The role of drift orbit bifurcations in energization and loss of electrons in the outer radiation belt

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2011ja016623, 2011 The role of drift orbit bifurcations in energization and loss of electrons in the outer radiation belt A. Y. Ukhorskiy, 1 M. I. Sitnov, 1 R. M. Millan, 2 and B. T. Kress 2 Received 7 March 2011; revised 27 April 2011; accepted 7 June 2011; published 10 September [1] Radiation levels in Earth s outer electron belt (L ^ 2.5) vary by orders of magnitude on the time scales ranging from minutes to days. Multiple acceleration and loss processes operate across the belt and compete in defining its global variability. One such process is the drift orbit bifurcation effect. Caused by coupling of the drift and bounce motions, it breaks the second adiabatic invariant of radiation belt electrons producing their transport in radius and pitch angle. In this paper we investigate implications of drift orbit bifurcations to the global state and variability of the outer electron belt. For this purpose we use three dimensional test particle simulations of electron guiding center motion in a realistic magnetic field model. We show that even at most quiet solar wind conditions bifurcations affect a broad range of the belt penetrating inside the geosynchronous orbit. This has an important practical implication for the analysis of experimental particle data: since the third adiabatic invariant is undefined for bifurcating orbit, the electron phase space density cannot be expressed in terms of three adiabatic invariants. We show that long term transport of electrons due to drift orbit bifurcations is a complex combination of large ballistic jumps and small amplitude diffusion in the second invariant and radial location. To model long term transport, we derive an empirical map of the second invariant and radial jumps at bifurcations. The map can also be implemented by other radiation belt models, which cannot directly account for the physics of drift orbit bifurcations. Drift orbit bifurcations can produce electron losses through the magnetopause escape and through pitch angle scattering into the atmospheric loss cone. Most electrons, however, can stay quasi trapped in the bifurcation regions for very long time periods. The pitch angle and radial transport due to drift orbit bifurcations lead to their meandering back and forth across the region producing mixing and recirculation of particle populations with different initial conditions. We show that this recirculation can greatly amplify electron energization by radial diffusion. Compared to the diffusion alone, the combined action of radial diffusion and drift orbit bifurcations can double electron energization at each recirculation cycle. Our results suggest that drift orbit bifurcations can play an important role in the buildup of increased electron fluxes in the storm recovery phase. Citation: Ukhorskiy, A. Y., M. I. Sitnov, R. M. Millan, and B. T. Kress (2011), The role of drift orbit bifurcations in energization and loss of electrons in the outer radiation belt, J. Geophys. Res., 116,, doi: /2011ja Introduction [2] Relativistic electron fluxes across the outer radiation belt (L ^ 2.5) at Earth vary dramatically in response to geomagnetic activity. During a geomagnetic storm fluxes may change by orders of magnitude producing either intensification or depletion of radiation levels in the belt [e.g., Reeves et al., 2003]. This nonlinear character of the outer belt response has been attributed to the existence of 1 Johns Hopkins University Applied Physics Laboratory, Laurel, Maryland, USA. 2 Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire, USA. Copyright 2011 by the American Geophysical Union /11/2011JA multiple mechanisms of electron acceleration and loss which compete in sculpting particle fluxes across the belt (see reviews by Friedel et al. [2002], Hudson et al. [2008], and Shprits et al. [2008a, 2008b]). Recent studies also suggest that individual mechanisms may be intrinsically complex. For example, local electron interaction with cyclotron waves can operate in both quasi linear and nonlinear regimes producing different signatures in particle acceleration and loss [e.g., Omura et al., 2007; Summers and Omura, 2007; Cattell et al., 2008; Albert and Bortnik, 2009]. Radial transport of electrons driven by various ultralow frequency waves can exhibit large deviations from quasi linear diffusion resulting in nonlinear response of the outer belt fluxes to similar geomagnetic storms [Ukhorskiy et al., 2006; Ukhorskiy and Sitnov, 2008]. Another effect which adds to 1of20

2 Figure 1. In a dayside compressed magnetic field of the inner magnetosphere charged particles can exhibit three types of drift bounce guiding center trajectories, if their first invariant is conserved (m = const). Donut like trajectories of stably trapped particles (cyan) lie on two dimensional tori defined by constant second (J) andthird(f) adiabatic invariants. Particles from the magnetopause drift loss cone (red line) intersect the magnetopause and are lost before completing a full drift orbit around Earth; only the second invariant J is conserved for these particles. Particles that exhibit drift orbit bifurcations (blue) violate the second invariant J at bifurcations. As a result, the third invariant is undefined for these particles. The trajectories were computed in the TS07D magnetic field at P dyn = 3 npa. Test particles were launched at r =( 8,0,0) with equatorial pitch angles of 80 (red), 59 (blue), and 20 (cyan). the complexity of outer belt dynamics is drift orbit bifurcations, also referred to as Shabansky orbits after Shabansky [1971]. Known for a long time [Northrop and Teller, 1960; Northrop, 1963; Roederer, 1970; Antonova et al., 2003], it recently received renewed attention [Öztürk and Wolf, 2007; Kim et al., 2008; McCollough et al., 2010; Wan et al., 2010] due to development of new particle tracing techniques and improved geomagnetic field models. In this paper we investigate the implications of drift orbit bifurcations for acceleration, transport and loss of the outer belt electrons. [3] In steady state, magnetized electrons in a dayside compressed magnetosphere can exhibit three types of trajectories (Figure 1). Stably trapped particles (shown in cyan color) participate in three distinct quasiperiodic motions: the gyromotion, the bounce motion and the gradient curvature drift around Earth, each associated with its own adiabatic invariant: m, J, and F (or L*) [e.g., Northrop, 1963]. Time scales for the different motions are separated by 1 to 3 orders of magnitude and all invariants are conserved. Particles from the magnetopause loss cone intersect the magnetopause and escape the belt before completing a full circle around Earth (red color). Since the drift trajectories are not closed in this case, only the first and the second invariants exist and are conserved (before particles are lost). Particles that undergo drift orbit