© The Authors 2024

1. Introduction

Magnetohydrodynamic (MHD) discontinuities in the heliosphere have been intensively studied since the first in situ observations in the interplanetary plasma. They are abrupt changes in the physical parameters of the solar wind, such as density, velocity, magnetic field direction, and temperature. Rotational (RD) and tangential (TD) discontinuities (together: directional discontinuities, DDs, or interplanetary discontinuities) are common types of these structures in the solar wind plasma. The latter type separates two distinct plasma regions with different properties that have no mass flow between them. In the case of RDs, there is no change in the magnetic field magnitude, but a constant normal component of the field and a finite mass flow are permitted across the discontinuity. TDs comove with the solar wind, while RDs move with the Alfvén speed with respect to the plasma. Although in the MHD approximation, these structures are abrupt changes (hence the name “discontinuity”), in reality, the change in plasma parameters is not instantaneous. There is a transition region of finite thickness between the upstream and downstream regions.

The spatial and temporal distribution of DDs can be used to investigate physical processes in the plasma, such as heating mechanisms and turbulence. Along with waves (commonly known as coherent structures), DDs play important roles in the development of the turbulent spectral break near the ion scales. Previous studies showed that the density of these discontinuities decreases with increasing distance from the Sun (Tsurutani & Smith 1979). Solar Orbiter and the Parker Solar Probe in the inner heliosphere present a great opportunity to study the spatial development of these structures (and turbulence in general) in the young solar wind in the innermost heliosphere, where they have never been studied before.

In the study of solar wind turbulence (Tu & Marsch 1995; Bruno & Carbone 2013), TDs and the associated current sheets appear as a property of intermittent turbulence. They represent a type of coherent structure that determines the exact shape of the turbulent fluctuation spectra close to the ion scales (the spectral breakdown). TDs are also important for understanding the mesoscale structure of the interplanetary magnetic field. In the so-called spaghetti paradigm of the solar wind (Schatten 1971; Mariani et al. 1973; Bruno & Carbone 2013; Borovsky 2008), the magnetic structures such as flux tubes, that originate in the photosphere, convect and expand together with the solar wind. They remain almost intact even at a heliographic distance of 1 AU and beyond. In this simplified picture, there are two types of TDs (current sheets): Some TDs separate the weakly interacting flux tubes, and others are dynamical, turbulent phenomena.

On the other hand, RDs are frequently observed at the edges of or embedded in Alfvén-wave trains, and two of the most popular theories about their formation are turbulent processes and Alfvén-wave steepening (Cohen & Kulsrud 1974; Malara & Elaoufir 1991). The former produce RDs with randomly distributed normals relative to the local magnetic field, while the latter tends to produce discontinuities with typically a large normal component (Vasquez et al. 2007). Thus, it can be decided which of the proposed formation processes is more consistent with the measurements.

Another categorization only focuses on the thickness of the current sheet and does not constrain the possible source or lifespan of the structures. The most abundant type of TDs is the thin current sheet (TCS), which has a thickness of about several proton gyroradii (Khabarova et al. 2021). Since there are many possible ways for a TCS to be created (e.g., as a surface between small-scale flux tubes or in the vicinity of high-speed streams; some of them are formed near heliospheric current sheets), it is expected that most of the TDs we find belong to this group.

There are many different ways to identify these structures in spacecraft data. The selection of discontinuities from observations is somewhat arbitrary, however, because the term “abrupt change” is poorly defined, and also because it is difficult not to select DDs from single-point measurements. Different methods use different time intervals and thresholds to define discontinuities. DDs can be considered planar compared to the scales measured by spacecraft and can therefore they can be described by their normal vector. The most commonly used method for determining the direction of the normal vector is minimum variance analysis (MVA). This is necessary to determine the discontinuity thickness and to obtain correct statistics of DD normals. Since DDs are spatial structures that move with the plasma, a measured time gradient of the angle variation in the magnetic field might introduce an error: The magnetic field is embedded in the solar wind flow, and the bulk speed of the plasma affects the magnitude of the temporal gradient. In order to overcome this problem, the flow velocity of the solar wind can be used to estimate the convection speed of solar wind structures, such as discontinuities, and thus, the measured temporal gradients can be converted into spatial gradients that truly characterize the DDs.

