text:
\section{Introduction}

Turbulence in galaxies stands at the crossroad of a wide variety of cosmic processes: star
and planet formation, galatic dynamos, magnetic reconnection, to name a few. It has a strong
interplay, not only with gravity and magnetic fields, but also with the microphysics of gas
because the scales at which it dissipates are close to the mean free path of atoms and
molecules. The turbulent cascade in galaxies therefore channels huge amounts of supra-thermal
kinetic energy from the large scales at which it is fed to those at which atoms and molecules
interact.

This property of the turbulent cascade has started to be exploited to elucidate the
long-standing puzzle raised by the large abundances of many molecular species that have
been detected in the diffuse medium: \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ \citep{Crane1995,Gredel1997,Weselak2008,
Falgarone2010,Godard2012}, \HCOp \citep{Lucas1996,Godard2010}, CO \citep{Sheffer2008,
Liszt2010}, SH and H$_2$S \citep{Neufeld2012}, \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi\ \citep{Godard2012}, CN, HCN, and
HNC \citep{Liszt2001,Godard2010}. Indeed these abundances cannot be understood in the
framework of UV-driven chemistry because their formation proceeds through highly
endoenergetic reactions that cannot be activated at the low temperature of the diffuse
interstellar medium (ISM). These findings are not specific to the Milky Way. Large
abundances of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ are now observed in the diffuse medium of external galaxies where
the \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi(1-0) line is observed in absorption against the bright dust continuum emission
of nearby starbursts: Arp 220 \citep{Rangwala2011}; Mrk231 \citep{van-der-Werf2010}; and
M82, Circinus, and NGC4945 (Falgarone et al., in prep.).

It has been proposed that the bursts of turbulent dissipation (i.e. its property of
space-time intermittency) sufficiently heat the diffuse gas locally to produce the
observed abundances \citep{Joulain1998}. This would explain why the diffuse ISM is,
occasionally, such a strong emitter in the pure rotational lines of \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\ \citep{
Falgarone2005,Ingalls2011}.
Godard et al. (2009, hereafter Paper I) have improved the model proposed by \citet{
Joulain1998} and set up global models of random sight lines across the medium that
sample {\it (i)} the active dissipative regions driving a non-equilibrium warm
chemistry, {\it (ii)} the regions in chemical and thermal relaxation that develop
after the extinction of the dissipation bursts, and {\it (iii)} the ambient phase
characterised by UV-driven chemistry. Although still idealised in some respects, these
models take the energetic constraints provided by the turbulent cascade into account.
Interestingly, it is found that less than a few percent of the turbulent energy, on
average, is sufficient to drive the warm chemistry at the observed level, the rest
being radiated away mostly in the pure rotational lines of \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\ and [CII] fine-structure
line emission.  Moreover, the comparison of the model predictions to observations
clearly favours a turbulent dissipation dominated by ion-neutral friction in structures
as small as $\sim$ 100 au.

In this paper, we further explore the models of Paper I and take advantage of the
impressive amount of new data obtained on light molecular hydrides in the diffuse
medium with {\it Herschel}. We compare the predictions of the TDR model to these
observations, and to older data sets on CO and \HCOp.
An overview of the information directly inferred from the absorption spectroscopy data
is given in Sect. \ref{sec-data}. The main characteristics of the model are summarised
in Sect. \ref{Sec-TDR}. In Sects. \ref{Sect-Diagnostics} to \ref{Sect-Cp}, we explore
the parameter space over large domains and use the model predictions to identify molecular
species (and pairs of species) that can be used as specific diagnostics of the turbulence
in the diffuse ISM. Finally, in Sect. \ref{Sect-discussion}, we discuss the overall
coherence of the parameters of the turbulence inferred from the light hydrid and CO
molecular abundances. The recent refinements of the dynamical and chemical prescriptions
of the TDR model are given in the Appendices.

\section{Observational constraints on the parameter space to explore}\label{sec-data}

\subsection{Gas density}
The medium discussed in the present paper is the cold phase of the neutral ISM, the cold
neutral medium (CNM). The thermal pressure of this medium has been measured with observations
of the [CI] line by \citet{Jenkins2011} to have a mean value $P_{th}/k_B=3800$ K \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi, with
most of the gas in the range $2\times 10^3$ to $10^4$ K\ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi. The lower temperature of the
CNM is reliably determined from HI absorption measurements $T_K \gtrsim 80$K, from which we
infer a range of upper limits for the gas densities $n_{\rm H} \lesssim 25$ \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ to 125
\ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi.  \citet{Goldsmith2013} has computed the excitation temperatures of CO molecules seen
through several absorption lines against nearby stars \citep{Sheffer2008}. He infers median
values of the density \ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi= 92 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ and 68 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ for gas temperatures 50 and 100 K, respectively,
and therefore thermal pressures well within the range of Jenkins and Tripp (2011). An independent
determination is based on [CII] and [CI] observations of the same sightlines as those where
the \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi(1-0) line has been detected \citep{Gerin2014}.  Self-absorption of
the [CII] line occurs over the same velocity range as that of the \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi(1-0) absorption (see
also \citealt{Falgarone2010a}). The excitation analysis of the [CII] and [CI] lines at these
velocities shows that the gas causing the \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi(1-0) absorption is indeed the CNM, with a
narrow range of densities from $n_{\rm H}= 50$ \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ to less than 200 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi.

\subsection{Gas column densities}
For reasons linked to their excitation conditions, most of the molecular lines observed in
the diffuse ISM are seen in absorption against continuum sources that are either quasars
and free-free emission of HII regions, at millimetre frequencies or warm dust thermal
emission in star forming regions, at submillimetre frequencies. One exception is the CO(1-0)
line which can be excited in gas densities as low as 50~\ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ and provide a detectable level
of line emission \citep{Liszt2012}. The parameters directly inferred from observations are
the line optical depth at a given velocity and the velocity distribution.

Absorption lines corresponding to long pathlengths across the Galaxy have broad line profiles
that are usually decomposed into individual Gaussian components. The Gaussian shape, the
number of components, and their width are somewhat arbitrary. This is not a major limitation,
because the total hydrogen column density (atomic and molecular) with which those of molecular
species are associated are computed over the same velocity components found in the profiles of
HI for the atomic gas, and CH and HF for \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\ (e.g. \citealt{Godard2012}). The column densities
plotted in most of the figures of the paper are those associated with these Gaussian velocity
components. This is the reason that the plotted total column densities of hydrogen are never
much larger than $10^{22}$\ifmmode{\,{\rm cm}^{-2}}\else{$\,{\rm cm}^{-2}$}\fi, in spite of the long pathlengths sampled in the Galactic plane
by the {\it Herschel} observations. Whether or not a well-defined ``cloud'' of gas is ascribed
to each of these components is a question that goes beyond the scope of our paper. Following
\citet{Rachford2002}, we stress that a {\it translucent line of sight} in the diffuse ISM is
definitely different from a {\it translucent cloud} since large column densities in general
result from the accumulation on the same line of sight of a large number of fragments, each
of low column density, and presumably poorly shielded from the ambient UV field.  This has
been well illustrated by \citet{Levrier2012} who show, using numerical simulations of
magnetohydrodynamical (MHD) turbulence, that the column density peaks do not correspond to
sightlines towards density peaks in the cube.

\subsection{UV shielding}
The very large scatter (by several orders of magnitude) of the \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\ fraction observed on
sightlines with total hydrogen column densities ranging between $3 \times 10^{19}$\ifmmode{\,{\rm cm}^{-2}}\else{$\,{\rm cm}^{-2}$}\fi\
and $5 \times 10^{20}$\ifmmode{\,{\rm cm}^{-2}}\else{$\,{\rm cm}^{-2}$}\fi\ \citep{Gillmon2006} is also an illustration that the gas on
such diffuse sightlines is highly fragmented and that there is not a systematic increase
in the UV-shielding with the gas column density sampled. Molecular hydrogen fractions are
computed in Appendix B using the Meudon PDR (photodissociation regions) code\footnote{
Version 1.4.4 available at \url{http://pdr.obspm.fr/PDRcode.html}.} \citep{Le-Petit2006}
for a range of densities and UV-shieldings. The \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\ self-shielding is so efficient that
the sharp transition from $f_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi} \sim 10^{-4}$ to 0.1 or more always occurs for very
low UV-shieldings, $A_V \lesssim 0.02$ mag, even at a density as low as 10 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi. This is
well below the range of observed UV-shieldings, $0.02 \lesssim A_V \lesssim 0.3$ mag,
that can be inferred from the above range of $N_{\rm H}$ assuming $N_{\rm H}= 1.8 \times
10^{21} A_V$\ifmmode{\,{\rm cm}^{-2}}\else{$\,{\rm cm}^{-2}$}\fi. The only way to reconcile what is known on the self-shielding of \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\
and the observed width of the transition region, given the low densities of the diffuse
ISM, is that this column density range is due to tens of fragments of very low column
density and probably low UV-shielding as well.

\subsection{The free parameters linked to turbulence in the diffuse ISM}

The CNM is a highly turbulent medium and its supra-thermal kinetic and magnetic pressures
provide it with the required support to reach quasi-static equilibrium in the gravitational
potential of the Galaxy \citep{Cox2005}. Its weak ionisation degree, carried by \Cp\ ions,
makes it not fully coupled to the magnetic fields. Turbulence in this cold phase of the
interstellar medium is therefore magnetised, compressible, and multifluid, with a high
Reynolds number. The power spectra of both the velocity and the column density inferred
from HI line emission are found to have the slope of the Kolmogorov incompressible turbulence
(e.g. in the North Celestial Loop complex, \citealt{Miville-Deschenes2003}). The energy
transfer rate estimated in the atomic turbulent cascade, is found to be the same as that
inferred in the diffuse molecular gas, traced by the optically thick CO(1-0) line emission,
suggesting that diffuse molecular and atomic gas are part of the same turbulent cascade
\citep{Hennebelle2012}. An interesting feature is the large range of fluctuations for this
energy transfer rate: it fluctuates by two orders of magnitudes above and below the average
value, $\ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi_{obs}=2 \times 10^{-25}$ \ifmmode{\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}$}\fi, and this is independent of the scale at which
these fluctuations are measured.
The non-thermal rms velocity dispersion at large scale (between 10 pc and 100 pc) ranges
between about 1 \kms\ and 8.5 \kms\ from the size-linewidth relation in diffuse molecular
gas \citep{Falgarone2009a}. The column density-weighted rms velocity of CNM HI Gaussian
components in the Solar neighbourhood is $\sim 7$ \kms\ \citep{Heiles2003} while the
distributions of HI linewidths for the coldest components are 4.9 \kms\ and 12 \kms\ at
larger scale in the Galaxy \citep{Haud2007}.  These quantities may be understood as due
to turbulence in the diffuse medium \citep{Kalberla2009,Kim2013}.

