\section{Introduction}
\label{section:Introduction}
After molecular hydrogen (H$_2$) and carbon monoxide (CO), the water molecule (H$_2$O)
can be one of the most abundant molecules in the interstellar medium (ISM) in galaxies.
It provides some important diagnostic tools for various physical and chemical
processes in the ISM \citep[e.g.][and references therein]{2013ChRv..113.9043V}.
Prior to the {\it Herschel Space Observatory} \citep{2010A&A...518L...1P}, in
extragalactic sources, non-maser {\hbox {H$_{2}$O}}\ rotational transitions were only detected by the
\textit{Infrared Space Observatory} \citep[\textit{ISO},][]{1996A&A...315L..27K} in the
form of far-infrared absorption lines \citep{2004ApJ...613..247G, 2008ApJ...675..303G}.
Observations of local infrared bright galaxies by {\it Herschel} have revealed a rich
spectrum of submillimeter (submm) {\hbox {H$_{2}$O}}\ emission lines (submm {\hbox {H$_{2}$O}}\ refers to
rest-frame submillimeter {\hbox {H$_{2}$O}}\ emission throughout this paper if not otherwise specified).
Many of these lines are emitted from high-excitation rotational levels with upper-level
energies up to $E_\mathrm{up}$/$k = 642\,$K \citep[e.g.][]{2010A&A...518L..42V,
2010A&A...518L..43G, 2012A&A...541A...4G, 2013A&A...550A..25G, 2011ApJ...743...94R,
2012ApJ...753...70K, 2012ApJ...758..108S, 2013ApJ...762L..16M, 2013ApJ...779L..19P,
2013ApJ...768...55P}. Excitation analysis of these lines has revealed that they are
probably excited through absorption of \hbox {far-infrared}\ photons from thermal dust emission in
warm dense regions of the ISM \citep[e.g.][]{2010A&A...518L..43G}. Therefore,
unlike the canonical CO lines that trace collisional excitation of the molecular
gas, these {\hbox {H$_{2}$O}}\ lines represent a powerful diagnostic of the \hbox {far-infrared}\ radiation field.

Using the {\it Herschel} archive data, \citet[][hereafter Y13]{2013ApJ...771L..24Y}
have undertaken a first systematic study of submm {\hbox {H$_{2}$O}}\ emission in local \hbox {infrared}\
galaxies. {\hbox {H$_{2}$O}}\ was found to be the strongest molecular emitter after CO within
the submm band in those \hbox {infrared}-bright galaxies, even with higher flux density than
that of CO in some local ULIRGs (velocity-integrated flux density of \htot321312 is
larger than that of \co54 in four galaxies out of 45 in the \citetalias{2013ApJ...771L..24Y}
sample). The luminosities of the submm {\hbox {H$_{2}$O}}\ lines (\hbox {$L_{\mathrm{H_2O}}$}) are near-linearly
correlated with total \hbox {infrared}\ luminosity ($L_\mathrm{IR}$, integrated over
8--1000\,$\mu$m) over three orders of magnitude. The correlation is revealed
to be a straightforward result of \hbox {far-infrared}\ pumping: {\hbox {H$_{2}$O}}\ molecules are excited
to higher energy levels through absorbing \hbox {far-infrared}\ photons, then the upper level
molecules cascade toward the lines we observed in an almost constant fraction
(Fig.\,\ref{fig:h2o-e-level}). Although the galaxies dominated by active galactic
nuclei (AGN) have somewhat lower ratios of \hbox {$L_{\mathrm{H_2O}}$}/\hbox {$L_{\mathrm{IR}}$}, there does not appear
to be a link between the presence of an AGN and the submm {\hbox {H$_{2}$O}}\ emission
\citepalias{2013ApJ...771L..24Y}. The {\hbox {H$_{2}$O}}\ emission is likely to trace the
\hbox {far-infrared}\ radiation field generated in star-forming nuclear regions in galaxies,
explaining its tight correlation with \hbox {far-infrared}\ luminosity.