It is important to note that one of the most difficult problems in a DD analysis is the classification into the two types. Our solution is to use three complementary methods to decide whether an event is an RD or a TD: the Smith method (Smith 1973) is only based on magnetic field measurements, while the Walén test in the spacecraft frame (Sonnerup et al. 2018) and in the deHoffmann–Teller frame (Paschmann & Daly 1998) needs both magnetic and plasma observations. After the classification, the radial distribution of normals, thicknesses and spreading angles of the DDs can be obtained, and their evolution can be followed. This allows us to choose between the different theories about their formation.

2. Data

We investigated the distribution of DDs using data from the ESA Solar Orbiter (launch in 2020) mission (Müller et al. 2020) and the NASA Parker Solar Probe (launch in 2018) mission (Fox et al. 2016). Their high-resolution plasma and magnetic field data make whole new temporal and spatial scales available in the inner heliosphere at a radial distance from the sun between 0.05-1.01 AU. In the case of the Solar Orbiter, we used magnetic field data of MAG (Horbury et al. 2020) and plasma data of the Proton Alpha Sensor (PAS), which is part of the Solar Wind Analyser (SWA) instrument (Owen et al. 2020). From Parker Solar Probe (PSP), the FIELDS MAG (Bale et al. 2016) magnetometer, the Solar Probe Cup (SPC) Faraday cup (Case et al. 2020), and the Solar Probe ANalyzer-Ions (SPAN-I) instrument (Livi et al. 2022) data were used. We used the Solar Orbiter MAG survey data, with a cadence of 8 Hz and PSP FIELDS MAG data downsampled to 16 Hz at the spacecraft encounters. A better time resolution for PSP is required because the magnetic field nearby the Sun is expected to be more strongly disturbed. The resolution of plasma data varies at about 0.2–3.0 Hz in both cases. The SPC and SPAN-I instruments of PSP are optimized for different heliocentric distances. Because it is a Faraday cup and it is on the sunward side of the spacecraft, SPC directly measures the solar wind radially. At the times of close encounters, when the tangential speed of the spacecraft is high, SPC cannot detect enough particles for precise measurements. On the other hand, SPAN-I is on the ram side of PSP and is designed to work in the near-Sun solar wind environment, but it is less reliable farther away from the Sun. We used these instruments to cover the largest possible radial interval. As a rule of thumb, SPC was typically used at r > 0.2 AU and SPAN-I at r < 0.2 AU, but in each case, we selected data with a satisfying quality. Observations from orbits 1–11 were used from PSP, and orbits 1–6 from Solar Orbiter. Since we needed plasma velocity data for the DD selection, we used time intervals for which both magnetic and plasma data were available.

3. Results

3.1. Directional discontinuities by Parker Solar Probe and Solar Orbiter

The applied selection method for finding the DDs was described in detail by Erdos & Balogh (2008). Here, we summarize the way of the analysis and indicate where we deviated from the cited study. First, the angles between every nth field vector were calculated. The window was 16 and 32 vectors for Solar Orbiter and PSP, respectively, which means a 2-s time leap for each spacecraft. By dividing this series of angles by the elapsed time between the measurements, we obtained the temporal attitude gradient. However, this attitude gradient is not very useful because it depends on the actual speed of the solar wind as the plasma transports the frozen-in magnetic field. Some earlier studies found that DDs are more common in the plasma of the fast solar wind because of this effect. To overcome this problem, we transformed the time gradient of the field angles into a spatial gradient by dividing it by the corresponding plasma speed. Although discontinuities are selected according to the spatial gradient of the angular change, in earlier studies, the conversion from time to distance in the frame of the solar wind was performed simply using the radial velocity of the solar wind. However, the high tangential velocity of the spacecraft does not allow this simplification, especially in the case of PSP, and we therefore used all the significant velocity components. As pointed out by Perez et al. (2021), the Taylor hypothesis is valid even for close encounters of PSP, and to determine the spatial gradient, the measured radial and tangential solar wind speed can therefore be used. Based on the velocity data, we determined the local maximum of the attitude gradient (a_max), and the event was regarded as a candidate for discontinuity when this maximum exceeded a certain ac threshold. As expected, the number of candidate events strongly depends on the chosen threshold. With trial and error, we set a_c = 0.02° /km, to find DDs with a thickness of about the gyroradius in the inner heliosphere.

The transition region of the DDs cannot be described with fluid models of the plasma, as described above, and the exact boundaries of this region are only poorly defined. First, we took measurement points on either side of the gradient peak, for which the attitude gradient decreased below half of (a_max). After this, the upstream and downstream boundaries of the DD candidate were determined as the local minima in the attitude gradient at the two sides of the maximum.

