Temperature Intermittent Structures in the Fast Solar Wind

We use the measurements from SolO between 2020 September and 2022 September. During this period, SolO was in its cruise phase at heliospheric distances between 0.35 and 1 au. The ion velocity distribution functions (VDF) and ion moments are provided by Solar Wind Analyser (SWA; Owen et al. 2020) instrument suite, specifically the Proton Alpha Sensor (PAS). The SWA-PAS measures the 3D VDF over an interval of 1 s every 4 s., which leads to the 4 s cadence of the ion data. The high time resolution of the VDF data provides us a very good opportunity to study the nature and kinetic effect of the intermittent structures at small scales in the inertial range. The SWA-PAS data are required to have a quality factor <0.2, in order to guarantee the reliability of data. The magnetic field data are obtained from the fluxgate magnetometer instrument (Horbury et al. 2020). The time resolution of the magnetic field data is 0.125 s in the normal mode. Following Opie et al. (2022), Opie et al. (2023), we average the magnetic field over each 1 s VDF measurement interval from SWA-PAS, so that the magnetic field data and the ion data have the same sampling time sequence.

From the 2 yr observations of SolO in the solar wind turbulence, we find eight fast-wind streams with the bulk flow velocity V_SW > 500 km s⁻¹. The beginning and ending times of the streams are listed in Table 1, together with other average parameters of them, including heliocentric distance, magnetic strength ∣ B ∣, bulk flow velocity V_SW, proton number density N_p , and proton temperature T_p . They were observed between 0.59 and 1.01 au, and each of the streams lasts about 1–4 days.

Figure 1 shows one of the selected fast-wind streams (stream No. 2 in Table 1) observed from 12:00:00 UT on 2021 July 6 to 00:00:00 UT on 2021 July 9, when SolO was located at 0.89 au. The top three panels are the time variations of the three components of magnetic field ( B , black) and bulk flow velocity ( V _SW, blue) vectors in radial, tangential, and normal (RTN) coordinates. Panel (d) shows the magnetic field strength (∣ B ∣, black) and proton number density (N_p , blue). During the 2.5 days interval, the bulk velocity V_SW keeps higher than 500 km s⁻¹, and neither interplanetary coronal mass ejections nor heliospheric CSs appear in the data set. So this interval is a typical fast-wind stream. Panel (e) shows proton temperature (T_p , gray) and its low-pass (<0.125 Hz) time series (${T}_{p}^{{\prime} }$, black). The low-pass proton temperature ${T}_{p}^{{\prime} }$ will be used in the following analysis.

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We perform a wavelet analysis on the proton temperature ${T}_{p}^{{\prime} }$, in order to illustrate the fluctuation energy at different time instants and scales. We calculate the wavelet coefficient ω(t, τ) of ${T}_{p}^{{\prime} }$ as a function of time t and scale τ within the scale domain from 16 to 100 s at 32 scales, which all belong to the inertial range. The mother wavelet used in this study is the Morlet wavelet (Torrence & Compo 1998; Horbury et al. 2008). Based on the wavelet coefficients, a variable called the local intermittency measure (LIM) is introduced to qualitatively illustrate intermittency (Farge 1992; Bruno et al. 2001; Perrone et al. 2016; Wang et al. 2019), which indicates that the energy is not evenly distributed in space. The variable LIM is defined as

$$\begin{eqnarray}&&\mathrm{LIM}(t,\tau )=\displaystyle \frac{| \omega (t,\tau ){| }^{2}}{{\left\langle | \omega (t,\tau ){| }^{2}\right\rangle }_{t}},\end{eqnarray} \tag{ 1 }$$

where ${\left\langle ...\right\rangle }_{t}$ denotes an ensemble time average. It corresponds to normalized wavelet energy. Panel (f) of Figure 1 shows the variable LIM of ${T}_{p}^{{\prime} }$ as a function of t and τ. It is clear that there appear lots of red stripes in the LIM plot, which correspond to large-amplitude fluctuations in the proton temperature. The stripes distribute intermittently in time, and each of them spread over several scales. The stripes indicate that the wavelet energy of the proton temperature is nonhomogeneously distributed, and the proton temperature is apparently intermittent in the plotted interval.

We also calculate the normalized partial variance of increments for the proton temperature (PVI_T ). Following previous studies (Marsch & Tu 1994; Greco et al. 2008; Osman et al. 2011; Wang et al. 2013), the parameter PVI_T as a function of time t and scale τ is given by

$$\begin{eqnarray}&&{\mathrm{PVI}}_{T}(t,\tau )=\displaystyle \frac{\delta T{{\prime} }_{p}(t,\tau )}{\sqrt{{\left\langle {\left|\delta T{{\prime} }_{p}(t,\tau )\right|}^{2}\right\rangle }_{t}}},\end{eqnarray} \tag{ 2 }$$

where ${T}_{p}^{{\prime} }(t)$ is the time series of the low-pass (<0.125 Hz) proton temperature, and ${T}_{p}^{{\prime} }(t)={T}_{p}^{{\prime} }(t+\tau )-{T}_{p}^{{\prime} }(t)$ is the low-pass temperature difference between the data points at two time instants with a temporal separation τ. It is worth noting that we use the low-pass proton temperature ${T}_{p}^{{\prime} }(t)$ instead of the original time series T_p , when calculating PVI_T . The purpose is to avoid the high-frequency noise of the measurements, so that the resulted signal can reflect the imbedded intermittency corresponding to the higher values of PVI_T more precisely. Here, the time lag τ is selected as 24 s, which belongs to the inertial range following Wang et al. (2013), Wang et al. (2023). In the following, we will refer to PVI_T (t, τ) as PVI_T (t) for simplicity without further specification. Panel (g) of Figure 1 shows the time series of PVI_T (t) for the plotted interval. Many spikes appear in the time series of PVI_T (t), which correspond to large-amplitude fluctuations imbedded in the background proton temperature.

In panel (h) of Figure 1, we show perpendicular (T_⊥) and parallel (T_∥) proton temperatures in green and yellow, respectively. The parallel and perpendicular indicate directions with respect to the magnetic field averaged over each 1 s VDF measurement interval as mentioned above. Following Opie et al. (2022), the perpendicular and parallel temperatures are obtained by rotating the proton pressure tensor in RTN frame to align with the 1 s averaged magnetic field. They both keep larger than 5 eV, and T_∥ is generally larger than T_⊥ at most of the time. Panel (i) of Figure 1 shows the time variations of the proton temperature anisotropy T_⊥/T_∥. We see that it is mostly smaller than 1 (horizontal dashed line) as expected.

