JOURNAL OF GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 118, 58 62, doi:10.1002/jgra.50098, 2013 Large deviations of the magnetic field from the Parker spiral in CRRs: Validity of the Schwadron model Edward J. Smith 1 Received 28 August 2012; revised 15 November 2012; accepted 23 December 2012; published 31 January 2013. [1] A difference in solar wind speed inside and outside a Coronal Hole Boundary (CHB) causes the field and solar wind to expand rapidly with distance to form a Corotating Rarefaction Region (CRR). In the Parker model, field lines inside the CRR form spirals determined by the local solar wind speed. Observationally, however, CRR fields often diverge from the Parker spiral by tens of degrees and in a more radial direction. In a model developed by Schwadron (2002), in addition to the velocity gradient, the field lines rotate differentially across the CHB. The solar wind speed then varies along the field lines rather than remaining constant as in the Parker model. The plausibility of the model is supported by calculations using nominal model parameters that reproduce the systematic deviations from the Parker spiral. The present study establishes the validity of the model by comparing observations of nine CRRs between 1 and 4 AU with model predictions. Estimates are obtained for the width, velocity gradient, differential velocity of the CRR, and the transit time of the magnetic field. These parameters are obtained at the surface near 10 solar radii where the pressure of the coronal magnetic field first becomes balanced. Citation: Smith E. J. (2013), Large deviations of the magnetic field from the Parker spiral in CRRs: Validity of the Schwadron model, J. Geophys. Res. Space Physics, 118, 58 62, doi:10.1002/jgra.50098. 1. Introduction [2] Large systematic deviations of the Heliospheric Magnetic Field (HMF) from the characteristic spiral angle associated with the Parker model of the solar wind [Parker, 1963] were first observed on the Pioneer 10 and 11 mission in early studies of Corotating Interaction Regions (CIRs) and their companion Corotating Rarefaction Regions (CRRs) [e.g., Smith and Wolfe, 1979]. Inside CRRs, the spiral angle, obtained from measured values of the radial and azimuthal field components, B r and B f, f B = atan (B f /B r ), was systematically more radial by tens of degrees than the Parker spiral, f P = invtan ( Ω r sin θ /V r ), where Ω, r,θ, and V r are the Sun s angular rotation rate, the radial distance, the colatitude angle (all quantities are in spherical polar coordinates), and the radial solar wind speed. Since the Parker model is such a good approximation in all other large-scale structures, why do these deviations occur and how does the model need to be modified? [3] Early observations of energetic solar particles inside CRRs also revealed a related anomaly. When the energetic particles within a CRR were extrapolated back to the Sun, they were found to originate at the same source longitude and were then called dwells [Roelof and Krimigis, 1973]. A further 1 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA. Corresponding author: E. J. Smith, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA. (Edward.J.Smith@jpl.nasa.gov) 2013. American Geophysical Union. All Rights Reserved. 2169-9380/13/10.1002/jgra.50098 finding was that the dwells coincided with the trailing edges of coronal holes, i.e., the Coronal Hole Boundary (CHB). [4] Using Ulysses data, Murphy et al. [2002] reviewed these observations and introduced a model developed by N. A. Schwadron that was capable of explaining the spiral angle departures of the HMF. A companion publication, Schwadron [2002], presented a more detailed description of the model and reproduced the departures qualitatively using nominal values of the model parameters. [5] The study reported here tests the validity of the Schwadron model by comparing observed and predicted values of f B. An additional objective, once the model validity is established, is to derive properties of the CRR near the Sun, specifically the width, the velocity gradient, and the time it takes the solar wind and magnetic field to cross the boundary. 2. The Schwadron Model and Its Origin [6] In the model, the Polar Coronal Hole (PCH) rotates at the solar equatorial rotation rate, Ω =2p / 25.4 days, as observed. Differential rotation causes the magnetic field to rotate through the coronal hole at the photospheric differential rotation rate in the corona. As the field crosses the CHB, the solar wind decreases from a fast speed, V F, inside to a slow speed, V S, outside, a change that occurs along the moving field line. It is this variation in solar wind speed along the field lines that produces the CRR and distinguishes this model from the Parker model in which the speed is constant along field lines. [7] The Schwadron model is an extension of a model developed by Fisk [1996]. In addition to the differential rotation, the Fisk model includes the non-radial expansion 58