Waves and wave-like events can also be found by searching for peaks in the attitude gradient. To avoid selecting these, we require the magnetic field to be ‘quiet’ at the two sides of the event candidate. First, we selected the upstream and downstream endpoints of the event (x_up and x_down) by finding the nearest local minima at the two sides of a_max that were smaller than a_max/2. Two intervals were determined upstream and downstream with a width of x_down − x_up, and when the attitude gradient was larger than a_max/2 anywhere in these intervals, the event candidate was dropped. A difference in the selection method used in this study compared to the original one was that we kept all DD candidates regardless of the spreading angle, that is, the angular difference between the upstream and downstream field vectors because we expect to observe many discontinuities with small spreading angles in this region.

An important aspect of the Erdos & Balogh (2008) method was the consideration of the possible deviation of the DD normal from the radial direction. In these cases, the thickness and therefore the measured gradients are overestimated for simple geometrical reasons. In order to correct for this effect and also to obtain realistic thicknesses, the normals of the DDs have to be determined. To determine the normals, we used the minimum variance analysis, and the variance matrices were calculated using the field components between the previously determined upstream and downstream vectors. The confidence of the MVA was accepted when the ratio of the intermediate and the lowest eigenvalue of the variance matrix λ_int/λ_min was large enough. We took the threshold for this ratio to be 20 based on Horbury et al. (2001), Knetter et al. (2004), Knetter (2005), and Liu et al. (2023), all of which were based on the triangulation technique using multispacecraft data. The reason for this strict constraint is the fact that below this ratio, the MVA can give incorrect normals for various reasons, such as surface Alfvén waves and geometric effects. Another study that was based on the triangulation method found no correlation between the MVA accuracy and the λ_int/λ_min ratio (Wang et al. 2024). Nevertheless, we relied on the use of the aforementioned strict condition, and trusted that accurate normal directions can be obtained in this way. The MVA normals and eigenvalues were initially calculated using the magnetic field data between the automatically found upstream and downstream endpoints. Since this interval may not be best suited for an MVA calculation, we moved the upstream and downstream boundaries by ±25% of the original event interval length. In this way, we had a number of normals and eigenvalues for each DD, and we kept the most reliable ones, that is, those with the highest λ_int/λ_min ratio. For most of the DDs, the λ_int/λ_min ratio increased when the width of the data interval decreased. From this event set, we selected the DDs with λ_int/λ_min > 20. This requirement does not only decrease the number of selected DDs, but also effects the statistics. As Liu et al. (2023) pointed out, the high MVA eigenvalue ratio in itself has a large effect on the relative numbers of RDs and TDs because more RDs have a high λ_int/λ_min ratio.

After we applied the strict selection criteria (the requirement of a relatively quiet upstream and downstream magnetic field), our selection method found more than 140 000 discontinuities in the combined data of the two spacecraft from 0.06 to 1.00 AU. We plot the resulting DD occurrence rate from the combined Solar Orbiter and PSP datasets in Figure 1. The horizontal axis is the distance of the observation from the Sun, and the vertical axis is the density of the DDs. Each marker shows the daily summarized DD number, and the color of the dots corresponds to the daily average radial plasma velocity. The distribution of DDs shows the increasing number of events near the Sun with a fitted power-law function with an exponent of 0.93, which is somewhat shallower than in previous PSP-based studies (Liu et al. 2021). Again, we highlight that this is based on findings in the plasma frame after correcting for the bias caused by the velocity of the solar wind and the orientation of the DDs. This behavior agrees with the findings of earlier studies (Tsurutani & Smith 1979; Erdos et al. 2001; Erdos & Balogh 2008) and also extends the validity region of the power-law decrease to the innermost heliosphere.

Fig. 1. Occurrence rate of the DDs measured by Solar Orbiter and PSP as a function of the distance from the Sun in a logarithmic plot. Each dot corresponds to a daily rate obtained from the spacecraft measurement, colored according to the average daily radial solar wind velocity. The black line marks the power-law function fit for the data.

