# Particle-in-cell simulations of Alfvén wave parametric decay in a low-beta
plasma

C.A. González 1<EMAIL_ADDRESS>Maria Elena Innocenti2
Anna Tenerani1  1Department of Physics, The University of Texas at Austin,
Austin, TX 78712, USA 2Institut für Theoretische Physik, Ruhr-Universität
Bochum, Bochum, Germany

###### Abstract

We study the parametric decay instability of parallel propagating Alfvén wave
in a low-beta plasma using one-dimensional fully kinetic simulations. We focus
for the first time on the conversion of the energy stored in the initial
Alfvén wave into particle internal energy, and on its partition between
particle species. We show that compressible fluctuations generated by the
decay of the pump wave into a secondary ion-acoustic mode and a reflected
Alfvén wave contribute to the gain of internal energy via two distinct
mechanisms. First, the ion-acoustic mode leads nonlinearly to proton trapping
and proton phase space mixing, in agreement with previous work based on hybrid
simulations. Second, during the nonlinear stage, a compressible front of the
fast type develops at the steepened edge of the backward Alfvén wave leading
to a field-aligned proton beam propagating backwards at the Alfvén speed. We
find that parametric decay heats preferentially protons, which gain about 50%
of the pump wave energy in the form of internal energy. However, we find that
electrons are also energized and that they contribute to the total energy
balance by gaining 10% of the pump wave energy. By investigating energy
partition and particle heating during parametric decay, our results contribute
to determine the role of compressible and kinetic effects in wave-driven
models of the solar wind.

##  1 Introduction

Alfvén waves are ubiquitous in magnetized plasmas and represent one of the
building blocks of many phenomena in space and in laboratory. Alfvén waves are
also thought to play a crucial role in coronal heating and solar wind
acceleration. By propagating almost undisturbed along the magnetic field, they
provide a means to transport energy from the photosphere to the corona and
further out (Del Zanna & Velli, 2002). Alfvén wave-like correlation of
magnetic and velocity field fluctuations are commonly observed in the solar
wind (Coleman Jr, 1967; Belcher & Davis Jr, 1971), providing support to wave-
driven wind theories (see, e.g., Cranmer (2012)). Understanding how the energy
carried by Alfvén waves is ultimately converted and released to the plasma in
the form of internal energy is therefore key to our understanding of coronal
heating and solar wind acceleration.

Large amplitude Alfvén waves at Magnetohydrodynamic scales are unstable to the
Parametric Decay Instability (PDI hereafter). In a low beta plasma (when the
thermal pressure is smaller than magnetic pressure), PDI leads to the decay of
the “pump” wave into a forward ion acoustic wave and a lower frequency
backward Alfvén wave (Galeev et al., 1973; Derby Jr, 1978). The PDI is of
great interest, first, because it provides a mechanism to generate reflected
Alfvén waves which are essential to trigger the turbulent cascade (Chandran,
2018; Malara et al., 2022) – the latter providing a possible mechanism to heat
and accelerate the plasma (e.g., Perez & Chandran (2013)). Second, the PDI
generates also compressible modes that in turn contribute to heat the plasma
via formation of shocks (Del Zanna et al., 2001) or, when proton kinetic
effects are retained, via particle trapping and phase space mixing (Araneda et
al., 2008; Matteini et al., 2010b), and proton energization at steepened
fronts (González et al., 2020, 2021). Since the growth rate increases with
decreasing plasma beta, theory and simulations predict that this instability
should contribute to the dynamics of the solar wind in the inner heliosphere,
preferentially in regions close to the sun where $\beta\ll 1$ (Tenerani &
Velli, 2013; Réville et al., 2018). Recently, signatures consistent with PDI
in the lower solar atmosphere have been reported (Hahn et al., 2022),
providing support to scenarios that invoke parametric decay, and in general
the generation of compressible modes, to drive fast wind streams (Suzuki &
Inutsuka, 2005; Shoda et al., 2019a, b; Verdini et al., 2019).

Despite the vast body of work investigating parametric instabilities of
parallel propagating (e.g., Sakai & Sonnerup (1983); Wong & Goldstein (1986a);
Nariyuki & Hada (2007); Jayanti & Hollweg (1993)) and oblique propagating
Alfvén waves Vasquez & Hollweg (1996); Del Zanna (2001); Matteini et al.
(2010a); Del Zanna et al. (2015), the heating processes resulting from the
coupling between Alfvén and compressible modes remain poorly understood from a
kinetic perspective. Among the most prominent issues still not fully
investigated is how much of the energy carried by the pump wave goes into
internal energy, and how the latter is partitioned between protons and
electrons. This is fundamental to understand the role of PDI in solar wind
heating and acceleration. Indeed, in-situ and remote-sensing observations have
found preferential heating of ions relative to the electrons (Wilson III et
al., 2018; Cranmer, 2020) and in general there is a well-known positive
correlation between solar wind speed and proton temperature (Elliott et al.,
2012), whereas the solar wind speed is anti-correlated with the coronal
electron temperature (Geiss et al., 1995). If PDI contributes significantly to
coronal plasma heating, it must be consistent with those constraints.