of the magnetic field and the solar wind caused by the nonuniform magnetic pressure of the dipole-like field inside the coronal hole. The non-radial expansion continues until the gradient in magnetic pressure is eliminated and the pressure balance restored. The solar wind flow is then radial as in the Parker model. Evidence of the expansion was obtained from Ulysses measurements reported in Smith et al. [1995] and cited in Fisk [1996]. [8] The principal goal of the Fisk model was to explain why low-energy particles accelerated by CIRs at latitudes below 30 could reach latitudes up to 80 and be observed by Ulysses. The model showed that field lines did not remain at fixed latitudes as in the Parker model but expanded equatorward and allowed particles from low latitudes to reach high latitudes. However, the evidence provided by CIR energetic particle data was not conclusive because an alternative explanation was available based on cross-field diffusion of the particles from low- to high-latitude field lines, e.g., Kota and Jokipii [1995]. [9] Direct evidence of the non-radial flow was provided by Ulysses measurements showing that the streamline magnetic flux, r 2 B r, was independent of latitude from the equator to midlatitudes in Smith et al. [1995] and then in Smith and Balogh [1995] when the absence of the latitude gradient was confirmed at all latitudes. Smith and Balogh [1995] independently concluded that the enhanced magnetic pressure gradient associated with the Sun s dipolar magnetic field was causing non-linear fields and flows near the Sun that eventually produced an equilibrium in pressure and constant r 2 B r beyond which the solar wind flow was radial in agreement with the Parker model. Theoretical estimates based on both potential field and MHD source surface models were that pressure equilibrium occurred within 5 10 solar radii [Suess and Smith, 1996]. [10] Further implications of the Fisk model were published in Fisk et al. [1999] that addressed the slow motions of the coronal fields and plasma at low latitudes that returned them to the polar hole and maintained the total magnetic flux. Such motion was attributed to the reconnection of the open field lines with adjacent closed field lines that resulted in the required global transport of the open flux and was described as a diffusive process in Fisk and Schwadron [2001]. The Fisk model then provided Schwadron [2002] with the basis to explain the behavior of CRRs. [11] The non-radial flow and pressure balance have important implications for the Schwadron model. In this report, the Pressure Balance Boundary will be abbreviated PBB (a boundary rather than a surface, in part, to avoid confusion with Pressure Balance Structures (PBS) observed in the solar wind). It is difficult to convert the location of the CHB to its location on the PBB without detailed knowledge of the magnetic field inside the PCH and details of the CHB. Furthermore, the model needed to refer to measurements in the solar wind so that the CRR measurements far from the Sun could be compared with the Parker spiral. These issues resulted in the Schwadron model referring the initial conditions to the PBB instead of the CHB. Consequently, the CRR parameters in the model that are sought in this study are appropriate to this solar wind boundary/pbb rather than to the CHB. [12] The spiral angle inside the expanding CRR derived in Schwadron [2002] is as follows: tanf BM ¼ B f =B r ¼ ½ ðω þ oþ r sinθš= ½V ðordv=vdf 0 ÞŠ: [13] At the PBB, the field crosses the CRR with a width, df 0, at longitudinal angular rate, o, and with solar wind decreasing from a fast speed, V F, to a slow speed, V S. The mean velocity is V = (V F +V S ) / 2. The change in speed, dv=v F V S, is assumed to be linear, the speed gradient is dv/df 0, and the time to cross the boundary is dt= df 0 / (Ω + o) (and