Weak Solar Radio Bursts from the Solar Wind Acceleration Region Observed by the Parker Solar Probe and Its Probable Emission Mechanism

The Parker Solar Probe (PSP) provides us with an unprecedentedly close approach to the observation of the Sun and hence the possibility of directly understanding the elementary process that occurs on the kinetic scale of particles' collective interaction in solar coronal plasmas. We report a type of weak solar radio burst (SRB) that was detected by PSP when it passed a low-density magnetic channel during its second encounter phase. These weak SRBs have a low starting frequency of ∼20 MHz and a narrow frequency range from a few tens of MHz to a few hundred kHz. Their dynamic spectra display a strongly evolving feature of the intermediate relative drift rate decreasing rapidly from above 0.01 s−1 to below 0.01 s−1. Analyses based on common empirical models of solar coronal plasmas indicate that these weak SRBs originate from a heliocentric distance of ∼1.1–6.1 R S (the solar radius), a typical solar wind acceleration region with a low-β plasma, and that their sources have a typical motion velocity of ∼v A (Alfvén velocity) obviously lower than that of the fast electrons required to effectively excite SRBs. We propose that solitary kinetic Alfvén waves with kinetic scales could be responsible for the generation of these small-scale weak SRBs, called solitary wave radiation.


Introduction
Solar radio bursts (SRBs) are the most direct manifestation of energetic electrons, which exist ubiquitously in the solar atmosphere, although their origin remains poorly understood.It has been commonly believed that ordinary type III and type V SRBs are generated by energetic electrons via the process of magnetic reconnection during solar flares.Type II and moving type IV SRBs are believed to be triggered by coronal mass ejection (CME)-associated shock-accelerated electrons.The dynamic spectra of SRBs can provide sensitive and rich information on solar energetic electrons, as well as on the background plasma in the emitting source regions (Wild 1950;Wild et al. 1954;Lin et al. 1973;Melrose 1980;Chen et al. 2017).For instance, a fast frequency drift of dynamic spectra of type III SRBs, characterized by a relative frequency-drift rate of D ≡ |(df/dt)/f| > 0.1 s −1 (where f is the emitting frequency), directly indicates the travel velocity of fast electron beams (FEBs) of emitting type III SRBs.The dynamic spectra of type II SRBs usually present a slow drift at D < 0.01 s −1 caused by the energetic electrons accelerated by the CME-driven shock.
Moreover, the dynamic spectra of moving type IV SRBs often display a very slow drift of D = 0.01 s −1 that can be attributed to the sub-Alfvénic motion of coronal loops, in which energetic electrons are trapped (Tan et al. 2019).
For type III SRBs, using observations made by the Geotail and Akebono satellites, micro-type III bursts characterized by short lifetimes and continuous and weak emission have been found by Morioka et al. (2007Morioka et al. ( , 2015)).They showed that micro-type III bursts have a distribution of emitted power flux that is different from that of ordinary type III bursts and concluded that they are not just weaker versions of the ordinary bursts.Owing to the Parker Solar Probe's (PSP) close distance to the Sun, PSP Integrated Science Investigation of the Sun (ISeIS) has observed a rich array of energetic particle events, which were not observed by the spacecraft at 1 au, during the first two orbits (McComas et al. 2019).Ma et al. (2022) also found that because of the radiation attenuation effect, many weak type III-like bursts with a higher cutoff frequency (and hence narrower bandwidth) clearly detected by PSP can hardly be observed by WIND when PSP approaches its perihelion.In the microwave frequency range, the radio bursts called solar microwave drifting spikes, which are typically characterized by a short lifetime (approximately tens of milliseconds), and an intermediate frequency-drift rate (∼a few hundred MHz s −1 ) have been detected by the Solar Broadband Radio Spectrometer of the National Astronomical Observatories of China.Wu et al. (2007) suggested that these solar microwave drifting spikes are probably produced by accelerated electrons trapped within solitary kinetic Alfvén waves' (SKAWs) potential well, and the intermediate frequency drifts are attributed to the SKAWs' propagation along the magnetic field.
