text:
\section{Introduction}\label{sec.1}

Plasma outflows released by the Sun in interplanetary space consist mainly of
electrons, protons and minor ions, among which the alpha particles are dominant
with an average abundance of about 5\% of the total number density of ions.
Owing to their mass density, typically 20\% of the total ion mass density,
alpha-ions may have important implications in the solar wind dynamics
\citep{robbins1970helium, Marsch-2006, kasper2007solar, Maruca2012,
Maneva-etal-2014}. The {\it in-situ} measurements in the fast wind and
regions not too distant from the Sun, i.e., $\lesssim 1$~AU, have revealed
alpha particle streams faster than the protons, with relative speeds of the
order of local Alfv{\'e}n speed \citep{Asbridge1976, Marsch-1982,
neugebauer1981observations}, and kinetic temperatures in general higher
than proton temperature \citep{Kohl1998}. This preferential energization of
heavier ions in collision-poor plasmas is attributed to the interaction with
plasma waves, via Landau or cyclotron resonances \citep{Marsch-2006}, and
is supported by observations \citep{marsch1991kinetic, vonSteiger1995}.

The observed velocity distributions display strong deviations from thermal
equilibrium, often with a dual structure combining a proton core and a
drifting alpha population called alpha beam \citep{Marsch-1982}.
Non-thermal features of the observed ion distributions are sources of free
energy that can locally generate instabilities and enhanced wave fluctuations
\citep{Garybook}. In our case, beaming alpha populations may destabilize the
plasma system via excitations associated with the Alfv\'en/ion cyclotron waves
and the fast magnetosonic/whistler modes \citep{Revathy-1978, Gary-etal-2000a,
Li-Habbal-2000}. In turn, these fluctuations regulate the ion VDFs through
the same wave-particle interactions, which determine a diffusion in velocity
space and a thermalisation and relaxation of the alpha beam
\citep{Marsch-Livi-1987, Gary-etal-2000b}. The observations confirm that
the alphas are heated in perpendicular direction and cool more slowly than
what would be expected from adiabatic expansion \citep{Reisenfeld-etal-2001,
Stansby-etal-2019}.

The investigations of alpha-ion beams have initially concluded that high
plasma beta conditions, e.g., $\beta \geqslant 1$, more typical to large
heliospheric distances ($\geqslant 1$~AU), can be favorable to magnetosonic
instabilities, with a maximum growth for propagation parallel to the
interplanetary magnetic field \citep{Gary-etal-2000a, Gary-etal-2000b}.
Later it was shown that magnetosonic instabilities were stimulated by an
internal anisotropic temperature $T_\perp/T_\parallel < 1$ of protons or
alpha particles ($\parallel, \perp$ being directions with respect to
the magnetic field), while isotropic beams, with, e.g.,
$T_{\alpha, \perp}/T_{\alpha, \parallel} \simeq 1$, may drive another
instability of the parallel-propagating Alfv\'en/ion-cyclotron waves
\citep{Verscharen-etal-2013}. The ion-beam instabilities are also highly
dependent on the plasma beta, which decreases towards the Sun and in
the outer corona ($\sim 0.3$~AU) becomes
$\beta_{p,\alpha}=8\pi n_{p,\alpha}T_{p,\alpha}/B_0^2<1$ for both
the protons (subscript $p$) or alpha particles (subscript $\alpha$)
\citep{Marsch-1982}. Here, $n_{p,\alpha}$ denote proton/alpha number
densities, $T_{p,\alpha}$ their respective temperatures (in the
unit of energy), and $B_0$ denotes the ambient magnetic field intensity.
The low-beta alpha/proton beams appear to be more
susceptible to Alfv\'enic  instabilities, but again, it is not clear
if these instabilities are driven by the alpha/proton drift or by
the anisotropic temperature $T_\perp/T_\parallel > 1$, as such a
condition is always assumed for protons or alpha particles in
the literature \citep{Li-Habbal-2000, Gomberoff-Valdivia-2003}.
In the outer corona an excess of temperature (kinetic energy) in
perpendicular direction may be associated only to the proton core,
while suprathermal populations including proton halo or beams of protons
and alpha particles rather show an opposite, relatively small anisotropy
$T_\perp/T_\parallel \lesssim 1$ \citep{Marsch-1982, Marsch-2006}.

