# Estimated Heating Rates Due to Cyclotron Damping of Ion-scale Waves Observed
by Parker Solar Probe

Niranjana Shankarappa Department of Physics, University of Arizona, Tucson, AZ
85721, USA  Kristopher G. Klein Lunar and Planetary Laboratory, University of
Arizona, Tucson, AZ 85721, USA  Mihailo M. Martinović Lunar and Planetary
Laboratory, University of Arizona, Tucson, AZ 85721, USA  LESIA, Observatoire
de Paris, Université PSL, CNRS, Sorbonne Université, Université de Paris,
92195 Meudon, France.  Trevor A. Bowen Space Sciences Laboratory, University
of California, Berkeley, CA 94720-7450, USA

###### Abstract

Circularly polarized waves consistent with parallel-propagating ion cyclotron
waves (ICWs) and fast magnetosonic waves (FMWs) are often observed by Parker
Solar Probe (PSP) at ion kinetic scales. Such waves damp energy via the
cyclotron resonance, with such damping expected to play a significant role in
the enhanced, anisotropic heating of the solar wind observed in the inner
heliosphere. We employ a linear plasma dispersion solver, PLUME, to evaluate
frequencies of ICWs and FMWs in the plasma rest frame and Doppler-shift them
to the spacecraft frame, calculating their damping rates at frequencies where
persistently high values of circular polarization are observed. We find such
ion-scale waves are observed during $20.37\%$ of PSP Encounters 1 and 2
observations and their plasma frame frequencies are consistent with them being
transient ICWs. We estimate significant ICW dissipation onto protons,
consistent with previous empirical estimates for the total turbulent damping
rates, indicating that ICW dissipation could account for the observed
enhancements in the proton temperature and its anisotropy with respect to the
mean magnetic field.

##  1 Introduction

Understanding the mechanisms that drive solar coronal heating and the
acceleration of the solar wind is a long-standing goal of space plasma
physics. This goal requires the identification of the processes that couple
electromagnetic fields to the protons and electrons. Dissipation of waves and
turbulent fluctuations (Matthaeus et al., 1999) can occur via many mechanisms
that can broadly be grouped into three classes: resonant mechanisms (e.g.
Landau and transit time damping, Barnes (1966), Leamon et al. (1999), Chen et
al. (2019)), non-resonant mechanisms (e.g. stochastic heating, Chen et al.
(2001), Chandran et al. (2010)), and intermittent dissipation in current
sheets and magnetic reconnection sites (Dmitruk et al., 2004; Matthaeus &
Velli, 2011). Multiple dissipation mechanisms can operate in the same
environment (He et al., 2015; Klein et al., 2020) making it necessary to
estimate the damping associated with each individually.

Cyclotron damping is a plausible wave-particle resonant mechanism for damping
circularly polarized waves, whose presence in the solar wind is supported by
spacecraft observations (Leamon et al., 1998; Kasper et al., 2013; Bowen et
al., 2022). Cyclotron damping of ion cyclotron waves (ICWs) has been proposed
to be a significant process leading to the anisotropic heating of protons that
enhances at close radial distances from the Sun (Hollweg & Isenberg, 2002).
Ion-scale waves are ubiquitous in the young solar wind (e.g. Bowen et al.,
2020b; Verniero et al., 2020; Zhao et al., 2021; Shankarappa et al., 2023) and
at 1 AU (e.g. Jian et al., 2009; Woodham et al., 2021).

Parker Solar Probe (PSP; Fox et al. (2015)) has been making in-situ
measurements of the solar wind thermal plasma (Kasper et al., 2016) and
electromagnetic fields (Bale et al., 2016) at unprecedented distances from the
Sun since its’ launch in 2018. In our previous work, Shankarappa et al. (2023)
(referred to as SKM23 hereon), we analyzed PSP observations from Encounters 1
and 2 between heliocentric distances 0.166 and 0.25 au and inferred the
feasibility of Landau damping to dissipate turbulent fluctuations. We further
inferred the presence of significant energy in circularly polarized ion-scale
waves–consistent with parallel propagating (with respect to the local magnetic
field) ICWs or fast magnetosonic waves (FMWs)–coincident with wavevector-
anisotropic background turbulence.

In this work, we expand upon that previous work by estimating the damping
rates of ion-scale wave energy onto protons and electrons via the parallel
cyclotron resonance. Such an estimation requires the identification of the
observed waves as ICWs or FMWs. Bowen et al. (2020) have found PSP electric
field observations on November 4, 2018 to be consistent with outward
propagating ICWs and FMWs. However, the accurate processing of the full
electric field vector measurement by PSP across entire encounters is limited
by the geometry of the antennas. Due to this limitation and inherent
difficulties with uniquely determining the plasma frame polarization of waves
using just magnetic field observations from a single spacecraft, the observed
ion-scale waves cannot be generally uniquely distinguished as ICWs or FMWs. To
compensate for this uncertainty, we employ the PLUME linear plasma dispersion
solver (Klein & Howes, 2015) to estimate the damping rates of the observed
waves as both ICWs or FMWs.

We find that consistent with Bowen et al. (2020b) and Liu et al. (2023), the
ion-scale waves are observed during $20.37\%$ of PSP Encounters 1 & 2 solar
wind observations. The plasma frame frequencies corresponding to the
frequencies where the waves are observed are consistent with them being
locally generated ICWs. We infer a significant possible ICW dissipation onto
protons that could explain the observed enhancement in the proton temperature
anisotropy at closer distances to the Sun (Mozer et al., 2023; Zaslavsky,
2023). The heating rates estimated in this work are in very good agreement
with the empirical estimates of total turbulent dissipation rates by
Bandyopadhyay et al. (2020).

The rest of this paper is organized as follows. The methodology employed to
evaluate heating rates via cyclotron damping of ion-scale waves is described
in Section 2. In Section 3 we quantify the duration and energy of ion-scale
waves observed in Encounters 1 and 2. We further discuss their transience and
compare their plasma frame frequencies to frequencies over which ICWs and FMWs
would be generated due to proton temperature anisotropy instabilities. Section
4 presents the proton heating rates estimated in this work and the ratio of
proton to total heating rates estimated by including previous work, followed
by a discussion in Section 5.

Figure 1: Overview of the estimation of turbulent and ion-scale-wave
dissipation for an example 15-minute PSP interval on 10th November 2018,
17:53:58 - 18:08:53, where ion-scale waves are observed: (a) The ion-scale
wave energy is distinguished from turbulent energy spectrum. (c,e) The
dissipation rates of turbulent energy via the Landau resonance were estimated
using a 1-D cascade model (SKM23). (b,d) The dissipation rates of ion-scale
wave energy via the cyclotron resonance as estimated in this work.

##  2 Methodology

The heating rates due to cyclotron damping of ion-scale waves are evaluated
using the following steps for each 15-minute interval of PSP observations
analyzed in Shankarappa et al. (2023) (SKM23).

###  2.1 Plasma and Magnetic Field Data

We use Level 2 magnetic field data (in spacecraft frame RTN coordinates) from
the Flux Gate Magnetometer (MAG) component of the PSP/FIELDS instrument suite,
which makes in situ observations of electromagnetic fields (Bale et al.,
2016). While the sampling rate of the MAG varies between $\sim$73.24 Sa
$s^{-1}$, 146.48 Sa $s^{-1}$, and 292.96 Sa $s^{-1}$ during Encounter 1, the
magnetic field is sampled at a constant rate of 146.48 Sa $s^{-1}$ during
Encounter 2. Measurements of the thermal plasma come from the PSP/SWEAP
instrument suite (Kasper et al., 2016). Huang et al. (2020) and Halekas et al.
(2020) have estimated parallel and perpendicular temperatures (with respect to
the direction of magnetic field) of protons ($T_{\parallel,p}$ and
$T_{\perp,p}$) and electrons ($T_{\parallel,e}$ and $T_{\perp,e}$),
respectively by applying bi-Maxwellian fits to velocity distribution functions
(VDFs) of protons evaluated from SWEAP/SPC (Case et al., 2020) and electrons
evaluated from SWEAP/SPANe (Whittlesey et al., 2020) measurements. We use
plasma parameters evaluated by averaging over 1 minute of SPC and SPANe
observations as inputs into the PLUME solutions.

In SKM23, we assessed 15-minute long intervals from Encounters 1 and 2 of
PSP/FIELDS and SWEAP observations, following the procedure described in
appendix A.1 of SKM23, where we obtained 2198 intervals from the available
FIELDS observations; further prescreening for the availability of SWEAP
observations yielded 2118 "good" intervals. For each of these intervals, we
evaluated interval-averaged values of the magnetic field amplitude ($B$),
proton number density ($n_{p}$), solar wind velocity
($\mathbf{V}_{\text{SW}}$), proton gyroradius ($\rho_{p}$), and Alfvén speed
$\left(v_{A}=\frac{B}{\sqrt{\mu_{0}n_{p}m_{p}}}\right)$ as described in
appendices A.2 and C of SKM23. Here $\mu_{0}$ is the permeability of free
space and $m_{p}$ is the proton mass.

