Alfvén wave solar model (AWSoM): proton temperature anisotropy and solar wind acceleration

Transcription

1 doi: /mnras/stv2249 Alfvén wave solar model (AWSoM): proton temrature anisotropy and solar wind acceleration X. Meng, B. van der Holst, G. Tóth and T. I. Gombosi Center for Space Environment Modeling, University of Michigan, Ann Arbor, MI 48109, USA Accepted 2015 September 28. Received 2015 September 25; in original form 2015 April 24 ABSTRACT Temrature anisotropy has been frequently observed in the solar corona and the solar wind, yet poorly represented in computational models of the solar wind. Therefore, we have included proton temrature anisotropy in our Alfvén wave solar model (AWSoM). This model solves the magnetohydrodynamic equations augmented with low-frequency Alfvén wave turbulence. The wave reflection due to Alfvén sed gradient and field-aligned vorticity results in turbulent cascade. At the gyroradius scales, the apportioning of the turbulence dissipation into coronal heating of the protons and electrons is through stochastic heating. This par focuses on the impacts of the proton temrature anisotropy on the solar wind. We apply AWSoM to simulate the steady solar wind from the corona to 1 AU using synoptic magnetograms. The Alfvén wave energy density at the inner boundary is prescribed with a uniform Poynting flux r field strength. We present the proton temrature anisotropy distribution, and investigate the firehose instability in the heliosphere from our simulations. In particular, the comparisons between the simulated and observed solar wind prorties at 1 AU during the ramping-up phase and the maximum of solar cycle 24 imply the importance of addressing the proton temrature anisotropy in solar wind modelling to capture the fast solar wind sed. Key words: methods: numerical solar wind. 1 INTRODUCTION In magnetized plasmas, the plasma temrature along and rndicular to the magnetic field can be different due to insufficient heat exchange between the parallel and rndicular thermal motions. This feature is referred to as temrature anisotropy or thermal pressure anisotropy, and commonly exists in space plasmas. A theoretical foundation is provided by the Chew Goldberger Low approximation (Chew, Goldberger & Low 1956). A large amount of in situ and remote observational data have shown that the plasma temrature is anisotropic in the solar corona (SC; Kohl et al. 1998, 2006; Lietal.1998; Antonucci, Dodero & Giordano 2000; Telloni, Antonucci & Dodero 2007) and the solar wind (Hundhausen 1968; Feldman et al. 1974; Marschetal. 1982; Gary, Skoug & Steinberg 2001; Gary, Goldstein & Neugebauer 2002; Kasr, Lazarus & Gary 2003; Marsch, Ao & Tu 2004; Hellinger et al. 2006). Since electrons transport much more quickly than protons, the proton temrature anisotropy is often much larger than the electron temrature anisotropy in the solar wind. Though being observed frequently, the xingm@umich.edu Present address: Jet Propulsion, 4800 Oak Grove Dr., Mail Stop , Pasadena, CA role of the temrature anisotropy, particularly the proton temrature anisotropy, in the solar wind is not fully understood. Solar wind modelling has become a very active research area (Riley, Linker & Mikić 2001; van der Holst et al. 2005; Feng, Zhou & Wu 2007; Kleimann et al. 2009; Watermann et al. 2009; Gressl et al. 2014). In particular, to address the proton temrature anisotropy in the solar wind, a lot of efforts have been made. The earliest work can be dated back to about 40 years ago. Leer & Axford (1972) develod a one-dimensional (1D) steady-state solar wind model with isotropic electron temrature and anisotropic proton temrature that allows for extended coronal proton heating. The model itself does not solve for the magnetic field, but the solar wind solutions are obtained for the cases of a purely radial and a spiral magnetic field, and the solutions at 1 AU are reasonable. At about the same time, Whang (1972) also develod a 1D steadystate solar wind model including proton thermal anisotropy, which assumes the same isotropic temrature for electrons and protons within a radial distance of 0.4 AU, while different temratures for electrons and protons, as well as proton temrature anisotropy beyond 0.4 AU. The calculated solar wind solution at 1 AU agrees well with observations. Both models are pioneers in simulating proton temrature anisotropy in the solar wind. More work has been done after 1990s. Hu, Esser & Habbal (1997) presented a 1D time-dendent fast solar wind model with isotropic electron temrature and anisotropic proton temrature. C 2015 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

2 3698 X. Meng et al. The model includes momentum and heat input to the solar wind by Alfvén waves, additional momentum input to the protons and additional heat input to both electrons and protons. The high-sed solar wind solutions they obtained match most of the empirical constraints from observations. This model was further extended to solve the 16-moment bi-maxwellian equations by Li (1999), who has found that the inclusion of proton parallel and rndicular heat flux densities greatly affects proton temrature anisotropy. More recently, Li et al. (2004) reported the first two-dimensional (2D) Alfvén wave turbulence-driven solar wind model with proton temrature anisotropy. They obtained solutions for both fast and slow solar wind, and found that the average proton temrature in the anisotropic case is lower than in the isotropic case. Chandran