The exact radial dependence of the occurrence rate might be a result of the combination of several effects. As pointed out earlier, if DDs were only present as surfaces between stationary but expanding flux tubes, two competing phenomena would govern their occurrence. First, the spacecraft would measure more TDs when the Parker angle is larger because more TDs are swept through the spacecraft with an increasing Parker angle. Since the latter grows with heliocentric distance, this mechanism alone would cause a decrease in DD density closer to the Sun. We verify the variation in the DD normal with radial distance below. Second, the cross-section of flux tubes is much smaller near the Sun, which makes a border crossing more likely. The occurrence rate can also be affected by the selection method. We used a threshold for the magnetic vector attitude gradient, and the gradient of a narrow DD is steeper than that of a thick DD with the same spreading angle. A DD like this is therefore more likely to be detected by our algorithm. Thus, if the DD thickness changes with radial distance, the selection method, which uses a constant thickness, could cause a bias. To prove (or rule out) the selection effect mentioned in the previous section, we calculated the thickness of the DD transition zones. Another important feature of the distribution is the dependence on the average radial plasma velocity. Figure 1 shows that at a given radial distance, periods with a slower solar wind tend to contain more discontinuities.

3.2. Analysis of directional discontinuities

In order to understand the physical properties and structure of the solar wind, we decided whether the selected events were TDs or RDs. A TD separates two different types of plasma, without any particle flux flowing across the discontinuity. It is therefore tempting to search for them by searching for DDs with a jump in density, temperature, or pressure. but these criteria require plasma measurements, which are much less reliable and have a much lower resolution than magnetic field measurements. In order to overcome this problem, several authors (Smith 1973; Neugebauer et al. 1984; Knetter et al. 2004) used only magnetic field data to classify the DDs. Smith’s method is based on two parameters: the normal component of the magnetic field inside the DD, denoted by B_n, and the change in the magnitude of the magnetic field vector upstream and downstream, denoted by [B]. Both are normalized with the maximum B magnitude for the entire event (B_max). A directional discontinuity with a large normal component B_n/_Bmax > 0.4 but a small change in magnitude [B]/B_max < 0.2, is considered a rotational discontinuity, and an event with a strong change in magnitude [B]/B_max and a small B_n/B_max is a tangential discontinuity. Events with B_n/B_max < 0.4 and [B]/B_max < 0.2 were first called “either discontinuity” (ED) because we were unable to determine them with certainty using this method. A DD, where [B]/B_max > 0.2 and B_n/_Bmax > 0.4, is called a ‘neither discontinuity’ (ND) because it is inconsistent with the jump conditions of RDs and TDs. Some of these events can be interplanetary shocks. The exact parameters of the classification vary in earlier studies, and it depends on the population of DDs in the analyzed data. Since many of the events were classified as ED (as in many earlier studies), we used another method, the Walén test, to decide whether these DDs are RDs with small B_n/B_max or TDs with small [B]/B_max.

One way to carry out the Walén test is to apply it in the deHoffman–Teller (HT) frame. In this frame, the convected electric field vanishes,

$$\begin{array}{c}\hfill \mathit{E}\mathbf{\prime }=\mathit{E}+{\mathit{V}}_{\mathbf{HT}}\times \mathit{B}=0,\end{array}$$(1)

where V_HT is the transformation velocity. We calculated the HT frame velocity for every discontinuity with λ_int/λ_min > 20 by minimizing the residual electric field (Paschmann & Daly 1998) using magnetic field and plasma velocity measurements from a one-minute time window around the DD and used them to calculate the −V × B as a proxy for the electric field. Only events were selected for further analysis whose correlation coefficient between the electric field calculated in the spacecraft and the HT frame (HT_cc) was larger than 0.9. Out of the 3479 DDs with λ_int/λ_min > 20, 19 702 met the HT_cc threshold criterion. After this the Walén tests were carried out in the HT frame which states that in the case of an RD, the $\mathit{V}-{\mathit{V}}_{\mathbf{HT}}{=}_{-}^{+}{\mathit{V}}_{\mathbf{A}}$ is satisfied, where V_A is the Alfvén velocity. In practice, we followed Paschmann & Sonnerup (2008) and Paschmann et al. (2013) and created a scatterplot of V − V_HT components versus local V_A components calculated from the magnetic field and the proton moment plasma density data. When the correlation coefficient (|W_cc|) was greater than 0.9, the slope (W_S) of the plot was calculated, which should be ±1 in the case of an ideal RD. In reality, events tend to have a much shallower slope. In our study, DDs with |W_S|> 0.5, were therefore considered an RD. When the correlation on the Walén scatterplot was above the threshold, but the slope did not meet the |W_S|> 0.5 condition, the event was selected as a TD. Figure 2 shows the distribution of events in terms of HT_cc, |W_cc| and W_S. The histograms show that most of the events that met the HT_cc > 0.9 condition also satisfied the |W_cc|> 0.9 requirement, which means that most of the selected DDs showed an Alfvénic nature, regardless of whether they were TDs or RDs, in agreement with earlier studies (Neugebauer et al. 1984; Paschmann et al. 2013). An interesting feature is the relatively large number of events with |W_S|> 1. This is hard to interpret. Maybe an acceleration takes place at these structures.