In this work, we further evaluate the parallel temperature disequilibrium
($T_{\parallel,p}/T_{\parallel,e}$), proton and electron temperature
anisotropies ($T_{\perp,p}/T_{\parallel,p}$ and
$T_{\perp,e}/T_{\parallel,e}$), parallel proton plasma beta
$\left(\beta_{\parallel,p}=\frac{n_{p}k_{B}T_{\parallel,p}}{(B^{2}\big{/}2\mu_{0})}\right)$,
and parallel proton thermal velocity
$\left(v_{\text{th},\parallel,p}=\sqrt{\frac{2k_{B}T_{\parallel,p}}{m_{p}}}\right)$.
Here $k_{B}$ is the Boltzmann constant. The proton and electron temperature
anisotropy estimates are discarded when their VDF measurements do not meet the
data quality criteria for bi-Maxwellian fitting (see appendices of Huang et
al. (2020)), further excluding 366 intervals in Encounter 1 and 2, leaving us
with 1752 total intervals for this study.

###  2.2  Overview of SKM23: Characterizing Turbulent Energy Spectra and
Estimating Heating Rates due to Landau Damping

Figure 1 shows the overview of the SKM23 method combined with the current work
to estimate dissipation rates of turbulence and ion-scale waves via the Landau
and cyclotron resonances, respectively for an example PSP interval from 10th
November 2018, 17:53:58 - 18:08:53 where ion-scale waves are observed.

In each interval analyzed in SKM23, we removed reaction wheel noise from the
FIELDS magnetic field observations, downsampled the data to 10 Hz by applying
an antialiasing filter, and evaluated the magnetic field wavelet energy
spectrum (green dashed curve in panel (a)) using a Morlet wavelet (Torrence &
Compo, 1998) across $f_{\text{log\\_bin}}$, the median bin values obtained by
splitting the DFT frequency domain of the 15 minute-B time series into
logarithmic bins.

We then distinguished the energy spectrum of turbulence (blue in panels (a,c))
from the energy spectra of ion-scale waves, $|\tilde{B}(f)|^{2}_{wave\\_psp}$
(black in panels (a,b)), by using a high degree of circular polarization of
the latter as a criterion. Circular polarization in the plane perpendicular to
the local magnetic field is quantified by spacecraft frame polarization,
$\sigma$ (Bowen et al., 2020b), which is evaluated using the wavelet-
transformed magnetic field components, $W_{x,y}(s,t)$, in a scale sensitive
coordinate system, $(X,Y,Z)$, where $\mathbf{Z}$ is parallel to the local
magnetic field at the scale under consideration, and $\mathbf{X}$ and
$\mathbf{Y}$ define the plane perpendicular to $\mathbf{Z}$,

$\sigma(s,t)=\dfrac{-2\text{Im}\big{[}W_{x}(s,t)W_{y}^{*}(s,t)\big{]}}{|W_{x}(s,t)|^{2}+|W_{y}(s,t)|^{2}}.$
(1)

Here $s$ is the inverse frequency and ∗ represents the complex conjugate.
Using our phase and Fourier conventions, in the plasma rest frame, $\sigma$ =
1 represents ion-resonant/left-handed waves (ICWs) and $\sigma$ = -1
represents electron-resonant/right-handed waves (FMWs) (see Appendix C for a
detailed mapping between helicity ($\sigma_{m}$) and polarization in relevant
frames). We applied a threshold of 0.7 on values of $|\sigma|$ for wave
selection (see Appendix B.4. of SKM23 for justification of this criterion).
Note that there is an additional minus sign in the definition of $\sigma$
(Equation 1) compared to SKM23. Our previous findings are unaffected by this
sign difference, as SKM23 used only the magnitude of $\sigma$ to identify and
exclude coherent wave power. Additionally, we called $\sigma$ cross-coherence
in SKM23, while we accurately call it spacecraft-frame polarization here. In
this work, we additionally filter out the energy of turbulent fluctuations
with random high values of polarization from $|\tilde{B}(f)|^{2}_{wave\\_psp}$
(described in Appendix A) and further distinguish right- (RH) and left-handed
(LH) spacecraft frame polarized wave energy spectrum,
$|\tilde{B}(f)|^{2^{LH/RH}}_{wave\\_psp}$.

To the turbulent energy spectra, we applied a 1-D cascade model described in
Howes et al. (2008). The model assumes a critically balanced (Goldreich &
Sridhar, 1995) distribution of low-frequency, Alfvénic turbulence from the
inertial to dissipation ranges, connecting the MHD and kinetic descriptions,
and calculates the linear Landau damping rates of kinetic Alfvén waves onto
protons and electrons (blue and orange, respectively in panel (e)) as a
function of the spatial scale perpendicular to the mean magnetic field,
producing a steady-state solution for a one-dimensional Batchelor (1953)-like
model. In intervals where the model energy spectra (red in panel (c)) fit well
with the observed turbulent energy spectra (as shown in panel (c)), we assume
the model to be accurate and evaluate heating rates via Landau damping.

In this work, we use linear cyclotron damping rates of ICWs onto protons and
FMWs onto electrons (blue and orange, respectively in panel (d)) to evaluate
the heating rates due to the dissipation of ion-scale waves.

Figure 2: Overview of methodology to estimate ion-scale wave dissipation in
the PSP interval (considered in Figure 1) on 10th November 2018 17:53:58 -
18:08:53 where both RH and LH polarized waves are observed: (a) The
spacecraft-frame polarization spectrum, $\sigma(\omega,t)$, (b) the frequency
ranges where LH and RH polarized waves are observed
($\omega_{\text{obs}}^{LH/RH}$, red and blue dash-dotted, respectively), and
(c) their corresponding energy spectra (red and blue) along with the total
energy spectrum (black). (d, e) The damping rates of ICWs and FMWs onto
protons and electrons, respectively as functions of spacecraft-frame
frequencies that are evaluated using PLUME for all possible Doppler shifts.

###  2.3 Removing non-wave Signatures and Identifying Ion-scale Wave
Frequencies in the Spacecraft Frame

In SKM23, we observed significant energy in ion-scale waves in the first two
PSP encounters, as has been previously reported, e.g. Bowen et al. (2020b),
Verniero et al. (2020), Zhao et al. (2021), and Liu et al. (2023). Panel (a)
of Figure 2 shows $\sigma(s,t)$ for the same example PSP 15-minute interval on
November 10, 2018, where ion-scale waves are observed. Persistent high values
of $|\sigma|$ associated with these waves are observed over frequency bands
near the proton cyclotron frequency, $f_{cp}=\frac{q_{p}B}{2\pi m_{p}}$, and
last for times varying from several seconds to a few hours. Here $q_{p}$ is
the charge of the proton. In the absence of ion-scale waves, non-persistent
random $\sigma$ values corresponding to the background turbulent fluctuations
are observed over timescales of $\sim$ 1/frequency, i.e. the time resolution
of the wavelet transform. By employing a time-smoothing routine on
$\sigma(s,t)$, we identify the frequency band, $f^{(RH/LH)}_{\text{obs}}$
where the ion-scale waves are observed (shown in panel (b) of Figure 2) and
evaluate their energy spectra, $|\tilde{B}(f)|^{2^{LH/RH}}_{wave\\_psp}$
(shown in panel (c) of Figure 2) as described in Appendix A. Here
$\omega^{(RH/LH)}_{\text{obs}}=2\pi f^{(RH/LH)}_{\text{obs}}$ is the angular
frequency band where the waves are observed.

###  2.4 Calculating Damping Rates at $\omega_{\text{obs}}$

The four MHD modes - the Alfvén, fast and slow magnetosonic waves, and the
entropy mode have kinetic counterparts in a collisionless plasma that can be
relatively lightly damped. The assumption of symmetric VDFs produces
degenerate solutions for the three propagating modes, and therefore the
damping rates of the forward and backwards solutions are identical. The impact
of the mode propagation direction is considered in the calculation of the
spacecraft to plasma frame conversion, see Eqn. 2. At small parallel scales,
Alfvén waves extend to ICWs and fast magnetosonic waves extend to whistler
waves. The PLUME dispersion solver (Klein & Howes, 2015) calculates the hot
linear plasma dispersion relation
$|D(\omega,\gamma;\mathbf{k},\mathcal{P})|=0$ for bi-Maxwellian VDFs of ions
and electrons in a uniform magnetized environment. The solutions of the
dispersion relation yield dimensionless angular frequencies and damping rates
$(\omega/\Omega_{p},\gamma/\Omega_{p})_{\texttt{PLUME}}$ in the plasma
frame–which are normalized to physical units via the observed proton
gyrofrequency, $\Omega_{p}=2\pi f_{cp}$–of different plasma modes as a
function of wavevector, ($k_{\perp}$, $k_{\parallel}$) and equilibrium plasma
parameters $\mathcal{P}$.

