FADING CORONAL STRUCTURE AND THE ONSET OF TURBULENCE IN THE YOUNG SOLAR WIND

Radial features exhibiting the "typical" solar wind behavior of constant-speed flow and conservation of mass are expected to maintain approximately constant brightness in Figure 2. To detail the above introductory discussion of expected brightness scaling, the radiance (surface brightness B) of a feature is just

$$\begin{eqnarray}&&{B}_{\mathrm{feat}}=\pi {R}_{\odot }^{2}{B}_{\odot }{\sigma }_{{\rm{t}}}\int \left({n}_{{\rm{e}}}\displaystyle \frac{1+{(\cos \chi )}^{2}}{{R}^{2}}\right){ds},\end{eqnarray} \tag{ 1 }$$

where R_⊙ is the solar radius, B_⊙ is the mean solar radiance, σ_t is the Thomson-scattering cross section, n_e is the local electron density, χ is scattering angle, R is distance from the scattering site to the Sun, and s is distance along a given line of sight (e.g., Howard & DeForest 2012). From conservation of mass in a constant-radial-flow solar wind, we observe that n_e can be written

$$\begin{eqnarray}&&{n}_{{\rm{e}}}={n}_{{\rm{e}}0}{R}_{0}^{2}{R}^{-2}\end{eqnarray} \tag{ 2 }$$

for a reference distance R₀ (e.g., 1 R_⊙) and a constant equivalent density n_e0 at that reference distance. Thus, Equation (1) simplifies in the constant-flow case to

$$\begin{eqnarray}&&{B}_{\mathrm{simple}}=\pi {R}_{\odot }^{2}{B}_{\odot }{\sigma }_{{\rm{t}}}{n}_{e,0}{R}_{0}^{2}\int (1+{(\cos \chi )}^{2}){R}^{-4}{ds}=k\,{b}^{-3},\end{eqnarray} \tag{ 3 }$$

where k is a constant of proportionality that includes all of the terms to the left of the integral and an additional geometrical factor, and b is the impact parameter of the line of sight relative to the Sun. Since the angles under consideration in the HI1 images are less than 30°, the small-angle formula applies and the relation b ≈ εR_obs holds (where ε is the observed separation angle between an image pixel and Sun center, and R_obs is the distance from the observer to the Sun). Thus, for simple flow,

$$\begin{eqnarray}&&{B}_{\mathrm{simple}}\,{\varepsilon }^{3}\approx {{kR}}_{\mathrm{obs}}^{-3},\end{eqnarray} \tag{ 4 }$$

which is constant against ε. We tracked several bright features to show that radiance is preserved as expected; two examples are given in Figure 3. The top row of panels shows a visually striking, compact CME that launched on 2008 December 16. The bottom row shows a single bright puff embedded inside a stria. Both features maintain approximately constant brightness in the radially scaled images.

By contrast, individual radial striae (at locations where there is no particular localized feature to track) fade faster than ε⁻³. The right-hand side of Figure 2 shows this effect in broad context: although the figure uses scaled brightness, the striae visibly fade before reaching the outer limit of the field of view. Figure 4 shows two particular examples of fading radial striae, extracted as in Figure 3 to follow the evolution of a particular patch of solar wind as it propagates outward.

Qualitatively, the striae (vertical stripes) in Figure 4 fade continuously from bottom to top in these tracked sequences, dropping below the noise floor. To quantify this fading and verify that it is not an artifact of the rising noise floor, we smoothed the final images in the vertical direction to bring down the noise floor still further and identify whether the radial stria is still present. To verify whether any identified structure was due to the smoothing direction, we also carried out the identical smoothing operation in the horizontal direction. The results are in Figure 5, which shows a clear remnant vertical structure at the smoothed noise floor after smoothing across 15 of radial extent or the equivalent (6°) in azimuthal extent (all at 20° elongation).

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Although (just) visible in the smoothed images, the remnant radial structure is attenuated by a factor of 3–5 at 20° elongation, compared to its scaled brightness at 6°. Note that the important parameter in Figure 5 is the peak-to-peak amplitude of the visible striations, not their absolute brightness: the necessary background subtraction and minsmooth unsharp-masking steps invalidate the zero-point measurement.