is positive since df 0 is negative). [14] The CRR starts at the Parker spiral angle of the fast solar wind and ends at the spiral angle of the slow wind. Inside the CRR, the field lines are nearly straight at large distances and depart significantly from the fast and slow Parker spirals in a more radial direction [Schwadron, 2002, Figure 2]. If o vanishes, the expression reduces to the Parker spiral, as required because the field lines remain inside or outside the coronal hole so the solar wind speeds are constant along the field lines. The model contains additional equations for other field properties such as B Y and the field magnitude, B. [15] Coincidentally, another model was published at about the same time that also assumes a change in solar wind velocity along field lines [Gosling and Skoug, 2001]. The intent of that model was to describe the decreasing speed profile and strictly radial magnetic fields that are observed occasionally in the solar wind. As such, that model is not relevant to this discussion since the Schwadron model addresses departures from the Parker spiral in general. It may be noted that the Schwadron model reduces to the Gosling and Skoug [2002] model with B f = 0 and the field radial when o = Ω, i.e., the field lines are not rotating. 3. Approach and Choice of Data [16] InSchwadron [2002, Figure 3], f B and f P are plotted vs. radial distance for nominal values of V F = 600 and V S = 300 km/s, o = 0.1 Ω and df 0 = 5. The approach adopted here is to test the model by a similar comparison of the measured f B with the model, f BM, assuming different values of o and df 0 while searching for a good fit. The single equation for f BM containing two unknowns means that different pairs of o and df 0 need to be assumed. [17] The model was tested using data acquired by Pioneers 10 and 11 in 1973 1974 near solar minimum. The two data sets contain a large number of CIR-CRRs. Another advantage is the absence of Interplanetary Coronal Mass Ejections (ICMEs) that would create a problem if unrecognized since they also exhibit a decrease in plasma velocity and a tendency to originate at a single source longitude. Pioneer measurements of the solar wind magnetic field, velocity, density, and temperature were used to sort out the few ICMEs that were present. The Pioneer vector helium magnetometer and plasma analyzer are described in Smith et al. [1975]and Wolfe et al. [1974]. 4. Analysis [18] The Pioneer 10 and 11 data are archived at omniweb. gsfc.nasa.gov/cgi/nx1/cgi. This site also includes software to tabulate and plot the data, make histograms, find averages and standard deviations, etc. The Pioneer data were inspected to identify CRRs and obtain averages of B f and 59
B r and values of f B. Hourly averages of the two field components inside the CRR were examined visually, an interval of about 2 days was chosen, and the components averaged over that interval to obtain f B. Rarefaction might have been expected to lead to a marked decrease in magnetic fluctuations attributed to waves, discontinuities, etc. However, visual inspection of many CRRs showed that the peak-peak amplitude of these magnetic fluctuations often appeared to be as large as or larger than averages of the components. The reason for that is not the subject of this analysis, but it is mentioned because it led to choosing intervals when this noise background was suitably low instead of averaging over the entire CRR. This procedure should not affect the comparison with f BM because the model assumes that the spiral angle is essentially constant inside the CRR. Furthermore, the results obtained justify this approach. [19] For each f B, the standard error, e, the ratio of the standard deviation to the square root of N which is the number of data in the averaging interval, was calculated. The standard deviation is the square root of the variance of f B including the variance in tan f B and the effect of taking the inverse tangent. The variances in the measurements, obtained over the same interval as the averages of tan f B, are s i 2 where i = 1,...9. Since the average, <tan f Bi > is the ratio of the averages, < B fi > and < B ri >, s i 2 is obtained from the variances of B f and B r, i.e., s i 2 = s i 2 (B f )/< B f > 2 + s i 2 (B r ]/ <B r > 2. Since f B is obtained by taking the inverse tangent, the variance is s i 2 (f B) = s i 2 [1/(1+ < tan f Bi > 2 ] 2 [Bevington, 1969, p.58]. The