\section{Overview of the TDR model} \label{Sec-TDR}

The model of turbulent dissipation regions (TDR) is built on the property of space-time
intermittency of turbulence, i.e. its dissipation is not evenly distributed in space but
concentrated in bursts. The purpose of the TDR model is to predict the molecular abundances
observed in the diffuse ISM taking into account those transiently formed in the warm
non-equilibrium chemistry driven by the bursts of turbulent dissipation, in addition to
those formed in the ambient medium, via UV-driven chemistry. The proper description of the
dissipation regions of compressible MHD turbulence being far beyond
the grasp of even the most powerful direct numerical simulations \citep{Kritsuk2007}, the
approach followed in the model is idealised, but relies on three universal properties of
turbulence. {\it (i) } The bulk of the kinetic energy in compressible turbulence lies in
incompressible modes because of the fast energy transfer from compressible to solenoidal
modes \citep{Porter2002,Vestuto2003,Kritsuk2010}. {\it (ii)} A significant fraction of the
turbulent energy in incompressible flow experiments is dissipated in coherent dynamical
structures, such as vortices \citep{Douady1991,Nagaoka2002, Mouri2007,Mouri2009}. Similar
structures are found in numerical simulations \citep{Moisy2004, Kritsuk2007,Uritsky2010,
Momferratos2014}. Their characteristics are set by those of the ambient turbulence
\citep{Jimenez1997,Tanahashi2004,Mouri2009}. {\it (iii) } Lagrangian intermittency is
observed in laboratory flows to be even more pronounced than in Eulerian statistics and
Lagrangian accelerations of the fluid cells are influenced by the large scale dynamics
(see the review of \citealt{Arneodo2008}).

\begin{table}[!ht]
\begin{center}
\caption{Parameters of the TDR model. The standard value and the range of explored values
are given in Cols. 3 and 4, respectively.}
\begin{tabular}{l l c c c}
\hline
             & unit        & standard            & range                                     & ref     \\
\hline
$\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi$      & \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi         & $50$                & $20 - 300$                                & $a$     \\
$\chi$       &             & 1                   & 0.5 - 10                                  & $b$     \\
$A_V$        & mag         & 0.4                 & 0.005 - 1                                 & $c$     \\
$\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$    & s$^{-1}$    & $10^{-16}$          & $3 \times 10^{-17}$ - $3 \times 10^{-16}$ & $d,e$   \\
\hline
$a$          & s$^{-1}$    & $3 \times 10^{-11}$ & $10^{-11}$ - $10^{-10}$                   & $f$     \\
$\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi$       & km s$^{-1}$ & $3.0$               & 1 - 7                                     & $g,h,i$ \\
$\ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi$      & \ifmmode{\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}$}\fi       & $10^{-24}$          & $10^{-25}$ - $10^{-23}$                   & $j$     \\
$E_{\tau_V}$ & \ifmmode{\,{\rm erg}\,{\rm cm}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-1}$}\fi        & $3 \times 10^{-13}$ & $10^{-13}$ - $10^{-11}$                   &         \\
\hline
\end{tabular}
\begin{list}{}{}
($a$) \citet{Snow2006},     ($b$) \citet{Mathis1983},    ($c$) \citet{Godard2012},
($d$) \citet{Dalgarno2006}, ($e$) \citet{Indriolo2012},
($f$) \citet{Godard2009},   ($g$) \citet{Tanahashi2004}, ($h$) \citet{Mouri2007},
($i$) \citet{Mouri2009},    ($j$) \citet{Hennebelle2012}
\end{list}
\label{Tab-Par}
\end{center}
\end{table}

Numerically, the TDR model is a three-step code (Paper I). It first computes the non-equilibrium
thermal and chemical Lagrangian evolutions of the gas during the active dissipation stage.
Then, after the end of the dissipation burst, the thermal and chemical relaxation of the
gas are computed as functions of time. In a third step, the column densities of all the
chemical species treated by the code are computed along a random line of sight across the
diffuse ISM, taking into account energetic constraints. We briefly summarise below the main
features of each of these steps, all the details being given in Paper I. The free parameters
of the model and the range of values explored in the present work are given in Table
\ref{Tab-Par}.

\subsection{The active dissipative stage}

This stage is treated in a hybrid way. Following the above, we adopt a modified Burgers vortex
\citep{Nolan1999} for the coherent structures responsible for intermittent dissipation. Such a
vortex results from the balance between the diffusion and the stretching of the vorticity by the
rate-of-strain. The vortex is defined by two independent parameters, the maximum orthoradial
velocity $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi$ and the turbulent rate-of-strain $a$ (see Table \ref{Tab-Par}). The vortex
equilibrium radius is $r_0^2=4 \nu/a$, where $\nu$ is the molecular viscosity. We first compute
the steady-state equilibrium of such a vortex of finite length threaded by a magnetic field,
initially parallel to the vorticity \citep{Mininni2006,Mininni2006a}. The steady-state
configuration is reached within $\sim 100$ yr with a slightly helical field and ion velocities
at least 10 times smaller than $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi$.

In this configuration, the gas is mechanically heated through viscous friction and
ion-neutral friction, at an average rate per unit volume $\overline{\Gamma}_{\rm turb}$
computed up to a radius $Kr_0$ ($K\sim 5$, see Paper I) where the turbulent heating stops
influencing the gas temperature and chemistry. The gas cools down through atomic and
molecular line emissions. The duration of this phase (i.e. the lifetime of the vortex),
$\tau_V$, is related to the total energy dissipated per unit length of the vortex\footnote{
The vortex is invariant along its axis.} over its lifetime and volume (hereafter called
the vortex dissipation integral),
\begin{equation} \label{Eq-Etau}
E_{\tau_V}=\pi (Kr_0)^2\overline{\Gamma}_{\rm turb} \tau_V,
\end{equation}
a free parameter also introduced in Paper I. The non-equilibrium coupled thermal and chemical
evolutions of the gas are computed along the Lagrangian trajectory of a fluid particle trapped
in the vortex. The code simultaneously solves the time-dependent evolution of 11 dynamical and
181 chemical variables, including the abundances of 163 species and of the first 18 rotational
levels of \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\ (see Appendix \ref{Append-Chemistry}). The rates of all collisional processes
(including chemical reactions) are computed at an effective temperature \citep{Flower1985},
\begin{equation} \label{Eq-Teff}
T_{\rm eff} = \frac{m_1 T_2 + m_2 T_1}{m_1 + m_2} + \frac{1}{k} \frac{m_1 m_2}{m_1 + m_2} u_D^2,
\end{equation}
where $m_1$, $m_2$, $T_1$, and $T_2$ are the masses and temperatures of the two colliders,
$u_D$ is their relative velocity, and $k$ is the Boltzmann constant.

\subsection{The relaxation stage}

\begin{figure*}[!ht]
\begin{center}
\includegraphics[width=17.0cm,angle=-0]{Temps-relax-TDR2_z1p1_Oxygen.eps}
\caption{Relaxation timescales (Eq. \ref{Eq-relax-time}) of OH (cyan crosses), \HdO\ (magenta
squares), \HCOp\ (red circles), CO (blue triangles), and \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ (orange upside-down triangles)
computed with the TDR model as functions of the shielding $A_V$ of the UV field (left panel)
and the density $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi$ of the gas (right panel), all the other parameters being set to their
standard values (see Table \ref{Tab-Par}).}
\label{Fig-relax-time}
\end{center}
\end{figure*}

By nature, an active dissipation burst has a short lifetime $\tau_V$. Once it ends, the gas
cools down and the chemical signatures imprinted by the active stage persist from a few
hundred years to a few 10$^4$ years depending on the destruction timescales of the different
species. An Eulerian approach is adopted here: the code computes the chemical and thermal
evolutions of 50 points evenly distributed along the radial axis of the vortex, now at rest,
assuming that the gas obeys an isochoric or isobaric equation of state. All the results
presented in the following are obtained for an isochoric relaxation. The column densities of
any species X integrated across the vortex, $N_R({\rm X},t)$, are therefore computed as
functions of time.

The relaxation timescales $\tau_R({\rm X})$ defined as
\begin{equation} \label{Eq-relax-time}
\int_0^{\tau_R({\rm X})} N_R({\rm X},t) dt = 0.9 \int_0^{\infty} N_R({\rm X},t) dt
\end{equation}
are shown in Fig. \ref{Fig-relax-time} as functions of the shielding of the UV field (left
panel) and the density of the gas (right panel) for \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ and four oxygen bearing species,
OH, \HdO, \HCOp, and CO. This figure illustrates the very large scatter among the chemical
relaxation timescales: from a few 100 yr for \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ and \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi\ to more than $10^4$ yr for CO.
Among all the species treated in our chemical network (see Appendix \ref{Append-Chemistry}),
CO has the largest relaxation timescale as long as $A_V \gtrsim 0.3$ mag and $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi \gtrsim
30$ \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi. This is due to the shielding of the molecule from the UV radiation field
(see Appendix \ref{Append-H2-CO}).

\subsection{Building of a line of sight} \label{Sec-build-los}

\begin{figure}[!hb]
\begin{center}
\includegraphics[width=8.5cm,angle=0]{Descript-code3.eps}
\caption{Schematic view of the TDR model. A random line of sight across the diffuse interstellar
medium intercepts the ambient UV-dominated gas (in green), a number of active vortices (in
red), and the corresponding number of relaxation stages (in blue). The chemical enrichment
(shown in the lower panel) only occurs on a small fraction of the entire line of sight
(a few percent, Paper I).}
\label{FigTDR2}
\end{center}
\end{figure}

\subsubsection{The three phases sampled by a random line of sight}

In Fig. \ref{FigTDR2} we display a schematic view of the turbulent diffuse interstellar
medium. In this conception, most of the gas volume is filled by a non-dissipative medium
(in green; hereafter called the ambient medium) where the chemical composition of
the gas solely results from its density $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi$, the strength $\chi$ of the UV radiation
field (expressed in Mathis' unit, \citealt{Mathis1983}), the shielding $A_V$ of the ISRF
(at visible wavelength) at each position, and the total CR ionisation rate of molecular
hydrogen $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$ (taking into account secondary ionisations). In addition to this
ambient medium, a random line of sight intercepts a number
$\mathscr{N}_V$ of active vortices (in red) and a number of relaxation stages (in blue).
Since each relaxation phase follows an active dissipation phase, their number along the
line of sight at any time is proportional to $\mathscr{N}_V$ and to $\tau_R({\rm X})/
\tau_V$ where $\tau_R({\rm X})$ is the relaxation timescale of species X. Introducing the
filling factor of the active vortices on a line of sight of length $L$,
\begin{equation}
f_L=\mathscr{N}_V \,2Kr_0/L,
\end{equation}
the column density of a species X writes\footnote{There is an unfortunate misprint in this
Equation in paper I.}
\begin{equation} \label{Eq-Coldens}
N({\rm X}) = N_{\rm amb}({\rm X}) + f_L N_V({\rm X}) + f_L {\tau_R({\rm X}) \over \tau_V}
{\overline N_R({\rm X})},
\end{equation}
where $N_{\rm amb}({\rm X})$ is its column density in the ambient medium, $N_V({\rm X})$
its column density in an active vortex, and ${\overline N_R({\rm X})}$ its averaged column
density during the relaxation phase. We note that according to Fig.
\ref{FigTDR2} (lower panel), $N_V({\rm X})$ and ${\overline N_R({\rm X})}$ are computed as
excesses\footnote{These quantities may therefore be negative.} of column densities compared
to $N_{\rm amb}({\rm X})$. We also note that $f_L$ being defined on any random line of sight,
it corresponds to a volume filling factor. It thus relates the dissipation rate averaged at
large scale \ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi\ to the same quantity averaged over the volume of a
vortex:
\begin{equation}
\ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi=f_L \,\overline{\Gamma}_{\rm turb}.
\end{equation}

Aside from \ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi, $\chi$, $A_V$, and $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$, the TDR model is therefore defined by four
independent parameters: the turbulent rate-of-strain $a$; the vortex maximum orthoradial
velocity $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi$ (which sets the ion-neutral velocity drift $\upsilon_{in}$);
the vortex dissipation integral $E_{\tau_V}$ (involving the vortex lifetime $\tau_V$);
and the average dissipation rate \ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi, related to the volume filling factor $f_L$ of
all the active vortices.

\subsubsection{Prescriptions for the shielding of the UV radiation field}

The UV radiation field being a main input of the gas phase chemistry, the prescription
for the radiative transfer is critical. In this paper, we adopt two different descriptions
of the variation of $A_V$ along the line of sight. In the first approach (standard case),
a homogeneous FUV shielding is adopted along the entire line of sight: the medium is
considered as highly fragmented, each fragment being evenly shielded by its environment.
In the second approach, we mimic PDR models: we consider a 1D slab illuminated from both
sides with the same UV radiation field (parametrised by $\chi$) and adopt a shielding
function that linearly increases from the edge to the mid plane of the slab and then
linearly decreases up to the opposite edge.

In the following, the predictions of the model will always correspond to those
obtained with the first approach, except when indicated otherwise.

\section{Chemical diagnostics of turbulent dissipation} \label{Sect-Diagnostics}

\subsection{Exploration of the parameter space} \label{Sect-explo-param}

The influence of the parameters of the TDR model on the vortex dynamics and lifetime, the
thermal evolution of the gas, and the distributions of the active and relaxation phases
along the line of sight have been extensively described in Paper I. In the present paper,
we broaden the parameter domain explored and focus on their impact on the chemical composition
of a random line of sight.

