Density Enhancement Streams in The Solar Wind

This Letter describes a new phenomenon on the Parker Solar Probe of recurring plasma density enhancements that have Δn/n ∼ 10% and that occur at a repetition rate of ∼5 Hz. They were observed sporadically for about 5 hr between 14 and 15 solar radii on Parker Solar Probe orbit 12 and they were also seen in the same radial range on both the inbound and outbound orbits 11. Their apparently steady-state existence suggests that their pressure gradient was balanced by the electric field. The X-component of the electric field component produced from this requirement is in good agreement with that measured. This provides strong evidence for the measurement accuracy of the density fluctuations and the X- and Y-components of the electric field (the Z-component was not measured). The electrostatic density waves were accompanied by an electromagnetic low-frequency wave, which occurred with the electrostatic harmonics. The amplitudes of these electrostatic and electromagnetic waves at ≥1 Hz were greater than the amplitude of the Alfvénic turbulence in their vicinity so they can be important for the heating, scattering, and acceleration of the plasma. The existence of this pair of waves is consistent with the observed plasma distributions and is explained as an oscilliton due to the nonlinear coupling between the kinetic Alfvén wave and the ion cyclotron mode, which belongs with the minor population of alpha particles.


Introduction
Low-frequency turbulence is thought to energize the solar wind plasma through a cascade process that is described by the power spectra of the fields.The magnetic field has generally been the parameter utilized in such studies (Chen et al. 2010(Chen et al. , 2020(Chen et al. , 2021;;Chen 2016;Bowen et al. 2020).In a few cases, the electric field and/or the plasma density spectra have been shown to have decreasing power in the kinetic range as compared to that in the inertial range (Salem et al. 2012;Chen et al. 2013;Mozer et al. 2020).The purpose of this Letter is to describe a new wave mode in which the power in the solar wind plasma density and electric field exceeded that of the local Alfvénic turbulence.
The measurements of interest were made on the Parker Solar Probe, whose X-Y plane, perpendicular to the Sun-satellite line, contains a two-component electric field and spacecraft potential measurement by antennas that are not much larger than the spacecraft (Bale et al. 2016).By fitting the measured spacecraft potential to the low-rate density measurements obtained from the SWEAP plasma measurements (Kasper et al. 2016;Whittlesey et al. 2020), higher-frequency estimates of the plasma density and density fluctuations (Mozer et al. 2022) have been obtained.

Data
Examples of power spectra obtained on Parker Solar Probe orbit 12 during a 15 s interval when the spacecraft was located about 15 solar radii from the Sun are given in Figure 1(a).The magnetic field spectrum (green) is typical of that often observed, with excess power at ∼1 Hz, a decrease in power of 3 orders of magnitude between 1 and 20 Hz, and a break at 10 Hz, near the ion gyrofrequency (denoted by the vertical dashed line).In normal turbulence, the electric power decreases by a similar factor over the given frequency range.However, the electric field (black) and density (red) spectra in the example of Figure 1 differ greatly from this expectation, having peaks near 1, 5, 11, and 16 Hz with powers more than 1 to 3 orders of magnitude greater than that expected from the magnetic field spectrum.Because there is power near 1 Hz in all three fields, there must be both an electrostatic and electromagnetic wave at this frequency.The peaks at higher frequencies must be due to an electrostatic wave whose characteristics are displayed in the waveform plots of Figures 1(b) and (d) as spiky pulses of electric field and density fluctuations, Δn/n, occurring at a frequency of several Hz.These density waves were observed sporadically for about 5 hr between 14 and 15 solar radii on this orbit and they were also seen in the same radial range on both the inbound and outbound orbits 11. (Other orbits have not been carefully searched to see if these waves are more wide-spread.)As summarized above, the electric field power in these waves is much greater than that in the Alfvénic turbulence at the same frequencies.
