Numerical simulation of the plasma acceleration process in a magnetically enhanced micro-cathode vacuum arc thruster

Jinyue GENG (耿金越),Yongcai CHEN (陈永财),Surong SUN (孙素蓉),3,Wendong HUANG (黄文栋) and Haixing WANG (王海兴),3

1 Beijing Institute of Control Engineering,Beijing 100094,People’s Republic of China

2 School of Astronautics,Beijing University of Aeronautics and Astronautics,Beijing 100191,People’s Republic of China

3 Authors to whom any correspondence should be addressed.

Abstract

Keywords:micro-cathode vacuum arc thruster,magnetic field,ion acceleration

1.Introduction

Increased launch opportunities and the rapid development of technology have paved the way for small satellites to become a more valuable platform for the scientific space community to conduct research and for the support of space mission operations.However,a large number of micro satellites and networking satellites are currently not equipped with a propulsion system due to the limitations of mass and available power.Recently,the development of the micro-cathode vacuum arc thruster on micro-propulsion systems applied to small satellites has received extensive attention.The advantages of the micro-cathode vacuum arc thrusters are their simple structure,light weight,low power consumption,and high performance.Therefore,it has become an excellent candidate for remedying the lack of propulsion in these small satellite systems [1].

As presented in figure 1,in general,the coaxial microcathode vacuum arc thruster is composed of a tubular ring geometry with an anode and cathode separated by an insulator[2].The main operating principle of this kind of thruster is to generate almost completely ionized metal vapor plasma on the small region of the cathode surface(cathode spot)through arc discharge,which expands to a vacuum at a very high speed,with the ions producing thrust.It has been reported that the density of plasma and the velocity of ions next to the cathode spots can respectively reach values of 1026m−3and 104ms−1[3-5].For the case with magnesium as the cathode material,the highest plasma velocity reported in the literature can reach 50 000 ms−1,which is very attractive for plasma propulsion [3].Therefore,the main reason to extend the application of the vacuum arc from the traditional application fields such as the vacuum arc disposition,remelting,etc[4,5]to the electric propulsion field is that the metal vapor emitted from the cathode spot has a quite high kinetic energy.

Figure 1.Schematic diagram of micro-thruster with coaxial electrode structure.

However,the metal vapor plasma jet formed at the cathode spots may expand in all directions.Previous studies have shown that for the case of arc currents less than 200 A,the self-induced electromagnetic field has little effect on the density,velocity,shape,and the distribution of the current density of the plasma jet[6].However,it has been shown that if an axial magnetic field can be applied outside the thruster head,it can significantly increase the axial ion velocity and suppress the plume divergence angle.This kind of external magnetic field thruster is called a magnetically enhanced vacuum arc thruster (MVAT).Keidar and co-workers conducted the first self-consistent two-dimensional numerical simulation of plasma generation and plasma flow in the presence of an applied magnetic field [6-9].The suppression of the radial velocity of the plasma jet by an external magnetic field was studied.It was shown that the applied magnetic field can significantly affect the plasma expansion process,especially downstream of the plasma jet.The shape of the plasma jet depends largely on the distribution of the applied magnetic field.Compared with the case of no applied magnetic field,when the applied magnetic field is 0.3 T,experimental studies have shown that the average ion velocity in the plasma jet increased 3-4 times[10,11].From the perspective of thruster design,a more detailed understanding of the acceleration mechanism of the plasma jet in a magnetic field and its dependence on magnetic field strength and configuration is required.Although there have been intensive studies[6,12-22]on the expansion process of the plasma jet,due to strongly coupling between the vacuum arc and the magnetic field structure,our understanding of its mechanism is not complete.

Figure 2.Sketch and computation domain of the micro-cathode vacuum arc thruster.

It is worth noting that most of the previous simulation studies on plasma jet expansion with applied magnetic field are based on the hydrodynamic method.In fluid models,both electrons and ions are described as fluids and usually the quasineutrality assumption is adopted.This kind of assumption can significantly reduce the amount of calculation required for numerical simulation.However,for a higher degree of ionization of the vacuum arc,the effects of an applied magnetic field on electrons and ions are quite different due to the mass difference.Due to small mass and Larmor radius,electrons are expected to be magnetized easily.While for the ion with a larger mass and Larmor radius,its movement is more complicated due to the complex interaction of the electron and the magnetic field.Therefore,a fully kinetic particle-in-cell (PIC) is a suitable tool to simulate the evolution,development,and acceleration of particles in the presence of an external magnetic field.

