JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. A12, PAGES 29,277-29,287, DECEMBER 1, 1998

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. A12, PAGES 29,277-29,287, DECEMBER 1, 1998 Comparison of macroscopic particle-in-cell and semikinetic models of the polar wind A. R. Barakat, H. Thiemann, and R. W. Schunk Center for Atmospheric and Space Sciences, Utah State University, Logan Abstract. The outflow of the polar wind along diverging geomagnetic field lines has been the subject of many modeling studies for the past 25 years. As the plasma drifts up and out of the topside ionosphere, it undergoe several transitions; for instance, its velocity changes from subsonic to supersonic and its velocity distribution changes from Maxwellian to non-maxwellian. The complexity of the flow led to the development of several modeling approaches, such as the generalized moment, the kinetic, and the semikinetic models. Recently, a "macroscopic" particlein-cell (PIC) model was adopted to study the polar wind. However, because one is always restricted to a finite number of particles, the validity of the approach must be established when it is applied to macroscopic flows. In this study the polar wind predictions obtained from a macroscopic PIC simulation were compared to those obtained from the more rigorous semikinetic model for steady state conditions. The study also shows the sensitivity of the PIC simulation to the adopted modeling parameters for both time-dependent and steady state conditions, including the number of simulation particles, the time step, the spatial bin size, etc. The study indicates that (1) the PIC model can be a powerful simulation tool if special attention is given to its potential pitfalls; (2) because of the finite number of particles. the PIC technique is subjecto a considerable amount of noise; (3) the noise level is higher for the higher-order moments, such as heat flow, and for the velocity distribution function; (4) the use of a time step that is too large leads to a modulation of the results; (5) an insufficient number of spatial bins yields a poor spatial resolution, while too many spatial bins leads to more noise; (6) the noise level can be reduced by averaging over time and/or space, but this affects the spatial and/or temporal resolution; (7) the bin size in velocity space must be carefully chosen to balance numerical noise and velocity space resolution; (8) in the steady state the PIC technique can achieve the same accuracy as the semikinetic model if all of the PIC modeling parameters are optimized; and (9) a few hundred thousand simulation particles, as used in some previous studies, are not adequate to resolve the tail of the velocity distribution, even in the steady state when time averaging is possible. For 106 particles the noise in the tail is still appreciable; and (10) when poor spatial resolution is used, important features can be missed, as was the case in some previous studies. 1. Introduction The polar wind corresponds to a thermal plasma outflow at high latitudes. As the plasma flows up and out of the topside ionosphere, it undergoes several transitions, including transitions from chemical dominance to diffusion dominance, from subsonic to supersonic flow, from heavy to light ion dominance, and from collision-dominated to collisionless regimes. In association with the latter transition the ion velocity distributions evolve from Maxwellian at low altitudes to highly non-maxwellian at high altitudes. Because of the complicated nature of the polar wind, numerous mathematical formulations have been used over the years to describe various aspects of the flow, including hydrodynamic, hydromagnetic, Copyright 1998 by the American Geophysical Union. Paper Number 98JA /98/98 JA generalized transport, kinetic, semikinetic, Monte Carlo, and hybrid particle-in-cell (PIC) techniques [Banks and Holzer, 1968, 1969; Raitt et al., 1975; Holzer et al., 1971; Schunk and Watkins, 1981, 1982; Mitchell and Palmadesso, 1983; Ganguli et al., 1987, 1992, 1994; Ganguli and Palmadesso, 1987, 1988; Barakat and Schunk, 1983, 1984; Demars and Schunk, 1989, 1991, 1994; Demars et al., 1993; Barghouthi et al., 1993; Blelly and Schunk, 1993; Schunk and Sojka, 1989; Barakat et al., 1990; Gombosi et al., 1985, 1992; Gombosi and Nagy, 1989; Ho et al., 1992, 1993; Horwitz et al., 1994; Lemaire and Scherer, 1970, 1972, 1978; Miller et al., 1993; Yasseen and Retterer, 1991; Yasseen et al., 1989; Wilson, 1992; Wilson et al., 1990]. All of the above mathematical formulations have both strengths and limitations when applied to polar wind situations. For example, with the transport formulations it is relatively easy to take account of multispecies, chemical reactions, collisions, time dependence, and multiple space Also at Arbeitsgruppe Weltraumforschung und -Technologie, Freiburg, Germany. 29,277 dimensions. They are limited, however, in that they are obtained by truncating the infinite hierarchy of moment equations, and in general, it may not be clear how the truncation affects the solution. The kinetic and semikinetic models are particularly suited to the collisionless, steady state polar wind. They have the advantage in that the full hierarchy

