11institutetext: Centre for mathematical Plasma Astrophysics, KU Leuven,
Celestijnenlaan 200B, B-3001 Leuven, Belgium
11email<EMAIL_ADDRESS>22institutetext: Department of Physics,
Catholic University of America, 620 Michigan Ave NE, Washington D.C. 20064,
USA
22email<EMAIL_ADDRESS>33institutetext: NASA Goddard Space Flight
Center, 8800 Greenbelt Rd, Greenbelt, Maryland 20771, USA
33email<EMAIL_ADDRESS>
# Relative drifts and temperature anisotropies of protons and $\alpha$
particles in the expanding solar wind – 2.5D hybrid simulations

Y. G. Maneva 1122 L. Ofman 2233 A. Viñas 33

(Received: June 13, 2014
Accepted: )

###### Abstract

Context. We perform 2.5D hybrid simulations to investigate the origin and
evolution of relative drift speeds between protons and $\alpha$ particles in
the collisionless turbulent low-$\beta$ solar wind plasma.

Aims. We study the generation of differential streaming by wave-particle
interactions and absorption of turbulent wave spectra. Next we focus on the
role of the relative drifts for the turbulent heating and acceleration of ions
in the collisionless fast solar wind streams.

Methods. The energy source is given by an initial broad-band spectrum of
parallel propagating Alfvén-cyclotron waves, which co-exists with the plasma
and is self-consistently coupled to the perpendicular ion bulk velocities. We
include the effect of a gradual solar wind expansion, which cools and
decelerates the minor ions. This paper for the first time considers the
combined effect of self-consistently initialized dispersive turbulent Alfvénic
spectra with differentially streaming protons and $\alpha$ particles in the
expanding solar wind outflows within a 2.5D hybrid simulation study.

Results. In the non-expanding wind, we find a threshold value of the
differential streaming $V_{\alpha p}=0.5V_{\mathrm{A}}$, for which the
relative drift speed remains nearly steady. For ions, streaming below the
threshold value, the waves act to increase the magnitude of the relative drift
speed. Ions, which stream faster than the threshold value become subject to
nonlinear streaming instability and as the system evolves their bulk
velocities decrease. We find that the solar wind expansion strongly affects
the relative drift speeds and significantly slows down both ion species for
all values of the relative drift speeds considered in this study.

###### Key Words.:

Sun: solar wind – Plasmas – Turbulence – Waves – Instabilities

##  1 Introduction

Remote sensing and in situ measurements in coronal holes and fast solar wind
show that the heavy ions are preferentially heated and accelerated, despite
the higher inertia and the stronger gravitational force, which they experience
in the vicinity of the solar surface (Marsch et al., 1982; Kohl et al., 1998;
Cranmer et al., 1999; Kohl et al., 2006). In situ observations of the
temperature anisotropies and the relative drift speed between protons and
$\alpha$ particles in collisionless fast solar wind streams indicate a
relation between the wave activity and the bulk properties of the ion species
in the range $0.3-1AU$ (Bourouaine et al., 2010, 2011; Maruca et al., 2011,
2012; Kasper et al., 2008, 2013; Bourouaine et al., 2013). Recent observations
indicate that doubly-ionized solar wind helium ions near the Earth can be
nearly 7 times hotter than protons on average (Kasper et al., 2013). What
causes such a strong heating for the $\alpha$ particles and what leads to the
onset of preferential acceleration and the generation of the observed
temperature anisotropies are still open questions, remaining to be solved by
future observations and models. Currently there are no measurements of helium
ions in the inner solar corona, but spectroscopic observations of other ions
like $O^{5+}$ and $Mg^{9+}$ in coronal holes imply high outflow speeds,
temperatures and anisotropies for the heavy species there (Kohl et al., 1997,
1999; Cranmer et al., 1999; Ofman et al., 2001, 2011a, 2013). This indicates
that the onset of the preferential heating, differential acceleration and
temperature anisotropies for the heavy ions takes place in the low-$\beta$
coronal plasma, where turbulence is important and wave amplitudes can be
higher than at 0.3 AU (Jian et al., 2009, 2010).

In this paper we perform 2.5D hybrid simulations to investigate the generation
of ion temperature anisotropies and differential streaming by broadband
turbulent spectra of low-frequency Alfvén-cyclotron waves. Further on, we
study the influence of ion relative drift speeds and the solar wind expansion
on the ion heating and differential acceleration. The results of this study
provide insight into the physical mechanisms responsible for the observed
preferential heating and acceleration of minor ions in the solar wind. The
work relates the particles properties to the existing solar wind wave spectra,
in terms of generation and absorption of waves via nonlinear wave-wave and
wave-particle interactions, and the evolution of the micro-turbulence in the
solar wind.

The scope of our paper is of broad interest to the solar plasma community and
extends various previous studies based on hybrid numerical modeling, e.g.,
Gary et al. (2003, 2006); Xie et al. (2004b); Ofman et al. (2005); Ofman &
Viñas (2007); Hellinger & Trávníček (2006); Perrone et al. (2013); Hellinger &
Trávníček (2013); Ofman et al. (2014), linear Vlasov kinetic instability
theory, Gary et al. (2001, 2002); Verscharen et al. (2013); Chandran et al.
(2013) and numerous observations, such as Reisenfeld et al. (2001); Gary et
al. (2002); Hellinger et al. (2006); Kasper et al. (2008); Bourouaine et al.
(2010, 2011); Maruca et al. (2012); Kasper et al. (2013); Bourouaine et al.
(2013). Podesta & Gary (2011) perform linear theory analysis to explore the
effects of the $\alpha$-proton differential streaming in the solar wind.
Hellinger & Trávníček (2006, 2013) discuss the role of $\alpha$-proton
relative drifts for destabilizing the plasma in nonlinear 2.5D hybrid
simulations. Verscharen et al. (2013), Chandran et al. (2013) and Bourouaine
et al. (2013) consider the limits of differential streaming and ion
temperature anisotropies from the point of view of linear kinetic
instabilities and observations. Hellinger & Trávníček (2013) have studied the
effects of the solar wind expansion on the plasma instabilities in high-
plasma-$\beta$ system, consisting of differentially streaming proton core,
beam and $\alpha$ particles in the heliosphere at distance $r>0.3AU$. In the
current paper we investigate the effect of the solar wind expansion for an
initially drifting $\alpha$-proton plasma in low plasma $\beta$ conditions
appropriate in particular to the inner heliosphere and the solar corona. We
initialize the system with an $\alpha$-proton population and let the ion beams
be self-consistently generated by the model ambient wave-spectra as the system
evolves in time. We load the particle populations with equal temperatures for
the two ion species and study the temporal evolution of their temperature
ratio as a function of the relative drift speed. With the goal of constructing
a more realistic model of the solar wind, we consider the effects of turbulent
initial wave spectra, which obey the three-species electron-proton-$\alpha$
cold-plasma dispersion relation. This allows for additional wave-particle and
wave-wave interactions, non-present in the absence of waves, and provides
essential information on the relation between plasma waves and the properties
of ions in collisionless low-$\beta$ solar wind. Recently, Ofman et al. (2014)
have considered the effect of the solar wind expansion for the cases of super-
Alfvénic initial relative drifts or $\alpha$ particle temperature
anisotropies, and a driven wave spectra. The waves are injected in time at one
side of the periodic boundaries with a prescribed power law consistent with
observations within a certain range in frequency space. While the study of
Ofman et al. (2014) is based on the introduction of time-dependent magnetic
fluctuations, in the present study the magnetic spectrum is introduced as an
initial value problem. By construction, the broad-band spectra preserves the
magnetic field divergence-free, conserves the sense of magnetic helicity and
obeys Parseval’s theorem for energy conservation. The waves represent
solutions of the cold plasma dispersion relation for a given narrow range in
wave-vector space, for which a $k^{-1}$ spectral slope is considered. Previous
1.5D and 2.5D hybrid simulations (Xie et al., 2004b; Araneda et al., 2009;
Ofman & Viñas, 2007; Ofman et al., 2011b; Maneva et al., 2013b, 2014; Ofman et
al., 2014) show that monochromatic pumps and broad-band spectra of waves can
preferentially heat and accelerate the $\alpha$ particles. In this 2.5D study
we have narrowed the spectral range of the initial turbulent fluctuations in
order to avoid the slow evolution of the very low-frequency MHD-type Alfvén
waves, which would not interact with the particles within the simulation time-
scales considered here (hundreds of gyroperiods). Details of the simulation
setup and the initial wave-spectra are presented in Section 2. Simulation
results are described in Section 3 and conclusions are given in Section 4.

