First Direct Imaging of a Kelvin–Helmholtz Instability by PSP/WISPR

The event was a slow CME (∼200 km s⁻¹; see its kinematic characterization later in this section) that propagated along a preexisting streamer and disrupted it. The CME entered the WISPR-I FOV at 16:33:20 UT on 19 November 2021. The evolution of the event can be seen in the movie available on the summary page of encounter 10 on the WISPR web page: https://wispr.nrl.navy.mil/wisprdata. The CME resembles a circular structure composed of concentric loops. To reconstruct the 3D CME flux-rope structure and estimate its direction of propagation we used the GCS model with combined observations obtained on November 19 at about 22:30 UT from the three remote-sensing instruments (i.e., SOHO/LASCO, STEREO-A/COR2, and WISPR-I).

From the GCS reconstruction, we estimated that the CME propagated radially in a direction with a Carrington longitude of 20° and latitude of 10°. The CME was slightly tilted by −7° with a half-angular width ⁴ of about 18°. The apex height was at 12.9 R_⊙ at 22:30 UT. The parameters we obtained with the GCS reconstruction are in agreement with the ones presented by McComas et al. (2023) using images one hour later (∼23:30 UT). According to McComas et al. (2023) the PSP spacecraft intercepted the southern flank of the CME trail during the period between November 20 at 23:00 UT and November 21 at 01:00 UT.

For the CME kinematics, we tracked two different features. They are marked in the left panel of Figure 1 as Point A, below the leading edge, and Point B, on the inner part of the CME at the same position angle as Point A (yellow and cyan asterisks, respectively). Point A is visible from November 19, 18:03:25 UT to November 20, 01:48:20 UT, and Point B from November 19, 21:18:20 UT to November 20, 03:33:20 UT.

Using the GCS reconstruction and the known heliocentric distance of the spacecraft (S/C) at the time of each observation, we can obtain the deprojected kinematics with a simple geometrical transformation. In the top panel of Figure 3, we display the resulting height-time (HT) plot for the features marked with the points A (diamonds) and B (circles). The blue, dashed lines delimit the spatial region where the eddy train was later observed (gray-shaded area). The red and black dots depict the HT measurements of a few such eddies (see Section 3.2 for a detailed explanation). The HT profiles for features A and B are rather similar, with initial speeds (acceleration) from a ballistic model of about 182 km s⁻¹ (2.0 m s⁻²) and 211 km s⁻¹ (1.8 m s⁻²), respectively. The models are depicted with the dark and light green lines, respectively. This result (combined with the visual examination of the CME development) indicates that the CME would be gradually contracting along the radial direction, without exhibiting any discernible deformation and/or distortion during the time period under analysis. In the lower panel of Figure 3, we display the magnitude of the average velocity vector of the two features as a function of distance (in orange color). The shaded band delimits the range of speeds between the speed profiles of each individual feature.

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The CME transit across the combined FOV of the two WISPR telescopes lasted for more than 24 hr. The slow speed and clear magnetic flux-rope morphology point toward a streamer-blowout event (Vourlidas et al. 2013; Vourlidas & Webb 2018). The CME propagated right through the center of the preexisting streamer disrupting it (see also Howard et al. 2022), as evidenced by the lack of emission behind the CME at that latitude. To shed light on the CME initiation, we examined wavelet-processed images ⁵ (Stenborg et al. 2008) from the Extreme UltraViolet Imager (EUVI; Wülser et al. 2007, one of the telescopes in the SECCHI suite) onboard STEREO-A. A filamentary structure emerging around November 18 at ∼04:00 UT from an area above the west limb of the solar disk was observed in the images. After almost 35 hours on November 19 at ∼15:00 UT this structure left the FOV of EUVI. These times are in agreement with the white-light observations from ST-A/COR2 where the broadening of the streamer appeared around 11:30 UT and the CME was not visible before November 19, 16:00 UT. This long-duration process is definitely not a surprising one and is typical for very slow streamer-blowout CMEs (Vourlidas & Webb 2018). Magnetograms and SDO/AIA observations reinforce our assessment that there is no visible active region on the surface of the Sun on the northwest part that could be associated with the CME.