bifurcations (blue color) violate the second invariant J at bifurcations, when the period of the bounce and the drift motions are no longer separated. This drives pitch angle and radial transport even when the magnetic field is constant [e.g., Antonova et al., 2003; Öztürk and Wolf, 2007; Wan et al., 2010]. Since the second invariant is not conserved, the drift orbits are not closed and the third invariant is undefined. [4] A detailed discussion of adiabatic invariance and its violations at drift orbit bifurcations is given in Appendix A. In the following we briefly discuss the mechanism of bifurcations illustrated in Figure 2, showing evolution of a stable (green) and a bifurcating (purple) drift trajectories. The bounce motion of a guiding center particle drifting around Earth is a one dimensional quasiperiodic motion in the effective potential defined by the magnetic field intensity along particle trajectory (B(s)). In a nonazimuthally symmetric field violation of strict periodicity is caused by changes in the B(s) profile due to particle drift around Earth. Along most of the drift bounce orbit B(s) is changing slowly compared the bounce period. Consequently the action variable I I sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J ¼ p k ds ¼ pi; I ¼ 1 Bs ðþ ds ð1þ is an adiabatic invariant of motion. In a constant magnetic field, the particle momentum p is conserved and conservation of J implies that the parameter I is also conserved. Throughout the rest of the paper I will be referred to as the second adiabatic invariant. From the conservation of p and the first adiabatic invariant it also follows that the magnetic field intensity at bounce points B m is constant along the drift orbit: ¼ p2? 2mB ¼ p2 ¼ const: 2mB m In the above expressions p k and p? are the parallel and the perpendicular components of particle momentum relative to the magnetic field direction. [5] In a dayside compressed geomagnetic field the distribution of B(s) along the field lines can exhibit two qualita- B m ð2þ 2of20

3 Figure 2. Schematic illustration of bifurcating (purple) and stably trapped (green) particle dynamics in a simplified case of a symmetric (day night and east west) dayside compressed magnetic field. (a j) Magnetic field profiles, (middle) the second invariant, and (k t) phase portraits along bifurcating and stable drift bounce trajectories. The magnetic field intensity at bounce points (B m ) is a constant of motion along with particle energy and the first invariant. Bifurcations occur when the profile of magnetic field intensity changes from (Figure 2a) U shape to (Figure 2b) W shape, and (Figure 2c) the magnetic field at local maximum at the equator becomes higher than B m. At bifurcations particle phase space trajectories cross a separatrix, and their second invariant exhibits jumps, which consist of two parts: an adiabatic factor of two (Figures 2l and 2m) decrease or (Figures 2m and 2n) increase due to the change in the area inside the phase space trajectory and a nonadiabatic change, whose magnitude depends on the location of the separatrix crossing. If, on the other hand, B m of a particle is higher than the maximum value of magnetic field at the equator (Figures 2f 2j), particle trajectory never crosses the separatrix. Its phase space orbit changes shape (Figures 2p 2t) but conserves the area within, corresponding to the second adiabatic invariant (~I 0 ). tively different profiles (Figures 2a 2c). On the night side, B(s) has a single minimum at the equator similar to a dipole field (Figures 2a, 2f, 2e, and 2j) (U profile). On the dayside, adjacent to the magnetopause, however, B(s) has a local maximum at the equator and two minima below and above the equator (Figures 2b 2d and 2g 2i) (W profile). Consider a particle initially bouncing across the equator from points with some value B m of the magnetic field intensity and gradient curvature drifting from the nightside to the dayside into the region where the magnetic field is compressed and has a local maximum at the equator. At some point of the drift trajectory the B(s) profile changes from the U to the W shape (Figures 2a and 2b). As the particle drifts further into the dayside, the height of the equatorial maximum in the W profile grows. If the magnetic field intensity at the maximum increases up to B m, the particle can no longer cross the equator and its drift orbit exhibits a bifurcation. To conserve the magnetic field intensity at the bounce points, the particle branches off the equator into one of the local B(s) minima pockets (Figures 2b and 2c). The trajectory traverses the dayside region either below or above the equator never crossing it until the point where the field at the equator decreases back to the B m value at the bounce points. The trajectory then bifurcates again and the particle resumes bouncing across the equator (Figures 2c and 2d). [6] At drift orbit bifurcations the particle phase space trajectory crosses a separatrix (Figures 2l and 2m), which divides the (p k, s) phase plane into three distinct regions. The region outside of the separatrix corresponds to the bounce motion across the equator, while two lobes connected at a saddle point correspond to trajectories trapped below and above the equator. While the particle approaches the separatrix its instantaneous bounce period increases logarithmi- 3of20

4 cally and in some small vicinity of the separatrix becomes comparable to the drift period. In this vicinity the quasiperiodic character of the bounce motion is broken, since the effective potential of the motion there is changing at the time scales of the instantaneous bounce period and can no longer be considered slowly varying. Close to the separatrix the second invariant is therefore not conserved. At two consecutive separatrix crossings corresponding to bifurcations off the equator and back, the invariant exhibits jumps. As a result by the time the particle resumes its motion across the equator it accumulates a nonzero change in the second invariant. Each bifurcation also leads to radial and pitch angle jumps. Consequently when the particle drifts back to its initial location on the nightside, the drift orbit does not close on itself as in the case of stably trapped particles (Figure 1). The third adiabatic invariant therefore is undefined for bifurcating orbits. [7] If, however, a particle bounces at points with B m larger than the maximum magnetic field intensity at the equator along the drift orbit (Figures 2f 2j), the drift orbit is stable. The shape of the particle phase space trajectory changes while the magnetic field profile changes from the U to the W shape along the orbit but its area and the second adiabatic invariants are conserved (Figures 2p 2t). [8] The jump of the second invariant at a separatrix crossing exhibits a sensitive dependence on the bounce phase, which determines location of the crossing. The closer it is to the separatrix saddle point, the larger is the jump in the invariant. The sign of the jump also