Fig. 2. Histograms of the quality parameters for the HT analysis and the Walén test. Left: histogram of the HT fit correlation coefficient for DDs with λint/λmin > 20 MVA normals. Middle: histogram of the Walén-scatterplot correlation coefficient magnitude of events with both λint/λmin > 20 and HTcc > 0.9. Right: distribution of Walén slope magnitude of DDs that met all the previous conditions.

Another realization of the Walén test can be made by using its jump condition form in the spacecraft frame. In this case, the form of the relation is $\mathrm{\Delta }V{=}_{-}^{+}\mathrm{\Delta }{V}_{\mathrm{A}}$ with ΔV = V_upstream − V_downstream and ΔV_A = V_A upstream − V_{A downstream}. The difficulty with this form is that the directional change (θ) and the magnitude change in the vector difference are both important. To simplify the classification, we used the single quality index Q, introduced by Sonnerup et al. (2018). This is defined by

$$\begin{array}{c}\hfill Q\pm =\pm (1-\left(\frac{|\mathrm{\Delta }V\mp \mathrm{\Delta }{V}_{\mathrm{A}}|}{|\mathrm{\Delta }V|+|\mathrm{\Delta }{V}_{\mathrm{A}}|}\right))\end{array}$$(2)

and the positive sign is used for cases when cosθ > 0 and the negative sign for cosθ < 0. The |Q| = 1 value corresponds to ideal Alfvénic events. Following Paschmann et al. (2020), we chose |Q| = 0.7 as the threshold for RDs. Figure 3 shows the |Q|−|W_S| scatterplot for the events with acceptable HT frames and Walén correlation. After the two-threshold filtering, we still had 16 860 potential RDs.

Fig. 3. Scatterplot of the Q factor from the jump version and the Walén slope from the HT version of the Walén test for the preselected DDs.

Based on the magnetic field classification and the Walén tests for the DDs, we combined the two methods for all discontinuities with HT_cc > 0.9. We used the Smith classification for DDs with a high B_n/B_max value (RDs) or with a large magnitude jump ([B]/B_max > 0.2, TDs). The Walén test was used for events that were originally classified as ED (B_n/B_max < 0.4) and ([B]/B_max < 0.2). The middle row of Table 1 shows the requirements for this combined classification scheme. When we did not find a good correlation in the Walén scatterplot or found it, but both the Walén slope and the Q factor were below the RD threshold (|W_s|< 0.5 and |Q|< 0.7), we marked the event as a TD. If the |W_cc|> 0.9 and both the |W_s|> 0.5 and |Q|> 0.7 conditions were met, the DD was considered an RD. In every other parameter combination of the classification, we dropped the DD in the ED+ND category.

In this way, we identified 16 746 RD candidates, only 268 TDs, and 780 ED+NDs. The very low number of TDs may suggest that the Walén test incorrectly classified many EDs as RDs. To check this possibility, we derived the normal component of the magnetic field for the different types of DDs we just classified. A possible bimodal or multimodal distribution of parameters within a single DD type could suggest some misclassification in the first round. Figure 4 shows the radial change in this parameter for RD candidates (left), TDs (middle), and the ED/ND class (right). The distribution of RD candidates is clearly bimodal: one peak with nearly field-aligned normals, and another with nearly perpendicular ones. Only a few events lie in the intermediate zone.

Fig. 4. Normal component of the magnetic field for the different type of DDs as a function of heliocentric distance. From left to right: RD candidates, TDs and EDs + NDs.

