A model for the magnetospheric effects of ELF/VLF modulation of the auroral electrojet using high-power ground-based high-frequency radio waves is developed. The effects of ducted topside whistler wave power flux, as generated from ELF/VLF auroral electrojet modulation, on the magnetospheric equatorial pitch angle diffusion coefficient are computed for a range of ELF/VLF modulation frequencies. We find that the precipitation lifetime in the presence of this whistler wave flux is smaller than for purely collisional processes. We find that aspects of the model are in agreement with recent experimental observations.
 Recently, VLF emissions from modulation of the auroral electrojet using the HF Active Auroral Research Program (HAARP) transmitter in Alaska (L = 4.9) have been measured in the conjugate hemisphere [Inan et al., 2004]. The one-hop signals were believed to be amplified and accompanied by triggered emissions. This has demonstrated the use of ELF/VLF waves, as generated by auroral electrojet modulation, as a technique for probing the magnetosphere.
 By modulating the ambient current in the auroral electrojet, it is possible to generate, using ground-based high-power HF transmitters [Stubbe et al., 1982; Papadopoulos et al., 1990; Milikh et al., 1999] ELF and VLF radiation. The basic mechanism is as follows. Ground-based high-power HF transmitters can heat, through collisional absorption, the ambient ionospheric electrons in the D region producing a temperature perturbation δTe. This temperature perturbation can generate a perturbation in the conductivity tensor δσ(Te). In conjunction with the auroral electrojet electric field E, an oscillating current can be generated J ≃ δσ · E from ELF to VLF frequencies. The Hall conductivity is primarily modified below approximately 75 km while the Pedersen conductivity perturbation dominates for altitudes greater than about 90 km. ELF/VLF radiation generated by this process has been observed on the ground [McCarrick et al., 1990; Villaseñor et al., 1996] in the near field, on the ground in the far field in the conjugate hemisphere [Inan et al., 2004] and in space [James et al., 1984; Inan and Helliwell, 1985; Kimura et al., 1994; Yagitani et al., 1994]. VLF radiation from modulation of the auroral electrojet have frequencies of approximately 1–4 kHz. Some of the VLF radiation from auroral electrojet modulation enters the Earth-ionosphere waveguide and some fraction can leak into the magnetosphere and propagate as a whistler wave. Recent satellite observations have shown [Bell et al., 2004] that nonducted electromagnetic waves generated via ELF/VLF modulation of the auroral electrojet can be recorded over large magnetospheric regions. Resonant interactions with whistler waves are known to be an important loss mechanism for the Earth's radiation belt [Lyons et al., 1972; Lyons and Thorne, 1973; Inan et al., 1984; Abel and Thorne, 1998; Albert, 2001]. VLF waves can also be launched using HF heater-induced ducts at low latitudes [Starks et al., 2001]. Much work has been performed [Villaseñor et al., 1996; Rowland et al., 1996; Papadopoulos et al., 1990] dealing with the ELF/VLF radiation entering the Earth-ionospheric waveguide. However a quantitative model for the magnetospheric effects of the topside whistler wave flux, generated by artificial auroral electrojet modulation using ground-based HF waves, has not been developed. This model requires a time-dependent ionospheric ELF/VLF generation model which is coupled to magnetospheric plasma. The objective of this work is to develop a quantitative model. This model would be useful in order to develop a predictive capability for the magnetospheric effects of ELF/VLF generation from ionospheric auroral electrojet modulation. In section 2 we present the simulation model used to compute the ELF/VLF waves from auroral electrojet modulation. In section 3 we discuss the magnetospheric effects of the topside whistler waves and compute the pitch angle diffusion coefficient for a range of ELF/VLF frequencies. Finally in section 4 we summarize our results.
 To model the ELF/VLF generation we use a time-dependent electromagnetic plasma fluid code using the following equations:
 Here j = electrons (e) or ions (i) with ne, ni the electron and ion densities, Ve, Vi the electron and ion velocities, E is the electric field, B is the magnetic field, and νij the electron and ion collision frequencies.