depends on the bounce phase as well as on the initial value of the invariant prior to the jump. If an ensemble of particles has a common and sufficiently large initial value of the second invariant I and is evenly distributed in drift phase, after one drift around Earth particles will exhibit semi random jumps in invariant spreading the invariants over their initial value in a diffusive manner: hdii =0,h(DI) 2 i 0. On the other hand, if the initial invariant value of the ensemble was close to zero, all particles will exhibit only positive jumps producing advection toward larger invariant values: hdii > 0. These two regimes of particle transport due to separatrix crossings are referred to as the diffusion and the ballistic regimes. As will be shown later in the paper, long term evolution of the second invariant produced by multiple crossings over many drift periods is determined by a complicated interplay of both the diffusion and the ballistic regimes. [9] The theory of separatrix crossings was previously invoked for understanding many important aspects of particle dynamics in the magnetosphere including chaotic scattering and acceleration of particles in stretched magnetotail geometries [e.g., Chen and Palmadesso, 1986; Büchner and Zelenyi, 1989; Chen, 1992; Ashour Abdalla et al., 1993; Holland et al., 1996] and local acceleration of particles by cyclotron waves in an inhomogeneous magnetic field [e.g., Albert et al., 1993; Albert and Bortnik, 2009]. The goal of this paper is to address implications of separatrix crossings at drift orbit bifurcations to long term transport, acceleration, and loss of the outer electron belt. [10] The rest of this paper has the following structure. In section 2 we outline the details of our 3 D test particle model, the primary tool used in this paper for the analysis of drift orbit bifurcations. We then discuss the extent of the outer belt region susceptible to drift orbit bifurcations. We show that even under most quiet solar wind conditions, drift orbit bifurcations affect a broad range of drift shells with the inner boundary inside geosynchronous orbit. In section 3 we discuss the details of the second invariant and radial jumps at drift orbit bifurcations. It is shown, in particular, that the invariant change over a drift orbit around Earth is dominated by nonadiabatic ballistic jumps at the first separatrix crossing followed by an approximate doubling of the invariant value at the second crossing. Section 4 describes long term radial and pitch angle transport in the outer belt due to multiple separatrix crossings over many drift periods. To model long term transport we use results of our test particle simulations to derive an algebraic map relating radial and second invariant jumps over one drift period to their initial values. It is shown that long term transport of relativistic electrons due to drift orbit bifurcations is a complex interplay of large ballistic jumps and small amplitude diffusive variations of their second invariant and radial position in the belt. It can produce electron losses via magnetopause escape and scattering into the atmospheric loss cone. Most of the outer belt electrons undergoing drift orbit bifurcations stay quasi trapped meandering back and forth across the bifurcating regions. This radial transport redistributes particles in both pitch angle and radial position. In section 5 we estimate transport rates due to drift orbit bifurcations and show that they are comparable to radial transport rates due to quiet time magnetic fluctuations. We also address how bifurcations affect other transport and energization mechanisms simultaneously operating in the belt. In particular, it is shown that the recirculation can greatly amplify the efficiency of electron acceleration by radial diffusion, which can play an important role in building up enhanced electron fluxes in storm recovery phase. The conclusions are given in section The Extent of Drift Orbit Bifurcations [11] To analyze properties of drift orbit bifurcations we use a three dimensional test particle model. The impact of drift orbit bifurcations on the overall state of the outer electron belt depends on the range of electron drift shells susceptible to the bifurcations, which is controlled by the magnetic field geometry. It is therefore important to use a realistic magnetic field model for computing electron drift orbits. In this paper we use the latest Tsyganenko Sitnov (TS07D) empirical model [Tsyganenko and Sitnov, 2007], which was thoroughly tested against experimental data [Sitnov et al., 2008, 2010]. The only solar wind parameter varied in this study was the solar wind dynamic pressure. All other model coefficients were computed for quiet time magnetospheric conditions at 05:00 UT on 8 March 2008 (a full set of model coefficients is given at [12] Since drift orbit bifurcations conserve the first adiabatic invariant, computations are conducted in the guiding center approximation. We use Hamiltonian equations of relativistic guiding center motion (Appendix B), which conserve the energy and adiabatic invariants better [e.g., Cary and Brizard, 2009; Wan et al., 2010] than classical guiding center formulations [e.g., Northrop, 1963]. While our Hamiltonian formulation is valid for all realistic magnetospheric conditions including large E B drift, all computations in this paper are conducted in a static magnetic field, in which 4of20

5 Figure 3. Even during most quiet conditions, drift orbit bifurcations affect a broad range of the outer electron belt. The plot shows equatorial Poincaré sections of electron drift orbits at two values of the solar wind dynamic pressure (P dyn ). Solid black curves correspond to stably trapped particles, while scattered dots indicate bifurcating trajectories. Solid red circles show the geosynchronous orbit. case the equation system reduces to the approximation derived by Brizard and Chan [1999]: 8 _p k ¼ >< >: B* rb B* k _R ¼ p k B* þ c ^b rb ; m B* k e B* k ð3þ where R is the guiding center position, e is the electron charge, g is the relativistic factor, and B* =B + cp k e r ^b. [13] We first determine the extent of drift orbit bifurcations in the outer electron belt. The range of drift shells affected by bifurcations depends on the solar wind and geomagnetic conditions controlling the geomagnetic field [Wan et al., 2010]. Bifurcations are most sensitive to the value of the solar wind dynamic pressure (P dyn ), which controls the daynight asymmetry of the field. We focus our analysis on quiet time conditions and low values of P dyn when the geomagnetic field does not exhibit large variability and the effects of drift orbit bifurcations are not masked by other transport and acceleration processes operating in the belt. To visualize bifurcating regions we use Poincaré sections in the equatorial plane. To simplify determination of the magnetic equator, these and all subsequent computations discussed in this paper were carried out for the zero tilt angle of the Earth s magnetic