Freeman, & Arya 1995). This could be interpreted as the ratio of specific heats for a gas of particles with n 4 degrees of

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1 The Astrohysical Journal, 595:L57 L1, 003 Setember The American Astronomical Society. All rights reserved. Printed in U.S.A. A THREE-DIMENSIONAL MODEL OF THE SOLAR WIND INCORPORATING SOLAR MAGNETOGRAM OBSERVATIONS I. I. Roussev, 1 T. I. Gombosi, 1 I. V. Sokolov, 1 M. Velli,,3 W. Manchester IV, 1 D. L. DeZeeuw, 1 P. Liewer, G. Tóth, 1,4 and J. Luhmann 5 Received 003 July 3; acceted 003 August 4; ublished 003 August ABSTRACT We resent a new comressible MHD model for simulating the three-dimensional structure of the solar wind under steady state conditions. The initial otential magnetic field is reconstructed throughout the comutational volume using the source surface method, in which the necessary boundary conditions for the field are rovided by solar magnetogram data. The solar wind in our simulations is owered by the energy interchange between the lasma and large-scale MHD turbulence, assuming that the additional energy is stored in the turbulent internal degrees of freedom. In order to reroduce the observed bimodal structure of the solar wind, the thermodynamic quantities for the initial state are varied with the heliograhic latitude and longitude deending on the strength of the radial magnetic field. Subject headings: interlanetary medium methods: numerical MHD solar wind Sun: evolution Sun: magnetic fields On-line material: meg animation 1. INTRODUCTION The global structure of the solar wind results from the comlex interaction between the exanding coronal lasma, magnetic fields, solar rotation, and interlanetary turbulence. The observed dichotomy between a fast solar wind and a slow solar wind (Phillis et al. 1995; Goldstein et al. 199) results from the toological structure of the solar magnetic fields that are divided into oen field and closed field regions by the boundaries of helmet-like streamers. Unlike oen field regions (associated with coronal holes), magnetic fields in helmet streamers are biolar, and their strength is large enough to revent the bulk exansion of coronal lasma. The simulation runs resented in this Letter are designed to reroduce the global structure of the helioshere under realistic conditions. The initial magnetic field at the solar surface is taken from synotic mas, and the field is reconstructed into the simulation volume through a otential field extraolation (Altschuler et al. 1977). The MHD solution is allowed to evolve from this initial configuration to a fully self-consistent, nonotential solution. Our model was insired by a multitude of revious numerical efforts ioneered by Pneuman & Ko (1971), who first studied the steady, axisymmetric exansion of coronal lasma from a shere with an embedded diolar magnetic field. Similar magnetic configurations were used in the majority of later studies in two dimensions (see Wang et al and Suess et al. 1999) and three dimensions (see Usmanov 1993 and Groth et al. 000), in order to model the solar wind conditions reresenting the helioshere at solar minimum. Most studies have successfully reroduced the global structure of the outer solar corona, 1 Center for Sace Environment Modeling, University of Michigan at Ann Arbor, 455 Hayward Street, Ann Arbor, MI ; iroussev@umich.edu. Jet Proulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA Deartment of Astronomy and Sace Science, University of Florence, Largo Enrico Fermi 5, Florence I-5015, Italy. 4 Deartment of Atomic Physics, Loránd Eötvös University, Budaest H- 1117, Hungary. 5 Sace Sciences Laboratory, University of California at Berkeley, Grizzly Peak at Centennial Drive, Berkeley, CA L57 but the bimodality of the solar wind could not be modeled in a quantitative manner without adding roer emirical sources to ower the wind in oen field regions. The incororation of Alfvén waves in the momentum balance and energetics of the solar wind was studied extensively in one-dimensional magnetohydrodynamics (see Holzer, Flå, & Leer 1983 and Mc- Kenzie, Axford, & Banaszkiewicz 1997). The effect of an Alfvén wave flux on the flow roerties of the solar wind, in a WKB aroximation, was recently studied in two dimensions by Usmanov et al. (000). These results are in good agreement with the Ulysses data normalized to 1 AU. The dissiation of an Alfvén wave sectrum at the ion-cyclotron frequency