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Electron temperature, electron heat flux and corresponding charge density. (a) Case 2B (m i =m e ¼ 100) at x pe0 t ¼ 200. (b) Case 2B (m i =m e ¼ 100) at x pe0 t ¼ 400.
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Grid-based Vlasov simulations are carried out to re-evaluate the one-dimensional collisionless plasma expansion into vacuum. The grid-based method eliminates the inherent statistical noise in particle-based methods and allows us to extend the solution beyond the self-similar expansion region and resolve small electron timescale wave perturbations....
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Context 1
... w ¼ v x À hv x i is the random electron velocity and ~ w ¼ w=v te0 . Figure 7 further correlates ~ T e ð~ xÞ, with the local electron heat flux ~ Q e ð~ xÞ, and the space charge density ~ n i ð~ xÞ À ~ n e ð~ xÞ for case 2B. Here, the electron heat flux ~ Q e ð~ xÞ is calculated from ...
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... the left-side boundary of region B propagates toward the upstream at the same speed as the group velocity of the electron Langmuir wave, ~ v ele g . Thus, the left boundary of region B coincides with that of the electron Langmuir wave front given by Eq. (25), ~ x ele À ð ~ tÞ ¼ À~ v ele g Á ~ t, and the location of region B is defined by À~ v ele Fig. 7 also shows that, while the plasma is overall quasi-neutral during the expansion process, there are small density oscillations on the electron timescale upstream of the expansion front. These small density oscillations correspond to the perturbations caused by the electron Langmuir wave. In region B, the electron temperature decreases ...
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... front does not affect those electrons that leave the bulk plasma at the start of the expansion but attracts some of the electrons to backflow during subsequent expansion. The stream of the backflow electrons creates the peak in the electron VDF in region D. This peak corresponds to the temperature enhancement shown in the ~ T e ð~ xÞ profile in Fig. 7. As the electron VDF at the expansion front does not spread out but has a non-Maxwellian component, the electron temperature will be estimated high from Eq. ...
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... The energy lost by the electrons sustains the propagation of the electron Langmuir wave. This resonant wave-particle interaction depletes the number of electrons with a velocity at ~ v $ À~ v ele g . Behind the wave front, the void in the electron VDF is gradually refilled. This leads to the slight decrease in ~ T e in region B as observed in Fig. 7. Previously, Medvedev 42 showed that an ion "cooling" wave with a spatial size much longer than the rarefaction wave perturbed region exists in the ion-ion plasma expansion. It was also speculated that the cooling observed should also exist FIG. 8. Local electron velocity distribution functions (VDFs) at selected locations for case 2B ...
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... The Vlasov-Poisson equation [1], which describes the collective kinetic behavior of collisionless plasmas, is one of the most fundamental models in plasma physics. It plays a significant role in various problems, such as plasma plume expansion [2,3], particle-beam interactions [4], plasma instabilities [5,6], and inertial electrostatic confinement fusion [7]. Many types of simulation methods for solving the Vlasov-Poisson equation have been developed, such as the fully kinetic particle-in-cell (PIC) method [8][9][10][11][12], the semi-Lagrangian method [13][14][15][16], and the Eulerian method [17][18][19][20]. ...
The Vlasov-Poisson equation is one of the most fundamental models in plasma physics. It has been widely used in areas such as confined plasmas in thermonuclear research and space plasmas in planetary magnetospheres. In this study, we explore the feasibility of the physics-informed neural networks for solving forward and inverse Vlasov-Poisson equation (PINN-Vlasov). The PINN-Vlasov method employs a multilayer perceptron (MLP) to represent the solution of the Vlasov-Poisson equation. The training dataset comprises the randomly sampled time, space, and velocity coordinates and the corresponding distribution function. We generate training data using the fully kinetic PIC simulation rather than the analytical solution to the Vlasov-Poisson equation to eliminate the correlation between data and equations. The Vlasov equation and Poisson equation are concurrently integrated into the PINN-Vlasov framework using automatic differentiation and the trapezoidal rule, respectively. By minimizing the residuals between the reconstructed distribution function and labeled data, and the physically constrained residuals of the Vlasov-Poisson equation, the PINN-Vlasov method is capable of dealing with both forward and inverse problems. For forward problems, the PINN-Vlasov method can solve the Vlasov-Poisson equation with given initial and boundary conditions. For inverse problems, the completely unknown electric field and equation coefficients can be predicted with the PINN-Vlasov method using little particle distribution data.
... Though never applied in surface flashover simulation, the continuum and kinetic simulation approach has been shown to be a supplementary solution for PIC model for a range of plasma conditions, and features some advantages if applied appropriately. The fact that kinetic simulation speed is not sensitive to particle number, and its noise-free data enables its implementation in a variety of industrial plasma simulations, such as plasma-surface interactions [43], plasma transport [44], capacitively coupled plasma [45,46], multipactor [47], glow discharge [48], in addition to fusion plasma as well as astrophysical plasma simulations [49][50][51][52]. In the present work, we propose for the first time to apply a 1D2V (one dimension in space, two dimensions in velocity) continuum and kinetic simulation in surface flashover modeling to add more insights into improved flashover modeling technics. ...