3.2.1. Spatial, Temporal, and Kinematic Characteristics

In Figure 4, we display six instances of the CME aftermath in the region delimited by the yellow rectangle in Figure 1. The yellow arrows in panels (C) through (E) point to the vortex-like structures we hypothesize are the signatures of a KHI. We see in the figure that these features, labeled as v_i (i = 1,...,6), appear in a wave-like pattern. To shed light on their physical nature, we measure (i) the relative distance between their centers, (ii) their sizes, and (iii) their speeds. The first plausible appearance of v₁ in the FOV of WISPR-I was at 01:33:20 UT (panel (B)). We say "plausible" because it could barely be spotted due to the high degree of signal contamination from dust debris (i.e., dust particles impacting the S/C and producing debris that is recorded by the detector; see, e.g., Zimmerman et al. 2021; Malaspina et al. 2022). In the frame at 01:48:20 UT (panel (C)), we distinguish the first two vortex-like features (v₁ and v₂) possibly at the maximum of their growth. In the frame at 02:18:20 UT (panel (D)), four more vortex-like features have appeared (v₃ to v₆). Features v₁ and v₂ appear to have deformed, which we hypothesize is an indication of the lifetime of the instability, their shapes now more resembling a typical Kelvin–Helmholtz vortex. The direction of the eddies, which is counterclockwise, matches, as expected, a situation with faster solar wind below and a slower CME flow above. In the following frame at 02:33:20 UT (panel (E)), the first four eddies are further deformed and, hence, it is difficult to discern them without considering the previous frame. Thus, it is not possible to make any measurements for the first four eddies in this frame. Finally, in the last displayed frame at 02:48:20 UT (panel (F)), all the eddies that were previously observable have now deformed or dissipated to the extent that they are no longer distinguishable at the resolution of the images. Instead, what remains is a thin, continuous region of plasma (indicated with two yellow arrows in each direction) that is no longer visible in the next frame.

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Numerous extended features are visible in the area surrounding the eddies, making it challenging to distinguish the boundaries between the CME and the ambient solar wind. This is a common issue for the interpretation of white-light images. However, in our case, several factors contributed to the identification of the position of the eddies with these boundaries. These factors include the 3D CME reconstruction with the GCS model, the CME's orientation as discussed earlier, the spacecraft's proximity to the CME, and the clear view below the CME (as discussed in Section 4.2). Moreover, within the CME itself, there are no velocity differences that are comparable to the velocity differences between the CME and the solar wind. In summary, the location of the thin line of plasma in the image results from a projection effect of their actual position in 3D space due to the optically thin nature of the recorded signal. Collectively, all these elements support our perspective that the eddies are located on the boundary of the CME.

Since all the features exhibited a rather elliptical shape, to characterize the typical scales involved, we measured the length of the major and minor axes (the major axis is along the propagation direction, while the minor axis is perpendicular to this direction). Briefly, to objectively measure the longitudinal and transversal size of the eddies (i.e., the length of the major and minor axes, respectively), we plotted (not shown here) the brightness profile along slits placed over the eddies in both selected directions. After detrending the brightness and deprojecting their location, we fitted them with an appropriate polynomial function (a seventh-degree polynomial function resulted in the smallest fitting errors while avoiding over-fitting). We adopt the width of the modeled profiles at half maximum as the sizes of the eddies. For each structure, we repeated the measurement process N times (N > 10) varying the slit position. The values obtained, which we report in Table 1, correspond to the average of all measurements for any given feature at any given time instance, with the standard deviation (inside parentheses) indicating the dispersion of the corresponding individual measurements. Note that in the frame (C) the first appearance of v₁ and v₂ occurs, while the other eddies are not visible yet, thus the corresponding cells in Table 1 are intentionally left empty. In frame (D), all the eddies are visible while, in frame (E), the deformation of the eddies v₁ to v₄ makes any attempt for a reliable measurement impossible. From the time-lapse considered, we estimate that the lifetime of the eddies (i.e., the temporal period) is less than 30 minutes.

Table 1. Average Sizes (in Mm) of the Minor (top row) and Major (lower row) Axis of Observed Eddies

Time	Frame	v1	v2	v3	v4	v5	v6
01:48:20 UT	(C)	80 (10)	89 (6)	⋯	⋯	⋯	⋯
		158 (16)	146 (15)	⋯	⋯	⋯	⋯
02:18:20 UT	(D)	114 (11)	117 (8)	52 (5)	40 (7)	39 (5)	36 (6)
		174 (15)	167 (13)	81 (14)	92 (8)	95 (10)	104 (14)
02:33:20 UT	(E)	⋯	⋯	⋯	⋯	54 (4)	47 (4)
		⋯	⋯	⋯	⋯	101 (10)	109 (12)

Note. The 1σ standard deviation is reported inside parenthesis.

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To measure the separation between the features, we analyzed the brightness profile in a slit placed (visually) along their radial direction of propagation. The procedure was repeated N times (N > 10), each time with a new visually selected slit position. For each instance, we plotted the brightness profile as a function of distance to the origin of the slit (the closest point to the inner edge of the WISPR-I FOV) and applied a low-pass filter to reduce the noise. In the left panel of Figure 5, we display one such instance, the black circles depicting the excess brightness and the blue line the low-pass filtered signal. The magenta circles pinpoint the relative maxima, which are representative of the centroids of the small-scale features. The actual positions of the centroids on the respective WISPR-I image in row and column coordinates are displayed on the right panel. Then, we estimated the separation of the eddies as the distance between adjacent centroids, d[v_i ,v_i+1], at each instance. In Table 2, we report the corresponding average values and their dispersion (between parentheses) as estimated by the standard deviation. The relative distance of the centroids ranges between 169 and 306 Mm with an average value equal to 237.5 Mm and a standard deviation of 58.6 Mm. The finding of varying distances between the centroids of the eddies is not a surprise. The specific spacing and arrangement of vortices in a Kelvin–Helmholtz instability are complex and can vary depending on various physical conditions and system parameters, including a combination of factors related to the fluid or plasma properties, the velocity shear, and external forces or magnetic fields (see, e.g., Thorpe 1971; Hwang et al. 2022). Varying distances between the vortices have been noted in observations (e.g., Möstl et al. 2013; Li et al. 2018) and in simulations (e.g., Möstl et al. 2013; Kieokaew et al. 2021).