The elemental abundances are set to their values observed in the Solar neighbourhood (SN,
see Table \ref{Tab-Ele} in Appendix \ref{Append-Chemistry}). We explore the parameter domain
of the code through a grid of $\sim$ 20,000 models\footnote{Since $\ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi$ and $E_{\tau_V}$
are only used for building a line of sight, there is no need to run additional models when we
change these parameters.} (corresponding to a total computing time of $\sim 500$ hours)
organised as follows:
\begin{itemize}
\item[$\bullet$] 5 densities $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi$ = 20, 30, 50, 100, and 300 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ to bracket the range
discussed in Sect. \ref{sec-data};
\item[$\bullet$] 4 radiation field scaling factors $\chi$ = 0.5, 1, 3, and 10, to take into
account variations from the SN to the inner Galactic disk (cf. Sect. \ref{Sect-Var-CHp});
\item[$\bullet$] 12 visible extinctions $A_V$ = 0.005, 0.01, 0.02, 0.05, 0.07, 0.1, 0.2,
0.3, 0.4, 0.6, 0.8, and 1.0 mag to explore media with very low UV shielding as discussed
in Sect. \ref{sec-data};
\item[$\bullet$] 3 cosmic ray ionisation rates $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi = 3 \times 10^{-17}$, $10^{-16}$,
and $3 \times 10^{-16}$ s$^{-1}$ to explore the largest average value recently derived in the
diffuse ISM \citep{Padovani2013,Indriolo2012};
\item[$\bullet$] 3 rates-of-strain $a$ = 1, 3, and 10 $\times 10^{-11}$ s$^{-1}$ (Paper I);
\item[$\bullet$] 9 maximum orthoradial velocities $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi$ = 1, 2, 2.5, 3, 3.5, 4, 5, 6, and
7 \kms\ following Sect. \ref{sec-data};
\item[$\bullet$] 5 average dissipation rates $\ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi$ = 0.1, 0.3, 1, 3, and 10 $\times 10^{-24}$
\ifmmode{\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}$}\fi\ following Sect. \ref{sec-data}; and
\item[$\bullet$] 5 vortex dissipation integrals $E_{\tau_V}$ = 0.1, 0.3, 1, 3, and 10
$\times 10^{-12}$ \ifmmode{\,{\rm erg}\,{\rm cm}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-1}$}\fi, i.e. below the value above which this parameter has no impact
(Sect. \ref{Sect-CO}) and keeping only dissipation timescales above the validity limit
of the model (see Paper I).
\end{itemize}
In the paper, we will often refer to a standard model defined by $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi = 50$ \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi, $\chi =
1$, $A_V$ = 0.4 mag, $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi = 10^{-16}$ s$^{-1}$, $a = 3 \times 10^{-11}$ s$^{-1}$, $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi
= 3$ \kms, $\ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi = 10^{-24}$ \ifmmode{\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}$}\fi, and $E_{\tau_V} = 3 \times 10^{-13}$ \ifmmode{\,{\rm erg}\,{\rm cm}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-1}$}\fi. We discuss
the influence of $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$ on the chemistry of the gas and justify our choice for the standard
value in Sect. \ref{Sect-zeta}. Unless indicated otherwise, all the parameters are
systematically set to their standard values.

\subsection{Method: ternary phase diagrams}

\begin{figure*}[!ht]
\begin{center}
{\bf \Large \underline{constant \ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi}}
\hspace{7.5cm}
{\bf \Large \underline{constant $A_V$}} \\
\vspace{0.5cm}
\includegraphics[width=7.5cm,angle=-0]{Triangles-paper-TDR2_z1p1-Av-1.eps}
\hspace{1.5cm} \rule{0.5mm}{6cm} \hspace{1cm}
\includegraphics[width=7.5cm,angle=-0]{Triangles-paper-TDR2_z1p1-nH-1.eps}
\includegraphics[width=7.5cm,angle=-0]{Triangles-paper-TDR2_z1p1-Av-2.eps}
\hspace{1.5cm} \rule{0.5mm}{7cm} \hspace{1cm}
\includegraphics[width=7.5cm,angle=-0]{Triangles-paper-TDR2_z1p1-nH-2.eps}
\caption{Ternary phase diagram of the TDR model. The projections of each point on the
medians of the equilateral triangle show the contributions of each phase (ambient medium,
active dissipative bursts, and relaxation stages) to the total column density of a given
species. In the left panels, the models are computed for several values of $A_V$:
0.1, 0.2, 0.3, 0.4, 0.6, 0.8, and 1.0 mag. In the right panels, the models are computed
for several values of $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi$: 20, 30, 50, 100, and 300 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi. All the other parameters are
set to their standard values (see Table \ref{Tab-Par}). In all panels, the size of the
symbols increases as the varying parameter increases. The beige, yellow, and orange zones
indicate the regions where the production of the different species is dominated by the
ambient medium, the active vortices, and the relaxation stages, respectively.}
\label{Fig-Triangles}
\end{center}
\end{figure*}

A helpful way to visualise the main results of the TDR model are the
ternary phase diagrams shown in Fig. \ref{Fig-Triangles}. They display
schematic representations of the contributions (see Eq. \ref{Eq-Coldens})
\begin{eqnarray}
f_{\rm amb}(X) & = & N_{\rm amb}({\rm X}) / N({\rm X}) \\[9pt]
f_V(X) & = & \mathscr{N}_V N_V({\rm X}) / N({\rm X}) \\[4pt]
f_R(X) & = & \mathscr{N}_V {\tau_R({\rm X}) \over \tau_V} {\overline N_R({\rm X})} / N({\rm X})
\end{eqnarray}
of the ambient medium, the active vortices, and the relaxation stages to the total column
densities of ten molecules. These contributions are given by the projections of each point
on the three triangle axis: for instance, the smallest symbol of CO (top left panel) reads
$f_{\rm amb}({\rm CO}) = 22$ \%, $f_V({\rm CO}) = 21$ \%, and $f_R({\rm CO}) = 57$ \%.
These diagrams illustrate two major features of the model.

First, the phase contributions of the species selected here are distributed over the
entire area of the triangles. It implies that each phase of the TDR model plays a role
on the global chemical composition of the gas, even if it fills only a small fraction
of the entire volume, and is therefore required to correctly model random diffuse
lines of sight. In particular, Fig. \ref{Fig-Triangles} shows for the first time the
critical impact of the relaxation times: the relaxation phase may account for 15
\% of the total column density of CH, 50 \% of those of OH and \HdO, 70 \% of \CtH, and
for more than 80 \% of the total column density of CO.

Second, except for CH, the production of which is dominated by the active phase at low
density and by the ambient medium as the density increases, and for OH and \HdO, the
productions of which may be dominated by the relaxation phase at low density, all the other
species appear to be confined in specific regions of the triangles: H, \ifmmode{\rm H_2}\else{$\rm H_2$}\fi, and HF are
almost entirely produced in the ambient medium; \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi, \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi, and \HCOp\ are predominantly
formed in the active phase; while \CtH\ and CO mostly originate from the relaxation phase.
We therefore anticipate that the species produced in
the active phase  can be used as signatures of the dynamics of the dissipation
regions; those formed in the relaxation phase are expected to be sensitive to the relative
timescales between the active dissipation and the relaxation; and those produced in the ambient
medium are species unaffected by turbulent dissipation.

In the next sections, we select several pairs of species and compare both their column
densities and column density ratios predicted by the TDR model to the observations of
hundreds of diffuse lines of sight. The goal of this approach is to address the following
questions: How do the model predictions compare with multiwavelength observations of atoms
and molecules in the diffuse medium across the Galactic disk? Conversely, how can we use
these observations to constrain the parameter space, hence infer the main properties of
turbulent dissipation?

\section{\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ and the turbulent dissipation rate} \label{Sect-CHp}

\begin{figure}[!hb]
\begin{center}
\includegraphics[width=8.5cm,angle=-0]{Compilation-CHp-NHtot-Hifi-and-visible_z1p1_epsilon.eps}
\caption{Observations ({\it top panel}) compared to the predictions of PDR ({\it middle
panel}) and TDR ({\it bottom panel}) models. Data (open circles) $-$ The \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ and total
hydrogen column densities inferred from visible absorption lines and extinction measurements
are shown in red and are from \citet{Crane1995}, \citet{Gredel1997}, and \citet{Weselak2008}.
The \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ and total hydrogen column densities inferred from the far-infrared absorption
lines of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi, HF, and CH and the 21 cm emission lines of HI are shown in blue and are
from \citet{Falgarone2010}, \citet{Gerin2010}, \citet{Neufeld2010}, \citet{Sonnentrucker2010},
and \citet{Godard2012}. The red and blue lines correspond to the mean ratio $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)/N_{\rm H}$
computed with the visible and submillimetre data, respectively. PDR and TDR model predictions
(filled symbols) $-$ Computed for several densities: 10 (crosses), 30 (triangles), 50 (squares),
and 100 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ (circles). The TDR models are computed for $\ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi = 10^{-24}$ (green) and
$10^{-23}$ (magenta) \ifmmode{\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}$}\fi, all the other parameters are set to their standard values.}
\label{Fig-CHp-NHtot}
\end{center}
\end{figure}

\subsection{Observational data} \label{Sec-obs-chp-H}

In the top panel of Fig. \ref{Fig-CHp-NHtot} we compare two datasets:
the column densities of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ derived from its visible absorption lines (in red) detected
towards nearby OB stars \citep{Crane1995,Gredel1997,Weselak2008} and those deduced from
its far-infrared rotational transitions (in blue) recently observed in absorption towards
the continuum of massive star forming regions with the \emph{Herschel}/HIFI instrument
\citep{Falgarone2010,Godard2012}. The total hydrogen column densities associated with the
visible data are inferred from extinction measurements, while those associated with
the far-infrared data are taken from \citet{Godard2012} who estimated the atomic hydrogen
column density from VLA observations of its $\lambda$21 cm transition and the \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\
column density from the far-infrared observations of two surrogate molecular species, CH
and HF \citep{Gerin2010,Neufeld2010,Sonnentrucker2010}. When both CH and HF are available,
both estimations are used to compute uncertainties on the total column density, hence the
large error bars on $N_{\rm H}$ for several far-infrared components.

Absorption spectroscopy against stars in the visible range is limited to the SN
 because of dust extinction, while that against star-forming regions in
the submillimetre domain does not have such a limitation (e.g. \citealt{Laor1993}). The red
data of Fig. \ref{Fig-CHp-NHtot} are thus associated with gas in the SN
while the blue data sample components located deeper in the inner Galactic disk (see
Table 3 of \citealt{Godard2012}). These two datasets share similar characteristics:
a large scatter of about one order of magnitude and the same linear
trend with \ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi. However, the column densities of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ observed in the inner Galactic
disk, with a mean ratio $\langle N({\rm CH}^+)/N_{\rm H} \rangle = 2.1 \times 10^{-8}$,
are found to exceed by a factor of 3 on average those observed in the SN,
where $\langle N({\rm CH}^+)/N_{\rm H} \rangle = 7.6 \times 10^{-9}$.

