# Heating and Acceleration of the Solar Wind by Ion Acoustic Waves-Parker Solar Probe

P. J. Kellogg, F. S. Mozer, M. Moncuquet, D. M. Malaspina, J. Halekas, S. D.

Bale, K. Goetz

► To cite this version:

P. J. Kellogg, F. S. Mozer, M. Moncuquet, D. M. Malaspina, J. Halekas, et al. Heating and Acceleration of the Solar Wind by Ion Acoustic Waves-Parker Solar Probe. The Astrophysical Journal, 2024, 964, 10.3847/1538-4357/ad029f. insu-04853468

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# Heating and Acceleration of the Solar Wind by Ion Acoustic Waves—Parker Solar Probe

P. J. Kellogg<fnref n="1" />, F. S. Mozer<fnref n="2" />, M. Moncuquet<fnref n="3" />, D. M. Malaspina<fnref n="4" />,<fnref n="5" />, J. Halekas<fnref n="6" />, S. D. Bale<fnref n="2" />, and K. Goetz<fnref n="1" />

Received 2022 December 7; revised 2023 July 31; accepted 2023 October 9; published 2024 March 15

## Abstract

The heating of the solar wind has been shown to be correlated with certain ion acoustic waves. Here calculations of the heating are made, using the methods used previously for STEREO observations, which show that the strong damping of ion acoustic waves rapidly delivers their energy to the plasma of the solar wind. It is shown that heating by the observed waves is not only sufficient to produce the observed heating but can also provide much or all of the outward acceleration of the solar wind.

Unified Astronomy Thesaurus concepts: Solar wind (1534)

## 1. Introduction

The electric field measurements of Parker Solar Probe (PSP) FIELDS (Bale et al. 2016) have detected a considerable number of interesting waveforms that bear on the principle goals of the PSP mission—understanding the heating and acceleration of the solar wind. Mozer et al. (2022a) have shown warming of the solar wind in connection with some of these waves, i.e., ion acoustic waves. In that paper, Mozer et al. (2022a) found that the heating of electrons at 20 eV above the expected temperature after correcting for the adiabatic cooling was due to the expansion of the solar wind. This heating was observed when ion acoustic waves were observed. As ion acoustic waves are strongly damped, their presence is a mystery but they are very common, and their energy is quickly transformed to particle energy by the damping. With some exceptions (O’Neil 1965; Gurnett & Frank 1978; Tu & Marsch 1995; Valentini et al. 2010), their study has been neglected in favor of the less damped modes, i.e., Alfvén waves and whistlers. In this work, their presence is shown to be equally important.

The waves are observed with four nearly orthogonal 2 m antennas, which are also orthogonal to the spacecraft's Sun line. The waves here include (1) waves of frequencies of the order of 0.1 Hz, (2) waves of frequencies of a few hertz, (3) sudden short bursts of a few hundred hertz, and (4) high-frequency waves consistent with the plasma frequency. The first three are ion acoustic waves. The waves of frequencies of a few hertz are identified as being in the ion acoustic mode as the potentials of the four antennas are nearly the same, indicating plasma fluctuations and the absence of correlated magnetic fields. The burst waves are also identified as being in the ion acoustic mode by the absence of correlated magnetic fields. These mode identifications imply that the waves are strongly damped and rapidly deliver their energy to the solar wind. The waves of hundreds of kilohertz are probably Langmuir waves at the

plasma frequency and will not be considered further here. They are visible in data from the Time Domain Sampler and with the Radio Frequency Spectrometer, both part of the FIELDS complex, (Bale et al. 2016) but are not thought to be ion acoustic waves.

Figure 1 shows the relative potentials of the four antennas, chosen as an exceptionally clear example of type (2), few-hertz, and type (3), bursts. The upper four lines indicate antenna potentials of the four antennas and the lower two are antenna differences indicating an electric field. It will be seen that the few hertz wave potentials are the same on all four antennas indicating that the signal is due to density fluctuations that can only be due to ion acoustic waves. Note that the antennas are biased to the plasma potential to overcome photoemission and that the potential sum is actually the spacecraft potential. The bursts are the obvious high-frequency elements in the lower two lines but can also be seen as slight wiggles in the upper four lines. It is also noted that the bursts occur when the antenna potentials are rising. As the potential is anticorrelated with electron density, this indicates an expansion and a decrease in density. This suggests that these bursts may be the expansion instability discussed in Kellogg (2022).