The plasma density is measured by the particle instruments on the Parker Solar Probe at an ∼1 Hz rate.To obtain the plasma density at a higher rate, the spacecraft potential (the average of the four biased antenna voltages) is utilized.It is measured at a high rate (∼500 Hz) and it is approximately proportional to the log of the plasma density (Mozer et al. 2022).The fit of the low-rate spacecraft potential to the log of the plasma density provides a least-squares equation that is then used with the high-rate spacecraft potential to obtain the highrate plasma density and its fluctuations.
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Figure 2 provides further information on the density waves, in which panel (a) gives the density fluctuation amplitude, Δn/n ∼ 0.1, while panels (b) and (c) give the two measured electric field components.The unmeasured EZ component is estimated in panel (d) by combining the two measured components with the assumption that the parallel electric field was zero.It was produced to show that the major electric field component was in the X-direction.Because this is an electrostatic wave, its k-vector must also have been largely in the same X-direction.Because the magnetic field components at this time were (0, −200, −600), the wave propagated nearly perpendicular to the background magnetic field.
The long duration of these density streams suggest that they are stable structures.According to the generalized Ohm's law, this stability requires an electric field that balances the pressure gradient.This pressure gradient electric field, in the X-direction, is computed as shown in Figure 2(e), for the observed temperature of 50 eV and an X-component solar wind speed of 200 km s −1 .(The correct speed to use is the solar wind speed plus the sound wave speed in the plasma frame.Because the sound wave speed is about 3 times smaller than the Alfvén speed (because beta ∼0.1), its speed is about 3 times smaller than the solar wind speed so it is ignored for this computation.) This estimate of EX, given in Figure 2(e), is in reasonable agreement with the measured electric field in Figure 2(b).This agreement offers evidence that the electric fields and the density streams were correctly measured and that the density structures were in pressure balance.
Also shown in Figures 2(f) and (g) are the four single-ended potentials of the electric field measurement.Their similarity and normal behavior suggest that the resulting fields and density structures were well measured and not associated with a spacecraft wake or similar non-physical perturbation.All data in Figure 2, other than that in panels (f) and (g), were high pass filtered at 1 Hz.
Variations of the proton and electron fluxes and spectra were uncorrelated with the times of the density streams.In addition, beta (1), the Alfvén speed divided by the solar wind speed (∼1), the ion temperature (50 eV), the wind speed (400-600 km s −1 ), and the Debye length (2 m), all had significant fluctuations but none of them correlated with the off-and-on nature of Δn/n.Power spectra of the electric field (black), plasma density (red), and magnetic field (green) in panel (a) at the time of the data in the remaining panels.The magnetic field spectrum appears normal with both a peak at ∼1 Hz (normally attributed to Alfvénic turbulence), a large decrease of power between 1 and 20 Hz, and a break in the spectrum near 10 Hz, at a frequency near the proton gyrofrequency (the dashed vertical line).The electric field and density spectra in the panel (a) display peaks at ∼1 Hz, 5 Hz, and harmonics, and have a much smaller decrease with frequency than the magnetic field.Because there is wave power in all three components near 1 Hz, this frequency signature must come from a mixture of electromagnetic and electrostatic waves, so it is not pure Alfvénic turbulence.The harmonic signatures at higher frequencies are purely electrostatic and the wave power is much greater than that expected for Alfvénic turbulence.The electric field, magnetic field, and density waveforms that produced these spectra are illustrated in panels (b), (c), and (d).
The electrostatic potential in these waves may be estimated from Ohm's law as where f is the potential in the wave.For the observed electron core temperature of 50 eV, the potential in the wave was ∼5 V.This potential is sufficiently large that it may be at least partially responsible for heating or accelerating the plasma by their trapping in the wave potential.The waveforms, being spiky, appear to be steepened.The time for a sound wave to steepen is about where τ is the steepening time, k is the initial wavenumber, and cs is the sound speed.Because k•Vsw ∼ ω, where ω is the observed angular frequency and Vsw/cs ∼ 3, the steepening time is the order of 1 s.This explains why the density waves are steepened to appear as shocklets because their lifetimes are observed to be greater than ∼1 s. Figure 3 presents another example of the density streams.In this case, there are combined electrostatic and electromagnetic signals at both 1 and 8 Hz, while there are harmonics of the electrostatic signal to frequencies greater than 50 Hz.It is again noted that the amplitude of the electric field is 1 order of magnitude greater than that expected from the turbulence described by the magnetic field.