The goal of this work is to investigate electron kinetic effects during the
parametric decay of a parallel-propagating Alfvén wave and to determine how
the initial energy stored in the pump wave is partitioned between the internal
energy of protons and electrons. So far, only a few works have studied
parametric instabilities with a fully kinetic (PIC) approach (Sakai et al.,
2005; Nariyuki et al., 2008). Those works have considered reduced mass ratio
($m_{i}/m_{e}=16$) and a smaller wavelength for the pump wave. However, in
order to understand if large scale processes can affect the electron
population, one has to maintain a large scale separation between electrons and
ions. A lower mass ratio would increase the relative electron gyroradius,
possibly introducing artificial electron heating, e.g., via unrealistic
particle scattering at magnetic structures. Here we perform PIC simulations of
a weakly dispersive circularly polarized Alfvén wave with a more realistic
electron-to-proton mass ratio in 1D; $m_{i}/m_{e}=400$ ensures that electron
and ion scales are well separated while at the same significantly reducing the
computational cost of resolving the electron scales (needed for the explicit
PIC simulation) with respect with a realistic mass ratio simulation with the
same size in terms of ion skin depths. We compare results from hybrid and
particle-in-cell models, and consider different proton-to-electron temperature
ratio and total plasma beta. For the PIC simulations we use the semi-implicit,
energy conserving ECsim code (Lapenta, 2017; Lapenta et al., 2017; Gonzalez-
Herrero et al., 2018) and the explicit VPIC code (Bowers et al., 2008b, a,
2009). The semi-implicit algorithm allows us to resolve only the scales of
interest without resolving the Debye length as in explicit PIC codes. This
feature will be useful in future work in 2D and 3D when explicit simulations
become prohibitively expensive. In addition, we run hybrid simulations with
the CAMELIA code (Matthews, 1994; Franci et al., 2018) as a reference for the
decay process and proton heating in the absence of electron kinetic effects.

In section 2 we describe the model and simulation setup and in section 3 we
present the main results. The conclusions of this study are summarized in
section 4.

##  2 Model and simulation setup

We consider a large amplitude, monochromatic Alfvén wave propagating parallel
to the mean magnetic field ${\bf B}_{0}$, taken along the $x$-axis. The
initial perturbation for the electromagnetic field is given by $\delta
B_{y}=\delta B\ \sin{(k_{0}x)}$, $\delta B_{z}=-\delta B\ \cos{(k_{0}x)}$,
$E_{y}=-\delta B\ \omega_{0}/(k_{0}c)\ \cos{(k_{0}x)}$ and $E_{z}=-\delta B\
\omega_{0}/(k_{0}c)\ \sin{(k_{0}x)}$. The current density is carried by both
species, and is initialized in order to satisfy Ampere’s law with an initial
drifting Maxwellian distribution with mean proton and electron velocities
given by (Sakai et al., 2005):

$\delta{\bf
u}_{i}=\left(\frac{(\omega_{0}+\Omega_{ce})(\omega_{0}^{2}-c^{2}k_{0}^{2})}{(\omega_{pe}^{2}+\omega_{pi}^{2})k_{0}}-\frac{\omega_{0}}{k_{0}}\right)\frac{{\delta\bf
B}}{B_{0}}$ (1a) $\delta{\bf
u}_{e}=\left(\frac{(\omega_{0}-\Omega_{ci})(\omega_{0}^{2}-c^{2}k_{0}^{2})}{(\omega_{pe}^{2}+\omega_{pi}^{2})k_{0}}-\frac{\omega_{0}}{k_{0}}\right)\frac{\delta{\bf
B}}{B_{0}},$ (1b)

where $\omega_{p\alpha}=\sqrt{e^{2}n_{\alpha}/\epsilon_{0}m_{\alpha}}$ and
$\Omega_{c\alpha}=q_{\alpha}\mathbf{B}/m_{\alpha}$ are the plasma frequency
and the gyro-frequency of the specie $\alpha=i,e$. The initial wave frequency
$\omega_{0}$ is obtained from the dispersion relation of parallel propagating
waves in a two-fluid plasma (Baumjohann & Treumann, 2012):

$\frac{k^{2}c^{2}}{\omega^{2}}=1-\frac{\omega_{pe}^{2}}{\omega(\omega+\Omega_{ce})}-\frac{\omega_{pi}^{2}}{\omega(\omega-\Omega_{ci})}.$
(2)

Run | $L_{x}(d_{i})$ | $\Delta x(d_{i})$ | $\Delta t(\Omega_{ci}^{-1})$ | ppc | $\beta$ | $\beta_{i}$ | $\beta_{e}$ | $T_{i}/T_{e}$
---|---|---|---|---|---|---|---|---
Hyb-TiTe1 | 102.4 | 0.1 | 0.005 | 10000 | 0.25 | 0.125 | 0.125 | 1
Vpic-TiTe1 | 102.4 | 0.0125 | 0.000125 | 500 | 0.25 | 0.125 | 0.125 | 1
ECsim-TiTe1 | 102.4 | 0.025 | 0.0005 | 500 | 0.25 | 0.125 | 0.125 | 1
ECsim-TiTe1B | 102.4 | 0.025 | 0.0005 | 500 | 0.1575 | 0.078125 | 0.078125 | 1
ECsim-TiTe4 | 102.4 | 0.025 | 0.0005 | 500 | 0.1575 | 0.03248 | 0.125 | 4
Table 1: Initial conditions for the simulations presented in this paper.

