Journal of Atmospheric and Solar–Terrestrial Physics 193 (2019) 105090

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New hiss and chorus waves diffusion coefficient parameterizations from theVan Allen Probes and their effect on long-term relativistic electronradiation-belt VERB simulationsHui Zhu a,∗, Yuri Y. Shprits b,a, M. Spasojevic c, Alexander Y. Drozdov a

a Department of Earth and Planetary Space Sciences, University of California, Los Angeles, CA, USAb Helmholtz Centre Potsdam, GFZ German Research Centre For Geosciences and University of Potsdam, Potsdam, Germanyc Department of Electrical Engineering, Stanford University, Stanford, CA, USA

A R T I C L E I N F O

Keywords:Inner magnetosphereRadiation beltsChorus wavesDiffusion coefficientsVERB code

A B S T R A C T

New wave frequency and amplitude models for the nightside and dayside chorus waves are built basedon measurements from the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS)instrument onboard the Van Allen Probes. The corresponding 3D diffusion coefficients are systematicallyobtained. Compared with previous commonly-used (typical) parameterizations, the new parameterizationsresult in differences in diffusion rates that depend on the energy and pitch angle. Furthermore, one-year3D diffusive simulations are performed using the Versatile Electron Radiation Belt (VERB) code. Both typicaland new wave parameterizations simulation results are in a good agreement with observations at 0.9 MeV.However, the new parameterizations for nightside chorus better reproduce the observed electron fluxes. Theseparameterizations will be incorporated into future modeling efforts.

1. Introduction

The outer electron radiation belt exhibits dramatic variability overtimescales from minutes to decades (Blake et al., 1992; Reeves et al.,1998; Li et al., 2001; Horne et al., 2005; Baker et al., 2013; Shpritset al., 2013b). The complex dynamics of electrons are controlled byvarious physical processes, which can be classified as either adiabaticor non-adiabatic (Shprits et al., 2008a,c). The adiabatic process relatingto the geomagnetic configuration changes can partially account for thedropout and buildup of electron fluxes during geomagnetically activetimes (McIlwain, 1966; Kim and Chan, 1997). Non-adiabatic processesare usually associated with resonant wave–particle interactions. Thedrift resonance driven by ultra low frequency (ULF) waves can causeradial diffusion of electrons violating the third adiabatic invariant (Elk-ington et al., 1999; Brautigam and Albert, 2000; Elkington et al., 2003;Zong et al., 2007; Shprits et al., 2008a; Degeling et al., 2014; Ozekeet al., 2014; Ali et al., 2016; Cunningham, 2016). Cyclotron or Landauresonance driven by very low frequency (VLF) waves, such as chorusand hiss, can drive pitch and energy diffusion of radiation belt electronsviolating the first and second adiabatic invariants (Summers et al.,2008; Thorne et al., 2013a; Ni et al., 2014; Ma et al., 2016b; Orlovaet al., 2016; Breneman et al., 2015; Abel and Thorne, 1998a,b; Shpritset al., 2006a; Li et al., 2007; Shprits et al., 2008c; Zhu et al., 2015;

∗ Corresponding author.E-mail address: zhuhui328@gmail.com (H. Zhu).

Agapitov et al., 2015, 2017; Li et al., 2016b,a; Agapitov et al., 2013,2016; Albert, 2017). Typically, chorus waves, predominantly observedoutside the plasmapause produce electron loss on the dayside but resultin electron acceleration on the nightside (Shprits et al., 2008c; Thorne,2010). Plasmaspheric hiss usually observed inside the plasmaspheremainly drives electron loss, and the waves are believed to result inthe formation of the slot region and slow decay of radiation beltelectrons (Lyons and Thorne, 1973; Abel and Thorne, 1998a; Meredithet al., 2007; Summers et al., 2008; Thorne et al., 2013a; Ni et al., 2014;Ma et al., 2016b; Orlova et al., 2016; Breneman et al., 2015; Zhu et al.,2019; Ripoll et al., 2019). The above-mentioned electromagnetic waveshave been incorporated in many radiation belt models (Shprits et al.,2006b, 2008a,b; Xiao et al., 2009; Su et al., 2010; Meredith et al., 2012;Horne et al., 2013; Glauert et al., 2014; Tu et al., 2014; Drozdov et al.,2015).

The 3D Versatile Electron Radiation Belt (3D VERB) code is builtbased on the Fokker–Planck equation, accounting for both global radialdiffusion and local diffusion (Shprits et al., 2008a, 2009; Subbotin andShprits, 2009; Subbotin et al., 2010, 2011; Kim et al., 2011, 2012;Kellerman et al., 2014; Shprits et al., 2015). The parameterizations ofnightside, dayside chorus and plasmaspheric hiss adopted in previousworks (Li et al., 2007; Shprits et al., 2007) are mainly obtained from

https://doi.org/10.1016/j.jastp.2019.105090Received 23 October 2018; Received in revised form 12 June 2019; Accepted 23 July 2019

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Table 1Adopted wave parameters for typical nightside chorus, new nightside chorus, typical dayside chorus, new dayside chorus, OS14 plasmaspheric hiss, new plasmaspheric hiss,lighting-generated whistlers and VLF transmitter.