Here, we report a kind of weak SRB, which was found in recent observations of PSP when it passed through a magnetic channel with low density at a heliocentric distance of ∼36 R S (the solar radius) during its second encounter with the Sun (Fox et al. 2016;Pulupa et al. 2017Pulupa et al. , 2020;;Ma et al. 2021).These weak SRBs can be characterized by a weak intensity, low starting frequency, narrow frequency range, and short lifetime and probably originated from some small-scale emitting sources.Their dynamic spectra display a strongly evolving feature in that the relative drift rate decreases rapidly from D > 0.01 to <0.01 s −1 when the emitting frequency drifts downward from a few tens of MHz to a few hundred kHz.Based on common empirical models of solar coronal plasmas (Mariani & Neubauer 1990;Leblanc et al. 1998;Wu & Fang 2003;Wu & Yang 2007), the observation of PSP shows that the emitting sources travel along the low-density magnetic channel (also called the equatorial coronal hole following Bale et al. 2019) from the heliocentric distance R ∼ 1.1 to ∼6.1 R S , a typical region of the solar wind acceleration.
Active region (AR) 12737, which is possibly associated with the low-density magnetic channel observed by PSP, does not have evident flare or jet activities during the corresponding time period.In addition, there is no obvious hard X-ray emission based on the observation of Fermi.Hence, it seems that the energetic electrons responsible for these weak SRBs do not come directly from flare or jet activities in AR 12737.However, in a low-β plasma with v v T A e > , where v A is the Alfvén velocity and v T e is the electron thermal velocity, SKAWs can travel at a velocity higher than v A , and their electric fields may accelerate electrons to a velocity much higher than v A and trap these energetic electrons within their potential wells.These trapped energetic electrons can generate coherent radio radiation via the electron cyclotron maser (ECM) instability.In particular, the propagation and evolution of SKAWs can reasonably explain the frequency-drifting feature in the dynamic spectra of these weak SRBs.Therefore, we propose that the kinetic-scale SKAWs can be responsible for the generation of the weak small-scale SRBs observed by PSP, called solitary wave radiation (SWR).The rest of the paper is organized as follows.In Section 2, the main observed properties of weak SRBs detected by PSP during its crossing of the low-density magnetic channel and the corresponding solar wind plasma parameters are presented.Then, combining the in situ measurement of PSP and the empirical model of solar atmospheres, we focus on discussing the generation mechanism of those weak SRBs in Section 3. Finally, the summary and discussion are given in Section 4.

PSP Observations and Data Analysis
NASA's PSP (Fox et al. 2016) was launched into a heliocentric orbit on 2018 August 12 and will fly closer to the Sun than any other spacecraft before it.The primary science goals for this mission are to trace the energy flow from the solar corona to the solar wind and to help us understand the solar corona heating and solar wind acceleration.In particular, PSP is the first spacecraft to do in situ measurements of the solar corona and the source region of the solar wind.To accomplish these science goals, four scientific instruments are carried by PSP: the Fields Experiment (FIELDS; Bale et al. 2016), the Solar Wind Electrons Alphas and Protons investigation (SWEAP; Kasper et al. 2016), ISeIS (McComas et al. 2016), and the Wide-field Imager for Solar Probe (Vourlidas et al. 2016).The data presented in this study are mainly made by FIELDS designed to do measurements of electric and magnetic fields.The radio data are obtained by the Radio Frequency Spectrometer (RFS; Pulupa et al. 2017) with the dual-channel receiver, the Low Frequency Receiver (10.5 kHz-1.7 MHz), and the High Frequency .
The weak SRBs reported in this work were observed by PSP during its second solar encounter (E02) from 2019 April 3 to 6, as shown in Figure 1, in which panel (a) displays the power spectral density (PSD) of the radio radiation, covering the frequency band from 10.5 kHz to 19.2 MHz.Panels (b) and (c) in Figure 1 present the magnetic field (B) and its three components in the R-T-N frame (B R , B T , and B N ), and panels (d) and (e) do the solar wind velocity (v p ) and its components (v pR , v pT , and v pN ).Plasma density (n e ) and temperature (T p ) are exhibited in panels (f) and (g), respectively, and panel (h) shows the heliocentric distance of PSP in units of solar radius, R S .From Figure 1, one can find that in the yellow region between 08:08:34 UT on 2019 April 3 and 13:17:09 UT on April 6, the magnetic field B strengthens mainly in its radial component B R from ∼80 nT at the edge to >100 nT in the center; on the contrary, the density n e significantly reduces from >400 cm −3 at the edge to ∼100 cm −3 in the center, as shown in panels (b) and (c) and panel (f), respectively.This implies that PSP was crossing a low-density magnetic channel at a heliocentric distance of ∼36 R S .On the other hand, panel (g) and panels (d) and (e) show that the plasma temperature (T p ) and flow velocity (v p , mainly its radial component v pR ) increase considerably from ∼10 eV and ∼250 km s −1 at the edge to ∼30 eV and ∼450 km s −1 in the center, respectively, implying that the plasma is heated and accelerated to form a solar wind stream in the center of this open magnetic channel.
