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Electron and ion kinetic effects on non-linearly driven electron plasma and ion acoustic waves
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10.1063/1.4794346
/content/aip/journal/pop/20/3/10.1063/1.4794346
http://aip.metastore.ingenta.com/content/aip/journal/pop/20/3/10.1063/1.4794346

Figures

Image of FIG. 1.
FIG. 1.

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.

Image of FIG. 2.
FIG. 2.

(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.

Image of FIG. 3.
FIG. 3.

Estimate for ratio of ion to electron frequency shift contributions for an IAW, assuming .

Image of FIG. 4.
FIG. 4.

Three estimates of the non-linear 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 non-linear dispersion , and (3) shift based on the numerically computed real root of the real non-linear dispersion .

Image of FIG. 5.
FIG. 5.

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.

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

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.

Image of FIG. 8.
FIG. 8.

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 .

Image of FIG. 9.
FIG. 9.

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.

Image of FIG. 10.
FIG. 10.

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 dashed-dotted curve uses that theory with the value of given by the fit to the data shown in Fig. 14 .

Image of FIG. 11.
FIG. 11.

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 dashed-dotted 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 .

Image of FIG. 12.
FIG. 12.

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 dashed-dotted blue line results from an iterative process that adjusts the phase velocity downward to account for the negative frequency shift as increases.

Image of FIG. 13.
FIG. 13.

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).

Image of FIG. 14.
FIG. 14.

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 (dashed-dotted 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 .

Image of FIG. 15.
FIG. 15.

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.

Image of FIG. 16.
FIG. 16.

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.

Image of FIG. 17.
FIG. 17.

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.

Image of FIG. 18.
FIG. 18.

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.

Image of FIG. 19.
FIG. 19.

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.

Image of FIG. 20.
FIG. 20.

(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

Generic image for table
Table I.

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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/content/aip/journal/pop/20/3/10.1063/1.4794346
2013-03-13
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Electron and ion kinetic effects on non-linearly driven electron plasma and ion acoustic waves
http://aip.metastore.ingenta.com/content/aip/journal/pop/20/3/10.1063/1.4794346
10.1063/1.4794346
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