et al. (2011) develod a 1D solar wind model including proton temrature anisotropy, pitch-angle scattering from mirror and firehose instabilities, and kinetic Alfvén wave turbulence, based on theories of linear wave damping and non-linear stochastic heating. They have found consistency between their model results and a number of measurements. All these studies provide valuable understanding to the modelling of solar wind with anisotropic proton temrature. However, these 1D or 2D models still cannot reveal the complete picture of the spatially and temporally varying solar wind, which could only be provided by time-dendent 3D global models of the corona and heliosphere. For the first time, we present the extension of our 3D global magnetohydrodynamic (MHD) Alfvén wave solar wind model AWSoM to include proton temrature anisotropy. The model assumes isotropic electron temrature; thus, it is a three-temrature (3T) model. Van der Holst et al. (2014) have presented the lower coronal modelling with the AWSoM, and found good agreement between simulated and observed multi-wavelength EUV images. The continuing work, being presented in this par, focuses on the solar wind simulations up to 1 AU. More scifically, we apply the model to simulate the corona and the inner heliosphere (IH) for two Carrington rotations (CRs) during the solar cycle 24. We look into the proton temrature anisotropy and plasma instabilities in the simulations, and validate the simulated solar wind prorties at 1 AU against the observed values made by the STEREO and WIND satellites, as well as from the OMNI data. By comparing the simulation results with the results from the two-temrature (2T) model, i.e. the model with isotropic proton and electron temratures, we find that including proton temrature anisotropy in the model leads to slower and more realistic solar wind seds. The following content of the par is divided into four sections. In Section 2 we describe the equations implemented, with emphasis on the treatment of proton pressure anisotropy limiting through instabilities. Sections 3 and 4 present the solar wind simulation results for CR2107 and CR2123, resctively. Section 5 concludes the par and discusses possible future work. The par also contains apndices about the plasma instabilities we considered in the model and on the electron pressure equation. 2 MODEL DESCRIPTION The governing equations for the AWSoM are ρ + (ρu) = 0, (1) t ρu + [ρuu + p I + p e I + (p p )bb t 1 μ 0 ( BB B2 2 I )] + ( ) w+ + w 2 = ρ GM r 3 r, (2) B t p t + ( u B) = 0, (3) + (p u) + 2p b (b )u = (γ 1) p e p τ ei + (γ 1)Q p + δp δt, (4) ( ) ( ) p p + t γ 1 γ 1 u + p ( u) + (p p )b (b )u = p e p + Q p, (5) τ ei ( ) ( ) + t γ e 1 γ e 1 u + p e ( u) w ± t = q e + p p e τ ei Q rad + Q e, (6) + (uw ± ± V A w ± ) w ± u = R w w + Ɣ ± w ±, (7) where ρ and u are the mass density and velocity, B is the magnetic field, b = B/ B is defined as the unit vector along the magnetic field and μ 0 is the rmeability of vacuum. G is the gravitational constant, M is the solar mass and r is the position vector relative to the centre of the Sun, with r = r. p represents the proton pressure component parallel with the magnetic field, while p is the rndicular component. p = (2p + p )/3 is the average scalar proton pressure. For simplicity, we do not have a subscript that stands for proton. p e represents the electron pressure that is assumed to be isotropic. w ± are the Alfvén wave energy densities, with the subscript + for the Alfvén waves propagating in the direction of B, and the subscript for the Alfvén waves propagating antiparallel to B. V A = B/ μ 0 ρ is the Alfvén velocity. τ ei represents the relaxation time for the heat exchanges between the electrons and the protons due to their collisions. The polytropic index γ is taken to be 5/3. We use a spatially varying electron polytropic index γ e in the electron pressure equation (6): γ e = γf S + γ H (1 f S ), (8) where 1 f S = (9) 1 + (r/r H ) 2 with r H = 5R describes the transition between the regions dominated by collisional and collisionless heat conduction, resctively, similar to the heat flux formulation of Chandran et al. (2011). In the collisionless region, r r H, the polytropic index becomes γ H = γ + α 1 + α, (10) which corresponds to the collisionless heat flux of Hollweg (1978) in the steady-state limit. We set α = 1.05 (Cranmer et al. 2009), resulting in γ H = See Apndix B for details. The collisional heat flux, q e = f S q e,s, is also weighted by the radius-dendent function f S because it is not valid in the

3 collisionless region (Chandran et al. 2011). The Spitzer & Härm (1953) collisional heat conduction is q e,s = κ e T 5/2 e b (b ) T e, (11) where κ e Wm 1 K 7/2 and T e is the electron temrature. The source term Q rad in equation (6) represents the optically thin radiative energy loss, primarily in the lower corona: Q rad = N e N h (T e ), (12) where N e and N h are the electron and hydrogen number densities, resctively, and (T e ) is the radiative cooling curve obtained from the CHIANTI version 7.1 tables (Landi et al. 2013). The reflection and dissipation of the Alfvén waves are addressed on the right-hand side of the wave energy density equation (7). The wave reflection rate R is written as R = min [R imb, max(ɣ ± )] ( ) 1 2 w w + if 4w w + 0 if (1/4)w <w + < 4w, (13) ) if 4w + w ( 2 w+ w 1 R imb = [(V A )logv A ] 2 + (b [ u]) 2, (14) where the subscript imb stands for imbalanced turbulence. The derivation