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Table 2. Relative Distance (in Mm) between the Centroids of the Eddies

Time	Frame	d[v1, v2]	d[v2, v3]	d[v3, v4]	d[v4, v5]	d[v5, v6]
01:48:20 UT	(C)	278 (10)	⋯	⋯	⋯
02:18:20 UT	(D)	296 (17)	211 (11)	185 (6)	306 (10)	173 (8)
02:33:20 UT	(E)	282 (21)	⋯	⋯	⋯	169 (6)

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The eddies labeled v₁ and v₂ are the only ones that could be tracked in three frames. The deformation of the eddies in the last frame influences the measurements of their size, including minor and major axis lengths. However, despite this deformation, the eddies remain visible for kinematic analysis. Therefore, the kinematics analysis of the small-scale features that we hypothesize are the signature of KHI was carried out only for these two eddies. In spite of the limited measurements, we applied a linear fit to their deprojected height-time measurements and obtained a speed of ∼370 km s⁻¹. The eddies were observed near the southern flank of the CME. This leads consequently to the logical assumption that if they are a signature of a KHI, the ambient solar wind beneath the CME had to be flowing with a different speed. In the PSP/WISPR movie for Encounter 10 (available on the WISPR home page), we observe density inhomogeneities (which we associate with tracers of the ambient solar wind) beneath the flank of the CME, clearly moving faster than the eddies. To characterize the kinematics of their development, we computed the average speed of the flow in this region by tracking a feature that is clearly and unambiguously visible a few hours later. This feature entered the FOV of WISPR-I on November 20 at ∼06:33 UT. We tracked it up to ∼08:33 UT (i.e., in nine frames) and found a linear, deprojected speed of ∼410 km s⁻¹. In brief, as recorded on WISPR-I images, the deprojected, average linear speed of the CME, train of eddies, and feature outside the post-CME flow (i.e., a tracer of the solar wind) were estimated at approximately 200 km s⁻¹, 370 km s⁻¹, and 410 km s⁻¹, respectively.

KHI has been observed primarily in EUV observations from SDO/AIA at low-corona distances during the early stages of CME formation (Foullon et al. 2011; Ofman & Thompson 2011; Möstl et al. 2013). Páez et al. (2017) discussed the possibility of KHI formation in the mid and outer corona, in particular at heliocentric distances between 4 and 30 R_⊙. In our work, we assess the possibility that WISPR actually observed a KHI in the wake of a CME, in particular, near its southern flank, at heliocentric distances between 7.5 and 9.5 R_⊙.

In our approach, we consider the case where the wavevector, $\hat{{\boldsymbol{k}}}$, is parallel to the flows, namely to both the CME and the ambient solar wind, which can be stabilized by the flow-aligned magnetic field. For our case, we need to study the magnetic field and density environment along the shear flow, for an incompressible plasma without viscosity and in a thin layer with an external magnetic field. Therefore, to evaluate where a KHI can develop, we utilize the KHI magnetic condition as proposed in Michael (1955) and Chandrasekhar (1961):

$$\begin{eqnarray}&&{[\hat{{\boldsymbol{k}}}\cdot ({{\boldsymbol{V}}}_{1}-{{\boldsymbol{V}}}_{2})]}^{2}\gt \displaystyle \frac{{n}_{1}+{n}_{2}}{4\pi {m}_{{\rm{p}}}{n}_{1}{n}_{2}}[{\left(\hat{{\boldsymbol{k}}}\cdot {{\boldsymbol{B}}}_{1}\right)}^{2}+{\left(\hat{{\boldsymbol{k}}}\cdot {{\boldsymbol{B}}}_{2}\right)}^{2}],\end{eqnarray} \tag{ 1 }$$

where V ₁ and V ₂ are the velocities, n₁ and n₂ are the number densities, B ₁ and B ₂ are the magnetic fields of the two magnetized plasma flows, m_p is the proton mass, and $\hat{{\boldsymbol{k}}}$ is the wavevector of the shear flow. Equation (1) represents the condition that is necessary for the development of the KHI in the boundary layer between the two flows, considering that the thickness of the boundary layer is negligible (i.e., Δ = 0), as long as the left-hand side is greater than the right-hand side.

In the following, we investigate the constraints on the 2021 November 19 CME magnetic field as a function of distance from the Sun that are required to create favorable conditions for the formation of a KHI. To this aim, we follow a similar methodology as in Páez et al. (2017).

