Analytical expressions have been obtained for three different types of cyclotron-resonant interaction of a test particle with a monochromatic electromagnetic wave: diffusion, phase bunching, and phase trapping. All three lead to changes in particle energy and pitch angle. They are evaluated for VLF waves and electrons at L = 5.5, to try to account for observed electron acceleration following magnetic storms. The results are that diffusion may be too slow, while phase bunching leads to deceleration. A maximal estimate of phase trapping, considered in isolation, leads to the acceleration of at least some 0.1 MeV electrons to ∼1 MeV in about 1 minute.
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 The general problem of the cyclotron-resonant interaction of a magnetized test particle with a prescribed, small monochromatic wave has a long history. Many of the key theoretical results are accessible from a direct analysis of the equations of motion [Bell and Inan, 1981; Bell, 1984, 1986], but Albert [1993, 2000] found a Hamiltonian formulation in slab geometry to be an efficient and powerful approach. The spatial variation of the background magnetic field, and resulting passage through resonance, were explicitly accounted for. The type of behavior was shown to depend on the ratio of the phase oscillation period at resonance to the timescale for passage through the resonance. When this ratio is large the phase at resonance is effectively random, resulting in pitch angle and energy diffusion. However, when this ratio is small the resonant interaction typically involves phase bunching, resulting in pitch angle and energy change with well-defined, definite signs, independent of the initial gyrophase and wave phase. Less often, when < 1 phase trapping can occur, potentially leading to large rates of systematic pitch angle and energy change.
2. Hamiltonian Formulation
 From the Hamiltonian for a magnetized test particle interacting with a small electromagnetic wave, a standard sequence of canonical transformations were made to isolate the resonances [Ginet and Heinemann, 1990; Ginet and Albert, 1991]. At each resonance the equations of motion were rewritten using the (unperturbed) distance along the field line z as the independent variable [Shklyar, 1986; Albert, 1993], and a corresponding “1–1/2 dimensional” Hamiltonian was found, with z playing the role of time. For cyclotron resonances (ℓ ≠ 0), this Hamiltonian was shown to have the form
with K1 proportional to the normalized wave amplitude ϵ = eEw/mcω. The variable I is proportional to the usual (relativistic) first adiabatic invariant μ, and ξ is a conjugate phase characteristic of the given resonance. Setting dξ/dt ≈ ∂K0/∂I = 0 recovers the standard resonance condition, which can be written as
Here P0 is |pz/mc|, where m is the rest mass, σz is the sign of the parallel velocity dz/dt, Ωc = |q|B/mc, and s is the sign of the particle charge,i.e., −1 for resonant electrons.The index of refraction is η and the wavenormal angle is α, with 0 ≤ α ≤ 180°. For the Landau resonance (ℓ = 0), I is constant and there is an analogous Hamiltonian of the form K0L (Γ, z) + K1L (Γ, z) sin ξ, where Γ is equal to γ2 + O(ϵ).
K has a time-dependent phase portrait which at each value of z (“time”) resembles that of a plane pendulum in the (I, ξ) plane. The island width W is 4[K1/(∂2K0/∂I2)]1/2, and the rotation frequency ω0 of particles deep inside the phase portrait island is (K1∂2K0/∂I2)1/2 [Lichtenberg and Lieberman, 1983]. The island itself moves at rate
with associated timescale T = W/(|dIres/dz|).
 When = 1/(ω0T) ≫ 1, the interaction of the particle and island occurs too quickly for ξ to adjust its arbitrary preexisting value, leading to a random, diffusive change in I [Shklyar, 1986]. This type of particle trajectory in the (I, ξ) plane is shown in Figure 6 of Albert . On the other hand, when = 1/(ω0T) ≪ 1, ∮ Idξ is an adiabatic invariant. Near the island a particle experiences phase bunching, typically circumnavigating the separatrix once before they separate, resulting in a well-determined net change in I. This type of particle trajectory in the (I, ξ) plane is shown in Figure 8 of Albert . Furthermore, under the right conditions the particle may enter the separatrix and become trapped there, subsequently moving with the island. When this happens, the value of I is dictated by the island motion. This type of particle trajectory is shown in Figure 11 of Albert . When ℓ = 0 there are analogous results in terms of K0L (Γ, z) and K1L (Γ, z).
