Coherent structures in one-dimensional Buneman instability nonlinear simulations Open Access

The linear stability of initially uniform plasma with relative drift velocity of electrons and ions $vde$ is a long-settled classic problem addressed in many textbooks. When $vde$ is smaller than the electron thermal velocity (⁠ $vte$⁠), instability can arise if the electron temperature $Te$ substantially exceeds the ion temperature $Ti$⁠;¹ it is then called “ion acoustic instability” (Ref. 2, Sec. 9.7). When, instead, $vde≳vte$ the usually more rapidly growing instability is called the “Buneman instability” in recognition of its earliest investigator. Buneman himself initially³ addressed situations when the ion temperature was negligible, but it is known that the instability occurs for a wide range of temperature ratios,⁴ including $Ti/Te>1$⁠.

Buneman unstable waves easily grow to amplitudes where nonlinear effects, such as particle trapping, saturation, and turbulence, become dominant. Analytical investigations of the nonlinear state have been pursued for many years, often based on Fourier treatments in terms of wave number and frequency, extending⁵ the linearized eigenmode analysis, and often invoking quasi-linear assumptions.⁶ The purpose of the present discussion is to address a complementary viewpoint on the nonlinear plasma state, namely, the identification of localized potential structures that act as persistent and approximately independent entities. Ion acoustic solitons⁷ are generally treated through fluid equations, and, in addition, to having fixed amplitude-speed relations, pass through one another with minimal interaction. The more general collisionless kinetic BGK (Bernstein, Greene, and Kruskal) structures⁸ are also important. An early Buneman instability experiment⁹ identified a BGK double layer near the electron beam injection boundary; but the present study addresses plasma behavior far from material boundaries. Such BGK potential structures often have regions of phase-space with depleted particle density and are hence often called electron or ion holes. Holes are of particular interest since they can persist self-consistently but also interact strongly with one another and with solitons.^10–16 Solitary potential structures are now observed widely in space plasmas, yet the processes by which they are formed are themselves rarely observable. One important open question, addressed here, is whether, and under what circumstances, Buneman instability produces electron or ion holes, and what distinctive character those Buneman-produced structures might have. We shall see that the answer is complicated.

Numerical simulation is particularly revealing of the formation and dynamics of localized and solitary potential structures. For Buneman instability, past simulations have been broadly of three types: (a) initialized with a preexisting velocity drift without applied electric field, usually in a periodic domain,^17–20 (b) applying a fixed current in an open domain,^15,21–23 (c) applying a fixed external field driving increasing current until instability develops.^24–29 The approaches (b) and (c) are particularly appropriate for investigating “anomalous resistivity” arising from the nonlinear turbulence whose level can continue to grow past initial saturation. Approach (a) effectively prescribes the free energy available for the nonlinear state, which allows its spatial average to become steady. Approach (a) is the focus of the present work, of which a major novelty is the construction of videos of high-resolution phase-space density evolution extending thousands of electron plasma periods into the saturated nonlinear state.

It is known from both observation and simulation that potential variation in multiple dimensions is important, especially for driven electric current. For example, linear instabilities propagating obliquely with respect to the magnetic field and electron drift (perpendicular wave-number $k⊥≠0$⁠) produce transversely localized perturbations. Lower hybrid waves, for example, may disrupt the continued electron runaway in electric-field driven situations. Even holes that are initially one-dimensional (1D) are known in simulations and theory to be often broken up by transverse instability¹⁶ of their non-linear equilibrium. Nevertheless, the present simulations are one-dimensional, which excludes all these effects. The main reasons for this choice include that the computational burden of multidimensional simulations reduces the parameter ranges that can reasonably be studied. Moreover, mitigation of the burden by the adoption of unphysical parameters such as ion/electron mass ratio certainly strongly affects nonlinear development, especially for ion-electron processes like Buneman instability. The present study avoids that distortion. In addition, the difficulty of diagnosing and visualizing the phase-space behavior becomes very much greater in multiple dimensions. The resulting multi-dimensional time-dependent complexity is then not readily susceptible to rendering in the form of (two-dimensional) videos, which is the present emphasis. One must nevertheless be conscious that multi-dimensional effects are deliberately being excluded here, and yet in many situations they may be vital for a complete understanding of the phenomena. The present studies provide only a first step.