2 29,2?8 BARAKAT ET AL.: PIC AND SEMIIrANT.TIC MODELS OF THE POLAR WIND of moment equations are implicit in the solution and multiple particle populations can be readily included. Some of their limitations are that they are difficult to apply to timedependent, multidimensional or collisional flows. Monte Carlo and PIC techniques have the advantage that you follow the motion of individual particles, and hence a lot of the important physics can be included self-consistently. Monte Carlo techniques are particularly useful for collisiondominated gases, and with the PIC approach, self-consistent electric fields can be easily taken into account. A disadvantage is that both techniques are computationally demanding, and therefore they cannot be easily extended to multidimensional simulations. They also are subject to numerical noise, which can affect the physics. With regard to polar wind applications, there have been several studies in which the results obtained from hydrodynamic, generalized transport, semikinetic, and Monte Carlo techniques have been compared for the same polar wind scenario. Also, within a given formulation the results obtained for different levels of approximation have been compared. The various levels of approximation and formulations have been compared for both steady state and time-dependent polar wind simulations (see above references). In contrast to the other mathematical techniques the strengths and limitations of the macroscopic PIC model of the polar wind have not been fully elucidated. This type of study is especially important in light of the increased popularity of the macroscopic PIC approach. Simulations that are based on the macroscopic PIC were used in order to study shocks in the polar wind [Demars et al., 1996a], as well as the effects of centrifugal acceleration [Demars et al., 1996b], trapped particles [Demars et al., 1998], and hot magnetospheric electrons (A. R. Barakat et al., Dynamic features of the polar wind in the presence of hot electrons, submitted to Journal of Geophysical Research, 1997, hereinafter referred to as Barakat et al., submitted manuscript, 1997) on the polar wind. Moreover, two-dimensional (2-D) and three-dimensional (3-D) versions of such a model are under construction which puts severe restrictions on the number of particles per bin. This, in turn, acts to increase the noise level in the results. (a) In this investigation we compared the results obtained from a macroscopic PIC simulation of the "steady state" polar wind with the corresponding results obtained from a semikinetic solution. We also conducted a quantitative study of the accuracy of the PIC technique for "time-dependent" simulations, where numerical noise can be a serious problem. A major component of this work is a critique of the macroscopic PIC approach. Consequently, all the pitfalls that can be encountered when applying this method are pointed out. However, this does not mean that it is not valid. In fact, the macroscopic PIC model (if applied on the proper set of problems and if its limitations are considered while interpreting the results) can be a valuable tool for simulating space plasmas. In section 2 we briefly describe our macroscopic PIC and semikinetic models of the polar wind. The accuracy of the PIC technique for time-dependent simulations is described in section 3, and a comparison of the macroscopic PIC and semikinetic simulations is presented in section 4. A summary and discussion are provided in section Theoretical Formulation The semikinetic models were developed and evolved over the seventies and eighties [e.g., Holzer et al., 1971; Barakat and Schunk, 1983, 1984; Li et al., 1988]. Here we present a brief description of their underlying theory. For more details the reader may refer to the aforementioned papers. In the semikinetic models (Figure la), a simple model is adopted for the ion exosphere. In this region, which extends above the exobase (r = to), the plasma is assumed to be collisionless. Below r o lies the barosphere, where the plasma is considered to be collision-dominated. A plasma, which is composed of electrons and protons, flows along a diverging magnetic flux -3 tube. The Earth's magnetic field is taken to be B - r e r where r is the geocentric distance and e r is a unit vector in the radial direction. The ions are assumed to have a drifting Maxwellian distribution fo(h +) at the exobase (to). To obtain a semikinetic solution, an initial electrostatic potential profile (b) g 8R e Particles Simulation Inject Ions Figure 1. Schematic representation of the simulation domain for the (a) semikinetic and the (b) macroscopic particle-in-cell (PIC) models. Both models include the effects of gravity g and the polarization electric field

