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Particle-in-cell simulation study of a lower-hybrid shock
Dieckmann, M. E., Sarri, G., Doria, D., Ynnerman, A., & Borghesi, M. (2016). Particle-in-cell simulation study ofa lower-hybrid shock. Physics of Plasmas, 23(6), [062111]. https://doi.org/10.1063/1.4953568
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Particle-in-cell simulation study of a lower-hybrid shock
M. E. Dieckmann,1 G. Sarri,2 D. Doria,2 A. Ynnerman,1 and M. Borghesi2
1Department of Science and Technology, Linkoping University, SE-60174 Norrkoping, Sweden2School of Mathematics and Physics, Queen’s University Belfast, BT7 1NN, UK
(Dated: May 24, 2016)
The expansion of a magnetized high-pressure plasma into a low-pressure ambient medium isexamined with particle-in-cell (PIC) simulations. The magnetic field points perpendicularly to theplasma’s expansion direction and binary collisions between particles are absent. The expandingplasma steepens into a quasi-electrostatic shock that is sustained by the lower-hybrid (LH) wave.The ambipolar electric field points in the expansion direction and it induces together with thebackground magnetic field a fast E cross B drift of electrons. The drifting electrons modify thebackground magnetic field, resulting in its pile-up by the LH shock. The magnetic pressure gradientforce accelerates the ambient ions ahead of the LH shock, reducing the relative velocity between theambient plasma and the LH shock to about the phase speed of the shocked LH wave, transformingthe LH shock into a nonlinear LH wave. The oscillations of the electrostatic potential have a largeramplitude and wavelength in the magnetized plasma than in an unmagnetized one with otherwiseidentical conditions. The energy loss to the drifting electrons leads to a noticable slowdown of theLH shock compared to that in an unmagnetized plasma.
PACS numbers: 52.35.Tc 52.65.Rr 52.35.Hr
I. INTRODUCTION
Collisions between unmagnetized clouds of electronsand ions at a speed in excess of the ion acoustic speedcan trigger the formation of electrostatic shocks. An elec-trostatic shock is an ion acoustic wave that has steepenedinto a sharp density jump. Ion acoustic waves and elec-trostatic shocks are sustained by the following process.
Consider a plasma with a spatially varying numberdensity. Thermal diffusion will let electrons flow froma region with a high plasma density towards one with alow density and their larger inertia implies that the ionscan not follow them. The current, which arises from theelectron redistribution, generates an electric field. Themean electric field is called the ambipolar electric field.Its amplitude value is such that the electrostatic forcebalances the electron’s thermal pressure gradient force,which inhibits a net flow of electrons.
The ambipolar electric field can accelerate ions froman interval with a high plasma density to one with a lowplasma density. The ion redistribution alters the plasmadensity gradients and thus the ambipolar electric field.The changes of the ambipolar electric field and of the iondensity distribution are out of phase and both oscillatearound an equilibrium distribution in the form of ionacoustic waves. We obtain almost undamped ion densityoscillations if the electrons are hotter than the ions.
The ion acoustic wave is the only wave mode in a non-relativistic setting and in the absence of a backgroundmagnetic field, which can modulate the density of theions. It is thus the only wave mode that can sustain anelectrostatic shock [1–8] in a collisionless unmagnetizedplasma unless the collision speed is high enough to yielda partially magnetic shock [9, 10]. The density gradi-ent at the electrostatic shock drives an ambipolar elec-tric field, which puts the downstream region behind the
shock on a higher positive potential than the upstreamregion ahead of it. This potential difference slows downthe inflowing upstream plasma. The therefrom resultingcompression of the plasma sustains the density gradient.Self-consistent and steady-state solutions of an ion acous-tic wave that steepened into an electrostatic shock exist,provided that the speed of the upstream plasma mea-sured in the shock frame does not exceed a few times theion acoustic speed [8].
Electrostatic shocks are now routinely produced inlaser-plasma experiments [11–18] and they attract con-siderable interest because they allow us to study in-situsome of the plasma processes that develop in remote as-trophysical environments. An example are the shocksthat ensheath the blast shells of supernova remnants [19].
A magnetized plasma supports several compressionalwave modes and, hence, different types of shocks. Magne-tohydrodynamic (MHD) shocks can be mediated by theAlfven wave, by the slow magnetosonic mode, by the in-termediate mode or by the fast magnetosonic mode. Theparticular mode is selected primarily by the angle be-tween the shock normal and the magnetic field and thespeed, with which the shock propagates. MHD shocksform on time scales that are longer than an inverse iongyrofrequency. The thickness of their transition layersexceeds the gyroradius of the inflowing upstream ions inthe shock’s magnetic field. The best-known examplesare probably the fast magnetosonic shock that forms be-tween the solar wind and the Earth’s magnetopause [20]and the solar wind termination shock [21] that separatesthe heliosphere from the interstellar medium.
A magnetized plasma does support more ion wavemodes than the aforementioned MHD waves. In whatfollows we consider waves that travel orthogonally to themagnetic field. A perpendicular magnetic field of suitablestrength will limit the electron mobility on spatial scales
2
in excess of their gyroradius. An ion density gradient willnevertheless drive an ambipolar electric field because theelectrons can still move on spatial scales below their gy-roradius. The ambipolar electric field has a componentthat is antiparallel to the ion density gradient, which en-forces an electron drift in the direction orthogonal to themagnetic field and to the density gradient.
