On the Dependency between the Peak Velocity of High-speed Solar Wind Streams near Earth and the Area of Their Solar Source Coronal Holes
Stefan J. Hofmeister, Astrid M. Veronig, Stefaan Poedts, Evangelia Samara, Jasmina Magdalenic
DDraft version June 2, 2020
Typeset using L A TEX default style in AASTeX62
On the dependency between the peak velocity of high-speed solar wind streams near Earth and the area of their solarsource coronal holes
Stefan J. Hofmeister, Astrid M. Veronig,
1, 2
Stefaan Poedts,
3, 4
Evangelia Samara,
3, 5 andJasmina Magdalenic Institute of Physics, University of Graz, Austria Kanzelh¨ohe Observatory for Solar and Environmental Research, University of Graz, Austria Centre for mathematical Plasma Astrophysics (CmPA), Department of Mathematics, KU Leuven, Belgium Institute of Physics, University of Maria Curie-Sk(cid:32)lodowska, Lublin, Poland Solar-Terrestrial Centre of Excellence - SIDC, Royal Observatory of Belgium, Brussels, Belgium
ABSTRACTThe relationship between the peak velocities of high-speed solar wind streams near Earth and theareas of their solar source regions, i.e., coronal holes, is known since the 1970s, but still physicallynot well understood. We perform 3D MHD simulations using the EUHFORIA code to show thatthis empirical relationship forms during the propagation phase of high-speed streams from the Sunto Earth. For this purpose, we neglect the acceleration phase of high-speed streams, and project theareas of coronal holes to a sphere at 0 . . − , similar to the observations. These findings implythat the empirical relationship between the coronal hole areas and high-speed stream peak velocitiesdoes not describe the acceleration phase of high-speed streams, but is a result of the high-speed streampropagation from the Sun to Earth. INTRODUCTIONThere are two well-known empirical relationships between the decisive properties of high-speed solar wind streams(HSS) and their solar source regions, i.e., coronal holes on the Sun: the relationship of the peak velocity of HSSsmeasured near Earth to (a) the area of coronal holes (Nolte et al. 1976), and to (b) the inverse flux tube expansionfactor of coronal holes (Wang, & Sheeley 1990). While the relationship of the HSS peak velocity to the inverse fluxtube expansion factor is usually interpreted in terms of the height at which energy is deposited to accelerate the solarplasma to form HSSs (Wang & Sheeley 1991), the relationship to the area of coronal holes is still physically not wellunderstood.The linear relationship to the areas of coronal holes has first been reported by Nolte et al. (1976, Fig. 1 a), andthereafter confirmed by numerous studies (Robbins et al. 2006; Vrˇsnak et al. 2007; Abramenko et al. 2009; Karachik, &Pevtsov 2011; Rotter et al. 2012, 2015; Tokumaru et al. 2017; Hofmeister et al. 2018; Heinemann et al. 2018, 2020). Thecorresponding Pearson’s correlation coefficients range from 0 .
40 to 0 .
80, depending on the composition of the datasets.The results of Robbins et al. (2006), who divided the Sun into three latitudinal regions of | ϕ | < ◦ , 30 ◦ ≤ | ϕ | ≤ ◦ , | ϕ | > ◦ , already suggested that the slope of this linear relationship depends on the latitude of the source coronal hole.Hofmeister et al. (2018) have investigated the dependence of this linear relationship on the position of the satellitetaking the in-situ measurements relative to the coronal hole center, i.e., on their latitudinal separation angle. Theauthors found that the slope of the relationship between coronal hole areas and HSS peak velocities depends linearlyon the latitudinal separation angle (Fig. 1 b). The slope is steepest at separation angles close to 0 ◦ , i.e., above thecenter of the coronal hole, continuously declines with increasing separation angle, and basically turns to zero at aseparation angle of 60 ◦ . Finally, Garton et al. (2018) have shown that the peak velocity of HSSs are similarly wellcorrelated to the longitudinal extension of coronal holes than to their areas, and that the peak velocities saturate at710 km s − for coronal holes with a longitudinal extension (cid:38) ◦ (Fig. 1 c). a r X i v : . [ a s t r o - ph . S R ] J u l Hofmeister et al.
Figure 1. a) Peak velocities of three recurrent HSSs, which were observed in total 15 times by the Helios satellites at 1 AU in1973, versus the areas of their source coronal holes (data from Nolte et al. 1976). b) Peak velocities of 115 HSS observations byACE, StereoA and B during 2010 to 2017, versus the areas of their source coronal holes. The absolute value of the latitudinalseparation angle between the satellites taking the HSS measurements and the center of mass of their source coronal holes arecolour-coded (data from Hofmeister et al. 2018). c) Peak velocities of 47 HSS observations by ACE in 2016 and 2017 versus thelongitudinal extensions of their source coronal holes; outliers are marked in red (data from Garton et al. 2018).
