Source-dependent Properties of Two Slow Solar Wind States

All data exploited in this study were obtained by the PSP (Fox et al. 2016). Data include magnetic field components in the radial tangential normal (R,T,N) coordinates with respect to the spacecraft (provided by the FIELDS instrument, Bale et al. 2016). Electron density and core temperature are provided by quasi-thermal-noise (QTN) spectroscopy (Meyer-Vernet et al. 2017) applied to FIELDS data (Moncuquet et al. 2020). Proton speeds were measured by the Solar Probe Cup (SPC), which is part of the SWEAP instrument (Kasper et al. 2016).

Data sets cover the period from 2019 March 31 at 00:00 UT to 2019 April 5 at 23:59 UT (during Carrington rotation 2215), during PSP's second encounter with the Sun. They are shown on Figure 1. During this time interval, the spacecraft traveled from 50.4–35.6 R_⊙, with the spacecraft perihelion on April 5 at 10:40 UT at 35.6 R_⊙. We used the time provided by the plasma QTN spectrum measured by the Radio Frequency Spectrometer (RFS)/FIELDS (electron densities and temperatures (Moncuquet et al. 2020) as a reference (most of the time, QTN data are provided every 7 s). We interpolated the magnetic field (FIELDS) and proton (SPC) data, all averaged on 1 s intervals, to the QTN time using the closest value approach.

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We manually selected time intervals with no magnetic field switchbacks, based on the selection of long enough time intervals (i.e., more than 30 minutes) without strong variation of the radial component of the magnetic field. These intervals were determined by visual inspection of the magnetic field data. An example of such a "calm" interval that occurred between two intervals of variable wind perturbed by magnetic switchbacks is shown in Figure 2. The periods of less variable solar wind are referred to as the "calm solar wind intervals." The calm solar wind intervals are shown in Figure 3.

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The aim of this study being to differentiate between different sources of the slow wind, we corrected the data to remove the radial variation of the different parameters. At first order and at constant speed, the density of the spherically expanding solar wind is expected to decrease as the inverse of the square of the radial distance from the Sun. We thus provide densities at a reference perihelion distance of 35.6 R_⊙ by multiplying values by ${\left(r/35.6\right)}^{2}$, where r is the radial distance of PSP from the Sun given in solar radii (R_⊙). The magnitude of the magnetic field is also related to its estimated value at 35.6 R_⊙, taking into account the decrease in 1/r² due to spherical expansion. The radial decrease of the core electron temperatures is directly taken from Moncuquet et al. (2020): ${T}_{c,e}\propto {\left(r/{R}_{\odot }\right)}^{-0.74}$.

The plasma β is defined as the ratio of the plasma gas pressure to the magnetic pressure. Thus, β describes the dominant pressure state in the plasma: β > 1 means that the gas dominates the magnetic pressure and vice versa. On 2019 April 3 at 07:55 UT, PSP encountered a boundary between two different plasma states, as can be seen in Figure 4. Only the electron β is considered here, defined as β_e = n_e k_B T_c,e /p_mag, with n_e the electron density, k_B the Boltzmann constant, T_c,e the core electron temperature, and p_mag the magnetic pressure: p_mag = ∣B∣²/2μ₀, where B is the magnetic field and μ₀ the vacuum permeability. The total β, including the proton, alphas, and heavier ions' contributions to the gas pressure, would be approximately twice as high as the electron β_e (if the fraction of the alpha particles and heavy ions compared to protons can be neglected). We see that the β_e mean value, computed on the calm solar wind intervals, is ≃0.7 in state 1 (standard deviation: 0.26 in state 1), which is close or superior to 0.5. Then, the β_e mean value computed on the calm solar wind intervals suddenly drops to ≃0.2 (standard deviation: 0.06 in state 2), which is much lower than 0.5. The total pressure is estimated as p_tot = 2n_e k_B T_c,e + p_mag and is displayed in the top panel of Figure 4. The total pressure is less variable in state 2 (the standard deviation is 0.95 in state 1 and 0.53 in state 2) and decreases slightly, by a factor of 0.25: in state 1, the median value of p_tot is ≃7.6 nPa, and it falls to 5.7 nPa in state 2. One may note here that p_tot,2 = n_e k_B T_c,e + n_p k_B T_p + p_mag, using SWEAP/SPC proton density and temperature, follows the same trend as p_tot but shows greater dispersion, due to calibration issues. However, a clear boundary is less visible in the total pressure, contrary to what is observed for the electron β_e (bottom panel of Figure 4), suggesting that the states identified here are roughly in pressure equilibrium at the boundary. When looking at the electron gas pressure and magnetic pressure separately (top panel of Figure 4), the transition is clearly visible for each pressure, and is especially sharp for the electron gas pressure.