In the rest of the paper we present our results using the following
normalization: the magnetic field and density fluctuations are expressed in
units of the guide field magnitude $B_{0}$ and of the initial plasma density
$n_{0}=n_{i_{0}}=n_{e_{0}}$, respectively. Lengths are normalized to the
proton inertial length $d_{i}=c/\omega_{p_{i}}$, time is expressed in units of
the inverse of proton gyrofrequency $\Omega_{ci}^{-1}$ and the velocities are
normalized to the proton Alfvén speed $v_{A}=B_{0}/\sqrt{4\pi n_{0}m_{i}}$.
The plasma beta for both protons and electrons is defined as
$\beta_{i,e}=2n_{0}k_{B}T_{i,e}/B_{0}^{2}$. The electron characteristic scales
are related to the proton scales through the mass ratio that we fixed to
$m_{i}/m_{e}=400$ for the PIC simulations. We considered an initial weakly
dispersive pump wave with a wave number $m_{0}=4$ and wave vector $k_{0}=2\pi
m_{0}/L$, $L$ being the size of the simulation domain. The initial wave
amplitude is $\delta B/B_{0}=0.5$ and the wave propagates with a phase
velocity $\omega_{0}/k_{0}=0.885v_{A}$, with $v_{A}/c=0.05$ and
$k_{0}d_{i}=0.2454$. A similar setup for the magnetic and the velocity fields
is used to initialize the hybrid simulation (in the limit
$\Omega_{ce}\gg\omega$). A summary of the numerical parameters for the
simulations presented here can be found in Table 1. We compare the results of
simulations with the same plasma beta and different temperature ratio between
electrons and protons (ECsim-TiTe1B and ECsim-TiTe4) with simulations with
same temperature ratio and different beta (ECsim-TiTe1 and ECsim-TiTe1B), to
separate the effects of the two parameters.

##  3 Results

###  I Overview

Figure 1 and Figure 2 provide an overview of the time evolution of the
parametric decay in the simulations listed in Table 1. We show, from top to
bottom, the variation of the normalized root mean square (rms) of the magnetic
and total density fluctuations, the correlation between velocity and magnetic
field fluctuations
($\rho_{vB}=\mathbf{v}\cdot\mathbf{B}/\lVert\mathbf{v}\rVert\lVert\mathbf{B}\rVert$),
the rms of the field aligned electric field $e_{x}$, and the variation of the
mean proton and electron temperatures normalized to their initial value. The
growth rate of the PDI is reported in the legend and corresponds to the slope
of the best fit indicated by the red solid line in the second panels.

Figure 1: Temporal evolution of the variation of the rms of magnetic field
fluctuations (top panels), rms of total density fluctuations (second panels),
the correlation between magnetic and velocity fluctuations $\rho_{VB}$ (third
panel), and rms of the field-aligned electric field (fourth panel). The last
two panels show the variation of the proton and electron mean parallel and
perpendicular temperatures. The left column shows a comparison between runs
Hyb-TiTe1, Vpic-TiTe1, and Ecsim-TiTe1. The vertical black dashed lines
represent the end of the linear stage of PDI for Ecsim-TiTe1.

In Fig. 1 we compare results from the hybrid simulation and from the explicit
(VPIC) and the semi-implicit (ECsim) PIC simulations, for $T_{i}=T_{e}$ and
$\beta_{e}=0.125$. The decay of the pump wave, marked by the exponential
growth of density fluctuations (second panel), proceeds similarly in the
hybrid and PIC simulations. In both models the decay starts at the same time,
around $t\Omega_{ci}\simeq 150$, with the density fluctuations growing at the
same rate. The saturation of the instability occurs when the backward
propagating Alfvén wave and forward propagating ion-acoustic wave are well
developed and density fluctuations reach a constant value of $\delta
n_{rms}/<n>\simeq 0.2$ at around $t\simeq 225\Omega_{ci}^{-1}$ (vertical
dashed line). A complete reflection of the pump wave then takes place, as can
be seen from the third panel, when the sign of $\rho_{VB}$ changes from $-1$
to $+1$. During this stage, a portion of the initial magnetic energy is
converted into kinetic and internal energy of both protons and electrons. In
particular, protons undergo strong parallel heating along the local magnetic
field with significant perpendicular heating (fifth and bottom panels), as
reported in previous studies (González et al., 2020). PIC simulations show
that electrons are not isothermal during the decay process, and that they are
also energized. The heating of electrons is on average isotropic and proceeds
at nearly a constant rate. This can be seen from the bottom panels of Fig. 1,
showing that $T_{e\parallel}\simeq T_{e\perp}$ and that temperatures keep
increasing during the non-linear stage (specifically after $t\Omega_{ci}\simeq
500$). We noticed a small difference in the electron temperature for the
explicit and semi-implicit simulations with a slightly larger final
temperature in the VPIC simulation. This might be due to numerical heating
associated with explicit codes, although the relative error of the total
energy is less than $1\%$ in both simulations.

