Abstract
A possible solution to the unexplained high intensity hard xray emission observable during solar flares was investigated via 3D fully relativistic, electromagnetic particleincell simulations with realistic ion to electron mass ratio. A beam of accelerated electrons was injected into a magnetised, Maxwellian, homogeneous, and inhomogeneous background plasma. The electron distribution function was unstable to the beamplasma instability and was shown to generate Langmuir waves, while relaxing to plateau formation. In order to estimate the role of the background density gradient on an unbound (infinite spatial extent) beam, three different scenarios were investigated: (a) a uniform density background; (b) a weak density gradient, ne , R /ne , L = 3; (c) a strong gradient case, ne , R /ne , L = 10, where ne , R and ne , L denote background electron densities on the left and right edges of the simulation box, respectively. The strong gradient case produced the largest fraction of electrons beyond 15vth . Furthermore, two cases (uniform and strong gradient background) with spatially localized beam injections were performed aiming to show drifts of the generated Langmuir wave wavenumbers, as suggested in previous studies. For the strong gradient case, the Langmuir wave power is shown to drift to smaller wavenumbers, as found in previous quasilinear simulations.
The authors are financially supported by the HEFCEfunded South East Physics Network (SEPNET) UK. D.T.’s research is supported by The Leverhulme Trust Research Project Grant No. RPG311 and STFC Grant Nos. ST/J001546/1 and ST/H008799/1.
I. INTRODUCTION
II. SIMULATION SETUP
III. UNBOUND BEAM INJECTION
A. Constant background plasma
1. Electric field evolution
1. Distribution function dynamics
C. Weak gradient case
1. Electric field evolution
2. Distribution function dynamics
D. Strong gradient case
E. Comparison
IV. LOCALIZED BEAM INJECTION
A. Constant background plasma
B. Strong gradient case
V. CONCLUSIONS
Key Topics
 Plasma waves
 51.0
 Plasma oscillations
 37.0
 Cumulative distribution functions
 10.0
 Maxwell equations
 10.0
 Beam plasma instabilities
 9.0
Figures
Timedistance plot for Ex for constant background density and uniform beam injection along the xdirection; .
Timedistance plot for Ex for constant background density and uniform beam injection along the xdirection; .
Ex (x) (solid) and ne (x) (dashed) at , zoomed in and arbitrary units on the yaxis for clarity.
Ex (x) (solid) and ne (x) (dashed) at , zoomed in and arbitrary units on the yaxis for clarity.
FastFourier transform of Ex for constant background density and uniform beam injection. The curves represent the dispersion relation for Langmuir waves, (“horizontal” curve), and the resonance condition for the beam plasma instability, .
FastFourier transform of Ex for constant background density and uniform beam injection. The curves represent the dispersion relation for Langmuir waves, (“horizontal” curve), and the resonance condition for the beam plasma instability, .
Snapshots of the electron distribution function (solid) with respect to energy for constant background density and uniform beam injection. Dotted curves track the initial distribution at t = 0.
Snapshots of the electron distribution function (solid) with respect to energy for constant background density and uniform beam injection. Dotted curves track the initial distribution at t = 0.
As in Fig. 1 but for the weak gradient case.
As in Fig. 3 but for the weak gradient case.
As in Fig. 4 but for the weak gradient case.
As in Fig. 1 but for the strong gradient case.
As in Fig. 3 but for the strong gradient case.
As in Fig. 4 but for the strong gradient case.
Fraction of electrons with energies higher than , as calculated by use of Eq. (2) , over time for constant background density (solid), weak gradient (dotted), strong gradient (dashed).
Fraction of electrons with energies higher than , as calculated by use of Eq. (2) , over time for constant background density (solid), weak gradient (dotted), strong gradient (dashed).
Left: Final simulation snapshots of Figs. 4 (solid), 7 (dotted), and 10 (dashed) plotted for comparison. Right: Zoomed in on high energy tail for clarity.
(a) [E(t)–E(0)]/E(0) for constant background (diamond), weak gradient (cross), strong gradient (asterisk). (b) Kinetic particle energy (solid) and Langmuir wave energy (dotted) for constant background. Langmuir wave energy multiplied by a factor 5 for clarity. (c) As top right but for weak gradient. (d) As top right but for strong gradient.
(a) [E(t)–E(0)]/E(0) for constant background (diamond), weak gradient (cross), strong gradient (asterisk). (b) Kinetic particle energy (solid) and Langmuir wave energy (dotted) for constant background. Langmuir wave energy multiplied by a factor 5 for clarity. (c) As top right but for weak gradient. (d) As top right but for strong gradient.
Localized beam injection with constant background density. (a)Timedistance plot for Ex component. (b) Timedistance plot for background electron density. (c) Ex (k) for (solid, black), (dashed, purple), (dasheddotted, blue), (dotted, green), (dotted with +, orange), (dotted with *, red). Yaxis in arbitrary units. (d) 2D Fourier transform of Ex (x,t). Note that for clarity in panel (d), the color scheme is inverse to the ones in the upper row.
Localized beam injection with constant background density. (a)Timedistance plot for Ex component. (b) Timedistance plot for background electron density. (c) Ex (k) for (solid, black), (dashed, purple), (dasheddotted, blue), (dotted, green), (dotted with +, orange), (dotted with *, red). Yaxis in arbitrary units. (d) 2D Fourier transform of Ex (x,t). Note that for clarity in panel (d), the color scheme is inverse to the ones in the upper row.
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