Analytic expressions for ULF wave radiation belt radial diffusion coefficients

Abstract

We present analytic expressions for ULF wave-derived radiation belt radial diffusion coefficients, as a function of L and Kp, which can easily be incorporated into global radiation belt transport models. The diffusion coefficients are derived from statistical representations of ULF wave power, electric field power mapped from ground magnetometer data, and compressional magnetic field power from in situ measurements. We show that the overall electric and magnetic diffusion coefficients are to a good approximation both independent of energy. We present example 1-D radial diffusion results from simulations driven by CRRES-observed time-dependent energy spectra at the outer boundary, under the action of radial diffusion driven by the new ULF wave radial diffusion coefficients and with empirical chorus wave loss terms (as a function of energy, Kp and L). There is excellent agreement between the differential flux produced by the 1-D, Kp-driven, radial diffusion model and CRRES observations of differential electron flux at 0.976 MeV-even though the model does not include the effects of local internal acceleration sources. Our results highlight not only the importance of correct specification of radial diffusion coefficients for developing accurate models but also show significant promise for belt specification based on relatively simple models driven by solar wind parameters such as solar wind speed or geomagnetic indices such as Kp.

Key points: Analytic expressions for the radial diffusion coefficients are presentedThe coefficients do not dependent on energy or wave m valueThe electric field diffusion coefficient dominates over the magnetic.

Figure 1

Azimuthal electric field PSD values derived from ground-based magnetometer measurements of the D-component magnetic field PSD at L = 7.94, 6.51, 5.40, 4.26, 4.21, 2.98, and 2.55. The dashed lines represent constant fits to these PSD values given by equation (9).

Figure 2

Median THEMIS and GOES compressional magnetic field PSD binned with Kp are represented by the circles. The analytic PSD fits from equation (10) are represented by the dashed lines.

Figure 3

Magnetic diffusion coefficients: The symbols represent magnetic diffusion coefficients derived directly from the THEMIS and GOES spacecraft PSD measurements assuming wave m values of m = 10 (squares) and m = 1 (triangles). The dashed green (m = 10) and blue (m = 1) lines represent the magnetic diffusion coefficients derived from the tabulated PSD fits in Ozeke et al. [2012a]. The solid red line represents the magnetic diffusion coefficients given by equation (10).

Figure 4

Electric diffusion coefficients: The symbols represent azimuthal electric field diffusion coefficients derived directly from the mapped ground PSD measurements (presented in Figure 1) and assuming wave m values of m = 10 (green squares) and m = 1 (blue triangles). The dashed green (m = 10) and blue (m = 1) lines represent the electric diffusion coefficients derived from the tabulated PSD fits in Ozeke et al. [2012a]. The red solid line represents the analytic electric field diffusion coefficients given by equation (9).

Figure 5

Comparison between our analytic electric and magnetic field diffusion coefficients, and , with the electromagnetic and electrostatic diffusion coefficients presented in Brautigam and Albert [2000]. The black solid, dashed, and dotted lines represent , , and , respectively. The solid red, blue, green, and cyan lines represent the electrostatic diffusion coefficient, , for electrons with M values of 20 MeV/G, 100 MeV/G, 500 MeV/G, and 2500 MeV/G, respectively. These M values correspond to a 1 MeV electron in a dipole model of the Earth's magnetic field at L = 1.84, 3.15, 5.38, 6.78, and 9.21, respectively.

Figure 6

(a) GOES East median (solid line) and upper and lower quartile (dashed lines) compressional magnetic field PSD for Kp = 1 (blue), Kp = 3 (green), and Kp = 6 (black); (b) GOES East PSD ratios of the upper quartile, median, and lower quartile showing a constant factor of 3 between quartiles independent of frequency and Kp; (c) Same format as Figure 6a for GOES West; (d) Same format as Figure 6b for GOES West.

Figure 7

(a) The median, upper quartile, and lower quartile D-component magnetic field PSD measured at GILL derived from our database of PSD values binned with Kp. The purple, green, and black curves illustrate the median PSD values for Kp = 1, Kp = 3, and Kp = 6, respectively. The dashed curves above and below the solid curves represent the upper and lower quartile values of the PSD. (b) The ratio of the upper and lower quartile and median PSD values for Kp values from 0 to 6.

Figure 8

The ratio of the upper and lower quartile and median D-component magnetic field PSD values for Kp = 0, 1, 2, 3, 4, 5, and 6 at each of the remaining six selected ground magnetometer stations (data for GILL is shown in Figure 6).

Figure 9

Comparison of the differential flux of 0.976 MeV electrons in 1990 measured by the CRRES Medium Electrons A (MEA) with those simulated using different diffusion coefficients and the electron lifetimes from Shprits et al. [2007] outside the plasmapause. Inside the plasmapause the electron lifetime is set to 10 days. (a) Time series of Kp; (b) CRRES MEA observed flux, with the plasmapause location represented by the white curve; (c) the diffusion model with 3 times higher D_LL values than the analytic diffusion coefficients shown in equations (20) and (23), corresponding to the upper quartile PSD values; (d) the diffusion model with the analytic diffusion coefficient shown in equations (20) and (23) derived from the fits to the median PSD values; (e) the diffusion model with D_LL values 3 times lower than the analytic diffusion coefficients shown in equations (20) and (23), corresponding to the lower quartile PSD values; (f) the diffusion model with the values from Brautigam and Albert [2000]; and (g) the diffusion model with the and values from Brautigam and Albert [2000].

Figure 10

Comparison of the differential flux of 0.976 MeV electrons in 1990 measured by the CRRES MEA with those simulated using different diffusion coefficients. Outside the plasmapause the electron lifetimes are set to twice the values from Shprits et al. [2007], and inside the plasmapause the electron lifetime is set to 10 days. (a) Time series of Kp; (b) CRRES MEA observed flux, with the plasmapause location represented by the white curve; (c) the diffusion model with 3 times higher D_LL values than the analytic diffusion coefficients shown in equations (20) and (23), corresponding to the upper quartile PSD values; (d) the diffusion model with the analytic diffusion coefficient shown in equations (20) and (23) derived from the fits to the median PSD values; (e) the diffusion model with D_LL values 3 times lower than the analytic diffusion coefficients shown in equations (20) and (23), corresponding to the lower quartile PSD values; (f) the diffusion model with the values from Brautigam and Albert [2000]; and (g) the diffusion model with the and values from Brautigam and Albert [2000].

Figure 11

Median electron flux measured by the CRRES MEA instrument within 0.1 R_E of L = 7 during each orbit. The symbols represent the electron flux measured at each energy channel at 10 day time intervals over a period of 90 days.