In order to better understand the relationship between the peak velocities of HSSs near Earth and the areas of theirsource coronal holes, we employ a simplified magnetohydrodynamic (MHD) simulation setup for a parameter study.By projecting the areas of coronal holes to a sphere at 0 . n the dependency between HSS velocities and CH areas Figure 2.
Simulation of a HSS in the inner heliosphere, derived by the EUHFORIA MHD code. a) and b) Snapshots of theradial velocity distribution in the solar equatorial and meridional plane. c) Snapshot of the properties of the HSS along a radialdirection in the equatorial plane: radial velocity component (black), density (orange), temperature (red), and longitudinal angleof the flow velocity to the radial direction (grey). The grey dashed horizontal line corresponds to a longitudinal angle of 0 ◦ , i.e.,a perfect radial flow. The black vertical line marks the position of the stream interface between the HSS and the preceding slowsolar wind stream. 2. SETUP OF THE SIMULATIONSThe European Heliospheric Forecasting Information Asset (EUHFORIA, Pomoell, & Poedts 2018) was developedto model and predict the solar wind and the evolution of coronal mass ejections (CMEs) in the inner heliosphere forspace weather forecasts. It consists of a simple, data-driven Wang-Sheeley-Arge (WSA) and Schatten-Current-Sheet(SCS) model of the solar corona which provides solar wind parameters at 21 . sun , and a 3D time-dependent MHDmodel of the inner heliosphere to propagate the solar wind from 21 . sun to 2 AU. CMEs can be injected in the modelruns at 21 . sun using a cone or a spheromak model (Verbeke et al. 2019). For furthers details, we refer to Pomoell,& Poedts (2018).In order to be independent of the not-well known acceleration phase of the solar wind from 1 to 21 . sun , we assumea circular coronal hole at the base of the solar corona, skip the coronal model, and project the area of the coronal holeradially to a sphere at 21 . sun , resulting in the foot-print of the corresponding HSS at this sphere. We then set thevelocity, temperature, and density of the HSS at this spherical shell to constant, homogeneous values, and propagate Hofmeister et al. the HSS through the inner heliosphere using the heliospheric MHD model of EUHFORIA. Based on this idealizedsetup, we perform a parametric study by varying the size of the coronal hole and its latitudinal position while keepingall other parameters constant, in order to study their effect on the in-situ measured peak velocities at 1 AU in thesolar equatorial plane.At the spherical shell with radius 21 . sun , which corresponds to the inner boundary conditions of our simulationswith EUHFORIA, we assume a bi-modal solar wind, and set the properties of the HSS and the surrounding slow solarwind for all simulations in the following way. For the HSSs, we set a uniform constant radial velocity of 650 km s − ,density of 150 cm − , and gas pressure of 3 . − , density of500 cm − , and gas pressure of 3 . − anddensities of 1 . − for HSS plasma at 1 AU, and 390 km s − and densities of 5 cm − for slow solar wind plasma at1 AU. Further, we assume a lateral pressure balance at 21 . sun between the HSS and ambient slow solar wind plasma.Since the plasma β , i.e., the ratio of thermal to magnetic pressure, is (cid:28)
1, i.e., the magnetic pressure dominates atthese heights, we assume a uniform absolute mean magnetic flux density of 217 nT both in the HSS and ambient slowsolar wind region, which corresponds to a mean magnetic flux density of 1 G at the solar surface. The magnetic field isassumed to be purely radial at the inner boundary, which also results in a radial direction of propagation of the HSSs.We added a heliospheric current sheet with a polarity inversion line specified by ϕ = − ◦ · cos λ , where λ and ϕ arethe solar longitude and latitude, respectively, and set corresponding opposite magnetic field polarities at the northernand southern hemisphere in the slow solar wind.We then propagate these inner boundary conditions in time and space using the heliospheric part of the 3D MHDEUHFORIA code (Fig. 2 a and b). We apply a fine spherical grid with a spatial resolution of 0 .