As seen in Figure 1(a), a large number of SRBs presented in the observation of PSP during its crossing of the low-density magnetic channel.We identified 428 SRBs with peak intensities between 10 −12.6 and 10 −16.3 V2 Hz −1 , well above the sensitivity of the RFS measurement, ∼10 −18 V 2 Hz −1 (Pulupa et al. 2017).Among these identifiable SRBs, only 43 bursts (∼10%) have peak intensities higher than 10 −15 V 2 Hz −1 , and the remaining 385 bursts have peak intensities lower than 10 −15 V 2 Hz −1 .These very weak SRBs have a rather high occurrence rate of ∼five bursts per hour, on average, and they can be regarded as point sources and should originate from small-scale emitting sources because of their weakness.In Figure 2(d), the starting ( f st ) and ending ( f lo ) frequencies of these weak SRBs are displayed by red and blue circles, respectively.The local plasma frequency f pe along the PSP orbit is presented by the black line.One can see that the lowest ending frequency ( f lo = 367 kHz) of these weak SRBs is much higher than the local plasma frequency f pe ∼ 100 kHz, implying that they had all stopped radiating before arriving at PSP.The starting frequencies of 247 weak SRBs (∼64%) can be definitely determined because their f st are all much lower than the upper-limit frequency of the RFS measurement (i.e., ∼19.2 MHz).Another about one-third (138 events) of the weak SRBs, however, have f st reaching the upper-limit frequency of the RFS measurement as shown in In order to find their real starting frequencies, we further compared the PSP observation with that by the Low-Frequency Array in Europe (LOFAR) during the same interval, which has an effective observation frequency range between 20 and 80 MHz.However, due to the limited observation time of LOFAR allocated to the solar and space weather division, only 151 of these 385 weak SRBs occurred in the solar observation windows of LOFAR.The result showed that only less than one-fourth (36 of the 151 SRBs) have f st > 20 MHz and hence were observed by LOFAR.In fact, only some strong SRBs with a peak intensity higher than 10 −15 V 2 Hz −1 can be seen by WIND, and these strong SRBs, in general, also have a higher starting frequency in the LOFAR observation.
The relative frequency-drift rate is an important and critical parameter that directly reflects the environment and motion of the emitting source in the background plasma.It has been used as a typically characteristic parameter to classify various kinds of SRBs.Here, we take event 5, marked by arrow in Figure 2(b), as an example to show the determination of the relative frequency-drift rate D. After subtracting the background noise, the dynamic spectrum of event 5 is shown in Figure 2(e), in which at low frequencies (∼1 MHz or below), the burst becomes more clear compared to the background.By finding the frequencies of maximal PSD (magenta dots) of event 5 at each time point, we can obtain the fitting curve (black curve) for the frequency-time relation in a polynomial form, lgf = at b + c, with the parameters a = 5.722, b = −0.704,and c = 5.474 for event 5, which are obtained by the least-squares fit.Then, the relative frequency-drift rate can be calculated by the expression of the fitting curve.The advantage of this method is to avoid limiting by lower time resolution around the highest frequencies (∼19 MHz) in the determination of drift rate.the result shows that their relative drifting rates have an initial value lower than and close to ∼0.1 s −1 and rapidly slow down from >0.01 to <0.01 s −1 as the emitting frequency drifts downward from a few tens of MHz to a few hundred kHz.This implies that their emitting sources probably experienced strong dynamical evolution.An interesting and puzzling question is where the energetic electrons responsible for these weak SRBs, which should exist rather ubiquitously, come from.
