# COSMIC RAY DRIVEN GALACTIC WINDS

SARAH RECCHIA  Université de Paris, CNRS, Astroparticule et Cosmologie,
F-75013 Paris, France,
<EMAIL_ADDRESS>
(Day Month Year; Day Month Year)

###### Abstract

Galactic winds constitute a primary feedback process in the ecology and
evolution of galaxies. They are ubiquitously observed and exhibit a rich
phenomenology, whose origin is actively investigated both theoretically and
observationally. Cosmic rays have been widely recognized as a possible driving
agent of galactic winds, especially in Milky-Way like galaxies. The formation
of cosmic ray-driven winds is intimately connected with the microphysics of
the cosmic ray transport in galaxies, making it an intrinsically non-linear
and multiscale phenomenon. In this complex interplay the cosmic ray
distribution affects the wind launching and, in turns, is shaped by the
presence of winds.
In this review we summarize the present knowledge of the physics of cosmic
rays involved in the wind formation and of the wind hydrodynamics. We also
discuss the theoretical difficulties connected with the study of cosmic ray-
driven winds and possible future improvements and directions.

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##  1 Introduction

Galactic winds (GWs) are powerful outflows that originate in the disk of
galaxies and may extend for hundreds of parsecs. The ejected material is
mainly composed of very hot dilute gas, often enriched of metals, dust and
portions of cooler gas, whose velocity can reach hundreds or even thousands
km/s, and, in some cases, the gas may even escape the galaxy and be injected
in the intergalactic medium (IGM). The induced mass loss can be comparable to
the mass formed in stars.

Since their first detection in the M82 starburst galaxy[1, 2], GWs are
ubiquitously observed both locally and at high redshift, especially in
galaxies that exhibit a high star formation rate (SFR) or an active galactic
nucleus (AGN)[3, 4, 5, 6]. Also the Milky Way (MW) might host GWs, as
suggested by several observations[7, 8]. GWs represent a most important
feedback process which deeply affects the evolution and the structure of
galaxies, star formation, and the chemical and physical properties of the
interstellar medium (ISM) and of the IGM.

The formation of GWs is not yet fully understood and it is actively
investigated in astrophysics. GWs may be powered by the energy and momentum
injected by stellar winds and supernova explosions[9] or by the energy
released during accretion onto supermassive black holes in AGN[10]. They may
also be produced by the radiation pressure induced by the
scattering/absorption of starlight radiation on dust grains and gas[11, 12].
Thermal and radiation pressure gradients are effective in driving winds in
very active galaxies, while in more quiescent galaxies, such as the MW, they
are probably not large enough, except maybe for the Galactic Center region.

On the other hand, cosmic rays (CRs) escaping from galaxies can effectively
push on the ISM, and eventually launch winds. In fact, CRs are known to be
produced in astrophysical sources, likely supernova remnants (SNRs), located
in the galactic disk and to undergo diffusive propagation in the ISM as a
result of the scattering off plasma waves. Such scattering provides a strong
coupling between CRs and the background plasma, which can allow CRs to produce
GWs. In the MW CRs are observed to be energetically in equipartion with the
thermal plasma and with the magnetic field, which suggests their importance in
the dynamics of the ISM. The relativistic CR gas exhibits a smaller adiabatic
index compared to the thermal plasma, so that the CR pressure can act at
relatively large distances from the disk, making possible the wind launching
also in situations where the thermal and radiation pressure would fail. This
aspect has been pointed out for the first time in Ref. Ipavich-1975, which
presented the first study of the CR impact on the launching of GWs in a MW-
like galaxy. The dynamical role played by CRs in the wind hydrodynamics has
been subsequently confirmed by many authors and widely investigated in the
stationary regime [14, 15, 16, 17], in time dependent approaches [18] and in
simulations [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32].

The possibility of CRs to launch winds is intrinsically connected with their
transport in the ISM, which in turn is affected by the presence of GWs. In
fact, CRs escaping from the source region, namely the disk, establish a
gradient in their distribution function whose consequences are manifold. First
of all, such gradient can lead to the excitation of plasma instabilities, most
notably of the streaming instability of Alfvén waves[33, 34], which in turn
affects the scattering of CRs, namely their diffusive motion, and their
coupling with the background gas. Moreover, the coupling with the ISM and the
CR gradient between the disk and the outer halo may induce the formation of
GWs, which in turn affect the advective transport of CRs. Finally, the damping
of the CR-induced plasma turbulence and the ionization losses of sub-GeV CRs
can heat the gas[35], further affecting the wind launching and the wind
structure. Thus, the dynamics of CR-driven winds is an intrinsically non
linear process and a multiscale phenomenon, in which the microphysics of the
CR transport, on scales of the Larmor radius of relativistic particles in the
galactic magnetic field, determines the CR distribution function and the wind
structure, which in turn affect the CR propagation itself.

The multiscale and nonlinear nature of CR-driven winds makes them particularly
difficult to study, especially when CRs are not treated macroscopically, as a
relativistic gas, but the details of the CR microphysics is taken into account
[36, 37, 38].

In this review we summarize the relevant properties and the physics involved
in the formation CR-driven winds. The review is organized as follows: 1) in
Sec. 2 we review the observations, formation mechanisms and the impact of GWs
in different types of galaxies; 2) in Sec. 3 we briefly introduce the main
aspects of CRs astrophysics; 3) in Sec. 4 we illustrate the relevant
ingredients of the CR transport in a CR-driven wind; 4) in Sec. 5 we review
models of GW hydrodynamics in the presence of CRs; 5) in Sec. 6 we show in
some detail the properties of CR-driven winds in the stationary regime, in
particular in a MW-like galaxy; 6) in Sec. 7 we drive some conclusions and
discuss future perspectives.

##  2 Galactic winds

Different gas phases coexist in GWs: dilute hot and warm phases (very hot
$\sim\,10^{8}$ K, hot $\sim\,10^{6}-10^{7}$ K and warm $\sim\,10^{4}$ K),
neutral atomic matter ($\sim\,10^{3}$ K), cold molecular material, dust
($\sim\,100$ K) and non-thermal particles. Both hot and warm/cool gas can
exhibit very high velocities (hundreds or thousands km/s), and produce mass
loss rates which can be as large as several solar masses per yr. How these
phases are accelerated to such high velocities is being actively investigated
in astrophysics and several aspects remain unclear. Moreover, the multiphase
nature of GWs and the fact that a large part of the ouflowing gas is hot and
very dilute, make their observation challenging.

In this section we briefly summarize the mechanisms responsible for the wind
formation in different types of galaxies and environments, and the
observational techniques employed to detect and study GWs. We conclude by
stressing the importance and implications of GWs in astrophysics and
cosmology. For further details we refer the reader to the excellent reviews
found in Refs. Veilleux-review,Zhang-review, Heckman-review and Rupke-review.

###  2.1 Production mechanisms

Hot winds are most likely accelerated in starburst regions or in AGN. The
mechanical energy injected in such regions and partially converted to thermal
energy through shocks, creates a bubble of very hot gas whose pressure can be
much larger than that of the surrounding ISM. The dynamical evolution of such
bubbles has been widely studied, starting form the seminal work of Ref.
Chevalier-Clegg, which showed that, under the assumption that radiative
losses, gravity and other effects can be neglected, the flow solution is self-
similar. The bubble may then be able to cross the disk height and eventually
break through the galactic halo. In general, in such energy-driven flows, the
energy input must be large enough, or radiative losses small enough, that the
wind is not hampered before it can reach the edge of the galactic disk.

Winds may also be driven by the mechanical momentum imparted by starburts and
AGN[11, 10] or by radiation. Radiation can heat the ISM via photoionization.
For instance, UV photons can create regions of ionized hydrogen with
temperatures of $\sim 10^{4}$ K. In addition, radiation pressure on dust
grains and gas[11, 12] can effectively push the ISM and contribute to the
generation of GWs. Radiation pressure may be important in different
environments, from massive stars to luminous AGNs, from starburst to rapidly
star-forming galaxies[39, 3, 4, 5].

A very important and still unclear aspect is how the warm and cold wind phases
interact with the hot phase and are themselves accelerated to velocities that
are comparable to that of the hot gas, and that can be as high as hundreds
km/s. A possible scenario is that portions of the warm and cold ISM, or even
clouds, are advected in the hot wind and accelerated by its strong ram
pressure[4, 5]. Another possibility is that the warm/cool gas in GW forms due
to the thermal instabilities induced in the gas by radiative cooling[40, 4, 5]

###  2.2 Observations

Given the multiphase nature of GWs, multiwavelength observations, of both
continuous and line emission and of absorption lines, spanning from radio to
X-rays, are necessary in order to identify and characterize the various flow
phases [39, 3, 6].
The flow velocity is strongly phase dependent, with the hotter phases showing
the highest speeds (up to thousands km/s), down to the $\sim 100$ km/s found
in the cold molecular and atomic gas phases[5]. The study of the line profile
of emission and absorption lines allows to estimate the fluid velocity, while
spatially resolved mapping of the emitting/absorbing material is used to
determine the wind morphology[39, 6].

The hot and very hot gas phases are mainly tracked through their X-ray
emission. In particular, hard X-rays are expected from the very hot and
tenuous gas heated by SNe and stellar winds, and from internal shocks in such
hot wind. When the wind collides with denser ambient material or with clouds,
the shock heated/evaporated material is expected to produce also soft
X-rays[39].

The interaction of a hot wind with a denser and cooler background gas,
possibly characterized by a mixture of phases, can produce radiative shocks
that compress and heat the ambient medium , thus producing the warm wind
phases. Such warm phases have been studied through the continuous emission in
the vacuum ultraviolet and through emission lines, in the optical band (e.g
$\rm H\alpha$, NII, OII, OIII, and SII) and in the near-IR band (e.g the $\rm
2.122\,\mu m$ line of $\rm H_{2}$, and the lines $\rm Pa\alpha$ and $\rm
Br\gamma$)[5, 3]. The presence of shocks is often revealed by the presence of
double-peaked emission with typical values of emission-line ratios like
SII/$\rm H_{\alpha}$, NII/$\rm H_{\alpha}$ and OII/$\rm H_{\alpha}$[39, 3].