The electron response to the ambipolar electric field isaltered by its gyro- and drift motion, which modifies inturn the dispersion relation of the electric field oscilla-tions. This effect plays an important role at frequenciesabove the ion gyrofrequency. The MHD approximationbreaks down at such high frequencies and it has to bereplaced by a two-fluid approximation. The two-fluidapproximation reveals the presence of the almost electro-static wave branch, which is known as the lower-hybrid(LH) mode. A kinetic model reveals that the LH modegoes over into the ion cyclotron waves if its wavelengthis no longer large compared to the ion’s thermal gyrora-dius. The dispersion relation of LH waves is discussed invarious approximations in Ref. [22].
LH oscillations can have a shorter wavelength and ahigher frequency than magnetosonic waves. A LH wavecan thus steepen into a shock faster and on a smallerspatial scale. A LH wave with a small wavelength ispractically electrostatic and we may expect that a shock,which is mediated by this wave mode, is electrostatic too.So far the LH waves have received attention with respectto instabilities close to shocks [23], but the observationof a LH shock per se has not yet been reported.
Collisions of magnetized plasma clouds have beenwidely examined by means of particle-in-cell (PIC) sim-ulations, but the collision speeds were always too high toyield shocks that could be sustained solely by the electriccross-shock potential. The fast shocks in the aforemen-tioned case studies formed primarily due to the magneticrotation and reflection of the colliding plasmas [24–28]and were caused to a lesser degree by the cross-shockelectric field. The latter has been observed in situ at theEarth’s perpendicular bow shock [29], which is otherwisea fast magnetosonic shock. See Ref. [30] for a recentreview of magnetized nonrelativistic shocks.
Here we show by means of PIC simulations that LHshocks exist and we discuss how such a shock differs fromelectrostatic shocks in an unmagnetized plasma. The LHshock we observe is a transient structure like its coun-terpart in unmagnetized plasma. The lifetimes of a LHshock and of an electrostatic shock are, however, limitedfor different reasons.
The narrow unipolar electric field pulse, which char-acterizes electrostatic shocks in unmagnetized plasma, istransformed into a broad shock transition layer by theion acoustic turbulence, which is generated upstream ofthe shock by the shock-reflected ion beam [4, 6, 31].
The LH shock is modified by the magnetic field it ispiling up as it expands into the upstream region. Themagnetic pressure gradient force it exerts on the ambi-ent ions in the upstream region is pre-accelerating them,
which reduces the ion velocity change at the shock toa value that is comparable to or below the phase speedof the LH wave. The structure changes in time from astrong LH shock into what appears to be either a weakershock or a nonlinear LH wave. A similar qualitative dis-tinction of shocks and nonlinear waves based on the shockspeed was given in Ref. [3].
While the LH shock compressed the ambient plasmato the same density as the unmagnetized electrostaticshock, the ambient plasma that crosses the magnetizedstructure at late times is hardly compressed. This struc-ture does, however reflect some of the incoming upstreamions, which is a signature of a collisionless shock. Thisnonlinear LH wave balances the ram pressure of the up-stream plasma primarily with the gradient of the mag-netic pressure and to a lesser degree with the thermalpressure of the downstream plasma.
The expanding blast shell piles up the magnetic fieldahead of the shock, thereby increasing the magnetic pres-sure in the ambient plasma. This suggests an the follow-ing long-term evolution of the plasma. Initially the am-bient plasma is the upstream medium and the blast shellis the downstream medium that expands into the am-bient plasma due to its thermal pressure. The gradualincrease of the magnetic pressure gradient force suggeststhat in the long term the magnetic field may dominatethe plasma dynamics. Eventually the ambient mediummay obtain a large enough magnetic pressure to becomethe downstream region of a magnetosonic shock and theblast shell provides the fast upstream flow.
This paper is structured as follows. Section 2 comparesthe solution of the linear dispersion of LH waves thatpropagate strictly perpendicularly to the backgroundmagnetic field with the noise spectrum that is computedby PIC simulation, exploiting the fact that the noise isstrongest when its wave number and frequency matchthat of a plasma eigenmode [32]. Section 3 shows bymeans of PIC simulations how a LH mode shock formsand how it changes with time into a nonlinear LH wave.The simulations reveal the electromagnetic signatures ofLH shocks, which should be detectable in laser-generatedplasma and which differ from those of the well-researchedelectrostatic shocks in unmagnetized plasma. We sum-marize our results in Section 4.