In order to complete the picture and provide a clear basic characterization
of these instabilities, in the present paper we reduce the complexity of
the low-beta beaming plasmas, and investigate alpha/proton drifting
populations with isotropic temperatures $T_\perp/T_\parallel = 1$.
The dispersion and stability properties of the plasma system are derived
on the basis of a general kinetic approach, which cover the full spectrum
of wave-frequencies, wave-numbers and angles of propagation with respect
to the magnetic field. The organization of our paper is the following:
In section~\ref{sec.2}, we derive the general dispersion relation for the
electromagnetic modes propagating at an arbitrary angle. Growth rates of
the unstable solutions are obtained for standard representations of the
particle velocity distribution as drifting-Maxwellians in section \ref{sec.3}.
The unstable solutions are discussed by numerical means in Section~\ref{sec.4},
providing also an analysis of their variation with the main plasma parameters.
Section~\ref{sec.5} summarizes our present results.

\section{Dispersion Relation}\label{sec.2}

In a collision-poor plasma the general linear dispersion relation
for electromagnetic modes propagating at an arbitrary angle $\theta$
with respect to the uniform background magnetic field
($\bm{B}_0=B_0\hat{\bm{z}}$)  is given by
\begin{equation}
0=\det \left|\epsilon_{ij}(\bm{k},\omega)
-\frac{c^2 k^2}{\omega^2}\left(\delta_{ij}
-\frac{k_ik_j}{k^2}\right)\right|,
\label{1}\end{equation}
with a dielectric tensor defined by
\begin{eqnarray}
\epsilon_{ij} &=& \delta_{ij}
+\sum_a\frac{\omega_{pa}^2}{\omega^2}
\int d{\bf v}\left[v_\parallel\left(
\frac{\partial}{\partial v_\parallel}
-\frac{v_\parallel}{v_\perp}
\frac{\partial}{\partial v_\parallel}\right)
b_jf_a\right.
\nonumber\\
&& +\sum_{n=-\infty}^\infty\frac{V_iV_j^*}
{\omega-n\Omega_a-k_\parallel v_\parallel}
\nonumber\\
&& \left.\times\left(\frac{n\Omega_a}{v_\perp}
\frac{\partial}{\partial v_\perp}+k_\parallel
\frac{\partial}{\partial v_\parallel}\right)f_a\right],
\label{HM}\end{eqnarray}
where $\omega$ is the wave frequency, $b_i=\bm{B}_0/B_0=\hat{\bm{ z}}$
is the unit vector along the direction of ambient magnetic field
vector, $f_a(v_\perp, v_\parallel)$ is the velocity distribution
function for particle species labeled $a$ ($a=e,p,\alpha$, etc.,
for electrons, protons, alpha particles, etc.),
$\omega_{pa}=~\sqrt{4 \pi n_a e^2/m_a}$ and
$\Omega_a=e_a B_0/m_a$ are the plasma and cyclotron frequencies
for species $a$, and
\begin{eqnarray}
V_i &=& \left(v_\perp\frac{nJ_n(b)}{b},\mbox{ }
-iv_\perp J'_n(b),\mbox{ }
v_\parallel J_n(b)\right)_i,
\nonumber\\
b &=& \frac{k_\perp v_\perp}{\Omega_a}.
\label{3}\end{eqnarray}
Here, $J_n$ is a Bessel function of the first kind, of order
$n$, with argument $b$. In the above, $e$, $n_a$, $m_a$, and $c$
stand for unit electric charge, ambient density for plasma
particle species $a$, their mass, and the speed of light
{\it in vacuo}, respectively.

