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Corresponding author: J. Krall, Plasma Physics Division, Naval Research Laboratory, Washington, DC 20375, USA. (email@example.com)
 The Naval Research Laboratory three-dimensional, first-principles simulation code SAMI3 (Sami3 is Also a Model of the Ionosphere) is used to model plasmasphere refilling. A time-dependent Volland-Stern-Maynard-Chen potential is used to model an idealized magnetic storm that erodes the plasmasphere to L<3. The potential is then relaxed to the prestorm state, and refilling is simulated for a range of L shells 3≤L≤5 over a period of 7 days. Refilling rates compare well to observed refilling rates. The model plasmasphere during this quiet period displays a day-to-day repetition in its morphology that has not been previously observed.
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 A significant body of work has been published on simulated plasmasphere refilling using first-principles models that in each case solve a coupled set of continuity, momentum, and energy equations. These typically include only a single flux tube [e.g., Singh et al., 1986; Rasmussen and Schunk, 1988; Tu et al., 2003], but some use two dimensions (a single magnetic longitude) [e.g., Sandel and Denton, 2007; Krall et al., 2008]. In all cases, these models feature one or more field lines, a model ionosphere, and a specified set of equations to be solved on each field line. We now simulate refilling using SAMI3 (Sami3 is Also a Model of the Ionosphere) [Huba and Krall, 2013], which is distinguished from previous studies by the inclusion of all three dimensions and cross-field ion transport in all directions. Here our aim is to demonstrate that SAMI3 generally reproduces previous refilling rates, both numerical and empirical, and to present a new dynamical result.
 To lowest order, the macroscopic dynamics of the plasmasphere results from the interaction of the sunward magnetospheric convection flow with the corotation flow of the inner magnetosphere. This leads to such phenomena as the duskside “bulge” [Nishida, 1966] that becomes a sunward-directed plume upon storm onset [Grebowsky, 1970; Sandel et al., 2001]. Observations of the combined corotation and convection effects are summarized by Darrouzet et al.  and Singh et al. , where such features as shoulders, fingers, notches, crenulation, and channels are described. We have seen analogs to many of these features in our model plasmasphere that includes the corotation potential, the Volland-Stern-Maynard-Chen potential, [Reinisch et al., 2009], and the wind-driven dynamo potential. For example, density profiles in Huba and Krall [2013, Figure 4] compare well to those shown in Singh et al. [2011, Figure 13].
 In this letter, we simulate the plasmasphere after the onset of an idealized magnetic storm using a time-dependent Volland-Stern-Maynard-Chen potential. At the end of the idealized storm, the plasmasphere has contracted to L<3. The potential is then relaxed to the prestorm state, and refilling is simulated for a range of L shells 3≤L≤5 over a period of 7 days. Refilling rates compare well to previous two-dimensional SAMI2 results at a fixed longitude and with observed refilling rates. The model plasmasphere during this quiet period displays a diurnally repeating morphology.
2 SAMI3 Model
 The Naval Research Laboratory SAMI3 code [Huba et al., 2008; Huba and Krall, 2013], which is based on the SAMI2 (Sami2 is Another Model of the Ionosphere) [Huba et al., 2000] code, was used in this study. SAMI3 includes wind-driven dynamo electric fields, solved as a two-dimensional electrostatic potential equation that is based on current conservation (∇·J=0). For dynamics along field lines, SAMI3 solves the continuity and momentum equations for seven ion species and the temperature equation for three ion species (H+,He+,O+) and for the electrons. SAMI3 includes 21 chemical reactions as well as photodeposition [Huba et al., 2000]. Cross-field transport is included as E×B drifts. Huba et al.  provides a good description of the equation set and of the potential solver.