II. ELECTROSTATIC WAVES INMAGNETIZED PLASMA
We consider here the approximate solution of the lin-ear dispersion relation of LH waves, which is based ona two-fluid approximation, that takes into account warmplasma effects and neglects electromagnetic effects. Itis valid for large wavenumbers k = |k| and for wavesthat move strictly perpendicularly to the magnetic fieldB with amplitude B0. The LH frequency
ωlh =[(ωceωci)
−1+ ω−2
pi
]−1/2
(1)
3
becomes a resonance frequency at low k, where thermaleffects are negligible. The electron’s thermal gyroradius
rce = vte/ωce, where vte = (kBTe/me)1/2
(kB, Te,me:Boltzmann constant, electron temperature and mass) isthe electron thermal speed and ωce = eB0/me is theelectron gyrofrequency. The ion’s thermal gyroradius isrci = vti/ωci. The plasma frequency of ions with thenumber density ni0, the charge qi and the mass mi is
ωpi = (q2i ni0/miϵ0)1/2
and ωci = qiB0/mi is their gy-rofrequency. The thermal speed of ions with the temper-
ature Ti is vti = (kBTi/mi)1/2
.We consider wave vectors k with k · B = 0, a tem-
perature ratio Te/Ti = 12.5 and fully ionized nitrogenions N7+ with the number density ni0 = 4 × 1013cm−3.The electron number density is 7ni0. The magneticfield strength is B0 = 0.85 T, yielding ωpi = 6ωlh andωlh = 60ωci. The ion composition and number den-sity are representative for the ambient plasma in labo-ratory experiments; more specifically, the ratio betweenthe plasma frequencies of the electrons and ions and thegyrofrequencies are comparable to those used in Ref. [18].The magnetic field strength is selected such that the fre-quency of the LH branch at large wavenumbers becomescomparable to the ion plasma frequency. Magnetic fieldeffects should develop fast enough to be detectable. Theplasma β ≈ 0.2 in the ambient medium implies that themagnetic pressure is high enough to balance the thermalpressure of the expanding dense medium.We can neglect magnetic field effects on the ion motion
if the wave frequencies are ω ≈ ωlh and approximatethe ion susceptibility as χi = ω2
pi/(3v2tik
2 − ω2). Theelectron susceptibility to LH oscillations is approximatedas χe = ω2
pe/(v2tek
2 + ω2ce). The dispersion relation is
1 + χi + χe = 0 or
ω2 = 3v2tik2 +
ω2pi(ω
2ce + v2tek
2)
ω2pe + ω2
ce + v2tek2
(2)
We compare the dispersion relation given by Eqn. 2to the electrostatic noise distribution, which is computedby the EPOCH PIC simulation code. The simulationresolves one spatial (x) direction and it has been initial-ized with the aforementioned plasma parameters. Thesimulation employs periodic boundary conditions and itresolves a box length of 16 mm or 96 rce by 3200 gridcells. We run the simulation for ts = 4 ns, which resolvesthe fraction ωcits/2π = 0.16 of an ion gyro-orbit. Theelectron temperature is set to Te = 2 keV.Electrostatic waves are polarized along the simulation
direction and they can be identified using the Ex com-ponent. We obtain the noise distribution by samplingthe electric field Ex(x, t), by taking its Fourier transformover space and time and by squaring the modulus of theresult. The power spectrum of the noise distribution ofa PIC simulation peaks at frequencies, which are eigen-modes of the plasma and it can thus be used to reveallinearly undamped or weakly damped wave branches.Figure 1 shows the result. Strong noise that follows
0 5 10 15 20 250
2
4
6
8
10
12
k vte
/ ωce
ω /
ωlh
−3
−2.5
−2
−1.5
−1
−0.5
0
FIG. 1: The 10-logarithmic power spectrum |Ex(k, ω)|2 ofthe electrostatic noise, which has been computed by a PICsimulation. Overplotted is the solution of the linear dispersionrelation. The vertical line corresponds to a wavelength of 0.1mm for the considered plasma parameters.
the solution of Eq. 2 is observed up to a wave numberkrce ≈ 20. We thus identify this mode as the LH mode.
We calculate the phase speed vph ≈ 2 × 105 m/s ofthe LH wave at the reference wavelength 0.1 mm usingthe noise distribution. We can compare the phase speedto the phase speed cs of the ion acoustic waves, whichwould be present if B0 = 0. The ion acoustic speed
cs = (kB(Te + 3Ti)/mi)1/2
that takes into account a one-dimensional adiabatic expansion of ions and isothermalelectrons amounts to ≈ 1.3 × 105 m/s for our plasmaparameters giving vph ≈ 1.5cs.
Electrostatic shocks, which are mediated by the ionacoustic wave, are routinely observed in laser-plasma ex-periments and we may assume that a magnetic field willnot affect the expansion speed of the laser-generated blastshell on time scales of the order of an inverse LH fre-quency. LH shocks should form in the presence of a per-pendicular magnetic field.
III. PIC SIMULATIONS OF LH SHOCKS INONE AND IN TWO DIMENSIONS
We perform PIC simulations with the aim to deter-mine if LH shocks can form and how we can distinguishthem from their unmagnetized counterparts based on thesampled field data. We compare the time-evolution ofthe particle and field distributions in an unmagnetizedplasma and in a magnetized plasma. The blast shell iscreated by letting a plasma with a high thermal pressureexpand into one with a low thermal pressure. The plas-mas in the high-pressure region and in the low-pressureregion are both spatially uniform and at rest in their
4
respective domains at the simulation’s start. Our sim-ulation setup thus differs from that in the related Ref.[3], which created the unmagnetized shock by letting aplasma beam collide with a reflecting wall and the mag-netized shock with a magnetic pressure gradient.
The low-pressure plasma consists of N7+ ions and elec-trons with the same density and temperature that wereconsidered in the previous section. We will refer to itas the ambient plasma. The high-pressure plasma has adensity, which exceeds that of the ambient plasma by thefactor 10. The electron temperature of the high-pressureplasma is 3 times higher than that of the ambient plasma,while the ion temperature is the same in both plasmas.The ambipolar electric field, which develops at the jumpof the thermal pressure between the high-pressure plasmaand the ambient plasma, forms a double layer [33, 34]that lets the plasma expand in the form of a rarefactionwave [35, 36] until a shock forms.