Surface flashover across insulator in vacuum is a destructive plasma discharge which undermines the behaviors of a range of applications in electrical engineering, particle physics, space engineering, etc. This phenomenon is widely modeled by the particle-in-cell (PIC) simulation, here the continuum and kinetic simulation method is first proposed and implemented as an alternative solution for flashover modeling, aiming for the prevention of the unfavorable particle noises in PIC models. A 1D2V (one dimension in space, two dimensions in velocity) kinetic simulation model is constructed. Modeling setup, physical assumptions, and simulation algorithm are presented in detail, and a comparison with the well-known secondary electron emission avalanche (SEEA) analytical expression and existing PIC simulation is made. Obtained kinetic simulation results are consistent with the analytical prediction, and feature noise-free data of surface charge density as well as fluxes of primary and secondary electrons. Discrepancies between the two simulation models and analytical predictions are explained. The code is convenient for updating to include additional physical processes, and possible implementations of outgassing and plasma species for final breakdown stage are discussed. The proposed continuum and kinetic approach is expected to inspire future modeling studies for the flashover mechanism and mitigation.
... However, the PIC method involves inherent numerical noise. Alternatively, the grid-based method which directly solves the kinetic equation in phase space [22][23][24][25][26][27][28][29] is attracting researchers as the development of the capacity of high-performance computers. We refer to this method as the direct kinetic method (DKM). ...
In this paper, the immersed finite element (IFE) coupled with the conservative semi-Lagrangian (CSL) kinetic scheme is developed for plasma-material interactions. The proposed method (IFE-CSL) enjoys respective advantages of the IFE and CSL, i.e, the flexibility and efficiency for treating complex boundary conditions, mass conservation and being free from the Courant-Friedrichs-Lewy (CFL) condition. In the current IFE-CSL, the IFE method based on structured interface-independent meshes is developed for the spatial discretization of the field equation, which provides an accurate approach with convenient implementations to solve the field problems with irregular interface. The CSL scheme combined with an immersed method termed half-way ghost-cell is proposed for the spatial discretization of the Vlasov equation, which enables the proposed method to exactly conserve mass and conveniently handle curved boundaries. Then the IFE and CSL method are coupled via the charge density depending on full kinetic or hybrid kinetic model, where the appropriate IFE solver is developed for the linear or nonlinear Poisson equation, respectively. In the current IFE-CSL, different geometries can be treated automatically for both IFE and CSL through the specific geometric information. As a result, the proposed CSL-IFE can be conveniently and efficiently applied to simulate the plasma-material interactions with structured interface-independent meshes. Finally, several numerical experiments are performed to demonstrate the capabilities of the proposed method.
... Though never applied in surface flashover simulation, the continuum and kinetic simulation approach has been shown to be a supplementary solution for PIC model, and features some advantages if applied in appropriate conditions. The fact that kinetic simulation speed is not sensitive to particle number, and its noise-free data enables its implementation in a variety of industrial plasma simulations, such as plasma-surface interactions [38], plasma transport [39], capacitively coupled plasma [40,41], multipactor [42], glow discharge [43], in addition to fusion plasma as well as astrophysical plasma simulations [44][45][46][47]. In the present work, we propose for the first time to apply the continuum and kinetic simulation in surface flashover modeling to add more insights into improved flashover modeling technics. ...
Surface flashover across insulator in vacuum is a destructive plasma discharge which undermines the behaviors of a range of applications in electrical engineering, particle physics, space engineering, etc. This phenomenon is widely modeled by the particle-in-cell (PIC) simulation, here the continuum and kinetic simulation method is first proposed and implemented as an alternative solution for flashover modeling, aiming for the prevention of the unfavorable particle noises in PIC models. The 1D2V (one dimension in space, two dimensions in velocity) kinetic simulation model is constructed. Modeling setup, physical assumptions, and simulation algorithm are presented in detail, and a comparison with the well-known secondary electron emission avalanche (SEEA) analytical expression and existing PIC simulation is made. Obtained kinetic simulation results are consistent with the analytical prediction, and feature noise-free data of surface charge density as well as particle fluxes of primary and secondary electrons. Discrepancies between the two simulation models and analytical predictions are explained. The code is convenient for updating to include additional physical processes, and possible implementations of outgassing and extra plasma species for final breakdown stage are discussed. The proposed continuum and kinetic approach is expected to inspire future flashover modeling studies for the understanding and mitigation.