of the reflection rate uses the incompressibility condition for the turbulent fluctuations. The wave dissipation rate can be expressed as Ɣ ± = 2 L w ρ, (15) where L is the transverse correlation length of the Alfvén waves and L B 1/2.WeuseL B = 150 km T 1/2 following Hollweg (1986) and Hollweg & Johnson (1988). The wave dissipation is caused by the non-linear interaction between the oppositely propagating waves, which triggers the transverse energy cascading to small scales around the proton gyroradius and below. The dissipated wave energy is separated to three parts that contribute to proton parallel temrature, proton average temrature and electron temrature, resctively. These are represented by Q p, Q p and Q e in equations (4), (5) and (6) based on kinetic Alfvén wave theory. The expressions for Q p, Q p and Q e can be found in Chandran et al. (2011) and van der Holst et al. (2014). The derivation of the wave energy equation (7), model consistency and overall energy conservation are presented in van der Holst et al. (2014). Here we focus on an important term related to temrature anisotropy. δp /δt in equation (4) is the pressure anisotropy relaxation term that limits the proton pressure anisotropy by the firehose (f), mirror (m) and ion cyclotron (ic) instability constraints in the unstable regions, with an option of globally limiting the anisotropy when necessary: δp δt ( p f p = max ( p m p min τ m τ f, p p ) or τ g, p ic p, p p τ ic τ g ). (16) p f, p m and p ic are the proton parallel pressures for the marginally unstable states, and τ f, τ m and τ ic are the relaxation times based on the instability growth rates (Apndix A). The three instabilities are the major mechanisms that reduce the pressure anisotropy in AWSoM: temrature anisotropy 3699 the corona and heliosphere. The global pressure anisotropy limiting term, when present, is applied everywhere in the computational domain to mimic the effect of some minor mechanisms that also drive the proton pressure towards isotropy. τ g is a constant parameter from input, which is normally set to be much larger than the local values of the instability-based relaxation times. This is to minimize the impact of this term in the regions with instabilities. The model described by equations (1) (7) has been implemented into the BATS-R-US MHD code(powell et al.1999; Meng et al. 2012b), and used to represent both the SC and IH components of the Space Weather Modeling Framework (SWMF; Tóth et al. 2005, 2012). For the model to be used in a rotating frame, we have also implemented additional terms due to centrifugal and Coriolis effects, i.e. we added ρ[ ( r) + 2 u] to the right-hand side of the momentum equation (2), where is the angular velocity of solar rotation. The inner boundary is at the top of the chromosphere, where the ion and electron temratures are assumed to be the same constant and uniform value. The ion temrature is isotropic at the boundary; in other words, the ion parallel temrature is the same as the ion average temrature and the electron temrature. This assumption is approximately valid due to the abundant particle collisions in the relatively dense plasma at the top of the chromosphere. 3 SIMULATION OF CR2107 The coronal simulation results for CR2107 (2011 February 16 through March 16) have been presented and analysed in van der Holst et al. (2014). In this section, we report the modelling results in the heliosphere and at 1 AU for the same CR. The simulation is computed with coupled SC and IH components of the SWMF to propagate the solar wind solution from the corona to 1 AU. Both components use the heliographic rotating frame. For the SC component, the computational domain is a sphere centred at the Sun with a radius of 24 solar radii (R S ). We use a stretched spherical grid with the radial resolution varying from about R S very close to the surface of the Sun to about 1 R S further away from the Sun. The angular cell size is about 1.4 inside 1.7 R S and 2.8 outside, in both meridional and azimuthal directions. The total number of cells is about 3 million. The inner boundary conditions are set identically as described in van der Holst et al. (2014). For the IH component, the computational domain is a square box surrounding the spherical domain of SC. The box extends from 250 to 250 R S along each of the X, Y and Z directions. We use an adaptive Cartesian grid with resolution varying from less than 0.5 R S to about 4 R S. The total number of cells is about 8 million. The global pressure anisotropy relaxation time is τ g = s in both SC and IH components. For comparisons, we also rform an isotropic MHD simulation, which is set up identically as the anisotropic MHD simulation except that the proton temrature is isotropic in the former. We will refer to the isotropic MHD simulation as 2T modelling, while the anisotropic MHD one as 3T modelling in the result analysis. Both simulations are first computed in the local time stepping mode for iterations with the SC component only. At the th step, we refine the grid near the current sheet and a shell with radial distance less than 1.2 R S to the centre of the Sun. We couple the IH component to the SC component once, and then switch off SC, and advance IH from the steady-state SC solution for 4000 iterations in the local time stepping mode. These numbers of iterations are chosen for the solutions to become approximately stationary. The results presented below are taken from the end of the simulations.

