Journal of Geophysical Research: Space Physics

MHD tomography using interplanetary scintillation measurement



[1] The MHD tomography analysis, an advanced version of the computer-assisted tomography (CAT) analysis for interplanetary scintillation (IPS) observational data, is presented. When combined with a MHD simulation, the CAT analysis can reconstruct the MHD-based three-dimensional solar wind structure within 1 AU from the IPS line-of-sight integrated velocity data. At the same time, the MHD code can simulate the actual conditions in the heliosphere beyond 1 AU by utilizing the results of the MHD tomography analysis and extrapolate to r ≥ 1 AU. The present MHD tomography aims to reconstruct the global structure of corotating solar wind, under the assumption that the global solar wind does not vary significantly in time during one Carrington rotation. The reconstructed three-dimensional solar wind variables including interplanetary magnetic field (IMF) are compared with the in situ measurement data made at the Ulysses spacecraft and nearby Earth, and good correlations are obtained. We anticipate that the combination of MHD simulation and the IPS tomography will enhance solar wind research and space weather prediction algorithms.

1. Introduction

[2] Radio wave are weakly scattered when they propagate through the solar wind, and as a result the radio flux observed at Earth scintillates. This radio wave scintillation is called interplanetary scintillation (IPS). The Solar-Terrestrial Environment Laboratory (STEL) of Nagoya University has four IPS observation sites that make synchronous IPS measurements. An estimate of the solar wind velocity is derived from the time lags of the IPS signal patterns from the four sites. At present, the IPS observations are unique ground-based methods to detect solar wind motion. The IPS observations at STEL sense the velocity of the already-accelerated solar wind at r ≥ 50 Rs, using a 327 MHz radio frequency.

[3] The velocity measured by IPS, however, does not directly represent the solar wind velocity at a particular point. It is an integrated value along the line-of-sight (LOS) with a weighting factor determined by the electron density fluctuation and diffraction [Young, 1971].

[4] Recently, a computer-assisted tomography (CAT) method was developed to reconstruct the three-dimensional distribution of solar wind from the IPS observations [Jackson et al., 1998]. Jackson et al. reconstructed the solar wind structure by using the integrated IPS data along all lines of sight and considering the conservation of momentum and mass. Kojima et al. [1998] developed another CAT analysis to make a fine-resolution map on a reference sphere centered on the Sun. In their method the IPS data made at various heliocentric distances and times are traced back along streamlines to the source positions on the reference sphere, under the assumption that the global solar wind does not vary drastically during one Carrington rotation. Then, using the data collected on the reference sphere, they succeeded in reconstructing a map of the global solar wind distribution with an angular resolution of 7.5 degrees. In order to guess the source positions of the observed volumes of solar wind, however, they used a ballistic model, where the solar wind plasma velocity was assumed constant along the streamlines. In these two IPS-CAT analyses, the nonlinear hydrodynamical (HD) or magnetohydrodynamical (MHD) processes of the solar wind are not directly taken into account. Therefore we increase the sophistication of the IPS-CAT analysis by incorporating an MHD treatment of the solar wind.

[5] Currently, MHD simulation is a rather common tool in the study of solar wind dynamics. The advantage of MHD simulation is literally that it can treat the solar wind in terms of the MHD equations. One MHD approach to calculate the global solar wind is to solve the initial boundary value problem of MHD equations and trace the time-relaxation process of the trans-Alfvénic flow at r ≥ 1 Rs [e.g., Steinolfson et al., 1982; Usmanov, 1993; Linker et al., 1999; Gibson et al., 1999]. In this case the boundary values of magnetic field on the solar surface are frequently used to assign the situations to be solved. Another MHD approach is to solve the steady state of the super-Alfvénic flow by the radius-increment calculation, assuming the inner boundary distribution [e.g., Han et al., 1988; Pizzo, 1994] or using the results of the simulation on the trans-Alfvénic flow [e.g., Usmanov, 1993; Usmanov et al., 2000]. These MHD approaches employ models and assumptions of the heating and acceleration of the solar wind such that the obtained solar wind is close to reality, and comparisons with spacecraft data show that they succeeded in reproducing several aspects of the solar wind. However, the actual solar wind structure is more complex than the model-based MHD simulation approaches can handle, and at present, there seem to be no models that produce the complexity very well.

[6] The advantage of the IPS-CAT approach over the MHD approach is that the CAT analysis uses the IPS velocity data representing real information about the already-accelerated solar wind. That is, having observational data of the end result of solar wind acceleration, we can avoid the use of models and directly treat the complex reality.