 The version of SAMI3 used here is identical to that used in Huba and Krall . Specifically, the grid includes magnetic apex heights from 90 km out to 8 RE. The grid is (nz,nf,nl) = (204, 124, 96) where nz is the number grid points along the magnetic field, nf is the number in “altitude,” and nl is the number in longitude. In this version of SAMI3, the magnetic field is a dipole aligned with Earth's spin axis, and the grid is fixed relative to the sun, so that a constant l index corresponds to constant magnetic local time (in some versions of SAMI3, the l index instead corresponds to magnetic longitude). To account for the rotation of the Earth within this grid, a corotation potential is specified, corresponding to exact corotation of the ionosphere and plasmasphere.
 To study refilling, we impose a Volland-Stern-Maynard-Chen potential [Reinisch et al., 2009] of the following form
and P(t) has the form shown in Figure 1. Other geophysical parameters are held artificially constant. These are the following: F10.7=120, F10.7A=120, Ap=4, Kp=6, and day-of-year =80.
 The results presented in Huba and Krall  correspond to the times t=t1 and t=t2in Figure 1. In the present study, we continued the simulation from t=t2 to t=t5 using P(t) shown in Figure 1. The convection potential was doubled which caused a further contraction of the “storm time” plasmasphere shown in Huba and Krall . The refilling study begins on day six of the SAMI3 run with this eroded plasmasphere, shown in the left panel of Figure 2 as a plot of the electron density ne in the magnetic equatorial plane at time t=t3. This plot is similar to the corresponding figure in Huba and Krall [2013, Figure 1 therein]. Here the plasmapause in the midnight sector is at an altitude of less than two Earth radii, corresponding to a magnetic L shell L<3. Figure 2, right panel, shows the electron density neat t=t4, after the amplitude of ΦVSMChas been reduced to one eighth of its previous value. The potential Φ, shown as black contour lines in Figure 2, includes ΦVSMC, the corotation potential, and the contribution from the wind-driven dynamo. With ΦVSMCreduced (right panel), a greater area is dominated by the closed contours of the corotation potential. This represents the region of the inner magnetosphere to be refilled.
 Figure 3 shows color contours of the electron density (log10ne) plotted every 8 h during 3 days of refilling; days seven to nine in Figure 1 are arranged so that plots in any one column are separated by 1 day of simulated time. Examination of the plots in any one column shows evidence of refilling, as expected. Perhaps more interesting is that, while the shape of the plasmasphere varies with time, it shows a clear diurnal cycle. In particular, all three plots at UT 0100 are similar in shape with a bulge near magnetic local time (MLT) 1600, plots at UT 0900 show a bulge near MLT 1430, and at UT 1700, the bulge is near MLT noon. We note that this effect does not hold during the storm when the plasmasphere is eroding. In Figure 2, for example, the plasmasphere morphology is affected by the strongly varying convection potential; it differs significantly from the morphologies seen a day later in the top row of Figure 3.
 Figure 4 shows the electron density ne, averaged over longitude, plotted at the magnetic equator for L=3, 4, and 5 (dashed curves). Similar to past results and analyses [e.g., Lawrence et al., 1999], refilling after the first day, commonly called late stage refilling, is approximately linear for a small number of days (<10) until it begins to show signs of saturation. For comparison to the L=3 curve, colored curves show the electron density ne at L=3 and fixed local time at dusk (red), midnight (dark blue), dawn (blue), and noon (yellow). The strong diurnal variation at fixed local time evident in the colored curves is related to the diurnal cycle seen in Figure 3. This diurnal signature at fixed local time is stronger than that of the longitude-averaged curves (Figure 4, dashed curves) or of fixed-longitude SAMI2 refilling curves [Krall et al., 2008, see Figures 1 and 3 therein].