The paper is organized as follows.In section 2,the numerical approach and the computational domain are presented.In section 3,the results of the numerical simulation of plasma acceleration under the applied magnetic field and the corresponding analysis and explanation are presented.In section 4,an analysis and a discussion are presented followed by the conclusions obtained in this study in section 5.

2.Modeling approach

2.1.Computation and boundary conditions

The geometry and computational domain of the micro-cathode arc thruster used in this study are shown in figure 2.A simplified structure of a vacuum arc thruster with coaxial electrodes is located at the left-hand side of the computation domain.The geometry and size of the thruster are taken from[10,23].As shown in figure 2,the anode is located on the axis of the calculation domain with a radius of 1 mm.The titanium cathode is arranged in an annular area separated from the anode by an insulating material with a thickness of 1 mm.The thickness of the housing shell is 2 mm.In the outermost circle,the thickness HI of the magnetic coil is 20 mm.As shown in figure 2,the size of the numerical simulation domain used in this study is set to be 0.08 ×0.04 m2,where AD is the axis of the symmetry.The cathode potential and anode potential are set to be zero and 50 V,respectively.The other boundaries except the axis of the symmetry AD in the calculation domain are taken as Neumann boundaries.

Metal vapor forms at the location of the cathode arc spot with strong dynamic characteristics on the order of micrometers,which has an extremely high current density,power density,and plasma density.In a real case,it was observed that the cathode spot may form at any position on the interface between the cathode and the insulating material,and rotates continuously during the discharge cycle [24-26].It is not an axisymmetric process in reality.However,for the case with the applied magnetic field,the cathode spot is in a rotating motion during the entire discharge process.The diffusion of the plasma will also cover the entire thruster channel area,so the discharge process can be simplified to a twodimensional axisymmetric process.Due to the extremely small size of the cathode arc spot and its highly unsteady state,theoretical analysis and measurements are quite difficult.Therefore,in this simulation,we do not consider the complex physical process at the cathode spot.We assumed that the metal vapor plasma emitted at the cathode spot enters the calculation domain from the entrance of the circular ring with a width of 1 mm on the cathode surface.

It has been reported that the average charge of titanium cathode ionization is 2.1 [15].Hence,in this simulation,the injected titanium particles are assumed to be secondary ionized ions.At the same time,the proportion of neutral atoms in the metal vapor emitted from the cathode surface is less than 1%,and the degree of ionization of the plasma is as high as 99%[27],so the plasma is assumed to approximately be a fully ionized plasma containing only electrons and doubly ionized titanium ions.According to reference[23],the electrons which are emitted from the titanium cathode spots usually have an initial kinetic energy of 2-3.2 eV.The initial velocity of ions emitted from the surface of the titanium cathode usually does not change with the applied magnetic field and remains constant.Therefore,in this study,it is assumed that both the electrons and ions obey the Maxwell distribution and the electron has an initial kinetic energy of 2 eV,while titanium ions have an initial velocity of 2×104ms−1[5,27,28],which is independent of the applied magnetic field.

2.2.Self-similarity method

In the PIC simulation,if the spatial scale is greater than the Debye length,the particle motion even on a fine scale cannot be accurately simulated due to the electrostatic shielding effect.Also,a large spatial scale may even cause a numerical self-heating phenomenon,leading to the calculation results being completely distorted.Therefore,the selection of the size of the calculation domain,grid scale,and time step are all related to the characteristics of the plasma.Based on the parameters of the plasma at the entrance,the estimated Debye length is approximately 10−5m [29,30].Moreover,duringthe expansion of the metal vapor plasma jet,the mass of the electrons is much smaller than that of the ions,and the velocity is higher than that of the ions,which will cause a charge separation.Charge separation and electron cyclotrons are both important forms of particle motion.The selection of the time step needs to be able to accurately resolve these processes.It is estimated that a time step of 10−12s is chosen to be able to distinguish the timescale of the physical process.According to this estimation,the number of grids in the calculation domain should be in the order of tens of millions and it will also take tens of millions of time step iterations.This means that the full-scale PIC simulation of the thruster requires a high computational cost.

Table 1.Summary of the scaling laws.