Besides detections of the {\hbox {H$_{2}$O}}\ lines in local galaxies from space telescopes,
redshifted submm {\hbox {H$_{2}$O}}\ lines in \hbox {high-redshift}\ lensed Ultra- and Hyper-Luminous InfraRed
Galaxies (ULIRGs, $10^{13}\,L_\odot > L_\mathrm{IR} \geq 10^{12}$\,{\hbox {$L_\odot$}}; HyLIRGs,
$L_\mathrm{IR} \geq 10^{13}$\,{\hbox {$L_\odot$}}) can also be detected by ground-based
telescopes in atmospheric windows with high transmission. Strong gravitational
lensing boosts the flux and allows one to detect the {\hbox {H$_{2}$O}}\ emission lines easily.
Since our first detection of submm {\hbox {H$_{2}$O}}\ in a lensed {\it Herschel} source at
$z = 2.3$ \citep{2011A&A...530L...3O} using the IRAM NOrthern Extended Millimeter
Array (NOEMA), several individual detections at \hbox {high-redshift}\ have also been reported
\citep{2011ApJ...738L...6L, 2011ApJ...741L..38V, 2011ApJ...741L..37B,
2012A&A...538L...4C, 2012ApJ...757..135L, 2013ApJ...779...67B,
2013A&A...551A.115O, 2013Natur.495..344V, 2013ApJ...767...88W,
2014ApJ...783...59R}. These numerous and easy detections of {\hbox {H$_{2}$O}}\ in \hbox {high-redshift}\
lensed ULIRGs show that its lines are the strongest submm molecular lines
after CO and may be an important tool for studying these galaxies.

We have carried out a series of studies focussing on submm
{\hbox {H$_{2}$O}}\ emission in \hbox {high-redshift}\ lensed galaxies since our first detection. Through the
detection of $J=2$ {\hbox {H$_{2}$O}}\ lines in seven \hbox {high-redshift}\ lensed Hy/ULIRGs reported by
\citet[][hereafter O13]{2013A&A...551A.115O}, a slightly super-linear correlation
between \hbox {$L_{\mathrm{H_2O}}$}\ and \hbox {$L_{\mathrm{IR}}$}\ (\hbox {$L_{\mathrm{H_2O}}$}\;$\propto$\;\hbox {$L_{\mathrm{IR}}$}$^{1.2}$) from local ULIRGs and
\hbox {high-redshift}\ lensed Hy/ULIRGs has been found. This result may imply again that \hbox {far-infrared}\
pumping is important for {\hbox {H$_{2}$O}}\ excitation in \hbox {high-redshift}\ extreme starbursts. The
average ratios of \hbox {$L_{\mathrm{H_2O}}$}\ to \hbox {$L_{\mathrm{IR}}$}\ for the $J=2$ {\hbox {H$_{2}$O}}\ lines in the \hbox {high-redshift}\ sources
tend to be $1.8\pm0.9$ times higher
than those seen locally \citepalias{2013ApJ...771L..24Y}. This shows that the
same physics with infrared pumping should dominate {\hbox {H$_{2}$O}}\ excitation in ULIRGs
at low and high redshift, with some specificity at \hbox {high-redshift}\ probably linked to
the higher luminosities.

Modelling provides additional information about the {\hbox {H$_{2}$O}}\ excitation. For example,
through LVG modelling, \cite{2013Natur.496..329R} argue that the excitation of the
submm {\hbox {H$_{2}$O}}\ emission in the $z \sim 6.3$ submm galaxy is \hbox {far-infrared}\ pumping dominated.
Modelling of the local {\it Herschel} galaxies of \citetalias{2013ApJ...771L..24Y}
has been carried out by \citet[][hereafter G14]{2014A&A...567A..91G}. They confirm
that \hbox {far-infrared}\ pumping is the dominant mechanism responsible for the submm {\hbox {H$_{2}$O}}\
emission (except for the ground-state emission transitions, such as para-{\hbox {H$_{2}$O}}\
transition \t111000) in the extragalactic sources. Moreover, collisional
excitation of the low-lying ($J \leq 2$) {\hbox {H$_{2}$O}}\
lines could also enhance the radiative pumping of the ($J \geq 3$) high-lying
lines. The ratio between low-lying and high-lying {\hbox {H$_{2}$O}}\ lines is sensitive to the
dust temperature ({\hbox {$T_{\mathrm{d}}$}}) and {\hbox {H$_{2}$O}}\ column density ($N_\mathrm{H_2O}$). From modelling
the average of local star-forming- and mild-AGN-dominated galaxies,
\citetalias{2014A&A...567A..91G} show that the submm {\hbox {H$_{2}$O}}\ emission comes from
regions with $N_\mathrm{H_2O} \sim (0.5\text{--}2) \times 10^{17}$\,cm$^{-2}$ and a
100\,$\mu$m continuum opacity of $\tau_{100} \sim 0.05\text{--}0.2$, where
{\hbox {H$_{2}$O}}\ is mainly excited by warm dust with a temperature range of $45\text{--}75$\,K.
{\hbox {H$_{2}$O}}\ lines thus provide key information about the properties of the dense
cores of ULIRGs, that is, their {\hbox {H$_{2}$O}}\ content, the infrared radiation field and the
corresponding temperature of dust that is warmer than the core outer layers
and dominates the far-infrared emission.

