Systematic Dynamic Fluid Kinetic (DyFK) model simulations are conducted to obtain a simulation-based formula representation of the O+ outflow flux level versus the energy flux of soft electron precipitation, and wave spectral density of transverse wave heating in the high-latitude auroral region. Based on results of 140 DyFK simulations of auroral outflows, we depict the O+ outflows at the ends of these two hour simulation runs in spectrogram form versus these parameters. We also approximate the results by the formula representation: = 8.8(3.0 × 105 + 2fe1.4 × 107)(tanh (8Dwave) + 0.2Dwave0.6), where is the O+ number flux in cm−2/s at 3 RE mapped to 1000 km altitude, fe is the electron precipitation energy flux in ergs cm−2/s, and Dwave is the wave electric field spectral density at O+ gyrofrequency at 1 RE in (mV)2 m−2 Hz−1. This formula representation provides a convenient way to set the boundary conditions of the ionosphere in global magnetosphere simulation since the ionosphere is the important plasma source of the magnetosphere.
 During the 1970's and 1980's, it became recognized that the ionosphere is the important plasma source of the earth's magnetosphere [e.g., Chappell et al., 1987; Moore and Delcourt, 1995]. Certain influences, including electron heating mechanisms causing elevated electron temperatures and consequently large ambipolar electric fields in the F region and topside ionosphere [e.g., Barakat and Schunk, 1983], centrifugal force [e.g., Horwitz et al., 1994], and electrostatic potential drops caused by photoelectron effects [e.g., Tam et al., 1995; Su et al., 1998], were recognized to potentially play important roles in propelling the heavy ion (O+) flows upward constantly over the nominal polar cap region. However, from the in situ observations of Dynamics Explorer-1(DE 1) [Lockwood et al., 1985; Waite et al., 1985] and Akebono [Abe et al., 1993], and other spacecraft, Horwitz and Moore  concluded that the O+ within the polar cap magnetosphere is primarily supplied by the cleft ion fountain, at least during southward IMF conditions.
 Among the various processes driving the auroral O+ outflow, soft electron precipitation is a strong candidate for the primary driver of many high-latitude F region and topside upflow situations [Horwitz and Moore, 1997]. Soft electron precipitation enhances F region plasma production, as well as increasing the electron temperature by Coulomb collisions between the soft electron precipitation and the thermal electron background, and/or other heat transfer processes. From the statistical results of Dynamics Explorer-2 (DE 2) observations at 850–950 km altitude, Seo et al.  found that the ion upflux was well-correlated with both the electron temperature and the soft electron precipitation energy flux, with correlation coefficients of approximately 0.97 and 0.72, respectively. In a systematic simulation investigation of the effects of soft electron precipitation, Su et al.  comprehensively studied the effects of soft electron precipitation with various characteristic energy and energy flux on electron temperature, ion temperature, ion upflow velocity, and ion upflow flux, and the temporal development of these plasma parameters during soft electron precipitation events in the F region and topside ionosphere.
 However, the enhanced O+ upflow plumes driven by soft electron precipitation still have insufficient energy to overcome the gravitational escape barrier in the absence of further energization at higher altitudes [Wu et al., 1999]. In terms of the ultimate level of the ionospheric plasma supply to the magnetosphere, the synergistic effects of soft electron precipitation and higher altitude energization mechanisms, such as wave-induced transverse ion heating, are potentially significant. Various mechanisms for the wave-particle interactions have been discussed for the transverse ion-heating [e.g., Norqvist et al., 1998]. The resonant heating of ions in the perpendicular direction produced by broadband extremely low frequency (BBELF) electrostatic waves has been widely used in some previous modeling studies [Wu et al., 1999; Zeng et al., 2006]. The transversely heated ions are subsequently energized in the parallel direction at higher altitudes by the mirror force associated with the diverging geomagnetic field. Using a sustained soft electron precipitation episode involving a Maxwellian precipitating electron energy spectrum with a peak at 100 eV and an energy flux of 1.0 ergs cm−2 s−1, and wave spectral density of 0.01 × 10−6 V2 m−2 Hz−1 at 6.5 Hz, Wu et al.  simulated the O+ dynamics and examined the outflux at 3 RE altitude for scenarios of soft electron precipitation only, wave heating of ions only, and synergistic effects of precipitation plus wave heating. The simulations showed that the synergistic effects of precipitation and wave heating could boost the net O+ outflow flux to 109 ions cm−2 s−1, which was approximately one order of magnitude higher than the flux attained in the absence of either the precipitation or wave heating.