In order to reduce the amount of numerical calculations while maintaining the fully kinetic character of the model,the size of the calculation domain in this study is scaled down using a self-similar method as reported in [31-36].The selfsimilar method was inspired by the similar design method often used in thruster design.Different from the method of changing the vacuum permittivity constant and the method of reducing heavy particle mass,the self-similar method can ensure that the important physical parameters and performance parameters remain unchanged and the physical processes are similar.This method can reduce the size of the computational domain.That is,the thruster size is reduced while keeping the self-similarity criterion number constant,and this can effectively decrease the time steps and the number of spatial grids.

In order to simulate the realistic physical processes of a micro-cathode vacuum arc thruster,important parameters and physical processes such as the specific impulse,the charge number density,and the ability of the binding of the applied magnetic field to the electrons should remain unchanged when the self-similar method is applied.According to this scaling rule,the characteristic length of the MVAT is reduced toξ(0 ＜ξ＜1) times of the original size,the related parameters such as potential,ion mass flow,current density,ion beam current,electric field,magnetic induction strength,thrust,specific impulse,and the relationship between the original physical parameters are summarized in table 1.

A well-reasoned selection of the scaling ratio is very important,because an inappropriate zoom ratio will lead to unrealistic physical results.The choice of the scaling ratio should conform to the following principles.First,the frequency of the plasma generated by the micro-arc thruster should be less than the frequency of the electron cyclotron.Second,the Debye length of the system should be a small amount compared to the entire calculation of the domain scale.If the scale of the system sheath is too large,it will affect the physical process of the plasma bulk region.Under these constraints,the scaling ratio of the system should be greater than 0.02[31,32].In this study,the sensitivity of this parameter to the simulation result has been checked with ζ=0.02,0.1.It was found that the numerical simulation results are very similar.In order to reduce the computation time,ζ=0.02 is also used in this numerical simulation,and then an orthogonal uniform grid of 320×160 can be derived and used in this study.

Figure 3.Distribution of the applied magnetic field for the case of 0.3 T.

The distribution of the applied magnetic field can be obtained from the coil structure and current.In this simulation,the geometry and parameters of the applied magnetic filed are the same as those in reference [10].The distribution of the magnetic field presented in figure 3 corresponds to the case of an applied magnetic field of 0.3 T.In order to facilitate a comparison with a subsequent simulation,the relative positions of the magnetic field coil and thruster are given in figure 3,while the range marked by the red-dashed line in figure 3 corresponds the adopted computational domain.It is noted that the magnetic field strength shown in the computational domain is lower than 0.3 T.This is because the maximum value of the magnetic field strength located at the center of the magnetic coil is not included in this computational domain.In order to be consistent with the experimental study,we refer to it as the case of 0.3 T.

3.Results

3.1.Code validation

The PIC code used in this numerical simulation was developed by our research group,so the verification of the code is a prerequisite for further numerical simulation.Firstly,we take a one-dimensional direct current sheath model [37]as a benchmark problem to check this code.Good agreement with available predictions or the reported results in [37]has been obtained;thus the feasibility of the code has been preliminarily demonstrated.

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Figure 4.Comparison between the predicted and measured axial distribution of electron temperature,plasma density,and mean ion velocity for a magnetically expanding plasma.

We then further validated the code by simulating the plasma acceleration process in a magnetic nozzle.In reference[38],the plasma flow expansion and detachment process of the plasma generated by the radio frequency (RF) plasma source were experimentally studied in the magnetic nozzle.In the experiment in [38],the plasma number density n and electron temperature Tewere measured by a swept RF-compensated Langmuir probe.The ion energy distribution function was measured by a four-grid retarding potential analyzer,and then the mean velocity of the ions can be obtained from the measured ion energy distribution function.The calculated and measured normalized electron temperature,number density,and ion velocity distributions along the axial direction are compared in figure 4.The result shows that the predicated distribution of the parameters is basically consistent with the experimental results.Therefore,it can be considered that the code can be used to simulate the plasma acceleration process in a micro-cathode vacuum arc thruster with an applied magnetic field.

In order to further verify the reliability of the numerical simulation approach and code used in this study,we compare the numerical results with the experimental results of microcathode vacuum arc thrusters reported in references [10,28].In their experiments,the average ratios of ion current over the arc current measured with different applied magnetic field strengths were presented and are taken as the input boundary condition in this study.A set of four-grid probes was used to measure the ion velocity of the micro-cathode vacuum arc plasma jet based on the time of flight method.Four-grid probes were placed 7.5,12.5,17.5,and 22.5 cm downstream of the cathode and isolator surfaces,respectively.

