Abstract
Twodimensional Vlasov simulations of nonlinear electron plasma waves are presented, in which the interplay of linear and nonlinear kinetic effects is evident. The plasma wave is created with an external traveling wave potential with a transverse envelope of width such that thermal electrons transit the wave in a “sideloss” time, . Here, v_{e} is the electron thermal velocity. The quasisteady distribution of trapped electrons and its selfconsistent plasma wave are studied after the external field is turned off. In cases of particular interest, the bounce frequency, , satisfies the trapping condition such that the wave frequency is nonlinearly downshifted by an amount proportional to the number of trapped electrons. Here, k is the wavenumber of the plasma wave and is its electric potential. For sufficiently short times, the magnitude of the negative frequency shift is a local function of . Because the trapping frequency shift is negative, the phase of the wave on axis lags the offaxis phase if the trapping nonlinearity dominates linear wave diffraction. In this case, the phasefronts are curved in a focusing sense. In the opposite limit, the phasefronts are curved in a defocusing sense. Analysis and simulations in which the wave amplitude and transverse width are varied establish criteria for the development of each type of wavefront. The damping and trappedelectroninduced focusing of the finiteamplitude electron plasma wave are also simulated. The damping rate of the field energy of the wave is found to be about the sideloss rate, . For large wave amplitudes or widths , a trappinginduced selffocusing of the wave is demonstrated.
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract number DEAC5207NA27344. This work was funded by the Laboratory Directed Research and Development Program at LLNL under project tracking code 08ERD031. We acknowledge informative discussions with A. B. Langdon, W. B. Mori, D. Pesme, H. A. Rose, W. Rozmus, D. J. Strozzi, L. Yin, and B. Winjum.
I. INTRODUCTION
II. APPROACH
III. EFFECTS OF TRAPPING AND DIFFRACTION ON PLASMA WAVE PROPAGATION
A. Nonlinear frequency shift due to electron trapping in two spatial dimensions
B. Wavefront bowing
IV. FOCUSING AND DAMPING OF FIELDS FROM TRAPPED ELECTRONS
V. CONCLUSIONS AND DISCUSSION
Key Topics
 Plasma waves
 72.0
 Boundary value problems
 19.0
 Radiosurgery
 17.0
 Backscattering
 11.0
 Self focusing
 10.0
Figures
(Color online) Nonlinear frequency shifts vs squareroot of wave amplitude , for laterally periodic boundary conditions. The squares are frequency shifts determined on axis. The blue triangles and red crosses are for different lateral positions. The blue data is for the wide driver case where the nonlinear shifts are dominant and the measurements were made for a time interval before a thermal transit time across the wave. The red crosses for a narrow driver run long enough so that the resonant electrons have the same distribution for all y. Thus, the frequency shift (red crosses) is nearly independent of the local field, .
(Color online) Nonlinear frequency shifts vs squareroot of wave amplitude , for laterally periodic boundary conditions. The squares are frequency shifts determined on axis. The blue triangles and red crosses are for different lateral positions. The blue data is for the wide driver case where the nonlinear shifts are dominant and the measurements were made for a time interval before a thermal transit time across the wave. The red crosses for a narrow driver run long enough so that the resonant electrons have the same distribution for all y. Thus, the frequency shift (red crosses) is nearly independent of the local field, .
(Color online) Plot of electric field contours at fixed times (a) and (b) with showing evolution of wavefront bowing from linear theory.
(Color online) Plot of electric field contours at fixed times (a) and (b) with showing evolution of wavefront bowing from linear theory.
(Color online) Plot of electric field contours at fixed times (a) and (b) with and initially on axis showing nonlinear wavefront bowing.
(Color online) Plot of electric field contours at fixed times (a) and (b) with and initially on axis showing nonlinear wavefront bowing.
(Color online) Plot of electric field contours at and 200 showing wavefront bowing from a simulation with and peak wave amplitude on axis which then steadily decreases.
(Color online) Plot of electric field contours at and 200 showing wavefront bowing from a simulation with and peak wave amplitude on axis which then steadily decreases.
(Color online) Plot of electric field contours at and 200 showing wavefront bowing from simulation with peak wave amplitude on axis which then decreases.
(Color online) Plot of electric field contours at and 200 showing wavefront bowing from simulation with peak wave amplitude on axis which then decreases.
(Color online) Plot of electric field contours at and 280 showing wavefront bowing from simulation with and peak wave amplitude on axis whose amplitude is relatively stationary near the axis.
(Color online) Plot of electric field contours at and 280 showing wavefront bowing from simulation with and peak wave amplitude on axis whose amplitude is relatively stationary near the axis.
(Color online) Comparison of the EPW field history for periodic and outgoing boundary conditions for the wide lateral shape, . The driving external field is turned off at . (a) The history of the EPW field energy, W_{E} , with periodic (black) and outgoing electron (red) boundary conditions. (b) The history of the peak EPW field with periodic (black) and outgoing electron (red) boundary conditions. (c) Transverse spatial dependence of E_{x} (x_{p} , y) at five times for the periodic electron distribution boundary conditions. (d) Transverse spatial dependence of E_{x} (x_{p} , y) at five times for the outgoing electron distribution boundary conditions. For (c) and (d), x_{p} is the position at which E_{x} (x, y = 0) is maximum. Electric fields are normalized such that .