 In order to provide the global simulation community a convenient and right way to set the boundary conditions in the global MHD simulations, such as for the Global Geospace Circulation Model, it is desirable to quantitatively represent the O+ outflow levels in terms of computationally-fast formulas or algorithms appropriate for various different geophysical conditions and outflow drivers. Recently, there have been efforts to obtain empirical relationships/correlations between the ion outflows and parameters at least related to different potential drivers [Strangeway et al., 2005; Zheng et al., 2005]. From the statistical analysis of outflows with possible influencing parameters for a geomagnetic storm observed by the FAST spacecraft, Strangeway et al.  obtained formulas to represent the correlative relationships between the outgoing ion flux and precipitation electron density or Poynting flux, respectively, at 4000 km altitude. The FAST observations were for the geomagnetic storm during 24–25 September 1998. The formulas obtained by Strangeway et al. were: When the ion flux was correlated with the electron precipitation, the flux was fi = 1.022 × 109±0.341 nep2.200±0.489, where fi is the ion flux in cm−2 s−1 and nep is the precipitating electron density. When the ion flux was also correlated with the Poynting flux, Strangeway et al. obtained the relationship fi = 2.142 × 107±0.242S1.265±0.445, where S is the Poynting flux at 4000 km altitude in mW m−2. Zheng et al.  proposed similar formulas based on 6000 km altitude observations of particle and fields by the Polar spacecraft during 2000.
 In this paper, we perform extensive simulations of the auroral ionospheric plasma transport and resulting outflows using the University of Texas at Arlington Dynamic Fluid Kinetic (DyFK) ionospheric plasma transport model and depict these in spectrogram form versus parametric representations of the soft electron precipitation and the transverse wave-driven ion heating. We approximate the results with a compact formula to represent the O+ outflows versus soft electron precipitation energy flux and a signature BBELF wave spectral density level, based on systematic simulations with different values of these parameters. A major purpose of this paper is to present a first compact formula representation of O+ outflow levels versus parameterizations of these two anticipated principal drivers of the outflows, for possible use in global magnetospheric dynamics modeling.
2. Model Description
 The UT Arlington DyFK model is a time-dependent, 1.5-dimensional high-latitude plasma transport model [Estep et al., 1999]. It couples a truncated version of the field line interhemisphere plasma (FLIP) model [Richards and Torr, 1990] to the generalized semi-kinetic (GSK) model [e.g., Wilson, 1992] across an overlapped boundary region. As illustrated in Figure 2 of Wu et al. , the simulated flux tube extended from 120 km to 3 RE altitude. In the lower, fluid-treatment portion (120 km–1100 km), the model calculates densities for H+, O+, other ion species and electrons, as well as field-aligned velocities and temperatures, by solving the relevant continuity, momentum, and energy-balance equations in the low-speed approximation. The upper boundary conditions for time-advancing the evolution in this fluid-treatment region, such as the ion density, parallel velocity, heat flux and the electron temperature, are provided at each time step by results of the advancing GSK treatment. In the altitude region 800 km – 3 RE of the flux tube, a generalized semi-kinetic (GSK) (or hybrid) treatment is used to advance the O+ and H+ gyrocenters. Simulation ions are injected at the lower boundary of the GSK portion using distributions based on the moment parameters resulting from the fluid-treatment results at that altitude for each alternating time step. The simulation ions are subject to the macroscopic forces of field-aligned electric field, gravity, geomagnetic mirror force and centripetal force. A more detailed description of the DyFK model has been given by Zeng et al.  and Wu et al. .
 In the DyFK model, the fluid-treatment portion incorporates the effects of precipitating auroral electrons on the ionospheric ions and electrons. The wave-induced transverse ion heating rate is calculated following the procedure of Crew et al.  as 2miDi⊥, where
in which ∣E∣2 is the electric spectral density at the ion gyrofrequency Ω(r). The factor η is the ratio of the left handed polarized wave components, which is assumed to be 0.125 [Chang et al., 1986]. The wave spectral density was approximated by
where ∣E0∣2 is the electric field spectral density at frequency ω0 and α is the power law index, which was assumed here to be 1.7 [Crew et al., 1990].