dipole in the GSM coordinate system, in which case the magnetic equator is defined by z = 0 plane. Figure 3 shows the Poincaré sections of electron drift orbits at two low values of the dynamic pressure. The sections are obtained by launching guiding center particles at midnight at 90 pitch angles from 18 radial locations distributed between 9 and 5 R E. Each crossing of the equatorial plane corresponds to a point of the section. Solid black contours indicate stable drift orbits, while scattered crossings correspond to bifurcating orbits. Red circles show the location of geosynchronous orbit. Even at the lowest value of the solar wind dynamic pressure, P dyn = 2 npa, the bifurcating region penetrates inside geosynchronous orbit. A modest increase of the pressure from P dyn = 2 npa to P dyn = 4 npa substantially widens the bifurcating region, pushing the nightside boundary to as low as 5.5 R E. The above results show that even during quiet solar wind conditions a broad range of the outer belt drift shells is susceptible to bifurcations. It is therefore necessary to further assess how the bifurcations affect transport and acceleration mechanisms, such as radial diffusion, simultaneously operating in the belt. [14] The third adiabatic invariant of a trapped particle is approximately equal to the magnetic flux F encompassed by its guiding drift shell [e.g., Roederer, 1970]: I F ¼ A dl ¼ B 0R 2 E L* ; ð4þ where A is the magnetic vector potential, B 0 is the magnetic field intensity on the Earth s surface at the equator, and L*is the generalized L parameter, which is commonly used for quantifying radial transport in the radiation belts [e.g., Schultz and Lanzerotti, 1974]. In the case of a dipole field L* is equal to the dimensionless radius L = r/r E of the equatorial cross section of the particle drift shell. Integration in the above equation is carried out along a closed contour which lies in the guiding drift shell of the particle. Since the bifurcations break the second adiabatic invariant, the drift shells of bifurcating particles are not closed. Consequently the third adiabatic invariant and L* are undefined in the bifurcating region. Thus the first important practical consequence of drift orbit bifurcations is that the phase averaged electron phase space density cannot be expressed in terms of adiabatic invariants 5of20

6 Figure 4. Nightside structure of the drift orbit bifurcation region. (a) Drift bounce trajectory of a bifurcating particle started at x = 7.5 RE at midnight with 90 equatorial pitch angle (blue). Red and cyan stripes show constant B surfaces along the midnight meridian (bounce points of the bifurcating trajectory lie on the red surface). (b) Boundaries of the drift orbit bifurcation region (grey lines) and (approximate) transport characteristics (blue lines), i.e., lines of Bm = const relating the radial and second invariant jumps at drift orbit bifurcations. Initial locations of ensembles of 200 test particles (red dots), their distribution after one drift orbit around Earth (black symbols), and average displacements of ensembles at different initial conditions (red segments). across the full radial extent of the outer belt as assumed by many radiation belt models [e.g., Schultz and Lanzerotti, 1974]. conserved we use the adiabaticity parameter [Kress et al., 2007]: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u ¼ t B i; j 3. The Structure and Mechanics of Drift Orbit Bifurcations [15] Since L* is undefined in the drift orbit bifurcation region, to quantify radial and pitch angle transport of electrons due to bifurcations we use the I LM space, where I is the second invariant (equation (1)), and LM is the radial distance to the drift shell at the magnetic equator at midnight. Here and everywhere else in the paper I and LM are expressed in Earth radii. This provides a dissection of particle drift orbits at different pitch angles along the midnight meridian. The structure of the bifurcating region is illustrated in Figure 4. This and all subsequent computations in this paper were carried out in a constant magnetic field computed with the TS07D model with Pdyn = 3 npa. In the absence of an electric field particle energy is conserved. Consequently, drift orbit bifurcations are energy independent. [16] The tailward extent of the bifurcating orbits is limited to the region where electrons are magnetized; that is, their first invariant is conserved. In a constant magnetic field the first invariant of relativistic electrons can be violated in the regions where the magnetic field varies on the spacial scales comparable to electron gyroradius, such as the regions of high magnetic field curvature in the tail [e.g., Birmingham, 1984]. To verify whether the first adiabatic invariant is ð5þ where r is the particle gyroradius. Particles are considered to be magnetized, if < 0.05 everywhere along their drift bounce trajectory. The tailward boundary of the region where electrons are magnetized depends on both particle energy and pitch angle. Under selected magnetic field conditions for 1 MeV particles with large equatorial pitch angles (aeq 90) the magnetization breaks at LM 9, which we consider to be the tailward boundary of the bifurcation region. [17] The range of the second invariant (equatorial pitch angle) values susceptible to drift orbit bifurcations depends on the radial location (see Figure 4b). The maximum value of the invariant (minimum value of the equatorial pitch angle) is defined by the boundary between the stably trapped and the bifurcating particles. On the other hand, the magnetopause loss cone (where exists) sets the minimum invariant (maximum equatorial pitch angle) values of bifurcating orbits. For instance, our test particle simulations show that at LM = 9, 8, and 7 particle orbits undergo drift orbit bifurcations if their equatorial pitch angles are the following ranges: aeq = [25, 40 ], [36, 60 ], and [51, 90 ], respectively. According to these geometric considerations a broad region of the outer belt phase space is affected by drift orbit bifurcation, which 6 of 20

