Semi-Empirical MHD Modeling of the Solar Wind Igor V. Sokolov, Ofer Cohen, Tamas I. Gombosi CSEM, University of Michigan Ilia I Roussev, Institute for Astronomy, University of Hawaii Thanks to: P.McNiece, W.Manchester, N.Lopez, R.Fraisin, Yang Li, N.Arge Igor V. Sokolov, Ofer Cohen, Tamas I. Gombosi CSEM, University of Michigan Ilia I Roussev, Institute for Astronomy, University of Hawaii Thanks to: P.McNiece, W.Manchester, N.Lopez, R.Fraisin, Yang Li, N.Arge

Model Description  Coupled MHD simulation for the solar corona (SC) and inner heliosphere (IH)  The three-dimensional model for the heliosphere from the low corona to 1 AU is based on the use of the solar corona magnetogram and the semi-empirical model for the solar wind velocity  Using the Bernoulli Integral, the semi- empirical values of the solar wind speed at 1 AU are related to the boundary condition for the “turbulent energy density” at the solar surface. To bridge from this boundary condition to the solar wind model in the solar corona, the varied polytropic index is used.  Coupled MHD simulation for the solar corona (SC) and inner heliosphere (IH)  The three-dimensional model for the heliosphere from the low corona to 1 AU is based on the use of the solar corona magnetogram and the semi-empirical model for the solar wind velocity  Using the Bernoulli Integral, the semi- empirical values of the solar wind speed at 1 AU are related to the boundary condition for the “turbulent energy density” at the solar surface. To bridge from this boundary condition to the solar wind model in the solar corona, the varied polytropic index is used. R SS  SS =1.1  Sun U WSA BI  (R,  )=a  SS +(1-a)  Sun, R,  a=( R-R sun )/ (R SS -R)

Magnetogram  Input: the file of spherical harmonic coefficients for the magnetic field potential, with the “standard header”  Standard header involves the information about: the Carrington longitude of the central meridian and the maximal order of harmonics N.  Mostly we use the processed MDI magnetogram (Yang Li), with N=90 or WSO harmonics file downloaded from the WSO web-cite, with N=29. No polar correction is applied. CCMC also use the original magnetogram from other observatories, but they process them to get the same kind of the harmonics file, as that from WSO. Presumably, the polar correction is applied.  Using the harmonic coefficients we calculate the components of the potential magnetic field B 0 [Gs] at the nodes of the uniform spherical grid by R, ,sin ,  being the heliographic (“Carrington”) longitude. To do this we automatically apply the longitudinal offset, take into account that different observatory conveniently use different units for the magnetic field and apply different file headers. The grid dimensions are as follows:  for WSO data, 30 points by sin of latitude (-1+1/30:1-1/30), 72 by longitude (0:355), 30 by radius (R sun :R SS =2.5R Sun )  for MDI mangnetogram: 90*90*90 grids. Forget about spherical harmonics, they are is not used any longer. Advantages of the use of this grid are computational efficiency and robustness.  Input: the file of spherical harmonic coefficients for the magnetic field potential, with the “standard header”  Standard header involves the information about: the Carrington longitude of the central meridian and the maximal order of harmonics N.  Mostly we use the processed MDI magnetogram (Yang Li), with N=90 or WSO harmonics file downloaded from the WSO web-cite, with N=29. No polar correction is applied. CCMC also use the original magnetogram from other observatories, but they process them to get the same kind of the harmonics file, as that from WSO. Presumably, the polar correction is applied.  Using the harmonic coefficients we calculate the components of the potential magnetic field B 0 [Gs] at the nodes of the uniform spherical grid by R, ,sin ,  being the heliographic (“Carrington”) longitude. To do this we automatically apply the longitudinal offset, take into account that different observatory conveniently use different units for the magnetic field and apply different file headers. The grid dimensions are as follows:  for WSO data, 30 points by sin of latitude (-1+1/30:1-1/30), 72 by longitude (0:355), 30 by radius (R sun :R SS =2.5R Sun )  for MDI mangnetogram: 90*90*90 grids. Forget about spherical harmonics, they are is not used any longer. Advantages of the use of this grid are computational efficiency and robustness.

MHD model : boundary condition for magnetic field at R=R Sun  For simplicity, state that we require the radial component of the magnetic field in the MHD model to be equal to the measured value at R=R Sun.  Actually we split the full magnetic field in the MHD equations for a static part and dynamical (time-dependent) part and take potential field described above as the static part. For the dynamical magnetic field we applied a zero boundary condition for the normal component at R=R Sun  For simplicity, state that we require the radial component of the magnetic field in the MHD model to be equal to the measured value at R=R Sun.  Actually we split the full magnetic field in the MHD equations for a static part and dynamical (time-dependent) part and take potential field described above as the static part. For the dynamical magnetic field we applied a zero boundary condition for the normal component at R=R Sun