Molecular gas is mapped through mm-wave (warm) and mid/near IR (cold)
spectroscopy. Atomic hydrogen is tracked by the 21 cm line, but also OI and
CII lines in the far IR are used to identify atomic gas. Finally, dust is
studied both in the IR and in the far-UV band[5].

Radio surveys reveal the presence of synchrotron emitting relativistic
electrons in magnetized winds[5].

Complementary to emission techniques, absorption-line techniques are also
massively employed, especially for the study of face-on systems and of distant
galaxies, where the diffuse emission from the wind is hard to detect. The most
used lines are in the UV band: Silicon (Si II $\lambda 1260$, $\lambda 1526$,
OI + SiII $\lambda 1303$), Carbon (CII $\lambda 1334$), Iron (FeII $\lambda
1608$) and Aluminum (Al II $\lambda 1670$), Oxygen (OVI $\lambda\lambda
1032,\,1038$), and in the optical band: Na I D ($\lambda\lambda 5890,\,5896$),
KI ($\lambda\lambda 7665,\,7699$)[3].

As for the Milky Way, the existence of GWs has not been firmly established.
However, the recent observation in the X-ray band of Oxygen OVII and OVIII
absorption lines in the spectra of distant quasars[41, 7, 8], implies the
existence around the Galaxy of a very extended ($\sim$ 100-200 kpc) halo
constituted of hot ($\sim~{}10^{6}$ K) dilute gas, with metallicity larger
than 0.3, and likely connected with GWs. Moreover, also the so called Fermi
Bubbles, giant structures detected in $\gamma-$rays that extends for several
kpc above and below the Galactic Center, might be the result of outflows[42,
43, 44].

###  2.3 Impact of galactic winds

GWs constitute a most relevant ingredient in the ecology of galaxies, and a
primary feedback mechanism.
In fact, GWs heat the ISM and drain hot gas and metals from the disk of the
host galaxy toward galactic halos or even in the intergalactic space. This
limits the amount of gas available to form stars. Moreover, the injected
material affects the surrounding space by changing the temperature, the degree
of ionization and the chemical composition, which especially influence the
cooling rate[45, 8], and so the time required for the ejected gas to fall back
to the disk. All this establishes a gas recycling and a regulation of the
SFR[3, 4, 5]. Models of galactic evolution that do not include GWs tend to
overestimate the amount of baryons and the SFR[46]. GWs are necessary in order
to explain the luminosity function of galaxies and the mass/metallicity
relation (see Ref. Zhang-review and references therein).
In the cases of dense outflows, star formation may even take place within the
outflow, with prominent consequences on the spheroidal component of galaxies
and on the chemistry of the circumgalactic and intergalactic medium[47].
The gas expelled by GWs in the IGM may represent a sizable fraction of the
baryonic matter of the Universe[48, 8] and has also been invoked as a possible
important contribution to the solution of the missing baryons problem in the
local Universe[49].
Finally GWs may also transport magnetic turbulence from the disk to the halo
and may contribute to shaping the large scale magnetic field in galaxies,
especially the out of disk components[50].

##  3 Cosmic rays in a nutshell

CRs are very high energy charged particles that originate in astrophysical
sources and reach our planet by propagating and interacting with the ISM.
Locally, CRs are revealed with balloon, satellite and air-shower detectors.
The observed CR flux is shown in Fig. 1. Remarkably, it extends as a nearly
perfect power-law in energy, $\rm\sim E^{-\gamma}$, from GeV energies up to
$\sim 10^{20}$ eV, with a corresponding energy density of $\rm\sim
1~{}eV/cm^{-3}$, a value comparable with the thermal and magnetic pressure
observed in the Galactic disk. Protons largely dominate the observed flux,
followed by Helium ($\sim 10\%$) and heavier nuclei ($\sim 1\%$). The leptonic
flux is about 100 times smaller than the hadronic flux, and electrons reaching
$\sim 20$ TeV have been measured so far[51, 52]. Notable spectral breaks occur
at: $\rm\sim 230\,GV$, more pronounced in the proton and Helium spectra, but
present also for heavier nuclei, where the slope of the proton spectrum
changes from $\gamma=2.89\pm 0.015$ to $\gamma=2.67\pm 0.03$[53, 54]; the
knee, at $\approx 3\times 10^{15}$ eV, where the spectral slope changes from
$\gamma\approx 2.7$ to $\gamma\approx 3.1$ and the chemical composition seems
to become heavier; the ankle, at $\approx 3\times 10^{18}$ eV, where the
spectral slope flattens to $\gamma\approx 2.7$[55].

Figure 1: CR spectrum at Earth as detected by different experiments[56]
(courtesy Tom Gaisser).

CRs are also tracked through the radiation produced by their interaction with
the ISM, most notably through $\gamma$-rays produced by the decay of $\pi^{0}$
mesons generated by the interaction of CR protons with the background gas, and
by bremsstrahlung and inverse Compton due to CR leptons, and by synchrotron
radiation produced by the interaction of CR leptons with the background
magnetic field.

Figure 2: Boron over Carbon ratio as a function of the kinetic energy per
nucleon[57].

Together with the flux, the chemical composition and the level of anisotropy
of the observed CRs, represent precious pieces of information. In particular,
the relative abundance of secondary spallation nuclei relative to their
primary nuclei (e.g the Boron/Carbon ratio, see Fig. 2, or the abundance of
unstable isotopes relative to stable isotopes (e.g $\rm{}^{10}Be/{}^{9}Be$),
represent the most reliable measurement of the amount of matter traversed by
CRs during their travel through the ISM, and so of the residence time of CRs
in the Galaxy. The estimated residence time decreases with energy and exceeds
by orders of magnitudes the ballistic propagation time. This strongly supports
the idea that CRs propagate diffusively in the magnetized ISM, as additionally
confirmed by the small observed anisotropy ($\sim~{}0.001$ at TeV
energies)[58, 59].

Nowadays, the most accredited picture of the origin of Galactic CRs, is the
so-called supernova remnant paradigm[60, 58, 59]. In this scenario, the bulk
of CRs is thought to be accelerated in the Galactic disk at the shocks of
supernova remnants (SNRs) through the mechanism of diffusive shock
acceleration (DSA). On the energetic ground, this requires that 3-10% of the
mechanical energy released by SNe is channeled into CRs[61, 62]. DSA is
rigidity dependent and produces CRs with approximately universal power law
energy spectra $\rm\approx E^{-2.0}$. Protons are thought to be accelerated up
to energies $\gtrsim 3\times 10^{15}$ eV, while nuclei with charge Z would
reach an energy Z times larger, namely a factor of 26 for Iron nuclei. In this
picture the knee would naturally result from the superposition of the energy
cut-offs of the different nuclear species. Radio, X-ray and $\gamma-$ray
observations confirmed SNR shocks as acceleration sites of CRs[63], however it
is still unclear whether they can accelerate CRs up to PeV energies (see Refs.
Blasi-2013-review, Gabici-2019-review-con-altri and references therein).

After leaving their sources, CRs are thought undergo diffusive propagation in
the turbulent Galactic magnetic filed. Magnetic inhomogeneities can scatter
CRs and produce their spatial diffusion. The scattering occurs on scales of
the Larmor radius $r_{L}$ of the particles and is likely of resonant nature,
namely CR with a given larmor radius are scattered by plasma waves whose wave
number $k$ satisfies the relation $k\sim 1/r_{L}$[64, 65]. The decrease of the
Boron/Carbon ratio with energy can be accounted for by a diffusion coefficient
that increases with energy. The spectral hardening at $\sim 230$ GV may be due
to a change in the type of turbulence relevant for CR scattering at $\sim
200-300$ GV [66, 67], but other possible interpretations have been proposed
(see Ref. [68]).

The origin of CRs beyond the ankle is probably extragalactic, since the Larmor
radius of such high energy particles in the typical Galactic magnetic field is
comparable to the size of the Galaxy. On the other hand, the energy of
transition between Galactic and extragalactic CRs is still largely debated[58,
59].

###  3.1 Cosmic ray propagation: building blocks

The propagation of CRs in the Galaxy is an intricate, non-linear, phenomenon
which depends on several processes that are often not understood. The details
of CR propagation are intimately connected with the formation of CR-driven
winds.

CRs are thought to undergo a diffusive motion during their escape from the
Galaxy due to the scattering on magnetic waves, as suggested by the measured
abundances of secondary nuclei and unstable isotopes and by the observed small
level of anisotropy. In addition, CRs may be advected out of the Galaxy due to
the plasma waves themselves and to large scale outflows, such as GWs. Finally,
other processes, such as ionization and Coulomb losses[35], bremsstrahlung,
synchrotron and inverse Compton, turbulent reaccelaration, nuclear
fragmentation and radioactive decay, proton-proton interactions, may affect
the transport, depending on the CR energy and species[60].

All these processes depend on the specific conditions of the ISM, such as the
density, temperature and degree of ionization of the background medium, the
magnitude and the level of turbulence of the magnetic field, the background
photon field, and, in the case of GWs, also on the Galactic gravitational
potential. Such conditions can vary from point to point in the Galaxy and CRs
can play an active role in determining the propagation parameters[38].