Section II presents the governing equations and their normalization units. The bulk of the text of this article aims to describe what is observed to occur in the videos referred to. Section III sets the scene with figures consisting of overall contour summaries and phase-space snap-shots for orientation and illustration. They are not intended to show everything that is observed and described. Viewing the videos is essential to understanding the detailed commentary covered in Sec. IV.

The analytic theory of steady nonlinear waves developed in Sec. V is aimed at modeling the structures observed. It seems consistent with some early features of the simulations when the electron distributions are not greatly modified, and the structures are approximately steady in some propagation frame. However, the analysis makes major simplifying approximations and should be considered a qualitative interpretive framework, rather than a precise representation, because it incorporates none of the longer-term non-linear modification of the background distributions.

The units of time are $ωpe−1=(ne2/ϵ0me)−1/2$⁠. Space units are $λDe=vte/ωpe$⁠, defined using the initial electron temperature $Te0$⁠, which sets the energy units. The velocity units are then $vte=Te0/me$ and charge units e, so $qj=±1$⁠. A total of typically 16 × 10⁶ each of pseudo-particle electrons and ions, $mi/me=1836$⁠, is used, in a periodic domain of length $L=255λDe$⁠, which permits typically dozens of Buneman wavelengths. Periodicity models a more extended plasma region, but imposes distant correlations that probably somewhat affect the details of behavior in the nonlinear phase. Cell size $Δx=1λDe$ and time step $Δt=0.5ωpe−1$ give sufficient resolution, as verified by tests using higher resolution that reproduce the results. Of course, the PIC particles are distributed across a continuum of velocities and positions. They are initially loaded with uniform Maxwellian distributions, the ions having zero velocity shift and the electron Maxwellian shifted by $vde$⁠, using a randomized “quiet start” algorithm to suppress initial long-wavelength noise. Diagnostics accumulate their phase-space density at time intervals down to $Δt$⁠, as two-dimensional histograms having 50 equal velocity intervals, and 200 equal space intervals. Velocity integrals provide electron and ion density $nj(x,t)=∫fjdv$⁠, whose initial values are normalized to unity. The potential, $ϕ(x,t)$⁠, provides the electric field $E=−∂ϕ/∂x$⁠, and allows average electric field energy density $〈E2〉=∫E(x,t)2dx/L$ to be integrated. A parallel magnetic field does not affect the one-dimensional dynamics, which is considered parallel to any B.

Figure 1 shows how the long-term nonlinear time development of Buneman instabilities depends on the ion temperature. In each subplot the top panel shows the time dependence of the electric field energy-density; the middle panel shows cube root (to expand dynamic range near zero) potential (⁠ $ϕ(t,x)1/3$⁠) contours in time and space; and the bottom panels show spatially averaged electron distribution function contours of the total number of simulation-particles in velocity range dv. [In detail, $fe(t,ve)dv$ denotes the spatial integral over the total domain length (L): $dv∫Lfe(t,x,ve)dx$⁠, where dv is the velocity width of a single histogram box, equal to the initial total velocity range divided by 50.] The different parameter values of the initial distributions in this paper are $Ti=0.04,1,4,25$ (times $Te$⁠) and electron drift $vde=−1.3,−1.5,−1.75,−2,−3$ (times $Te/me$⁠), which appropriately cover the interesting range for formation of coherent structures. Higher drift velocities lead to large amplitude, mostly incoherent, potential fluctuations, and rapidly flattened electron distributions.