Our simulations will show that the actual shock formsin the ambient plasma well ahead of the expanding high-pressure plasma and far away from the location wherethe rarefaction wave was launched. The formation pro-cess should thus be independent of the idealized initialconditions.
The simulation box spans the interval -10 mm < x < 10mm along the x-direction and this interval is subdividedinto 4000 simulation cells. The high-pressure plasma islocated in the interval -4 mm < x < 4 mm and it is sur-rounded by the ambient plasma. We use periodic bound-ary conditions that connect the ambient plasma at -10mm with that at 10 mm and consider only the shock thatforms in the half-space x > 0. The simulation is stoppedbefore the shock-accelerated ions reach the boundaries.
We introduce an initial magnetic field B = B0z withB0 = 0.85 T in one simulation, while the plasma in thesecond one is unmagnetized. Both simulations use 6.4×107 computational particles (CPs) to represent the high-pressure electrons and ions, respectively. The ambientelectrons are represented by a total of 1.6× 107 CPs andthe same holds also for the ambient ions. We normalizetime to the inverse of the ion plasma frequency ωpi =1.55× 1010 rad s−1 of the ambient medium and one timeunit thus corresponds to 6.5 ps. We compare the resultsof both one-dimensional simulations at the times t1 =1.8, t2 = 7.4, t3 = 17.5 and t4 = 32.7.
Time t1 = 1.8 : The ion density distributions in bothsimulations and the electric fields are compared in Fig.2. The ion density distributions in both simulations arepractically identical; the magnetic field has not affectedthe ion expansion at this time. The matching distribu-tions of the electric field Ex(x, t) support this conclusion.An overlap layer, which is formed by the ions of bothplasmas, is observed close to x = 4.07 mm. This overlaplayer is the first stage of the shock formation [17].
Time t2 = 7.4 : The ion density hump has spread outin space, forming a plateau between 4.1 mm and 4.25mm in Fig. 3. A rarefaction wave with a density thatdecreases with increasing x is still present in the interval
4 4.05 4.1 4.15 4.2 4.250
5
(a) x (mm)
n i
4 4.05 4.1 4.15 4.2 4.250
5
(b) x (mm)
n i
4 4.05 4.1 4.15 4.2 4.25−5
0
5
10
15
(c) x (mm)
Ex
FIG. 2: Ion density normalized to ni0 and electric field nor-malized to 107 V/m at t1 = 1.8: Panel (a) shows the densitydistributions of the magnetized and unmagnetized plasmas.The contributions of the high-pressure- and of the ambientplasma are displayed separately and the latter is located tothe right. The cumulative ion distributions are shown in (b).Panel (c) shows the electric field distributions. Solid curvescorrespond to the magnetized plasma and dashed curves tothe unmagnetized one.
x < 4.1 mm (not shown). The plateau is bound to theright by a shock, which is followed by the oscillations thatare known to trail collisionless shocks [37]. The densitypeak is located at x ≈ 4.35 mm in the unmagnetizedplasma and at x ≈ 4.33 mm in the magnetized plasma.The electric field distribution confirms that the blast shellhas expanded farther in the unmagnetized plasma thanin the magnetized one.
Figure 4(a) compares the phase space density distribu-tion fi(x, vx) of the ions in the magnetized plasma withthat of the ions in the unmagnetized plasma at the timet = t2. The phase space density distributions are iden-tical besides the lag of 20 µm of the shock in the mag-netized plasma relative to that in the unmagnetized one.The simulation time t2 corresponds to two percent of anion gyroradius in the field B0 since ωci/ωpi = 2.6×10−3.The ion’s gyromotion is negligible at this time and theslowdown of the magnetized shock is not caused by theion’s rotation in the magnetic field.
In Fig. 4 we find the ambient ions at x > 4.4 mm andvx ≈ 0. A small fraction of the ions is reflected by theshock and they feed the shock-reflected ion beam at x >4.4 mm and vx ≥ 106 m/s. The remaining ions make itinto the downstream region and form the hot populationin the interval 4.25 mm < x < 4.35 mm and 4×105m/s <vx < 6 × 105m/s. The high-pressure plasma’s ions formthe cool dense ion beam at 5×105m/s in the interval x <4.25 mm. These ions do not mix with the hot downstreampopulation and their density is negligibly small already atx = 4.3 mm (See Fig. 3(a)). The high-pressure ions thusserve as a piston but they do not mix with the ions close
5
4.1 4.2 4.3 4.4 4.5 4.60
2
4
(a) x (mm)
n i
4.1 4.2 4.3 4.4 4.5 4.60
2
4
(b) x (mm)
n i
4.1 4.2 4.3 4.4 4.5 4.6
0
5
10
(c) x (mm)
Ex
FIG. 3: Ion density normalized to ni0 and electric field nor-malized to 107 V/m at t2 = 7.4: Panel (a) shows the densitydistributions of the magnetized and unmagnetized plasmas.The contributions of the high-pressure- and of the ambientplasma are displayed separately and the latter is located tothe right. The cumulative ion distributions are shown in (b).Panel (c) shows the electric field distributions. Solid curvescorrespond to the magnetized plasma and dashed curves tothe unmagnetized one.