In the left panel of Figure 2, we show the 2D distribution of temperature anisotropy T_⊥/T_∥ and parallel plasma beta β_∥ = 8π N_p k_B T_∥/∣ B ∣² for the selected eight fast streams. Here, N_p and B are the 4 s cascade proton number density and magnetic field, respectively, and k_B is the Boltzmann constant. The solid and dashed black curves denote the thresholds of MM instability and OF instability, respectively. It is known that the MM instability competes with the Alfvén ion cyclotron (A/IC) instability, which dominates at lower ion plasma beta values (Gary 1992). Therefore, the instability thresholds for A/IC (solid gray curve) and fast-magnetosonic/whistler (FM/W, dashed gray curve, Verscharen et al. 2016) are also plotted for reference. The thresholds are adopted from Hellinger et al. (2006), Verscharen et al. (2016). They can be obtained from the following analytical relation

$$\begin{eqnarray}&&\displaystyle \frac{{T}_{\perp }}{{T}_{\parallel }}=1+\displaystyle \frac{a}{{({\beta }_{\parallel }-{\beta }_{0})}^{b}},\end{eqnarray} \tag{ 3 }$$

where for the MM instability, a = 0.77, b = 0.76, and β₀ = − 0.016; for the OF instability, a = − 1.4, b = 1.0, and β₀ = − 0.11; for A/IC instability, a = 0.43, b = 0.42, and β₀ = − 0.0004;for FM/W instability, a = − 0.497, b = 0.566, and β₀ = 0.543. The maximum growth rate for the instability thresholds is ${\gamma }_{\max }={10}^{-3}{{\rm{\Omega }}}_{p}$, where Ω_p is the proton gyrofrequency. The contours are also superposed on the 2D distribution. We see that most of the data points are located at the red-patch area with 0.5 < T_⊥/T_∥ < 1 and 0.7 < β_∥ < 2. In general, the proton temperature anisotropy in our observations in the fast streams is constrained by the MM instability (solid black line) and the OF instability (dashed black line), which is consistent with previous studies (e.g., Hellinger et al. 2006; Bale et al. 2009; Chen et al. 2016; Opie et al. 2023).

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Based on the data from SolO, the plasma conditions for proton temperature-anisotropy-driven MM and OF instabilities to occur in the solar wind are investigated by Opie et al. (2022), Opie et al. (2023). It is evident that a large number of data points are well beyond the instability threshold (i.e., Figure 2 in Opie et al. 2022; and Figure 1 in Opie et al. 2023). Nevertheless, only a few pixels are marginally above the threshold in our study as shown in the left panel of Figure 2. The main reason for the difference is that we only analyze the events found in the fast-wind streams with the bulk flow velocity V_SW > 500 km s⁻¹ here, and do not investigate the events in the slow wind for the moment. The different feature of the fast and slow streams in the [T_⊥/T_∥, β_∥] plane has been illustrated in Hellinger et al. (2006), in which it is clear that the 2D distribution of [T_⊥/T_∥, β_∥] is more concentrated in the fast streams than in the slow ones.

In order to identify the temperature intermittency, we demonstrate the probability distribution function (PDF) of PVI_T for stream No. 2 in Figure 3. The black curve is a Gaussian fit for the PDF. It is clear that the profile of the PDF deviates significantly from the Gaussian distribution, and has long tails when the absolute value of PVI_T increases. The Gaussian distribution is located between the PVI_T range [−3, 3]. Beyond this range, the observed PDF curve exhibits long tails, and the tails extend beyond [−7, 7]. The long tails of the non-Gaussian PDF profile indicate the existence of intermittency. Then, the PVI_T range [−3, 3] is used to select the temperature intermittency (in red), which corresponds to the non-Gaussian tails of the PDF.

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We calculate the flatness for the distribution as follows: $F=\langle {({\mathrm{PVI}}_{T}{(t))}^{4}\rangle /\langle ({\mathrm{PVI}}_{T}(t))}^{2}{\rangle }^{2}$, where 〈...〉 denotes an ensemble average in the given stream. The value of the flatness is marked in the upper right corner of Figure 3, as F = 7.94. It is much larger than 3 (characteristic of a standard Gaussian distribution). It again indicates the temperature fluctuations are highly intermittent.

The criterion ∣PVI_T (t)∣ > 3 is applied for the basic identification of temperature intermittency. The procedure used here to find the temperature intermittency is similar to that used to find the magnetic intermittent structures described in Wang et al. (2023). First, we find the time instants that satisfy the condition as follows: ∣PVI_T (t)∣ > 3. Only if the number of the continuous instants is not smaller than 3, they are chosen for the following study. A continuous series of the intermittent instants is considered as an intermittent case. Moreover, if the number of the instants between two adjacent cases is smaller than 3, the two cases and the time instants between them are merged together and are seen as one "long-lived" case. For a given case, we use t_B and t_E to denote the beginning time and ending time of the case, respectively. An interval between [t_B − 150 s, t_E + 150 s] is called an intermittent interval, and will be illustrated below. The interval between [t_B −150 s, t_E + 150 s] is chosen, since it is a suitable duration to present the profile of time variations. During this interval, the fluctuations of the intermittent structure imbedded in the middle can be seen clearly, and the two wings are relatively calmer. After the constrain of 10% data gap, we find 498 temperature intermittent intervals from the eight fast streams.

In the right panel of Figure 2, we show the normalized distribution in the T_⊥/T_∥ − β_∥ plane for all the selected temperature intermittent cases. In each pixel, the color denotes the number of data points related to temperature intermittency in the T_⊥/T_∥ − β_∥ plane normalized by the number of the corresponding pixel in the left panel. We find that a majority of the intermittency is located very close to the thresholds of proton MM instability. Some other data points are located at the low-beta boundary of the 2D distribution. One possible mechanism for the temperature anisotropy constraint in the solar wind with low beta β_∥ < 1 is Coulomb collisions (Vafin et al. 2019). Another possible mechanism could be α A/IC instability induced by alpha beams from private communication with Dr. Liang Xiang (the related work is still under review). Next, we will focus on studying the nature and kinetic effect of the selected temperature intermittency.

