Equatorial Pitch Angle Research Articles

Bounce‐averaged advection and diffusion coefficients for monochromatic electromagnetic ion cyclotron wave: Comparison between test‐particle and quasi‐linear models

The electromagnetic ion cyclotron (EMIC) wave has long been suggested to be responsible for the rapid loss of radiation belt relativistic electrons. The test‐particle simulations are performed to calculate the bounce‐averaged pitch angle advection and diffusion coefficients for parallel‐propagating monochromatic EMIC waves. The comparison between test‐particle (TP) and quasi‐linear (QL) transport coefficients is further made to quantify the influence of nonlinear processes. For typical EMIC waves, four nonlinear physical processes, i.e., the boundary reflection effect, finite perturbation effect, phase bunching and phase trapping, are found to occur sequentially from small to large equatorial pitch angles. The pitch angle averaged finite perturbation effect yields slight differences between the transport coefficients of TP and QL models. The boundary reflection effect and phase bunching produce an average reduction of >80% in the diffusion coefficients but a small change in the corresponding average advection coefficients, tending to lower the loss rate predicted by QL theory. In contrast, the phase trapping causes continuous negative advection toward the loss cone and a minor change in the corresponding diffusion coefficients, tending to increase the loss rate predicted by QL theory. For small amplitude EMIC waves, the transport coefficients grow linearly with the square of wave amplitude. As the amplitude increases, the boundary reflection effect, phase bunching and phase trapping start to occur. Consequently, the TP advection coefficients deviate from the linear growth with the square of wave amplitude, and the TP diffusion coefficients become saturated with the amplitude approaching 1 nT or above. The current results suggest that these nonlinear processes can cause significant deviation of transport coefficients from the prediction of QL theory, which should be taken into account in the future simulations of radiation belt dynamics driven by the EMIC waves.

Statistical model of electron pitch angle diffusion in the outer radiation belt

We calculated the bounce averaged electron pitch angle diffusion coefficients using the statistical characteristics of lower band chorus activity collected by the Cluster mission from 2001–2009. Nine years of Cluster observations provide the distributions of the θ angle between wave vectors and the background magnetic field; and the distributions of the wave total intensity Bw2 for relatively wide ranges of the magnetic latitude λ, the magnetic local times, and the Kp indices. According to Cluster observations, the probability of observing a larger Bw2 increases with λ and depends upon the magnetic local time and Kp. We compared the obtained results with the diffusion coefficients that were calculated under an assumption of parallel whistler wave propagation with a constant intensity Bw2 = 104 pT2. The last calculations substantially underestimated pitch angle diffusion for the small equatorial pitch angles, αeq, but likely overestimates for αeq > 60°. An important increase in for αeq < 30° can be explained by a large mean value for the θ distribution and by an increase of Bw2 at λ > 15°. We took the probability density distribution of the wave mean amplitude into consideration instead of the averaged value. The obtained distribution of the diffusion coefficients indicated that approximately 20% of the most intense waves can provide the main portion of pitch angle diffusion for the dawn/day sector. For the dusk/night sector, wave intensity was significantly weaker and the relative importance of intense waves was not clearly pronounced.

Resonance zones and quasi-linear diffusion coefficients for radiation belt energetic electron interaction with oblique chorus waves in the Dungey magnetosphere

The resonance regions for resonant interactions of radiation belt electrons with obliquely propagating whistler-mode chorus waves are investigated in detail in the Dungey magnetic fields that are parameterized by the intensity of uniform southward interplanetary magnetic field (IMF) Bz or, equivalently, by the values of D=(M/Bz,0)1/3 (where M is the magnetic moment of the dipole and Bz,0 is the uniform southward IMF normal to the dipole’s equatorial plane). Adoption of background magnetic field model can considerably modify the determination of resonance regions. Compared to the results for the case of D = 50 (very close to the dipole field), the latitudinal coverage of resonance regions for 200 keV electrons interacting with chorus waves tends to become narrower for smaller D-values, regardless of equatorial pitch angle, resonance harmonics, and wave normal angle. In contrast, resonance regions for 1 MeV electrons tend to have very similar spatial lengths along the field line for various Dungey magnetic field models but cover different magnetic field intervals, indicative of a strong dependence on electron energy. For any given magnetic field line, the resonance regions where chorus-electron resonant interactions can take place rely closely on equatorial pitch angle, resonance harmonics, and kinetic energy. The resonance regions tend to cover broader latitudinal ranges for smaller equatorial pitch angles, higher resonance harmonics, and lower electron energies, consistent with the results in Ni and Summers [Phys. Plasmas 17, 042902, 042903 (2010)]. Calculations of quasi-linear bounce-averaged diffusion coefficients for radiation belt electrons due to nightside chorus waves indicate that the resultant scattering rates differ from using different Dungey magnetic field models, demonstrating a strong dependence of wave-induced electron scattering effect on the adoption of magnetic field model. Our results suggest that resonant wave-particle interaction processes should be implemented into a sophisticated, accurate global magnetic field model to pursue comprehensive and complete models of radiation belt electron dynamics.