On the other hand, combining the remote-sensing observations by the Hinode EUV Imaging Spectrometer and the Solar Dynamics Observatory Atmospheric Imaging Assembly, Harra et al. (2021) found that AR 12737 is possibly associated with the low-density magnetic channel observed by PSP.AR 12737, however, does not have evident flare or jet activities during the corresponding time period.Based on the observation of Fermi, there is also no obvious hard X-ray emission.Therefore, it seems to be unlikely that the FEBs responsible for these weak SRBs come directly from flare or jet activities in AR 12737.In addition, there is an extended blueshifted outflow region inside AR 12737, and its expanding behavior probably leads to the formation of the low-density magnetic channel and the solar wind stream.Moreover, the solar wind magnetic field is dominated by the radial component B R and follows a similar Parker spiral structure (Parker 1958).Therefore, it is a reasonable inference that the magnetic channel extends from AR 12737 in the solar corona to the solar wind.Accompanied by the acceleration and heating of the solar wind, the emitting sources of these weak SRBs were formed and traveled outward along this magnetic channel (Reiner & Kaiser 1999;Ma et al. 2021).In particular, the rapid decrease of their relative drifting rates indicates that their emitting sources should have experienced strong dynamic evolution in the solar wind acceleration region.

Model of Plasma Parameters and Generation Mechanism
of Weak SRBs

Model of Plasma Parameters
Both the emitting source and the emission mechanism of these weak SRBs sensitively depend on the local plasma parameters in the source region.For example, their emitting frequencies are closely associated with the characteristic frequencies of the local plasma in the source regions, such as the plasma frequency f n 8.98 e pe = kHz for the plasma emission and the electron cyclotron frequency f ce = 2.8B 0 MHz for the ECM emission, where the ambient plasma density n e and magnetic field B 0 are in units of cm −3 and gauss, respectively.The relative frequency-drift rate D of the dynamic spectra, in general, can be determined by the moving velocity of the emitting source and the variation of the characteristic frequencies ( f pe or f ce ), which are given directly by the ambient plasma density (n e ) or magnetic field (B 0 ) along the propagation path of the weak SRBs.
A widely adopted model for the radial distribution of the average plasma density from the solar corona at ∼1.8 R S to the solar wind at ∼1 au is the polynomial distribution proposed by Leblanc et al. (1998) , in which r is the heliocentric distance in units of the solar radius R S ; the first and third terms proportional to r −6 and r −2 are dominant in the solar corona and solar wind, respectively; the second term proportional to r −4 is used to fit the transition between the corona and the wind; and the coefficient a can be determined by the measured density value at 1 au or other distances.In the low corona, however, the density gradient is very steep in the exponential fall way.In order to fit the density of ∼10 10 cm −3 at the base of the corona, Wu & Fang (2003) introduced an exponential function with a scale height h ∼ 0.02 R S , 10 10 e −50( r−1) cm −3 , to model the density distribution in the low corona.In addition, a density dilution factor, d r e 1 9 º + -- -, may be invoked to describe the low-density feature of the magnetic channel (Esser & Sasselov 1999;Esser et al. 1999;Young et al. 1999;Teriaca et al. 2003;Wu & Yang 2007).In consequence, the radial distribution of the electron density along the magnetic channel can be fitted by On the other hand, some numerical two-fluid models of highspeed streams from the corona to the solar wind (Hu et al. 1997) showed that the radial distribution of the electron temperature can be characterized by a quick increase from ∼5 × 10 5 to ∼1.5 × 10 6 K within r < 3, then a slow decrease by a factor of ∼10 at a radial distance of about a few tens of R S , followed by a slower decrease in the interplanetary space.Such temperature behavior may be characteristically described by The radial distribution of the magnetic field along the magnetic channel can be modeled by the combination of a dipole field ∝r −3 in the corona and a monopole field ∝r −2 in the solar wind as (Mariani & Neubauer 1990) Here, the averaged electron density (n e ∼ 120 cm −3 ), temperature (T e ∼ 25 eV), and magnetic field (B 0 ∼ 100 nT) in situ measured by PSP at ∼36.6 R S (Halekas et al. 2020) have been used to fit the coefficients in Equations (1)-(3).
Based on the empirical models mentioned above, Figure 3(a) shows the radial distributions of the electron cyclotron ( f ce = 2.8 × 10 6 B 0 (r) Hz; solid line) and plasma ( f pe =8.98 ń r 10 Hz; e 3 ( ) dashed line) frequencies.From Figure 3(a), it can be found that the low-density magnetic channel extends from ∼1.1 to ∼6.1 R S , in which the electron cyclotron frequency is evidently higher than the plasma frequency, that is, f ce > f pe , and the corresponding range of the characteristic frequencies between ∼30 MHz and ∼300 kHz covers well the emitting frequency band of the weak SRBs.This further confirms that their emitting sources can be well located within the extended solar corona from ∼1.1 to ∼6.1 R S , which is a low-density magnetic channel or an equatorial coronal hole with outflows and open magnetic fields.Usually, this extended coronal region is extensively believed to be the important region of the solar wind origin and acceleration.=  and v ez = 4v sw ; 7.66v A .This can provide an efficient acceleration mechanism for the local generation of energetic electrons that is required to excite the emissions of the weak SRBs.

