Testing the Flux Expansion Factor – Solar Wind Speed Relation with Solar Orbiter data

(Solar Orbiter Nugget #44 by J-B. Dakeyo^1,2 , A.P. Rouillard², V. Réville², P. Démoulin^1,3, M. Maksimovic¹, A. Chapiron², R.F. Pinto², P. Louarn²)

I. Introduction

The presence of “open” magnetic field lines in the solar atmosphere is associated with magnetic flux tubes evolving into solar wind streams subject to the “frozen-in” condition of plasmas.
A series of observational studies, well-discussed in the solar and space scientific communities [1,2,3], show that magnetic flux tubes diverge faster than the expected spherical expansion outwards from the Sun, especially in the high corona, in a region called "super-expansion", as shown in Figure 1. The expansion factor f, sketched in Figure 1, is commonly used to quantify the divergence of flux tubes. The typical value of the expansion factor varies with the type of coronal structures (streamers, pseudo-streamers, corridors, coronal holes) and can affect the final wind speed [2, 3]. Inferred global trends, combining coronal magnetic field reconstruction calculations (PFSS algorithm), and observations at 1 au show an anti-correlation between wind bulk speed v and expansion factor f. As shown in Figure 1, this can be explained both by the conservation of mass flux, causing the solar wind speed at 1 au to decrease for larger flux tube expansions, and by the fact that the heating and acceleration by wave interactions are expected to be stronger for the low expansion factor case, leading to an increase in speed for low expansion factors [1, 3, 4]. 
In this study, we statistically test the v - f anticorrelation, exploiting the Solar Orbiter capability to sample a broader range of radial distances compared to previous studies.

Figure 1. Sketch highlighting the two scenarios of large and small-super expansion, their implications on mass conservation and their consequences on the relevant heating/acceleration processes. [3,4]; Left : Large expansion factor; Right : Small expansion factor. Spherical expansion with grey area, and super expansion with green area.

To obtain statistically significant results, this method was applied to all HRIEUV pixels. The pixels are categorized as either event pixels or Quiet Sun (QS) pixels. Figures 1a and 1b show the extraction of the light curves from the AIA channels and HRIEUV for one of the event pixels. Light curves are then cross-correlated between pairs of AIA channels, to obtain the maximum correlation and its associated time lag. We applied this algorithm for nine AIA pairs, to cover a large temperature range, spanning from 0.5 to 8 MK.

II. Magnetic connectivity

We first investigate the magnetic connectivity of the measurements with the solar source. To this end, we compute the magnetic connectivity of a solar wind streamline with a two-step process entailing interplanetary magnetic field modeling and near-Sun magnetic field modeling.
In such computation, we include both wind acceleration and corotational effects. For this, we use a Parker-like solar wind model, the two-fluid "isopoly" model [5], treating the solar wind radial propagation assuming isothermal evolution close to the Sun then polytropic further away from a given “isothermal radius”, free parameter of the model. Modeling the corotational effects results in a tangential component of the solar wind velocity, due to the fact that the escaping wind is still magnetically attached to its source (i.e. corotate) below the Alfvèn radius [6,7].Both the radial and tangential components of the resulting velocity are used to compute the plasma streamline from Solar Orbiter location sunward to the solar “source surface” called rss , located at 2.5 solar radii.  The second stage consists in employing a Potential Field Source Surface (PFSS) to locate the photospheric footpoints.

III. Statistical solar wind backmapping results

Figure 2 (below) presents the results of the statistical study. In particular, the plot on the left-hand side of the Figure reveals that the v-f anticorrelation holds only for solar wind originating from high latitude structures (mainly coronal holes). Indeed, when only high-latitude data is considered, the v-f correlation coefficient is of -0.51 for high latitudes data, while only -0.27 considering all data together. Further, the existence of a new type of fast wind emerges from this study. Such wind  originates from low-latitude coronal regions, with large expansion factors. n periods of solar minima, where the low latitude of the solar atmosphere is more representative of slow wind source regions (steamers magnetic structures), the fact that such source regions could also generate fast wind observed further out in the interplanetary medium, raise some questions about the relation between the acceleration processes and the expansion factor.