Additional properties associated with this wave mode are illustrated in Figure 4, which presents data obtained over the 1 hr interval that included the density streams.Figures 4(a) and (b) give wavelet plots of the electric field and density fluctuations, which show that the ∼5 Hz power of the density streams and the associated ∼5 Hz electric field fluctuations occurred sporadically during the time interval.In addition to the ∼5 Hz power in the electric field, there was power at 1 Hz, which was due to the electric field of the simultaneously occurring electromagnetic wave.As expected, this power was absent in the density fluctuations of Figure 4(b).Figure 4(c) presents the core proton perpendicular temperature divided by the core proton parallel temperature (T perp /T par ), which varied but was as large as 10 sporadically, but not necessarily at the time of the density streams.This result is seen inside the three pairs of vertical dashed lines that border three of the about eight regions of enhanced fields and density fluctuations.somewhat larger than the electron beta.Figure 4(e) gives the ratio of the alpha to proton densities, which are seen to maximize inside the three pairs of vertical dashed lines.
To summarize the experimental data: 1. Streams of spiky, enhanced plasma density occurring at a rate of ∼5 Hz and harmonics have been observed along with the electric field required for pressure balance in the plasma.2. The spectra of the density streams and the associated electric field were comprised of many frequency harmonics.3.These electrostatic waves occurred along with a lowfrequency electromagnetic wave whose harmonics decreased in amplitude more rapidly than the electric field or density harmonics.4. The amplitudes of these electrostatic and electromagnetic waves were greater than the amplitude of the nearby Alfvénic turbulence.
5. The k-vector of the density stream electric field was perpendicular to the background magnetic field.6.The core proton temperature distribution was anisotropic with T perp /T par as large as 10 at times in the vicinity of the density streams.7. The proton plasma beta was somewhat larger than the electron plasma beta but both were near 0.1.8.The alpha to proton density ratio was maximum in the vicinity of the density streams.

Discussion
Previous work on magnetoacoustic (magnetosonic) waves has shown that a low-frequency electromagnetic ion cyclotron wave (EMIC) can be accompanied by harmonics that are electrostatic (Zhu & Chen 2019;Gao et al. 2021).This suggests that magnetoacoustic waves may be associated with the observed density streams and electromagnetic wave.As Figure 3. 10 s of data in panel (a) that illustrate the power spectra of the electric field (black), the density fluctuations (red), and the magnetic field (green).The 1 and 7 Hz peaks contain electric field, magnetic field, and density maxima, indicating that both an electrostatic and electromagnetic wave existed at these frequencies.It is noted that the amplitude of the electrostatic wave at all frequencies was large compared to that of the Alfvénic turbulence expected from the magnetic field spectrum.Thus, this new wave mode may be an important contributor to the heating, scattering, and acceleration of the plasma.described in a recent paper by Sauer & Dubinin (2022), EMIC waves are interpreted as stationary, nonlinear waves (oscillitons).These arise due to the energy and momentum coupling between the right-hand polarized electromagnetic wave (R-mode) and the cyclotron mode of the minor ion population whereby mainly quasi-parallel wave propagation has been considered.As a first step, the dispersion analysis deals with the consequences of the coupling of two wave modes in the ω-k range where their phase velocities coincide.The associated effects, as the existence of spatially growing waves, have been investigated for a number of different wave modes.After the first description of oscillitons in a two-ion plasma (Sauer et al. 2001), other types of stationary nonlinear waves have been considered, for example in the range of whistler waves (Sauer et al. 2002;Agapitov et al. 2018) whose coupling with Langmuir waves results in Langmuir-whistler oscillitons (Sauer & Sydora 2011).Further, nonlinear kinetic Alfvén waves (KAWs) have been considered by Dubinin et al. (2005).
