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Chapter 3 BOUNCE-RESONANCE TEST-PARTICLE MODEL The guiding center equations developed in Chapter 2 are adapted in this chapter for use in the test-particle simulation. Motion of particles in a model field line resonance (FLR) are expressed using geomagnetic dipole coordinates [Streltsov and Lotko, 1997]. Dipolar coordinates are chosen to make use of the precomputed FLR fields provided by Streltsov et al. [1998]. The dipole model in equation (3.1) is used to represent a background geomagnetic field. It is noted that the actual background magnetic field in the LLBL usually differs significantly from that of a simple dipole model, so the model results presented in subsequent chapters may be compared with observations only in a qualitative sense. The first section of this chapter defines the dipolar coordinate system. The second specifies the constraints placed on the model. The third section describes the phenomena of bounce-resonance acceleration. The explicit differential equations pertaining to this model, defined in dipole coordinates are derived in the final section. 3.1 DIPOLAR COORDINATE SYSTEM Dipolar coordinates are a locally orthogonal system with respect to an ambient, dipolar magnetic field. The precomputed FLR fields used in the calculations found in Chapter 6 are expressed in dipole coordinates. This coordinate system is derived from the dipole representation of the geomagnetic field, 17 18 RE3 Br , 31,000 3 rˆ 2 cos θˆ sin nT. r 3.1 The geomagnetic field is expressed in terms of the spherical polar coordinates r, and unit vectors rˆ , θˆ . Here, r is the geocentric radial position; the mean radius of the earth, RE, is 6371 km; and θ, measured from the north dipole axis, is sometimes referred to as the magnetic colatitude. Streltsov and Lotko [1997] define nondimensional dipole coordinates and their respective metric factors, h, in terms of spherical polar coordinates as, L r , RE sin 2 θ hL RE sin 3 θ 1 3 cos 2 θ μ φ, , h r sin θ , RE2 cos θ r2 h 3.2 r3 RE2 1 3 cos 2 θ . 3.3 The dipolar coordinate system is illustrated in Figure 3-1. The unit vector μ̂ , points along the dipolar magnetic field lines; L̂ and φ̂ represent perpendicular outward “radial-like” and azimuthal directions, respectively. If ζ represents any one of the dipole coordinates, then the unit vectors ζ̂ along each coordinate axis may be derived from the expression, ζˆ ζ ζ . Using the definitions for dipole coordinates given in 3.2 and 3.3, the dipolar unit vectors in terms of spherical polar coordinates are: ˆ ˆ rˆ sin θ 2 cos , L 1 3 cos 2 φ̂ , μˆ rˆ 2 cos θˆ sin 1 3 cos 2 . 19 Figure 3-1. Dipolar computational domain defined by Streltsov and Lotko [1997]. 3.2 MODEL CONSTRAINTS The particle motion is based on guiding center equations in the presence of an ambient dipole magnetic field and precomputed dispersive Alfvén wave fields [Streltsov et al., 1998]. The two-dimensional dispersive field line resonance model from Streltsov et al. [1998] specifies the parallel electric field ( E ), the meridonal perpendicular electric field ( EL ) and the azimuthal magnetic field ( B ) as functions of (L, , t). Because the fields are periodic in time, it is sufficient to know, for the test-particle integration, the spatiotemporal dependence of the fields for one wave period only. The specific particles of interest here are electrons. In subsequent equations, the particle charge is specified as the electron charge, q e , and the particle mass is the 20 electron mass, m me . The magnetic moment, M, is conserved as mentioned in the last chapter. The electrons here are non-relativistic, because the electron energies calculated in this model are less than 1 keV. Collisions between particles are neglected. Self-consistent contributions to the electromagnetic fields by the evolving electron populations are also neglected. To assess the effects of wave-particle interaction on electrons in a field line resonance, the electron is initialized at the magnetic equator in the wave electromagnetic field with a velocity chosen at random from a Maxwellian distribution with thermal energy of 100 eV. The phase of the time-periodic wave field at the initial time is chosen randomly with uniform weight over the wave period. The motion of 600,000 electrons, which drift and accelerate according to equations (2.8) and (2.9), is recorded to develop a statistical picture of the aggregate electron dynamics. The electron population is initially placed in the boundary layer, presumed to be located at L = 7.5. This location was chosen in order to make use of the precomputed FLR specified by Streltsov et al. [1998], which straddles L = 7.5. Although it is unusual for the LLBL to be observed at L = 7.5, the bounce-resonance criterion given in the next section is shown to depend weakly on the L-value and, more significantly, on the Alfvén speed in the layer. Therefore, the results on bounce-resonance acceleration discussed later in Chapter 6 are not expected to change qualitatively from those for a FLR located, for example, at L = 10, corresponding to a more typical radial location for the LLBL. 21 3.3 BOUNCE-RESONANCE ACCELERATION The physical picture of bounce-resonance