3. Simulation and Results
 For the purposes of obtaining a formula representation of the O+ outflow flux driven by soft electron precipitation and wave-induced transverse heating, we simulated the plasma outflows along the flux tube with varying soft electron precipitation energy flux and wave spectral density values. We arbitrarily chose the geophysical conditions on 20 August 1998 as the input parameters of our model, i.e., setting the planetary activity index Ap = 17, the solar radio activity index F10.7 = 142, and its 3-month average was ∼133. The geographical location of the foot point of the simulated flux tube was at 77.58°S, 290.19°E, and the corresponding geomagnetic latitude and longitude coordinates were 66.56°S, 0.59°E. We assumed the precipitating soft electron had a Maxwellian energy spectrum. For different simulation runs, the characteristic energy of the soft electron precipitation spectrum was selected to peak at either 100 eV or 50 eV, and the energy fluxes of the precipitating soft electrons varied from 0.1 to 6.0 ergs cm−2 s−1 at 800 km altitude, and the wave spectral densities were altitudinal constant in the wave heating region, i.e., 1600 km – 2 RE. The wave spectral density at 6.5 Hz, which is the O+ gyrofrequency at 1 RE altitude, varied from 0 to 2.5 × 10−6 V2 m−2 Hz−1 for different simulation runs.
 In each simulation run used here, the flux tube was initially evolved (in the absence of auroral effects) to a quasi-steady polar wind state, and then subjected to sustained soft electron precipitation and wave-induced transverse ion heating for two hours. Figure 1 shows the simulated O+ flux spectrogram versus altitude and time after the initiation of electron precipitation and wave heating, where the energy flux of the precipitating soft electron was 1.0 ergs cm−2 s−1 with characteristic energy 100 eV, and the electric field wave spectral density at 6.5 Hz was 0.3 × 10−6 V2 m−2 Hz−1. It can be seen from Figure 1 that the effects of electron precipitation and wave heating on O+ flux enhancement were very noticeable: the O+ flux increased from ∼1.4 × 107 to ∼1.0 × 109 cm−2 s−1 at 1000 km altitude, and increased from ∼1.5 × 105 to ∼1.7 × 107 cm−2 s−1 at 3 RE altitude during the two hour auroral processes stage. Figure 1 also shows that the O+ flux experienced a dramatic temporal evolution during the first ten minutes (dependent on the altitude, with a longer time for higher altitudes) after the initiation of soft electron precipitation and wave heating, then gradually became stable and remained nearly constant until the termination of the precipitation and wave heating [Zeng et al., 2006; Su et al., 1999]. For our formula representation, we sampled the O+ outflow flux at 3 RE altitude in the flux tube after the precipitation and wave heating had been on for two hours, and then mapped to 1000 km altitude for the convenience to compare with other simulations and observations. Therefore, the O+ flux represented by the formula based on our simulations should be suitable for the continual ionospheric plasma source, such as dayside cleft ion fountain (CIF). It has been shown for such simulations that at 3 RE all O+ ions are distributed in the v∥ > 0 region [Zeng et al., 2006], which means that at 3 RE, the O+ flux is a net outflow provided for circulation within the magnetosphere.
Figure 2a displays the O+ outflow flux at 3 RE (mapped to 1000 km altitude) versus wave spectral density at 6.5 Hz and electron precipitation energy flux based on the simulation results of 140 DyFK model runs. As the spectrogram color changes from dark blue to red, the O+ outflow flux level varies from ∼5 × 105 cm−2 s−1 to ∼2.5 × 109 cm−2 s−1. The electron precipitation energy spectrum was assumed to have a Maxwellian distribution and peak energy at 100 eV. It may be seen that at all wave spectral levels O+ flux increased monotonically with electron energy flux, which is consistent with the results of Seo et al.  and Su et al. . However, when the wave spectral density was lower than 0.005 (mV)2 m−2 Hz−1, the O+ flux remained below 108 cm−2 s−1, and didn't change with increasing wave spectral level until 0.005 (mV)2 m−2 Hz−1, the increase in wave spectral density resulted in rapid increase of the outflowing O+ flux, which attained values as high as 2.5 × 109 cm−2 s−1.
 From Figure 2a, we may infer that the wave heating process may be regarded as functioning as a “valve” on the net O+ outflows. If wave spectral density is below a certain threshold (0.005 (mV)2 m−2 Hz−1 for the conditions here), the wave heating is insufficient, and the produced O+ outflows are limited. If wave spectral density exceeds a value of about 0.02 (mV)2 m−2 Hz−1 for the conditions here, most of the entering O+ ions from below are energized to escape energies and the number flux attains a roughly asymptotic level.
 The results shown in Figure 2a may be approximated in a formula representation of O+ flux versus electron precipitation energy flux and wave spectral density in the following expression:
where is the O+ number flux in cm−2 s−1 at 3 RE mapped to 1000 km altitude; fe is the electron precipitation energy flux in ergs cm−2 s−1, and Dwave is the wave spectral density at 6.5 Hz in (mV)2 m−2 Hz−1.