In the present analysis we adopt the cold plasma approximation for
the real frequency, combined with warm plasma growth/damping rate
expression in the context of weak growth/damping formula, or equivalent,
the dissipative instability formalism. The weak growth or dissipative
instability theory is also known as the weak kinetic instability
theory, as opposed to the reactive or fluid instability theory. We take
the general cold plasma dielectric tensor elements given
in component form by
\begin{align}
\epsilon_{xx} &=1-\frac{\omega_{pe}^2}{\omega^2-\Omega_e^2}
-\sum_a\frac{\omega_{pa}^2}{\omega^2-\Omega_a^2}=\epsilon_{yy},
\nonumber\\
\epsilon_{xy} &=-i\,\frac{\Omega_e}{\omega}
\frac{\omega_{pe}^2}{\omega^2-\Omega_e^2}
+i\sum_a\frac{\Omega_a}{\omega}
\frac{\omega_{pa}^2}{\omega^2-\Omega_a^2}=-\epsilon_{yx},
\nonumber\\
\epsilon_{xz} &=\epsilon_{zx}=0
=\epsilon_{yz}=\epsilon_{zy},
\nonumber\\
\epsilon_{zz} &=1-\frac{\omega_{pe}^2}{\omega^2}
-\sum_a\frac{\omega_{pa}^2}{\omega^2},
\label{a1}\end{align}
and further approximate the situation by considering two species
ions, namely, protons and alpha particles, and ignore the
displacement current, that is, the unity on the right-hand side of
Eq.\ \eqref{a1}, which is valid for low frequency waves. We also assume
$\omega^2\ll\Omega_e^2$, in order to simplify the dielectric tensor elements,
\begin{align}
\epsilon_{xx} &\approx-\frac{\omega_{pp}^2}{\omega^2-\Omega_p^2}
-\frac{\omega_{p\alpha}^2}{\omega^2-\Omega_\alpha^2},
\nonumber\\
\epsilon_{xy} &=i\,\frac{\omega_{pp}^2}{\omega\Omega_p}
\left(\frac{n_0}{n_p}
+\frac{\Omega_p^2}{\omega^2-\Omega_p^2}\right)
+i\,\frac{\Omega_\alpha}{\omega}
\frac{\omega_{p\alpha}^2}{\omega^2-\Omega_\alpha^2},
\nonumber\\
\epsilon_{zz} &=-\frac{m_p}{m_e}\frac{n_0}{n_p}
\frac{\omega_{pp}^2}{\omega^2},
\label{a5}
\end{align}
where $\Omega_p=eB_0/(m_pc)$, $\Omega_\alpha=\Omega_p/2$,
$\omega_{pp}^2=4\pi n_pe^2/m_p$, $\omega_{pe}^2=m_pn_0\omega_{pp}^2/(m_e n_p)$,  and
$\omega_{p\alpha}^2=(n_\alpha/n_p)\omega_{pp}^2$. Note that in $\epsilon_{zz}$ the other ion terms can be neglected compared to $m_p/m_e\gg1$.

Substituing the dielectric tensor elements \eqref{a5} to the
dispersion relation \eqref{1}, it can be shown that the
linear dispersion relation is given in dimensionless form by
\begin{equation}
q^2=\frac{sx}{(1+\mu^2)x
\mp\sigma\sqrt{(1+\mu^2)^2x^2+st\mu^2}},
\label{a8}\end{equation}
where
\begin{equation}
q=\frac{ck}{\omega_{pp}},\quad
x=\frac{\omega}{\Omega_p},\quad
\mu=\cos\theta,\quad
\delta=\frac{n_\alpha}{n_p},
\end{equation}
are dimensionless wave number, frequency, cosine of the wave
propagation angle, and the alpha-to-proton density ration, and
other quantities are defined by
\begin{eqnarray}
s &=& \frac{2(xQ_-+\delta P_-Q_+)
(xQ_++\delta P_+Q_-)}{D},
\nonumber\\
\sigma &=& \frac{|D|}{D},\quad
D=Q_+Q_-+\delta P_+P_-,
\nonumber\\
t &=& \frac{2P_+P_-Q_+Q_-}{D},
\nonumber\\
P_\pm &=& x\pm1,\qquad Q_\pm=x\pm\tfrac{1}{2}.
\label{a9}\end{eqnarray}

   \begin{figure}[t!]
   \begin{center}
   \includegraphics[width=\columnwidth]{fig1}
   \end{center}
   \caption{Dispersion relation plotted as $q^2=c^2k^2/\omega_{pp}^2$
   versus $x=\omega/\Omega_p$ curves for three different values
   of $\theta$ corresponding to $1^\circ$ (solid line), $60^\circ$ (dash-dotted), and
   $\theta=80^\circ$ (dotted).}
   \label{f1}
\end{figure}

Figure \ref{f1} plots $q$ versus $x$ for three values of $\theta$
corresponding to $1^\circ$, $60^\circ$, and $80^\circ$.
The dispersion curves show that they are composed of three
distinct branches. The first branch (topmost, color coded in red)
denotes the low frequency mode that begins as Alfv\'en wave for
low frequency ($\omega\ll\Omega_p$), but gradually turns into
the resonant mode (where $k\to\infty$) at the alpha cyclotron
frequency, $\omega=\Omega_\alpha=\Omega_p/2$.
For quasi parallel propagation angle ($\theta\to0$), this branch
is left-hand circularly polarized.
The second brach (color coded in blue for $\omega<\Omega_p$)
starts off in low frequency regime ($\omega\ll\Omega_p$) as
fast/magnetosonic mode, and this mode is right-hand circularly
polarized for quasi parallel angle of propagation.
For nonzero $\theta$, this mode switches over to the proton
cyclotron mode, which becomes resonant ($k\to\infty$) at the
proton cyclotron frequency ($\omega=\Omega_p$). The switchover
takes place in the vicinity of the alpha cyclotron
frequency. A third brach that begins as the alpha cyclotron
mode turns into the fast/magnetosonic mode above $\Omega_\alpha$
and beyond. The third branch is plotted with red color.
In general, modes designated with red color corresponds to the
lower sign in the dispersion relation \eqref{a8}, while the
blue curves belong to the upper sign.

