volume yields a net change of zero for $W_{s}$ . The third term may be integrated by parts in velocity to yield the result

(3.6) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}W_{s}}{\unicode[STIX]{x2202}t}=-\int \,\text{d}x\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}x}\int \,\text{d}v\,q_{s}vf_{s}=\int \,\text{d}x\,j_{s}E, & & \displaystyle\end{eqnarray}$$

where the current density for species $s$ is given by $j_{s}=\int \,\text{d}v\,q_{s}vf_{s}$ .

Since the total current density $j=\sum _{s}j_{s}$ , we may integrate (3.3) over space and combine it with (3.6) summed over species to obtain an expression for the conservation of energy in a 1D-1V electrostatic plasma,

(3.7) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\int \,\text{d}x\left(\frac{E^{2}}{8\unicode[STIX]{x03C0}}\right)+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\mathop{\sum }_{s}\int \,\text{d}x\int \,\text{d}v\frac{1}{2}m_{s}v^{2}f_{s}=0. & & \displaystyle\end{eqnarray}$$

Therefore, the conserved Vlasov–Poisson energy $W$ for electrostatic fluctuations in a collisionless, unmagnetized plasma is given by

(3.8) $$\begin{eqnarray}\displaystyle W=\int \,\text{d}x\frac{E^{2}}{8\unicode[STIX]{x03C0}}+\mathop{\sum }_{s}\int \,\text{d}x\int \,\text{d}v\frac{1}{2}m_{s}v^{2}f_{s}. & & \displaystyle\end{eqnarray}$$

We also define the electrostatic field

(3.9) $$\begin{eqnarray}\displaystyle W_{\unicode[STIX]{x1D719}}\equiv \int \,\text{d}x\frac{E^{2}}{8\unicode[STIX]{x03C0}}, & & \displaystyle\end{eqnarray}$$

such that the total conserved Vlasov–Poisson energy for a single ion species plasma is given by

(3.10) $$\begin{eqnarray}\displaystyle W=W_{\unicode[STIX]{x1D719}}+W_{i}+W_{e}. & & \displaystyle\end{eqnarray}$$

3.2 Energy transfer via collisionless wave–particle interactions

To illuminate the secular transfer of energy between the electrostatic field and the particles via resonant wave–particle interactions, it is instructive to examine more closely the different contributions to the change in the particle energy, $W_{s}$ . As (3.7) mandates, in the Vlasov–Poisson system, the energy gain by the particles must be equal to the energy lost from the electrostatic field, $\sum _{s}\unicode[STIX]{x2202}W_{s}/\unicode[STIX]{x2202}t=-\unicode[STIX]{x2202}W_{\unicode[STIX]{x1D719}}/\unicode[STIX]{x2202}t$ . We may express the rate of energy exchange (gain or loss) for species $s$ by

(3.11) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}W_{s}}{\unicode[STIX]{x2202}t}=\int \,\text{d}x\int \,\text{d}v\frac{1}{2}m_{s}v^{2}\frac{\unicode[STIX]{x2202}f_{s}}{\unicode[STIX]{x2202}t}. & & \displaystyle\end{eqnarray}$$

To progress further, we decompose the distribution function into an equilibrium and perturbed component,

(3.12) $$\begin{eqnarray}\displaystyle f_{s}(x,v,t)=f_{s0}(v)+\unicode[STIX]{x1D6FF}f_{s}(x,v,t), & & \displaystyle\end{eqnarray}$$

where the equilibrium distribution function $f_{s0}(v)$ is assumed to be uniform in space and static in time. We also make the additional assumption that $f_{s0}(v)$ is an even function of velocity so that the equilibrium has no bulk plasma flow (first moment), but it need not be a Maxwellian. Not that for solar wind measurements, we may transform to the frame of reference flowing with the solar wind plasma (see § 7.1 for more discussion of the frame of reference of measurements).

We emphasize here that we have made no ordering assumptions on the magnitude of $\unicode[STIX]{x1D6FF}f_{s}$ relative to $f_{s0}$ , so the nonlinear evolution of the distribution function described by this form is not limited in any way. The term $\unicode[STIX]{x1D6FF}f_{s}$ contains the entire (nonlinear) perturbation, not just the lowest-order (linear) perturbation. Of course, the physical limitation

(3.13) $$\begin{eqnarray}\displaystyle f_{s}(x,v,t)=f_{s0}(v)+\unicode[STIX]{x1D6FF}f_{s}(x,v,t)\geqslant 0 & & \displaystyle\end{eqnarray}$$

must always be satisfied, so this means that $\unicode[STIX]{x1D6FF}f_{s}(x,v,t)\geqslant -f_{s0}(v)$ for all values of velocity $v$ . Practically, this does lead to constraints on the allowable time step in numerical simulations to maintain a physically realizable $f_{s}(x,v,t)\geqslant 0$ everywhere.

Substituting (3.12) into the Vlasov equation (3.1), we obtain

(3.14) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}}{\unicode[STIX]{x2202}t}=-v\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}}{\unicode[STIX]{x2202}x}+\frac{q_{s}}{m_{s}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}f_{s0}}{\unicode[STIX]{x2202}v}+\frac{q_{s}}{m_{s}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}}{\unicode[STIX]{x2202}v}. & & \displaystyle\end{eqnarray}$$

In this form, on the right-hand side, the first term is the (linear) ballistic term, the second term is the linear wave–particle interaction term and the third term is the nonlinear wave–particle interaction term. Next, we substitute (3.14) into (3.11), yielding

(3.15) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}W_{s}}{\unicode[STIX]{x2202}t}=\int \,\text{d}x\int \,\text{d}v\frac{1}{2}m_{s}v^{2}\left[-v\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}}{\unicode[STIX]{x2202}x}+\frac{q_{s}}{m_{s}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}f_{s0}}{\unicode[STIX]{x2202}v}+\frac{q_{s}}{m_{s}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}x}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}}{\unicode[STIX]{x2202}v}\right]. & & \displaystyle\end{eqnarray}$$

We now evaluate the influence of each of these terms on the evolution of the microscopic kinetic energy, $W_{s}$ .

The first term, the ballistic term, may be written as a perfect differential in $x$ , thereby yielding a value of zero upon integration over $x$ for periodic or infinite boundaries. The second term, the linear wave–particle interaction term, may be written in the form

(3.16) $$\begin{eqnarray}\displaystyle \int \,\text{d}x\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\left\{\frac{q_{s}\unicode[STIX]{x1D719}}{2}\left[\int \,\text{d}vv^{2}\frac{\unicode[STIX]{x2202}f_{s0}}{\unicode[STIX]{x2202}v}\right]\right\}=0. & & \displaystyle\end{eqnarray}$$

Since we have chosen $f_{s0}$ to be an even function of $v$ , then its derivative $\unicode[STIX]{x2202}f_{s0}/\unicode[STIX]{x2202}v$ is an odd function, so the integrand of the velocity integral is an odd function evaluated over an even interval, yielding zero. In addition, because $f_{s0}$ is not a function of $x$ , everything in the braces is also a perfect differential, so this term will vanish upon integration over $x$ for any choice of $f_{s0}$ , not just even functions of $v$ .

For the third term, the nonlinear wave–particle interaction term, we may integrate by parts in velocity to yield the final result for the rate of change of microscopic kinetic energy for species $s$ ,

(3.17) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}W_{s}}{\unicode[STIX]{x2202}t}=-\int \,\text{d}x\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D719}}{\unicode[STIX]{x2202}x}\int \,\text{d}vq_{s}v\unicode[STIX]{x1D6FF}f_{s}=\int \,\text{d}x\,j_{s}E. & & \displaystyle\end{eqnarray}$$

Therefore, the secular change of particle energy in the Vlasov–Poisson system occurs via the nonlinear wave–particle interaction term in (3.14). Furthermore, the perturbations in the distribution function arising from the collisionless transfer of energy from fields to particles are generated by this term, making it possible to separate the fluctuations in velocity space due to resonant wave–particle interactions from the (generally larger-amplitude) fluctuations generated by the ballistic term and the linear wave–particle interaction term.

Note that linearization of the kinetic system leads to dropping the nonlinear wave–particle interaction term, the third term on the right-hand side of (3.15). But this term is necessary to describe the change in the energy of the particles $W_{s}$ . So, although a linearized system will describe the collisionless Landau damping of the electrostatic waves of the Vlasov–Poisson system (Landau Reference Landau1946), the Vlasov–Poisson energy $W$ given by (3.8) is not conserved in a linearized system. The nonlinear wave–particle interaction term must be retained in order to achieve conservation of $W$ , or an alternative Kruskal–Obermann energy (Kruskal & Oberman Reference Kruskal and Oberman1958; Morrison Reference Morrison1994) can be devised to achieve a conserved quadratic invariant of the linearized system.