to the shock. The LH waves close to the shock shouldthus obey the wave dispersion relation of the ambientplasma shown in Fig. 1. The speed, with which the high-density plasma is pushing the ambient plasma, is morethan twice the phase speed of the LH wave estimated inSection 2 for a wavelength of 0.1 mm.Figure 3(c) evidences a strong electrostatic field Ex in
both simulations. The magnetic field with the strengthBz = 0.85 T in the magnetized simulation will yield aE × B-drift of the electrons relative to the ions. Fig-ure 5 reveals the magnitude of the electron drift. Theelectron phase space density distribution fe(x, vy) showsa clear modulation close to the location x = 4.34 mmof the magnetized shock. The mean speed ⟨vy⟩(x) =∫vyfe(x, vy)dvy reveals that the drift speed is a sizeable
fraction of vte ≈ 1.9×107 m/s over a wide spatial interval.The peak drift speed is ≈ vte/2 or about 107 m/s.The speed gain of the electrons at the shock exceeds
that of the ions at the shock by the factor 20. The en-ergy gain of the electrons is thus not negligible comparedto that of the ions. The thermal pressure gradient ofthe dense plasma, which drives the shock, is the samein both simulations at this time and we may attributethe slower speed of the magnetized shock to this electronacceleration along the y-direction. The drastic change ofthe electron distribution at x ≈ 5 mm marks the front ofthe shock-reflected ion beam, where the ion density jumpresults in a jump of the electrostatic potential.Figure 6 reveals two important differences between
both simulations at the time t3 = 17.5. Firstly, the iondensity in the magnetized simulation and in the region
FIG. 4: Panel (a) shows the phase space density distributionfi(x, vx) of the unmagnetized ions and panel (b) that of themagnetized ions at the time t2 = 7.4. The color scale is 10-logarithmic, the densities are normalized to their peak valueand velocities are normalized to 105 m/s. The vertical dashedline shows the position x = 4.36 mm of the unmagnetizedshock and the solid vertical line the position x = 4.34 mm ofthe magnetized shock.
FIG. 5: The electron drift at t2 = 7.4. Panel (a) shows the 10-logarithmic electron phase space density distribution fe(x, vy)and panel (b) the electron drift speed ⟨vy⟩(x) expressed inunits of 107 m/s. The vertical line x = 4.34 mm is overplottedin both panels.
4.6 mm < x < 4.9 mm is well below that in the simulationwith B0 = 0. The density oscillates around 3ni0 in theunmagnetized simulation and around 2ni0 in the mag-netized one. The density oscillations in the magnetizedsimulation have a larger wavelength and amplitude thanthose behind the unmagnetized shock. The oscillations
6
of the electrostatic potential will thus be much larger.The wavenumber, which corresponds to the wavelength
4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.20
1
2
3
4
5
(a) x (mm)
n i
4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2
−2
0
2
4
6
(b) x (mm)
Ex
FIG. 6: Ion density normalized to ni0 and electric field nor-malized to 107 V/m at t3 = 17.5: Panel (a) shows the cu-mulative distributions of the magnetized and unmagnetizedplasmas. Panel (b) shows the electric field distributions.Solid curves correspond to the magnetized plasma and dashedcurves to the unmagnetized one.
of the oscillations ≈ 0.1 mm in the magnetized plasma,is indicated in Fig. 1 and we find only LH waves at lowfrequencies in this interval.The electron Bernstein modes and the upper-hybrid
mode are fast electronic modes and such waves can notinteract resonantly with ions that move at speeds thatare much lower than vte [38] and therefore these high-frequency waves are not destabilized by the ion beam.Figure 7 demonstrates that the trailing waves in both
simulations are strong enough to visibly modulate theion distribution. The large amplitude of the electrostaticoscillations affects in particular the downstream ion pop-ulation of the magnetized structure. The amplitude ofthe velocity modulation at x ≈ 4.8 mm is larger than thethermal spread of the ions and it is close to that neededfor the formation of ion phase space vortices [39].The magnetized structure and the electrostatic shock
in Figs. 6 and 7 show several important differences. Acollisionless shock is characterized by a ramp with strongelectric fields. The ramp of the unmagnetized shock islocated in the interval 4.87 mm < x < 5 mm. The electricfields in the magnetized simulation are largest for 4.83mm < x < 4.87 mm and naturally we would associatethis interval with the ramp.The electric field in the magnetized simulation reaches
out beyond x = 5 mm. If the magnetized structure werea shock, the upstream region with the weak electric fieldwould be its foot. A foot is a feature of collisionlessshocks. It is created by the ions, which have been re-flected by the shock. A foot usually stretches out byabout an ion gyroradius if the shock is perpendicular.
FIG. 7: Panel (a) shows the phase space density distributionfi(x, vx) of the unmagnetized ions and panel (b) that of themagnetized ions at the time t3 = 17.5. The color scale is 10-logarithmic, the densities are normalized to their peak valueand velocities are normalized to 105 m/s.
The gyroradius of the reflected ions with a speed vx ≈ 106
m/s and B0 = 0.85 T will be 25 mm and the foot is thusstill developing at this time.