Wavetype

Wave intensity 𝐵𝑤 (pT) 𝜆max Density model Wave spectrum MLT averagingratio

Wave normal anglemodel

Typicalnightsidechorus

𝐾𝑝 ≤ 2+

15◦ Sheeley et al. (2001)

𝜔𝑚∕𝛺𝑒 = 0.35

25%

𝜃𝑚 = 0◦

50(2 ⋅ 100.73+0.91𝐾𝑝∕3319.2)0.5 𝜔𝑤∕𝛺𝑒 = 0.15 𝜃𝑤 = 30◦

2 < 𝐾𝑝 ≤ 6 𝜔𝑙𝑐∕𝛺𝑒 = 0.05 𝜃𝑙𝑐 = 0◦

50(2 ⋅ 102.5+0.18𝐾𝑝∕3319.2)0.5 𝜔𝑢𝑐∕𝛺𝑒 = 0.65 𝜃𝑢𝑐 = 45◦

Newnightsidechorus

(101.8821+0.3637𝐾𝑝−0.05015𝜆)0.5 15◦ Sheeley et al. (2001)

𝜔𝑚∕𝛺𝑒 = 0.25

50%

𝜃𝑚 = 0◦

𝜔𝑤∕𝛺𝑒 = 0.175 𝜃𝑤 = 30◦

𝜔𝑙𝑐∕𝛺𝑒 = 0.05 𝜃𝑙𝑐 = 0◦

𝜔𝑢𝑐∕𝛺𝑒 = 0.5 𝜃𝑢𝑐 = 45◦

Typicaldaysidechorus

𝐾𝑝 ≤ 2+

35◦ Sheeley et al. (2001)

𝜔𝑚∕𝛺𝑒 = 0.2

25%

𝜃𝑚 = 0◦

100.75+0.04𝜆(2 ⋅ 100.73+0.91𝐾𝑝∕3319.2)0.5 𝜔𝑤∕𝛺𝑒 = 0.1 𝜃𝑤 = 30◦

2 < 𝐾𝑝 ≤ 6 𝜔𝑙𝑐∕𝛺𝑒 = 0.1 𝜃𝑙𝑐 = 0◦

100.75+0.04𝜆(2 ⋅ 102.5+0.18𝐾𝑝∕3319.2)0.5 𝜔𝑢𝑐∕𝛺𝑒 = 0.3 𝜃𝑢𝑐 = 45◦

Newdaysidechorus

(101.0561+0.4095𝐾𝑝+0.0351𝜆)0.5 35◦ Sheeley et al. (2001)

𝜔𝑚∕𝛺𝑒 = 0.2

50%

𝜃𝑚 = 0◦

𝜔𝑤∕𝛺𝑒 = 0.15 𝜃𝑤 = 30◦

𝜔𝑙𝑐∕𝛺𝑒 = 0.05 𝜃𝑙𝑐 = 0◦

𝜔𝑢𝑐∕𝛺𝑒 = 0.4 𝜃𝑢𝑐 = 45◦

OS14 hiss See Orlova et al. (2014) 45◦ Denton et al. (2006)

𝑓𝑚 = 550 Hz

62.5% See Orlova et al. (2014)𝑓𝑤 = 300 Hz𝑓𝑙𝑐 = 100 Hz𝑓𝑢𝑐 = 2000 Hz

New hiss See Spasojevic et al. (2015) 45◦ Denton et al. (2006) See Li et al. (2015) See Spasojevicet al. (2015)

See Orlova et al. (2016)

Lightning-generatedwhistlers

7 ∗ 𝐾𝑝∕4 45◦ Carpenter and Anderson (1992)

𝑓𝑚 = 4500 Hz

100%

𝜃𝑚 = 45◦

𝑓𝑤 = 2000 Hz 𝜃𝑤 = 22.5◦

𝑓𝑙𝑐 = 2500 Hz 𝜃𝑙𝑐 = 22.5◦

𝑓𝑢𝑐 = 6500 Hz 𝜃𝑢𝑐 = 67.5◦

VLFtransmitter17.1 kHz

0.8 45◦ Carpenter and Anderson (1992)

𝑓𝑚 = 17100 Hz

2.4 × 4%

𝜃𝑚 = 45◦

𝑓𝑤 = 50 Hz 𝜃𝑤 = 22.5◦

𝑓𝑙𝑐 = 17000 Hz 𝜃𝑙𝑐 = 22.5◦

𝑓𝑢𝑐 = 17200 Hz 𝜃𝑢𝑐 = 67.5◦

VLFtransmitter22.3 kHz

0.8 45◦ Carpenter and Anderson (1992)

𝑓𝑚 = 22300 Hz

2.4 × 4%

𝜃𝑚 = 45◦

𝑓𝑤 = 50 Hz 𝜃𝑤 = 22.5◦

𝑓𝑙𝑐 = 22200 Hz 𝜃𝑙𝑐 = 22.5◦

𝑓𝑢𝑐 = 22400 Hz 𝜃𝑢𝑐 = 67.5◦

Combined Release and Radiation Effects Satellite (CRRES) measure-ments (Vampola et al., 1992). The simulated relativistic radiation beltwith VERB using those parameterizations have shown good agreementwith the electron fluxes observed by CRRES (Subbotin et al., 2011;Kim et al., 2011). A significant limitation of the CRRES plasma waveexperiment was that it only provided measurements of a single electricfield and the amplitude of the wave magnetic field was derived fromthe wave electric field assuming a cold plasma dispersion relation forparallel propagating waves (e.g. Meredith et al., 2004). The launch ofVan Allen Probes spacecraft (Mauk et al., 2013) which provides thecomplete six components of electromagnetic waves provides a goodopportunity to obtain a more comprehensive wave dataset. Spasojevicet al. (2015) using two-years Electric and Magnetic Field InstrumentSuite and Integrated Science (EMFISIS) (Kletzing et al., 2013) databuilt a global hiss intensity model and Orlova et al. (2016) accordinglycreated a parameterized loss model for relativistic electrons. Spasojevicet al. (2015) also performed an extensive comparison between the VanAllen Probes and CRRES and found that the CRRES 𝑬 to 𝑩 conversionresults in an overestimation of the total hiss magnetic field and alatitudinal dependence that is inconsistent with findings from the VanAllen Probes. In this work, new statistical models of wave frequencyand intensity are developed for both the nightside and dayside choruswaves. We compare the corresponding scattering rates with typicalrates based on CRRES measurements. VERB code simulations of rela-tivistic radiation belt electron are conducted from October 1, 2012 toSeptember 30, 2013.