The heating to be calculated is the transformation of wave energy to plasma thermal energy. The wave energy is calculated from the Rayleigh expression, Equation (1). The two sides are numerically equal energies that occur alternately. The left pressure side is used in this section. Figure 2 shows the same presentation for another period, i.e., 2021 January 19 at 00:00–06:00 featured in Figure 6 of Mozer et al. (2022a). It will be seen that the few-hertz waves are at a lower frequency of about 1 Hz.

Figure 6 in Mozer et al. (2022a) shows bursts, like those of Figure 1 here. They cannot be seen here, however, because their frequency is about 610 Hz, while the Nyquist frequency, the upper limit of the Digital Fields Board sensor, is 292 Hz. These bursts will be treated in detail in Section 5.

Identification of the waves shown in Figures 1 and 2 (and later, Figure 3) as ion acoustic waves is essential to the calculations presented as the rapid decay, faster than other modes, determines the heating rate. First, it will be seen that the signals on the four antennas are nearly the same. This signal is

<figure><img src="image_3.png" /></figure>

Original content from this work may be used under the terms of the [Creative Commons Attribution 4.0 licence](https://creativecommons.org/licenses/by/4.0/). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹Minnesota Institute of Astrophysics, University of Minnesota, Minneapolis, MN, USA</footnote>

<footnote n="2">²Space Science Laboratory, University of California, Berkeley, CA, USA</footnote>

<footnote n="3">³LESIA Observatoire de Paris, 5, Place J. Janssen, F-92195 Meudon, France</footnote>

<footnote n="4">⁴Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO, USA</footnote>

<footnote n="5">⁵Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO, USA</footnote>

<footnote n="6">⁶Department of Physics and Astronomy, University of Iowa, Iowa City, IA, USA; jasper.halekas@uiowa.edu</footnote>
<figure><img src="image_4.png" /></figure>

Parker S P read_PSP_DFB3_pub.ps, from some version 22 Feb 2022, output at Thu Oct 27 20:40:57 2022

<caption>**Figure 1.** 3.495 s of antenna potentials (upper four curves) and derived electric fields showing types (2) and (3) waves, which are ion acoustic waves to be used to calculate the heating and acceleration of the solar wind plasma.</caption>

then not directional and therefore is due to density changes. The antenna potential is set by electron flux and therefore electron density. Of the other magnetohydrodynamic modes, Alfvén waves have essentially no density fluctuation. Whistler fast-mode waves have density fluctuations which, however, are directional. A second distinguishing characteristic is a weak magnetic field. Ion acoustic waves propagating parallel to the ambient magnetic field have no associated magnetic field but oblique propagation leads to a weak field. Again, Alfvén waves, and particularly, whistlers have larger magnetic fields. Mozer et al. (2022a, 2022b) looked at the magnetic field shown in Figure 2 to verify that these are ion acoustic waves. This was also been done for Figure 1 but is not shown.

## 2. Wave Energy and 2 Hz Waves

The treatment of the data to determine the heating due to the waves described above is similar to the treatment described in Barnes (1968, 1969) and Kellogg (2020). The acoustic wave energy of the few-hertz waves is calculated from the pressure half (left) of the Rayleigh formula (Rayleigh 1894)

$$W = \text{pressure} + \text{velocity} = \int_V \frac{\delta p^2}{2\rho_0 V_s^2} dV + \int_V \frac{\rho_2 \delta v^2}{2} dV. \quad (1)$$

In this equation, the two terms are equal but present energy alternately. In the left-hand term, the pressure term, $\delta\rho$, is the deviation of the average pressure, averaged over sets of 220 s. $\rho_0$ and $V_s$ are the average density and the ion sound speed for the set. $dV$ is taken as $1 m^3$ as the waves involved in the calculations all have wavelengths much larger than 1 m. The right-hand side is referred to as the velocity term. In terms of measured parameters, $\delta p$ is a familiar expression: $\text{density} * M_p * (kT_p + kT_e)$. Because $\delta\rho$ is the difference in pressure from some average, the computed energy is sensitive to the length over which the average is calculated. The chosen length is rather short, so the calculation of energy and heating to be presented are lower limits and could be larger.

