Laboratory laser experiments offer a novel approach to studying magnetized collisionless shocks, and a common method in recent experiments is to drive shocks using a laser-ablated piston plasma. However, current experimental capabilities are still limited to spatiotemporal scales on the order of shock formation, making it challenging to distinguish piston and shock dynamics. We present quasi-1D particle-in-cell simulations of piston-driven, magnetized collisionless shock formation using the code PSC, which includes a model of laser-driven plasmas that can be well-matched to experimental conditions. The simulations cover a range of upstream and ablation parameters and yield several robust signatures of shock formation, which can provide a reference for experimental results.

Collisionless shocks are ubiquitous nonlinear phenomena that involve intense plasma heating and acceleration in space and astrophysical plasmas. They act to thermalize the kinetic energy of supersonic plasma flows, forming a quasi-discontinuous boundary between regions of low (upstream) and high (downstream) entropy. In collisionless plasmas, this process is mediated by electromagnetic effects over characteristic length scales that are much shorter than the particle mean free path. Importantly, above a critical Mach number, the shock layer is mediated in part by the reflection of incoming ions. These supercritical magnetized shocks, in which the shock forms in a pre-existing magnetic field, are the most commonly observed, ranging from planetary bow shocks1–4 to the heliospheric termination shock5 to supernova remnants and gamma ray bursts.6 

The Earth's bow shock was the first collisionless shock to be discovered,1 and it has remained for decades a natural laboratory to study collisionless shocks with spacecraft. While satellite observations have yielded a wealth of data and understanding of shock physics,7,8 spacecraft are limited by their 1D trajectories and statistical sampling of non-stationary conditions. This has led to the re-emergence of laboratory experiments, and in particle laser-based experiments, to generate and study shocks. Laser experiments are attractive because they can drive fast, supersonic plasma flows in a wide range of controlled conditions.9–11 Laser-driven shocks have taken two forms in recent experiments: piston-driven and obstacle-driven. In piston-driven setups, the laser-produced plasma acts as a supersonic piston that sweeps up and accelerates a pre-existing, magnetized ambient plasma. The accelerated ambient plasma then drives a traveling shock as it streams through unperturbed upstream plasma, similar to the expansion of supernovae remnants.9 An important aspect of these setups is the mechanisms that couple energy and momentum between the piston and ambient plasmas.12 In obstacle-driven setups, the laser plasma impinges a magnetic obstacle and creates its own counter-streaming flow that drives a standing shock, similar to a planetary bow shock.

Most laboratory experiments, to date, have utilized the piston-driven method to study shocks. The first laser-driven, magnetized collisionless shock experiments focused on subcritical (low-Mach-number) shocks,13–15 which resulted in the first observations of collisionless shock formation16 since pinch experiments in the 1960s17 (which were also piston-driven, except the piston was the imploding magnetic field). More recent experiments have observed high-Mach-number magnetized collisionless shocks by utilizing significantly higher ambient plasma densities and larger magnetic fields.18 In both setups, direct observations of the piston-ambient coupling were made.19,20 Several obstacle-driven shock experiments have been performed, but these either have been in a collisional regime21 or have not had sufficient energy to drive a fully formed shock.22 

One of the primary tools for interpreting laser-plasma experiments is numerical simulations. Most experimentally relevant magnetized shock simulations are initialized with either uniform counter-streaming flows or a uniform region of denser plasma that subsequently expands into a more tenuous plasma.23–27 Such simulations are concerned with timescales long after the shock has formed, when the dynamics of the shock is independent of its creation. This is useful for predicting or understanding effects primarily driven by shocks; however, experiments are currently limited to regimes in which shocks are forming or have newly formed, where observable features may be dominated by the formation mechanisms rather than the shock. To be of greatest utility to experiments, it is, therefore, valuable to have simulations that include and focus on the shock formation stage. Recent particle-in-cell (PIC) simulations28 have explored this regime for obstacle-driven shock experiments. While several studies were carried out using hybrid (kinetic ions, fluid electrons) simulations to understand low-Mach-number piston-driven shock formation,24,29 these simulations did not attempt to model laser-ablated piston plasmas and were conducted in a regime in which the piston lifetime is comparable to the shock formation time. As a result, the interaction between the piston and ambient plasmas is difficult to compare directly with experimental observations.

By modeling ablation plasmas that can be matched to experiments, the 3D PIC code PSC30,31 offers several advantages over previous simulation efforts for studying piston-driven shock formation in laser experiments. Recent simulations using PSC have shown how shocks form between expanding and background plasmas in a large 2D plasma plume expansion under experimental conditions and have generated observables of shock formation for comparison with experiment.31,32 In particular, traditional metrics for classifying shocks, such as the Rankine–Hugoniot (RH) jump conditions, are challenging to apply when the piston dynamics are dominant since there is no well-defined downstream region. Instead, PSC simulations can elucidate new observables that provide a reference for experiments constrained to the formation regime, including spatial and temporal scales over which a shock forms; typical profiles for density, magnetic and electric fields, and temperature; the evolution of particle velocity distributions during shock formation; and signatures of piston-shock separation. These observables also complement those investigated in previous works9,33 that have discussed laboratory shock criteria.

In this paper, we report quasi-1D PSC simulations of piston-driven, magnetized collisionless shock formation that scan over a wide range of ablation and upstream parameters. The simulations model the supersonic expansion of a laser-ablated piston plasma through a uniform, magnetized ambient (upstream) plasma in a strictly perpendicular magnetic geometry. We focus on conditions relevant to high-energy-density (HED) laser experiments, in which the Alfvénic Mach number MA>1,βablation1, and βupstream1 (β=2μ0neTe/B2). The results indicate robust shock formation under a wide variety of plasma parameters and provide a set of criteria and characteristics by which shock formation in the absence of the RH conditions is evaluated, which are seen to apply at late times when a sufficient downstream region has developed.

This paper is organized as follows: Sec. II describes the setup of the simulations and the parameters considered in this study. Section III discusses the main results, including a model for evaluating the piston and shock speeds in the simulations and experiments (Sec. III A), criteria for judging piston-driven shock formation (Sec. III B), characteristics of the early and late time evolution of piston-driven shocks (Sec. III C), the extension to multi-species plasmas (Sec. III D), and the effect of collisions (Sec. III E) and mass ratio (Sec. III F) on the results.

The simulations were performed using PSC, a fully electromagnetic, relativistic, and massively parallel 3D PIC code.30,31 PSC implements a plasma ablation model, which allows fully kinetic simulations of the coronal plasma relevant for piston-driven shocks and other laboratory astrophysics and HED phenomena.31 In the simulations, a piston plasma is “ablated” from the target and expands through a pre-initialized uniform magnetized ambient plasma. To do so, these simulations utilize a localized heating operator to mimic the effect of laser ablation, which is different from classical shock simulations that are initiated with counter-streaming plasmas, moving walls, or a uniform dense plasma expanding into a tenuous plasma (see, for example, recent reviews by Marcowith et al.34 and Treumann35 and references therein). This allows a strongly driven piston plasma to be included self-consistently and more closely resembles laser experiments. The resulting hydrodynamic evolution of the ablation plasma in PSC is generally in good agreement with radiation-hydrodynamic predictions at the same parameters.31 The kinetic ablation model can be run with reduced electron parameters (reduced electron-ion mass ratio and reduced speed-of-light to electron-thermal speed), while simultaneously keeping the ion-scale parameters well-matched to experiments, as discussed in Ref. 31. This reduces computational requirements for ion-dominated phenomena like collisionless shocks and the large, multi-ion-gyroradius system sizes encountered in laboratory HED experiments.