This section discusses only constant initial electron velocity shift (⁠ $−2vte$⁠). For cold ions (a) $Ti0=0.04$(⁠ $×Te0$⁠), the initial sinusoidal unstable wave(s) have speed slightly exceeding the cold ion sound speed $cs=Te0/mi$ propagating (relative to ions) in the same (negative) direction as the electron distribution shift. As the potential perturbations approach unity amplitude (and energy saturation) however, they accelerate. This acceleration might be caused partially by the spreading of the electron distribution, but also as will be shown later, higher phase speeds are associated with greater potential amplitude and longer wavelengths. The highest potential peaks $ψ$ have electron holes trapped within the ion perturbations whose signature is rapid oscillation of the peak position, observable in Fig. 1 during $t∼$ 500–1000 (units of $ωpe−1$⁠), with oscillation speeds a substantial fraction of electron thermal speed. The speed of the structures averaged over oscillations is typically up to 6  $cs$⁠. The oscillations appear to generate positive-propagating peaks after about $t=$ 800, giving rise to diagonal cross-hatching of the potential contour plot, which persists thereafter. The potential peaks move at $∼±3cs$ in this later phase (⁠ $t≳3000$⁠).

When (b) $Ti0=1$ the behavior is fairly similar, though with fewer positive-propagating peaks being generated. However, when (c) $Ti0=4$⁠, there are hardly any positive-propagating peaks, and the oscillatory behavior persists throughout, with the number of distinct peaks gradually decreasing through mergers; their speeds remain higher: $∼6cs$⁠. In addition in (b) and (c) there are much thinner, steeper, mostly downward streaks at near electron thermal speed, which detailed phase-space portraits show to be mostly small electron holes spawned near larger vortex separatrices.

When (d) $Ti0=25$⁠, the initial perturbation growth is very slow, and short wavelength waves moving at near the acoustic speed dominate until time 1000. The growth time fitted line in this case does not correspond to a linear phase. Instead its algorithm chooses the following period when the wave peaks simultaneously grow, accelerate, and merge, so that by 1500 the field energy saturates at close to the lower- $Ti0$ cases' levels, and is concentrated in just three potential peaks by 2200. These are clearly electron holes, traveling at (minus) almost half the initial electron thermal speed (relative to ions), approximately $18cs$⁠. In this case, the average electron distribution function has been broadened only in a limited velocity extent around zero, unlike the low- $Ti0$ cases which show flattening over much of the final $fe$ spread.

The full phase-space of the simulations summarized in Fig. 1 is imaged as a function of time in videos of which the contour plots Figs. 2–4 are individual frames. These videos add persuasive phase-space portraits to enable the dominant time-dependent phenomena to be identified. (The cases and their case letters, in sorted order are listed in Table I in Sec. VI).

Case (g) $Ti0=1/25$⁠, whose video can be viewed at https://youtu.be/lSc8ZlhoYas, and of which Fig. 2 is a frame, shows rapid early growth of an unstable wave, large enough by time 200 to trap electrons and form vortices in the electron phase-space. Velocity units are initial electron thermal velocities for both panels. The contour units of f are the number at this time step of simulation particles per pixel of the phase-space, thus giving a measure of statistical uncertainty. Where indicated, the base-10 logarithm of $fi$ is contoured to reveal small-amplitude structure. Spatially averaged initial and instantaneous velocity distributions, rescaled to constant peak height, are shown at right. Time of frame is printed top right. Adjacent potential peaks merge with each other so that the electron (species 1) vortices along with their associated ion velocity perturbations grow larger and fewer. By t = 500 the number of vortices is about half its initial number, and the largest potential peak has reached 2. By 750 another approximate halving of the vortex numbers has occurred with peak potentials up to 4. The electron and ion density perturbations at this stage are of comparable magnitude and no longer in antiphase as they had been during most of time till 400. Merging into discrete vortices has mostly stopped by time 1000 and thereafter the ion and electron densities, and potential perturbations are all in phase, but with the ion perturbation amplitude exceeding the electron. The trapped phase-space $fe$ depression of the electrons is only a modest contributor to potential perturbations. The electron vortices continue to merge and divide under the dominant influence of ion density fluctuations. To call them electron holes in these late stages would be misleading, considering their minimal electron phase-space depletion.