Figure 7(b) shows that the ambient ions are alreadyaccelerated in the foot well before the ramp arrives. Theambient ions at the front of the ramp at x = 4.9 mmhave reached a speed ≈ 3×105 m/s and the velocity gapbetween the ions of the downstream plasma and thoseof the upstream plasma has been reduced substantially.The plasma compression by the structure at x ≈ 4.83mm in Fig. 6(a) is weaker than that observed for its un-magnetized counterpart, suggesting that the magnetizedstructure depicted in Fig. 7(b) at the same position iseither a weak shock or a propagating nonlinear LH wave.In what follows we refer to it as nonlinear LH wave.
We want to determine the process that pre-acceleratedthe ambient ions. Any ion acceleration in a collisionlessplasma must be tied to an electric field and this accel-eration mechanism does apparently not work in the un-magnetized plasma.
An upstream electric field can be driven by the cur-rent of the shock-reflected ions. A perpendicular mag-netic field limits the electron’s mobility in the ambientplasma and thus their ability to react to this field. Hencethe consequences of the electric field will be different inboth simulations. It is, however, not this electric fieldthat pre-accelerates the ambient ions because such a fieldshould reduce the net ionic current ahead of the shock.The acceleration of ambient ions towards positive x will,however, enhance it and we can rule out this process.
The E×B drift of the electrons in Fig. 5 will modulatethe magnetic field. Figure 8(a) compares the spatial dis-
7
tributions of Bz at the times t2 = 7.4 when the LH shockwas strong and t3 = 17.5 when it had transformed into aweaker shock or a nonlinear wave. The expansion of the
3 3.5 4 4.5 5 5.5 6
0.6
0.8
1
1.2
(a) x (mm)
Bz
3 3.5 4 4.5 5 5.5 6
−10
−5
0
5x 107
(b) x (mm)
For
ce d
ensi
ty
FIG. 8: The spatial distribution of magnetic Bz(x) obtainedfrom the simulation with the magnetized plasma is shown inpanel (a) at the times t2 =7.4 (dashed curve) and t=17.5(solid curve). Panel (b) shows a moving average over 10 gridpoints of the associated force density FB = −(2µ0)
−1dB2z/dx.
The vertical line shows the location x=4.82 mm of the mag-netized structure at t3 = 17.5
high-pressure plasma redistributes the uniform magneticfield Bz = 0.85 T into a sine-like pulse that expands inspace and time. The spatial extent of the pulse has grownby a factor 3 in space and the gradient has increased byabout the same factor during the time interval t3 − t2.The strong gradient gives rise to a gradient of the mag-netic pressure FB = −(2µ0)
−1dB2
z/dx. The force densitychanges sign at the nonlinear LH wave and it is of theorder ∼ 107N/m3 ahead of it in an x-interval with thesize ∼ 1 mm. The force density accelerates ions with amass density mini0 ∼ 10−6kg/m3, giving an accelerationaB ∼ 1013 m/s2. The acceleration time ta ∼ 10−8 s ofthe ambient ions is given by the spatial extent of the ac-celeration zone ∼ 1 mm divided by the speed of the mag-netized structure ∼ 105 m/s. The product aBta ∼ 105
m/s is comparable to the speed gain of the ambient ionsin Fig. 7(b) that led to the weakening of the shock.The ion beam displayed in Fig. 7(b) at x > 4.9 mm
and vx > 7 × 105 m/s is a shock signature. The densitycompression in Fig. 6(a), which is weak compared tothat we observe at its unmagnetized counterpart, rulesout that the structure in the simulation with B0 = 0 isa shock that balances the ram pressure of the inflowingupstream medium solely with the downstream’s thermalpressure.Figure 9 compares the phase space density distribu-
tions of the ions at the time t4 = 32.7. The unmagne-tized shock remains qualitatively unchanged comparedto its counterpart at the time t3. The ions from the
FIG. 9: Panel (a) shows the phase space density distributionfi(x, vx) of the unmagnetized ions and panel (b) that of themagnetized ions at the time t4 = 32.7. The color scale is 10-logarithmic, the densities are normalized to their peak valueand velocities are normalized to 105 m/s. The vertical lineshows the position x = 5.57 mm of the unmagnetized shock.
high-pressure plasma are visible at x < 5.1 mm andvx ≈ 5.5 × 105 m/s. They still act as a piston thatpushes the ambient ions to increasing values of x. Theshock is located at x = 5.57 mm and it separates thehot downstream ions from the cool ambient ions and theshock-reflected ion beam.
The ion phase space density distribution in the mag-netized simulation in Fig. 9(b) resembles qualitativelythat of the unmagnetized shock. We observe the pre-acceleration of ambient ions and a beam of ions that isreflected by the nonlinear LH wave. The nonlinear LHwave has overtaken the unmagnetized shock and it is lo-cated about 0.1 mm ahead of it. The reflected ion beam isstill accelerated at x=6.5 mm; the magnetic pressure gra-dient force accelerates all ions far upstream of the nonlin-ear LH wave. We observe strong oscillations downstreamof this structure, which are ion charge density oscillationsand they are thus tied to the electrostatic LH waves.
Figure 9(b) raises an important question. How can themagnetic pressure gradient force, which is a MHD force,sustain a nonlinear LH wave structure on a scale that issmall compared to rci and develop on a time-scale thatis small compared to ω−1
ci : The ions have completed justabout 1.4 % of a gyro-orbit at the time t4.