006 AU in the radialdirection from 0 . . ◦ in the latitudinal and longitudinal direction. We relax the initial conditions over17 days, which yields a steady, rotating solar wind solution. The solar wind properties are then extracted from thesimulations using virtual in-situ spacecraft placed into the heliocentric equatorial plane at 0 . . v r of theHSSs at 1 AU in the solar equatorial plane, and we derive the radial velocity v SI of the stream interface between theHSS and preceding slow solar wind plasma by its radial displacement with time.Since the HSSs in our simulations propagate radially away from the Sun, it follows that the virtual spacecraft is inthe latitudinal center of the HSS whenever the center of mass of the coronal hole is located in the solar equatorialplane. Shifting the center of mass of the coronal hole to higher solar latitudes also shifts the direction of propagationof the HSS to higher heliospheric latitudes; consequently, the virtual spacecraft located in the solar equatorial planewill take its measurements farther in the flanks of the HSS. Furthermore, since the plasma in HSSs propagates aboutradially away from the rotating Sun, the radial velocity component also approximates well the absolute value of theplasma velocity. The angle of propagation of the plasma to the radial direction does usually not exceed 10 ◦ . Finally,due to the very high magnetic Reynolds numbers in the solar wind, the magnetic field is frozen in the plasma, with theconsequence that HSS plasma and the preceding slow solar wind plasma cannot mix. Instead, at the stream interface,a strong gradient in the solar wind properties is observable. The preceding slow solar wind plasma gets piled up bythe faster HSS plasma, whereas the HSS plasma itself thermalizes and accumulates in the back of the stream interface(Fig. 2 c). With increasing distance to the Sun, this stream interaction region develops into a forward and reverseshock pair. Since, at Earth distance, the magnetic field has an inclination of 40 to 60 ◦ to the radial direction due tothe rotation of the Sun, the slower preceding solar wind plasma is deflected by the faster impinging HSS plasma atthe stream interface to the west, i.e., in direction of the solar rotation, whereas the radially propagating HSS plasmain its back is deflected to the east, i.e., along the direction of the Parker Spiral. Following Borovsky & Denton (2010),we define the zero-crossing of the longitudinal deflection angle in this shearing region as the position of the streaminterface. RESULTSSince we want to compare our results with the empirical relationships derived from observations, we set up the studyin a similar way. To reflect the evolution of an individual coronal hole and the resulting HSSs, we set a coronal holeto a given latitude and vary its size A CH in 16 non-uniform steps from 1 · to 12 · km . To reflect the varietyof latitudes at which coronal holes appear, we vary the latitudes ϕ of the center of masses of the coronal holes from 0to 12 ◦ at steps of 3 ◦ . n the dependency between HSS velocities and CH areas Figure 3.
Peak solar wind velocities as measured by the virtual spacecraft versus the area of the coronal holes in the simulations.The virtual spacecraft was set to 1 AU in the solar equatorial plane. The latitudinal position of the coronal holes on the Sun iscolour-coded. The velocity of the ambient slow solar wind at 1 AU is 390 km / s. The resulting peak solar wind velocities measured in the solar equatorial plane by our virtual spacecraft versus thecoronal hole areas are plotted in Figure 3, the latitudes of the coronal holes are colour-coded. These results matchqualitatively the observations as described in Section 1, and naturally show less spread than the observations due tothe idealized setup. For each given coronal hole latitude, the HSS peak velocity increases linearly with the area of thecoronal hole up to a given size, and then saturates at a constant value of 730 km s − . Thereby, the slopes of the linearrelationships depend on the latitude of the coronal holes. For coronal holes located at the solar equator at ϕ = 0 ◦ , thepeak velocities rise rapidly with increasing area up to the maximum velocity of 730 km s − , whereas the slopes are lesssteep for coronal holes located at higher latitudes.Since for all simulations we used the same properties for the HSSs at the inner boundary and only varied the latitudesand sizes of the source coronal holes, only the propagational evolution and the latitudinal direction of the HSSs canaffect the peak velocities as measured by the virtual spacecraft at 1 AU in the heliospheric equatorial plane. Thesefindings strongly suggest that the relationship between the HSS peak velocities and coronal hole areas is a direct resultof the evolution of HSSs during their propagation in the inner heliosphere and the location of the spacecraft withinthe HSS. In the following sections, we further scrutinize the fundamental parameters of this relationship.3.1. Peak velocities of the HSSs in their latitudinal center
First, we investigate the effect of the longitudinal width ∆ λ CH and latitudinal width ∆ ϕ CH of the coronal hole onthe peak velocity of the HSSs in their center at 1 AU. To this aim, in this section, we set the center of masses ofthe coronal holes to the solar equator, and change its shape to rectangular. Since our virtual spacecraft is locatedin the solar equatorial plane and therefore in the latitudinal center of the HSS, it measures the maximum velocity ofthe overall stream. For the first set of simulations, we fix the latitudinal width at a large width of 30 ◦ , and vary thelongitudinal width from 2 to 30 ◦ (Fig. 4 a, red). For the second set of simulations, we fix the longitudinal width at30 ◦ , and vary the latitudinal width from 2 to 30 ◦ (Fig. 4 a, blue).We find that increasing the longitudinal width of the coronal hole steadily increases the maximum velocity of theHSS over the whole range of the longitudinal widths, whereby the velocity increase with increasing longitudinal widthslowly flattens out. In contrast, increasing the latitudinal width of the coronal hole steeply increases the maximumvelocity of the HSS up to a width of about 10 ◦ , and then sharply saturates. Therefore, we may assume that up tolatitudinal widths of 10 ◦ , erosion effects at the flanks of the HSSs affect the speeds in the center of the HSS. At largerwidth > ◦ , the virtual spacecraft sees solar wind plasma which is undisturbed by its flanks.Therefore, both the longitudinal and latitudinal width of the coronal hole affect the maximum velocity of the overallHSS. Increasing the longitudinal width increases the maximum speeds of HSSs in their center at 1 AU up to large widthof > ◦ , whereas the effect of increasing the latitudinal width on the peak velocities in the center of HSSs saturatesat a latitudinal width of 10 ◦ . Hofmeister et al.