For the type of the bump cases, we can see a clear signal of a local temperature increase when the temperature intermittency happens. Among these cases, some of them show little variation of magnetic field direction on the two sides of the intermittency, which are considered here to be associated with linear magnetic holes (LMHs; "bump-LMH" hereafter for simplicity). While some of them show a direction reversal, which are identified to be related to CS ("bump-CS" hereafter).

3.1.1. Bump-LMH

Figure 4 shows a typical bump-LMH case observed by SolO at 08:05:04 UT–08:10:37 UT on 2022 January 2. Panel (a) shows the time series of the original proton temperature T_p (black) and the low-pass proton temperature ${T}_{p}^{{\prime} }$ (magenta). Panel (b) shows the time series of PVI_T calculated from Equation (2). The two vertical dotted lines mark the beginning time (t_B = 08:07:32 UT) and ending time (t_E = 08:08:04 UT) of the temperature intermittent case, respectively. We see that, between the two vertical lines, ∣PVI_T ∣ is larger than 3 (horizontal dashed lines in panel (b)) during the first 12 s and the ending 12 s. There are only two data points between them that have ∣PVI_T ∣ < 3. According to our criteria of the intermittency election as mentioned in Section 2.2, the data between t_B and t_E are considered as a typical temperature intermittent case. The horizontal dotted line in panel (b) denotes PVI_T = 0.

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From the profile of the PVI_T time series shown in magenta in panel (b), we see clearly that it has a "bipolar" feature between the two vertical dashed lines. The bipolar magenta curve crosses the horizontal dotted line (PVI_T = 0) once at the magenta dot. When it crosses the zero line, its slope has a negative value. The bipolar feature and the once zero-crossing with the negative slope are resulted from the "bump" appearing in the curve of the proton temperature in panel (a), which is a signal of local temperature increase related to the background plasma. Here, the background proton temperature outside the intermittency T_pout is recorded as 33.9 eV, and is calculated from the average of the proton temperature at the upstream ([t_B − κ d, t_B ]) and downstream ([t_E , t_E + κ d]) of the intermittency, where d = t_E − t_B corresponds to the width of the intermittency, and κ = 0.2 ∼ 1.0. The value of κ is set according to the length of the adjacent quiet period with no large variations. In most of the cases, we set κ as 1. The temperature inside the intermittent interval T_pin is recorded as 42.8 eV, which corresponds to the proton temperature of the time instant (denoted by the short vertical dashed line in panel (a)) that has maximum value of ${T}_{p}^{{\prime} }$ between [t_B , t_E ]. Therefore, in this case, an "once-zero-crossing" profile of PVI_T with negative slope when crossing corresponds to a signal of the local proton temperature increase from T_pout = 33.9 eV, to T_pin = 42.8 eV. The times of "zero-crossing" for the PVI_T curve are recorded as N_T = −1, where "−" represents the sign of the slope when the PVI_T curve crosses the horizontal zero line.

In order to reveal the nature of the intermittency, we look at the fluctuations in the magnetic field fluctuations during the plotted interval. Panels (c)–(e) of Figure 4 present the time variations of the three components of the magnetic field vector (black) and the proton velocity vector (blue) in RTN coordinates. Panel (f) shows the time variations of the magnetic strength (∣ B ∣, black) and proton number density (N_p , blue). We apply minimum variance analysis (MVA; Sonnerup & Cahill 1967) on the magnetic field vectors between [t_B − κ d, t_E + κ d] (two short vertical dashed lines in panel (g), and κ = 1, here). The magnetic field components in the maximum-intermediate-minimum variance (LMN) coordinates are presented in panel (g). For the magnetic field component in the maximum variance direction (B_L , purple), we also calculate the normalized partial variance of increments (PVI _B ) from

$$\begin{eqnarray}&&{\mathrm{PVI}}_{{\boldsymbol{B}}}(t,\tau )=\displaystyle \frac{\delta {B}_{L}(t,\tau )}{\sqrt{{\left\langle {\left|\delta {B}_{L}(t,\tau )\right|}^{2}\right\rangle }_{t}}},\end{eqnarray} \tag{ 4 }$$

and show it in panel (h). It is clear that the PVI _B profile also has a "bipolar" feature, and the curve also crosses the horizontal zero line once at the purple dot. The times of "zero-crossing" for the PVI _B profile are recorded as N_B = 1. It is not necessary to put the sign "+" or "−" in front of it as N_T , since the magnetic field is a vector instead of a scalar like the proton temperature. This characteristic of N_B = 1 is due to a "bump" appearing in the B_L profile from −4.1 to −0.8 nT and then back to −4.5 nT, but the direction of B_L does not change. It indicates a small-angle variation of the magnetic field vector, instead of a sudden jump. Meanwhile, the magnetic strength shown in panel (f) has a dip.

The behaviors of B_L (i.e., N_B = 1) and ∣ B ∣ also indicate that this case could be an LMH as discussed in many previous studies (e.g., Turner et al. 1977; Winterhalter et al. 1994; Zhang et al. 2008; Xiao et al. 2010; Volwerk et al. 2021), and it is accompanied by the local proton density increase (blue curve in Figure 4(f)) and proton temperature increase (i.e., N_T = −1, see Figures 4(a), (b)).

For a thorough understanding of the relation between temperature fluctuations and magnetic holes, we investigate the intrinsic properties (including scale, depth, and rotation) of the bump-LMH-type case following previous studies on magnetic holes (e.g., Sperveslage et al. 2000; Burlaga et al. 2007; Zhang et al. 2008; Yao et al. 2017; Huang et al. 2021; Karlsson et al. 2021; Wu et al. 2021). In Figure 4, the width of the LMH is d = t_E − t_B = 32 s, corresponding to 90.2 ρ_i , where ρ_i is the average ion gyroradius in the plotted interval [t_B − 150 s, t_E + 150 s]. It is a typical scale of a magnetic hole at 1 au in the solar wind as mentioned in Burlaga et al. (2007). The depth of a magnetic hole is defined as $A=| {\boldsymbol{B}}{| }_{\min }/| {{\boldsymbol{b}}}_{0}|$, where $| {\boldsymbol{B}}{| }_{\min }$ is the minimum field magnitude inside a hole [t_B , t_E ]. The depth of the case shown in Figure 4 is 0.2. The rotation angle ω across the hole is as small as 27, which is calculated from the angle between the average fields in the upstream [t_B − d, t_B ] and downstream [t_E , t_E + d] of the hole. The depression of the magnetic field magnitude and the little variation of the magnetic field direction are typical characteristics of LMH as mentioned in Burlaga et al. (2007), Karlsson et al. (2021).