Observations of the submm {\hbox {H$_{2}$O}}\ emission, together with appropriate modelling and
analysis, therefore allows us to study the properties of the \hbox {far-infrared}\ radiation
sources in great detail. So far, the excitation analysis combining both
low- and high-lying {\hbox {H$_{2}$O}}\ emission has only been done in a few case studies.
Using {\hbox {H$_{2}$O}}\ excitation modelling considering both collision and \hbox {far-infrared}\ pumping,
\cite{2010A&A...518L..43G} and \cite{2011ApJ...741L..38V} estimate the sizes of
the \hbox {far-infrared}\ radiation fields in Mrk\,231 and APM\,08279+5255 (APM\,08279 hereafter),
which are not resolved by the observations directly, and suggest their AGN dominance
based on their total enclosed energies. This again demonstrates that submm {\hbox {H$_{2}$O}}\
emission is a powerful diagnostic tool which can even transcend the angular resolution
of the telescopes.

The detection of submm {\hbox {H$_{2}$O}}\ emission in the {\it Herschel}-ATLAS\footnote{The
{\it Herschel}-ATLAS is a project with {\it Herschel}, which is an ESA space
observatory with science instruments provided by European-led Principal Investigator
consortia and with important participation from NASA. The {\it H}-ATLAS website is
\url{http://www.h-atlas.org}.} \citep[][{\it H}-ATLAS hereafter]{2010PASP..122..499E}
sources through gravitational lensing allows us to characterise the \hbox {far-infrared}\ radiation
field generated by intense star-forming activity, and possibly AGN, and learn the
physical conditions in the warm dense gas phase in extreme starbursts in the early
Universe. Unlike standard dense gas tracers such as HCN, which is weaker at \hbox {high-redshift}\
compared to that of local ULIRGs \citep{2007ApJ...660L..93G}, submm {\hbox {H$_{2}$O}}\ lines are
strong and even comparable to high-$J$ CO lines in some galaxies
\citepalias{2013ApJ...771L..24Y, 2013A&A...551A.115O}. Therefore, {\hbox {H$_{2}$O}}\ is an
efficient tracer of the warm dense gas phase that makes up a major fraction of the
total molecular gas mass in \hbox {high-redshift}\ Hy/ULIRGs \citep{2014PhR...541...45C}. The successful
detections of submm {\hbox {H$_{2}$O}}\ lines in both local \citepalias{2013ApJ...771L..24Y} and the
\hbox {high-redshift}\ universe \citepalias{2013A&A...551A.115O} show the great potential of a systematic
study of {\hbox {H$_{2}$O}}\ emission in a large sample of \hbox {infrared}\ galaxies over a wide range in redshift
(from local up to $z\sim4$) and luminosity ($\hbox {$L_{\mathrm{IR}}$} \sim10^{10}$--$10^{13}$\,{\hbox {$L_\odot$}}). However,
our previous \hbox {high-redshift}\ sample was limited to seven sources and to one $J=2$ para-{\hbox {H$_{2}$O}}\ line
($E_\mathrm{up}$/$k = 100$--$127\,$K) per source \citepalias{2013A&A...551A.115O}. In
order to further constrain the conditions of {\hbox {H$_{2}$O}}\ excitation, to confirm the dominant
role of \hbox {far-infrared}\ pumping and to learn the physical conditions of the warm dense gas phase
in \hbox {high-redshift}\ starbursts, it is essential to extend the studies to higher excitation lines.
We thus present and discuss here the results of such new observations of a strong $J=3$
ortho-{\hbox {H$_{2}$O}}\ line with $E_\mathrm{up}$/$k = 304\,$K in six strongly lensed {\it H}-ATLAS
galaxies at z\,$\sim$\,2.8--3.6, where a second lower-excitation $J=2$ para-{\hbox {H$_{2}$O}}\ line
was also observed (Fig.\,\ref{fig:h2o-e-level} for the transitions and the
corresponding $E_\mathrm{up}$).

