Characterizing Fluid and Kinetic Instabilities using Field-Particle Correlations on Single-Point Time Series

TL;DR

This paper applies a modified field-particle correlation technique to single-point time series from electrostatic simulations to diagnose kinetic instabilities and understand energy transfer mechanisms in plasma systems.

Contribution

It introduces a modified correlation method suitable for single-point data, enabling analysis of kinetic instabilities in systems where integrated quantities are inaccessible.

Findings

Correlation characterizes physical mechanisms driving instabilities

Method effectively diagnoses counter-streaming and bump-on-tail instabilities

Potential application to turbulent magnetized plasmas and solar wind

Abstract

A recently proposed technique correlating electric fields and particle velocity distributions is applied to single-point time series extracted from linearly unstable, electrostatic numerical simulations. The form of the correlation, which measures the transfer of phase-space energy density between the electric field and plasma distributions and had previously been applied to damped electrostatic systems, is modified to include the effects of drifting equilibrium distributions of the type that drive counter-streaming and bump-on-tail instabilities. By using single-point time series, the correlation is ideal for diagnosing dynamics in systems where access to integrated quantities, such as energy, is observationally infeasible. The velocity-space structure of the field-particle correlation is shown to characterize the underlying physical mechanisms driving unstable systems. The use of this correlation in simple systems will assist in its eventual application to turbulent, magnetized plasmas, with the ultimate goal of characterizing the nature of mechanisms that damp turbulent fluctuations in the solar wind.

Tables

Table 1. Table 1: Electron Bulk Parameters

	$n_{e1}/n_i$	$v_{d1}/v_{te}$	$n_{e2}/n_i$	$v_{d2}/v_{te}$
Case 1	$0.5$	$0.75$	$0.5$	$-0.75$
Case 2	$0.5$	$1.75$	$0.5$	$-1.75$
Case 3	$0.9$	$0.00$	$0.1$	$4.25$

Equations

$$\frac{\partial f_s}{\partial t}+v\frac{\partial f_s}{\partial x}-\frac{q_s}{m_s}\frac{\partial \varphi }{\partial x}\frac{\partial f_s}{\partial v}=0,$$

$$\frac{{\partial }^2\varphi }{\partial x^2}=-4\pi \sum _s{q}_s{\int }_{-\infty }^{\infty }dv{f}_s$$

$$F_{s0}(v)=\frac{n_{sj}}{\sqrt{2\pi }}\exp \left[\frac{-{\left(v-v_{dsj}\right)}^2}{2v_{ts}^2}\right]$$

$$\underline{\underline{D}}\left(\omega ,k\right)=k^2{\lambda }_{De}^2+\sum _j{\left(\frac{q_s}{q_e}\right)}^2\frac{n_{sj}}{n_i}\frac{T_e}{T_s}\left[1+{\xi }_{sj}{Z}_0({\xi }_{sj})\right]$$

$$W_{\mathrm{total}}=\int dx\frac{E^2}{8\pi }+\sum _s\int dx\int dv\frac{m_s{v}^2}{2}{f}_s$$

$$C_E(x_0,v,t_i,N)=-\frac{1}{N}\sum _{j=i}^{i+N}\frac{q_s{v}^2}{2}\frac{\partial f_s(x_0,v,t_j)}{\partial v}E(x_0,t_j).$$

Full text

I Introduction

A significant goal of plasma physics research is the characterization of mass, momentum, and energy transport in a wide variety of complex systems. In particular, the question of what mechanisms mediate the transfer of energy between turbulent fields and distributions of plasma particles, leading to the eventual damping and dissipation of turbulence, is open. One system which displays turbulent behavior is the solar wind, a hot, diffuse emanation from the Sun’s surface that fills the heliosphere. While lacking the precise control over conditions afforded in a laboratory setting, the large volume of in situ measurements of the solar wind over the last half century has led to the accrual of observations of turbulence with a wide variety of plasma parameters. Such measurements have proven useful to the study of phenomena in magnetized turbulence.Bruno:2005

