Radiation Belt Electron Dynamics in the Presence of ULF Waves#

A cut-away illustration showing the outer radiation belt (yellow - orange).

One of the outstanding questions in the physics of the magnetosphere surrounding our planet concerns the origin and dynamics of the Van Allen radiation belts. These are roughly torus-shaped regions surrounding the Earth, comprised of extremely high energy electrons and ions, extending from about three to eight Earth radii in the equatorial plane. High fluxes of these particles can cause damage to satellites and are harmful to humans in space. This places a high practical value on understanding the physics - especially because this region overlaps with the geostationary orbit path at 6.6 Earth radii, which is crucial for satellite based telecommunications and navigation. The radiation belts are also particularly interesting in their own right. For example, they may be regarded as a naturally occurring synchrotron particle accelerator, representing a local (and hence more easily measurable) example of acceleration processes thought to produce extremely high energy particles in more exotic astrophysical objects, such as stellar magnetic environments surrounding pulsars.

Introduction: Modeling Electron Dynamics in the presence of Global ULF Waves#

Fig. 1. Integrated electron flux data (for energies above the indicated threshold value) from the CRRES satellite mission, showing the variability of energetic electrons in the outer radiation belt, and also the formation of an inner radiation belt following a geomagnetic storm in March,1991.

Satellite measurements show that the size and shape of the outer radiation belt, and the density of relativistic populations of charged particles it contains, is strongly affected by geomagnetic storms. Significant enhancements in relativistic electron flux are often correlated with ULF wave activity within the magnetosphere during and following these events. A causal link between the electron flux and ULF wave observationsis postulated to be the drift resonance mechanism, whereby ULF waves strongly interact with electrons drifting through the magnetosphere at the ULF wave phase fronts. In doing so, the ULF wave does work by adiabatically transporting the electrons into a region of higher geomagnetic field strength, increasing their kinetic energy.

We model the effect of ULF waves on an initial distribution of relativistic electrons representing an unperturbed state of the radiation belts. An underlying assumption is that the density of relativistic electrons is sufficiently low that any work done on them is much smaller than the energy density of ULF waves. We can then neglect feedback of the electron dynamics on the ULF waves themselves. This assumption enables the ULF wave-fields to be calculated a-priori for use in the electron dynamics part of the model. We are therefore able to divide the problem of modelling radiation belt dynamics into two sub-problems: (a) ULF waves, and (b) test-particle electron dynamics, which may each be dealt with using models of varying complexity.

Modeling ULF Waves in the Magnetosphere#

The calculation of ULF wave-fields in a realistic magnetospheric geometry is difficult. Currently, in FDAM we have developed two types of model in which a ULF MHD fast wave incident from the magnetopause boundary couples energy to a field line resonance within the magnetosphere (see Appendix 1). These two models are summarized below.

The first model describes ULF wave fields in the magnetospheric equatorial plane. It is used in conjunction with a test-particle model that describes equatorially mirroring electron dynamics. Considering only equatorially mirroring electrons allows some simplifying assumptions about the wave field structure in the field-aligned direction to be made. For example, the only information regarding field-aligned structure that affects these electrons is the field-aligned gradient of the wave fields evaluated on the equatorial plane. A reasonable first step simplification is to treat the geomagnetic field as a cylindrical box model rather than a dipolar structure. In this model, field lines are considered straight (defining the z-direction in cylindrical polar coordinates (r,f,z) ), and are bounded by planar "ionospheres" at the top and bottom of the cylinder. A useful feature of the model is that simplifications in the field aligned direction enables the inclusion other important effects at a lower numerical cost. For example, this model includes the effect of day/night asymmetry and the presence of the dayside magnetopause in the wave field calculations. An example from this model is shown in the figure below, which displays an equatorial slice of the magnetosphere, with the dayside magnetopause (shown by contours of constant B) on the right. The animation colours (red is positive, blue is negative) show the radial electric field component of a 2 mHz ULF wave, which is continuously excited from a source in the afternoon sector of the magnetopause, and propagates anti-sunward along the dusk flank of the magnetosphere, exciting a field line resonance (FLR) closer to the Earth.