[7] In this context we embed the MHD simulation code in the CAT analysis. This CAT analysis incorporating the MHD simulation code is hereafter called the MHD tomography analysis. Using the observational IPS data to constrain the MHD simulation, the MHD tomography analysis can provide the MHD-based and IPS-based three-dimensional solar wind structure in the range of heliocentric distance from 50 Rs to 1 AU, to which STEL-IPS observation are sensitive. In addition, by including the magnetic field, temperature and plasma density, it can provide a complete description and expand results derived for 50 Rsr ≤ 1 AU outward beyond 1 AU.

[8] The derived solar wind structures are compared with in situ measurements from the Ulysses (r > 1 AU) and near-Earth satellites (IMP8, WIND, ACE, and ISEE-3) (r = 1 AU) to demonstrate the capabilities of this new reconstruction method.

2. Methods

[9] The MHD tomography analysis is based on the concept of CAT analysis developed by Kojima et al. [1998]. However, to introduce MHD code into the CAT analysis, some additional procedures and assumptions are required.

[10] The region to be treated is outside of a sphere with a radius of 50 Rs. This choice of the inner boundary distance is made for two reasons. One is that the IPS measurements at the STEL radio frequency (327Mhz) has sensitivity for r ≥ 50 Rs and the other is that the MHD simulation code becomes very simple when only the super-Alfvénic region is treated.

[11] The MHD tomography analysis is an iterative procedure modifying the solar wind velocity distribution on the inner boundary sphere. The modification of the boundary velocity is made in three steps. In the first step, MHD simulation of solar wind is carried out using a provisional distribution on the inner boundary at 50 Rs, and the steady state of the three-dimensional MHD-based solar wind at r ≥ 50 Rs is calculated (described in section 2.1). In the second step (described in section 2.2), the IPS velocity observations are simulated in this numerical solar wind, and the discrepancies between the velocities from the IPS simulations and the actual IPS observations are calculated for each LOS. In the last step (described in section 2.3), the velocity distribution on the inner boundary surface is modified such that the discrepancies between numerical and actual IPS velocities are reduced.

[12] The iterations can be started from any arbitrary boundary distribution. We use a uniform solar wind speed distribution of 500 km/s. The process of modification is repeated until the IPS velocity discrepancies become sufficiently small. The three-dimensional numerical solar wind obtained is MHD-based and best matches IPS observations.

2.1. MHD Simulation of Quiet Global Solar Wind

2.1.1. Basic MHD Equations and Coordinate System

[13] It is assumed that the ideal single-fluid MHD equations govern the system of solar wind to be solved. A spherical coordinate system is used to denote the position and vector components. The coordinate frame is assumed to rotate with the sidereal angular velocity equation image of the rotation around the solar rotation axis (θ = 0 or θ = π) so that the boundary conditions at the inner surface remain fixed in the computation.

[14] Taking the rotation of the frame and gravity into account, the basic equations of the time-dependent MHD simulations are written as

equation image
equation image
equation image


equation image

where variables t, r, ϱ, Pg, equation image, equation image and equation image are time, heliocentric distance, mass density, plasma gas pressure, a vector of plasma velocity viewed in the rotating frame, a vector of magnetic field, and a vector of gravity force of the Sun, respectively. The proton and electron temperature are assumed to be equal and T = Tp = Te = Pgmp/2ϱkB, where mp and kB are mass of proton and the Boltzmann constant, respectively, and the mass of electron is neglected.

[15] The specific heat ratio, or polytropic index, γ is a crucial parameter in the present MHD simulation. If there were no additional energy or momentum input, this parameter would be 5/3 and the solar wind speed must decrease at distant regions. In reality, however, the solar wind speed does not significantly decrease even at several AU, and there must be thermal energy and momentum addition to the solar wind there. The likeliest source of the heating and acceleration is the decay of Alfvén waves, and the orthodox MHD approach might be to treat explicitly the dynamics of energy addition, for example, the Alfvén wave transport and decay [cf., Usmanov et al., 2000] and to involve two-fluid MHD code [cf., Suess et al., 1999]. In this study, however, instead of treating the additional energy and momentum process explicitly, we assume that γ is constant everywhere in the simulation regions and equal to 1.46, employing the empirical polytropic index derived from Helios data by Totten et al. [1995].

2.1.2. MHD Simulation Code

[16] In the simulation region the velocity decrease of solar wind plasma flow is negligibly small, while the velocity of magneto-acoustic waves decreases with distance. Therefore if the solar wind is super-Alfvénic everywhere on the inner boundary sphere, the solar wind is safely super-Alfvénic everywhere in the simulation region. In this case no information can propagate toward the Sun, and the property of the downstream solar wind is determined only by the upstream conditions. Because of this, the present MHD tomography analysis uses a simpler MHD code to calculate the steady state of the solar wind for r ≥ 50 Rs. We use the radius-increment method, instead of tracing the time-relaxation process with a time-increment simulation.