The averaged energies from this equation for the 6 hr periods shown in Figures 1 and 2, as well as those of a later figure, are shown in Figure 4. These are averages of approximately 224 s subsamples of data like those in these figures and are energies per cubic meter as in Equation (1). If these waves were monochromatic, the heating rate would be this energy times the damping—the imaginary part of the frequency. As they are broadband, the heating is to be calculated by a convolution of the spectra of damping and energy.

The spectrum of one of these energies is shown in Figure 5. The full spectrum is in black and the red curve represents a
<figure><img src="image_5.png" /></figure>

Parker S P read_PSP_DFB3_pub.ps, from some version 22 Feb 2022, output at Thu Oct 27 17:17:43 2022

<caption>**Figure 2.** A similar presentation including the period analyzed in Mozer et al. (2022a).</caption>

spectrum obtained after processing the data with a bandpass filter with a pass from 0.5–10 Hz, to separate the type (2) driver waves of Figure 1 and its harmonics. The 2 Hz component of the energy seems important as can be seen in Figures 1 and 2 but here it is seen as only a fraction of the energy. It will be seen that the spectrum of the few-hertz waves is diffuse and not very dominant. This is, however, a result of the smoothing that has been done for further use of the calculations. More importantly, the presence of the very low-frequency waves is also apparent and their frequency is of the order of 1/10 Hz and below. These low frequencies, part of the solar wind cascade, were originally expected to contribute less to the heating because of their low damping rate, but in fact, if the calculation is extended to include frequencies down to zero, the heating rate is larger by nearly an order of magnitude. Hence, the heating used here is for the full spectrum. Nevertheless, the energy in the few-hertz waves will be calculated separately.

## 3. Data Treatment

As described, the data are analyzed in a way similar to that in Kellogg (2020). Although the results of the data treatment have already been shown, some details of the treatment will be given in this section. The transformation of wave energy to plasma thermal energy is calculated from the Rayleigh expression, Equation (1). The pressure term is used in this section. However, the vast differences in the sampling rate for the

various components required in the calculation (electric potentials sampled 2048 times faster than ion temperature and density 8192 times faster than electron temperature) on PSP have required some modification of data treatment from that used in Kellogg (2020). The cadence of the SWEAP plasma measurements (Kasper et al. 2016) is too slow for even the few-hertz waves shown in Figures 1 and 2, so the density fluctuations are calculated from the spacecraft potentials relative to the antenna potential. As the antenna and spacecraft potentials are determined mostly by the flux of ambient electrons, these potentials can be used to determine the electron density. This procedure has been much used at 1 au but a simple application does not work well for PSP, probably because the spacecraft is not directly exposed to the Sun, but is shielded by the carbon thermal protection shield whose electric connection to the spacecraft is not direct. As demonstrated by Ergun et al. (2010), secondary emission also plays a role. Mozer et al. (2022b) developed a procedure to overcome this difficulty. This procedure involves a least squares fit of the measurements of relative antenna potential as a function of a sequence of measured densities, measured either by the SWEAP instruments or from quasi-thermal noise. Here the quasi-thermal noise is used. The antenna potential is smoothed to a pace matching that of the quasi-thermal noise treatment (Kellogg 1981; Meyer-Vernet & Perche 1989; Moncuquet et al. 2020) of the PSP Radio Frequency Spectrometer (Pulupa et al. 2017) for which the available data intervals are 3.495 s. The ion
<figure><img src="image_6.png" /><figcaption>Figure 3. A 3.495 s sample of waves near perihelion 10, at about 12 $R_\odot$ from the Sun.</figcaption></figure>

and electron temperatures are then obtained by interpolating the SWEAP data, at 3.495 s intervals, using the IDL code INTERPOL to match the antenna potentials. The result may be a small overestimate due to the anticorrelation between density and the ion temperature, but will not be an error by an order of magnitude.

The damping that delivers this energy to the plasma is frequency dependent. A proper calculation requires a convolution between a Fourier transform of frequency and energy spectra, as explained in Kellogg (2020). The inverse results of this convolution, in watts per kilogram, are shown in Figure 6 for 2021 January 16 and the averages for the full spectrum and the 2 Hz section are $1.02 \times 10^{10}$ and $9.07 \times 10^8$ W kg$^{-1}$ with the filtered result in red. The result of a similar calculation for the period used in Figure 4 of Mozer et al. (2022b), a work that inspired the present work, is shown in Figure 6. This period is for 6 hr beginning on 2021 January 19 at 00:00 and the corresponding results are $1.09 \times 10^{10}$ and $8.78 \times 10^8$ W kg$^{-1}$. The average heating rates for the two periods just described are shown by black and red lines.