For this work, we consider the parameter space
$\mathcal{P}=\left[\beta_{\parallel,p},\frac{T_{\parallel,p}}{T_{\parallel,e}},\frac{T_{\perp,p}}{T_{\parallel,p}},\frac{T_{\perp,e}}{T_{\parallel,e}},\frac{v_{\text{th},\parallel,p}}{c}\right]$,
representing a proton-electron plasma with temperature anisotropies for both
species. Additionally, the eigenfluctuations of the distribution function and
the electromagnetic fields corresponding to the modes are evaluated. For each
mode, using the perpendicular electric field eigenfunctions and components of
proton and electron susceptibility tensor, the damping rates onto the species
normalized to frequency, $(\gamma_{p,e}/\omega)_{PLUME}$ are evaluated,
following the prescription described in Quataert (1998). Values of
$\gamma_{PLUME}$ are negative for a damped wave, and positive when a wave is
linearly unstable, which can arise in this work over narrow bands of
$k_{\parallel}$ driven by proton temperature anisotropy (see Sections 3.1 and
5).

For each interval, we evaluate the values of $\mathcal{P}$ using PSP
observations, input them into PLUME, identify the four least damped modes at
MHD scales ($k_{\perp}\rho_{p}$ = $k_{\parallel}\rho_{p}$ = $10^{-3}$), follow
their dispersion solutions to smaller parallel scales up to
$k_{\parallel}\rho_{p}=10^{2}$ with $k_{\perp}\rho_{p}$ held constant. Left
panels of Figure 3 show the plasma-frame angular frequencies (black-solid),
$\omega_{\text{plasma}}=2\pi f_{\text{plasma}}$, and damping rates (black-
dashed) of the ICW (panel (a)) and FMWs (panel (c)) that are evaluated using
PLUME for the example interval considered in Figure 2. For the observed values
of $\mathcal{P}$ in Encounters 1 and 2, we accurately identify the forward and
backward ICWs and FMWs and distinguish between the electron- and ion-resonant
modes via the procedure described in Appendix B.

The plasma frame frequencies are then Doppler shifted to the spacecraft frame
via

$\omega_{sc}=\omega_{\text{plasma}}+\mathbf{k}\cdot\mathbf{V_{\text{SW}}},$
(2)

where $\omega_{sc}$ is the spacecraft-frame frequency. Here we adopt the
convention that the plasma-frame wave frequencies, $\omega_{\text{plasma}}$,
are positive definite, and their direction of propagation is determined by
wavevector k. We assume that the observed ion-scale waves propagate along the
magnetic field. The validity of this assumption is underlined by the first,
median, and third quartiles of the distribution of
$\theta_{\mathbf{k}\mathbf{B}}$, the acute angle between the wavevector and
the mean magnetic field, for all intervals where ion-scale waves are observed
being $5.2^{\circ}$, $9.7^{\circ}$, and $18.8^{\circ}$, respectively. Here,
the wavevector direction is estimated by calculating the direction of minimum
variance of B (Sonnerup & Cahill, 1967). Thus we rewrite Equation 2 as

$\omega_{sc}=\omega_{\text{plasma}}\pm
k_{\parallel}\hat{\mathbf{b}}\cdot\mathbf{V_{\text{SW}}},\hskip
28.45274pt\hat{\mathbf{b}}=\frac{\mathbf{B}}{|\mathbf{B}|}.$ (3)

Here, plus (minus) sign of the
$k_{\parallel}\hat{\mathbf{b}}\cdot\mathbf{V_{\text{SW}}}$ term corresponds to
waves propagating along (opposite to) B. The sign of the
$k_{\parallel}\hat{\mathbf{b}}\cdot\mathbf{V_{\text{SW}}}$ term is further
determined by the angle between B and $\mathbf{V}_{\text{SW}}$. The sign of
the $k_{\parallel}\hat{\mathbf{b}}\cdot\mathbf{V_{\text{SW}}}$ term is thus
determined by the direction of wave propagation with respect to the parallel
component of $\mathbf{V}_{\text{SW}}$,
$\mathbf{V}_{\text{SW}\parallel}=\hat{\mathbf{b}}\cdot\mathbf{V}_{\text{SW}}$.
For a wave propagating along (opposite to) the
$\mathbf{V_{\text{SW}\parallel}}$ direction i.e. outward (inward) with respect
to the Sun, the sign of
$k_{\parallel}\hat{\mathbf{b}}\cdot\mathbf{V}_{\text{SW}}$ term is positive
(negative) and thus the wave is positive (negative) Doppler shifted,

$\omega_{sc}=\omega_{\text{plasma}}\pm k_{\parallel}V_{\text{SW}\parallel}.$
(4)

Figure 3: All possible Doppler shift cases for an ICW (panels a,b) and FMW
(panels c,d) for the PSP interval considered in Figure 2 where both RH and LH
polarized waves are observed. The behavior of Doppler-shifted ICWs and FMWs
depends on their intrinsic direction of propagation, k, with respect to
$\mathbf{V}_{\text{SW}\parallel}$, the component of $\mathbf{V}_{\text{SW}}$
parallel to B and the relative magnitudes of plasma frame frequencies,
$\omega_{\text{plasma}}$, and $k_{\parallel}V_{\text{SW}\parallel}$. Each
Doppler shift case is described in the text of section 2.4. Here PLUME
dispersion solutions where waves are heavily damped, $\gamma/\omega\geq
e^{-1}$ are discarded, where $e=2.718$.

Consequently, depending on the relative magnitudes of the
$\omega_{\text{plasma}}$ and $k_{\parallel}V_{\text{SW}\parallel}$ terms on
the R.H.S of Eqn. 4, the sign of $\omega_{sc}$ is either positive or negative.
A positive (negative) $\omega_{sc}$ value corresponds to retaining (reversing)
the plasma-frame direction of propagation and polarization of the wave when
Doppler shifted to the spacecraft frame. Figure 3 shows the possible Doppler
shift behaviors of ICWs (panels (a,b)) and FMWs (panels (c,d) for the example
PSP interval considered in Figure 2. The magnitudes of the
$\omega_{\text{plasma}}$ (black-solid) and
$k_{\parallel}V_{\text{SW}\parallel}$ (light blue) terms, and their ratio
(orange) are plotted as functions of parallel scale in panels (a,c). Further,
the resultant spacecraft frame frequencies with LH ($\omega_{\text{sc}}^{LH}$,
red) and RH ($\omega_{\text{sc}}^{RH}$, blue) polarizations upon positive
(solid) and negative (dashed) Doppler shifts are plotted as functions of
parallel scale in panels (b,d). For the ICW and FMWs at a particular parallel
scale, $k_{\parallel}$, there are three possible outcomes due to Doppler
shifting;

  1. 1.

An ICW (FMW) propagating along $\mathbf{V_{\text{SW}\parallel}}$ is positively
Doppler shifted by the solar wind, retaining its plasma frame polarization and
is observed LH (RH) polarized in spacecraft frame.

  2. 2.

An ICW (FMW) propagating opposite to $\mathbf{V_{\text{SW}\parallel}}$ is
negatively Doppler shifted by solar wind and if:

    1. (a)

$|\omega_{\text{plasma}}|>k_{\parallel}V_{\text{SW}\parallel}$, it retains its
plasma frame polarization and is observed LH (RH) polarized in spacecraft
frame.

    2. (b)

$|\omega_{\text{plasma}}|<k_{\parallel}V_{\text{SW}\parallel}$, it reverses
its plasma frame polarization and is observed RH (LH) polarized in spacecraft
frame.

These behaviors are depicted in the right panels of Figure 3 and summarized in
Table 1.

Plasma frame polarization | Spacecraft frame polarization
---|---
\+ Doppler shift | \- Doppler shift
$|\omega_{\text{plasma}}|>$ | $|\omega_{\text{plasma}}|<$
$k_{\parallel}V_{\text{SW}\parallel}$ | $k_{\parallel}V_{\text{SW}\parallel}$
ICW (LH) | LH | LH | RH
FMW (RH) | RH | RH | $\text{LH}^{*}$
Table 1: Three possible Doppler shifted polarizations for ICWs and FMWs. The
shaded Doppler shift doesn’t occur in Encounters 1 and 2. ∗ Negative Doppler
shifted FMW frequencies overlap with observed frequencies over two distinct
bands which are represented with a suffix $\\_1,2$. Figure 4: Contour maps of
PLUME damping (red) and growth (blue) rates as a function of heliocentric
distance and $f_{obs}$, the frequencies where ion-scale waves that are
observed by PSP during Encounters 1 and 2 for the positive and negative
Doppler-shifted ICW cases.

