By three-dimensional hybrid simulations, proton heating is investigated starting from a monochromatic large-amplitude Alfvén wave with left-handed circular polarization launched along the mean magnetic field in a low-beta plasma. We find that the perpendicular scattering is efficient in three dimensions and the protons are heated by the obliquely propagating waves. The thermal core proton population is heated in three dimensions as well in the longitudinal and parallel directions by the field-aligned and obliquely propagating sound waves out of the parametric decay. The astrophysical context is discussed.

Early in situ measurements at 1 AU from the VELA satellite

Parametric instabilities play an important role in the dissipation of the
large-amplitude Alfvén waves with parallel or quasi-parallel propagation
with respect to the mean magnetic field and in
plasma heating by means of the ion Landau damping mechanism.
Parametric instabilities, including decay, modulational, and beat
instabilities, have been extensively analyzed by theoretical studies

Here we address the question “Is the stochastic ion heating stronger in a
3-D parametric decay?” Our question is motivated by two preceding studies.
First,

We perform a 3-D hybrid plasma simulation for the parametric decay, and track
the time evolution of the proton distribution functions. We find that the
stochastic heating (i.e., pitch-angle scattering) occurs more quickly and the
ions are heated most strongly
in the 3-D treatment.
Our finding that the particles can be more quickly heated by the 3-D
parametric decay can be tested by in situ measurements by the upcoming
heliospheric missions such as Parker Solar Probe

Parametric decay of a parallel-propagating Alfvén wave (“

We perform hybrid simulations with the AIKEF hybrid code

The parametric decay modeled in the actual study is a three-wave process
starting from a large-amplitude monochromatic Alfvén pump wave propagating
parallel to the mean magnetic field

The protons are treated in the hybrid scheme as particles, while the
electrons are considered as a massless fluid. The values of the

The simulated magnetic field, density, and bulk velocity fluctuations are
first averaged in the real space over one of the perpendicular directions
(

Figure 2a shows the time evolution of the antisunward-propagating Alfvén
pump (10,0), the compressional daughter (18,0), and the sunward-propagating
Alfvén daughter (

The root-mean-squared (rms) density fluctuations

When the decay instability is set on, the density fluctuations start to
increase and the cross-helicity starts to decrease. At a time just before

The power spectrum of magnetic field and density fluctuations is given in Fig. 2b at time

While Fig. 2 based on analyzing the wave spectrum brings evidence of the
decay instability, Fig. 3 reports the particle heating process. The upper
panels of Fig. 3 present the particle distribution functions in the phase
space

The lower panels of Fig. 3 report the proton-reduced velocity distribution
functions constructed in the (

Due to the transversal wave field imposed as the initial condition, the
velocity distribution functions at time

The dashed lines at time

It is straightforward to notice by comparing the distribution functions at
the intermediate (

Here we will check whether field-aligned propagating waves can resonate and
whether scatter protons by pitch-angle diffusion mechanisms or obliquely
propagating waves are necessary to explain the plateau levels observed in
Fig. 3. The resonance velocity defined in Eq. (2) is plotted by a solid line
along the wavenumber axis in Fig. 4. The right vertical axis shows the values
of the resonance velocity in terms of Alfvén velocity

By solid and dashed lines are given the normalized resonance
velocity (solid line) of ions in dependence with the wavenumber according to
Eq. (2) (see right axis). By dashed line is presented the dispersion relation
for cold plasma (see left axis). Overplotted in a gray-coded scale, the
frequency–wavenumber spectrum of the magnetic field fluctuations is given at
time

The important role of the obliquely propagating daughter waves in perpendicularly heating the particles can be alternatively emphasized by the comparison with the results obtained from additional simulations carried out by decreasing the dimensionality from the 3-D down to the 2-D and 1-D configurations, while all the other physical and numerical parameters are basically maintained the same. In the 1-D box spatial variations are allowed only in directions parallel and antiparallel to the background magnetic field, whereas in the 2-D simulation spatial variations are allowed in both the parallel/antiparallel direction and one perpendicular direction. Figure 5 shows the time evolution of the parallel and perpendicular proton temperatures obtained in the actual and additional 2-D and 1-D setups. The temperatures are determined by computing the thermal velocities (as the second-order velocity moment) obtained by subtracting the bulk velocity from the full particle velocities according to the definition of kinetic temperature. The time evolution of the parallel temperature of protons is similar for all the simulation runs. In contrast, the perpendicular temperature for the 3-D setup starts to increase and becomes larger by a factor of 4 with respect to the temperatures obtained in the downgraded configurations. The three simulations carried out with the same physics in one dimension, two dimensions, and three dimensions demonstrate that the 3-D simulation yields the strongest proton heating.

With respect to this result we have done a
quantification of the differences between the results of the 2-D and 3-D
setups in what the amplitude and slope of the oblique modes mean. At the
linear stage of the instability growth, the Alfvén oblique mode (

Time evolution of the parallel (

The used spatial resolution for the field quantities (magnetic field,
electric field, and velocity moments) is close to the ion inertial length and
the proton gyroradius (

We performed for the first time a 3-D hybrid simulation study on the plasma heating problem associated with the parametric decay. The analysis of wave–wave coupling driven by decay instability and the time evolution of the main wave modes and cross-helicity bring evidence that obliquely propagating waves are excited early by the field-aligned Alfvén pump wave.

We draw the following conclusions.

The pitch-angle scattering is efficient in three dimensions
and the plasma becomes heated more quickly
by the obliquely propagating daughter waves.
The stochastic heating can be verified by
the upcoming solar wind measurements
by Parker Solar Probe and Solar Orbiter.
A temperature rise by a factor of about 4 is obtained
in our 3-D hybrid simulation study by a time of
300 to 600 ion gyroperiods. When applying the mapping
of the elapsed time to the radial distance from the Sun
advected by the solar wind

Thermal core particle population is effectively
heated in three dimensions as well in the longitudinal (to the wavevector)
and parallel (to the mean magnetic field) directions by the field-aligned and
the obliquely propagating sound waves out of
the parametric decay, confirming
the lessons from the earlier studies

Needless to say, our conclusions are limited to a beta parameter of 0.01.
Three-dimensional hybrid simulations provide more
realistic predictions for the wave-heating problem in nonlinear space plasma
dynamics. We propose to study the following items to extend the 3-D
simulations.

Data from our hybrid simulations are stored at the Institut fuer Theoretische Physik – Technische Universitaet Braunschweig. Data can be obtained by writing to the following email addresses: h.comisel@tu-braunschweig.de or comisel@spacescience.ro.

HC carried out the simulations, calculation, and writing. Yasuhir. Nariyuki came out with analysis and opinion. Yasuhit. Narita contributed with writing and conclusive remarks. UM handled the discussion and finalization.

The authors declare that they have no conflict of interest.

This work is financially supported by a grant of the Deutsche Forschungsgemainschaft (DFG grant MO539/20-1). HC acknowledges the hospitality at the University of Toyama for hosting the research visit. We acknowledge the John von Neumann Institute for Computing (NIC) for providing computing time on supercomputer JURECA at Juelich Supercomputer Centre (JSC). Edited by: Yoshizumi Miyoshi Reviewed by: three anonymous referees