Figure 2: Temporal evolution of the variation of the rms of magnetic field
fluctuations (top panels), rms of total density fluctuations (second panels),
the correlation between magnetic and velocity fluctuations $\rho_{VB}$ (third
panel), and rms of the field-aligned electric field (fourth panel). The last
two panels show the variation of the proton and electron mean parallel and
perpendicular temperatures. Results for runs Ecsim-TiTe4 (dot-dashed line),
Ecsim-TiTe1 (dashed line) and Ecsim-TiTe1B (dotted line). The vertical black
dashed lines represent the end of the linear stage of PDI for Ecsim-TiTe4 and
Ecsim-TiTe1B. The gray dashed lines mark the times when distribution functions
are shown in Fig. 3.

The Fig. 2 shows results for simulations performed with ECsim with different
$\beta$ and proton to electron temperature ratio $T_{i}/T_{e}$. As can be seen
from these plots, the total plasma $\beta$ determines the growth rate and the
overall decay of the pump wave in agreement with fluid theory (Wong &
Goldstein, 1986b; Hollweg, 1994), which predicts that higher beta plasmas have
a lower growth rate. This dependence can be seen by comparing runs ECSim-TiTe4
(dot-dashed) and TiTe1B (dotted), which have different electron temperature
but same total plasma beta, and by comparing ECSim-TiTe1B and TiTe1,
corresponding to different values of the plasma beta. We observe that the
higher the initial $T_{e}$, the higher the electric field, in agreement with
the dependence of the electric field on $\beta_{e}$ from the generalized Ohm’s
law.

The most significant difference between the simulations with same beta but
different $T_{i}/T_{e}$ is registered in the rms of the electric field,
(second column, fourth panel: dotted vs. dot-dashed line). We observe lower
rms values of $e_{x}$ when $T_{i}>T_{e}$. This is consistent with Landau
damping of the ion-acoustic mode, which is expected to be stronger when
$T_{i}>T_{e}$ than in the case where $T_{i}=T_{e}$. Linear Landau damping
however is not strong enough to affect the growth of the instability.

Figure 3: Reduced particle VDF for protons (top) and electrons (bottom)
integrated over the entire domain. We show, from left to right, the VDFs for
Ecsim-TiTe4, Ecsim-TiTe1 and Ecsim-TiTe1B, at $t=400\Omega_{ci}^{-1}$,
$t=560\Omega_{ci}^{-1}$ and $t=400\Omega_{ci}^{-1}$, respectively.

To conclude our overview of the decay process, in Fig 3 we show the contour
plots of the velocity distribution functions (VDF) in velocity space,
$f(v_{\parallel},|v_{\perp}|)$, integrated in $x$. We plot the VDF of protons
(top panels) and electrons (bottom panels) when the system has achieved the
nonlinear stage for runs ECSim-TiTe4, ECSim-TiTe1 and ECSim-TiTe1B. This time
is marked by the vertical gray dashed lines in the right panel of Fig. 2, at
$t=400\Omega_{ci}^{-1}$, $t=560\Omega_{ci}^{-1}$ and $t=400\Omega_{ci}^{-1}$
for each simulation, respectively. As will be discussed in the next sections,
protons develop an anisotropic distribution function with a clear field-
aligned beam propagating backward, and an energized field-aligned population
at the (positive) ion-acoustic speed (see Fig. 6 for more details). Electrons
instead remain on average isotropic, and their heating is much less pronounced
than proton heating.

###  II Energy conversion and particle heating

The energy lost by the pump wave powers the formation of the daughter waves
and results into thermal and parallel bulk energy gain for protons and
electrons. PIC simulations show that, after the decay of the pump wave, most
of the particle internal energy gain goes into proton internal energy. In Fig.
4, we analyze the pump wave energy conversion and partition by breaking down
the total energy (per unit volume) into its contributions from Alfvén wave
energy (top panel), parallel kinetic energy (thus including the contribution
from the ion acoustic wave; second panel) and internal energy of protons and
electrons (third and bottom panels). All quantities are expressed as
variations from their initial value, and they are normalized to the initial
pump wave energy $E_{w0}=E_{B_{\bot}}(0)+E_{v_{\bot}}(0)$, where

$E_{B_{\bot}}(t)=1/2<B_{y}(t)^{2}+B_{z}(t)^{2}>$

is the magnetic energy density and

$E_{v_{\bot}}(t)=\sum_{\alpha}1/2<m_{\alpha}n_{\alpha}(u_{y_{\alpha}}(t)^{2}+u_{z_{\alpha}}(t)^{2})>$

is the bulk kinetic energy and the brackets denote spatial average. The
$x$-component of the kinetic energy is defined as