4 3700 X. Meng et al. The magnetic field information near the inner boundary of SC is obtained from the Solar Dynamic Observatory (SDO)/Helioseismic and Magnetic Imager, shown as the input magnetogram in Fig. 1. There are some active regions that can be identified by several pairs of large red and blue dots representing the strong magnetic field regions with bipolarity. 3.1 Proton temrature anisotropy and firehose instability First, we examine the spatial distribution of the proton temrature anisotropy in our model. The top panel in Fig. 2 shows the proton pressure anisotropy distribution in the Y = 0 and Z = 0 planes, from both SC and IH solutions. Overall, we find highly rndicular pressure close to the inner boundary, and the pressure anisotropy ratio p /p gradually reduces away from the Sun. This feature is more prominent in the two polar regions than close to the equatorial region as seen in the Y = 0 plane. The polar regions are dominated with on field lines, which can result in adiabatic focusing, i.e. a more rndicular pitch-angle distribution towards the stronger magnetic field region along the magnetic field lines. In the Z = 0 plane, proton pressure anisotropy exhibits corotational structure. The pressure anisotropy distribution can be related to the solar wind sed distribution in the heliosphere beyond several tens of solar radii. The bottom panel in Fig. 2 displays the flow sed from the same 3T simulation. Comparing the anisotropy and the flow sed, we find that the fast solar wind originating from high-latitude regions is characterized with highly parallel proton pressure, while the slow solar wind coming from low-latitude regions is featured with almost isotropic or slightly rndicular proton pressure. In the Z = 0 plane, the highly rndicular pressure region away from the Sun is near the interface between regions with slow and fast solar wind, i.e. the corotating interaction region (CIR). To better understand the proton pressure anisotropy variation in the simulation, we extract the proton pressures and density along the positive Z-axis, which are used to calculate the parallel and rndicular temratures along this direction. The profiles along the Z-axis provide information about the fast solar wind. The temrature and temrature anisotropy profiles are shown in Fig. 3. For the temrature profiles shown in the left plot, outside 10 RS, the parallel temrature decreases slower than the rndicular temrature with increased distance from the Sun along both axes, so that the parallel temrature exceeds the rndicular temrature at around 40 RS. The temrature profiles of the fast solar wind along the Z direction can be compared to the results obtained from the 2D MHD model with proton anisotropy (fig. 2 c in Li et al. 2004) and 1D MHD model with proton anisotropy (fig. 4 in Chandran et al. 2011). Our temrature profiles are very close to theirs in terms of the shas and magnitudes, as well as the measured values between 1 and 3 RS shown in Chandran et al. (2011). Furthermore, Chandran et al. (2011) obtain isotropic proton temrature at a radial distance of about 40 RS, which is the same as our results for CR2107. The right-hand panel of Fig. 3 displays the proton temrature anisotropy ratio T /T along the Z-axis, together with the firehose, mirror and ion cyclotron instability criteria. The firehose instability limits T /T from below, while the other two instability limits the ratio from above. The anisotropy is bounded by the ion cyclotron instability between 2 and 20 RS, and by the firehose instability beyond 200 RS. A global view of the firehose unstable regions in the simulated heliosphere is shown in Fig. 4. The coloured regions are firehose unstable, and the colour scale represents the value of the instability growth rate. The slower the instability grows, the longer τ f is, and the less the anisotropy is limited. The value of τ f also reflects the strength of the instability, or by how much T exceeds the marginal firehose unstable state. Comparing the figure with the flow sed plots in Fig. 2, we find that the firehose instability mostly exists in the simulated interface between the fast and slow solar wind in the Y = 0 plane, and its strength gradually reduces away from the Sun. 3.2 Solar wind prorties at 1 AU As the model validation, we compare the simulated solar wind prorties at 1 AU to satellite observations. Fig. 5 shows the comparison of the simulations with the WIND measurements. We extract the magnetic field strength, flow sed and number density along the WIND trajectory, which is very close to the Earth orbit shown in Fig. 2, from both 3T and 2T simulations. In this way, we can not only evaluate the 3T model, but also compare the 3T and 2T modelling results. Although WIND is at the L1 point instead of the Earth, we neglect this difference since the distance between the L1 point and the Earth is small compared to the heliospheric scale. Also, the grid resolution in our simulation is about 4 RS near 1 AU, which is even larger than the distance between the L1 point and Earth (less than 3 RS ). The comparison indicates that the 3T simulation is similar to the 2T simulation in matching the WIND observations. However, the 3T simulation seems to reproduce the observed fast solar wind sed during March 1 and March 5 better than the 2T simulation. The bottom panel in Fig. 5 compares Figure 1. The input radial magnetic field for CR2107 simulations.