The last aspect is particularly relevant in the case of CR-driven GWs. In
fact, the CR density gradient that exists between the Galactic disk, where the
sources of the bulk of CRs are probably located, and the halo, together with
the fact that CRs are observed to be roughly in equipartition with the thermal
plasma and with the background magnetic field, suggest that CRs may play a
relevant dynamical role.
It is known that spatial gradients in the CR density can lead to the
excitation of Alfvén waves, at the CR Larmor radius scale, through the so
called streaming instability[69, 34], which propagates along the magnetic
field lines in the direction of the decreasing CR density. The scattering of
CRs on such waves affects their diffusive propagation and contribute to
determine the diffusion coefficient, together with other sources of magnetic
turbulence. The level of plasma waves is limited due to different damping
processes, such as the turbulent cascading[70, 71], the non linear Landau
damping[72, 73], the turbulent damping[74, 75] and the ion-neutral
damping[76], depending on the conditions of the background medium. A larger CR
gradient leads to a higher level of turbulence, thus to a stronger CR
scattering and to a smaller diffusion coefficient. Moreover, in the case of
self-generated Alfvén waves, that move in the direction opposite to the CR
gradient, CRs experience also convection with such waves at roughly the Alfvén
speed. It has been shown[77, 66] that streaming instability of Alfvén waves
may dominate the CR propagation up to $\sim 200$ GeV, while at higher energies
other sources of turbulence are probably at work[67]. The nature of such
additional types of turbulence is still debated. In fact, for a long time
Alfvénic turbulence injected on large scale ($10-100$ pc) by astrophysical
sources, such as SNe, and cascading to the resonant scales, was thought to be
responsible for the CR scattering. However, subsequent studies showed that
Alfvénic turbulence cascading to small scales becomes anisotropic and very
inefficient at scattering CRs[78, 79, 80]. This problem does not affect self-
generated Alfvén waves, which are already produced at the resonant scale.
Recently it has been suggested that compressive MHD waves could provide a
relevant contribution to CR scattering[75, 81, 82].

In most cases relevant for galactic propagation, the level of magnetic
turbulence on scales resonant with the Larmor radius of CRs is
$\rm\langle\delta B\rangle/B\ll 1$ and diffusion can be treated in the
framework of quasi-linear theory[60, 65]. This implies that CRs diffusion is
anisotropic, with particles diffusing preferentially along rather than across
the magnetic field lines, which translates in a parallel and perpendicular
diffusion coefficient $D_{\parallel}\gg D_{\perp}$. In this case, the CR
transport perpendicular to the mean background field may be provided by the
wandering of field lines[64, 83, 84].

The coupling established by CRs and the ISM by scattering and the presence of
a CR gradient between the disk and the halo, imply that CRs exert a force on
the background plasma directed toward the Galactic halo. If this force is
large enough as to win against the gravitational force, a GW may be launched.
The hydrodynamics of such a wind is determined by CRs. On the other hand, the
advective transport of CRs is influenced by the presence of a wind.

Since CR protons are the most abundant species, all these phenomena, that
depend on the CR density gradient, are mostly determined by protons. Moreover,
the wind launching depends on the CR pressure, which is dominated by 1-10 GeV
protons.

####  3.1.1 Basic model of proton transport

The propagation phenomenology described above is very rich and complicated.
However it is possible to understand some basic aspects of CR propagation
relevant in the case CR-driven winds, with the help of a simplified stationary
model of proton transport. In such basic model the CR propagation is assumed
to be 1D and to occur only perpendicular to the Galactic disk and solely along
a constant background magnetic field. The diffusion coefficient is assumed to
be spatially homogeneous and is treated as a fitting parameter. The specific
nature of the turbulence responsible for the CR scattering is ignored. The
injection of CRs occurs in the Galactic disk, assumed to be infinitely thin,
while also the advection velocity (assumed to be constant) and the size of the
propagation region are preassigned. Other possible effects, such as ionization
and Coulomb losses, are neglected.
In this scenario the stationary CR advection-diffusion equation reads[58]

$-\frac{\partial}{\partial z}\left[D\frac{\partial f}{\partial
z}\right]+u\frac{\partial f}{\partial
z}-\frac{d\,u}{d\,z}\frac{p}{3}\frac{\partial f}{\partial p}=Q(z,p),$ (1)

where $f(z,p)$ is the CR distribution function, which is function of the
distance from the Galactic disk $z$ and of the particle momentum $p$. Under
the following assumptions this equation is easily solved: the diffusion
coefficient $D(p)\propto p^{\delta}$ is a power-law in momentum, the injection
occurs in the disk $Q(z,p)=Q_{0}(p)\delta(z)$ with a power law injection
spectrum $Q_{0}(p)\propto p^{-\gamma}$ (as expected for DSA[58]), the
advection velocity $u$ is constant and directed away from the disk
($\frac{d\,u}{d\,z}=2u\delta(z)$), and CRs freely escape at $\rm|z|=H$, namely
$f(\pm H,p)=0$. The free escape boundary H, namely the size of the propagation
region, if fixed, and it must be not infinity in order to guarantee the
existence of a stationary solution.

A standard way to solve Eq. 1 is to integrate it around the disk, between
$z=0^{-}$ and $z=0^{+}$, and between $z=0^{+}$ and $z$, with the free-escape
boundary condition. The solution reads

$\displaystyle
f(z,p)=f_{0}(p)\frac{1-e^{-\zeta\left(1-\frac{|z|}{H}\right)}}{1-e^{-\zeta}}$
(2) $\displaystyle
f_{0}(p)=\frac{3}{2\,u}\int_{p}^{\infty}\frac{dp^{\prime}}{p^{\prime}}Q_{0}(p)\exp\left[\int_{p}^{p^{\prime}}\frac{d\,p^{\prime\prime}}{p^{\prime\prime}}\frac{3}{\lambda(p^{\prime\prime})}\right],$
(3)

where $\zeta(p)=\frac{u\,H}{D(p)}$ and $\lambda(p)=1-e^{\zeta(p)}$.
For particle momenta for which advection dominates, namely
$\frac{u\,H}{D(p)}\gg 1$, the CR spectrum becomes

$f_{0}^{adv}(p)=\frac{Q_{0}(p)}{2\,u}\sim p^{-\gamma}.$ (4)

Thus, in this regime the equilibrium spectrum has the same slope of the
injection spectrum.
When diffusion dominates, namely $\frac{u\,H}{D(p)}\ll 1$, the CR spectrum
becomes

$f_{0}^{diff}(p)=\frac{Q_{0}(p)\,H}{2\,D(p)}\sim p^{-\gamma-\delta}$ (5)

and the equilibrium spectrum is steeper than the injection spectrum due to the
energy dependent diffusive propagation. Since, as inferred, for instance, from
the the Boron/Carbon ratio, the diffusion coefficient is an increasing
function of momentum, advection(diffusion) dominates for low(high) enough
particle momenta. Typical values used to fit CR data are: slope of the
injection spectrum $\gamma\sim 4.1-4.4$, slope of the diffusion coefficient
$\delta\sim 0.3-0.6$ and normalization $\sim 3\times 10^{28}$ cm2/s at $\sim
1$ GeV, halo size of $H\sim 4$ kpc and advection velocity of $\sim 20$
km/s[58, 67].

It is interesting to notice that in a GW scenario the size of the diffusive
region emerges naturally, without the need of imposing a free-escape boundary.
This is due to the fact that, as we will see in the remaining of this review,
in a stationary GW the wind velocity increases with the distance from the disk
and eventually tends to a constant value. This means that, for a particle of
momentum $p$, advection starts to dominates over diffusion at a distance from
the disk, $s^{*}(p)$, given by

$\frac{s^{*2}(p)}{D(s^{*},p)}=\frac{s^{*}(p)}{U(s^{*})}.$ (6)

Since $D$ generally increases with momentum, the size of the diffusion region
also increases with momentum. Above $s^{*}(p^{*})$ particles of momentum
$p^{*}$ are basically advected out of the Galaxy. All this may have relevant
consequences on the CR spectrum, anisotropy and production of secondaries[36,
37].

##  4 Cosmic ray driven winds: cosmic ray transport

The inclusion of the CR transport in models of GWs is made extremely difficult
by its non linear nature and by the fact that the scales relevant for the CR
propagation (Larmor radius of the particles) are much smaller than the those
relevant for the wind ($\gtrsim$ kpc). For this reason, in most wind models
and simulations CRs are treated as a fluid. The first attempt to account for
the effects of a stationary CR driven wind on the CR transport was by Ref.
Ptuskin-wind-transp, where the authors used a simplified model of the wind
structure (as inferred from hydrodynamic calculations) in order to deduce some
general implications for the observed CR spectrum. More recently Ref. Recchia-
winds-I. presented a semi-analytic, self-consistent calculation, of the wind
formation together with the propagation of CR protons in the wind. The work is
restricted to the stationary regime, and includes the self-generation of
Alfvén waves by CR streaming instability, the damping of such waves and the
resulting gas heating, the CR scattering parallel to the field lines on the
self-generated waves, and the CR advection with such waves and with the wind.
The CR pressure contributes to the wind formation and is dominated by CR
protons, which are the most abundant CR species. The authors also performed a
parametric study of the wind launching and CR distribution in different
galactic environments[38].
Based on such studies, here we briefly summarize the main features of the CR
transport in a CR-driven wind that have been explored in the literature, and
how they are linked to the wind hydrodynamics, which is described in the next
section. Possible future improvements of the present models are illustrated in
the conclusions.