We briefly review the ion-scale parameters and introduce the notation used here; for more details, the reader is referred to Ref. 31. In addition to the electron ablation density ne,ab and electron ablation temperature Te,ab, the charge state Zab and ion mass mi define the fundamental length and time scales of the simulation through the ion inertial length di,ab=(mi/ne,abZabe2μ0)1/2 and time tab=di,ab/Cs,ab, where Cs,ab=(ZabTe,ab/mi)1/2. It is also convenient to define a fundamental magnetic field Bab=μ0ne,abTe,ab and electric field Eab=Cs,abBab=Te,ab/di,ab. The simulations presented here include a uniform background magnetic field B0 and uniform ambient plasma of charge state Z0, electron density ne0, ion density Z0ni0=ne0, temperature Te0=Ti0, and sound speed Cs0=(Z0Te0/mi)1/2. The addition of a background field allows us to define an ablation beta βab=2μ0ne,abTe,ab/B02 and upstream beta β0=2μ0ne0Te0/B02. These background parameters also provide a more convenient set of length and time scales for shocks by defining an upstream ion inertial length di0=(Zab2ni,ab/Z02ni0)1/2di,ab and time t0=(di0/vA)=ωci01=(Zab/Z0)βabtab, where vA=B0/(μ0ni0mi)1/2 is the Alfvén speed.

Coulomb collisions are included in the simulations and are modeled with the Takizuka-Abe binary collision operator.30 This introduces an additional timescale and free parameter and the electron-ion collision time τei,ab=νei,ab1. The collisionality is matched with experiments through the dimensionless parameter λab=ωce,ab/νei,ab=λmfp,ab/de,ab, where ωce,ab is the electron gyrofrequency at Bab and λmfp,ab=Tab/me/νei,ab is the electron-ion mean free path evaluated at Tab and ne,ab=1. This scaling ensures that dimensionless quantities such as the magnetic Reynolds number are correct (see Ref. 31), but electron collisionality relative to global scales (e.g., νei,abtab) is only quantitatively matched at physical mass ratios.

Multiple simulations were performed covering a range of ablation and upstream parameters for both single-species and multi-species ion compositions. The simulations are “quasi-1D,” with only a few cells and uniform driving conditions in the transverse direction, so that the ablation is effectively planar. This is sufficient for the purposes of this study since shock formation physics is dominantly 1D. Examples of 2D shock simulations using PSC can be found in Ref. 31. The simulations presented here are also in a strictly perpendicular magnetic geometry (angle θB0=90° between the piston expansion and the background magnetic field). This was chosen because it is the most efficient geometry for generating shocks relative to a fixed domain size and therefore the easiest to achieve experimentally. Quasi-perpendicular PSC simulations (45°<θB0<90°) of collisionless shocks are being considered separately.36 

The simulations were gridded in the xz plane, with B0=B0x̂ and the primary ablation direction along ẑ. The simulation domain was centered along z, so that particles are ablated in both directions. Only the z0 domain is considered here. Both the x and z boundaries were periodic, but the simulations were stopped before significant numbers of particles reached the +z boundary. The number of macro particles per cell was defined to be 1000 at ne,ab=1. The simulations consisted of a box of 30 000 cells in z and 12 cells in x, corresponding to a domain size of 9000 × 5 de,ab. The initial ambient plasma and magnetic field occupied the region z>2di,ab. For 0<z2di,ab, we placed a “target” of density 2.5ne,ab. As particles were ablated, the heating operator continuously added new particles to the target to maintain this target density. Additionally, the target surface smoothly reaches the desired ablation temperature Te,ab on the timescale of tab, significantly longer than the simulation time step such that transient effects due to the application of heating are avoided. The simulations were run for 400 000 timesteps, and the heating operator was applied for the lifetime of the simulation. Outside of the target region, energy was conserved to within 2% over the lifetime of the simulations, with deviations primarily due to numerical heating in the cold upstream plasma. The simulations were carried out with a reduced proton-to-electron mass ratio of μp=100 (di,ab=10de,ab) and a reduced speed of light set by the ratio Te,ab/mec2, which can be written relative to the sound speed as c=μp/Te,abCs,ab. The single-species runs consisted of H piston and ambient plasmas (Z =1), while the multi-species runs consisted of a co-equal CH mix for both the piston and ambient plasmas (average Z =3.5 and average ion mass ratio μi=650).

All runs used a fixed ablation density, but covered a range of ablation temperatures Te,ab=0.030.134mec2. They also covered a range of ablation betas βab=3011840 and upstream betas β0=0.0083.2 through varying the upstream field and density, as well as runs with B0=0 and ne0=0. For all runs, the electrons were modestly collisional (ωce,ab/νei,ab20). Section III A describes how the shock speed vsh of the ambient plasma can be extracted from the ablation and upstream parameters. Using this speed, the runs also covered a range of upstream Alfvénic Mach numbers MA=vsh/vA= 3 to 57 and magnetosonic Mach numbers Mms=vsh/vA2+Cs02= 3 to 27.

Parameters for a representative run are listed in Table I, along with a set of corresponding physical values relevant to HED laboratory experiments. Relative to this reference run, subsequent runs modified a single variable (Te,ab, B0, ne0, or Z) to explore its effect on piston-driven shock formation and evolution. A summary of the simulations is plotted in Fig. 1 as a function of βab and MA.

TABLE I.

Parameters for a representative quasi-1D run, with the corresponding simulation values in code units and one possible set of experimentally relevant physical values.