For ICWs (panels (a,b)), $|\omega_{\text{plasma}}|$ is usually less than
$k_{\parallel}V_{\text{SW}\parallel}$ at all $k_{\parallel}$ (see orange
curve), and they have LH (RH) polarization upon positive (negative) Doppler
shift which are plotted as red-solid (blue-dashed) curves. However, for FMWs
(panels (c,d)), $|\omega_{\text{plasma}}|$ become strongly dispersive at small
parallel scales gaining larger phase velocities, allowing them to exceed
$k_{\parallel}V_{\text{SW}\parallel}$ in magnitude (see orange curve). While a
positive Doppler-shifted FMW has RH polarization (blue-solid curve), a
negative Doppler-shifted FMW can switch polarizations from LH (red-dashed
curve) to RH (blue-dashed) moving from smaller to larger $k_{\parallel}$.
Consequently, for these waves, the magnitude of $\omega_{sc}$ increases with
$k_{\parallel}$, reaches a maximum, and then decreases before reversing sign
(see the red-dashed curve). The Doppler shift routine employed in this work
and the behaviors of ICWs and FMWs upon Doppler shifting are consistent with
Bowen et al. (2020), Verniero et al. (2020), and McManus et al. (2024).

In an interval where an ion-scale wave with a RH or LH polarization is
observed such as the one depicted in panel (a) of Figure 2 and panels (b,d) of
Figure 3, its occurrence can be ascribed to the three discussed Doppler shift
cases each for either polarization. We check for overlaps (as seen in panels
(b,d) of Figure 3) of $\omega_{\text{obs}}$ (blue/red shaded region) with
$\omega_{sc}$ (blue/red, solid and dashed curves) for all the Doppler shift
cases and consider damping rates corresponding to the overlapping
$\omega_{sc}$. For a negative Doppler shifted FMW (like the red dashed curve
in panel (d) of Figure 3), there could be two bands of $\omega_{sc}$
overlapping with $\omega_{\text{obs}}$ which are named with suffixes
${}_{1},_{2}$ in the rest of this paper (e.g. Figure 6). To estimate energy
dissipation rates in such cases, we identify the band of $\omega_{sc}$ that
encompasses or is nearest to $k_{\parallel}\rho_{p}=0.3$ as the relevant
region and use the corresponding damping rates. This selection criterion is
justified as the most likely wavevector region that supports the emission of
FMWs from the parallel firehose instability (Gary, 1993; Verscharen et al.,
2019) is typically around $k_{\parallel}\rho_{p}\sim 0.3$. The PLUME damping
rates, $\gamma_{p,e\\_PLUME}(\omega_{sc})$ are then linearly interpolated at
$\omega_{\text{obs}}$ when the overlaps occur. Because heavily damped waves
cannot propagate, PLUME dispersion solutions with $\omega\leq e\times\gamma$
are discarded, where $e=2.718$. As a result of this selection, occasionally,
$\gamma_{p,e\\_PLUME}$ is unavailable at higher frequencies of
$\omega_{\text{obs}}$. We extend the higher end value of
$\gamma_{p,e\\_PLUME}$ to these frequencies. Panels (d) and (e) of Figure 2
show $\gamma_{p,e\\_PLUME}(\omega_{\text{obs}})$ corresponding to all Doppler
shift cases where the overlaps occur.

Figure 4 shows $\gamma_{p,e\\_PLUME}$ as a function of $f_{\text{obs}}$, the
frequency bands where ion-scale waves are observed in Encounters 1 and 2 and
heliocentric distance for the positive and negative Doppler-shifted ICW cases.
The $\gamma_{p,e\\_PLUME}$ values corresponding to all the FMW Doppler shift
cases are 6 orders of magnitude lower than the $\gamma_{p,e\\_PLUME}$ values
shown in Figure 4. The insignificance of FMW cyclotron damping rates is
consistent with the resonance between FMW’s rotating electric field and
gyrating electrons being negligible at proton scales where the waves are
observed and are assumed to dissipate. We see a considerable fraction of
intervals with unstable waves, where growth rates are comparable to damping
rates. However, the supported region of unstable modes occurs over a narrower
frequency band compared to $f_{obs}$, which can lead to a net damping of
cyclotron waves rather than emission, as discussed in Sec. 5.

###  2.5 Evaluating Energy Dissipation Rates onto Protons and Electrons

We transform the observed solar wind magnetic field fluctuations to velocity
units ([m/s]) and evaluate the energy spectra of observed ion-scale waves in
SI units as

$|\tilde{B}(f)|_{wave\\_psp}^{2}[\text{m}^{2}\text{s}^{-2}\text{Hz}^{-1}]=\frac{|\tilde{B}(f)|_{wave\\_psp}^{2}[\text{T}^{2}\text{Hz}^{-1}]}{\mu_{0}m_{p}n_{p}[\text{m}^{-3}]}.$
(5)

In each interval where ion-scale waves are observed, the dissipation rates are
estimated as the sum over the product of $\gamma_{p,e}$ and the magnetic power
spectrum over the identified frequency range

$\begin{split}Q_{p,e}[\text{m}^{2}\text{s}^{-3}]=&-\sum\left(|\tilde{B}(\omega_{\text{obs}})|^{2^{RH/LH}}_{wave\\_psp}[\text{m}^{2}\text{s}^{-2}\text{Hz}^{-1}]\times\right.\\\
&\left.\Delta
f(\omega_{\text{obs}})[\text{Hz}]\times\gamma_{p,e\\_PLUME}(\omega_{\text{obs}})[\text{Hz}]\vphantom{|^{2}_{wave\\_psp}}\right),\end{split}$
(6)

where $\Delta f$ is the array of widths of logarithmic frequency bins at which
energy spectra are evaluated, and $Q_{p,e}$ is in units of energy rate per
unit mass. This method of estimating heating rates is similar to the method
employed in SKM23.

##  3 Occurrence of ion-scale waves in the young solar wind

Figure 5: The moving averages of: panel (a) the fraction of observation time
during which LH (red), RH (blue) polarized waves, and either/both of them
(green) are shown as a function of heliocentric distance along with their
means over all E1 and E2 intervals (dashed). Panel (b) the energy in LH (red),
RH (blue) polarized waves, and their sum (green) are plotted as a function of
heliocentric distance.

In all intervals where ion-scale waves are observed, we estimate their
duration, $T_{wave}$, by evaluating the total time length of the
$(f-t)_{coherent}$ domain, the region where a high spacecraft frame
polarization of $|\sigma|\geq 0.7$ is observed at at least one frequency in
$f_{\text{log}\\_\text{bin}}$ after time averaging the spacecraft frame
polarization spectrum, $\sigma(s,t)$ as described in subsection 2.3. We then
evaluate the percentage of interval observational time (i.e. $\sim$ 15
minutes) during which waves are observed (their moving averages shown in panel
(a) of Figure 5). We find that persistent ion-scale waves are present during
20.37$\%$ of Encounters 1 and 2 observations, of which LH polarized (in the
spacecraft frame) waves are dominant and are observed during 85.86$\%$ of the
time (17.49$\%$ of total observational time), and RH waves are observed during
15.8$\%$ of the time (3.22$\%$ of total observational time). Both the RH and
LH waves are observed simultaneously during 0.34% of total observational time.
The dominance of LH waves is consistent with Bowen et al. (2020b) and Liu et
al. (2023). We further examine the dependence of wave detection on the angle
between B and $\mathbf{V}_{\text{SW}}$ by subdividing Encounters 1 and 2
magnetic field observations into $\sim$0.873 second intervals (corresponding
to the cadence of SPC measurements) and evaluating the acute angle between B
and $\mathbf{V}_{\text{SW}}$, $\theta_{BV_{\text{SW}}}$. We find that the
first, median, and third quartiles of the $\theta_{BV_{\text{SW}}}$
distribution in intervals with waves are 17.3, 25.4, and 34.4, respectively,
demonstrating that (consistent with Bowen et al. (2020b) and Woodham et al.
(2021)), ion-scale waves are preferentially observed when B and
$\mathbf{V}_{\text{SW}}$ are aligned. When $\theta_{BV_{\text{SW}}}$ is
oblique, the wave occurrence is undetermined and the wave signatures could be
dominated by anisotropic turbulence in which case the percentages reported
above would be a lower bound on the ion-scale wave occurrences.