Bounce‐averaged diffusion coefficients due to resonant interaction of the outer radiation belt electrons with oblique chorus waves computed in a realistic magnetic field model

We present the results of calculations of the bounce‐averaged pitch angle, mixed, and momentum diffusion coefficients in a dipole and two realistic field models (the T01s model for quiet and storm conditions). We consider resonant interactions of the outer radiation belt electrons with oblique chorus waves. We demonstrate that on the dayside, the use of a realistic magnetic field versus a dipole field only makes a significant difference for small equatorial pitch angles at energies larger than E = 1 MeV. On the nightside, the differences between the scattering rates calculated in the Tsyganenko and dipole models can reach several orders of magnitude at various equatorial pitch angles for electrons with E ≥ 0.5 MeV. The most significant changes in the scattering rates computed in the realistic and dipole magnetic fields occur during the geomagnetically active conditions. On the nightside, for E ≥ 0.5 MeV, the diffusion coefficients calculated in the Tsyganenko field show significant scattering near the edge of the loss cone that can produce loss of electrons to the atmosphere, while in the dipole model there is no scattering at small equatorial pitch angles. Our computations in the realistic field show that resonant interactions between electrons with E ≤ 1 MeV and chorus waves can be an effective net loss mechanism on both the dayside and the nightside. To explain the differences in the scattering rates associated with a change in the magnetic field model, we present the contribution of various resonant harmonics to the diffusion and examine the changes in the resonance condition.

Bounce-averaged Fokker-Planck diffusion equation in non-dipolar magnetic fields with applications to the Dungey magnetosphere

Abstract. We perform a detailed derivation of the bounce-averaged relativistic Fokker-Planck diffusion equation applicable to arbitrary magnetic field at a constant Roederer L. The form of the bounce-averaged diffusion equation is found regardless of details of the mirror geometry, suggesting that the numerical schemes developed for solving the modified two-dimensional (2-D) Fokker-Planck equation in a magnetic dipole should be feasible for similar computation efforts on modeling wave-induced particle diffusion processes in any non-dipolar magnetic field. However, bounce period related terms and bounce-averaged diffusion coefficients are required to be computed in realistic magnetic fields. With the application to the Dungey magnetosphere that is controlled by the intensity of southward interplanetary magnetic field (IMF), we show that with enhanced southward IMF the normalized bounce period related term decreases accordingly, and bounce-averaged diffusion coefficients cover a broader range of electron energy and equatorial pitch angle with a tendency of increased magnitude and peaking at lower energies. The compression of the Dungey magnetosphere can generally produce scattering loss of plasma sheet electrons ~100 keV on a timescale shorter than that in a dipolar field, and induce momentum diffusion at high pitch angles closer to 90°. Correspondingly, the strong diffusion rate drops considerably as a product of changes in both the equatorial loss cone and the bounce period. The extent of differences in all the parameters introduced by the southward IMF intensification also becomes larger for a field line with higher equatorial crossing. With the derived general formulism of bounce-averaged diffusion equation for arbitrary 2-D magnetic field, our results confirm the need for the adoption of realistic magnetic fields to perform accurate determination of electron resonant scattering rates and precise multi-dimensional diffusion simulations of magnetospheric electron dynamics.

