An empirical model of the global distribution of plasmaspheric hiss based on Van Allen Probes EMFISIS measurements

Using wave measurements from the EMFISIS instrument onboard Van Allen Probes, we investigate statistically the spatial distributions of the intensity of plasmaspheric hiss waves. To reproduce these empirical results, we establish a fitting model that is a third‐order polynomial function of L‐shell, magnetic local time (MLT), magnetic latitude (MLAT), and AE*. Quantitative comparisons indicate that the model's fitting functions can reflect favorably the major empirical features of the global distribution of hiss wave intensity, including substorm dependence and the MLT asymmetry. Our results therefore provide a useful analytic model that can be readily employed in future simulations of global radiation belt electron dynamics under the impact of plasmaspheric hiss waves in geospace.


Introduction
Plasmaspheric hiss is a broadband incoherent whistler mode emission with frequencies ranging from ~20 Hz to ~2 kHz that is observed predominantly inside the plasmasphere or high density plumes (Thorne et al., , 1979Meredith et al., 2004;Li W et al., 2013Li W et al., , 2015aNi BB et al., 2014;Shi R et al., 2017). It can persist under quiet conditions; its fluctuations are positively correlated with the level of solar activity Thorne et al., 1973Thorne et al., , 1974. Broadband amplitudes of plasmaspheric hiss range from a few pT to as high as 100 pT Smith et al., 1974;Meredith et al., 2004). Plasmaspheric hiss is mainly observed at a broad range of wave normal angle, e.g., its propagation near the geomagnetic equator is predominantly fieldaligned, but oblique at higher latitudes (Santolík et al., 2001;Bortnik et al., 2008).
It has been well recognized that electron scattering by plasmaspheric hiss acts as a dominant contributor to formation of the slot region, which separates the radiation belts into inner (1.2 < L < 2) and outer (3 < L < 7) parts (Lyons et al., 1972;Lyons and Thorne, 1973;Albert, 1994;Abel and Thorne, 1998a, b;Meredith et al., 2004Meredith et al., , 2006a. Resonant interactions with plasmaspheric hiss result in electron scattering losses with decay time scales varying from less than 1 hour to tens or hundreds of days. Decay times are closely dependent on electron energy, ambient magnetic field, and plasma density, as well as on wave amplitude, spectral intensity, and wave normal angle (WNA) distribution of the hiss waves as a function of spatial location and geomagnetic activity level (Meredith et al., 2007;Ni BB et al., 2014;Yu J et al., 2017). As reported in Summers et al. (2007a, b), the pitch angle diffusion coefficient of resonant electrons in quasi-linear formalisms is proportional to the wave amplitude. Thus, the development of a reliable global model of hiss wave amplitude is essential to facilitate quantification of hiss-induced electron scattering rates and resultant global variations of radiation belt electron distribution.
B 2 w A number of empirical models of hiss wave global distribution have been constructed in previous studies (Meredith et al., 2004;Kim et al., 2015;Li et al., 2015b). Data from the Combined Release and Radiation Effect Satellite (CRRES) are utilized to investigate the features of plasma waves. Meredith et al. (2004) presented elaborate global maps of hiss wave intensity and described its distribution features. Orlova et al. (2014) produced empirical quadratic fittings of as a function of Kp, L, and geomagnetic latitude (λ) for the daytime and nighttime sectors respectively. However, the CRRES wave instrument has limitations in space and frequency band, in particular it records only electric field data; accordingly, some important wave information such as magnetic field spectral intensity was estimated, based on theoretical assumptions. Characteristics of whistler mode waves have also been analyzed by Agapitov et al. (2013) with Cluster data, which separate chorus from hiss merely by f ce (the electron cyclotron frequency), i.e., chorus frequency above 0.1f ce and hiss below 0.1f ce , though their frequencies may actually overlap. Subsequently Agapitov et al. (2014) parameterized the hiss wave activity at L < 2 based on Akebono spacecraft observations. In addition, combined observations of Cluster (Agapitov et al., 2013) and Polar (Tsurutani et al., 2015) showed that hiss can be widely distributed over the magnetic latitude, extending to λ > 45°.
Recently, using data from Van Allen Probes, Spasojevic et al. (2015) and Yu J et al. (2017) constructed one-dimensional fitting models of hiss wave amplitude as a function of AE, L, MLT, and Kp, L, MLT, respectively. Highlighting the dependence of hiss amplitude on single variables, those studies focus on the fluctuations of the fitting curves associated with each variable in their hypothetical formulae and do not reproduce the observed latitudinal variations of hiss wave amplitude. Since the dependence on MLAT cannot be neglected in accurate predictions of hiss amplitude, as previous studies have done, the present investigation considers the combined effects of L, MLT, MLAT, and AE*, and intends to establish a more comprehensive empirical model of plasmaspheric hiss amplitude as a function of these four input parameters.