In the assumption that the CR transport is dominated by the scattering on
self-generated Alfvén waves and in the presence of a GW, the CR diffusion-
advection equation reads[34]

$\frac{\partial f}{\partial
t}+(\vec{u}+\vec{v}_{A})\cdot\vec{\nabla}f=\vec{\nabla}\cdot\left[D\vec{\nabla}f\right]+\left[\vec{\nabla}\cdot(\vec{u}+\vec{v}_{A})\right]\frac{p}{3}\frac{\partial
f}{\partial p}+Q_{c}$ (7)

where $f(p,\vec{r})$ is the CR distribution function and $D(p,\vec{r})$ the CR
diffusion coefficient, while $\vec{u}(\vec{r})$ and $\vec{v_{A}}(\vec{r})$ are
the background plasma velocity (the wind velocity in this context) and the
Alfvén velocity. In the case of self-generation, Alfvén waves move
preferentially along the magnetic field lines and in the direction of the
decreasing CR density, so that CRs will be advected at a speed which is
roughly $\vec{u}+\vec{v}_{A}$. In the case of an extrinsic turbulence, waves
would move along the field lines in both directions and CRs would advect at
$\vec{u}$[34].
The CR diffusion coefficient is given by the Alfvén wave energy density
$W(k,\vec{r})$, determined by the CR streaming instability and by the possible
sources of wave damping. In the context of galactic propagation the level of
turbulence is weak at the scale resonant with CRs and the diffusion
coefficient can be expressed using quasi-linear theory[60, 65] :

$D(\vec{r},p)=\left.\frac{1}{3}\frac{v(p)r_{L}(\vec{r},p)}{k_{\rm res}W(k_{\rm
res},\vec{r})}\right|_{k_{\rm res}=1/r_{L}},$ (8)

where $v$ and $r_{L}$ are the particle velocity and Larmor radius, and the
wave energy density is calculated at the resonant wavenumber $k_{\rm
res}=1/r_{L}$.
The Alfvén wave transport equation, taking into account the velocity of the
background medium, wave generation by CR streaming instability and the
relevant damping processes, is given by[85]

$\frac{\partial W}{\partial
t}+\vec{\nabla}\cdot\left[\left(\frac{3}{2}\vec{u}+\vec{v}_{A}\right)W\right]=\frac{1}{2}\vec{u}\cdot\vec{\nabla}W-[\Gamma_{\rm
CR}+\Gamma_{\rm damp}]W$ (9)

where $\Gamma_{\rm CR}$ is the wave growth rate due to CR streaming
instability (the $\delta-$function encloses the resonance condition):

$\Gamma_{\rm
CR}=\frac{16\pi^{2}}{3}\frac{\vec{v}_{A}}{kWB^{2}}\cdot\int_{0}^{\infty}dpp^{4}v(p)\delta\left(p-\frac{qB}{kc}\right)\vec{\nabla}f,$
(10)

and $\Gamma_{\rm damp}$ is the wave damping rate.
The damping of waves depends on the conditions of the background plasma and on
the wave density itself. In a hot ionized medium the most relevant processes
are the turbulent cascade, the non linear Landau damping and the turbulent
damping (see Sec. 3.1).
The turbulent cascade of waves to smaller scales is a nonlinear process that
can be described as a diffusion in wavenumber space, with a diffusion
coefficient given by[70, 71]

$D_{kk}=5.2\times 10^{-2}v_{A}k^{7/2}W^{1/2}.$ (11)

At small enough scales waves will eventually dissipate and heat the gas. The
typical rate for this process is $\Gamma_{\rm casc}\sim D_{kk}/k^{2}$.
Non linear Landau damping is due the interaction between the beat of two
Alfvén waves and the thermal ions, of temperature $T$, in the background
plasma. The damping rate is given by[72, 73]

$\Gamma_{\rm NLL}=\frac{1}{2}\sqrt{\frac{\pi}{2}\frac{k_{\rm
Bol}\,T}{m_{p}}}\frac{k\,W}{r_{L}}.$ (12)

Turbulent damping is due to the interaction of the self-generated Alfvén waves
with the background turbulence. Such turbulence can be injected by
astrophysical sources at scales, $\rm L_{inj}$, much larger than that of the
Larmor radius of CR particles, and with a turbulent velocity $\rm v_{T}$. For
waves resonant with CRs with a given Larmor radius, the damping rate is given
by[74, 75]

$\Gamma_{\rm turb}=\left(\frac{v_{T}^{3}/L_{inj}}{r_{L}\,v_{A}}\right)^{1/2}.$
(13)

In a cold/warm partially ionized medium, a most relevant damping process is
the ion-neutral damping [33, 76, 86]. For the typical conditions found in the
disk of the MW, this process is very effective, so that the damping timescale
for waves resonant with CRs of momentum $\lesssim 100$ GeV is of the order of
tens of years. This means that, probably, the bulk of CRs stream very fast in
the disk and is weakly coupled to the ISM. In the halo, instead, the fraction
of neutrals is negligible and CRs can be scattered. The galactic halo is
probably the actual diffusion region. In this scenario, the transport of CRs
in the disk region is unclear[87, 88]. On the other hand, another possibility
is that the neutral phases are mostly confined in clumps/clouds, while most of
the disk volume is filled with warm/hot ionized gas[89], so that CR diffusion
could work also in the near-disk region. This aspect is very relevant for the
CR-driven wind formation, since the coupling between CRs and the ISM provided
by the CR scattering on plasma waves, is necessary to launch winds[14, 37].

Starting from the CR and wave distribution functions one can define the
corresponding hydrodynamic energy density and pressure that determine the wind
hydrodynamics:

$\displaystyle\epsilon_{c}(\vec{r})=\int_{0}^{\infty}dp\;4\pi
p^{2}\,T(p)\,f(p,\vec{r}),$ (14) $\displaystyle
P_{c}(\vec{r})=\int_{0}^{\infty}dp\;\frac{4\pi}{3}p^{2}\,pv(p)\,f(p,\vec{r})$
(15)
$\displaystyle\epsilon_{w}(\vec{r})=\int_{k_{0}}^{\infty}\frac{{B}^{2}}{4\pi}W\,(k,\vec{r})\,dk$
(16) $\displaystyle P_{w}(\vec{r})=\epsilon_{w}/2,$ (17)

where $T(p)=\sqrt{p^{2}c^{2}+(mc^{2})^{2}}-mc^{2}$ is the particle kinetic
energy.

##  5 Cosmic ray driven winds: hydrodynamics

The hydrodynamics of CR-driven winds is described by the equations for the
conservation of mass, momentum and energy for the gas, by the evolution of the
background magnetic field and by the equations for the CR and magnetic wave
pressure. The last are derived by appropriate integrations, respectively in
momentum and in wave number, of the transport equation of CR and of the waves
(see Sec. 4). The equations[14] read

$\displaystyle\frac{\partial\rho}{\partial
t}+\vec{\nabla}\cdot\left(\rho\vec{u}\right)=q$ (18)
$\displaystyle\frac{\partial\rho\vec{u}}{\partial
t}+\vec{\nabla}\cdot\left(\rho\vec{u}\vec{u}+P_{tot}\mathbf{I}-\frac{\rho\vec{B}\vec{B}}{4\pi}\right)=\rho\vec{g}+\vec{m}$
(19) $\displaystyle\frac{\partial e_{tot}}{\partial
t}+\vec{\nabla}\cdot\vec{S}=\rho\vec{u}\cdot(\vec{g}+\vec{m})+\epsilon$ (20)
$\displaystyle\frac{\partial\vec{B}}{\partial
t}=\nabla\times\left(\vec{u}\times\vec{B}\right)$ (21)
$\displaystyle\vec{\nabla}\cdot\vec{B}=0$ (22)
$\displaystyle\frac{\partial}{\partial
t}\left(\frac{P_{c}}{\gamma_{c}-1}\right)+\vec{\nabla}\left[\frac{\gamma_{c}}{\gamma_{c}-1}\left(\vec{u}+\vec{v_{A}}\right)P_{c}-\frac{\bar{D}}{\gamma_{c}-1}\vec{\nabla}P_{c}\right]=\left(\vec{u}+\vec{v_{A}}\right)\cdot\vec{\nabla}P_{c}+Q_{c}$
(23) $\displaystyle\frac{\partial e_{w}}{\partial
t}+\vec{\nabla}\left[\left(\frac{3}{2}\vec{u}+\vec{v_{A}}\right)e_{w}\right]=\vec{u}\cdot\vec{\nabla}P_{w}-\vec{v_{A}}\cdot\vec{\nabla}P_{c}+Q_{w},$
(24)

where

$\displaystyle P_{tot}=P_{g}+P_{c}+P_{w}+\frac{B^{2}}{8\pi}$ (25)
$\displaystyle e_{tot}=\frac{1}{2}\rho
u^{2}+\frac{P_{g}}{\gamma_{g}-1}+\frac{P_{c}}{\gamma_{c}-1}+e_{w}+\frac{B^{2}}{4\pi}$
(26)
$\displaystyle\vec{S}=\left(\frac{1}{2}u^{2}+\frac{\gamma_{g}}{\gamma_{g}-1}\frac{P_{g}}{\rho}\right)\rho\vec{u}+\frac{1}{\gamma_{c}-1}\left[\gamma_{c}\left(\vec{u}+\vec{v_{A}}\right)P_{c}-\bar{D}\vec{\nabla}P_{c}\right]$
(27) $\displaystyle\hskip
10.0pt+\left(\frac{3}{2}\vec{u}+\vec{v_{A}}\right)e_{w}+\frac{\vec{E}\times\vec{B}}{4\pi}.$

Here $\rho$, $\vec{u}$, $P_{g}$ and $\gamma_{g}$ are the gas density,
velocity, pressure and adiabatic index, respectively. $P_{c}$, $\gamma_{c}$
and $\bar{D}$ are the CR pressure, adiabatic index and momentum averaged
diffusion coefficient. $\vec{B}$ is the large scale background magnetic field
and $e_{w}=2P_{w}=\langle\delta\vec{B}^{2}\rangle/4\pi$ is the Alfvén wave
energy density, whose speed is

$\vec{v}_{A}(\vec{r})=\frac{\vec{B}(\vec{r})}{\sqrt{4\pi\rho(\vec{r})}}.$ (28)

These equations may also include possible additional sources and sinks of
mass, momentum and energy for the wind, deriving, for instance, from
supernovae explosions and stellar winds, radiative cooling and the damping of
plasma waves. $g(\vec{r})$ represents the galactic gravitational acceleration.

