The effect of electron and ion temperature on the refractive index surface of 1–10 kHz whistler mode waves in the inner magnetosphere

Abstract

Whistler mode waves in the magnetosphere play an important role in the energy dynamics of the Earth's radiation belts. Previous theoretical work has been extended to include ions in the fully adiabatic warm plasma theory. Using a finite electron and ion temperature of 1 eV, refractive index surfaces are calculated for 1–10 kHz whistler mode waves in the inner magnetosphere (L ≲ 2.5). For the frequencies of interest, a finite ion temperature is found to have a greater effect on the refractive index surface than the electron temperature and the primary effect is to close an otherwise open refractive index surface. Including a finite ion temperature is especially important when the wave frequency is just above the local lower hybrid resonance frequency. For wave frequencies more than ∼1 kHz above the local lower hybrid resonance frequency, including the ion temperature has a negligible effect on the refractive index surface calculation. The results are used to assess previous conclusions on whether in situ whistler mode sources can be realistically used to precipitate energetic electrons. It is found that the number of in situ sources needed to illuminate the inner plasmasphere (L≲2.5) with whistler mode energy may be greater than previously predicted.

Oblique Whistler-Mode Waves in the Earth’s Inner Magnetosphere: Energy Distribution, Origins, and Role in Radiation Belt Dynamics

“…such that the resonance condition can still be fulfilled near the maximum of the Bessel function corresponding to a given n, which becomes progressively impossible as N increases above ∼ 300). It can therefore be used only when the θ distribution of very oblique waves (of similar-mean-magnetic amplitudes) can be roughly approximated by a step function with only very strongly damped (vanishing) waves between θ ∼ θ(N = N max ) and θ r , as actually expected from previous studies (see Hashimoto et al 1977;Horne and Sazhin 1990;Kulkarni et al 2015, and the detailed discussion in Sect. 4.2).…”

“…Besides, Kulkarni et al (2015) have recently shown that including a finite ion temperature (equal to the electron temperature) could modify the full warm-plasma N more sensibly than the finite electron temperature alone when the wave frequency is just above the local lower hybrid resonance frequency ω LH = √ Ω ci Ω ce . For wave frequencies ω > 1.8 − 2ω LH , however, the ion temperature was found to have a negligible effect on the refractive index N .…”

Magnetospheric whistler mode ray tracing in a warm background plasma with finite electron and ion temperature

“…The basis of our numerical code is the VLF raytracer developed at Stanford University [ Golden et al , ; Golden , ] which implements numerical ray tracing by solving the Haselgrove equations[ Haselgrove , ; Kimura , ] at each time step. The background plasma properties such as density, magnetic field intensity, and temperature enter via the refractive index, μ , which is solved for from the dispersion relation and is modified by the finite temperature physics as described by Kulkarni et al []. Under the ideal cold plasma assumption, the dispersion relation is a fourth‐order equation With finite temperature, the dispersion relation becomes sixth order with where T s and m s are the temperature and mass of the charged species and s and k B and c are the Boltzmann constant and speed of light, respectively.…”

“…If simultaneous temperature effects on both electrons and ions are imposed, then the dispersion relation is a summation over the species: In equations , and the parameters A 1 , B 1 , and C 1 are warm plasma parameters that are functions of wave normal angle, plasma frequency, and cyclotron frequency. The parameters A 0 , B 0 , and C 0 are cold plasma parameters that are functions of wave normal angle and Stix parameters[ Stix , ].Further details of the formulation are provided by Kulkarni et al [].…”

“…Most relevant to a wide range of whistler mode phenomena, Kulkarni et al [] recently predicted modifications to the refractive index surface for 1–10 kHz whistler mode waves with the inclusion of finite electron and ion temperature. A key feature of the refractive index surface is whether it is “open” or “closed.” The motion of ions in addition to electrons is one physical effect that closes the refractive index surface [ Hines , ].…”