Data and Methodology
Van Allen Probes were launched on September 8, 2012 with perigee of ~1.1R E (radius of the Earth), apogee of ~5.8R E , inclination of 10°, and an orbital period of ~9 hours. Equipped with identical electromagnetic detectors, the EMFISIS instruments onboard Van Allen Probes can acquire high quality measurement of whistlermode waves in the inner magnetosphere . The waveform receiver (WFR) on EMFISIS provides wave power spectral densities ranging from 10 Hz-12 kHz with a temporal resolution of 6 s. The high frequency receiver (HFR) records electric spectral information between 10 and 400 kHz, in which the traces of upper hybrid resonance (UHR) frequency can be used to estimate the ambient plasma density (Mosier et al., 1973) and identify the location of the plasmapause that separates the regions outside and inside the plasmasphere (e.g., He F et al., 2011He F et al., , 2013He F et al., , 2016He F et al., , 2017Katus et al., 2015;Verbanac et al., 2015;Zhang XX et al., 2017a, b).
According to the wave characteristics of hiss emissions, in this study we identify a hiss event with the ellipticity > 0.7 (i.e., righthand polarized) and the frequency band within 10-2000 Hz in the plasmasphere. By doing so, we distinguish the hiss waves from chorus waves outside the plasmasphere and from magnetosonic waves with nearly linear polarization. Subsequently, we integrate the wave spectral intensity in the determined frequency band to calculate the wave amplitude (B w ). In our following analysis we concentrate on the hiss emissions with B w ≥ 5 pT. The adopted database consists of observations made during the period from September 8, 2012 to June 30, 2017: L-shell (L), magnetic local time (MLT), magnetic latitude (MLAT), AE* (averaged value of the AE index in the previous hour), and hiss wave amplitude (B w ) with temporal resolution of 6 s. We then implement two methods to construct our empirical global model of hiss wave amplitude. In each method, we divide the entire database data into two groups, i.e., the training group and the test group. The training group is used as the baseline to obtain the fitting coefficients in the regression, while the test group is regarded as a comparison to examine the model's performance and verify its reliability.
|w obs − w mod | w obs As tabulated in Table 1 Taking L, MLT, MLAT, AE*, and B w as known quantities and F(i, j, k, l) as unknown, we implement the fits of a third-order polynomial function as follows: where the parameters A, B, C, and D are quantified as L/10, MLT/24, MLAT/20, and AE*/500, respectively. Following previous studies of Spasojevic et al. (2015) and Yu et al., (2017), we set the maximum values of i, j, k, and l as 3, 2, 2, and 2, respectively; consequently we have 108 coefficients for the fit to each training group.