might additionally lead to referential heating of the ions in oen field regions (Axford & McKenzie 199). We are still taking the first stes in modeling the global structure of the helioshere for realistic conditions derived from solar magnetogram data (Linker et al. 1999; Riley, Linker, & Mikić 001). In this Letter, we resent further attemts in this direction, which emloy a fully three-dimensional and comressible MHD in our simulations. We allow for the olytroic index to change with radial distance from the Sun (Wu et al. 1999), but in our model, a form of the total energy is conserved.. MODEL In order to simulate the large-scale structure of the corona roerly, one has to consider the solar lasma as a gas with a olytroic index g close to unity (Steinolfson & Hundhausen 1988), say g 1.1, close to the Sun. Near the Earth, however, observations indicate that the olytroic index of the solar wind lasma has a value of g0 1.5 (Phillis et al. 1995; Totten, Freeman, & Arya 1995). This could be interreted as the ratio of secific heats for a gas of articles with n 4 degrees of freedom since, in general, g0 (n )/n. For a fully ionized lasma, such as the solar wind, there are almost no secies with standard internal degrees of freedom such as those arising from molecule vibrations, rotations, and bound-bound and bound-free electron transitions in atoms/ions. However, esecially close to the Sun, a significant amount of energy is stored in waves and turbulent fluctuations. We therefore adot a mod-

2 L58 ROUSSEV ET AL. Vol. 595 Fig. 1. Comarison between Ulysses data normalized to 1 AU (dotted curves in black) and simulation results for a nontilted, rotating magnetic diole (solid curves in red). eling aroach in which low values of g are associated with such turbulent internal degrees of freedom, in a way analogous to the work of Zeldovich & Raizer (00) in artially ionized lasmas. Within some temerature intervals, the ratio of secific heats for artially ionized lasmas is very close to unity, or, equivalently, the secific heat (g 1) n/ is very large recisely because energy can be stored in a form of the ionization otential of decouled electrons, and it can be deosited in the course of lasma recombination as these electrons rejoin the ions. The contribution of the secific heat due to internal degrees of freedom is (g 1) (g0 1) (n 4)/, and the additional internal energy may be exressed as E st [P(n 4)/](T /T). Here P is the lasma ressure, and T is st the temerature in the internal degrees of freedom, which generally differs from the kinetic lasma temerature, T. The full energy balance in this case is given by one more equation for E st describing the energy exchange between the classical and st internal degrees of freedom in terms of temerature relaxation, namely, P(n 4)/ (energy eq. with g g ), (1) 0 P(n 4)/ (UE st), () t where t rel is the relaxation time. If this time tends to infinity, the stored and classical internal energies are conserved indeendently, while in general only the total is conserved. Following this aroach, we assume that the energy of turbulent motions near the Sun can also be treated as the extra energy stored in the lasma, in a manner analogous to equations (1) and (). Note that henomenological models of turbulence based on hydrodynamical equations for the turbulent energy t rel t rel Fig.. Left: Three-dimensional view of the magnetic field geometry near the Sun. The solid lines are magnetic field lines, where the false-color code visualizes the wind seed in kilometers er second. Right: Three-dimensional view far from the Sun showing isosurfaces of a wind seed of 470 km s (shaded in green); a wind seed of 70 km s (orange); and a zero radial magnetic field (urle). This figure is also available as an meg animation in the electronic edition of the Astrohysical Journal.

3 No. 1, 003 THREE-DIMENSIONAL MODEL OF SOLAR WIND L59 Fig. 3. Mercator ma rojection of the steady state wind seed in kilometers er second (background image) and the radial magnetic field (contours) at 1 AU. The osition of the current sheet is shown as the dashed contour in white. The magnetic field strength is given in units of nanotesla. Fig. 4. Mercator ma rojection of the steady state wind temerature in kelvins (image) and the number density in units of m 3 (contours) at 1 AU. Both quantities are shown in a logarithmic scale. The dashed contour line in black marks the minimum value of number density.