standard error is then e i = s i (f B ) / (N) 1/2.The solar wind velocity along with the orbital parameters, r and θ, was used to calculate f P corresponding to f B. [20] Figure 1 contains f B, e, and f P between 1.8 and 4.0 AU. The two observables do not overlap so that f B is statistically well separated from f P. An unanticipated feature is the U-shaped variation in f B with distance that is only slightly present in f P. The likely cause is a similar systematic variation in velocity, a time dependence, evident in Figure 2, a plot of V vs. distance. The increase of f B with V is contrary to the 1/V dependence in the Parker spiral and is a point in favor of the model. The figure also contains the slow speeds, V S =V dv/2, in part, as a way of showing the variation in dv, an important measureable appearing in the Schwadron equation. [21] Figure 3 is a comparison of tan f B with tan f BM based on the nominal values of o = 0.1 Ω and df 0 = 5. The tangents of f BM and f B were used in the figure in order to avoid any non-linearity associated with the inverse tangents in favor of a linear analysis. [22] The two profiles are similar, but tan f B appears to be shifted above tan f BM by an equivalent 10 to 20. The Schwadron model appears capable of reproducing the deviations from the Parker model, but values of o and df 0 need to be found that will produce a better agreement between the data and the model. [23] The usual Goodness of Fit between a model and data, the Chi-squared test, was used to find the best-fit pairofthe values of o and df 0. The usual definition is w 2 = Σ(y i y(x i )) 2 / s i 2 where both y(x i ), the model value of tan f BM (o, df 0 ), and y i =tanf B, the measured value, are at the same x i =r.thes i 2 are the variances in the measurements over the same interval as the averages of tan f Bi as described above. [24] The minimum in w 2 was found by trial and error while inspecting simultaneous plots of tan f BM and tan f B for the improvement in fit that could easily be seen as w 2 decreased. The fit associated with the minimum w 2 = 0.772 is shown in Figure 4. The improvement compared to Figure 3, for which w 2 = 16.82, is obvious. The best-fit w 2 value may not be the Figure 1. Variation in the observed and Parker spiral angles with distance. Pioneer 10 and 11 data in 1973 1974 were used to obtain the observed field spiral angles between 1 and 4 AU. The corresponding Parker spiral angles were computed from the usual expression given in the text using the measured solar wind speed. The large discrepancies with the observed fields that are significantly more radial than predicted are the subject of this report. Figure 2. The average velocity observed inside the CRRs. The average velocities, V = (V f +V s ) / 2, are shown for the nine test CRRs. The systematic upward curvature reproduces that in the observed Phi angles in Figure 1. This correspondence is consistent with the model dependence on V. The slow speeds, V S =V dv / 2, are included essentially to provide a record of dv. 60
Figure 3. Comparison of the tangents of the observed and model spiral angles. The observed angles are the same as in Figure 1, but tan f B =(B f /B r ) is shown. The model values, tan Φ BM, use measured values of V and dv and the nominal values of o / Ω = 0.1 and df 0 = 5 published in Schwadron [2002]. The tangents are used instead of the angles to avoid possible non-linearity associated with taking the inverse tangents especially when finding the Chi-square goodness of fit. The model predicts values that are systematically too low. Figure 4. A fit of the model to the data that satisfies the Chisquare test for goodness of fit. The tangents of the observed and model spiral angles are shown as in Figure 3 in order to avoid the non-linear effect of taking the inverse tangents. This fit was obtained by trial and error while varying o and df 0 and calculating w 2 at each step. The sequence converged rapidly with w 2 decreasing to 0.772 (the result shown here). The agreement between the model and the data is statistically significant at a confidence level of 99%, establishing the validity of the model and o / Ω =0.20anddf 0 = 2.5. absolute minimum but must be close, and in any case, that doesn t matter because it shows unequivocally that the model fits the data with a high level of confidence. [25] Statistics books typically contain tables that list the probability, P (w 