A limitation of these in situ observations is that they are taken at a single-point in space at a given time.111Notable exceptions to the single-point limitation are the are the Cluster,Escoubet:2001 THEMIS,Angelopoulos:2008 and Magnetospheric Multiscale (MMS)Burch:2016 missions, which are comprised of four or five spacecraft arranged in particular geometric configurations. This raises at least two significant complications; one must disentangle the dynamics associated with spatial and temporal variation, and track the evolution of spatially integrated quantities, such as the energy content of a field or distribution of charged particles, given only single-point measurements. The first of these complications is addressed by invoking Taylor’s Hypothesis,Taylor:1938 the conjecture that for single-point measurements of sufficiently fast flows, the time evolution is essentially frozen and the measurement traces out the spatial structure of the turbulence; a review of the application of Taylor’s Hypothesis to solar wind observations can be found in Klein et al 2014.Klein:2014b

To address the second complication of inaccessible spatially integrated quantities, one may consider the dynamics of the phase-space energy density rather than the total energy. A technique has been recently proposed to measure the local-in-phase-space energy transfer between fields and plasma distributions using single-point time series of simple plasma systems.(Klein:2016a, ; Howes:2016, ) By correlating the product of the electric field and velocity derivative of the particle distribution measured at a single point in space, the velocity structure of the transfer of energy between fields and particles is obtained. By averaging this correlation over a selected time interval, the oscillatory energy transfer between the fields and particles is removed, leaving only the secular energy transfer. The mechanisms responsible for this energy transfer can be identified by the velocity space structure of the field-particle correlation. Initial work applied this correlation to systems that damp via the Landau resonance.Landau:1946 Here, we consider the transfer of energy in linearly unstable systems, and show that field-particle correlations are able to identify the presence of both fluid and kinetic instabilities in such systems.

The unstable systems under consideration are reviewed in Section II. In Section III, we present the nonlinear numerical code employed in this work, VP, as well as the three simulations under consideration. In Section IV, the field-particle correlation is presented and applied to the three simulations. Analysis and discussion related to the correlations are found in Section V. Application of field-particle correlations to simple, homogeneous, linearly unstable systems enables the construction of signatures of basic energy transfer mechanisms. Combined with signatures for all relevant energy transfer mechanisms, such correlations may be usefully employed to diagnose the behavior of more complex systems.

II Linearly Unstable Systems

The 1D1V electrostatic systems of interest in this work are governed by the Vlasov and Poisson equations

[TABLE]

and

[TABLE]

which evolve the distribution of species $s$, $f_{s}(x,v,t)$ and the electric field $E(x,t)=-\partial\phi(x,t)/\partial x$. Recent work has applied field-particle correlations to electrostatic stable systems as a means of extracting the velocity dependent signature of Landau damping from single-point time series.(Klein:2016a, ; Howes:2016, ) Here, we consider electrostatic systems unstable to either fluid or kinetic instabilities, and characterize the signature of the associated growth using field-particle correlations. For fluid instabilities, the behavior of the plasma depends on the bulk parameters of the system and the energy transfer is not organized by characteristic velocities such as the resonant wave phase speed; energy transfer for kinetic instabilities depends on such characteristic velocities, as the electrostatic field acts to exchange particles of higher and lower kinetic energies near the field’s resonant velocity. For both types of instabilities, one or more of the distributions lose energy, while the fields and other distributions gain energy, resulting in an inverse damping. General discussion of plasma instabilities, as well as particular treatments of the electron drift instabilities of interest in this work, can be found in many plasma textbooks.Krall:1973 ; Stix:1992 ; Hazeltine:2004

To highlight the distinct velocity space structure of energy transfer in unstable systems, we consider three cases, each with distinct sets of parameters for one population of ions and two populations of electrons. The equilibrium distributions for the three populations take the Maxwellian form

