Abstract
Fully nonlinear kinetic simulations of electron plasma and ion acoustic waves (IAWs) have been carried out with a new multispecies, parallelized Vlasov code. The numerical implementation of the Vlasov model and the methods used to compute the wave frequency are described in detail. For the first time, the nonlinear frequency of IAWs, combining the contributions from electron and ion kinetic effects and from harmonic generation, has been calculated and compared to Vlasov results. Excellent agreement of theory with simulation results is shown at all amplitudes, harmonic generation being an essential component at large amplitudes. For IAWs, the positive frequency shift from trapped electrons is confirmed and is dominant for the effective electrontoion temperature ratio, Z with Z as the charge state. Furthermore, numerical results demonstrate unambiguously the dependence [R. L. Dewar, Phys. Fluids 15, 712 (1972)] of the kinetic shifts on details of the distribution of the trapped particles, which depends in turn on the conditions under which the waves were generated. The trapped particle fractions and energy distributions are derived and, upon inclusion of harmonic effects, shown to agree with the simulation results, completing a consistent picture. Fluid models of the wave evolution are considered but prove unable to capture essential details of the kinetic simulations. Detrapping by collisions and sideloss is also discussed.
We acknowledge valuable discussions with B. I. Cohen, C. Riconda, D. Pesme, and I. Y. Dodin. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DEAC5207NA27344. This work was partially funded by the Laboratory Directed Research and Development Program at LLNL under Project Tracking Code Nos. 08ERD031 and 12ERD061.
I. INTRODUCTION
II. ANALYTIC DERIVATION OF NONLINEAR FREQUENCY SHIFTS RESULTING FROM RESONANT WAVEPARTICLE INTERACTION AND HARMONIC GENERATION
A. Nonlinear frequency shifts for EPWs
B. Nonlinear frequency shifts for IAWs
1. IAW frequency shift from resonant electron and ion wave interaction
2. IAW frequency shift from harmonic generation
III. VLASOV SIMULATION MODEL, NUMERICAL IMPLEMENTATION, AND ANALYSIS METHODS
A. Simulation model
B. Numerical implementation
1. The semiLagrangian method for solving the Vlasov equation
2. Sources to Poisson's equation
3. Poisson solver
C. Extracting the frequency shift using a Hilbert transform method
IV. VLASOV SIMULATION RESULTS
A. Nonlinear frequency shifts of EPWs
1. EPW: Intermediate wavenumber
2. EPW: Larger wavenumber
B. Nonlinear frequency shifts of IAWs
1. IAW case
2. IAW case
3. IAW case
C. Summary of frequency shifts
D. Fraction of trapped electrons and ions in IAWs
1. IAW case
2. IAW case
3. IAW case
E. Distribution in resonant region as function of energy
V. MODELING THE DRIVEN ION ACOUSTIC WAVE
VI. CONCLUSIONS
Key Topics
 Electrostatic waves
 78.0
 Plasma waves
 70.0
 Adiabatic theorem
 49.0
 Dispersion relations
 27.0
 Acoustic waves
 14.0
Figures
Distribution in resonant region seen from the wave frame. Phase space trajectories corresponding to two neighboring energy levels W and are shown in the trapped (red) as well as in the forward ( ) and backward ( ) passing regions (green). Corresponding small phase space volumes (dashed), which undergo filamentation over time in the sudden case, are also pointed out. The separatrix (black), given by , delimits the trapping from the passing regions.
Distribution in resonant region seen from the wave frame. Phase space trajectories corresponding to two neighboring energy levels W and are shown in the trapped (red) as well as in the forward ( ) and backward ( ) passing regions (green). Corresponding small phase space volumes (dashed), which undergo filamentation over time in the sudden case, are also pointed out. The separatrix (black), given by , delimits the trapping from the passing regions.
(a) Electrons in an EPW or ions in an IAW: trapping in tail of distribution. (b) Electrons in an IAW: trapping in bulk of distribution.
(a) Electrons in an EPW or ions in an IAW: trapping in tail of distribution. (b) Electrons in an IAW: trapping in bulk of distribution.
