The interplay of the collisionless nonlinear thin-shell instability with the ion acoustic instability
aa r X i v : . [ phy s i c s . p l a s m - ph ] J a n MNRAS , 1–10 (0000) Preprint 18 September 2018 Compiled using MNRAS L A TEX style file v3.0
The interplay of the collisionless nonlinear thin-shellinstability with the ion acoustic instability
M. E. Dieckmann ⋆ D. Folini R. Walder Department of Science and Technology (ITN), Link¨oping University, 60174 Norrk¨oping, Sweden. Universit´e de Lyon 1, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69007, Lyon, France
ABSTRACT
The nonlinear thin-shell instability (NTSI) may explain some of the turbulent hy-drodynamic structures that are observed close to the collision boundary of energeticastrophysical outflows. It develops in nonplanar shells that are bounded on either sideby a hydrodynamic shock, provided that the amplitude of the seed oscillations is suf-ficiently large. The hydrodynamic NTSI has a microscopic counterpart in collisionlessplasma. A sinusoidal displacement of a thin shell, which is formed by the collision oftwo clouds of unmagnetized electrons and protons, grows and saturates on timescalesof the order of the inverse proton plasma frequency. Here we increase the wavelengthof the seed perturbation by a factor 4 compared to that in a previous study. Likein the case of the hydrodynamic NTSI, the increase in the wavelength reduces thegrowth rate of the microscopic NTSI. The prolonged growth time of the microscopicNTSI allows the waves, which are driven by the competing ion acoustic instability, togrow to a large amplitude before the NTSI saturates and they disrupt the latter. Theion acoustic instability thus imposes a limit on the largest wavelength that can bedestabilized by the NTSI in collisionless plasma. The limit can be overcome by binarycollisions. We bring forward evidence for an overstability of the collisionless NTSI.
Key words: plasmas – instabilities – shock waves – methods: numerical
The boundary between an energetic large-scale astrophys-ical outflow and an ambient medium like the interstellarmedium (ISM) is prone to a plethora of hydrodynamic in-stabilities, most notably the Rayleigh-Taylor instability, theKelvin-Helmholtz instability and thin-shell instabilities.The Rayleigh-Taylor instability can disrupt the bound-ary between the ISM and the blast shell of a type Ia su-pernova (Gamezo et al. 2003) or of a type II supernova(Chevalier et al. 1992). It can also develop at the bound-ary between a pulsar wind and a supernova blast shell(Blondin et al. 2001; Porth et al. 2014).The Kelvin-Helmholtz instability limits the growth ofthe fingers that develop during the nonlinear stage of theRayleigh-Taylor instability (Chevalier et al. 1992) and itmight be important for radiation- and cosmic ray generationin the shear boundary layers of jets (Stawarz & Ostrowski2002). A recent numerical study of this instability is per-formed by Palotti et al. (2008).Linear thin-shell instabilities can form at the col-lision boundary between the blast shell of a super- ⋆ E-mail: [email protected] nova and the ISM (Vishniac 1983; van Marle et al. 2011;Sanz et al. 2011; van Marle & Keppens 2012; Michaut et al.2012; Edens et al. 2010). A dense shell forms at the front ofthe blast shell, where it sweeps up the ISM. Initially only theouter boundary between the thin shell and the ISM is a hy-drodynamic shock. The inner boundary between the denseshell and the blast shell material changes into a shock at alater time. The linear thin-shell instability can develop priorto the formation of the reverse shock.A shell that is bounded by two shocks is linearly stable.Vishniac (1994) showed however that such a shell is unsta-ble against a sufficiently strong sinusoidal perturbation of itsshape and hence it is called the nonlinear thin-shell insta-bility (NTSI). This instability results in turbulent flow in-side the shell (Folini & Walder 2006; Folini et al. 2014) andmay play an important role in the thermalization of collidingwinds (Walder & Folini 2000).The large time scales over which hydrodynamic astro-physical instabilities develop imply that we can observe onlysnapshots of their evolution. Some hydrodynamic instabili-ties can be studied in denser material. A high density of thematerial compresses the time scale over which the instabil-ity evolves and we can observe it from its onset through itsnonlinear evolution to its final stage. If we understand the c (cid:13) M. E. Dieckmann et al. evolution of an instability and know how its density