 Numerous determinations of empirical refilling rates for quiet times can be found in the literature [Denton et al., 2012, and references therein]. Such determinations, typically presented as a rate of electron density ne increase that varies with L, are problematic because they are almost always based on multiple events (data is sparse) and because there is not yet an underlying reason to assume that the refilling rates for any two given events vary with L in the same way. In a comprehensive analysis of IMAGE satellite data, refilling rates are generally found to be lower than past empirical results [Denton et al., 2012, see Figure 1 therein], and the scatter in the various refilling rates (e.g., rates at solar min versus solar max) is almost as large as the differences between the rates [Denton et al., 2012, see Figure 11 therein]. Following the suggestion of R. E. Denton (private communication, 2013), we compare our first two days of refilling to the median refilling rate reported in the abstract of Denton et al. . The empirical refilling rates for L=3, 4, and 5, 75.1, 43.6, and 19.9 cm−3 day−1, respectively, are represented as long-dashed lines in Figure 4. We see that the SAMI3 result agrees very well with empirical rate at L=3 and that the SAMI3 rates are smaller than the empirical rates at L=4 and 5. Late stage refilling rates, based on days seven and eight, are 74.3, 26.1, and 4.0 cm−3 day−1 for L=3, 4, and 5, respectively. These vary more strongly with L than the empirical rate.
 Because the IMAGE satellite observes He+ rather than the electron density ne, He+ refilling is of interest. Following the format of Figure 4, Figure 5 shows shows the He+ density averaged over longitude at L=3, 4, and 5 (dashed curves). At L=3, colored curves show He+ at dusk (red), midnight (dark blue), dawn (blue), and noon (yellow). Again, the curves at fixed local time show a stronger diurnal variation than the longitude-averaged curves. This result is consistent with variations in column-density versus local time found by Galvan et al. [2008, see, e.g., Figure 6], with weaker refilling at dawn and stronger refilling at dusk. The fixed-local-time curves show a clearer dawndusk asymmetry in He+ than was seen in the electron density in Figure 4.
 Empirical refilling rates for He+are not available, but measurements have been made. Sandel and Denton [2007, see Figure 3], for example, find 24, 17, and 9.6 cm−3 day−1for L=3,4, and 5, respectively, during 2001 June 27–30. Late-stage refilling rates found here, 9.2, 3.6, and 0.8 cm−3 day−1 for L=3,4, and 5, respectively, are lower than that result but compare well with our previous refilling study using SAMI2 [Krall et al., 2008]. The ratio on day 13 is 0.11, 0.14, and 0.14 at L=3,4, and 5, respectively, which compares very well with IMAGE/EUV column density ratios [Sandel, 2011].
 In this idealized SAMI3 simulation of plasmasphere refilling during a geomagnetically quiet period, we find that SAMI3 refilling rates compare well to empirically-determined refilling rates. Here the agreement is generally within a factor of two, except at L=5, where the difference was about a factor of five. Where differences exist, the SAMI3 curves show slower refilling than the empirical results. Also, we have shown that the oblong shape of the rotating, dynamic plasmasphere is such that the plasmasphere morphology displays a remarkable day-to-day repetition. That is the shape of the plasmasphere at a given time is similar to that seen one day earlier or later, a result not seen elsewhere. We speculate that weak convection fields during refilling allow other processes, such as the wind-driven dynamo, to noticeably affect global plasma transport. We also see a strong diurnal signature in the refilling curves at fixed local time, consistent with measurements by Galvan et al. . If the refilling is plotted at a fixed longitude or is averaged over all longitudes, then this diurnal variation is reduced.
 In comparing refilling rates, three caveats should be considered. First, this SAMI3 simulation differs from a typical period of quiet refilling in that there are no solar wind variations to affect transport within the plasmasphere during refilling. The significance of this solar wind “noise” is not yet known. Second, SAMI3 does not include an energization model for O+outflow at high latitudes. Developing such a model would be a significant step forward and would enable comparisons to mass-density refilling measurements [e.g., Obana et al., 2010]. And finally, the electron temperature equation used in SAMI3 considers a simplistic model of photoelectron heating [Bailey and Balan, 1996]. Since the electron, as well as ion temperatures, are expected to be important in controlling the refilling rates, a better photoelectron heating model should be used. For example, a physically based photoelectron heating model has been developed for SAMI2 by Varney et al. . This model will be used in future studies to assess the impact of photoelectron heating on plasmasphere refilling.
 This research was supported by NRL Base Funds and the NASA LWS Program. We thank Richard Denton for helpful discussions.
 The Editor thanks Mark Moldin and Phil Richards for their assistance in evaluating this paper.