Emission Mechanism of Weak SRBs
It is worth noting that the low-density magnetic channel has similar plasma conditions to the source region of the terrestrial auroral kilometric radiation (see, e.g., Mozer et al. 1980;Bryant 1990;Wu & Chao 2004), in which the electron cyclotron f ce is higher than the plasma frequency f pe , and the Alfvén velocity v A is larger than the electron thermal velocity v T e .The condition f ce > f pe indicates that the coherent radio radiation at the cyclotron frequency and its harmonics can be effectively excited by the ECM instability.
The solar wind acceleration region, located in the extended solar corona at ∼1.1-10 R S , is a complex dynamical transition region, where the coronal plasma is heated and accelerated into the solar wind.Alfvén wave (AW) turbulence, originating from the photospheric turbulence and convection, plays an important role in the coronal heating and solar wind acceleration (Cranmer & van Ballegooijen 2005;Cranmer et al. 2007).The AW turbulence, via the anisotropic turbulent cascade, period doubling, and wave breaking (Goldreich & Sridhar 1995;Tsurutani et al. 2018), can extend into the kinetic scales of particles and become KAWs.Meanwhile, in a low-β plasma of v v T A e > , nonlinear solitary wavelets of KAWs (i.e., SKAWs) may be formed effectively because KAWs can be free from the heavy Landau damping when their phase speed (>v A ) is considerably larger than the electron thermal speed (v T e ).SKAWs and their associated structures have been observed and identified extensively in near-Earth space plasmas, such as the auroral plasma.A number of studies in both observation and theory have shown that the nonlinear KAWs can play an important role in the field-aligned acceleration of electrons and the crossing-field heating of ions in the aurora plasma (Louarn et al. 1994;Wahlund et al. 1994;Wu et al. 1995;Chaston et al. 1999;Stasiewicz et al. 2000;Wu & Chao 2004).For the case of the solar corona, it is also found that SKAWs can be responsible for the anomalous anisotropic energization of minor heavy ions discovered by the Solar and Heliospheric Observatory in the extended solar corona of ∼1.5-5 R S (Wu & Yang 2007;Wu 2012;Wu & Chen 2020).
On the other hand, the field-aligned electric field of SKAWs has a typical dipole structure, which can accelerate electrons along the magnetic field and trap these energetic electrons inside the potential well.Some recent works have shown that the presence of AW turbulence may significantly influence the ECM instability (Wu et al. 2012;Wu 2014;Zhao et al. 2015;Chen et al. 2017Chen et al. , 2021)).Recently, Kasper et al. (2021) found that the strong AW turbulence exists not only in super-Alfvénic streams such as the solar wind but also can present in the low-β solar corona with a sub-Alfvénic stream.They reported the spectrum of Alfvén turbulence measured by PSP during its eighth solar encounter (E08), which has a typical PSD from ∼106 to 10 3 km 2 s −2 Hz −1 and a spectral index of ∼−3/2 for the inertial turbulence range from 0.002 to 0.2 Hz.An ambient Alfvén velocity v A ∼ 450 km s −1 indicates the relative energy density of the Alfvén turbulence δ B ∼ 2 × 10 −2 (Kasper et al. 2021).However, this relative strength expresses only an average turbulence level of AWs, while the actual relative strength can vary dependently on cases in a wide range, say, 0.01-0.1, in the solar corona (Cranmer & van Ballegooijen 2005).
Following Wu et al. (2012), the velocity distribution function of beam electrons with a characteristic beam velocity v b under the influence of AW turbulence with the relative strength δ B can be modeled by the so-called crescent-shaped distribution, is the pitch angle of the electron velocity, v T is the velocity spread of the beam electrons, and + ^is the pitch-angle spread of the beam electrons.In particular, the crescent-shaped distribution can effectively excite the ECM emission (Wu et al. 2012;Wu 2014;Zhao et al. 2015).Figure 4 shows the growth rates of the ECM instability versus the frequency ratio ω pe /ω ce , where the parameters v b = 0.35c, v T = 0.05v b , and δ B = 0.05 have been used.As shown in Figure 4, the excited modes depend considerably on the frequency ratio ω pe /ω ce , in which the most easily excited emission is the fundamental ordinary mode (O1) in the low-β plasma of ω pe /ω ce < 1, while for the case of 1 < ω pe /ω ce < 2, the harmonic waves of the ordinary (O2) and extraordinary (X2) modes also may be excited but at a much lower growth rate.However, the fundamental extraordinary mode (X1) can be excited only in the extreme condition of ω pe /ω ce = 1 because of its higher cutoff frequency than that of the ordinary mode.