In summary, using measurements of the fluctuations in the particle distribution function $\unicode[STIX]{x1D6FF}f_{s}(x,v,t)$ and the electric field $E(x,t)$ , we may calculate the rate of transfer of energy from the fields to the particles by either form of the following expression,

(3.18) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}W_{s}}{\unicode[STIX]{x2202}t}=-\int \,\text{d}x\int \,\text{d}vq_{s}\frac{v^{2}}{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}(x,v,t)}{\unicode[STIX]{x2202}v}E(x,t)=\int \,\text{d}x\int \,\text{d}v\,q_{s}v\unicode[STIX]{x1D6FF}f_{s}(x,v,t)E(x,t).\qquad & & \displaystyle\end{eqnarray}$$

3.3 Diagnosing secular energy transfer

The key challenge in diagnosing the collisionless damping of fluctuations in a turbulent plasma is to separate the often small-amplitude signal of the secular energy transfer from the generally much larger-amplitude signal of the oscillating energy transfer. The arguments of the preceding section suggest that by integrating over all space – or taking a spatial average, as is done in quasilinear theory – we can average out the zero net effect of the oscillating energy transfer and extract the smaller secular energy transfer that is associated with collisionless damping. In this case, the spatial integration eliminates the contribution from the ballistic and linear wave–particle interaction terms in (3.14), isolating the non-zero net effect of the secular energy transfer due to the nonlinear wave–particle interaction term.

Of course, in a numerical simulation, where all of the spatial information is known, spatial integration can be used to isolate the secular energy transfer. But such spatial information is not accessible observationally, where spacecraft measurements are typically made at only a single point (or at most, a few points) in space. Furthermore, numerical simulations of plasma turbulence provide strong evidence that energy dissipation is often highly localized in space (Wan et al. Reference Wan, Matthaeus, Karimabadi, Roytershteyn, Shay, Wu, Daughton, Loring and Chapman2012; Karimabadi et al. Reference Karimabadi, Roytershteyn, Wan, Matthaeus, Daughton, Wu, Shay, Loring, Borovsky and Leonardis2013; TenBarge & Howes Reference Tenbarge and Howes2013; Wu et al. Reference Wu, Perri, Osman, Wan, Matthaeus, Shay, Goldstein, Karimabadi and Chapman2013; Zhdankin et al. Reference Zhdankin, Uzdensky, Perez and Boldyrev2013; Zhdankin, Uzdensky & Boldyrev Reference Zhdankin, Uzdensky and Boldyrev2015b ); so, even in numerical simulations, integration over a volume larger than the region of strong dissipation may obscure the details of the local dissipation mechanism, making it more difficult to identify the physical mechanism responsible. Here we aim to develop a method that can be used to analyse the secular energy transfer using single-point measurements of the electromagnetic fields and velocity distribution functions, enabling an improved analysis of the collisionless damping of turbulent fluctuations in both spacecraft measurements and kinetic numerical simulations.

Let us define the phase-space energy density for a particle species $s$ by $w_{s}(x,v,t)=m_{s}v^{2}f_{s}(x,v,t)/2$ . Note that, for the 1D-1V Vlasov–Poisson system, this is the energy density per unit length per unit velocity. The integral of $w_{s}(x,v,t)$ over all velocity yields the spatial energy density of the plasma species as a function of position, which is the usual meaning of the term energy density. Subsequent integration over the spatial volume yields the total microscopic kinetic energy of the species, $W_{s}$ .

If we want to understand how the phase-space energy density evolves in time, we can take the time derivative of $w_{s}(x,v,t)$ and replace $\unicode[STIX]{x2202}f_{s}/\unicode[STIX]{x2202}t$ using the Vlasov equation to obtain an expression for the instantaneous change of the phase-space energy density,

(3.19) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}w_{s}(x,v,t)}{\unicode[STIX]{x2202}t}=-\frac{1}{2}m_{s}v^{3}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}}{\unicode[STIX]{x2202}x}-q_{s}\frac{v^{2}}{2}\frac{\unicode[STIX]{x2202}f_{s0}(v)}{\unicode[STIX]{x2202}v}E(x,t)-q_{s}\frac{v^{2}}{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}(x,v,t)}{\unicode[STIX]{x2202}v}E(x,t). & & \displaystyle\end{eqnarray}$$

From the analysis of the energy conservation equation (3.15), we know that, if this equation is integrated over all velocity and all physical space, only the third term contributes to the secular energy transfer from fields to particles (or vice versa). However, in the absence of these integrations, all three terms contribute to the instantaneous phase-space energy density change at each point in $(x,v)$ phase space.

3.4 Field–particle correlations

The form of the term responsible for the secular energy transfer in (3.19) suggests that the product of $(-q_{s}v^{2}/2)(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}(x,v,t)/\unicode[STIX]{x2202}v)$ and $E(x,t)$ provides a direct measure of the rate of energy transfer (possibly including an oscillating component). To isolate the small secular component, we take the unnormalized field–particle correlation, $C_{\unicode[STIX]{x1D70F}}(-q_{s}v^{2}/2\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}/\unicode[STIX]{x2202}v,E)$ , at a single point in space over a selected correlation interval $\unicode[STIX]{x1D70F}$ . By correlating the two signals over a sufficiently long interval (more than a linear wave period, $\unicode[STIX]{x1D70F}\gtrsim 2\unicode[STIX]{x03C0}\unicode[STIX]{x1D714}$ ), the oscillating energy transfer is averaged out, extracting the smaller signal of the secular energy transfer.

For measurements of the velocity distribution and the electric field at a single point in space, $x=x_{0}$ , this correlation is a function of velocity, time and the correlation interval,

(3.20) $$\begin{eqnarray}\displaystyle C_{E}(v,t,\unicode[STIX]{x1D70F})=C_{\unicode[STIX]{x1D70F}}\left(-q_{s}\frac{v^{2}}{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}(x_{0},v,t)}{\unicode[STIX]{x2202}v},E(x_{0},t)\right). & & \displaystyle\end{eqnarray}$$

Therefore, the general idea of diagnosing the energy transfer at each point in $(x,v)$ phase space reduces, due to the observational constraints of single-point measurements, to the case of determining the distribution of the energy transfer rate in velocity space. A key advance with this method is that determining how the secular energy transfer rate varies in velocity space provides valuable new information about the physical mechanism responsible for the collisionless damping of the fluctuations. Different mechanisms, such as Landau damping or stochastic ion heating, are likely to have distinct signatures of this damping in velocity space. In this paper, we illustrate this field–particle correlation analysis method using the case of the Landau damping of Langmuir waves in a 1D-1V Vlasov–Poisson plasma, but the concept of using field–particle correlations to diagnose collisionless energy transfer is extremely general, and it can be also applied to examine the damping of turbulence in heliospheric plasmas using single-point spacecraft measurements.

There are two issues that merit further discussion regarding the construction of field–particle correlations. First, because the product of the two terms that are correlated is a direct measure of the instantaneous collisionless energy transfer (specifically, just the component due to the nonlinear wave–particle interaction term), the field–particle correlation is to be performed in the following way: (i) the mean values of the two correlated variables are not subtracted before multiplication; and (ii) the correlation is not normalized by the variances of each of the correlated variables. Although a normalized correlation can be performed and indeed contains information about the nature of the collisionless wave–particle interactions, the process of normalization effectively removes the amplitude variation of the energy transfer rate as a function of velocity, a vital piece of information provided by this analysis. A simple model that predicts the form of the normalized correlation for an exact linear eigenfunction in terms of the normalized damping rate $-\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D714}$ is presented in appendix B.

The second issue regards the applicability of the particular field–particle correlation given in (3.20) to spacecraft observations. The often low velocity-space resolution of particle measurements by spacecraft instrumentation and the corruption by noise for low particle count rates mean that accurate computation of the necessary derivative $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}(x,v,t)/\unicode[STIX]{x2202}v$ may be difficult, or even impossible. But, as shown in § 3.2, upon integration over velocity space, both $(-q_{s}v^{2}/2)(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}(x,v,t)/\unicode[STIX]{x2202}v)$ and $q_{s}v\unicode[STIX]{x1D6FF}f_{s}(x,v,t)$ yield the same result for the total species current, $j_{s}(x,t)$ . Therefore, we propose an alternative field–particle correlation

(3.21) $$\begin{eqnarray}\displaystyle C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})=C_{\unicode[STIX]{x1D70F}}(q_{s}v\unicode[STIX]{x1D6FF}f_{s}(x_{0},v,t),E(x_{0},t)) & & \displaystyle\end{eqnarray}$$

that may be more suitable for implementation with low resolution spacecraft measurements of particle velocity distribution functions. Different collisionless damping mechanisms are likely to produce distinct signatures of any chosen form of field–particle correlation in velocity space. So, although the form $q_{s}v\unicode[STIX]{x1D6FF}f_{s}(x,v,t)E(x,t)$ does not correspond directly to the energy transfer rate at a point in $(x,v)$ phase space as given by (3.19), this alternative field–particle correlation may still be valuable in distinguishing one collisionless damping process from another.