 \begin{figure}[htbp]
	 \begin{center}
 \includegraphics[scale=0.441]{e_level}
 \caption{
          Energy level diagrams of {\hbox {H$_{2}$O}}\ and {\hbox {H$_2$O$^+$}}\ shown in black and red,
          respectively. Dark blue arrows are the submm {\hbox {H$_{2}$O}}\ transitions we
		  have observed in this work. Pink dashed lines show the \hbox {far-infrared}\ pumping
		  path of the {\hbox {H$_{2}$O}}\ excitation in the model we use, with the wavelength
		  of the photon labeled. The light blue dashed arrow is the transition
		  from para-{\hbox {H$_{2}$O}}\ energy level $2_{20}$ to $2_{11}$ along the cascade
		  path from $2_{20}$ to $1_{11}$. Rotational energy levels of {\hbox {H$_{2}$O}}\
		  and {\hbox {H$_2$O$^+$}}, as well as fine structure component levels of {\hbox {H$_2$O$^+$}}\, are
		  also shown in the figure.
         }
 \label{fig:h2o-e-level}
     \end{center}
 \end{figure}

We describe our sample, observation and data reduction in Section \ref{section:Sample
and Observation}. The observed properties of the \hbox {high-redshift}\ submm {\hbox {H$_{2}$O}}\ emission are
presented in Section \ref{Results}. Discussions of the lensing properties, \hbox {$L_{\mathrm{H_2O}}$}-\hbox {$L_{\mathrm{IR}}$}\
correlation, {\hbox {H$_{2}$O}}\ excitation, comparison between {\hbox {H$_{2}$O}}\ and CO, AGN contamination will
be given in Section \ref{Discussion}. Section \ref{htop} describes the detection of
{\hbox {H$_2$O$^+$}}\ lines. We summarise our results in Section \ref{Conclusions}. A flat $\Lambda$CDM
cosmology with
$H_{0}=71\,{\rm km\,s^{-1}\,Mpc^{-1}}$, $\Omega_{M}=0.27$, $\Omega_{\Lambda}=0.73$
\citep{2003ApJS..148..175S} is adopted throughout this paper.

\section{Sample and observation}
\label{section:Sample and Observation}
Our sample consists of eleven extremely bright \hbox {high-redshift}\ sources with
$F_\mathrm{500\mu m}>200$\,mJy discovered by the {\it H}-ATLAS
survey \citep{2010PASP..122..499E}. Together with the seven similar
sources reported in our previous H$_2$O study \citepalias{2013A&A...551A.115O},
they include all the brightest \hbox {high-redshift}\ {\it H}-ATLAS sources ($F_\mathrm{500\mu m}>170$\,mJy),
but two, imaged at 880\,$\mu$m with SMA by \citet[][hereafter B13]{2013ApJ...779...25B}.
In agreement with the selection according to the methods of
\citet{2010Sci...330..800N}, the detailed lensing modelling
performed by \citetalias{2013ApJ...779...25B} has shown that
all of them are strongly lensed, but one, G09v1.124 \citep[][see below]{2013ApJ...772..137I}.
The sample of our present study is thus well representative
of the brightest \hbox {high-redshift}\ submillimeter sources with $F_\mathrm{500\mu m}>200$\,mJy
(with apparent total infrared luminosity $\sim 5\text{--}15 \times 10^{13}$\,{\hbox {$L_\odot$}}\
and $z \sim 1.5\text{--}4.2$) found by {\it H}-ATLAS in its equatorial ('GAMA')
and north-galactic-pole ('NGP') fields, in $\sim 300$\,deg$^2$ with a density
$\sim 0.05$\,deg$^{-2}$. In our previous project \citepalias{2013A&A...551A.115O},
we observed {\hbox {H$_{2}$O}}\ in seven strongly lensed \hbox {high-redshift}\ {\it H}-ATLAS galaxies from the
\citetalias{2013ApJ...779...25B} sample. In this work, in order to observe
the high-excitation ortho-\htot321312 line with rest frequency of 1162.912\,GHz
with the IRAM/NOEMA, we selected the brightest sources at 500\,$\mu$m with
$z \gtrsim 2.8$ so that the redshifted lines could be observed in a reasonably
good atmospheric window at $\nu_\mathrm{obs} \lesssim 300$\,GHz. Eight sources
with such redshift were selected from the \citetalias{2013ApJ...779...25B}
{\it H}-ATLAS sample.