ParameterSim. valuePhys. value
Simulation x size Lx 0.5 di,ab 4.7 μ
Simulation z size Lz 900 di,ab 8.4 mm 
Simulation time τsim 220 tab 10.9 ns 
Ablation    
Charge state Zab 
Electron density ne,ab 1.25 6×1020 cm−3 
Electron temperature Te,ab 0.092 mec2 470 eV 
Collision frequency νei,ab 0.009 tab 0.43 ps 
Sound speed Cs,ab 0.030 c 210 km/s 
Piston speed vp 0.104 c 730 km/s 
Shock speed vsh 4.6 Cs,ab 980 km/s 
Upstream    
Magnetic field B0 0.01 mec2 7 T 
Charge state Z0 
Electron density ne0 0.01 ne,ab 4.8×1018 cm−3 
Temperature T0 0.002 mec2 10 eV 
Ion inertial length di0 11.2 di,ab 104 μ
Ion gyroperiod ωci01 33.9 tab 1.5 ns 
Dimensionless  Value  
Mass ratio mi/me 100  
Speed of light csim/cphys 0.02  
Ablation beta βab 1150  
Upstream beta β0 0.2  
Collisionality λab 20  
Ion mean free path λmfp/di0 350  
Alfvén Mach number MA 14  
Magnetosonic number Mms 13  
ParameterSim. valuePhys. value
Simulation x size Lx 0.5 di,ab 4.7 μ
Simulation z size Lz 900 di,ab 8.4 mm 
Simulation time τsim 220 tab 10.9 ns 
Ablation    
Charge state Zab 
Electron density ne,ab 1.25 6×1020 cm−3 
Electron temperature Te,ab 0.092 mec2 470 eV 
Collision frequency νei,ab 0.009 tab 0.43 ps 
Sound speed Cs,ab 0.030 c 210 km/s 
Piston speed vp 0.104 c 730 km/s 
Shock speed vsh 4.6 Cs,ab 980 km/s 
Upstream    
Magnetic field B0 0.01 mec2 7 T 
Charge state Z0 
Electron density ne0 0.01 ne,ab 4.8×1018 cm−3 
Temperature T0 0.002 mec2 10 eV 
Ion inertial length di0 11.2 di,ab 104 μ
Ion gyroperiod ωci01 33.9 tab 1.5 ns 
Dimensionless  Value  
Mass ratio mi/me 100  
Speed of light csim/cphys 0.02  
Ablation beta βab 1150  
Upstream beta β0 0.2  
Collisionality λab 20  
Ion mean free path λmfp/di0 350  
Alfvén Mach number MA 14  
Magnetosonic number Mms 13  
FIG. 1.

Summary of simulation runs in terms of the ablation beta βab and Alfvénic Mach number MA. Equation (3) (dashed line) marks the approximate upper bound of MA for a given βab.

FIG. 1.

Summary of simulation runs in terms of the ablation beta βab and Alfvénic Mach number MA. Equation (3) (dashed line) marks the approximate upper bound of MA for a given βab.

Close modal

In this section, we use our scan over a range of plasma parameters to help model relevant speeds of interest, in particular, a characteristic piston speed vp and shock speed vsh. This is necessary because we initialize our simulations with an ablative expansion. As a result, there is no single initial plasma speed that could be used to define either the initial ambient flow or an upstream inflow speed in the shock frame (and thus a shock speed). Indeed, the piston (ablated) ions can be expected to cover a wide range of velocities. This can be seen through the ablation density, which acquires a classic scale-free profile37 of the form ne=(ne,abne0)exp[(zz0)/z0]+ne0, where z=vt0 and v is an expansion speed. For these simulations, z02.5di,ab measured at time t0=5tab, implying z00.5Cs,abt0. Solving for v yields the expansion speed at a given density,

v=Cs,ab2[1ln(nene0ne,abne0)].
(1)

In the presence of a background magnetic field, the piston plasma will sweep out the field, creating a magnetic cavity near the target and compressing the field near the leading edge of the piston plume. If there is an ambient plasma, ambient ions will likewise be swept up by the initial electrostatic fields near the target surface. This leads to a comparable density cavity and compression closely aligned with the magnetic ones. An example of these features can be clearly seen in Fig. 2. We thus define a characteristic piston speed vp by tracking the peak field linked to the magnetic cavity, which is the dominant feature in the absence of shock formation and often easily observable in laboratory experiments. The results for a range of ambient densities ne0 are shown in Fig. 3(a) (here, ne,ab1 in all cases, and there is no appreciable dependence on B0). The velocities are well matched by Eq. (1) for ne1.35ne0, meaning that the interaction between the piston and the background magnetic field primarily occurs where the piston density is about 1.3 times the ambient density. In the absence of an ambient plasma, the piston speed asymptotes to vp6.5Cs,ab.

FIG. 2.

Streak plot of the Bx component of the magnetic field for the run in Table I, with the corresponding piston vp and shock vsh speeds labeled.

FIG. 2.

Streak plot of the Bx component of the magnetic field for the run in Table I, with the corresponding piston vp and shock vsh speeds labeled.

Close modal
FIG. 3.

(a) Piston speed vp normalized to the ablation sound speed as a function of upstream electron density ne0 (squares). Also shown is Eq. (1) for ne1.3ne0 (dashed). (b) Shock speed normalized to the piston speed as a function of the normalized upstream Alfvén speed (squares). Also shown is Eq. (2) (dashed).

FIG. 3.

(a) Piston speed vp normalized to the ablation sound speed as a function of upstream electron density ne0 (squares). Also shown is Eq. (1) for ne1.3ne0 (dashed). (b) Shock speed normalized to the piston speed as a function of the normalized upstream Alfvén speed (squares). Also shown is Eq. (2) (dashed).

Close modal

The same analysis can be used to define the shock speed vsh, which corresponds in Fig. 2 to the faster-moving magnetic compression associated with the shocked ambient plasma. The results for a range of upstream Alfvén speeds with approximately the same vp are shown in Fig. 3(b). The shock speeds asymptote to about 4/3 times the piston speed at low background fields and then increase with increasing B0, which is consistent with the RH relations. This follows because the piston speed constrains the downstream shock speed by effectively becoming the downstream speed in the frame at rest with the upstream. In the frame moving with the shock, the RH relations determine the change in speed across a perpendicular shock according to

v1v2=[MA2+β0¯]+[(MA2+β0¯)2+8MA2]1/2,
(2)

where the subscripts 1 and 2 refer to the upstream and downstream speeds, respectively, and β0¯=5/2(1+β0). Note that in the shock frame, v2=v1vp and v1=MAvA=vsh. In the limit of large MA (vA0), vsh/vp4/3, as seen in Fig. 3(b).

Because the shock speed is related to the piston speed, MA and βab are also necessarily related, through the ablation sound speed. Using Eqs. (1) and (2), βab can be written in terms of MA as

βab=Z0ne,abZabne0MA2(1f(MA)1)2(vpCs,ab)2,
(3)

where f(MA) is the RHS of Eq. (2). The maximum MA attainable for a given βab is plotted in Fig. 1.

Given current limitations in spatial and temporal scales in laboratory laser experiments, there is often insufficient space or time for a collisionless shock to clearly separate from the driving piston and develop properties, such as a well-defined downstream, classically associated with shocks. One consequence of this is that the RH jump conditions, derived from MHD theory and often invoked to classify whether a shock has formed, are not easily applied since they describe the structure of shocks on length scales much larger than the shock width. Instead, for these types of experiments, the dominant structure downstream of the shock, if it has separated at all, is the piston, which means that the shock and its upstream and downstream regions are not an isolated system. As a result, it is important to determine additional criteria by which to judge whether a shock is forming in these conditions. In this section, we discuss relevant timescales and criteria for piston-driven shock formation and then discuss associated plasma and field properties that may be experimentally observable.