[TABLE]

where $v_{dsj}$ is the population’s drift velocity. The linear normal mode behavior for such equilibria is governed by solutions of the dispersion relation

[TABLE]

where $\underline{\underline{D}}$ is a function of wavenumber $k$ and complex frequency $(\omega,\gamma)$, $q_{s}$ is the species charge, $Z_{0}$ is the plasma dispersion functionFried:1961 with argument $\xi_{sj}=\left(\omega/\omega_{pe}k\lambda_{De}\right)\left(\sqrt{T_{e}m_{s}/2T_{s}m_{e}}\right)-v_{dsj}/v_{ts}$, with the sum taken over the three plasma populations $j$. The electron plasma frequency $\omega_{pe}\equiv\sqrt{4\pi\sum_{j}n_{ej}q^{2}/m_{e}}$ and Debye length $\lambda_{De}=\sqrt{T_{e}/4\pi\sum_{j}n_{ej}q^{2}}$, both defined using the total electron density, normalize the time and length scales of our system. Values for the normalized density of electron population $j$, $n_{ej}/n_{i}$, and bulk velocity, normalized by the electron thermal velocity $v_{te}=\sqrt{T_{e}/m_{e}}$, are given in Table 1. For all three cases, the electron populations have equal temperatures; the ions are singly ionized, and initialized with $T_{i}=T_{e}$, $m_{i}=100m_{e}$, and $v_{di}=0$. Case 1 is stable to the effects of the counter-streaming electrons, while the increase in $|v_{dej}|$ for case 2 yields the classic counter-streaming instability. Case 3 is unstable to the bump-on-tail instability. Cases 2 and 3 serve as examples of fluid and kinetic instabilities respectively.

In Fig 1, solutions to Eqn 4 for the three cases are presented. In panel a, the complex frequency solutions $(\omega,\gamma)/\omega_{pe}$ are given for fixed $k\lambda_{De}=0.2$. For case 1 (black circles) all modes are shown to be damped. For case 2 (red triangles), the increase in $|v_{dej}|$ leaves the frequency and damping rate of the Langmuir modes, the modes with $|\omega|\approx\omega_{pe}$, largely unaffected. The pair of least damped acoustic modes from case 1 are now both unstable, having $\omega=0$ and two distinct growth rates, $\gamma>0$. The parametric path from stability ($|v_{dej}|=0.0$, open triangles) to instability ($|v_{dej}|=1.75$, filled triangles) for these two modes is illustrated as a function of complex frequency in panel b. For case 3, the $\omega<0$ Langmuir mode and the least damped acoustic modes are negligibly affected by the bump distribution. By parametric variation of $v_{de2}$ from $0.0$ (open diamonds in panel c) to $4.25$ (filled diamonds), it is observed that the growing mode for the bump-on-tail instability arises from an acoustic mode which is strongly damped in the stable regime (solid line), while the $\omega>0$ Langmuir mode (dashed line) becomes heavily damped. The dispersion relations for the least damped and/or fastest growing Langmuir (acoustic) modes as a function of $k\lambda_{De}$ for the three cases are presented in panels d and f (e and g) illustrating the wavelengths for which linear instabilities arise for cases 2 and 3 (solid lines in panel g).

III Numerical Simulations

To evaluate the time evolution of these systems, we have extended the Vlasov-Poisson solver VPHowes:2016 to allow for the inclusion of an arbitrary number of drifting plasma populations. VP solves the nonlinear Vlasov-Poison system using second-order finite differencing for spatial and velocity derivatives and a third-order Adams-Bashforth scheme in time. As a test of this extension, we compare numerical solutions of Eqn. 4 for the three cases described in section II to frequencies and damping rates extracted from time traces of the electrostatic field energy from linear and small-amplitude nonlinear VP simulations for a range of wavevectors, given as points in Fig. 1. Agreement between the dispersion relation and VP are close as long as the mode of interest is not heavily damped and unstable modes supported by the system do not grow too quickly with respect to the damped modes.