The second ULF wave model is more comprehensive in that it describes waves excited in a 3D volume representing the inner magnetosphere. It is useful in application against bounce-average and equatorially mirroring electron dynamics studies. This model does not include day/night asymmetry, but instead it includes the time dependence of the ULF wave source, as well as the coupling between ULF wave polarizations and field aligned eigenfunctions. An example from this model is shown in the figures below, which display a meridinal slice of space in the magnetosphere. Inner and outer boundaries in the model are defined by magnetic field lines, which connect the northern and souther ionospheres (the Earth is located at the extreme left in each figure and is not shown) and pass through the equator a radial distance from the Earth of 3 and 10 RE respectively. The animations show the time dependent amplitudes of radial (left) and azimuthal (right) perturbations to magnetic field lines associated with 2 mHz ULF waves launched from the outer boundary (to the right of each figure). The left figure mainly shows the fast wave propagation across field lines, while the right figure shows that some of this energy resonantly excites standing waves along magnetic field lines, called field line resonances (FLRs).

Electron Energisation by ULF Waves#

Electron trajectories in the frame of reference of the wave, showing the drift resonance (closed orbits).

The relativistic electrons in the radiation belt have three characteristic, cyclic motions: 1) Gyration about magnetic field lines, 2) Bounce motion between mirror points along magnetic field lines, and 3)Orbital drift motion around the earth. We are interested with the interaction between these electrons and global scale ULF waves with frequencies in the pc-5 range (1 to 20 mHz). This frequency range is much lower than the ocsillation frequencies associated with gyration and bounce motion (1 and 2 above), allowing these motions to be averaged over when computing the electron dynamics, and represented by constants of motion known as the first and second adiabatic invariants. However, the drift motion around the earth for these electrons occurs at the mHz range of frequencies, and enables the possibility of drift resonance.

The drift resonance effect between electrons and ULF waves occurs when the orbital drift speed of the electrons (given by the above equations of motion) nearly matches the phase speed of the wave. In this case the electrons experience the same sign electric field for time intervals much longer than a wave period, enabling them to significantly gain energy as they travel with the wave. Also, if the ULF wave frequency is well defined these electrons may become trapped between consecutive ULF wave fronts, and behave coherently with other trapped electrons (that is, be accelerated and decelerated at the same time) to produce peaks in the local population density of electrons as a function of energy.

Equatorially Mirroring Electrons#

A useful way to view the bounce motion of an electron along a magnetic field line is to think of a particle trapped in a 1D potential well (where in this case the potential is proportional to the magnetic field strength, which increases towards both poles). A particle with a given total energy oscillates, as it accelerates towards the bottom of the well, overshoots, then decelerates as it climbs the potential on the other side, and so on. If the magnetic field is north-south symmetric, then this minimum occurs on the equatorial plane. The lowest energy particles in this picture are confined to the bottom of the well - the equatorial plane. These are the equatorially mirroring electrons in the magnetosphere. Their dynamics are much simpler to analyze because they have no momentum parallel to the msagnetic field and, since they lie at the bottom of the potential well, are confined to the equatorial plane. For this reason, they are particularly useful as a test case for gaining an understanding of basic physical processes thought to be involved in radiation belt electron energisation.

Averaging over the electron gyro-motion, the equations of motion for electrons mirroring in the equatorial plane (using polar coordinates (r, f)) are:

where q is the electron charge, E_r and E_f are the radial and azimuthal components of the electric field, B is the magnitude of the magnetic field, M is the magnetic moment (the first adiabatic invariant), g is the relativistic correction factor, m_e and p are the electron mass and momentum, and c is the speed of light.