2, n), that for a set of random data, w 2 will exceed a specific value associated with the number of degrees of freedom (n =9 2 = 7, in this case). Table C-4 in Bevington [1969] that lists the reduced value, w 2 / n, shows that 0.772/7 = 0.110 will be exceeded 99% of the time for data that are randomly distributed. Alternatively, the level of confidence of the goodness of fit exceeds 99%, or there is only a chance of 1 in 100 that the data fit is random. [26] It is also significant that the fit is not only based on r, the independent parameter, but also depends on large variations in the average velocity, V (332 625 km/s), and the difference between fast and slow winds, dv (25 to 180 km/s). In addition to errors in measurements of B f and B r, the residuals in Figure 4 could contain contributions from differences in o and df 0 from one coronal hole to another. The goodness of fit implies that the associated pair of values is a reasonable approximation to average values over the nine examples. [27] The fit in Figure 4 yields o = 0.20 Ω and a longitudinal width df 0 = 2.5 at the PBB. These values differ from the nominal values adopted in Schwadron [2002]. Disregarding possible latitudinal motions, the transit time of the field lines across the CRR at the PBB is dt= df 0 / (Ω + o) = 3.52 h. The velocity gradient dv/df 0, varies between 10.0 and 120 km/s deg. 5. Discussion [28] The preceding analysis shows that the Schwadron [2002] model accounts for the large departures from the Parker spiral angles inside CRRs. It explains why the magnetic field inside the CRR characteristically turns away from the Parker spiral angle toward a more radial direction. It also provides valuable information about the CHB although relative to the CHB shadow on the Pressure Balance Boundary/PBB rather than in the corona. The model fit yields two previously unknown parameters, o, the differential angular velocity, and dj 0, the angular width of the shadow CHB. The measured dv along with dj 0 yields the longitudinal speed gradient, and o and dj 0 lead to the transit time to cross the shadow CHB. [29] The success of the model improves our understanding of the solar origin and structure of corotating rarefaction regions that dominate the heliosphere throughout the solar minimum phase. It provides modifications within CRRs of the Parker model, the gold standard that has proven so successful in describing the solar wind and HMF throughout most of the heliosphere. [30] The distinctive feature of the Schwadron model is the change in speed of magnetic field lines as they cross the trailing coronal hole boundary. The model is based on the solar wind-magnetic field model of Fisk [1996] that incorporates both the differential field line motion and the excessive magnetic pressure in the coronal hole that causes the field and solar wind to expand in latitude and longitude until the pressure balance boundary is reached beyond which the solar wind flows radially outward and obeys the Parker model. At the CHB, the solar wind velocity changes from very fast to very slow in a narrow longitude range but then expands first to the PBB and by many tens of degrees with increasing distance to form the usual CRR that occupies a large fraction of the heliosphere. 61
[31] It has long been known that CRRs are associated with a velocity gradient at the coronal hole source, i.e., a decrease in V from fast to slow and when extrapolated back to the solar source map back to a very limited range of longitudes. The Parker model was originally thought to produce this result while ignoring the differential motion across the CHB and assuming that the velocity of the field lines changes from V F inside to V S outside the coronal hole with V F and V S remaining constant as the wind propagates outward. However, the field lines inside the CRR then conform to the Parker spiral and gradually change from the spiral appropriate to V F to that appropriate to V S. That behavior proved to be contrary to observation. [32] Assuming for simplicity, and in the absence of evidence to the contrary, that the velocity gradient is linear, the Schwadron model connects the spirals conforming to V F and V S at the edges of the CRR by nearly straight field line segments that deviate toward the radial direction as observed. The success of the model is a further demonstration that the field lines near the Sun are not fixed in latitude and longitude but