If the RDs are produced during Alfvén-wave steepening, they should not have B_n/B_max ≪ 1. On the other hand, if they are formed in Alfvénic turbulence, the normals should be distributed more randomly (Vasquez et al. 2007). To understand these findings, we should consider that waves propagating along the surface of TDs can also produce Walén test results showing Alfvénic behavior. Therefore, we assumed that the positive Walén tests in these cases are due to surface waves superposed on TDs. To verify this hypothesis, we examined the velocities in the HT frame ((V − V_HT)). For surface waves, this vector should be in (close to) the DD surface, for real RDs this velocity is quasi-perpendicular to the DD surface. Thus, when both B_n/B_max and ((V − V_HT)⋅n)/|V − V_HT| are small, the candidate is a TD with surface waves; when both parameters are large, it is a real RD. Figure 5 shows a scatter plot of the two quantities for the RD candidates in the combined classification. It is clear that two very distinct populations are present: one with low B_n/B_max and low ((V − V_HT)⋅n)/|V − V_HT| values, and one with both parameters close to unity. We therefore conclude that the DDs in the former group are, in reality, TDs with surface Alfvén waves propagating along the discontinuity plane. On the other hand, the latter are real RDs propagating relative to the solar wind plasma with the Alfvén velocity. Only a few hundred events are scattered between these two groups, and we can therefore use one of these parameters, for example, B_n/B_max, to classify these DDs (former RD candidates). We chose B_n/B_max > 0.4 as a threshold for our second round of classification. We have shown that below this threshold, all DDs are TDs, regardless of the jump in field magnitude. We list in the right column of Table 1 the final classification scheme and thresholds. This changes the relative number of DD types dramatically: there are 6992 TDs, 10 855 RD, no EDs, and only 9 ND events.

Fig. 5. 2D histogram for DDs originally selected as RD candidates in the first round of classification based on the Walén test. The horizontal axis shows the normalized normal component of the magnetic field vector across the DD, and the vertical axis marks the normalized normal component of the fluctuation velocity in the HT frame.

Earlier studies have found that even with the application of the triangulation method, it is not possible to distinguish between a TD and an RD at a heliocentric distance of 1 AU if the RD has small B_n/B_max parameter (Knetter et al. 2004; Wang et al. 2024). This is mainly due to the low Alfvén velocity at 1 AU, which prevents the triangulation method from detecting whether a DD is propagates relative to the solar wind. We argue that in the vicinity of the Sun, the high propagation velocity of the DDs that are finally classified as RDs is a solid proof of their RD nature. This follows from the requirement that RDs must have a good Walén correlation above the threshold, which (because the Alfvén speed is high near the Sun) also implies a high propagation speed relative to the solar wind. This also means that the uncertainty of the Walén classification decreases closer to the Sun. The clear separation of the two populations shown in Figure 5 strongly suggests that the small B_n/B_max DDs are mostly TDs everywhere, and thus our new classification scheme, which only uses the magnetic field data, is able to distinguish the two types even for larger solar distances. An important step toward understanding the origin and evolution of the interplanetary discontinuities would be a thorough investigation of their internal structure. Artemyev et al. (2019) demonstrated at a heliocentric distance of 1 AU that the internal structure of DDs is dual, with a thin intense current layer embedded within a thick current sheet for which the Walén relation holds. They stated that this structure endows the discontinuity with both RD and TD-like properties simultaneously, but we argue that these DDs are in fact TDs on which surface Alfvén fluctuations are superposed. This can explain both the jumps in the plasma parameters that (Artemyev et al. 2019) found in the thin current sheets and the RD-like behavior of the outer layer.

3.3. Properties of rotational and tangential discontinuities

After selecting and classifying DDs, we calculated the thickness of the transition zone for each type. Just as was the case with the upstream and downstream borders, the exact borders of the transition zone are hard to find automatically. Our algorithm selected the first points where the attitude gradient is lower than a_max/2 on either side of a_max as the position of the transition zone borders. To determine the thickness of an event, this time duration was multiplied with the convection velocity V_SW − V_spacecraft in the case of TDs, and as V_HT − V_spacecraft when it is an RD, and it was then corrected for the DD normal direction.

The second classification seems to be justified in Figures 6 and 7. The left panel of Figure 6 shows the thicknesses of RDs as a function of the distance from the Sun. An interesting feature of the distribution is the decrease of the thicknesses to approximately 0.3 AU, and then perhaps a small increase. In the case of TDs (right panel), the trend is more direct: An increase can be seen all the way from 0.1 AU (with a few hundred km) to 1.0 AU (where, similarly to the case of RDs, TDs have thicknesses of about 500–1000 km).

Fig. 6. Calculated thickness of the DDs using the second classification. The horizontal axis shows the radial distance from the Sun, and the vertical axis is the DD thickness in kilometers.