and mo-mentum are distributed at each evolution stage, then we canrelate the observed astrophysical gas and plasma structuresto the instabilities that created them. Laboratory experi-ments thus provide essential support for the interpretationof astrophysical observations.Laboratory experiments have addressed the Kelvin-Helmholtz instability (Amatucci 1999) and the Rayleigh-Taylor instability. Sharp (1984); Piriz et al. (2006) providea description of the Rayleigh-Taylor instability and refer-ences to experiments. Edens et al. (2010) have observed thelinear thin-shell instability at the boundary between a laser-generated blast shell and an ionized ambient medium.The hydrodynamic (Vishniac 1994; Blondin & Marks1996; Lamberts et al. 2011) and magnetohydrodynamic(Heitsch et al. 2007; McLeod & Whitworth 2013) NTSIshave been examined by analytic means and through simula-tion experiments but, to the best of our knowledge, neitherof them has been studied in the laboratory. Its observationin a controlled laboratory experiment would strengthen thecase for its existence in astrophysical flows and laboratorystudies of its time evolution would shed further light on thetopology of the flow patterns it drives.The basic mechanism of the NTSI can be described asfollows. The flow velocity vector of a fluid, which crosses ahydrodynamic shock at an oblique angle, is rotated awayfrom the shock normal because only the velocity componentalong this normal is decreased by the shock crossing. A fluidflow across a corrugated shock will result in a rotation an-gle of the velocity vector that is a function of the positionalong the shock boundary and the inflowing material andthe momentum it carries will thus be spatially redistributedin the downstream region. This redistribution amplifies thethin shell’s initial corrugation.The particle-in-cell (PIC) simulation study byDieckmann et al. (2015c) showed that an analogue tothe hydrodynamic NTSI exists in a collisionless plasma.The velocity vector of the ions that flow into the shell isrotated by the ambipolar electric field, which is antiparallelto the density gradient at the shell’s boundaries.Here we examine in more detail the evolution and thesaturation of the NTSI in collisionless plasma by means ofa particle-in-cell (PIC) simulation. The purpose is to deter-mine if it can develop on a larger scale and for stronger elec-trostatic shocks than in the simulation by Dieckmann et al.(2015c). A broad range of unstable wavelengths and shockstrengths would imply that this instability can grow for awide range of initial conditions, which is a prerequisite forit to be astrophysically relevant and detectable in labora-tory plasma. A coupling of the shell’s perturbations fromthe small collisionless scale to larger collisional scales wouldalso imply that the rapidly growing collisionless NTSI couldprovide the strong seed perturbations that let its large-scalecollisional counterpart grow.Our paper is structured as follows. Section 2 summarizesthe PIC simulation method and the initial conditions thatwe have used for the simulation. Section 2 also describes thedouble layers and electrostatic shocks (Hershkowitz 1981)that enclose the thin shell in the collisionless plasma and itsummarizes related experimental studies. Section 3 presentsour simulation results and we discuss them in Section 4.
Particle-in-cell (PIC) simulation codes are based on the ki-netic theory of plasma. The ensemble of the plasma particlesthat belong to the species i is represented by a phase spacedensity distribution f i ( x , v , t ), where x and v are the posi-tion and velocity coordinates and t is the time. We do nottake into account binary collisions in our simulation. Theplasma evolution is determined exclusively via the collectiveelectromagnetic fields and x and v are thus independentcoordinates. The phase space density distribution describescharged particles and its time-evolution is determined byexternal or self-generated electromagnetic fields, which wecompute by Amp`ere’s law and by Faraday’s law µ ǫ ∂ E ∂t = ∇ × B − µ J , (1) ∂ B ∂t = −∇ × E . (2)The electromagnetic PIC code EPOCH (Arber et al. 2015)we use solves Eqns. 1 and 2 on a numerical grid. The timestep is ∆ t . It fulfills ∇· E = ρ/ǫ and ∇· B = 0 as constraints.Maxwell’s equations require the knowledge of the cur-rent density J and of the charge density ρ of the plasma.The phase space density distribution of each plasma speciesis evolved separately. We obtain the charge contribution ofspecies i from the zero’th moment of its phase