7 Figure 5. (a) Invariant jumps of an ensemble of 400 particles after one drift orbit around Earth as a function of their initial bounce phase. Particles were initiated at L M0 = 7 with I 0 = The blue curve corresponds to the TS07 magnetic field model, and the red curve shows invariant jumps in a modified TS07 model with forced east west symmetry. Sharp peaks in the invariant distributions correspond to separatrix crossings in the saddle point vicinity. (b and c) Comparisons of two drift bounce trajectories of the ensemble with the same initial bounce phase in realistic (blue) and symmetric (red) magnetic field models. reinforces the importance of addressing their role in dynamic variability of radiation belt fluxes. [18] Drift orbit bifurcations produce radial and pitch angle jumps of bifurcating electrons. In the I L M space they are constrained to the curves defined by constancy of the magnetic field intensity at bounce points as follows from conservation of energy and the first invariant: B m ði; L M ; I 0 ; L M0 Þ ¼ const; ð6þ where I 0 and L M0 are the initial values of the second invariant and radial location prior to the jumps. We will refer to these curves as transport characteristics. Thus, for a given drift bounce trajectory the change in the radial position of a bifurcating electron is uniquely related to the change in its second invariant. [19] To compute the radial and pitch angle transport produced by two consecutive orbit bifurcations, i.e., after an ensemble of particles completed one full drift orbit around Earth, we simulated the guiding center motion of 200 test particles. In each simulation particles were initiated at L M = 8 with a common value of the second invariant and were distributed in the bounce phase. For this purpose particles of the ensemble were positioned along the magnetic field line piercing the equatorial plane at L = 8 at midnight and were distributed at equal intervals of the arc length (ds) between the conjugate bounce points corresponding to the given value of the second invariant. The results are shown in Figure 4b for different initial values of the second invariant. The initial conditions are indicated by red dots. Distributions of I and L M values after one drift orbit are shown with black symbols. Average displacements are shown with red line segments connecting the initial locations with ensemble averaged positions after one drift. Straight line fits into the distributions are shown with blue lines. It follows from Figure 4b that smaller initial values of the second invariant (larger pitch angles) correspond to larger spreads as well as larger average displacements of the ensemble, as expected from shifting initial conditions from the diffusion to the ballistic regimes. For the ensemble of particles initially located on the edge of the loss cone it takes only one drift around Earth to populate almost the entire range of drift shells susceptible to bifurcations. The average position of this ensemble shifts toward larger invariant values, i.e., further into the tail. [20] From Figure 4b it also follows that transport characteristics in the I L M space are approximately straight lines. Depending on the initial values (I 0, L M0 ) the characteristics can intersect both the tailward boundary of the bifurcating region and the boundary separating it from the drift loss cone. Drift orbit bifurcations can therefore produce two types of electron losses: (1) the outward electron transport followed by their pitch angle scattering at the regions of high magnetic field curvature and loss into the atmosphere and (2) the inward radial transport causing an increase in particle pitch angle which places them into the magnetopause drift loss cone leading to their escape through the magnetopause on the time scales less than one drift period. Both of these loss mechanisms are addressed in section 4. [21] In a realistic magnetic field lacking north south and east west symmetries a new value of the second invariant after one drift orbit around Earth and two (outside in and inside out) separatrix crossings can be written as 8 I 1 ¼ ϰ 1 >< 2 I 0 þ DI þ ð 1 ; I 0 ; L M0 Þ >: I 2 ¼ 2 I 1 þ DI ð 2 ; I 1 ; L M0 Þ; ϰ 2 where the first term in the sum on the right hand side of both equations corresponds to a new invariant value after the crossing, which reflects the adiabatic change in the invariant, whereas the second terms correspond to the invariant jumps due to violation of the adiabaticity of the bounce motion in the separatrix vicinity (see Appendix A for details). The coefficients ϰ 1 and ϰ 2 reflect the north south asymmetry of the magnetic field at orbit bifurcations, i.e., the ratio of areas inside two separatrix lobes. For a symmetric magnetic field ϰ 1 = ϰ 2 = 1. Drift orbit bifurcations produce the largest effect on particles with small initial values of the invariant I 0 0 (ballistic regime). A single ð7þ 7of20

8 Figure 6. Distributions of the second invariant values (I k+1 ) after one drift orbit around Earth (two drift orbit bifurcations) at different initial values of the invariant (I k ) and different locations in the bifurcating region (L M ). Distributions at three different locations have a qualitatively similar shape. The magnitude of the jumps increases with increase in L M, i.e., increase in the day night asymmetry of the magnetic field along the drift orbit. separatrix crossing in this case can produce a large jump DI + > 0 in the invariant, such that the second crossing will correspond to the diffusion regime producing only a small jump in the invariant: I 1 DI + DI. The total change in the invariant for these particles can be therefore estimated as I 2 I 0 2 DI þ ð 1 ; I 0 ; L M0 Þ: ð8þ ϰ 2 [22] From the above qualitative considerations it follows that most substantial changes in the second invariant values of an ensemble of particles after a drift orbit around Earth are caused by nonadiabatic jumps at the first separatrix crossing followed by approximate doubling of the invariant value at the consecutive crossing. This contradicts the conclusions of Wan et al. [2010], who suggested that the dominant effect is caused by the east west asymmetry in the magnetic field which produces large adiabatic (independent of the bounce phase) changes in the invariant. To validate our qualitative conjecture we ran the following numerical experiments illustrated in Figures 5 and A2a. We first computed evolution of an ensemble of 400 guiding center particles in the TS07D model at P dyn = 3 npa over one drift period around Earth. Particles were initially at L M0 = 7 with I and evenly distributed in the bounce phase. Figure A2a shows the distribution of the second invariant as a function of the initial bounce phase of the ensemble after the first (blue line) and the second (red line) drift orbit bifurcations. The obtained distributions are in a good agreement with the above qualitative considerations (equation (8)). After the first jump the invariant distribution exhibits a sharp peak, indicative of invariant jumps at separatrix crossings in the vicinity of the saddle point. As predicted the invariant values after the second bifurcation are approximately twice large than after the first crossing. [23] To address the role of asymmetry in the field the above simulation was compared with a similar simulation in a modified magnetic field model obtained from the TS07D by forcing the east west symmetry, i.e., the field west half of the magnetosphere was equated to the field on the eastern half: B(y) B( y). The results are shown in Figure 5. Neither the distribution of the second invariant after two bifurcations (Figure 5a) nor the individual particle trajectories (Figures 5b and 5c), show any substantial qualitative or quantitative differences between the two cases. The results suggest that in the considered cases the east west asymmetry does not play a substantial role in modifying the invariant changes at separatrix crossings and that the largest changes in the second invariant are caused by nonadiabatic jumps in the ballistic regime. 