The fading of the radial striae indicates a real attenuation of their emission compared to a radially propagating, mass-conserving solar wind whose speed is independent of radius. The result is strong: by starting with the flat-fielded Level 1 data, avoiding any motion filtering beyond simple star-field removal, performing a control analysis by tracking compact features, and demonstrating persistence of the features despite anisotropic smoothing, we have eliminated most credible causes of potential confusion or loss-of-signal artifacts.

In parallel with the radial fading of the long, anisotropic radial striae in the corona, smaller bright features become more prominent, giving the plasma a puffy or flocculated appearance. The Sheeley et al. (1997) blobs and the Viall et al. (2010) puffs arise from the low corona and do not fade in scaled brightness as they propagate, becoming part of the flocculated gestalt. In addition, there are small puffs with similar size scales that fade in from invisibility in the moving frame of reference of the wind itself and are therefore visible primarily in the outer field of view. We refer to these brightening puffs as "flocculae." Figure 6 illustrates two of them. The flocculae are in general fainter and larger than the puffs visible in the lower portion of the field of view, appear to grow in scaled brightness as they propagate, and can best be detected only outside of ∼10°, fading in even as the striae fade out with altitude.

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It is this combination of the fading striae and the growing flocculae that gives rise to the flocculated appearance of the outer HI1 field of view in properly background-subtracted image sequences. In particular, if the HI1 image quality were degrading sufficiently to obscure the striae, the flocculae and other localized features would be equally obscured. The fact that particular classes of feature, propagating with the solar wind, grow in amplitude, even as others fade, reveals that simple loss of contrast or related image degradation is not causing the change in image texture across the field of view: the textural shift reflects a shift in the spatial form of density fluctuations across this field of view.

In addition to anecdotal analysis of particular striations in the corona, we characterized the evolution of image structure across the HI1 field of view using structure functions (Schulz-Dubois & Rehberg 1981; Cover & Thomas 1991, p. 175), which summarize the variability of a data set at different scales and directions. Given a three-dimensional data set such as our HI1 image sequence, h(x, y, t), the second-order structure function is a six-dimensional functional in both the independent variables and their offsets:

$$\begin{eqnarray}&&{S}_{{xyt}}(x,y,t,{\rm{\Delta }}x,{\rm{\Delta }}y,{\rm{\Delta }}t)\\ &&\quad \equiv {(h(x,y,t)-h(x-{\rm{\Delta }}x,y-{\rm{\Delta }}y,t-{\rm{\Delta }}t))}^{2}.\end{eqnarray} \tag{ 5 }$$

This S_xyt is useful because it can be averaged across neighborhoods of the primary variables' domains to yield a location-dependent measure of structure versus scale in the data sets. We set Δt = 0 and average over t to arrive at a four-dimensional time-averaged structure function $\langle {S}_{{xy}}\rangle$, which characterizes the average local variability of the data set as a function of location and scale:

$$\begin{eqnarray}&&\langle {S}_{{xy}}\rangle (x,y,{\rm{\Delta }}x,{\rm{\Delta }}y)\\ &&\quad \equiv \langle {(h(x,y,t)-h(x-{\rm{\Delta }}x,y-{\rm{\Delta }}y,t))}^{2}\rangle ,\end{eqnarray} \tag{ 6 }$$

where the angle brackets represent averaging over time. We carried out the computation of the Δt = 0 cut of S_xyt at each time and averaged the result across all HI1 frames, prepared as in the middle panel of Figure 2, from 2008 December 15–29, to yield $\langle {S}_{{xy}}\rangle$ throughout the data set, with image x ranging over azimuthal angle α and image y ranging over log(ε). To further reduce noise, we smoothed the computed values of $\langle {S}_{{xy}}\rangle$ by convolution in the x and y dimensions with a kernel 27 wide in α and having the equivalent size in log(ε). Further, to analyze the radial evolution of "typical" striae and other wind features without contamination from the potentially anomalous central stripe that is visible at α = 270°, we selected two azimuths, α = 250° and α = 290°, on opposite sides of the 270° line and averaged $\langle {S}_{{xy}}\rangle$ between them. This further reduced the data set to three dimensions:

$$\begin{eqnarray}&&S(\varepsilon ,{\rm{\Delta }}\alpha ,{\rm{\Delta }}(\mathrm{log}(\varepsilon ))\\ &&\quad \equiv \overline{\langle {S}_{{xy}}\rangle (\{250^\circ ,290^\circ \},\varepsilon ,{\rm{\Delta }}\alpha ,{\rm{\Delta }}(\mathrm{log}(\varepsilon ))),}\end{eqnarray} \tag{ 7 }$$

where the overbar represents the averaging over neighborhoods in α and ε. We used the two separate values of α to symmetrize S as a function of Δα, while still eliminating the structure near 270°.