2. VERB code

The evolution of relativistic radiation belt electrons can be describedby the Fokker–Planck equation (Schulz and Lanzerotti, 1974a; Subbotin

and Shprits, 2009)

𝜕𝑓𝜕𝑡

= 𝐿2 𝜕𝜕𝐿

|

|

|

|𝜇,𝐽

1

𝐿2𝐷𝐿𝐿

𝜕𝑓𝜕𝐿

|

|

|

|𝜇,𝐽

+1

𝑝2𝜕𝜕𝑝

|

|

|

|𝛼0 ,𝐿𝑝2

(

𝐷𝑝𝑝𝜕𝜕𝑝

|

|

|

|𝛼0 ,𝐿𝑓 +𝐷𝑝𝛼0

𝜕𝜕𝛼0

|

|

|

|𝑝,𝐿𝑓

)

+1

𝑇 (𝛼0) sin(2𝛼0)𝜕𝜕𝛼0

|

|

|

|𝑝,𝐿

⋅ 𝑇 (𝛼0) sin(2𝛼0)

(

𝐷𝛼0𝛼0

𝜕𝜕𝛼0

|

|

|

|𝑝,𝐿𝑓 +𝐷𝛼0𝑝

𝜕𝜕𝑝

|

|

|

|𝛼0 ,𝐿𝑓

)

−𝑓𝜏

(1)

where 𝑓 is the electron phase space density (PSD), 𝜇 and 𝐽 are thefirst and second adiabatic invariants, 𝛼0 is the equatorial pitch angle,𝑝 is the electron momentum, and 𝐿 is radial distance from the centerof the Earth to the equatorial crossing of magnetic field lines. 𝑇 (𝛼0)is a function associated with the electrons’ bounce time (Schulz andLanzerotti, 1974b). The term 𝑓∕𝜏 represents the loss into bounceloss cone, where 𝜏 is the typical electron bounce lifetime, set to beone quarter electron bounce time inside the loss cone and infiniteoutside the loss cone. 𝐷𝐿𝐿 is the radial diffusion coefficient, whichcharacterizes the effect of wave–particle interactions with ULF waves.Here we adopt the 𝐾𝑝-driven magnetic radial diffusion coefficientsfrom Brautigam and Albert (2000) without loss of continuity withour previous studies (Subbotin et al., 2011; Drozdov et al., 2015).Recently some updated radial diffusion coefficient models (Ozeke et al.,2014; Liu et al., 2016) have been developed based on a larger ob-servational database. However, Drozdov et al. (2017a) showed thatthe radial diffusion model by Ozeke et al. (2014) produces essen-tially identical results to simulations with magnetic 𝐷𝐿𝐿 of Brautigam

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Fig. 1. The normalized power spectral density as a function of frequency for nightsidechorus (left) and dayside chorus models (right) averaged over all 𝐿−shell. The blacksolid lines represent the power spectral density profiles obtained from Van Allen Probes.The black dashed lines represent the typical wave models and the red dashed linesrepresent the new wave models. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

and Albert (2000). 𝐷𝛼0𝛼0 , 𝐷𝑝𝑝 and 𝐷𝛼0𝑝 = 𝐷𝑝𝛼0 are Magnetic LocalTime (MLT) averaged and bounce-averaged pitch angle, momentumand cross-pitch-angle-momentum diffusion coefficients, respectively,which quantify the efficiency of cyclotron wave–particle interactionson electrons driven by VLF waves, such as whistler-mode chorus andplasmaspheric hiss. The diffusion coefficients are calculated by the FullDiffusion Code (FDC) discussed in the next section.

The Fokker–Planck equation is numerically solved by the VERBcode. In this work, the computational grid has 46 × 101 × 91 pointsin 𝐿, 𝐸𝑘 and 𝛼0, respectively. 𝐿 ranges from 1 to 5.5, with linear grid

points. The kinetic energy 𝐸𝑘 ranges from 10 keV to 10 MeV at 𝐿 = 5.5and the grid points are distributed logarithmically. To obtain the gridat the other values of 𝐿, we assumed the first and second invariants areconserved along with 𝐿 such that 𝐸𝑘 at 𝐿 = 1 ranges from 895.44 keVto 134.9 MeV. The equatorial pitch angle 𝛼0 ranges from 0.3◦ to 89.7◦on a linear grid. The adoption of 𝛼0 range [0.3◦, 89.7◦] rather than [0◦,90◦] is in order to avoid the potential singularity in the Fokker–Planckequation, which contains the term 1∕ sin(2𝛼0).

We solve the steady-state radial diffusion equation with 𝐾𝑝 = 2 andlifetime 𝜏 = 10 days as suggested by Subbotin et al. (2011) and Drozdovet al. (2015) in order to obtain the initial electron distribution as afunction of 𝐿 for all pitch angles and energies. The initial PSD atouter boundary 𝐿 = 5.5 is determined by the long-time averagedspectrum (Subbotin et al., 2011). Since there are three variables in theFokker–Planck equation we need to specify six boundary conditions.Here we adopt the same boundary conditions as described by Drozdovet al. (2015). PSD at the 𝐿 lower boundary is set to zero, whichrepresents loss into the atmosphere. PSD at the 𝐿 upper boundarycondition is determined by the Magnetic Electron Ion Spectrometer(MagEIS) (Blake et al., 2013) electron flux observed by the Van AllenProbes at 𝐿 = 5.5. PSD at the lower 𝐸𝑘 boundary is set to be constant,which is determined by the initial state which represents the balancebetween the source and loss of the seed populations. PSD at the upper𝐸𝑘 boundary condition is set to zero because the observed flux athigher energy is very low. PSD at the lower 𝛼0 boundary condition isset to zero, representing the loss in the loss cone. To reproduce a flatpitch angle distribution near 90◦, the gradient of PSD at the upper 𝛼0boundary condition is set to zero.

