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Solar Physics (2005) 229: 161–174 DOI: 10.1007/s11207-005-5376-9

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Springer 2005

ON THE MAGNETIC CORRESPONDENCE BETWEEN THE PHOTOSPHERE AND THE HELIOSPHERE C. E. DEFOREST, D. M. HASSLER and N. A. SCHWADRON Southwest Research Institute, 1050 Walnut Street, Suite 400, Boulder, CO, U.S.A. (e-mail: [email protected])

(Received 10 May 2005; accepted 19 January 2005)

Abstract. The solar magnetic field maps every point in the corona to a corresponding place on the solar surface. Identifying the magnetic connection map is difficult at low latitudes near the heliospheric current sheet, but remarkably simple in coronal hole interiors. We present a simple analytic magnetic model (‘pseudocurrent extrapolation’) that reproduces the global structure of the corona, with significant physical advantages over other nearly analytic models such as source-surface potential field extrapolation. We use the model to demonstrate that local horizontal structure is preserved across altitude in the central portions of solar coronal holes, up to at least 30 Rs , in agreement with observations. We argue that the preserved horizontal structure may be used to track the magnetic footpoint associated with the location of a hypothetical spacecraft traveling through the solar corona, to relate in situ measurements of the young solar wind at ∼10–30 Rs to particular source regions at the solar surface. Further, we discuss the relationship between readily observable geometrical distortions and physical parameters of interest such as the field-aligned current density.

Happy is he who can trace effects to their causes. — Virgil

1. Introduction The solar magnetic field threads from the surface of the Sun through the heliosphere. Neglecting transient plasmoids, the open field maps each point in the inner heliosphere to a particular open-field location on the surface of the Sun. This mapping is important to quantify because near-surface conditions are thought to determine the mass flux and other parameters of the solar wind. Furthermore, the mapping is required in order to make use of data from the Ulysses, ACE, and Wind spacecraft and future in situ missions such as NASA’s Solar Probe and ESA’s Solar Orbiter, by associating the in situ sampled wind with its sources in particular features on the surface of the Sun. In this article, we discuss the feasibility of such association, and develop a simple analytic, morphological model of the solar magnetic field. We use the model to demonstrate that it is feasible to associate a particular in situ field line with a particular network flux concentration based on measured patterns of density in the in situ data and at the solar surface.

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Associating heliospheric events and features in the solar wind with their counterparts on the solar surface is difficult because of the complex magnetic field topology between the solar surface and the outer corona. Transient events due to coronal mass ejections (e.g. Webb et al., 2000) and high speed wind streams due to low-latitude coronal holes (Zirker, 1977) may be associated simply because there is generally only one appropriate feature on the surface of the Sun at a given time; but many corotating interaction regions and other wind features cannot be reliably mapped back to a particular location on the surface. This difficulty is a symptom of working within the slow solar wind, near the heliospheric current sheet. In contrast, the interiors of large coronal holes retain a simpler and more persistent morphology than the low-latitude corona and heliosphere, so that structures within the high-speed wind should be easily traceable to the surface. The difficulty of mapping the heliosphere to the surface can be understood in terms of the magnetic field mapping MF : R2 → R2 that maps (longitude, latitude) at the photosphere to (longitude, latitude) at the top of the corona. MF associates points that lie on the same magnetic field line, as illustrated in Figure 1. The domain of MF is the open field regions on the surface of the Sun; its range is the entire surface of a large sphere at (for example) 10 Rs . Tracing magnetic features from the inner heliosphere to the solar surface is equivalent to exploring the inverse mapping MF −1 . The difficulty of associating low-latitude wind and heliospheric features with surface features is easily explained in terms of the structure of MF −1 . The lowlatitude open field is thought to originate in myriad small open loci within the low-

Figure 1. Bifurcations in the surface – corona mapping. This illustration shows several flux configurations at various latitudes. The heliospheric current sheet exists between field lines D and E, but bifurcations without high-altitude current sheets exist between C and D, and between E and F. At bifurcations, small changes in latitude and longitude high in the corona correspond to large jumps in latitude and longitude at the surface. Many bifurcations exist at low latitudes in the map MF −1 from the heliosphere to the surface; but at high latitudes, the mapping is approximately continuous due to the predominance of open field (as between lines A–C, and F–G).