space den-sity distribution ρ i = q i R f i ( x , v , t ) d v and its current con-tribution from the first moment J i = q i R v f i ( x , v , t ) d v .The total charge- and current densities are ρ = P i ρ i and J = P i J i .The phase space density distribution f i ( x , v , t ) ofspecies i is approximated by an ensemble of computationalparticles (CPs). The j ’th CP of species i is characterized bythe position x j and by the momentum p j . The electromag-netic fields are interpolated from the grid to the position ofeach CP and a suitably discretized form of Eqn. 3 updatesits momentum. d p j dt = q j ( E + v j × B ) . (3)A discretized form of d x j /dt = v j updates the particle’sposition. After these updates the current density of each CPis interpolated to the grid, summed up and used to updatethe electromagnetic fields via Eqns. 1 and 2. This cycle isrepeated for every time step. The plasma, which is composed of protons and electronswith the correct mass ratio m p /m e = 1836, has initiallya constant temperature and density n everywhere. Theplasma frequency of the electrons is ω pe = ( n e /m e ǫ ) / ,where e is the elementary charge. The plasma frequency ofthe protons is ω pi = ω pe / √ T e = 1 keV and T p = T e / c s =( k B ( γ e T e + γ p T p ) /m p ) . , where m p is the proton mass and k B the Boltzmann constant. Its value is c s = 5 × m/s ifwe assume that γ e = 2 and γ p = 3, which implies 2 degrees MNRAS , 1–10 (0000) he collisionless NTSI of freedom for the electrons and one degree of freedom forthe protons.The electrons with their low inertia are easily scat-tered by the thermal fluctuations in the PIC simulation(Dieckmann et al. 2004). The fluctuating electrostatic fieldsare predominantly polarized in the simulation plane. Thescattering of electrons by the electrostatic field fluctuationscouples the two velocity components in the simulation plane,which thus has a similar effect as binary collisions (Bret2015), yielding two degrees of freedom for the electrons.We express space in units of the electron inertial length λ s = c/ω pe where c is the speed of light. We resolve thespatial interval − . ≤ x ≤ . ≤ y ≤ .
54 by 250 grid cells. The boundaryconditions along y are periodic, the boundary condition at x = − . x = 16 . x B ( y ) = A sin ( k y y )with the wave number k y = 2 π/λ p . The wave length λ p =6 .
54 of the seed perturbation equals the box length alongy. The amplitude of the seed perturbation is A = 0 .
114 or A = 0 . λ p . The value of A has been selected such thatthe initial oscillation amplitude is significantly larger than agrid cell while being small compared with λ p .Each cloud has a mean speed that is spatially uniform.The plasma cloud 1 in the interval x ≤ x B ( y ) has the pos-itive mean speed v = 1 . × m/s equalling v = 3 . c s along x, while the plasma cloud 2 in the interval x > x B ( y )is initially at rest with v = 0. The plasma is initially free ofany net charge and current and we set all electromagneticfields to zero at the simulation’s start t = 0. The clouds start to interpenetrate for t >
0. A thin shell ofplasma like that depicted in Fig. 1(a) forms at the initial con-tact boundary x B ( y ) and expands towards increasing valuesof x at the speed v . The density of the plasma in the thinshell is 2 n as long as the protons of both clouds do not inter-act electromagnetically. We refer to the area covered by thethin shell as the downstream region. A boundary on eachside separates the downstream region from the respectiveplasma cloud. We refer with upstream region to the parts ofthe plasma cloud that have not yet crossed this boundary.Thermal diffusion will lead to a net flow of downstreamelectrons into the dilute upstream region. A negative chargelayer builds up outside of each boundary while the escapingelectrons leave behind a positively charged layer just insideof each boundary. A unipolar electrostatic field pulse growsat each boundary of the thin shell between the positive andnegative charge layers, which puts the downstream plasmaon a higher electric potential than the upstream one. Thisambipolar electric field grows and saturates on electron timescales. The field accelerates the protons and it adapts to theirchanging density distribution.Figure 1(b) shows the lower boundary of the thin shell,which remains initially close to x B ( y ) because it representsthe boundary of the plasma cloud that is at rest. The dashedvectors show the trajectories of three protons that enter thethin shell. Protons 1 and 3 are slowed down as they cross Figure 1.