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When plotting the electron β_e against the proton bulk speed (provided by SWEAP/SPC), which is used as a proxy for plasma speed, a clear difference appears from state 1 to state 2, as shown in Figure 5 (top panels). In the top panels of Figure 5, the electron β_e computed at the distance of 35.6 R_⊙ is plotted against the proton speed for calm solar wind intervals (top left panel) and for the rest of the data set (top right panel), in red (magenta) for state 1 and in blue (cyan) for state 2. In each of the two top panels, the difference between the two plasma states is obvious. In both states, the solar wind can be considered as a "slow" solar wind, even if the plasma reaches much higher speed values (almost twice higher) in state 2. In state 1, the plasma gas pressure dominates the magnetic pressure. In state 2, on the contrary, the magnetic pressure dominates over the gas pressure.

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Figures 1, 3, and 4, reveal that the parameter with the greatest change between the two states is the plasma density. Figure 5 shows the electron density scaled to 35.6 R_⊙ (middle panels), with the error bars provided by the plasma QTN spectrum measured by RFS/FIELDS (Moncuquet et al. 2020), which are used to compute error bars on the β_e , as the other quantities are not provided with errors. In the left middle panel, data from calm solar wind are plotted in red for state 1 and in blue for state 2. Those calm solar wind intervals are representative of a "background" solar wind, plotted against the proton speed, which is representative of the plasma bulk speed. The electron density is higher across the speed range in state 1 than in the second one. In state 1 (in red), the faster the plasma is, the higher is the electron density. In state 2 (in blue), an opposite behavior is observed: the electron density decreases when the plasma speed increases. This statement is supported by linear least square fit, which provides a positive slope of +0.31 in state 1 and a negative one of −0.23 in state 2. The anticorrelation is strongest on speeds between 260 and 370 km s⁻¹. Beyond 350 km s⁻¹, the linear least squares on state 2 data shows that the density is constant as the speed increases. The unscaled electron density shows the same trends.

Magnetic field is stronger in state 2. There is no particular relationship between the magnetic field and the plasma speed, as can be seen in Figure 5, bottom panels. Figure 5 (bottom panels) shows the magnetic field intensity scaled to 35.6 R_⊙. There is a clear difference between the two plasma states when regarding the calm solar wind intervals (left bottom panel), essentially due to the fact that the magnetic field is stronger in the plasma state encountered after 2019 April 3 at 07:55 UT. The rest of the data set (right bottom panel) reveals that the presence of magnetic field fluctuations (including magnetic switchbacks) make the two states less distinct between each other from the point of view of the magnetic field intensity. The variability of the magnetic field is much higher in state 1 than in state 2.

Electron core temperature is relatively constant over all the data set under consideration in this study (see third panel in the top of Figure 1), i.e., from 2019 March 31, 00:00 UT to 2019 April 5, 23:59 UT. Electron core temperature is thus not responsible for the two different plasma states. However, small differences in terms of temperatures are still noticeable from one state of plasma to the other.

When considering temperature–speed diagrams for the core electron temperature T_c,e and proton speed V_p (used as a proxy of plasma speed) in Figure 6, it is interesting to compare the global temperature–speed (T–V) diagrams (for all data points, top panels), in state 1 in the middle panel and in state 2 in the bottom panel, and note that the "gun-shaped" distribution is actually made up by the sum of two distributions, corresponding respectively to the data in state 1 (middle panels, in red-to-black colors) and data in state 2 (bottom panels, in blue-to-black colors). The T–V diagrams considered separately present slightly different slopes from one state of plasma to the other (on calm solar wind intervals, the slope is 0.04 ± 0.08 in state 1 and 0.02 ± 0.07 in state 2), resulting in the gun-shaped global distribution. Moreover, T–V diagrams were plotted (but not displayed) for both core electron temperature T_c,e and proton temperature T_p , showing the expected difference between electron and proton temperatures: when electron temperature is relatively constant with plasma speed, the proton temperature actually increases with the plasma speed. The speed at which protons become hotter than electrons is around 325 km s⁻¹ in state 1 and around 380 km s⁻¹ in state 2. The slope of the T–V distribution for proton temperature seems to be steeper in state 1. Although the major change in electron density from state 1 to state 2 could affect the absolute temperature measurements. A complete study of the T–V diagrams is beyond the scope of this paper, but should be investigated in order to establish how the different slopes could be relevant to different heating/propagation mechanisms.