Fig. 7. Same calculated thicknesses of the DDs in ion inertial length units.

Expressing these thicknesses in terms of some characteristic length of the plasma can shed light on the underlying physics. An obvious choice is the ion inertial length, d_inertial, which measures the spatial scale at which the ions are demagnetized and the magnetic field is frozen into only the electron fluid, MHD therefore no longer describes the behavior of the plasma on this scale. The ion inertial length is inversely proportional to the square root of the plasma ion density. A problem with the comparison of the DD width and the ion scale data is the uncertainty in the determination of both. First, our automatic selection of the transition zone is quite arbitrary and we therefore only expect the values to be in the right order of magnitude. Plasma measurements, which are required for calculating the ion inertial length, can also have relatively large uncertainties. Figure 7 shows the same plot of thicknesses and radial distance as Figure 6, but with ion inertial length units for the thickness. In this length unit, the RD thickness decreases rapidly until the distance from the Sun reaches 0.3 AU. After this, the thickness (in d_inertial units) is constant, which means that it scales very well with the ion inertial length, and most of the events stay in the [0, 20] interval. This can be understood when we assume that these structures emerge from the steepening of large-amplitude Alfvén waves. In this case, we expect that the RD thickness decreases until kinetic processes, such as ion-wave dispersion, stop this decrease. Cohen & Kulsrud (1974) found that this change in behavior occurs at about 0.2 AU, which is compatible with our results. We return to this explanation in the next paragraph. As expected, TD thicknesses (right panel) seem to be more correlated to d_inertial at all the radial distances, most of them are within a few dozen of local ion inertial lengths.

Lotekar et al. (2022) reported that the typical thickness of current sheets in the near-Sun solar wind (between 0.17 and 0.24 AU) is in the range from about 0.1–10 ion inertial lengths. This is very similar to our results for TDs, especially when we take into account that the data used in the aforementioned study have a much higher temporal resolution (varying between 73 and 290 Hz), and that their selection method is fine-tuned to find the thinnest coherent structures that are still resolvable by PSP magnetic field measurements, by selecting for strong and very fast variances of the magnetic field. This very specificity also means that their method does not target the other DD population, however, which we have found to be also present in the near-Sun region. Figures 6 and 7 show that this other population (the RDs) is characterized by a much higher thickness near the Sun, and thus, it does not show up in a method that searches for thin current sheets. In view of our findings, the thin and strong current sheets found by Lotekar et al. (2022) are probably associated with tangential discontinuities, and we have shown that these structures have typical thicknesses of about the ion inertial length not only near the Sun, but in the entire inner heliosphere.

The classification also provides the opportunity of investigating the angular distribution of the DD normals in the inner heliosphere. Lukács & Erdős (2013) found that most of the TDs in the 0.3–1.0 AU region tend to have normals that are normal to the local Parker spiral direction. Liu et al. (2021) reported similar results while investigating the inner heliosphere 0.12–0.80 AU and concluded that RD normals are likely be aligned with the local Parker direction. Using PSP and Solar Orbiter data from 0.06 to 1.01 AU, we repeated the same analysis, but with the correction for the plasma flow direction, which is a combination of the solar wind flow direction and the spacecraft velocity direction. The local Parker spiral direction was determined using a 900 s time window centered on the DDs. These results are shown in Figure 8. It is clear that the RD normals are reasonably well aligned with the Parker direction, but the normals of TDs have a large angle with respect to the local Parker spiral direction. The fact that the two populations show such a distinctly different behavior reconfirms our DD classification. This result agrees with the idea that the majority of TDs are boundaries between flux tubes because the flux tube boundaries are aligned with the Parker spiral on average. It is still debated whether these flux tubes originate from the solar photosphere, as the simplified ‘spaghetti solar wind’ paradigm suggests, or if turbulent mixing creates them. Based on our observation, we cannot rule out any of these possibilities. The RD normals, on the other hand, are mostly well aligned with the local magnetic field, which agrees with the Alfvén-wave steepening mechanism.

Fig. 8. Distribution of the angles between the DD normals combined from PSP and Solar Orbiter data, and the direction of the calculated local magnetic field. The left panel shows the distribution corresponding to the RDs, and the right panel shows the TD distribution.

4. Conclusions

1. The number of these structures increases as we approach the Sun. This trend is apparently not affected by solar wind speed or other plasma parameters, but an important feature is that there tend to be more DDs in the slow solar wind than in the fast streams.