##  2 Simulation setup

In this section we introduce the normalization units and the characteristic
parameters used for the simulations. The expanding box model and the
construction of the initial wave spectra are presented in separate
subsections.
The simulation time is given in units of proton gyro-frequency
$\Omega_{p}^{-1}$ and the velocities are normalized to the local Alfvén speed,
$V_{\mathrm{A}}\equiv\frac{B_{0}}{\sqrt{\mu_{0}n_{e}m_{p}}},$ as defined by
the magnitude of the homogeneous background magnetic field $B_{0}$, the
electron number density $n_{e}$ and the proton mass $m_{p}$. The length of the
simulation box is the same in both spatial directions. In units of the proton
inertial length it reads $L_{x}=L_{y}=384V_{\mathrm{A}}/\Omega_{p}.$ The
simulations are performed with 256$\times$256 cells and 127 particles per cell
per species, or in total $>16.6$ million particles. The time step used is a
small fraction of the proton gyroperiod, $\Delta t=0.02\Omega^{-1}_{p}$.

###  2.1 Initial wave spectra

To initialize the simulations we assume that the magnetic field consists of a
constant background and a fluctuating part, which corresponds to a broad-band
spectrum of low-frequency parallel propagating Alfvén-cyclotron waves:

$\mathbf{B}=\mathbf{B_{0}}+\mathbf{B_{\perp}}.$ (1)

To construct the initial spectra for the two-dimensional simulations we
perform a one-dimensional parallel propagating broad-band reconstruction,
based on the procedure described in Viñas et al. (2014), and distribute it
homogeneously at each point in the transverse direction until the entire
simulation domain is filled with waves. For the one-dimensional reconstruction
we assume that the ambient magnetic field is along the $x$ axis,
$\mathbf{B_{0}}=B_{0}\hat{x}$, and the fluctuating part
$\mathbf{B_{\perp}}(x)$ represents left-hand polarized transverse waves
propagating along the background field. For the magnetic fluctuations we
consider plane waves decomposition:

$\mathbf{B_{\perp}}=\textit{Re}\left[\sum_{k>0}C_{k}^{L}\exp(i\phi_{k})(\hat{y}+i\hat{z})\exp(ikx)\right],$
(2)

where the phases $\phi_{k}$ are randomly set between $\pm[0,2\pi]$. The
Fourier coefficients for left-hand polarized waves $C_{k}^{L}$ are computed
based on the following physical conditions: 1) a divergence-free magnetic
field $\nabla\cdot\mathbf{B}=0$, 2) preservation of the sense of magnetic
helicity, and 3) conservation of energy between configuration and Fourier
spaces, also known as Parseval’s theorem. For left-hand circularly polarized
waves the conservation of energy can be expressed as:

$E=\frac{1}{2}\int{|B_{\perp}(x)|}^{2}dx=\frac{1}{2}\textit{Re}\sum_{k>0}{|C^{L}_{k}|}^{2}.$
(3)

The total magnetic helicity for all fluctuations, defined as the scalar
product of the vector potential $\mathbf{A}$ with the fluctuating magnetic
field $\mathbf{B}$, can be represented as a superposition of the spectral
magnetic helicity $H_{mk}$ for each mode

$H_{m}=\frac{1}{2}<\mathbf{A}\cdot\mathbf{B}>=\frac{1}{2}\sum_{k}H_{mk}.$ (4)

For a spectrum, which consists only of left-hand polarized waves, the spectral
magnetic helicity can easily be expressed in terms of the Fourier amplitudes
of the fluctuations as follows:

$H_{mk}=\frac{1}{4k}\sum_{k}({|C^{L}_{k}|}^{2}).$ (5)

To construct a magnetic field power spectra, which resembles solar wind
observations we assume a power-law dependence for the amplitude of the waves
${|C^{L}_{k}|}^{2}=C^{2}k^{-\alpha}$, where $\alpha$ is a prescribed power
spectral index. The constant $C$, depends on the total amplitude of the
magnetic field fluctuations
$|\delta\mathbf{B}_{\perp}|=\sum_{k>0}{|C^{L}_{k}|}^{2}$, the spectral slope
and the selected spectral range

$C^{2}=\frac{{\delta B}^{2}}{\sum_{k>0}k^{-\alpha}}.$ (6)