Case (h) $Ti0=1$⁠, viewable at https://youtu.be/-LcFx--4SgE (and with $f2$ contoured linearly at https://youtu.be/USz-k3gEBUg) is largely similar, but the substantial initial ion velocity spread $Ti0=Te0$ allows the formation of spatially localized high speed filamentary ion streams accelerated to both positive and negative velocities, which then move (and shear) in accordance with their phase-space positions in both positive and negative directions. Figure 3 shows a frame. The ion density enhancements are then greatest where both positive and negative velocity streams are present at the same position, that is at the intersection of the opposite slope diagonal lines in Figs. 1(a) and 1(b). Again, ions dominate the longer-term potential fluctuations, and electron vortices bounce around under the ions' influence.

Case (i) $Ti0=4$⁠, viewable at https://youtu.be/K-3p7rprikY, shows greater persistence of the electron vortex structures. They bounce around quite rapidly in the background ion density perturbations, but the relative importance of ion contribution to potential is lower, because the higher ion temperature phase-mixes $fi$ perturbations more rapidly. The electron vortices that remain, toward the end of the simulations, still have substantial $fe$ and $ne$ depression, and self-identity, even though their buffeting by the ions causes them to decay in magnitude (and number) fairly rapidly. The final average electron distribution is far less flattened than before.

Case (j) $Ti0=25$⁠, viewable at https://youtu.be/pawoTH5rGUQ, and of which Fig. 4 is a single frame, takes a long time for the instability to grow to a level where nonlinear effects are obvious. Its initial wavelength is considerably longer. The spurt of energy growth starting at t = 1000 accompanies a group of peaks that accelerate (in the negative direction) rapidly and devour the smaller, slower peaks; so that by 1500 only five peaks with their vortices are left, all moving fast. Two further merges occur so that by 2200 only three peaks remain; see Fig. 4. These are very obviously persistent electron holes for which the ion density perturbation influence is small, in part because of the high ion temperature, but also because the holes move much faster than the ion thermal speed, also reducing the ion response. The final distortion of the electron average distribution $f1$ appears mostly as two regions of reduced gradient, attributable to the holes themselves, and not the background between them.

When the distribution shift is greater (3  $vte$⁠), the case (l), $Ti0=25$ is illustrated in Fig. 5 and viewable at https://youtu.be/G2GZgww-87Y. The initial growth is no longer significantly slowed. However, like the corresponding $2vte$ shift case, strong acceleration of the potential peaks accompanies saturation, leading to electron–hole merger, and finally by time 1500 to a rapidly moving single hole with large amplitude $ψ∼8$⁠. It experiences occasional reflections from the potential perturbations produced by slowly evolving ion density perturbations, but persists beyond time 5000 with only gradual diminution of its amplitude caused mostly by detrapping associated with the reflections. The background electron velocity distribution is extensively flattened, but the spatially averaged ion distribution is only weakly perturbed.

For case (k) $vde=3$ and $T=1$ (video at https://youtu.be/yL3jLr1br4k), the initial growth time is extremely fast, only 14, and by time 1000 electron holes have mostly dissipated during a period of high perturbation amplitude $ψ∼10$⁠, leaving counter-propagating ion-generated potential peaks with speed $∼6cs$⁠, together with extended ion phase-space streams, and $ψ∼5$ by time 2500.

Figures 6–8 illustrate cases with smaller velocity shifts $vde$ of the initial electron distribution. When $vde=−1.3$ (and $Ti0=1$⁠), case (b), which can be considered at the threshold³² of equal-temperature Buneman instability [Fig. 6(a) video at https://youtu.be/WpRx-5CUzZ8], the potential perturbation grows weakly with modestly accelerating peaks, and the total field energy saturates at a low level. Even during this early phase, the electron and ion density perturbations are of comparable magnitude. Then the peaks merge, increasing their height to $ψ≃0.15$ without significant total field energy growth, and accelerate somewhat more. Ion and electron density perturbations are of comparable amplitude, both positive at the potential peaks, with modest trapped $fe$ depression. By time 4400, only five peaks are left and merging has become rare. A (barely detectable) narrow region of reduced slope at zero velocity is present on the spatially averaged $fe$⁠. The peaks' speed, $∼−4cs$⁠, can probably be interpreted as approximately the nonlinear ion acoustic speed for the corresponding shifted electron distribution as will be discussed later. A simulation Fig. 6(b) (video at https://youtu.be/fwLCVTGD6ZE) with lower ion temperature case (a) $vde=−1.3$⁠, $Ti0=0.04$ reaches considerably higher field energy density (⁠ $∼3×10−3$⁠) and peak potential $∼0.8$⁠. Its potential peaks are on average slower than those of Fig. 6(a). They also have oscillating coupled electron holes, which sometimes temporarily escape the ion density perturbations. In neither case is a clear linear growth fit of the early stages persuasive.