... The plasma parameter is small and electromagnetic fields become important for particle dynamics. Any instabilities that emerge in the domain can propagate quickly between the two regions at the electron thermal speed, far exceeding the expansion front speed that characterizes the baseline plume dynamics [10]. ...
Laser ablation plasma thrusters are an emerging space propulsion concept that provides promise for lightweight payload delivery. Predicting the lifetime and performance of these thrusters hinges on a comprehensive characterization of the expansion dynamics of the ablated plasma plume. While state-of-the-art techniques for simulating plasmas are often particle-based, a grid-based direct kinetic solver confers advantages in such a transient and inhomogeneous problem by eliminating statistical noise. A direct kinetic solver including interparticle collisions is employed on a plume expansion model problem spanning one dimension each in configuration and velocity space. The high degree of thermodynamic nonequilibrium inherent in plume expansion is characterized, justifying the need for a kinetic rather than a hybrid or fluid solver. Thruster-relevant metrics such as the momentum flux are also computed. The plume dynamics are observed to be highly inhomogeneous in space with insufficient time for thermalization in the region preceding the expansion front, and the theoretical possibility of reducing the local grid resolution by up to two orders of magnitude at the far end of the domain is established. These grid-point requirements are verified via the employment of nonuniform grids of various expansion ratios, several of which also employ coarsening in velocity space. Longer domain lengths are explored to characterize thruster-scale phenomena and larger ambient pressures are simulated as a testbed to probe facility effects due to collisions with background particles.
... The plasma parameter is small and the background electromagnetic fields become important for particle dynamics. Any instabilities that emerge in the domain can propagate quickly between the two regions at the electron thermal speed, far exceeding the expansion front speed that characterizes the baseline plume dynamics [10]. ...
Laser ablation plasma thrusters are an emerging space propulsion concept that provides promise for lightweight payload delivery. Predicting the lifetime and performance of these thrusters hinges on a comprehensive characterization of the expansion dynamics of the ablated plasma plume. While state-of-the-art techniques for simulating plasmas are often particle-based, a grid-based direct kinetic solver confers advantages in such a transient and inhomogeneous problem by eliminating statistical noise. A direct kinetic solver including interparticle collisions is employed on a plume expansion model problem spanning one dimension each in configuration and velocity space. The high degree of thermodynamic nonequilibrium inherent in plume expansion is characterized, justifying the need for a kinetic rather than a hybrid or fluid solver. Thruster-relevant metrics such as the momentum flux are also computed. The plume dynamics are observed to be highly inhomogeneous in space with insufficient time for thermalization in the region preceding the expansion front, and the theoretical possibility of reducing the local grid resolution by up to two orders of magnitude at the far end of the domain is established. These grid-point requirements are verified via the employment of nonuniform grids of various expansion ratios, several of which also employ coarsening in velocity space. Longer domain lengths are explored to characterize thruster-scale phenomena and larger ambient pressures are simulated as a testbed to probe facility effects due to collisions with background particles.
... In addition, a fourth kind of simulator has emerged in recent years called Vlasov simulators. 20 Vlasov simulators solve the Vlasov equation directly and thus include kinetic effects. However, the collisionless assumption is usually included in the design of the simulator and would need further development to include charge-neutral particle collisions. ...
We present the first set of particle-in-cell simulations including Monte Carlo collisions between charged and neutral particles used to simulate a cylindrical Langmuir probe in the electron saturation regime with a collisional electron sheath. We use a setup focused on the E-region ionosphere; however, the results of these simulations are analyzed in a general sense using dimensionless values. We find that the electron currents get enhanced as the collision frequency for electrons increases and the values of 𝜆𝑒/𝜆𝐷→1, where λe is the electron mean free path and λD is the electron Debye length. In addition, we apply the simulation results to a sounding rocket experiment and show how we can correct the currents for the Investigation of Cusp Irregularities-4 sounding rocket due to collisions while it flies through the E-region.
... Previously, we presented a sequential one-dimensional grid-based electrostatic Vlasov solver, Vlasolver (Vlasov Solver) and applied it to study one-dimensional collisionless plasma expansion [16][17][18]. Recently, a parallel version of Vlasolver was developed [19]. ...
... Here, since the plasma has a drifting velocity, we modify the equation by fitting the parameter α as the work [18] and adding a constant from the drifting of plasma. The positions of the dotted lines in fig. ...
View Video Presentation: https://doi.org/10.2514/6.2022-1618.vid We present the development progress of a parallel, multi-dimensional grid-based Vlasov solver, Vlasolver. The Vlasov equation is solved using the third order Positive Flux Conservation (PFC) scheme, and the parallelization of the code is implemented using domain decomposition and MPI. The capability of Vlasolver is demonstrated by an application in 2-dimensional plasma plume expansion. We find that the grid-based Vlasov method is advantageous over particle-based methods as it eliminates the inherent statistical noise in particle-based methods in the low density region.