The spectra is constructed in Fourier space and inverse Fourier transform is
used to provide the magnetic and velocity field fluctuations in configuration
space. A detailed description of the method used to construct the initial
broad-band spectra for a general combination of left-hand and right-hand
circularly-polarized parallel propagating waves can be found in Viñas et al.
(2014). For the power spectral slope in our simulation study we choose
$\alpha=-1$. We should note that the observed solar wind power spectra close
to the Sun are in frequency domain. Thus the observed power spectral slope at
large scales $\omega^{-1}$ can be directly related to $k^{-1}$ spectrum only
for the low-frequency linear part of the dispersion relation, where
$\omega=V_{\mathrm{A}}k$.
In order to self-consistently couple the initial ion velocities to the
reconstructed magnetic field one needs to prescribe a certain dispersion
relation for the waves and solve it to find the relation between the ion
motion and the prescribed electromagnetic fluctuations. Complementary to the
wave-spectra initialization in our previous 1.5D work (Maneva et al., 2013b),
in the present 2.5D study we construct the initial ion bulk velocity
fluctuations as solutions of the 3-fluid cold plasma dispersion relation, see
Eq. 11 and Fig. 1. In other words, we take into account the fact that the
given spectrum of parallel propagating Alfvén-cyclotron waves imposes separate
transverse bulk velocity fluctuations for the protons and the $\alpha$
particles, cf. with Eq. 12. Using a consistent initialization for both ion
species is important to avoid artificial pressure imbalances and
instabilities, which can affect the resulting ion heating and acceleration. We
should note that for the chosen range of wave-numbers in the low plasma
$\beta$ regime considered here (see Table 1), the solution of the cold plasma
dispersion relation for parallel wave propagation in drifting isotropic plasma
is practically identical to the solution of the full Vlasov linear theory
dispersion relation, as there is no damping for the Alfvén-cyclotron branch at
wave-numbers $k_{0}<0.5\Omega_{p}/V_{\mathrm{A}}$. Thus the chosen spectra are
stable with respect to linear plasma instabilities, which become important at
higher wave-numbers $k_{0}>0.5\Omega_{p}/V_{\mathrm{A}}$, as shown for example
by (Ofman & Viñas, 2007). To derive the cold plasma dispersion for parallel
wave propagation, we start with the combined set of multi-fluid-Maxwell
equations, which govern the behavior of low-frequency waves and particles in a
drifting isotropic plasma:

$\displaystyle\frac{\partial n_{s}}{\partial
t}+\nabla\cdot(n_{s}\mathbf{V}_{s})=0$ (7)
$\displaystyle\frac{\partial\mathbf{V}_{s}}{\partial
t}+\mathbf{V}_{s}\cdot\nabla\mathbf{V}_{s}-\frac{q_{s}}{m_{s}}(\mathbf{E}+\mathbf{V}_{s}\times\mathbf{B})+\frac{\nabla
P_{s}}{m_{s}n_{s}}=0$ (8)
$\displaystyle\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}$
(9) $\displaystyle\nabla\times\mathbf{B}=-\mu_{0}\mathbf{J}.$ (10)

The solution of this system for parallel wave-propagation is given by the
dispersion relation (Smith & Brice, 1964; Sonnerup & Su, 1967; Araneda et al.,
2009):

$k_{0}^{2}-\frac{\mu_{0}}{B^{2}_{0}}\sum_{i\neq
e}n_{i}m_{i}\frac{{\left(\omega_{0}-V_{i}k_{0}\right)}^{2}}{1-\omega_{0}/\Omega_{i}+V_{i}k_{0}/\Omega_{i}}=0,$
(11)

where quasi-neutrality and current conservation are assumed to eliminate the
electron contribution and the summation is carried over the ions. Since we are
interested in low-frequency waves in non-relativistic plasma, we have
neglected the vacuum electromagnetic fluctuations described by the
$\omega_{0}^{2}/c^{2}$ term. The dispersion relation for warm drifting multi-
species plasma can be found for example in (Davidson & Ogden, 1975; Gary,
1993; Xie et al., 2004a). For the velocity fluctuations of the ion species we
obtain the relation:

$\mathbf{V}_{\perp
i}=-\frac{\omega_{0}/k_{0}-V_{i}}{1-(\omega_{0}-V_{i}k_{0})/\Omega_{i}}\delta\mathbf{B}_{\perp}/B_{0},$
(12)

where $V_{i}$ is the bulk velocity speed for each species, which determines
the differential streaming between the protons and the $\alpha$ particles:
$V_{\alpha p}=V_{\alpha}-V_{p}.$ The velocity and the magnetic field
fluctuations are coupled, so that the initially isotropic ions acquire a bulk
transverse motion, determined by the properties of the wave. When no drifts
are considered the corresponding velocity fluctuations for the $\alpha$
particles are much higher than the ones for the protons for the same given
choice of initial parallel Alfvén-cyclotron waves, see e.g., Ofman et al.
(2005). In the presence of relative drifts the bulk velocity fluctuations for
the protons and the $\alpha$ particles change. For $V_{\alpha
p}=0.2V_{\mathrm{A}}$ the transverse velocity fluctuations for the 2 species
become of the same order and for $V_{\alpha p}=0.5V_{\mathrm{A}}$ the proton
velocity fluctuations are dominant.
The initial spectrum consists of 16 modes laying on the left-hand circularly-
polarized Alfvén-$\alpha$-cyclotron branch (with a frequency below the
$\alpha$-cyclotron frequency). As noted above, the spectra is chosen from the
low-frequency solution of the dispersion relation (see Figure 1), which is
stable with respect to warm plasma linear Vlasov instability theory and
contains no significant wave amplification, nor damping. The selected
normalized wave-number space given in terms of the ion inertial length as
given by the local proton gyrofrequency $\Omega_{p}$ and the local Alfvén
speed $V_{\mathrm{A}}$ is $k_{0}\in[0.26-0.52]\Omega_{p}/V_{\mathrm{A}}$ or
equivalently by the ratio of the proton plasma frequency to the speed of light
$\omega_{p}/c.$ The corresponding frequencies for the individual runs are
presented in Table 1. Note that in a previous 1.5D study, (Maneva et al.,
2013b), we find that spreading the wave power within a rather wide range of
frequencies and wave-numbers, including very low-frequency MHD-type non-
dispersive waves, leads to significant reduction in the ion heating and
particularly acceleration rate. This is due to the slow direct cascade and the
low left-over power at the relevant kinetic scales. To investigate the ion
heating and acceleration, in the present study we have considered initial wave
spectra within the intermediate frequency range given above (in between the
low-frequency MHD type Alfvén waves and the resonant ion-cyclotron waves),
where dispersive effects facilitate nonlinear cascade and energy transfer
towards the relevant scales for wave-particle interactions. For this test
study, the simulations are performed with a total magnitude for the entire
initial wave spectra chosen to be $20\%$ of the magnitude of the external
background magnetic field $|\delta\mathbf{B}_{\perp}|=0.2B_{0}$. If the
constituent waves had the same phase (which is randomly selected in our
model), this would translates to amplitudes of the individual waves of about
$1\%$ of the ambient magnetic field. Recent in situ observations of
monochromatic Alfvén-cyclotron waves with similar amplitudes at 1AU have been
presented in (Jian et al., 2009). The authors of that paper argue that larger
amplitudes should be expected at smaller heliocentric distances, closer to the
Sun. Large-amplitude magnetic fluctuations, presumably of Alfvénic nature,
have also been detected by Helios at various distances in the inner
heliosphere (Bourouaine et al., 2011). Wind observations of large-amplitude
Alfvén waves at 1AU have recently been reported in Wang et al. (2012).