\begin{table*}[htbp]
\small
\setlength{\tabcolsep}{0.42em}
\caption{Observation log.}
\centering
\begin{tabular}{lcllllllll}
\hline
\hline
{IAU Name}                                        & {Source}      & {$\mathrm{RA}$}              & {$\mathrm{DEC}$}                &  {$\mathrm{RA_{pk}}$} &  {$\mathrm{DEC_{pk}}$} &  {{\hbox {H$_{2}$O}}\ line}             &{$\nu_\mathrm{obs}$}     & {Beam}                         &{$t_\mathrm{on}$} \\
                                                  &               &  (J2000)                     & (J2000)                         &   (J2000)             &  (J2000)               &                           &  (GHz)                  & ($''$)                         &  (h)                 \\
\hline
{{\it H}-ATLAS J083051.0$+$013224}                & {G09v1.97}    & {08:30:51.02}                & {$+$01:32:24.88}                &  {08:30:51.17}        &  {$+$01:32:24.39}      &  \t211202                 &  {162.286}              & {5.6$\times$3.3}               &  {3.5}               \\
                                                  &               &                              &                                 &  {08:30:51.17}        &  {$+$01:32:24.09}      &  \t321312                 &  {250.952}              & {2.6$\times$1.1}               &  {3.1}               \\

{{\it H}-ATLAS J113526.3$-$014605}                & {G12v2.43}    & {11:35:26.36}                & {$-$01:46:05.56}                &  {11:35:26.27}        &  {$-$01:46:06.44}      &  \t202111                 &  {239.350}              & {2.3$\times$1.0}               &  {6.9}               \\
                                                  &               &                              &                                 &  {11:35:26.28}        &  {$-$01:46:06.43}      &  \t321312                 &  {281.754}              & {2.2$\times$1.1}               &  {1.5}               \\

{{\it H}-ATLAS J125632.7$+$233625}                & {NCv1.143}    & {12:56:32.70}                & {$+$23:36:24.86}                &  {12:56:32.56}        &  {$+$23:36:27.92}      &  \t211202                 &  {164.739}              & {3.1$\times$2.9}               &  {1.5}               \\
                                                  &               &                              &                                 &  {12:56:32.56}        &  {$+$23:36:27.69}      &  \t321312                 &  {254.745}              & {2.1$\times$1.0}               &  {1.5}               \\

{{\it H}-ATLAS J132630.1$+$334410}                & {NAv1.195}    & {13:26:30.12}                & {$+$33:44:09.90}                &  {13:26:30.14}        &  {$+$33:44:09.11}      &  \t202111                 &  {250.045}              & {2.0$\times$1.7}               &  {3.8}               \\
                                                  &               &                              &                                 &  {13:26:30.14}        &  {$+$33:44:09.09}      &  \t321312                 &  {293.334}              & {1.0$\times$0.9}               &  {3.1}               \\

{{\it H}-ATLAS J132859.3$+$292327}                & {NAv1.177}    & {13:28:59.29}                & {$+$29:23:27.07}                &  {13:28:59.25}        &  {$+$29:23:26.18}      &  \t202111                 &  {261.495}              & {1.9$\times$1.7}               &  {2.3}               \\
                                                  &               &                              &                                 &  {13:28:59.25}        &  {$+$29:23:26.34}      &  \t321312                 &  {307.812}              & {1.6$\times$0.9}               &  {2.3}               \\