5.1 Case I: moderately damped standing Langmuir wave

According to the procedure outlined in § A.1, we initialize a standing Langmuir wave pattern in VP with $k\unicode[STIX]{x1D706}_{de}=0.5$ using an electron density perturbation $\unicode[STIX]{x1D6FF}n_{e}/n_{0}=0.1$ . For this wavenumber, a numerical solution of the linear dispersion relation for Langmuir waves gives $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{pe}=1.43$ and $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D714}_{pe}=-1.59\times 10^{-1}$ , yielding a resonant phase velocity $v_{p}/v_{te}=\unicode[STIX]{x1D714}/(kv_{te})=2.86$ . For this simulation, $L=4\unicode[STIX]{x03C0}\unicode[STIX]{x1D706}_{de}$ , $v_{max}=5v_{te}$ , and $f_{CFL}=0.05$ .

Figure 1. Energy evolution for the moderately damped Case I up to $\unicode[STIX]{x1D714}_{pe}t=20$ . Plotted are the total perturbed energy $\unicode[STIX]{x1D6FF}W$ (black), field energy $W_{\unicode[STIX]{x1D719}}$ (green), perturbed ion energy $\unicode[STIX]{x1D6FF}W_{i}$ (red) and perturbed electron energy $\unicode[STIX]{x1D6FF}W_{e}$ (blue). Linear (a) and logarithmic (b) scales are both presented.

In figure 1, we plot the evolution of the perturbed energies of the system. The perturbed energy for a species $s$ is given by

(5.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}W_{s}=\int _{-L/2}^{L/2}\,\text{d}x\int _{-\infty }^{\infty }\,\text{d}v\frac{1}{2}m_{s}v^{2}\unicode[STIX]{x1D6FF}f_{s}, & & \displaystyle\end{eqnarray}$$

where we include all contributions to the perturbation, $\unicode[STIX]{x1D6FF}f_{s}=\unicode[STIX]{x1D6FF}f_{sB}+\unicode[STIX]{x1D6FF}f_{sWl}+\unicode[STIX]{x1D6FF}f_{sWn}$ . The total perturbed energy $\unicode[STIX]{x1D6FF}W=W_{\unicode[STIX]{x1D719}}+\unicode[STIX]{x1D6FF}W_{i}+\unicode[STIX]{x1D6FF}W_{e}$ , with an initial value of $\unicode[STIX]{x1D6FF}W=0.2453(n_{0}T_{e}\unicode[STIX]{x1D706}_{de}^{3}/2)$ , is conserved to within 0.05 %. From figure 1, it is clear that the electrostatic electric field energy $W_{\unicode[STIX]{x1D719}}$ (green) is converted primarily to microscopic kinetic energy of the electrons $\unicode[STIX]{x1D6FF}W_{e}$ (blue), with less than 0.2 % of the energy transferred to the ions $\unicode[STIX]{x1D6FF}W_{i}$ (red). The normalized period of the standing Langmuir wave pattern with $k\unicode[STIX]{x1D706}_{de}=0.5$ is $\unicode[STIX]{x1D714}_{pe}T=4.39$ , so 99 % of the electrostatic energy in the wave pattern is secularly transferred to the electrons in approximately 3 periods.

Figure 2. For Case I, (a) total electron distribution function $f_{e}$ (magenta), equilibrium electron distribution function $f_{e0}$ (black) and total perturbed electron distribution function $\unicode[STIX]{x1D6FF}f_{e}$ (cyan). (b) The separated components of the perturbed electron distribution function: (i) $\unicode[STIX]{x1D6FF}f_{eB}$ (green), (ii) $\unicode[STIX]{x1D6FF}f_{eWl}$ (blue) and (iii) $\unicode[STIX]{x1D6FF}f_{eWn}$ (red). Dashed vertical black lines denote the resonant velocities $v=\pm \unicode[STIX]{x1D714}/k$ .

Next we explore the different components of the perturbed electron distribution function for this nonlinear simulation at time $\unicode[STIX]{x1D714}_{pe}t=19.64$ and position $x=0$ . Note that the position $x=0$ is a maximum (anti-node) of the electric field standing wave pattern. In panel (a) of figure 2, we plot the total electron distribution function $f_{e}$ (magenta), the equilibrium electron distribution function $f_{e0}$ (black) and the total perturbed electron distribution function $\unicode[STIX]{x1D6FF}f_{e}=\unicode[STIX]{x1D6FF}f_{eB}+\unicode[STIX]{x1D6FF}f_{eWl}+\unicode[STIX]{x1D6FF}f_{eWn}$ (cyan). We can gain deeper insight into the nature of the fluctuations in the electron distribution function by separating out the components arising from the different terms in (3.14), shown in panel (b): (i) the ballistic term yields $\unicode[STIX]{x1D6FF}f_{eB}$ (green), (ii) the linear wave–particle interaction term yields $\unicode[STIX]{x1D6FF}f_{eWl}$ (blue) and (iii) the nonlinear wave–particle interaction term yields $\unicode[STIX]{x1D6FF}f_{eWn}$ (red). As shown in § 3.2, it is only the nonlinear wave–particle interaction term that leads to a secular transfer of energy from the electrostatic field to the electrons, and this component of the perturbed electron distribution function is generally much smaller than the ballistic and linear wave–particle interaction components. The primary aim of this paper is to devise a procedure to isolate this small component of the fluctuations in the distribution function using only single-point measurements of the particle velocity distribution functions and the electrostatic field.

Figure 3. For Case I, (a) total electron distribution function evaluated at position $x=0$ , $f_{e}(0,v,t)$ (thin magenta), the spatially averaged, quasilinear distribution function $f_{eQL}(v,t)$ (thick magenta) and the equilibrium electron distribution function $f_{e0}$ (black). (b) The nonlinear wave–particle interaction component of the perturbed electron distribution function at position $x=0$ , $\unicode[STIX]{x1D6FF}f_{eWn}(0,v,t)$ (thin red), the quasilinear perturbed distribution function $\unicode[STIX]{x1D6FF}f_{eQL}(v,t)$ (thick red). The quasilinear flattening of the distribution function at the resonant velocities $v=\pm \unicode[STIX]{x1D714}/k$ (dashed black) is apparent in the total quasilinear distribution function $f_{eQL}(v,t)$ (thick magenta), but this signature is obscured in the total electron distribution function evaluated at position $x=0$ , $f_{e}(0,v,t)$ (thin magenta).

If full spatial information about the fluctuations is available, then a spatial average (or integration) over the volume eliminates the large fluctuations associated with the oscillatory energy transfer. This is the standard approach in quasilinear theory, taking a spatial average of the distribution function over the volume to obtain $\unicode[STIX]{x1D6FF}f_{eQL}(v,t)=(1/L)\int \,\text{d}x\,\unicode[STIX]{x1D6FF}f_{e}(x,v,t)$ . In panel (a) of figure 3, we plot the total electron distribution function evaluated at position $x=0$ , $f_{e}(0,v,t)$ (thin magenta) and the spatially averaged, quasilinear distribution function $f_{eQL}(v,t)$ (thick magenta) at time $\unicode[STIX]{x1D714}_{pe}t=19.64$ . It is clear that spatial average eliminates almost all of the fluctuations of the distribution function away from the equilibrium $f_{e0}$ (black). To better see the deviations of the quasilinear distribution function from the equilibrium, in panel (b) we plot an expanded view of the nonlinear wave–particle interaction component of the perturbed electron distribution function at position $x=0$ , $\unicode[STIX]{x1D6FF}f_{eWn}(0,v,t)$ (thin red) and the quasilinear perturbed distribution function $\unicode[STIX]{x1D6FF}f_{eQL}(v,t)$ (thick red). The total quasilinear distribution function $f_{eQL}(v,t)$ (thick magenta) in this bottom plot shows a quasilinear flattening of the distribution function at the resonant velocities $v=\pm \unicode[STIX]{x1D714}/k$ , where the flattening occurs at both positive and negatives values of the Langmuir wave phase velocity because the standing wave pattern consists of equal-amplitude Langmuir waves propagating in opposite directions. Note that, when only single-point measurements at $x=0$ are observed, the large perturbations of the distribution function associated with the oscillating energy transfer (due to the ballistic and linear wave–particle interaction terms) obscure this quasilinear flattening effect when the total electron distribution function evaluated at position $x=0$ , $f_{e}(0,v,t)$ (thin magenta), is plotted.

The key point of figure 3 is that the perturbed component of the electron distribution function due to the nonlinear wave–particle interaction term measured at a single point in space, $\unicode[STIX]{x1D6FF}f_{eWn}$ (thin red), agrees closely with the spatially averaged, quasilinear perturbed distribution function $\unicode[STIX]{x1D6FF}f_{eQL}(v,t)$ (thick red). Thus, if we can devise a procedure to isolate the perturbation due to the nonlinear wave–particle interaction term from the larger fluctuations, shown in figure 2(b), then single-point measurements can be used to determine the secular energy transfer associated with the collisionless damping of the electrostatic Langmuir waves. We will show in § 6 that the correlation given by (3.20) can accomplish this isolation of the secular energy transfer.