\subsection{Predictions of UV-dominated chemistry} \label{Sec-carb-chem}

The middle panel of Fig. \ref{Fig-CHp-NHtot} displays the column densities of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ predicted
by the Meudon PDR code for densities of 30, 50, and 100 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi, a total column density $\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi=
0.18$, $1.8$, and $18$ $\times 10^{21}$ \ifmmode{\,{\rm cm}^{-2}}\else{$\,{\rm cm}^{-2}$}\fi, and an incident radiation field scaling factor
$\chi=1$. In the diffuse ISM, the destruction of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ occurs through hydrogenation (see Fig.
\ref{Fig-Network-CHp} in Appendix \ref{Append-Networks}), an exothermic process with a very
short timescale $\sim 1$ yr $f_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi} (50 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi / \ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi)$ \citep{McEwan1999}, where $f_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$
is the molecular fraction defined as $f_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi} = 2n(\ifmmode{\rm H_2}\else{$\rm H_2$}\fi) / \ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi$. Conversely, in the absence
of a suprathermal energy source to activate the endothermic reaction
\begin{equation} \label{Reac-CHp-text}
\Cp +\ifmmode{\rm H_2}\else{$\rm H_2$}\fi \rightarrow \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi + {\rm H} \quad (\Delta E / k \sim 4640 {\rm K}),
\end{equation}
the formation of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ is initiated by the radiative association
\begin{equation}
\Cp +\ifmmode{\rm H_2}\else{$\rm H_2$}\fi \rightarrow \ifmmode{\rm CH_2^+}\else{$\rm CH_2^+$}\fi + \gamma,
\end{equation}
a process with a very long timescale
$\sim 2 \times 10^6$ yr $f_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}^{-1} (50 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi / \ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi)$ \citep{Herbst1985}. With the lack of
an efficient production pathway, PDR models thus systematically underestimate the observed
column densities of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ by one to three orders of magnitude (Fig. \ref{Fig-CHp-NHtot}).

\subsection{Predictions of the TDR model} \label{Sect-Meas-epso}

In the bottom panel of Fig. \ref{Fig-CHp-NHtot}, we compare the observational data with the
predictions of the TDR models. With a homogeneous reconstruction of the line of sight (see
Sect. \ref{Sec-build-los}), all the column densities computed by the code are simply
proportional to the total hydrogen column density $N_{\rm H}$. For a given $N_{\rm H}$, the
dependence of $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)$ on the other parameters is then the result of several effects. First,
\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ is predominantly produced in the active stages (see Fig. \ref{Fig-Triangles}); its
column density is thus proportional to the number of active vortices, which is proportional
to the average dissipation rate $\ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi$, inversely proportional to the square of the density
$\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi$ and independent of $E_{\tau_V}$ (see Paper I). Second, the rate-of-strain $a$ and the
maximum orthoradial velocity $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi$ control the heating by viscous friction and ion-neutral
friction, respectively, hence set the effective temperature (Eq. \ref{Eq-Teff}) at which
ion-neutral chemical reactions proceed. If $a$ or $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi$ increase, the effective temperature
in an active vortex increases and so does the rate of reaction \ref{Reac-CHp-text}, hence the
production of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi. When integrated over the line of sight, however, this effect is balanced
by the number of vortices which decreases as the local dissipation rate increases (Paper I).
Third, since \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ is formed through the hydrogenation of \Cp\ and the photodissociation of
\CHtp\ (see Fig. \ref{Fig-Network-CHp} of Appendix \ref{Append-Networks}), its
abundance, unlike most of the molecules, increases with the UV radiation field. With the
analysis of all the models we find that the $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)/N_{\rm H}$ column density ratio
scales as
\begin{equation} \label{scaling-CHp}
N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi) / N_{\rm H} \sim 1.5 \times 10^{-9}\, \overline{\varepsilon}_{24} \left(
\frac{\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi}{50 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi} \right)^{-2.2} \left( \frac{A_V}{0.4 {\rm mag}} \right)^{-0.3} \chi^{0.5}
\end{equation}
(with $\overline{\varepsilon}_{24} = 10^{24} \,\ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi $ \ifmmode{\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}$}\fi)
for $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi \lesssim 300$ \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi, and
is almost independent of $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi$ for $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi \gtrsim 2.5$ \kms, i.e. as long as the gas rotation
induces a sufficient ion-neutral drift to activate reaction \ref{Reac-CHp-text}.

The above relation shows a weak dependence of $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)$ on both $A_V$ and $\chi$; $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)$
varies by only a factor of 3 over the ranges covered by these two parameters which are
inferred from the molecular fractions of the gas (see Sect. \ref{sec-data}) and from its
positions in the Galactic disk (see Sect. \ref{Sect-Var-CHp}). Equation \ref{scaling-CHp}
reveals stronger dependences of $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)$ on both \ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi\ and \ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi, the latter reflecting
the fast destruction of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ by collisions with hydrogen. This result somewhat hampers
the interpretation of the observations: (1) there is a degeneracy between \ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi\ and
\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi\ from the sole observations of $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)$ and $\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi$ and (2) the steep dependence on
\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi\ implies that the observations of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ preferentially sample the lowest densities
of the diffuse medium.

The lowest values of $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)/\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi$ in Fig. \ref{Fig-CHp-NHtot} correspond to detection
limits and thus cannot be used to infer a lower limit on the dissipation rate \ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi\ or
an upper limit on the gas density. In contrast, we can derive an upper limit of the turbulent
dissipation rate from the largest observed \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ abundances. The largest values of $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)
/ \ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi$ observed in the local ISM (red points in Fig. \ref{Fig-CHp-NHtot}) and the inner
Galactic disk (blue points in Fig. \ref{Fig-CHp-NHtot}) give $\overline{\varepsilon}_{24}
\times (\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi / 50\ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi )^{-2.2} \lesssim 20$ and 70, respectively. Assuming that these upper
limits are associated with the lowest density components of the diffuse matter ($\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi \sim
20$ \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi, \citealt{Fitzpatrick1997}, see also Sect. \ref{sec-data}), we find $\ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi \lesssim
10^{-23}$ \ifmmode{\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}$}\fi\ for the components located in the inner Galactic disk, and about 3 times
less in the SN.

\subsection{Variation of $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)/\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi$ across the Galactic disk} \label{Sect-Var-CHp}

It is interesting to analyse the increase by a factor of three of the \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ abundance between
the SN and the inner Galaxy in the light of what is known about the Galactic dynamics and
the star formation rate. Although the far-infrared lines of sight intercept the spiral arms
of the Galaxy, we assume that, even in the inner Galaxy, the density of the gas rich in \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\
remains low because of the fast destruction rate of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ by collisions. The increase in the
carbon elemental abundance towards the inner Galaxy accounts for an increase in the \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\
column density by only a factor of 1.2 at $R_g=6.5$ kpc because the trend is only of 0.037
dex kpc$^{-1}$ \citep{Daflon2004}. The increase in the mean ISRF intensity towards the inner
Galaxy has an influence \citep{Porter2005}. Using the SKY star distribution model \citep{
Wainscoat1992}, \citet{Moskalenko2006} estimate that the mean UV radiation field intensity
increases by a factor of three from $R_g=8.5$ to 4 kpc. This sole increase, however, seems
insufficient to explain the observed variation of the $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)/N_{\rm H}$ ratio because most
of the observed components beyond the SN lie in the range 5.5 $-$ 7.5 kpc \citep{Godard2012}.
What is left is an increase in the turbulent dissipation rate \ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi\ that follows the
turbulent injection rate. The latter scales with the supernovae explosion rate and the
differential rotation of the Galaxy. These two processes are expected to increase \ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi\ by
a factor of 2.2 \citep{Lumsden2013,Urquhart2014} and 1.7 (assuming a flat Galactic rotation
curve, \citealt{Fich1991}) from $R_g=8.5$ kpc to $R_g=6.5$ kpc. The increase in both the
mean UV ISRF and the turbulent energy injection rate are therefore likely to be at the
origin of the increase in the $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)/\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi$ ratio from the SN to the inner Galaxy.

\section{The \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ to \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi\ ratio and the ion-neutral velocity drift} \label{Sect-SHp}

\begin{figure}[!hb]
\begin{center}
\includegraphics[width=7.5cm,angle=-0]{Compilation-SHp-vs-CHp_z1p1.eps}
\caption{Observations ({\it top panel}) compared to the predictions of TDR models computed
for three values of the maximum ion-neutral velocity drift: starting from the second panel,
$u_i-u_n=2.0$, $3.0$, and $5.0$ \kms. Data (open circles) $-$ The \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\
and \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi\ column densities and the least-squares power-law fit of their ratio (black line)
are from \citet{Godard2012}. Model predictions (filled symbols) $-$ Computed for $A_V=
0.1$ (red), $0.2$ (magenta), and  $0.4$ (blue) mag and for several densities: 20 (crosses),
30 (triangles), 50 (squares), and 100 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ (circles). Along each curve, the rate-of-strain
$a$ varies between 10$^{-11}$ and $10^{-10}$ s$^{-1}$ from right to left.}
\label{Fig-SHp-CHp}
\end{center}
\end{figure}

We now select two species mainly formed in the active phases of the TDR model: CH$^+$
and SH$^+$ (see Fig. \ref{Fig-Triangles}).

\subsection{Observational data}

As the first step of the sulfur hydrogenation chain, the sulfanylium ion \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi\ has long
been sought without success in the local diffuse ISM through UV absorption spectroscopy
of its electronic transitions \citep{Millar1988,Magnani1989,Magnani1991}. Only recently,
thanks to the development of submillimetre and infrared telescopes (APEX, \emph{Herschel}),
has interstellar \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi\ finally been observed through its ground-state rotational transitions
\citep{Menten2011,Godard2012,Nagy2013}. Among these new data, those of \citet{Godard2012}
are particularly interesting because they show the first unambiguous detection of \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi\
in diffuse interstellar components. As illustrated by the top panel of Fig. \ref{Fig-SHp-CHp},
the $\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi/\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi$ abundance ratio measured in the diffuse gas covers a broad range
of values from less than 0.01 to more than 0.1 and appears to be proportional to $(N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)/
\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi)^{-1.4}$. In this section we discuss the implications of this relation in the framework
of UV-dominated and turbulence-dominated chemistries.

\subsection{Sulfur chemistry}

In many ways, the analysis of the sulfur hydrides provides unique constraints on chemical
models applied to the interstellar medium. As quoted by \citet{Neufeld2012}, the sulfur
differentiates itself from any other abundant element because its hydrides and hydride
cations have small bonding energies. In consequence, the chemical reactions ${\rm X} + \ifmmode{\rm H_2}\else{$\rm H_2$}\fi
\rightarrow {\rm XH} + {\rm H}$ are highly endothermic for ${\rm X} = {\rm S}$, ${\rm SH}$,
$\Sp$, $\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi$, and $\ifmmode{\rm SH_2^+}\else{$\rm SH_2^+$}\fi$. The hydrogenation chain of sulfur therefore displays very
distinctive chemical patterns when compared with that of carbon, both in a UV-driven and
turbulence-driven regimes (see Fig. \ref{Fig-Network-CHp} in Appendix \ref{Append-Networks})

In a UV-driven chemistry, the sulfur hydrogenation is quenched by two reactions:
\begin{equation} \label{Reac-SHdp-rad}
\Sp + \ifmmode{\rm H_2}\else{$\rm H_2$}\fi \rightarrow \ifmmode{\rm SH_2^+}\else{$\rm SH_2^+$}\fi + \gamma, \,\, {\rm and}
\end{equation}
\begin{equation} \label{Reac-SHtp-rad}
\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi + \ifmmode{\rm H_2}\else{$\rm H_2$}\fi \rightarrow \ifmmode{\rm SH_3^+}\else{$\rm SH_3^+$}\fi + \gamma.
\end{equation}
With their very long characteristic timescales, these two reactions limit the production of
SH and, by extension, the production of \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi\ that originates from the photodissociation of
\ifmmode{\rm SH_2^+}\else{$\rm SH_2^+$}\fi\ rather than the photoionisation of SH. In addition, and unlike \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ and \ifmmode{\rm CH_2^+}\else{$\rm CH_2^+$}\fi\
which are destroyed by exothermic reactions with \ifmmode{\rm H_2}\else{$\rm H_2$}\fi, the destructions of \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi\ and \ifmmode{\rm SH_2^+}\else{$\rm SH_2^+$}\fi\
mainly occur through
\begin{equation} \label{Reac-SHp-rec}
\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi + e^- \rightarrow {\rm products}, \,\, {\rm and}
\end{equation}
\begin{equation} \label{Reac-SHdp-rec}
\ifmmode{\rm SH_2^+}\else{$\rm SH_2^+$}\fi + e^- \rightarrow {\rm products}.
\end{equation}
At equilibrium the $\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi/\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi$ abundance ratio writes
\begin{equation}
\frac{n(\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi)}{n(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)} \propto \frac{n(\Sp)}{n(\Cp)} \frac{n(\ifmmode{\rm H_2}\else{$\rm H_2$}\fi)^2}{n_e^2}
\end{equation}
and is independent from the extinction $A_V$ as long as \Cp\ and \Sp\ are the main ion
carriers and the major reservoirs of carbon and sulfur. It results that the $\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi/\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi$
column density ratio integrated over PDR slabs is constant for a total extinction varying
between 0.5 and 3 magnitudes and densities varying between 30 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ and 500 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi. This ratio
is equal to 0.01 \citep{Godard2012}, i.e. between one and two orders of magnitude smaller
than the largest observed value.