The intermediate velocity $vde=−1.5$⁠, $Ti0=1$ case (d) [Fig. 7, video at https://youtu.be/ThRDFiP4FFo and a shorter linear $fi$-contours version at https://www.youtube.com/watch?v=70lJ814aUr4] has no substantial early electron–hole oscillations and has weak average $fe$ perturbations and final peak heights $ψ≃0.8$⁠, after substantial mergers over times of several thousand $ωpe−1$⁠. The potential in the early nonlinear stages is generated approximately equally by opposite polarity electron and ion density perturbations, giving what can be considered a train of electron holes coupled steadily to soliton-like ion density modulations. However as peak mergers raise their amplitude and speed, the electron phase space, Fig. 7(b), shows that the electron density is actually somewhat greater in the potential peaks than outside, and there is only weak depletion of $fe$ in the trapped region. The ion density enhancement is greater (sustaining the potential peak) and arises mostly from local expansions of the ion distribution toward negative velocities, accompanied by some ion streams. Thus, these very long-lived structures have a mostly soliton character but their speed, $∼4.5cs$⁠, is enhanced by the substantial electron distribution shift, as will be analyzed later. It is notable in the video that each merging process consists of a larger amplitude peak overtaking a smaller amplitude peak. That is consistent with soliton speed increasing with amplitude. However, merging is not consistent with historic analysis of KdV fluid solitons, which pass through one another and emerge with their identities and amplitudes intact. Thus, again, these solitary waves should probably be thought of as CHS (coupled hole-solitons) structures where the trapped electron dynamics is important.

For this shift, $vde=−1.5$⁠, lowering $Ti0$ to 0.04 case (c) (video at https://youtu.be/75MhqpX0djE and shorter linear $fi$-contour version at https://youtu.be/_MHqEAPYTzE) raises the peak amplitude to $ψ∼2$⁠, from which it later relaxes to $∼1$⁠, restores coupled electron–hole oscillations, substantially widens the $fe$⁠, produces a few long fast ion streams which emerge from high electric field regions, and produces occasional streams of electron holes that are rapidly captured and usually dissipated.

Figure 8, case (e), has $vde=−1.75$⁠, $Ti0=Te0$ (video https://youtu.be/_xFMa-8cWHY, linear at https://youtu.be/97wzpGqSCVo). It is similar to the corresponding $vde=−2$ case except that none of the forward-propagating ion structures are of clear CHS structure. Instead, the faint cross-hatching in Fig. 8(a) appears to be caused by the high-speed ion streams of positive velocity. It has a substantially flattened $fe$ and the potential peaks approach $ψ≃2$ at time 1000, with deep trapped $fe$ depressions and negative $ne$ excursions. These are electron holes trapped but unstable in CHS phenomena, and their average negative speed is $∼5cs$⁠. Thereafter the peaks gradually decay to $ψ∼0.8$ by time 4000, as the trapped electrons are phase-mixed and the trapped $fe$ depressions are smoothed away, with the result shown in Fig. 8(b).

At this $vde=1.75$⁠, but with high ion temperature $Ti0=25Te0$⁠, case (f) the video at https://youtu.be/wUnVmEBEJ7A shows formation of electron phase-space vortices, merging to give larger electron holes (up to $ψ=2.7$⁠) with noticeable central $fe$-depression, and moving at high speed (⁠ $vp∼20cs$⁠, very similar to the high- $Ti0$⁠, $vde=2$ case of Fig. 4.