The heating rate in watts per cubic meter for the period shown in Figure 6 is $6.54 \times 10^{-12}$ W m$^{-3}$. From this, an effective damping frequency can be obtained by dividing this rate by the energy shown in Figure 4 to get an effective damping frequency of 1.2 Hz. The frequency of the waves in Figure 1 is about 3.7 Hz, so the damping frequency, the imaginary part of the frequency for propagation parallel to the magnetic field, one-fifth of the real part would be 0.75 Hz. This

effective damping frequency is about twice the expectation for propagation parallel to B, indicating that the waves are somewhat oblique to the magnetic field, as would be expected from Kellogg (2020).

The average heating rate from the described calculations, utilizing the full spectrum from the lowest frequency, is $1.05 \times 10^{10}$ W kg$^{-1}$ for the 6 hr periods.

Because of the expansion, heating per kilogram has been used. For comparison, with the 1 au heating (Kellogg 2020), the heatings per cubic meter are $6.3 \times 10^{-12}$ W m$^3$ for 2021 January 16 and $5 \times 10^{-12}$ for 2021 January 19. For 1 au (Kellogg 2020) the heating was calculated to be 3 or $9 \times 10^{-17}$ W m$^3$. The energy available for heating, density times ($T_e + T_i$), is about $10^4$ times greater for PSP, closer to the Sun, than for STEREO at 1 au.

## 4. Less than 1 Hz Waves (Less than Zero)

The nature of these waves, an important part of the energy delivered to the plasma, had not been clearly established previously. The usual determination of whether waves are ion acoustic involves a comparison of their electric fields with magnetic fields. Such a comparison for a section of 230 s of antenna potential data and magnetic field is shown in Figure 7. It can be seen that there is little correlation. Further, the four antenna potentials show the same signal, indicating the density fluctuation characteristic of ion acoustic waves, which are weak in other modes. A more rigorous test is to compare the density
<figure><img src="image_7.png" /></figure>

psp_energy_paper.ps, version 17 Oct 2022, output at Wed Oct 19 14:55:37 2022

<caption>**Figure 4.** Average wave energies, from Equation (1), of approximately 56 s subsamples of data like those in these figures.</caption>

fluctuation and the electric field. The competitors of ion acoustic mode turbulence are whistlers and Alfvén waves, for which there is little density variation and significant electric and magnetic fields. The detection of the waves considered here, the measurement of changes in antenna potential, depends on density changes. Ion acoustic waves, because the densities of electrons and ions are nearly equal and have positions that nearly cancel their electric fields, have weak fields. Alfvén

waves and whistlers, on the other hand, have weak density perturbations and strong electric fields. The average rms for antenna potential fluctuations is shown in the upper right corner of each panel in Figures 1 and 2. The average rms variation shown in Figure 1 is 78 and that in Figure 2 is 37. The rms of the electric field shown in the lower two panels in Figure 1 is 17 and in Figure 2 is 3.7, in accordance with expectations that the electric field of ion acoustic waves is weak. For the waves
<figure><img src="image_8.png" /></figure>

PSP Energy, spectrum, version 20 Nov 2022, output at Sun Nov 20 21:07:04 2022

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Calculated heating rate for the period of 2021 January 16 at 06:00–12:00.</figcaption></figure>

shown in Figure 8, the average potential variation is 139 mV, while the electric fields' average is 23 mV. It seems clear that these less than zero waves have a large ion acoustic component and that furthermore, they are an important part of the cascade, which is usually considered to consist of magnetic turbulence. This justifies the inclusion of the spectrum down to the lowest meaningful frequency in calculating the heating.

However, in Kellogg (2020) it was shown that at 1 au the ion acoustic energy was slightly larger than the magnetic energy in the cascade. This is not true at $25 R_\odot$ and closer to the Sun where the magnetic energy is considerably larger than the ion acoustic pressure energy. Ultimately, the energization of the solar wind must come from this magnetic energy and the ion acoustic waves are just a stop toward this energization.