As shown in Figure 3(a), in the low-density magnetic channel between ∼1.1 and ∼6.1 R S , the exciting condition for the ECM emission, ω pe /ω ce < 1, can be satisfied, and the frequency range between ∼35 and ∼0.25 MHz covers well the emitting frequencies of the weak SRBs.Meanwhile, the low-β condition of v v T A e > in the emitting source region indicates that SKAWs with a dip density soliton can propagate at a super-Alfvén velocity v sw > v A along the magnetic field (Wu 2012;Wu & Chen 2020).In particular, their field-aligned electric field E ∥ can efficiently accelerate electrons to form an oscillating energetic electron beam with a higher velocity v ez > v sw .For SKAWs with a normalized inner density n m < 1, the energetic electron beam can have a characterized beam velocity v b = v ez (Wu 2012;Wu & Chen 2020), that is, We propose that the ECM emission excited by the energetic electron beam, trapped in the potential well of the SKAW, can be responsible for the weak SRBs, called SWR, while the frequency drift of SWR is caused by the travel of the SKAW at the velocity . Specifically, a single small-scale weak SRB is attributed to a single SKAW.
Combining the relative frequency-drift rate, D, presented in Figure 2(f) with the radial distribution of the electron cyclotron frequency, f ce (i.e., D = (df ce /dr)(v S /f )), Figure 5 shows the moving velocity of the emitting sources v S /c versus the heliocentric distance (solid lines) for the several typical weak SRBs in Figure 2(b), where the local Alfvén velocity v A /c is presented by the dashed line for the sake of comparison, with a typical value v A ∼ 0.05c in the source region.The sources have initial velocities v S ∼ 0.1c ∼ 2v A and then rapidly descend to v S < v A .Here, we point out that these weak SRBs were excited by the energetic electrons with velocity v b ∼ 0.35c ∼ 7v A that are trapped in the potential well of SKAWs with velocity v sw = v S ∼ 0.1c ∼ 2v A (for n m = 0.2).From Figure 5, it can be found that the motion of these sources has a common evolutionary feature, that is, a rapid deceleration with the heliocentric distance from initial v S > v A closer to the Sun to v S < v A at further distances, implying that the emitting sources experience an evident and lasting deceleration as they travel outward.This common feature can be reasonably explained by the evolving property of SKAWs due to the dissipation in lowβ plasma (Wahlund et al. 1994;Voitenko & Goossens 2000;Wu et al. 2007).
In the initial stage of SKAW formation, energetic electrons with the characteristic oscillating velocity v b , accelerated by the SKAW electric field, are trapped within the SKAW potential well and travel at a velocity v sw together with the SKAW (Wu et al. 2007).It is these trapped energetic electrons that trigger SWR via the ECM mechanism, in which the transverse free energy required by exciting the ECM instability can be provided by the wave-particle scattering of AW turbulence.As propagating, the SKAW may lose energy due to various possible dissipations, such as the Coulomb collision or ionacoustic turbulence (Wahlund et al. 1994;Voitenko & Goossens 2000;Wu et al. 2007).This leads to the deceleration of the SKAW, i.e., the emitting source, because the propagating velocity v sw decreases with the dissipation of the SKAW.In the meantime, accompanying the dissipation, the symmetry of the SKAW potential well is deformed and evolves into a shocklike structure, and in consequence, a part of the energetic electrons may escape from the SKAW potential well until the SKAW is exhausted when the inner density n m → 1 (Wahlund et al. 1994;Wu & Chao 2004).The energetic electrons escaping from the dissipated SKAW gradually merge into the local plasma environment with strong AW turbulence, and the corresponding SWR has a wider spectral band and a slower drifting velocity and ultimately travels together with the accelerated solar wind at a velocity much lower than the local Alfvén velocity as shown in Figure 5. > , SKAWs have a characteristic width λ ⊥ ∼ 2πλ e (i.e., k ⊥ λ e ∼ 1), where λ e ≡ c/ω pe is the electron inertial length (Chaston et al. 1999).In consequence, the damping rate is γ ∼ 1.58 × 10 −3 Hz, implying that the dissipation time of SKAWs is typically about 10 minutes (i.e., τ ∼ γ −1 ∼ 600 s).However, the actual damping rate in general can be higher