The hydrodynamic CR and wave equations can be derived from the corresponding
kinetic transport equations by integration in momentum and wavenumber,
respectively[37]. The expression for the momentum averaged diffusion
coefficient is given by,

$\bar{D}(\vec{r})=\frac{\int_{0}^{\infty}dp\;4\pi
p^{2}\,T(p)D(p,\vec{r})\vec{\nabla}f}{\int_{0}^{\infty}dp\;4\pi
p^{2}\,T(p)\vec{\nabla}f},$ (29)

while the CR adiabatic index is defined as $\epsilon_{c}=P_{c}/(\gamma_{c}-1)$
and $\gamma_{c}=4/3(5/3)$ for relativistic(non relativistic) particles. It can
be also shown[37] that the hydrodynamic counterpart of the streaming
instability term in the wave transport equation, $\Gamma_{\rm CR}W$, is given
by $\vec{v}_{A}\cdot\vec{\nabla}P_{c}$, which appears in the RHS of Eq. 24.

In many cases relevant for CR propagation, the wave damping, also in the hot
ionized medium of galactic halos, is fast enough that waves produced by
streaming instability are practically damped locally and one can safely ignore
the actual wave transport[16]. In this case, the level of magnetic turbulence
resonant with CR particles is determined by the balance between the growth and
damping rate of waves, $\Gamma_{\rm CR}=\Gamma_{\rm damp}$. The resulting wave
pressure is usually orders of magnitude smaller than that of CRs and of the
background plasma and its contribution to the wind hydrodynamics can be
neglected. However such turbulence is still necessary in order to scatter CRs
and thus to couple them to the ISM. In addition, the dissipated waves heat the
gas. This effect is taken into account by adding the the term
$\vec{v}_{A}\cdot\vec{\nabla}P_{c}$ to the the equation for the gas energy
density, since all the energy injected by CRs into waves is transferred to the
gas.

The hydrodynamic of CR-driven winds has been widely investigated in the
literature, starting from the relatively computationally simpler and faster
stationary approaches[14, 15, 17, 16, 37, 38], to time-dependent calculations
up complex, refined simulations[19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32].

##  6 Stationary cosmic ray-driven winds in the Milky Way

So far, the only self-consistent solutions of the coupled system of equations
of the CR transport and of the CR-driven wind hydrodynamics, have been
obtained solely in the stationary regime and under some simplifying
assumptions, for instance on the large scale structure of the galactic
magnetic field, and neglecting radiative cooling[37, 38]. Despite these
limitations, the study of stationary winds provides precious information on
their properties and on their impact on the CR transport. Moreover, being much
easier and faster than simulations and of more complex time-dependent
approaches, it allows to quickly explore wide parameter ranges[16, 38].

In this section we report a common model of stationary winds, illustrating the
main physical ingredients, the relevant launching parameters in the MW and
some possible implications on the observed CR spectrum.

###  6.1 The geometry of the magnetic field lines

Radio observations show the presence of galactic magnetic fields, usually of
the order of $\rm\mu G$ and with a coherence length that can exceed $\sim$
kpc, both in the MW and in other galaxies[90, 91, 92]. Their origin is
commonly attributed to complex dynamo processes (see Refs. Widrow-2002-B-gal-
origin, Hanasz-2017 and references therein). In the disk region, field lines
are mainly parallel to the disk surface, while also X-shaped, out of disk
components, have been detected[94]. The activity from several classes of
astrophysical sources, such SNe explosions of single stars and cluster of
stars, and stellar winds, which also contribute to the formation of GWs, can
push the field lines in the disk and make them break through the halo. The
pressure support provided by these events may last $10^{6}-10^{7}$ yr, but the
overlap of the effect of multiple such events, randomly taking place in the
galactic disk, may keep the field line opening up to distances from the disk
of the order of the disk radius and for time scales of $\gtrsim 10^{8}-10^{9}$
yr. Moreover, in the presence of GWs, field lines attached to the outflowing
material can be further transported outward. The rotation of the galaxy
additionally contribute to shape the large scale magnetic field[14, 17].

In principle, the magnetic field geometry, which is also essential to CR
propagation, should be determined self-consistently with the wind structure,
which represents a big complication in models of CR-driven winds. However a
good approximation, adopted in several stationary approaches[14, 16, 37], is
to pre-assign the magnetic flux tube shape in a way that mimics the geometries
found in more complex simulations, giving its area as a function of the
distance from the disk. With such approach the equations also becomes one-
dimensional and only depend on the coordinate perpendicular to the galactic
disk. A typically adopted geometry, that intuitively encompass the ingredients
illustrated above and confirmed by simulations, is the mushroom-like shape in
which the magnetic flux tube is roughly cylindrical up to a given distance
from the disk (comparable with the disk radius) and then opens up in a roughly
cylindrical way.

###  6.2 Stationary equations in the given flux tube geometry

In the stationary regime and assuming a given flux-tube geometry, as explained
above, the equations greatly simplify and become one-dimensional[14, 37]. Be
$z$ the coordinate perpendicular to the disk, and $A(z)$ the area of the
mushroom-like flux tube

$A(z)=A_{0}\left[1+\left(\frac{z}{Z_{b}}\right)^{\alpha}\right],$ (30)

where $Z_{b}\sim$ disk radius and $\alpha=2$ for spherical opening.
The hydrodynamic equations and the CR transport equation become[14, 37]

$\displaystyle\rho uA=\textnormal{const},$ (31) $\displaystyle
AB=\textnormal{const},$ (32)
$\displaystyle\frac{du}{dz}=u\frac{c_{*}^{2}\frac{1}{A}\frac{dA}{dz}-\frac{d\Phi}{dz}}{u^{2}-c_{*}^{2}},$
(33)
$\displaystyle\frac{dP_{g}}{dz}=\gamma_{g}\frac{P_{g}}{\rho}\frac{d\rho}{dz}-(\gamma_{g}-1)\frac{v_{A}}{u}\frac{dP_{c}}{dz}$
(34)
$\displaystyle\frac{dP_{c}}{dz}=\gamma_{eff}\frac{P_{c}}{\rho}\frac{2u+v_{A}}{2(u+v_{A})}\frac{d\rho}{dz},$
(35) $\displaystyle
c_{*}^{2}=\gamma_{g}\frac{P_{g}}{\rho}+\gamma_{eff}\frac{P_{c}}{\rho}\left[1-(\gamma_{g}-1)\frac{v_{A}}{u}\right]\frac{2u+v_{A}}{2(u+v_{A})},$
(36)
$\displaystyle\frac{\gamma_{eff}}{\gamma_{eff}-1}=\frac{\gamma_{c}}{\gamma_{c}-1}-\frac{\overline{D}}{(\gamma_{c}-1)(u+v_{A})P_{c}}\frac{dP_{c}}{dz}$
(37)
$\displaystyle\frac{u^{2}}{2}+\frac{\gamma_{g}}{\gamma_{g}-1}\frac{P_{g}}{\rho}+\Phi+\frac{\gamma_{eff}}{\gamma_{eff}-1}\frac{P_{c}}{\rho}\frac{u+v_{A}}{u}=\textnormal{const}$
(38)

and

$\frac{\partial}{\partial z}\left[A\,D\frac{\partial f}{\partial
z}\right]-A\,(u+v_{A})\frac{\partial f}{\partial
z}+\frac{d\,A\,(u+v_{A})}{dz}\frac{1}{3}\frac{\partial f}{\partial\ln
p}+A\,Q=0,$ (39)

where $\Phi(z)$ is the galactic gravitational potential. It has been assumed
that Alfvén waves produced by CR streaming instability are damped locally and
heat the gas[16, 37]. $Q(z,p)$ represents the CR injection, which, assuming
that the sources are mainly SNRs, can be approximated as a power-law in
momentum and to be concentrated in a thin disk $Q(z,p)\approx
Q_{0}(p)\delta(z)$. Moreover, $Q_{0}\propto\xi_{CR}\mathcal{{R}_{SN}}$, where
$\xi_{CR}$ is the CR acceleration efficiency and $\mathcal{{R}_{SN}}$ is the
SN explosion rate.

Eq. 33 is the so-called wind equation, which represents the acceleration of
the wind fluid. It contains the generalized sound speed $c_{*}$, which
includes the sound speed, $\sqrt{\gamma_{g}P_{g}/\rho}$, modified by the CR
pressure through a ”CR sound speed” that depends on the Alfvénic Mach number
$u/v_{A}$. It the case in which Alfvén waves are not damped very quickly, one
would have an equation for waves and a ”wave sound speed” in the definition of
$c_{*}$[14]. $\gamma_{eff}$ is an effective adiabatic index that takes into
account the CR diffusivity. It has been shown that in most cases
$\gamma_{eff}\sim 1.1-1.2$, while $\gamma_{c}\sim 4/3$[38]. The wave heating
is encompassed by the term $-(\gamma_{g}-1)v_{A}/u\,dP_{c}/dz$ in Eq. 34.

The wind equation 33 presents different classes of solutions depending on
whether its numerator or its denominator, or both, vanish[13, 14] at a given
position $z$. The location where both vanish is called the sonic point and it
corresponds to the position where the flow velocity equals the compound sound
speed, $u=c_{*}$.
The solution relevant for GWs is the one for which the flow is subsonic near
the galactic disk, at the position $z_{0}$ where the boundary conditions are
specified, then it is accelerated outward and smoothly transit, at the sonic
point $z_{c}$, to the supersonic regime. At large enough distance from the
disk, all pressures tend to zero and the wind reaches its terminal velocity,
$u_{f}$. The boundary conditions to be given at $z_{0}$ are, for instance, the
gas density and temperature, the CR pressure and the value of the background
magnetic field, but not the launching velocity $u_{0}$. In fact, the launching
velocity is determined by imposing the passage through the sonic point. Since
all other quantities are fixed at $z_{0}$, this also fixes the mass and energy
fluxes.