4. Long Term Transport Due to Multiple Bifurcations [24] The second invariant jumps at drift orbit bifurcations exhibit sensitive dependence on the bounce phase, the invariant values prior to the bifurcations, as well as on the proximity of drift orbits to the magnetopause. Figure 6 shows distributions of the invariant values after two drift orbit bifurcations (I k+1 ), i.e., a full drift orbit around Earth, at different initial values (I k ) at three different locations in the bifurcating regions (L M ) obtained from the guiding center simulations. Each initial value was sampled with an ensemble of 200 test particles. The thick red line connects average invariant values, while thin red lines show the lower and the upper envelopes. Invariant jumps at all radial locations exhibit similar qualitative dependence on the initial conditions. At small initial values, the jumps are predominantly ballistic: hdii = hi k+1 I k ih(di) 2 i 1/2, shifting to the diffusion regime at larger values: hdii 0, h(di) 2 i 0. Interestingly, at intermediate initial values the average change in the invariant can be negative. The magnitude of jumps in the invariant increases with increase in L M ; the larger the daynight asymmetry of the magnetic field, the larger the jumps of the invariant at orbit bifurcations. [25] From the obtained profiles of the invariant jumps at different initial conditions it can be expected that long term transport of particles due to multiple bifurcations is an intermittent combination of ballistic jumps and diffusion. For instance, if an ensemble of particles has a small initial value of the invariant, after one drift period all particles of the 8of20

9 Figure 7. Radial transport of radiation belt electrons due to drift orbit bifurcations quantified by h(dl(t)) 2 i of ensemble of particles at different initial values of the second invariant. (a) Weak diffusive transport at large initial value of the second invariant; (b) combination of diffusive and ballistic transport at the intermediate value of the second invariant; and (c) large ballistic jumps followed by diffusion at small initial value of the second invariant. ensemble experience large positive jumps in the invariant accompanied by radial outflow distributing particles over the diffusion region. Subsequent bifurcations result in particle diffusion both in the invariant and in radial position. Depending on the location in the belt, some of the particles may escape from the region either through the tail boundary or through the drift loss cone and get lost either in the atmosphere or through the magnetopause. Some of the particles eventually reacquire small invariant values and then experience large jumps returning them into the diffusion regime. [26] Statistical analysis of long term transport of radiation belt electrons due to drift orbit bifurcations requires simulating dynamics of large number ( 10 6 ) of particles over long periods of time ( 100 drift orbits). Motivated by minimizing computational time as well as deriving a simplified description of radial and pitch angle transport due to bifurcations which can be used in radiation belt models to capture physics of the bifurcation process, we sought a reduced description of transport due to orbit bifurcations. The changes in the invariant I and the radial position L M of a particle due to two drift orbit bifurcations over the course of one drift orbit depend only on three parameters: their initial values and the bounce phase value prior to the bifurcations. Particle transport due to drift orbit bifurcations can therefore be described by a dynamical map relating the change in three state variables (, I, L M ) over a drift orbit. To derive such a map we conducted a series of test particle simulations of the guiding center motion of ensembles of particles over one drift orbit around Earth. The initial values of I and L M for the ensembles where varied to cover the entire bifurcating regions. The initial values of covered the full range of the bounce phase values between 0 and p. The obtained results were used to derive the following empirical map: 8 I kþ1 ¼ I k þ DIð k ; I k ; L Mk Þ >< L Mkþ1 ¼ L Mk þ DLð k ; I k ; L Mk Þ ð9þ >: kþ1 ¼ k þ D mod 2; where D is a random shift in the bounce phase between 0 and 2p. The jumps DI and DL in the first two equations of the map are derived by numerical interpolation in the (, I, L M ) space, while the third equation assumes that after a drift period the bounce phases of an ensemble of particles are randomized due to bifurcations and phase mixing. Due to its simple algebraic structure the above map can be efficiently used in modeling long term transport of electrons due to drift orbit bifurcations. [27] Figure 7 shows examples of radial transport due to drift orbit bifurcations of particle ensembles initiated at L M0 = 7 and three different values of the second invariant calculated with the use of map (9) over 100 drift periods. Note the difference in scale of the vertical axis showing the extent of radial expansion of particle populations with different initial conditions. Simulations shown in Figure 7a correspond to the initial second invariant value close to the boundary between the bifurcating and stably trapped orbits. Obit bifurcations in this case produce very small jumps in the invariant and radial position leading to a weak diffusion. After as many as 100 drift orbits particles have expanded only over DL The opposite limit is shown in Figure 7c, which corresponds to a very small initial value of the invariant. During the first orbit particles exhibit a semi coherent outward jump into the diffusion region. Subsequent bifurcations result in broadening of particle distribution along their transport characteristic. After the particle population expands across the whole bifurcation region, the h(dl) 2 i width stops growing, while the individual particles keep meandering back and forth along the characteristic. Figure 7b corresponds to an intermediate value of the invariant initial value where the collective transport is affected by both ballistic jumps and radial diffusion producing transport rates varying with time. According to these results long term transport due to orbit bifurcations cannot be reduced to either diffusion or advection but rather is a complex time varying interplay of both of them. [28] As was discussed in section 3 transport characteristics can intersect the tailward and drift loss cone boundaries of the bifurcating region producing atmospheric and magnetopause losses of the radiation belt particles. To quantify 9of20