Figure 2.  Relationship between the measured velocity and the final expansion factor value computed with PFSS from the back-mapping applied to Solar Orbiter data. Panel (a) Colored by photospheric magnetic field intensity (Pearson correlation coefficient of -0.27). Panel (b) : Same as the panel (a) but colored in black for low unsigned latitudes (< 45°) and in orange for high latitude footpoints (> 45°) for each mapped observation. Typical range of values from Wang et Sheeley (1990) in green bars. Pearson correlation coefficient of -0.51 and -0.24 for high and low latitude data respectively. The mapping results cover the time interval from 01/08/2020 to 17/03/2022.

Moreover, the fast wind originating from large expansion factor regions shows similar interplanetary properties (proton bulk velocity, temperature and density) to the more “classical” fast wind originating high-latitude coronal holes (with low expansion factor). This property implies that a physical process must counteract the expected lack of acceleration. We suggest that such mechanism is the DeLaval Nozzle effect [9] (illustrated in Figure 3 below), commonly used in rocket science to enhance the gas velocity outcome. Indeed, such winds may become supersonic during the super radial expansion (below rss), inducing an acceleration rate theoretically governed by a positive correlation v-f, as shown in the lower left case of the Figure 3.

Figure 3. Illustration of the DeLaval nozzle effect for a flow in a diverging or converging flow tube, (right and left case respectively), depending on the subsonic or supersonic speed flow regime (upper and lower case respectively).

IV. Physical implications for the DeLaval Nozzle effect

The consideration of nozzle effect in the solar wind modeling leads to two types of solutions that are referred to as “f-subsonic” and “f-supersonic”, respectively subsonic and supersonic in the super expansion region below rss. The inclusion of the expansion factor in isopoly modeling [5], allows the Solar Orbiter observations to be classified with these two types of wind solutions using isopoly modeling. Figure 4 shows that more than half of the winds are of f-supersonic type (panel a), and that they present higher typical height of expansion (panel b).

Figure 4. Wind speed and typical expansion properties of the f-subsonic and f-supersonic solutions. Panel (a) : Same as panel (a) of Figure 2 but colored by the type of identified solutions (f-subsonic or f-supersonic).The isopoly parameters are interpolated from the 5 populations isopoly parameters of Dakeyo et al. (2022). The modeled observations include 42% and 58% of f-subsonic and f-supersonic solutions, respectively. Panel (b) : Typical expansion radius (strongest local expansion) of the data presented on panel (a) classified by solution type.

All these results indicate that the characteristics of the expansion factor profile are of primary importance for a better understanding of the wind acceleration in the solar atmosphere. Wind acceleration is sensitive to the local temperature conditions as well as the local flow regime (subsonic or supersonic) in the super expansion region. There are also implications for the improvement of solar wind prediction models, which to a large extent take into account the anticorrelation wind speed expansion factor, but do not consider the possibility of a positive correlation associated to f-supersonic flows. Possible improvements to solar wind forecasting could be achieved by determining the flow regime in the high corona and using it as a parameter to modulate the sign of the v-f correlation.       

Acknowledgements

This research was funded by the European Research Council ERC SLOW_SOURCE (DLV-819189) project. This research was supported by the International Space Science Institute (ISSI) in Bern, through ISSI International Team project #463 (Exploring The Solar Wind In Regions Closer Than Ever Observed Before) led by L. Harra. This work was supported by CNRS Occitanie Ouest and LESIA. D.V. is supported by STFC Consolidated Grant ST/W001004/1. This work made use of the Magnetic Connectivity Tool provided and maintained by the Solar-Terrestrial Observations and Modelling Service (STORMS). This work utilizes data produced collaboratively between AFRL/ADAPT and NSO/NISP. We recognize the collaborative and open nature of knowledge creation and dissemination, under the control of the academic community as expressed by Camille Noûs at http://www.cogitamus.fr/ indexen.html. We thank the instrumental Solar Wind Analyser team (SWA) for valuable discussions. We acknowledge very valuable discussion with Miho Janvier, who has opened the problematic leading to this study

Affiliations

[1] LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris, 5 place Jules Janssen, 92195 Meudon, France
[2] IRAP, Observatoire Midi-Pyrénées, Université Toulouse III – Paul Sabatier, CNRS, 9 avenue du Colonel Roche, 31400 Toulouse, France 
[3] Laboratoire Cogitamus, 75005 Paris, France  
 

Citations

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