The 2021 November 19 CME was a slow event, slower than the ambient solar wind, and as a result, no shock or sheath region around the CME was observed (Howard et al. 2022). In Figure 6, we present a cartoon with a simplified view of the CME and the fast ambient solar-wind environment (top panel), and the magnetic configuration of the KHI on the CME flanks (lower panel). In particular, in the lower panel, we display a vertical cut of the helical magnetic field structure of the CME where the dark blue arrows represent the outward and inward magnetic field polarities on the plane of the image. For our case study, in the region between the CME flanks and the ambient solar wind, Equation (1) becomes:

$$\begin{eqnarray}&&{[\hat{{\boldsymbol{k}}}\cdot ({{\boldsymbol{V}}}_{\mathrm{SW}}-{{\boldsymbol{V}}}_{\mathrm{CME}})]}^{2}\gt \displaystyle \frac{{n}_{\mathrm{SW}}+{n}_{\mathrm{CME}}}{4\pi {m}_{{\rm{p}}}{n}_{\mathrm{SW}}{n}_{\mathrm{CME}}}[{\left(\hat{{\boldsymbol{k}}}\cdot {{\boldsymbol{B}}}_{\mathrm{SW}}\right)}^{2}+{\left(\hat{{\boldsymbol{k}}}\cdot {{\boldsymbol{B}}}_{\mathrm{CME}}\right)}^{2}].\end{eqnarray} \tag{ 2 }$$

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The term $\hat{{\boldsymbol{k}}}\cdot {{\boldsymbol{B}}}_{\mathrm{CME}}$ on the right-hand side of Equation (2) can be decomposed in two terms to account for the poloidal, ${{\boldsymbol{B}}}_{\mathrm{CME}}^{\mathrm{Pol}}$, and toroidal, ${{\boldsymbol{B}}}_{\mathrm{CME}}^{\mathrm{Tor}}$, components of the CME magnetic field, i.e.,:

$$\begin{eqnarray}&&\hat{{\boldsymbol{k}}}\cdot {{\boldsymbol{B}}}_{\mathrm{CME}}\,=\,\hat{{\boldsymbol{k}}}\cdot {{\boldsymbol{B}}}_{\mathrm{CME}}^{\mathrm{Pol}}+\hat{{\boldsymbol{k}}}\cdot {{\boldsymbol{B}}}_{\mathrm{CME}}^{\mathrm{Tor}}.\end{eqnarray} \tag{ 3 }$$

As shown in Figure 6 (lower panel), ${{\boldsymbol{B}}}_{\mathrm{CME}}^{\mathrm{Pol}}$ is perpendicular to the shear flow $\hat{{\boldsymbol{k}}}$. Therefore, this term is zero, and hence it does not affect the formation of the KHI (Chandrasekhar 1961). On the other hand, since ${{\boldsymbol{B}}}_{\mathrm{CME}}^{\mathrm{Tor}}$ is parallel to $\hat{{\boldsymbol{k}}}$, the term $\hat{{\boldsymbol{k}}}\cdot {{\boldsymbol{B}}}_{\mathrm{CME}}^{\mathrm{Tor}}$ is the relevant one for the formation of the KHI. Then, we can take this radial component equal to the CME radial magnetic field, i.e.,

$$\begin{eqnarray}&&\hat{{\boldsymbol{k}}}\cdot {{\boldsymbol{B}}}_{\mathrm{CME}}\,=\,\hat{{\boldsymbol{k}}}\cdot {{\boldsymbol{B}}}_{\mathrm{CME}}^{\mathrm{Tor}}\,=\,{B}_{\mathrm{CME},r}.\end{eqnarray} \tag{ 4 }$$

Since both V _SW and V _CME are parallel to $\hat{{\boldsymbol{k}}}$, the left side of Equation (2) can be defined as a shear function, S(r):

$$\begin{eqnarray}&&S{(r)=| {V}_{\mathrm{SW}}-V(r)}_{\mathrm{CME}}| ,\end{eqnarray} \tag{ 5 }$$

S(r) represents the shear flow between the CME and the ambient solar wind. In our case, the solar-wind speed, as estimated from a solar-wind tracer, was assumed to be constant (∼410 km s⁻¹, see Section 3.2.1). On the other hand, the CME speed, as estimated from two particular features of the bulk of the CME (features A and B in Figure 1; see Section 3.1) was found to exhibit a slight acceleration trend during its development, varying from ∼175 km s⁻¹ at a distance of 5 R_⊙ up to ∼290 km s⁻¹ at 25 R_⊙.