Figure 5.Comparison between the predicted and measured distribution of the ion velocity with different applied magnetic field strengths.

The distances of the four-grid probes from the cathode and isolator surface were 7.5.12.5,17.5,and 22.5 cm respectively.The velocity was derived from the time difference of the ion current initial peak between every two adjacent grid probes.Figure 5 presents the comparison between the predication and experimental results of the distribution of ion velocity with different magnetic field strengths.It can be seen that generally the predicted axial distribution of the ion speeds agrees well with the corresponding experimental results and this provides evidence of the reliability of the code.

As shown in figure 5,the experiments illustrate that adding an external magnetic field can indeed increase the axial velocity of ions.When the applied magnetic field is set to be 0.3 T,the particle speed is increased by three times compared with the case without an applied magnetic field.It is worth noting that for titanium as the cathode material,when there is no external magnetic field,the typical ion velocity is about 2×104ms−1.This velocity is also taken as the inlet velocity in this simulation.

3.2.Time evolution of the distribution of the electron and ion density

In order to investigate the acceleration process of the electrons and ions in the presence of an applied magnetic field,the magnetic field strength of 0.3 T is chosen to analyze the electron and ion motion processes,as shown in figure 6.Figure 6 presents the time evolution of electron and ion behavior at different time steps.It can be seen that the electron mass is lighter and it moves faster,so it can move to downstream of the calculation domain in a short time.Under the influence of the applied magnetic field,the electrons are captured by the magnetic field lines and make a spiral movement around the magnetic field line,and rotate with the Larmor radius.As a result of the spiral motion electrons,a high electron density band is formed,which can be called an‘electron channel’.Upstream of plasma inlet,the magnetic field lines are relatively dense and the corresponding magnetic field intensity is relatively large.The Larmor radius of the electrons is small which leads to the relative high density of the electrons.Downstream of the plasma jet,the distribution of the magnetic field lines is more divergent and the corresponding magnetic field strength is relatively small and thus the Larmor radius of the electron increases.Hence,the electron channel gradually becomes wider,and the electron number density decreases.As the ions move downstream,the electron channel also tends to approach the axis.The main reason is that the ions are mainly distributed near the axis and a large amount of positive charge is accumulated along the axis.The positively charged ions attract the electrons in the electron channel,so that the electrons move around the magnetic field line closer to the axis and this also widen the electron channel.

The average axial velocity of ions at different times along the axial direction is presented in figure 7.It is noted that at 0.50 μs,the velocity of injected ions can be increased to about 60 kms−1.With the increase of time,more ions are injected into the computational domain,and subsequent incoming ions are continuously accelerated,but the trend of acceleration gradually slows down.Although the following ions can also eventually accelerate to 60 kms−1,this acceleration process takes longer time and can be achieved until the downstream of the computing domain.

If we take a closer look at the acceleration process of ions with the evolution of electrons with the time shown in figure 6,it is found that in the initial stage of 0-0.5 μs,the charge separation phenomenon is relatively significant,especially at the upstream of computational domain,so it can be speculated that the electric field caused by charge separation has a relatively large gradient.Therefore,ions can be accelerated to very high speeds in a short period of time.As time increases,more ions enter the computation and weaken the charge separation phenomenon.The electric field caused by charge separation is extended downstream,so the acceleration process of ions takes a long time to achieve.Based on these analyses,it can be seen that the acceleration of heavy particles is closely related to the electric field caused by charge separation.

Figure 6.Distribution of the electron number density (left) and ion number density (right) at different times.

Figure 7.The axial distribution of the average ion velocity at different times.

The time evolution of the distribution of the electric potential is given in figure 8.In the initial stage,without the injection of electrons and ions,the electric potential in the computational domain is only formed by the voltage between the electrodes.With the injection of electrons,the difference in the electric potential in the computational domain increases rapidly,even exceeding the difference in the electric potential between the electrodes.It can be noted from figures 8(a)-(d)that the region of the electron channel formed by the magnetized electrons is accompanied by a significant electric potential drop.Especially near the center of the calculation domain,the axial electric potential gradient plays an important role in accelerating ions.As the ions move downstream,the electric potential gradient gradually decreases.This is because ions gradually keep up with the movement of the electrons,and the charge separation phenomenon gradually weakens,and this leads to the decrease of the axial electric potential gradient.