In all intervals, we further evaluate the energy in ion-scale waves,
$E_{\text{wave}}^{RH/LH}=\sum\left(|\tilde{B}(\omega_{\text{obs}})|^{2^{RH/LH}}_{wave\\_psp}\Delta
f(\omega_{\text{obs}})\right)$ (shown in panel (b) of Figure 5). We find that
LH (RH) polarized waves constitute 90.05$\%$ (9.95$\%$) of the total energy in
ion-scale waves in Encounters 1 and 2 observations.

Note that ion-scale waves are observed during a higher fraction i.e. 29.27% of
observational time in intervals that were discarded due to the unavailability
of SWEAP observations or temperature anisotropy estimations. However, the
statistics of ion-scale wave duration and energy remain similar to Figure 5
upon inclusion of the discarded intervals.

###  3.1 Observed ion-scale waves are consistent with transient ICWs

Figure 6: The plasma-frame frequency ranges corresponding to the spacecraft
frame frequencies at which waves are observed
($f_{\text{plasma}}^{\text{wave}}$, red shaded), the frequencies in the plasma
frame where ICWs/FMWs grow due to proton temperature anisotropy instabilities
($f_{PLUME}^{wave\\_growth}$, blue shaded), and the local proton cyclotron
frequency ($f_{cp}$, green dotted) as functions of heliocentric distance are
shown. Figure 7: The ratio of the decay time, $\tau_{\text{decay}}$ to the
nonlinear turbulent timescale at proton scale, $\tau_{nl}$ corresponding to
the positive and negative Doppler-shifted ICW cases.

Using PLUME solutions ($f_{PLUME}=\frac{\omega_{PLUME}}{2\pi}$) we evaluated
in all the intervals where ion-scale waves are observed, the plasma-frame
frequency ranges corresponding to $f_{\text{obs}}^{RH/LH}$, the spacecraft
frame frequencies at which the waves are observed for all Doppler shift cases,

$\begin{split}f_{\text{plasma}}^{\text{wave}}=&\left[\text{min}(f_{\text{plasma}}^{\text{wave},RH}\cup
f_{\text{plasma}}^{\text{wave},LH})\right.,\\\
&\left.\text{max}(f_{\text{plasma}}^{\text{wave},RH}\cup
f_{\text{plasma}}^{\text{wave},LH})\right].\end{split}$ (7)

Temperature anisotropy instabilities (Gary, 1993) play a major role in
constraining the temperature anisotropies of solar wind plasma species
(Hellinger et al., 2006; Bale et al., 2009; Huang et al., 2020). The ion
cyclotron (occurs when $\frac{T_{\perp,p}}{T_{\parallel,p}}>1$) and parallel
firehose (occurs when $\frac{T_{\perp,p}}{T_{\parallel,p}}<1$) instabilities
generate ICWs and FMWs, respectively. We find that local (i.e. 15-minute
interval-averaged) plasma parameters satisfy the linear instability criteria
in 38% (3.3%) of the analyzed intervals for the ion cyclotron (parallel
firehose) instability. In intervals where $\gamma_{PLUME}>0$, we evaluated
$f_{PLUME}^{wave\\_growth}$, the frequencies in the plasma frame where
ICWs/FMWs are unstable and grow. Figure 6 shows
$f_{\text{plasma}}^{\text{wave}}$ (red-shaded), $f_{PLUME}^{wave\\_growth}$
(blue-shaded), and the local $f_{cp}$ (green) as functions of heliocentric
distance. For ICWs (panels (a) and (d)), $f_{\text{plasma}}^{\text{wave}}$
overlaps well with $f_{PLUME}^{wave\\_growth}$ and both the frequency ranges
scale with the local $f_{cp}$. However, for FMWs (panels (b), (c), (e), and
(f)), $f_{\text{plasma}}^{\text{wave}}$ does not overlap with
$f_{PLUME}^{wave\\_growth}$ band that is determined by either local plasma
parameter values or by the values of plasma parameters elsewhere in Encounters
1 and 2. This analysis suggests that the observed ion-scale waves are locally
generated ICWs, consistent with Bowen et al. (2020). Bowen et al. (2020) have
found spacecraft-frame RH waves to be consistent with outward propagating
FMWs. These FMWs could be driven by a proton beam instability, which we expect
to be driven at frequencies consistent with the observed waves. The effects of
secondary populations will be considered in future work.

In all intervals where ion-scale waves are observed, we evaluated
$\gamma^{\text{max}\\_\text{wave}}$, the damping rate corresponding to the
frequency in $f_{obs}$ at which the fraction of wave energy is maximum. In all
intervals where $\gamma^{\text{max}\\_\text{wave}}<0$, that is the observed
waves damp at the frequency where their energy is maximum, we estimated the
timescales over which the waves decay by evaluating
$\tau_{\text{decay}}=\frac{1}{\gamma^{\text{max}\\_\text{wave}}}$. We then
evaluated the turbulent non-linear timescale
$\tau_{nl}=\frac{k_{\perp}^{-2/3}k_{0}^{-1/3}}{v_{A}}$ at proton scales
($k_{\perp}\rho_{p}\sim 1$) by assuming critical balance (Goldreich & Sridhar
(1995)). Here $k_{0}=\frac{2\pi f_{\text{break}}}{V_{\text{SW}}}$ is the outer
scale evaluated by subdividing Encounter 1 and 2 observations into 6-hour
intervals and then calculating $f_{\text{break}}$, the spectral break
frequency at which the transition from injection to inertial scales occurs.
Figure 7 shows the ratio $\frac{\tau_{\text{decay}}}{\tau_{nl}}$ for all the
positive and negative Doppler shifted ICW cases. For ICWs
$\tau_{\text{decay}}$ and $\tau_{nl}$ are comparable and both have median
values of 2-3 seconds. We further estimated the distance,
$l_{prop}=\tau_{\text{decay}}v_{\text{phase}}^{\text{max}\\_\text{wave}}$ that
the observed waves propagate during the time their amplitudes damp by a factor
of e = 2.718. Here $v_{\text{phase}}^{\text{max}\\_\text{wave}}$ is the phase
velocity at the frequency where the wave energy fraction is maximum. We find
that the median value of $l_{prop}/\rho_{p}$ is $\sim 9$. Hence these ICWs are
local and very short-lived.

Figure 8: The PLUME damping (solid)/growth (dashed) rate as functions of
parallel wavevector corresponding to
$(\beta_{\parallel,p},\frac{T_{\perp,p}}{T_{\parallel,p}})$ values of 1-minute
intervals (shades of blue) along with the damping/growth rates corresponding
to the 15-minute averaged plasma parameters (red) for two intervals where the
average parameters are stable (a) or unstable (b).

These waves are short-lived compared to both the cadence of the temperature
anisotropy estimations used in this work (1 minute estimations from Huang et
al., 2020) and the time length of the intervals considered in this analysis
(15 minutes), and could be driven unstable by short intervals of temperature
anisotropy beyond the appropriate threshold. The width of the observed wave
frequencies in an interval could be a result of the aggregation of distinct
waves from different regions with different plasma parameters and solar wind
velocities–and hence Doppler shifts– compared to their interval-averaged
values. The observed transient ICWs can be intermittently generated in
unstable regions (Qudsi et al., 2020) and damp at nearby stable regions,
mediating the local spatial transport of free energy associated with
temperature anisotropy. To demonstrate this, we illustrate in Fig. 8 the ICW
growth and damping rates calculated using both the 15 minute average plasma
parameters compared against rates determined from the 1-minute values. For
both intervals, the one-minute averages demonstrate some stable and unstable
behavior, even though the 15 minute averages are either stable or unstable.
Across all wavevectors, we see the damping rates for the stable waves
determined by the 15-minute average are qualitatively similar to the 1-minute
values, suggesting that the assumption that the waves damp at rates associated
with the plasma parameters from the 15 minute averages is a reasonable
statistical estimation. Therefore, we argue that the estimates of cyclotron
damping rates in stable regions as well as the estimates of
$f_{PLUME}^{wave\\_growth}$, $\tau_{\text{decay}}$, and $l_{prop}$ that are
evaluated using interval averaged temperature anisotropy values are
reasonable.

Higher cadence temperature anisotropy observations (e.g. at the 7s cadence in
Verniero et al. (2020) and McManus et al. (2024)) could be used to better
analyze the correlation between observed temperature anisotropies and waves
and to determine the specific instabilities that could drive the observed
waves and obtain a more precise, localized measurement of the wave-particle
interaction.

##  4 Significant ICW dissipation onto young solar wind protons

Figure 9: Estimated dissipation rates of observed ion-scale waves via
cyclotron damping (this work) and turbulence via Landau damping (SKM23) in all
the analyzed intervals from PSP Encounters 1 and 2. The dissipation rates of
positive (red) and negative (blue) Doppler shifted ICWs onto protons, the
turbulent dissipation rates onto protons via Landau damping (green), and the
inertial scale energy cascade rates evaluated using PSP observations by
Bandyopadhyay et al. (2020) (blue star) are shown. Panel (a) shows moving
means of all $Q_{p}$ values and Panel (b) shows moving medians of net damped
intervals $Q_{p}<0$. Here dots (circles) represent magnitudes of energy
damping (growing) rates i.e $-Q_{p}$ ($Q_{p}$).

