Modeling 1 AU solar wind observations to estimate azimuthal magnetic fields at the solar source surface



[1] A recent two-dimensional (radial distancer and solar longitude ϕ) model for the solar wind is driven using 1-hour average data from the Wind spacecraft. We extend the treatment of the Sun's magnetic field to allow a nonzero azimuthal componentBϕ at the source surface, assumed to be at the photosphere where r = Rs, in addition to the radial component Br. We find nonzero azimuthal magnetic fields at the source surface with important consequences for the more distant heliosphere. The averages ∣Bϕ(Rs)∣ and ∣Br(Rs)∣ inferred over solar cycle 23 are 0.44 ± 0.48 μT (4.4 ± 4.8 mG) and 120 ± 30 μT (1.2 ± 0.3 G) at the photosphere, respectively. Both components vary with time by more than an order of magnitude, with ∣Bϕ(Rs)∣ ≤ ∣Br(Rs)∣. While the surface magnetic field is closely radial on average it is sometimes 20° from radial. Both Bϕ and Br vary smoothly on periods of 10 hours, with evidence for relatively narrow current sheets, and vary with the solar cycle: Br(Rs) is correlated with the sunspot number, but with a time lag of 20 months, while Bϕ(Rs) has a two level behavior, decreasing near solar maximum and increasing near solar minimum. Our results and model can account naturally for non-Parker-like magnetic field directions at 1 AU since theBϕ fields inferred at the source surface lead to the Bϕ and Br fields at 1 AU having similar average magnitudes and large variability.

1. Introduction

[2] Solar phenomena like sun spots, flares, coronal mass ejections and production of coronal plasma vary with the solar cycle. Outflow of heated ions and electrons from the Sun forms the solar wind, a plasma with a high electrical conductivity and a magnetic Reynolds number much greater than unity. Thus, the magnetic field is frozen-in to the plasma and transported into interplanetary space from an effective “source surface” located at a heliocentric distancer = rs.

[3] Different models have been developed to relate the magnetic fields of the photosphere, chromosphere, and corona to interplanetary magnetic fields (IMF). The oldest is Parker's [1958] spiral model which assumes a purely radial wind with constant speed at all r and a purely radial magnetic field at the source surface. (In this paper without loss of generality the source surface occurs at the photosphere where r = rs = Rs.) Recent work [Cairns et al., 2009] suggests that the solar wind starts very low in the corona near 1 Rs.

[4] The Parker model implies that in the solar equatorial plane the azimuthal angle between B(r) and r should be close to 45° or 225° at 1 AU, with the azimuthal component entirely due to solar rotation. It is known that B is well described by the Parker model for suitably long time averages (over many solar rotations) but rarely instantaneously [e.g., Forsyth et al., 1996], instead varying greatly in direction and magnitude.

[5] Intrinsic changes in the magnitude of a radial vfrom a specific point on the Sun can lead to non-Parker directions, as shown [Gosling and Skoug, 2002; Riley and Gosling, 2007] in the context of near-radialB observed at 1 AU and beyond [Neugebauer et al., 1997; Jones et al., 1998]. Similarly underwinding of the Parker spiral in CIRs due to velocity shear is proposed [Smith and Bieber, 1991; Smith et al., 2000], but should be primarily beyond 1 AU where CIR effects are strong. Motion of magnetic foot points, thereby imparting nonzero vϕand so non-radialv, is also proposed to explain non-ParkerB at large heliolatitudes in Ulysses data [Fisk, 1996; Zurbuchen et al., 1997; Fisk and Schwadron, 2001]. Note that wind models with strong turbulence effects typically assume Parker magnetic fields [e.g., Usmanov et al., 2011].

[6] The foregoing ideas for non-ParkerBall involve either non-radialv or variations in the magnitude of v. While each is reasonable and has some quantitative support, a new and distinct model is considered here: that non-ParkerBcan arise due to an intrinsic non-radial, azimuthal, component ofB at the solar source surface. This generalizes the previous assumption Bϕ = 0 and B = Br/r at the source surface.

[7] At one level, exploration of the assumption Bϕ(Rs) ≠ 0 is long overdue given the strong evidence in TRACE, STEREO, SDO, and other datasets for non-radialB from the photosphere to the corona [Petrie and Patrikeeva, 2010; De Pontieu et al., 2011]. On another level, intrinsic nonzero Bϕ(Rs) offers a direct interpretation for non-ParkerB in the solar wind. For instance, near radial B could result from an intrinsic Bϕ at the source surface that cancels the ϕ component resulting from solar rotation and convection of the radial component from the source surface. Finally, assuming Bϕ(Rs) ≠ 0 is as reasonable as assuming vϕ ≠ 0 or varying v, so the consequences of assuming Bϕ(Rs) ≠ 0 with constant radial v should be examined as thoroughly as previous models with Bϕ(Rs) = 0.