3 BARAKAT ET AL.: PIC AND SEMIIONETIC MODEI_.S OF THE POLAR WIND 29,279 is assigned. Using the Liouville theorem, in combination with the adopted ion distribution function at r o, it is possible to compute the ion distribution function at any altitude in the exosphere. By taking the moments of the H + distribution, its density n(h +) can be obtained, which is equal to the electron density. The Boltzmann relation is then used to find an improved electrostatic potential profile. The previous steps are repeated until the solution converges to a consistent one. Then it is straightforward to take the moments of the H + velocity distribution function to find the profiles of the H + density, drift velocity, temperature, and heat flow vectors. A basic implicit assumption for the validity of the semikinetic approach is that the number of particles in a Debye sphere (, necks) tends to infinity, where k D is the plasma Debye length. In our polar wind problem the Debye length varied from 38 cm at low altitudes to 12 m at high altitudes. At all altitudes the number of particles in the "Debye sphere" exceeded 106. The macroscopic PIC model considered here was first applied to space physics by Wilson et al. [1990] and was then applied to different ionospheric and magnetospheric problems in subsequent investigations by several authors [e.g., Ho et al., 1992; Miller et al., 1993; Demars et al., 1996a, b; Demars et al., 1998; Barakat et al., submitted manuscript, 1997]. For this model the simulation region consists of a magnetic flux tube that extends from the exobase (r o = 1.7 RE) to several earth radii (as shown in Figure 1). This tube is divided into a few hundred cells (Nc) of length Ar each. The motion of the ions is represented by a large number of simulation particles N. In particular, each simulation particle is randomly assigned a position r and two components of velocity parallel (vii) and perpendicular (vñ) to the geomagnetic field B. The contribution of each particle to the different spatial cells is then computed according to the finite-size particle approach, and consequently, the plasma density profile is computed. Applying the quasi-neutrality condition (hi = n E) and assuming that the electrons obey the Boltzmann relation, the electrostatic field E is then found. Subsequently, the motion of each simulation ion is followed for a small time interval At, under the influence of gravitational, electrostatic, and mirror forces, and hence its position and velocity are incremented. The new positions of the ions are used to update the density profile. The ions that leave the simulation domain, from the top or bottom boundaries, are replaced by injecting ions at the lower boundary. The injected ions are randomly assigned velocities that are consistent with the drifting Maxwellian ion velocity distribution at the exobase. The previous steps are repeated at each time step At. At any instant of time t the velocities of the ions in a given spatial cell can be used to compute the velocity moments (e.g., bulk drift velocity u d, parallel and perpendicular temperatures (Tii, and Tñ), etc.) at the center of that cell. Moreover, the ions in a given spatial cell can be binned further via a 2-D mesh in velocity space in order to find the ion velocity distribution function. The above description of the macroscopic PIC model shows its close similarity to the traditional PIC models, such as those used by Okuda and Ashour-Abdallah [1981]. In particular, both of them are based on pushing simulation particles in a simulation domain that is divided into cells. However, the macroscopic PIC model uses large cells (i.e., Ar >> Debye length k D) in contrast with the earlier PIC models where smaller cells were used (i.e., Ar < XD). This feature is necessary to enable the macroscopic PIC model to simulate large-scale problems, such as magnetic flux tubes that are several Earth radii long. Hence we cast the term macroscopic PIC for such models rather than the terms "PIC" or "semikinetic" which have been used to designate other models for tens of years. 3. Effect of Modeling Parameters on Macroscopic PIC Simulation From the description of the macroscopic PIC model in the previous section it is clear that it includes several adjustable parameters, such as the time step At, the cell size Ar, and the number of simulation ions, N. Such parameters should be assigned values that strike a balance between the required accuracy and the available computer resources. In this section we vary these values over wide ranges in order to investigate their effects on the results. The ranges of parameters were chosen to cover the typical values used in previous studies. For example, Wilson et al. [1990] adopted the following values in simulating the polar wind plasma: the simulation magnetic flux tub extended from the exobase (r o = 1.7 RE) to 8 RE; there were 200 cells with each km long; the number of simulation ions (N) was -105; the time step At was 2.5 s; and the ion temperature r o was 3000 K Number of Simulation Particles A critical parameter is the number of simulation particles (N) because an insufficient number of particles can lead to numerical noise, which, in turn, can affect the results. Therefore three simulations were carried out in order to investigate the effect of N on the results. The simulations started with an enhanced plasma density in the flux tube, and the subsequentemporal evolution was followed until a steady state outflow was achieved. All the parameters were identical for the three runs except for N, which was given low, moderate, and high values. In Figure 