3. New parameterizations

In this study, we aim to investigate the effects of updated param-eterization on the long-term trapped relativistic electrons dynamics.

Fig. 2. Bounce-averaged pitch angle (first row), momentum (second row), absolute values of mixed and sign of mixed diffusion coefficients for new nightside chorus and typicalnightside chorus and energy 𝐿 = 4.5 and Kp = 4. The third column is the normalized difference between the new one and typical one. Red represents new coefficients larger thantypical ones and vice versa. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 3. Same as Fig. 2 except for new dayside chorus and typical dayside chorus.

The quasi-linear diffusion coefficient (Albert, 2005; Glauert and Horne,2005) is a powerful tool to quantify the effect of cyclotron resonanceon radiation belt electrons, which is a key part of the Fokker–Planckequation. The Full Diffusion Code, developed by Ni et al. (2008)and Shprits and Ni (2009), is capable of obtaining accurate diffusioncoefficients for different waves modes, e.g. chorus, plasmaspheric hiss,electromagnetic ion cyclotron (EMIC) and magnetosonic waves (Niet al., 2008; Shprits and Ni, 2009; Orlova and Shprits, 2010; Orlovaet al., 2012; Shprits et al., 2013a). FDC uses a parallel architecture andcalculations are performed on NCAR’s YellowStone supercomputer andUCLA’s Hoffman2 Cluster.

Whistler-mode chorus emissions are frequently observed outside theplasmasphere located in the frequency range from 0.1 to 0.8 of theequatorial electron gyrofrequency (𝜔𝑒) (Tsurutani and Smith, 1974,1977; Meredith et al., 2001; Li et al., 2011). The magnetic intensityof chorus waves is in the range of 1−100 pT and is strongly affected bygeomagnetic activity. Resonant interactions driven by chorus emissionsare widely considered to contribute to the loss and acceleration of rela-tivistic radiation belt electrons (Horne et al., 2005; Shprits et al., 2008b;Thorne et al., 2013b). Due to the dependence of chorus wave features,e.g. frequency spectrum, wave amplitude and latitudinal extent onMLT, the chorus scattering effects on electrons are different for differentMLT sectors (Li et al., 2007). For example, in the nightside sector (18 ≤MLT ≤ 6) energy diffusion due to chorus waves dominates and produceselectron acceleration while in the dayside sector (6 ≤ MLT ≤ 18) pitch-angle diffusion dominates and produces electron loss. Nightside chorusand dayside chorus parameterizations have been previously built basedon measurements from CRRES (Li et al., 2007; Shprits et al., 2007) andhave been widely adopted in many works (Xiao et al., 2009; Subbotinet al., 2011) (referred to as the typical chorus waves hereinafter).

Here, new spectrum and amplitude models of the nightside anddayside lower band chorus are developed based on data from the EMFI-SIS instruments onboard the Van Allen Probes. Following a procedure

similar to Spasojevic et al. (2015), the chorus wave amplitude andspectra are obtained from the diagonal elements of the magnetic cross-spectral matrix in the EMFISIS Level 2 Waveform Receiver data forall intervals outside the manually determined plasmapause boundary.Since the chorus waves are observed outside the plasmapause weuse the same way described by Spasojevic et al. (2015) to determinethe location of plasmapause. In the paper, the authors describe themanual plasmapause location determination in detail. Waves inside theplasmapause are considered to be plasmaspheric hiss and outside theboundary are considered to be chorus. The chorus wave amplitude isdetermined by integrating the magnetic power spectral density from0.05 to 0.5 of the estimated equatorial electron gyrofrequency, 𝜔𝑒. Theequatorial gyrofrequency is estimated by using measurements of thelocal gyrofrequency from the spacecraft magnetometer and scaling itassuming a dipolar variation along the field line. Here, equivalently toCRRES, two years of RBSP-A data from October 1, 2012 to September30, 2014 is used, resulting in 1.39 million measurements of daysidechorus (6 ≤ MLT ≤ 18) amplitude and spectral distribution and 1.81million observations of nightside chorus (18 ≤ MLT ≤ 6). In orderto create a model of the time-averaged wave intensity of chorus, wecompute the average value of 𝐵2

𝑤 in an 18 × 10 grid of Kp and magneticlatitude 𝜆, separately for dayside and nightside chorus. Because thedependence of chorus power on 𝐿-shell is not significant, here we makeaveraged over 𝐿 to obtain the power distribution. We then apply simplefirst-order linear regression to the logarithm of 𝐵2

𝑤. We also computethe average spectrum of dayside and nightside chorus and fit thisaverage to a Gaussian frequency distribution. The obtained frequencyspectrum and wave intensities are listed in Table 1.

Fig. 1 illustrates the frequency spectrum and wave intensities fornightside chorus (left panel) and dayside chorus (right panel). For thenew nightside chorus model (dashed red lines), the peak of frequencyspectrum shifts towards the lower frequency compared with the typical

Journal of Atmospheric and Solar-Terrestrial Physics 193 (2019) 105090

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Fig. 4. Same as Fig. 2 except for new plasmaspheric hiss, and OS14 plasmaspheric hiss.

Fig. 5. Profiles of bounce-averaged pitch angle (a) and momentum (b) diffusion coefficients for new nightside chorus (black), new dayside chorus(red) and new hiss (blue) wavesas functions of equatorial pitch angle at 0.9 MeV for Kp = 1 (solid lines), Kp = 4 (dotted lines) and Kp = 6 (dashed lines). (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

frequency spectrum (dashed black lines). This significant shift affectsthe resonant interaction and further affects the shape and distributionof the corresponding diffusion coefficients. For the new dayside chorusmodel, the wave frequency spectrum slightly widens, which is expectedto produce a variation in the diffusion coefficient shape. The waveintensities for new nightside chorus are comparable to the typicalmodel.