Thus S contains a separate image in the independent variables Δα and Δε for each separate value of ε. Several of these structure images are shown in Figure 7. The transformed data use a logarithmic scaling for ε, so these corresponding structure images are most naturally measured in degrees of azimuth and decibels (dB) of elongation angle. One dB corresponds to a factor of roughly 1.26 in ε, so in the ε = 11° panel at the center of Figure 7, a value of 0.8 dB in the radial separation corresponds to a separation of 22. Because the underlying images are conformal to the original sky plane, the structure function planes are also conformal, and any apparent anisotropy reflects a real anisotropy in the original images.

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Each image plot across the top row of Figure 7 shows a single two-dimensional (2D) slice of S: it is the squared difference (averaged across azimuthal selections and time) between the value of a single pixel at a given apparent radius and each selected azimuth and the corresponding value of nearby pixels with the corresponding offset throughout the plotted image. The image values drop to zero at the origin because the average squared difference between a single pixel and itself is identically zero. The tall, narrow structure of the trough at central radii of 57, 81, and 11° reflects the striation of the "typical" corona at those altitudes: there is more variation of image value in the α direction than in the ε direction. The more rounded structure at central radii of 15° and 20° reflects the increasing isotropization of the brightness variation at higher altitudes.

The plots in the bottom row of Figure 7 are horizontal and vertical cuts through the origin of the 2D structure function slices. They reveal the increasing isotropy of the image as the wings of the vertical cuts rise with altitude and the wings of the horizontal cuts drop with altitude. It is immediately apparent by inspection that the shift in the structure function is not due to residual noise in the images: uncorrelated noise would produce a sharp rise from the origin, rapidly transitioning to a shallower slope at scales dominated by image features rather than by background noise. The cuts grow monotonically with distance from the origin and show no sign of a noise-related break at the scales of interest. The horizontal cuts grow more shallow with altitude, indicating fading of the radial striae, even as the radial cuts grow steeper.

Note that we have followed the typical definition of S, which is asymmetric rather than being symmetrized around a central point. This means that S can in principle be, and in practice is, asymmetric with respect to the Δ variables at particular locations. For example, there is a slight systematic asymmetry visible in the vertical (radial) structure functions in Figure 7. This is because there is a systematic variation in the structure function with ε, and point pairs taken in the positive-going Δε direction are centered at a different location than point pairs taken in the negative-going Δε direction.

To better understand the radial evolution, we also plotted the structure function in quasi-spatial coordinates. These are plotted in Figure 8. We made use of the known but nontrivial observing geometry to make the conversion, which is somewhat approximate. In particular, there is a variable (±5% to ±40%) cross-scale mixing effect, increasing with b_eff  across the field of view, due to the shifting perspective along varying lines of sight. Further, because we are observing a large fraction of the corona rather than a compact feature such as a CME, the conversion from elongation ε to spatial distance from the Sun uses an "effective impact parameter" b_eff, rather than the simple trigonometric impact parameter b, for each line of sight. Both of these effects are described at length in the Appendix.

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While the horizontal (azimuthal) structure function in the images clearly expands faster than the overall radial expansion of the solar wind (Figure 7), the vertical (radial) structure function in the images appears roughly constant as a function of b_eff. Figure 9 shows the evolving trend. Because, as we demonstrated in Section 3.1, the average scaled brightness of the wind does not change with altitude, the separation distance ${\rm{\Delta }}{b}_{\mathrm{eff},\mathrm{th}}$ for S to exceed a set threshold S_th is a valid measure of the changing hardness of S with solar distance. That is plotted in Figure 9 for ${S}_{\mathrm{th}}=0.5\times {10}^{-22}{B}_{\odot }^{2}\,{\deg }^{6}$. Throughout the measurable range of offsets from the Sun, this thresholding distance remains approximately constant: the radial cuts do not soften as do the lateral cuts. This behavior is consistent with fully developed turbulence in the radial direction.