2a we present the temporal evolution of the normalized H + density profiles for the different values of N. The small (104), medium (105), and large (106 ) N cases are presented from left to right, respectively}) For N = 1 6 (right) the noise level is very small, and the temporal behavior of the H + density profile is clear. In particular, the plasma is first depleted at lower altitude, and then the depletion expands to higher altitudes as time progresses. For a lower value of N (105 ) the noise level increases, but the temporal evolution of the H + density profile can still be clearly seen. For an even lower value of N (104) the noise level increases further, and it tends to obscure the temporal evolution of the density profile (left). In general, the noise level decreases as N increases. This phenomenon was expected since it is well known that the noise level depends on the number of simulation ions per cell. However, the effect of numerical noise in macroscopic simulations has not been discussed previously. The behavior of the higher-order moments was investigated in a similar way. These moments are: bulk drift velocity u d, parallel (Tii) and perpendicular (Tñ) temperatures, and parallel (qll) and perpendicular (qñ)heat fluxes. All these moments display the general trend mentioned above; that is, the noise level increases as N is decreased. Moreover, the noise level was found to be greater for the higher-order moments. This

4 -- 29,280 BARAKAT ET AL.: PIC AND SEMIKINETIC IVlODELS OF TI-IE POLAR WIND Time [s] = 0 ( ), 750(... ) (... )! i [ illill 104 i! part. i IllIll 1! I Ill 1 ' [ [! illill [ i I IllIll I I I Illll 10 5 part. \. i i IllIll I I I I I I I II1,1 ' 10 $ part. - ;" 'z. ' - I I I I!111 i i I i i i iii 11 '2 1 1,olc i i i iiiii1 i i i iiiii1 i i i III 11 '2 1 ' )ø1,-3 1( -2 1( -1 ß 0 Norm. Density Norm. Density Norm. Density Figure 2a. Comparison of the tem P40ral evolution of the density profile. as predicted by the macroscopic PIC model, for the cases of (left) N =10,(middle) N =10 s, and (right) N =106. The curves correspond to t = 0 (solid), t = 750 (dotted), and t = 10,000 s (dashed) The following ionospheric conditions and modeling parameters are adopted: u o = 11 km s -1, T,o = Tño = 7; E = 3000 K, N c = 200, and At = 2.5 s. trend can be explained as follows. The high-order moments are more sensitive to the ions in the tail off(h+). Since these "tail" particles are relatively rare they suffer from more noise, and therefore the high-order moments display higher noise levels. In order to elucidate the above point we present the temporal evolution of qn in Figure 2b. For N = 104 the noise function at 5 R E for four cases, namely, N = 10, ' 106, and level overwhelms the temporal variation (left). Even for N = l0 s, it is not possible to draw reliable conclusions about the considered the additional case of 10-particles. This behavior of q" (middle). In fact, it is only possible to obtain exceptionally high number of simulation particles N enables reliable qll profiles if N is increased significantly. The case of us to resolve the features off(h +) down to <1% of its maximum N = 106 is presented at the right, where the noise reduced to value. Figure 3d shows the distribution function for N= 107, an acceptable level. We conclude that in order to resolve the dynamic behavior of the heat flow q and higher-order moments, at least 106 simulation particles are needed for the polar wind. Even more simulation particles ( 107) are needed to study the finer dynamic features of the velocity distribution function f(h+), as will be shownext. If only steady state results are required, time averaging can be used, and hence a smaller number of simulation particles are needed. As explained earlier, the ion velocity distribution at a certain altitude (spatial cell) can be found by binning the ions in velocity space. This further reduces the number of simulation ions per bin and hence increases the numerical noise level, especially in the tail of the velocity distribution. +. Figure 3 shows snapshots of the H velocity distribution 107. Since the distribution function is ver prone to noise we which suffers from the least level of noise. The distribution function is composed of a primary component, with a maximum at v,-- 15 km s -], and a second_ ry component in the form of an enhanced tail at Vii '-- 8 km s The noise level in the secondary component (tail) is higher because it contains fewer simulation particles. Also, the inner contours of the primary component (near its maximum) tend to be less noisy than the other contours because they contain more particles. Time Is] [ ' I 0 4 part. ), 750 (... ) (... ) I [ ' ] I part. ' 1 ' ' [ I ' [ I 106 part.,, I I I i"d' ' 1'- i i - : ' -20, I o qll [10-8 g cm3s-3] q]] [10'8 g cm3s '3] I i i q I [10'8 g cmos 'a] II Figure 2b. Comparison of the temporal evolution of the parallel heat l ux (q) profile, as predicted by the macroscopic PIC model, for the cases of (left) N = 104, (middle) N =10", and (right) N =106. The curves correspond to t = 0 (solid), t = 750 (dotted), and t = 10,000 s (dashed). The adopted ionosphericonditions and modeling parameters are similar to those for Figure 2a.