[8] Accordingly, this paper presents and applies to Wind spacecraft data an analytic equatorial solar wind model with Bϕ(Rs) ≠ 0 and radial vat the source surface. By extrapolating smoothed IMF data back to the source surface with MHD-like equations, we demonstrate the existence and extraction of nonzeroBϕ(Rs) and show the consequences for Bat the photosphere and in the solar wind out to 1 AU. The variability of the source surface magnetic fields over a solar rotation period is examined, finding modulation on spatial scales similar to supergranulation cells. The model and results can explain naturally non-Parker-like magnetic fields at 1 AU close to the ecliptic plane. A strong solar cycle 23 variation of ∣Bϕ(Rs)∣ and ∣Br(Rs)∣ is shown. A discussion of the results and the identification of future improvements complete the paper.

2. Two-Dimensional Solar Wind Model

[9] The starting points for the theory of a magnetized wind are ∇ ⋅ B = 0 and the frozen-in equationE = − u × B, where E is the electric field [e.g., Parker, 1958; Schatten, 1971; Jokipii and Kóta, 1989; Fisk, 1996]. With ∇ × E = 0 for a static magnetic field,

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For simplicity we assume Bθ = uθ = 0 in spherical coordinates (r, θ, ϕ) around the solar rotation axis, i.e., u and B only have radial and azimuthal components. Equation (1) leads to three component equations of which the θ-component leads to nothing useful for the present analysis. Conservation of magnetic flux in the wind implies

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with the assumption that Bϕ is constant versus ϕ, either for all ϕ or in small patches in which ∂Bϕ/∂ϕ can be neglected. Inserting (2) into the ϕ-component of(1) leads to

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Here Ωs = 2π/(27 days) is the (equatorial) angular rotation frequency of the Sun and ur is the constant radial wind speed. From (2) and (3) Br ∝ r−2 and Bϕ ∝ r−1 at large r. Assuming B to be radial at the source surface, so that Bϕ(Rs, ϕs) = 0, (3) reduces to the Parker solution

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Two ways exist to find Br(Rs, ϕs) using observations of B(r) at a specific r = rob, either by rearranging (2) to find

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or (4) to obtain

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The quantities rob, Rs, ϕob, ϕs at r = Rs, the observation time t, and the start time t0 of the interval [t0, t0 + 27 days] when the longitude meridian ϕs faces Earth are related by

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These approaches yield different Br(Rs, ϕs) and indeed different B(r) for all r ≠ rob if the Parker solution does not hold exactly. Florens et al. [2007] used these two approaches to find two sets of Br(Rs, ϕs) and separately model Br(r) and Bϕ(r), because they found the Parker solution to be a poor approximation. This approach is obviously not self-consistent, thus necessitating more realistic modeling of the magnetic field.

[10] Here we allow Bϕ(Rs, ϕs) ≠ 0 and find unique values for Br(Rs, ϕs) and Bϕ(Rs, ϕs) by using (5) and writing (3) as

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Equations (2), (3), (5), and (7)(9) determine B(r,ϕ) at the source surface and for all r and ϕself-consistently. This extended model uses centered sliding boxcar 11-point averages of 1-hour averaged data from Wind spacecraft observations at 1 AU as a function of timet, essentially to smooth out turbulence. The solar wind quantities at Rs as a function of ϕs are identified by assuming that the wind sources are constant over a solar rotation period. Time is then converted to longitude ϕs on the Sun using the time average of the wind speed. The spacecraft's heliographic latitude is used for θob. Since the Wind spacecraft orbits near the solar equatorial plane, where θob ≈ 90°, the θ-dependence in our model is negligible. The plasma number densityne, speed ur, and ion and electron temperatures vary with r and ϕ as by Florens et al. [2007].

3. Predictions for the Solar Source Surface and the Solar Wind

[11] We demonstrate the process and results of the new model by extracting the radial and azimuthal magnetic field components over the solar rotation period 1 to 27 June 2010 (Figure 1). Nonzero ∣Bϕ(Rs)∣ are found in the range 10−6 − 10−8 T, with ∣Br(Rs)∣ in the range 10−4 − 10−6 T, and average values of order 10−7 T (1 mG) and 10−4 T (1 G), respectively. The radial fields agree well with previous measurements for the mean surface field [e.g., Severny et al., 1970]. A sign change in Bϕ from + to − means a change from clockwise to anticlockwise direction (as seen from the north pole of the ecliptic) of Bϕ, while Br changing from + to − means a flip from outward to inward orientation of Br.