When $|vde|>1.5$⁠, high $Ti0≫Te0$ favors formation of electron holes that are accelerated to speeds much greater than the ion thermal and sound speeds. When the free energy released by the instability is high, they are occasionally reflected from ion perturbations, but without being trapped by a coherent ion density peak. More moderate temperature ratios $Ti0≲4Te0$ allow soliton-like ion perturbations to trap electron holes within themselves, forming coupled hole-solitons (CHS), often with hole oscillations. At ion temperatures $Ti0≲1$⁠, the CHS structures mostly disassemble by approximately time 1000, giving rise to counter-propagating individual potential peaks attributable to ion density perturbations moving at speeds several times $Te0/mi$⁠.

Higher distribution shifts (⁠ $vde=−3$⁠) give rise to higher potential peaks and produce ion fluctuations that, even at $Ti0=25$⁠, are large enough to reflect the electron holes.

Lower distribution shifts decrease the peak potential height, avoid generating forward-propagating structures, and reduce the $fe$ flattening. Their potential peaks have speeds of order $4cs$ showing they are neither pure electron holes nor standard ion acoustic solitons.

No obvious ion holes: ion phase-space vortex structures in negative polarity potential valleys have been observed in any case. Instead, influential ion streams in phase-space are generated when the potential peaks are high enough, and initial ion temperature is $∼1$ or 4. The streams extend to speeds $|vi|$ several times $vti$⁠, but remain relatively incoherent, becoming turbulently randomized or elongated, preventing a coherent ion vortex from being completed.

The purpose of this section is to present a highly idealized analysis based on adopting model electron and ion distributions that are a function only of energy, in a frame of reference in which the steady potential form is stationary, hence satisfying Vlasov's equation. Certain constraints and relationships between the distribution and the potential structure's velocity relative to them emerge from calculation of the resulting self-consistent potential shape $ϕ(x)$⁠. The analysis here parallels the treatment of Refs. 33 and 34, but avoids small-argument expansion of $n(ϕ)$⁠, and directly calculates wavelength and spatial profile numerically, based on a specific choice of distribution shapes. The advantage for the present purpose is more transparent algebra and comprehensive quantitative results. However, it does not take account of variable phase-space depletion of trapped particles, so the approach is not a treatment appropriate for classic electron or ion holes.

The model distribution of ions is, for algebraic simplicity, a single velocity stream (cold plasma fluid) whose speed v in the potential structure's rest frame is given by constant energy $E=12mv2+qϕ=const$⁠. In dimensionless units, the stream velocity is then $2(Ei−ϕ)/mi=2(Ei−ϕ) cs$ (since $Te0=1$⁠). Adopting the ion speed at zero potential as the primary reference (to a good approximation the initial simulation ion velocity), the structure (phase) velocity in the ion frame is $vp=2Ei$⁠. The sign of $vp$ and of $vde$ is taken positive in this analysis section to avoid frequent minus signs. The (assumed steady) ion density is then $ni(ϕ)=ni(0)/1−ϕ/Ei$⁠, valid only as long as there is no ion reflection (⁠ $ϕmax≡ψ<Ei$⁠).

The electron distribution $fep(ve)$ in the structure frame of reference for untrapped (passing) particles, $Ee=ve2/2−ϕ>ϕmin=0$⁠, is taken as a Maxwellian of temperature $Te0(=1)$⁠, shifted from the ions by a drift velocity $vde$⁠. Their drift velocity relative to the potential structure frame is then $vde−vp$ and $fep=exp[−(ve−vde+vp)2/2]/2π$⁠. In this hypothetical steady state, trapped electrons must have a symmetric distribution, and are taken to have a distribution function independent of velocity: $fe(Ee<0)=fep(Ee=0)$⁠. This form of distribution is called flat-trapped. The corresponding density $ne(ϕ)$ cannot be expressed compactly through standard functions but can quickly be numerically evaluated and has been plotted elsewhere. (Ref. 13, Fig. 10) Electron holes have trapped distribution deficit relative to flat-trapped. However, many of the electron vortices observed in the videos have small deficit; so adopting flat-trapped electrons is reasonable approximation, even though often only a fairly crude one. We are not here addressing the structure of the observed electron holes, but of the wave peaks arising from electron–ion interactions.