5 BARAKAT ET AL.: PIC AND SEMIKINETIC MODELS OF THE POLAR WIND 29,281 i! - b - c. d I ' 104 part., I I -10 v [kin s ' ] 105 part. 106 part. I I I I i I I I V. [krn s'l] V. [kin s'l] 10 7 part. I V. [kin s' ] Figure 3. Comparison of the ion distribution function, a predicted by the macroscopic PIC model, for the cases of (a) N = 104, (b) N = 105, (c) N = 106, and (d) N = 10. Snapshots off(h +) at 5 R E altitude and t = 3125 s are shown. The distribution function is pre nted by equivalued contours plotted in velocity (vii, vñ)space. The contour levels decrease by a factor of 10. The adopted onosphenc conditions and modehng parameters are similar to those for Figure 2a. For N = 106 the noise level is greater; that is, the outer contours of the main component display significant noise levels, and the features of the secondary component can be seen only in a qualitative manner. As N decreases further (N 105), the noise overwhelms the result, and eventually (N = 104), the secondary component disappears entirely, and the primary component becomes too noisy to resolve any of its characteristics Number of Spatial Cells Another important modeling parameter is the number of cells (Nc) in the simulation domain. The effect of N c (and consequently the cell size Ar) is illustrated in Figure 4, where the temporal evolution of the parallel temperature profile is presented for different values of N c, namely, 20, 200, and Although the computational resources do not present a direct constraint on the choice of Nc, there are other factors that restrict the range of optimum values of N c. For values of N c that are too small the altitude resolution becomes poor, and consequently, a systematic error is introduced into the solution. Figure 4a shows a representative case of this problem (N c = 20), where the gradient of Tii at low altitudes i s less steep than it should be. As Nc increases (with the total number of simulation particles held constant), the above mentioned problem is reduced, but another problem starts to appear. The increase of No and the consequent decrease of the number of ions per cell, causes the noise level to increase. It was found that the results reach an optimum accuracy for a range of N c ( ). This is shown in Figure 4b (N c = 200), where the combination of both kinds of error (systematic and random noise) is minimized. A similar behavior was found for the higher-order moments, with the expected increase in the noise level for these moments, as noted in the previous subsection. The results for these other moments are not presented here Time Step We also considered the effect of the time step At on the accuracy of the results. Ideally, At should be infinitesimal, but this would increase the required computing time beyond the Time [s] = 0 ( ), 750 (... ) (... ) ] I [ I [ I - a 20 bins 200 ] I [ bins I ] I ' "oo'o"oo'o"o' r[] IF] ' I ' r]] [K] r][ [K] Figure 4. The effect of the number of cells N c on the ;/'11 profile. Three cases are considered, namely (a) N c = 20, (b) N c = 200, nd(c) N c = The curves correspond to t = 0 (solid), t = 750 (dotted), and t = 10,000 s (dashed). N = 10" simulation particles is used, while the rest of the ionospheric conditions and modeling parameters are similar to those for Figure 2a.

6 29,282 BARAKAT ET AL.: PIC AND SEMIKINETIC MODELS OFTHE POLAR WIND available resources. Therefore we performed several numerical experiments for a wide range of At. For time steps as large as 25 s the error due to the finite value of At was found to be less than the errors from other sources. For At 3 50 s the effect of a finite At started to be noticeable in the form of a modulation in the velocity-moment profiles. When At was increased, both the amplitude and the length of the modulation wave increased. Figure 5 shows the finite-at effect on the different moment profiles for the extreme case of At = 250 s. Note that the modulation waves in the different moment profiles display similar characteristics. Also, the error for the higher-order moments is greater than that for the lower-order moments Time Averaging Ai discussed earlier, the reduction of the noise level by an increase in the number of simulation particles is restricted by the available computational resources. The noise level can be further reduced by integrating the results over a range of altitudes and/or over a longer time interval. However, the improvement comes at the expense of the spatial and/or temporal resolution. On the other hand, when the model reaches the steady state, the results can be integrated over very long time intervals, and this greatly reduces the noise level. Unfortunately, this only works for steady state situations, which can be simulated more efficiently with other models such as the semikinetic ones. It is worthwhile to point out that the noise at two instants (say, t 1 and t2) can be considered independent only if It 1 - t21 is comparable to or exceeds the auto correlation time (x). An estimate of x can be taken as the time it takes half the population of a given cell to leave it. Therefore, in computing the steady state we picked a long integration time (from 7500 to 12,500 s), which was sampled every 20 time steps. The results were then averaged over these samples to get the improved results. The steady state profiles (with and without time averaging) were investigated for different levels of N (104, 105, and 106). We present the results of such a comparison for the qll profiles in Figure 6. Since the higherorder moments suffer from higher