[17] Writing the vector-matrix presentation form of time-dependent MHD equations (1), (2), (3), and (4) as

equation image

the equations of steady state that the radius-increment method is going to solve are given as

equation image

where equation image is a column vector containing the eight variables of MHD flow; the matrices Mr,Mθ, and Mϕ are 8 × 8 Jacobian matrices with respect to the three coordinate directions; and the column vector equation image contains the gravitational, centrifugal, and Coriolis force terms and the apparent surplus terms due to the spherical coordinate presentation.

[18] In this study we employ the two-step method to increment the heliocentric distance, where after the provisional values are once evaluated by

equation image

the variables at r + Δr are obtained as

equation image

[19] Note that this two-step method is equivalent to a Taylor expansion of the second-order accuracy with respect to r;

equation image

[20] Centered differencing is employed to calculate the derivatives ∂equation image/∂θ and ∂equation image/∂ϕ and some of the trigonometric functions, sin θ and cos θ, appearing in the basic equations are evaluated by the differencing of ∂(−cos θ)/∂θ and ∂(sin θ)/∂θ in accordance with the configuration of the numerical cells. The increment of radius Δr is controlled by a condition similar to the Courant condition:

equation image

where λθ and λϕ are the eigenvalues of the multiplied matrices Mr−1Mθ and Mr−1Mϕ, respectively, and Δθ and Δϕ are the angle intervals of numerical lattice along latitude and longitude directions, respectively. In this study the two angular interval Δθ and Δϕ are taken to be uniform and equal to π/32. The Lapidus type artificial viscosity [Lapidus, 1967] is used to control the computational instability.

[21] The solar rotation axis (θ = 0 or π) is singular and, at the same time, the computational boundary of the θ-ϕ space where centered differencing cannot be used. We simply assume that the vectors of plasma velocity and magnetic field have only radial component along the solar rotation axis, and the density, gas pressure, and radial component of plasma flow along the rotational axis can be calculated as the average of the values at the neighboring grids on the same spherical layer. The radial component of magnetic field is calculated from the divergence free condition, ∇· equation image = 0. The periodic boundary condition is used for the azimuthal direction at ϕ = 0 and ϕ = 2π.

2.1.3. Boundary Values at 50 Rs Density and Temperature

[22] In order to determine the density n and temperature T at 50 solar radii, we derive empirical functions for n(V) and T(V) from Helios data obtained within 0.5 AU from December 1974 to June 1981. The restriction to heliocentric distances is to avoid the effects of the solar wind evolution due to the stream-stream interactions. Helios data of number density and proton temperature are normalized to the reference heliocentric distance at 50 Rs by the power-law relations Vr0(constant), Tr−1, and nr−2. These power-law relations are the same as for the Parker solution [Parker, 1958] of the case γ = 1.5, and seem appropriate for the present study where γ = 1.46. Next, assuming

equation image
equation image

we derive the coefficients of n1 = 62.98, n2 = 866.4, Vn = 154.9, T1 = −0.455, T2 = 0.1943, and the power index Pn = −3.402 by least-square fit, denoting n in count/cm3, V in km/s and T in 106K. These two functions are drawn in Figure 1 together with the normalized Helios data.

Figure 1.

The empirical functions n(V) and T(V) (solid lines) and the normalized Helios data (dots) at 50 solar radii. Radial Component of Magnetic Field

[23] We use observed solar photospheric magnetic field data to determine the magnetic field distribution on the inner boundary surface. Using the photospheric magnetic field data observed at the Wilcox Solar Observatory, we determine the magnetic field at 50 Rs by the following procedure.

[24] First, we calculate the potential field in the solar corona [Altschuler and Newkirk, 1969] with the a radial boundary condition [Wang and Sheeley, 1992] and construct a map of the magnetic field on a source surface at 2.5 solar radii, B1st(2.5 Rs,θ,ϕ). Next we calculate the extended source surface magnetic field at 50 solar radii (B2nd(50 Rs,θ,ϕ)), assuming that the magnetic field strength decreases with r−2 and its global distribution rotates rigidly eastward.

equation image

In this study the eastward shift angle ϕ′ is determined to be 14 degrees from the approximate estimation of the travel time from the solar surface to 50 solar radii with a constant speed of 400 km/s.