5 AWSoM: temrature anisotropy 3701 Figure 3. Simulated profiles of proton temratures and temrature anisotropy along the positive Z-axis for CR2107. The dotted lines represent the instability thresholds. the simulated and observed proton pressure anisotropy. Similar to the observation, the model shows relatively small deviation from isotropy, which is consistent with the high similarity between 3T and 2T simulation results. A few aks with large p /p in the observation are missed by the model. Here we point out that the sudden jump in the pressure anisotropy during February 18 and 20, together with the simultaneous jumps in the magnetic field strength and number density, is because of an interplanetary coronal mass ejection (ICME), which is not modelled in our simulations presented here. We also compare the 3T and 2T simulations with the STEREO-A and STEREO-B data in Figs 6 and 7, resctively. As shown in the bottom-right panel of Fig. 2, the STEREO-A satellite orbits a little Figure 2. The simulated proton pressure anisotropy ratio p /p in the Y = 0 and Z = 0 planes (top panel) for CR2107. The bottom panel displays the simulated solar wind sed in the two planes, and the plot on the right also shows the trajectories of the Earth, STEREO-A and B satellites projected to the Z = 0 plane. Note that these trajectories are not closed loops in the 3D space. The asterisks mark the locations of the Earth, STEREO-A and B satellites at the beginning of CR2107.

6 3702 X. Meng et al. Figure 4. The simulated firehose unstable region coloured by the growth-rate-based relaxation time τ f in the Y = 0andZ = 0 planes from IH for CR2107. The unit for the axes is solar radii. Figure 5. The simulated solar wind prorties along the WIND orbit and the WIND data during CR2107. The ICME (not modelled) time riod is highlighted in grey. bit closer to the Sun, and the STEREO-B satellite orbits further away from the Sun, compared to the trajectory of the Earth. Similar to the comparison with the WIND data, comparisons with the STEREO-A and STEREO-B measurements show that the 3T simulation captures the fast solar wind sed slightly better than the 2T simulation does. Other than that, the 3T and 2T simulation results are very close. The 1 AU validation for CR2107 implies that the 3T model produces a slower fast solar wind than the 2T model. Since the solar wind accelerates close to its terminal sed within tens of solar

7 AWSoM: temrature anisotropy 3703 Figure 6. The simulated solar wind prorties along the STEREO-A trajectory compared with the actual data from STEREO-A for CR2107. The grey areas mark ICMEs observed by STEREO-A. Figure 7. The simulated solar wind prorties along the STEREO-B trajectory compared with the actual data from STEREO-B for CR2107. The grey areas mark ICMEs observed by STEREO-B.