With the above constraints, we can transform Equation (2) into a parametric form. To that aim, as we do not know the actual values of the magnetic field and density of both the CME and ambient solar wind, we define the ratios ξ_n = $\tfrac{{n}_{\mathrm{CME}}}{{n}_{\mathrm{SW}}}$ and ${\xi }_{B}\,=\,\tfrac{{B}_{\mathrm{CME},r}}{{B}_{{SW},r}}$. Here, the solar-wind magnetic field is all in the radial direction, so, B_SW,r = B_SW. Using these ratios in Equation (2) and solving it for B_CME,r , we obtain the parametric form:

$$\begin{eqnarray}&&{B}_{\mathrm{CME},r}^{\mathrm{KH}}(r)\lt \sqrt{4\pi {m}_{{\rm{p}}}{n}_{\mathrm{SW}}(r)\cdot \displaystyle \frac{{\xi }_{n}\cdot {\xi }_{B}^{2}}{(1+{\xi }_{n})\cdot (1+{\xi }_{B}^{2})}}\cdot S(r).\end{eqnarray} \tag{ 6 }$$

Equation (6) shows that by just expressing the unknown CME field and density values in units of the solar-wind counterparts, we can estimate the magnetic field values of the CME, ${B}_{\mathrm{CME},r}^{\mathrm{KH}}$, that might be favorable for the development of a KHI. In particular, we note that ${B}_{\mathrm{CME},r}^{\mathrm{KH}}$ depends on (1) the density of the ambient solar wind, n_SW(r); (2) the shear flow function, S(r); and (3) the CME relative magnetic field strength and density with respect to those of the solar wind, ξ_B and ξ_n . To estimate the density of the ambient solar wind, n_SW(r), we use the Leblanc density model (Leblanc et al. 1998):

$$\begin{eqnarray}&&{n}_{\mathrm{SW}}(r)=d* (3.3\cdot {10}^{5}{r}^{-2}+4.1\cdot {10}^{6}{r}^{-4}+8.0\cdot {10}^{7}{r}^{-6}),\end{eqnarray} \tag{ 7 }$$

where d is the scaling factor. The scaling factor is used to normalize the Leblanc density model to the instantaneous value of the in situ density measurement obtained from the Advanced Composition Explorer Solar Wind Electron, Proton, and Alpha Monitor (ACE SWEPAM; McComas et al. 1998) instrument. This allowed us to approximate the density value close to the Sun following Thernisien & Howard (2006). Since the CME event was associated with a streamer blowout, we scale the model as in Thernisien & Howard (2006; i.e., d = 6.0). The values resulting from this approach are within the density range estimated from several electron density models close to a coronal streamer (see, e.g., Figure 1 in Raouafi 2011 and references therein).

The CME observed is denser than the ambient solar wind (n_CME > n_SW). Hence, for the following parametric analysis, we assume ξ_n = 2 (this choice is discussed in Section 4), and ξ_B = j, with j = 1, 2, 3, q (q ≫ 1). Note that with ξ_B = 1 (ξ_B = q), we are assuming that the CME has a magnetic field similar to (much larger than) that of the solar wind. This represents a lower (upper) limit, for the ξ_n = 2 scenario, hence constraining the range of possible ${B}_{\mathrm{CME},r}^{\mathrm{KH}}$. In the top panel of Figure 7, we plot the output of the parametric analysis (Equation (6)) for the cases mentioned above and the density model considered. The continuous lines delineate ${B}_{\mathrm{CME},r}^{\mathrm{KH}}$ at the limit cases when the two sides of Equation (6) are equal. For comparison, we also plot with the dashed, magenta line the CME magnetic radial evolution assuming a power-law curve of the form B_r ∝ 1/r² (see, e.g., Patsourakos & Georgoulis 2016, and references therein) from a PSP/FIELDS in situ measurement of the radial component of the magnetic field (∼550 nT, magenta star) when PSP was at ∼14.1 R_⊙ and intersected with the leg of the CME (November 20 23:00–November 21 01:00, McComas et al. 2023). It must be noted that this in situ magnetic field measurement was not taken at the time/location when/where the eddies were observed but rather one day later. This estimate of the CME magnetic field at lower coronal heights is a valid approach. However, it should not be regarded as the actual magnetic field of the CME at the boundary layer where the eddies were observed. We also include two cases representative of different scenarios: the very simple case assuming ξ_n = ξ_B =1 (light green line) and the extreme case where ξ_n = ξ_B =q, with q ≫1 (light blue line). These two cases serve as the boundaries of our analysis, representing the lower and upper limits of the outputs from Equation (6). The area delimited by the two vertical, blue dashed lines depicts the distance range where the train of eddies was observed, i.e., between 7.5 R_⊙ and 9.5 R_⊙. Therefore, the graded, shaded area (ξ_B >1) indicates the range of ${B}_{\mathrm{CME},r}^{\mathrm{KH}}$ values that support our hypothesis (i.e., that the train of eddies observed is a signature of a KHI instability). The graph shows, in brief, that the assumptions taken lead to a case that can explain the occurrence of the KHI. In fact, the top panel of Figure 7 clearly indicates an increase in instability with larger radii. At 5 R_⊙, the magenta dashed curve is positioned above all unstable curves, indicating stability under every tested condition. However, at 25 R_⊙, it is situated below all curves except for ξ_n = ξ_B = 1, signifying instability under all conditions except that one.

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To put our result in context, in the lower panel of Figure 7 (same color and labeling code), we display the output of the parametric analysis considering the Leblanc density model scaled by d = 0.54 to match the in situ density values measured by PSP one day later at ∼14.1 R_⊙. In this case, the graph shows that, with this scaling of the density model, the conditions would not be favorable for the development of a KHI.