[25] Ulysses observed the uniform magnetic field distribution in the fast wind regions during its travel at high heliographic latitude regions in 1995. This observational fact implies that magnetic flux is rearranged. Usmanov et al. [2000] showed with two-dimensional MHD simulation that the magnetic flux rearrangement is due to the tangential component of plasma motion toward heliospheric current sheet (HCS) produced by latitude-dependent solar wind heating process. Because the present analysis method cannot treat the dynamics of magnetic field rearrangement at r < 50 Rs directly, we employ a power-law modification of boundary magnetic field B2nd that pushes flux closer to the current sheet, creating a more uniform flux distribution away from the current sheet and a steeper gradient across the current sheet. The power-law employed in this study is given as

equation image

where Pm is a positive power index less than 1, Bm is the multiplying factor to adjust the source surface magnetic field strength to the actual interplanetary magnetic field strength, and the coefficients I+ and I are determined so that the total magnetic flux of each polarity is preserved,

equation image

We define Pm = 0.33 and Bm = 6 in this study. The distribution B3rd is used as boundary magnetic field. Figure 2 shows maps of B2nd and B3rd, with contour lines demonstrating the magnetic field rearrangement toward HCS.

Figure 2.

The maps of magnetic field at 50 Rs, B2nd (upper maps) and B3rd (lower) for CR 1844 (left column) and 1895 (right). Thick lines are the magnetic neutral line. Countour lines are at equal interval of 25 nT. Azimuthal and Latitudinal Components of Velocity and Magnetic Field

[26] The two components of the plasma flow vector, Vθ and Vϕ, are determined so that the plasma seems to flow only in the radial direction when observed in the rest frame:

equation image

[27] The two magnetic field vector components, Bθ and Bφ, are determined from the requirement of the steady state of magnetic field and the magnetic solenoidality [e.g., Yeh and Dryer, 1985]. For this study we simply employ the free condition of the electric field (equation image = −equation image × equation image = 0) to determine Bθ and Bϕ:

equation image
equation image

2.2. Simulation of IPS Velocity Measurement in Numerical Solar Wind

[28] The IPS velocity measurement is simulated in the numerical three-dimensional solar wind obtained by the MHD simulation, and the discrepancy between the actual and simulated IPS velocities is evaluated. If the discrepancy is not small enough, the discrepancy amount is used to modify the flow speed distribution on the inner boundary surface.

[29] The IPS simulation is made with the integration [Kojima et al., 1998],

equation image

where equation image is the velocity vector of the solar wind plasma flow observed in the rest frame, equation image is the position vector of a point on the LOS at the distance z from Earth, and w′ is the normalized weight. The square of electron density fluctuation (δne)2 is the weighting factor of radio diffraction, and the other weighting factor w(z) is made by using the Born approximation and taking the Fresnel propagation filter and the size of the radio sources into account [Young, 1971]:

equation image

where q is a wavenumber, Λ is the wavelength of observation (92 cm at STEL IPS observation, corresponding to 327 MHz), θ0 is the angular size of radio source assumed to be 0.1 arcsecond, and a cubic power-law spectrum q−3 for the fluctuations is assumed.

[30] To complete equation (17), the electron density fluctuation must be determined. Although the density fluctuation in reality may depend on not only the local solar wind conditions but also its origin in the solar corona, we utilize the power-law dependence of the density fluctuation on the density and flow speed found by Asai et al. [1998]. Although Asai et al. classified the data into three flow speed classes and derived power-law dependences for each class, the present analysis assumes the power index is constant and equal to the average of Asai et al.'s power-law indices,

equation image

The LOS integration of equation (17) is made for the regions within 1 AU, because the weighting factors at r ≥ 1 AU are sufficiently small to be negligible.

2.3. Modification of Boundary Map of Solar Wind Flow Speed

[31] The modification of the flow speed on the inner boundary, Vr(50 Rs,θ,ϕ), is made such that the discrepancy of IPS velocity, ΔVVsimVobs, will be reduced. To assign the position and guess the amount of modification of boundary velocity, the discrepancy ΔV and the normalized weight w′(z) on LOS are traced back to the inner boundary sphere along the MHD-based streamline with Runge-Kutta method. Thus projected ΔV and w′(z) are restored in the numerical lattice around the foot point of the streamline together with other weights wspread and werr.

[32] The weight wspread is a Gaussian spread function around the source position,

equation image

where η is the angular distance from the foot point of the streamline to the concerned numerical lattice. The typical angular size η0 is determined from the transverse travel distance of an acoustic wave. Assuming the typical distance from LOS to the inner boundary ΔRtypical to be 80 solar radii, the typical sound speed ctypical to be 50 km/s, and typical solar wind speed Vtypical to be 400 km/s, the angular size of domain of influence on the inner boundary sphere ηd.i. is approximated as

equation image

Since the numerator and denominator of ctypical/Vtypical are in positive correlation, ηd.i. is rather insensitive to the conditions. We use this value as the constant half-width of the spread function. This angular width is approximately twice as large as the angular interval of the computational grid and the spread function helps to avoid numerical instability in the MHD simulation.