We also investigate the parameterization of plasmaspheric hisswaves. Orlova et al. (2014) developed a quadratic model of plasmas-pheric hiss amplitude as functions of 𝐾𝑝, 𝐿 and magnetic latitude basedon CRRES wave observation (referred to as OS14 hiss hereinafter).Furthermore, Spasojevic et al. (2015) built an improved hiss amplitudemodel using the Van Allen Probes. Li et al. (2015) also investigated thefrequency spectral characteristics of plasmaspheric hiss and used non-Gaussian functions to fit the spectrum. Orlova et al. (2016) combinedthe frequency model and intensity model from the Van Allen Probes’

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Fig. 6. Solar wind parameters and geomagnetic indices during the 365 day period starting on October 1, 2012.: (a) magnitude of interplanetary magnetic field; (b) 𝑧 componentof interplanetary magnetic field; (c) solar wind plasma flow velocity; (d) Dst index (black solid line) and AE index (red solid line); (e) Kp index; (f) daily averaged electron fluxat 𝐸𝑘 = 0.9 MeV and 𝛼0 = 85◦, observed by MagEIS onboard the Van Allen Probes. The white dashed line in panel (f) indicates the location of plasmapause. (For interpretation ofthe references to color in this figure legend, the reader is referred to the web version of this article.)

measurement and built an updated parameterization of hiss waves(referred to as new hiss hereinafter). The authors discussed in detailthe difference between the new hiss model and the OS14 hiss model interms of lifetime. Lightning-generated whistlers and two bands of VLFtransmitter waves, which assist the hiss waves to scatter the radiationbelt electrons in the slot region, are also included in our simula-tion (Kim et al., 2011). In this study, the other electromagnetic wavesinside the magnetosphere such as EMIC waves, electrostatic cyclotronharmonic (ECH), the upperband chorus waves and magnetosonic wavesare not included. EMIC waves are considered to produce the rapidloss of ultra-relativistic radiation belt electrons (Shprits et al., 2013b;Usanova et al., 2014; Shprits et al., 2016) but do not contribute tothe long-term loss of relativistic (∼ 1 MeV) electrons (Drozdov et al.,2015). ECH is electrostatic emission which is frequently observed inthe plasma sheet from 2100 to 0600 MLT (Meredith et al., 2009). ECHwaves mainly contribute to the scattering loss of ∼ few keV plasmasheet electrons and the formation of diffusive aurora combining withchorus (Thorne et al., 2010) but could not significantly affect radiationbelt electrons. Upperband chorus (0.5𝛺𝑐𝑒 < 𝜔 < 𝛺𝑐𝑒) are considered toresonate with few or several keV electrons due to their high frequencyfeature which is not our interest. Magnetosonic waves have attractedmuch attention due to their potential acceleration effect on radiationbelt electrons (Horne et al., 2007; Shprits et al., 2013a; Ma et al., 2016a;Zhu et al., 2019a). However, their averaged amplitudes are smallerthan those of chorus waves and are not expected to play a significantrole in the longtime scales (Shprits et al., 2013a).

In order to obtain MLT-averaging diffusion coefficients, it is nec-essary to specify several parameters, such as the wave frequency spec-trum, wave intensity, latitude coverage, wave normal angle distributionand background electron number density. All parameters of the wavemodels are shown in Table 1. In this work, for most of wave types thefrequency spectrum is assumed to be a Gaussian distribution (Glauertand Horne, 2005), with peak angular frequency 𝜔𝑚 or frequency 𝑓𝑚,width 𝜔𝑤 or 𝑓𝑤, the lower cutoff 𝜔𝑙𝑐 or 𝑓𝑙𝑐 and the upper cutoff 𝜔𝑢𝑐 or𝑓𝑢𝑐 . As to OS14 plasmaspheric hiss waves the frequency is defined asnon-Gaussian distribution based on Van Allen Probes observation (Liet al., 2015; Spasojevic et al., 2015; Orlova et al., 2016). The wavenormal angle is also assumed to be a Gaussian distribution (Glauertand Horne, 2005; Albert, 2017), with the peak 𝜃𝑚, width 𝜃𝑤, thelower cutoff 𝜃𝑙𝑐 and the upper cutoff 𝜃𝑢𝑐 . Particularly, in two hisswave models (OS14 hiss and new hiss), the peak, the width, the lowercutoff and the upper cutoff vary with latitude according to the raytracing method estimation (Ni et al., 2013; Orlova et al., 2014, 2016).Electron density model which essentially affects the dispersion relationis an important part in diffusion coefficients. For dayside and nightsidechorus waves which are always observed outside the plasmapause, weadopt the plasmatrough density model by Sheeley et al. (2001). ForVLF transmitters and lightning-generated whistler which are alwaysobserved inside the plasmapause, we follow the previous studies (Kimet al., 2011; Subbotin et al., 2011) to adopt the plasmasphere densitymodels by Carpenter and Anderson (1992). For plasmaspheric hisswaves, we adopt the updated model (Denton et al., 2006) similar

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Fig. 7. Electron flux at 𝐸𝑘 = 0.9 MeV and 𝛼0 = 85◦ (a) MagEIS observation onboard the Van Allen Probes (b) VERB simulation with typical (CRRES) nightside chorus (c) normalizeddifference between MagEIS observation flux and VERB simulation with typical nightside chorus (d) VERB simulation with new (RBSP) nightside chorus (e) normalized differencebetween MagEIS observation flux and VERB simulation with new nightside chorus.

to Orlova et al. (2014, 2016). All waves are assumed to be obliquewithin the maximum of latitude coverage 𝜆𝑚𝑎𝑥. In the dayside sectorchorus waves are able to propagate towards high latitude (e.g. 40◦)because of weak Landau damping. The equatorial confinement of VanAllen Probes orbits could not allow good data coverage for daysidechorus characteristics particularly at higher latitude. Thus for newdayside chorus model the intensities in latitude range [0◦, 20◦] aresame to one described by Table 1 and the intensities in latitude range[20◦, 35◦] are assumed to be same as the intensity at 𝜆 = 20◦. MLTaveraging ratio is a factor multiplying the bounce-averaging diffusioncoefficients to obtain MLT-averaging coefficients due to the limitedMLT coverage of waves.