###  2.2 Expanding box model

The expanding box model used in our simulation setup follows the original work
by (Grappin & Velli, 1996) and (Liewer et al., 2001), which has been further
utilized by many authors, for example Hellinger et al. (2005); Hellinger &
Trávníček (2006); Ofman et al. (2011b); Hellinger & Trávníček (2013); Maneva
et al. (2013b); Ofman et al. (2014). The expanding box equations for a two-
dimensional hybrid simulation setting read:

$\frac{\mathrm{d}x^{{}^{\prime}}}{\mathrm{d}t^{{}^{\prime}}}=v_{x}^{{}^{\prime}},\quad\frac{\mathrm{d}y^{{}^{\prime}}}{\mathrm{d}t^{{}^{\prime}}}=\frac{1}{a(t)}v_{y}^{{}^{\prime}}$
(13)

where $x$ is the direction of radial expansion and the external magnetic
field, $a(t)=1+U_{0}t/R_{0}$ and the ion velocities in the expanding (prime)
frame and at rest are related by the magnitude of the constant solar wind bulk
speed $U_{0}$ and the initial distance from the Sun $R_{0}$

$v_{x}^{{}^{\prime}}=v_{x}-U_{0},\quad
v_{y}^{{}^{\prime}}=v_{y}-U_{0}/R_{0}y^{{}^{\prime}},\quad
v_{z}^{{}^{\prime}}=v_{z}-U_{0}/R_{0}z^{{}^{\prime}}.$

The electric field in the expanding (prime) frame of reference is computed
from the momentum equation for the massless fluid isothermal electrons

$n_{e}m_{e}\frac{\mathrm{d}\mathbf{V}^{{}^{\prime}}_{e}}{\mathrm{d}t}=-en_{e}\mathbf{E}^{{}^{\prime}}+\frac{(\nabla^{{}^{\prime}}\times\mathbf{B}^{{}^{\prime}})\times\mathbf{B}^{{}^{\prime}}}{4\pi}-\frac{\mathbf{J^{{}^{\prime}}_{i}}\times\mathbf{B^{{}^{\prime}}}}{c}-\nabla^{{}^{\prime}}P_{e}=0,$

where $\mathbf{J_{i}^{{}^{\prime}}}$ is the ion current in the co-moving
reference frame, $P_{e}=n_{e}k_{\mathrm{B}}T_{e}$ is the electron pressure and
the transformations for the electric field and the bulk velocities for all
species are respecively
$\mathbf{E}^{{}^{\prime}}=\mathbf{E}+\mathbf{U_{0}}\times\mathbf{B}$ and
$\mathbf{V_{s}^{{}^{\prime}}}=\mathbf{V_{s}}-\mathbf{U_{0}}.$ The evolution of
the magnetic field is modified by the expansion as follows

$\frac{\partial\mathbf{B}^{{}^{\prime}}}{\partial
t}=-c(\nabla^{{}^{\prime}}\times\mathbf{E}^{{}^{\prime}})-\frac{U_{0}}{R(t)}\hat{\mathbf{L}}\cdot\mathbf{B}^{{}^{\prime}},$
(14)

where $\hat{\mathbf{L}}=(2,1,1)\delta_{ij}$ is the transformation matrix which
accounts for the effect of the slow expansion, determined by the expansion
parameter $\varepsilon t\equiv U_{0}/R_{0}\ll 1,$ where $t$ is given in units
of $\Omega_{p}^{-1}$. In the considerations above all second and higher order
terms of the type $O(\varepsilon^{2})$ have been neglected. In the present
hybrid model we solve the above system of equations conserving the net charge
and the total current. The charge neutrality for the 3 species plasma is set
by $n_{e}=n_{p}+2n_{\alpha}$ and the simulations are initialized with zero net
current, which is conserved throughout the dynamical evolution of the system.
As an initial state we consider isotropic plasma with equal ion temperature
for the protons and the $\alpha$ particles, $T_{\alpha}=T_{p}$, and different
value of the parallel relative drift speed between the 2 species $V_{\alpha
p}$. The parallel components of the corresponding ion plasma beta
$\beta_{\parallel i}=v^{2}_{\mathrm{th},\parallel i}/V^{2}_{\mathrm{A}},$
defined in terms of the parallel ion thermal speed $v_{\mathrm{th},\parallel
i}$ and the local Alfvén speed $V_{\mathrm{A}}$ is $\beta_{\alpha}=0.02$ for
the He++ ions and $\beta_{p}=0.08$ for the protons. The $\alpha$ particles
constitute $5\%$ of the background electron density, $n_{\alpha}=0.05n_{e}$
and the plasma beta of the isothermal fluid electrons is
$\beta_{e}=2\mu_{0}n_{e}k_{\mathrm{B}}T_{e}/B_{0}^{2}=0.05$. Most simulations
are computed until $t=700\Omega_{p}^{-1}$, whereas the simulations with higher
expansion factor (see Table 1) are computed until $t=600\Omega_{p}^{-1}$ to
comply with the approximation used in the expanding box model, $\varepsilon
t<<1$.