$E_{v_{\parallel}}(t)=\sum_{\alpha}1/2<m_{\alpha}n_{\alpha}u_{x_{\alpha}}(t)^{2}>,$

and the internal energy of species $\alpha$ as

$E_{th,\alpha}=1/2<Tr(P_{ij,\alpha})>.$

Energy balance is quantitatively similar in all our simulations. The top panel
of Fig. 4 shows that the energy contained in the Alfvén waves decreases of
$|\Delta E_{w}|/E_{w0}\simeq 60\%$. Notice that this does not correspond to
the total pump wave energy decrease. The pump wave fully decays in the
simulations considered here (see the correlation $\rho_{vb}$ in Fig. 2). After
the decay ($\rho_{vb}=+1$ after $t\Omega_{ci}\gtrsim 500$), $40\%$ of the pump
wave energy goes into a reflected Alfvén wave, and the remaining $60\%$ goes
in an ion-acoustic mode and other forms of internal energy. The second panel,
as well as the trend of $\rho_{rms}$ reported in Fig. 2, shows that at the end
of the linear stage (indicated by the vertical dashed line for the case
$T_{i}/T_{e}=4$), $\Delta E_{\parallel}$ has increased due to the growth of
the ion-acoustic mode. The subsequent peak of $E_{\parallel}$ however is due
to the ongoing particle trapping by the ion-acoustic mode that generates a
population of protons moving at about the ion-sound speed, as discussed in
section III and, e.g., in Matteini et al. (2010b). Particle trapping and
subsequent phase space mixing ultimately lead to the dissipation of the ion-
acoustic mode and particle heating. After the complete decay of the pump wave,
most of the Alfvén wave energy has gone into proton internal energy. In
particular, we find that about $50\%$ of the initial pump wave energy goes
into proton internal energy and $10\%$ to electron internal energy. Notice
that, unlike protons, electrons keep heating at a constant rate the
instability has saturated and the pump wave has decayed. At the end of the
simulation, the electron internal energy is a factor of 2 larger than the
energy at the end of the complete decay ($t\Omega_{ci}\gtrsim 500$). Still,
the good energy conservation in the simulations and similar results between
explicit and semi-implicit simulations suggest that the electron heating is
indeed physical and not a numerical artifact. Such an energy balance appears
to be unaffected by the total plasma beta and only slightly by $T_{i}/T_{e}$
and $m_{i}/m_{e}$ (not shown).

Figure 4: Energy conversion and partition. Variation of the wave energy (top
panel), of the field-aligned bulk kinetic energy (second panel), and the
internal energy variations for protons and electrons (third and fourth panels,
respectively). Each panel shows results for runs Ecsim-TiTe4 (dashed), Ecsim-
TiTe1 (solid) and Ecsim-TiTe1B (dotted). The vertical dashed line indicates
the end of the liner stage for the case $T_{i}/T_{e}=4$. Figure 5: Left
panels: contour plot in the $(t,x)$ plane of the $b_{y}$ component (panel
(a)), the magnitude of the magnetic field (panel (b)), the field-aligned
component of electric field (c), and the scalar agyrotropy and heat flux for
protons and electrons (panel (d) and (e)) and (panel (f) and (g)),
respectively. Results are shown for ECSim-TiTe4. Right panels: rms of the
divergence of the parallel heat flux for protons (top panel) and electrons
(second panel), and variance of the agyrotropy for protons (fourth panel) and
electrons (bottom panel) for all simulations.

###  III Phase space dynamics

Compressibility plays an important role in proton and electron energization
and in providing paths to particle heating. In the low-beta simulations
presented in this work, we were able to identify and distinguish two types of
compressible fluctuations generated during parametric decay: an ion-acoustic
mode and a compressible steepened front of the fast-mode type propagating at
the Alfvén speed. Such signatures were difficult to discern in higher beta
simulations (González et al., 2020). Both types of compressible fluctuations
contribute to the increase of internal energy. However, they lead to
fundamentally different phase space dynamics and to different kinetic
features. The ion-acoustic mode contributes nonlinearly to phase space mixing
via particle trapping. On the other hand, the development of the steepened
front with non-constant $|{\bf B}|$ leads to the formation of strong parallel
heat fluxes and ultimately causes the acceleration of a population of protons
into a well-defined beam of particles and the enhancement of proton
perpendicular heating. In the following, we analyse the signatures of these
processes in the fields and particle VDFs.

In Fig. 5, left column, we show contour plots in the $(t,x)$ plane of $B_{y}$
(top panel), of the magnetic field magnitude $|{\bf B}|$ (second panel) and of
the electric field $e_{x}$ (third panel). We also plot the scalar pressure
agyrotropy (Aunai et al., 2013; Cazzola et al., 2016),