 To simulate the ELF/VLF radiation source we impose a time-dependent current source at and near an altitude of z = 75 km of the form J(x, y, z, t) = J sin (ω0t). It has been shown [Papadopoulos et al., 1990] that the maximum conductivity perturbation is located in the altitude region of approximately 70–80 km. Assuming an ambient electrojet electric field of E = 30 mV/m and Pedersen and Hall conductivity perturbations given by Papadopoulos et al.  we find that J ≃ 3 × 10−9 A/m2. Here ω0/2π is the ELF/VLF driving frequency. This current perturbation gives for the effective radiative current moment M ≃ 2 × 104A − m. The ionospheric simulation volume extends to 1000 km (z direction), 200 km (x direction), and 100 km (y direction). The background ionospheric density profile used in this study is shown in Figure 1 and is taken from the International Reference Ionosphere (IRI) [Bilitza, 2001] and is representative of nighttime quiet conditions at L = 4.9. The collision frequencies are taken from previous studies [Hanson, 1965; Nicolet, 1953]. We work in a coordinate system with the geomagnetic field in the z direction.
Equations (1)–(6) are solved on a staggered mesh using finite difference techniques. The boundary conditions are outflow at the left and right boundaries. For the lower (ground) boundary we impose a conductivity of 10−3 mho/m which is typical of land areas [Wait, 1970]. We have found that the ground-reflected wave is smaller in amplitude, at high altitudes above the source region, than the source generated wave for the range of frequencies studied. For topside altitudes where Ωe ≫ ω ≫ Ωiequations (1)–(6) give the whistler dispersion relation ω2 + ω(ωpe2/Ωe) − kz2c2 = 0 with damping length ki−1 = 2(c/ωpe)[Ωe3/ω(νen + νei)2]1/2.
 Because of computational limitations we cannot extend the simulation volume to the magnetospheric equatorial region. Instead we adopt the following approach. The ELF/VLF Poynting flux is computed at the top boundary of the simulated ionospheric region at 1000 km. We then use a model [Inan et al., 1984] to compute the wave magnetic field amplitudes (Bw,eq) in the magnetospheric equatorial region, that is, Peq = (Ωe,eq/Ωe,1000)P1000 where P is the wave power density and Ωe is the electron gyrofrequency. We use a model for the equatorial plasma density [Angerami and Thomas, 1964; Chiu et al., 1979]. The background magnetic field model is computed using the International Geomagnetic Reference Field [Peddie, 1982] and the results of Inan et al. .
 The magnetospheric equatorial wave magnetic fields are then used to compute the pitch angle diffusion coefficient and associated precipitation lifetime for a range of ELF/VLF frequencies and ionospheric current source amplitudes. The magnetospheric wave-particle resonance given by ω − k∥v∥ = −nΩe/γ where ω is the wave frequency, k∥, v∥ are the components of the wave number and velocity parallel to the local magnetic field, n = 0, ±1, ±2, . is the harmonic resonance number, Ωe is the electron gyrofrequency, and γ is the relativistic Lorentz factor. Neglecting radial diffusion, the general equation for pure pitch angle scattering can be written [Lyons et al., 1972; Abel and Thorne, 1998] as
where f0 is the distribution function, α0 is the equatorial pitch angle, T gives the approximate variation of electron bounce time with pitch angle [Schulz and Lanzerotti, 1974] T(α0) = 1.38–0.31(sin α0 + sin1/2 α0) and Dαα is the pitch angle diffusion coefficient.
