Abstract
To understand the possible destabilization of twodimensional current sheets, a kinetic model is proposed to describe the resonant interaction between electrostatic modes and trapped particles that bounce within the sheet. This work follows the initial investigation by Tur et al. [Phys. Plasmas 17, 102905 (2010)] that is revised and extended. Using a quasiparabolic equilibrium state, the linearized gyrokinetic Vlasov equation is solved for electrostatic fluctuations with period of the order of the electron bounce period. Using an appropriated Fourier expansion of the particle motion along the magnetic field, the complete time integration of the nonlocal perturbed distribution functions is performed. The dispersion relation for electrostatic modes is then obtained through the quasineutrality condition. It is found that strongly unstable electrostatic modes may develop provided that the current sheet is moderately stretched and, more important, that the proportion of passing particle remains small (less than typically 10%). This strong but finely tuned instability may offer opportunities to explain features of magnetospheric substorms.
I. INTRODUCTION
II. THEORETICAL FORMALISM
A. Description of the equilibrium state
B. Equation for the perturbed distribution functions
C. Charge density perturbation
D. Dispersion relation for electrostatic modes
III. EVALUATION OF THE PERTURBED DISTRIBUTION FUNCTIONS
A. Ion distribution function: Local approximation
B. A simple model for electron motion along the field lines
C. Nonlocal electron distribution function
IV. DIMENSIONLESS DISPERSION RELATION
A. Electron contribution to the dispersion relation
B. Ion contribution to the dispersion equation
C. Some numerical solutions of the dispersion relation
D. Inclusion of passing electrons
V. CONCLUSION
Key Topics
 Electrostatics
 23.0
 Dispersion relations
 15.0
 Magnetic fields
 15.0
 Particle distribution functions
 15.0
 Electrostatic waves
 14.0
Figures
Geometry of the 2D current sheet and notations used in this study. The magnetic field lines are bent curves in the (x, z)plane with length within a plasma sheet of half thickness L. The curvilinear coordinate, linked to the Bfield line, is denoted (ψ, y, ). The considered electrostatic perturbations propagate in the vicinity of a given magnetic surface . The origin of the curvilinear coordinate is at the lower end of the field line. The equatorial plane (x, y) cuts the field line at . The magnetic field at the center of the current sheet is B _{0} while its asymptotic value is B _{1}. The ratio varies typically between 0.01 and 0.3 in the Earth magnetotail.
Geometry of the 2D current sheet and notations used in this study. The magnetic field lines are bent curves in the (x, z)plane with length within a plasma sheet of half thickness L. The curvilinear coordinate, linked to the Bfield line, is denoted (ψ, y, ). The considered electrostatic perturbations propagate in the vicinity of a given magnetic surface . The origin of the curvilinear coordinate is at the lower end of the field line. The equatorial plane (x, y) cuts the field line at . The magnetic field at the center of the current sheet is B _{0} while its asymptotic value is B _{1}. The ratio varies typically between 0.01 and 0.3 in the Earth magnetotail.
Ion term A_{i} as a function of the normalized perpendicular wavenumber K as given by Eq. (47) . Several values of the stretching parameter are chosen: 0.2 (bottom curve), 0.3 (middle curve), and 0.4 (top curve).
Ion term A_{i} as a function of the normalized perpendicular wavenumber K as given by Eq. (47) . Several values of the stretching parameter are chosen: 0.2 (bottom curve), 0.3 (middle curve), and 0.4 (top curve).
Imaginary part of electron term A_{e} as a function of the normalized frequency q_{r} for and various stretching parameters ranging from 0.22 to 0.27. For these values of ε, the imaginary part of the electron term vanishes at a real frequency leading to existence of stable standing electrostatic oscillations of the current sheet. If ε is lower than 1/5, the imaginary part does not vanish for any q_{r} and if ε is higher than, say, 1/3, the real frequency making the imaginary part zero becomes too high to be valid in the simplified frame of the model. The frequency should stay indeed much less than the ion cyclotron frequency, typically of the order of 10.
Imaginary part of electron term A_{e} as a function of the normalized frequency q_{r} for and various stretching parameters ranging from 0.22 to 0.27. For these values of ε, the imaginary part of the electron term vanishes at a real frequency leading to existence of stable standing electrostatic oscillations of the current sheet. If ε is lower than 1/5, the imaginary part does not vanish for any q_{r} and if ε is higher than, say, 1/3, the real frequency making the imaginary part zero becomes too high to be valid in the simplified frame of the model. The frequency should stay indeed much less than the ion cyclotron frequency, typically of the order of 10.
Dispersion curves. Top panel: imaginary part of frequency q_{i} versus real frequency q_{r} . Bottom panel: wavenumber K versus q_{r} for various stretching parameters ε and a temperature ratio .
Dispersion curves. Top panel: imaginary part of frequency q_{i} versus real frequency q_{r} . Bottom panel: wavenumber K versus q_{r} for various stretching parameters ε and a temperature ratio .
Real part of the electron term A_{e} given by Eq. (45) as a function of frequency q_{r} for various stretching parameters ε using the same color code as in Figure 4 . The real part is computed along the dispersion line plotted on top panel of Figure 4 .
Tables
Spatial and temporal scales characterizing particle dynamics in the Earth plasma sheet with , L = 3200 km, , and .
Spatial and temporal scales characterizing particle dynamics in the Earth plasma sheet with , L = 3200 km, , and .
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