<figure><img src="image_10.png" /><figcaption>**Figure 7.** Calculated heating rate for the period of 2021 January 19 at 0:00–06:00.</figcaption></figure>

In both this work and that of Kellogg (2020), there has been no rigorous separation of ion acoustic waves from the turbulence. There is a mixture but the use of antenna potential, which is driven by density variations, favors ion acoustic waves in detection, and this was shown more carefully in Kellogg (2020). Further, the heating calculations are done on the averages of the antenna potentials, which tend to exclude electric fields. It is therefore concluded that the turbulence of the solar wind cascade contains a large contribution of ion acoustic waves, as others have done (Tu & Marsch 1995, especially), and then that the absorption of these ion acoustic waves provides a large part of the energy of the solar wind and must also be considered an important part of the cascade.

As mentioned, the resulting data are used to calculate the wave energy according to the Rayleigh formula. The wave energies for three 6 hr sections of data are shown in Figure 5. These energies are deposited in the plasma at a time corresponding to the damping time of ion acoustic waves. Because the frequency range is from 0–10 Hz, the damping time to use is quite uncertain. Following Kellogg (2020), this filtered wave energy is then convolved with the wave damping frequency to give this rate.

The electron temperature for one of the regions of interest is well above the ion temperature, which is known to decrease the damping, the imaginary part of the frequency. The damping rate was therefore calculated from the Vlasov equations for warm plasma and the result is that the imaginary frequency is one-fifth of the real part, rather than one-third to one-quarter as it is for equal temperatures.

This work began with the same idea presented in Kellogg (2020) that the heating counteracted the adiabatic cooling. But now it is seen that the heating is considerably more than is required for heating at 20 eV. In Kellogg (2020) it was also seen that the heating was larger than the heating measured by Coleman (1968) and by Gazis & Lazarus (1982). In that case, it was shown that the damping of oblique ion acoustic waves could be up to 20 times slower than one-quarter of the real frequency so the calculated heating could be consistent with the observed heating. The same factor here cannot account for the large heating here, as the damping, and therefore the heating rate can be reduced by, at most, a factor of about 20 due to
<figure><img src="image_11.png" /><figcaption>Figure 8. Comparison of antenna potential signals and magnetic fields for a 230 s sample of their respective data.</figcaption></figure>

obliquity. The heating is genuinely large and results in a surprising result that presents a new view of the solar wind.

## 5. Bursts

As can be seen in Figures 1 and 2, there are also short bursts of ion acoustic waves at a considerably higher frequency. The bursts shown in Figure 1 are at about 200 Hz but the bursts shown in Figure 6 of Mozer et al. (2022a) are at about 610 Hz. These 610 Hz bursts cannot be seen in Figure 2, as the Nyquist frequency of the data used is only 293 Hz. As can also be seen in Figure 1, the bursts are found where the antenna potentials are increasing. As density and potential are anticorrelated, the plasma is expanding, which suggests that these are examples of expansion instability (Kellogg 2022). The signals created by the expansion instability differ from ion acoustic signals only in that the real part of the expansion wave frequency is zero, a difference that is of little consequence for the calculations under discussion here.

The electric field observed in bursts is more prominent than density fluctuations, so the velocity term on the right-hand side of the Rayleigh formula (Equation (1)) is used to calculate the wave energy, instead of the density term on the left-hand side. The necessary flow speeds for the velocity side of the Rayleigh formula must be obtained by integrating the acceleration due to the observed electric field. The motion of a particle in an

electric field is

$$dv/dt = eE/m. \qquad (2a)$$

So that

$$v = eE/\omega m. \qquad (2b)$$

The Rayleigh formula then implies that the larger contribution to the energy is due to the electrons. Evaluation of the velocity term on the right-hand side of Equation (1) for the burst of Figure 9 accordingly gives an energy of $2.8 \times 10^{-8}$ J in $dV$ of $1 \text{ m}^2$, or $1.2 \times 10^{10} \text{ J kg}^{-1}$.