8 3704 X. Meng et al. Figure 9. The input radial magnetic field for CR2123 simulations. radii, the difference in the fast solar wind sed from the 3T and 2T simulations can be seen clearly in the SC component, shown in the top panel of Fig. 8. The 3T model simulated flow in the high-latitude regions, which forms the fast solar wind, is considerably slower than the 2T model simulated one, escially beyond 10 RS. The corresponding profiles of the solar wind along the positive Z-axis, with extension to the IH component, are shown in the bottom-left plot. Close to 1 AU, the solar wind sed in the 3T simulation is about 200 km s 1 less than that in the 2T simulation. This is most likely due to the p I + (p p )bb term in the momentum equation of the 3T model, while the corresponding term in the 2T model is p I. The separation between the parallel and rndicular pressures in the 3T model allows the two pressure components to evolve differently. The bottom-middle plot of Fig. 8 shows the proton pressure anisotropy in the Y = 0 plane from the 3T simulation. Note that the colour scale is saturated, and the highest value for p /p is more than 10 near the Sun. The plot indicates p > p in the high-latitude regions within the SC domain. Thus, the protons are heated less in the direction parallel to the magnetic field than in the direction rndicular to the magnetic field. This feature Figure 8. The top panel shows the solar wind sed in the Y = 0 plane from the SC component in the 2T and 3T simulations. The bottom-left plot shows the solar wind sed profiles extracted along the Z-axis from the top plots, with extension to the IH domain. The bottom-middle plot displays the proton pressure anisotropy ratio in the Y = 0 plane from SC in the 3T simulation. The colour scale of p /p is saturated. The bottom-right plot compares the scalar pressure and parallel pressure along the Z-axis from the 2T and 3T simulations, resctively.

9 AWSoM: temrature anisotropy 3705 cannot be described by the 2T model, which assumes isotropic proton pressure, i.e. p = p. In the 3T simulation, the ratio p /p decreases with increased radial distance in the polar regions outside a few solar radii. The scial case along the positive Z-axis is shown in the bottom-right plot. Beyond about Z = 2 RS, the parallel pressure gradient in the 3T simulation is smaller than the scalar pressure gradient in the 2T simulation. The smaller pressure gradient in the 3T simulation provides less force to drive the fast solar wind propagating outwards along the on field lines, which leads to a smaller flow sed eventually. 4 S I M U L AT I O N O F C R In this section, we apply the model to simulate CR2123 from 2012 April 28 through May 25, which is during solar maximum and has more active regions with stronger magnetic fields on the solar surface compared to CR2107. The simulation is set up in an identical way as for CR2107, except with a different input magnetogram, which is taken from the National Solar Observatory s Global Oscillation Network Group, shown in Fig. 9. The simulated proton pressure anisotropy and flow sed with the 3T model are shown in Fig. 10. Similar to the results of the CR2107 simulation, highly parallel proton pressure is found in the fast solar wind region, while large p /p occurs close to the Sun Figure 11. The simulated proton pressure anisotropy along the WIND trajectory against the WIND data during CR2123. and the CIR. The slow solar wind is featured with nearly isotropic or slightly rndicular pressure. We extract the proton pressure anisotropy ratio along the Earth trajectory (shown in Fig. 10) from the 3T simulation, and compare it with the WIND satellite measurement. The comparison is shown Figure 10. The simulated proton pressure anisotropy ratio p /p and solar wind sed in the Y = 0 and Z = 0 planes for CR2123. The trajectories of the Earth, STEREO-A and B satellites are projected to the Z = 0 plane in the bottom-right plot. Again, these trajectories are not closed loops in the 3D space. The asterisks mark the locations of the Earth, STEREO-A and B satellites at the beginning of CR2123.

10 3706 X. Meng et al. Figure 12. The simulated solar wind prorties along the Earth orbit and the OMNI data during CR2123. The grey area marks an ICME occurred during this CR. Figure 13. The simulated solar wind prorties along the STEREO-A trajectory compared with the actual data from STEREO-A for CR2123. The grey areas mark ICMEs observed by STEREO-A.