Figure 9 further presents the time evolution of the distribution of the electric field in the computational domain.As mentioned above,there is a high electric field strength upstream of the calculation domain in the initial stage which leads the ions to be accelerated to a very high speed in a short time,as shown in figure 7.As time increases,as the ions move downstream in the computational domain,the electric field caused by the charge separation also moves downstream.This infers that the acceleration process of ions is closely related to the evolution of the distribution of the electric field.

3.3.Effect of the axial distribution of the applied magnetic field on the plasma acceleration process

As given in figure 3,the applied magnetic field is usually installed at the exit of the thruster.The relative positions of the applied magnetic fields and thruster presented in figure 3 are consistent with the data in references [10,28],and is referred to as Case 1 as follows.Note that in figure 3,the coil end of the applied magnetic field coincides with the end of the thruster anode.For the case shown in figure 10,the applied magnetic coil is moved back 20 mm relative to the thruster anode,and is referred to as Case 2.Note that the strength of the magnetic field in Case 2 shown in figure 10,is close to 0.3 T near the upstream axis of the calculation domain,which is higher than the strength of the magnetic field at the corresponding position shown in figure 3.In order to save computational time,we take the range of the red-dashed line as the computational domain in figure 10.

Figure 11 shows the distribution of the electron and ion densities under the condition of Case 2 at a time of 1.5 μs.Compared with the distribution of the electron number density at the same time as in Case 1 as given in figure 6,it is found that,because the magnetic field strength is large near the upstream axis of the calculation domain,the electrons are magnetized almost immediately near the entrance.Magnetic fields have a stronger constraint on electrons.As a result,a narrower electron channel compared to that of Case 1 is formed in the computing domain.At the downstream of the computational domain,the magnetic field diverges,and the magnetized electron channel also deviates from the axis.

Due to the increase of the number density of electrons inside the narrow electron channel,the phenomenon of charge separation is more significant than that of Case 1.The electric field caused by the accumulation of negative charges is more obvious in the traction of ions,so the direction of ion movement is also closer to the electron channel.As shown in figure 11(b),at the downstream of the computing domain,as the magnetized electron channel deviates from the axis,the ion movement also gradually deviates from the axis.Compared with the corresponding cases of Case 1 shown in figure 6,the hollowing of the plume is more obvious [39].

Figure 12 compares the effects of two different distributions of applied magnetic field on the ion acceleration.It can be seen that compared with Case 1,the backward movement of the coil relative to the thruster has a significant effect on the acceleration of ions.The main reason is that the applied magnetic field has a strong binding effect on electrons,and the ions are subjected to the radial electric field caused by the charge separation when moving downstream.The radial component of the ion velocity increases,so the increase in the axial velocity is much lower than in Case 1.

4.Discussion

Figure 8.Time evolution of the distribution of the electric potential with the injection of electrons and ions at (a) 0.25 μs,(b) 0.50 μs,(c) 1.00 μs,and (d) 1.50 μs.

Figure 9.Time evolution of the distribution of the electric field strength at (a) 0.25 μs,(b) 0.50 μs,(c) 1.00 μs,and (d) 1.50 μs.

Figure 10.The position of the applied magnetic field relative to the thruster and the calculation domain used in this study.

The acceleration process and mechanism of the ions of the micro-cathode vacuum arc thruster are not clearly understood.Previous experiments and theoretical studies have found that ion flux has a relatively high energy,and the average potential of ions is much larger than the arc voltage [40].Based on these observations,the potential hump theory is proposed to explain the acceleration mechanism in a vacuum arc [41].This theory proposes that the electrons generated by the cathode spot diffuse in all directions.Because the diffusion speed of an electron is faster than that of an ion,a potential peak,or hump,is formed immediately downstream of the cathode surface.The existence of a potential hump has been verified by some experiments [42]and a recent simulation[43].Davis et al [42]further developed this theory.It was considered that neutral particles near the cathode spot are ionized by collision with electrons,resulting in a large number of ions.Therefore,the ion concentration rapidly increases near the cathode,which creates a potential maximum(hump)in this region.However,in a series studies by Beilis [44],it is pointed out that the formation of the potential hump is not the cause of the development and expansion of plasma,but the consequence of its formation.The development and expansion of plasma are caused by gasdynamic acceleration in which the enthalpy of the plasma jet converts into kinetic energy.The previous study [9]showed that the metal vapor generated by ablation near the cathode spot is accelerated by the aerodynamic force due to the high pressure.The ion energy depends on the ratio of the ion flux and electron flux and the ratio is further determined by the rate of the erosion of the cathode material [45].It is worth noting that these two theories use different assumptions.In potential hump theory,it is assumed that all ions are generated at the same potential,while in gas-dynamic acceleration theory,all ions are assumed to be generated with the same flow velocity [40].A further investigation based on a onedimensional hydrodynamic two-fluid model showed that there may be three mechanisms of ion acceleration,ion pressure gradient,electric field,and electron-ion friction.These three mechanisms roughly contribute the same magnitude to the final ion kinetic energy [46].