##  3 Results

Figure 1 describes the shift of the dispersion relation branches due to the
presence of relative drift speed between the protons and the $\alpha$
particles within the cold quasi-neutral plasma approximation considered here.
The red line represents the right-hand polarized fast mode, excited by the
electrons. The blue line corresponds to the left-hand polarized proton-
cyclotron branch and the orange curve represents the left-hand polarized
Alfvén-cyclotron waves, related to the heaviest $\alpha$ particles in the
three-species plasma considered here. The selected initial wave-spectra
belongs to the latter low-frequency Alfvén-alpha-cyclotron branch and is
overplotted in brown color. The relative drift leads to asymmetries in the
forward and backward propagating Alfvén-cyclotron branches and increases the
forbidden bands between the proton- and the $\alpha$-cyclotron branches found
in the linear dispersion relation. Note, that the relative drift has no
visible influence over the initially selected frequency range for our
simulations. Nevertheless, it significantly changes the bulk velocity
fluctuations for the different ions, see Eq. 12 and Figure 2. The relative
drifts also influence the turbulent evolution of the initial wave spectra as
shown later on Figures 10-13.
Figure 2 shows the Alfvénic correlation between the reconstructed initial ion
bulk velocities and the magnetic field fluctuations as a function of the
simulation box length in “$x$” direction, parallel to the background magnetic
field. For the sake of simplicity the figure shows only one component of the
transverse velocities (given in red) and magnetic field fluctuations
(described by the blue lines). The other transverse components exhibit
identical correlation. The left panel shows the correlation for protons and
the right panel describes the $\alpha$ particles. The plots illustrate the
change in the magnitude of the velocity fluctuations for the different ion
species depending on the value of the initial relative drift speed. The top
panels correspond to non-drifting distributions. The middle panels show the
case with $V_{\alpha p}=0.2V_{\mathrm{A}}$ and the lower panels stand for the
case with $V_{\alpha p}=0.5V_{\mathrm{A}}.$ The figure demonstrates how the
transverse velocity fluctuations for the $\alpha$ particles decrease with
increasing relative drift speed. Since the current conservation condition in a
frame co-moving with the electrons implies $V_{\alpha}>>V_{p}$, the proton
velocity fluctuations are in the opposite direction to the $\alpha$’s and
remain nearly unaffected by the values of the relative drift considered here.
Figure 3 shows the temporal evolution of the ion temperature anisotropies and
the relative drift speed. Overplotted with dashed lines is the effect of a
gradual solar wind expansion with $\varepsilon t_{0}=10^{-4}$, where
$t_{0}=1\Omega_{p}^{-1}$. The initial temperature anisotropy for both species
is apparent and is caused by the non-thermal transverse velocity fluctuations,
induced by the waves. The onset of the differential streaming between the
protons and the $\alpha$ particles is caused by the acceleration of the proton
beam in the system with zero net currents. The close to realistic expansion
factor considered here leads to a minor change in the bulk ion properties.
Figure 4 and Figure 5 also describe the temporal evolution of the ion
anisotropies and relative drift speed, but for the case of initially drifting
distributions with $V_{\alpha p}=0.2V_{\mathrm{A}}$ and $V_{\alpha
p}=0.5V_{\mathrm{A}}$, respectively. The effect of the gradual expansion
varies with the value of the initial relative drift speed. It leads to less
than $2\%$ change in the ion temperature anisotropies and a $4.5\%$ change in
the relative drift speed for the case of $V_{\alpha p}=0.2V_{\mathrm{A}}$. As
we increase the initial relative drift speed, the solar wind expansion leads
to about $17\%$ reduction for the temperature anisotropy of the $\alpha$
particles and a $8.9\%$ reduction of the proton temperature anisotropy,
whereas the evolution of the relative drift speed stays almost unchanged.
In order to understand the relation between the initial wave spectra and the
apparent temperature anisotropy of the ions we should take into account
several factors. To begin with, the simulations are initialized with equal
temperatures for both ion species. The normalized parallel ion temperature for
each species is set by half the product of their mass ratio and the parallel
plasma $\beta$, i.e., $T_{\parallel,i}=m_{i}\beta_{\parallel,i}/2m_{p},$ so
that $T_{\parallel,p}=T_{\parallel,\alpha}=0.04$. Since we assume initially
isotropic plasma, the initial perpendicular temperatures for both ion species
are also equal. What appears on the plots instead are the apparent
temperatures and anisotropies, which are calculated based on the thermal
velocities of the ions and the non-thermal ion motions, imposed by the initial
wave spectra, see Eq. 12. Since we consider only parallel waves their effect
on the ions is only in perpendicular direction. Therefore the larger non-
thermal component of the velocity fluctuations induce higher apparent
temperature anisotropies. This picture changes for oblique wave propagation,
where the waves affect also the parallel motion of the ions and induce
parallel bulk velocity fluctuations as well [Maneva et al., in preparation].
For the initial wave spectra used in our simulations the magnitude of the
velocity fluctuations of the $\alpha$ particles and the protons constitutes
from $42\%$ to almost twice the value of their initial perpendicular thermal
speeds ($v_{\mathrm{th},\perp\alpha}=\sqrt{0.02}V_{\mathrm{A}}$ and
$v_{\mathrm{th},\perp p}=\sqrt{0.08}V_{\mathrm{A}}$). The ratio of the average
wave-induced perpendicular bulk velocity fluctuations to the thermal
velocities of the ions at the beginning of the simulations as a function of
the relative drift speed between the two species is presented in Table 1. At
zero drifts the motion of the $\alpha$ particles is strongly affected by the
wave-spectra, see Figure 2, and the average of their perpendicular velocity
flucutations for the chosen broad-band spectra is
$<V_{\perp\alpha}>=0.26V_{\mathrm{A}}$ vs. $<V_{\perp p}>=0.15V_{\mathrm{A}}$
for the protons. As the relative drift speed increases to $V_{\alpha
p}=0.2V_{\mathrm{A}}$ the selected wave-spectra affects the $\alpha$ particles
and the protons in a similar manner and the imposed velocity flucutuations for
the two species are of the same order: $<V_{\perp\alpha}>\approx<V_{\perp
p}>\approx 0.16V_{\mathrm{A}}.$ For higher drifts, $V_{\alpha
p}=0.5V_{\mathrm{A}}$, the influence of the electromagnetic field of the wave
spectra on the $\alpha$ particles is negligible
$<V_{\perp\alpha}>=0.06V_{\mathrm{A}},$ whereas its effect on the protons
remains almost unchanged $<V_{\perp p}>=0.17V_{\mathrm{A}}.$ We should note
that for a fixed type of waves the relation between the velocity fluctuations
and the relative drift speed strongly depends on the selected spectral range,
as a shift in $\omega-k$ space would change the resonant condition according
to Eq. 12. This relation is only valid for parallel wave propagation within
the cold isotropic homogeneous plasma limit and it would change once oblique
waves and anisotropic ions or plasma inhomogeneities are introduced.
Figures 6–8 describe the temporal evolution of the parallel and perpendicular
temperature components for protons and $\alpha$ particles for the three sets
of values for the initial relative drift speed, starting with a non-drifting
distributions and increasing the drift to $V_{\alpha p}=0.2V_{\mathrm{A}}$ and
$V_{\alpha p}=0.5V_{\mathrm{A}}.$ Each figure consists of two panels which
demonstrate the difference between the ion heating and cooling in the non-
expanding wind, panel (a), and the effect of a gradual expansion with
$\varepsilon t_{0}=10^{-4}$, panel (b). The perpendicular component of the ion
temperature is given by the solid red lines, whereas the parallel component is