For the evaluation of the field-particle correlation, we perform three nonlinear simulations corresponding to the three cases from section II. For these simulations, we add a sinusoidal perturbation to the ion’s equilibrium distribution of the form $0.1F_{i0}(x,v)\sin(k_{0}x)$ with $k_{0}\lambda_{de}=0.2$. 256 (128) points in velocity (coördinate) space are resolved over the interval $v/v_{ts}\in[-8,8]$ $(x/\lambda_{De}\in[-5\pi,5\pi])$. Each simulation is run for longer than $t=40\omega_{pe}^{-1}$ and the total energy in the system

[TABLE]

is conserved to better than a few tenths of a percent. Changes in the electrostatic energy $W_{\phi}=\int dxE^{2}/8\pi$ as well as energy in the three plasma populations, $W_{j}=\int dx\int dvm_{j}v^{2}f_{j}/2$, from their initial values are shown in Fig. 2, with energy normalized by $T_{e}$. In run 1, most of the energy damped from the electric field is equally partitioned between the two electron populations, with little energy transferred to the ions. For run 2, an instability is clearly triggered, with significant losses of energy from the electrons and both $W_{\phi}$ and $W_{i}$ growing from their initial values. A more virulent instability is triggered in run 3, with the bump electron population losing a significant fraction of its energy to the core electron population, with little energy transferred to the ions.

IV Field-Particle Correlations

While tracking the change in energy is sufficient to identify instabilities in systems with complete knowledge of spatial and velocity structure, we seek the signature of unstable behavior given limited, single-point measurements of a system of the type available to spacecraft in the solar wind. The application of field-particle correlations to such measurements allows for a local observation of secular energy transfer.

We define the field-particle correlation for a discrete set of measurements of $f_{s}(x,v,t)$ and $E(x,t)$ with timestep $dt$, taken at a single point $x=x_{0}$ as

[TABLE]

The correlation averages the field-particle interaction term in the Vlasov equation, the third term in Eqn. 1, over a time interval of length $\tau=Ndt$. As the ballistic term, the second in Eqn. 1, does not lead to net energy transfer,Howes:2016 the product in $C_{E}$ represents the energy density transferred at one point in phase space between the electric field and velocity distribution. By averaging over a selected time interval $\tau$, oscillatory energy transfer between $E$ and $f_{s}$ is removed, leaving only the secular energy transfer. To track the accumulated change of the phase-space energy density, we integrate the correlation over time, defining $\Delta w_{s}(x,v,t,N)\equiv\int_{0}^{t}dt^{\prime}C_{E}(x,v,t^{\prime},N)$.

Unlike Eqn. 2 in Klein & Howes 2016,Klein:2016a we include the full distribution function in our definition of $C_{E}$, rather then only the perturbed component $\delta f_{s}$. Previous studies had focused on particular cases where the equalibria $F_{s0}$ were even with respect to $v=0$, ensuring that they would not contribute to net energy transfer. For the cases under consideration in this work, the equilibrium electron distributions have odd components and therefore may contribute to a secular transfer of energy between the fields and distributions. To show that the two forms of the correlation obtain similar results when $F_{s0}$ is even, we apply both correlations to single-point field and distribution data from a Landau damped counter-propagating Langmuir wave simulation, case 1 in Klein & Howes 2016, and plot $\Delta w_{e}$ at $x=0$ with $\tau\omega_{pe}=6.28$ in Fig. 3. We see that both correlations produce qualitatively similar structure in the accumulated phase-space energy density, especially in regards to production of a plateau surrounding the resonant velocities of the system, $|v_{\rm res}|=2.86v_{te}$, which serves as the key velocity-space signature of Landau damping.