Figure 9 illustrates the solution process of the resulting Vlasov-Poisson system, with electron shift $vde=1.5$⁠, structure phase velocity $vp=3cs$ (both in the ion frame), and peak height $ψ=0.5$ (relative to $ϕmin=0$⁠). These parameters give the corresponding $ne(ϕ)$ and $ni(ϕ)$ shown in the top panel. The self-consistent solution of Poisson's equation can be found by obtaining the first integral $−12(dϕdx)2=V(ϕ)=∫ϕminϕqini+qenedϕ$⁠, which is called the pseudo-potential. It is shown in the middle panel together with the contributions of electrons $∫−ne(ϕ)dϕ+ne0ϕ$⁠, and ions $∫ni(ϕ)dϕ−ne0ϕ$ [using shorthand notation $n0≡n(ϕmin)$], in which the addition and subtraction of $ne0ϕ$ serves to enable plotting on a convenient scale. The sum of the electron and ion contributions is the pseudo-potential $V=∫ρdϕ$⁠. It must be negative over the potential range (whose upper end is $ϕmax=ψ$ where $V=0$⁠), to ensure $dϕ/dx$ is real. Then, the potential variation with position is found implicitly from $∫xminxdx=∫ϕminϕdϕ/−2V$ shown in the bottom panel. This procedure is familiar in the analysis of solitons and electron-holes (see, for example, Refs. 13, 35 and 36) where truly solitary structures require that the second derivative of $ϕ$ which is proportional to the charge density, becomes zero at $ϕmin$⁠: $ne0=ni0$⁠. However, the present context includes approximately periodic (wave-like) structures. For finite wavelength, no such second derivative requirement arises. Instead all that is formally required is that the total charge contained in a half-period be zero: $V(ϕmax)=V(ϕmin)=0$⁠. The wave is no longer quasi-neutral at $ϕmin$⁠, and instead, we require $ne0>ni0$ to enforce $d2ϕdx2|ϕmin>0$⁠.

For chosen electron and ion distributions, there is generally no guarantee that V remains negative between $ϕmin$ and any chosen $ϕmax$⁠. If it does not, no valid solution for that $ϕmax$ exists, but valid solutions with small enough $ϕmax$ do exist provided $dn̂i/dϕ>dn̂e/dϕ$ at $ϕmin$ and there exists some positive potential at which $V̂e+V̂i=0$⁠. The top panel of Fig. 9 shows the electron and ion densities determined by the requirement that $V(ψ)=0$ with $ψ≡ϕmax=0.5$⁠, which needs $ni0=0.974ne0$⁠. The lowest panel is the resulting spatial dependence of the potential for half a wavelength, in which the ordinate is the position x relative to the potential peak. The periodic structure's wavelength is $L=2×10.5=21$ (Debye lengths) in this case.

The range of half wavelengths (⁠ $L/2$⁠) for a specified electron velocity shift obtained by running the solver for many $ψ$ and $vp$ values is shown in Fig. 10. For this electron drift speed (1.5), we see that the wavelength L increases as $vp$ increases, with gradient increasing as $ψ$ increases. This behavior is consistent with the PIC simulation observations of simultaneous acceleration, peak-potential growth, and wavelength growth. Remember, though, that this theory does not account for the coupled electron–hole effects that are observed in the simulations. Phase velocities exceeding about $4cs$ are prevented for $vde=1.5$ by the reduction of the ion response, and strong wavelength growth, for $ψ>1$⁠. These are attributable to reduction of $dni/dϕ$ toward $dne/dϕ$ eventually preventing the positive crossing of their densities. It should be remarked that these large wavelengths have minima narrower than their maxima. In other words, in the limit they tend to solitary negative potential valleys, rather than solitary peaks.

Increasing to $vde=2$ gives rise to a noticeably negative electron density gradient $dne/dϕ$⁠, and permits flatter ion response and higher $vp$ and $ψ$ as illustrated in Fig. 11. The analytic treatment then permits simultaneous growth of $vp$⁠, as well as $ψ$ and L, as has been seen in the corresponding simulations. However, the electron density dependence on potential for these analytic structures has $dne/dϕ$ negative. Early times in the corresponding simulations do have negative $ne$ valleys at the potential peaks, but later, when the spatially averaged $fe$ has an extended flat region, the electron density perturbations reverse their polarity with respect to the potential, and disagree with the analytic predictions. This is presumably because the actual electron distribution has been substantially changed from what the analytic approximation assumes.