noise levels they display the effect of time averaging clearly. Figure 6a compares the averaged and nonaveraged q" profiles for the case of N = 104. The noise level for the nonaveraged profile obscures the spatial features of the profile. The time averaging reduces the noise to low, yet noticeable, levels, and hence the characteristics of the steady. state qll profile become clear. When N is increased to 105 (Figure 6b), the noise level is reduced for both the averaged and nonaveraged profiles. The noise in the averaged profile is ve[y small, although it is still visible. For the case of N = 10" the noise level is reduced t rther, as shown in Figure 6c Velocity Binning In order to calculate the velocity distribution function f(h +) at a given altitude a velocity space mesh must be used. The choice of the velocity bin size /Sv of the mesh critically determines the accuracy of the resulting f(h+). The adoption of a/sv that is small improves the resolution of f(h +) but reduces the number of particles per bin, which acts to increase the noise level. At the other extreme, if a large value of /Sv is adopted, the noise level is reduced at the expense of losing the resolution of f(h +) in velocity space. Figure 7 shows the effect of/sv on the producedistribution ß + function. Figure 7 gives three representations of f(h ) at 5 R E and t = 10,000 s for the case of 10 simulation particles. The only difference between Figures 7a, 7b, and 7c is the velocity space mesh used to bin the simulation particles. Figure 7a corresponds to a relatively large bin size (/Sv = 2 km s-i). It clearly shows the loss of resolution in velocity space and the small sampling noise. Figure 7c represents the other extreme where/sv = 400 m s- Here the resolution is very good, but the contours are noisy, especially the outer ones. A case of intermediate bin size (6v = 800 m s- ) is presented in Figure 7b, where a compromise between the two kinds of errors (noise and resolution loss) is achieved. We found that the optimum value of 6v depends on the number of simulation particles, on the distribution function f(h+), and on the nature of the features that need to be investigated. 6 Time [s]: 1 ' ' ' I ' 0 ( ), 7s0(... ) 0000(... ) [ U½ [krn s ' ] Figure 5. The evolution of some velocity moment profiles, as predicted by the macroscopic PIC model, for a large time step (At = 250 s). The moments presented here are (a) bulk drift velocity, u d, (b) parallel temperature Tii, and (c) parallel heat flux qll. The curves correspond to t = 0 (solid), t = 750 (dotted), and t = 10,000 s (dashed). N = 10 v simulation particles is used, while the rest of the ionospheric conditions and modeling parameters are similar to those for Figure 2a.

7 BARAKAT ET AL.: PIC AND SEMIKINETIC MODELS OF THE POLAR -WIND 29,283 [ [ i! [ i ] part. '!! part. ]!! part..,r:,,..._ _ i i i i i i i g cmos 'ø] 10 I i! i i! 1 0 [10-8 g cmos-o] lo i! i i I i! i i i ii i i o qll [10'8 g cmos '3] lo Figure 6. Effect of time averaging on the parallel heat flux q]] profile, as predicted by the macroscopic PIC model, for steady state conditions. The solid curve is for the case where the profiles are averaged over the interval from 7500 to 12,500 s, while the dotted curve is for the nonaveraged case at 10,000 s. Three different values of Nare considered, namely' (a) N= 104, (b) N=105, and (c) N= 1. The rest of the ionospheric conditions and modeling parameters are similar to those for Figure Boundary Conditions 4. Comparison of Macroscopic PIC and Semikinetic The choice of the ion distribution function fo at the exobase Models critically controls the evolution of the plasma in the polar wind. This was clearly pointed out by Miller et al. [ 1993] and will be discussed here briefly. The two cases presented in There are two major differences in the theoretical basis underlying the semikinetic and macroscopic PIC models. First, the semikinetic model is based on the Vlasov equation, Figure 8 are identical, except for the drift u o of the Maxwellian which implicitly assumes that there is a large number of ions distribution at the lower boundary fo' Figure 88 corresponds to per Debye sphere, while the available computational resources 1 the case of u o = 11 km s', while Figure 8b corresponds to the limit the number of simulation particles (N) that can be used in case of u o = 0. Only the evolution of Tll is presented here as an the macroscopic PIC model. Second, the macroscopic PIC example. However, similar differences occur in the other model divides the simulation domain into a relatively small moments. Although the initial Tii profiles were chosen to be similar (Tii = 3000 K), the temporal evolution of the Tii profile follows a different path, depending on the adopted distribution function at the boundary. number of cells (N c -, 100), which limits the spatial resolution. In contrast, the semikinetic model can compute the results at any altitude r. These factors should be considered when we compare the results of the two models. - a I 5v=2 km s I 5v=0.8 km s t ' 5v=0.4 km s - I i I I I I,, I I I V _ [km S -1] Vj_ [km S '1] V.L [km S -1] Figure 7. Effect of the adopted velocity bin size ( Sv), on the velocirf( distribution function f(h+) We considered the cases of 52, = (a) 2, (b) 0.8, and (c) 0.4 km s-'. Snapshots of H +) at 5 R E altitude and t'= fb,000 s are presented. N = 10" simulation particles is used, while the rest of the plotting format, the ionospheric conditions, and the modeling parameters are similar to those for Figure 3.