In the growth rate calculation to be discussed subsequently,
we will denote instabilities operative on the topmost branch as
the ``alpha cyclotron'' instability, while instabilities operative
on the middle, blue-colored mode will be designated as the
``proton cyclotron'' instability. It turns out that the third,
bottommost curve -- that is, the fast/magnetosonic mode --
remains stable.

\section{Growth Rate}\label{sec.3}

Assuming weak growth/damping rate ($|\gamma|\ll\omega$), and following
the standard method of derivation \citep{Melrose}
we obtain an explicit expression for the growth/damping rate,
\begin{eqnarray}\label{eq14}
\gamma &=& \sum_a\frac{\pi\omega_{pa}^2}
{\left[1-|{\bf e}({\bf k})\cdot\hat{\bf k}|^2\right]
\left[\partial(\omega^2N^2)/\partial\omega\right]}
\nonumber\\
&& \times\int d{\bf v}\sum_{n=-\infty}^\infty
\left|{\bf e}({\bf k})\cdot{\bf V}\right|^2
\delta(\omega-n\Omega_a-k_\parallel v_\parallel)
\nonumber\\
&& \times\left(\frac{n\Omega_a}{v_\perp}
\frac{\partial}{\partial v_\perp}+k_\parallel
\frac{\partial}{\partial v_\parallel}\right)
f_a\left(v_\perp, v_\parallel\right),
\end{eqnarray}
where $N=ck/\omega$ is the index of refraction,
${\bf V}$ is defined in Eq.\ \eqref{3}, and ${\bf e}$
represents the unit electric field vector, which is
discussed in Appendix A. Making use of the
dispersion relation \eqref{a8} one may also compute
$\partial(\omega^2N^2)/\partial\omega$ explicitly, which
is given in Appendix B. In Eq.\ \eqref{eq14}
$f_a=f_a(v_\perp^2,v_\parallel)$ represents the velocity
distribution function for ion species labeled $a$.
The following is the resulting explicit expression for the
growth/damping rate after taking into account the
dispersive wave properties associated with the
low frequency modes:
\begin{eqnarray}\label{Rs}
\frac{\gamma}{\Omega_p} &=& \sum_{a=p,\alpha}
\frac{n_a}{n_p}\frac{m_p}{m_a}
\frac{2\pi^2}{(1+M^2x^2\mu^2)R}
\int_0^\infty dv_\perp v_\perp^3
\nonumber\\
&& \times\sum_{n=-\infty}^\infty
\left|M^2x^2\frac{n^2J_n^2}{b^2}
+(J'_n)^2-2Mx\frac{nJ_nJ'_n}{b}\right|
\nonumber\\
&& \times\left(\frac{n\Omega_a}{k v_\perp\mu}
\frac{\partial f_a(v_\perp^2,v_r)}{\partial v_\perp}
+\frac{\partial f_a(v_\perp^2,v_r)}{\partial v_r}\right),
\nonumber\\
v_r &=& \frac{\omega-n\Omega_a}{k\mu},
\end{eqnarray}
where the argument of the Bessel function $b=k_\perp v_\perp/\Omega_a$
applies for proton ($a=p$) and alpha particles ($a=\alpha$),
and $M$ and $R$ are defined in Eqs.\ \eqref{KTM} and \eqref{a48}, respectively.

