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

Abstract

Abstract. We perform a detailed derivation of the bounceaveraged 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 <∼4 keV and radiation belt 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.

Parametric validations of analytical lifetime estimates for radiation belt electron diffusion by whistler waves

“…To quantify this effect, some lifetime calculations will also be performed taking into account the measured variation of B w (θ) in the day sector for 3.5 < L < 5.5: the wave power at λ > 20 • is then simply divided by 2 to 4 for θ > 60 • . Although the effects of a non-dipolar magnetic field could be important in the nightside region for large L ≥ 6 (see, e.g., Orlova and Shprits, 2010;Ni et al, 2011;Ma et al, 2012;Ni et al, 2012;Orlova et al, 2012), we restrict our considerations to a dipole field B = B 0 1 + 3 sin 2 λ/ cos 6 λ for averaging diffusion coefficients over electron bounce oscillations (Lyons et al, 1972). To estimate lifetimes, we use the expression in Eq.…”

Effect of plasma density on diffusion rates due to wave particle interactions with chorus and plasmaspheric hiss: extreme event analysis

“…To model wave-particle interactions, with a numerical code known as WAPI (wave-particle interaction, Boscher et al, 2007) for example, several input parameters must be properly defined including the characteristics of the waves (intensity, propagation angle) and that of the ambient plasma (electron density). The magnetic field configuration also plays an important role in wave-particle interactions (Orlova and Shprits, 2010;Ni et al, 2012;Artemyev et al, 2013), but this is not the subject of the present investigation and is not considered further here.…”

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

“…In the companion paper (Ni et al, 2012), we demonstrate that for 2-D non-dipole magnetic field models it is reasonable to use a bounce-averaged Fokker-Planck diffusion equation similar to that for PSD evolution in a dipole field, but with modified bounce period related terms and bounce-averaged diffusion coefficients. In the present study we choose the nightside Dungey magnetic field model at L = 6 to simulate the influence of a southward IMF, and focus on the effects of a non-dipole magnetic field on the Earth's diffuse auroral scattering due to upper band chorus (UBC) and lower band chorus (LBC).…”

“…The equations for resonant particle diffusion in pitch angle and energy were first developed by Lyons (1974a,b). The bounce averaged Fokker-Planck equation that describes evolution of phase space density f , using any 2-D magnetic field B = B(λ) at fixed L is given as (e.g., Schulz, 1976;Schulz and Chen, 1995;Summers, 2005;Ni et al, 2012):…”

“…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 nondipole 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.…”