Figure 1.

Magnetic components Bϕ(Rs) and Br(Rs) for the period 1–27 June 2010 obtained from the solar wind model and Wind spacecraft data. The vertical scale is −δlog10Bi∣, with field Bi (i = ϕ or r) measured in tesla and δ = ± 1 for Bi > 0 and <0, respectively.

[12] Both components vary with time by more than an order of magnitude, with Bϕ less than or equal in magnitude to Br. While the surface magnetic field is closely radial on average, as assumed previously, it is sometimes almost 25° from radial (Figure 2), thereby justifying the assumption Bϕ(Rs, ϕs) ≠ 0 and qualitatively consistent with photospheric and chromospheric data from SOLIS VLM [Petrie and Patrikeeva, 2009]. They show a field structure over 5 years that is nearly radial in the photosphere with a tilt angle θBr = 1.8° ± 10.8°, which lies between B and er, and which expands in a greater variety of directions at chromospheric heights (θBr = 5.5° ± 35°). Similarly, de Pontieu et al. [2011]show non-radialB from the chromosphere to the corona.

Figure 2.

Inferred angles θBr between the magnetic field BM(r, ϕ) and the radial direction er at the source surface and at 50 Rs, and θP(50Rs) for 1 to 27 June 2010.

[13] The Bϕ fields vary smoothly on periods of order 10 hours with sharp transitions implying relatively narrow current sheets. The radial field changes polarity on the same time scale, sometimes correlated with the azimuthal field, again with sharp transitions via current sheets and evidence for magnetic sectors, which are observable in the IMF as well (Figure 3). The very similar qualitative properties of the Br and Bϕ fields, and the consistency of the Brfields with well-accepted measurements [e.g.,Severny et al., 1970], provide strong arguments that the extracted Bϕ fields are real and intrinsic to the Sun and/or source surface.

Figure 3.

New two-dimensional (r, ϕ) solar wind model for 1 to 27 June 2010 driven by Wind spacecraft data. The Sun is at (X,Y) = (r cos ϕ, r sin ϕ) = (0,0) and the Earth is at the red “1” on 1 June and then moves clockwise to the red “2” in one day and around the white circle in 27 days. (top) Br, the different azimuthal components Bϕ for Bϕ(Rs) = 0 and ≠ 0, respectively and the relative difference Q of the magnetic field between Florens et al.'s [2007] model and our new model. (bottom) ne, ur, and θBr for our model, and the Parker model prediction θP. Note the different ranges of the color bars for θBr and θP.

[14] The ≈10 hour period corresponds to a distance of about 5 Mm on the Sun, on mapping time to arc length along Earth's orbit. Comparing this with supergranulation (≈30 Mm in diameter and a lifetime of ≈24 hours) and granulation (≈1 Mm in diameter and a lifetime of 20 minutes) cells, these spatiotemporal changes in the signs and magnitudes of Br and Bϕ appear reasonable. For instance, Jokipii and Kóta [1989] show that these cells move magnetic foot points and should generate Bϕ and Bθ fields greater than 10−3Br for (r < 2Rs) and even comparable to Br(Rs) at the solar surface.

[15] Predictions for the near equatorial magnetic quantities are shown in Figure 3 for r ≤ 1.1 AU. These include Br(r, ϕ), the different azimuthal components Bϕ(r, ϕ) predicted for Bϕ(Rs) = 0 and ≠ 0, and the relative difference Q(r, ϕ) = ∣BF − BM∣/BM in total magnetic field between Florens et al.'s [2007] model BF(r, ϕ) which assumes Bϕ(Rs) = 0 and our new model BM(r, ϕ) which allows Bϕ(Rs) ≠ 0. Figure 3 shows strong variations in the solar wind with r and ϕ, with clear evidence for magnetic sectors and regions where B does not follow the Parker spiral, i.e., where Br is not opposite in sign to Bϕ and (4) does not hold accurately. Quantified by Q, differences in the total field magnitude are due to the inferred nonzero Bϕ(RS) and are strongest close to the Sun. In addition, Figure 3 shows ne and ur(providing clear evidence for CIR-like structures), the angleθBr between BM(r, ϕ) and er, and the Parker model prediction θP for the angle between B(r, ϕ) and er, calculated using the same average ur and the Br(1 AU, ϕob) data. The angles θBr and θP are modulo 180° with zero corresponding to the radial direction and positive clockwise. Clearly θP(r, ϕ) is very different from θBr(r, ϕ), failing to account for the observed B directions at 1 AU.