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The two different plasma states show clearly different fluxes, approximated here as n_e (35.6 R_⊙)V_p (in cm⁻² s⁻¹), as shown in Figure 7. The higher speed in state 2 does not compensate the sudden drop in plasma density, thus the flux in state 2 is lower. This is true for both calm solar wind intervals and for variable solar wind intervals with numerous switchbacks. Plotted against the square of the magnetic field intensity (representative of magnetic pressure) two plasma states appear distinct both in terms of the gas and magnetic field properties. There is a slow solar wind characterized by low magnetic pressure and high flux and another slow wind with low flux and high magnetic pressure (see Figure 7).

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In the previous sections, several macroscopic properties of the solar wind were compared before and after 2019 April 3 at 07:55 UT (plasma density, speed, temperature, flux, beta, and magnetic field intensity). Those differences suggest that PSP crossed, successively, two different states of slow solar wind, for at least 3 days before 2019 April 3 at 07:55 UT (state 1) and 3 days after (state 2). As mentioned in Section 1, this date of April 3 at 7:55 UT was first investigated in a study of white-light images of the corona taken by SOHO and STEREO-A. It revealed that PSP escaped streamer flows to sample solar wind likely released from deeper inside a coronal hole (Rouillard et al. 2020a). The latter study did not investigate the presence of such a coronal hole, which we address here in greater detail. A more precise study of PSP connectivity against white-light Carrington maps was therefore pursued in the present study in order to characterize potential sources of the slow wind and is presented in the top panels of Figure 8.

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We assumed that PSP was connected to the upper corona by a Parker spiral defined by the speed of the solar wind measured in situ. We then connected the upper corona to the photosphere by using a PFSS extrapolation of the photospheric magnetograms (Wang & Sheeley 1992). There is considerable uncertainty associated with determining the exact configuration of the solar magnetic field in order to determine how a spacecraft connects magnetically to the different regions of the solar atmosphere (Rouillard et al. 2020b). This is related to the inherent assumptions of the model such as a spherical shape of the source surface or the assumption of a current-free corona. Any model of the solar atmosphere is also dependent on the quality of the solar magnetograms and the choice of free parameters. In the PFSS model, this is the height of the source surface (see e.g., Badman et al. 2020; Panasenco et al. 2020). White-light observations from the SOHO-LASCO C2 and C3 instruments can be exploited to extract the location of the streamer belt and to compare this location with that of the polarity inversion line obtained with each PFSS realization. We developed a new technique to segregate between PFSS realizations obtained for different choices of solar magnetograms and source surface heights based on how well the derived neutral line position matched the location of the streamer belt. The streamer belt is reduced to a single curve defined by the locus of maximum brightness, hereafter called streamer maximum brightness (SMB) line. For most longitudes, the SMB lies at the center of the streamer belt. The angular distance between the SMB line and the neutral line from the PFSS model can be measured. We then consider that a PFSS realization fits best the white-light observations when the distance between the SMB and the neutral line averaged over all relevant longitudes is minimal. This comparison is prioritized in the range of Carrington longitudes magnetically connected to PSP via the Parker spiral. The height of the source surface was further adjusted until the small equatorial coronal hole observed in EUV was reproduced by the PFSS model (data from SDO-AIA, 193 Å). In this iterative process, a compromise had to be found in order to keep a PFSS neutral line locally consistent with the white-light observations. The best-fit PFSS model was obtained for a GONG-Air Force Data Assimilative Photospheric Flux Transport (ADAPT) magnetogram (realization n° 2) of 2019 March 24 at 00:00 UT and a source height of 2.1 R_⊙. The GONG magnetogram was processed by the ADAPT model, which simulates the motion of photospheric magnetic fields (see Arge et al. 2003). The magnetic connectivity from the source surface (at 2.1 R_⊙) was then extended using the Parker spiral model (Parker 1958) for the solar wind speed measured in situ at PSP.

From these maps, it is clear that PSP left the boundary of the streamer around 2019 April 3, before reentering it at around April 6. Our modeling of PSP's connectivity around 2019 April 3 reveals that the spacecraft has intersected magnetic field lines rooted deep inside an equatorial coronal hole, clearly visible (as a dark area) on EUV light maps (middle left panel in Figure 8) during this period. This small coronal hole remains visible in the EUV map during the entire time interval considered here, though varying slightly over time.