Equation (3) succinctly displays the synergistic effects of soft electron precipitation and transverse wave heating. The first term on the right side, 3.0 × 105 + 2fe1.4 × 107, contains the effects of soft electron precipitation on driving O+ outflows. It includes a power law relationship between O+ outflow flux and precipitation energy flux, and has some similarity with the correlation expressions of Strangeway et al.  and Zheng et al. , although both of these investigator teams used electron number density to parameterize electron precipitation. The differences of the power law indices of equation (3), Strangeway et al. , and Zheng et al.  might result from the different sampling altitudes of the O+ flux. Another difference of equation (3) from Strangeway et al.  and Zheng et al.  is that equation (3) has a weak O+ outflow baseline, i.e., the term 3.0 × 105, which represent the small portion of O+ ions with escape energy under the combined driving of ambipolar field, frictional heating, mirror force, and centrifugal force. The second term on the right side, tanh (8Dwave) + 0.2Dwave0.6, contains the wave spectral density “valve” effect associated with the wave heating processes, as alluded to above. When Dwave is sufficiently small, tanh (8Dwave) + 0.2Dwave0.6 ≈ 0, and there should be no apparent wave heating effects on the outgoing O+ flux. Then, as Dwave increases, the term tanh (8Dwave ) + 0.2Dwave0.6 also increases rather dramatically, but the main term associated with tanh (8Dwave) changes dramatically from 0 to 1 over a limited range in Dwave.
Figure 2b displays the O+ outflow flux versus wave spectral density at 6.5 Hz and electron precipitation energy flux represented by equation (3). Comparing Figures 2a and 2b, we find that the spectrogram display of the formula in equation (3) agrees closely with our simulation results at various wave spectral density and electron precipitation energy flux levels, although the O+ fluxes represented by equation (3) are slightly lower than the simulation results when the precipitation electron energy flux is lower than 0.5 ergs cm−2 s−1 and wave spectral density is higher than 0.3 mV2 m−2 Hz−1. As noted above, equation (3) and the associated spectrogram in Figure 2b also adequately expresses the “valve” effect from the wave heating process observed in the simulations.
 In the formula representation of the O+ flux we have distilled from simulations here, we considered two major controllers of the O+ outflow flux: soft electron precipitation and wave-induced transverse ion heating. As for the soft electron precipitation, equation (3) only exhibits the relationship between O+ flux and electron precipitation energy flux. This is from those runs where the precipitation characteristic energy was fixed at 100 eV. However, the characteristic energy of the electron precipitation is also a factor controlling the O+ outflow flux. We performed a separate set of 140 DyFK model runs in which the characteristic energy of the precipitating electron energy spectrum was reduced from 100 eV to 50 eV, and found the O+ flux increased about 1.7 times for various other conditions on the energy flux and wave spectrum. However, the response of O+ flux to the characteristic electron energy is more complicated and will require a more extensive series of simulations to properly codify for general purposes. This will be done in future work, which should lead to an additional dependence on this parameter in the resulting approximate formula which would extend beyond equation (3).
 With regard to the above noted “valve” effect of the wave heating process, we interpret this results as follows: When the transverse wave-driven ion heating is sufficiently strong and extended, presumably the majority of the entering O+ ions are sufficiently energized to overcome the gravity barrier. Therefore, the net O+ number flux should be expected to saturate, and further increases of wave spectral density cause no significant further increase in O+ number flux. Of course, the O+ energy flux, not considered here, should continue to increase with increases in wave power. The O+ flux saturation in this “valve” effect shows up in our simulation and formula representation, within the wave spectral density (at 6.5Hz) range of 0–2.5 (mV)2 m−2 Hz−1. This range already covers the upper limit of the wave spectral density observed by DE 1, Freja and Viking spacecraft [Gurnett and Inan, 1988; André et al., 1998; Oscarsson and Ronnmark, 1990].
 As an important plasma source of magnetosphere, the ionospheric O+ outflow is also dependent on many other geophysical factors, such as KP index, solar radiation flux, season, and location of auroral oval. We have incorporated effects of two of the most important outflow drivers for the simulations here and the resulting equation (3). This formula representation of O+ flux may provide useful ionospheric outflow boundary conditions for certain types of global magnetosphere simulation models.
 This work was completed under financial support by NASA grant NNG05GF67G and NSF grant ATM-0505918 to the University of Texas at Arlington.