In a chemistry driven by turbulence, the release of energy due to dissipation is sufficient
to activate the endothermic reactions producing the sulfur hydride cations such as
\begin{equation} \label{Reac-SHp-text}
\Sp +\ifmmode{\rm H_2}\else{$\rm H_2$}\fi \rightarrow \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi + {\rm H} \quad (\Delta E / k \sim 9860 {\rm K}).
\end{equation}
The associated timescales decrease and become smaller than those of reactions \ref{Reac-SHdp-rad}
and \ref{Reac-SHtp-rad}, yet still larger than those of reactions \ref{Reac-SHp-rec} and
\ref{Reac-SHdp-rec}. Hydrogenation reactions therefore dominate the productions of \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi,
\ifmmode{\rm SH_2^+}\else{$\rm SH_2^+$}\fi, and \ifmmode{\rm SH_3^+}\else{$\rm SH_3^+$}\fi, while their destruction remain driven by the dissociative recombinations.
From this simple scheme (see Fig. \ref{Fig-Network-CHp} in Appendix \ref{Append-Networks}),
and using the endothermicities given in Eqs. \ref{Reac-CHp-text} and \ref{Reac-SHp-text},
the $\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi/\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi$ abundance ratio writes
\begin{equation}
\frac{n(\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi)}{n(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)} \propto \frac{n(\Sp)}{n(\Cp)} \frac{n(\ifmmode{\rm H_2}\else{$\rm H_2$}\fi)}{n_e} {\rm exp} \left( -\frac{5220 {\rm K}}{T_{\rm eff}} \right)
\end{equation}
at chemical equilibrium. It follows that $n(\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi)/n(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)$ strongly depends on the effective
temperature $T_{\rm eff}$ of reactions \ref{Reac-CHp-text} and \ref{Reac-SHp-text}, thus on
the velocity drift between $\Cp$ (or $\Sp$) and the molecular hydrogen (see Eq. \ref{Eq-Teff}).

\subsection{Influence of the TDR model parameters}

The three bottom panels of Fig. \ref{Fig-SHp-CHp} display a comparison between the
observational data and the predictions of the TDR models computed for $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi=$ 2, 3, and
5 \kms, a density ranging from 20 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ to 100 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi, a shielding varying between 0.1 mag
and 0.4 mag, and a total dissipation rate $\ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi = 3 \times 10^{-24}$ \ifmmode{\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}$}\fi, i.e. within
the range of values deduced in the diffuse ISM from the $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)/\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi$ ratio (cf. Sect.
\ref{Sect-Meas-epso}).

As expected from the large difference between the \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ and \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi\ endothermicities, the
$N(\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi)/N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)$ column density ratio computed with the code is highly sensitive to the
ion-neutral velocity drift set by the $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi$ parameter. An increase in $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi$ by a factor
of two induces a rise of $N(\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi)/N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)$ from one to three orders of magnitude. Detection
limits on $N(\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi)$ prevent us from deriving lower limits on \ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi. In contrast, we infer
$\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi \lesssim 3.5$ \kms\ regardless of $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi$ and $A_V$. For $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi = 3$ \kms, the code
not only explains the most extreme values of the $N(\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi)/N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)$ column density ratio,
but also naturally reproduces its trend with $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)/\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi$. The dependence of $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)$
and $N(\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi)$ on the other parameters indicates that the matter traced by both \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ and
\ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi\ has a density $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi \lesssim 50$ \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ in agreement with the results of the previous
section.

\section{The CO to \HCOp\ ratio and the dissipation timescale} \label{Sect-CO}

We now focus on \HCOp and CO, i.e. on two species which, according to the ternary phase
diagrams (Fig. \ref{Fig-Triangles}), originate from the active and relaxation stages,
respectively. Because these two species are closely related by their chemistry and belong
to two different phases of the TDR model, they offer a unique opportunity to obtain
information on the relative value of the relaxation and active dissipation timescales.

\subsection{Observational data}

\begin{figure*}[!ht]
\begin{center}
\includegraphics[width=16.0cm,angle=-0]{Compilation-CO-vs-HCOp_z1p1-article.eps}
\caption{{\it Left panels}: column densities of selected species integrated across a single vortex
as functions of time during the isochoric relaxation phase. The models are computed for two
values of the vortex dissipation integral $E_{\tau_V}=1.4 \times 10^{-11}$ \ifmmode{\,{\rm erg}\,{\rm cm}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-1}$}\fi\ ({\it top
panels}) and $E_{\tau_V}=1.4 \times 10^{-13}$ \ifmmode{\,{\rm erg}\,{\rm cm}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-1}$}\fi\ ({\it bottom panels}). The corresponding
vortex lifetimes are indicated with vertical straight lines. All the other parameters are
set to their standard values. {\it Right panels}: comparison of observations with the
predictions of TDR models. The observed CO and \HCOp\ column densities (open circles) are
from \citet{Lucas1996,Liszt1998,Liszt2004a}. The predictions of the TDR models (filled
symbols) are shown for $A_V= 0.2$ (red), $0.4$ (magenta), and $0.8$ (blue) mag,
$\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi = 30$ (triangles), $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi = 50$ (squares), $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi = 100$ (circles), and
$\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi = 300$ \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ (upside-down triangles), and a total column density $\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi = 1.8
\times 10^{21} \ifmmode{\,{\rm cm}^{-2}}\else{$\,{\rm cm}^{-2}$}\fi$; along each curve, the rate-of-strain $a$ varies between
10$^{-11}$ (right) and $10^{-10}$ s$^{-1}$ (left).}
\label{Fig-CO-HCOp}
\end{center}
\end{figure*}

Two different data sets are used: (1) the \HCOp(1-0) and CO(1-0) lines observed in absorption
on the continuum of quasars \citep{Lucas1996,Liszt1998,Liszt2004a} and (2) the CO and \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\
lines observed in the visible and UV ranges in absorption towards nearby OB stars (e.g.
\citealt{Burgh2007,Sheffer2008}).

In the first set of data (Fig. \ref{Fig-CO-HCOp}), the selected lines of sight sample all
Galactic latitudes (\citealt{Liszt2010}, Appendix). The observed column densities of CO
and \HCOp\ show a non-linear correlation $N({\rm CO}) \propto N(\HCOp)^{1.4}$ with an
averaged abundance ratio varying between $5 \times 10^2$ and $8 \times 10^{3}$. The total
column densities associated with each data point are similar and close to one magnitude of
matter (Paper I).

In the second set of data, displayed in the top panel of Fig. \ref{Fig-CO-NHtot}, the
sightlines sample diffuse components located in the SN, i.e. within $\sim 1$ kpc from
the sun. No correlation is observed between $N({\rm CO})$ and $N(\ifmmode{\rm H_2}\else{$\rm H_2$}\fi)$. Unlike in the
previous data set, the (unknown) total hydrogen column density associated with the CO
data likely spans a large range, not only because the \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\ column densities vary by two
orders of magnitude, but also because the \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\ fraction is expected to fluctuate by a
large factor (Sect. \ref{sec-data}).

\subsection{Oxygen chemistry} \label{Sec-oxy-chem}

In diffuse interstellar gas, CO is expected to be destroyed by photodissociation and
formed by the dissociative recombination of \HCOp, regardless of
turbulent dissipation (see Fig. \ref{Fig-Network-CO} of Appendix \ref{Append-Networks}).
At chemical equilibrium, the CO/\HCOp\ abundance ratio writes
\begin{equation} \label{Eq-CO-HCOp}
\frac{n({\rm CO})}{n(\HCOp)} = \frac{n_e k_e}{k_\gamma},
\end{equation}
where $k_e = 2.46 \times 10^{-7} (T/300 \,\,{\rm K})^{-0.69}$ cm$^{3}$ s$^{-1}$
\citep{Florescu-Mitchell2006}. Adopting the photodissociation rate $k_\gamma$ (in s$^{-1}$)
of CO given in Fig. \ref{Fig-Photo-CO} (Appendix \ref{Append-H2-CO}), $n({\rm CO})/n(\HCOp)$
is found to vary between $10^2$ and $10^5$ for typical diffuse gas conditions ($30 \lesssim
\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi \lesssim 300$ \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi, $0.01 \lesssim A_V \lesssim 1$ mag), i.e. to entirely cover the
observed range (Fig. \ref{Fig-CO-HCOp}). Consequently, any model computing the chemical
equilibrium of the gas should be able to explain the observed abundance of CO, provided
that it produces enough HCO$^+$.

In a UV-dominated chemical scheme (see Fig. \ref{Fig-Network-CO} of Appendix
\ref{Append-Networks}), the formation of \HCOp\ proceeds though the oxygen
hydrogenation chain followed by the two reactions: ${\rm OH} +
\Cp \rightarrow \COp +{\rm H}$ and $\HdO + \Cp \rightarrow \HCOp +{\rm H}$.
The bottleneck process of this pathway is the conversion of O into \Op\ via
the charge exchange with ionised hydrogen. The abundance of \HCOp\ therefore
depends on that of \Hp, the production of which is driven by the cosmic ray ionisation
of H and \ifmmode{\rm H_2}\else{$\rm H_2$}\fi. To investigate this effect, we have run the Meudon PDR code along
the grid presented in Appendix \ref{Append-H2-CO} assuming two values of the
cosmic ray ionisation rate of molecular hydrogen : $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi = 3$ and $30 \times
10^{-17}$ s$^{-1}$. In all cases, and despite the strong dependence of $n(\HCOp)$
on $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$, we find that the PDR models systematically underestimate the column
densities of \HCOp\ by more than one order of magnitude for $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi = 3 \times
10^{-17}$ s$^{-1}$ and by more than a factor of 5 for $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi = 3 \times 10^{-16}$
s$^{-1}$. These results are in line with those of Paper I and \citet{Levrier2012}
who found similar discrepancies by comparing the predictions of the Meudon PDR code
with the observed column densities of CO.

In an active vortex (bottom panel of Fig. \ref{Fig-Network-CO} of Appendix \ref{Append-Networks})
it is not the hydrogenation chain of O that matters, but that of C. Indeed it enhances the
production of the very reactive cation, \CHtp, which swiftly reacts with O to form \COp\
and \HCOp. With the opening of this additional formation pathway, the abundance of \HCOp\
rises by several orders of magnitude and that of CO by a factor of up to 100 (see Fig.
3 of Paper I).  If these behaviors appear to be in contradiction with Eq. \ref{Eq-CO-HCOp},
it is because in dissipative regions the chemistry is not at equilibrium and the electronic
fraction is substantially smaller than that in the ambient gas. After integration over the
entire line of sight, we showed in Paper I  that the sole contributions of the active phases
could explain the observed column densities of \HCOp\ for small values of the rate-of-strain
$10^{-11} \lesssim a \lesssim 10^{-10}$ s$^{-1}$, but fail to reproduce those of CO by more
than one order of magnitude. Because \HCOp\ and CO only differ by the timescales associated
with their respective destruction processes, we suggested in Paper I that CO may in fact trace
the relaxation period of turbulent dissipation (Fig. \ref{Fig-Triangles}).