2. We used two versions of the Walén test to classify the EDs (either discontinuities – DDs, which former classification methods could not classify). The Walén test gave similar results for the Q factor in the spacecraft frame or the Walén slope in the HT frame. The large number of events with positive Walén tests and small B_n/B_max indicates that this method has to be used carefully to avoid confusing RDs and TDs with surface Alfvén waves. After a thorough examination of EDs, we found that they are indeed mostly TDs with small [B]/B_max on which surface waves are superimposed. This finding is supported by the distributions of the field normals and the remaining velocity in the HT frame and also by the alignment of DDs with respect to the local magnetic field.

3. Based on these results, we find that the RDs and TDs in the inner heliosphere can be clearly separated based on the magnetic field measurements alone. We therefore suggest a very simple classification with only three categories (no need for EDs): all DDs with B_n/B_max < 0.4 are TDs, those with B_n/B_max > 0.4 and [B]/B_max < 0.2 are RDs, and a few NDs remain that are defined by B_n/B_max > 0.4 and [B]/B_max > 0.2.

4. We determined the thickness of the RDs, and found that up to 0.3 AU the thickness tends to decrease as the RDs move away from the Sun. From 0.3 AU to 1.0 AU, a decrease and then an increase can be seen. This suggests that these DDs are probably produced by Alfvén-wave steepening, which occurs in the innermost heliosphere. After the thickness reaches the ion inertial scale, ion kinetic effects balance the structure, and thus, its thickness scale with the characteristic ion inertial length. In the case of the TDs, this scaling can be seen at all radial distances that we studied. The directional distribution of the DDs classified as TDs in the innermost region (0.06–0.30 AU) supports the idea that these are mostly boundaries of flux tubes that approximately follow the theoretical Parker-spiral. We found that the TD normals tend to be normal to the calculated local magnetic field direction throughout the inner heliosphere. The normals of RDs are well aligned with the local magnetic field direction, which agrees well with their origin as steepening Alfvén-waves.

Acknowledgments

We would like to acknowledge the Solar Orbiter and Parker Solar Probe instrument teams for providing the dataset used in this study. Data was obtained through ESA Solar Orbiter Archive, NASA Space Physics Data Facility (https://nssdc.gsfc.nasa.gov), and AMDA science analysis system provided by the Centre de Données de la Physique des Plasmas (CDPP) supported by CNRS, CNES, Observatoire de Paris and Université Paul Sabatier, Toulouse. The study was funded by the National Research, Development and Innovation Office – NKFIH, grant FK128548.

References

All Tables

All Figures

	Fig. 1. Occurrence rate of the DDs measured by Solar Orbiter and PSP as a function of the distance from the Sun in a logarithmic plot. Each dot corresponds to a daily rate obtained from the spacecraft measurement, colored according to the average daily radial solar wind velocity. The black line marks the power-law function fit for the data.
In the text

	Fig. 2. Histograms of the quality parameters for the HT analysis and the Walén test. Left: histogram of the HT fit correlation coefficient for DDs with λint/λmin > 20 MVA normals. Middle: histogram of the Walén-scatterplot correlation coefficient magnitude of events with both λint/λmin > 20 and HTcc > 0.9. Right: distribution of Walén slope magnitude of DDs that met all the previous conditions.
In the text

	Fig. 3. Scatterplot of the Q factor from the jump version and the Walén slope from the HT version of the Walén test for the preselected DDs.
In the text

	Fig. 4. Normal component of the magnetic field for the different type of DDs as a function of heliocentric distance. From left to right: RD candidates, TDs and EDs + NDs.
In the text

	Fig. 5. 2D histogram for DDs originally selected as RD candidates in the first round of classification based on the Walén test. The horizontal axis shows the normalized normal component of the magnetic field vector across the DD, and the vertical axis marks the normalized normal component of the fluctuation velocity in the HT frame.
In the text

	Fig. 6. Calculated thickness of the DDs using the second classification. The horizontal axis shows the radial distance from the Sun, and the vertical axis is the DD thickness in kilometers.
In the text

	Fig. 7. Same calculated thicknesses of the DDs in ion inertial length units.
In the text

	Fig. 8. Distribution of the angles between the DD normals combined from PSP and Solar Orbiter data, and the direction of the calculated local magnetic field. The left panel shows the distribution corresponding to the RDs, and the right panel shows the TD distribution.
In the text