10 Figure 8. Magnetopause escape and atmospheric losses due to current sheet scattering produced by drift orbit bifurcations. (a) Total number of particles in the system normalized to the number of particles at the simulation start (blue line); normalized loss rates (red line). (b) Relative contributions of the magnetopause escape and current sheet scattering to the losses at different locations in the bifurcating region after 50 drift periods. (c) Loss time scales (measured in number of drift orbits) at different locations in the region and different initial values of the second invariant. Losses affect the regions adjacent to the tailward and drift loss cone boundaries. Particles from the bulk of the bifurcation regions stay quasi trapped, meandering back and forth across the region along their transport characteristics. losses due to drift orbit bifurcations the following simulation was conducted. A set of initial conditions (I 0, L M0 ) was constructed to densely cover the full extent of the bifurcating region. Ensembles of 200 test particles each were initiated at each initial condition of the set. Radial and pitch angle transport due to orbit bifurcations were modeled with the use of map (9) over 200 drift periods (iterations of the map), bookkeeping the magnetopause and atmospheric losses separately. The results are summarized in Figure 8. Figure 8a shows the total number of particles in the system normalized to the number of particles at the beginning of the simulation (blue line) and the normalized loss rates (red line) as function of time measured in drift periods around Earth (N D ). It follows from Figure 8a that most of the losses take place within the first several drift periods which points to their association with large ballistic jumps and escape either through the tailward or the drift loss cone boundary. After 50 drift periods the system reaches a steady state when only ]1% of particles are lost over each subsequent drift period. From the entire bifurcating region 20% of particles are lost after 100 drift orbits. [29] The relative contributions of the magnetopause drift loss cone escape and magnetotail losses due to current sheet scattering are shown in Figure 8b. Figure 8b shows fractions of particles lost from different radial locations across the region after first 50 drift orbits around the Earth. Not surprisingly, the drift shells adjacent to the tailward boundary are dominated by losses through the magnetotail, while the central parts of the bifurcating region are dominated by losses through the magnetopause drift loss cone. [30] The lifetimes t of particles in the bifurcating region with different initial conditions are shown in Figure 8. The lifetime at given location (I, L M ) of the bifurcating region was defined as the minimal number of drift periods that takes all particles of the ensemble initiated at this point of the region to escape either through the drift loss cone or the magnetotail boundaries. As can be seen from Figure 8 the losses mostly affect the regions adjacent to the boundaries while particles from the bulk of the bifurcating regions stay quasi trapped. 5. Recirculation and Energization [31] From our numerical simulations discussed in previous sections it follows that drift orbit bifurcations produce jumps in particle pitch angle and radial position along transport characteristics. Long term pitch angle and radial transport due to drift orbit bifurcations has a complex character defined by a combination of ballistic and diffusive transports. It can result in particle loss both through the atmospheric precipitation and the magnetopause escape. Most of the bifurcating particles however are quasi trapped in the bifurcating region meandering back and forth across the region along transport characteristics. To understand whether this recirculation process can be important for global variability of the belt, the recirculation rates must be compared to transport rates due to other mechanisms simultaneously operating in the belt. [32] Broadband small amplitude fluctuations in the magnetic field can produce radial diffusion of radiation belt electrons across their drift shells [e.g., Schultz and Lanzerotti, 1974]. Since magnetic field fluctuations are present even during most quiet conditions due to the variable nature of the solar wind driving, to asses the importance of transport due to drift orbit bifurcations we compare their recirculation rates with the magnetic diffusion rates. For this purpose we use the magnetic diffusion coefficient derived by Brautigam and Albert [2000] from CRRES spacecraft data and parameterized by the level of geomagnetic activity using the Kp index: D B LL ¼ 100:506Kp 9:325 L 10 ½days 1 : ð10þ [33] To make a comparison, we calculated transport due to drift orbit bifurcations over 10 drift periods for ensembles of particles starting at different initial conditions in the bifur- 10 of 20

11 Figure 9. Radial transport due to drift orbit bifurcations quantified by root mean square measured of radial jumps of ensembles of 200 particles initiated at different locations in the bifurcating regions after 10 drift periods. (a) Full distribution, (b)positive (outward) jumps only, and (c) negative (inward) jumps only. cating region and then computed the following root meansquare measure for each ensemble rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X DL rms ði 0 ; L M0 Þ ¼ ðl Mk ðt ¼ 10T D Þ L M0 Þ 2 ð11þ k in three different ways: (1) using the full ensemble of particles L Mk at each initial conditions (Figure 9a); (2) using only the particles that moved outward: D (+) L : L Mk L M0 >0 (Figure 9b); and (3) particles that moved inward: D ( ) L : L Mk L M0 < 0 (Figure 9c). The largest root mean square values of >1.5 are attributed to outward expansion of ensembles with small initial values of the second invariant around L M0 7.5 produced by large ballistic outward jumps. [34] The root mean square spread of an ensemble of particles due radial diffusion can be estimated from pffiffiffiffiffiffiffiffiffiffiffiffiffi the definition of the radial diffusion coefficient: DL rms B = 2TD B LL. Taking T =10T D, where T D is the drift period of 1 MeV particles in a dipole magnetic field at L = 7.5, from expression (10) we obtain: DL rms B 1atKp = 1 and DL rms B 2atKp = 2, which are similar to the root mean square values of radial transport due to bifurcations. From the above estimates it follows that drift orbit bifurcations produce radial and pitch angle transport at the rates comparable to other processes simultaneously acting in the belt. It is therefore important to determine whether the bifurcations can modify the action of other processes in sculpting the outer radiation belt. [35] In the following we analyze how recirculation of the outer belt electrons across the L shells susceptible to drift orbit bifurcations can amplify acceleration produced by radial diffusion. From the conservation of the first invariant (m = p 2? /2mB) it follows that the acceleration of a guiding center electron in the process of its inward radial transport from point 1 to point 2 in the belt is limited to the ratio of the magnetic field intensities B 1 and B 2 in these pffiffiffiffiffiffiffiffiffiffiffiffi points, thus for a highly relativistic electron: g 2 /g 1 B 2 =B 1. To attain