Note that if we rearrange Equation (6) with the magnetic field and density on the same side (and write ${B}_{\mathrm{CME},r}^{\mathrm{KH}}\,=\,{B}_{r}$ and n_SW = n), we get the relation between the Alfvén speed and the shear flow function S(r):

$$\begin{eqnarray}&&\displaystyle \frac{{B}_{r}(r)}{\sqrt{4\pi {m}_{p}n(r)}}\equiv {V}_{A}(r)\lt \sqrt{\displaystyle \frac{{\xi }_{n}\cdot {\xi }_{B}^{2}}{(1+{\xi }_{n})\cdot (1+{\xi }_{B}^{2})}}\cdot S(r).\end{eqnarray} \tag{ 8 }$$

Equation (8) provides another way to examine the favorable conditions for the development of the KHI by comparing the Alfvén speed and the shear flow function S(r) as a function of the ratios ξ_n and ξ_B . If we use the PSP in situ measurements at ∼14.1 R_⊙ (i.e., B_r ∼ 500 nT and n ∼ 950 cm⁻³) we get an Alfvén speed of about 350 km s⁻¹, which is much higher than the right part of Equation (8). This shows that the formation of a KHI is rather sensitive to local conditions. This explains why we cannot observe the eddies one day later (see Section 4.2).

According to McComas et al. (2023), PSP crossed the leg of the CME between November 20, 23:00 UT and November 21, 01:00 UT, i.e., for a period of about 2 hr. PSP in situ measurements of the electron density by the Solar Wind Electrons Alphas and Protons (SWEAP) Investigation (Kasper et al. 2016) before this interaction averaged a number density of ∼950 cm⁻³. Inside the CME, the density amounted to ∼ 1800 cm⁻³ (i.e., ξ_n ≈ 2). Note that the PSP measurements were obtained at a distance of ∼14.1 R_⊙, almost a day after the observations of the eddies; therefore, they can be considered only as a rough estimate of the density conditions at the location of the eddies.

Among other cases, in Figure 7, we plotted Equation (6) for various ξ_B considering ξ_n = 2. In particular, we explored the CME magnetic field constraints for the development of KHI assuming a scaling factor in the Leblanc density model as in Thernisien & Howard (2006; i.e., d = 6.0; Figure 7 top panel). Next, we considered a scaling factor d ≈ 0.5 to match the in situ PSP/SWEAP density measurements when PSP crossed the CME leg (Figure 7 lower panel). Now, we explore the case when the density model is scaled such that the in situ PSP/FIELDS measurements of the CME magnetic field match the magnetic field constraints from Equation (6). The average in situ PSP/FIELDS values of the magnetic field before the interaction of PSP and the CME was ∼500 nT, and slightly higher at ∼550 nT while inside the CME. By considering this approach, we are assuming that the in situ PSP measurements of the magnetic field (although they were taken one day later in a region of space far away from the eddies location), serve as a ground truth. Note, however, that, while this assumption is speculative and not representative of the conditions where the eddies were observed (see also Section 3.3), it might represent a plausible approximation. These two values of the magnetic field indicate ξ_B ≈ 1 while ξ_n = 2 (this set is represented by the red line in Figure 7). We extrapolated them to shorter distances assuming flux conservation, resulting in values 1915/1190 nT of the magnetic field in the distance range where the eddies were observed at 7.5/9.5 R_⊙, respectively. To infer the minimum density required for instability under these conditions, the Leblanc density model must be scaled accordingly so that this red line matches the extrapolated magnetic field of the CME (i.e., the magenta dashed) at the distance where the eddies were observed (i.e., between the vertical blue dashed lines). To achieve this, the scaling factor becomes d ∼ 14.6. This new case is illustrated in Figure 8. From our approach, considering ξ_n = 2 and ξ_B = 1, we have that the KHI criterion is satisfied for CME magnetic fields less than 1915/1350 nT at 7.5/9.5 R_⊙, respectively. The newly scaled density model results in very high density values, so we shed light on this fact by comparing it to instances of high density values recorded by PSP. For example, the highest density measured by PSP up to Orbit 15 (as obtained using the simplified quasi-thermal noise spectroscopic technique or SQTN; Moncuquet et al. 2020) was ∼2.13 · 10⁴ cm⁻³. This measurement occurred on 2022 June 2 at approximately 14:10 UT when the S/C was at ∼15.8 R_⊙. This case is considered here as a special case and definitely not assumed as typical of a background solar wind. The newly scaled density model (d ∼ 14.6) predicts, at the corresponding distance, a density value of ∼2.0 · 10⁴ cm⁻³, which is a comparable estimate (∼7% lower) and certainly is not a nonphysical estimate. As the CME under study was associated with a streamer blowout, we expect that the compression of the already dense plasma from the streamer becomes denser as the CME develops. Consequently, this results in a higher local density, favoring the formation of a KHI.