Fig. 2 presents the diffusion coefficients calculated by the FDC asa function of equatorial pitch angle and energy for the new nightsidechorus (panels a–d) and typical nightside chorus (panels e–h) at 𝐿 = 4.5and 𝐾𝑝 = 4, and normalized difference (panels i–k). The normalizeddifference for diffusion coefficients ND is calculated by

ND(𝛼𝑒𝑞 , 𝐸𝑘) =⟨𝐷⟩new(𝛼𝑒𝑞 , 𝐸𝑘) − ⟨𝐷⟩typical(𝛼𝑒𝑞 , 𝐸𝑘)

(

⟨𝐷⟩new(𝛼𝑒𝑞 , 𝐸𝑘) + ⟨𝐷⟩typical(𝛼𝑒𝑞 , 𝐸𝑘))

∕2× 100%. (2)

where ⟨𝐷⟩new and ⟨𝐷⟩typical are updated and typical diffusion coeffi-cients, respectively.

Fig. 2 illustrates a difference between diffusion coefficients for thenew and typical nightside chorus models which becomes pronouncedat lower energies. This is due to the difference of the frequency uppercutoff, which is 0.6𝜔𝑒 in the typical model but 0.5𝜔𝑒 in the new model.Fig. 2j shows that the energy diffusion rates of the typical model are

larger than the new ones in the range 𝐸𝑘 > 300 keV and high pitchangle 𝛼0 > 70◦ while at median pitch angle range and the same energy,the new rates are larger than the typical ones. Hence, the differencelooks complex and it is non trivial by visual inspection to decide whichone may produce stronger acceleration.

Fig. 3 is similar to Fig. 2 except it illustrates diffusion coefficientsfor the dayside chorus. It shows that the shape of diffusion coefficientsof new and typical dayside chorus are roughly similar, which is ex-pected since their center frequencies are the same and their frequencywidths are close. However, the magnitudes of diffusion distributionsare quite different due to the difference in the wave intensities. Sincethe dayside chorus waves mainly cause electron loss the comparison ofpitch angle diffusion is important. The normalized difference of pitchangle diffusion coefficients in Fig. 3i shows that new diffusion ratesare higher than typical ones at several 100 keV but smaller at > 500keV. Such difference is probably caused by the difference of the lowerand upper cutoffs. Such difference of pitch angle diffusion distributionsuggests that new dayside chorus produces a stronger loss of ∼ 100 keVelectrons but less loss of > 500 keV electrons.

The diffusion coefficients for OS14 hiss and new hiss are plottedin Fig. 4. It is clear that for both new and typical hiss wave mod-els, the pitch angle diffusion rates are larger than energy diffusionrates by several orders of magnitude, which is consistent with thefact that plasmaspheric hiss contributes to the electron loss. From thecomparison, we can see their distributions are quite different. Thisis because the upper cutoffs of the frequency spectrum are different:2000 Hz in OS14 and 4000 Hz in the new model. The normalizeddifference of pitch angle diffusion coefficients shows that the new hiss

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Fig. 8. Same as Fig. 6 except panel (b) represents VERB simulation with typical (CRRES) dayside chorus wave.

model produces more scattering of 𝐸𝑘 ∼ 100 keV electrons near theloss cone while the OS14 hiss produces more scattering of relativisticelectrons with 𝐸𝑘 > 300 keV. Moreover, the profiles of bounce-averageddiffusion coefficients at 0.9 MeV for three new waves are plotted inFig. 5. Three different Kp values (1,4 and 6) are selected to showthe Kp-dependence of diffusion coefficients. The significant differencesbetween the diffusion coefficients driven by three different waves aresignificant.

4. Comparisons between simulations and observations

To test the influence of the above-mentioned parameterizations onthe long-term trapped relativistic electrons, we use the VERB model tonumerically simulate the radiation belt dynamics for a 365-day periodstarting on October 1, 2012. This long-term simulation is necessaryto systematically assess the deviations of the simulation from obser-vations, which cannot be accomplished with the analysis of a singlestorm.

Fig. 6 presents hourly averaged interplanetary solar wind param-eters, geomagnetic indices and daily averaged electron flux at 𝐸𝑘 =0.9 MeV and 𝛼0 = 85◦ for a 365-day period starting on October1, 2012. The solar wind data and geomagnetic indices are obtainedfrom CDAweb-OMNI database. We can see that all solar wind andgeomagnetic parameters varied significantly over the course of year.Several (at least 16) moderate geomagnetic storms (−100 < 𝐷𝑠𝑡 <−50 nT) and 4 strong geomagnetic storms (−250 < 𝐷𝑠𝑡 < −100 nT)occurred during this period. Some of the intense storm events have beenreported, analyzed and numerically simulated (Baker et al., 2014; Xiao

et al., 2014; Shprits et al., 2015). Fig. 6f indicates radiation electronflux at 𝐸𝑘 = 0.9 MeV and 𝛼0 = 85◦ observed by the MagEIS instrumentonboard the Van Allen Probes. We can see that during the mainphase of most geomagnetic storms, radiation-belt electrons experiencedrapid dropout and recovered gradually, while during the geomagneticquiet time, the radiation-belt electron flux gradually decreased dueto loss mechanisms. Hence, this 365 day period, which covers bothgeomagnetic active and quiet conditions, is a good period to examinethe validity and accuracy of various parameterization used in the VERBcode between the observed and simulated fluxes.