In contrast to EMIC waves, the observed density enhancements occur at quasi-transverse propagation.They obviously arise below the proton cyclotron frequency due to the interaction of the KAW with the cyclotron wave (α-IC) of the alpha particles.Such a situation is illustrated in Figure 5(a), where results of Vlasov kinetic dispersion analysis are shown.The Fortran code applied here has been used in past for several other purposes, such as for studies of the excitation of whistler (Sauer & Sydora 2010) and EMIC waves (Sauer & Dubinin 2022).A propagation angle of θ = 85°, electron and proton temperatures given by β e = β p = 0.1, and a temperature anisotropy of T p⊥ /T p|| = 6.5 were chosen.The two features to note are the splitting of both wave modes marked by the red circle in the top (ω versus k) and middle panel (phase velocity versus k) of Figure 5(a) and the instability of the KAW at kc/ω p ∼ 4.5, shown in the bottom panel (growth rate versus k).
The subsequent theoretical investigations are carried out on the basis of the Hall MHD, which ultimately enables the study of nonlinear interactions.Figure 5(b) shows the already described effect of mode splitting which leads to the formation of an ω-k point of maximum phase velocity (marked by red circles as in Figure 5(a)) and thus suggests the existence of a stationary growing wave there.This property can be seen in Figure 5(c), where the (complex) wavenumber k = (k r , k i ) and the associated frequency ω = k r U are shown as a function of the "oscilliton" velocity U.The spatially growing wave occurs at U ∼ 0.1V Ap (V Ap : Alfvén velocity of protons).The related frequency is just the gyrofrequency of the alpha particles), i.e., ω ∼ 0.5Ω p (Ω p : proton gyrofrequency).
The Figures 5(d)-(i) result from the stationary, nonlinear Hall-MHD theory (Sauer & Dubinin 2022), which is applied to the interaction of the KAW and the alpha-particle ion cyclotron wave (α-ICW).The hodograms in Figures 5(d   observations, in particular with regard to spectral data in Figure 3(a).
After presenting the theoretical results, which are summarized in Figure 5, a few remarks about kinetic Alfvén (KA) oscillitons and oscillitons in general should be added: the KA oscillitons represent independent identities, which differ in their properties from the unstable KA waves driven by a proton temperature anisotropy (see Figure 4(c)).Such a difference is a general characteristic of oscillitons that arises from the nonlinear coupling of two different wave modes.The unstable wave takes on the triggering function and ultimately becomes saturated.The oscillitons, on the other hand, can survive without the presence of instability.Such a behavior has been described in the kinetic simulations of Langmuir-whistler oscillitons by Sauer &Sydora (2011), andFigure C1 in Sauer &Dubinin (2022).A similar connection between instability and nonlinear wave has also been pointed out in the captions of Figure 4.It is written that the density enhancements are present in the vicinity of the large proton temperature anisotropy, but not exactly at the time of their appearance.Especially, in the interval around 11:20 UT marked in Figure 4, pronounced density increases occur, without sufficient proton temperature anisotropy of T p⊥ /T p|| > 4 being present.
In summary of the theoretical analyses, an unstable KA wave mode is described that is consistent with the experimental requirements that the wave must propagate close to perpendicular to the background magnetic field in an environment having plasma betas close to 0.1, a proton temperature anisotropy that is as large as 8, and an enhanced alpha particle density.The arising electromagnetic field combined with large electron density disturbances and electrostatic fields is interpreted as a KA oscilliton due to the nonlinear coupling between the KAW and the ion cyclotron mode (α-ICW) that belongs with the minor population of alpha particles, which are a maximum at times of the density wave, as shown in Figure 4(e).

Figure 1 .