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The striae themselves fade faster with respect to radius than would be expected from bright features in a constant-speed, conservative solar wind, eventually dropping below our processed images' noise floor, where they can be seen by further, more aggressive spatial averaging of the HI1 images to further beat down the noise floor, at the cost of further reducing spatial resolution. There is no large-scale breakup of the striae as might be expected from large-scale instabilities.

We are confident in this fundamental observational result because we have eliminated the most plausible observational scenarios that might mimic fading of the coronal striae. In particular, by demonstrating that the striae, while faint, can be observed even under the reduced noise floor from processed HI1 data, we have shown that noise and background effects alone are not responsible for the apparent fading of the striae, and this result is corroborated by the opposite brightness trend in the flocculae, which have spatial scales and scaled brightness values similar to the striae themselves.

The striae are rendered visible by Thomson scattering, and therefore the features' geometry relative to the Thomson surface—the locus of the sphere whose diameter extends from the observer to the Sun—becomes important. The Thomson surface marks the point of greatest apparent surface brightness per unit density along a given line of sight, and therefore purely radial structures may gain or lose apparent surface brightness relative to the applied ε⁻³ scaling, depending on this geometry. In fact, Vourlidas & Howard (2006) proposed to use this effect to locate features in three-space by looking for anomalous dimming as the features propagate. However, the Thomson surface is surrounded by the "Thomson plateau" (Howard & DeForest 2012), a broad range of out-of-image-plane angles over which feature radiance is very nearly independent of geometry on a given line of sight.

We eliminate the possibility that the fading is due to scattering physics, by noting that no striae appear to fade in with altitude, only to fade out. Because our data set spans over half a solar rotation and striae are observed to exist across a wide range of coronal longitudes, Thomson plateau effects should cause cross-fading between different striae and not a uniform attenuation of all striae. That is because, in the overall population of dense radial features, approximately the same number of radial features should be outside the Thomson plateau at low altitudes and enter it at high altitudes as exist within the Thomson plateau at low altitudes and exit it at high altitudes. Moreover, because the field of view is smaller than the angular extent of the Thomson plateau, it is expected that most visible striae remain within the plateau for their entire visible length.

Turning to more subtle solar geometric effects: the "pancake effect" of azimuthal expansion in the solar wind, coupled with perspective or projection effects, can cause fading of features that are compact in the lower corona and separated radially. Features of that description can apparently merge in the image plane as they propagate outward, even when no physical merging or mixing occurs; this is illustrated with a simple geometric argument by Viall et al. (2010). Similar effects can cause apparent merging in blobs that pass one another along the line of sight, as quantified for the "Sheeley blobs" by Sheeley & Rouillard (2010). These effects impact variations along the radial direction and may affect the "flocculated" appearance. Importantly for the current work, these effects work to blend individual features and produce an illusion of a radial stria, rather than the opposite, so they cannot account for the general fading of the striae. To cause a stria to fade, this mechanism requires a very rare, coincident coalignment to cause an illusive stria close to the Sun and reveal puffs farther from the Sun.

The fading of the striae, in the scaled brightness parameter ${B}_{\odot }\,{\deg }^{3}$, is surprising because it varies from the expected behavior of a mass-conserving, constant-speed radial solar wind. The two most obvious physical mechanisms to explain it are (A) that the wind might not be constant speed or (B) that the features themselves might not conserve visible mass.

We eliminate acceleration effects (possibility A) by noting that, to explain the observed fade of a factor of more than 3 in scaled brightness, the wind inside the striae would have to accelerate by a comparable factor across the field of view. No such acceleration is measured, and the outflow speed of bright features varies by less than a factor of 2 across the entire image (and considerably less along a given radius).

Regarding conservation effects (possibility B), conservation of bright structure is not necessarily conservation of mass. The images are produced using sunlight that is Thomson-scattered off of free electrons. Photometric mass estimates in the heliosphere (DeForest et al. 2014) or corona (Vourlidas et al. 2000) are produced by assuming the free electrons are part of a neutral, 100% ionized plasma with approximately coronal composition (specifically, proton-to-helium ratio). Recombination of ions and free electrons could in principle cause the striae to fade, but this effect would require more than two-thirds of the plasma to neutralize. That effect is implausible: although electrons in the solar wind typically suffer a few tens of collisions enroute from the Sun (Salem et al. 2003), by the altitude range under consideration (40–100 R_⊙), collisions are very rare (Hundhausen et al. 1968; Owocki & Scudder 1983). Further, the high kinetic temperature of order 10⁵–10⁶ K prevents recombination (Gibson 1973, p. 274).