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latitude closed field regions of the Sun (e.g. DeForest, 1995; Wang and Sheeley, 1995), or at the edges of the polar coronal holes (Suess et al., 1998), where MF −1 is neither smooth nor even continuous. This lack of continuity is apparent as bifurcations in MF −1 , or, equivalently, cusps in the magnetic field structure. Bifurcations may be marked by current sheets or field reversals (like the central current sheet depicted between field lines D and E in Figure 1), or may have no local manifestation at all in the heliosphere (like the bifurcations between lines C and D, and between E and F, in Figure 1). Even in places where there is no bifurcation, the expansion ratio varies greatly at the edge of the coronal hole (Suess et al., 1998), so that surface structures between lines B and C or between lines F and G in Figure 1 will be greatly distorted by MF . In contrast, the interiors of coronal holes show smooth, uniform expansion and a marked lack of cusp-like features, as noted through imaging studies (Munro and Jackson, 1977; Fisher and Guhathakurta, 1995; DeForest et al., 1997; DeForest, Plunkett, and Andrews, 2001) and cross-correlation of features at different altitudes (DeForest, Lamy, and Llebaria, 2001). The coherence of the expansion may be expressed by noting that MF −1 is smooth and nearly continuous in coronal holes. One may expect, firstly, that simple extrapolation techniques can provide a good approximation of MF and locate the approximate surface location associated with any point in the extended coronal hole (Munro and Jackson, 1977; Suess et al., 1998); and, secondly, that patterns and shapes on the surface of the Sun will extend more or less intact into the solar wind, where they may be used as fiducial marks to pinpoint the exact associated location to any particular position in the corona, given a local map of the density structure of the corona both at high altitude and near the surface of the Sun. While there is significant time delay in the propagation of many density structures into the heliosphere, most polar plumes and other large-density structures last longer than that and hence serve as useful fiducials. The question of how well small features are preserved from the surface to the middle-to-upper corona is of particular interest due to the possibility of placing a combination of remote sensing and in situ instruments onboard a Solar Probe spacecraft to fly through the corona and investigate the solar wind in situ at or just above its main acceleration region. Such a mission could easily investigate the horizontal structure of the corona near the flight path by tomographic inversion (Hick and Jackson, 2004) of data from an onboard hemispheric or other wide field imager (Jackson, Buffington, and Hick, 2001; Buffington et al., 1998) as it observes the corona from a variety of angles. Spacecraft in situ data could then be associated with features at the bottom of the corona, enabling careful study of the relationship between surface activity and heliospheric wind streams. We sought to investigate whether mid-scale features in the upper corona could be used as fiducial markers for such an experiment, identifying recognizable patterns that could be used to determine the mapping MF −1 and thereby allow detailed comparison between near-spacecraft structure and the surface structures of interest.

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We have modeled a hypothetical solar corona with solar wind, and used that model to test the idea that the solar surface structure remains imprinted on the superradially expanding corona over the large polar coronal holes. We developed a simple, analytic magnetic field model that reproduces a global morphology and current sheets more effectively than the potential source-surface models currently in use. By adding a radial term to the analytic potential field equation, we reproduce the observed superradial expansion profile of polar plumes, the shape of helmet streamers, and the near-equatorial current sheet of the quiet, rising, or declining phase corona. We then applied a simple 1D solar wind to each magnetic field line in the model, and generated simulated data to represent results from a local plasma imager and a downward-looking surface imager onboard a hypothetical Solar Probe spacecraft, demonstrating that the surface structure is preserved over the bulk of the coronal hole in our analytic model.

2. Magnetic Model The corona’s characteristic shape is formed by the interplay of magnetic forces that tend to relax the corona to a nearly potential configuration, and plasma dynamic pressure that tends to stretch the corona into a nearly radial configuration. This force balance is usually achieved with a source-surface model: a potential field extrapolation that includes prescribed source terms within the Sun and also on a surface 1–3 solar radii away from the surface of the Sun. The model can include kinked field lines (e.g. Rowse and Roxburgh, 1981) or an outer source distribution that forces the field direction to be radial at the outer source surface (e.g. Hoeksema and Scherrer, 1986; Hoeksema, Wilcox, and Scherrer, 1982). Source-surface models have difficulty representing nuances of the field geometry, such as behavior near cusps and the formation of multiple current sheets at different altitudes in the corona. To overcome these limitations, we developed a new simplified morphological model, a “pseudocurrent” model of the coronal morphology. The pseudocurrent approximation uses a quasi-potential field model that represents the plasma dynamic pressure as a perturbative term in the usual potential field calculation. The model field is the vector sum of a potential field produced by a set of internal magnetic sources below the photosphere, and a radial wind pressure term. The field at the location r is given by:  r) = B(