interaction in a standing dispersive Alfvén wave is shown in Figure 3-2. Four half-cycles of the parallel electric wave-field, E||, from Streltsov et al. [1998] are plotted along a magnetic field line. An electron is initially near its northern hemispheric mirror point in the first frame and is allowed to interact with the parallel electric field. The electric field points from positive to negative and the electron accelerates in the opposite direction towards the equator. If the wave E|| has changed its direction (one-half wave cycle later) after the electron has reached the equator, it continues to see a negatively directed parallel electric field and is then accelerated toward its southern hemispheric mirror point. Upon reaching its southern hemispheric mirror point, the antisymmetric E|| has again evolved one-half cycle; the electron now sees a positive E|| that accelerates it back towards the equator. One-half wave cycle later when the electron again reaches the equator, the E|| remains positive along the electron’s trajectory and continues to accelerate the electron back to its northern hemispheric mirror point. The electrons are energized in this manner, causing them to travel farther which lowers their mirror points towards the atmosphere. Simultaneously, their bounce periods shorten, until they become sufficiently short, causing the electrons to slip out of phase with the wave E||. 22 e t t + 1/2 w E|| (V/m) t+w t + 3/2 w Distance along the field line (RE) Figure 3-2. Illustrates the wave-particle interaction due to bounce-resonance in the presence of a FLR parallel potential and dipole magnetic field. The top panel represents time t; the second frame is one-half wave period later; the third panels depicts time at one wave period later; and the last panel is at one full wave period and a half. The red dot represents an electron traveling along the field line depicted as the horizontal axis in earth radii. 23 The parallel energization of the electron by wave-particle interaction is optimal when the phase of electron motion is synchronized with the wave phase as illustrated in figure 3-2. The electron bounce period, B , is twice the wave period, w according to the illustration. The bounce period in a dipole magnetic field without the perturbation from the wave is [Baumjohann and Treumann, 1997], τ B LRE 3.7 1.6 sin eq W / m 1 / 2 . Here, W represents the total energy of the particle and eq is the equatorial pitch angle. The wave period, τ w , scales as the field line length divided by the Alfvén speed, vA, at the equator. For large values of L the field line length is approximated by 2.4LRE , so that, τ w 2.4LRE v A . Equating the expressions, τ B 2 τ w , results in the following, 3.7 1.6 sin eq 1 W me ve2 me v A2 2 4.8 ve v A 2 3.7 1.6 sin eq 3.4 3.4 The term within the absolute value brackets in equation (3.4) varies from 0.62 to 1.1 as the equatorial pitch angle varies from 2 to 0 radians, respectively. Therefore, bounce resonance occurs when the equatorial electron velocity is essentially the Alfvén speed in the LLBL. The motion of electrons in the LLBL with velocities significantly exceeding vA will be only weakly perturbed by the presence of the wave parallel electric field. Electrons with velocities much less than vA are initially accelerated nonresonantly by the slowly varying parallel electric field until their velocities become comparable to vA at which point a bounce resonance interaction occurs. This interaction is confirmed by the numerical studies of Chapter 6. 24 3.4 DIPOLE EQUATIONS OF MOTION The guiding center equations (2.8) and (2.9) are rewritten in terms of the orthogonal curvilinear dipole coordinate system described in Section 3.1. The FLR fields computed by Streltsov et al. [1998] are used in the test-particle calculations to evaluate the drift and acceleration motion of the electrons. This is facilitated using their dipole forms in equations (2.8) and (2.9). The orthogonal dipolar coordinate system (L, , ) represents the general orthogonal curvilinear coordinates ( e1 , e2 , e3 ) defined in Appendix A. Evaluating the drift terms in equation (2.8) explicitly with the vector components provided by the FLR model by Streltsov and Lotko [1997] and the ambient dipole magnetic field gives: vD E Lˆ E μˆ B φˆ B μˆ L B B 2 0 2 0 μˆ E L B Lˆ E B φˆ E L B0 3.5 B2 B02 v μˆ v L Lˆ v φˆ . where, v EL B B 2 B02 , vL E B B2 B02 , v EL B0 B2 B02 . The parallel acceleration in Chapter 2, equation (2.9), becomes, v e M 1 B0 E . m m h μ 3.6 The parallel gradient of the dipole magnetic field, B0, required to evaluate the mirror force in equation (3.6) is calculated analytically in terms of the spherical polar 25 coordinates r, θ . Representing the parallel gradient as μ̂ and defining the μ̂ unit vector in terms of spherical polar coordinates yields: 2 cos θ sin θ B0 . μˆ B0 2 2 r θ r 1 3 cos θ 1 3 cos θ Evaluating the partial derivative using magnitude B0 from (3.1) then provides, 3BE cos θ 1 B0 4 h μ RE r RE sin 2 θ 2 . 2 1 3 cos θ 3.7 Finally inserting (3.7) into equation (3.6) gives the explicit form of the parallel acceleration for use in the numerical calculations of Chapter 4, v e M 3BE cos θ E m m RE r RE 4 sin 2 θ 2 . 1 3 cos 2 θ 3.8 26