Upon evaluating dissipation rates of ion-scale waves by employing the routine
described in section 2 for all intervals from PSP Encounters 1 and 2 between
0.166 and 0.25 au that were analyzed in SKM23, we find that ICWs can dissipate
onto protons at significant rates that could explain the anisotropic heating
of protons which enhances at closer distances to the Sun (Hellinger et al.,
2011, 2013). Figure 9 shows the dissipation rates onto protons, $Q_{p}$ by
cyclotron damping of ion-scale wave estimated in this work and Landau damping
($Q_{p,Landau}$, green) of turbulence estimated in SKM23. Here dots represent
net damping rates and open circles represent net growth rates. Panels (a,b)
show $Q_{p}$ via cyclotron damping of positive (red) and negative (blue)
Doppler-shifted ICWs and the corresponding moving averages/medians (solid
lines). The mean $Q_{p}$ values via cyclotron damping are usually an order of
magnitude higher than $Q_{p}$ values via Landau damping and are in good
agreement with the inertial scale energy cascade rates estimated at 36 and 54
solar radii by Bandyopadhyay et al. (2020) using PSP observations (blue
stars). That the cyclotron damping rate is similar to the turbulent cascade
rate is consistent with Bowen et al. (2024), who show that ion-scale waves are
correlated with signatures of sub-ion-scale turbulence and suggest that the
waves play an active role in dissipating turbulence. Note that for $Q_{p}$ via
cyclotron resonance the mean is an order of magnitude higher than the median
while for $Q_{p}$ via Landau resonance the mean and median are similar. We
infer that the proton cyclotron damping is intermittent and bursty, while the
proton Landau damping is constant and continuous. We find that in 11.86% of
the intervals where waves are observed, there is a net ICW energy growth rate
(open circles in panel (a)). This fraction is lower than the fraction of
intervals (38%) where the interval averaged values of
$\frac{T_{\perp,p}}{T_{\parallel,p}}$ determine a growing wave at any
wavevector. This discrepancy arises because ion-scale waves are observed over
a wide effective $k_{\parallel}$ range, while the instability due to
temperature anisotropy arise over a narrower $k_{\parallel}$ band (as seen in
Figure 4). Integrating the growth and damping rates over the observed scales
often results in a net damping rate.

We define a total possible heating rate due to cyclotron damping onto protons,
$Q_{p,cyc}$, as the sum of the red and blue dots in panel (a). In intervals
where both LH and RH polarized waves are observed, two distinct oppositely
propagating ICWs can be positive and negatively Doppler shifted to
independently produce both the observed polarizations, and there is no double
counting in the evaluation of $Q_{p,cyc}$ by adding heating rates of both
polarizations.

Moreover, we find that consistent with expectation, the possible dissipation
of FMWs onto electrons at the observed scales is insignificant compared to
$Q_{p,cyc}$ as well as $Q_{e,Landau}$, the turbulence dissipation rates onto
electrons via Landau damping that were estimated in SKM23.

It is important to note that in this work we have assumed the proton velocity
distribution to be a bi-Maxwellian. However, the observed proton velocity
distributions have beams. While accounting for the beams in the fits of VDFs
would enhance the proton core temperature anisotropy estimates, (Huang et al.,
2020; Halekas et al., 2020) the presence of beams can drive instabilities
(Montgomery et al., 1975; Klein et al., 2018; Martinović et al., 2021) that
are not considered in this work. We subsequently do not consider the
interaction of proton beams with FMWs. However, FMW-ion interaction is
relevant to the pitch angle diffusion of the ion beams (Verniero et al., 2022;
Martinović & Klein, 2023). Accounting for beams in this kind of analysis is a
topic for future work.

Figure 10: Dominant heating of protons: (a) $Q_{p}/(Q_{p}+Q_{e})$ evaluated by
calculating the moving averages of $Q_{p}$ and $Q_{e}$ by considering proton
heating via Landau resonance (light green dashed), cyclotron resonance (medium
green dash-dotted) and both the resonances (dark green solid) in intervals
where waves are observed are shown (green shaded in Table 2). Further
$Q_{p}/(Q_{p}+Q_{e})$ evaluated due to Landau resonance alone (red) in
intervals where waves are not observed (red shaded in Table 2) as functions of
heliocentric distance are shown. The fit of $Q_{p}/(Q_{p}+Q_{e})$ along with
error bars estimated by Bandyopadhyay et al. (2023) is shown (cyan) for
comparison. Additionally, the moving medians of (b) wave energy and (c)
$\beta_{p}$ as functions of heliocentric distance are shown.

###  4.1 $Q_{p}/(Q_{p}+Q_{e})$ consistent with the dominant proton heating
that is observed near the Sun

Because cyclotron damping onto electrons is negligible, we evaluate
$Q_{p}/(Q_{p}+Q_{e})$ by considering just $Q_{e,Landau}$ for electrons while
we consider both $Q_{p,cyc}$ and $Q_{p,Landau}$ for protons. Moreover the
estimations of $Q_{p/e,Landau}$ and $Q_{p,cyc}$ are possible in intervals
where the cascade model employed in SKM23 is accurate and waves are present,
respectively. Table 2 shows the percentages of the analyzed intervals where
$Q_{p/e,Landau}$ and $Q_{p,cyc}$ estimations are available.

|  | Landau damping model
---|---|---
| accurate | inaccurate
Waves |  present | 26.54 | 46.06
absent/no overlap | 16.04 | 11.36
Table 2: Percentages of intervals where Landau and cyclotron damping rates are
available. Landau damping rates are available when the cascade model employed
in SKM23 is accurate. Cyclotron damping rates are available when waves are
observed and the frequencies at which they are observed, $\omega_{\text{obs}}$
overlap with the Doppler shifted PLUME frequencies, $\omega_{sc}$.

We consider intervals where both $Q_{p/e,Landau}$ and $Q_{p,cyc}$ available
(i.e. 26.54% of intervals, green shaded in Table 2) and evaluate moving
averages of $Q_{p}$ and $Q_{e}$ and then evaluate $Q_{p}/(Q_{p}+Q_{e})$ (shown
in panel (a) of Figure 10) corresponding to $Q_{p,cyc}$ (medium green dash-
dotted), $Q_{p,Landau}$ (light green dashed), and their sum (dark green solid)
as functions of heliocentric distance. We infer that cyclotron damping greatly
enhances the dissipation onto protons, producing much higher rates than those
possible by Landau damping alone. By using PSP observations, Bandyopadhyay et
al. (2023) have extended to near Sun distances the $Q_{p}/(Q_{p}+Q_{e})$
estimation made by Cranmer et al. (2009) using Ulysses and Helios
observations. They estimate that protons receive $\sim$ 80% of plasma heating
(cyan in panel (a) of Figure 10). Our estimates of $Q_{p}/(Q_{p}+Q_{e})$ (dark
green) are comparable to theirs except in a narrow region between 0.16 and
0.18 au. We hypothesize that this inconsistency occurs because this region has
both extremely low $\beta_{p}$ for E1 and E2 observations (see panel (c)),
thus inhibiting the proton Landau damping as the protons don’t have large
enough thermal velocities for a significant population to undergo Landau
resonance with kinetic Alfvén waves, as well as relatively low wave energy
(panel (b)), suppressing the cyclotron heating leading to inadequate proton
heating via both mechanisms included in our model. These short regions of
low-$\beta$ values are likely statistically anomalous, and we expect the
plasma parameters to revert to a mean radial distribution (Maruca et al.,
2023) with the inclusion of data from additional orbits. We expect this
reversion to produce proton heating rates comparable to the elevated levels in
nearby distances. This extension will be considered in future work.

In 46.06% of intervals, ion-scale waves are observed but the cascade model
employed to estimate $Q_{e,Landau}$ is inaccurate. Note that such intervals
constitute $\sim$two-thirds of the intervals where waves are observed and
$Q_{p,cyc}$ in these intervals have not been considered in calculating dark
solid and medium green dash-dotted curves in Figure 10. The ratio of the
average of the $Q_{p,cyc}$ values when the Landau damping model is inaccurate
compared to the average of the $Q_{p,cyc}$ values when the Landau damping
model is accurate is 2.4, suggesting that the ratios involving $Q_{p,Cyc}$ in
Fig. 10 are a lower bound on an estimated $Q_{p}/(Q_{p}+Q_{e})$. In 16.1% of
analyzed intervals, there is no overlap between $\omega_{\text{obs}}$ and
$\omega_{sc}$ which occurs because of discarding heavily damped PLUME
solutions.