Estimate for ratio of ion to electron frequency shift contributions for an IAW, assuming .
Estimate for ratio of ion to electron frequency shift contributions for an IAW, assuming .
Three estimates of the nonlinear frequency shifts of an EPW with : (1) the approximate analytic solution given by Eq. (19) , (2) shift based on the real part of the numerically computed root of the complex nonlinear dispersion , and (3) shift based on the numerically computed real root of the real nonlinear dispersion .
Three estimates of the nonlinear frequency shifts of an EPW with : (1) the approximate analytic solution given by Eq. (19) , (2) shift based on the real part of the numerically computed root of the complex nonlinear dispersion , and (3) shift based on the numerically computed real root of the real nonlinear dispersion .
Time evolution of the electrostatic field amplitude for a standing EPW when the charge density is initialized to an amplitude and wavenumber of (a) and (b) , respectively. In both cases, a fit to the early time behavior provides a damping rate and frequency in close agreement with the solution to the complex linear dispersion relation. A fit to the time history after the initial decay provides however a reduced frequency as well as an essentially undamped wave.
Time evolution of the electrostatic field amplitude for a standing EPW when the charge density is initialized to an amplitude and wavenumber of (a) and (b) , respectively. In both cases, a fit to the early time behavior provides a damping rate and frequency in close agreement with the solution to the complex linear dispersion relation. A fit to the time history after the initial decay provides however a reduced frequency as well as an essentially undamped wave.
EPW case with . Frequency shifts as a function of measured from simulations. Both initial value (blue circles) and driven waves (red triangles) results are shown and compared to theoretical estimates given by Eq. (19) in the sudden (blue full line) and adiabatic (red dashed line) wave generation limit.
EPW case with . Frequency shifts as a function of measured from simulations. Both initial value (blue circles) and driven waves (red triangles) results are shown and compared to theoretical estimates given by Eq. (19) in the sudden (blue full line) and adiabatic (red dashed line) wave generation limit.
Initial value EPW case with . Frequency shifts as a function of measured from simulations (blue circles). Comparison with theoretical estimate (blue full line) given by Eq. (19) in the sudden wave generation limit.
Initial value EPW case with . Frequency shifts as a function of measured from simulations (blue circles). Comparison with theoretical estimate (blue full line) given by Eq. (19) in the sudden wave generation limit.
Driven IAW case . (a) Internal electric field and (b)phase of the wave as a function of time t. The frequency and average amplitude are measured over the interval . The slope of the linear fit to the phase allows the frequency to be extracted. The driver electric field is .
Driven IAW case . (a) Internal electric field and (b)phase of the wave as a function of time t. The frequency and average amplitude are measured over the interval . The slope of the linear fit to the phase allows the frequency to be extracted. The driver electric field is .
Driven IAW case . Subthermal electron velocity distribution f(v) at time and position corresponding to the maximum of the electrostatic potential field . The wave phase velocity and the location of the separatrices at are indicated by vertical fine dashed lines. The initial Maxwellian distribution is shown as a bold dashed line.
Driven IAW case . Subthermal electron velocity distribution f(v) at time and position corresponding to the maximum of the electrostatic potential field . The wave phase velocity and the location of the separatrices at are indicated by vertical fine dashed lines. The initial Maxwellian distribution is shown as a bold dashed line.
Driven IAW case . Normalized frequency shifts from the Vlasov simulations [black triangles for helium (He, Z/A = 1/2) and blue circles for hydrogen (H, Z/A = 1)] are shown as a function of along with the theoretical prediction (solid line) assuming adiabatic electron and sudden ion wave generation. In this case of large , only the positive electron contribution is in fact significant. The dashed curve includes the frequency shift from the harmonic theory. The dasheddotted curve uses that theory with the value of given by the fit to the data shown in Fig. 14 .