Panel (a) illustrates the shape of the thin shell. Thelower boundary is determined by the front of the protons that areat rest and the upper one by the protons that move at the speed v along x . Both boundaries are displaced relative to an aver-age boundary (dashed line). Panel (b) shows the lower boundary,the (solid) electric field vectors and the (dashed) trajectories ofprotons that move to increasing x at the speed v . Panel (c)sketches the proton phase space distribution in the x, v x -plane.The dashed vertical lines denote the interval around the lowerboundary where we find a nonzero electric field. The abbrevia-tions ES and DL stand for electrostatic shock and double layer. the boundary but their direction is unchanged. Proton 2is slowed down and deflected by the boundary crossing thatdecreases only its velocity component along the electric field.Proton 2 is deflected towards an extremal point of x B ( y ).Figure 1(c) sketches out the proton phase space den-sity distribution in the phase space plane x, v x parallel tothe trajectories of the protons 1 or 3. The vertical dashedlines enclose the spatial interval, in which the electric field isnonzero. The protons of cloud 1, which moves to increasingvalues of x , are found to the left and their mean speed along x is v . These protons are slowed down by the electric field ofthe lower boundary when they enter the thin shell. An elec-trostatic structure that slows down inflowing upstream pro-tons is called electrostatic shock (Forslund & Shonk 1970b;Forslund & Freidberg 1971). The protons of the stationarycloud 2 are found to the right at a speed ≈
0. Their thermalspread implies that some of the protons enter the spatialinterval with the nonzero electric field. These protons areaccelerated towards the upstream direction and such a struc-ture is called a double layer. Raadu (1989) gives a review ofdouble layers in astrophysical plasma.Electrostatic shocks and double layers can coexist ina collisionless plasma in the form of a hybrid structure(Hershkowitz 1981). We will use this term to denote the
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M. E. Dieckmann et al. nonlinear electrostatic structure that encloses the thin shellunless we discuss its components.
The potential difference between the upstream and thedownstream plasma is set by the density jump, which is ofthe order of n , and the electron temperature that does usu-ally not vary much across a hybrid structure. If the kineticenergy of the inflowing upstream protons in the shell’s restframe is large compared with the potential energy changeat the shell boundary then these protons are hardly sloweddown. The colliding clouds will interpenetrate without form-ing a well-defined dense and localized thin shell. The maxi-mum Mach number of such a shell is thus limited. Our col-lision speed v = 3 . c s brings us into the regime where thevelocity gap between the counterstreaming proton clouds inthe shell is initially comparable to v . A gradually increasingcompression of the plasma in the thin shell and the associ-ated growth of the potential difference between the upstreamand downstream plasmas reduces in time the gap betweenthe beam velocities (Dieckmann et al. 2013a).Another property of the hybrid structure is that theinflowing upstream protons are not fully thermalized whenthey enter the downstream region (See Fig. 1(c)). A thermal-ization is eventually achieved by the electrostatic turbulence(Dum 1978; Bale et al. 2002; Dieckmann et al. 2013b) thatis driven by an instability between the counterstreaming pro-ton beams (Forslund & Shonk 1970a). In what follows wecall it the proton-proton beam instability.The plasma parameters, which we have selected, arecomparable to those in the experiment performed byAhmed et al. (2013). A thin plasma shell was created in thisstudy by the collision of a blast shell, which was ejected bya laser-ablated solid target, with an ambient medium. Thesource of the ambient medium was the residual gas, whichwas contained in the plasma chamber prior to the ablationof the target and got ionized by secondary X-ray emissionsfrom the ablated target. The ultraintense laser pulse and ob-servational time scale that was of the order of 100 picosec-onds implied that effects caused by binary collisions betweenplasma particles were negligible. It may thus be possible toreproduce the NTSI in a collisionless laboratory plasma.Binary particle collisions would establish a Maxwellianvelocity distribution of the protons in the downstream re-gion. Only few protons are fast enough in such a distribu-tion to catch up with the hybrid structure and be acceler-ated upstream by its double layer component. Those thatmake it will collide with the inflowing upstream particlesand they will be pushed back to the hybrid structure. Aswe increase the collisionality of the plasma the hybrid struc-ture will gradually change into a fluid shock. The degree ofcollisionality in a laboratory plasma experiment depends onthe intensity of the laser pulse and on the observational timescale. Hansen et al. (2006) observed a collisional shock.It is of interest to establish with PIC simulations therange of parameters for which the collisionless NTSI can de-velop and to test if it can develop in a collisionless laser-plasma experiment. Here we examine if the collisionlessNTSI can destabilize a wavelength that exceeds that inDieckmann et al. (2015c) by a factor of 4. Further exper- Table 1.
The multiplier for the normalized quantities for threevalues of the electron density n expressed in units cm − . n x t E B µ s 96.2 V/m 320 nT10
168 m 0.56 µ s 3 kV/m 10 µ T10 Table 2.