plotted with dash-dotted blue lines. The expected perpendicular cooling as
predicted by the double-adiabatic CGL model is described by the solid green
lines. In all cases the expansion leads to significant perpendicular cooling
for both ion species and minor cooling in parallel direction. We observe about
16-20% decrease in the perpendicular temperature for the $\alpha$ particles
and about 6-11% decrease in the perpendicular temperature for the protons. The
parallel cooling is less prominent for both protons and $\alpha$ particles,
and it decreases with increase of the initial relative drift speed. Thus the
highest reduction of the ion temperature anisotropy for the $\alpha$ particles
due to the effect of the gradual expansion is at $V_{\alpha
p}=0.5V_{\mathrm{A}}$, refer to Figure 5. As discussed above, the presence of
relative drifts decreases the non-thermal component of the transverse velocity
for the minor ions, which reduces their apparent perpendicular temperature.
Despite the substantial perpendicular cooling caused by the expansion, the
perpendicular temperature decrease for the minor ions is less than what is the
expected by the CGL model and preferential heating for the $\alpha$ particles
is present in all simulation cases. The protons on the other hand are less
affected by initial and induced waves throughout the simulations, and their
evolution is close to the double-adiabatic expectations.
In Figure 9 we show the contour plot of the two-dimensional ion velocity
distribution function at the final stage of the simulations for the case of
initially drifting plasma with $V_{\alpha p}=0.2V_{\mathrm{A}}$. The parallel
velocity component is along the $x$ axis and the perpendicular one is along
the ordinate. The ion acceleration and anisotropic heating lead to deviation
from the initial isotropic distributions. Ion acceleration associated with
prominent ion beam formations occurs for both species. The beams are caused by
the turbulence-generated ion-acoustic fluctuations, similar to the parametric
instability scenario described in Araneda et al. (2009); Maneva et al. (2009).
The perpendicular heating is a combination of the quasilinear diffusion in
phase-space from the forced bouncing caused by the initial wave-spectra and
additional resonant scattering with the ion-cyclotron waves generated during
the non-linear evolution of the system. To facilitate the comparison of their
distributions, in this figure we use the same isotropic range in $v_{x}$ and
$v_{y}$ for both protons and the $\alpha$ particles. The 2D velocity
distributions for both ion species are symmetric in the phase-space formed by
the transverse velocity components $v_{y}$ and $v_{z},$ due to the circular
polarization of the waves and the ion gyro motion. This makes the particle
velocity distributions in $v_{x},v_{y}$ and $v_{x},v_{z}$ identical. The
velocity distribution functions, corresponding to the non-drifting plasma
case, show similar features for both ion species. However, in this case the
proton beams are stronger and the $\alpha$ particles are heated to higher
temperatures in both parallel and perpendicular direction. The stronger proton
beams and parallel minor ion heating is likely related to Landau damping of
the turbulence-generated ion density fluctuations, whose resonant absorption
depends on the relative drift speed.
Figure 10 and Figure 11 show the power spectral density of the magnetic field
fluctuations in Fourier space as a function of frequency and wave-numbers.
Figure 10 corresponds to a non-drifting non-expanding plasma, whereas Figure
11 refers to a drifting non-expanding plasma with $V_{\alpha
p}=0.2V_{\mathrm{A}}$. The top panels describe the redistributed wave power
for the duration of the simulations as a function of the parallel wave-number,
as projected along the ambient magnetic field. In both simulation cases, with
and without relative drifts, the power spectra is asymmetric with respect to
the direction of propagation. There are strong forward-propagating ion-
cyclotron modes at $k_{\parallel}\leq 0.35\Omega_{p}/V_{\mathrm{A}}$, remnants
from the initial broad-band spectra, together with powerful forward-
propagating fast modes, which are excited in the course of evolution. The
initial waves with $k_{\parallel}>0.35\Omega_{p}/V_{\mathrm{A}}$ quickly decay
and are absorbed by the plasma. A broad-spectra of powerful backward
propagating higher order proton-cyclotron and some $\alpha$-cyclotron modes
are also generated and intersect with the backward-propagating
Alfvén-$\alpha$-cyclotron branch. The parallel dispersion relation is
calculated by fixing a certain point in perpendicular $y$ direction and making
a fast Fourier transform of the magnetic field fluctuations in time $t$ and
along the parallel spatial coordinate $x$. The transverse dispersion relation
is shown on the bottom panels of Figures 10-11. It is calculated following a
similar procedure, but the fast Fourier transform was performed in $t$ and
$y$, for a fixed point in $x$. The perpendicular power spectra indicate the
presence of oblique modes, propagating across the ambient magnetic field. The
oblique waves have higher intensity in the non-drifting case. The magnetic
power spectra for both figures are computed over a large time interval,
starting at the beginning of the simulations at $t=0$ and ending towards the
end of the simulations at $t=600\Omega_{p}^{-1}$. In this way the plots
illustrate the turbulent evolution of the fluctuations and the wave coupling
at the fully nonlinear stage of the simulations.
Figure 12 and Figure 13 show the power spectral density of the magnetic field
fluctuations (wave power) as a function of the parallel and perpendicular wave
numbers at the end of the simulations for the same cases as described in
Figures 10-11 above. These complementary plots show the nonlinear anisotropic
cascade of wave energy from parallel towards perpendicular wave-numbers. The
two-dimensional spectra are computed with a two-dimensional Fourier transform
(along both spatial coordinates) of the fluctuating component of the magnetic
field at the end stage of the simulations. The figures demonstrate the role of
the differential streaming for the perpendicular energy cascade and oblique
mode generation. The initial state for both cases is a broad-band spectrum of
parallel Alfvén-cyclotron waves with parallel wave-numbers within the range
$k_{0}=k_{\parallel}=[0.26-0.52]\Omega_{p}/V_{\mathrm{A}}$. In the case of
non-drifting plasma the initial spectrum is strongly depleted above
$k_{\parallel}\leq 0.35\Omega_{p}/V_{\mathrm{A}}$ and parallel waves at larger
wavelengths are being generated, so that the initial spectra gets shifted
towards lower parallel wave-numbers. In the drifting plasma case, the initial
spectra for $k_{\parallel}>0.35\Omega_{p}/V_{\mathrm{A}}$ is not fully
depleted and the generated oblique waves are slightly stronger. For both cases
the strongest parallel waves are concentrated within
$|k_{\parallel}|\in[0.1-0.3]\Omega_{p}/V_{\mathrm{A}}$, whereas the power of
the generated oblique waves is concentrated within the limited range
$|k_{\perp}|\in[0-0.1]\Omega_{p}/V_{\mathrm{A}}.$ Should we think of the
spectra as a superposition of individual monochromatic waves, then we could
try to understand the turbulent evolution in terms of superposition of the
daughter waves generated by the parametric instabilities of the individual
constituent pump waves. If one makes such an analogy, we could say that there
are strong modulational and decay instabilities (Kauffmann & Araneda, 2008;
Maneva et al., 2009). There is an interplay between the modulational and the
decay-like instabilities in the non-drifting case and the drifting case. We
should note that the analogy with linear superposition of parametrically
unstable waves is given only as an illustration of a possible way to introduce
nonlinear effects. Realistic treatment of the solar wind turbulence requires
fully nonlinear wave-wave couplings and consequent wave-particle interactions
achieved in the present hybrid model.