4 L0 ROUSSEV ET AL. Vol. 595 density with some equation of state for the latter are widely used in alied hysics (see Saurel, Chinnayya, & Renaud 003 and the references therein). The role of the energy interchange deends on the relation between the relaxation time, t rel, and the characteristic hydrodynamical time, t hyd. If thyd k trel, then P(n 4)/, and the resulting energy equation (the sum of eqs. [1] and []) is for g (n )/n. In the oosite limit, thyd K trel, equation (1) is valid for g g0, and equation () is not needed. In our numerical model, we choose t rel to be as small as ossible (of the order of the time ste). We further assume that the internal degrees of freedom deend on the kinetic lasma temerature as n(t) n 0 nt/t 1 0, where we choose n 0 4 and n1 9 following the discussion above. Here T0 is some reference value of temerature at the Sun that will be defined later. Thus, we emloy the full energy equation in our comutations with a olytroic index g [n(t) ] /n(t), which is now a function of temerature describing the additional energy density associated with turbulent motions. We note that this technique is an emirical one insired by the hidden internal degrees of freedom. This is only a model we are using to bridge from a olytroe, which is nearly isothermal, to a fully fledged energy equation with wave couling and heating. In the static initial state, the lasma temerature as a function of radial distance from the Sun, R, can be obtained from the differential equation of state, namely, d {g(t)/ [g(t) 1]} r dt, and the equation for hydrostatic equilibrium in the absence of magnetic field, i.e., d GM, rd(1/r). (We work in units where the universal gas constant equals the mean molecular weight.) As a result of simle integration from R, to R, we obtain the following quadratic equation for u(r) T(R)/T 0 : ( ) ( ) 4 n 0 4 n0 n1 R, u 1 u 1 0. (3) n n 4 R 1 1 Here the reference temerature T 0 is chosen such that the asymtotic behavior of u(r) at large distances is 1/R, so that u r 0 as R r. Furthermore, this choice is consistent with solar wind observations showing that r 1/R and T 1/R for g 1.5. The exlicit choice for T0 reads T0 GM,/[R,(1 n 0/ n 1/4)]. Once the temerature is derived from equation (3), the lasma ressure can be obtained from d ln (/ 0) {g(t)/[g(t) 1]}d ln (T/T 0) [1 n 0/ nu/]d 1 ln u, following a direct integration. (Here 0 is the reference value for ressure at R,, where T T 0.) As a result, ] n 1(1 u) (1 n 0/) u 0 ex [. (4) From the equation of state, the mass density is r 4 r 0(/ 0)/u, where r0 0/T0 1 (in units of 8.3 # 10 kg m 3 ). For n 0 4 and n1 9, the value of T0 becomes 4.40 # 10 K, and u(r) T(R)/T0 ranges from 1 at R 3 R, to # 10 at R 1 AU. Accordingly, g(r) ranges from at the solar surface to at 1 AU. In order to reroduce the observed bimodal structure of the solar wind, we imose a temerature variation on the solar surface deending on the strength of the radial magnetic field. This is chosen as T [R,, B R(v,f)] T0 L [B R(v, f)] T 0[1 (L t 1)B R(v, f)/b 0]. Here BR is the strength of the radial magnetic field at the surface (by absolute value), B 0 is the maximum value of BR, and L t ( 1) is the ratio between the temerature where BR B0 (magnetic oles in the case of diole field) and that where BR 0 (magnetic equator). From this definition, L(B 0) Lt and L(0) 1 are the two extremes. With this modification in lace, all of the above derivations still aly, but n1 has to be relaced by n1l [B R(v, f)] everywhere. Since the ressure at the surface, [R,, B R(v, f)] 0(B R), scales as T 0(B R), the following relation alies: 0(B R)/ 0(0) T 0(B R)/T 0(0) [1 n0/ n1l(0)/4]/[1 n0/ n1l(b R) / 4] Y(B R) 1. The mass density at the solar surface can be obtained from the equation of state, which gives r 0(B R) r 0(0)Y(B R)/L(B R). In order to achieve good agreement between model results and solar wind data, as will be shown in 4, it is necessary to choose L t.10, which gives Y(0)/Y(B 0) 1.47, and so r 0(0)/r 0(B 0) Note also that for BR 0, g(r) ranges from at the solar surface to at 1 AU, while for BR B0, these values are and 1.474, resectively. Once derived for the initial temerature distribution, the value of g at each oint of the three-dimensional volume is maintained throughout the simulations. 3. SIMULATION SETUP The governing equations of comressible MHD are solved in conservative form using the Block Adative-Tree Solar-wind Roe-tye Uwind Scheme (BATS-R-US) code (Powell et al. 1999). We emloy the Artificial Wind method (Sokolov et al. 00) as an aroximate Riemann solver. An exlicit, twostage time-steing algorithm is used to advance the solution in time. The solenoidal constraint B 0 is achieved using the eight-wave scheme of Powell et al. (1999). The simulation domain is a cube with a length of 440 R,, and the numerical grid is nonuniform, consisting of self-similar blocks with 4 3 cells. After five levels of uniform initial refinement, we alied seven more levels of body-focused refinement in order to achieve the highest satial resolution near the Sun. Thus, the total number of comutational cells becomes 3,147,4. The smallest grid size corresonds to 0.09 R,, while the largest cells (near the outer boundaries) are 18 times bigger than this. The boundary conditions describe an imenetrable and highly conducting sherical inner body, laced at R R, 1, and oen outer boundaries. In order to achieve line-tying of the magnetic field at the inner boundary, all velocity comonents are maintained at zero throughout the simulation runs, with the excetion of solar rotation (which is assumed to be uniform with latitude). The ressure, turbulent energy density, E st, and mass density are fixed at the inner body. This ensures a dynamical evolution of the system toward a steady state. Floating conditions are alied to the radial and tangential comonents of the magnetic field. The oen boundary conditions at the outer surface of the comutational domain are imlemented by alying a zero gradient to all hysical variables. 