Evolution of the plasma sheet electron pitch angle distribution by whistler-mode chorus waves in non-dipole magnetic fields

Abstract. We present a detailed numerical study on the effects of a non-dipole magnetic field on the Earth's plasma sheet electron distribution and its implication for diffuse auroral precipitation. Use of the modified bounce-averaged Fokker-Planck equation developed in the companion paper by Ni et al. (2012) for 2-D non-dipole magnetic fields suggests that we can adopt a numerical scheme similar to that used for a dipole field, but should evaluate bounce-averaged diffusion coefficients and bounce period related terms in non-dipole magnetic fields. Focusing on nightside whistler-mode chorus waves at L = 6, and using various Dungey magnetic models, we calculate and compare of the bounce-averaged diffusion coefficients in each case. Using the Alternative Direction Implicit (ADI) scheme to numerically solve the 2-D Fokker-Planck diffusion equation, we demonstrate that chorus driven resonant scattering causes plasma sheet electrons to be scattered much faster into loss cone in a non-dipole field than a dipole. The electrons subject to such scattering extends to lower energies and higher equatorial pitch angles when the southward interplanetary magnetic field (IMF) increases in the Dungey magnetic model. Furthermore, we find that changes in the diffusion coefficients are the dominant factor responsible for variations in the modeled temporal evolution of plasma sheet electron distribution. Our study demonstrates that the effects of realistic ambient magnetic fields need to be incorporated into both the evaluation of resonant diffusion coefficients and the calculation of Fokker-Planck diffusion equation to understand quantitatively the evolution of plasma sheet electron distribution and the occurrence of diffuse aurora, in particular at L > 5 during geomagnetically disturbed periods when the ambient magnetic field considerably deviates from a magnetic dipole.

The role of Shabansky orbits in compression‐related electromagnetic ion cyclotron wave growth

Electromagnetic ion cyclotron waves at high L values near local noon are often found to be related to magnetospheric compression events. These waves arise from temperature anisotropies in trapped warm plasma populations. There are several possible mechanisms that can generate these temperature anisotropies, including both energizing and nonenergizing processes. In this work we investigate a nonenergizing process arising from dayside bifurcated magnetic field minima. There are two kinds of behavior particles undergo in the presence of bifurcated minima: particles with high initial equatorial pitch angles (EPAs) are forced to execute so‐called Shabansky orbits and mirror at high latitudes without passing through the equator, while those with lower initial EPAs will pass through the equator with higher EPAs than before; as a result, perpendicular energies near the equator increase at the cost of parallel energies. By using a 3‐D particle tracing code in a tunable analytic compressed‐dipole field, we explore the effects of Shabansky orbits on the anisotropy of the warm plasma and contrast with the anisotropy resulting from drift shell splitting. We show that Shabansky orbits are an independent source of temperature anisotropy for warm dayside ions.

Efficient diffuse auroral electron scattering by electrostatic electron cyclotron harmonic waves in the outer magnetosphere: A detailed case study

This paper is a companion to a paper by Liang et al. (2011) which reports a causal connection between the intensification of electrostatic ECH waves and the postmidnight diffuse auroral activity in the absence of whistler mode chorus waves at L = 11.5 on the basis of simultaneous observations from THEMIS spacecraft and NORSTAR optical instruments during 8–9 UT on February 5, 2009. In this paper, we use the THEMIS particle and wave measurements together with the magnetically conjugate auroral observations for this event to illustrate an example where electrostatic electron cyclotron harmonic (ECH) waves are the main contributor to the diffuse auroral precipitation. We use the wave and particle data to perform a comprehensive theoretical and numerical analysis of ECH wave driven resonant scattering rates. We find that the observed ECH wave activity can cause intense pitch angle scattering of plasma sheet electrons between 100 eV and 5 keV at a rate of >10−4 s−1 for equatorial pitch angles αeq < 30°. The scattering approaches the strong diffusion limit in the realistic ambient magnetic field to produce efficient precipitation loss of <∼5 keV electrons on a timescale of a few hours or less. Using the electron differential energy flux inside the loss cone estimated based upon the energy‐dependent efficiency of ECH wave scattering for an 8‐s interval with high resolution wave data available, the auroral electron transport model developed by Lummerzheim (1987) produced an intensity of ∼2.3 kR for the green‐line diffuse aurora. Separately, Maxwellian fitting to the electron differential flux spectrum produced a green‐line auroral intensity of ∼2.6 kR. This is in good agreement with the ∼2.4 kR green‐line auroral intensity observed simultaneously at the magnetic foot point (as inferred using the event‐adaptive model of Kubyshkina et al. (2009, 2011)) of the location where the in situ observations were obtained. Our results support the scenario that enhanced ECH emissions in the central plasma sheet (CPS) can be an important or even dominant driver of diffuse auroral precipitation in the outer magnetosphere. This paper is an important compliment to recent work that has shown lower band and upper band chorus to be mainly responsible for the occurrences of diffuse aurora in the inner magnetosphere.