From our simulations, we find three important timescales in the initial evolution of piston-driven shocks, which we label t1,t2, and t3, along with the corresponding location of the shock at those times, z1,z2, and z3. The first is the onset of shock formation (t1). The second is the formation of the shock on ion scales and its separation from the piston (t2). The third is the development of a well-defined downstream region and the emergence of the shock structure on MHD scales much larger than the shock width (t3). For the purposes of terminology, we label structures before t2 as shock precursors and after t2 as shocks.

To define the first time, we first note that in most HED experiments, the laser plasmas are moving fast enough to drive supercritical shocks (MA,crit3 for perpendicular geometries35) in which ion reflection off the shock front is a key method of dissipating energy across the shock. In the absence of well-defined downstream and associated RH jump conditions, a population of reflected ions is a key hallmark of shock formation. For piston-driven shocks, this specifically refers to the reflection of ambient ions. In the upstream reference frame, this corresponds to ambient ions moving faster than the shock. Let fa(v,z,t) be the 1D velocity distribution of the ambient ions along z, Na,tot(t)=0dz+fadv the total number of ambient ions, Na,refl(t)=0dzvsh+fadv the total number of ambient ions moving faster than the shock speed, fa,refl(z,t)=vshfadv the distribution of ambient ions moving faster than the shock speed, F(z,t)=fa,refl/Na,tot, and G(t)=Na,refl/Na,tot. We then define t1 to be the time at which the change in the reflected ambient ion fraction over all space is the greatest, i.e., dG/dt is the maximum, and z1 to be the location at time t1, where the change in the ambient ion fraction is the greatest, i.e., dF(t1)/dz is the maximum. We find that t1 and z1 are robust metrics of the onset of shock formation over our range of simulations. This process is illustrated in Fig. 4 for the run in Table I.

FIG. 4.

(a) Change in the fraction of reflected ambient ions as a function of time relative to the upstream gyroperiod ωci01. The peak change at time t1 indicates when shock formation begins. (b) Change in the spatial distribution of reflected ambient ions at time t1 relative to the upstream directed gyroradius ρi0=vp/ωci0. The peak change at position z1 indicates where the shock forms. The functions F and G are defined in the text.

FIG. 4.

(a) Change in the fraction of reflected ambient ions as a function of time relative to the upstream gyroperiod ωci01. The peak change at time t1 indicates when shock formation begins. (b) Change in the spatial distribution of reflected ambient ions at time t1 relative to the upstream directed gyroradius ρi0=vp/ωci0. The peak change at position z1 indicates where the shock forms. The functions F and G are defined in the text.

Close modal

The onset of shock formation generally occurs within the envelope of the piston plasma. This is illustrated in Fig. 5(b), which shows the ion phase space and electron density and magnetic field profiles corresponding to the time t1 for the run in Table I. As a result, while at time t1 the structure of the shock precursor exists on kinetic scales and exhibits reflected ions, it is not yet independent of the piston. Instead, the shock forms once the shock precursor clearly separates from the piston, which can be easily tracked through the compressed magnetic field profile (see, for example, Fig. 2). We define this shock formation time t2 (z2) to be the time (location) at which the peak magnetic field associated with the shock precursor has separated by 1/4ρi0 from the peak field associated with the piston, where ρi0=vp/ωci0 is the upstream directed gyroradius for an ion moving at the piston speed vp. This is shown in Fig. 5(c) at time t2, where the newly formed shock is at z29di0 and the piston is at z25di0. As the newly formed shock propagates further, a downstream region develops behind it. We define time t3 (z3) to be the time (location) by which the shock has separated 1ρi0 from the piston and, hence, the time at which the size of the downstream region is much larger than the shock width (ρi0di0). This is shown in Fig. 5(d) at time t3, where the shock is at z56di0 and the piston is at z46di0.

FIG. 5.

Illustration of the initial evolution of piston-driven shock formation. The first panel is an early time step before a shock forms, while the remaining panels correspond to the key timesteps t discussed in the text. The timesteps are in units of the upstream gyroperiod ωci01 for the run described in Table I. Shown are the total ion v-z space (where v is relative to the ablation sound speed Cs,ab) and profiles of magnetic field Bx (black) and total electron density ne (red) relative to their upstream values.

FIG. 5.

Illustration of the initial evolution of piston-driven shock formation. The first panel is an early time step before a shock forms, while the remaining panels correspond to the key timesteps t discussed in the text. The timesteps are in units of the upstream gyroperiod ωci01 for the run described in Table I. Shown are the total ion v-z space (where v is relative to the ablation sound speed Cs,ab) and profiles of magnetic field Bx (black) and total electron density ne (red) relative to their upstream values.

Close modal

Examining Fig. 5 leads to seven experimentally relevant criteria to determine shock formation for times t<t3,

  1. super-magnetosonic (Mms=v/vA2+Cs2>1),

  2. collisionless (L/λii>1),

  3. large density compression (ne/ne0>2),

  4. large magnetic field compression (B/B0>2),

  5. steep ramp (dB/dz and dne/dzdi0),

  6. presence of magnetically reflected ambient ions, and

  7. separation from the piston.

For piston-driven shocks in our simulations, the first six criteria are satisfied by time t1, while all seven criteria are satisfied by time t2. Thus, following our earlier terminology, structures that satisfy the first six criteria are shock precursors. Shock precursors that additionally satisfy criteria 7 are shocks.

The first two criteria are trivially associated with the definition of a collisionless shock. The density and field compression criteria are required to distinguish shocks from interpenetrating flows, where the increase in the density or field amplitude is due to the superposition of each flow. Note that these compression ratios are only a guide since, at these early times, only the peak density or field compression in the shock (or precursor) layer is generally observable. Because compressions in the shock layer can be associated with overshoots for high-Mach-number shocks, they are distinct from the compression ratios defined by the RH jump conditions, which require a well-defined downstream region that develops at later times (t3). This RH ratio can asymptote to a factor of 4 for large MA in a perpendicular magnetic geometry;35 alternatively, for very low MA, the compression ratio can be less than 2, though this is difficult to achieve in practice with collisionless HED laser plasmas. Also, note that large field and density compressions can be associated with piston-driven plasmas that do not generate shocks.20 

The structure of magnetized shocks is dominated by ion dynamics, with dissipation in the shock layer occurring at ion scales or smaller.35 Thus, the fifth criterion stipulates that the density and field compressions of this structure are sufficiently steep—on the order of di0—to indicate that a shock is forming on ion kinetic scales.