\subsection{Influence of the TDR model parameters} \label{Sec-oxy-TDR}

To illustrate the  influence of the relaxation period for CO, we display the column densities
of several species computed across the diameter of a single vortex as functions of time in
the left panels of Fig. \ref{Fig-CO-HCOp}. All parameters are set to their standard values
except for $E_{\tau_V}$ which is set to $1.4 \times 10^{-11}$ \ifmmode{\,{\rm erg}\,{\rm cm}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-1}$}\fi\ (top panels) and $1.4
\times 10^{-13}$ \ifmmode{\,{\rm erg}\,{\rm cm}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-1}$}\fi\ (bottom panels). The corresponding values of the vortex lifetime (see
Eq. \ref{Eq-Etau}) are indicated by straight vertical lines. Because the vortex is stationary,
the column densities it produces are constant over its entire lifetime ($t \leqslant \tau_V$).
Once the vortex has destabilised ($t > \tau_V$), the column density of each species decreases
towards that produced by the equivalent amount of matter of ambient gas with a characteristic
timescale that depends on the species.

Since the computed column density of a given species takes its time-dependent decrease
into account during the relaxation phase (Eq. \ref{Eq-Coldens}), Fig. \ref{Fig-CO-HCOp}
illustrates the importance of the relative values of the durations of the active and
relaxation phases and their dependence on the vortex dissipation integral $E_{\tau_V}$.
For large values, $E_{\tau_V} = 1.4 \times 10^{-11}$ \ifmmode{\,{\rm erg}\,{\rm cm}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-1}$}\fi, i.e. large values of $\tau_V$,
the relaxation phase has no impact on the time-averaged column density of any species. For
small values, $E_{\tau_V} = 1.4 \times 10^{-13}$ \ifmmode{\,{\rm erg}\,{\rm cm}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-1}$}\fi, i.e. small values of  $\tau_V$, the
relaxation phase dominates the time-averaged column densities of CO, OH, and \HdO, and may
start to have an impact on those of \HCOp\ and CH.

The resulting total column densities of CO and \HCOp\ are displayed in the right panels of
Fig. \ref{Fig-CO-HCOp}. We find that the TDR model can account for the observed ranges of
column densities of both CO and \HCOp\ if $E_{\tau_V}$ is sufficiently small because
$E_{\tau_V}$ affects CO and \HCOp in different ways: $N({\rm CO})$ increases by a factor
of 60 as $E_{\tau_V}$ decreases by a factor of 100, while $N(\HCOp)$ only increases by a
factor of 3. Further reducing $E_{\tau_V}$ increases both $N({\rm CO})$ and $N(\HCOp)$.
These effects are due to the large difference between the relaxation times of \HCOp\ and
CO that affect differently the total column densities, depending on the value of $\tau_V$
(Eq. \ref{Eq-Coldens}). The best range of vortex integral dissipations is  $10^{-13}
\lesssim E_{\tau_V} \lesssim 10^{-12}$ \ifmmode{\,{\rm erg}\,{\rm cm}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-1}$}\fi, which corresponds to a vortex lifetime
ranging between 100 yr and 1000 yr (for the standard vortex).

Interestingly, \HCOp\ and CO depend differently on the other parameters. Both $N({\rm CO})$
and $N(\HCOp)$ decrease as $a$ increases, \HCOp\ is most sensitive to $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi$, but is almost
independent of $A_V$, while the reverse is true for CO. Adopting the best range for $E_{\tau_V}$,
we find that the TDR model reproduces the observations for all the values $10^{-11} \leqslant
a \leqslant 10^{-10}$ s$^{-1}$, in agreement with our previous work (Paper I). It is predicted
that the detections of \HCOp\ and CO trace a medium with a density comprised between 30 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\
and 300 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ and a shielding $A_V \gtrsim 0.2$ mag, i.e. a gas phase denser and more shielded
from the UV photons than that inferred from the observations of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ and \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi.

\subsection{Highly fragmented versus PDR-like medium }

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=8.5cm,angle=-0]{Compilation-CO_z1p1-article.eps}
\caption{Observations of the column densities of CO and \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\ (open circles, \citealt{Sheffer2008})
compared to the predictions of the TDR model. The model (filled symbols) are computed for several
densities, $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi = 20$ (crosses), 50 (squares), and 300 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ (upside-down triangles) and total
column densities, $\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi = 1.1$ (red), $2.2$ (orange), $4.5$ (magenta), $9.0$ (cyan), and $18$
(blue) $\times 10^{20}$ \ifmmode{\,{\rm cm}^{-2}}\else{$\,{\rm cm}^{-2}$}\fi. {\it Top panel:} the lines of sight are reconstructed with a
homogeneous shielding. The value adopted for $A_V$ varies along each curve from left to right
from 0.005 mag to $A_{V,{\rm max}}$, where $A_{V,{\rm max}}$ is the shielding equivalent to a
column density of matter of $\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi/2$. {\it Bottom panel:} the lines of sight are reconstructed
with a PDR-like shielding function of $\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi$ (Sect. \ref{Sec-build-los}), hence $A_V$ is no
longer a free parameter.}
\label{Fig-CO-NHtot}
\end{center}
\end{figure}

The predictions of the TDR model are now compared with the column densities of CO and \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\
observed in the SN, i.e. two species sensitive to the porosity of the gas to UV radiation
(Fig. \ref{Fig-CO-NHtot}). The model parameters are set to their standard values. We adopt
the two prescriptions for the extinction of the UV field discussed in Sect. \ref{Sec-build-los}:
a homogeneous shielding along the line of sight (top panel) and a PDR-like shielding increasing
with $\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi$ (bottom panel). In both cases, we explore the impact of very small UV shielding
(as small as 0.005 mag) on the column densities of CO predicted by the model.

The results shown in Fig. \ref{Fig-CO-NHtot} confirm the results reported in the previous
section and extend towards low column densities. They are in good  agreement with the
observations even for $N(\ifmmode{\rm H_2}\else{$\rm H_2$}\fi)$ as small as $10^{19}$ \ifmmode{\,{\rm cm}^{-2}}\else{$\,{\rm cm}^{-2}$}\fi. They also illustrate the influence
of the fragmentation of the gas on $N({\rm CO})$ and $N(\ifmmode{\rm H_2}\else{$\rm H_2$}\fi)$.

The predictions obtained with a highly fragmented medium (top panel) reproduce both the
general increase in $N({\rm CO})$ with $N(\ifmmode{\rm H_2}\else{$\rm H_2$}\fi)$ and the large scatter of the data (about a
factor of 10), whatever the other parameters. In this case, large column densities of CO
at small $N(\ifmmode{\rm H_2}\else{$\rm H_2$}\fi)$ are explained by the accumulation of many fragments, each weakly shielded
from the UV  field. We stress that $A_V$ as low as 0.005 mag combined with large \ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi\ might
reproduce the largest CO column densities at low $N(\ifmmode{\rm H_2}\else{$\rm H_2$}\fi)$. A PDR-like reconstruction of
the line of sight (bottom panel) reproduces the increase of $N({\rm CO})$ on $N(\ifmmode{\rm H_2}\else{$\rm H_2$}\fi)$ and
especially the flattening at small $N(\ifmmode{\rm H_2}\else{$\rm H_2$}\fi)$. The $N({\rm CO}) / N(\ifmmode{\rm H_2}\else{$\rm H_2$}\fi)$ column density
ratio is almost independent of the gas density. In this case, the observed scatter can
only be explained by invoking variations of the other parameters over the acceptable
ranges inferred in the previous sections.

Turbulent dissipation is therefore a crucial process to explain the CO abundances observed
in the diffuse gas. The fragmentation of the gas and its permeability to UV photons has an
important impact on the column densities of CO and molecular hydrogen. The two prescriptions
discussed here (top and bottom panels of Fig. \ref{Fig-CO-NHtot}), however, are plausible
scenarios, although the arguments developed in Sect. 2.3 on the broad width of the H/\ifmmode{\rm H_2}\else{$\rm H_2$}\fi\
transition region tend to favour a high level of fragmentation of the diffuse matter.

\section{[CII] line emission versus far-infrared dust emission} \label{Sect-Cp}

\subsection{Observational data}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=8.5cm,angle=-0]{Compilation-Cp-vs-FIR.eps}
\caption{Intensities of the [CII] and of the far-infrared dust
emission observed by \citet{Ingalls2002} at high Galactic latitudes (black empty circles)
and comparison with the predictions of PDR ({\it middle panel}) and TDR ({\it bottom panel})
models. The black curves ({\it top panel}) are the PDR predictions of \citet{Ingalls2002}
assuming a photoelectric heating efficiency of 7 \%, 4.3 \%, and 2 \% from top to bottom.
PDR and TDR models (blue and red filled symbols) are computed for $a= 10^{-11}$ s$^{-1}$
and $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi= 30$ (triangles) and 100 \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi\ (circles), all other parameters are set to their
standard values.}
\label{Fig-Cp-FIR}
\end{center}
\end{figure}

In this section we focus on the 158 $\mu$m [CII] line observations performed by \citet{
Ingalls2002} towards 101 transluscent lines of sight at high Galactic latitudes. Their
observational results are displayed in Fig. \ref{Fig-Cp-FIR} as a function of the far-infrared
dust brightness. They show that (1) the [CII] line and the interstellar dust emissions are
linearly correlated $I_{\rm [CII]} = (2.5 \pm 0.9) \times 10^{-2} I_{\rm FIR}$, in agreement
with the results of the COBE Galactic survey \citep{Wright1991,Bennett1994}; (2) this relation
breaks for $I_{\rm FIR} \gtrsim 2.5 \times 10^{-4}$ \ecqs\ indicating a slower increase in
[CII] emission at high values of far-infrared brightness (i.e. at high column densities of
matter); and (3) the observed scatter is real.

\citet{Ingalls2002} interpreted these data in terms of a variable efficiency of the dust
photoelectric heating in diffuse gas. Assuming that the cooling by [CII] entirely balances
the heating by the UV photons, as expected in PDR environments, they deduced the heating
rate in order to match the [CII] emission. Their best-fit scenario (thick black curve on
the top panel of Fig. \ref{Fig-Cp-FIR}) is obtained for an efficiency of 4.3 \% and a
radiative energy flux of 1.6 times the standard value in the SN \citep{Habing1968}. While
plausible, this interpretation raises an issue: the best-fit value is larger than the 3 \%
obtained by \citet{Bakes1994} with the COBE Galactic survey and the range of efficiencies
needed to reproduce the scatter exceeds the 5 \% limit they prescribed.

\subsection{Predictions of PDR models}

The middle panel of Fig. \ref{Fig-Cp-FIR} displays a comparison between the observations
and the predictions of the Meudon PDR code computed for two densities $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi=30$ and
$\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi=100$ \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi, a radiation scaling factor $\chi=1$, and a total amount of matter varying
between 0.3 and 2 magnitudes. We adopt the conservative value of 3 \% for the photoelectric
heating efficiency. As found by \citet{Ingalls2002}, a PDR-type medium naturally explains
the slower increase in \Cp\ emission at large $I_{\rm FIR}$ because at large visible extinction
($A_V \gtrsim 1$ mag) the gas temperature is too low to excite the $^2P_{3/2} \rightarrow
^2P_{1/2}$ transition of \Cp\ and because \Cp\ undergoes the \Cp/C/CO transition. Once the
transition is completed, the integrated emission of \Cp\ across the slab tends towards a
constant value regardless the total amount of matter along the line of sight. Yet, the
comparison with the observations shows that PDR models computed with a photoelectric effect
efficiency of 3 \% underestimate by a factor 2 to 3 the observed \Cp\ emission for
$I_{\rm FIR} \gtrsim 2 \times 10^{-4}$ \ecqs.