 The condition ≪ 1 is essentially the so-called second-order resonance condition, d2ξdz2 ≈ 0. It is straightforward to show that is proportional to (dB/dz)/ϵ for any ℓ, so that small is favored by large wave amplitude and small parallel gradient of B.
3. Effects of a Single Resonance
 Interaction with each resonance produces a change in I (or Γ, if ℓ = 0). Regardless of the nature of this interaction, there will be an associated change in equatorial pitch angle PA and in energy E, the latter given by δE = δI/sℓ (when ℓ ≠ 0) or (when ℓ = 0).
 When > 1, the change in I at a resonance with ℓ ≠ 0 is
where σa is the sign of ∂2K0/∂z∂I [Shklyar, 1986; Albert, 2000], and similarly for δΓ if ℓ = 0. Since ξ at resonance is effectively random, this leads to diffusion. Albert  showed that the pitch angle diffusion coefficient agrees with the monochromatic limit of the bounce-averaged quasilinear expression for broadband waves [Lyons, 1974], provided ω ≪ Ωc. (A broadband wave spectrum was also assumed by Summers and Ma .) Typically, pitch angle diffusion dominates over energy diffusion but this is less true for decreasing index of refraction, which for VLF waves is favored by lower cold electron density, i.e., outside the plasmasphere.
3.2. Phase Bunching
 At a resonance where < 1, the change in I is
where σr is the sign of dIres/dz at resonance. There is an analogous expression for δΓ when ℓ = 0.
 Since δE = δI/sℓ when ℓ ≠ 0, equation (5) shows that δE has the same sign as −σzσr/sℓ. If Ωc/ω ≫ γ, equation (2) shows that this is the same as the sign of σr cos α. From estimates of K0 to evaluate σr, it was shown in section 6.1 of Albert  that at each resonance where equation (5) applies, δE has the same sign as −(dB/dz) cos α at the resonance. Similarly, when ℓ = 0, δE has the same sign as ∂2K0L/∂z∂Γ, which can be estimated as the sign of −(dB/dz) cos α as well. Thus in a dipole-like field, δE is proportional to −z cos α.
 These estimates show that at resonances where phase bunching (but not trapping) occurs, the energy change of the particle will be negative if the VLF wave is directed away from the geomagnetic equator (0 ≤ α < 90° when z > 0 and 90° < α ≤ 180° when z < 0). Repeated passages through the resonance will have the same result (combining linearly with time), quite likely overwhelming any energy gain due to diffusion (proportional to) at resonances where > 1. For phase bunching to cause energy gain, the wavevector must be directed towards the equator, or else the geomagnetic field must be so distorted that dB/dz < 0 (but not both).
3.3. Phase Trapping
Albert  found that particles initially outside the island could cross the separatrix at resonance and become trapped inside the moving island for long periods of time. This is phase trapping, as distinguished from phase bunching. For this to occur seems to require both ∼ 1 and decreasing at resonance, so that the particle crosses into the island but is adiabatically confined once inside. Even when had the right behavior, numerical simulation showed that whether a particle actually became trapped depended on its initial phase far from the island. Thus phase trapping at favorable resonances may be considered to occur probabilistically. Also, changes in the wave parameters (amplitude, frequency, wavenormal angle) are being ignored, but clearly have a large effect on the occurence of trapping. (Bell  estimated the variation of the wavenormal angle with latitude for VLF waves launched by transmitters on the ground or at low altitude.) Furthermore, if perfect north-south symmetry is assumed, the change in I (hence E and PA) will reverse as the phase-trapped particle crosses the equator; at the conjugate resonance, the particle will detrap, with no net change of I. This will be avoided if the wave parameters have changed during the motion.