 r − xj r |r |n S(r ),  3 + W   r − xj j

(1)

where xj is the location of the jth magnetic field source, W is a wind scaling coefficient, n is a wind scaling exponent, and S(r ) is a sign term that is 1 or −1 depending on the sign of the radial component of the potential field. The W r |r |n S (“wind”) term is an effective additional pressure that models the effect of the plasma

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dynamic pressure on the magnetic field direction: the wind term always bends the local field vector toward the nearest radial direction because of the S coefficient, and scales according to the selected n scaling law. The S term changes sign at locations where the potential field is horizontal, giving rise to a current sheet wherever the radial field changes sign. We obtained the best qualitative fit to the morphology of LASCO and EIT images from SOHO with the parameters n = 1 and W = 1.0D, where D is the total dipole moment of all the magnetic source terms. Setting n = 1 makes the wind perturbation grow linearly with altitude, so that regions near the solar surface are nearly potential, but in the middle- and upper corona the field becomes increasingly radial and current sheets between these domains grow stronger. The wind term is divergence free only in the n = −2 case, so the computed field is not, prima facie, a valid physical magnetic field configuration. The divergent nature of the model field is acceptable provided that (1) it is used only to find the  and (2) one always integrates field lines outward direction, and not magnitude, of B; from the Sun to find the coronal morphology. Figure 2 illustrates the field line structure of a global pseudocurrent model. True field lines (shown in bold) originate at the surface of the Sun and may be integrated along loops or outward to the heliosphere. The true lines are shaped by false field lines (shown fine) that arise from the divergent wind term. By construction, true field lines maintain the same morphology as a divergence-free field that is shaped by an additional non-magnetic pressure. Just as it’s better to pet a dog along the fur than against the fur, it is better to integrate pseudocurrent field lines in the outward direction: inward integration is unstable over long distances because of the nonzero divergence. The advantages of the pseudocurrent model over a source-surface model are the lack of an artificial seam at a source-surface altitude (smooth mapping from the

Figure 2. Modeled field divergence replaces gas pressure in the “pseudocurrent” model we implemented to describe coronal morphology. True field lines (bold) originate at sources on the solar surface and trace the coronal morphology. The effect of plasma pressure on the force balance is simulated by magnetic pressure from false field lines (faint) from the divergent wind-pressure term in Equation (1).

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surface to the heliosphere); and the ability to represent multiple streamer-belt cusps and current sheets at varying altitudes. We assembled a model solar magnetic field from a balanced collection of monopole sources under the photosphere: a global finite dipole located near the center of the Sun, several finite active-region dipoles located at about 0.9 Rs , and several hundred small monopoles sprinkled over the polar regions at 0.98 Rs . The small monopoles were meant to represent small network flux concentrations and were formed into clusters to represent network structures such as form the bases of polar plumes. The global dipole and the polar flux distribution each had a dipole moment near unity, with five active region dipoles varying in moment from 0.05 to 0.1. Individual field lines are found by shooting outward through the analytically defined field. The advantage of this model is that it preserves superradial expansion of the open field, and wind-related current sheets such as the top of helmet streamers, while being computationally efficient.

3. Preservation of Horizontal Structure We demonstrated the preservation of horizontal structure from the surface of the Sun out to 30 Rs , above the main acceleration region for the high speed solar wind. We rendered individual extrapolated field lines, to test how much the horizontal structures at the surface were distorted by the polar superradial expansion. The field lines were extrapolated using a pseudocurrent model with n = 1 and W = D, as described earlier. We used an arbitrary but recognizable pattern of sources to model the approximately unipolar collection of sources in the polar coronal holes; that pattern is shown in Figure 3. The polar flux concentrations were placed in a squared-off, random arrangement in the transverse Mercator projection, then their coordinates were deprojected onto the pole of the model Sun in three dimensions. The point-like sources were placed 0.02 Rs under the surface, and each had a source strength of 5 × 10−3 D/Rs (where D is the total dipole moment of the model Sun). On the actual Sun, magnetic flux concentrations are not placed in squaredoff arrangements that are so easily recognizable. But the polar coronal holes are dotted with recognizable pattern of magnetic flux concentrations that lie within the supergranular network. The pattern changes on about the supergranulation time scale but individual patterns of flux are different enough to identify particular supergranules. In the first test, we calculated the shape of the field lines anchored immediately above each small source, out to 30 solar radii from the origin. As in coronal observations of the real Sun (DeForest, Lamy, and Llebaria, 2001), we found that all open structures are essentially radial above 10 Rs . Figure 4 shows an orthographic perspective view of the extrapolation; it has been limited to the innermost 10 Rs for