\subsection{Predictions of TDR models}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=8.5cm,angle=-0]{TDR-D1e2-Cp-VAct-paper-tdr2.eps}
\includegraphics[width=8.5cm,angle=-0]{TDR-D1e2-Cp-paper-tdr2.eps}
\caption{{\it Top panel}: \Cp\ relative abundance as a function of the distance from the
vortex axis during the active stage of the TDR model. {\it Bottom panel}: cooling rate
associated with the $^2P_{3/2} \rightarrow ^2P_{1/2}$ transition of \Cp\ as a function of
time during the relaxation stage of the TDR model. The model is computed for $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi = 100$
\ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi, $a= 10^{-11}$ s$^{-1}$, and several values of the visible extinction $A_V$: $0.1$
(red), $0.2$ (magenta), $0.4$ (cyan), and $1.0$ (blue) mag.}
\label{Fig-Cp-TDR}
\end{center}
\end{figure}

To illustrate the impact of turbulent dissipation on the excitation of the [CII] line,
we display in Fig. \ref{Fig-Cp-TDR} the relative abundance of \Cp\ as a function of the
position in an active vortex of the TDR model (top panel) and the cooling rate due to
the [CII] line during the relaxation period (bottom panel) for a point located at the
$r_0$ radius of a vortex (see Sect. \ref{Sec-TDR} and Paper I). We note that in this
figure, the time axis is expressed in log units to allow the visualisation of the [CII]
line cooling evolution. The origin at $t= 10$ yr is somewhat arbitrary and is meant
to allow a display of the [CII] emission close to the origin of the relaxation phase.

In an active vortex, the ignition of reaction \ref{Reac-CHp-text} opens a fast chemical
loop starting from \Cp\ that successively increases the production of \ifmmode{\rm CH_2^+}\else{$\rm CH_2^+$}\fi, \CHtp, and
of neutral carbon (see Fig. \ref{Fig-Network-CHp} of Appendix \ref{Append-Networks}).
The bottleneck of this loop being the slow photoionisation of C, the abundance of \Cp\
decreases across the active vortex (top panel of Fig. \ref{Fig-Cp-TDR}). The intensity
of the [CII] line emitted during the active phase ($I_{\rm [CII]}$ at the shortest time
(or before) in the bottom panel of Fig. \ref{Fig-Cp-TDR}), is therefore smaller than that
emitted by the ambient medium ($I_{\rm [CII]}$ at times larger than a few $10^4$ yr in
the bottom panel of Fig. \ref{Fig-Cp-TDR}). Once the active stage stops, the abundance
of \Cp\ quickly increases up to its initial value, while the thermal relaxation lasts
much longer. The outcome is a period during which the abundance of \Cp\ is high and the
temperature still high enough to enhance the [CII] emission. This period lasts for more
than $\sim 10^4$ yr while the active phase lasts only for a few hundred to one thousand
years (see Sect. \ref{Sec-oxy-TDR}). Turbulent dissipation is therefore responsible for
an enhancement of [CII] emission above that of the ambient gas.

The resulting [CII] intensities computed with TDR models combined with a PDR-like reconstruction
of the line of sight are shown in the bottom panel of Fig. \ref{Fig-Cp-FIR}. For small values
of the rate-of-strain, we find that the predictions of the model are in good agreement
with the observations, that a PDR-like shielding function along the line of sight accounts
for the slower increase in \Cp\ emission at large $I_{\rm FIR}$ and that the scatter of
observational data could be explained by two indistinguishable scenarios: either an
increase in the fragmentation of the gas or variations of \ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi, $E_{\tau_V}$, or $a$.

The [CII] line emission of diffuse and transluscent gas is therefore enhanced by the
heating of the gas due to turbulent dissipation compared to PDR predictions.

\section{Discussion} \label{Sect-discussion}

\subsection{The turbulent dissipation scenario inferred from chemistry}

\begin{figure*}[!ht]
\begin{center}
\includegraphics[width=8.0cm,angle=-0]{Compilation-OH-vs-HCOp_z1p1_article.eps}
\includegraphics[width=8.0cm,angle=-0]{Compilation-H2O-vs-HCOp_z1p1_article.eps}
\includegraphics[width=8.0cm,angle=-0]{Compilation-C2H-vs-HCOp_z1p1_article.eps}
\includegraphics[width=8.0cm,angle=-0]{H2-Diagram_z1p1_article.eps}
\caption{Observations compared to TDR model predictions. Data (open circles): column
densities of \HCOp\ and OH from \citet{Lucas1996} ({\it top left panel}), \HdO\ from
\citet{Flagey2013} ({\it top right panel}), \CtH\ from \citet{Lucas2000} ({\it bottom left panel}),
and the excited levels of \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\ from \citet{Gry2002} and \citet{Lacour2005} ({\it bottom right
panel}). For clarity, in the bottom right panel the points for a given level are slightly
shifted on the abscissa. Models (filled symbols): computed for $A_V=0.2$ (red), 0.4
(magenta), and 0.6 (blue) mag and $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi = 20$ (crosses), 30 (triangles), 50 (squares),
100 (circles), and 300 (upside-down triangles) \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi. Except for the bottom right panel,
$a$ increases along each curve from $10^{-11}$ to $10^{-10}$ s$^{-1}$ from right to left.
In the bottom right panel, the models are shown for $A_V=0.1$ mag (red) and $0.4$ mag (blue)
and single values of the rate-of-strain $a = 3\times 10^{-11}$ s$^{-1}$ and the
density $\ifmmode{n_{\rm H}}\else{$n_{\rm H}$}\fi = 100$ \ifmmode{\,{\rm cm}^{-3}}\else{$\,{\rm cm}^{-3}$}\fi.}
\label{Fig-add-TDR}
\end{center}
\end{figure*}

The previous sections have shown that, despite unavoidable degeneracies, the comparison of
the TDR model results with the observations identifies optimal ranges of three of the free
parameters related to turbulent dissipation: $\ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi \lesssim 10^{-23}$ \ifmmode{\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}}\else{$\,{\rm erg}\,{\rm cm}^{-3} {\rm s}^{-1}$}\fi\ from the
largest abundances of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi, $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi \lesssim 3.5$ \kms\ from the \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi/\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ abundance ratio,
and $ 10^{-13} \lesssim E_{\tau_V} \lesssim 10^{-12}$ erg cm$^{-1}$ from the CO/\HCOp\
abundance ratio. We found in Paper I that the chemical properties of the diffuse ISM favour
small rates-of-strain with the optimal range  $10^{-11} \lesssim a \lesssim 10^{-10}$
s$^{-1}$ that has been used in the present paper. Without fine tuning, these parameters
reproduce the column densities of three additional molecular species OH, \HdO, and \CtH,
and their link with those of \HCOp. They also reproduce the populations of the first five
rotational levels of \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\ observed in the local diffuse matter through UV absorption
spectroscopy \citep{Gry2002,Lacour2005} (Fig. \ref{Fig-add-TDR}). These results are in
good agreement with those of Paper I, even though the chemical network has been updated
(see Appendix \ref{Append-Chemistry}) because the local abundances of all these hydrides
both in UV-driven and turbulence-driven chemistries are determined by only a handful of
chemical reactions (see Appendix \ref{Append-Networks}), the rates of which are well
constrained by laboratory experiments and theoretical calculations of reaction dynamics.

The outcome of the whole study is therefore an ensemble of four parameter ranges, related
to turbulent dissipation, that have been determined independently from different molecular
species, or pairs of species, and that satisfactorily reproduce the abundances of many other
species observed in the diffuse ISM. We discuss in Sect. \ref{Sect-consist} how these values
compare with the known properties of turbulence in the diffuse ISM.

\subsection{Influence of the cosmic ray ionisation rate on the TDR model predictions} \label{Sect-zeta}

\begin{figure}[!ht]
\begin{center}
\includegraphics[width=8.5cm,angle=-0]{CH-OH-H2O_vs_zeta_article.eps}
\caption{Column densities of CH, \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi, \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi, OH, \HdO, \HCOp, and CO predicted with the
TDR model for a cloud of total column density $\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi = 1.8 \times 10^{21} \ifmmode{\,{\rm cm}^{-2}}\else{$\,{\rm cm}^{-2}$}\fi$ as functions
of the \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\ ionisation rate $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$. All the other parameters of the code are set to
their standard values (see main text).}
\label{Fig-zeta}
\end{center}
\end{figure}

Since the low-energy cosmic ray fluxes cannot be observed from Earth (partly because of the
interplanetary magnetic field) the current estimations of the cosmic ray ionisation rate
of atomic or molecular hydrogen\footnote{$\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi = 2.3 / 1.7 \times \ifmmode{\zeta_{\rm H}}\else{$\zeta_{\rm H}$}\fi$ \citep{
Glassgold1974}.} ($\ifmmode{\zeta_{\rm H}}\else{$\zeta_{\rm H}$}\fi$ and $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$, respectively) mostly rely on the observations
of molecular tracers. Such methods are, however, uncertain because they strongly depend on
the chemical models used to interpret the observations. Studies of dense clouds with OH,
\HCOp, and HD indicate that $\ifmmode{\zeta_{\rm H}}\else{$\zeta_{\rm H}$}\fi$ is in the range $1 - 5$ $\times 10^{-17}$ s$^{-1}$
with an average of $\sim 3 \times 10^{-17}$ s$^{-1}$ \citep{van-der-Tak2000}. Similarly,
early studies of diffuse clouds with the same species have led to $\ifmmode{\zeta_{\rm H}}\else{$\zeta_{\rm H}$}\fi \sim 5 \times
10^{-17}$ s$^{-1}$ in the direction of $\zeta$ Oph, $o$ Per, $\chi$ Oph, $\zeta$ Per, and
$\xi$ Per \citep{van-Dishoeck1986,Federman1996a}. However, the recent detections of \Htp\
\citep{McCall2004,Indriolo2007,Indriolo2012}, \OHp, and \HdOp\ \citep{Gerin2010a,Neufeld2010a,
Indriolo2014} in the diffuse gas now point towards values considerably larger. Indeed the
comparison of these data with chemical models solely driven by the UV radiation and the
cosmic ray particles shows that $\ifmmode{\zeta_{\rm H}}\else{$\zeta_{\rm H}$}\fi \sim 2 \times 10^{-16}$ s$^{-1}$ on average (e.g.
\citealt{Hollenbach2012}).

In this paper, we have therefore systematically run the TDR model for three different values
of $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$: $3 \times 10^{-17}$, $10^{-16}$, and $3 \times 10^{-16}$ s$^{-1}$. In addition,
and in order to further explore the impact of this parameter, 20 models were run covering the
range $10^{-17} \leqslant \ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi \leqslant 10^{-15}$ s$^{-1}$; all the other parameters were
set to their standard values (see Sect. \ref{Sect-explo-param}). The standard model, $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi =
10^{-16}$ s$^{-1}$, corresponds to the best-fit scenario of all the observational data (column
densities and column density ratios) with the TDR model. We discuss here the sensitivity
of our results to $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$.

Figure \ref{Fig-zeta} shows that most of the species considered in this work are not sensitive
to the ionisation rate of \ifmmode{\rm H_2}\else{$\rm H_2$}\fi. The column densities of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi, \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi, \HCOp, CO, and \CtH\ vary
by less than a factor of four when $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$ increases by two orders of magnitude. It follows
that $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$ has a limited impact on the determination of the properties of turbulent
dissipation (\ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi, \ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi, and $E_{\tau_V}$, see Sects. \ref{Sect-CHp}, \ref{Sect-SHp}, and
\ref{Sect-CO}) and thus weakly affects the conclusions of this paper. In contrast, Fig.
\ref{Fig-zeta} shows that CH, OH, and \HdO\ are particularly sensitive to $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$. The
column density of CH decreases by a factor of 10 and those of OH and \HdO\ increase by two
orders of magnitude when $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$ increases from $10^{-17}$ to $10^{-15}$ s$^{-1}$ because
$\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$ boosts both the productions of H and \Hp\ which respectively destroy CH (via CH +
H $\rightarrow$ C + \ifmmode{\rm H_2}\else{$\rm H_2$}\fi) and initiate the oxygen hydrogenation chain (via O + \Hp\
$\rightarrow$ \Op\ + H).

In Fig. \ref{Fig-add-TDR} we find that the column densities of OH and \HdO\ predicted with
the TDR model are in good agreement with the observations for $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi = 10^{-16}$ s$^{-1}$.
It is worth noting that the column density of CH predicted for this value ($N({\rm CH}) \sim
2 \times 10^{13}$ \ifmmode{\,{\rm cm}^{-2}}\else{$\,{\rm cm}^{-2}$}\fi) is in agreement with those observed in the local diffuse medium along
lines of sight with total column density $\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi = 1.8 \times 10^{21} \ifmmode{\,{\rm cm}^{-2}}\else{$\,{\rm cm}^{-2}$}\fi$ \citep{Sheffer2008}.