$D_{ng}=\sqrt{8\left(P_{xy}^{2}+P_{xz}^{2}+P_{yz}^{2}\right)}/(P_{\parallel}+2P_{\perp}),$

and the parallel heat flux ($q_{\parallel}$) for protons (fourth and fifth
panels) and electrons (sixth and bottom panels), respectively. The thermal
energy flux is obtained by calculating the total energy flux for each specie
($\mathbf{Q}_{s}=\frac{1}{2}m_{s}\int
f_{s}(\mathbf{x},\mathbf{v},t)v^{2}\mathbf{v}d^{3}\mathbf{v}$) and subtracting
the energy flux components in the Eulerian frame
($\mathbf{q}_{s}=\mathbf{Q}_{s}-\mathbf{H}_{s}-\mathbf{K}_{s}$), namely the
contribution from the bulk energy flux
($\mathbf{K}_{s}=\mathbf{u}_{s}\frac{\rho u_{s}^{2}}{2}$) and the enthalpy
flux ($\mathbf{H}_{s}=P_{s}\cdot\mathbf{u}_{s}+\mathbf{u}_{s}U_{th}$). This
requires the calculation of the lower moments of the velocity distribution
(see details in Lapenta et al. (2020)). We show results from ECSim-TiTe4 as a
reference, since ECSim-TiTe1 and ECSim-TiTe1B are similar. In the right column
of Fig. 5 we show the rms of the divergence of the parallel heat flux
(normalized to its initial value) and the standard deviation of the scalar
agyrotropy for both species and for all simulations.

From Fig. 5, left, one can see that a dominant backward propagating mode forms
at $t\Omega_{ci}\simeq 200$, when the PDI has reached saturation. The
signature of the forwards ion acoustic mode can be seen in the plots of
$e_{x}$ in the time interval $t\Omega_{ci}=200-400$, before it dissipates,
corresponding to a wavevector $k_{s}=7\times 2\pi/L_{x}$. At about
$t\Omega_{ci}\simeq 400$, a steepened front propagating backwards at the
Alfvén speed also develops. This steepened front corresponds to a localized
increase of $|\mathbf{B}|$, positively correlated with a density enhancement.
As discussed in our previous work (González et al., 2020, 2021), compressive
fluctuations in $|{\bf B}|$ are associated to a field-aligned electric field
$e_{x}$. Enhancements of $e_{x}$ at the steepened front can also be seen in
the third panel. The strongest non-thermal features (agyrotropy and heat
fluxes) of particles develop at such a steepened front and within the ion-
acoustic mode, as shown in Fig. 5, from the third to the bottom panels on the
left.

The cumulative contribution of non-Larmor particle motion produces non-zero
values of the off-diagonal terms of the pressure tensor, quantified by
$D_{ng}$, where local values of up to $|D_{ng,e}|\simeq 0.3$ and
$|D_{ng,i}|\simeq 0.65$ are reached. The variance of $D_{ng}$ for both species
is reported in the third and fourth panels on the right, and the rms of the
divergence of the parallel heat flux in the first two panels. Agyrotropy in
particular is considered one of the VDF based dissipation measures (Pezzi et
al., 2021). From Fig. 5, right panels, one can see that agyrotropy first
builds up during the linear stage of the PDI for both species in
correspondence with the ion-acoustic mode, before decaying during the
nonlinear stage as the ion-acoustic mode is dissipated and, correspondingly,
the plasma is heated. However, protons, unlike electrons, undergo a two-step
heating process. Besides particle trapping, protons subsequently interact with
the steepened front, where they develop a second peak of agyrotropy.
Interaction of protons with a field-aligned discontinuity and electric field
causes both proton scattering in phase space, contributing to heating, and the
acceleration of protons into a field-aligned beam (González et al., 2021).
This is consistent with the observed time evolution of proton agyrotropy and
parallel heat flux (see right top and third panels in Fig. 5). Electrons,
which have a smaller gyroradius than protons, display on average $D_{ng,e}\ll
D_{ng,i}$ (bottom right panel) and, accordingly, they do not gain much
internal energy.

Figure 6: The phase-space ($x,v_{\parallel}/v_{\bot}$) for protons (left) and
electrons (right). The phase space is shown at $t\Omega_{ci}=400$ for Ecsim-
TiTe4. The magnetic field’s magnitude (red line) and $B_{y}$ (green line) are
also presented. The inset in the top right panel represents the electron
distribution function integrated in space around the discontinuity (blue) and
outside the discontinuity (orange and green).

Kinetic features in phase space corresponding to the dynamical evolution
described above can be seen in Fig. 6, where we show the proton and electron
VDFs in the $(x,v_{\parallel})$ and $(x,v_{\perp})$ space, top and bottom
panels, respectively, at $t\Omega_{ci}=400$. We show results for EcsimTiTe4
only since similar features develop in the other simulations. Protons in phase
space are shown on the left, and electrons on the right. For reference, we
plot $B_{y}(x)$ (green color) and $|\mathbf{B}(x)|$ (red color) at the same
time to show that the particle beam forms at the steepened front. The contour
plot of $f(x,v_{\parallel})$ shows the two sources of the proton internal
energy increase discussed above. First, saturation of the growth of the ion-
acoustic mode is determined by nonlinear particle trapping. The latter causes
the formation of a second proton population that propagates forwards at the
ion-acoustic speed, and develops well-known signatures of phase space mixing
(vortexes). Second, a proton beam is generated locally at the backward
propagating steepened Alfvén front. The latter can be identified in Fig. 6 at
around $x\simeq 85\ d_{i}$, and has a size of a few $d_{i}$. Both particle
trapping and beam formation cause the large increase of the parallel proton
temperature and parallel heat flux. One can see in the bottom left panel that
also perpendicular heating takes place at the steepened edge, where particles
with high perpendicular velocities are observed. The perpendicular
energization is due to particle scattering at those discontinuities (González
et al., 2021). The electrons are also characterized by a local deviation from
Maxwellian at the steepened front. The inset in the top right panel shows the
electron VDF integrated in $x$ over an interval $\Delta x=3\ d_{i}$, at
different positions in the simulation domain: centered at the location of
discontinuity $x=85\ d_{i}$ (blue line) and outside the discontinuity, at
$x=60\ d_{i}$ and $x=95\ d_{i}$ (orange and green lines), where the VDF is
Maxwellian. The electrons at the discontinuity show a flat-top distribution
with enhanced number of electrons in that region as to maintain the quasi-
neutrality of the system (we do not find evidence of charge-space separation
in the simulations). We do not observe any electron beam in the simulations,
and the electrons undergo on average isotropic heating.