For the high-Mach-number shocks considered here, a key dissipation mechanism is (ambient) ion reflection off the shock front, which is our sixth criterion. Ion reflection is also the most direct method for observing the formation of a shock, especially if phase-space measurements are available. However, it is worth emphasizing that this population of magnetically reflected ambient ions is distinct from the populations of fast (vvp) piston or piston-accelerated ambient ions that may also be present. In particular, ambient ions accelerated by the ambipolar electric fields in the piston should not be confused with ambient ions reflected by the forming shock. These two populations of ambient ions can be distinguished primarily by the location where the acceleration or reflection occurs. This can be seen in Fig. 5(a), where the ambient ions are primarily swept up and accelerated to large speeds around z/di04 by the piston, which can occur even in the absence of magnetic fields (see Fig. 7). In contrast, the start of magnetic reflection occurs around z/di07, where the magnetic field is compressed, and swept-up ambient ions begin to interact with upstream ambient ions. Note that this region is associated with large density and magnetic field compressions (4× upstream) and strong ion heating (50× upstream) even though no shock has yet formed.

While the development of a significant reflected ambient ion population signifies the onset of shock formation (t1), this generally occurs while the piston plasma is still dominant. This can be seen in Fig. 5(b), where the forming shock at z/di012 is only slightly distinguished from the piston at z/di011. This necessitates the last criterion, which enforces that magnetically reflected ambient ions (and associated field and density compressions) are dynamically independent of the piston (which continues to pileup behind the forming shock). This is shown in Fig. 5(c) and is representative of the time t2 that a shock forms on ion scales. At later times t3 [Fig. 5(d)], the shock separates even further from the piston, allowing the development of a well-defined downstream region on scales much larger than the shock layer.

In our simulations, these key shock formation times t are nearly constant for a wide range of conditions, with little dependence on the Alfvénic Mach number MA or ablation beta βab, as shown in Fig. 6. Over the range of our simulations, we find the onset formation time t11ωci01, the piston separation time t22.5ωci01, and the downstream development time t35ωci01. Similarly, the corresponding locations z are only weakly sensitive to MA or βab, with z11ρi0,z22.5ρi0, and z356ρi0. The formation time is also not sensitive to the duration of the heating operator, as long as heating is applied for at least 1ωci01. This is because the shock is driven by the interaction of the piston and ambient ions, which primarily occurs near the leading edge of the piston plasma. Behind this edge, all the ambient ions and magnetic flux are swept out, and so there is no further interaction.

FIG. 6.

Shock onset time t1 and location z1, piston separation time t2 and location z2, and downstream development time t3 and location z3 as discussed in the text. The times and locations are plotted vs Alfvénic Mach number MA and ablation beta βab. The formation time is relative to the upstream gyroperiod ωci01, and the formation location is relative to the upstream directed gyroradius ρi0=vp/ωci0 for ions moving at the piston speed vp.

FIG. 6.

Shock onset time t1 and location z1, piston separation time t2 and location z2, and downstream development time t3 and location z3 as discussed in the text. The times and locations are plotted vs Alfvénic Mach number MA and ablation beta βab. The formation time is relative to the upstream gyroperiod ωci01, and the formation location is relative to the upstream directed gyroradius ρi0=vp/ωci0 for ions moving at the piston speed vp.

Close modal

It is worth emphasizing that while a shock satisfies all the criteria outlined above at time t2, it does not satisfy the RH jump conditions until it has sufficiently separated from the piston for a well-defined downstream region to develop (t3, see Sec. III C 2). At intermediate times such as t2, the newly formed shock only exists on kinetic scales that are not captured by the RH conditions and therefore may not exhibit shock physics, such as particle heating or energization, which is seen in shocks at later times.

1. Early time evolution

It is of interest to determine what characteristic features are associated with the formation of a magnetized collisionless shock by an ablation-driven piston, with an eye toward experimental observables. In Sec. III B, we discussed the criteria necessary to establish that a piston-driven shock has formed. In this section, we examine in more detail the evolution and dynamics of the piston and ambient plasmas within the first few upstream gyroperiods after ablation.

The basic dynamics of piston-driven shock formation are illustrated in Fig. 7 for the run in Table I. The figure shows five timesteps leading to the development of a shock precursor and the onset of shock formation. The first and second rows of panels are ambient and piston ion phase space plots, respectively, for the z velocity component relative to the shock speed vsh. The third row consists of the electron phase space, along with profiles of the magnetic field (black) and the total electron density (red). At early times, the ablated piston plasma expands into the magnetized ambient plasma and acts like a “snowplow,” sweeping up ambient ions and magnetic flux (region I in Fig. 7). The coupling between the piston and ambient plasma is primarily through the in-plane pressure-gradient (ambipolar) electric field Ez in the middle of the piston plume, seen in the first time step in Figs. 8(b) and 8(c). There is also a comparable-magnitude Ey field due to the streaming piston ions, which leads to complementary “Larmor” coupling.12 The sweeping up of magnetic flux generates the corresponding magnetic cavity,38 the edge of which is associated with strong density gradients and electron heating (since the electrons are collisional in these simulations, the electron temperature remains isotropic at all times). Following the discussion in Sec. III B, also note that in this region, some of the ambient ions are accelerated to large speeds. If not considered carefully, this feature (along with the field compression and heating) may appear to indicate shock formation if observed in experiments, whereas, in fact, these ions are piston-accelerated, not shock-reflected.

FIG. 7.

Series of timesteps showing initial shock formation by an ablation-driven magnetized piston for the run in Table I. 1st row: Ambient ion v-z space, where the velocity is relative to the shock speed vsh. 2nd row: Piston ion v-z space. 3rd row: Electron v-z space, along with the magnetic field Bx (black) and the total (red) electron density ne relative to their upstream values. Regions of interest are marked with Roman numerals or shaded and are discussed in the text.

FIG. 7.

Series of timesteps showing initial shock formation by an ablation-driven magnetized piston for the run in Table I. 1st row: Ambient ion v-z space, where the velocity is relative to the shock speed vsh. 2nd row: Piston ion v-z space. 3rd row: Electron v-z space, along with the magnetic field Bx (black) and the total (red) electron density ne relative to their upstream values. Regions of interest are marked with Roman numerals or shaded and are discussed in the text.

Close modal
FIG. 8.

Temperatures and electric field components at two timesteps for the run in Table I. Also shown for the fields are contributions from terms in the generalized Ohm's law. 1st row: Total electron and ion temperature profiles relative to their upstream values, taken over both piston and ambient populations. For each species, the temperature is divided into the field-perpendicular component (T) and field-parallel component (T||). 2nd row: Total out-of-plane electric field Ey (black) and the contribution to Ey from ui×B0 (red), where ui is the mass-weighted ion flow speed. 3rd row: Total electric field Ez (black) and the contributions to Ez from ui×B0 (red), J×B0 (green), and ·Pe (blue), where J is the total current density and Pe is the electron pressure. The fields are normalized to E0=vpB0.

FIG. 8.

Temperatures and electric field components at two timesteps for the run in Table I. Also shown for the fields are contributions from terms in the generalized Ohm's law. 1st row: Total electron and ion temperature profiles relative to their upstream values, taken over both piston and ambient populations. For each species, the temperature is divided into the field-perpendicular component (T) and field-parallel component (T||). 2nd row: Total out-of-plane electric field Ey (black) and the contribution to Ey from ui×B0 (red), where ui is the mass-weighted ion flow speed. 3rd row: Total electric field Ez (black) and the contributions to Ez from ui×B0 (red), J×B0 (green), and ·Pe (blue), where J is the total current density and Pe is the electron pressure. The fields are normalized to E0=vpB0.