Long-term radiation belt simulation with the VERB 3-D code: Comparison with CRRES observations

[1] Highly energetic electrons in the Earth’s radiation belts are hazardous for satellite equipment. Fluxes of relativistic electrons can vary by orders of magnitude during geomagnetic storms. The evolution of relativistic electron fluxes in the radiation belts is described by the 3-D Fokker-Planck equation in terms of the radial distance, energy, and equatorial pitch angle. To better understand the mechanisms that control radiation belt acceleration and loss and particle flux dynamics, we present a long-term radiation belt simulation for 100 days from 29 July to 6 November 1990 with the 3-D Versatile Electron Radiation Belt (VERB) code and compare the results with the electron fluxes observed by the Combined Release and Radiation Effects Satellite (CRRES). We also perform a comparison of Phase Space Density with a multisatellite reanalysis obtained by using Kalman filtering of observations from CRRES, Geosynchronous (GEO), GPS, and Akebono satellites. VERB 3-D simulations include radial, energy, and pitch angle diffusion and mixed energy and pitch angle diffusion driven by electromagnetic waves inside the magnetosphere with losses to the atmosphere. Boundary conditions account for the convective source of electrons and loss to the magnetopause. The results of the simulation that include all of the above processes show a good agreement with the data. The agreement implies that these processes are important for the radiation belt electron dynamics and therefore should be accounted for in outer radiation belt simulations. We also show that the results are very sensitive to the assumed wave model. Our simulations are driven only by the variation of the Kp index and variations of the seed electron population around geosynchronous orbit, which allows the model to be used for forecasting and nowcasting.

Energetic particle phase space densities at Saturn: Cassini observations and interpretations

[1] Saturn's magnetosphere has been studied extensively by the Cassini spacecraft during the last 6 years. We present mission-averaged energetic proton and electron measurements obtained by the MIMI/LEMMS instrument onboard Cassini in an energy range from several 10 keV to several 10 MeV separated by equatorial pitch angle. We discuss the resulting radial profiles and energy spectra. The measured intensities are converted to phase space densities. The distribution of energetic particles is governed by a large variety of processes. For instance, moons absorb energetic particles, creating macrosignatures or microsignatures. We have found that the moon Rhea is partly responsible for a change in gradient of electron phase space densities. We show that, in contrast to larger distances, the particle distribution for L < 8 is not driven by radial diffusion alone. There, the particle profiles are significantly modified due to Saturn's Neutral Torus, plasma environment, E ring, injection events, and cosmic ray albedo neutron decay. Large parts of our analysis are focused near L = 7. There, protons are lost within the Neutral Torus and not the E ring. For electrons, we find that these two losses are of comparable rate but have discovered that neither process is the dominant driver of loss. We point out that intensity measured by a energy channel, such as in a particle instrument, can actually increase in the region of ring and torus instead of decrease. The importance of injection events is shown to be at least of similar importance as radial diffusion.

Resonant scattering of plasma sheet electrons leading to diffuse auroral precipitation: 2. Evaluation for whistler mode chorus waves

[1] Using the statistical wave power spectral profiles obtained from CRRES wave data within the 0000–0600 MLT sector under different levels of geomagnetic activity and a modeled latitudinal variation of wave normal angle distribution, we examine quantitatively the effects of lower band and upper band chorus on resonant diffusion of plasma sheet electrons for diffuse auroral precipitation in the inner magnetosphere. Whistler mode chorus-induced resonant scattering of plasma sheet electrons is geomagnetic activity dependent, varying from above the strong diffusion limit (timescale of an hour) during active times (AE* > 300 nT) with peak wave amplitudes of >50 pT to weak scattering (timescale of a day) during quiet conditions (AE* ∼2 keV) plasma sheet electrons in the inner magnetosphere. Efficient scattering by the combination of active time lower band and upper band chorus can cover a wide energy range from ∼100 eV to >100 keV and a broad interval of equatorial pitch angle, thereby accounting for the formation of observed electron pancake distribution. Decreased chorus scattering during less disturbed times can also modify the magnetic local time distribution of plasma sheet electrons. Compared to the effects of chorus waves, electron cyclotron harmonic wave-induced resonant diffusion coefficients are at least 1 order of magnitude smaller and are negligible under any geomagnetic condition, indicating that chorus waves act as the major contributor dominantly responsible for diffuse auroral precipitation in the inner magnetosphere. Chorus-driven momentum diffusion and mixed diffusion are also important. Lower band and upper band chorus can cause strong momentum diffusion of plasma sheet electrons in the energy ranges of ∼500 eV to ∼2 keV and ∼2 keV to ∼3 keV, respectively, which can significantly result in energization of the electrons and attenuation of the waves.