[16] The angles θBr at the source surface and at 50 Rs are shown in Figure 2 for 1 to 27 June 2010. In comparison θP(Rs) = 0° and θP(50Rs) = 10° for this period. While B(RS, ϕ) is usually inclined at small angles to the radial direction (with θBr(Rs) = 0° ± 2.5° on average from 1 to 27 June 2010 and θBr(Rs) = 0° ± 4° on average over solar cycle 23) and at 50 Rs, these angles often reach ± 20° and ± 90° at Rs and 50 Rs, respectively. As shown by Petrie and Patrikeeva [2009], the measured photospheric field is usually within about 12° of being radial, while the chromospheric field expands in all directions. Similarly, θBr(1 AU) = 32° ± 41° over solar cycle 23 while θP(1 AU) = 45°. Thus the Parker prediction is often not a good approximation.

[17] It is shown next that the new model and typical ratios Bϕ(Rs)/Br(Rs) extracted from IMF data can account naturally for non-Parker field directions at 1 AU. For constantur we define the variability in magnitude of the azimuthal field at 1 AU by ΔBϕ(1 AU) = Bϕ(1 AU) + Br(1 AU)(1 AU − Rs)Ω/ur ≈ Bϕ(Rs)Rs/1 AU using (3). The relation Br(1 AU) = Br(Rs)(Rs/1 AU)2 from (2), the experimental result ∣Bϕ(Rs)∣ ≈ ∣Br(Rs)∣/250 on average over solar cycle 23 (Figure 4), and r = 1 AU ≈ 215 Rs then imply

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rather than the Parker predictions ΔBϕ(1 AU) = 0 and Bϕ(1 AU) ≈ − Br(1 AU). Thus the values of Bϕinferred at the solar surface lead to the radial and azimuthal components at 1 AU having similar average magnitudes and large variability. It appears that non-Parker magnetic fields at 1 AU can often be due to nonzeroBϕ at the source surface.

Figure 4.

Monthly means of B(Rs) (blue curve) and ∣Bϕ(Rs)∣ (red curve) over solar cycle 23 (May 1996 to December 2008) inferred from Wind spacecraft data with (5) and (9). The green curve shows monthly sunspot numbers (SSN) from the Solar Influences Data Analysis Center at

[18] To examine the long term variability of surface magnetic fields inferred from the IMF data we compare their dependence during solar cycle 23 with sunspot numbers (SSN). Figure 4 shows the monthly means for B(Rs), ∣Bϕ(Rs)∣ and SSN from April 1996 to December 2008. Both ∣Br(Rs)∣ and B(Rs) have very strong and essentially identical (not shown) solar cycle dependences and magnitudes, peaking near solar maximum. The correlation between B(Rs) and SSN is evident and B(Rs) has a time lag of about 20 months. In contrast ∣Bϕ(Rs)∣ appears to have a two level behavior, having a larger average value from solar minimum to ≈4 months after solar maximum, and a smaller average value for the rest of the cycle. The averages ∣Bϕ(Rs)∣ and ∣Br(Rs)∣ inferred over solar cycle 23 were 0.44 ± 0.48μT (4.4 ± 4.8 mG) and 120 ± 30 μT (1.2 ± 0.3 G) at the photosphere, respectively.

[19] The analysis reported above use a 11-hour sliding average for the data. Different data-smoothing periods lead to similar results but with the expected effects of varying the smoothing periods, i.e., lower average magnetic field magnitudes for longer smoothing periods since the peak positive and negative values are smoothed out (not shown).

4. Discussion and Conclusions

[20] The solar cycle variations of Br(Rs) and Bϕ(Rs) in Figure 4 may be related to the strong meridional plasma flows that caused an unusual sunspot minimum during solar cycle 23 [Nandy et al., 2011]. They found that very deep minima in SSN are correlated with weak polar fields since the meridional plasma flow speed is around 20 m/s on average and therefore the polar field needs around 20 months to be transported to the equator, not inconsistent with the 20 month time lag between Br(Rs) and SSN found here. In 2008 the meridional flows extended almost to the poles and returned at a slower rate than in previous years. These flows typically occur in loops that reach two thirds of the distance to the poles. At a speed of 20 m/s, it takes about 11 years to make one circuit.