4. STEADY-STATE SOLAR WIND DRIVEN BY SOLAR MAGNETOGRAMS The model described above was tested for a nontilted, rotating magnetic diole (with the field strength at the oles chosen as 3.87 G). We used this simle case to exlore the consequences of the source terms we introduced in equations (1) and (), as well as to filter out the right choice of arameters that roduces the most realistic solar wind. The results of this comarison are shown in Figure 1. The solid curves in this figure corresond to simulation results, while the dotted curves reresent Ulysses data obtained for the eriod from 1994 Setember through 1995 July. These data are normalized to 1 AU following Goldstein et

5 No. 1, 003 THREE-DIMENSIONAL MODEL OF SOLAR WIND L1 al. (199). The set of arameters for this simulation run is as given above. As seen in the figure, the bimodal structure of the modeled solar wind is well reroduced. However, there is a lack of shar transition between the two tyes of wind because of the insufficient numerical resolution. Simulation results for a realistic magnetic field configuration at the Sun are shown in Figures, 3, and 4. To obtain the bulk solar magnetic fields, we followed the Potential Field Source Surface method by Altschuler et al. (1977). This method assumes that the magnetic field is a otential field and that the magnetic scalar otential is estimated as a series of sherical harmonics. The coefficients in the series are chosen to fit real magnetogram data obtained from the Wilcox Solar Observatory. The source surface (where the scalar otential is equal to zero) is laced at radial distance Rss.5 R, from the Sun. The function L [B R(v, f)] is chosen as {1 (L t 1) min [1, B R(v, f)/b 0]}, where B G as above. In this Letter, we resent simulation results for Carrington rotation 1881 (with a central meridian on 1994 Aril ). The left anel of Figure shows a three-dimensional view of the field geometry near the Sun under steady state conditions. The shere is laced at R R,, and the color code reresents the solar wind seed. A three-dimensional view far from the Sun is shown in the right anel of Figure. As seen in Figures 3 and 4, the fast wind ( 790 km s ) correlates with oen field regions, whereas the slow wind ( 45 km s ) originates from helmet-tye coronal streamers. 7 The dense ( m # 10 ) and cool ( 1.1 # 10 K) outflow is associated with closed field regions at the Sun, whereas in coronal holes we observe a rarefied ( 5.0 # 10 m 3 ; see the 5 dashed contour line in black in Fig. 4) and hot ( 3.1 # 10 K) They can be obtained from htt://solar.stanford.edu/ wso/forms/rgs.html. solar wind. These results agree with the observed solar wind characteristics at 1 AU, both in a qualitative and a quantitative manner, desite the relatively course satial resolution in our simulations ( R, ) at this radial distance from the Sun. 5. CONCLUSIONS We used the concet of energy interchange between the lasma and turbulence, in terms of temerature relaxation, to describe the thermal roerties of lasma and flow acceleration in the inner helioshere. We develoed an emirical model in which the low values of g near the Sun are associated with the turbulent internal degrees of freedom; we allowed for g to change with radial distance from the Sun, but a form of the total energy was conserved. By means of this aroach, we were able to roduce a steady state solar wind in simulations involving magnetic fields derived from observations, and we succeeded in obtaining results that are consistent with observed wind roerties at 1 AU. We believe that the comutational techniques resented in this Letter oen new doors to incororating magnetic observations into numerical simulations aimed at modeling the global helioshere and sace weather. In a follow-u study, we intend to make a detailed comarison between MHD results and observational data, ranging from solar minimum to solar maximum conditions. The authors thank T. G. Forbes, B. C. Low, T. Holzer, and an unknown referee for their comments. This research work was suorted by DoD MURI grant F , NSF CISE grant ACI , and NASA AISRP grant NAG5-940 at the University of Michigan. G. Tóth acknowledges the suort by OTKA grant T REFERENCES Altschuler, M. D., Levine, R. H., Stix, M., & Harvey, J. 1977, Sol. Phys., 51, 345 Axford, W. I., & McKenzie, J. F. 199, in Solar Wind Seven: Proc. Third COSPAR Colloq., ed. E. Marsch & R. Schwenn (Tarrytown: Pergamon), 1 Goldstein, B. E., et al. 199, A&A, 31, 9 Groth, C. P. T., DeZeeuw, D. L., Gombosi, T. I., & Powell, K. G. 000, J. Geohys. 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