For drifting Maxwellian distributed plasmas we replace the arbitrary
distribution function $f_a$ in the growth rate expression \eqref{Rs} by
\begin{equation}
f_a(v_\perp,v_\parallel)=\frac{1}{\pi^{3/2}v_{Ta}^3}
\exp\left(-\frac{v_\perp^2}{v_{Ta}^2}
-\frac{(v_\parallel-V_a)^2}{v_{Ta}^2}\right),
\end{equation}
where $v_{Ta}=(2T_a/m_a)^{1/2}$ stands for thermal speed, and
$V_a$ represents the average drift speed.
After some straightforward mathematical manipulations it can be
shown that the growth rate expression reduces to
\begin{eqnarray}\label{e21}
\frac{\gamma}{\Omega_p} &=& -\sum_a\frac{n_a}{n_p}
\frac{m_p}{m_a}\frac{\pi^{1/2}}{(1+M^2\mu^2)R}
\sum_{n=-\infty}^\infty
\nonumber\\
&& \times\biggl(\frac{n(1+M^2x^2)}{2}\,[I_{n-1}(\lambda_a)
-I_{n+1}(\lambda_a)]e^{-\lambda_a}
\nonumber\\
&& +(Mnx+\lambda_a)\,[2I_n(\lambda_a)-I_{n-1}(\lambda_a)
\nonumber\\
&& -I_{n+1}(\lambda_a)]e^{-\lambda_a}\biggr)
\xi_a\exp\left[-(\zeta_n^a)^2\right],
\end{eqnarray}
where $I_n$ is the modified Bessel function of
the first kind of order $n$, and
\begin{align}
&\xi_a=\frac{\omega-kV_a\mu}{kv_{Ta}\mu},
\nonumber\\
& \zeta_n^a=\frac{\omega-n\Omega_a-kV_a\mu}{kv_{Ta}\mu},
\nonumber\\
&\lambda_a =\frac{k^2v_{Ta}^2(1-\mu^2)}{2\Omega_a^2}.
\end{align}

\section{Numerical Analysis}\label{sec.4}

The growth rate of low frequency modes is a function of frequency
$\omega$ and angle of propagation $\theta$. It also depends implicitly
on alpha-proton number density ratio $n_\alpha/n_p$,
alpha particle drift velocity $V_\alpha$ (we assume zero drift
for the protons, $V_p=0$), plasma beta parameters $\beta_p$ and
$\beta_\alpha$, where $\beta_p=8\pi n_pT_p/B_0^2$ and
$\beta_\alpha=8\pi n_\alpha T_\alpha/B_0^2$, respectively.
In the fast solar wind alpha particles possess an average density of 5\%
of the total number density and are drifting with respect to the
protons with a typical speed on the order of local Alfv\'en speed,
$V_\alpha = v_A$ \citep{Marsch-1982, Reisenfeld-etal-2001}.
Consequently, in the present analysis, for alpha-proton relative
number density we consider $n_\alpha/n_p=0.05$, and for alpha drift velocity,
we fix the value at $V_{\alpha}=v_A$. For low corona the beta
values are relatively low. We thus consider low value of $\beta_\alpha=0.01$
and a slightly higher proton beta of $\beta_p=2\beta_\alpha$, as an example.
Note that while $\beta_\alpha$ and $\beta_p$ are comparable, this
actually represent much high alpha particle temperature, since
the alpha particle number density is much lower. This is
consistent with observation.

   \begin{figure*}[t!]
   \begin{center}
      \includegraphics[width=\textwidth]{fig2}
   \end{center}
   \caption{Dispersion surfaces corresponding to alpha cyclotron
    modes [left], and proton cyclotron mode, which includes
    the magnetosonic mode branch [right]. The color-coded growth
    rates for each mode is shown, for $n_\alpha/n_p=0.05$,
    $T_\alpha/T_p=2$, and $\beta_\alpha=0.01$. Note that the
    alpha cyclotron instability is an order of magnitude higher
    than that for the proton cyclotron instability.}
   \label{f2}
\end{figure*}

Figure \ref{f2} displays on the left, the dispersion surface
or manifold, corresponding to Alfv\'en-alpha cyclotron mode, while
the right-hand panel plots the dispersion surface, depicting the
proton cyclotron branch (which also includes fast/magnetosonic mode
brach in the lower frequency regime).
Vertical axis represents normalized wave number, $q=ck/\omega_{pp}$,
while the two horizontal axes denote normalized (real) frequency
$x=\omega/\Omega_p$, and wave propagation angle $\theta$, respectively.
We indicate the region of wave growth on each surface as well as the
magnitude by color scheme. As indicated by color bars, however, it is
apparent that the alpha cyclotron instability growth rate is almost
an order of magnitude higher than that of proton cyclotron branch.
The instability for both branches take place over narrow bands
of frequencies and along extended domains of propagation angles.
However, only the most unstable alpha cyclotron modes take place along
quasi parallel direction, while the peaking growth rates of the proton
cyclotron instability peak appear at oblique angles.