The magnetic pressure gradient force also acceleratesthe electrons. The force density operates on electronswith the number density 7ni0 = 2.75× 1020m−3 and thedivision of FB by 7ni0 gives us the force per electron. Theelectrons are accelerated along x and their current drivesan electric field that counteracts the effects of FB . Theelectron acceleration stops once a force balance in the x-
8
direction is established and FB/7eni0 = Ebx, where Ebx
is the saturation field along x.Figure 10 shows Bz(x) at the time t4 = 32.7 and it
compares Ebx to the electric field, which is measured inthe simulation. We observe that Ebx matches the electric
2 3 4 5 6 7 80.4
0.6
0.8
1
1.2
(a) x (mm)
Bz
4.5 5 5.5 6 6.5 7−2
−1
0
1
2x 107
(b) x (mm)
Ex a
nd E
bx
FIG. 10: The spatial distribution of magnetic Bz(x) obtainedfrom the simulation with the magnetized plasma is shownin panel (a) at the time t4 = 32.7. Panel (b) compares theelectric field Ebx (blue curve with low-amplitude oscillations)that would balance the magnetic pressure gradient force tothe electric field computed by the magnetized simulation.
field Ex(x) in the foot region of the nonlinear LH wavestructure in the interval x > 5.8 mm.The electric field driven by the magnetic pressure gra-
dient force replaces the ambipolar electric field, which istied to the thermal diffusion of electrons, with respect tosustaining the shock. For lower values of x, Ex(x) = Ebx.The mean value of Ex(x) is in fact close to zero for 4.5mm < x < 5.5 mm. The absence of an electric fieldimplies that the magnetic pressure is balanced by thethermal pressure in this region, yielding a vanishing netforce on the particles. Strong LH waves form in regionswith a strong gradient of Ebx and, thus, in an intervalwith a strong magnetic pressure gradient.One goal of our simulations is to determine how the
field signature of an LH shock or a nonlinear LH wave dif-fers from those of the well-researched electrostatic shocksin unmagnetized plasma. We have seen that an elec-trostatic shock approximately maintains its speed in theabsence of a magnetic field and in one spatial dimen-sion. Its magnetized counterpart is initially slower, butit overtakes the unmagnetized shock at a later simulationtime. The structures in both simulations will also differin their electric field distribution. A diagnostic techniquethat measures electromagnetic fields in laboratory plas-mas should be able to distinguish both shocks.A comprehensive overview of the electric field distribu-
tion is given by Fig. 11. The electric field distribution ofthe shock in the unmagnetized plasma reveals a localized
FIG. 11: The time-evolution of the electric field Ex(x, t) inboth simulations: Panel (a) and (b) show the field distributionin the unmagnetized plasma and in the magnetized plasma,respectively.
unipolar electric field pulse at all times. The pulse speedgradually decreases in time. The ion acoustic waves thattrail the shock co-move with the shock and they keeptheir amplitude and wavelength constant.
The unipolar pulse in the magnetized simulation hasinitially the same amplitude and width as its unmagne-tized counterpart. The pulse amplitude, which character-izes the LH shock, decreases with time and it has almostvanished at t4 = 32.7. The nonlinear steepening of theLH waves, which gave rise to the LH shock, is thus nolonger sustained by the expanding plasma, substantiat-ing that the LH shock is weakening or vanishing.
We can not exclude that the LH shock will eventuallyreform. However, the time-scale of a cyclic reformationof fast magnetized shocks is of the order of the inverseion gyrofrequency [24], exceeding by far our simulationtime. Even if the reformation period is shorter for LHshocks than for their faster counterparts the instabilitybetween the two ion beams in the upstream would haveto be taken into account. We also note that the pilingup of the magnetic field ahead of the LH shock (See Fig.10) may eventually result in a dynamic confinement ofthe expanding blast shell by the magnetic pressure inthe ambient plasma. We leave a study of that long-termevolution to future work.
The waves that trail the shock increase their wave-length in time and they keep the electric amplitude un-changed. The electrostatic potential, which is associatedwith these oscillations, thus increases in time. The differ-ence between the distribution of the electrostatic poten-tial downstream of an unmagnetized and of a magnetizedshock should be detectable. The same may hold for thedifferent velocities of both shocks.
We must verify that the results obtained from the 1D
9
PIC simulations are valid also in a more realistic two-dimensional geometry. Electrostatic shocks in more thanone dimension are eventually destroyed by instabilitiesbetween the ambient ions and the shock-reflected ones.Their evolution is well-documented [4, 6, 31] and we willnot consider further the unmagnetized case.
Drift instabilities can develop in the magnetizedplasma due to the substantial E×B-drift of the electronswith respect to the ions. The LH and electron cyclotrondrift instabilities will drive waves with wavevectors thatare aligned with the electron drift speed [31, 40, 41].These drift instabilities are thus suppressed by the one-dimensional simulation geometry.
We thus perform a two-dimensional PIC simulationwith plasma parameters and with a box size along thex-direction that are identical to their counterparts in themagnetized simulation, which we have discussed above.We resolve the y-direction by 400 grid cells. The lengthof the box along y is 2 mm and the grid cell size along yis the same as the one along x. The number of CPs percell is reduced by the factor 50 compared to that in theone-dimensional simulation.