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In this work, we investigate the possibility that a train of small vortex-like structures, detected in the aftermath of a CME, are the signatures of Kelvin–Helmholtz instability. To assess the likelihood of such a physical phenomenon, we compare the measured dynamical and kinematic properties of the eddies against theoretical expectations under certain assumptions.

The relative distance between the centroids of the vortexes is considered to be the characteristic length scale for the associated projected wavelength, or spatial period λ. In many KHI simulations or observations in solar coronal physics, the wavelength estimates are on the order of a few megameters. In particular, Foullon et al. (2011) reported a projected wavelength of λ = 18 ± 0.4 Mm. Möstl et al. (2013) mentioned that the vortex-like structures, which were observed at the northern side of the embedded filament boundary of the 2011 February 24 CME, had a wavelength of approximately λ ∼14.4 Mm. Ofman & Thompson (2011) reported a wavelength of λ = 7 Mm based on the size of the initial ripples. Li et al. (2018) observed a KHI in a solar blowout jet with a λ ∼5 Mm. All of the aforementioned studies estimated the wavelength based on EUV observations at low coronal heights (< 160 Mm). Kieokaew et al. (2021) reported a KHI analysis based on in situ measurements from Solar Orbiter. Their wavelength estimates, λ = 66.4 ± 8.4 Mm, are larger than the coronal one. We estimate a wavelength average of λ = 237.5 (58.6) Mm based on the distances between the eddies (Table 2).

A key discriminant for a KHI is the wavelength of the fastest growing mode. The fastest growing mode excited by a KHI at the interface should have a spatial period, or wavelength λ, between 2πΔ and 4πΔ (Miura & Pritchett 1982), where Δ is the thickness of the boundary layer. Equation (6) was obtained under the assumption of a boundary layer of zero thickness. We can now relax this assumption. From the WISPR-I images, we can estimate both Δ and λ independently. If the eddies are signatures of KHI, they should satisfy the relation 2πΔ < λ < 4πΔ. If not, then the eddies are not a KHI signature. In general, it is challenging to estimate Δ from imaging observations due to the small scales involved. Thanks to the proximity to the CME, WISPR-I resolves many small-scale details. In particular, at 02:48:20 UT all the vortexes are fully deformed, creating a thin line of plasma (Figure 4, panel (F)). We take the width of this line, Δ_obs = 28 (7) Mm as the upper limit of the boundary layer thickness. Therefore the wavelength of the fastest growing mode should lie between 2πΔ = 176 Mm and 4πΔ = 352 Mm. From the observations, we estimate λ as the relative distance between the vortexes (169–306 Mm, see also Table 2), with an average value of 237.5 Mm. Thus, the observed λ satisfies the condition of the fastest growing KH mode for most cases, which, in turn, supports our hypothesis that the observed eddies are the results of the development of a KHI.

Another physical parameter to check for the plausibility of a KHI is the growth rate. The linear growth rate of a KHI in incompressible, inviscid fluid layers with equal density, equal field strength, separated discontinuously within a "vortex sheet" (Chandrasekhar 1961; Frank et al. 1996), associated with, and which leads to, the KHI criterion (Equation (1)), is given by:

$$\begin{eqnarray}&&\gamma \,=\,\displaystyle \frac{1}{2}| {\boldsymbol{k}}\cdot {\rm{\Delta }}\,{\boldsymbol{V}}| {[1-{\left(2{V}_{{\rm{A}}}\hat{{\boldsymbol{k}}}\cdot \hat{{\boldsymbol{B}}}\right)}^{2}/{\left(\hat{{\boldsymbol{k}}}\cdot {\rm{\Delta }}\,{\boldsymbol{V}}\right)}^{2}]}^{1/2},\end{eqnarray} \tag{ 9 }$$

where the k is the wavevector whose magnitude is equal to 2π/λ, Δ V is the velocity difference between the two layers (i.e., the shear flow function S(r) defined in Equation (5)) and V_A is the Alfvén speed (see Equation (8)). In the case of an interface region with finite thickness, the growth rate is even smaller than the predicted value according to Equation (9) (see, e.g., Miura & Pritchett 1982; Ofman & Thompson 2011). Compressible plasma also tends to reduce the growth of the instability. Based on the parameters we estimated before we can compute the upper and lower limit values of the linear growth rate of the KHI using Equation (9). If we use an Alfvén speed range between 20 and 65 km s⁻¹ (see Section 4.2 for details), k = 2π/λ with λ the average value of 237.5 Mm, and ΔV ∼200 km s⁻¹, we get, from Equation (9), upper and lower γ_KH values of 0.0027 and 0.0022 s⁻¹, respectively. If we assume that nonlinear saturation occurs on a timescale of a few γ⁻¹ (see, e.g., Kieokaew et al. 2021), which ranges in our case from ∼6.1 to 7.6 minutes, the observed timescale evolution of the eddies recorded in WISPR-I images (less than 30 minutes) falls within the expected values. Table 3 compares various KHI parameters from past works against the estimates described in this section.