Eliminating (A) and recombination effects leaves the possibility of intrinsic variation of the entrained mass within each stria. Such mass loss would be due to diffusion or other effects that transport material from the dense structure to the intervening spaces.

One possibility for structural mass loss is misalignment at high altitudes of quasi-planar structures that may, by coincidence, align with the line of sight when seen closer to the Sun. This possibility is important because many streamers and pseudostreamers are thought to have quasi-planar geometry rather than being compact structures (Wang et al. 2007). But the same argument applies to these effects as to effects from the geometry of Thomson scattering: if the striae were primarily due to chance alignments with quasi-planar density structures (and loss of alignment were the reason most observed striae fade at large radii), then chance alignments at those farther distances should be expected to cause a roughly equal number of striae to fade in as out, which is not observed. We conclude that the fading striae are not due to chance alignments and subsequent misalignments of smoothly expanding quasi-planar density structures in the solar wind.

Our discussion so far has focused on eliminating various observational effects that could in principle explain the apparent radial evolution seen in the images and diagnostic parameters shown in Section 3. It remains to offer a physical scenario to explain the fading of striations and emergence of puffs and flocculation and the isotropization of the computed structure functions of the images. We now consider the effects of the turbulent fluctuations observed at 1 au, extrapolated inward to the observed range of altitudes, and whether they can account for the inferred lateral transport of mass.

We begin by recalling the general changes in nonlinear behavior moving outward from the Sun. Below the Alfvén surface, where the solar wind speed exceeds the radial speed of fast-mode MHD waves, the plasma is also low β, and the dynamics are strongly influenced by the magnetic field. This gives rise to a strong correlation (or spectral) anisotropy relative to the magnetic field direction (Zank & Matthaeus 1992; Cranmer et al. 2015; Oughton et al. 2015). Under these conditions, magnetic flux tubes strongly align with the large-scale magnetic field, and magnetic fluctuation gradients concentrate in the perpendicular direction. The variability of density across flux tubes then gives rise to strong anisotropic density gradients that, in turn, appear as striae in the imaging data. In the sub-Alfvénic coronal regions, the dynamical emergence of this anisotropy is in part due to an active anisotropic spectral transfer (Shebalin et al. 1983; Oughton et al. 1994) that amplifies the transverse gradients relative to the radial ones. Furthermore, in the corona this turbulence is possible because inward-propagating fluctuations overcome the outward flow of the solar wind and interact with their outward-propagating counterparts (MHD wave–wave interactions).

Above the Alfvén surface, both of these effects change. First, inbound (in the comoving frame) waves are advected outward (in the solar system stationary frame), reducing the overall strength of the wave field and delaying development of active Alfvénic turbulence. Second, above the related β = 1 surface, the magnetic field strength is no longer dominant over either thermal or convective effects. On the other hand, the state of the plasma flowing into this critical region near the Alfvén and β = 1 surfaces is already highly anisotropic, having been shaped by the well-ordered and nearly radial average magnetic field. This accounts for the dominant appearance of striae, interrupted by discrete features, including CMEs, in images from coronagraphs and our HI1 data set. Therefore, the observed striae seen at or above the β = 1 surface are consequences of factors operating at lower altitudes. Although the striae persist into our field of view, the factors that build and maintain the coronal anisotropy are no longer present in the young solar wind.

One may now ask whether the existing turbulence field in the solar wind at 150 Gm (1 au), extrapolated inward to 38 Gm (b_eff for lines of sight close to 12° from the Sun), account for the breakup of the striae that we observe there. At 150 Gm, typical values of the turbulent velocity Z(150 Gm) and correlation length L(150 Gm) are 25 km s⁻¹ and 1 Gm, respectively (Ruiz et al. 2014; Isaacs et al. 2015). Both of these quantities are approximately log-normally distributed, so extreme outliers are to be expected. Observations suggest that Z ∝ r^−1/2 in the inner heliosphere (Verma & Roberts 1993), so Z(38 Gm) ≈ 12 km s⁻¹. Further, taking the correlation length to scale between L ∝ r and L ∝ r^1/2 (Breech et al. 2008), at 38 Gm (0.25 au) the correlation scale should be L(0.25 au) ≈ 0.25–0.5 Gm if the turbulent field observed at 150 Gm (1 au) persists inward to this extent.