The parameters of the fitted sine curveTime t i : t t t t t Time value : 268 536 1100 1600 2100Amplitude A i : 1.0 1.6 2.3 2.6 2.8Offset x i : 0.8 1.6 3.15 4.73 6.23Speed v i : 0.51 0.51 0.49 0.51 0.51Growth speed: ∆ A i : 0.19 0.21 0.1 0.07 iments and PIC simulation studies can then examine howthe NTSI evolves in collisional plasma. We present and discuss the proton density distribution andthe distributions of the in-plane electric field and of the out-of-plane magnetic field at several times. In what follows wenormalize time as t = ˜ tω pe where ˜ t is expressed in SI units.We select the times t = 268, t = 536, t = 1 . × , t = 1 . × , t = 2 . × and t = 2 . × . The protondensity distribution n p is normalized to n , the in-plane elec-tric field E p ( x, y ) = ( E x ( x, y ) + E y ( x, y )) / is normalizedto m e ω pe c/e and the out-of-plane magnetic field B z ( x, y ) isnormalized to m e ω pe /e .The Maxwell equations can be normalized with theaforementioned normalization of the electric and magneticfield if we use λ s to normalize space and ω − pe to normalizetime (See Dieckmann et al. (2008) for details). The Maxwellequations and the particle equations of motion do not de-pend explicitely on the value of n in their normalized form,as long as binary collisions between particles are not im-portant. The value of n does not influence in this case theplasma dynamics and n only becomes important when wescale the simulation results to SI units. Space and time scalewith n − / and the electric and magnetic field amplitudeswith n / . Table 1 presents the numerical values of the fac-tors we have to multiply to the positions, times and fieldamplitudes for several values of n .Snapshots of n p ( x, y ) and E p ( x, y ) are displayed in Fig.2. The curves x i ( y ) = x i + A i sin (2 πy/ .
54) are overplot-ted at the centres of the thin shells at the times t i with1 ≤ i ≤
5. The offset x i is expressed in units of λ s andthe amplitude A i is normalized to A . We calculate thenormalized speed v i = x i / ( t i v ) and the normalized speed∆ A i = ( A i − A i − ) / ( v [ t i − t i − ]) with which the amplitudegrows at the extrema of the oscillation. Table 2 shows theirvalues.The normalized speed v i ≈ v / v / x as we expect from the global conservation of momentumand the equal cloud densities. The amplitude A i grows from MNRAS , 1–10 (0000) he collisionless NTSI Figure 2.
The proton density distribution n p ( x, y ) and the in-plane electric field distribution E p ( x, y ) multiplied by a factor 10 . Thefirst and the third row correspond to n p ( x, y ). The second and the fourth row show E p ( x, y ). The electric field distribution belonging toa proton density distribution is shown underneath the latter. Panels (a,d) correspond to the time t = 268. Panels (b,e) correspond to t = 536. Panels (c,f) correspond to t = 1 . × . Panels (g,j) correspond to t = 1 . × . Panels (h,k) correspond to t = 2 . × .Panels (i,l) correspond to t = 2 . × . A sine wave is fitted to the centre of the thin shell for the times t to t . Table 2 shows thevalues of its amplitude and offset along x . t to t at an average value of 0 . v or 0 . c s . Its growth ratedecreases for t > t . The increase of the amplitude from A to 2 . A demonstrates that the thin shell is unstable againstthe initial spatial displacement.The thickness of the thin shell is about 0.7 at t in Fig.2(a). The positive potential of the thin shell slows down theinflowing upstream plasma. The ensuing pile-up of the pro-tons increases the plasma density within the shell to a valueabove 2. The proton density has not reached anywhere thevalue n p ( x, y ) ≥ x B ( y ). The thin shell hasthus merely expanded along x.Protons have accumulated close to the extrema of thethin shell’s oscillation at y ≈ . y ≈ . v / y = 0 and y ≈ . v x ≈ v at y ≈ . v x ≈ y ≈ .
9, whichalters the momentum balance between both clouds at theextremal points and amplifies the oscillation via a change ofthe mean speed of the thin shell. Indeed the amplitude of
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M. E. Dieckmann et al. the oscillation has increased to 1 . A . The proton densityhas increased to a value n p ≈ . x and the density oscillates along the thin shellwith an amplitude of about 0 . n p ≈ . E p ( x, t ) in Fig. 2(f) demon-strates that the narrow unipolar electric field bands, whichare the characteristic of hybrid structures, are strongly mod-ulated along y. Their amplitude peaks at the concave sec-tions, which thus provide the largest density gradients. Theamplitude A i of x B ( y ) has grown further to a value 2.3.The large density value n p ≈ . x ≈ . y ≈ . x ≈ . y ≈ .