According to Equation (8), the formation of a KHI is possible if ΔV >2 · V_A (Miura & Pritchett 1982), where ΔV is the shear flow function S(r) (Equation (5)) and V_A is the Alfvén speed (Equation (8)), assuming equal densities (i.e., ξ_n = 1) and magnetic fields (i.e., ξ_B = 1). With existing instrumentation, it is impossible to measure the actual density and magnetic field, and by extension V_A , at the heliocentric distances where the plausible KHI was observed (7.5 R_⊙–9.5 R_⊙). Since we know S(r) for this region (lower panel of Figure 3), we can, provide an upper limit on V_A < $\tfrac{1}{2}$ S(r)∼100 km s⁻¹ for the onset of a plausible KHI. This value appears to be low for such heliocentric distances, especially for a CME-related feature. However, we note that this speed corresponds to the component parallel to the flow. It is conceivable that the Alfvén speed perpendicular to the flow is much higher, leading to a higher overall V_A. An obvious scenario would be the propagation of a curved magnetic structure in a parallel flow. Figure 1 (and the movie available on the web at the WISPR home page) indicate that the structures surrounding the area of interest are indeed curved as they seem to form the lower extensions of the helical structures comprising the CME magnetic flux rope.

An alternative way to explore the constraints of the Alfvén speed is to use the output of a model for this specific date. To that aim, we utilize the thermodynamic MHD code "Magnetohydrodynamic Algorithm outside a Sphere" (MAS), which was developed by and is maintained at Predictive Science Inc. (see, e.g., Riley et al. 2012, and references therein). The V_A as estimated with the MAS model for Carrington rotation 2251 at a distance of 8.43 R_⊙ (i.e., at a distance within the distance range where the eddies were observed) is displayed in the top panel of Figure 9. The propagation direction of the CME on 2021 November 20 at ∼02:00 UT (i.e., during the time of the eddies) as estimated with the GCS model (Carrington longitude and latitude of 20° and 10°, respectively; Section 3.1) is indicated with the magenta star symbol. We notice in the figure that V_A for the ambient solar wind near the CME is <30 km s⁻¹. The propagation direction of the CME is almost above the location of the heliospheric current sheet (HCS) and as a result, the very low Alfvén speeds at these positions are reasonable. This result provides another supporting evidence for our hypothesis.

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Recently, Verbeke et al. (2023) showed that the error in determining the longitude and latitude of a CME direction using the GCS model when two or three viewpoints is approximately 2°. Thus, to assess the effect of our estimate of the CME direction, we consider this error in the lower panel of Figure 9. For all the combinations of these two coordinates available, we created the V_A radial profiles for the ambient solar wind, which we display in the lower panel of Figure 9 with the light blue lines. In this case, V_A is ⪅75 km s⁻¹ at 7.5 R_⊙ and ⪅55 km s⁻¹ at 9.5 R_⊙. For a more ample case, we also assumed ad hoc a larger error (5°, represented by the gray lines). The red line represents the radial profile considering the direction of propagation of the CME, and the dashed green line represents the limit condition V_A = $\sqrt{\tfrac{1}{3}}$ S(r) for ξ_n = 2 and ξ_B = 1 (Equation (8)). The magenta curve represents the V_A assuming ξ_n = ξ_B =1 (as in Miura & Pritchett 1982). The black star symbol represents the Alfvén speed calculated using the extrapolated magnetic field B_r and density values similar to the values used in Figure 8. We note that, in the range where the eddies are observed (delimited by the dashed blue lines), all the V_A radial profiles satisfy Equation (8), regardless of the error, and hence support our hypothesis.

This condition perhaps offers an indication of why KHI observations are rare at these heights. From our analysis, it becomes clear that the development of a KHI (and its observation in white light) requires a combination of conditions to be in place:

• 1.  
  two flows with varying speeds;
• 2.  
  an unusually low Alfvén speed or an interaction geometry leading to a weak parallel and strong perpendicular V_A components relative to the flow to satisfy the Equation (8);
• 3.  
  short LOS imaging observations to resolve small scales (PSP is the only platform capable of providing this vantage point so far; therefore, it is unsurprising that white-light signatures of KHI have not been detected before).
• 4.  
  favorable viewing conditions, in addition to short LOS. The location of PSP below the CME plane offered a favorable viewing angle for observing the complex CME structure across a wide longitude and further avoiding LOS overlap. Furthermore, PSP was transiting over a coronal hole region at the time, and hence the WISPR-I LOS was crossing through a partially depleted region, resulting in a "cleaner" LOS. Stenborg et al. (2023) recently showed that the presence of even small equatorial Coronal Holes (CHs) can directly affect the overall brightness observed by WISPR-I. This is another argument in support of the assumption that the ambient solar wind below the CME has a larger speed than the CME.

As a final thought, the 2021 November 19 CME was associated with a streamer blowout (Howard et al. 2022), and originated in a quiet-Sun region. Therefore, it must have had lower magnetic field strengths than CMEs associated with active regions (Vourlidas & Webb 2018). Thus, the weak magnetic obstacle in combination with the orientation of the magnetic field could have been favorable for the formation of a KHI.