In order to link the magnetic field configuration obtained from the optimized PFSS model and the properties of plasma states 1 and 2, we use the magnetohydrodynamic "single fluid" MULTI-VP code (Pinto & Rouillard 2017), which computes full solar wind solutions (surface to high corona) taking into account heating and cooling processes in the corona. In MULTI-VP, as explained in Pinto & Rouillard (2017) and Griton et al. (2020), the mechanical heating directly depends on the photospheric magnetic field. We associate the mechanical heating flux F_h (or, equivalently, the heating rate Q_h = −∇ · F_h ) to a function that represents the effects of the coronal heating processes:

$$\begin{eqnarray}&&{F}_{{\rm{h}}}={F}_{{\rm{B}}0}\left(\displaystyle \frac{{A}_{0}}{A\left(s\right)}\right)h\left(s\right),\end{eqnarray} \tag{ 1 }$$

with $A\left(s\right)$ being the flux tube's cross-sectional area at a given curvilinear coordinate s, A₀ the flux tube's cross-sectional area at the solar surface, and the coefficient F_B0 proportional to the basal magnetic field amplitude $\left|{B}_{0}\right|$ (with for basal field amplitudes consistent with typical Wilcox Solar Observatory magnetograms, ${F}_{{\rm{B}}0}=8\times {10}^{5}\left|{B}_{0}\right|\ \mathrm{erg}\,{\mathrm{cm}}^{-2}\ {{\rm{s}}}^{-1}$). The coronal heating profile $h\left(s\right)$ is

$$\begin{eqnarray}&&h\left(s\right)=\exp \left[-\displaystyle \frac{s-{R}_{\odot }}{{H}_{{\rm{f}}}}\right],\end{eqnarray} \tag{ 2 }$$

which takes into account a damping scale height H_f, which is anticorrelated with the superradial expansion ratio in the low corona, as in Pinto & Rouillard (2017).

Thus, MULTI-VP computes the profiles of contiguous one-dimensional solar wind streams from the surface of the Sun up to about 30 R_⊙. The individual streams are guided along individual magnetic flux tubes, which are obtained from the optimized PFSS model: we manually identify the magnetic field tube A as representing the edge of streamer tubes connected to PSP from 2019 March 31 to April 3, and the magnetic field tube B for the edge of the equatorial coronal hole. A projection of the field-line trajectories into a meridional plane at Φ = 18° Carrington longitude is displayed in the left panel of Figure 9, where magnetic field lines A and B are highlighted in red and blue.

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The middle and right panels Figure 9 present the results of the MULTI-VP simulations along five flux tubes around tube A (red profiles) and five flux tubes around tube B (blue profiles), separated by 1° of spatial resolution at the source surface at 2.1 R_⊙ from the center of the Sun. The simulations reproduce many properties measured in situ (and usually produced by MULTI-VP simulations Pinto & Rouillard 2017): the solar wind along tube A, representative of plasma state 1, is indeed slower and denser compared with the solar wind along tube B, representative of plasma state 2 and emerging from the equatorial coronal hole. On the logarithmic scale, a kink lies at about h = 1 R_⊙ (i.e., at a radial distance of 2.0 R_⊙ from the center of the Sun), still slightly below the source-surface radius at 2.1 R_⊙. This difference of almost 0.1 R_⊙ is correct and occurs because the PFSS extrapolation produces sometimes very sharp field-line bends just before they connect to the radial field above. The numerical code requires that all quantities are spatially resolved and are smooth enough (i.e., third order differentiable). For safety, a thin interface zone is introduced in between the two coronal regions, on which the maximum gradients of the field-line profiles are constrained. That interface region is 0.1 R_⊙ wide, and only acts effectively on rather extreme field-line geometries. The overall representation of such field lines remains faithful to that of the underlying extrapolation. The presence of a pseudo-streamer (visible on the cut on the left panel of Figure 9) bends both flux tubes in such a way that the expansion of both tubes A and B exert a significant deviation from the radial direction around an altitude of 0.5 R_⊙, with an effect on the number density and speed profiles at this altitude. Temperature profiles provide an explanation for the differences between the two states of solar wind. As we said, the heating depends on the photospheric magnetic field but also on the geometry of the magnetic flux tube. Heating more concentrated lower down (due to the geometry) makes the lowest and densest layers expand (that is, increase scale-height, or "evaporate"). There are then more particles per "unit of energy," although the thermal driving is lower. As a result, along streamer-like flux tubes, density, temperatures, and speeds are lower higher in the corona for a similar energy flux.