than this collision damping, and hence the observed lifetime of SWR may be shorter than the estimation here.A single SKAW can contribute to a single burst, and the energetic electrons trapped in the potential well of the SKAW can gain energy by the acceleration of the SKAW electric field.The power of the SKAW may be estimated by the production of the Joule heating rate (J • E) in the SKAW and the SKAW volume, that is, Q jE 2.9 10 ~are the current density and the electric field in the SKAW.Here, the parallel scale of SKAW λ ∥ ∼ 10 3 λ ⊥ , the dimensionless parameters n m = 0.2 (hence, v b ; 7.66v A ∼ 0.35c) and E m = 3.5 (Wu 2012;Wu & Chen 2020) have been assumed.The magnetic field B 0 ; 1.12 G is obtained from Equation (3) in the normalized heliocentric distance r = 2.5.On the other hand, the rate of energy loss due to the radio emission may be estimated as , where the effective antenna length L eff ∼ 1 m, capacitive gain factor Γ = 0.32, and impedance of free space Z 0 = 377Ω have been used (Pulupa et al. 2017;Jebaraj et al. 2023).Here, taking event 5, for example, we have adopted the average PSD P ∼ 10 −16 V 2 Hz −1 , frequency width !f ∼ 10 6 Hz, and distance between PSP and emitting source R pb ∼ 33.5 R S .If we further consider the fact that the coherent radio radiation is often a strong anisotropic emission, the radiation will only be concentrated within a small flare angle whose cross section is possibly much smaller than R pb 2 .In consequence, Q R will decrease further and become much smaller than Q H .In fact, the energy loss via radio radiation is only a very small part of the energy loss of the SKAW; the major energy loss of the SKAW is due to the dissipation caused by the classical or abnormal collision.

Summary and Discussion
In summary, we reported a kind of weak SRB observed by PSP when crossing a low-density magnetic channel during its E02 phase.These weak SRBs have a weak intensity lower than 10 −15 V 2 Hz −1 , a relatively low starting frequency (∼20 MHz), a narrow frequency range from a few tens of MHz to a few hundred kHz, and an evolving and intermediate relative frequency-draft rate D ≡ |(df/dt)/f| from D > 0.01 to <0.01 s −1 .They can occur quite frequently (five bursts per hour), and the nature of their weak intensity indicates that they originate from small-scale emitting sources.Based on the common empirical models for the solar coronal plasma, these small-scale emitting sources lie in the heliocentric distance between ∼1.1 and 6.1 R S , a typical solar wind acceleration region.We proposed that SKAWs in kinetic scales, which are formed easily in the solar wind acceleration region with a lowβ plasma environment, can be responsible for the small-scale emitting sources of these weak SRBs, called SWR.
Although the radio radiation of SWR has an insignificant impact on the space plasma environment, the kinetic-scale characteristic of their emitting sources has important implications for the dynamics of magnetic plasmas in the solar wind acceleration region.One of the unsolved problems in solar physics is the heating and acceleration mechanism of coronal plasmas into the solar wind in the extended corona from 1.1 to 10 R S .The complexity of the extended coronal plasma in both the kinetics and dynamics is the result of the plasma density decreasing with the heliocentric distance, as well as the complicated magnetic topology in coronal plasmas.The decrease of the density leads to the transition of the coronal plasma from a collisionally dominated plasma to a nearly collisionless one.As a result, the kinetic wave-particle interaction process plays an important role in the heating and acceleration of the coronal plasmas.On the other hand, the fully ionized state of the hydrogen, the major solar atmospheric component, results in the impossibility of gaining the physical information associated with the acceleration and heating processes via spectral line observations of the hydrogen, which is a main method of inferring the physical situation and processes in the photosphere and chromosphere.Alternately, however, radio observations, especially the observations of radiation originating from small-scale emitting sources, can provide us rich information on energetic electrons and their kinetic processes, as well as the ambient magnetic plasmas in the solar wind acceleration region.