For this case, we check the MVA eigenvalue ratios, as well as the directions between the eigenvectors and local mean magnetic field. The local mean magnetic field b ₀ is determined as the mean field in the upstream ([t_B − d, t_B ]) and downstream ([t_E , t_E + d]) periods of the case. The ratio between the eigenvalues in the minimum and intermediate variance directions is ${\lambda }_{\min }/{\lambda }_{\mathrm{int}}=0.04$, which indicates that the minimum variance direction is well defined. In previous studies (e.g., Sonnerup & Cahill 1967; Lepping & Behannon 1980; Sonnerup & Scheible 1998; Knetter et al. 2004), a higher limit of ${\lambda }_{\min }/{\lambda }_{\mathrm{int}}=0.5$ is often used when estimating the discontinuity normally by MVA. If the ratio ${\lambda }_{\min }/{\lambda }_{\mathrm{int}}$ is too high, the estimate of the normal could be impaired by superposed fluctuations. The ratio between the eigenvalues in the intermediate and maximum variance directions is ${\lambda }_{\mathrm{int}}/{\lambda }_{\max }=0.32$, implying the presence of 2D fluctuations according to Perrone et al. (2020). Namely, the magnetic hole has a cylinder topology as illustrated in Figure 18(b) of Perrone et al. (2016). The angle between b ₀ and the maximum variance direction L (θ_BL) is obtained as 658, which means the case is compressible, while the minimum N and intermediate M variance directions are found to be both nearly perpendicular to b ₀ with θ_BN = 8838, and θ_BM = 8863.

Panel (i) of Figure 4 shows the time variations of the temperature anisotropy T_⊥/T_∥ (black) and the parallel plasma beta β_∥ (blue). We notice that, within the intermittent case, there is a clear bump in the profile of T_⊥/T_∥. It is about 0.87 for the background plasma (at the upstream ([t_B − d, t_B ]) and downstream ([t_E , t_E + d]) of the intermittency), and increases 35.6% to 1.18 inside the intermittent case. The profile of β_∥ shows a similar behavior as that of T_⊥/T_∥. The value of β_∥ is about 2.0 for the background plasma, and increases dramatically to 78.8 inside the intermittent case. The large values of T_⊥/T_∥ and β_∥ are typical characteristics of MMs (Chandrasekhar et al. 1958; Hasegawa 1969; Tsurutani et al. 1982; Southwood & Kivelson 1993). Accordingly, we speculate that the LMH case is likely related to MMs.

Panel (k) and panel (l) of Figure 4 show the $\mathrm{LIM}(t,\tau )$ plot of the proton temperature (LIM_T ) and magnetic field (LIM_B ) fluctuations, respectively. It is clear that the interval between the beginning time t_B and ending time t_E in the middle of the plotted time sequences is characterized by red stripes, and the stripes cover a wide scale range between [16, 100 s]. It means that the fluctuations of temperature and magnetic field are coherent structures rather than waves that often appear as a narrow horizontal band in the $\mathrm{LIM}(t,\tau )$ plot (Alexandrova et al. 2013; Lion et al. 2016; Perrone et al. 2016, 2017, 2020). No clear wave cases are found among the selected temperature intermittent structures. The reason could be that, here, we focus on the fast solar wind streams and at the inertial range [16, 100 s]. In this context, the most common wave observed should be Alfvén wave, which is incompressible and is generally not accompanied by temperature variations.

We then check if the data points inside the intermittent case are unstable to the MM instability. Panel (a) of Figure 5 shows the data points inside the intermittent case shown in Figure 4 in T_⊥/T_∥ − β_∥ parameter space in multicolored dots. The dots in purple and in red correspond to the beginning and ending of the intermittent case, respectively. The background gray contours are adopted from the left panel of Figure 2, which are the contours of the 2D distribution of T_⊥/T_∥ and β_∥ for the whole eight fast streams. The solid and dashed black lines are also the same as in the left panel of Figure 2, which denote the thresholds of MM and OF instabilities, respectively. We notice that, at the beginning, the data point (purple dot) is located near the center of the contours, and the plasma is well stable, which indicates that it has similar characteristics as the background plasma. When in the middle of the intermittent case, two data points (green dots) are located beyond the threshold of the MM instability. It tells us that the structure could be related to the MM instability. Then, at the end of the case, the data point (red dot) goes back to the center of the contours. The plasma become stable as the background wind, again. Therefore, we suggest the temperature intermittent case with N_T = −1, and N_B = 1 shown in Figure 4 is an LMH associated with the MM instability.

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In addition, in panel (j) of Figure 4, we show the time variations of total pressure (P, black) and proton specific entropy (S_p , blue). The total pressure includes proton kinetic pressure and magnetic pressure. We see that the proton kinetic pressure and magnetic pressure are balanced inside the intermittent case, so the total pressure keeps nearly constant during the whole plotted interval. The proton specific entropy is calculated from ${S}_{p}={T}_{p}/{N}_{p}^{2/3}$, which could be a parameter for detecting nonadiabatic heating (Marsch et al. 1983; Schwartz & Marsch 1983; Whang et al. 1989; Borovsky & Denton 2011; Borovsky & Steinberg 2014). The value of S_p is slightly larger inside the intermittent case with the maximum value of S_pin = 21.3 eV cm² than the background plasma with S_pout = 18.0 eV cm². It indicates that this temperature intermittent case is accompanied by the possible signal of nonadiabatic heating.