Analysis Results
Based on the two methods described above, we do the fits for the training groups, compute the model results with the obtained polynomial fitting functions, and perform quantitative comparisons with the statistical observations of the test groups.
First, we follow Method #1 to establish the empirically analytic model of plasmaspheric hiss intensity, the results of which are shown in Figures 1−4. Figures 1 and 2 show the global distributions of time-averaged root-mean-square (RMS) hiss wave amp-litudes in the equatorial plane and the meridian plane, at resolution of 0.5L × 1MLT and 1° × 0.1L, respectively, obtained using Database 1 (the period from 2012/9/8−2015/9/8), and comparisons between these statistical observations and results of the empirical model.
The first-row subplots in Figure 1 reveal that the hiss wave amplitude strengthens significantly when AE* increases and that the waves on the dayside are much stronger than those at the nightside. Overall, hiss wave activity shows a strong dependence on Lshell, MLT, and geomagnetic level. Correspondingly, the secondrow panels illustrate model results of hiss wave amplitude for the same time period as in Figures 1a−1c, obtained using the fitted three-order polynomial function (i.e., Equation (1)). It is evident  It is clearly seen that during most bins the relative differences are less than 15%, though the differences are larger in some limited regions of L = 4.5-6 and MLT ~23-06. Figure 2 results are similar to those in Figure 1 but as a function of L and MLAT in the indicated four MLT sectors. It can be seen clearly that, besides its L-shell and MLT dependence, the hiss wave amplitude also depends on MLAT but in a less significant manner. Figures  While the wave observations are slightly different from those in Database 1, the dominant trends are favorably similar. In order to validate the feasibility of the fitting function, we compare the observations from Database 2 (Figures 3a−3c) with the model results for the same period that are obtained using the empirically analytic function model as a function of L, MLT, MLAT, and AE* derived using Database 1 (Figures 3d−3f). Apparently, large differences between the observational and model results occur especially during geomagnetically moderate and active times, showing that the value of R d can be well above 1 in association with large deviation of the model results from the observations. In addition, it is distinct in Figure 4 that the model results become much less reliable compared to the observations, especially for the interval of MLT = 03-09. These discrepancies between model results and observations during the time period exclusive to the test group tell that the empirical model obtained using Method #1 has limitations that make it unsuited to the task of obtaining a global hiss wave distribution.
Accordingly, we adopt Method #2 to establish an empirical model of the global distributions of hiss intensity; that is, the data during   Some discrepancies at higher latitudes may be attributed to hiss wave amplitude enhancements during geomagnetically disturbed periods, which, however, are difficult to be captured by the fits possible with third-order polynomial functions.
Overall, this study establishes an empirical model of the global distribution of hiss wave intensity by implementing third-order polynomial function fits to the long-term Van Allen Probes EMFISIS wave data, which is determined by Equation (1) and the polynomial coefficients tabulated in Tables S1-S12 in the Appendix for the four considered MLT sectors (i.e., 03−09, 09−15, 15−21, and 21−03) under three geomagnetic conditions (i.e., quiet-time: AE* < 100 nT; moderate-time: 100 nT ≤ AE* ≤ 300 nT; active-time: AE* > 300 nT).

Concluding Remarks
In the present study, we have used approximately five years of Van Allen Probes EMFISIS wave data to analyze statistically the global distribution of plasmaspheric hiss intensity at L = 2−6 under various conditions of geomagnetic activity. During quiet times or times of weak substorm activity (AE*<100 nT), the hiss wave amplitude distribution is not dependent on MLT. However, when substorm activity intensifies, the hiss wave amplitude distribution is more dependent on MLT and the intense hiss events are concentrated at high L (L>5) on the morning side and the nightside (MLT ~ 00:00−06:00). Furthermore, the observation results (the first row of our Figures 2, 4, 6, and 8) suggest that the hiss amplitude distribution is slightly dependent on MLAT in the data on which the present study is based, which means the distribution is mainly controlled by L and MLT. The hiss wave amplitude is very intense inside the dayside plasmasphere (the first row of Figures  Two different methods have been implemented to separate the training group and the test group, as shown in Table 1. Our analysis indicates that Method #2, which uses the odd-number days as the training group and the even-number days as the test group, is superior in deriving well-performing fitting functions for the global hiss wave model. One possible reason can be that the training and test groups in Method #2 may reflect the phases of a solar cycle more reasonably. Several other factors that may affect the modeling results should also be considered: first, the latitudin- al distribution of the number of samples is dependent on the MLAT, which could be another reason why Method #1 is worse than Method #2; second, Databases 1 and 2 have more differences in the MLAT distribution, which may also be the cause of differences between the observations and model results. These unsolved questions require further investigation in future studies. In summary, it is well demonstrated, by quantitative comparisons between the statistical observations and model results, that the empirical model in terms of fitted functions can reflect favorably the major features of the global distribution of hiss wave intensity, including the MLT asymmetry, substorm dependence, and latitudinal variations (Xiang Z et al., 2017). Because both the energy spectra and pitch angle distributions of radiation belt electrons are critically affected by hiss wave scattering (e.g., Ni BB et al., 2013Zhao H et al., 2019;Li LY et al., 2008;Hua M et al., 2019), and because the scattering rates increase proportionally to the square of hiss amplitude for the near-resonance cases, our results therefore provide an empirically useful analytic model to be readily used for numerical quantification of hiss-driven electron diffusion coefficients and global simulations of resultant modulation of radiation belt electron dynamics in response to varying conditions of solar wind and geomagnetic activities.