Effects of energy and pitch angle mixed diffusion on radiation belt electrons

Understanding the dynamics of the Earth’s radiation belts is important for modeling and forecasting the intensities of energetic electrons in space. Wave diffusion processes are known to be responsible for loss and acceleration of electrons in the radiation belts. Several recent studies indicate pitch angle and energy mixed-diffusion are also important when considering the total diffusive effects. In this study, a two-dimensional Fokker Planck equation is solved numerically using the Alternating Direction Implicit method. Mixed diffusion due to whistler-mode chorus waves tends to slow down the total diffusion in the energy-pitch angle space, particularly at smaller equatorial pitch angles. We then incorporate the electron energy and pitch angle mixed diffusions due to whistler-model chorus waves into the 4-dimensional Radiation Belt Environment (RBE) model and study the effect of mixed diffusion during a storm in October 2002. The 4-D simulation results show that energy and pitch angle mixed diffusion decrease the electron fluxes in the outer belt while electron fluxes in the slot region are enhanced (up to a factor of 2) during storm time.

Asymmetries in Saturn's radiation belts

We present both energetic proton and electron data from the main radiation belts of Saturn. When organized by L shell and equatorial pitch angle, data from Cassini's Magnetospheric Imaging Instrument reveal proton radiation belts that are highly symmetric in local time. The energetic electron radiation belts are asymmetric in two principal ways. Using data from two close passes of the planet, we find for energies near a few MeV, electron intensity levels are different between noon and midnight. Furthermore, when Cassini was inbound to Saturn, electron fluxes dropped precipitously before the spacecraft reached the main rings. Outbound, electron fluxes returned to high levels when the spacecraft moved outward of the main rings. In this paper, we suggest that electron flux level asymmetries are due in part to the presence of local time stationary particles in Saturn's inner magnetosphere. We also consider possible mechanisms to account the drop off of electrons near the ring edge on the dayside.

Low altitude energetic electron lifetimes after enhanced magnetic activity as deduced from SAC-C and DEMETER data

Abstract. When flux enhancements of energetic electrons are produced as a consequence of geomagnetic storm occurrence, they tend to vanish gradually when the magnetic activity calms down and the fluxes decay to quiet-time levels. We use SAC-C and DEMETER low altitude observations to estimate the energetic electron lifetimes (E=0.16–1.4 MeV, L=1.6–5, B=0.22–0.46 G) and compare the decay rates to those observed at high altitude. While crossing the radiation belts at high latitude, the SAC-C and DEMETER instruments sample particles with small equatorial pitch angles (αeq&lt;18° for L&gt;2.5) whereas the comparison is done with other satellite data measured mainly in the equatorial plane (for αeq&gt;75°). While in the inner belt and in the slot region no significant lifetime differences are observed from the data sets with different αeq, in the outer belt, for the least energetic electrons (&lt;500 keV), the lifetimes are up to ~3 times larger for the electrons with the equatorial pitch-angle close to the loss cone than for those mirroring near the equator. The difference decreases with increasing energy and vanishes for energies of about 1 MeV.

Three‐dimensional VERB radiation belt simulations including mixed diffusion

The evolution of relativistic electron fluxes in the radiation belts is described by the modified Fokker‐Plank equation in terms of the radial distance, energy and equatorial pitch angle. In this study we present numerical solutions of the two‐dimensional (2‐D) and 3‐D Fokker‐Planck equation including mixed diffusion terms. We use finite differences method with implicit numerical scheme, which is stable for any given time step. We evaluate the importance of the mixed diffusion in 2‐D and 3‐D cases of the Fokker‐Planck diffusion equation for radiation belts simulations. In both cases the mixed diffusion tends to inhibit local acceleration and results in lower relativistic electron fluxes, as compared to the simulation without mixed diffusion. The effect of the mixed diffusion terms is most significant at small pitch angles. The inclusion of mixed diffusion also tends to delay the formation of the peak in phase space density in the recovery phase of a storm. We also perform sensitivity simulation to the assumed wave models, which indicates that an accurate knowledge of the wave parameters is the most important factor.