For launching velocities smaller than $u_{0}$ the flow never becomes
supersonic and there is a point where the numerator of the wind equation
vanishes. These solutions are called breezes and are characterized by a non
vanishing gas pressure at infinity. For launching velocities larger than
$u_{0}$ one may either have unphysical solutions, where only the denominator
vanishes at some point, or purely supersonic flows.

Figure 3: De Laval nozzle: the flow is constantly and smoothly accelerated all
along the duct, from the subsonic regime (converging portion), to the
supersonic regime (diverging portion). The sonic point coincide with the
throat.

The wind equation is analogous to that of the Parker model for the Solar
wind[13, 95] and to that of the De Laval nozzle[38]. In fact, if one defines
$a(z)\equiv\frac{1}{A}\frac{dA}{dz}$ and $g(z)\equiv\frac{d\Phi}{dz}$ in the
wind equation, the last becomes formally identical to that of the De Laval
nozzle

$\displaystyle\frac{1}{u}\frac{du}{dz}=\frac{a(z)-\frac{g(z)}{c^{*2}}}{M^{2}-1}\;\;\;(\textnormal{Wind
equation})$ (40)
$\displaystyle\frac{1}{u}\frac{du}{dz}=\frac{\mathfrak{a}(z)}{M^{2}-1}\;\;\;(\textnormal{De
Laval nozzle equation})$

where $M=u/c^{*}$ is the Mach number and
$\mathfrak{a}(z)\equiv\frac{1}{A}\frac{dA}{dz}$ for a nozzle of area $A(z)$,
as shown in Fig. 3. The nozzles is composed by a duct that presents a
converging section followed by a diverging section. The choke point
corresponds to $\mathfrak{a}(z)=0$. The geometry of the nozzle allows a smooth
transition from the subsonic to the supersonic regime. The flow begins as
subsonic in the converging portion, with $\mathfrak{a}(z)<0$ and $M<1$, so
that $du/dz>0$. At the throat $\mathfrak{a}(z)=0$ and $M=1$ and the flow
becomes transonic. Note that only in this way $du/dz$ can remain smooth. In
the diverging portion $\mathfrak{a}(z)>0$ and $M>1$ so that $du/dz$ remains
positive and the flow continues to be accelerated (at least until a shock
appears).
In the case of a wind the numerator $a(z)-\frac{g(z)}{c^{*2}}$ plays the role
of the duct area in a De Laval nozzle. In order to keep accelerating the fluid
away from the disk:

  * •

$u^{2}<c^{*2}$ (subsonic flow)

the gravitational term $g(z)$ has to dominate. It behaves as a sort of
”converging duct” term,

  * •

$u^{2}>c^{*2}$ (supersonic flow)

the area term $a(z)$ has to dominate. It behaves as a sort of ”diverging duct”
term.

This analysis illustrates that if the area of the flux tube is constant,
$a(z)=0$, a stationary wind solution is not possible. Moreover, in an
expanding flux tube, $a(z)>0$, as it happens in the case of a wind, the
gravity allows for a stationary solution.

###  6.3 The role of cosmic rays and the cosmic ray spectrum

The possibility to launch a (stationary) wind and its properties strongly
depends on the conditions, such as the gas density and temperature, the CR
density and the magnetic field at the launching point near the galactic disk.
Moreover it also depends on the galactic gravitational potential (and in
particular on the Dark Matter halo) and on the geometry of the field
lines[38]. As discussed above, the spatial dependent CR spectrum affects the
wind launching and is in turn affected by the presence of a wind. All these
conditions depend on the type of galaxy and on the specific location in a
galaxy.

A parametric study of the wind launching is beyond the scope of this review,
and the reader is referred to Refs. Everett-2008 and Recchia-winds-II, which
performed a thorough investigation in the case of the MW. Moreover, Refs.
Recchia-winds-I studied how the CR spectrum in the disk gets modified when the
launching conditions change and how this compares to the observed CR spectrum
at Earth. Here we briefly summarize the main findings of these studies,
focusing specifically on the case of MW and on winds launched near disk region
at the galactocentric distance of the Sun.

####  6.3.1 Wind hydrodynamics

In general, stationary winds can exist only for closed intervals of the
launching parameters. In the model presented here, the relevant quantities
near the disk are the ISM density, $n_{0}$, and temperature, $T_{0}$, the CR
pressure, $P_{c0}$, and the magnetic field, $B_{0}$.
Winds probably originate from the less dense and hotter phase of the ISM,
namely from the HIM[14, 16], which means for the MW, $n\approx
10^{-3}-10^{-2}\rm\,cm^{-3}$ and $T\approx\rm 10^{6}\,K$[89]. The CR pressure
is $P_{c}\sim 4\times 10^{-13}\,\rm erg/cm^{3}$[60]. The galactic magnetic
field is observed to be of the order of few $\rm\mu G$[94]. As for the flux
tube area, $Z_{b}\approx 15$ kpc (the disk radius) and $\alpha\approx 2.0$[14,
38].
The gravitational potential of the MW is mainly contributed by the bulge, the
disk and the DM halo. The last is especially relevant for GWs, since it
extends for hundred kpc from the disk and affects the wind structure up to
such large distances[38].
Wave damping is also a relevant ingredient in determining the possibility to
launch winds[14, 16, 37]. In fact, in the absence of damping, Alfvén waves
produced by CR streaming instability can grow undisturbed, such that their
pressure becomes comparable to that of CRs and contribute to push the gas.
This would make the wind launching easier, also in the case of denser and
cooler gas. In the presence of wave damping the parameter space is
additionally reduced by the fact that wave heating reduces the gas pressure
gradient along $z$, and if it is too strong it may even stall the outflow[16].

It as been shown that, with launching parameters close to those mentioned
above, stationary winds can exist. An example is reported in Fig. 4, 5 and 6,
where the chosen launching parameters are listed in Table 6.3.1.

Example of wind launching parameters that are compatible with the observed
conditions in the local Galactic environment and that allow to fit reasonably
well the observed CR proton spectrum below $\sim 200$ GeV. The resulting
Alfvén and wind speeds at the launching point are $v_{A0}\sim 28$ km/s and
$u_{0}\sim 3$ km/s, respectively. The resulting final velocity is $u_{f}\sim
404$ km/s, while the mass loss rate per unit area is $\dot{m}\approx 4.7\times
10^{-4}\rm M_{\odot}kpc^{-2}yr^{-1}$.  Parameter Symbol Value gas density
$n_{0}$ $6\times 10^{-3}$ gas temperature $T_{0}$ $2\times 10^{6}$ K magnetic
field $B_{0}$ $1\,\mu G$ magnetic flux tube scale height $Z_{b}$ 15 kpc
magnetic flux tube opening index $\alpha$ 2.0 CR pressure $P_{c0}$ $4\times
10^{-13}$

The resulting Alfvén speed is $v_{A0}\sim 28$ km/s, the wind launching
velocity is $u_{0}\sim 3$ km/s and the wind final velocity is $u_{f}\sim 404$
km/s, while the mass loss rate per unit area is $\dot{m}\approx 4.7\times
10^{-4}\rm M_{\odot}kpc^{-2}yr^{-1}$.
This set of launching parameters are compatible with the observed conditions
in the local Galactic environment and lead to a reasonably good fit of the
observed CR proton spectrum below $\sim 200$ GeV, as illustrated in what
follows.

Near the disk, the gas pressure dominates over the CR pressure and the flow is
sub-Alfvénic. A sizable fraction of the CR momentum is employed to produce
Alfvén waves, whose fast damping contribute to keep the gas hot up $\approx
100$ kpc and makes the gas temperature decrease with $z$ much slower than the
gas density above $\sim 10$ kpc. On the other hand, CRs have a smaller
adiabatic index than the gas ($\gamma_{c}=4/3$, $\gamma_{eff}\approx 1.1-1.2$,
$\gamma_{g}=5/3$) so that the CR pressure decreases slower with $z$ than the
gas pressure. This has prominent consequences on the wind launching: the
thermal pressure is effective at driving winds near the disk, but at larger
$z$ CRs continue pushing the gas, often making an outflow possible also in
situations where the thermal pressure would not have been sufficient[16].

Figure 4: Example model: wind velocity $u_{wind}$, Alfvén velocity $v_{A}$ and
compound sound speed $c^{*}$, as function of the distance from the disk. The
sonic point is where $u_{wind}$ crosses $c^{*}$. Figure 5: Example model: gas
pressure $P_{g}$, CR pressure $P_{c}$ and wave pressure $P_{w}$, as function
of the distance from the disk. Due to damping, the wave pressure is much
smaller than the $P_{g}$ and $P_{c}$. Since the CR adiabatic index is smaller
than the gas adiabatic index, $P_{c}$ drops with $z$ more slowly than $P_{g}$.
Figure 6: Example model: gas density $n$ and temperature $T$, as function of
the distance from the disk. Due to wave heating, the gas temperature drops
quite slowly with $z$. In fact the wind is roughly isothermal up to $\sim 100$
kpc from the disk.

It is interesting to notice that the mass loss rate per unit area reported
above $\dot{m}\approx 4.7\times 10^{-4}\rm M_{\odot}\,kpc^{-2}\,yr^{-1}$, if
present on the whole Galactic disk, would lead to a mass loss rate of
$\dot{M}\approx 0.36\,\rm M_{\odot}\,yr^{-1}$, which is comparable with the
Galactic star formation rate ($\sim\rm M_{\odot}$/yr; see e.g Ref.
Robitaille-2010). This confirms the importance of CR-driven winds in MW-like
galaxies.