 While a particle is phase trapped, its value of I changes at a rate given by equation (3), or the ℓ = 0 equivalent. Note that in phase bunching the particle moves in I (or Γ) opposite to the island, while a phase-trapped particle moves with the island.
4. Evaluation and Discussion
 The expressions for δE and δPA were evaluated numerically. The wave quantities ω, α, and Ew were fixed; η was determined using the dispersion relation of [Lyons, 1974], which requires a value of the cold electron density ne (all values were taken constant along a field line, except the geomagnetic field B). For a particle with given E and PA, the resonances along the slab field line were found, was evaluated, and the advective or diffusive changes were converted to rates using the bounce time τb, e.g., VE = δE/τb (if < 1) and DE = δE2/2τb (if > 1). These rates were summed over ℓ from −10 to 10.
 The top row of Figure 1 shows the diffusion rates for a range of E and PA, evaluated with L = 5.5, ω/2π = 1500 Hz, α = 10°, Ew = 1 mV m−1, and ne = 20 cm−3 (so that η ≈ 17, Bw ≈ 56 pT, and ϵ ≈ 6 × 10−5). All parameters were taken to be the same in both hemispheres (positive and negative z), except that α = 170° in the Southern hemisphere (i.e., wave directed away from the equator in both hemispheres). The middle row of Figure 1 shows the corresponding phase bunching results. A particle trajectory specified by (E, PA) may encompass several resonances, which may result in both D and V terms being nonzero. However, generally speaking, particles with large pitch angle and low energy (hence low v∥, so small dB/ds) experience phase bunching while other particles diffuse. As expected from the previous section, wherever VE due to phase bunching is nonzero, it is negative.
 If instead the waves were directed towards the equator, with all parameters otherwise unchanged, both VE and VPA would change sign. VE would be positive over a large area, and the large region of positive VPA would confine particles there. The phase bunched particles would quickly be accelerated up to greater than 1 MeV.
 Phase trapping is evaluated in the following ad hoc manner. At a resonance with 1/2 < ≤ 1, (z) is evaluated at a few (five) values of z during an island-crossing “time” T = W/(|dIres/dz|). If (z) is strictly decreasing, the particle is assumed to become phase trapped. The particle energy and pitch angle are then evaluated at successive locations z of the island, until either exceeds 1 or the mirror point or the equator is reached. This is repeated for all ℓ, and the maximum value of E experienced is used to compute VE (and corresponding VPA). The results are shown in the bottom row of Figure 1.
 The calculated values of D and V vary widely with E and PA. However, as a rough estimate, consider the values at E = 0.1 MeV, maximized over PA. With DE ∼ 2 × 10−7 MeV2/s, the timescale for diffusion to 1 MeV is 1/2DE ∼ 30 days, which seems too slow to account for the observed energization. With VE (phase bunching) ∼ −1.6 × 10−3 MeV/s, the deceleration timescale is ∼10 minutes, while with VE (phase trapping) ∼2.5 × 10−2 MeV/s, the acceleration timescale is less than one minute. This acceleration seems unrealistically rapid, but the trapping efficiency (≪1) has not been taken into account. This, combined with opposition from phase bunching, might account for the factor of ∼103 needed to obtain the observed timescale of ∼2 days. To evaluate this possibility further calls for a more detailed model of the trapping process, combined with simulation of the phase space evolution.
 In summary, analytical expressions have been obtained for three different regimes of cyclotron-resonant interaction with a monochromatic VLF wave. Numerical evaluation suggests that diffusion may be too slow to account for the observations, while nonlinear phase bunching leads to deceleration. Phase trapping, while subject to large modeling uncertainties, shows promise as a significant contributor to outer zone acceleration.
 This work was supported by the Space Vehicles Directorate of the Air Force Research Laboratory and by the Boston College Institute for Scientific Research under USAF contract F19628-00-C-0073.