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Figure 3. Small flux concentrations are placed in an arbitrary but recognizable pattern near the pole of the model Sun, as seen in this orthographic projection from a B angle of 80◦ . The grid lines show 10◦ increments in longitude and latitude.

Figure 4. Three-dimensional rendering of field lines from selected regions over the whole surface of the model Sun. Field lines extend from 1.0 to 12 Rs . Darker lines are closer to the viewer. Only lines that extend from immediately above a flux concentration are shown, to illustrate the correspondence between horizontal structures at the surface and in the outer corona. The surface pattern in Figure 3 is preserved within the coronal hole even at 10 Rs , though features at the boundaries of the coronal hole are severely distorted (as seen at lower left).

clarity. Within the core of the polar coronal hole, the expansion is nearly uniform, and local structure is preserved well enough that the eye can clearly pick out the correspondence between features in Figures 3 and 4; however, near the edge of the coronal hole, the expansion is nonuniform and local structure is severely distorted.

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An example of the distortion can be seen at the lower left of Figure 4, where the square array of a local cluster of flux concentrations is distorted beyond recognition.

4. Coronal Rendering In the second test, we generated a model of the coronal density structure and used it to render both the appearance and the local structure of the corona. The reason is that the magnetic field itself is not visible; but the density structures within it are. To generate a morphologically realistic model corona, we filled the locus around each individual field line with a modified hydrostatic, single-fluid, isothermal plasma model:  w Di (r ) = Di (1) e−r/rscale,i + 2 , (2) r where Di is density as a function of radius along the ith field line, r is measured in units of solar radii, rscale,i is the scale height of the plasma in the ith field line, and w is a wind weighting coefficient (0.05 in our simulation). The basal densities Di (1) were generated with a scaling law similar to the observed scaling relation between the magnetic field and injected power (Schrijver et al., 2004). Following the observation by Wang (1994) that basal power deposition and density are related in open magnetic structures, we gave all field lines a base plasma density proportional to the computed magnetic field strength on the field line at the photosphere, multiplied by a random factor that varied between 0.75 and 1.5 to account for coronal variability. Open field line densities were attenuated by a factor of 3. Open field lines were given a constant√scale height of 0.26 Rs , and closed field lines were given a scale height of 0.26 + α B where B is the basal magnetic field strength and α is a conversion coefficient with value 5 × 10−3 Rs G −0.5 , to simulate higher temperatures in active regions. The corona was rendered by shooting a field line from each square 0.5◦ of photosphere (5 × 104 in all) and calculating the density along the length of the line, then transforming the 3D line coordinates for each point along the field line to screen coordinates as seen from a particular location in space. The coordinate transformation was accomplished via the PDL::Transform package in the Perl data language (Glazebrook et al., 2003), using transformation from 3D Cartesian coordinates to observer-centric spherical coordinates, followed by gnomonic projection to simulate a simple camera. A light spot was rendered on the output plane at each projected location, proportional to the density of the plasma on the field line and to r −2 , to approximate the effect of Thomson scattering of photospheric light. The spots were scaled in size according to a local linearization of the screen projection transform (DeForest, 2004). Figure 5 shows a view of the model K-corona in Thomson-scattered white light, as seen from 30◦ North heliospheric latitude at a distance of 20 Rs . Other types of

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Figure 5. The solar corona as rendered in three dimensions with a pseudocurrent model and a simple collection of 1D plasma models. Open field lines support Parker-like flow and closed field lines support hydrostatic plasma. Direct emission from the photosphere is not rendered, giving a physically impossible view of the Thomson-scattered corona against the disk. The viewpoint is from a hypothetical Solar Probe-like spacecraft at 53◦ north latitude and an altitude of 20 Rs .