{{\it H}-ATLAS J133008.4$+$245900}                & {NBv1.78}     & {13:30:08.56}                & {$+$24:58:58.30}                &  {13:30:08.56}        &  {$+$24:58:58.55}      &  \t321312                 &  {282.878}              & {1.7$\times$1.1}               &  {4.2}               \\
\multirow{2}{*}{{\it H}-ATLAS J084933.4$+$021443} & {G09v1.124-W} & \multirow{2}{*}{08:49:33.36} & \multirow{2}{*}{$+$02:14:42.30} &  {08:49:33.59}       &  {$+$02:14:44.68}     & \multirow{2}{*}{\t211202} &\multirow{2}{*}{220.537} &\multirow{2}{*}{1.8$\times$1.2} &\multirow{2}{*}{8.4}  \\
                                                  & {G09v1.124-T} &                              &                                 &  {08:49:32.95}       &  {$+$02:14:39.70}     &                           &                         &                                &                      \\
{{\it H}-ATLAS J085358.9$+$015537}                & {G09v1.40}    & {08:53:58.90}                & {$+$01:55:37.00}                &  {08:53:58.84}        &  {$+$01:55:37.75}      &  \t211202                 &  {243.425}              & {1.8$\times$1.0}               &  {1.9}               \\
{{\it H}-ATLAS J091043.1$-$000321}                & {SDP11}       & {09:10:43.09}                & {$-$00:03:22.51}                &  {09:10:43.06}        &  {$-$00:03:22.10}      &  \t202111                 &  {354.860}              & {1.9$\times$1.5}               &  {3.8}               \\
{{\it H}-ATLAS J125135.4$+$261457}                & {NCv1.268}    & {12:51:35.46}                & {$+$26:14:57.52}                &  {12:51:35.38}        &  {$+$26:14:58.12}      &  \t211202                 &  {160.864}              & {2.9$\times$2.6}               &  {7.7}               \\
{{\it H}-ATLAS J134429.4$+$303036}                & {NAv1.56}     & {13:44:29.52}                & {$+$30:30:34.05}                &  {13:44:29.46}        &  {$+$30:30:34.01}      &  \t211202                 &  {227.828}              & {1.7$\times$1.7}               &  {2.3}               \\
\hline
\end{tabular}
\tablefoot{
        RA and DEC are the J2000 {\it Herschel} coordinates which were taken as the centres of
        the NOEMA images displayed in Fig.\,\ref{fig:map-all}; RA$_\mathrm{pk}$ and
        DEC$_\mathrm{pk}$ are the J2000 coordinates of the NOEMA dust continuum image peaks;
		$\nu_\mathrm{obs}$ is the central observed frequency. The rest-frame frequencies
		of para-{\hbox {H$_{2}$O}}\ \t202111, \t211202 and ortho-{\hbox {H$_{2}$O}}\ \t321312 lines are: 987.927\,GHz,
		752.033\,GHz and 1162.912\,GHz, respectively (the rest-frame frequencies are taken from the JPL catalogue: \url{http://spec.jpl.nasa.gov}); $t_\mathrm{on}$ is the on-source
		integration time. The source G09v1.124, which is not resolved by SPIRE, is a cluster that
		consists of two main components: eastern component W (G09v1.124-W) and western component
		T (G09v1.124-T) as described in \cite{2013ApJ...772..137I} (see also Fig.\,\ref{fig:map-3}).
}
\label{table:obs_log}
\end{table*}
\normalsize

\citetalias{2013ApJ...779...25B} provide lensing models, magnification
factors ($\mu$) and inferred intrinsic properties of these galaxies and
list their CO redshifts which come from \citet{2012ApJ...752..152H};
Harris et al. (in prep.); Lupu et al. (in prep.); Krips et al. (in prep.)
and Riechers et al. (in prep.).

In our final selection of the sample to be studied in the \htot321312
line, we then removed two sources, SDP\,81 and G12v2.30, that were
previously observed in {\hbox {H$_{2}$O}}\ (\citetalias{2013A&A...551A.115O}, and
also \citealt{2015ApJ...808L...4A} for SDP\,81), because the $J=2$
{\hbox {H$_{2}$O}}\ emission is too weak and/or the interferometry could resolve
out some flux considering the lensing image. The observed \hbox {high-redshift}\ sample
thus consists of two GAMA-field sources: G09v1.97 and G12v2.43, and
four sources in the {\it H}-ATLAS NGP field: NCv1.143, NAv1.195,
NAv1.177 and NBv1.78 (Tables\,\ref{table:obs_log} and
\ref{table:previous_obs_properties}). Among the six remaining
sources at redshift between 2.8 and 3.6, only one, NBv1.78, has
been observed previously in a low-excitation line, para-\htot202111
\citepalias{2013A&A...551A.115O}. Therefore, we have observed both para-{\hbox {H$_{2}$O}}\
line \t202111 or \t211202 and ortho-\htot321312 in the other five sources,
in order to compare their velocity-integrated flux densities.