6. The boundaries of the thin shellthus have a double-layer component and the boundaries areindeed hybrid structures.Figure 2(g) evidences that the density of the thin shellhas equilibrated. The electric field in Fig. 2(j) has a practi-cally constant amplitude along both boundaries and its dis-tribution shows a piecewise linear shape. The electric fieldof the hybrid structure, which is determined by the densitygradient at the shell’s boundary, should still deflect mostprotons towards the extrema of x B ( y ). The absent densityaccumulation at the extrema suggests that a second processis counteracting this mass flow.The density distribution in Fig. 2(b) is the one ex-pected from Fig. 1(b). The density distribution has equi-librated sometime between t (Fig. 2(c)) and t (Fig. 2(g))and the equilibration time scale ∆ t is thus between t − t and t − t or 500 < ∆ t < . Let us assume that thedensity oscillates along a planar part of the thin shell,which has a length of ≈ λ s . The wavelength of the os-cillation is thus k = 2 π/ λ s . The ion acoustic speed is c s = 5 × m/s. One ion acoustic oscillation takes placeduring ˜ t s = 2 π/ ( k c s ), where ˜ t s is given in seconds. We canrewrite this expression as t s = ˜ t s ω pe = 3 c/c s , which gives t s ≈ . < ∆ t /t s < . A i at this time. The amplitude A has grownonly by ≈ . t and t and ∆ A = ∆ A /
2. Thiscoincidence suggests that the density equilibration is respon-sible for the decrease of the growth rate, which would implythat the collisionless NTSI is overstable.The amplitude growth of the shell’s spatial displace-ment slows down further as we go from t to t in Fig. 2(h)and we measure the largest value A ≈ . n p ( x, y ) peaks now at y ≈ . y ≈
0, which is the opposite of what we expect from theproton deflection by the hybrid structure. This distributioncan be explained in terms of an overshoot of the ion acous-
Y in λ s X i n λ s Figure 3.
The thin shell at the time t . The contour lines corre-spond to 0.4 times the peak value of E p ( x, y ). The horizontal linedenotes x = x . The two vertical lines delimit the spatial inter-val from which we will sample the velocities of the computationalparticles. The diagonal line is oriented at an angle of 20 ◦ relativeto x = x at y ≈ . tic wave, which is further evidence for an oscillation of theproton density distribution along the thin shell.The shell remains thin during the entire simulation timeand it does thus hardly accumulate material. The slow ex-pansion of the thin shell is favorable for a continuing growthof the oscillation amplitude A i . However, the thin shell startsto break up at the extremal points of the spatial oscillation.The distribution of n p ( x, y ) in Fig. 2(h) within the thin shelland that of E p ( x, y ) in Fig. 2(k) at its boundaries are bothfragmented. The same is true for the proton density distri-butions in both upstream regions. These density oscillationsare the result of a proton-proton beam instability inside theshell and in the upstream region close to it (See Fig. 1(c)).This instability ultimately seals the fate of the thin shellby giving rise to the growth of strong electrostatic fieldswith potential variations that are comparable to the poten-tial jump between the thin shell and the inflowing plasma.The destruction of the thin shell by ion acoustic waves isevidenced by Fig. 2(i) and the electric field in Fig. 2(l).The mechanism that results in the hydrodynamic NTSIis the transport of material towards the extrema of the thinshell’s spatial oscillation. The rotation of the fluid veloc-ity vector by the oblique crossing of a hydrodynamic shockalways results in a flow towards the extremal positions,because the fluid is trapped within the thin shell. A hy-brid structure can, however, not trap protons within thethin shell. Once the protons reach the opposite side of thethin shell, they are reaccelerated by the double layer andpropagate upstream. The thin-shell instability in collision-less plasma is thus only similar to the NTSI if a significantfraction of the protons is indeed moving to the extremal po-sitions of the thin shell at y ≈ . y ≈ .
9. We mustcompare the flow direction of the protons within the shellwith the direction of the thin shell.We estimate with Fig. 3 the angle between x = x (Ta-ble 2 at the time t ) and the direction of the thin shell at y ≈ . ◦ . Protons that move along this direction MNRAS , 1–10 (0000) he collisionless NTSI −0.2 0 0.200.20.40.60.81 (a) v y / v v x / v −0.2 0 0.200.20.40.60.81 (b) v y / v v x / v Figure 4.