As the electric field of ion acoustic waves is strongly aligned along the magnetic field, the magnetic field can be ignored in determining the acceleration due to the electric field. Further, the Parker antennas are perpendicular to the Sun line, while the desired electric field is along the magnetic field, which has a large parallel component. The relevant electric field is then the measured transverse field multiplied by the ratio $|B|$ to B transverse. This implies that the actual electric field is somewhat larger than the measured field. The radial field was 160 nT, while the tangential field was 50 nT, so the true field is 3.2 times the observed field, i.e., $3.8 \times 10^{10} \text{ J kg}^{-1}$.

In order to remove low-frequency signals, which tend to dominate, it was necessary to subtract the average and also to apply a high-pass filter with a lower bound of 100 Hz, which did not change the size of the oscillations much.
<figure><img src="image_12.png" /></figure>

PSP VxVy.ps, in PSP_June24_300_VX_VY.pro, version 04 Sep 2022, output at Mon Sep 19 13:31:48 2022

<caption>Figure 9. Details of one of the bursts in a Figure 6 in Mozer et al. (2022a). The frequency of this burst is about 610 Hz.</caption>

As these burst signals are also in the ion acoustic mode (Mozer et al. 2022a, 2022b) or in the equivalent expansion instability mode, they also rapidly decay and deliver their energy to the plasma. Because the electron temperature is 50–60 eV, while the ion temperature is only 20 eV, the decay rate of ion acoustic waves is slowed. Again the imaginary frequency of one-fifth of the real part has been used. As noted in the STEREO calculations (Kellogg 2020), the decay for oblique propagation could be considerably slower, implying a lower heating rate. Extreme obliquity leading to very slow decay would also result in a correlated magnetic field, which is not seen, so only modest obliquity corresponds to the observations. The bursts shown in Figure 1 have been treated similarly. A single burst has been isolated and its energy is calculated using the Rayleigh formula, a sum of the squared electron speed, obtained by integrating the electric field force. However, the sampling rate of these bursts is close to the Nyquist frequency. It was not possible to apply a high-pass filter as the number of points was only about 120 and the filter removes twice this number to form the filter. However, the bursts shown in Figure 1 occur more frequently than those that occurred on 2021 January 19, at the rate of about 15 bursts in 3.495 s. Their amplitude is a little lower than the burst shown in

Figure 1 and their frequency is three times slower, implying slower heating but otherwise their heating should be similar.

Bursts are episodic and therefore not as useful in the discussion of the acceleration and heating of the solar wind as a whole. There are estimated to be about 400–600 such bursts in the 6 hr period shown in Figure 2. Substitution in the Rayleigh formula then gives energy for a collection of 500 such bursts of $5.44 \times 10^{12} \text{ J kg}^{-1}$ in 6 hr or at a rate of $2.52 \times 10^8 \text{ W kg}^{-1}$. It is clear, however, that an episode of bursts can cause the short-term heating described in Mozer et al. (2022a, 2022b).

The result is that heating by bursts may account for short-term heating but is not included in the calculations as a major part of the longer-term heating and acceleration of the solar wind.

## 6. Nearer to the Sun

The heating closer to the Sun of the plasma has been assumed above to be something like the heating at $25 R_\odot$, so it is of interest to consider the observations. The closest data available at the time of this writing is perihelion 10, at $13 R_\odot$, on 2021 November 21 at 08:23:23. These waveforms near the exact perihelion are similar to those shown in Figures 1 and 2 but with stronger signals. Heating during two 6 hr periods, one
<figure><img src="image_13.png" /></figure>

PSPHeating_plot_kg.ps, from PSP_heating_Timeline_11, version 11 Sep 2022, output at Mon Nov 28 11:06:33 2022

<caption>**Figure 10.** Calculation of 6 hr of heating containing perihelion 10, including the waves shown in Figure 3.</caption>

(Figure 10) including the exact perihelion, are shown in Figures 10 and 11.

Figure 3 shows a short, 87 s section of data at perihelion, taken on 2021 November 21 at 08:23, showing that the data are similar to the data in Figures 1 and 2. The average rms for antenna potential was 159 mV and the rms for the electric field was 28 mV, a ratio quite comparable to that of known ion acoustic waves, so these are ion acoustic waves. In order to compare better with Figures 1 and 2, the wave energy for the 6 hr period around perihelion is shown as the green line in Figure 4. It will be seen that the wave energy is of the same order as at 25 $R_\odot$. The heating program was run for the same 6 hr period, i.e., 2021 November 21 at 06:00–12:00 with the results shown in Figure 10. The heating rates were $1.60 \times 10^{10} \text{ W kg}^{-1}$ or $6.03 \times 10^{-11} \text{ W m}^{-3}$.