Figure 4. Plots of the full $(x,v)$ phase space at time $\unicode[STIX]{x1D714}_{pe}t=19.64$ for Case I, showing (a) the total perturbed electron distribution function $f_{e}$ , (b) the ballistic contribution $\unicode[STIX]{x1D6FF}f_{eB}$ , (c) the linear wave contribution $\unicode[STIX]{x1D6FF}f_{eWl}$ and (d) the nonlinear wave contribution $\unicode[STIX]{x1D6FF}f_{eWn}$ . Green horizontal lines indicate the resonant velocities.

Although the aim of this paper it is determine what insight can be gained from single-point measurements of the electromagnetic fields and particle velocity distribution functions, it is instructive to examine how the different terms in the Vlasov equation contribute to the evolution of the distribution function throughout 1D-1V phase space. In figure 4, for Case I at $\unicode[STIX]{x1D714}_{pe}t=19.64$ , we plot on the full $(x,v)$ phase space (a) the total perturbed electron distribution function $\unicode[STIX]{x1D6FF}f_{e}$ , (b) the ballistic contribution $\unicode[STIX]{x1D6FF}f_{eB}$ , (c) the linear wave contribution $\unicode[STIX]{x1D6FF}f_{eWl}$ and (d) the nonlinear wave contribution $\unicode[STIX]{x1D6FF}f_{eWn}$ . These plots demonstrate that it is quite difficult to see any features at the resonant velocities (green lines) in the total perturbed electron distribution function $\unicode[STIX]{x1D6FF}f_{e}$ because, focusing on the amplitudes of each of these plots, the ballistic contribution $\unicode[STIX]{x1D6FF}f_{eB}$ and linear wave contribution $\unicode[STIX]{x1D6FF}f_{eWl}$ dominate the perturbation to the distribution function. Only the nonlinear wave term $\unicode[STIX]{x1D6FF}f_{eWn}$ shows any obvious feature at the resonant velocities, generally showing an increase in particle energy just above the resonance at $|v|>|\unicode[STIX]{x1D714}/k|$ and a decrease just below the resonance at $|v|<|\unicode[STIX]{x1D714}/k|$ , qualitatively consistent with the quasilinear flattening of the distribution function at the resonance. The full phase-space plot shows that the nonlinear wave term $\unicode[STIX]{x1D6FF}f_{eWn}$ is not spatially uniform in $x$ . In fact, the amplitude variation of the nonlinear perturbation in $x$ reflects the amplitude variation of the electric field standing wave pattern in $x$ : as noted in § A.1, the standing wave pattern is initialized with an electron density perturbation $\unicode[STIX]{x1D6FF}n_{e}\propto \sin (kx)$ , leading to an electric field pattern $E\propto \cos (kx)$ ; the interaction of the standing electric field pattern leads to a pattern of the nonlinear wave term $\unicode[STIX]{x1D6FF}f_{eWn}$ that is also modulated in amplitude by $\cos (kx)$ , as evident in panel (d).

Also notable in these phase-space plots is absence of any strongly nonlinear features, such as trapping in phase-space vortices. These features do not arise because the perturbation amplitude is too small for such features to develop over the time scales simulated here (up to $\unicode[STIX]{x1D714}_{pe}t=40$ ). As shown in figure 3, the evolution is indeed well described by the quasilinear average, further reinforcing that the nonlinear evolution is a higher-order (long-time) correction to the linear evolution. This is the appropriate regime for the intended application of this new field–particle correlation technique to heliospheric turbulence, such as the solar wind. In the solar wind, the magnetic field fluctuations at the ion kinetic length scales have typical amplitudes $\unicode[STIX]{x1D6FF}B/B\lesssim 0.1$ ; the fluctuations of the velocity distribution are expected to be approximately the same order of magnitude, $\unicode[STIX]{x1D6FF}f(v)/f_{0}(v)\sim \unicode[STIX]{x1D6FF}B/B\lesssim 0.1$ , so the amplitudes chosen for the test case presented here are indeed relevant to the turbulent heliospheric plasmas of interest.

Figure 5. (a) Plot of $-q_{e}(v^{2}/2)\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{e}(x_{0},v)/\unicode[STIX]{x2202}vE(x_{0})$ (black) and $q_{e}v\unicode[STIX]{x1D6FF}f_{e}(x_{0},v)E(x_{0})$ (red) evaluated at time $\unicode[STIX]{x1D714}_{pe}t=11.29$ and position $x_{0}=0$ . (b) Plot of the even (in $v$ ) components of the same curves in panel (a).

As a final note, we compare the product of the correlated quantities given in (3.20) and (3.21) as a function of the velocity. In figure 5(a), we plot $F(v)=-q_{e}(v^{2}/2)\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{e}(x_{0},v)/\unicode[STIX]{x2202}vE(x_{0})$ (black) and $F^{\prime }(v)=q_{e}v\unicode[STIX]{x1D6FF}f_{e}(x_{0},v)E(x_{0})$ (red) evaluated at time $\unicode[STIX]{x1D714}_{pe}t=11.29$ and position $x_{0}=0$ . Recall that the form $-q_{e}(v^{2}/2)\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{e}(x_{0},v)/\unicode[STIX]{x2202}vE(x_{0})$ is direct computation of the rate of change of phase-space energy density due to the nonlinear particle interaction term in (3.19), and this is the only term that yields a non-zero secular energy transfer. When integrated over velocity, both of these forms yield the same rate of change of the spatial energy density at $x_{0}$ (since they are equivalent through an integration by parts in velocity). Although these two quantities indeed have very different structures as a function of velocity, they are indeed related. The zero crossings of $-q_{e}(v^{2}/2)\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{e}(x_{0},v)/\unicode[STIX]{x2202}vE(x_{0})$ correspond closely (though not exactlyFootnote ¹ ) with the minima and maxima of $q_{e}v\unicode[STIX]{x1D6FF}f_{e}(x_{0},v)E(x_{0})$ . Thus, although the form given by the correlation (3.21) does not correspond directly to the rate of change of phase-space energy density, this more observationally accessible quantity can still provide valuable information about the distribution of the secular energy transfer from fields to particles as a function of the particle velocity.

The correspondence between the two curves in figure 5(a) can be made a little more clear by taking just the component of these products that have an even parity in velocity, $F_{even}(v)=[F(v)+F(-v)]/2$ . Because the odd component, $F_{odd}(v)=[F(v)-F(-v)]/2$ , yields no net phase-space energy density change when integrated over velocity, plotting only the even component in panel (b) more clearly shows the variation of the energy transfer rate about the resonant velocity (dashed vertical black) – specifically, a loss of energy at $|v|<|\unicode[STIX]{x1D714}/k|$ and a gain of energy at $|v|>|\unicode[STIX]{x1D714}/k|$ .

5.2 Case II: weakly damped standing Langmuir wave

We initialize a standing Langmuir wave pattern in VP, using the procedure outlined in § A.1, with $k\unicode[STIX]{x1D706}_{de}=0.25$ and an electron density perturbation $\unicode[STIX]{x1D6FF}n_{e}/n_{0}=0.025$ , yielding an initial total perturbed energy of $\unicode[STIX]{x1D6FF}W=0.1256(n_{0}T_{e}\unicode[STIX]{x1D706}_{de}^{3}/2)$ . For this wavenumber, a numerical solution of the linear dispersion relation for Langmuir waves gives $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{pe}=1.11$ and $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D714}_{pe}=-2.03\times 10^{-3}$ , yielding a resonant phase velocity $v_{p}/v_{te}=\unicode[STIX]{x1D714}/(kv_{te})=4.4$ . For this simulation, $L=8\unicode[STIX]{x03C0}\unicode[STIX]{x1D706}_{de}$ , $v_{max}=6v_{te}$ , and $f_{CFL}=0.05$ .

Figure 6. Quasilinear electron distribution function $f_{eQL}$ (magenta) and equilibrium distribution function $f_{e0}$ (black). Dashed vertical black lines denote the resonant velocities $v=\pm \unicode[STIX]{x1D714}/k$ .

In this simulation, the collisionless damping of the Langmuir wave is very weak because the resonant velocity occurs far out in the tail of the electron distribution function at $v_{p}/v_{te}=4.4$ , so there are few electrons to interact resonantly with the electric field fluctuations. Thus, it takes very little electrostatic field energy to flatten the electron distribution function at the resonant velocity, eliminating further Landau damping. We illustrate this in figure 6, where the total quasilinear electron distribution function $f_{eQL}$ (magenta) is plotted at $\unicode[STIX]{x1D714}_{pe}t=28.63$ along with the equilibrium distribution function $f_{e0}$ (black). Here the value of the distribution function must be plotted logarithmically to see the perturbations to the distribution function, which occur at an amplitude four orders of magnitude smaller than the peak of the velocity distribution. This minute secular transfer of energy from the electric field to the electrons is likely to be unobservable given the instrumental noise levels for spacecraft particle measurements.