Figure 4. a) Peak velocities of the HSSs as measured by the virtual spacecraft in the solar equatorial plane at 1 AU versus thewidth of rectangular coronal holes. In the first set (red), the latitudinal width of the rectangular coronal hole was fixed at 30 ◦ while the longitudinal width was varied. In the second set (blue), the longitudinal width was fixed at 30 ◦ while the latitudinalwidth was varied. b) Latitudinal velocity profile of HSSs at a distance of 1 AU to the Sun, for HSSs originating from circularcoronal holes with areas of 1 (blue), 2 (red), 4 (green), and 8 · km (yellow). Latitudinal velocity profile of HSSs
Next, we investigate the latitudinal velocity profile of HSSs at 1 AU for coronal holes with areas of 1, 2, 4, and8 · km , using again circular shaped coronal holes (Fig. 4 b). For the large coronal hole of 8 · km , we findthat the latitudinal velocity profile shows a broad plateau at 730 km s − around the center of the HSS, and that thevelocities fall off steeply at the flanks to slow solar wind velocities. This plateau reflects the maximum velocity of theHSSs: further increasing the latitudinal width of the coronal hole, while keeping the longitudinal width constant, doesnot affect the maximum velocity of the HSS in its center, but only increases the width of the plateau. By decreasingthe size of the coronal hole, we find that the plateau vanishes, and the maximum velocities in the center decrease. Thegradual disappearance of the plateau in combination with the steep, but not abrupt decrease of the velocities acrossthe flanks is a sign of erosion effects in the flanks of the HSSs.Note that even for the small coronal hole of 1 · km , the maximum velocity of the HSS, measured in its center,is still 615 km s − , i.e., changing the size of a coronal hole by a factor of 8 only reduces the maximum velocity ofits associated HSS in its center at 1 AU by 115 km s − . In contrast, changing the position of the satellite from thecenter to the flanks of the HSS strongly affects the peak velocities measured. By moving the latitudinal location ofthe virtual spacecraft by only 10 ◦ from the end of the plateau into the direction of the ambient slow solar wind, thepeak velocities measured by the satellite change from large peak velocities close to the center of the HSS to small peakvelocities close to that of the ambient slow solar wind. Therefore, the peak velocities measured are mostly affected n the dependency between HSS velocities and CH areas Figure 5.
Velocities in the latitudinal centers of HSSs, belonging to coronal holes with areas of 1 (blue), 2 (red), 4 (green), and8 · km (yellow). a) Snapshots of the radial velocity profile, taken at 1 day (dotted), 2 .
25 days (dashed), and 3 . v SI derived by itsradial displacement with time versus the distance of the stream interface to the Sun. c) Peak velocity v p of the HSS versus theradial distance of the location of the peak velocity to the Sun. by the relative location of the satellite within the HSSs. This relative location, again, is dependent on the latitudinalseparation angle of the satellite to the center of the coronal hole, and on the area of the coronal hole.3.3. Radial evolution of HSSs
In the previous sections, we studied how the HSS peak velocities measured in the solar equatorial plane at 1 AUdepend on the area and latitude of the source coronal hole. Here, we investigate where these relationships form. InFigure 5 a, we show three snapshots of the velocity profile in the center of HSSs along a radial direction, taken 1 day,2 .
25 days, and 3 . · km , stays almost constant. Hofmeister et al.
Figure 6.