The velocity distribution at the time t and y ≈ . x . The beam with v x ≈ v corresponds to theprotons of cloud 1, which flow towards the thin shell. The lowerbeam is composed of protons of the cloud 2 that left the thin shellat the opposite side. Panel (b) shows the proton distribution inthe centre of the thin shell. The velocity vectors of both beamshave been rotated by an angle of approximately 20 ◦ , which isindicated by the overplotted line. in the rest frame of the shell remain inside the shell.We sample the in-plane velocity components v x and v y from the protons that are located in the spatial interval,which is delimited by the two vertical lines in Fig. 3. Thevelocity distribution of the protons with 2 . < x < . . < x < . v x ≈ ≈ ◦ around the pivot point v x = v / v y = 0.The velocity distribution inside the thin shell demonstratesthat the majority of the protons flow along the thin shell.These protons will eventually reach the extremal positionsof the shell’s oscillation at y ≈ . y ≈ . v / v . Figure 5.
Projections of the phase space density distribution ofthe protons of cloud 1 onto the x, v x plane (a) and onto the x, v y plane (b) at t = t . The colour scale is linear. The protons thus leave the thin shell at the extrema of itsoscillation, feeding the collimated outflow seen in Fig. 2(c).Figure 5 shows the phase space density distribution ofthe protons of cloud 1 averaged over the y-interval, whichis delimited by the vertical lines in Fig. 3. The thin shell islocated at x ≈ . v x ≈ . v at x ≈ x inFig. 5(a) and their mean speed along the y-direction reaches v y ≈ − . v in Fig. 5(b). Figure 5(a) shows that the protonsare reaccelerated by the double layer at x ≈ . x ≈
5. Some of the protons are reflected by the electrostaticshock at x ≈ . x ≈ v x ≈ .
2. These protons fall behind the thin shell, whichmoves at the mean speed v / x > < x < . x > .
8. The growth in timeof the thin shell’s potential relative to the upstream resultsin an increasing proton compression within the shell, whichreduces temporally the number of protons that exit the thinshell via the double layer.Figure 6 shows the projections of the proton phase spacedensity distribution onto the x, v x plane and onto the x, v y plane at the time t = t . The phase space density distribu-tions are qualitatively similar to those at the previous timebut they are more turbulent. The phase space density in theinterval 4 < x < v x ≈ v varies in Fig. 6(a). Thedensity changes are correlated with changes in the meanspeed in Fig. 6(b). We attribute these localized changes ofthe proton’s mean speed and density to the ion acousticwaves, which we observe in Fig. 2(h).According to Fig. 1(b), the electric field deflects theprotons towards the extrema of the shell’s oscillation by de-celerating them along the normal of the shell’s boundary.Electrons that enter the shell should be accelerated alongthe normal by this field due to their opposite charge. Figure7 demonstrates that this drift generates magnetic fields. Themagnetic field modulus peaks at y = 0 and y = 3 . MNRAS , 1–10 (0000)
M. E. Dieckmann et al.
Figure 6.
Projections of the phase space density distribution ofthe protons of cloud 1 onto the x, v x plane (a) and onto the x, v y plane (b) at t = t . The colour scale is linear. value of x = x i . The magnetic field amplitude grows and themagnetic field patches expand until t = t .The potential difference between the shell plasma andthe upstream plasma determines the drift velocity betweenthe electrons and protons that enter the shell and, thus, thenet current. A growth of this potential difference through anincrease of the plasma density within the shell thus resultsin the growth of the magnetic field energy, as long as theelectric fields are well-defined unipolar pulses.The magnetic field weakens once the thin shell starts tobe fragmented by the ion acoustic instability at t = t and allthat remains at t = t are small-scale magnetic fluctuations.The temporal correlation between the magnetic field collapseand the destruction of the thin shell demonstrates that thelatter is the driver of the magnetic field. We have examined the collision of two clouds of electronsand protons at a speed that exceeded the ion acoustic speedby a factor 3.5. Their initial contact boundary was sinu-soidally displaced along the collision direction. The displace-ment of the contact boundary resulted in a sinusoidally cor-rugated thin shell that was formed by the interpenetratingplasma clouds and this corrugation seeded the NTSI. Wehave confirmed that a wavelength of the seed perturbation,which exceeded that used in the previous simulation studyby Dieckmann et al. (2015c) by a factor of 4, is unstable. Awide range of wavenumbers of the seed perturbation is thussubjected to the NTSI.We have identified here the proton-proton beam in-stability as the process that limits the life-time of thethin