3.1.2. Bump-CS

Figure 6 shows a case with a clear signal of local proton temperature increase, which is similar to that shown in panels (a), (b) of Figure 4, but this case has a different kind of magnetic field configuration. The beginning time and ending time of this case are t_B = 00:40:16 UT and t_E = 00:40:48 UT, respectively, denoted by the two vertical dashed lines. Between the two lines, the value of ∣PVI_T ∣ shown in magenta in Figure 6(b) is larger than 3 at the first three and last three time instants, and satisfies the criterion of the temperature intermittency as mentioned in Section 2.2. Accordingly, the width of this case is recorded as d = t_E − t_B = 32 s, corresponding to 135.5 ρ_i . From the profile of the PVI_T , we see clearly that it has a "bipolar" feature between the two vertical dashed lines, and the times of "zero-crossing" for the PVI_T curve are also N_T = − 1 as the case shown in Figure 4. As seen from Figure 6(a), the background proton temperature outside the intermittent case T_pout is recorded as 19.5 eV, and is also calculated from the average of the proton temperature at the upstream ([t_B − d, t_B ]) and downstream ([t_E , t_E + d]) of the intermittent case. The proton temperature inside the case T_pin increases 28.2% to 25.0% eV, which corresponds to the proton temperature of the time instant (short vertical dashed line in panel (a)) that has the highest value of ${T}_{p}^{{\prime} }$. We also see the vertical red-stripe feather in both the ${\mathrm{LIM}}_{T}(t,\tau )$ plot (panel (k)) and the ${\mathrm{LIM}}_{B}(t,\tau )$ plot (panel (l)) between [t_B , t_E ]. It again means that the fluctuations of temperature and magnetic field correspond to coherent structures.

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The big difference between the case in Figure 6 with that shown in Figure 4 is the magnetic field configuration. The magnetic components in RTN coordinate frame in panels (c)–(e) all present a big jump, and there is a slight decrease in the magnetic strength (black and gray curves in Figure 6(f)). So we conjecture that this case may be associated with CS.

In order to verify the conjecture, we check the magnetic field components in the LMN frame. From panel (g) of Figure 6, we see that there is a big jump appearing in the component in the maximum variance direction (B_L , purple). From panel (h) of Figure 6, we see that the PVI _B profile has a "dip" feature without "zero-crossing," instead of a "bipolar" feature as shown in Figure 4(h). Therefore, the times of "zero-crossing" for the PVI _B profile are recorded as N_B = 0 for this case. The minimum variance direction is well defined for this case with ${\lambda }_{\min }/{\lambda }_{\mathrm{int}}=0.08$. The eigenvalue ratio ${\lambda }_{\mathrm{int}}/{\lambda }_{\max }$ is as small as 0.05, so it is a linearly polarized structure. The maximum variance direction L is nearly perpendicular to b ₀ with θ_BL = 8177. For the component in the minimum variance direction (i.e., normal component B_N , green), we find that it is very small comparing to the magnetic strength with ∣B_N ∣/∣ B ∣ = 0.05. The small value of the ratio indicates that the magnetic field component in the normal direction of the discontinuity is nearly negligible with respect to the magnetic strength, and only the tangential components exist on the two sides of the jump. Accordingly, we identify this type of case with N_T = − 1, and N_B = 0 as CS.

In panel (i) of Figure 6, we also present the time sequences of β_∥ (blue) and T_⊥/T_∥ (black). During the local temperature increase, the temperature anisotropy T_⊥/T_∥ increases from 0.81 to 1.21, and the parallel plasma beta first increases from 2.0 to 3.9 then decreases to 1.1. We then check if the data points inside this case are unstable to the instabilities as shown in panel (b) of Figure 5. We see that it is different from panel (a) of Figure 5, and the plotted data points are all stable.

Figure 6(j) shows the time variations of total pressure (P, black) and proton specific entropy (S_p , blue). There is a dip in the total pressure near the end of the case, which means the reduction in the magnetic pressure is not balanced by the local increase in the kinetic pressure. It could be replenished by electron kinetic pressure to some extent. For S_p , a local increase from 9.5 to 12.0 eV cm² happens, which is a possible signal of nonadiabatic heating.

Beside the local temperature increase with N_T = −1, the temperature intermittency includes another kind of case, i.e., a temperature step with N_T = 0 as shown in Figure 7. Figure 7 is plotted in the same format as Figures 4 and 6. From panel (a) of Figure 7, we see a step in the time series of the proton temperature, and it changes from 12.7 eV at the upstream of the step to 16.2 eV at the downstream of the step, which results in a "bump" profile of PVI_T without crossing the horizontal zero line (N_T = 0). The beginning time and ending time of this case are t_B = 18:45:53 UT, and t_E = 18:46:17 UT, respectively, denoted by the two vertical dashed lines. The width of the intermittent structure is 24 s, corresponding to 60.5 ρ_i . During this interval, the value of ∣PVI_T ∣ keeps larger than 3 as shown in Figure 7(b).

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We call the temperature interface a step. When the step happens, we notice that both the magnetic components and the magnitude show a jump in their time series. The ${\mathrm{LIM}}_{T}(t,\tau )$ plot (panel (k)) and the ${\mathrm{LIM}}_{B}(t,\tau )$ plot (panel (l)) are also both characterized by a vertical red stripe. After performing the MVA, we see from panel (g) that there is a big jump in B_L (purple). The eigenvalue ratios (${\lambda }_{\min }/{\lambda }_{\mathrm{int}}=0.08$ and ${\lambda }_{\mathrm{int}}/{\lambda }_{\max }=0.02$) are both very small. It tells us that the MVA result is reliable, and the structure has 1D fluctuations. The component in the minimum variance direction (i.e., normal component B_N , green) is found to be very small comparing to the local mean magnetic strength with ∣B_N ∣/∣ b ₀∣ = 0.08, which indicates that the magnetic field component in the normal direction is nearly negligible, and only the tangential components exist on the two sides of the step. Meanwhile, the magnetic strength on the two sides changes a lot with Δ∣ B ∣/∣ b ₀∣ = 0.88, where Δ∣ B ∣ is the absolute value of the magnitude change between the upstream and the downstream of the step. The ratios fulfill the conditions ∣B_N ∣/∣ b ₀∣ < 0.2 and Δ∣ B ∣/∣ b ₀∣ > 0.2 that are often used to identify TD (Smith 1973; Tsurutani & Smith 1979; Wang et al. 2013). Moreover, the minimum variance direction N is found to be nearly perpendicular to b ₀ with θ_BN = 8515.

From panel (h) of Figure 7, we see that the PVI _B profile has a "dip" feature without "zero-crossing." Therefore, the times of "zero-crossing" for the PVI _B profile are recorded as N_B = 0 for this case. We also notice the interface in the variations of T_⊥/T_∥ (black) and β_∥ (blue) in panel (i), while the total pressure is balanced between the upstream and the downstream as shown in black in panel (j). Then, we identify the step case with N_T = 0 and N_B = 0 as TD that is considered as separating two different parcels of plasma.