However, it was found that this 6 hr period is higher than usual in electromagnetic waves. In general, it is found that ion acoustic waves make up about half of the cascade (Kellogg 2020) but this period is weaker in ion acoustic waves. A slightly later 6 hr period of 18–24 hr on 2021 November 12 contains a period of particularly strong ion acoustic waves and the calculation of the heating for this period is shown in Figure 11.

The heating and acceleration of the solar wind must be sustained along the whole distance from close to the Sun to nearly 1 au. Full investigation of this long region must await further work but we have presented measurements at two widely separated places and it is likely that they are fair examples of the entire solar wind.

## 7. Comparison of Observation and Calculation Results

Several areas of the solar wind are analyzed here for heating. The two areas shown in Figures 1 and 2 and whose heating is shown in Figures 6 and 7 are at some distance ($25 R_\odot$) from the Sun. Figure 1 illustrates an area of particularly low density, which allows the data to show bursts, short periods of high-frequency waves, as the frequency, 200 Hz, falls below the Nyquist frequency. Figure 2 contains data from Figure 6 of

<figure><img src="image_14.png" /><figcaption>**Figure 11.** Calculation of 6 hr of heating just beyond Encounter 10.</figcaption></figure>

Mozer et al. (2022a). Mozer et al. (2022a) observed heating estimated at 20 eV, which is correlated with the ion acoustic waves under study here. This 20 eV heating is relative to the temperature expected from adiabatic cooling. The energy to increase the temperature of 1 kg of hydrogen plasma by 20 eV, including the electrons, is $3.3 \times 10^9 \text{ J}$. As discussed in Section 5, a burst can provide $1.2 \times 10^{10} \text{ J kg}^{-1}$. They repeat in 0.7 s so the heating rate is $1.7 \times 10^{10} \text{ W kg}^{-1}$ or eV s$^{-1}$ kg$^{-1}$. This can augment the heating by waves when they are present. Bursts are sufficient to supply this observed 20 eV heating of electrons with some left over.

As mentioned, the result of a major part of the calculation presented here is of heating rate, which requires an estimate of time. The longest time would be the time for a cloud to travel from the Sun to the spacecraft at 25 $R_\odot$ with continuous heating all the way. If the cloud underwent uniform acceleration for this distance, implying that the average speed was $100 \text{ km s}^{-1}$, the travel time to reach 25 $R_\odot$ would be about $1.7 \times 10^5 \text{ s}$. The observed speeds are, however, not very uniform, suggesting a shorter abrupt heating is happening. A shorter period may be obtained from an abrupt heating on April 29 in Figure 1 of Mozer et al. (2022a), where there is an appreciable heating of 20 eV in 1 hr. This is not, of course, a genuine heating observation. It is the time for a temperature gradient to be conducted past the spacecraft but is suggestive of heating in about an hour. The energy then required for 20 eV heating, $3.8 \times 10^9 \text{ J kg}^{-1}$, is the rate of $1.1 \times 10^6 \text{ W kg}^{-1}$ for 1 hr.

As there is considerable heating due to the steady heating by the more continuous waves, it is of interest to calculate this on the heating and acceleration of the full solar wind. The energy required has several components. The plasma considered at 25 $R_\odot$ moves out from the Sun at $200 \text{ km s}^{-1}$. The electron and ion temperatures are respectively 53 and 18 eV and the density is $1350 \text{ cm}^{-3}$, which gives a thermal energy of $5.9 \times 10^5 \text{ J kg}^{-1}$. However, the loss of energy due to the Sun's gravity must also be compensated. In traveling to $24 R_\odot$, the change in energy of 1 kg of plasma is 95% of the potential at the surface of the Sun, $GM^{-1}R^{-1}$ ($M$ and $R$ are the mass and radius of the Sun) so about $1.81 \times 10^{14} \text{ J kg}^{-1}$. This is the major energy requirement. When this plasma was part of the transition region, at about $2 \times 10^6 \text{ K}$ or 172 eV, the thermal
energy of 1 kg was $2.48 \times 10^{10}$ J. This large energy is not nearly enough to drive the plasma to $25 R_\odot$.