##  4 Conclusions

We have investigated the parametric decay instability of a large-amplitude
Alfvén wave with large-scale particle-in-cell simulations with an explicit
(VPIC) and semi-implicit (ECsim) PIC code. Results from the PIC model are
compared with those from a state-of-the-art hybrid model (CAMELIA) that
employs an isothermal electron closure. Our main findings can be summarized as
follows:

  * •

The growth rate of the decay is determined by the total plasma beta in
agreement with fluid theory. The electron temperature affects compressibility
in a way that is consistent with the generalized Ohm’s law. The higher
$T_{e}$, the larger the rms of the field aligned electric field, and the lower
the rms of density fluctuations.

  * •

Compressible effects lead not only to the generation of the ion acoustic mode,
but also to the development of a fast-mode steepened front propagating with
the backward Alfvén wave. Particle interaction with those two types of
compressible fluctuations leads to the increase of particle internal energy.

  * •

Proton internal energy gain has contributions from: $(i)$ phase space mixing
due to particle trapping by the ion-acoustic wave; $(ii)$ beam acceleration
aligned to the mean magnetic field at the steepened front; $(iii)$ particle
scattering at the steepened front. Protons are overall heated preferentially
in the parallel direction due to the contributions of particle trapping and
beam formation. Electrons are also energized at the steepened front. Contrary
to protons, electrons are heated isotropically.

  * •

Analysis of the energy balance shows that after the complete decay of the pump
wave $40\%$ of the pump wave goes into reflected wave, $\sim 50\%$ into proton
internal energy (perpendicular and parallel) and $\sim 10\%$ into electron
internal energy.

In conclusion, we have clarified that parametric decay can heat the plasma via
two processes that in prior work (Araneda et al., 2008; Matteini et al.,
2010b; González et al., 2020) were not as clearly separated as in this set of
simulations. The processes are proton trapping and beam acceleration at
steepened fronts. While proton trapping causes the energization of a proton
population which propagates at the ion-sound speed, it is the steepened front
that generates an actual beam at the Alfvén speed. Since the steepened front
propagates backward due to the complete reflection of the pump wave, the beam
also propagates backward in this set of simulations. Our results also provide
the first energy balance analysis of parametric decay with fully kinetic
simulations. In particular, we find that parametric decay preferentially heats
protons, although electrons also gain a fraction of the energy stored in the
pump wave. Heating triggered by parametric decay is consistent with the
observed preferential proton heating with respect to electrons in the corona.
In this context, our results provide a promising starting point to assess the
role of (kinetic) compressible effects in wave-driven theories of the solar
wind.

###### Acknowledgements.

This research was supported by NASA grant #80NSSC18K1211. We also acknowledge
NSF award 2141564 and the Texas Advanced Computing Center (TACC) at The
University of Texas at Austin for providing HPC resources that have
contributed to the research results reported within this paper. URL:
http://www.tacc.utexas.edu. We thank G. Lapenta and the ECSim development team
for granting us access to the code and . M.E.I. acknowledges support from the
German Science Foundation DFG within the Collaborative Research Center
SFB1491.

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indicates wave evolution based on the interaction region properties. Using a
1D MHD model it is possible to model the steepening of a wave packet in a
fluid with non-negligible viscosity and resistivity. It shows that the plasma
environment is large enough for the wave packet to steepen to the values of
skewness observed in the magnetic field observations. Comparison of the model
parameters and the measured wave properties allows to infer the viscosity of
the plasma (_Ostaszewski et al._ , 2020).

Some of these structures can also be detected inside the diamagnetic cavity
(_Masunaga et al._ , 2019; _Hajra et al._ , 2017). There, the magnetic field
of the structure remains close to zero, but the density and ion flux are
similar to the steepened waves upstream of the boundary. This indicates that
while the magnetic field remains zero, the diamagnetic cavity boundary is
permeable to the heavy ions observed in the inner coma. The exact mechanism of
the transmission of those wave packets through the boundary is not yet clear
and requires further modelling and analysis.