Close modal

The first signs of shock formation are visible as deformations in the piston and ambient ion flows, as shown in region II. As the piston-accelerated ambient ions gyrate back into the +z direction, they form a secondary density compression approximately one gyroradius farther upstream. This creates another potential ramp that then starts the secondary acceleration of upstream ambient ions, i.e., the beginning of ambient-ambient ion interaction. The deformation of the ambient ion flow also causes a further compression of the magnetic field (>4× the upstream value), which, in turn, begins to deform the piston flow as piston ions pileup behind the compressed field (care must be taken not to confuse this magnetic compression with the magnetic overshoot of a shock). A population of fast ions, remnants of the piston-driven ambient ions that have not gyrated, leads to the appearance of strong perpendicular ion heating ahead of the field and density compressions (again, care must be taken to not confuse these ions with shock-reflected ions). This process continues in the third panel (t·ωci010.75), with additional acceleration of upstream ambient ions and larger field and density compressions.

Region III demonstrates the full onset of shock formation per the criteria outlined in Sec. III B. The population of ions moving faster than the shock speed now represent the magnetically reflected ions (in the upstream reference frame), and the shock precursor begins to separate from the piston, which takes the form of distinct localized density and field compressions and ion heating for the ambient and piston ion populations. As seen in the second time step in Figs. 8(b) and 8(c), the electric fields are now dominated by contributions from the moving ions, with a large potential ramp associated with gyrating reflected ambient ions in a proto-foot region ahead of the shock layer. A significant Larmor electric field continues to be associated with the piston ions.

The piston-shock separation becomes clear by the last time step in Fig. 7, with distinct regions for the forming shock (IV) and the piston (V). Region V also includes a large population of ambient ions that constitute a proto-downstream region although, at this stage, the density is dominated by the piston. While there are still density and field compressions (and some ion heating) associated with the piston, electron heating is only associated with the developing shock, and the strongest ion heating is observed in the region of reflected ions [see Fig. 8(a)]. Note that the parallel and perpendicular ion temperatures, and ion and electron temperatures, do not equilibrate until well within the piston region. This is largely why a downstream shock region is difficult to define and why the RH conditions are not yet applicable.

The piston-ambient ion-ion interactions are highly collisionless due to their large flow speeds. In contrast, the upstream electrons are modestly collisional at initialization (ωce,0/νei,01), and on ion timescales, they remain so throughout the simulation (ωci/νei<1). The beginning of shock formation is evidenced by a localized broadening of the electron velocity distribution (see, for example, region II), ahead of which exists the broad proto-foot region. The steepening of the shock layer results in distinct peaks in the electron distribution, which appear at early times largely due to the pileup of piston ions and are readily apparent at later times as two sets of peaks when the shock separates from the piston (regions IV and V).

Figure 9 compares shock formation for five simulations under different conditions. The first three panels show different Alfvénic Mach number expansions at the same dimensionless time step t·ωci01.5. In dimensionless units, the structures for all three are qualitatively the same although the shock-piston separation is most pronounced at lower Mach numbers. The last two panels represent simulations without a background magnetic field (B0=0) or without an ambient plasma (ne0=0) and correspond to timesteps that are equivalent (relative to the ablation time tab) to the first panel. Without a background magnetic field, the piston plasma is still able to initially accelerate ambient ions, but no secondary acceleration process takes place and no local density compression forms. Without any magnetic flux to sweep up, there is no magnetic compression, and there is no strong ion heating. Without an ambient plasma, there are, of course, no ambient ion structures, nor any appreciable density or magnetic field compressions.

FIG. 9.

Comparison of runs with different Alfvén Mach numbers MA, as well as cases with no upstream magnetic field or ambient density. The first three columns are taken at time t·ωci01.5, and z is relative to the upstream directed gyroradius ρi0=vp/ωci0. The last two columns are taken at a time step equivalent to the first panel, and z is relative to the ablation ion inertial length di,ab. Top row: Ambient ion v-z space for the first four columns, piston ion v-z space for the last column, where the velocity is relative to the piston velocity vp. Bottom row: Ambient electron v-z space for the first four columns, piston electron v-z space for the last column. Also shown are the magnetic field (black) and total electron density (red) profiles relative to their upstream values in the first three columns and relative to their ablation values in the last two columns.

FIG. 9.

Comparison of runs with different Alfvén Mach numbers MA, as well as cases with no upstream magnetic field or ambient density. The first three columns are taken at time t·ωci01.5, and z is relative to the upstream directed gyroradius ρi0=vp/ωci0. The last two columns are taken at a time step equivalent to the first panel, and z is relative to the ablation ion inertial length di,ab. Top row: Ambient ion v-z space for the first four columns, piston ion v-z space for the last column, where the velocity is relative to the piston velocity vp. Bottom row: Ambient electron v-z space for the first four columns, piston electron v-z space for the last column. Also shown are the magnetic field (black) and total electron density (red) profiles relative to their upstream values in the first three columns and relative to their ablation values in the last two columns.

Close modal

Soon after the shock begins to form (t·ωci01), it separates from the piston (since vsh>vp), as shown in Fig. 10 for the run in Table I. While there are small density and magnetic field perturbations associated with a proto-foot region where reflected ions overlap with upstream ions, initially most of the shock structure is contained in large-amplitude density and magnetic compressions at the shock layer. However, the proto-foot is sufficient to almost immediately begin a process of cyclic reformation,39 in which the shock front reassembles farther upstream approximately every 1.5 ωci01. This can be seen in the first three panels of Fig. 10, where the field and density compressions temporarily relax around t·ωci02 as the foot region expands, before reappearing at t·ωci02.5 as the shock layer re-steepens. The process then begins again.

FIG. 10.

Evolution of the shock as it separates from the piston for the run in Table I. For each time step, the ambient ion v-z space (where v is relative to the ablation sound speed Cs,ab) is shown, along with profiles of the magnetic field Bx (black) and total electron density ne (red) relative to their upstream values.

FIG. 10.

Evolution of the shock as it separates from the piston for the run in Table I. For each time step, the ambient ion v-z space (where v is relative to the ablation sound speed Cs,ab) is shown, along with profiles of the magnetic field Bx (black) and total electron density ne (red) relative to their upstream values.

Close modal

Once the shock separates from the piston, a downstream region begins to develop, which is shown in Fig. 11 for two different Mach number runs. This separation process takes several additional gyroperiods (5ωci01) to reach a state where the downstream is well-defined, such that the RH jump conditions can be applied. This can be seen in the last panel of Fig. 11(a), where the density and magnetic field compressions in the downstream have relaxed to values predicted by the RH conditions by 10 ωci01, while exhibiting overshoots at the shock layer. At this stage, the shock is well-separated from and dynamically independent of the piston, and its continuous evolution and characteristics are similar to those previously observed in many simulations by others.