Figure1.Power spectra of the electric field (black), plasma density (red), and magnetic field (green) in panel (a) at the time of the data in the remaining panels.The magnetic field spectrum appears normal with both a peak at ∼1 Hz (normally attributed to Alfvénic turbulence), a large decrease of power between 1 and 20 Hz, and a break in the spectrum near 10 Hz, at a frequency near the proton gyrofrequency (the dashed vertical line).The electric field and density spectra in the panel (a) display peaks at ∼1 Hz, 5 Hz, and harmonics, and have a much smaller decrease with frequency than the magnetic field.Because there is wave power in all three components near 1 Hz, this frequency signature must come from a mixture of electromagnetic and electrostatic waves, so it is not pure Alfvénic turbulence.The harmonic signatures at higher frequencies are purely electrostatic and the wave power is much greater than that expected for Alfvénic turbulence.The electric field, magnetic field, and density waveforms that produced these spectra are illustrated in panels (b), (c), and (d).
Figure 4(d)  gives the proton and electron plasma betas whose values were around 0.1 with the proton beta being

Figure 2 .
Figure 2. Density streams during s interval (panel (a)), measured X-and Y-components of the electric field (panels (b) and (c)), and an estimate of the unmeasured EZ (panel (d)), obtained from the two measured electric field components and the assumption that the parallel electric field was zero.The EX electric field of panel (e) was computed from the Ohm's law requirement that the pressure gradient due to the density fluctuations of panel (a) was balanced by the electric field.It is noted the EX was larger than the other two electric field components which shows that the electrostatic k-vector was in the X-direction, which was perpendicular to the background magnetic field.Also shown (panels (f) and (g)) are the four individual antenna voltages whose average is the spacecraft potential.All data in this figure, other than that in panels (f) and (g), are high pass filtered at 1 Hz.
) and (e) indicate a large magnetic field component, B y , (B y /B 0 ∼ 0.07) and a large electric field component, E x , (E x /E 0 -0.07 with E 0 = V Ap B 0 ).The spatial structures (oscilliton profiles) of the electron density N e and of the electric field strength E x are shown in Figures 5(f) and (g).The maximum density variation is about 20%.Using V Ap = 500km/ and B 0 = 650nT, the maximum field strength of E x /E 0 ∼ 0.1 corresponds to an electric field of E x ∼ 33 mV m −1 , which agrees quite well with the observed values in Figures 1(b), 2(b), and 3(b).The spatial

Figure 4 .
Figure 4. (a) gives the electric field as a function of time, (b) gives the spectrum of density fluctuations,(c)  gives the instantaneous perpendicular to parallel proton temperature ratio as the black dots and their 30 s averages as the red dots, (d) gives the electron and proton betas as computed from the quasi-thermalnoise, and (e) gives the alpha to proton density ratio.Note that the spike at 11:38 is an artifact.Three pairs of vertical dashed lines denote three of the about eight regions in which the proton and density fluctuations in panels (a) and (b) were large.These regions occurred near but not necessarily in regions where the temperature anisotropy in panel (c) was larger than 6 and in regions where the alpha to proton density ratio was maximum.

Figure 5 .
Figure 5. (a), (b) Dispersion of kinetic Alfvén waves in a proton plasma with an abundance of 5% alpha particles.From top to bottom: the normalized frequency ω/ Ω p , the phase velocity V ph normalized to the proton Alfvén velocity V Ap , and the growth rate γ/Ω p vs. the normalized wavenumber kc/ω p are shown; ω p is the proton plasma frequency.(a) Vlasov approach with β e = 0.15, β p = 0.15.β e and β p are the electron and proton plasma beta, respectively.The temperature anisotropy that drives the kinetic Alfvén instability is T perp /T par = 6.5.The propagation angle is θ = 85 °.(b) Frequency and phase velocity from Hall-MHD theory.(c) Dispersion of (linear) stationary waves k = k(U) related to the waves in (b).From top to bottom: real part k r , imaginary part k i , and frequency ω = k r U, U: velocity of the moving structure.(d)-(i) Results of the oscilliton theory: (d) Magnetic field hodogram B z -B z0 vs. B y , (e) electric field hodogram E x vs. E y , (f) electron density profile N e (x), (g) electric field profile E x (x), (h) electric power spectrum |E x (k)| 2 , and (i) magnetic power spectrum |B y (k)| 2 .