In Figure \ref{f3} we plot the maximum growth rate for the unstable
alpha cyclotron modes, which was determined by surveying the entire
frequency and angle space for a given set of input parameters
$\beta_p$ and $\beta_\alpha$. We have then systematically varied
both $\beta_p$ and $\beta_\alpha$, for fixed $\delta=0.05$ and
$V_\alpha=v_A$, until we covered the two dimensional parameter
space $(\beta_p,\beta_\alpha)$.
Figure \ref{f3} shows that the alpha cyclotron beam instability
becomes more unstable as $\beta_\alpha$ decreases, for fixed
value of $\beta_p$. Of course, one cannot indefinitely decrease
$\beta_\alpha$, since the present weak growth/damping rate formalism
assumes that the distribution function has a relatively mild
of velocity space gradient,
$\partial f_\alpha/\partial v_\perp$ and
$\partial f_\alpha/\partial v_\parallel$. For very low $\beta_\alpha$
values, the distribution will have a sufficiently high velocity
derivative so that the assumption of weak growth rate is violated.
Such a caveat notwithstanding, it is interesting to note that
the beam-driven alpha cyclotron instability becomes more unstable
for lower beta values for alpha particles. Figure \ref{f3} shows
that the instability is suppressed as the proton beta (or, equivalently,
proton temperature) increases.

Figure~\ref{f4} exhibits the maximum growth rate for proton cyclotron
instability in the same format as Fig.\ \ref{f3}. Conditions relevant
for a low-beta solar wind are in general confined around the solid blue line
that corresponds to the sample case assumed in Figure~\ref{f2}. Note that the maximum
growth rate is lower in magnitude than that of the alpha cyclotron
instability by an order of magnitude in an overall sense.
It is interesting to note that the maximum proton cyclotron beam
instability has a peak value around $\beta_\alpha\sim10^{-1}$,
but for both higher and lower $\beta_\alpha$,
the maximum growth rate decreases. This behavior is in contrast to that
of the alpha cyclotron instability, where the alpha cyclotron
instability monotonically increases in magnitude of the maximum
growth rate as $\beta_\alpha$ is decreased (until, presumably, the
assumption of weak growth rate is eventually violated).
Note that the proton cyclotron instability is also suppressed by
increasing proton beta, which is similar
to that of the unstable alpha cyclotron mode.

\begin{figure}[t!]
   \begin{center}
   \includegraphics[width=\columnwidth]{fig3}
   \end{center}
   \caption{Maximum growth rate for alpha cyclotron mode branch
   versus $\beta_p$ and $\beta_\alpha$. Other parameters are fixed,
   $n_\alpha/n_p=0.05$ and $V_\alpha=v_A$.} \label{f3}
\end{figure}

To summarize, we find that the relative proton-alpha beam driven
cyclotron instabilities of both the alpha cyclotron and proton cyclotron
mode branches are generally confined to low beta regime, which in general
conforms with measurements in the solar wind at low altitudes
\citep{Matteini2013, Maruca2012}. Of the two modes, however, the
dominant instability is that of the alpha cyclotron branch, so that
in the nonlinear stage, we expect that the alpha cyclotron mode will
dominate the dynamics. For higher beta values, the beam driven cyclotron
instabilities are generally suppressed, which might explain why in the
literature, the proton-alpha drift instabilities in the high beta regime
have been typically studied in combination with temperature
anisotropies of either the protons or alpha particles.

\section{Conclusions}\label{sec.5}

The alpha/proton beam instabilities can play an important role in constraining
the beaming velocity of alpha particles, and may explain the deceleration of
alpha particles in the solar wind with increasing distance from the Sun.
Previous studies have explored in much detail the high beta plasma regime of
these instabilities, clarifying the role of internal temperature anisotropies
of protons or alpha particles, which may switch from Alfv\'enic instabilities
for isotropic beams to an instability of fast-magnetosonic/whistler mode
if alpha beam exhibit $T_\perp / T_\parallel < 1$.