Here we use the normalized difference to quantify the compari-son between observation and simulation (Subbotin et al., 2010). Thenormalized difference can be calculated as

ND(𝐿, 𝑡) =𝐽model(𝐿, 𝑡) − 𝐽obs(𝐿, 𝑡)

maxover 𝐿 at each 𝑡

(

𝐽model(𝐿, 𝑡) + 𝐽obs(𝐿))

∕2× 100% (3)

where 𝐽model(𝐿, 𝑡) is the modeled electron flux produced by the VERBcode and 𝐽obs(𝐿) is the observed electron flux by the Van Allen Probes.We can see the normalized difference is a function of 𝐿 and 𝑡. Thisparameters estimate how accurate the simulation results are comparedwith observation. When it is zero, the simulation fluxes are exactlyequal to the observed fluxes.

Fig. 7a shows the electron flux at 𝐸𝑘 = 0.9 MeV and 𝛼0 = 85◦ ob-served by the MagEIS instrument onboard the Van Allen Probes. Fig. 7bshows the electron flux at 𝐸𝑘 = 0.9 MeV and 𝛼0 = 85◦ in the VERBsimulation with typical nightside chorus, new dayside chorus and newplasmaspheric hiss. Fig. 7c shows the normalized difference betweenthe MagEIS observation and the VERB simulation in Fig. 7b. Fig. 7b is

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Fig. 9. Same as Fig. 6 except panel (b) represents VERB simulation with OS14 (CRRES) plasmaspheric hiss wave.

similar to Fig. 7d except using new nightside chorus parameterizationand Fig. 7e shows the corresponding normalized difference. For bothcases, the simulation fluxes are relatively similar to the observed fluxes.The normalized differences are acceptable, whose maximum can reachup to 200%. These results are consistent with previous VERB simulationresults (Drozdov et al., 2015), suggesting that our typical or newparameterization can basically account for the effect of wave–particleinteraction in the radiation belts near 0.9 MeV. The normalized differ-ence shown in Fig. 7c is red above 𝐿 = 4, indicating that the simulationwith typical nightside chorus diffusion coefficients overestimates theobservations. Considering the new nightside chorus parameterization,the overestimate in Fig. 7e is smaller than that in Fig. 7c. Hence,the new nightside chorus parameterization can be considered to betterreproduce the radiation belt evolution.

Fig. 8 shows similar results as Fig. 7 with typical and new daysidechorus parameterizations. The VERB simulation and correspondingnormalized difference with typical dayside chorus parameterizationare shown in Fig. 8b and c. The VERB simulation and correspondingnormalized difference with new dayside chorus parameterization areshown in Fig. 8d and e. The simulated fluxes and the normalizeddifference distributions are similar. Our results suggest that the consid-eration of new dayside chorus parameterization does not significantlychange the VERB simulation results. The 2-D diffusion coefficients oftypical and new dayside chorus shown in Fig. 3 present a remarkabledifference while the variations of the corresponding VERB simulationsare not significant. The likely reason for this result is that our VERBsimulations are implemented as a 3D simulation in pitch angle, energyand 𝐿. Diffusion depends on the diffusion coefficients also on the

gradients of PSD in 𝐿, pitch angle and energy. As it is a 3D system,the response of such system is complex. In addition, the intensificationin the dayside chorus will enhance both the loss and accelerationof MeV electrons. Thus, the VERB simulation with different daysideparameterizations may result in similar results. This also suggests thatthe typical parameterization has been adopted in a number of previousstudies (Subbotin et al., 2011; Kim et al., 2011; Drozdov et al., 2015,2017b) is reliable.

The comparison of VERB simulations using different hiss parameter-izations is shown in Fig. 9. The results show that the VERB simulationswith OS14 and new hiss parameterization are similar. Fig. 10 presentsthe comparison between old (based on CREES) and new (based onRBSP) parameterization. As we mentioned before, the adoption of newparameterization significantly decreases the overestimate of electronfluxes by old parameterization.

In order to comprehensively investigate the influence of new param-eterizations on radiation belt electrons, the similar VERB simulationsat 𝐸𝑘 = 0.5 and 2 MeV are produced (not shown here). For 0.5 MeV,the VERB simulations are similar to those at 0.9 MeV, in which theinclusion of new nightside chorus significantly improves the resultswhile the inclusions of new dayside chorus and new hiss do not improvemuch. For 2.0 MeV electron, all simulations overestimate the observa-tion. It is most likely that at 𝐿 > 4 region for these energies neitherdayside chorus nor hiss could provide sufficient loss (Ripoll et al.,2017). The incorporation of EMIC waves (Shprits et al., 2013b; Drozdovet al., 2015; Shprits et al., 2016) will help improve the estimation ofultra-relativistic electrons which is beyond the scope of this study. Inaddition, though only a single pitch angle 85◦ results are shown in ourwork, our main conclusions are valid for the other pitch angles.

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Fig. 10. Same as Fig. 6 except panel (b) represents VERB simulation with typical (CRRES) nightside chorus, typical (CRRES) dayside chorus wave and OS14 (CRRES) plasmaspherichiss wave.

5. Discussion

The new parameterization in this study is built up based on VanAllen Probes measurement while old parameterizations were built upfrom CRRES observation. It should be noted that the CRRES satellitewas launched on 25 July 1990 and failed on 11 October 1991, resultingin data that spans only the maximum of solar cycle 22. The Van AllenProbes were launched on August 2012 and were operable throughoutthe study period which spans the maximum of the solar cycle 24.Hence, the datasets about CRRES and Van Allen Probes are comparablein terms of the solar cycle phase. In addition, the corresponding param-eterizations are not limited just within the maximum of the solar cycle.That is because that the geomagnetic index 𝐾𝑝 rather than the solar orinterplanetary parameters, such as sunspot number, is adopted to scalethe wave activities. Though the datasets were collected in the maximumof solar cycle, the sufficient data samples in high 𝐾𝑝 levels as well aslow 𝐾𝑝 levels allow the extended adoption of wave parameterizationsin the different solar cycle phases.