We also check if the data points inside this case are unstable to the instabilities as shown in panel (c) of Figure 5. We see that the plotted data points are almost stable, and there are clearly two different parcels of plasma. However, some of the data points are very close to the MM instability threshold, and we speculate that there may be a magnetic hole at the downstream of the step. Therefore, the step-type case could be the interface of two adjacent flux tubes.

The temperature intermittency could also be associated with the local temperature decrease as shown in Figure 8. From panel (a), a dip appears in the time series of T_p between t_B = 16:24:43 UT and t_E = 16:25:07 UT on 2022 January 2 from 28.9 to 20.1 eV at the time instant marked by the short vertical line. The width of it is d = t_E − t_B = 24 s, corresponding to 81.5 ρ_i . During this interval, the ∣PVI_T ∣ curve shows a "bipolar" feature and crosses the horizontal zero line once. When crossing the zero line, the ∣PVI_T ∣ curve has a positive slope. So the times of the "zero-crossing" for this case are recorded as N_T = + 1. The ${\mathrm{LIM}}_{T}(t,\tau )$ plot (panel (k)) and the ${\mathrm{LIM}}_{B}(t,\tau )$ plot (panel (l)) are also both characterized by a vertical red stripe.

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During the plotted interval, we see in panel (f) that the magnetic strength shows a small local increase, and the proton number density shows a slight dip, which lead to a local decrease in the total pressure shown in panel (j) considering the distinct local temperature decrease. The parallel plasma beta β_∥ decreases, while the temperature anisotropy T_⊥/T_∥ shows a peak between the two vertical dashed lines. It indicates that the "cooling" happens mainly in the parallel direction.

The eigenvalue ratios are ${\lambda }_{\min }/{\lambda }_{\mathrm{int}}=0.35$ and ${\lambda }_{\mathrm{int}}/{\lambda }_{\max }\,=0.07$, which indicates that the MVA result is not perfect, and the structure has 1D fluctuations. The minimum variance direction N is found to be nearly parallel to b ₀ with θ_BN = 843, and the intermediate and the maximum directions are both perpendicular to b ₀ with θ_BM = 8717, and θ_BL = 8206.

In the T_⊥/T_∥ − β_∥ plane shown in Figure 5(d), we see that the data points are well constrained by the thresholds. Though, there is a clear local increase of the temperature anisotropy with T_⊥/T_∥ = 1.4 (see Figure 8(i)). Therefore, we consider that the temperature dip, here, with N_T = + 1 is possibly just recovered from the MM instability. However, the quasiparallel θ_BN angle and quasiperpendicular θ_BL angle are not consistent with the conjecture of the MM instability. The contradiction could be attributed to the accuracy of MVA, the determination of the local mean field, and/or the mixture of the fluctuations at different scales.

All of the cases mentioned above have ∣N_T ∣ ≤ 1. Here, we present the situation about the case with PVI_T crossing the zero line more than once. As shown in Figure 9(a), there are three "bumps" appearing in the proton temperature. From PVI_T , the beginning time and ending time of this case are determined as t_B = 02:55:38 UT, and t_E = 02:57:44 UT on 2021 October 21, respectively. The width of this case is d = t_E − t_B = 126 s, corresponding to 510.7 ρ_i . The three "bumps" of the proton temperature result in the multiple zero-crossings of PVI_T , and the times are recorded as N_T = 5. Due to the alternate signs of the slope when crossing, we only use the number 5 and do not put the sign "+" or "−" in front of it. This kind of case is called "chain." In the ${\mathrm{LIM}}_{T}(t,\tau )$ plot (panel (k)) and the ${\mathrm{LIM}}_{B}(t,\tau )$ plot, we see red patches that cover the wide scale range [16, 100 s].

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When we look at the magnetic strength ∣ B ∣ in Figure 9(f), we notice that its variation is anticorrelated with the variation of T_p . It results in a nearly steady total pressure as shown in black in panel (j). The variation in the proton number density is not clear.

From B_L (purple in Figure 9(g)), we see that the time variations could be separated into three parts. In the first and third parts, the profile of B_L shows a dip and a bump, respectively. And the direction of B_L does not change in either of them. Simultaneously, a dip happens in ∣ B ∣ in the both parts. In the middle part, there is also a dip in ∣ B ∣, but B_L shows a big jump, and its direction changes. During the intermittent interval between [t_B , t_E ], PVI _B crosses the zero line for four times (N_B = 4), which are denoted by the purple dots in panel (h).

The eigenvalue ratios for this case are ${\lambda }_{\min }/{\lambda }_{\mathrm{int}}=0.39$ and ${\lambda }_{\mathrm{int}}/{\lambda }_{\max }=0.04$. The minimum variance direction N and the maximum variance direction are found to be both perpendicular to b ₀ with θ_BN = 8901, and θ_BL = 8368. The angles are similar to the characteristics of CS.

The signals mentioned above indicate that the temperature intermittency case shown in Figure 9 could be related to LMHs in the first and third parts, and related to CS in the middle part. The idea is confirmed by the variations of T_⊥/T_∥ and β_∥. From Figure 9(i), T_⊥/T_∥ and β_∥ both show a bump in the first and third parts. In the T_⊥/T_∥ − β_∥ plane shown in Figure 5(e), we notice that a blue dot and a yellow dot are located very close to and beyond the MM threshold, respectively, which correspond to the peaks in panel (i) in the first and the third parts, respectively. The data points corresponding to the other dots in the plane are all stable. These phenomena associated with the middle part and the other two parts are consistent with the CS discussed in Section 3.1.2 and the LMH discussed in Section 3.1.1, respectively. Additionally, in each part, the specific entropy S_p shows a clear enhancement, which indicates possible signals of nonadiabatic heating. Therefore, this type of temperature intermittency with N_T > 1 and N_B > 1 is related with the combination of LMHs and CSs.

In order to distinguish between planar and vortex-like structures, we perform MVA (Sonnerup & Cahill 1967) on all the selected temperature intermittent cases. The statistical results of the MVA eigenvalue ratios and the angles between eigenvectors and local mean magnetic field b ₀ are shown in Figure 10.

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The left panels of Figure 10 show the distribution of the angles between eigenvectors and local mean magnetic field b ₀ for the different groups, respectively, with the bump-LMH in blue, bump-CS in magenta, step in green, dip in yellow, and chain in cyan. From top to bottom, the panels correspond to the distributions of θ_BN , θ_BM, and θ_BL, respectively. The right panels show the distributions of the MVA eigenvalue ratios. From top to bottom, the panels present the distributions of ${\lambda }_{\mathrm{int}}/{\lambda }_{\max }$, ${\lambda }_{\min }/{\lambda }_{\max }$, and ${\lambda }_{\min }/{\lambda }_{\mathrm{int}}$, respectively.