For values of the launching parameters much larger or smaller than the one
quoted above, winds might not exist or there could be some type of outflow but
it would not be stationary. For instance, if the ISM is too dense and/or cool,
the available energy could not be enough to drive an outflow, while for very
hot and dilute gas the outflow may resemble more to an evaporation.
In general, an increase of the gas density leads to winds that have a smaller
launching velocity and a smaller final velocity, and overall, a smaller mass
loss rate. An increase of the gas temperature and of the CR pressure lead to
an increase of $u_{0}$, and, with fixed $n_{0}$, also to an increase of the
mass loss rate. However, multiplying $T_{0}$ and $P_{c0}$ by the same factor,
the resulting increase in $u_{0}$ is larger in the case of an increase of
$T_{0}$. This is due to the fact that momentum and energy deposited before the
sonic point can affect both $u_{0}$ (and $\dot{m}$) and $u_{f}$, while beyond
the sonic point they affect only $u_{f}$, since $u_{0}$ is determined by the
passage through the critical point. As illustrated above, the larger adiabatic
index of the gas compared to CRs implies that the gas pressure get ”exhausted”
closer to the disk, while the CR pressure continues to be effective also at
larger distances. Thus, the thermal pressure is generally more efficient at
increasing the $u_{0}$ (and $\dot{m}$). Moreover, the effect of the CR
pressure also depends on whether the flow is sub-Alfvénic or super-Alfvénic.
In fact, in these two limits the CR pressure gradient becomes[16, 38]:

$\displaystyle\lim_{u<<v_{A}}\frac{dP_{c}}{dz}=\frac{\gamma_{eff}}{2}\frac{P_{c}}{\rho}\frac{d\rho}{dz}$
(42)
$\displaystyle\lim_{u>>v_{A}}\frac{dP_{c}}{dz}=\gamma_{eff}\frac{P_{c}}{\rho}\frac{d\rho}{dz}.$
(43)

Thus, the effective CR adiabatic index is reduced in the sub-Alfvénic case,
making CRs less efficient at driving the wind in this regime. The behavior of
$u_{f}$ is more complex and in general an increase of $T_{0}$ and $P_{c0}$
does not necessarily lead to an increase of $u_{f}$. In fact, $u_{f}$ is
computed from the energy conservation equation 38, considering that at large
$z$ all other quantities drop to zero, and the total wind energy is solely
kinetic:

$\frac{u_{f}^{2}}{2}=\frac{u_{0}^{2}}{2}+\frac{\gamma_{g}}{\gamma_{g}-1}\frac{P_{g0}}{\rho_{0}}+\Phi(z_{0})+\frac{\gamma_{eff}}{\gamma_{eff}-1}\frac{P_{c0}}{\rho_{0}}\frac{u_{0}+v_{A0}}{u_{0}}.$
(44)

The CR term depends on the Alfvénic Mach number $M_{A0}=u_{0}/v_{A0}$ through
the factor $(1+M_{A0})/M_{A0}$, which is large for $M_{A0}\ll 1$ and becomes
unity for $M_{A0}\gg 1$. An increase of either $T_{0}$ or $P_{c0}$ increases
$u_{0}$ and, thus $M_{A0}$. Correspondingly, the $(1+M_{A0})/M_{A0}$ decrease.
If such decrease is compensated by the increase of $P_{g0}$ or $P_{c0}$,
$u_{f}$ will increase, otherwise $u_{f}$ will decrease. In the limit in which
the flow is super-Alfvénic $u_{f}$ increases with $T_{0}$ or $P_{c0}$[38].

####  6.3.2 Radiative cooling

In the model illustrated here radiative cooling is not included. However it
may be a key ingredient in shaping or hampering a wind and deserves some
attention.
For a hot ISM phase ($T\approx 10^{6}$ K), the dominant cooling process is the
emission of forbidden lines and soft X-rays, at a rate given by[45]

$\frac{1}{T}\frac{d\,T}{dt}\sim\Lambda(T)n^{2},$ (45)

where the cooling function is $\Lambda(\rm T\,\sim\,10^{6})\sim 2\times\rm
10^{-23}erg\,cm^{3}/s$ for Solar metallicity[45]. The cooling rate increases
with the gas density.
Radiative cooling can be included in the stationary wind model, without
affecting the topology of the possible solutions[97] by modifying the equation
of the gas pressure, Eq. 34, as

$\frac{dP_{g}}{dz}=\gamma_{g}\frac{P_{g}}{\rho}\frac{d\rho}{dz}-(\gamma_{g}-1)\frac{v_{A}}{u}\frac{dP_{c}}{dz}-(\gamma_{g}-1)\Lambda(T)\frac{n^{2}}{u},$
(46)

and the wind equation, Eq. 33, as

$\frac{1}{u}\frac{du}{dz}=\frac{c_{*}^{2}\,a(z)-g(z)+(\gamma_{g}-1)\Lambda(T)\frac{n}{u}}{u^{2}-c_{*}^{2}}.$
(47)

Comparing the timescale for radiative cooling and wave heating for the case
shown in Fig. 6, one gets $\tau_{cool}\sim 10^{7}$ yr and $\tau_{heat}\sim
10^{8}$ yr[37, 38]. In general, in the absence of additional heating sources
close to the disk, where the gas density is larger, radiative cooling might be
so intense as to hamper the wind launching. A possible heating agent are SN
explosions which heat and ionize the surrounding medium and also inject
magnetic turbulence that may eventually dissipate and additionally heat the
gas[98]. Also ionization and Coulomb losses of CRs may represent an important
contribution to the gas heating[35]. Finally, radiative cooling may limit the
”stationarity” of GWs, making the outflowing material to fall back to the
disk[7, 8], and establishing a gas recycling in the galaxy.

####  6.3.3 Cosmic ray spectrum

When dealing with GWs launched at the galactocentric distance of the Sun, one
has to keep in mind that, if on the one side the wind is determined by the CR
pressure, the last depends on the CR spectrum, which is affected by the wind
and by self-generated waves in a highly non-linear way, and in the near-disk
region should match the observed CR spectrum.

When we speak about the ”observed CR spectrum”, it is important to stress that
the fluxes detected at Earth, for instance those reported by the AMS
experiment[54] are largely affected by the solar modulation at energies below
$\sim 10$ GeV[99]. On the other hand, the Voyager I spacecraft has traveled
far enough from the Sun that the measured spectrum, which is reported down to
$\sim 3$ MeV, is no more affected by solar modulation[100, 101]. On the other
hand, the CR pressure is largely dominated by $\sim$ GeV CR protons, which
represent the most abundant CR species (see Sec. 3). For this reason , the
wind launching is mainly affected by CRs whose local measurement is affected
by the solar activity. Thus it is important to compare the computed spectrum
in the wind both with the Voyager I data and with AMS data, provided that
solar modulation is taken into account.

Sub-GeV CRs may still affect the wind through the heating produced by
ionization and Coulomb losses[35]. The importance of low-energy CRs is not
restricted to that. In fact, such CRs are widely recognized as the main
ionization agent of the interior of molecular clouds (see e.g Refs.
Phan-2018-ionization, Recchia-2019-carrot and references therein), with
important implications on the interstellar chemistry and on star formation
(see e.g Krumholz-2011-ion-MC-star). Moreover, they play a crucial role in the
nucleosynthesis of light elements (lithium, beryllium, and boron)[105].

The requirement that the calculated spectrum matches the observed spectrum
provides a stringent constraint on the wind parameters[37, 38]. In fact, in
many cases, winds launched with parameters compatible with the local ISM lead
to CR spectra much different from the observed one. In most cases such spectra
result harder than observations at low energy, below 100-1000 GeV, and much
steeper than observations at higher energies[37, 38]. In Refs. Recchia-winds-I
and Recchia-winds-II it is shown that, assuming that CR sources (say SNRs)
inject a power-law spectrum in momentum, $Q_{0}(p)\propto p^{-\gamma}$, with
$\gamma\approx 4.0-4.4$[58, 67], the overall normalization of the injection
spectrum can be chosen in two ways, by suitably adjusting the SNe explosion
rate (in the case of SNRs) and/or the CR acceleration efficiency: i) the
calculated and observed proton spectra at Earth coincide at a specific
particle momentum. Such momentum should be taken between 10-20 GeV since below
that energy the spectrum detected at Earth is affected by solar modulation;
ii) the total CR pressure computed from the proton spectrum, using Eq. 16,
matches the observed one, $P_{c\,\odot}\sim 4\times\rm 10^{-13}erg/cm^{3}$.
Notice that, since the CR pressure derives from an integral in momentum of the
CR distribution, different CR spectra may lead to the same pressure. Moreover,
since the CR spectrum is steeper than $\propto p^{-4}$, the specific spectral
shape above $\sim 10$ GeV barely affects the total CR pressure.

With both prescriptions on the normalization of the CR injection spectrum
there is no guarantee at all that the computed spectrum matches the observed
spectrum at all energies[38]. This is mainly due to two factors which can be
better understood keeping in mind the benchmark propagation model illustrated
in Sec. 3 and again assuming a power-law CR injection spectrum as illustrated
above. For particle momenta for which advection dominates the spectral slope
of the calculated spectrum will be close to the injection slope, while, where
diffusion dominates, the spectral slope will be steeper. In the model
presented in this section, the CR diffusion coefficient is solely due to self-
generated Alfvén waves, whose growth rate, that depends on the CR density
gradient, decreases with energy (the CR spectrum is steeper that $\propto
p^{-4}$). This lead to a diffusion coefficient which sharply rises with
energy, with a slope larger than the $\sim 0.3-0.6$ typically assumed in
standard models of CR propagation[60], as shown in Fig. 9.