[33] Another weight werr is a function of observational error Verr that is evaluated at the time when the observational IPS velocity is calculated from the time lags of IPS signals among four STEL IPS sites. The typical Verr is 10 ∼ 20 km/s for a typical solar wind of 500 km/s. In this study this weight is defined as a function of the ratio of Verr to the observed IPS velocity Vobs,

equation image

It should be mentioned here that IPS velocity data with Verr/Vobs ≥ 0.3 are not used in the analysis.

[34] This procedure is done for all actual IPS observations during the time period concerned, and the amount of velocity modification at each lattice (j, k) is calculated as

equation image

where the integer m and l denote the digit position on LOS and the IPS observation number, respectively.

[35] Finally, the flow speed at the lattice (j, k) is modified by

equation image

The positive factor α(≤1) prevents overshoot of the solution. We set α = 0.8 in the present analysis.

[36] To examine the degree of agreement between the derived and observed IPS data, the average of the discrepancy

equation image

and the standard deviation

equation image

are checked during the analysis. Table 1 tabulates the values of equation image and σ(ΔV) obtained in the analysis for Carrington rotation (CR) 1844, as an example. It is observed that equation image rapidly decreases and approaches zero and σ(ΔV) decreases to 10% or better level of the average IPS velocity equation imageobs. Since the discrepancies evaluated in the final iteration are generally sufficiently small, we can say that the derived three-dimensional solar wind structure matches the IPS observations well.

Table 1. Average and Standard Deviation of the Discrepancy Between the Actual and Synthetic IPS Velocitiesa
Trial Time12341020
  • a

    The averaged actual IPS velocity equation image is also tabulated for reference. These values (in km/s) are derived from the analysis for Carrington rotation 1844, as an example.

equation image16.650.96−3.00−1.12−0.04−0.04
$\ \overline{V_{\rm obs}}$438.55438.55438.55438.55438.55438.55

[37] The standard deviation σ(ΔV) is also used in the data selection. Because IPS observation detects not only the steady component of solar wind but also transient components like propagating disturbances and because the actual solar wind varies during one Carrington rotation, some of IPS data are not suitable for the present purpose of reconstructing the quiet component of solar wind. First, we carry out a provisional analysis with five iterations and then discard the IPS velocity data whose discrepancy is greater than 1.5-sigma,

equation image

After this data selection, the iterative process is repeated.

3. Experimental Analysis

3.1. Reconstruction Test

[38] If the MHD tomography analysis is a self-consistent reconstruction technique, it should be able to reconstruct an a priori known solar wind structure. To examine this, we carry out experimental analyses where an artificial simple-structured solar wind is to be reconstructed.

[39] The artificial three-dimensional solar wind is made by the radius-increment MHD simulation code identical to the one used in the real analysis using a boundary distribution with a sinusoidal-shaped slow wind given by

equation image


equation image

Then the artificial IPS velocities are calculated by LOS integration with equation (17) in this synthetic solar wind.

[40] We prepare two artificial IPS velocity datasets: one is made using the same lines of sight as for the actual IPS observations in CR 1846 (case A) and the other for CR 1841 to 1849 (case B). The number of IPS velocity data is 392 in case A and 3474 in case B. Using these two artificial IPS velocity datasets, we carry out the MHD tomography analyses. Figures 3a1–3a3 and 3b1–3b3 demonstrate how the MHD tomography approach the target map (y) from the initial uniform map (x) in cases A and B, respectively.

Figure 3.

The flow speed on the inner boundary surface derived at third, tenth, and twentieth modification in case A (a1–a3) and case B (b1–b3). The uniform distribution given at the initiation of analysis and the target artificial distribution with sinusoidal-shaped slow wind region are shown at box (x) and (y), respectively.

[41] Two points are noticed. The first is that in both cases, the derived boundary flow speed map smoothly approaches the target map and the sinusoidal shape of slow speed regions is retrieved. The reconstructed boundary flow speeds are equal to 400 and 690 km/s, agreeing well with the targets (400 and 700 km/s). The other point is that except for the error levels (∼20 km/s in case A and 10 km/s in case B), no critical differences are seen between the results of case A and case B. Since the difference between the simulation for case A and B is merely the number of artificial IPS data used in the input, this result confirms that the number of the IPS velocity data made at STELab during one Carrington rotation is sufficient for the reconstruction of the global solar wind.

[42] From these two points we conclude that the MHD tomography analysis and the IPS observations have sufficient ability to reconstruct the solar wind structure during a particular Carrington rotation.