In this paper we have formulated a dissipative (kinetic) theory for the
dispersion and stability of alpha/proton beams, and applied to low
beta regime specific to solar wind conditions at low heliospheric
distances closer to Sun. In this case early studies assume an intrinsic
thermal anisotropy $T_\perp / T_\parallel$ for protons or alpha particles,
which stimulates Alfv\'enic instabilities and prohibits any other
instabilities to develop. Contrary to that, here we assumed alpha and
proton beams with isotropic temperatures $T_\perp/T_\parallel = 1$,
in order to provide basic insights on the dispersion and stability
of alpha/proton beams.

\begin{figure}[t!]
   \begin{center}
   \includegraphics[width=\columnwidth]{fig4}
   \end{center}
   \caption{Maximum growth rate for proton cyclotron mode branch
   versus $\beta_p$ and $\beta_\alpha$. Other parameters are fixed,
   $n_\alpha/n_p=0.05$ and $V_\alpha=v_A$.} \label{f4}
\end{figure}

In Section~\ref{sec.2}, we have derived the general dispersion
relations of electromagnetic waves propagating at arbitrary angle
($\theta$) with respect to the magnetic field. In Section \ref{sec.3},
we formulated the weak growth rate theory for plasma particles that are
assumed to be distributed according to standard (Maxwellian) statistics,
with protons and alpha particles considered to be counter-drifting Maxwellians.
In section~\ref{sec.4} we have examined a sample growth rate calculation
associated with the electromagnetic modes: fast-magnetosonic/whistler,
proton-cyclotron and alpha-cyclotron waves, for a given set of alpha-to-proton
density ratio, $n_\alpha/n_p$, alpha and proton beta's, $\beta_\alpha$ and
$\beta_p$, and alpha-proton relative drift speed, $V_\alpha$, which is fixed at $v_A$.
The results unveiled a potential competition between instabilities
of proton and alpha cyclotron modes, but the sample calculation also
showed that the alpha cyclotron mode may reach growth rates one order of
magnitude higher than that of the proton cyclotron mode.
We have then proceeded with the calculation of maximum growth rates for
each mode as we continuously varied $\beta_\alpha$ and $\beta_p$.
It was shown that for the low beta solar wind conditions in the outer corona,
the alpha cyclotron instability is the dominant mode, with growth rates increasing with
decreasing $\beta_\alpha$. In contrast, the less unstable proton
cyclotron mode has a local peak associated with the maximum growth
rate around $\beta_\alpha=10^{-1}$, but the mode is suppressed for
either decreasing or increasing $\beta_\alpha$ around this peaking value.
Both modes, however, are stabilized by
increasing proton beta, or equivalently, proton temperature.

The consequence of the excitation of these unstable modes on the
solar wind proton and alpha particle dynamics cannot be understood
purely on the basis of linear theory. In order to address such an
issue, we will carry out quasilinear analysis in the near future.
The impact of instability excitation on the radially expanding
solar wind condition can also be studied in the future where,
the the effects of radial expansion can be counter balanced by
the wave-particle relaxation by instabilities. Such a task
is a subject of our ongoing research.

\begin{acknowledgments}
The authors acknowledge support from the Katholieke Universiteit
Leuven (Grant No. SF/17/007, 2018),
Ruhr-University Bochum, and Alexander von Humboldt Foundation.
These results were obtained in the framework of the projects SCHL~201/35-1
(DFG--German Research Foundation), GOA/2015-014  (KU Leuven),
G0A2316N (FWO-Vlaa- nderen), and C 90347 (ESA Prodex 9).
M.A.R. acknowledges Punjab Higher Education Commission (PHEC) Pakistan
for granted Postdoctoral Fellowship FY 2017-18.
S.M.S. gratefully acknowledges support by a Postdoctoral Fellowship
(Grant No. 12Z6218N) of the Research Foundation Flanders (FWO-Belgium).
P.H.Y. acknowledges NASA Grant NNH18ZDA001N-HSR and
NSF Grant 1842643 to the University of Maryland,
and the BK21 plus program from the National Research Foundation
(NRF), Korea, to Kyung Hee University.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


Here's a quick guide on how to integrate your 360 product spins in Wix. The process is pretty straightforward but you will need to have a separate hosting account for uploading the product spins. If you don't have an extra hosting space outside of Wix, we would be happy to upload a couple of your 360 product views to our optimized Amazon servers - just let us know.
1) Enter Edit mode in your Wix dashboard and select a page where you plan to add a product spin. Click Add button (a circle with a plus sign) in the menu on the left to expand available widgets and then press More in the menu to bring the advanced options.
2) Select Embed a Site (under the Embeds section) and then resize and align the widget box on your page according to your site layout.
To create your own 360 product views, use WebRotate 360 SpotEditor to import your existing images (360 product photography or rendered in a 3d design software) and publish a finalized interactive viewer package on your computer.
Make sure to select the "Mobile full page" template on the Publish form in SpotEditor so that your template is fully responsive and can contract and expand with your Wix widget on any device.
Once published, upload the contents of the 'published' folder that the software creates in your project directory to a web hosting of your choice (you may simply share it with us and we will host it for you free of charge for all your testing needs - contact us). Note the URL of the .html file in the uploaded folder and use this URL inside the HTML Code widget under Website Address as we show in Step 3 above.