Relativistic electron loss timescales in the slot region

Recent observations show that the decay rate of relativistic electrons measured at low altitudes in the slot region at L = 2 is an order of magnitude shorter than theoretical estimates based on CRRES wave data. Here we compare the decay rates of 2–6 MeV electrons measured at low altitudes by the SAMPEX spacecraft with those derived from CRRES wave observations. We show that pitch angle scattering by plasmaspheric hiss (0.1 &lt; f &lt; 2 kHz) is the dominant process responsible for electron loss in the outer slot region (2.4 &lt; L &lt; 3.0), but hiss alone cannot account for the observed loss timescales at lower L. Although SAMPEX samples small equatorial pitch angles (αeq ≈ 18°), this is not the dominant reason for the different timescales. We find that the decay of 2–6 MeV electrons measured by SAMPEX in the inner slot region (2.0 &lt; L &lt; 2.4) is most likely due to the combined effects of hiss and guided whistlers propagating with small wave normal angles. Unguided whistlers have little or no effect on the loss timescales. Magnetosonic waves may be as important as guided whistlers for electron loss under active conditions. Guided whistlers and fast magnetosonic waves increase the diffusion rates in a “bottleneck region” near αeq = 75°, enabling electrons with larger pitch angles to diffuse into the loss cone more effectively and hence the entire distribution function decays more rapidly. Even though the power of guided whistlers and magnetosonic waves may be two orders of magnitude less than hiss, they play a very important role in electron loss in the inner slot region.

The coupling of quasi-linear pitch angle and energy diffusion

Wave–particle interactions are described in the quasi-linear formulation by bounce-averaged diffusion coefficients for equatorial pitch angle α 0 and for momentum p , along with a mixed or cross diffusion coefficient D α 0 p . Because the cross terms complicate the associated time-dependent diffusion equation, and because D α 0 p has a somewhat unfamiliar character (e.g., it may be negative), it has frequently been omitted in numerical simulations of multidimensional diffusion. Generally, D α 0 p becomes increasingly important for small values of f pe / f ce , as does D pp , and for small widths of the wave frequency and wavenormal angle distributions. Here we use very simple models of diffusion to investigate the effect of the cross terms, and to demonstrate numerical problems associated with them.

Estimates of lifetimes against pitch angle diffusion

We consider timescales on which particle distributions respond to pitch angle diffusion. On the longest timescale, the distribution decays at a single rate independent of equatorial pitch angle α 0 , even though the diffusion coefficient, and the distribution itself, may vary greatly with α 0 . We derive a simple integral expression to approximate this decay rate and show that it gives good agreement with the full expression. The roles of both the minimum and loss cone values of the diffusion coefficient are demonstrated and clarified.

Effects of latitudinal distributions of particle density and wave power on cyclotron resonant diffusion rates of radiation belt electrons

Abstract We evaluate cyclotron resonant interactions of radiation belt electrons with VLF chorus, plasmaspheric ELF hiss and electromagnetic ion cyclotron (EMIC) waves. We assume that the Earth’s magnetic field is dipolar and that each wave mode has a Gaussian spectral density. The dependence of the resonant electron diffusion rates on the latitudinal distributions of particle density and wave power is examined. We find that while the diffusion rates can be sensitive to the latitudinal distributions of density and wave power, in general the sensitivity depends on wave mode, equatorial pitch-angle, electron energy and L-shell. We determine the effects of the latitudinal distributions of density and wave power on the electron precipitation loss timescale due to combined scattering by VLF chorus, ELF hiss and EMIC waves. Accurate modeling of radiation belt electron dynamics requires observational data on the global distributions of particle number density and wave power.

Stochastic modeling of multidimensional diffusion in the radiation belts

A new code for solving radiation belt diffusion equations has been developed and applied to the 2‐D bounce‐averaged energy pitch angle quasi‐linear diffusion equation. The code uses Monte Carlo methods to solve Itô stochastic differential equations (SDEs) which are mathematically equivalent to radiation belt diffusion equations. We show that our SDE code solves the diffusion equation with off‐diagonal diffusion coefficients in contrast to standard finite difference codes which are generally unstable when off‐diagonal diffusion coefficients are included. Our results are in excellent agreement with previous results. We have also investigated effects of assuming purely parallel propagating electromagnetic waves when calculating the diffusion coefficients and find that this assumption leads to errors of more than an order of magnitude in flux at some equatorial pitch angles for the specific chorus wave model we use. Further work is needed to investigate the sensitivity of our results to the wave model parameters. Generalization of the method to 3‐D is straightforward, thus making this method a very promising new way to investigate the relative roles of pitch angle, energy, and radial diffusion in radiation belt dynamics.