3.2. Dependence on Initial Model

[43] The results of the MHD tomography should not depend on the boundary flow speed distribution given at the initiation of the analysis. To confirm this, we tested three different initial distributions of 300, 500, and 700 km/s. Figure 4 shows the boundary velocity maps obtained at tenth iteration starting from theses three different flow distributions with the same actual IPS velocity dataset. The largest and average difference among the derived flow speed distributions are less than 15 km/s and less than 5 km/s, respectively. Therefore the dependence on the choice of the boundary flow speed at the initiation of analysis is not significant.

Figure 4.

The boundary velocity maps derived at the tenth modification of the analyses beginning with three different initial flow speeds, from the left, 300, 500, and 700 km/s.

4. Comparisons With Spacecraft Observations

[44] The three-dimensional solar wind structure is calculated in the distance range from 50 Rs up to 1350 Rs (6.3 AU) with the MHD tomography method and compared with in situ measurements by Ulysses and satellites nearby Earth. The solar wind at 50 Rsr ≤ 1 AU is reconstructed by the MHD tomography analysis and is expanded beyond 1 AU with the MHD code.

[45] We made comparisons for three Carrington rotations, 1844, 1894, and 1895. The first, CR 1844, in 1991, is the period when Ulysses spacecraft traveled at 3.6 AU near the solar equatorial plane before the Jupiter swing-by. Choosing this period, we can use the in situ data obtained simultaneously at two different heliocentric distances, near Earth and at 3.6 AU, to see the evolution of the solar wind structure with distance. The others, CR 1894 and 1895 in 1995, are the periods during which Ulysses made the fast latitudinal scan at about 1.4 AU and observed the complex-structured slow solar wind at low-latitude regions and the uniform fast solar wind at the northern high-latitude regions [e.g., Philips et al., 1995; Smith et al., 1995]. By comparing with the Ulysses and nearby-Earth data made during this period, we can make a simultaneous comparison at two different latitudes. It should be noted here that in 1995, the IPS observation facilities at STEL began its annual observations at the middle of CR 1894, and we filled the gap of the IPS observation in the earlier half of CR 1894 with observations made in the earlier half of CR 1895.

[46] We use the daily averaged near-Earth data at the OMNI database (constructed using IMP8, ISEE-3, WIND, and ACE data) and Ulysses data at COHO database. These two databases are constructed at NSSDC of NASA. The choice to use the daily averaged data is made considering the longitudinal angular resolution of the present MHD simulation code that corresponds to a temporal resolution of about 12 hours.

[47] The simulated solar wind variables are sampled at the site of the in situ measurements in the simulated solar wind. As an example, the simulated solar wind structure at CR 1844 is shown in Figure 5. Figure 5 contains the latitude-longitude maps of the density, flow speed, and temperature distribution at various heliocentric distances. The radial evolution of solar wind structures such as the corotating spiral structures and the walls of enhanced density and temperature at the trailing edge of the slow flow regions are observed.

Figure 5.

Radial component of plasma flow Vr (center column, b0–b3), density n (left, a1–a3), and temperature T (right, c1–c3) at four different heliocentric distances 50 Rs, 1 AU, 3 AU, and 5 AU, derived by the MHD simulation in the MHD tomography analysis for CR 1844. The temperature and density at 50 Rs are given by the empirical functions of velocity (equations (10) and (11)). The thick dotted lines represent the HCS where Br = 0. The grayscales for Vr are identical, while those for n and T are controlled by the maximum and minimum values in each box.

[48] Figure 6 shows the comparisons between the simulated solar wind variables and the actual in situ measurement data. The gray scale maps show the solar wind velocity derived from the MHD tomography, on which are superposed the foot points of streamlines that have passed through the in situ measurement sites (nearby-Earth by circles, Ulysses spacecraft by triangles). The line drawings show the simulated solar wind variables (solid lines) and the in situ data (dashed line). The density, temperature, magnetic field, and plasma flow speed are compared at Ulysses (upper four panels) and near Earth (lower four panels). The abscissa of the graphs is the Carrington longitude of the source positions at 50 Rs so that the unique longitude in the spiral-structured solar wind can be defined. The magnetic field is represented by the radial component Br, except for the comparison at Ulysses spacecraft for CR 1844 where the azimuthal component Bϕ (denoted as Bp in plot) is used. The vertical lines attached to the data points in the line graphs for the simulated flow speed represent the errors evaluated from the residual discrepancy at the foot point, ±(∣δVj, ϕk)∣ + 30) km/s.

Figure 6.