FIG. 11.

Late-time shock evolution for two runs with (a) MA=7.1 and (b) MA=3.2. For each time step, profiles of the electron density ne (black), magnetic field Bx (red), and perpendicular ion temperature Ti (green) are shown, relative to their upstream values. Also shown in the last time step is the compression ratio (gray) predicted by the RH conditions for each MA.

FIG. 11.

Late-time shock evolution for two runs with (a) MA=7.1 and (b) MA=3.2. For each time step, profiles of the electron density ne (black), magnetic field Bx (red), and perpendicular ion temperature Ti (green) are shown, relative to their upstream values. Also shown in the last time step is the compression ratio (gray) predicted by the RH conditions for each MA.

Close modal

Laboratory experiments often utilize multi-species targets, such as plastic (CH), which result in multi-species plasmas. The addition of multiple ion species to the simulations tends not to change the overall results seen for single-species runs, but the observable features are weighted toward the dynamics of the more massive element. This is illustrated in Fig. 12, which shows a series of timesteps for a multi-species CH quasi-1D run. Here, the timesteps are in units of the upstream gyroperiod ωci01 evaluated for the average ion mass ratio. The first two rows show ambient H and C ion phase space plots, respectively, and the third row shows the ambient electron phase space along with the magnetic field (black) and total electron density (red) profiles. The fourth row displays the profiles of the total ion density (black) and its constituent parts contributed by the piston and ambient components for each species. Likewise, the fifth row shows the total perpendicular ion temperature (black) and the individual perpendicular H (red) and C (green) ion temperatures.

FIG. 12.

Series of timesteps showing initial shock formation for a run with multi-species CH plasmas. 1st row: Ambient H ion v-z space, where the velocity is relative to the shock speed vsh. 2nd row: Ambient C ion v-z space. 3rd row: Ambient electron v-z space with profiles of the magnetic field (black) and total electron density (red) relative to their upstream values. 4th row: Total (black) ion density ni, ambient (red) and piston (green) H ion densities, and ambient (blue) and piston (purple) C ion densities, all relative to their upstream values. 5th row: Perpendicular electron temperature (black) and contributions to the perpendicular ion temperature from the H (red) and C (green) ions, all relative to their upstream values. Regions of interest are marked with Roman numerals and discussed in the text.

FIG. 12.

Series of timesteps showing initial shock formation for a run with multi-species CH plasmas. 1st row: Ambient H ion v-z space, where the velocity is relative to the shock speed vsh. 2nd row: Ambient C ion v-z space. 3rd row: Ambient electron v-z space with profiles of the magnetic field (black) and total electron density (red) relative to their upstream values. 4th row: Total (black) ion density ni, ambient (red) and piston (green) H ion densities, and ambient (blue) and piston (purple) C ion densities, all relative to their upstream values. 5th row: Perpendicular electron temperature (black) and contributions to the perpendicular ion temperature from the H (red) and C (green) ions, all relative to their upstream values. Regions of interest are marked with Roman numerals and discussed in the text.

Close modal

Since the ions are effectively collisionless, each species evolves largely independently. As in the single species runs, shocks begin to form for each species at approximately one upstream gyroperiod, where the gyroperiod is evaluated for that specific species. So the ambient H ions begin to form a shock first 1ωci0,H10.5ωci01 after ablation (region I in Fig. 12), followed by the C ions at 1ωci0,C1ωci01 (region II). The distinct structures persist as they separate from the piston, as seen in regions III and IV in the last panel of Fig. 12. While shocks form for both species, the magnetic field and electron density profiles are primarily dictated by the C ions since the C ions form more relatively localized structures due to their smaller ion inertial lengths. As a result, in experiments, it would be difficult, for example, to discern whether the H shock has formed from these profiles alone.

Unlike the HED plasmas modeled here, astrophysical plasmas are highly collisionless for both the electron and ion populations. However, the lack of collisionality does not appear to significantly impact piston-driven shock formation. This is shown in Fig. 13, which compares identical simulations with (first column) and without (second column) collisions at the onset of shock formation (t·ωci0=1.1). The parameters for the case with collisions are given in Table I, where the electron collisionality λab20 is matched to typical HED experiments, although this implies that the ion collisional mean free path is overestimated by a factor of μphysical/μsim4 for H plasmas. The first two rows show the total (piston and ambient) ion and electron phase space plots, respectively. Magnetic field (black) and electron density (red) profiles are also shown in the second row. The parallel and perpendicular components of the electron and ion temperatures are shown in the third row, and perpendicular components of the electric field E are shown in the fourth row.

FIG. 13.

Comparison of shock formation with and without collisions for the run in Table I and its collisionless equivalent (i.e., λab) at t·ωci0=1.1. 1st row: Total (piston and ambient) ion v-z space, where the velocity is relative to the shock speed vsh. 2nd Row: Total electron v-z space with profiles of the magnetic field (black) and total electron density (red) relative to their upstream values. 3rd row: Parallel and perpendicular components of the electron and ion temperatures relative to their upstream values. 4th row: Perpendicular components of the electric field relative to E0=vpB0.

FIG. 13.

Comparison of shock formation with and without collisions for the run in Table I and its collisionless equivalent (i.e., λab) at t·ωci0=1.1. 1st row: Total (piston and ambient) ion v-z space, where the velocity is relative to the shock speed vsh. 2nd Row: Total electron v-z space with profiles of the magnetic field (black) and total electron density (red) relative to their upstream values. 3rd row: Parallel and perpendicular components of the electron and ion temperatures relative to their upstream values. 4th row: Perpendicular components of the electric field relative to E0=vpB0.

Close modal

As might be expected, the addition of collisions smears out phase space structures smaller than di0 compared to the collisionless case. There is also more collisional electron heating of the ions in the piston plasma behind the shock. But the main features discussed earlier are still present in both cases since the piston-ambient and ambient-ambient interactions between relative streaming populations of ions remain strongly collisionless due to the high flow speeds (λmfp/di01, where λmfp is the mean free path of ions moving at vsh), and the electrons do not play a significant role, at this stage, in shock formation. Note that while, in this simulation, we overestimate λmfp/di0 by a factor of 4 due to the reduced mass ratio, we find that the observed collisionless ion-ion interactions are robust to changes in the mass ratio and hence effective collisionality; this is illustrated in Sec. III F. In both the collisional and collisionless cases, a similar pileup of piston ions and separation of the ambient-mediated shock are seen although the separation is more distinct in the collisionless case as reflected in the magnetic field and electron density profiles. Larger differences are seen in the temperature profiles, where with collisions, there is significantly more perpendicular ion heating ahead of the newly formed shock (although the heating is similar at the edge where the piston ions pileup). Without collisions, the electrons are preferentially heated in the perpendicular direction through the shock, before equilibrating with the parallel temperature in the piston-dominated plasma. In contrast, with collisions, the electrons retain an isotropic temperature throughout and exhibit strong collisional heating ahead of the shock due to the counterstreaming ions.40 In neither case is there appreciable parallel ion heating near the shock layer. The perpendicular electric fields are also similar in both cases, with the same structure as discussed in Fig. 8.