Figure 2
Figure 2(d), which indicates that their actual starting frequencies are possibly higher than 19.2 MHz.In order to find their real starting frequencies, we further compared the PSP observation with that by the Low-Frequency Array in Europe (LOFAR) during the same interval, which has an effective observation frequency range between 20 and 80 MHz.However, due to the limited observation time of LOFAR allocated to the solar and space weather division, only 151 of these 385 weak SRBs occurred in the solar observation windows of LOFAR.The result showed that only less than one-fourth (36 of the 151 SRBs) have f st > 20 MHz and hence were observed by LOFAR.Figure2further shows the comparison between observations of LOFAR (a), PSP (b), Figure 2(d), which indicates that their actual starting frequencies are possibly higher than 19.2 MHz.In order to find their real starting frequencies, we further compared the PSP observation with that by the Low-Frequency Array in Europe (LOFAR) during the same interval, which has an effective observation frequency range between 20 and 80 MHz.However, due to the limited observation time of LOFAR allocated to the solar and space weather division, only 151 of these 385 weak SRBs occurred in the solar observation windows of LOFAR.The result showed that only less than one-fourth (36 of the 151 SRBs) have f st > 20 MHz and hence were observed by LOFAR.Figure2further shows the comparison between observations of LOFAR (a), PSP (b),

Figure 1 .
Figure1.The solar wind plasma parameters observed by PSP between 2019 April 1 and 9. From top to bottom, the panels are the PSD of the radio radiation (a), the magnetic field (b) and its components (c), the solar wind velocity (d) and its components (e), the plasma density (f) and temperature (g), and the heliocentric distance of PSP in units of solar radius, R S (h).A low-density magnetic channel is clearly displayed between 08:08:34 UT on 2019 April 3 and 13:17:09 UT on April 6 (yellow region).

Figure 2 .
Figure 2. The weak SRBs observed by PSP during its crossing of the low-density magnetic channel.The top three rows show PSD measured by LOFAR (a), PSP (b), and WIND (c) in two intervals, 08:13-09:03 on 2019 April 3 (left) and 05:23-09:23 on 2019 April 5 (right).(d) The starting ( f st ; red circles) and ending ( f lo ; blue circles) frequencies for the 385 weak SRBs and the local plasma frequency f pe (back line) along the PSP orbit.(e) The fitting curve (black curve) of maximal PSD for event 5 marked by an arrow in panel (b) through the least-squares fit, where the magenta dots are the location of maximal PSD in the dynamic spectrum.(f) The relative frequency-drift rate for the several typical bursts as marked by arrows in panel (b).
Figure 2(f) shows the relative frequency-drift rate D versus the central frequency for the several typical examples marked by arrows in Figure 2(b), and

Figure 3
(b) presents the local Alfvén velocity v A (solid line) and the electron thermal velocity v T e (dashed line) normalized by the light velocity c.From Figure3(b), one has v v region of these weak SRBs (∼1.1-6.1 R S ), implying that there is a low-β plasma of β < m e /m p in the source region, where m e and m p are the electron and proton masses, respectively.Based on the theory of kinetic Alfvén waves (KAWs; Wu 2012; Wu & Chen 2020), a low-β SKAW, accompanied by a dip density soliton with an inner density n em < n 0 , can propagate at a super-Alfvén velocity v magnetic field, where n m ≡ n em /n 0 is the inner density normalized by the ambient density n 0 .In particular, the field-aligned electric field E ∥ of SKAW can efficiently trap and accelerate electrons to a typical velocity v than v A for n m = 1(Wu 2012;Wu & Chen 2020).For example, for n m = 0.2

Figure 3 .
Figure 3.The radial distributions of characteristic frequencies and velocities.(a) The radial distributions of the electron cyclotron ( f ce ) and plasma ( f pe ) frequencies.(b) The radial distributions of the Alfvén (v A ) and electron thermal (v Te ) velocities.
The dissipation time of SKAWs may be approximately estimated by the inverse of the damping rate k 0the coronal plasma(Voitenko & Goossens 2000; Wu et al.  2007collision frequency and lnL is the Coulomb logarithm.Assuming the typical electron density n e ∼ 10 5 cm −3 and temperature T e ∼ 150 eV in the source region, one has ln 20 L ~and ν e ∼ 3.16 × 10 −3 Hz.In a low-β plasma of v v T A e

Figure 5 .
Figure5.The radial distributions of the emitting source velocities for the typical events of weak SRBs v S /c (solid lines) and the Alfvén velocity v A /c (dashed line) along an open magnetic field region in the low-density magnetic channel, where all velocities have been normalized by the light speed c.