[21] The solar cycle variation of Br(Rs) appears to agree with the variation of the radial magnetic component measured by Ulysses [Fisk and Zhao, 2009]. They found that the distance-normalized radial component varies over the solar cycle, increasing by a factor of ≈2 near solar maximum and being smallest near solar minimum. This field component is a measure of the Sun's open magnetic flux. The faster meridional plasma flow in the first half of solar cycle 23 and the slower flow in the second half in models [Fisk and Schwadron, 2001; Fisk and Zhao, 2009; Nandy et al., 2011] are consistent with a larger average magnitude of Bϕ(Rs) from solar minimum to just after solar maximum, and then a smaller average value for the rest of the cycle.

[22] The extracted azimuthal surface fields are large enough to perturb the distant polar field (r ≫ 1 AU), since Bϕr−1 while Br ∝ r−2, and lead to several important consequences for solar modulation of cosmic rays [e.g., Fisk, 1996; Jokipii and Kóta, 1989; Hitge and Burger, 2009]. Since the polar entry of cosmic rays into the inner solar system depends on the magnitude and direction of the B, this entry may be strongly affected by the nonzero Bϕ(Rs) fields found here and the different component Bϕ(r, ϕ) predicted for Bϕ(Rs) ≠ 0.

[23] The analyses above establish that a solar wind model with intrinsic Bϕ ≠ 0 at the solar source surface yields reasonable, broadly self-consistent, results forBr and Bϕat the source surface and within 1 AU and can explain naturally non-Parker field directions in the solar wind. It is not claimed here thatBϕ(Rs) ≠ 0 is the unique explanation for non-Parker field directions. Indeed, analysis based on the complementary assumptionsBϕ(Rs) = 0 and either vϕ(Rs) ≠ 0 or non-constant radialv(r) [e.g., Fisk, 1996; Smith and Bieber, 1991; Zurbuchen et al., 1997; Smith et al., 2000; Gosling and Skoug, 2002; Riley and Gosling, 2007; Fisk and Zhao, 2009] yield some attractive results. Indeed, the most plausible and general boundary conditions at the source surface involve nonzero azimuthal components of both B and v, not one or the other.

[24] Collected in one place, the arguments for regarding intrinsic Bϕfields at the source surface as both real and an attractive model for non-Parker fields in the heliosphere include: the frequent observation of non-radial solar magnetic fields, the characteristics of the extracted fieldsBϕ(Rs) and Br(Rs) and their consistency with previous Br(Rs) data, the simplicity and generality of the explanation for non-Parker fields in the solar wind,Section 2's analytic results, and the solar cycle variations of the inferred Bϕ(Rs) and Br(Rs). Understanding theoretically the origin and persistence of these azimuthal fields and their persistence are important areas for future work; their characteristic timescales suggest a link with supergranulation cells. Future analysis should also determine the relative importance of nonzero Bϕ, vϕ, velocity shear, and turbulence effects. This could be done via suitable multi-dimensional MHD simulations (with appropriate boundary conditions) and theories that include turbulent heating and transport [e.g.,Usmanov et al., 2011].

[25] In conclusion, the extended treatment of magnetic fields in Florens et al.'s [2007] solar wind model and its application to Wind spacecraft data let us demonstrate the existence of nonzero azimuthal magnetic fields at the solar surface on the order of 10−3 − 10−2of the radial surface component. Hence, we can naturally explain non-Parker magnetic field configurations at 1 AU using our model and its results, since the azimuthal components inferred at the solar surface lead to the radial and azimuthal components at 1 AU having similar average magnitudes and large variability. Sharp transitions via current sheets are evident in both magnetic field surface components from polarity changes and the time scales for variability are similar to the lifetimes of supergranular motions. In the equatorial region of the Sun, the direction of the magnetic fieldB and its magnitude change by typically small but significant amounts compared to Florens et al.'s [2007] model, both being substantially different from the nominal Parker model. A strong solar cycle dependence of SSN and surface magnetic fields is shown, with ∣Br(Rs)∣ and B(Rs) having essentially identical solar cycle dependences, peaking near the solar maximum with a time lag of 20 months to the SSN. In contrast ∣Bϕ(Rs)∣ appears to have a two level behavior, decreasing near solar maximum and increasing near solar minimum.


[26] We thank the Australian Research Council for financial support.

[27] The Editor thanks two anonymous reviewers for their assistance evaluating this paper.