8 29,284 BARAKAT ET AL.: PIC AND SEMIKINETIC MODELS OF THE POLAR WIND w I w I w I I Figure 9 shows the different moment profiles for both the co semikinetic model (solid) and the macroscopic PIC model a (dotted). Figure 9a compares the normalized density profiles to for the two models. The density profiles predicted by the two models display an excellent agreement, such that the two curves (solid and dotted) are hardly distinguishable. The same excellent agreement is displayed for the profiles of u d Tii, and Tñ. Even for the high-order moments, qll and qñ,'the two - -.,. - models are in very good agreement. - 7 ;/.- _ A comparison of the distribution function f(h ) at 5 R E is ß,!., presented in Figure 10. There is a close similarity between the distribution functions produced by the semikinetic (Figure xl' a) and the macroscopic PIC (Figure 10b) models. This i s _.. _ consistent velocity moments. with the agreement However of we the point two out models the for following all the ' '.-., - differences. First, even with 1; 6 simulation particles and time - _.., - averaging, the outer contours of the PIC result show some cxi I I,"' "-]"... I minor semikinetic noise signatures, result. We an also effect notice that the is entirely effect of absent a finite bin the size (/Sv) in both results. For the semikinetic model the effect L I I I I is mesh an artifact, in the plotting and it can routine. b easily We deliberately removed by used using the a same finer co i "'"".,,..,... ' b - compared. velocity mesh In in contrast, the two models for the so macroscopic they can be PIC more model, easily r tj x... - refining the velocity mesh is limited because it causes a earlier. to ; i I, -- corresponding The comparison increase of the semikinetic the noise level, and PIC as distribution mentio I:' functions in Figure 10 shows that if a large number of [ ',,,-,. simulation particles are used, if the spatial and velocity bin sizes are optimized, and if the results are time-averaged, the ø - - PIC technique can produce steady state results that are -- comparable to those obtained from the semikinetic model. Xl'l -.L.- However, visible noise appears in the tail of the distribution, starting from the contour corresponding to values of 1% of the _ t.!,' - maximum. If we need a better signal-to-noise ratio, more -- _ simulation particles should be used, or the results should be,. averaged over a longer period of time. It should also be noted that time averaging is only effective in the steady state. Time averaging during dynamic (time-dependent) PIC simulations can significantly affecthe results and care must be exercised if [[I [K] Figure 8. Effect of the boundary condition the Tii profile. We considered Maxwellian distribution functions at the lower boundstry (to) that are nondrifting (u o = 0) an drifting (u o = 11 km s' ). The drifting case is shown in Figure 8a, and the nondrifting case is shown in Figure 8b. The curves correspond to = 0 (solid), t = 750 (dotted), and t = 10,000 s (dashed). N = 10 simulation particles is used, while the rest of the adopted ionospheric conditions and modeling parameters are similar to those for Figure 2a. In this section the results of the two models are compared -1 for the same ionospheric conditions; namely, u o = 11 km s, Tii o = Tño = T E = 3000 K. We ran the macroscopic PIC simulation until it reached a steady state in order to facilitate comparison with the semikinetic model results. Also, in order to reduce the effect of the "granulation" due to a finite N we used a large number of PIC simulation particles (N = 106), and we averaged the steady state results over a long period of time. this done. 5. Summary and Discussion In this study we investigated the characteristics of two different models, namely, the semikinetic and the macroscopic PIC models. Both models were applied to the problem of exospheric polar wind outflow along a divergent geomagnetic flux tube. The results of the two models were compared in order to elucidate the strengths and weaknesses of each approach. The following are basic differences between the two models. 1. The semikinetic model implicitly assumes that the number of ions per Debye sphere is much greater than unity. This assumption is required in order to derive the Vlasov equation, which ignores the interparticle correlation. In our polar wind problem this condition is definitely satisfied. In contrast, the finite number of simulation particles used in the macroscopic PIC model causes a strong interparticle correlation and hence granulation in the results. 2. The semikinetic model requires only a small fraction of the computational resources (i.e., CPU time and memory)