11 AWSoM: temrature anisotropy 3707 Figure 14. The simulated solar wind prorties along the STEREO-B trajectory compared with the actual data from STEREO-B for CR2123. The grey areas mark ICMEs observed by STEREO-B. in Fig. 11, from which we can tell that the simulation roughly captures the observed anisotropy ratio during this CR, though with less variations. Fig. 12 displays the magnetic field magnitude, the solar wind sed, the number density and the proton average temrature along the Earth orbit from both 3T and 2T simulations, as well as the OMNI data, which contains the solar wind and initial mass function (IMF) conditions at 1 AU. Overall, both 3T and 2T simulations capture the general trends in the observed flow sed, density and temrature during most of the time. The poor rformance in reproducing the observed magnetic field is largely due to the inaccuracy of the synoptic magnetograms near the polar regions that are the main source of the magnetic flux in the solar wind. The difference between the 3T and 2T simulations is small. In particular, the fast solar wind sed from the 3T simulation seems slightly lower than the 2T simulation, which fits our discussion at the end of Section 3.2. In the last two data-model comparisons for CR2123 shown in Figs 13 and 14, we compare the solar wind prorties along the STEREO-A and STEREO-B trajectories with the actual data. For both 3T and 2T simulations, the comparisons with STEREO observations are not as good as the comparison with the OMNI data. This is because that the synoptic magnetogram is centred for the Earth, yielding the most realistic simulation results at the Earth. 5 CONCLUSION We have successfully develod a 3D 3T Alfvén wave-driven solar wind model AWSoM. The model assumes anisotropic proton pressure and isotropic electron pressure. The Alfvén wave energy dissipates into three parts: proton parallel, proton rndicular and electron temratures. At the top of the chromosphere where the inner boundary of the model lies, we assume isotropic proton temrature. To limit the proton pressure anisotropy in the model, we have introduced the pressure anisotropy relaxation term in the parallel proton pressure equation, which constrains the anisotropy based on the growth rates of the firehose, mirror and proton cyclotron instabilities, as well as an optional global anisotropy relaxation term. The model is applied to simulate the steady states of two CRs during the solar cycle 24. In the simulations of both CR2107 and CR2123, we find highly rndicular proton pressure close to the Sun, and the anisotropy ratio p /p reduces away from the Sun. There are also corotational structures in the proton pressure anisotropy distributions. In the heliosphere beyond about R S, the fast solar wind is characterized with p /p less than 1, while the slow solar wind is with p /p close to or slightly over 1. Near the CIRs, p /p is typically the highest and larger than 1. We have also obtained reasonable agreement in the comparisons with in situ observational data for the two CRs. At 1 AU, the difference between the 3T and 2T simulations is very small. Interestingly, the 3T simulations predict slightly smaller and better fast solar wind sed than the 2T simulations do, which is primarily due to smaller pressure gradient along the magnetic field to accelerate the fast solar wind in the 3T simulations. Note that we have validated the IMF magnitude instead of its individual components at 1 AU, because the current generation of predictive solar wind models, AWSoM included, are incapable of providing reliable IMF components due to the missing physics in the models and the limited accuracy of magnetograms. In the future when AWSoM is sufficiently advanced, we could validate IMF components, escially B z, which is the most relevant to space weather predictions.

12 3708 X. Meng et al. This is the first time that the proton temrature anisotropy is included in a 3D solar wind model. The model does not offer much improvement over the isotropic pressure 2T AWSoM for the solar wind at 1 AU, unlike the notable improvement over the 2T model for the SC presented in van der Holst et al. (2014). This is excted, given that the temrature anisotropy is more prominent near the Sun than further away. In the heliosphere where the temrature anisotropy is small or even negligible, the 3T model naturally reduces to the 2T model. Nevertheless, our model extends the current capability of 3D solar wind modelling. As next steps, the model can be further improved by incorporating some missing physics. One factor, which also impacts the solar wind acceleration, is that the momentum equation needs to be modified by Alfvénic turbulence with proton temrature anisotropy. More scifically, the Alfvén wave pressure gradient term [the last term on the left of equation (2)] will contain the proton temrature anisotropy after the modification. This will be addressed in the future. ACKNOWLEDGEMENTS This work was supported by the NSF grant AGS We would like to acknowledge high-rformance computing support from Yellowstone (ark:/85065/d7wd3xhc) provided by NCAR s Computational and Information Systems Laboratory, sponsored by the NSF. We also thank J. H. King, N. Papatashvilli, K. Ogilvie at AdnetSystems, NASA GSFC and CDAWeb for providing the OMNI, WIND, STEREO-A and B data. The near- Earth ICMEs were identified based on the WIND ICME list ( and the Richardson/Cane ICMElist. The STEREO-observed ICMEs were obtained from Dr Lan Jian s ICME list. REFERENCES Antonucci E., Dodero M. A., Giordano S., 2000, Sol. Phys., 197, 115 Barnes A., 1966, Phys. Fluids, 9, 8 Chandran B. D. G., Dennis T. J., Quataert E., Bale S. D., 2011, ApJ, 743, 197 Chandrasekhar S., Kaufman A. 