Based on the estimates above, correlated density or flow features in the outer reaches of the solar wind observed in Figure 2 have an estimated expected size of 0.25–0.5 Gm and subtend approximately 025–05 of azimuth. Features of this scale are close to the resolution limit of the present observation, but large-scale outliers, extending to several times the expected correlation scale, should be observed if the near-Earth turbulent field happens to be already well developed in the observed range of altitudes.

In fact, such displacements appear to be observed. Figure 10 shows the lateral evolution of the stria in the bottom row of Figures 4 and 5 as it propagates outward. This stria is typical, and several other striae show similar behavior. The image shows the average brightness evolution, averaged vertically across 10% of image height, in a horizontal band extending across each of the subfield images in the Figure 4 data set. The overlaid points reveal lateral evolution of the center of the stria in the comoving frame of the solar wind. Each point shows the horizontal location of maximum brightness at the corresponding time and apparent distance (elongation ε). We determined the error bars by noting that the rms noise level at ε = 14° is 4 × 10⁻¹³ R_⊙ deg³ and finding the location where the radiance in each horizontal cut falls from the maximum by at least that amount. The points are calculated only for every third frame of the data set, to avoid the effects of the three-frame temporal averaging we used during the processing: each point represents a true independent sample.

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While the variations in azimuth do not exceed the error bars for most pairs of points, the coherence of the lateral motion renders the motion statistically significant. The amplitude is approximately 05 in azimuth, in good agreement with the estimate above.

The lateral motions that we observe are inconsistent with direct radial motion and consistent either with scaling of the fully developed turbulence detectible near Earth, with local onset of lateral turbulent motion due to hydrodynamic instabilities (such as the Kelvin–Helmholtz instability), or with isotropization of MHD turbulence present in the corona itself. Lateral convective transport from these processes must occur to account for the fading of the striae, but there is no clear and direct sign of the postulated breakup. Therefore, we conclude that the breakup must occur on scales smaller than the effective spatial resolution of ∼0.5–2.5 Gm.

In counterpoint to the fading of the striae, flocculae "fade in" smoothly from elongation angles of ∼8° out to 20°. The radial length scale of the flocculae is comparable to the length scale in the Viall puffs and Sheeley blobs that enter the HI1 field from the lower corona: 3–10 Gm. Two explanations spring immediately to mind: (1) the flocculae could be local interaction regions, zones of density enhancement caused by variation of outflow speed along a given wind flow streakline⁶  or (2) the flocculae could be a symptom of the same instabilities that are seen to offset and fade the striae.

To investigate the hypothesis that the flocculae are caused by local interactions between different-speed streams, we consider the divergence of the velocity field ∇ · ${\boldsymbol{v}}$ necessary to cause the observed growth rate of the density in particular flocculae (as in Figure 6). To determine this divergence, we must estimate the proportional density enhancement represented by the brightness growth of the flocculae.

Because of the way that coronal and heliospheric images are created, there is no "absolute zero" to unpolarized Thomson-scattering images in general. This is because the images are produced by subtracting a steady or nearly steady background model in the fixed focal plane (coronagraphs) or the celestial sphere (heliospheric images); there is no intrinsic difference between a steady signal from the F corona or stray light and a steady signal from a stationary or quasi-stationary electron density structure in the field of view. Hence, essentially all unpolarized Thomson-scattered images from current instruments report "feature excess" brightness, and hence inferred densities are only "feature excess" densities, compared to an unknown steady background.