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text:
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Advertise your Business on What's On in Dublin and join the other Businesses that advertise with us. Explore our website for local businesses and you will see Restaurants, Shopping, Pubs, Sightseeing, Sports, Events, Nightclubs, & Accommodation. Also see the available trades people for all your domestic or commercial work.
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https://www.whatsonindublin.net/wp-content/uploads/2020/12/WOI.mp4
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redpajama_set_name: RedPajamaCommonCrawl


text:
Le concours Aftalion est un concours eurorégional d'économie parrainé par le Crédit mutuel Nord Europe. Il a été créé en 1995 en hommage à l'économiste français Albert Aftalion. Il regroupe les étudiants de 12 facultés d'économie.

Le concours est divisé en deux catégories :
– épreuve de macroéconomie (niveau licence) ;
– épreuve de science économique et management (niveau master 1).

Le concours consiste en une dissertation de cinq heures évaluée par un jury de spécialistes et de personnalités.

Le concours est doté de prix d'une valeur de  euros répartis comme suit :

 Niveau licence :
  :  euros
  : 500 euros
  : 200 euros

 Niveau master 1 :
  :  euros
  : 500 euros
  : 200 euros

Aftalion
Aftalion
Fondation en 1995

redpajama_set_name: RedPajamaWikipedia


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'Yellowstone': Meet 'Riggins,' the Man Who Attacked the Dutton Family
Billy Dukes Published: November 17, 2021
Yellowstone's John Dutton (Kevin Costner) may not know the guy who attacked him, but we do. "Riggins" appears in a photograph — a mugshot, actually — during Ep. 3 of Season 4, titled "All I See Is You."
"He knows you," Mo (actor Mo Brings Plenty) says to the confused Dutton patriarch, who's holding a file with more information.
The Paramount Network program seems poised to introduce a new character soon, albeit one in Montana's state prison. Riggins is the guy who hired the militia to attack John, Kayce (Luke Grimes) and Beth Dutton (Kelly Reilly), as well as the ranch, but his motives are still unknown.
Actor Bruno Amato is the man photographed, as all but confirmed on his Instagram page. Amato is a veteran actor with more than 100 credits to his name. He's also a Navy veteran and a current candidate for the U.S. House of Representatives.
"This guy looks very familiar," he captions.
IMDB points to shows including Lucifer on Netflix, American Housewife on ABC and the Will & Grace reboot as Amato's recent credits. His 20-year resume also includes small roles on soap operas, Brooklyn Nine-Nine and How I Met Your Mother. The Bakersfield Californian refers to his service in the U.S. Navy, where he served as a torpedoman. Presently, the Democrat is trying to unseat Kevin McCarthy in the U.S. House or Representatives.
While not yet listed as part of the Yellowstone cast, there's further circumstantial evidence pointing to his involvement on Instagram and Twitter. The character of Riggins couldn't be more enigmatic, however — keep reading only if you don't mind a spoiler alert.
After the indigenous tribe is cleared as a possible attacker, and the Market Equities team seems clueless about who attacked the ranch, most started to look at Jamie Dutton and his birth father as potential planners. Neither John nor Kayce Dutton believe that's true, although Beth Dutton sure does, and promises she'll kill him one day for it. The slow unpacking of who hired the militia is the primary plot point driving Season 4 of the show, although hardly the only one.
On Thursday (Nov. 18), look for Ep. 2 of Dutton Rules: A Yellowstone and 1883 Podcast, in which the various threads will be untangled and laid out to see where the show may be headed. Find the podcast on Apple Podcasts and Spotify.
10 'Yellowstone' Facts You Probably Didn't Know
How big of a fan of Yellowstone are you? These 10 facts about the Paramount Network show are sure to stump even the most dedicated viewers. They're almost all about the cast members and their real-life passions and roles. John's kids? Beth's accent? Rainwater's guitar playing? Tate's spoilers? It's all part of this list of 10 Facts You Probably Didn't Know About Yellowstone.
Source: 'Yellowstone': Meet 'Riggins,' the Man Who Attacked the Dutton Family
Filed Under: Yellowstone
Categories: Television