additional acceleration without breaking the first adiabatic invariant, the electron must be transported back outward from point 2 to point 1 along a different path in the phase space without loosing all of its acquired energy and then be radially transported again inward from point 1 to point 2 along the original path. The idea of amplifying the efficiency of electron acceleration by adding such recirculation process to ordinary radial diffusion was first proposed by Fujimoto and Nishida [1990]. The outward leg of the recirculation loop in their model was suggested to be the low altitude outward diffusion due to interaction with the electric field fluctuations with average amplitudes of mv/m. The efficiency of this recirculation scenario was questioned by subsequent analysis largely due to the lack of a stable source of high amplitude electric field fluctuations capable of scattering particles across L shells at low amplitudes. [36] Drift orbit bifurcations naturally produce radial recirculation without any additional energy source and even more importantly without altering particle energy. Such recirculation process can greatly amplify the efficiency of particle acceleration produced by inward radial transport. Consider for example a particle with a large value of the second adiabatic invariant which is transported from the geosynchronous orbit at L M = 6.6 to L M = 6 directly by radial diffusion. The adiabatic acceleration for ap relativistic particle in this case is limited to the factor of ffiffiffiffiffiffiffiffiffiffiffiffi B 2 =B If, however, the particle has a small value of the second adiabatic invariant, it also exhibits drift orbit bifurcations which over the course of several drift orbits may increase its second invariant value and produce an outward radial jump putting it as far as L M =9 without changing particle energy. If the particle then gets transported again inward to L M = 6 by radial diffusion it gets accelerated by a much larger factor of 2.6 compared to the direct acceleration by radial diffusion. [37] To validate the above qualitative scenario we modified map (7) to include the effects of radial diffusion. Since radial diffusion does not violate the second adiabatic invariant the first equation of the map remained the same. To the second equation, describing radial jumps at each step of the map, we added a random walk term of the magnitude dl k =(2D B LL T D ) 1/2 due to radial diffusion, with the diffusion coefficient specified by equation (10) with Kp = 2. We also added another equation 11 of 20

12 Figure 10. Pitch angle recirculation due to drift orbit bifurcations can greatly amplify the efficiency of electron energization by radial diffusion. Here snapshots of transport and energization of an ensemble of MeV particles initially at L M0 = 6.6 and I 0 = 0.07 (a eq = 80 ) due to simultaneous action of radial diffusion and drift orbit bifurcations are shown. Particle energy is indicated with color. (a) Three, (b) 10, and (c) 100 drift periods. The lower left corner corresponds to the inward boundary and maximum energization attained by radial diffusion alone (1.3 MeV), while the upper left corner is populated because of combination of diffusion and recirculation resulting in increased energization of 2.5 MeV. describing the adiabatic (m = const) change of particle energy due to radial diffusion: 8 I kþ1 ¼ I k þ DIð k ; I k ; L Mk Þ >< L Mkþ1 ¼ L Mk þ DLð k ; I k ; L Mk ÞþL k BL ð Mkþ1 Þ kþ1 ¼ BL ð Mkþ1 L k Þ 2 k 1 1=2 þ 1 >: kþ1 ¼ k þ D mod 2; ð12þ where h is a random variable equal to either 1 or+1,which defines the direction of random walk steps due to radial diffusion. [38] We used map (12) to run a long term simulation of radial, pitch angle, and energy transport of radiation belt electrons due to a combined effect of radial diffusion and drift orbit bifurcations. An ensemble of particles was initiated at L M0 = 6.6, with a small value of the second invariant I 0 = 0.07 (a eq = 80 ), kinetic energy of 1 MeV and evenly distributed over the bounce phase. Particle transport was computed over 100 drift orbits around Earth. Figure 10 shows three snapshot of the ensemble distribution in the I L M space at different times of the simulation process. Particle energy is shown in color. Figure 10a shows a snapshot of the system after 3 drift orbits with most of the particles still bunched around the initial location with exception of a few particles who participated in large outward ballistic jumps along bifurcating transport characteristics outward in radius and toward larger values of the second invariant. The snapshot also illustrates how particles propagate out of the bifurcating regions into the magnetopause loss cone and in the stable orbit regions due to radial diffusion. Figure 10b shows a snapshot after 10 drift periods when particles further expanded along transport characteristics filling most of the bifurcating region. By that time some of the particles not substantially affected by bifurcations, i.e., the particle remaining at small values of the second adiabatic invariant, reached the inward radial boundary of the simulation region at L M = 6.0 and thus got energized to the maximum limit allowed by purely adiabatic heating reaching 1.3 MeV. [39] Figure 10c shows a snapshot of the system after 100 drift orbits around Earth. The highest energy particles populate the upper left corner of Figure 10c corresponding to large values of the second adiabatic invariant. Particle energy reaches as high as 2.5 MeV increasing the efficiency of energization by radial diffusion by almost factor of 2. Only a combined action of radial diffusion and drift orbit bifurcations can transport particles into this part of the phase space. Drift orbit bifurcations produce outward transport and an increase in the second invariant, while radial diffusion transports particles inward energizing them. To elucidate this process we show three examples of particle trajectories which reached the inward boundary of the simulation (Figure 11). While in all three cases particles participate in multiple bifurcations, they reached different final energy and second invariant values. Thus energization levels produced by radial diffusion combined with drift obit bifurcation has a probabilistic character due to random nature of radial diffusion and sensitive dependence of the bifurcation process on the bounce phase. Only a small percentage of particles can acquired energies much exceeding purely adiabatic acceleration. While our results clearly demonstrate that such acceleration is possible, more detailed and realistic analysis is required to determine whether, where and when this process can be responsible for the observed increased radiation levels across the outer belt. 6. Conclusions [40] In a dayside compressed inner magnetosphere drift orbits of energetic particles can exhibit bifurcations leading to their transport in radius and pitch angle. Drift orbit bifurcations are controlled by the magnetic field geometry. 12 of 20