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C., 1972, ApJ, 178, 221 APPENDIX A: FIREHOSE, MIRROR AND ION CYCLOTRON INSTABILITIES The firehose, mirror and ion cyclotron instabilities are three tys of instabilities resulted from, and thus constrain, proton pressure anisotropy in magnetized plasmas (Chandrasekhar, Kaufman & Watson 1958; Barnes1966; Kennel & Petschek 1966; Gary et al. 1976). These instabilities, escially the firehose instability, have been frequently observed in the solar wind. Below we briefly introduce each instability and list their criteria, p at marginally unstable states, as well as the relaxation times applied in our model. More detail can be found in Meng et al. (2012a). The firehose instability arises when the parallel pressure is sufficiently large that a small rturbation rndicular to the magnetic field grows. This can be viewed as an analogue of the violent motion of a firehose with water flowing through rapidly. The instability criterion is p > 1 + B2. p μ 0 p (A1) Hence, the parallel pressure in marginally firehose unstable plasmas is given by p f = p + 2B2. (A2) 3μ 0 The maximum growth rate of the firehose instability γ f, m determines τ f in equation (16): τ f = 1 = 2 p (p p /4), γ f,m i p p B 2 /μ 0 where i is proton gyrofrequency. (A3)

13 The mirror and ion cyclotron instabilities occur when the rndicular pressure is large enough. The mirror instability is excited when part of the particles are trapd in small magnetic mirrors formed by the rturbed magnetic field. A typical observational feature is the anti-phased variations in the magnetic field strength and the plasma number density. The criterion is p p > 1 + B2 2μ 0 p. When marginally unstable, ( p m = 1 B2 B 2 ) 2 + 6p + 12B2 p + 9p 3 μ 0 μ 2. 0 μ 0 (A4) (A5) The mirror instability involves kinetic effects, and thus cannot be fully described by anisotropic MHD. The MHD analysis may still be employed to find τ m, because the wavelength corresponding to the maximum growth rate γ m,m is much larger than the ion gyroradius. τ m is approximated as τ m = 1 = 3 5 γ m,m 4 i p 2(p p ) B 2 p /(2μ 0 p ). (A6) The ion cyclotron instability is a kinetic and resonant instability that occurs when the electric field vector of the ion cyclotron wave rotates synchronously with the particle gyration. Its criterion can be written as p p > 1 + C 1 ( B 2 2μ 0 p ) C2, (A7) where C1 andc2are constants that vary with different maximum growth rates. In our model, we use C 1 = 0.3 and C 2 = 0.5. The marginally unstable p then becomes p ic = 1 B B2 + 2p. (A8) 2 μ 0 μ 0 The analytical expression for the maximum growth rate of the ion cyclotron instability can only be obtained through kinetic analysis and is very complicated. We use the following approximate expression for τ ic : τ ic = 1 γ ic,m = 102 i. (A9) APPENDIX B: ELECTRON PRESSURE EQUATION In this apndix, we show how the modified polytropic index defined in equation (8) can be chosen so that it reproduces both the collisional heat conduction (Spitzer & Härm 1953) near the Sun and the collisionless heat flux formulation of Hollweg (1978) at large distances from the Sun. The latter limit is strictly true for steady-state conditions. We start from equations (6) (9): ( ) ( ) + t γ e 1 γ e 1 u + p e ( u) where = q e + p p e τ ei Q rad + Q e (B1) γ e = γf S + γ H (1 f S ) (B2) AWSoM: temrature anisotropy f S = 1 + (r/r H ). (B3) 2 Note that this form of the electron pressure equation describes the time evolution of the electron internal energy p e /(γ e 1) for an arbitrary function of the polytropic index γ e. We choose f S and γ e so that the above electron pressure equation reproduces the effects of the collisional and collisionless heat conductions in the appropriate limits. In the collisional limit r r H,sothatf S = 1, γ e = γ = 5/3 and q e = q e, S is the Spitzer heat flux (equation 11), which results in the usual form of the electron pressure equation with collisional heat conduction. In the collisionless limit r r H,sothatf S = 0, γ = γ H and q e = 0 and we get t ( γ H 1 ) ( ) + γ H 1 u + p e ( u) = p p e Q rad + Q e. (B4) τ ei Taking the steady-state limit and dropping the electron ion heat exchange and radiative cooling terms that are negligible and not essential in the following derivation, the equation reduces to ( ) γ H 1 u + p e ( u) = Q e. (B5) This can be rewritten as ( ) γ 1 u + p e ( u) [ ( 1 = p e u γ H 1 1 )] + Q e. (B6) γ 1 Substituting γ H = (γ + α)/(1 + α) from equation (10) gives ( ) [ ] γ 1 u 1 + p e u = γ 1 αp eu + Q e. (B7) The first term on the right-hand side is the divergence of the collisionless heat flux αp e u of Hollweg (1978). Note that the electron pressure equation with a modified polytropic index is indendent of the frame of reference; therefore, the steady-state solution in the corotating frame is also a valid (timedendent) solution in the inertial frame. As long as the solution is approximately steady state in the corotating frame, the modified polytropic index γ H gives the same result as Hollweg s collisionless heat conduction. This relationship was also pointed out by Hollweg (private communication). This par has been tyset from a TEX/LATEX file prepared by the author.