Driven IAW case . Normalized frequency shifts from the Vlasov simulations [black triangles for helium (He, Z/A = 1/2) and blue circles for hydrogen (H, Z/A = 1)] are shown as a function of along with the theoretical prediction (solid line) assuming adiabatic electron and sudden ion wave generation. In this case of large , only the positive electron contribution is in fact significant. The dashed curve includes the frequency shift from the harmonic theory. The dasheddotted curve uses that theory with the value of given by the fit to the data shown in Fig. 14 .
Driven IAW case . Normalized frequency shifts from the Vlasov simulations (triangles) are shown as a function of along with the theoretical prediction assuming an adiabatic wave generation for both electrons and ions. The straight solid red curve is obtained from Eq. (24) considering constant phase velocity obtained from . The dashed (red) curve includes the contribution from the harmonic. Using an iterative process, one finds the dasheddotted red curve for which the phase velocity is adjusted upward to account for the positive frequency shift as increases. The data value pointed out with an arrow corresponds to a simulation considered in more detail in Sec. IV E .
Driven IAW case . Normalized frequency shifts from the Vlasov simulations (triangles) are shown as a function of along with the theoretical prediction assuming an adiabatic wave generation for both electrons and ions. The straight solid red curve is obtained from Eq. (24) considering constant phase velocity obtained from . The dashed (red) curve includes the contribution from the harmonic. Using an iterative process, one finds the dasheddotted red curve for which the phase velocity is adjusted upward to account for the positive frequency shift as increases. The data value pointed out with an arrow corresponds to a simulation considered in more detail in Sec. IV E .
Driven IAW case . Normalized frequency shifts from the Vlasov simulations (triangles) are shown as a function of along with the theoretical prediction assuming adiabatic electron and sudden ion (solid straight blue line) and adiabatic ion (dashed straight red line) wave generation. The dashed blue line includes the second harmonic contribution from the fluid theory. In this case of low , the negative contribution from ions to the frequency shift is dominant. The dasheddotted blue line results from an iterative process that adjusts the phase velocity downward to account for the negative frequency shift as increases.
Driven IAW case . Normalized frequency shifts from the Vlasov simulations (triangles) are shown as a function of along with the theoretical prediction assuming adiabatic electron and sudden ion (solid straight blue line) and adiabatic ion (dashed straight red line) wave generation. The dashed blue line includes the second harmonic contribution from the fluid theory. In this case of low , the negative contribution from ions to the frequency shift is dominant. The dasheddotted blue line results from an iterative process that adjusts the phase velocity downward to account for the negative frequency shift as increases.
Driven IAW case . The fraction of trapped electrons from the Vlasov simulations [black triangles for Helium (He, Z/A = 1/2) and blue circles for Hydrogen (H, Z/A = 1)] is shown as a function of the wave amplitude . The theoretical predictions in both the sudden (solid blue line) and adiabatic (dashed red line) wave generation limits and assuming a sinusoidal field is plotted for comparison. The theoretical estimate in the adiabatic limit that uses the electrostatic field from the simulation is also shown for a few simulation cases (red crosses).
Driven IAW case . The fraction of trapped electrons from the Vlasov simulations [black triangles for Helium (He, Z/A = 1/2) and blue circles for Hydrogen (H, Z/A = 1)] is shown as a function of the wave amplitude . The theoretical predictions in both the sudden (solid blue line) and adiabatic (dashed red line) wave generation limits and assuming a sinusoidal field is plotted for comparison. The theoretical estimate in the adiabatic limit that uses the electrostatic field from the simulation is also shown for a few simulation cases (red crosses).
Driven IAW case . (a) The internal electrostatic field from simulations for a small amplitude ( , dashed line) and large amplitude ( , full line) wave. An exact sinusoid is shown for comparison (dasheddotted line). All wave fields are normalized to a maximum of one. (b) The amplitude of the harmonic is shown as a function of the fundamental from the simulations (crosses) and (dashed line) from the fluid theory of Sec. ??? . The simulation data are fit best with the power law .
Driven IAW case . (a) The internal electrostatic field from simulations for a small amplitude ( , dashed line) and large amplitude ( , full line) wave. An exact sinusoid is shown for comparison (dasheddotted line). All wave fields are normalized to a maximum of one. (b) The amplitude of the harmonic is shown as a function of the fundamental from the simulations (crosses) and (dashed line) from the fluid theory of Sec. ??? . The simulation data are fit best with the power law .