5.3 Case III: moderately damped single Langmuir wave mode

In this simulation, a single Langmuir wave with $k\unicode[STIX]{x1D706}_{de}=0.5$ and amplitude $-q_{e}\unicode[STIX]{x1D719}_{1}/T_{e}=0.297$ propagating in the $+x$ direction is initialized using the solution of the linear dispersion relation for the Langmuir wave mode, as detailed in § A.1. Both the electron and ion distribution functions are initialized according to (A 9) using the numerically determined complex frequency from a linear dispersion relation solver, $\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{pe}=1.43$ and $\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D714}_{pe}=-1.59\times 10^{-1}$ . To minimize transients in the simulation initialization, these initialized eigenfunctions are smoothed in velocity space using a Crank–Nicholson diffusion operator in velocity space $N_{s}=15$ times with a diffusion coefficient $\unicode[STIX]{x1D708}/(\unicode[STIX]{x0394}v)^{2}=1.5625\times 10^{-3}$ . For this wave, the resonant phase velocity $v_{p}/v_{te}=\unicode[STIX]{x1D714}/(kv_{te})=2.86$ . For this simulation, $L=4\unicode[STIX]{x03C0}\unicode[STIX]{x1D706}_{de}$ , $v_{max}=5v_{te}$ , and $f_{CFL}=0.05$ .

Figure 7. Energy evolution for the moderately damped single Langmuir wave with $k\unicode[STIX]{x1D706}_{de}=0.5$ up to $\unicode[STIX]{x1D714}_{pe}t=20$ . Plotted are the total perturbed energy $\unicode[STIX]{x1D6FF}W$ (black), field energy $W_{\unicode[STIX]{x1D719}}$ (green), perturbed ion energy $\unicode[STIX]{x1D6FF}W_{\unicode[STIX]{x1D6FF}f_{i}}$ (red) and perturbed electron energy $\unicode[STIX]{x1D6FF}W_{\unicode[STIX]{x1D6FF}f_{e}}$ (blue). Linear (a) and logarithmic (b) scales are both presented.

In figure 7, we plot the evolution of the perturbed energies of the system. Total perturbed energy $\unicode[STIX]{x1D6FF}W=W_{\unicode[STIX]{x1D719}}+\unicode[STIX]{x1D6FF}W_{i}+\unicode[STIX]{x1D6FF}W_{e}$ , with an initial value of $\unicode[STIX]{x1D6FF}W=0.008659(n_{0}T_{e}\unicode[STIX]{x1D706}_{de}^{3}/2)$ , is conserved to within 0.1 %. This figure shows that the electrostatic field energy $W_{\unicode[STIX]{x1D719}}$ (green) is converted primarily to microscopic kinetic energy of the electrons $\unicode[STIX]{x1D6FF}W_{e}$ (blue).

Figure 8. (a) Total electron distribution function $f_{e}$ (magenta), equilibrium electron distribution function $f_{e0}$ (black) and total perturbed electron distribution function $\unicode[STIX]{x1D6FF}f_{e}$ (cyan). (b) The separated components of the perturbed electron distribution function: (i) $\unicode[STIX]{x1D6FF}f_{eB}$ (green), (ii) $\unicode[STIX]{x1D6FF}f_{eWl}$ (blue) and (iii) $\unicode[STIX]{x1D6FF}f_{eWn}$ (red). The dashed vertical black line denotes the resonant velocity $v=\unicode[STIX]{x1D714}/k$ .

For the evolution of this single Langmuir wave, we plot in figure 8 the perturbed electron distribution function for this nonlinear simulation at time $\unicode[STIX]{x1D714}_{pe}t=19.64$ and position $x=0$ . In panel (a), we plot the total electron distribution function $f_{e}$ (magenta), the equilibrium electron distribution function $f_{e0}$ (black) and the total perturbed electron distribution function $\unicode[STIX]{x1D6FF}f_{e}=\unicode[STIX]{x1D6FF}f_{eB}+\unicode[STIX]{x1D6FF}f_{eWl}+\unicode[STIX]{x1D6FF}f_{eWn}$ (cyan). The separate components arising from the different terms in (3.14) are shown in panel (b): (i) the ballistic term yields $\unicode[STIX]{x1D6FF}f_{eB}$ (green), (ii) the linear wave–particle interaction term yields $\unicode[STIX]{x1D6FF}f_{eWl}$ (blue) and (iii) the nonlinear wave–particle interaction term yields $\unicode[STIX]{x1D6FF}f_{eWn}$ (red). In this simulation, the perturbations of the distribution function are highly localized around the resonant velocity (dashed black). Again, as in the case of the Landau damping of the standing wave pattern, the perturbations due to the ballistic and linear wave–particle interaction terms are much larger than that due to the nonlinear wave–particle interaction term, making the isolation of this term a significant challenge.

Figure 9. (a) Total electron distribution function evaluated at position $x=0$ , $f_{e}(0,v,t)$ (thin magenta), the spatially averaged, quasilinear distribution function $f_{eQL}(v,t)$ (thick magenta) and the equilibrium electron distribution function $f_{e0}$ (black). (b) The nonlinear wave–particle interaction component of the perturbed electron distribution function at position $x=0$ , $\unicode[STIX]{x1D6FF}f_{eWn}(0,v,t)$ (thin red) and the quasilinear perturbed distribution function $\unicode[STIX]{x1D6FF}f_{eQL}(v,t)$ (thick red) coincide exactly here. The quasilinear flattening of the distribution function at the resonant velocity $v=\unicode[STIX]{x1D714}/k$ (dashed black) is apparent in the total quasilinear distribution function $f_{eQL}(v,t)$ (thick magenta).

In figure 9(a), we compare the spatially averaged, quasilinear distribution function $f_{eQL}(v,t)$ (thick magenta) at time $\unicode[STIX]{x1D714}_{pe}t=19.64$ with the total electron distribution function evaluated at position $x=0$ , $f_{e}(0,v,t)$ (thin magenta). For the small amplitude of the initialized Langmuir wave, the deviation of $f_{eQL}(v,t)$ from the equilibrium $f_{e0}$ (black) is difficult to see at this scale, so, in panel (b), we plot the total quasilinear distribution function $f_{eQL}(v,t)$ (thick magenta) and the equilibrium $f_{e0}$ (black), showing the quasilinear flattening at the resonant velocity $v=\unicode[STIX]{x1D714}/k$ (dashed black). The total electron distribution function evaluated at position $x=0$ , $f_{e}(0,v,t)$ (thin magenta), has much larger fluctuations, dominated by the ballistic and linear wave–particle interaction terms, that obscure the signature of the quasilinear flattening. We also plot the perturbed distribution function $\unicode[STIX]{x1D6FF}f_{eQL}(v,t)$ (thick red) and the nonlinear wave–particle interaction component of the perturbed electron distribution function at position $x=0$ , $\unicode[STIX]{x1D6FF}f_{eWn}(0,v,t)$ (thin red) – these two curves coincide almost exactly in this case, providing motivation that single-point measurements can indeed be used to determine the secular energy transfer from the electrostatic field to the electrons.

Figure 10. Plots of the full $(x,v)$ phase space at time $\unicode[STIX]{x1D714}_{pe}t=19.64$ for single Langmuir wave Case III, showing (a) the total perturbed electron distribution function $f_{e}$ , (b) the ballistic contribution $\unicode[STIX]{x1D6FF}f_{eB}$ , (c) the linear wave contribution $\unicode[STIX]{x1D6FF}f_{eWl}$ and (d) the nonlinear wave contribution $\unicode[STIX]{x1D6FF}f_{eWn}$ . The green horizontal line indicates the resonant velocity for this single propagating Langmuir wave.

In figure 10, for Case III at $\unicode[STIX]{x1D714}_{pe}t=19.64$ , we plot on the full $(x,v)$ phase space (a) the total perturbed electron distribution function $f_{e}$ , (b) the ballistic contribution $\unicode[STIX]{x1D6FF}f_{eB}$ , (c) the linear wave contribution $\unicode[STIX]{x1D6FF}f_{eWl}$ and (d) the nonlinear wave contribution $\unicode[STIX]{x1D6FF}f_{eWn}$ . Although all of these perturbations to the electron distribution functions only have significant amplitudes at positive velocities, concentrated somewhat around the resonant velocity (green line), only the nonlinear wave contribution $\unicode[STIX]{x1D6FF}f_{eWn}$ is highly localized in velocity around the resonant velocity, with the characteristic qualitative signature of quasilinear damping (phase-space energy density loss at $v<\unicode[STIX]{x1D714}/k$ and gain at $v>\unicode[STIX]{x1D714}/k$ ). Note also that the effect of the resonant wave–particle interaction that transfer energy from the electric field to the electrons does not show the significant variation along $x$ : the reason is that electric field associated with the propagating Langmuir wave travels across the domain, having crossed the domain more than four times by $\unicode[STIX]{x1D714}_{pe}t=19.64$ , leading to the energization of the electrons (at the expense of the electric field energy) being spread fairly evenly across $x$ .