(Color online) Comparison of the EPW field history for periodic and outgoing boundary conditions for the wide lateral shape, . The driving external field is turned off at . (a) The history of the EPW field energy, W_{E} , with periodic (black) and outgoing electron (red) boundary conditions. (b) The history of the peak EPW field with periodic (black) and outgoing electron (red) boundary conditions. (c) Transverse spatial dependence of E_{x} (x_{p} , y) at five times for the periodic electron distribution boundary conditions. (d) Transverse spatial dependence of E_{x} (x_{p} , y) at five times for the outgoing electron distribution boundary conditions. For (c) and (d), x_{p} is the position at which E_{x} (x, y = 0) is maximum. Electric fields are normalized such that .
(Color online) Comparison of the EPW field history for periodic and outgoing boundary conditions for a narrow lateral shape, . The driving external field is turned off at . (a) The history of the peak EPW field with periodic (black) and outgoing electron (red) boundary conditions. (b) The history of the EPW field envelope at with periodic (black) and outgoing electron (red) boundary conditions. (c) Transverse spatial dependence of E_{x} (x_{p} , y) at five times indicated for the outgoing electron distribution boundary conditions. Here, x_{p} is the position for which E_{x} (x, y = 0) is a maximum.
(Color online) Comparison of the EPW field history for periodic and outgoing boundary conditions for a narrow lateral shape, . The driving external field is turned off at . (a) The history of the peak EPW field with periodic (black) and outgoing electron (red) boundary conditions. (b) The history of the EPW field envelope at with periodic (black) and outgoing electron (red) boundary conditions. (c) Transverse spatial dependence of E_{x} (x_{p} , y) at five times indicated for the outgoing electron distribution boundary conditions. Here, x_{p} is the position for which E_{x} (x, y = 0) is a maximum.
(Color online) The EPW field for a lateral shape . The driving external field is turned off at . (a) The transverse shape of the EPW at the axial position, x_{p} , where the field is a maximum for five times after the driving field is off. (b) Twodimensional filled contour maps of the EPW field at the time of maximum focusing. (c) Twodimensional filled contour maps of the EPW field after diffraction has overtaken focusing. There are 13 equally spaced contours in the range [−0.03, 0.03] for both figures (b) and (c).
(Color online) The EPW field for a lateral shape . The driving external field is turned off at . (a) The transverse shape of the EPW at the axial position, x_{p} , where the field is a maximum for five times after the driving field is off. (b) Twodimensional filled contour maps of the EPW field at the time of maximum focusing. (c) Twodimensional filled contour maps of the EPW field after diffraction has overtaken focusing. There are 13 equally spaced contours in the range [−0.03, 0.03] for both figures (b) and (c).
The distribution of electrons shown in Fig 9. F(V_{x} , Y) at the position of maximum potential for the EPW field and F(V_{x} , X) at y = 0: (a) and (b) , (c) and (d) , (e) and (f) . The distribution is integrated over the transverse velocity. The phase velocity is 3.6 V_{te} .
The distribution of electrons shown in Fig 9. F(V_{x} , Y) at the position of maximum potential for the EPW field and F(V_{x} , X) at y = 0: (a) and (b) , (c) and (d) , (e) and (f) . The distribution is integrated over the transverse velocity. The phase velocity is 3.6 V_{te} .
(Color online) Lineouts of the distribution of electrons at the position of maximum potential for the EPW field shown inFig. 9. and various y positions, F(V_{x} ) vs V_{x} at (a) , (b) , (c) , and (d) . The distribution is integrated over the transverse velocity. The phase velocity is 3.6 V_{te} . Note, the distributions for and are nearly the same.
(Color online) Lineouts of the distribution of electrons at the position of maximum potential for the EPW field shown inFig. 9. and various y positions, F(V_{x} ) vs V_{x} at (a) , (b) , (c) , and (d) . The distribution is integrated over the transverse velocity. The phase velocity is 3.6 V_{te} . Note, the distributions for and are nearly the same.
(Color online) Time evolution of wave front curvature resulting from linear dispersion of EPWs.
(Color online) Time evolution of wave front curvature resulting from linear dispersion of EPWs.
(Color online) Electrostatic field at time t _{max} where the negative curvature bowing resulting from the linear evolution is maximum.
(Color online) Electrostatic field at time t _{max} where the negative curvature bowing resulting from the linear evolution is maximum.
(Color online) Critical curve in the plane delimiting the region corresponding to the linear regime with negative curvature of the wavefront for times from the nonlinear regime with positive curvature. Critical curves corresponding to the wavenumbers , 0.3, and 0.4 are shown, having used the Bohm–Gross dispersion and assumed an adiabatic generation of the EPWs.
(Color online) Critical curve in the plane delimiting the region corresponding to the linear regime with negative curvature of the wavefront for times from the nonlinear regime with positive curvature. Critical curves corresponding to the wavenumbers , 0.3, and 0.4 are shown, having used the Bohm–Gross dispersion and assumed an adiabatic generation of the EPWs.
Tables
Transit damping rates.
Transit damping rates.
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