Near Earth, the typical electron density of the slow solar wind is 3–10 cm⁻³. Extrapolating inward to 10° elongation from the Sun (a line-of-sight angle whose weighted-average distance from the Sun is 28 Gm; see the Appendix), we estimate that the "typical background" electron density at 10° elongation in our images is 100–300 cm⁻³ along an effective line-of-sight distance of 50 Gm (0.33 au). Taking the lower figure and following Equation (5) of Howard & DeForest (2012), we infer that

$$\begin{eqnarray}&&{B}_{\mathrm{bk}}(\varepsilon =10^\circ )\approx \displaystyle \frac{{B}_{\odot }{\sigma }_{{\rm{t}}}\pi {r}_{\odot }^{2}}{((1\,\mathrm{au}){(\sin (10^\circ ))}^{2}}(100\,{\mathrm{cm}}^{-3})(50\,\mathrm{Gm}),\end{eqnarray} \tag{ 8 }$$

where B_bk is the calculated background radiance from Thomson scattering, B_⊙ is the mean solar radiance, σ_t is the Thomson-scattering cross section of 4.0 × 10⁻³⁰ m², and r_⊙ is the mean solar radius of 6.96 × 10⁸ m. This evaluates to B_bk(ε = 10°) = 4.2 × 10⁻¹⁴B_⊙. Multiplying by ε³ yields the elongation-independent scaled background radiance ${B}_{\mathrm{bk}}^{\prime }\ =4.2\times {10}^{-11}{B}_{\odot }\,{\deg }^{3}$. This ${B}_{\mathrm{bk}}^{\prime }$ remains approximately equal throughout all values of ε that we observed.

The parameter ${B}_{\mathrm{bk}}^{\prime }$ is comparable to the measured feature-excess brightness in typical flocculae, as seen in the rightmost panels of Figure 6. The calculated ${B}_{\mathrm{bk}}^{\prime }$ exceeds this feature brightness by a factor of 3–10. However, a typical floccule does not extend along the entire line of sight. Taking the flocculae in Figure 6 to have roughly the same extent out of the screen as in azimuth, they subtend roughly 10° relative to Sun center, rather than the approximately 78° subtended by the effective line of sight. Therefore the excess density associated with each floccule is larger than the calculated ratio of ${B}_{\mathrm{floccule}}^{\prime }(\varepsilon )/{B}_{\mathrm{bk}}^{\prime }$ by a factor of ∼8. Hence the floccule's excess density is roughly equal to the density of the background flow at the same positions on the line of sight, at the highest elongations in Figure 6 (∼20° from the Sun). This implies that, between altitudes of roughly 40 R_⊙ and 80 R_⊙, the density in a floccule approximately doubles compared to a quiescent propagating wind.

The flocculae have typical radial sizes of 3–6 Gm, so if the density enhancement arises from a simple negative divergence of the bulk flow, the material at the visible edges must propagate inward relative to the mean bulk flow, crossing the width of the floccule during a transit across 40 R_⊙ (30 Gm). Thus, the average Δv between the leading and trailing sides of the floccule must be about 10%–20% of the wind speed, corresponding to a ±5% to ±10% antisymmetric deviation from the mean radial outflow speed in the vicinity of each floccule.

In light of the existence of highly structured radial striae, cross-field evolution due to instabilities or onset of turbulence must play a large role in the development of the flocculae. If the flocculae were generated by purely compressive processes due to temporal variation in the speed of individual flux tubes' coronal wind, as described above, then their azimuthal size should be comparable to the observed scale of individual wind structures, that is, striae. Instead, we observe the flocculae to be, typically, over three times larger in azimuthal extent than typical striae. This implies a collective behavior on scales larger than the natural azimuthal scale on which the wind is injected from the corona itself.

An alternative explanation is that the flocculae are density enhancements caused by turbulent perturbations to the flow field rather than by pileup in a smooth flow. In this view, turbulent motions within the flow give rise to density perturbations on length scales comparable to the correlation scale of the turbulence. We note, however, that the radial length scale of the flocculae is about 10× larger, and their azimuthal length scale is perhaps 30× larger, than the scaled correlation length calculated in Section 4.1. In this regard, one might explain the flocculae as density enhancements associated with the very largest and strongest velocity perturbations, associated with the onset of turbulence in the inner heliosphere. The moderately frequent occurrence of dynamically active fluctuations appears to be reasonable, based on the observed long tails on the distributions of correlation lengths (approximately log normal) seen in Helios data (Ruiz et al. 2014). It is also possible that the flocculae emerge as a manifestation of so-called f⁻¹ noise, which is observed in situ at very low frequencies in solar wind density observations (Matthaeus et al. 2007).

A detailed analysis of how and whether exceptional turbulent fluctuations, f⁻¹ noise, or some other collective phenomenon may account for the present observations is a topic for future work.