##  4 Discussion and Conclusions

We have performed 2.5D hybrid simulations investigating the role of initially
imposed broad-band wave spectra in a drifting and expanding solar wind plasma.
We have studied the relaxation and evolution of the initial turbulent spectra
and their effect on the ions. The results from the 2.5D simulations yield
similar heating and acceleration rates for the ion species as the ones
observed in 1.5D simulations (Maneva et al., 2013a, b, 2014). The threshold
value for the differential acceleration, $V_{\alpha p}=0.5V_{\mathrm{A}}$
agrees with previous results from 1.5D hybrid simulations with single
monochromatic pump waves (Maneva et al., 2014) and the power spectra of the
magnetic field fluctuations shows that the initial parallel waves are evolving
and growing predominantly parallel to the background magnetic field with less
wave activity in the perpendicular direction. The combination of those effects
suggests that the parametric instabilities and the turbulent evolution of low-
frequency finite-amplitude Alfvén-cyclotron waves in 1.5D studies could serve
as a starting point for understanding parallel wave propagation in two-
dimensional turbulent problems.
Direct comparison between the nonlinear evolution of turbulence generated by
parametrically unstable pump waves and a broad-band initial turbulent wave
spectra based on a 1.5D hybrid simulation study shows similar spectral slopes
for the late-stage magnetic field and density fluctuations. Nevertheless the
resulting minor ion heating and differential acceleration are significantly
different due to the different fluctuations at the early stage of their
evolution, cf. with Fig. 3 from (Maneva et al., 2013a). This suggests that the
two processes (of initial pump waves and initial turbulent wave spectra)
should be considered complementary to each other, as they can be both realized
in the solar wind, but have different consequences on the ions.
For the plasma parameters considered here the linear Vlasov theory predicts
much higher threshold value for the relative drift speed required for the
onset of the two-streaming instability, $V_{\alpha p}>1.8V_{\mathrm{A}}.$ In
this respect the observed deceleration of the $\alpha$ particles at $V_{\alpha
p}>0.5V_{\mathrm{A}}$ is a nonlinear phenomenon, which requires higher-order
analytical treatment or direct numerical simulations. In this study we have
constructed a self-consistent initial wave spectra of parallel Alfvén-
cyclotron waves and computed the related velocity fluctuations in the presence
of differential streaming. We have estimated the apparent temperature
anisotropies corresponding to the non-thermal velocity components generated by
the initial wave spectra. We have demonstrated the effect of the differential
streaming on the apparent temperature anisotropies for the minor ions. In a
non-drifting plasma the $\alpha$ particles are strongly influenced by the
initial spectra. They acquire high transverse bulk velocity fluctuations, with
higher magnitude than the corresponding transverse component of the minor ion
thermal speed. These fluctuations are significantly reduced in the presence of
relative drifts. The protons are much less affected by the selected wave
spectra and their non-thermal velocity component constitutes a fraction of the
relevant proton thermal speed for all simulation cases. For the sake of
clarity we should note that 2.5D simulations with higher initial relative
drifts $V_{\alpha p}>0.5V_{\mathrm{A}}$ clearly show that the relative drift
speed decreases in time. Furthermore, 1.5D simulations show similar decrease
of the relative drift speed for $V_{\alpha p}\leq 1.8V_{\mathrm{A}}$ \- i.e.
throughout the entire linearly stable regime of the magnetosonic instability
(without waves) for the given plasma parameters (see Fig. 9 from (Maneva et
al., 2014)). For larger drifts the system is prone to the linear theory
streaming instability, which again causes the decrease of the relative drift
speed in time.
Although one-dimensional simulations can provide useful hints for
understanding the solar wind properties, they fail to capture other factors
like oblique wave generation, anisotropic turbulence, plasma structures, etc.
The solar wind expansion by construction is strongly influenced by the degrees
of freedom in the system and its role can be stronger in two-dimensional
systems. In addition, many of the observed solar wind magnetic fluctuations
have oblique nature, whose scattering and interaction with the co-existing
parallel waves require more than one spatial dimension. The present 2.5D
simulation study provides a further step towards a proper treatment of the
dynamic ion properties in the solar wind and the anisotropic evolution of the
solar wind micro-turbulence, which cannot be captured by 1.5D modeling.
We find that the presence of relative drift speed prolongs the lifetime of the
parallel waves at short wavelengths, as new waves with $|k_{\parallel}|\geq
0.35\Omega_{p}/V_{\mathrm{A}}$ are generated and the wave power is replenished
after the initial spectra is damped. This implicates that the turbulent
evolution of the plasma waves and the nonlinear wave-wave interactions are
affected by the differential streaming. In addition the relative drifts enter
the cyclotron damping resonant condition and Doppler-shift the frequency of
the Alfvén-cyclotron waves, required for resonant wave-particle interactions
with the minor ions. The selected initial spectra is far from resonance with
the protons for all values of the relative drift speeds considered in this
study. In addition the direct turbulent cascade is not strong enough to bring
sufficient wave power at the proton-cyclotron scales during the short
(kinetic) timescale of the simulations. Hence there is no perpendicular
heating for the protons at any time throughout the simulations. The
differential streaming also suppresses the parallel wave generation at large
wavelengths $|k_{\parallel}|<0.125\Omega_{p}/V_{\mathrm{A}}$ and facilitates
the oblique mode generation with enhanced energy transfer in the perpendicular
direction. In both drifting and non-drifting plasmas the oblique waves appear
confined within the range of $|k_{\perp}|\in[0,0.3]\Omega_{p}/V_{\mathrm{A}},$
with most of their power concentrated in
$|k_{\perp}|\in[0-0.1]\Omega_{p}/V_{\mathrm{A}}.$
The close-to-realistic values of the solar wind expansion factor in the inner
heliosphere used here, $\varepsilon t_{0}=10^{-4}$, lead to only a few percent
change in the evolution of the differential streaming on the temporal scales
considered here. Still, the expansion plays significant role in decreasing the
ion temperature anisotropies and the perpendicular ion temperatures – up to
20% for the $\alpha$ particles. As the minor ions are preferentially heated by
the existing waves in the system, their perpendicular temperature remains
higher than the double adiabatic prediction. The protons on the other hand
experience no wave heating and their cooling in time is practically given by
the CGL model. The slight additional cooling comes from the lack of exact
energy conservation in the system, where the total electromagnetic energy is
preserved within $14\%$, the kinetic energy of the protons is preserved with
$1.5\%$ and the kinetic energy for the alpha particles is preserved within
$11\%$. Although the energy is quasi-conserved, there is still some energy
loss in the system as all the above energies slightly decrease in time. If we
account for this numerical effect we would expect even higher heating for the
minor ions.
We should note that the gradual solar wind expansion considered here would
have stronger net effect as we follow the plasma evolution over a longer time
interval. Assuming expansion factor of $\varepsilon t_{0}=5\times 10^{-4}$
already leads to a substantial reduction of the relative drift speed (up to
$40\%$) and the ion temperature anisotropies (up to 51%), as visible from
Table 1. The faster expansion also changes the nonlinear evolution of the
initial wave spectra, affecting the wave-particle interactions and the
resulting shape of the ion velocity distribution functions. Since the wave-
particle interactions operate much faster than the expansion time-scales,
though, a pure increase in the expansion factor might not properly describe
the plasma evolution. A self-consistent description of the evolution of the
turbulent fast solar wind plasma requires long simulation times and is
computationally very expensive even in 2.5D hybrid kinetic models. It is out
of the scope of this paper to investigate the turbulent cascade at very large
MHD scales. However the present study is able to capture the dynamics at the
ion scales and demonstrate the back-reaction of the ions on the cascade
processes and the evolution of the turbulent wave spectra at the intermediate
ion scales and some limited part of the larger MHD spectra.

Figure 1: Initial parallel propagating Alfvén-$\alpha$-cyclotron waves spectra
– solution of the coupled set of cold quasi-neutral multi-fluid plasma and the
non-relativistic Maxwell equations. The top panel shows the solution for non-
drifting plasma. The middle panel shows the solution for relative drift speed
$V_{\alpha p}=0.2V_{\mathrm{A}}$. The bottom panel shows the solution for the
special case of relative drift speed $V_{\alpha p}=0.5V_{\mathrm{A}}$. The
initial wave spectra was selected from the Alfvén-$\alpha$-cyclotron branch
and its $[\omega_{0},k_{0}]$ range is shown in solid brown.