To investigate the effect of the mass ratio on shock formation in our simulations, we performed two identical runs (shown in Table I) with mass ratio μ1=mi/me=100 and μ2=400, shown in Fig. 14 at time t·ωci0=1.1. Overall, the results are comparable, indicating that the shock formation physics, especially for ions, has already converged. The ion phase space and peak density and magnetic field compressions at the leading edge are all very similar between the two cases. Following the discussion in Sec. III B, the shock formation onset time and location are also similar for both cases: t1=1.1ωci01 and z1=1.1ρi0 for μ1 and t1=1.1ωci01 and z1=1.2ρi0 for μ2. This results in part because the electric fields near the compression front are comparable (see the 4th row of Fig. 14), which accelerates similar populations of reflected ions. The electrons, which are more sensitive to changes in the effective collisionality, show larger overall heating at larger mass ratios although this is mostly in the piston-dominated region behind the shock. Ion heating is more similar for both cases, with a slightly lower peak ion temperature at the leading edge and more upstream heating for the larger mass ratio. The comparable ion heating and similar shock structures at these two mass ratios further indicate that ion-ion collisions do not play a significant role in shock formation, as discussed in Sec. III E.

FIG. 14.

Comparison of shock formation for two different mass ratios at t·ωci0=1.1 for the run in Table I. 1st row: Total (piston and ambient) ion v-z space, where the velocity is relative to the shock speed vsh. 2nd row: Total electron v-z space with profiles of the magnetic field (black) and total electron density (red) relative to their upstream values. 3rd row: Parallel and perpendicular components of the electron and ion temperatures relative to their upstream values. 4th row: Perpendicular components of the electric field relative to E0=vpB0.

FIG. 14.

Comparison of shock formation for two different mass ratios at t·ωci0=1.1 for the run in Table I. 1st row: Total (piston and ambient) ion v-z space, where the velocity is relative to the shock speed vsh. 2nd row: Total electron v-z space with profiles of the magnetic field (black) and total electron density (red) relative to their upstream values. 3rd row: Parallel and perpendicular components of the electron and ion temperatures relative to their upstream values. 4th row: Perpendicular components of the electric field relative to E0=vpB0.

Close modal

In this paper, we have detailed a new class of PIC simulations and their application to piston-driven, magnetized collisionless shock formation. The simulations are scaled to dimensionless ion parameters and include a model of laser ablation, making them well-suited to studies of laser-driven HED plasmas and well-matched to experimental conditions. Multiple quasi-1D runs were performed, covering a range of Alfvénic Mach numbers, upstream beta, and ablation beta. The goal is to provide robust signatures of collisionless shock formation, and potential caveats to laboratory measurements, in piston-driven experiments, where traditional characterizations of shocks, such as the RH jump conditions, are not easily applied.

The simulation results indicate that in uniform ambient conditions, there are three key timescales in the evolution of shock formation. The first, the onset of shock formation, occurs approximately one upstream gyroperiod ωci01 after the piston begins expanding and about one directed gyroradius ρi0=vp/ωci0 from the piston target. Note that while some compression is necessary, large-amplitude density and magnetic field compressions and particle heating are unreliable indicators of shock formation at these times since these can appear as the piston sweeps up ambient plasma well before the shock has fully formed. The second key time, the clear separation of the shock from the piston, occurs at approximately 2.5ωci01 once the shock precursor separates a distance 1/4ρi0 from the piston. Only by the third time at approximately 5ωci01 has the shock sufficiently separated from the piston (ρi0) for a downstream region to develop, consistent with the RH jump conditions.

The most straightforward method to determine shock formation is to observe the ion phase space. The presence of a reflected ion population is a key signature of collisionless shocks, and by the time of shock onset, these reflected ions are well distinguished in phase space from piston and piston-accelerated ambient ions (see Fig. 7). In typical laboratory HED plasmas, this can be measured, for example, by collective optical Thomson scattering.20 A representative ion phase space plot near the shock onset time is shown in Fig. 15. Compared to unmagnetized or piston-only plasmas, the interaction between the piston and ambient ions will drive significant flow deformations as upstream (region I) ambient ions are accelerated (region II) and the piston ions pileup (region III). The formation of a shock will then result in a population of magnetically reflected ambient ions (region V) although care must be taken to both distinguish magnetically reflected ambient ions from piston-accelerated ions (region VI) and distinguish the piston-ambient interface (region VII) from the forming shock layer (region IV). In the absence of phase space measurements, the key signature of shock formation is the separation of the steep (ion-scale) density and magnetic compressions of the shock from the compressions associated with the piston.

FIG. 15.

Representative total (piston and ambient) ion v-z space showing key signatures of piston-driven shock formation. The red contour represents 20% of the maximum value of the ambient ion velocity distribution. Regions of interest are labeled with Roman numerals and discussed in the text.

FIG. 15.

Representative total (piston and ambient) ion v-z space showing key signatures of piston-driven shock formation. The red contour represents 20% of the maximum value of the ambient ion velocity distribution. Regions of interest are labeled with Roman numerals and discussed in the text.

Close modal

Due to the space and time scaling discussed above, shock formation is more advantageous at higher charge to mass ratio Z/m for fixed spatial or temporal boundaries since the shock will evolve more quickly and closer to the target. Similarly, for a given Z/m, slower shocks (lower MA) are favorable as the shock will form closer to the target. These results continue to hold in mixed-species plasmas, with each species evolving on scales commensurate with its own Z/m, although the field, density, and temperature profiles are dominated by the species with the largest m/Z.

Even though collisions are enabled, the ion-ion interactions remain dominantly collisionless in the simulations due to the large flow speeds. In contrast, the electrons are modestly collisional and maintain robust Maxwellian velocity distributions at all times. While these collisions destroy many of the small-scale electron phase space structures seen in collisionless simulations, the effect on shock formation appears to be muted in our simulations.

Finally, we note that our quasi-1D simulations only capture a subset of potentially interesting effects that could be seen in shock experiments. Laser-driven plasmas expand approximately hemispherically and could thus drive shocks over a range of non-perpendicular magnetic geometries. Similar 2D effects have been shown to self-generate magnetic fields,31 which can further affect the shock structure and dynamics. At sufficiently high Mach numbers, electromagnetic instabilities can affect the shock, as observed in recent 3D simulations.41 We have also not considered here the effect of non-uniform background magnetic fields or ambient densities, which may be common in experiments. These topics, including a more detailed study of the effect of collisionality on shock formation, will be explored in future work.

The PIC simulations were conducted on the Titan supercomputer at the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, supported by the Office of Science of the DOE under Contract No. DE-AC05-00OR22725. Time at the Omega facility was provided by the Department of Energy (DOE) through Grant No. DE-NA0003613. This research was also supported by the DOE under Grant Nos. DE-SC0008655 and DE-SC0016249 and by NASA under Grant No. 80NSSC19K0493.

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