The $1.81 \times 10^{14}$ J of energy must be replaced in the time for $1 \text{ m}^3$ of the solar wind to leave the Sun and pass the observation points shown in Figures 1 and 2, estimated above as $1.7 \times 10^5$ s, so the required heating rate is $1.06 \times 10^9 \text{ J kg}^{-1} \text{ s}^{-1}$. The calculated heating of the 6 hr samples shown in Figures 1 and 2 is $1.02 \times 10^{10}$ W, implying that the heating may provide not only the 20 eV heating but the entire heating and acceleration of the solar wind to $25 R_\odot$, estimated above as $1.7 \times 10^5$ s, so the required heating rate is $1.1 \times 10^9 \text{ W kg}^{-1}$. The available heating at $1 R_\odot$, as shown in Figures 10 and 11, is 2.5 and $2.9 \times 10^9 \text{ W kg}^{-1}$.

The energy shown in Figure 10 is for Encounter 10. Further investigation showed that the immediate region surrounding the perihelion was primarily filled with electromagnetic mode waves. There are some ion acoustic waves, however, and an example is shown in Figure 3. A slightly later 6 hr period is filled with unusually large amplitude ion acoustic waves and the heating rate during this period is shown in Figure 11.

It is assumed that heating and acceleration persist through most of the region between the regions shown in Figures 1 and 2 and Figures 10 and 11. A detailed calculation of the acceleration necessary to maintain the solar wind over the whole distance is beyond the scope of this paper but the present work provides some assurance that the waves must be there to maintain the solar wind.

It was calculated above that $1.1 \times 10^9 \text{ W kg}^{-1}$ is required for continuous heating as the packet of plasma leaves the Sun. The heating by absorption of ion acoustic waves calculated here is $2.5 \times 2.9 \times 10^9 \text{ W kg}^{-1}$, so it then it appears that the energy provided by the damping of the observed ion acoustic waves can provide the observed heating and acceleration of the solar wind.

## 8. Summary and Conclusions

A calculation is presented of the heating rate and acceleration of the solar wind due to the absorption of observed ion acoustic waves. It is the rate of energy transfer that is calculated. As the energy transfer of ion acoustic waves to the plasma is frequency dependent, the rate must be calculated using the convolution of the Fourier transforms.

Detailed calculations of heating from various components of the wave spectrum are presented. The wave spectrum has been divided into waves of a few hertz, bursts, and a full spectrum down to sample length limits. The calculated rates are given for a few hertz (Sections 2 and 3), bursts (Section 5), and the full spectrum in Section 2 and in Section 6 for $13 R_\odot$.

To account for the 20 eV heating observed in Mozer et al. (2022a) the heating must be $3.3 \times 10^9 \text{ J}$ from a single burst.

The heating from a burst displayed in Figure 6 of Mozer (2022a) is calculated to be $1.189 \times 10^{10}$ J, more than sufficient to match the observed heat.

The total heating rates for two 6 hr periods near Encounter 10 at $13 R_\odot$ were $2.5 \times 2.9 \times 10^9 \text{ pW kg}^{-1}$. This is enough to provide the $1.1 \times 10^9 \text{ W kg}^{-1}$ necessary to carry a solar wind packet from the transition region of the Sun to the observation points.

The conclusion is that the heating rate from the full ion acoustic wave decay process is sufficient for the heating observed by Mozer et al. (2022a) and also sufficient to accelerate and heat the plasma to form the solar wind observed at $25 \text{ and } 13 R_\odot$.

The process that gives rise to these ion acoustic waves is not yet understood and remains the next step in our attempts to understand the solar wind and by extension, the winds of similar stars.

## Acknowledgments

The work of P.J.K. was supported by (STEREO) 80NSSC20K0876 and (PSP) NNN06AA01C.

## ORCID iDs

P. J. Kellogg https://orcid.org/0000-0001-5223-689X

F. S. Mozer https://orcid.org/0000-0002-2011-8140

M. Moncuquet https://orcid.org/0000-0002-9621-0365

D. M. Malaspina https://orcid.org/0000-0003-1191-1558

J. Halekas https://orcid.org/0000-0001-5258-6128

S. D. Bale https://orcid.org/0000-0002-1989-3596

K. Goetz https://orcid.org/0000-0003-0420-3633

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