With our correlation defined, we next select an appropriate correlation interval $\tau$ for the three simulations. By averaging over a particular interval, we remove the transfer of energy between the fields and particles which oscillates with frequency of the order $\omega\sim 2\pi/\tau$, leaving the secular, or non-oscillatory, component. Plotted in Fig. 4 are velocity integrated correlations, $\int dvC_{E}(x,v,t,\tau)$, between $E$ and the three plasma populations for a range of correlation lengths $\tau\omega_{pe}\in[0,30]$ at a single spatial location $x=0$.

There is significant oscillatory transfer for small $\tau$ correlations for run 1, panels a-c of Fig. 4. This oscillatory transfer is reduced for longer intervals, with nearly all of the oscillations removed for $\tau\omega_{pe}=5.64$, a correlation length corresponding to the period of Langmuir waves supported by the system with frequency $\omega=1.11\omega_{pe}$. This $\tau$ leaves the velocity integrated correlations for all three distribution functions nearly monotonic, while slightly longer correlations reintroduce some oscillatory behavior. As the Langmuir wave is the least damped, finite frequency linear mode supported by this system, correlating over its period is physically justified. For run 2, panels d-f, correlating over the interval $\tau\omega_{pe}=5.15$, which corresponds to the Langmuir wave frequency $\omega=1.21\omega_{pe}$, removes significant oscillatory energy transfer. Averaging over the time scale associated with the least damped modes, rather than the unstable modes, is motivated by the fact that the unstable modes of this system have zero frequency; see Fig 1. For run 3, we see that there does not exist a single value of $\tau$ for which the oscillatory energy transfer is completely removed. This due to the fact that the system supports both a weakly damped Langmuir wave as well as a finite-frequency unstable acoustic mode. Correlating over the Langmuir period retains some of the acoustic oscillations, while correlating over the acoustic period retains some of the Langmuir oscillations. We choose $\tau\omega_{pe}=7.11$, corresponding to the period of the growing acoustic mode with $\omega=0.88\omega_{pe}$ and acknowledge that some oscillatory contributions from Langmuir waves will persist.

V Velocity-Space Structure of Energy Transfer

With an appropriate interval $\tau$ selected, we calculate the velocity dependent field-particle correlation for the three simulations. By retaining the velocity dependence, we are able to address the question of where the energy transfer occurs in phase space, and use the structure of this energy transfer to characterize the nature of the underlying instability.

For the stable, counter-streaming electron case, run 1, the phase-space energy transfer obtained from the field-particle correlation is a fairly regular function of velocity. At a given point in coördinate space, for example $x=0\lambda_{De}$ shown in the first row of Fig. 5, one of the electron populations gains energy from, while the other population loses energy to, the electric field. The energy transfer to the ions from the electric field is an odd function of velocity, meaning that when the correlation is integrated over $v$, there is no net energy transfer to the ions. There is no evidence for the dependence of the phase-space energy transfer on drift velocities (black vertical dashed lines) or the resonant velocities of either the Langmuir (green lines) or acoustic (magenta) modes for any of the three populations.

Increasing the speed of the counter-streaming electrons for run 2, shown in panels e-h of Fig. 5, the structure of the energy transfer is altered. The sign of the field-particle correlation changes at $v_{dej}$ due to a change in sign of $\partial f_{ej}/\partial v$. Unlike for case 1, this change in sign occurs for electrons which exchange significant energy with the electric field. For the ions, a small, even component in the field-particle correlation arises, yielding a net transfer of energy from the fields to the ions, as seen in panel h. This transfer of energy to the ions, which does not depend on either the Langmuir or acoustic resonant velocities, serves as a phase-space signature for the fluid instability that arises for this system.