##  5 Discussion

We identify and analyze the abundantly observed ion-scale waves in Encounters
1 and 2 of PSP measurements and account for their dissipation via cyclotron
damping. Along with our previous work, which accounts for the dissipation of
turbulence via Landau damping, our findings help to better quantify the
mechanisms that heat the young solar wind as it expands.

We find that ion-scale waves are observed over 20.37$\%$ of PSP Encounters 1
and 2 observational time of which the LH polarized (in spacecraft frame) waves
are dominant constituting 85.86$\%$ of the time during which waves are
observed. The plasma-frame frequencies corresponding to the observed waves
indicate that the waves are locally generated ICWs. We further infer that
these ICWs are transient with decay times comparable to the turbulent non-
linear interaction times at proton scales. ICWs among the observed waves could
thus mediate the local transport of free energy associated with temperature
anisotropy. We find pronounced dissipation of ICWs onto protons that could
account for the enhanced anisotropic heating of protons near the Sun and the
estimated rates of dissipation are in good agreement with empirical estimates
of turbulent energy cascade rates by Bandyopadhyay et al. (2020). We further
infer dominant proton heating and find the ratio of proton-to-total heating
estimated in this work comparable to the estimate by Bandyopadhyay et al.
(2023).

Intermittent ion-scale instabilities are a likely mechanism to generate ICWs.
We find a non-linear correlation of wave occurrence ($T_{wave}$) and
$\frac{T_{\perp,p}}{T_{\parallel,p}}$ (Spearman correlation coeffecient =
0.37). As the waves are short-lived, these 15-minute averaged
$\frac{T_{\perp,p}}{T_{\parallel,p}}$ values may not well represent the
anisotropy of the shorter intervals where the waves would be generated,
preventing a conclusive inference from this correlation.

Spectral steepening is often observed in Encounters 1 and 2 (Bowen et al.
(2020a); Shankarappa et al. (2023)) at ion gyroscales. We find a strong
correlation of $T_{wave}$ with the steepening of the energy spectrum. Bowen et
al. 2024 (in review) find that this steepening is strongly associated with LH
ion-scale waves. Furthermore, Squire et al. (2022) have argued that the
imbalanced turbulent flux creates a “helicity barrier” at ion gyroscales,
inhibiting further cascade to smaller $k_{\perp}$, leading to an inverse
cascade for $k_{\perp}>1$ channeling the turbulent energy to oblique ICWs that
dissipate onto protons leading to a steeper energy spectrum near the proton
gyroscale. Although we do not find a notable correlation between $T_{wave}$
and cross-helicity extracted from our 15 minute intervals, this is an area of
future research.

A significant caveat in this work is the assumption that bi-Maxwellians well
represent the proton and electron VDFs. The presence of proton beams can drive
instabilities which is unaccounted for in this work. Moreover, the structure
of VDFs can differ significantly from Maxwellian (Marsch et al., 1982; Bowen
et al., 2022; Walters et al., 2023); consideration of more accurate solar wind
particle VDFs and their impact on the dissipation and emission of waves is a
topic for future work. Additionally, we have not taken into account the
possibility of proton velocity distributions reaching marginal stability and
thus being unable to further exchange energy with the ion-scale waves
(Isenberg et al., 2000).

KGK and NS were supported by NASA Grants 80NSSC19K0912 and 80NSSC20K0521. MMM
and KGK were supported by NASA grant 80NSSC22K1011. TAB acknowledges support
from NASA via grant 80NSSC24K0272. Parker Solar Probe was designed, built, and
is now operated by the Johns Hopkins Applied Physics Laboratory as part of
NASA’s Living with a Star (LWS) program (contract NNN06AA01C). Support from
the LWS management and technical team has played a critical role in the
success of the Parker Solar Probe mission.

##  Appendix A Identifying $\sigma(s,t)$ of Ion-scale Waves by Removing non-
wave signatures

We identify ion-scale waves from turbulence due to the persistent high values
of spacecraft frame polarization, $\sigma$ of the former as compared to the
random short-lived low $\sigma$ values due to the latter. Turbulent
fluctuations can often have random high values of $\sigma$ and can lead to
their erroneous identification as waves when $\sigma$ is used as a criterion.
In order to remove these short-lived random contributions to the $\sigma$ and
retain only the persistent high-polarized bands, we perform a moving window
time averaging on $\sigma(s,t)$ with a window width of
$20\times\frac{1}{\text{frequency}}$ at all $f_{\text{log\\_bin}}$ frequencies
ranging from 0.005 Hz to 10 Hz. The averaging factor of 20 is empirically
determined by identifying the minimum window width required to reduce
magnitudes of averaged $|\sigma(s,t)|$ to less than 0.7 at all times and
frequencies in intervals where ion-scale waves are not observed. After time
averaging, we identify domains in frequency-time space where
$|\sigma(s,t)|\geq 0.7$ (the choice of the threshold value of 0.7 is discussed
in Appendix B.4. of SKM23), $(f-t)_{coherent}$, which we attribute to the
presence of coherent waves (depicted in panel (b) of Figure 2). The waves are
often observed over a wide range of frequencies.

Furthermore, in SKM23, each analyzed 15-minute interval was extended by
considering 7.5-minute peripheral intervals on either end to limit finite-
length errors (see Appendix B.3.2 of SKM23). We employed a method to remove
reaction wheel noise (see Appendix B.1 of SKM23) that was based on the
relative magnitudes of energies in the central 15-minute interval and the
peripheral 7.5-minute intervals. The noise-removal method chooses between the
following two routines: removing the noise in the whole extended interval, or
removing the noise in central and peripheral intervals separately and then
joining them. This method is not effective in a handful of intervals where the
energies of central and peripheral intervals are similar. In many intervals, a
narrow frequency band, $f_{noise}$, with high $\sigma$ values corresponding to
the remnant noise, persists across a region in $(f-t)_{coherent}$ despite time
averaging. The frequencies of $f_{noise}$ are usually higher than and thus do
not overlap with, the ion-scale wave frequencies and have lower amplitude
values of their $\sigma$ compared to the latter. We identify $f_{noise}$ as
the frequencies in $(f-t)_{coherent}$ with the time-averaged spacecraft frame
polarization value, $\left<|\sigma(f,t)|\right>_{t}$ is less than an empirical
threshold of 0.0078. We can then remove $f_{noise}$ and retain noise-removed
$(f-t)_{coherent}$. The noise-frequency filtering additionally removes the
edge frequencies in $(f-t)_{coherent}$ that are far from the core region of
ion-scale wave frequencies. However, we have found that the energies in these
edge frequencies are negligible and don’t affect our analysis.

We consider the wavelet energy spectrum (evaluated as in SKM23, see Appendix
B.3) corresponding to RH and LH polarization at $(f-t)_{coherent}$ and sum
these domains over time to evaluate $|\tilde{B}(f)|^{2^{LH/RH}}_{wave\\_psp}$
(shown in panel (c) of Figure 2). We consider the frequency range of
$(f-t)_{coherent}$ domain as the frequencies where waves are observed in the
spacecraft frame, $f^{(RH/LH)}_{\text{obs}}$, and evaluate the corresponding
angular frequencies, $\omega^{(RH/LH)}_{\text{obs}}$. Note that choosing a
$\sigma$ threshold value of > 0.7 in Section 2.3 causes an insignificant
difference in the evaluation of $f_{obs}^{LH/RH}$.

##  Appendix B Identification of modes in PLUME

We accurately identify the four modes and distinguish the Alfvén ICW from FMW
by applying the following criteria in succession,

  1. 1.

For a plasma with no drifting components, the entropy mode is non-propagating
at large scales, i.e. $\omega=0$ for $k_{\parallel}\rho_{p}\ll 1$, allowing
immediate identification of the entropy mode.

  2. 2.

For the three propagating modes, the variation in the parallel group velocity,
$v_{g,\parallel}=\frac{\partial\omega}{\partial k_{\parallel}}$ offers a
measure to distinguish between solutions. The values of $v_{g,\parallel}$
increase, decrease, and remain roughly constant for the fast, Alfvén, and slow
waves, respectively with increasing $k_{\parallel}$; see Equations 8, 14, 16,
and 18 in Howes et al. (2014) for analytic approximations of $\omega(k)$ for
Alfvén and FMWs at inertial and parallel kinetic scales. At small parallel
scales, the ICWs damp heavily onto ions, resulting in a decrease in
$v_{g,\parallel}$, while the FMWs don’t damp onto ions; their frequency scales
as $\propto kk_{\parallel}$ resulting in an increase in $v_{g,\parallel}$. We
calculate $\Delta v_{g,\parallel}$, the change in the parallel group velocity
between $k_{\parallel}\rho_{p}=0.003$ and $k_{\parallel}\rho_{p}=1$ for the
three modes. The mode with the maximum positive $\Delta v_{g,\parallel}$ value
is identified as the FMW, and the one with maximum negative
$\Delta\frac{\partial\omega}{\partial k_{\parallel}}$ value is identified as
the Alfvén mode.