Superimposed plots of the in situ data (dotted lines) and the synthetic values (solid lines) obtained from the MHD tomography analysis at the position of Ulysses (the upper four panels) and Earth (the lower four panel) and the derived boundary flow speed map at 50 Rs (middle). The Carrington rotation number is denoted on the top of grayscale map. The abscissa of the line graph is the Carrington longitude of the source positions, and the ordinate of the grayscale map is sine of heliographic latitude. The source positions, defined as the foot points of the streamlines traced from the in situ measurement sites, are marked with triangles (Ulysses) and circles (nearby-Earth) on the grayscale velocity map.

[49] Table 2 shows the correlation coefficients and averages of the compared variables. Since the increase in the heliocentric distance of Ulysses spacecraft during one Carrington rotation is small, the correlation analysis ignores the dependence of n, T, and B on r. To examine if the magnetic flux rearrangements mentioned in section worked adequately, the averages of the magnetic field are calculated using the absolute value of the representative magnetic field component.

Table 2. Correlation Coefficients, the Means of the Synthetic Solar Wind Variables, and the Daily In Situ Dataa
CRR1844 equation imageequation imageR1895 equation imageRequation image1894 equation imageequation image
  • a

    The correlation coefficients R, the means of the synthetic solar wind variables, equation image, and the daily in situ data, equation image for variables, flow speed V, density n, temperature T, and magnetic field B are tabulated. The magnetic field is represented by the radial component Br, except for the comparison at Ulysses spacecraft for CR 1844 (marked by dagger (†)). The correlations of the magnetic field are made using the raw positive and negative quantities, and the means of the magnetic field are made using the absolute values. The asterisks (*) are attached to the correlation coefficients for CR 1895, when Ulysses traveled in the uniform flow at the northern high-latitude region. The averages are given in units km/s, cm−3, 103K, and nT.

N0.810.520.53* 0.661.251.250.712.102.67
B† 0.88† 0.90† 1.04*−0.401.351.500.491.241.67
V0.52506.59494.18* 0.37763.03756.930.85645.60650.57

4.1. Comparison for CR 1844

[50] The positive correlation coefficients in Table 2 shows that good matches between the derived solar wind and the spacecraft measurements are obtained. The panels at the left column of Figure 6 demonstrate the longitudinal dependence of the match of the solar wind variables.

[51] Since the heliographic latitudes of the two spacecraft measurement sites differ, we can examine the solar wind evolution with distance. The labels (a) and (a') in the panel show that the MHD simulation is able to produce good extrapolation of the solar wind for the outer heliosphere (r ≥ 1 AU) if the solar wind in the inner part of the heliosphere (r ≤ 1 AU) is well reconstructed. The labels (b) and (b′) weakly support this, and the labels (c) and (c′) imply the opposite; that is, the bad guess at r ≤ 1 AU leads to poor extrapolation at r ≥ 1 AU.

[52] We easily recognize that labels (a′) and (b′) correspond to the regions of the longitudinally expanding streamlines at the trailing edges of the fast wind regions and the leading edges of the slow wind regions, and the label (c′) corresponds to the region of the compressed streamlines, namely the corotating interaction region (CIR). Therefore the discrepancy of flow speed at the longitude around label (c) resulted in the larger discrepancy at the positions of the Ulysses measurement around label (c′).

[53] Nevertheless, the density enhancement from label (x) to (c′) is well reconstructed. We think that the good agreement of density variation results from two factors. The first is that the flow speed at the both sides of the label (c′) are well reproduced, and the second is that the measurement-based empirical function n(V) determining the density on the inner boundary is rather insensitive to V at V > 400 km/s. The first factor implies that the flow speeds beside the region (c′) are well reconstructed from 50 Rs to 3.6 AU as well, and the second implies that the density flux that is to be involved in the density enhancement is well evaluated at 50 Rs. Therefore the density enhancement at (c′) is simulated well. The coincidence of the temperature enhancement from label (x) to (c′) supports this explanation.

4.2. Comparison for CR 1894

[54] The match of the flow speed around the label (d) in the top panel of Figure 6 indicates that the transition of the flow speed from the slower complex structured solar wind at low latitude to the faster uniform wind at the high latitude, as observed by Ulysses, is well reproduced.

[55] Because we have replaced the gap of IPS observation during the earlier half of CR 1894 with the data from the earlier half of CR 1895, the derived flow speed distribution for the longitude range of 180 to 360 degrees on the inner boundary mainly represents that of CR 1895. This might be one reason of the discrepancy for flow speed at Ulysses spacecraft around the label (e).

4.3. Comparison for CR 1895

[56] The correlation coefficients calculated for the nearby-Earth comparison are all positive, and the averages of derived and actual parameters agree well both at Ulysses and nearby Earth.