For the bump-on-tail instability, case 3 shown in panels i-l of Fig. 5, the velocity-space structure of the field-particle correlation is significantly different. As we have correlated over the unstable acoustic period, the oscillatory structure from the weakly damped Langmuir mode is evident. We also see in the bump electron population that the energy transfer changes sign across the acoustic resonance and at later time across the Langmuir resonance. This resonant structure is the signature of the transfer of energy from the bump population to the core electron population as mediated by the electric field. We note that the resonant structure is maintained for other choices of $\tau$, including an interval corresponding to the Langmuir wave period, not shown. The velocity-integrated correlation confirms that the core population has a net gain of energy, and as expected for this instability, we see that the electrons that receive the energy are near the resonant velocities. This explicit dependence of the phase-space energy transfer on resonant coupling between the fields and particles serves as a distinct signature of kinetic instabilities when compared to the fluid instability in run 2.

The field-particle correlations presented in Fig. 5 were calculated at a single point in coördinate space. An obvious question arises as to how the correlation and the associated phase-space energy transfer change as a function of position within the simulation. To asses this question, we calculate $C_{E}$ at five other points within the simulation and present the velocity integrated accumulated change of the phase-space energy density, $\int dv\Delta w_{s}$, in Fig. 6. For the stable case, run 1, we see that the energy transfer to or from the two electron populations changes sign and amplitude as a function of the position in the simulation, with the transfer passing through zero at the two nodes of the initial standing wave pattern at $\pm 7.9\lambda_{De}$. The correlation with the ions maintains an even velocity space structure such that the ions continue to neither lose nor gain net energy from the electric field regardless of spatial position. For the unstable counter-streaming electrons, run 2, the same pattern of shifting sign and amplitude for the energy transfer to and from the electron holds. The ions gain energy from the electric field regardless of which electron population is gaining or losing energy. The amplitude of the ion energy gain changes with the amplitude of the electron field-particle correlation, going to zero at the nodes and having its largest value at the anti-nodes. This phase space structure serves as a distinct signature of growing fluid instabilities. For the bump-on-tail instability, run 3, there is no regular spatial variation in the transfer of energy between the beam and core electron populations; the core gains energy at the expense of the beam, with the energy density accumulating at nearly the same rate regardless of spatial position, as expected for a kinetic instability.

VI Conclusion

Field-particle correlations of the form defined in Eqn. 6, modified from Klein & Howes 2016Klein:2016a to account for drifting equilibrium distributions, are applied to a set of simulations where fluid and kinetic instabilities are present. The structure of the resulting correlations, which can be interpreted as the secular phase-space energy density transferred between the fields and distributions, can be used to identify the presence of instabilities as well as to characterize the mechanisms driving the unstable growth. The form of the correlation allows for such characterizations to be made from observations at a single point, or a few points, in coördinate space, as opposed to requiring knowledge of spatially integrated quantities typically not accessible to experimental measurements.

We consider simplified 1D-1V electrostatic systems in an attempt to characterize field-particle correlations as measurements of secular energy transfer in advance of future work applying such correlations to systems of higher dimensionality, where magnetization, turbulence, and inhomogeneities may complicate the interpretation of the correlation. By determining signatures of basic plasma physics phenomena responsible for energy transfer between fields and distributions, such as Landau damping and one-dimensional instabilities, we lay the foundation for the determination of the velocity distribution signatures of more complex interactions, such as cyclotronStix:1992 and transit time damping,Barnes:1966 heating by large amplitude, stochastic fluctuations,Chandran:2010a and magnetic reconnection,Yamada:2010 which have all been proposed to play a role in the dissipation of turbulent fluctuations. Future work will also consider the effects of solar wind expansion, electron conduction, and other inhomogeneous mechanisms on the plasma to clearly identify the role of such energy transfer mechanisms in the solar wind. By constructing the correlation to be obtained from single-point measurements, we allow for the identification of such damping mechanisms from in situ observation of the solar wind on current and future missions including Deep Space Climate Observatory (DSCOVR), MMS,Burch:2016 and Solar Probe Plus.Fox:2015

Acknowledgements.

The author would like to thank Gregory Howes, Justin Kasper, and Jason TenBarge for insightful discussions regarding aspects of this work. This research was supported by the NASA HSR grant NNX16AM23G.