  3. 3.

To verify the above identification, we utilize two additional wave metrics,
magnetic helicity (see Equation C1 in the following section) which quantifies
the helicity of the wave mode, and the ratio of magnitudes of parallel
electric field to total electric field, $\frac{E_{z}}{E}$. The forward
propagating FMW and Alfvén waves have a constant $\sigma_{m}$ value of +1 and
-1, respectively for $k_{\parallel}\rho_{p}>0.01$, while the slow wave has
$\sigma_{m}$ values varying between +1 and -1. Only the slow mode has a
$\frac{E_{z}}{E}$ value of $\approx 1$ at larger parallel scales, while both
Alfvén and FMWs have a value of $\approx 0$ for the considered range of
wavevectors. We verify that the identified FMW and Alfvén waves have expected
magnitudes of both the wave metrics at two values of parallel scales,
$k_{\parallel}\rho_{p}=0.1$ and $k_{\parallel}\rho_{p}=2$, further ensuring
that there is no numerical mode interchange near $k_{\parallel}\sim\rho_{p}$.

##  Appendix C Quantifying circular Polarization of waves

ICWs and FMWs are circularly polarized at scales of order $\rho_{p}$ and in
their plasma dispersion relations,
$\left(\frac{k}{\omega}\right)^{2}_{\text{ICW},\text{FMW}}$ is a function of
$\left(\frac{1}{\omega-\mathbf{k}\cdot\mathbf{v}-\Omega_{p,e}}\right)$, where
$\Omega_{p,e}=\frac{eB}{m_{p,e}}$ is the proton (electron) cyclotron
frequency. Hence, The ICWs (FMWs) propagating both along and opposite to B
resonate with protons (electrons) in the plasma rest frame when beams are not
considered. Note that the handedness of light has conventionally been defined
with respect to the observer’s point of view (i.e. source or receiver). A
rotating magnetic field of a wave can appear clockwise (anti-clockwise) when
viewed from the source (receiver) point of view. However, in this work
(following Bowen et al. (2020b)), we avoid this ambiguity by considering
proton (electron)-resonant polarization to be left (right)-handed irrespective
of the observer’s point of view. Hence, ICWs (FMWs) propagating both along and
opposite to B are left (right)-handed i.e. LH (RH).

Two interrelated metrics quantify circular polarization in waves: magnetic
helicity ($\sigma_{m}$) and polarization/reduced helicity spectrum ($\sigma$).

  * •

Magnetic helicity, $\sigma_{m}=\int d^{3}x\mathbf{A}\cdot\mathbf{B}$ is an
invariant of ideal MHD equations, where A is the vector potential of B.
$\sigma_{m}$ is a good measure of circular polarization when B is wavelike. In
the wavevector space, normalized magnetic helicity for $k\eqsim k_{z}$ is
written as, (Matthaeus & Goldstein, 1982; Howes & Quataert, 2010),

$\sigma_{m}(k)=\frac{k}{|B(\mathbf{k})|^{2}}i\left(\frac{B_{x}(k)B^{*}_{y}(k)-B^{*}_{x}(k)B_{y}(k)}{k_{z}}\right)=\frac{-2\text{Im}\left[B_{x}(k)B_{y}^{*}(k)\right]}{|B_{x}(k)|^{2}+|B_{y}(k)|^{2}}..$
(C1)

Here $B_{x,y}(k)$ are spatial Fourier transforms of fluctuations of the
magnetic field, $\delta B$.

  * •

Polarization, $\sigma$ is the ratio of third-to-zeroth Stokes parameters and
is written as (Bowen et al., 2020b),

$\sigma(f)=\dfrac{-2\text{Im}\big{[}B_{x}(f)B_{y}^{*}(f)\big{]}}{|B_{x}(f)|^{2}+|B_{y}(f)|^{2}}\
\ \ \ \ \text{or}\ \ \ \ \
\sigma(s,t)=\dfrac{-2\text{Im}\big{[}W_{x}(s,t)W_{y}^{*}(s,t)\big{]}}{|W_{x}(s,t)|^{2}+|W_{y}(s,t)|^{2}}.$
(C2)

Here $B(f)/W_{x,y}(s,t)$ are temporal Fourier/wavelet transforms of $\delta
B$. Polarization, effectively is the reduced magnetic helicity spectrum along
a single direction (Matthaeus & Goldstein, 1982). We can measure only
polarization and not magnetic helicity with single spacecraft measurements of
magnetic field observations.

In the plasma frame, magnetic helicity quantifies the rotation of B with
respect to space at a fixed time while polarization quantifies the rotation of
B with respect to time at a fixed spatial point (Narita et al., 2009).
Magnetic helicity is defined with respect to the wavevector, k, while
polarization is not. Hence, in the plasma rest frame, magnetic helicity
(polarization) has the opposite (same) sign for counter-propagating waves with
the same intrinsic polarization. This difference is depicted in Figure 11
which is a variation of Figure 1 from Narita et al. (2009) (where a RH wave is
depicted, as opposed to the LH wave in our example). Panels (a1) and (b1) show
forward and backward propagating example ICWs in the plasma rest frame where
black arrows show the direction of wave propagation. Here XYZ is a right
coordinate system where the mean magnetic field, $\mathbf{B_{o}}$ is in the Z
direction. The position of $\mathbf{\delta
B_{\perp}}=\mathbf{B_{x}}+\mathbf{B_{y}}$ in the plane perpendicular to
$\mathbf{B_{o}}$ is shown as colored arrows at 5 consecutive points along Z.
When $\mathbf{\delta B_{\perp}}$ points towards X or Y directions, the arrows
are colored red or green, respectively.

Figure 11: Spatial and temporal behavior of magnetic field components in the
plane perpendicular to the direction of propagation of circularly polarized
counter-propagating ICWs in the plasma rest frame.

When the $\mathbf{\delta B_{\perp}}$ of waves is observed with respect to time
at a fixed spatial point, the observer sees it in the order t1-t5 for both
cases and the rotation of $\mathbf{B_{\perp}}$ (as shown in panels (a4,b4)) is
observed to be proton-resonant (LH) for both the waves. The corresponding
oscillations of $B_{x}$ and $B_{y}$ with respect to time (shown in panels
(a3,b3)) are identical which is consistent with the sign of polarization for
both the forward and backward ICWs being the same. On the other hand, panels
(a2,b2) show the oscillations of $B_{x}$ and $B_{y}$ with respect to space (z)
at a fixed time, and the variation of $B_{y}$ relative to $B_{x}$ is inverted
for the forward and backward waves which is consistent with the sign of
magnetic helicity being reversed.

The expressions for magnetic helicity (polarization) (Equations C1 and C2)
evaluate the phase difference between $B_{x}$ and $B_{y}$ corresponding to
their spatial (temporal) variations. This can be shown by expressing the
oscillating magnetic field of forward/backward propagating electromagnetic
wave as complex numbers $B_{x,y}\propto e^{i(kz\mp\omega t+\theta_{x,y})}$,
where $\theta_{x,y}$ are phases of $B_{x,y}$. In length or time space,

$\dfrac{-2\text{Im}\big{[}B_{x}B_{y}^{*}\big{]}}{|B_{x}|^{2}+|B_{y}|^{2}}=-\text{Im}\left(e^{i(kz\mp\omega
t+\theta_{x})}e^{-i(kz\mp\omega
t+\theta_{y})}\right)=\sin(\theta_{y}-\theta_{x}).$

This phase difference between $B_{x}$ and $B_{y}$ is conserved in the Fourier
wavevector (frequency) domain. The values of magnetic helicity and
polarization can be calculated for the example forward/backward ICWs by using
expressions for $B_{x}$ and $B_{y}$ as $B_{x}\propto\cos{(kz\mp\omega t)}$,
and $B_{y}\propto\cos{(kz\mp\omega t\mp\pi/2)}$, respectively. Table 3
summarizes the values of $\sigma_{m}$ and $\sigma$ for the forward and
backward ICW and FMWs in the plasma frame. Polarization observed by spacecraft
is Doppler-shifted and can change signs due to the relative motion of the
spacecraft with respect to the solar wind, as explained in Section 2.4.

|  | magnetic helicity | polarization
---|---|---|---
ICW | Forward | -1 | 1
Backward | 1 | 1
FMW | Forward | 1 | -1
Backward | -1 | -1
Table 3: Magnetic helicity ($\sigma_{m}$) and polarization($\sigma$) values
for forward and backward propagating (with respect to B) ICW and FMWs in
plasma rest frame.

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