shell. This instability is known to destroy planar dou-ble layers and electrostatic shocks (Karimabadi et al. 1991;Kato & Takabe 2010; Dieckmann et al. 2015a) and here wehave shown that it also affects the nonplanar ones.The amplitude of the shell’s spatial oscillation grew be-cause the NTSI introduces a spatially varying velocity of thethin shell in the reference frame that moves with the mean speed of the shell. The modulus of the velocity peaked at theextrema of the shell’s oscillation and the velocity at thesepositions reached 70% of the ion acoustic speed. The ampli-tude of the thin shell’s spatial displacement grew during thesimulation to almost three times its initial value before theshell was destroyed by the proton-proton beam instability.Our simulation data hints at a possible coupling of theNTSI with ion acoustic oscillations along the thin shell. Wehave explained the change of the NTSI’s growth rate at latetimes in terms of these oscillations, which would make thecollisionless NTSI overstable. Such an overstability has alsobeen observed for the hydrodynamic linear thin-shell insta-bility (Vishniac 1983).The ambipolar electric field at the boundaries of thethin shell deflected the inflowing upstream electrons and pro-tons into different directions. The relative drift of the elec-trons and the protons resulted in a net current and, thus, inthe growth of magnetic fields.We can obtain additional qualitative insight into thecollisionless NTSI by comparing the simulation results wehave obtained here with those discussed in related work.The shorter wavelength of the seed perturbation in thesimulation by Dieckmann et al. (2015c) resulted in two im-portant differences. Firstly, the shorter wavelength of theseed perturbation used in that previous work implied thatthe ratio of the amplitude of the shell’s corrugation tothe wavelength of the corrugation could grow to a muchlarger value before the proton-proton beam instability setin. The low maximum ratio that can be reached for longwave lengths of the seed oscillation probably implies thatit will be more difficult to observe their growth. Secondly,the larger proton density gradients within the thin shell thatdeveloped during the growth phase of the NTSI in the sim-ulation by Dieckmann et al. (2015c) resulted in ambipolarelectric fields along the thin shell that were strong enoughto drive nonlinear plasma structures within the thin shell.The lower density gradients within the thin shell that werereached in the present simulation resulted in density oscilla-tions along the thin shell that remained in the linear regime.The peak amplitude of the magnetic field strength inthe present simulation is four times that in the simula-tion by Dieckmann et al. (2015c) and the field patches ex-tended far upstream. The weak magnetic fields observed byDieckmann et al. (2015c) remained practically confined tothe thin shell. The longer wavelength of seed oscillation wehave used here thus generates magnetic fields with a largerenergy than those found by Dieckmann et al. (2015c).The size of the magnetic field patches we found here wascomparable to those in the simulation by Dieckmann et al.(2015b), where the curved shell was created by a spatial vari-ation of the collision speed. The magnetic field in the presentsimulation and in that in (Dieckmann et al. 2015b) dampedout when the thin shell was destroyed by the proton-protonbeam instability, evidencing that the hybrid structures wereresponsible for its growth.It is possible to introduce a collision operator into aPIC simulation that emulates the effects of binary collisionsbetween particles. Collisions affect the growth rate of theproton-proton beam instability and if they occur frequentlythey can thermalize the protons within the thin shell andscatter the proton beam that moves back upstream before
MNRAS , 1–10 (0000) he collisionless NTSI Figure 7.
The out-of-plane magnetic field distribution B z ( x, y ) multiplied by the factor 100. Panel (a) corresponds to the time t = 268,panel (b) to t = 536, panels (c) to t = 1 . × , panel (d) to t = 1 . × , panels (e) to t = 2 . × and panels (f) to t = 2 . × . an instability sets in. The hybrid structure will probablychange into a fluid shock if collisions are frequent.The maximum speed, which parts of a hydrodynamicthin shell can reach in the shell’s rest frame, is just below thesound speed (Vishniac 1994). The sound speed is the hydro-dynamic equivalent of the ion acoustic speed in a collisionlessplasma, which suggests that we can go from the collisionlessto the hydrodynamic limit discussed by Vishniac (1994) byincreasing the collisionality of the plasma. We will test thishypothesis in future work. Acknowledgements:
The simulation was performedon resources provided by the Swedish National Infrastruc-ture for Computing (SNIC) at HPC2N (Ume˚a).
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