These results differ from those of \citet{Indriolo2012}, \citet{Hollenbach2012}, and
\citet{Indriolo2014} who found $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi \sim 3 \times 10^{-16}$ s$^{-1}$ with PDR-type
models and the recent observations of \Htp, \OHp, and \HdOp. The main reason for this
difference can be found in Fig. \ref{Fig-Triangles} which shows that the ambient medium
accounts for only about 50 \% of the column densities of OH and \HdO\ in the framework
of the TDR model. Indeed, it implies that the ionisation rate inferred from the TDR model
is about two times smaller than that inferred from PDR-type models. In addition, as shown
by \citet{Wolfire2008} and \citet{Hollenbach2012}, the predictions of PDR-type models on
carbon and oxygen chemistry depend on the details of the modeling of the polycyclic aromatic
hydrocarbons (PAH) in the computation of the chemical reaction rates and in particular
their size, geometry and abundances. The difference between our optimal value of $\ifmmode{\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}}\else{$\zeta_{\ifmmode{\rm H_2}\else{$\rm H_2$}\fi}$}\fi$
and that deduced by \citet{Hollenbach2012} could therefore be due to the fact that we model
PAHs as spherical particles of radius 6.4 \AA\ (or equivalently $\sim 120$ carbon atoms)
and relative abundance $\sim 4 \times 10^{-7}$ while they consider planar PAHs containing
100 carbon atoms and with a relative abundance of $2 \times 10^{-7}$.

\subsection{Consistency with known properties of turbulence in the diffuse ISM} \label{Sect-consist}

The upper limit of \ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi\ was derived from the largest values of $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi)/\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi$ detected in
the diffuse medium towards the inner Galaxy. The parameter \ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi\ is the injection rate of
turbulent energy at the large scales, $\rho \sigma_l^3/l$, where $\rho$ is the gas volumic
mass and $\sigma_l$ the rms velocity dispersion at scale $l$. This quantity is difficult to
determine observationally in the atomic gas because it involves a dynamical time $l/\sigma_l$
and therefore a sizescale $l$, while the available data are only Gaussian velocity components.
As discussed in Sect. \ref{sec-data}, this value has been estimated in the atomic medium in
only one region of the SN \citep{Miville-Deschenes2003}, so no statistics are available. In
diffuse molecular gas traced by CO(1-0), large statistics are now available (\citealt{Hennebelle2012})
and the upper limit on \ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi\ is of the order of the largest values of $\ifmmode{\overline{\varepsilon}}\else{$\overline{\varepsilon}$}\fi_{obs}$ measured
at scales between 10 pc and 50 pc that can be considered as relevant large scales for turbulence
in the diffuse medium. Moreover, these values are themselves averages measured at each of these
large scales. The fact that \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ is not detected with average relative abundances larger than
$10^{-7}$ confirms previous estimations of the maximum turbulent energy density contained in
the diffuse ISM. The lower limit of $N(\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi) /\ifmmode{N_{\rm H}}\else{$N_{\rm H}$}\fi$ cannot be exploited because it comes from
the sensitivity limit of the observations. The observed large scatter by a factor of $\sim 10$
of the \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ abundance (Fig. 4) could be interpreted by the fact that the formation of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\
is highly intermittent: the \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ abundance traces subsets of the CNM where the turbulent
transfer rate is above average, hence driving an above-average dissipation rate. The above-average
dissipation rate needs nonetheless to heat the gas to temperatures that compare with the
endothermicity threshold of the \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ formation.

The upper limit on the ion-neutral drift velocity rules out values larger than 3.5 \kms. In
the model, this velocity drift is set by the maximum tangential velocity of the vortex, which
is itself linked to the velocity dispersion of the ambient turbulence, i.e. a property of
turbulence discussed in Sect. \ref{Sec-TDR}. It is noteworthy that this value of 3.5 \kms\
is close to the large-scale velocity dispersion of the CNM seen in the HI line \citep{
Miville-Deschenes2003,Haud2007} and is also representative of the internal velocity dispersion
of the diffuse molecular gas at scales of 10 -- 50 pc. The parameters relevant to the ambient
turbulence, inferred independently from the \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ and \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi\ abundances, are therefore broadly
consistent with our knowledge of turbulence in the diffuse medium.

\subsection{Internal coherence}
We now turn to the parameters controlling the dynamics of the dissipative regions and
compare the different associated timescales. The optimal range of rates-of-strain given
above corresponds to timescales $a^{-1}$ varying between 300 yr and 3000 yr. The independent
determination of the burst lifetime, $\tau_V$, is in the range  100 -- 1000 yr for the
reference vortex (see Sect. \ref{Sect-CO}). Finally, the turnover timescale of a vortex,
$2\pi r_0/\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi$ is of the order of $10^3$ yr for $r_0=100$ au and $\ifmmode{u_{\theta,{\rm M}}}\else{$u_{\theta,{\rm M}}$}\fi=3$ \kms. All these
timescales are therefore commensurate, confirming the picture of a highly non-equilibrium
chemistry driven by turbulent dissipation bursts over timescales as short as 100 yr and size
scales of $\sim 100$ au. It is interesting to note that the small values of $a$, favoured
by the chemistry, lead to timescales longer than the vortex lifetime, i.e. the action of the
rate-of-strain lasts at least as long as the vortex lifetime.

\subsection{Limitations of the TDR model}

Some of the limitations of the TDR model have already been discussed in Paper I. In particular,
the use of only one type of dissipation burst was mentioned, while dissipation is known to be
distributed among a large number of bursts of different strengths \citep{Mouri2009}. The warm
chemistry being driven by endoenergetic reactions with exponential thresholds, the abundances
and abundance ratios of several species greatly depend on the strength of the burst as long as
the corresponding thresholds are not overcome (see Sect. \ref{Sect-SHp}). A distribution of
bursts could thus have an important impact. Implementing such a distribution in the TDR code
would require defining additional parameters that cannot be constrained with the current
observations of the diffuse ISM.

Our model is idealised, but the results rely on two robust properties of gas-phase
interstellar chemistry: the large differences of endothermicities and the orders of
magnitude differences between relaxation times due to the various destruction mechanisms
(photodissociation, recombination, collisions with H and \ifmmode{\rm H_2}\else{$\rm H_2$}\fi). Hence, the results of the
TDR model presented here do not depend on the details of the treatment of the regions of
dissipation. Its originality  relies in the exploitation of the fact that the chemical
timescales in the diffuse medium are commensurate with the smallest dynamical times of
turbulence, at which its huge reservoir of energy is dissipated.

\subsection{Turbulent dissipation in vortices and shear layers or C-shocks?}

The problem raised by the high abundances of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ in the diffuse medium was recognised
several decades ago. Additional gas heating mechanisms were invoked and modelled to
reproduce the observed column densities: gas heating behind shocks associated with HII
regions expansion \citep{Elitzur1978}, ion-neutral friction in shocks with magnetic
precursors, i.e. C-shocks \citep{Draine1986a,Pineau-des-Forets1986}, or in Alfv\'en
waves that also involve the streaming of \Cp\ relative to \ifmmode{\rm H_2}\else{$\rm H_2$}\fi\ \citep{Federman1996}.

To favour one scenario over the others from the sole observations of chemical abundances
and line profiles is difficult. For instance \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ could be explained by one C-shock
with a velocity of 10-12 \kms\ leading to a predicted velocity shift between \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ and
CH of 2 \kms\ at most, i.e. if the shock propagates along the line of sight \citep{
Flower1998c}. Yet, it is more likely that several low velocity shock waves will be present
along the line of sight, with different orientations and velocities. \citet{Gredel2002}
proposed such a model with $\sim 20$ C-shocks of velocities of $\sim 8$ \kms\ per $A_V$,
leading to a velocity shift between CH and \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ smaller than 1 \kms.

However, it is now understood that shocks are not the only structures where dissipation is
concentrated in compressible turbulence. In transonic hydrodynamic turbulence, \citet{
Porter2002} found that the interaction of shocks generates vorticity, resulting in only
a small fraction of the kinetic energy left in compressional modes. In supersonic MHD
turbulence, the dynamic alignment of velocity and magnetic fields tends to Alfv\'enize
turbulence, a process at the origin of the low ratio of the energy density in compressional
to solenoidal modes \citep{Kritsuk2010}. Similar conclusions were found in other simulations
of MHD turbulence where incompressible modes carry more than 70 \% of the total kinetic energy
\citep{Vestuto2003}. It is for these reasons that we chose to investigate the chemical
signatures of turbulent dissipation in vortices.

\section{Conclusions} \label{Sect-conclusion}

The vast diversity of the chemical formation and destruction pathways of atoms and molecules,
and of their timescales in particular, provide unique clues to the non-equilibrium physical
processes driving the thermal and chemical evolution of the diffuse ISM that have never been
investigated before. In this paper we have presented a comprehensive study of the chemical
composition of diffuse matter in the framework of UV-driven and turbulence-driven chemistries.
The analysis focuses on 16 different atomic and molecular species observed towards several
hundred sightlines sampling the SN and the inner Galaxy diffuse medium.

We have distinguished the species that are the most sensitive to turbulent dissipation
(the abundances of \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ and \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi, and the populations of the $J=3,4,5$ rotational levels
of \ifmmode{\rm H_2}\else{$\rm H_2$}\fi) from those that are not and trace the interaction between the UV radiation field
and the interstellar matter (H, \ifmmode{\rm H_2}\else{$\rm H_2$}\fi, HF, and the $J=0,1,2$ excited levels of \ifmmode{\rm H_2}\else{$\rm H_2$}\fi). The
column densities of CO and \HCOp and the intensity of the [CII] line emission are also
found to be sensitive to turbulent dissipation.

The chemical differences among all these species are so pronounced that robust results on
the turbulent cascade in the diffuse ISM can be inferred from the observations of specific
pairs of species. In the framework of the TDR model, we find that (1) the \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ abundance
is proportional to the average turbulent dissipation rate, thus to the average energy transfer
rate in the turbulent cascade; (2) the \ifmmode{\rm SH^+}\else{$\rm SH^+$}\fi/\ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\ abundance ratio sets an upper limit to the
ion-neutral velocity drift in the active dissipation regions because of the large difference
of endothermicities involved in the formation of these two species; and (3) the \HCOp/CO
abundance ratio sets the lifetime of the active dissipation bursts because of the large
difference of destruction timescales of these two species. We find that the increase in \ifmmode{\rm CH^+}\else{$\rm CH^+$}\fi\
abundance in the inner Galaxy compared to the value derived in the local ISM could be due to
an increase in the turbulent injection rate towards the inner Galaxy. This set of independently
determined properties provides a coherent scenario in which the regions of active dissipation
of turbulence in the diffuse ISM are small-scale (a few $100$ au) short-lived ($\sim 100 -
1000$ yr) bursts in which dissipation is mainly due to ion-neutral decoupling ($\sim 3$ \kms)
rather than viscous friction. These results are in good agreement with independent determinations
of the turbulence properties in the diffuse ISM and with the fundamental properties of turbulence
known from laboratory experiments.

We also show for the first time that turbulent dissipation plays a crucial role in the
formation of diffuse interstellar CO and stress the influence of the fragmentation of
the gas on the observed correlation between CO and molecular hydrogen: large column
densities of CO at low $N(\ifmmode{\rm H_2}\else{$\rm H_2$}\fi)$ may result from the accumulation on the line of sight
of fragments, each weakly shielded from the UV radiation field. Finally, we show that
turbulent dissipation enhances the [CII] line emission in transluscent environments by
a factor of $\sim 2$ compared to the value obtained if the gas heating was solely driven
by UV photons.

These results shed a new light on the interpretation of atomic and molecular data and
provide templates for the analysis of future observations of diffuse environments in Galactic
and extragalactic sources.

\begin{acknowledgements}

We are most grateful to James G. Ingalls for providing the data used in Fig. \ref{Fig-Cp-FIR}.

\end{acknowledgements}

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