Semi-empirical WSA model  The WSA model stems from the observation that:  The solar wind originating from smaller coronal holes is slower than that originating from the larger holes  The slow(er) solar wind originates from the coronal hole boundaries  Quantitative parameters to characterize these effects, for a given magnetic field line, are the expansion factor and the angular distance from the coronal hole boundary  b. At the coronal hole boundary the expansion factor turns to infinity and  b =0. For smaller coronal holes the expansion factor is larger and  b is constrained. The WSA model provides the dependence of the velocity, u  of the solar wind, at the infinitely distant point of the given magnetic field line, on the expansion factor and  b for this line.  For any point laying in between R sun and R SS we define u  This can be done by constructing the line of the potential magnetic field, passing through this given point. Then we can EITHER calculate the expansion factors for thus obtained magnetic field lines and apply for the WSA semi- empirical relationship OR find in any other way the WSA velocity in the mapping point at the source surface. Note that, first, u  has nothing to do with the local plasma velocity and, second it is constant along the line of the potential magnetic field.  The WSA model stems from the observation that:  The solar wind originating from smaller coronal holes is slower than that originating from the larger holes  The slow(er) solar wind originates from the coronal hole boundaries  Quantitative parameters to characterize these effects, for a given magnetic field line, are the expansion factor and the angular distance from the coronal hole boundary  b. At the coronal hole boundary the expansion factor turns to infinity and  b =0. For smaller coronal holes the expansion factor is larger and  b is constrained. The WSA model provides the dependence of the velocity, u  of the solar wind, at the infinitely distant point of the given magnetic field line, on the expansion factor and  b for this line.  For any point laying in between R sun and R SS we define u  This can be done by constructing the line of the potential magnetic field, passing through this given point. Then we can EITHER calculate the expansion factors for thus obtained magnetic field lines and apply for the WSA semi- empirical relationship OR find in any other way the WSA velocity in the mapping point at the source surface. Note that, first, u  has nothing to do with the local plasma velocity and, second it is constant along the line of the potential magnetic field.

Varied polytropic index  We employ the model of a single fluid with the varied polytropic index, which is the function of coordinates at R<12.5 R s :  We use the full energy equation, in which  is a function of coordinates.  Physical sense: we assume that the total energy per particle minus the kinetic energy is much greater than the internal gas energy at small heliocentric distances:.  The extra energy is assumed to be stored in turbulence. While the solar wind reaches the heliocentric distance of 12.5 R S the extra energy is entirely converted to the solar wind energy. This approach allows us to avoid the contradiction between the high particle energy in the solar wind at 1 AU and much lower internal energy of a plasma in the solar corona.  We employ the model of a single fluid with the varied polytropic index, which is the function of coordinates at R<12.5 R s :  We use the full energy equation, in which  is a function of coordinates.  Physical sense: we assume that the total energy per particle minus the kinetic energy is much greater than the internal gas energy at small heliocentric distances:.  The extra energy is assumed to be stored in turbulence. While the solar wind reaches the heliocentric distance of 12.5 R S the extra energy is entirely converted to the solar wind energy. This approach allows us to avoid the contradiction between the high particle energy in the solar wind at 1 AU and much lower internal energy of a plasma in the solar corona.

The Bernoulli integral  Assume the approximate conservation of the Bernoulli integral along the magnetic field line in the solar corona:  For a given magnetic field line at infinity the integral tends to the kinetic energy of the solar wind  Hence, the semi-empirical model allows us to calculate the value of the Bernoulli integral at the given magnetic field line.  Mapping this line towards the solar surface, we can calculate the boundary value for the turbulent energy density, or, in our model, for the polytropic index:  Assume the approximate conservation of the Bernoulli integral along the magnetic field line in the solar corona:  For a given magnetic field line at infinity the integral tends to the kinetic energy of the solar wind  Hence, the semi-empirical model allows us to calculate the value of the Bernoulli integral at the given magnetic field line.  Mapping this line towards the solar surface, we can calculate the boundary value for the turbulent energy density, or, in our model, for the polytropic index:

Boundary condition for the “turbulent energy density”  The derived distribution of the polytropic index over the solar surface  The assumed constant value for the corona base temperature is  The derived distribution of the polytropic index over the solar surface  The assumed constant value for the corona base temperature is

Towards more realistic model for a turbulence  Definitely the ad hoc description of the turbulence in terms of varied polytropic index is not sufficient.  On the other hand, the model sensitivity to the way the energy transferred to the plasma from turbulence requires to specify not only the physics of the turbulent motions (Alfven turbulence?) but also to quantify their dumping (cyclotron resonance?) with the energy transfer to the bulk solar wind. The way to include the semi-empirical model is described by Sudzuki (2006)  To describe quantitatively the SEP acceleration and transport, the description of the turbulence should be quantitative and detailed, including the turbulence spectrum, polarization and anisotropy (see Sokolov et al, (2006 May 1) ApJ, 642, L81- L84 )  Definitely the ad hoc description of the turbulence in terms of varied polytropic index is not sufficient.  On the other hand, the model sensitivity to the way the energy transferred to the plasma from turbulence requires to specify not only the physics of the turbulent motions (Alfven turbulence?) but also to quantify their dumping (cyclotron resonance?) with the energy transfer to the bulk solar wind. The way to include the semi-empirical model is described by Sudzuki (2006)  To describe quantitatively the SEP acceleration and transport, the description of the turbulence should be quantitative and detailed, including the turbulence spectrum, polarization and anisotropy (see Sokolov et al, (2006 May 1) ApJ, 642, L81- L84 )

Conclusion  We included the semi-empirical model data into coupled three-dimensional MHD models for the solar corona and the inner heliosphere