The magnetic moment is given by x , where x is the momentum perpendicular the magnetic field associated with the electron gyro-motion (note: x for equatorially mirroring electrons). Since M is a constant of motion, if the electron moves into a region where B has increased, then x must also increase. This means that the kinetic energy x is simply a function of magnetic field strength for equatorially mirroring electrons, given by:

Top: The time-evolution of local phase space density (PSD) versus L along a meridinal slice showing the coherent transport of electrons in response to a ULF wave pulse that decays in 20 wave periods; Bottom: The time-evolution of PSD averaged over f versus L* showing the formation of a local peak.

These equations for equatorially mirroring electrons have been used to understand the basic physical processes involved in the wave-particle interaction between electrons an ULF waves.

Azimuthal localisation of the ULF wave field can be described in terms of a sum of waves of constant amplitude with the same frequency and differing azimuthal wave numbers. Each of these waves has a different phase velocity, and therefore electrons with the same M will be drift-resonant with these waves at different locations in the magnetosphere. For low amplitude waves, these resonances may be considered independently, however if the wave amplitude is greater than a threshold value, the resonances interact with one another, causing electron transport - and hence energisation - over a much wider range than is possible with the individual resonances.

Observations by satellites of peaks in the number density of energetic electrons when plotted as a function of the third adiabatic invariant (L*) have been cited in the literature as strong evidence that electrons are accelerated locally by VLF waves, rather than adiabatically transported by ULF waves. One of the foundations of this argument is that most models for energisation by ULF waves involve diffusive transport. We show that a burst of narrow band ULF waves can cause coherent (or convective, rather than diffusive) radial transport, and result in the formation and growth of a peak in electron phase space density plotted against L*.

References:

1. The Effect of ULF compressional modes and field line resonances on relativistic electron dynamics,A. W. Degeling, R. Rankin, K. Kabin, R. Marchand and I. R. Mann, Planet. Space Sci., 55, 731-742, doi:10.1016/j.pss.2006.04.039,2007
2. The Generation of Peaked Relativistic Electron Distributions by Drift Resonance with Pc-5 ULF Waves, submitted to Journal Geophys. Res, 2007

Bounce Averaged Electron Dynamics#

The advantage of bounce averaging is that the number of dimensions required to model the dynamics of an electron can be reduced from six to two, with two constants of motion. This is reasonably straightforward if the following assumptions can be made: 1) any perturbations are electrostatic, 2) there is no electric field component parallel to the magnetic field. Unfortunately, while the second assumption holds for ULF waves described by ideal MHD, the first does not.

Northrop (1963) has developed a general bounce-averaging procedure that is applicable in this case. We use the approach he developed to calculate the bounce-average drift velocities for radiation belt electrons in the presence of ULF waves. Then Liouville's theorem can be used to construct the dynamics of the entire distribution function of electrons. This theorem is the cornerstone of all kinetic simulations, and states that the distribution of particles (i.e. the number of particles per unit phase space volume) acts like an incompressible fluid in phase space, provided only conservative forces are acting. Since the adiabatic invariants are constants of motion, each pair of invariants represents a slice through the distribution function that acts independently of any other slice. This enables parallelization of our model, which takes advantage of the adiabatic invariants, in a straightforward manner to recover the full distribution function of electrons.

The increased efficiency of this approach opens the way to comprehensively study the interaction between ULF waves and energetic electrons, by allowing meaningful comparisons between observations and simulations of events where correlations have been observed, and also by enabling studies designed to test the relative importance of the interaction with ULF waves against other possible energisation mechanisms.

In the ideal MHD model for ULF waves in the magnetosphere, electromagnetic perturbations can be described solely in terms of perturbations to the magnetic field lines themselves (see Appendix 1). That is, the magnetic field lines act as if they are coupled, vibrating strings. The bounce-averaging calculation requires integrations to be carried out along these field lines. Northrop shows that this can be achieved in a natural way by describing the electron dynamics using time dependent Euler potential coordinates. Euler potential coordinates have the property that they remain constant on a given magnetic field line (i.e. they are field line labels), hence any field line motion is taken into account by this choice of coordinate system.