light (such as direct photospheric emission) are not rendered. The disk of the Sun is shown as a circle. The streamer belt and familiar superradially expanding polar coronal holes are clearly visible, showing that the model reproduces the familiar observational structure of the corona. A 2D radial filter has been applied, to partially compensate for the gradient in brightness. Figure 5 is particularly interesting because the wider field of view highlights perspective effects that might be seen by a wide-field imager. In particular, the polar plumes in the top half of the figure appear to bend slightly together with altitude; this is a perspective effect due to the wide angle of the field of view through the simulated camera. Likewise, the streamer at lower right is especially bright both because of its intrinsic brightness and because of the narrow angle between the streamer itself and the line of sight through it. The latter perspective effect gives particularly long lines of sight through the streamer, enhancing its brightness.

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Figure 6. A horizontal ‘cut’ through the modeled solar corona in Figure 5, showing the preservation of horizontal density structure at 20 Rs . The noisy character of the image is due to randomization of the individual 1D field models used for each field line. This sort of cut-plane might be directly imaged by a spacecraft such as the planned Solar Probe, using tomographic techniques and Thomson-scattered light collected during a coronal fly through.

Tomographic “cuts” through the model corona were produced by finding the intersection of each field line with a cut-plane and calculating the plasma density at the intersection point of the field line with the plane of interest. We calculated a particular tomographic cut corresponding to the solar-horizontal shell at the observer position in Figure 5, showing the small-scale density structure in the vicinity of the spacecraft (Figure 6). The associated network-scale surface pattern remains clearly defined, demonstrating that, within the polar coronal hole, surface structures are recognizable and can be used to determine the mapping MF −1 between the spacecraft location and associated features in the corona. Despite strong inhomogeneities in the magnetic field at the surface of the Sun, the surface density pattern is preserved at scales smaller than a supergranule, suggesting that structures as small as the transient local maxima observed in the bases of polar plumes (DeForest et al., 1997) may be preserved in the outer corona.

5. Discussion The result that small structures are preserved near the center, but not the edges, of open field regions is not dependent on the pseudocurrent model used here: many field expansion observations and models reproduce the approximately linear expansion required to preserve structure. The small-scale behavior of MF in nonbifurcated regions (i.e. where MF is continuous) can be understood in terms of

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its Taylor expansion about the center of a small structure near the solar surface.  on the solar surface and considering the mapped locations Writing x = x0 + x MF(x)  in the heliosphere, the expansion (in coordinate index notation) is: ∂MFi MF(x)  i ≈ MF(x0 )i + x j + · · · , (3) ∂x j where j is a summation index. Because of the relationship between field line density and field strength, we can understand the terms in a simply connected region in terms of field strength. The first term simply relates how the center of the structure is translated in latitude and longitude between the two altitudes related by MF. The first-order coefficient Ji j = ∂MFi /∂ x j relates the horizontal shape of the structure at the surface to its shape in the heliosphere in first order. It can be decomposed more usefully, according to the singular value theorem:  σ 0  T r , (4) J = η R1 S R2 = η R r 0 σ −1 where J is (still) the Jacobian of MF, η is an expansion scalar, R1 , R2 , R, and r are rotation matrices, S is the diagonal matrix of singular values, normalized to a determinant of unity, and σ is an anisotropy coefficient. Then J may be represented by four scalar parameters with immediate descriptive meaning: η, the overall expansion factor of MF; θ R , the rotation angle (associated with R); σ 2 , the ratio of maximum to minimum expansion factor across direction; and φr , the direction of maximum expansion (associated with r ). For small structures to be preserved, σ 2 should be near unity (making the transformation approximately conformal); for the large-scale pattern to be evident, θ R should also be small (making the transformation approximately irrotational) at moderate altitudes because there is no observed overall helical structure to the coronal holes until the Parker spiral structure becomes important at distances of several tens of solar radii. The descriptive terms may also be understood in terms of the magnetic field’s behavior. From the construction of field lines, η is just the ratio of radial field strengths at the photosphere and the outer plane, while the rotation angle θ R is the normalized line-integral of the current helicity along the field line:

Bphot 2  ; and θ R = ds B · ( × B)/B , (5) η(x)  = Bhel. while the anisotropy term σ 2 measures nonuniformity of expansion. Given the interpretation of the Jacobian matrix J , it is easy to see why small structure is preserved in coronal holes: because the plasma β parameter in the low corona is very small, the magnetic field is very uniform above a thin canopy layer (as seen by the “blunt pencil” shaped ends of polar plumes, e.g. DeForest et al. (1997), so that MF has few cusps (and MF −1 has few bifurcations) above bright features seen near the base of the coronal hole, so that Taylor expansion of the mapping is appropriate at nearly all locations. Because of the simple topology, there can

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be little field-aligned current along the open field: any helical perturbations are free to propagate outward at the Alfvén speed, rapidly returning the field topology to a near-helicity-free state near the Sun. This places a strong constraint on the rotation angle θ R , which must be smaller than l/tevo va , the ratio of the distance from the top of the corona to the photosphere to the Alfvén propagation length scale of photospheric evolution. Taking va = 106 m/s, l = 1010 m, and tevo = 48h (two supergranulation times) yields θ R < 3◦ . Finally, the coronal holes are measured to have nearly isotropic expansion near their centers (Munro and Jackson, 1977; DeForest, Lamy, and Llebaria, 2001; Fisher and Guhathakurta, 1995), limiting the size of the anisotropy term σ 2 to within a few percent of unity. It is also easy to see why the preservation of structure fails near the edges of the polar coronal holes. While the edge of each coronal hole undergoes a large expansion in the latitude direction (perpendicular to the hole boundary) to fill the area left at high altitudes by closed flux systems in the streamer belt, there is no room to expand in the longitudinal direction (parallel to the hole boundary). The outward curvature of the outermost field lines supports a gradient in magnetic pressure away from the center of the hole, permitting greater expansion in that direction than in the circumferential direction along the boundary. Over mostly-closed regions, the mapping MF −1 is neither continuous nor stable as exchange reconnection swaps field line locations; this breaks the assumptions inherent in Equation (3) and prevents preservation of structure through the mostlyclosed low-latitude corona into the equatorial region of the heliosphere.

6. Conclusions We have demonstrated that the shape of small horizontal magnetic structures should be well preserved from the surface of the Sun through the outer corona, under the superradial expansion of the open magnetic field. If such small structures can be detected via tomography or in situ measurement, they may be used as fiducial ‘landmarks’ to pin down the location on the solar surface associated with a particular field line as measured in the inner heliosphere. We anticipate easy identification of the particular network flux concentration associated with the field line at the spacecraft’s location. While the propagation time of wind structures to 30 solar radii is comparable to the supergranulation time, it is short compared to the lifetime of polar plumes, the strongest density structures in the coronal holes. Should evolution effects be large enough to make feature recognition difficult, one may compare surface images from an earlier time to in situ data from a later time, to account for the solar wind propagation time to the spacecraft altitude. Near the edges of the coronal hole, we find that individual small structures are not preserved; in this thin region, the expansion ratio changes rapidly with effective magnetic latitude, causing a distortion of individual features, beyond recognition.

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Likewise, structures are also not preserved in the portion of the heliosphere that is connected to the streamer belt and predominantly closed corona. The preservation of structure may readily be understood in qualitative terms, by considering both the conditions under which the geometric mapping between the photosphere and the heliosphere is continuous and differentiable, and the physical meaning of that mapping’s structure in regions where it is continuous and differentiable. We have also used pseudocurrent modeling to demonstrate the preservation. Pseudocurrent modeling is a computationally inexpensive way to model the global coronal magnetic field, with some advantages over source-surface models in the presence of current sheets. Pseudocurrent models work by replacing conventional plasma pressure with a parametrized pseudo-magnetic pressure, at the expense of the strict divergence-free condition normally imposed on the magnetic field. The modeled magnetic field has physical meaning when used to shape individual field lines that are launched from known locations near the Sun, outward toward the heliosphere; and when used to determine the magnetic field direction (but not strength) at an arbitrary location. Pseudocurrent models can represent current sheets and related local structures in the corona that are difficult to reproduce with sourcesurface potential extrapolation.

Acknowledgements Thanks to NASA’s Solar Probe Science Definition Team for suggesting this line of inquiry, and to Karel Schrijver for illuminating discussion and feedback. Our model was coded in PDL, a free data analysis language (http://pdl.perl.org). This work was funded internally by Southwest Research Institute.

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