Figure 2: Magnetic field (blue lines) and transverse velocity fluctuations
(red lines) for the protons and $\alpha$ particles, corresponding to the
initial broad-band spectra of Alfvén-cyclotron waves. The plot shows the
strong dependence of the induced transverse velocity fluctuations for the
minor ions on the value of the initial relative drift speed. Figure 3:
Temporal evolution of the ion temperature anisotropies and relative drift
speed, showing the generation of differential streaming between protons and
$\alpha$ particles. The solid lines indicate simulations without expansion and
the dashed lines show the results when a gradual expansion with $\varepsilon
t_{0}=10^{-4}$ was considered. The apparent initial anisotropies are caused by
the waves. Figure 4: Temporal evolution of the ion temperature anisotropies
and relative drift speed for the case of differentially streaming $\alpha$
particles with $V_{\alpha p}=0.2V_{\mathrm{A}}$. The dashed lines denote the
ion bulk properties in the case of a gradual expansion with $\varepsilon
t_{0}=10^{-4}$. Figure 5: Temporal evolution of the ion temperature
anisotropies and relative drift speed for the case of differentially streaming
$\alpha$ particles with $V_{\alpha p}=0.5V_{\mathrm{A}}$. The solid lines
indicate simulations without expansion and the dashed lines denote the results
with $\varepsilon t_{0}=10^{-4}$. Figure 6: Temporal evolution of the parallel
and perpendicular components of the ion temperatures for initially non-
drifting distributions. Panels (a) shows a non-expanding case. Panel (b)
illustrates the expansion with $\varepsilon t_{0}=10^{-4}$. The top rows
describe the $\alpha$ particles, whereas the bottom ones depict the protons.
The solid red lines denote the perpendicular and the dash-dotted blue lines
denote the parallel temperature components. The solid green lines denote the
perpendicular temperature decrease as predicted by the CGL model. Figure 7:
Temporal evolution of the individual ion temperatures for the case of
initially drifting plasma with $V_{\alpha p}=0.2V_{\mathrm{A}}.$ Similar to
Figure 6, the right panel shows the effect of a gradual expansion with
$\varepsilon t_{0}=10^{-4}$. Figure 8: Temporal evolution of the individual
ion temperatures for the case of initially drifting plama with $V_{\alpha
p}=0.5V_{\mathrm{A}}.$ The right panel shows the effect of the gradual
expansion with $\varepsilon t_{0}=10^{-4}$. The notations are the same as on
Figure 6 and Figure 7.

Figure 9: Snapshot of the two-dimensional velocity distribution functions for
protons (left) and $\alpha$ particles (right) for the case of initially
drifting isotropic and isothermal plasma with $V_{\alpha
p}=0.2V_{\mathrm{A}}.$ The reduced distributions in the lower panel show the
number of particles as a function of the parallel component of the ion thermal
speeds. A best-fit Maxwellian velocity distribution is overplotted with the
dotted line.

Figure 10: Dispersion diagram $\omega$ vs. $k$, representing the wave
intensity as a function of wave frequency vs parallel (up) and perpendicular
(bottom) wave-number space. The graph corresponds to the case of non-drifting
initial distributions.

Figure 11: Wave intensity as a function of wave frequency vs parallel (up) and perpendicular (bottom) wave-number for the case of a drifting non-expanding plasma with $V_{\alpha p}=0.2V_{\mathrm{A}}$. Figure 12: Wave power as a function of the parallel vs. perpendicular wave-number for the case of a non-drifting non-expanding plasma at the end stage of the simulations $\Omega_{p}t=700$. Figure 13: Wave power as a function of the parallel vs. perpendicular wave-number at $\Omega_{p}t=700$ for the case of a drifting non-expanding plasma with $V_{\alpha p}=0.2V_{\mathrm{A}}$. Table 1: Summary of the parameters used in the 2.5D hybrid simulation study. In column 2 and 3 we present the range of the normalized wave frequencies and wave-numbers, followed by the expansion parameter $\varepsilon t_{0}$ in column 4. Next are the ratios of the average of the wave-induced ion perpendicular bulk velocity fluctuations to the perpendicular component of the ion thermal speeds for each species, given in columns 5 and 6. The last four columns show the initial $V_{\alpha p,\mathrm{in}}$ and the final stage values $V_{\alpha p,\mathrm{fin}}$ of the relative drift speed, followed by the ion temperature anisotropies for protons and $\alpha$ particles, calculated at the end of the simulations. Case # |  $\omega_{0}$ [$\Omega_{p}$] |  $k_{0}$ [$\Omega_{p}/V_{\mathrm{A}}$] |  Exp. param. $\varepsilon t_{0}$ | $<V_{\perp\alpha}>/v_{\mathrm{th},\perp\alpha}$ | $<V_{\perp p}>/v_{\mathrm{th},\perp p}$ | $V_{\alpha p,\mathrm{in}}[V_{\mathrm{A}}]$ | $V_{\alpha p,\mathrm{fin}}[V_{\mathrm{A}}]$ | $T_{\perp p}/T_{\parallel p}$ | $T_{\perp\alpha}/T_{\parallel\alpha}$
---|---|---|---|---|---|---|---|---|---
1 | [0.21-0.36] | [0.26-0.52] | 0 | 1.84 | 0.53 | 0 | 0.074 | 1.2 | 2.2
2 | [0.21-0.36] | [0.26-0.52] | $5\times 10^{-4}$ | 1.84 | 0.53 | 0 | 0.068 | 0.94 | 2.2
3 | [0.21-0.36] | [0.26-0.52] | $10^{-4}$ | 1.84 | 0.53 | 0 | 0.05 | 0.7 | 2.1
4 | [0.22-0.386] | [0.26-0.52] | 0 | 1.15 | 0.58 | 0.2 | 0.218 | 1.14 | 2.25
5 | [0.22-0.386] | [0.26-0.52] | $5\times 10^{-4}$ | 1.15 | 0.58 | 0.2 | 0.208 | 1.09 | 2.2
6 | [0.22-0.386] | [0.26-0.52] | $10^{-4}$ | 1.15 | 0.58 | 0.2 | 0.174 | 0.84 | 1.3
7 | [0.22-0.39] | [0.26-0.52] | 0 | 0.42 | 0.60 | 0.5 | 0.491 | 1.46 | 1.56
8 | [0.22-0.39] | [0.26-0.52] | $5\times 10^{-4}$ | 0.42 | 0.60 | 0.5 | 0.488 | 1.33 | 1.31
9 | [0.22-0.39] | [0.26-0.52] | $10^{-4}$ | 0.42 | 0.60 | 0.5 | 0.477 | 0.96 | 0.78

###### Acknowledgements.

This work was supported by NASA, grant NNX10AC56G. Y. G. Maneva would like to
thank the $F^{+}$ fellowship at KU Leuven for the partial support. A. F. Viñas
would like to thank the Wind/SWE MO/DA grant for the support. Fruitful
discussions with P. Hunnana are highly appreciated.

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