## 1. Introduction

[2] The study of electromagnetic wave propagation and wave-particle interactions in the Earth's radiation belts has received a great deal of interest [*Carpenter and Anderson*, 1992; *Carpenter et al.*, 2003; *Bell et al.*, 2004; *Platino et al.*, 2005]. Along with electromagnetic waves launched from ground based VLF transmitters, naturally occurring VLF radiation such as whistlers injected by lightning discharges and hiss and chorus emissions generated by the energetic magnetospheric plasma have been shown to influence the populations of these highly energetic electrons that reside within the Earth's radiation belts [*Abel and Thorne*, 1998]. It has been recently proposed [*Inan et al.*, 2003], that space-based transmitters may be used as in situ wave-injection instruments for the purpose of mitigating unwanted and harmful enhancements of energetic electron fluxes in the inner radiation belt. As suggested by *Albert* [2001], the dominant mechanism behind the precipitation of these energetic particles is pitch angle diffusion in the course of cyclotron resonant wave-particle interactions by whistler mode waves.

[3] Whether operating as transmitting or receiving elements, electric dipole antennas in a magnetoplasma are surrounded by an electrostatic sheath. This sheath can significantly alter the antenna properties (both near and far field) relative to those which would be in effect if the plasma remained uniform near the antenna surface. For receiving purposes, the sheath is on the order of a few Debye lengths and is well approximated by existing analytical theory. However, when used for transmit applications requiring the driving of the transmitting element at large voltages far in excess of the surrounding plasma potential, the sheath is highly nonlinear and its structure is generally not well known.

[4] Since the pioneering work of *Langmuir* [1929] and later by *Bohm* [1949] which formed the basis of the sheath models found in most literature on the subject, there has been considerable work performed in the areas of theory, simulation and experiment, some of which is now discussed.

### 1.1. Sheaths and Electric Dipole Antennas

[5] Early attempts at modeling the sheath effects on the terminal properties of dipole antennas include *Mlodnosky and Garriott* [1963] who used small signal analysis coupled with a fixed-capacitor analogy to derive closed-form expressions for the sheath radius, capacitance and resistance of a VLF dipole antenna moving through an ionospheric plasma. *Shkarofsky* [1972] extended the analysis of *Mlodnosky and Garriott* [1963] to include large signal excitation and the effects of an induced electromotive force (emf) resulting from the drift motion of the antenna at orbit speed (i.e., due to **v** × **B**_{0}). The following year *Baker et al.* [1973], using the same linear theory, incorporated a DC bias into their model resulting from spacecraft charging between the antenna and the satellite body on which the antenna was mounted. *Mlodnosky and Garriott* [1963], *Shkarofsky* [1972], and *Baker et al.* [1973] all used very crude first order approximations of the current and voltage on the antenna and greatly simplified the description of the sheath region through approximations such as uniform charge density and a simple exponential voltage dependence through the sheath. More recently, *Song et al.* [2007] used a theoretical formulation based upon that of *Shkarofsky* [1972] to analytically determine the terminal properties and sheath characteristics surrounding electrically short dipole antennas in the inner magnetosphere at large drive voltages relative to the ambient plasma potential. However, *Song et al.* [2007] ignored the ion current to the antenna, which is crucially important as we show later in the paper.

[6] In general, analytical sheath models are only valid under the assumption that *q*Φ/*k*_{B}*T* ≪ 1 where *q* is the charge of the particle, Φ is the potential, *T* is the temperature and *k*_{B} is Boltzmann's constant. This assumption allows for an implicit linearization of the set of fluid equations providing the steady state equilibrium distribution of the electrons within the sheath modified by the Boltzmann factor: *n*_{e} = *n*_{0} exp(−*q*_{e}Φ/*k*_{B}*T*), where *n*_{e} is the density variation of the electrons, *n*_{0} is the ambient density of the quasi-neutral bulk plasma, and the quantity exp(−*q*_{e}Φ/*k*_{B}*T*) is the Boltzmann factor for electrons.

### 1.2. Numerical Simulation Work

[7] When nonlinear behavior is prevalent and the simplifying assumptions underlying an analytical treatment are no longer justified, numerical simulation provides an invaluable tool for determination of antenna behavior in a plasma. Numerical methods generally fall into the categories of kinetic and fluid approaches.

[8] Particle In Cell (PIC) codes are used when wave-particle interactions are of interest since a fluid code by its nature cannot, in general, properly describe the influence of single particles. In a fluid approach, this individual particle motion is averaged out into collective behavior. A number of authors have examined the sheath dynamics and related phenomena using a PIC approach.

[9] Time-dependent sheath dynamics resulting from both positive and negative step function voltage changes on an electrode in a collisionless nonmagnetized plasma were also studied for both cylindrically and spherically symmetric geometries [*Calder and Laframboise*, 1990; *Calder et al.*, 1993]. The magnitude of the drive potentials used by *Calder and Laframboise* [1990] and *Calder et al.* [1993] were on the order of 10^{3} times the background plasma potential. Langmuir oscillations amplified by the electron-ion two-stream instability were evident in these simulations, which as noted by *Calder and Laframboise* [1990] can also be treated with a fluid description. However it was also suggested by *Calder and Laframboise* [1990] that plasma ringing exists due to the abrupt voltage changes which can affect the transient current collection on the electrodes for many plasma periods that cannot be accounted for in a fluid treatment. A similar analysis was made by *Borovsky* [1988] using a PIC approach in which he varied the potential on the electrode and noted the plasma ringing effects which were also amplified by the electron-ion two-stream instability.

[10] Despite their potential deficiencies, fluid models have successfully been applied to the sheath problem with good comparisons with PIC techniques. Some of the most pertinent works were in relation to the recent Space Power Experiments Aboard Rockets (SPEAR) program. This work includes *Ma and Schunk* [1989, 1992a, 1992b] and *Thiemann et al.* [1992], who used a two-moment fluid analysis to study the temporal evolution of particle fluxes on high-voltage spheres in a collisionless nonmagnetized plasma noting abrupt changes to the current collection as a result of the initial sheath formation. For large negative voltages, *Ma and Schunk* [1992a] and *Thiemann et al.* [1992] were able to reproduce the transient plasma ringing found in earlier PIC codes such as *Borovsky* [1988], *Calder and Laframboise* [1990], and *Thiemann et al.* [1992] (who performed a PIC-fluid comparison). *Labrunie et al.* [2004] performed a comparison between a one-dimensional Vlasov-Poisson kinetic simulation and a three-moment fluid code by studying ion-acoustic waves in a collisionless plasma. These authors highlight that fluid codes, even in the collisionless limit, can be very accurate, provided that certain conditions are met. The most relevant of these conditions is that the characteristic speeds of the phenomena of interest are not on the same order as the particle thermal velocities in which Landau damping is of concern.

[11] In this paper, we have developed an electrostatic simulation tool to examine the dynamics of the collisionless sheath using a two-species plasma fluid formulation. The antennas of interest here are located at magnetospheric points corresponding to *L* = 2 and *L* = 3 where the plasma consists of a fully ionized electron-proton plasma. Our paper utilizes a Finite-Volume (FV) method with the electrostatic fields provided through solution of Poisson's equation. Whereas past work has primarily involved the study of antennas using linear analysis, or in the case of the sheath formation has considered only DC potentials applied to two-dimensional symmetric geometries, we extend this past analysis to include AC applied potentials and three-dimensional geometries using fully nonlinear formulations. This paper thus presents significant contributions in the area of antenna-plasma coupling, most notably on the subject of sheath dynamics surrounding electric dipole antennas.