In this study, using Van Allen Probe data we build a new parameter-ization for nightside and dayside chorus waves, which is different fromthe typical one (from CRRES observation). However, there are still lim-itations on our models, which require further improvements. Thoughtwo-years Van Allen Probe data from October 1, 2012 to September30, 2014 are used to build the chorus and hiss wave models, we willcomplement the models in the future by treating more years (fourmore years until now) to improve the statistical resolution, in particular

for rare events. Due to the limited latitudinal coverage of Van AllenProbes (< 20◦), the hiss and chorus wave information at high latitudeis absent. Thus we assume that the wave frequency and wave amplitudeare consistent at the higher latitude, equaling to those at 𝜆 = 20◦.The including of the other satellites, e.g. CLUSTER and MagnetosphericMultiscale Mission, could help understand the wave features at higherlatitude regions. The wave normal angle distributions of new choruswaves and new hiss waves are adopted from the typical models and theray tracing results, respectively. However, Van Allen Probes provide thewave normal angle feature by polarization analysis. The realistic wavenormal angle distribution (including mean value and width) based onthe statistics of Van Allen Probes data could improve our understandingon chorus and hiss waves’ effect on electrons. In the calculation of dif-fusion coefficients, we use Sheeley et al. (2001) model as plasmatroughdensity model and we use Carpenter and Anderson (1992) and Dentonet al. (2006) models as plasmasphere density models, which seemsinconsistent with each other. The reason is that our previous studiesseparately adopted those models for different calculation of diffusioncoefficients (Subbotin et al., 2011; Orlova et al., 2014, 2016). We willimprove the adoption of density models in future work. Moreover,recent studies (Malaspina et al., 2016, 2018) suggested that the hisspower distribution depends on the density and the relative distanceto plasmapause rather than 𝐿−shell, which could be potential futureimprovement.

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6. Conclusions

In this work, we develop new chorus wave frequency and am-plitude models, which are statistically obtained from the two-years(from October 1, 2012 to September 30, 2014) EMFISIS data onboardthe Van Allen Probes. The corresponding 3D diffusion coefficients asfunctions of equatorial pitch angle, energy and 𝐿 have been calculated.The comparisons between these new and typical diffusion coefficientssuggest that for nightside chorus waves the distributions and intensitiesof rates are significantly different, while for dayside chorus wavesthe distributions are similar but the intensities change. Moreover, thediffusion coefficient between old (based on CRRES) and new (basedon RBSP) are shown for comparison. We then perform long-term 3Dradiation belt simulations using the VERB code to investigate how thelong-term electron flux at 0.9 MeV depends on the assumed parame-terizations of waves. The simulation period is from October 1, 2012to September 30, 2013. The simulation results show that the VERBcode can reproduce the radiation belt electron evolution, indicatingthat both the typical parameterizations and new parameterizations areable to rather accurately reproduce the dynamics of the radiation belt.Moreover, the VERB simulations with new parameterizations are muchcloser to the observation: particularly important were nightside choruswaves. These results indicate that new parameterizations based onthe Van Allen Probes measurements can more accurately account forthe local diffusion. Simulations with new dayside chorus, and plas-maspheric hiss parameterizations were similar to typical CRRES-basedparameterizations, indicating that they still provide reliable results inlong-term simulations at 0.9 MeV.

Acknowledgments

The authors used geomagnetic indices provided by OMNIWeb (http://omniweb.gsfc.nasa.gov/form/dx1.html). We would like to acknowl-edge high-performance computing support from Yellowstone(ark:/85065/d7wd3xhc) provided by NCAR’s Computational and Infor-mation Systems Laboratory, sponsored by the National Science Founda-tion. This work used computational and storage services associated withthe Hoffman2 Shared Cluster provided by UCLA Institute for Digital Re-search and Educations Research Technology Group. This research wassupported by the NSF grant AGS-1203747, NASA grant NNX13AE34G,NASA grant NNX16AF91G, and NASA grant NNX15AI94G. This projectalso received support from Progress Horizon2020 grant #637302.

Appendix. MLT-dependent chorus parameterization

In the main manuscript, we have used a new parameterization forboth nightside and dayside chorus waves in which MLT-dependencewas simply averaged (therefore unresolved), since the feature is moreimportant for 4-D parameterization. However, we provide here to thereader the full chorus wave parameterization including MLT, 𝐿, MLATand Kp dependent chorus wave power:

𝐵2𝑤 = 𝑓 (𝑎, 𝑏, 𝐿, 𝜆)𝑔(Kp)

where

𝑓 (𝑎, 𝑏, 𝐿, 𝜆) = −3266.7 + 546.23𝑎 − 670.18𝑏 + 1150𝐿

+ 87.776𝜆 + 199.54𝑎𝑏

+ 228.42𝑏𝐿 − 77.367𝐿2 − 54.412𝑎𝜆 − 10.025𝑏𝜆 − 18.864𝐿𝜆

𝑎 = cos(MLT∕24 ∗ 2𝜋)

𝑏 = sin(MLT∕24 ∗ 2𝜋)

𝑔(Kp) = (75.532Kp − 45.677Kp2

+ 50.591Kp3 − 6.265Kp4 + 0.53362Kp5)∕1478.1

where 𝐵2w is unit of pT2 and 𝜆 is unit of degree. One can calculate the

bounce-averaged diffusion coefficient at 1 𝐿 only (e.g., 𝐿 = 4.5) as othervalues are obtainable through the scaling of the wave amplitude (𝐿, Kpand MLT).

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