are driven by the spatial variation of the magnetic field pressure within the coronal hole as represented by the parameter. [33] The availability of a more realistic model provides opportunities for further investigation of the CRR structure and its properties at the PBB. The complete set of Schwadron equations contains other parameters of interest such as the latitudinal angular velocity of the solar wind and magnetic field. The study of the same CRR at two or more locations has the potential to determine specific values of o and df 0 separately as well as other properties for individual CRRs that could demonstrate how they vary from one CRR to another. Although not the subject of this analysis, the random field variations inside CRRs appear to be larger than might be expected in the presence of the large on-going expansion. As such, they deserve further inquiry to explore their strength, nature, and origin. [34] Acknowledgments. Prior to 2002, in a private conversation on the departures inside CRRs, Gene Parker suggested that a change in solar wind speed along the field line could be the cause. In a subsequent discussion of Ulysses CRRs, Tim Horbury of Imperial College made the same suggestion. However, it was Nathan Schwadron who independently came to the same conclusion and derived the equations describing this phenomenon. Joyce Wolf generated large numbers of CRR plots for this study and assisted in preparing the final figures. The Heliospheric COHO website at the GSFC National Space Science Data Center provided the magnetic field and solar wind data, and their companion software was useful in the analysis. The results reported here represent one aspect of research carried out by the California Institute of Technology Jet Propulsion Laboratory for the National Aeronautics and Space Administration. References Bevington, P. R. (1969), Data Reduction and Error Analysis for the Physical Sciences, McGraw- Hill Book Co., New York. Fisk, L. A. (1996), Motion of the footpoints of heliospheric magnetic field lines at the Sun: Implications for recurrent energetic particle events at high heliographic latitudes, J. Geophys. Res., 101(A7), 15,547 15,553. Fisk, L. A., T. H. Zurbuchen, and N. A. Schwadron (1999), On the coronal magnetic field: Consequences of large-scale motions, Astrophys. J., 521, 868. Fisk, L. A., and N. A. Schwadron (2001), The behavior of the open magnetic flux of the sun, Astrophys. J., 560, 425. Gosling, J. T., and R. M. Skoug (2001), On the origin of radial fields in the heliosphere, J. Geophys. Res., 107, doi:10.1029/2002ja009434. Kota, J., and J. R. Jokipii (1995), Corotating variations of cosmic rays near the south heliospheric pole, Science, 268, 1024. Murphy, N., E. J. Smith, and N. A. Schwadron (2002), Strongly underwound magnetic fields in co-rotating rarefaction regions: Observations and implications, Geophys. Res. Lett., 29, NO.22, doi:10.1029/2002gl015164. Parker, E.N. (1963), Interplanetary Dynamical Processes, p. 138, Interscience Publishers, New York. Roelof, E. C., and S. M. Krimigis (1973), Analysis and synthesis of coronal and interplanetary energetic particle, plasma and magnetic field observations over three solar rotations, J. Geophys. Res., 78, 5375. Schwadron, N. A. (2002), An explanation for strongly underwound magnetic field in co-rotating rarefaction regions and its relationship to footpoint motion on the sun, Geophys. Res. Lett., 29, NO. 14, doi 10.1029/ 2002GL0150028. Smith, E. J., and J. H. Wolfe (1979), Fields and plasmas in the outer solar system, Space Sci. Rev., 23, 217 252. Smith, E. J., B. V. Connor, and G. T. Foster (1975), Measuring the magnetic fields of Jupiter and the outer solar system, IEEE Trans. Magn., MAG-11, 962. Smith, E. J., M. Neugebauer, A. Balogh, S. J. Bame, R. P. Lepping, and B. T. Tsurutani (1995), Ulysses observations of latitude gradients in the heliospheric magnetic field: Radial component and variances, Space Sci. Rev., 72, 165. Smith, E. J., and A. Balogh (1995), Ulysses observations of the radial magnetic field, Geophys. Res. Lett., 23, 3317. Suess, S. T., and E. J. Smith (1996), Latitudinal dependence of the radial IMF component: Coronal imprint, Geophys. Res. Lett., 23, NO.22, 3267 3270. Wolfe, J. H., J. D. Mihalov, H. R. Collard, D. D. McKibben, L. A. Frank, and D. S. Intriligator (1974), Pioneer 10 observations of the solar wind interaction with Jupiter, J. Geophys. Res., 79, 3489. 62