##### Ion Bernstein Waves

A closer inspection of the steepened wave plasma density observations reveals
a substructure of wave activity in the descending, longer part of the
steepened wave. A corresponding signature in the magnetic field was found to
be of lower amplitude and phase shifted by $90^{\circ}$. These were
tentatively attributed to ion bernstein waves, which is an electrostatic wave
mode that can be excited by a ring/ring-beam instability (_Odelstad et al._ ,
2020). In Figure 12f the current (plasma density), magnetic field and
coherence of these two parameters is shown. There are clear signatures in the
coherence at around $100\,$Hz for several minutes at a time.

##  5 Summary and Outlook

In this chapter we have shown the richness of the processes that arise when
the cometary ion cloud interacts with the charged solar wind. The cometary
plasma environment is not only highly variable in time, but also in spatial
dimension. Depending on the properties of the nucleus and its distance to the
Sun, the extension of the coma can vary by four orders of magnitude. It
therefore is an ideal laboratory to explore processes at different scale sizes
and cross-scale interactions in a multi-ion plasma.

Simply put, the environment is created by the ions and electrons of cometary
origin and the interaction with the solar wind distributes energy and momentum
in this plasma to achieve a stable state, where the solar wind plasma and the
cometary plasma are fully mixed. This fundamental process is often referred to
as mass-loading and manifests itself in different ways, depending on the scale
sizes of the interaction region and the particles in the plasma. Most
importantly, the ion gyroradius and collision length scales determine how the
solar wind particles and the cometary particles are interacting with each
other and amongst each other. At a high activity comet, a more fluid-like
behavior is common, while a kinetic approach is preferred at low activity
comets and in the inner coma of any comet.

For all comets except the most active, the large gyroradius means that the
solar wind ions are not just slowed down, but also deflected. Eventually the
cometary ions are picked up and substitute the solar wind ions in the flow.
Often, two distinct cometary ion populations are detected: accelerated pick-up
ions and slow, newborn ions.

On small scales, the electrons and ions decouple and electrons are accelerated
into the inner coma by an ambipolar field. There they can be trapped by
collisions and cooled. Therefore three electron populations are present: cold,
warm, and hot. The interplay of electric fields and collisions changes the
electron energy.

Transient solar wind events like ICMEs and CIRs increase not only the solar
wind pressures but also the compression factor of the cometosphere and the
electron impact ionisation rate. This leads to an increase in the cometary ion
density that is greater than the changes of the solar wind parameters itself.
This also increases the magnetic field to unprecedented values.

The interplanetary magnetic field is piled up and draped around the inner
coma, creating an induced magnetosphere. Nested draping creates current
sheets, and the small gyroradius can lead to draping in a different direction
due to ions being deflected. Thus, while the ions are not directly tied to the
magnetic field, they still have an influence on it via the electrons. The
plasma tail is structured, with cometary rays and density enhancements being
visible from Earth remotely. Magnetic reconnection at current sheets could
potentially cause tail disconnection events.

There are multiple boundaries that form depending on the gas production rate.
The first is a bow shock that is broad and weak due to the mass-loaded nature
of the plasma. Here, small scale processes, like electron impact ionisation
and charge exchange, can affect the bow shock standoff distance, demonstrating
the importance of cross scale coupling. While a cometopause seems to exist at
most comets, a solar wind ion cavity also appears and is the more obvious
boundary. Collisionopauses are broad regions where different collisional
processes dominate.

The diamagnetic cavity at comets seems highly unstable, asymmetric and the
boundary is often dominated by surface waves. While a diamagnetic cavity
exists at both comet 1P/Halley and at the lower activity comet 67P, the
mechanism that forms this region seems entirely different.

The free energy that is added to the plasma by the creation of heavy water
ions modulates the overall wave activity that is observable in the environment
as waves are a way to distribute energy. Often, the existence of certain wave
modes allows us to learn more about the plasma in which they were generated.
Steepening of waves can be used to diagnose the resistivity and viscosity of
the plasma and how waves evolve when they travel through it.

All of the phenomena described here can be investigated in their own right,
but the coupling between processes of different temporal and spatial scales
necessitates that a more rounded approach is taken. While the comet nucleus is
quite small, the plasma environment can extend up to millions of km, with the
plasma tail spanning multiple AU at times. Therefore a large parameter space
is covered and a multitude of processes can be observed to have effects on the
plasma environment.

In the corresponding chapter of Comets II (_Ip_ , 2004), high hopes were put
on the results from the Rosetta mission. However, in hindsight, it is very
difficult to compare Rosetta results with those from previous flyby missions
to more active comets. It turned out that the plasma environment at 67P was
very different in terms of collisionality and gyro radius effects. Instead of
answering the questions posed in Comets II, Rosetta has provided a whole new
set of results that expand our knowledge of the plasma environment of low to
medium activity comets.

Many open questions remain with regards to this topic and future missions to
gather a more complete data set are necessary. First and foremost among these
should be a multi-spacecraft mission that will be able to provide spatial and
temporal coverage at the same time and allow us to disentangle the influence
of the different contributions (solar wind processes and internal processes)
to the plasma. Only then can we take full advantage of this intriguing plasma
laboratory that presents itself to us any time a comet is explored.

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