For the situation with the presence of an applied magnetic field,reference [24]pointed out that different applied magnetic field configurations and strengths may enhance or suppress ion acceleration.The acceleration mechanism of ions can be described by a hydrodynamic model.That is,during the plasma expansion process,the pressure of electrons and ions drops by many orders of magnitude.This process interacts with electrons and ions to drive the ions to accelerate.However,as pointed out in reference [2],the ion velocity emitted from the cathode spot is independent of the magnetic field,indicating that the particle acceleration process near the cathode spot is hardly affected by the strength of the applied magnetic fields.Therefore,according to this phenomenon,it has been pointed out that in the case of a micro-cathode arc thruster,in the presence of an external magnetic field,acceleration is caused by the Lorentz force acting on the ions in the field divergence region [2,27].

Figure 11.Distribution of(a)electron and(b)ion densities at 1.50 μs for Case 2.

In view of the state of the art as summarized above,this main question of ion acceleration with the presence of an applied magnetic field still remains partly open.As shown in this simulation,the electron is easily magnetized due to the small mass and Larmor radius.Therefore,after the electrons are injected into the flow field,they are bound by the magnetic field lines to form a narrow electron channel.The large mass and Larmor radius of ions lead to the movement of ions taking a long time to keep up with the electron movement.It can be inferred that at the downstream of the magnetically enhanced vacuum arc thruster,the divergent magnetic field strengthens the charge separation phenomenon.This electric field formed by charge separation is the main reason for the ion acceleration derived from this simulation.The charge separation phenomenon has also been found in recent PIC simulations of a vacuum arc thruster [30,35,36].It is noted that the magnetic field due to charge separation evolves over time,making measurement and verification very difficult.Fortunately,a new method for measuring electric fields with a high time resolution has recently been proposed[47].In this method,a rapidly changing electric field can be measured by the electric field induced second harmonic generated by a nanosecond pulse duration laser.It is expected that this method may be extended to measure the evolution of the electric field in magnetic enhanced vacuum arc thrusters and provide evidence for ion acceleration mechanism.

It is noted that there are still some limitations in this simulation and some assumptions are used.For example,at the inlet of the computational domain,the plasma is assumed to be completely ionized,and the physical processes near the cathode spot are completely ignored.The neutral particles are neglected,and inelastic collision processes such as charge exchanges,ionization,and recombination are neglected.In spite of these restrictions,this study still helps improve the understanding of the acceleration process of magnetic enhanced micro-cathode vacuum arc thrusters.

Figure 12.The axial distribution of the average axial velocity of ions for different magnetic distributions.

5.Conclusion

The effect of an applied magnetic field on the acceleration process of plasma in a micro-cathode vacuum arc thruster is investigated by PIC simulation in this study.The geometry and operating parameters of the thruster are the same as those reported in the literature.The reliability and reasonability of the code are verified by comparing the experimental measurement of electron temperature,plasma density,and mean ion velocity for a magnetically expanding plasma and the experimental measurement results of the ion velocity of the micro-cathode arc thruster with different applied magnetic field strengths.

The time evolution of electron and ion behavior at different time steps shows that the electrons are magnetized after the electrons are injected into the flow field and are bound by the magnetic field lines to form a narrow electron channel.The ions take a long time to keep up with the electron movement.The applied magnetic field strengthens the charge separation phenomenon and the electric field caused by charge separation which is the main reason for the ion acceleration concluded in this simulation.The electric field caused by charge separation evolves with time.In a further study,an electric field measurement method with a higher time resolution can be used to verify the rationality of this acceleration mechanism.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Nos.11735004,11575019,and 11702021),National Postdoctoral Program for Innovative Talents (BX20180029),and Defense Industrial Technology Development Program (JCKY2018203B029).