Velocities in the flank of the HSS belonging to the coronal hole with an area of 8 · km . The velocities weredetermined at latitudinal displacements of 6 ◦ (yellow), 9 ◦ (orange), 12 ◦ (brown), and 15 ◦ (dark brown) to its latitudinal center.a) Snapshots of the radial velocity profiles in the the flank, taken at 1 day (dotted), 2 .
25 days (dashed), and 3 . v p of the HSS in the flank versusthe radial distance of the location of the peak velocity to the Sun. We note that the velocity of propagation of the overall HSS is not given by its peak velocity, but by the velocity ofthe stream interface between the HSS and the preceding slow solar wind plasma, which is shown in Figure 5 b. Sincethe peak velocities of HSSs are faster than the velocity of the stream interface, the fastest plasma regions in the centerof HSSs ultimately collide and are decelerated by the stream interaction region. Then, the new peak velocity is givenby the fastest plasma regions which did not interact with the stream interaction region yet. For very extended HSS,which show an extended plateau in the radial velocity profile, this results in only the leading region of the plateaubecoming decelerated. Consequently, the position of the peak velocity in the HSS shifts from the front further toits back, while the peak velocity itself stays roughly constant (Fig. 5 a and c, yellow lines). For more usual HSSsnot having an extended plateau, this results in a continuous impingement of the fastest plasma cells into the streaminteraction region, and consequently a decrease of the peak velocity with increasing distance to the Sun (Fig. 5 c, red,blue, and green lines).Finally, in Figure 6, we show the radial evolution of the peak velocities in the flanks of the HSS that result fromthe large coronal hole of A CH = 8 · km . The peak velocities are derived at latitudinal displacements of 6 ◦ , 9 ◦ ,12 ◦ , and 15 ◦ to the center of the HSS. Besides the decrease of peak velocities with radial distance, it is apparentthat already very close to the Sun the peak velocities in the flanks of the HSS are much smaller. This is in particularnoteworthy since we have set the initial velocity of the HSS homogeneously to 650 km s − . Therefore, the initial chosenvelocity step function at the boundary between the HSS and the ambient slow solar wind erodes very quickly close tothe Sun, forming the latitudinal velocity profile of the HSS. DISCUSSION AND CONCLUSIONS n the dependency between HSS velocities and CH areas . sun defining the cross-sections of the associated HSSs, set the plasma and magnetic properties ofall resulting HSSs in the same way, and propagated these streams to 1 AU using EUHFORIA MHD simulations. Wefound that: • the peak velocities of the HSSs, measured by the virtual spacecraft in the solar equatorial plane at 1 AU,scale linearly with the areas of the source coronal holes for each of the presumed coronal hole latitudes ϕ =0 ◦ , ◦ , ◦ , ◦ , ◦ , • the slope of this linear relationship decreases with increasing latitude of the source coronal hole, • the peak velocities saturate at a constant value of about 730 km s − .The empirical relationship between the peak velocities of HSSs and the areas of their source coronal holes can thereforebe reproduced by our simulations. Since we have neglected the acceleration phase of HSSs and only modelled theirpropagation from the Sun to Earth, it follows that this empirical relationship describes the propagational evolution ofHSSs on their way from the Sun to Earth. Consequently, this empirical relationship is not related to the accelerationof HSSs. By looking for the cause of this dependency, we found that: • the maximum velocities of the overall HSSs, determined in their center, depend on the latitudinal width of thesource coronal hole up to a width of 10 ◦ due to erosion effects from the flanks which appear very close to theSun, • the maximum velocities further depend on the longitudinal width of the coronal hole, which can be explainedby the dispersion of the streams during their propagation away from the Sun, and ”absorption” of the fastestplasma regions by the stream interaction region.Despite the simplicity of our simulation setup, we can reproduce and explain basic observational relationships betweenthe area and latitude of the source coronal holes on the Sun and the HSS peak velocities in the solar equatorial planeat 1 AU. However, we also note that our study lacks a few details. First, for getting the cross-section of the HSS atthe inner boundary at 21 . sun by simple projection of the coronal hole area, we have assumed a constant expansionfactor of 1. However, in general, each coronal hole could have its own expansion factor dependent on its magnetic fluxdensity and the global magnetic field configuration. Second, we have assumed a radial magnetic field at 21 . sun ,resulting in a radially propagating HSS. In fact, the magnetic field at this height is inclined due to the interaction withthe heliospheric current sheet. This results in the HSSs being bent towards the solar equatorial plane, which affectsthe relative position of the HSS with respect to the measuring satellite. This also means that the very low coronalhole latitudes < ◦ Hofmeister et al.