The electric field distributions Ex(x, y, t2) andEx(x, y, t3) are displayed in Fig. 12. They are similar to
FIG. 12: The electric field Ex(x, t) computed by the 2D PICsimulation: Panel (a) and (b) show the field distribution forthe time t2 = 7.4 and t3 = 17.5, respectively.
those in the Figs. 3(c) and 6(b). The electric field showsplanar structures that are aligned with the y-axis andthe dynamics of the plasma is thus one-dimensional. Ifa drift instability develops, then the resulting wave fieldsare too weak to be detectable and to affect the plasmadynamics. We do not show the electric field along they-direction, because it consists solely of noise.
IV. DISCUSSION
We have compared the expansion of a high-pressureplasma into a dilute ambient plasma with and withouta perpendicular background magnetic field. The plasmaparameters were comparable to those we find in laser-generated plasma. We found strong electric field pulses inboth simulations, which did correspond to the ambipolarelectric field across a sharp plasma density change. Thesepulses expanded into the ambient plasma and acceleratedit. The density of the ambient plasma was compressedby a factor 3 when it crossed the pulse and a fraction ofthe ambient plasma was reflected back upstream.
The pulse propagated in the simulation with no mag-netic field into the ambient medium at the practicallyconstant speed ≈ 2.5cs, where cs is the ion acoustic speedin the latter. The electric field pulse and the densitychange were thus an electrostatic shock.
The introduction of the perpendicular magnetic fieldin the second simulation suppressed the ion acousticwave. The lower-hybrid (LH) wave branch emerged andits phase speed at large wavenumbers was comparableto the value of cs in the unmagnetized plasma for ourplasma parameters. The electric field pulse and the den-sity jump in the simulation with the magnetized plasmamoved at a speed above the phase speed of LH wavesat large wavenumbers and the pulse corresponded to aLH wave shock. The structure of the LH shock resem-bled that of the electrostatic shock in the unmagnetizedplasma apart from is slightly lower expansion speed. Wehave attributed the lower speed to the additional resis-tance imposed on the shock by the E × B-drift of theelectrons.
The gradient of the magnetic pressure gave rise toa pre-acceleration of the ambient ions and the relativespeed between the LH shock and these upstream ionsdecreased to a value below the phase speed. The LHshock changed into what appeared to be a nonlinear LHwave, which balanced the ram pressure of the inflowingambient medium with the magnetic pressure and only toa lesser degree with the thermal pressure of the down-stream medium.
The increasing magnetic pressure in the ambientplasma suggests that eventually we may obtain a magne-tosonic shock that separates the fast-moving and weaklymagnetized (upstream) blast shell plasma from a slow-moving and strongly magnetized ambient (downstream)plasma. The electrostatic thermal-pressure gradientdriven LH shock that separated the (downstream) blastshell plasma from the (upstream) ambient plasma wouldthus be only be a transient structure and its main effectwould be to mediate the development of a magnetohy-drodynamic shock.
A two-dimensional PIC simulation demonstrated thatat least during the initial expansion phase the plasmadynamics remained one-dimensional close to LH shock.More specifically, the electron E × B-drift current wasnot strong enough to drive LH wave turbulence close to
10
the shock.The LH wave has been invoked as a means to accel-
erate ions in the foreshock regions of magnetized shocks[23]. Boundary layers in a magnetized plasma that sepa-rate different ion populations have previously been foundin hybrid simulations [18, 42, 43], which examined thedemagnetization of an ambient plasma by an expandingplasma plume. Given the right plasma conditions, suchboundary layers could steepen into an LH shock.The LH shock is trailed by LH waves with a larger
electric field amplitude and wavelength than its unmag-
netized counterpart. It should be possible to distinguishin laboratory experiments like the one performed in Ref.[18] LH shocks from unmagnetized shocks based on thepotential distribution that is trailing the shock.
Acknowledgements: G. Sarri wishes to acknowledgeEPSRC (Grant number: EP/N022696/1. The simulationwas performed on resources provided by the Swedish Na-tional Infrastructure for Computing (SNIC) at HPC2N(Umea). We thank the referee for the simple yet accuratelinear dispersion relation for the LH wave.
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0 5 10 15 20 250
2
4
6
8
10
12
k vte
/ ωce
ω / ω
lh
−3
−2.5
−2
−1.5
−1
−0.5
0
4 4.05 4.1 4.15 4.2 4.250
5
(a) x (mm)
ni
4 4.05 4.1 4.15 4.2 4.250
5
(b) x (mm)
ni
4 4.05 4.1 4.15 4.2 4.25−5
0
5
10
15
(c) x (mm)
Ex
4.1 4.2 4.3 4.4 4.5 4.60
2
4
(a) x (mm)
ni
4.1 4.2 4.3 4.4 4.5 4.60
2
4
(b) x (mm)
ni
4.1 4.2 4.3 4.4 4.5 4.6
0
5
10
(c) x (mm)
Ex
4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.20
1
2
3
4
5
(a) x (mm)
ni
4.3 4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2
−2
0
2
4
6
(b) x (mm)
Ex
3 3.5 4 4.5 5 5.5 6
0.6
0.8
1
1.2
(a) x (mm)
Bz
3 3.5 4 4.5 5 5.5 6
−10
−5
0
5x 10
7
(b) x (mm)
Forc
e d
ensity
2 3 4 5 6 7 80.4
0.6
0.8
1
1.2
(a) x (mm)
Bz
4.5 5 5.5 6 6.5 7−2
−1
0
1
2x 10
7
(b) x (mm)
Ex a
nd E
bx