9 BARAKAT ET AL.: PIC AND SEMIKINETIC MODELS OF THE POLAR WIND 29,285 a Norm. Density b Semikinetic I I I b m U d [km s ' ] c c:) c:) Macroscopic PIC I I I V_L [km S -1] Figure 10. Comparison of f(h +) at 5 R E produced by the (a) semikinetic and (b) macroscopic PIC models. The steady state results of the latter model were obtained by averaging over the time from 7500 to 10,000 s. The plotting format is similar to that for Figure 3, and the adopted ionosphericonditions and modeling parameters are similar to those for Figure 9. required by the macroscopic PIC model. For instance, a macroscopic PIC model takes as much as 10, ,000 Heat Flux [10'8 g cm 3 s -3] times more CPU time than the semikinetic model. Because of the computationally intensive nature of the macroscopic PIC approach, substantial computing resources are needed in order Figure 9. Comparison of the semikinetic (dotted) and the to apply this approach to 2-D and 3-D macroscopic problems. steady state macroscopic PIC (solid) models. For the PIC 3. The accuracy of the results (especially the higher-order model the profiles were averaged over the time from 7500 to moments and distribution function) of the macroscopic PIC 12,500 s. The moments presented here are (a) normalized ion model is limited because of sampling errors. In contrast, it is density, (b) drift velocity Ud,, (c) parallel and perpendicular relatively easy to compute f(h +) and its moments with a very temperatures II (Tii and 1 T1), and d) parallel and perpendicular heat fluxes (q and q ). N = 10 simulation particles is used, high accuracy using the semikinetic code. while the ionospheric and modeling parameters are similar to 4. The macroscopic PIC model is time-dependent, while the those for Figure 2a. semikinetic model can handle only time-independent

10 29,286 BARAKAT ET AL.: PIC AND SEMIKINETIC MODELS OF THE POLAR WIND problems. However, a large number of PIC simulation particles must be used in order to keep the noise at acceptable levels. The number required for our problem was particles. 5. It is relatively straightforward to modify the macroscopic PIC model to incorporate the influence of collisions, chemical interactions, etc., which is not the case with the semikinetic model. 6. The two models can produce similar results for steady state collisionless conditions if the PIC parameters are chosen carefully. A systematic study was also performed in order to estimate the effect of the different modeling parameters on the results of the macroscopic PIC model. The following conclusions were drawn: (1) The noise level is higher for the higher-order velocity moments, such as the heat flux. (2) Although the densitqv profiles can be resolved for moderate values of N (510"), reliable profiles of the high-order moments (e.g., heat flux) and ion distributions can be obtained only for N 106. (3) In order to produce reliable results the time step At should not exceed 25 s. Larger values of At introduce an error in the form of modulations of the profiles. (4) An optimum value for the number of cells was found to be Nc ' This value results from a balancing of the error due to a low spatial resolution, which dominates for small N c, and the increase in noise level, which dominates at large N c. (5) The noise level can be reduced by integration over time and/or space, however, this improvement comes at the expense of the temporal and/or spatial resolution. (6) The bin size ( Sv) of the velocity space mesh should be chosen to balance two kinds of error, namely, the large noise level associated with small values of/sv and the loss of resolution in velocity space associated with large values of /Sv. (7) After the model reaches steady state conditions the noise level can be reduced by integrating the results over a long time interval. This method applies only to time-independent results, which can be achieved with other (more efficient) models such as the semikinetic model. We conclude that each modeling approach (the ones discussed here and the others) has strengths and weaknesses. These strong and weak points should be assessed by applying the approach on a well-understood problem and/or by comparing its results with those of the other models. The successful solution of a given space plasma problem depends on utilizing the proper model (which can differ from one problem to another) and interpreting the results in the light of its limitations. With regard to the limitations associated with the PIC Ganguli, S. B., T. Chang, F. Yasseen, and J. M. Retterer, Plasma model our results indicate that even when 106 simulation transport modeling using a combined kinetic and fluid approach, in particles are used and time averaging is done, the steady state velocity distributions still display noise in the tail of the distribution. Therefore it is not clear how smooth contours in the deep tail were obtained in some previous studies when only 200,000 simulation particles were used. Also, when a poor spatial resolution is used, important features can be missed, as was recently shown by Barakat et al. [1995]. Acknowledgment. This research was supported by NASA grant NAG and NSF grant ATM to Utah State University. The Editor thanks T. E. 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