6.1 Case I: moderately damped standing Langmuir wave

Before presenting the results of the field–particle correlation technique applied to the problem of the collisionless damping of electrostatic Langmuir waves, we begin with a plot of the single-point measurements used for this analysis. For the case of the moderately damped standing Langmuir wave pattern presented in § 5.1, we plot in figure 11(a) the total perturbed electron distribution function $\unicode[STIX]{x1D6FF}f_{e}(0,v,t)$ (colour map) and (b) the electric field $E(0,t)$ measured at $x=0$ as a function of normalized time $\unicode[STIX]{x1D714}_{pe}t$ . Here the electric field is normalized to $E_{0}=T_{e}/(q_{e}\unicode[STIX]{x1D706}_{de})$ . The data plotted in figure 11 correspond to observable quantities that can be derived from single-point spacecraft measurements.

Figure 11. For Case I, (a) the total perturbed electron distribution function $\unicode[STIX]{x1D6FF}f_{e}(0,v,t)$ (colour map) and (b) electric field $E(0,t)$ measured at $x=0$ as a function of normalized time $\unicode[STIX]{x1D714}_{pe}t$ .

In figure 12, we plot the products of the quantities used in the correlations $C_{E}(v,t,\unicode[STIX]{x1D70F})$ and $C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})$ : (a) $(-q_{e}v^{2}/2)(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{e}(0,v,t)/\unicode[STIX]{x2202}v)E(0,t)$ and (b) $q_{e}v\unicode[STIX]{x1D6FF}f_{e}(0,v,t)E(0,t)$ as a function of velocity $v/v_{te}$ and time $\unicode[STIX]{x1D714}_{pe}t$ . Note that the regions of velocity space where these functions have a significant amplitude are not especially well correlated with the resonant velocities (dot-dashed green). Without taking the correlation of these quantities over an appropriate correlation interval $\unicode[STIX]{x1D70F}$ (typically one or more periods of a wave), the small-amplitude signal of the secular energy transfer is masked by the much larger-amplitude oscillating energy transfer, making it difficult to determine resonant nature of the collisionless damping.

Figure 12. For Case I, plots of (a) the $C_{E}(v,t,\unicode[STIX]{x1D70F})$ correlation quantity, $(q_{s}v^{2}/2)(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}(0,v,t)/\unicode[STIX]{x2202}v)E(0,t)$ and (b) the $C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})$ correlation quantity, $q_{s}v\unicode[STIX]{x1D6FF}f_{s}(0,v,t)E(0,t)$ , at $x=0$ as a function of velocity $v/v_{te}$ and time $\unicode[STIX]{x1D714}_{pe}t$ . The dot-dashed vertical green lines denote the resonant velocities $v=\pm \unicode[STIX]{x1D714}/k$ .

A key part of this field–particle correlation analysis is the selection of an appropriate correlation interval $\unicode[STIX]{x1D70F}$ to isolate successfully the small-amplitude secular energy transfer in the presence of a much larger-amplitude oscillating energy transfer. For this case, the standing wave period is $\unicode[STIX]{x1D714}_{pe}T=4.39$ . In figure 13(a,b), we plot the correlation $C_{E}(v_{0},t,\unicode[STIX]{x1D70F})$ from (3.20) over a range of correlation intervals $0\leqslant \unicode[STIX]{x1D714}_{pe}\unicode[STIX]{x1D70F}\leqslant 16$ at two velocity values, (a) off-resonance at $v_{0}=0.08v_{te}$ and (b) on-resonance at $v_{0}=2.85v_{te}$ . Note that the $\unicode[STIX]{x1D70F}=0$ curve (dark blue) corresponds to a vertical slice along figure 12(a) at the specified velocity $v_{0}$ . The impact of increasing the correlation interval $\unicode[STIX]{x1D70F}$ is clear. As the correlation interval increases, the large-amplitude oscillations of the $\unicode[STIX]{x1D70F}=0$ case (blue), which are dominated by the oscillating energy transfer, are averaged out, isolating the smaller-amplitude secular energy transfer for correlation intervals longer than the wave period, $\unicode[STIX]{x1D70F}>T$ (red). Note that the amplitude of the energy transfer rate – estimated by the correlation $C_{E}(v_{0},t,\unicode[STIX]{x1D70F})$ – at $v_{0}=0.08v_{te}$ is two orders of magnitude smaller than at $v_{0}=2.85v_{te}$ , so only the latter would likely be observable, given limitations to the sensitivities of particle instruments aboard spacecraft.

Figure 13. The correlation $C_{E}(v_{0},t,\unicode[STIX]{x1D70F})$ from (3.20) over a range of correlation intervals $0\leqslant \unicode[STIX]{x1D714}_{pe}\unicode[STIX]{x1D70F}\leqslant 16$ at two velocity values, (a) off-resonance at $v_{0}=0.08v_{te}$ and (b) on-resonance at $v_{0}=2.85v_{te}$ . Also, the change in the electron phase-space energy density $\unicode[STIX]{x0394}w_{e}(x_{0},v,t)$ as a function of time $\unicode[STIX]{x1D714}_{pe}t$ at the same two velocity values, (c) off-resonance at $v_{0}=0.08v_{te}$ and (d) on-resonance at $v_{0}=2.85v_{te}$ . Units of the energy transfer rate (a,b) and change in phase-space energy density (c,d) are arbitrary, but consistent from one panel to another.

As shown in § 3, the correlation $C_{E}(v_{0},t,\unicode[STIX]{x1D70F})$ in (3.20) is a direct calculation of the rate of energy transfer between the electrostatic field and the plasma particles. Therefore, to determine the accumulated energy transfer to the electrons, we can simply integrate this correlation over time. Thus, we obtain the total accumulated change in the electron phase-space energy density at $x=x_{0}$ , $\unicode[STIX]{x0394}w_{e}(x_{0},v,t)$ , given by

(6.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x0394}w_{e}(x_{0},v,t)=\int _{0}^{t}C_{E}(v,t^{\prime },\unicode[STIX]{x1D70F})\,\text{d}t^{\prime }. & & \displaystyle\end{eqnarray}$$

In figure 13(c,d), we plot the time-integrated correlation, giving the change in the electron phase-space energy density $\unicode[STIX]{x0394}w_{e}(x_{0},v,t)$ as a function of time $\unicode[STIX]{x1D714}_{pe}t$ at the same two velocity values, (c) off-resonance at $v_{0}=0.08v_{te}$ and (d) on-resonance at $v_{0}=2.85v_{te}$ . Note again that the change in phase-space energy density is several orders of magnitude larger for the resonant case at $v_{0}=2.85v_{te}$ . The take away lesson here is that, by selecting a correlation interval $\unicode[STIX]{x1D70F}\gtrsim 1.5T$ , the large-amplitude oscillating energy transfer rate can be averaged out, isolating the signal of the secular energy transfer rate to the electrons associated with the collisionless damping of the electrostatic field.

Another point worth mentioning is that the total accumulated change in the electron phase-space energy density, $\unicode[STIX]{x0394}w_{e}$ , decreases as the correlation interval is increased in figure 13(d). Since the envelope of the electric field oscillation decreases in amplitude monotonically in time, as shown in figure 11(b), the rate of energy transfer given by the form of $C_{E}$ will also have an envelope that decreases in time. For a longer correlation interval $\unicode[STIX]{x1D70F}$ , the average over a decreasing function will yield ever smaller values, leading to the decrease in the estimate of the total accumulated phase-space energy density change seen in figure 13(d). Therefore, one should not choose a correlation interval that is much longer than the lifetime of a damped wave. In a turbulent plasma, however, the nonlinear interactions underlying the turbulent cascade of energy from large scales will continually feed energy into the fluctuations at the scales where collisionless damping can occur, so this constraint is likely to be far less important for diagnosing collisionless damping in a turbulent plasma.

Figure 14. (a) The field–particle correlation $C_{E}(v,t,\unicode[STIX]{x1D70F})=C_{\unicode[STIX]{x1D70F}}(q_{s}v^{2}/2\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}/\unicode[STIX]{x2202}v,E)$ at $x=0$ using a correlation interval $\unicode[STIX]{x1D714}_{pe}\unicode[STIX]{x1D70F}=6.28$ . (b) The time-integrated correlation $\int _{0}^{t}C_{E}(v,t^{\prime },\unicode[STIX]{x1D70F})\,\text{d}t^{\prime }$ , showing a clear resonant signature of the secular energy transfer about the resonant velocities $v=\pm \unicode[STIX]{x1D714}/k$ (dashed green).