For the bump-LMH-type cases (in blue), we see that the minimum variance direction N of them is mostly (∼80%) perpendicular to b ₀ with θ_BN ≥ 75°. The distribution of θ_BM shows a similar behavior as θ_BN. The distribution of θ_BL peaks at the parallel angular bin [0°, 15°], which means the structures are compressible, and more than 60% of the cases have θ_BL < 45°. From the right panels, we see that 56% of the cases are linear polarized with 1D fluctuations (${\lambda }_{\mathrm{int}}/{\lambda }_{\max }\lt 0.2$). The rest of the 44% have $0.2\leqslant {\lambda }_{\mathrm{int}}/{\lambda }_{\max }\lt 0.8$, which indicates that these magnetic holes have 2D fluctuations corresponding to a cylinder topology as illustrated in Figure 18(b) of Perrone et al. (2016).

We investigate the intrinsic properties (including scale, depth, and rotation) of the bump-LMH-type cases. Statistically, the width of them ranges from 40.4 ρ_i to 411.8 ρ_i . The depth of the holes ranges from 0.03 for the deepest hole to 0.87 for the shallowest hole. The smallest and largest rotation angles are 11 and 342, respectively, and the average angle is 125. In some previous studies (e.g., Turner et al. 1977; Zhang et al. 2008), the holes with the rotation angle larger than 30° are called rotational holes, instead of LMHs. Among the 46 bump-LMH-type cases, we only find three of them have the rotation angle larger than 30°, but smaller than 342. They could be contaminated by adjacent weak discontinuities. Since the times of "zero-crossing" for the PVI _B profiles of them are recorded as ∣N_B ∣ = 1, the fluctuations could be dominated by the linear hole feature. Therefore, here, we still consider them as LMHs.

For the bump-CS-type cases (in magenta), we see that N and L are both nearly perpendicular to b ₀ with θ_BN ≥ 75° and θ_BL ≥ 75°. The intermediate variance direction M is close to the direction of b ₀, and 80% of the cases have θ_BM < 45°. The quasiperpendicular feature of both θ_BN and θ_BL and quasiparallel feature of θ_BM are well consistent with the characteristics of CSs as mentioned in previous studies (Burlaga & Ness 2011; Perrone et al. 2016, 2017). Moreover, from the right panels, we see that most of the bump-CS-type cases are linearly polarized with 93% of the cases owning ${\lambda }_{\mathrm{int}}/{\lambda }_{\max }\lt 0.2$, and the minimum variance direction N is always well determined with ${\lambda }_{\min }/{\lambda }_{\mathrm{int}}\lt 0.2$. The width of them ranges from 51.1 ρ_i to 412.0 ρ_i .

The statistical characteristics associated with MVA for the step-type cases (in green) are similar to that of the bump-CS-type cases (in magenta as described above). The most significant difference between them is the temperature variations, with a temperature step for the former and a temperature bump for the latter. In addition, the N direction for the former is not determined as well as the latter with only 37% of the cases owning ${\lambda }_{\min }/{\lambda }_{\mathrm{int}}\lt 0.2$. The width of them ranges from 59.4 ρ_i to 318.2 ρ_i . According to the step feature of both the temperature and the magnetic field (N_T = 0, and N_B = 0), we preliminarily speculate that the step-type cases would be associated with TD that separate two different parcels of the plasma (Bruno et al. 2001; Knetter et al. 2004; Borovsky 2008). Therefore, we check the jump conditions of the magnetic fields of the step-type events to see if they can satisfy the TD criteria ∣B_n ∣/∣ b ₀∣ < 0.2 and Δ∣ B ∣/∣ b ₀∣ > 0.2 as described in Smith (1973), Tsurutani & Smith (1979), and Neugebauer et al. (1984); hereafter "Smith-TD criteria" for simplicity. Here, B_n is the magnetic field component in the minimum variance direction, and Δ∣ B ∣ is the absolute value of the magnitude change between the two sides of the jump. We then find that, among the 21 step-type cases, only three of them can strictly satisfy the Smith-TD criteria. However, the large change of the field magnitude (Δ∣ B ∣/∣ b ₀∣ > 0.2) is a sufficient but unnecessary condition for TD (Knetter et al. 2004; Liu et al. 2021). Therefore, based on the condition of only a vanishing normal component, we here identify eight step-type cases with ∣B_n ∣/∣ b ₀∣ < 0.2 as TDs. For the rest of the 13 cases, they are characterized by the temperature jump together with the nonnegligible normal component ∣B_n ∣, which could be considered as shock-like structures.

For the dip-type cases (in yellow), we see that the minimum variance direction N is mostly (∼70%) close to b ₀ with θ_BN < 30°, while the intermediate and maximum directions are both quasiperpendicular to b ₀ with the angle larger than 75°. Their size varies between 66.5 ρ_i and 360.2 ρ_i . From the distribution of T_⊥/T_∥ and β_∥, we find that some of the dip-type cases are associated with the mirror instability. However, the quasiperpendicular θ_BL angle means an absence of compressive structures, so it is not consistent with the conjecture of the mirror instability. The contradiction could be attributed to the accuracy of MVA, the determination of the local mean field, and/or the mixture of the fluctuations at different scales. As shown in Rees et al. (2006), the magnetic solitons have a preference for the minimum variance direction to be parallel/antiparallel to the mean field. Therefore, the dip-type cases accompanied by the enhancement in the magnetic strength are also probably associated with solitons. Accordingly, we classify the dip-type cases as the MMs or the magnetic solitons based on the value of the temperature anisotropy as mentioned in Baumgärtel (1999). Among them, 21 cases with T_⊥/T_∥ > 1 are more likely to be associated with the mirror instability, and the rest of the 16 cases with T_⊥/T_∥ ≤ 1 may prefer the soliton scenario.