with P∥,m = (Ωe + γω)mec/μωcosθ with P∥ the electron parallel momentum and μ the index of refraction. Here λ, λm, α, θm, δθ, ωm, δω are the latitude, mirroring latitude, local pitch angle, wave normal angle, spread in wave normal angle, wave frequency, spread in wave frequency, x = tan θ, and erf is the error function. In addition,
Here ψ(θ) = exp[−(tanθ − tanθm)2/tan2δθ] and ϕn(θ) = 0.25[(1 + cosθ)Jn+1 + (1 − cosθ)Jn−1]2. We have modeled the dependence of the wave magnetic field as Bw2(ω, θ) ∼ exp−((ω − ωm)/δω)2 exp−((x − xm)/δx)2 with δx = tanδθ, xm = tanθm. From the simulation code we have computed the temporal and spatial Fourier transform of Bw(x, t), that is, Bw(k, ω) with k2 = k∥2 + k⊥2 and found that δω/ωm ≃ 0.1 and θm ≃ 0° with δθ ≃ 12° for the wave normal angle.
with Ne the equatorial electron density which is computed using a model [Chiu et al., 1979].
 We apply a perturbing source current which is Gaussian shaped and centered at z = 75 km. Figure 2 shows the wave By evolution at 500 μs for a frequency of f = 2.5 kHz. The maximum amplitude is 4 pT which is in general agreement with ground-based VLF observations [Villaseñor et al., 1996]. Figure 3 gives the By evolution at 3 ms after current turn on for the frequency of f = 2.5 kHz. The maximum amplitude is 0.5 pT which is consistent with satellite measurements [Kimura et al., 1991, 1994].
 We assume ducted propagation and compute the VLF wave magnetic field in the magnetospheric equatorial plane [Inan et al., 1984]. Figure 4 gives a summary, from several simulations, of the wave magnetic field in the equatorial plane for a range of perturbation current magnitudes in the auroral electrojet. The magnitude of the predicted wave magnetic field is an increasing function of the perturbing current for the range of currents studied. Figure 5 gives the wave magnetic field in the equatorial plane as a function of the VLF perturbing modulation frequency. The predicted wave magnetic field frequency dependence is consistent with previous experimental observations [Villaseñor et al., 1996; Stubbe et al., 1982]. The magnitudes of the predicted equatorial wave magnetic fields are consistent with recent observations using the HAARP ionospheric research facility at L = 4.9 [Inan et al., 2004]. Figure 6 shows the computed resonant electron energy as a function of equatorial pitch angle at a fixed VLF modulation frequency. As can be seen the resonant energy increases with equatorial pitch angle. The typical resonant energies are tens to a few hundred keV. Figure 7 gives the computed pitch angle diffusion coefficient using equations (7)–(8) for a range of pitch angles at several fixed VLF modulation frequencies. We have found that the n = −1 resonance gives the dominant contribution and that the typical resonant energies are a few to several tens of keV. The computed pitch angle diffusion coefficients are smaller than those from lightning, plasmaspheric hiss, and ground-based VLF transmitters [Abel and Thorne, 1998]. We have found that the magnetospheric equatorial wave magnetic fields and associated pitch angle diffusion coefficients depend on several factors, that is, ELF/VLF current magnitudes, the ionospheric plasma density and electron collision frequency, and the background magnetospheric plasma density and magnetic field together with the wave frequency. Figure 8 gives the approximate precipitation lifetime for a range of energies together with the lifetime associated with purely collisional processes. We have found that the whistler-induced precipitation lifetime is smaller than pure collisional processes.
 We have presented a quantitative model for the magnetospheric effects of ducted whistler waves generated by artificial ELF/VLF auroral electrojet modulation by ground-based high-power HF waves. This was accomplished using a time-dependent electromagnetic plasma fluid code which is driven by an imposed current source in the D region ionosphere. Using the magnetospheric wave magnetic field we have computed the pitch angle diffusion coefficient for a range of VLF frequencies. We have found that the whistler-induced precipitation lifetime is smaller than for purely collisional processes. In this study we have considered only ducted propagation whereas nonducted waves can have a greater effect on magnetospheric processes. As a result, the basic conclusions of this study have only limited generality. We will treat the nonducted case in a future report.
 This work was supported by the Office of Naval Research and the High Frequency Active Auroral Research Program (HAARP) office.