Now that we have determined an appropriate value for the correlation interval, we apply the field–particle correlation $C_{E}(v,t,\unicode[STIX]{x1D70F})$ , given by (3.20), to the observable data in figure 11 using a correlation interval $\unicode[STIX]{x1D714}_{pe}\unicode[STIX]{x1D70F}=6.28$ . The resulting value of the correlation $C_{E}(v,t,\unicode[STIX]{x1D70F})$ as a function of velocity $v/v_{te}$ and time $\unicode[STIX]{x1D714}_{pe}t$ for Case I is shown in figure 14(a). The selection of an appropriately long correlation interval has eliminated the large oscillations seen in figure 12 at $v/v_{te}<2$ , showing that the remaining rate of secular energy transfer has a significant amplitude that is much more localized around the resonant velocity of $v/v_{te}=2.86$ . Further, we can time integrate this correlation to obtain the secular change in the electron phase-space energy density, $\unicode[STIX]{x0394}w_{e}(x_{0},v,t)$ , shown in figure 14(b). Panel (b) is the primary result of this field–particle correlation technique, showing the net change in electron phase-space energy density is tightly correlated with the resonant velocity. The loss of energy at $v<\unicode[STIX]{x1D714}/k$ and gain of energy at $v>\unicode[STIX]{x1D714}/k$ corresponds physically to a flattening of the distribution function at the resonant velocity, consistent with the quasilinearly averaged electron distribution function shown in figure 3.

As mentioned in § 3.4, it may be impractical to compute the derivative of the perturbed distribution function, $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}(x,v,t)/\unicode[STIX]{x2202}v$ , using spacecraft measurements that are affected by noise and typically have limited velocity-space resolution. Therefore, the alternative correlation $C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})$ , given by (3.21), may be more suitable for the analysis of single-point spacecraft measurements. Therefore, we repeat the field–particle correlation analysis, starting with the observable single-point measurements given in figure 11, and using the alternative form of the correlation $C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})$ with the same correlation interval $\unicode[STIX]{x1D714}_{pe}\unicode[STIX]{x1D70F}=6.28$ . In figure 15, we plot (a) the correlation $C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})$ as a function of velocity $v/v_{te}$ and time $\unicode[STIX]{x1D714}_{pe}t$ and (b) the time-integrated correlation $\int _{0}^{t}C_{E}^{\prime }(v,t^{\prime },\unicode[STIX]{x1D70F})\,\text{d}t^{\prime }$ for Case I. This alternative form $C_{E}^{\prime }$ of the field–particle correlation analysis does not represent the rate of change of phase-space energy density, although its velocity-integrated value does yield the same net local energy transfer as correlation $C_{E}$ . Nonetheless, this alternative form still yields a signature that is highly peaked at the resonant velocities $v=\pm \unicode[STIX]{x1D714}/k$ , indicating that the physical mechanism of the secular energy transfer is a resonant process. In addition, of course, the velocity-integrated rate of energy change is the same for both forms of the correlation, so the total secular transfer of energy from collisionless interactions between the fields and particles can still be determined from the alternative form of the correlation $C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})$ .

Figure 15. For Case I, (a) the alternative field–particle correlation $C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})=C_{\unicode[STIX]{x1D70F}}(q_{s}v\unicode[STIX]{x1D6FF}f_{s}/\unicode[STIX]{x2202}v,E)$ at $x=0$ using a correlation interval $\unicode[STIX]{x1D714}_{pe}\unicode[STIX]{x1D70F}=6.28$ . (b) The time-integrated correlation $\int _{0}^{t}C_{E}^{\prime }(v,t^{\prime },\unicode[STIX]{x1D70F})\,\text{d}t^{\prime }$ . Vertical dashed green lines denote the resonant velocities $v=\pm \unicode[STIX]{x1D714}/k$ .

Figure 16. (a) The net local energy transfer rate, computed using the velocity-integrated correlation $\int \,\text{d}v\,C_{E}(v,t,\unicode[STIX]{x1D70F})$ (red dashed) and alternative correlation, $\int \,\text{d}v\,C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})$ (blue dashed). Also plotted on the right-hand vertical axis is the electric field amplitude as a function of time, $E/E_{0}$ (green). (b) The net accumulated transfer of spatial energy density, $\int \,\text{d}v\,\unicode[STIX]{x0394}w_{ej}(x_{0},v,t)=\int \,\text{d}t^{\prime }\int \,\text{d}v\,C_{j}(c,t^{\prime },\unicode[STIX]{x1D70F})$ for both $C_{E}$ (red dotted) and $C_{E}^{\prime }$ (blue dashed) and local electrostatic spatial energy density, $|E(x_{0},t)|^{2}/8\unicode[STIX]{x03C0}$ (green).

To show that this field–particle correlation method indeed recovers the local energy transfer rate, in figure 16(a) we plot the velocity-integrated correlation $\int \,\text{d}v\,C_{E}(v,t,\unicode[STIX]{x1D70F})$ (red dotted), which provides a measure of the local (at position $x_{0}=0$ ) rate of transfer of spatial energy density from the electrostatic field to the electrons. We also plot the velocity-integrated value of the alternative correlation, $\int \,\text{d}v\,C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})$ (blue dashed), to demonstrate that these two alternative forms, when integrated over velocity, indeed yield the same result for the rate of transfer of spatial energy density between the fields and particles. Also plotted is the electric field amplitude as a function of time, $E/E_{0}$ (green), where the electric field is normalized to $E_{0}=T_{e}/(q_{e}\unicode[STIX]{x1D706}_{de})$ . In figure 16(b), we plot the net accumulated transfer of spatial energy density, $\int \,\text{d}v\,\unicode[STIX]{x0394}w_{ej}(x_{0},v,t)=\int \,\text{d}t^{\prime }\int \,\text{d}v\,C_{j}(c,t^{\prime },\unicode[STIX]{x1D70F})$ for both $C_{E}$ (red dotted) and $C_{E}^{\prime }$ (blue dashed). Also plotted is the local electrostatic spatial energy density, $|E(x_{0},t)|^{2}/8\unicode[STIX]{x03C0}$ (green), showing that energy is lost from the electrostatic field while it is gained by the electrons. Note, however, that the accumulated local energy changes in the electrons and the electrostatic field do not sum to a constant value – this is because the ballistic and linear wave–particle interaction terms in (3.19) are non-zero locally. Only upon integrating over space is total energy conserved, as shown in figure 1.

6.2 Case II: weakly damped standing Langmuir wave

For the weakly damped standing Langmuir wave pattern described in § 5.2, we can perform the same field–particle correlation analysis. In figure 17, we plot (a) the field–particle correlation $C_{E}(v,t,\unicode[STIX]{x1D70F})$ using a correlation interval $\unicode[STIX]{x1D714}_{pe}\unicode[STIX]{x1D70F}=6.28$ , and (b) the time-integrated correlation $\int _{0}^{t}C_{E}(v,t^{\prime },\unicode[STIX]{x1D70F})\,\text{d}t^{\prime }$ for the weakly damped Case II. Note that the colour map magnitude of this figure is the same as that in figure 14, showing that the secular energy transfer for this weakly damped case is very small compared to the moderately damped Case I. The rate of energy transfer in panel (a) indeed shows some signal associated with the oscillating energy transfer that is not eliminated using a correlation interval of $\unicode[STIX]{x1D714}_{pe}\unicode[STIX]{x1D70F}=6.28$ , but the time integration of the correlation in panel (b) effectively eliminates any net energy transfer accumulating over time. Note that this field–particle correlation technique produces only a very-small-amplitude signature in this weakly damped case, a signature that is unlikely to be observable given realistic instrumental limitations on the measurement of small-amplitude fluctuations in the velocity distribution function.

Figure 17. (a) The field–particle correlation $C_{E}(v,t,\unicode[STIX]{x1D70F})=C_{\unicode[STIX]{x1D70F}}(q_{s}v^{2}/2\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}/\unicode[STIX]{x2202}v,E)$ for Case II at $x=0$ using a correlation interval $\unicode[STIX]{x1D714}_{pe}\unicode[STIX]{x1D70F}=6.28$ . (b) The time-integrated correlation $\int _{0}^{t}C_{E}(v,t^{\prime },\unicode[STIX]{x1D70F})\,\text{d}t^{\prime }$ , showing a negligible signature of secular energy transfer at the resonant velocities $v=\pm \unicode[STIX]{x1D714}/k$ (dashed green).

Figure 18. (a) The net local energy transfer rate, computed using the velocity-integrated correlation $\int \,\text{d}v\,C_{E}(v,t,\unicode[STIX]{x1D70F})$ (red dashed) and the electric field amplitude as a function of time, $E/E_{0}$ (green). (b) The net accumulated transfer of spatial energy density, $\int \,\text{d}v\unicode[STIX]{x0394}w_{e}(x_{0},v,t)=\int \,\text{d}t^{\prime }\int \,\text{d}v\,C_{E}(c,t^{\prime },\unicode[STIX]{x1D70F})$ (red dotted) and local electrostatic spatial energy density, $|E(x_{0},t)|^{2}/8\unicode[STIX]{x03C0}$ (green).