[57] The agreement of the magnetic field strength at two different heliographic latitudes confirms that the determination of magnetic field at 50 solar radii can substitute for the actual evolution of magnetic field in the trans-Alfvénic solar wind regions over a wide range of latitudes. The agreement of the average flow speed shows that the IPS observations can sense the solar wind motion even at the high-latitude region and the MHD tomography can reconstruct it.

[58] As the poor correlations of the comparison with Ulysses data shows, it is difficult to reconstruct the spatial variations of the uniform fast wind regions at the high heliographic latitude regions. The uncertainty of the MHD tomography analysis is generally about 100 km/s, and the solar wind flow speed of the Ulysses data was approximately 750 ± 50 km/s. Therefore the spatial variation detected by Ulysses spacecraft was not retrieved.

4.4. Azimuthal Angle of Magnetic Field

[59] The comparison of the azimuthal angles of magnetic field at Ulysses spacecraft in 1995 is made to examine whether the boundary conditions of the radius-increment MHD code at the singularity along the solar rotational axis worked appropriately or not. To examine them at the Ulysses' closest approach to the solar rotational axis as well, the period is extended to CR 1901. Figure 7 shows the azimuthal angles of the daily averaged magnetic field calculated as equation image. In terms of the averaged angle, general good agreement of the azimuthal angle is obtained, inferring that the spiral geometry at the high heliographic latitude regions is actually reconstructed by the MHD code.

Figure 7.

Comparison of the azimuthal angle of IMF at the northern polar region with the Ulysses data from CR 1894 to 1901 in 1995. The angle is defined as equation image, and the daily average value of Br and Bϕ is used to calculate the angle of in situ data. Carrington number is denoted at the bottom together with its commencement date, the heliocentric distance (in AU), and the angle from the northern axis to Ulysses in situ measurement site.

5. Summary

[60] We have developed the MHD tomography analysis algorithm to reconstruct the MHD-based three-dimensional solar wind structure at 50 Rs-1 AU using the IPS observations. In addition, the MHD simulation expands the solar wind streams to the outer heliosphere. The solar wind obtained in this way was compared with in situ measurements by Ulysses and near-Earth spacecraft, and several good matches were obtained. We again stress that the combination of the remote-sensing IPS observation technique and MHD tomography enables us to obtain a observation-based three-dimensional solar wind for a wide range of heliocentric distance and heliographic latitude.

[61] Since the previous version of the CAT method did not treat any solar wind parameters other than velocity and did not consider the nonlinear MHD process, there is the possibility that results from previous and present method are significantly different despite using the same IPS data. To see the difference, these two analyses are carried out using the same IPS observations. Figure 8 shows the boundary velocity maps derived from the previous IPS-CAT analysis [Kojima et al., 1998] and from the MHD tomography analysis. Although the values are not identical in detail, it is notable that the major aspects of velocity distribution are not so much different. As a whole, it can be said that the previous version of the CAT analysis using the ballistic model can be safely used as long as the other variables are not necessary.

Figure 8.

The solar wind velocity map derived with the previous version of CAT analysis by Kojima et al. [1998] (a) and with the present MHD tomography (b). The general aspects of the distribution are not so much different between the two analyses.

[62] We anticipate that the presented analysis will enhance the study of the space weather prediction by providing our quiet solar wind data to other models, especially time-dependent MHD simulation models that can handle the perturbed solar wind more accurately [cf., Wu et al., 1983; Smith and Dryer, 1990; Dryer et al., 1991; Detman et al., 1991; Usmanov and Dryer, 1995; Riley et al., 1997]. In addition, the extrapolation of solar wind at r > 1 AU by our analysis will be also useful for the study of, for example, merged interaction regions (MIRs) [e.g., Burlaga and Ness, 2000], the poleward diffraction of the stellar wind at several hundred AU [Sakurai, 1985; Pizzo, 1994], and the interactions with the interstellar medium, under the conditions close to reality.


[63] The authors wish to thank NSSDC of NASA for the use of their data set of the spacecraft measurement data of the solar wind, distributed at OMNI and COHO database. We also thank Wilcox Solar Observatory, Stanford University for the use of the photospheric magnetic field data in the synoptic map format. The computations in this work are partially made with the VPP5000 system at the Computer Center of Nagoya University. We would like to thank R. Woo for his valuable comments on this paper during his stay in STELab. We would like to thank B. V. Jackson, P. P. Hick, and T. Ohmi for their valuable comments on this work. We would also like to acknowledge engineering support from Y. Ishida, K. Maruyama, and N. Yoshimi. This work was partially supported by the Japan Society for the Promotion of Science (grant 12440130).

[64] Shadia Rifai Habbal thanks Paul L. Hicks and another referee for their assistance in evaluating this paper.