For the chain-type cases (in cyan), we see that the minimum variance direction N is also close to b ₀ with θ_BN < 30°, and the intermediate and maximum directions are both quasiperpendicular to b ₀ with the angle larger than 75°. The general alignment of minimum variance directions and mean field directions is consistent with the MVA results in the solar wind fluctuations in many previous studies (e.g., Burlaga & Turner 1976; Solodyna & Belcher 1976; Matthaeus et al. 1986; Klein et al. 1991). The size of them varies from 108.6 ρ_i to 1028.9 ρ_i . As shown in the case study in Figure 9, some of the chain-type cases are the superposition of different types of structures, like LMHs and CSs. It is also reasonable to explain the magnetic field as well as the temperature signatures shown in Figure 9 with a simple picture where the spacecraft approaches and leaves the interior of the current layer during the CS crossing. The quasiperpendicular θ_BN and θ_BL angles as well as the small eigenvalue (${\lambda }_{\mathrm{int}}/{\lambda }_{\max }=0.04$) in this case are the typical characteristics of CS. We check the 38 chain-type cases, and find that among them there are three other cases that have similar features as that shown in Figure 9. They can satisfy the conditions θ_BN > 60°, θ_BL > 60°, and ${\lambda }_{\mathrm{int}}/{\lambda }_{\max }\lt 0.2$. However, we notice from the right panels that 45% of the chain-type cases have $0.2\leqslant {\lambda }_{\mathrm{int}}/{\lambda }_{\max }\lt 0.8$, which indicates that this kind of the structures have 2D fluctuations. In addition, we have also checked that they are generally accompanied by the changes in the field magnitude, which means that they are compressive structures. The characteristics described above (including the multiple "zero-crossing" of PVI _B , the quasiperpendicular θ_BL, the 2D fluctuations, and changes in the field magnitude) coincide with the typical properties of compressive vortex-like structures proposed by Perrone et al. (2016). Therefore, we speculate that 45% of the chain-type cases could be related to compressive vortex-like structures. For the 17 vortex-like structures, we calculate the normalized cross helicity σ_c of them between [t_B − κ d, t_E + κ d], and find that they have high Alfvénicity with 0.79 ≤ ∣σ_c ∣ ≤ 0.96. It is also consistent with the observed high Alfvénicity in the fast solar wind (Marsch & Tu 1990; Tu & Marsch 1995; Wang et al. 2012; Bruno & Carbone 2013). Further study needs to be done to verify if the vortex model (Petviashvili & Pokhotelov 1992; Alexandrova 2008) can explain the observations, although the models are for an ideal incompressible system.

We then study the kinetic effects for the different five types of temperature intermittency statistically. In Figure 11, we present the normalized 2D distribution of T_⊥/T_∥ and β_∥ for the different types of temperature intermittency as listed in Table 2, separately. In each panel, the upper and lower curves correspond to the thresholds of proton mirror and proton OF instability, respectively, which are adopted from Hellinger et al. (2006). In each pixel, the number is normalized by the number of the corresponding pixel shown in the left panel of Figure 2 for the whole data set of the eight streams.

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For the bump-LMH-type intermittency with N_T = − 1 and N_B = 1 (see Figure 11(a)), the data points are mainly located close to the threshold of proton mirror instability, which means the bump-LMH-type cases are related to the MM instability. For the bump-CS-type intermittency with N_T = − 1 and N_B = 0 (see Figure 11(b)), the data points concentrate at the area with 0.63 < T_⊥/T_∥ < 1.58 and 3.2 < β_∥ < 22. For the step-type intermittency with N_T = 0 and N_B = 0 (see Figure 11(c)), the data points are located around T_⊥/T_∥ ∼ 1 and 2 < β_∥ < 10. For the dip-type intermittency (see Figure 11(d)), the data points are also located close to the threshold of proton mirror instability as the bump-LMH type. When comparing panel (d) with panel (a), we see that the bump-LMH-type and the dip-type intermittency occupy the lower right part and upper left part along the threshold of proton mirror instability, respectively, in the T_⊥/T_∥ − β_∥ plane. Therefore, the difference between them is that the bump-LMH-type intermittency corresponds to a higher value of β_∥ and lower value of T_⊥/T_∥ (1 < β_∥ < 100 and 1.0 < T_⊥/T_∥ < 1.58) comparing to the dip-type intermittency (0.1 < β_∥ < 0.63 and 1.3 < T_⊥/T_∥ < 3.0). For the temperature intermittency with multiple "zero-crossing" (N_T > 1 and N_B > 1) in panel (e), we see there are two patches in the 2D distribution plane. One is the upper right part, which is along the threshold of proton mirror instability. The other one is the lower left part with lower beta 0.1 < β_∥ < 1. The two patches probably tell us that the temperature intermittency with N_T > 1 and N_B > 1 could be associated with multiple mechanisms: one is related to the proton mirror instability, and another one may be related to Coulomb collisions (Vafin et al. 2019) and/or α A/IC instability from a private discussion with Dr. Liang Xiang, which are reported to responsible for the constraint of temperature anisotropy at the low-beta condition.

In order to reveal the inherent signatures of different types of the temperature intermittent structures, we perform a superposed epoch analysis on the fluctuations of T_p , N_p , T_⊥/T_∥, β_∥, and ∣B∣. The result is shown in Figure 12. From left to right, the panels are for bump-LMH-type, bump-CS-type, step-type, and dip-type structures, respectively. From the result, we can conclude that, for the bump-LMH-type structures (panels (a1)–(a5)), statistically, the mean T_p shows a local increase (∼20%) at the center of the holes. At the same time, the proton number density and T_⊥/T_∥ both show a slight enhancement. However, the magnetic strength gets weaker by about 25%, which leads to the clear bump in the plasma beta. These are all consistent with the typical properties of LMHs. For the bump-CS-type structures (panels (b1)–(b5)), the mean T_p also has a local increase (∼25%) at the epoch zero, accompanied also by the weaker ∣B∣ and larger β_∥, while no clear variations appear in N_p and T_⊥/T_∥ statistically. For the step-type cases (panels (c1)–(c5)), we see a jump of about 20% in T_p . The same trend happens in N_p and β_∥, while an opposite trend is found in ∣B∣ in order to balance the thermal pressure. No clear trend is found in T_⊥/T_∥. For the dip-type cases (panels (d1)–(d5)), a local decrease of T_p (∼15%) happens, together with a local increase in T_⊥/T_∥ and a sight increase in ∣B∣. The characteristics again imply that some of the dip-type cases are related with MM instability. The analysis is not performed for the chain-type cases, since the variations of them are mixed and complicated. It could not be easy to find a systematic trend for them from the superposed epoch analysis.

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