Figure 7: CR proton spectrum as a function of the particle kinetic energy in
the near disk region, as compared to observations. The green curve shows the
spectrum corresponding to the wind model of Fig. 4, with a total advection
velocity near the disk of $\sim 30$ km/s. The red curve shows the same
spectrum with the application of solar modulation[99]. The blue curve shows a
spectrum obtained with a total advection velocity near the disk of $\sim 100$
km/s, which is too hard to fit data. The magenta curve shows qualitatively the
effect of an additional, Kolmogorov-like, extrinsic turbulence that scatter
CRs. This leads to a much harder spectrum at energies $\gtrsim 100-1000$ GeV,
as compared to streaming instability. Figure 8: CR distribution as function of
the particle momentum $p$, for different distance from the disk. The
corresponding wind structure is shown in Fig. 4. Figure 9: CR diffusion
coefficient as function of the particle momentum $p$, for different distances
from the disk. The corresponding wind structure is shown in Fig. 4. Notice the
sharp momentum dependence of $D(p)$, as compared to the $\propto p^{0.3-0.6}$
usually assumed in standard CR propagation scenarios[60, 58]. This is a normal
consequence of streaming instability, for which the level of waves depends on
the CR density. Figure 10: Effective diffusive halo size as function of the
particle momentum $p$. The corresponding wind structure is shown in Fig. 4.
Particles of momentum $p$ are basically advected out of the galaxy for
$z>S^{*}(p)$. The effective diffusive halo size increases with energy, as
shown in Sec. 3

As illustrated above, winds are mainly launched from relatively hot and dilute
phases of the ISM. This leads to a CR total advection velocity, $u+v_{A}$,
close to the disk which can easily exceed $\sim 100$ km/s. Such value
increases with $z$. The resulting CR transport will be dominated by advection
up to hundreds GeV, resulting in a spectrum harder than observations at these
energies. A smaller advection speed can be obtained, at fixed $T_{0}$, by
considering launching parameters still compatible with observations but with
larger gas density and smaller $B_{0}$. This can be seen in Fig. 7, where we
show the spectrum obtained with the launching parameters: $n_{0}=0.003$ cm-3
and $B_{0}=2\,\mu$G, which gives $u_{0}+v_{A0}\sim 100$ km/s; $n_{0}=0.006$
cm-3 and $B_{0}=1\,\mu$G, which gives $u_{0}+v_{A0}\sim 30$ km/s (the last
values have been used in Fig. 4, 5 and 6, and are reported in Table 6.3.1). In
the case of larger density and smaller $B_{0}$, the predicted spectrum matches
reasonably well observations, while with the other set of parameters the
predicted spectrum is too hard up to hundreds GeV. This is a rather general
feature of CR-driven winds, in which launching parameters compatible with
observations and for which a wind can exist, may lead to a CR spectrum
qualitatively different from the observed one, even if the corresponding CR
pressure is close to observations.
For this reasons, the measurement of the CR spectrum at energies below few
hundreds GeV represents a precious constraint on the wind parameters. This is
surely true for winds launched at the galactocentric distance of the Sun,
where we have a direct measurement of the CR spectrum, but it is applicable
also to the cases in which an indirect indication of the CR spectrum is
available, for instance thanks to $\gamma-$ray observations. This is the case
for CRs interacting with the ISM in the Galactic disk at different
galactocentric distances[106], or for $\gamma-$ray observations in other
galaxies[107, 108].

At higher CR energies ($E\gtrsim 100-1000$ GeV), the steep momentum dependence
of the CR-induced diffusion coefficient, additionally enhanced by the opening
of the flux tube, and only mildly compensated by the progressive increase of
the effective diffusion region with energy (see Sec. 3 and Fig. 10), gives a
very steep spectrum. This result cannot be changed by a change of the
launching parameters. On the other hand, it has been shown by several authors
(see e.g Ref. Aloisio-2015 and Evoli-2018-halo), that additional sources of
turbulence, other than self-generated Alfvén waves, maybe injected by SNe
explosions and characterized by a Kolmogorov-like spectrum (which would
correspond to a diffusion coefficient $D\propto p^{0.3}$), may be relevant at
those energies and could even revert the spectral trend illustrated here and
lead to a hardening[67]. Such hardening is in fact measured in the CR spectrum
(see Sec. 3). The possible effect of such an additional turbulence is
qualitatively shown in Fig. 7.
In Fig. 8 we show the CR distribution as a function of particle momentum $p$
for different distances from the disk. Low energy particles, below $\sim 100$
GeV, experience a diffusive halo smaller than $\sim 10$ kpc. Above that
distance they are advected out of the galaxy in a progressively larger flux
tube. This results in a fast decrease of the CR density with $z$ and in a
progressively harder spectrum. Higher energy particles experience a much
larger diffusive halo (see Fig.10). This leads to a smaller decrease of the CR
density with $z$. For such energies, the opening of the flux tube is partially
compensated by a decrease of the diffusion coefficient at large $z$, also
shown in Fig. 9. This effect is the result of the non-linear nature of the
streaming instability combined with the decrease of the background magnetic
field. If fact, if on the one side the CR density is small at high energies
and slightly decreases with $z$, the background magnetic field also decreases,
so that $\rm\langle\delta B\rangle/B$ in Eq. 8 may become larger. This may in
fact lead to an overall decrease of the diffusion coefficient, despite the
small amount of produced waves[106].

##  7 Conclusions and perspective

GWs represents a most important feedback process in the life of galaxies, and
their study constitutes an active research area in astrophysics, both from the
observational and theoretical point of view. GWs may be powered by several
mechanisms, depending on the specific environment in which they get formed.

The key role of CRs in launching GWs, especially in MW-like galaxies, has been
confirmed by many hydrodynamic studies. The coupling between CRs and the ISM,
necessary for CRs to effectively push on the background plasma, is mediated by
the scattering of CRs off plasma waves at the very small scales (well below 1
pc, compared to the multi kpc scale of the wind structure) of the Larmor
radius of the CR particles in the galactic magnetic field. Such scattering is
also responsible for the diffusive motion of CRs. The dynamics of CR-driven
winds is a multiscale non-linear phenomenon intrinsically connected with the
microphysics of the CR transport. In fact, the CR distribution responsible for
the wind launching is often affected by CR-induced plasma instabilities, by
the (non-linear) damping of plasma waves, and by the presence of the wind
itself. Moreover, the details of the CR propagation in different galactic
environments is also not well understood. For these reasons, CR-driven winds
are particularly difficult to study, and most of current models only focuses
on the wind hydrodynamics, treating CR as a relativistic gas and neglecting
the kinematics of the CR propagation. So far, the only self-consistent
modeling of the CR propagation together with the wind structure has been
obtained solely for CR protons, in the stationary regime, and under the
assumption that only self-generated waves contribute to diffusion and with a
pre-assigned geometry of the large-scale magnetic field. Moreover possible
relevant effects, such as radiative cooling, the effect of extrinsic sources
of turbulence, and ionization and Coulomb losses for sub GeV CRs, have been
neglected.
Despite these limitations, all these studies allowed to investigate the wind
structure in a variety of galactic environments and to drive some general
conclusions on the CR spectrum expected in the presence of GWs. In particular
it has been shown that the measurement of the CR spectrum below few hundreds
GeV represents an important constraint on the wind structure, since winds
launched with parameters compatible with a given galactic environment
generally lead to spectra different from observations.

Future improvements of the state of the art should focus on several aspects.
First of all, it is of primary importance a better understanding of the
sources and types of plasma turbulence that may scatter CRs, in addition to
self-generated Alfvén waves, together with a deeper study of the possible
damping processes. For instance, the plasma turbulence injected by sources in
the disk and cascading to smaller scales may represent a relevant contribution
to CR scattering[98], whose transport in the halo may be enhanced by GWs. This
would require to solve also the wave transport equation. The relevance of such
investigation is obviously not restricted to CR-driven winds, and is central
to the physics of CRs in general.
Ionization and Coulomb losses are important in determining the shape of the CR
spectrum below GeV and may represent and important source of heating in the
near-disk region[35], which may mitigate the possible effects of radiative
cooling[14, 37].
Another important aspect is the transport of heavier nuclei in the wind, with
possible relevant implications for the production of secondaries, and thus on
observables such as the B/C ratio. This would require the inclusion of
spallation and nuclear decay terms in the equations.
An other interesting possibility could be the reacceleration of CRs at the
wind termination shock, analogous to the CR acceleration at SNR shocks[109,
110].

Second, of great importance is to treat the full, hydrodynamic and CR
transport problem, in time-dependent calculations. In fact, while in some
interesting cases the assumption of stationarity may be justified, it is
inadequate in describing interesting outflows such as ”galactic fountains”,
where the ejected material cools down and falls back to the disk, or outbursts
produced in active galaxies, possibly with the contribution of CRs. Non
stationary winds may originate from the Galactic center region, due e.g to a
periodic activity of the central supermassive black hole, and possibly
connected with the Fermi Bubbles[43]. Moreover, in time-dependent winds, the
formation of multiple shocks may additionally contribute to the CR
reacceleration. Of great interest is also the role of CRs in shaping the large
scale, rotating, galactic magnetic field, which in turn affects the wind
structure and the CR propagation[50].
In general, the development of a fully time-dependent, self consistent, CR-
driven wind model, would allow to investigate such winds and the resulting CR
distribution in detail and in a large variety of environments, from MW-like
galaxies, to starburst[107, 108] and AGN galaxies.

Third, GWs populate the galactic halos with hot plasma up to hundreds of kpc
from the Galactic disk[41, 7, 8], which act as a target for the interaction of
CRs, with the consequent production of $\gamma-$rays and neutrinos. This may
lead to a relevant contribution to the isotropic $\gamma-$ray background[111]
and to the diffuse neutrino flux detected by Ice-Cube[112]. Such and extended
$\gamma-$ray halo has indeed been detected in the Andromeda galaxy[113].

As a final remark, it is important to stress the possible prominent role
played by CRs in the evolution of galaxies. In this review we showed the
effect of CR driving in MW-like galaxies, where the mass loss induced by CR-
driven winds can be comparable to the mass formed in stars. Despite the role
of CRs in starburst and AGN galaxies is not that clear, it may also be
relevant. This, together with the other possible effects of CRs on the
galactic environments discussed above, shows the importance of including CRs
in models of galactic evolution.

## Acknowledgments

The author acknowledge support from Agence Nationale de la Recherche (grant
ANR- 17-CE31-0014).

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