Driven IAW case . The fraction of trapped (a) ions and (b) electrons from the Vlasov simulations (black triangles) is shown as a function of the wave amplitude and , respectively. The theoretical predictions in both the sudden (solid blue line) and adiabatic (dashed red line) wave generation limits with a sinusoidal field are plotted for comparison. The theoretical estimate in both the sudden (blue circles) and adiabatic limits (red crosses) that use the electrostatic field from the simulation is also shown for a few simulation cases.
Driven IAW case . The fraction of trapped (a) ions and (b) electrons from the Vlasov simulations (black triangles) is shown as a function of the wave amplitude and , respectively. The theoretical predictions in both the sudden (solid blue line) and adiabatic (dashed red line) wave generation limits with a sinusoidal field are plotted for comparison. The theoretical estimate in both the sudden (blue circles) and adiabatic limits (red crosses) that use the electrostatic field from the simulation is also shown for a few simulation cases.
Driven IAW case . The fraction of trapped (a) ions and (b)electrons from the Vlasov simulations (black triangles) is shown as a function of the wave amplitude and , respectively. The theoretical predictions in both the sudden (solid blue line) and adiabatic (dashed red line) wave generation limits with a sinusoidal field are plotted for comparison.
Driven IAW case . The fraction of trapped (a) ions and (b)electrons from the Vlasov simulations (black triangles) is shown as a function of the wave amplitude and , respectively. The theoretical predictions in both the sudden (solid blue line) and adiabatic (dashed red line) wave generation limits with a sinusoidal field are plotted for comparison.
Driven IAW case with amplitude (pointed out with an arrow in Figs. 11 15(a) and 15(b) ). The electron and ion distribution functions at time as a function of the particle energy in the trapping regions. The distributions for the sudden and adiabatic approximations with the purely sinusoidal field and the field from the simulation are also shown.
Driven IAW case with amplitude (pointed out with an arrow in Figs. 11 15(a) and 15(b) ). The electron and ion distribution functions at time as a function of the particle energy in the trapping regions. The distributions for the sudden and adiabatic approximations with the purely sinusoidal field and the field from the simulation are also shown.
The time history of the wave amplitude according to the model equations (black curve) and from the Vlasov simulation (blue dashed curve) for the parameters and . Also shown is the time dependence of the driver as well as the model's frequency shift . For the model curve, the parameter was chosen equal to the adiabatic value of 0.544, as suggested by results in Fig. 10 . In the above plots, the driver amplitude is normalized to a maximum of 0.01 for convenience of illustration. (a) for and afterwards. (b) for and afterwards.
The time history of the wave amplitude according to the model equations (black curve) and from the Vlasov simulation (blue dashed curve) for the parameters and . Also shown is the time dependence of the driver as well as the model's frequency shift . For the model curve, the parameter was chosen equal to the adiabatic value of 0.544, as suggested by results in Fig. 10 . In the above plots, the driver amplitude is normalized to a maximum of 0.01 for convenience of illustration. (a) for and afterwards. (b) for and afterwards.
Similar to Fig. 18 but for . For the model curve, both and were chosen to be equal to the adiabatic value , as suggested by results in Fig. 11 . (a) for and afterwards. (b) for and afterwards.
(a) Collisional detrapping in the tail of the distribution. Case of resonant electrons in an EPW or of ions in an IAW. (b) Collisional detrapping in the bulk of the distribution. Case of resonant electrons in an IAW.
(a) Collisional detrapping in the tail of the distribution. Case of resonant electrons in an EPW or of ions in an IAW. (b) Collisional detrapping in the bulk of the distribution. Case of resonant electrons in an IAW.
Tables
Summary of the EPW and IAW frequencies and frequency shifts and the coefficients of their dependence on powers of from theory and simulations.
Summary of the EPW and IAW frequencies and frequency shifts and the coefficients of their dependence on powers of from theory and simulations.
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