The field–particle correlation method indeed shows a negligible amount of local secular transfer of energy from fields to particles on the time scale of the wave period, as shown by plotting the velocity-integrated correlation $\int \,\text{d}v\,C_{E}(v,t,\unicode[STIX]{x1D70F})$ (red dotted) in figure 18(a) and the net accumulated transfer of spatial energy density, $\int \,\text{d}v\,\unicode[STIX]{x0394}w_{e}(x_{0},v,t)=\int \,\text{d}t^{\prime }\int \,\text{d}v\,C_{E}(c,t^{\prime },\unicode[STIX]{x1D70F})$ in figure 18(b). Instrumental constraints likely would render any such small energy transfer rate unobservable.

6.3 Case III: moderately damped single Langmuir wave mode

Finally, we apply the field–particle correlation analysis to Case III, the propagating single Langmuir wave mode. The observable single-point measurements at $x=0$ are plotted in figure 19, showing (a) the total perturbed electron distribution function $\unicode[STIX]{x1D6FF}f_{e}(0,v,t)$ (colour map) and (b) the electric field $E(0,t)$ as a function of normalized time $\unicode[STIX]{x1D714}_{pe}t$ . Note that the perturbed electron distribution function is highly localized in velocity near the resonant velocity, $v/v_{te}=\unicode[STIX]{x1D714}/(kv_{te})=2.86$ .

Figure 19. For Case III, (a) the total perturbed electron distribution function $\unicode[STIX]{x1D6FF}f_{e}(0,v,t)$ (colour map) and (b) electric field $E(0,t)$ measured at $x=0$ as a function of normalized time $\unicode[STIX]{x1D714}_{pe}t$ .

For this case, we plot in figure 20 the products of the quantities used in the correlations $C_{E}(v,t,\unicode[STIX]{x1D70F})$ and $C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})$ : (a) $(-q_{s}v^{2}/2)(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}(0,v,t)/\unicode[STIX]{x2202}v)E(0,t)$ and (b) $q_{s}v\unicode[STIX]{x1D6FF}f_{s}(0,v,t)E(0,t)$ as a function of velocity $v/v_{te}$ and time $\unicode[STIX]{x1D714}_{pe}t$ . In this case, even without the time average over the correlation interval, the product has a definite net sign on either side of the resonant velocity $v=\unicode[STIX]{x1D714}/k$ (dash-dotted green). Note that in this section we also plot the resonant velocity of the counterpropagating Langmuir wave, $v=-\unicode[STIX]{x1D714}/k$ , because a very slight signature does appear in the opposite direction, likely due to an imperfect initialization of the single propagating Langmuir wave. Note that this minor amount of energy in the counterpropagating Langmuir wave does not impact the results of this study.

Figure 20. For Case III, plots of (a) the $C_{E}(v,t,\unicode[STIX]{x1D70F})$ correlation quantity, $(q_{s}v^{2}/2)(\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}(0,v,t)/\unicode[STIX]{x2202}v)E(0,t)$ and (b) the $C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})$ correlation quantity, $q_{s}v\unicode[STIX]{x1D6FF}f_{s}(0,v,t)E(0,t)$ , at $x=0$ as a function of velocity $v/v_{te}$ and time $\unicode[STIX]{x1D714}_{pe}t$ . The dot-dashed vertical green line at $v=\unicode[STIX]{x1D714}/k$ denotes the resonant velocity of the initialized Langmuir wave mode, and we also plot $v=-\unicode[STIX]{x1D714}/k$ since nonlinear couplings may excite the counterpropagating Langmuir wave mode.

Figure 21. For Case III, (a) the field–particle correlation $C_{E}(v,t,\unicode[STIX]{x1D70F})=C_{\unicode[STIX]{x1D70F}}(q_{s}v^{2}/2\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}f_{s}/\unicode[STIX]{x2202}v,E)$ at $x=0$ using a correlation interval $\unicode[STIX]{x1D714}_{pe}\unicode[STIX]{x1D70F}=6.28$ . (b) The time-integrated correlation $\int _{0}^{t}C_{E}(v,t^{\prime },\unicode[STIX]{x1D70F})\,\text{d}t^{\prime }$ , showing a clear resonant signature of the secular energy transfer about the resonant velocity $v=\unicode[STIX]{x1D714}/k$ (dashed green).

In figure 21, we present the results of the field–particle correlation $C_{E}(v,t,\unicode[STIX]{x1D70F})$ , given by (3.20), applied to Case III, the propagating single Langmuir wave mode using a correlation interval $\unicode[STIX]{x1D714}_{pe}\unicode[STIX]{x1D70F}=6.28$ . We plot (a) the field–particle correlation $C_{E}(v,t,\unicode[STIX]{x1D70F})$ , and (b) the time-integrated correlation $\int _{0}^{t}C_{E}(v,t^{\prime },\unicode[STIX]{x1D70F})\,\text{d}t^{\prime }$ for Case III. In both (a) the rate of secular energy transfer and (b) the time-integrated net change in phase-space energy density, the resonant signature of Landau damping of the Langmuir wave is very clear, with the loss of energy at $v<\unicode[STIX]{x1D714}/k$ and the gain of energy at $v>\unicode[STIX]{x1D714}/k$ corresponding to a flattening of the distribution function at the resonant velocity. Finally, we plot in figure 22 the results of the alternative form of the correlation $C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})$ , with (a) the correlation $C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})$ as a function of velocity $v/v_{te}$ and time $\unicode[STIX]{x1D714}_{pe}t$ and (b) time-integrated correlation $\int _{0}^{t}C_{E}^{\prime }(v,t^{\prime },\unicode[STIX]{x1D70F})\,\text{d}t^{\prime }$ for Case III. Again, we see that this alternative form has a peak in the energy transfer rate and net transferred energy at the resonant velocity $v=\unicode[STIX]{x1D714}/k$ (dotted green), indicating clearly the resonant nature of the mechanism of collisionless energy transfer.

Figure 22. For Case III, (a) the alternative field–particle correlation $C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})=C_{\unicode[STIX]{x1D70F}}(q_{s}v\unicode[STIX]{x1D6FF}f_{s}/\unicode[STIX]{x2202}v,E)$ at $x=0$ using a correlation interval $\unicode[STIX]{x1D714}_{pe}\unicode[STIX]{x1D70F}=6.28$ . (b) The time-integrated correlation $\int _{0}^{t}C_{E}^{\prime }(v,t^{\prime },\unicode[STIX]{x1D70F})\,\text{d}t^{\prime }$ . Vertical dashed green line denotes the resonant velocity $v=\unicode[STIX]{x1D714}/k$ .

Figure 23. (a) The net local energy transfer rate, computed using the velocity-integrated correlation $\int \,\text{d}v\,C_{E}(v,t,\unicode[STIX]{x1D70F})$ (red dashed) and alternative correlation, $\int \,\text{d}v\,C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})$ (blue dashed). Also plotted is the electric field amplitude as a function of time, $E/E_{0}$ (green). (b) The net accumulated transfer of spatial energy density, $\int \,\text{d}v\,\unicode[STIX]{x0394}w_{ej}(x_{0},v,t)=\int \,\text{d}t^{\prime }\int \,\text{d}v\,C_{j}(c,t^{\prime },\unicode[STIX]{x1D70F})$ for both $C_{E}$ (red dotted) and $C_{E}^{\prime }$ (blue dashed), and local electrostatic spatial energy density, $|E(x_{0},t)|^{2}/8\unicode[STIX]{x03C0}$ (green).

In figure 23, we show the local (at position $x_{0}=0$ ) rate of transfer of spatial energy density from the electrostatic field to the electrons by plotting (a) the velocity-integrated correlation $\int \,\text{d}v\,C_{E}(v,t,\unicode[STIX]{x1D70F})$ (red dotted) and the alternative correlation, $\int \,\text{d}v\,C_{E}^{\prime }(v,t,\unicode[STIX]{x1D70F})$ (blue dashed). Also plotted is the electric field amplitude as a function of time, $E/E_{0}$ (green). In figure 23(b), we plot the net accumulated transfer of spatial energy density, $\int \,\text{d}v\,\unicode[STIX]{x0394}w_{ej}(x_{0},v,t)=\int \,\text{d}t^{\prime }\int \,\text{d}v\,C_{j}(c,t^{\prime },\unicode[STIX]{x1D70F})$ for both $C_{E}$ (red dotted) and $C_{E}^{\prime }$ (blue dashed). Also plotted is the local electrostatic spatial energy density, $|E(x_{0},t)|^{2}/8\unicode[STIX]{x03C0}$ (green), showing that energy is lost from the electrostatic field while it is gained by the electrons.