The simulation observations are summarized in a highly simplified form in Table I, which records the observed phase speed of peak propagation $|vp|/cs$⁠, and the peak height $ψ$⁠, arising from different initial simulation parameters $|vde|$ and $Ti0$⁠. It also shows for the corresponding PIC parameters the analytic model's upper limit of $vp/cs$ that permits a solution, which is determined by exploring a range values of $vp$⁠. The value of $ψ$ chosen for this exploration changes the limit very little; but the limit is very sensitive to the value of $|vde|$⁠. In particular, there is a very rapid transition in the range $1.5<|vde|<1.75$ between a value $∼4cs$ that is of approximately the same magnitude as observed in the simulations, to a much higher analytic upper limit for larger $|vde|$⁠. In short, when $|vde|≤1.5$ there is an equilibrium limit to $vp$⁠, while for $|vde|>1.75$ the simulation phase velocity is not limited by nonlinear equilibrium, because $dne/dϕ$ is negative. The higher $|vde|$ simulations appear to be saturated by the strong electron distribution flattening when the potential exceeds $∼vde2$⁠, which is sufficient to make the electron phase-space vortices reach beyond $vde$⁠. The $Ti0=Te0$ simulations of Tavassoli et al.²⁰ observed a threshold for generation of counter-propagating potential peaks they call “backward waves,” between $vde$ values of 1.5  $cs$ and 1.75  $cs$⁠, which they attributed to the increase in the averaged distribution flattening as the peak potential increases, and the consequent reduction in backward waves' theoretical linear damping rate in this distribution. The present observations agree, except that we observe the backward waves emerging from coupled electron–hole oscillations, so there is more to the story. Clearly, a lot is changing in this range of drift speeds.

Only the high- $Ti0$ cases (f), (j), and (l) give predominant electron holes, rather than slower ion-dominant structures. Electron holes of various amplitudes are created in the lower ion temperature cases, but are trapped or reflected by the potentials generated by ion density perturbations. The result is sometimes a quiescent CHS, sometimes an oscillatory CHS, and sometimes buffeting small electron holes incoherently between different potential peaks.

Future studies would very naturally address the important remaining question of how the present 1D results are affected by multidimensional physics and transverse spatial dependence. Some expensive 3D simulations have already been done in the context of anomalous resistivity during reconnection (see, e.g., Ref. 38 and references therein). In them, coherent field structures are observed but their formation has not been analyzed in detail. A difference is also that the simulations are driven by imposed electric field, unlike the present case, and they are carried out for greatly reduced mass ratio $mi/me=100$⁠, which changes the nonlinear effects in Buneman instability phenomena in ways that are hard to predict. Therefore, much remains to be done.

In summary, the nonlinear evolution of perturbations in an initially Buneman unstable one-dimensional plasma is revealed to have many quasi-coherent features that are not well represented by random-phase quasi-linear analysis, but can be fruitfully (though incompletely) understood in terms of compound entities such as electron holes, coupled hole-solitons, and non-linear wave peaks. These structures move faster relative to the ions than classic ion-acoustic solitions, but with the exception of free electron holes not faster than about 6  $cs$⁠, or even 4.5  $cs$ for electron drift speeds less than approximately 1.75 times the electron thermal speed $Te0/me$⁠. Their general trend is that phase speed and spatial period increases with potential height. When such structures are solitary, they do not retain their identity when they overtake one another as do KdV solitons. Instead, they combine to form a peak of greater height. Electron holes are generated only when $Ti0≫Te0$⁠. Ion hole formation does not occur.

It is hoped that easy access provided to the detailed simulation videos will inspire in other investigators further insights into the nonlinear phases of this classic instability, and the formation of persistent potential structures.

See the supplementary material for corresponding YouTube videos.

This work was not supported by any external funding.

The author has no conflicts to disclose.

I. H. Hutchinson: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.