\begin{table*}[htbp]
\setlength{\tabcolsep}{2.8pt}
\small
\caption{Previously observed properties of the sample.}
\centering
\begin{tabular}{cllllcccrccc}
\hline
\hline
Source          &  $z$                   & $F_{250}$                  &  $F_{350}$                 &  $F_{500}$                 & $F_{880}$                     & $r_\mathrm{half}$ & $\Sigma_{\mathit{SFR}}$                       & $f_{1.4GHz}\;\;\;$&  {\hbox {$T_{\mathrm{d}}$}}     &  $\mu$         & $\mu$\hbox {$L_{\mathrm{IR}}$}\         \\
                &                        & (mJy)                      &  (mJy)                     &  (mJy)                     & (mJy)                         & (kpc)             & (10$^{3}\,$M$_\odot$\,yr$^{-1}$\,kpc$^{-2}$)  & (mJy)\;\;\;  &  (K)    &                & (10$^{13}$\,{\hbox {$L_\odot$}}) \\
\hline
G09v1.97        &  3.634                 &  $260\pm7$                 &  $321\pm8$                 &  $269\pm9$                 & $85.5\pm4.0$                  & $0.85$     & $0.91\pm0.15$                                 &     $\pm0.15$ & $44\pm1$ &$\;\;6.9\pm0.6$ &    $15.3\pm4.3$  \\
G12v2.43        &  3.127                 &  $290\pm7$                 &  $295\pm8$                 &  $216\pm9$                 & $48.6\pm2.3$                  &     --     &            --                                 &     $\pm0.15$ &       -- &           --   &\;\;($8.3\pm1.7$) \\
NCv1.143        &  3.565                 &  $214\pm7$                 &  $291\pm8$                 &  $261\pm9$                 & $97.2\pm6.5$                  & $0.40$     & $2.08\pm0.77$                                 & $0.61\pm0.16$ & $40\pm1$ &   $11.3\pm1.7$ &    $12.8\pm4.3$  \\
NAv1.195        &  2.951                 &  $179\pm7$                 &  $279\pm8$                 &  $265\pm9$                 & $65.2\pm2.3$                  & $1.57$     & $0.21\pm0.04$                                 &     $\pm0.14$ & $36\pm1$ &$\;\;4.1\pm0.3$ & $\;\;7.4\pm2.0$  \\
NAv1.177        &  2.778                 &  $264\pm9$                 & $310\pm10$                 & $261\pm10$                 & $50.1\pm2.1$                  &     --     &            --                                 &     $\pm0.15$ &       -- &           --   &\;\;($5.5\pm1.1$) \\
NBv1.78         &  3.111                 &  $273\pm7$                 &  $282\pm8$                 &  $214\pm9$                 & $59.2\pm4.3$                  & $0.55$     & $1.09\pm1.41$                                 & $0.67\pm0.20$ & $43\pm1$ &   $13.0\pm1.5$ &    $10.7\pm3.9$  \\
\hline
G09v1.124-W$^a$ & \multirow{2}{*}{2.410} & \multirow{2}{*}{$242\pm7$} & \multirow{2}{*}{$293\pm8$} & \multirow{2}{*}{$231\pm9$} & \multirow{2}{*}{$50.0\pm3.5$} &    --      & --                                            &     $\pm0.15$ & $40\pm1$ &             1  & $\;\;3.3\pm0.3$  \\
G09v1.124-T$^a$ &                        &                            &                            &                            &                               &    --      & --                                            &     $\pm0.15$ & $36\pm1$ &    $1.5\pm0.2$ & $\;\;2.7\pm0.8$  \\
G09v1.40        &  2.089$^b$             &  $389\pm7$                 &  $381\pm8$                 &  $241\pm9$                 & $61.4\pm2.9$                  & $0.41$     & $0.77\pm0.30$                                 & $0.75\pm0.15$ & $36\pm1$ &   $15.3\pm3.5$ & $\;\;6.5\pm2.5$  \\
SDP11           &  1.786                 &  $417\pm6$                 &  $378\pm7$                 &  $232\pm8$                 & $30.6\pm2.4$                  & $0.89$     & $0.22\pm0.08$                                 & $0.66\pm0.14$ & $41\pm1$ &   $10.9\pm1.3$ & $\;\;6.2\pm1.9$  \\
NCv1.268        &  3.675                 &  $145\pm7$                 &  $201\pm8$                 &  $212\pm9$                 & $78.9\pm4.4$                  & $0.93$     & $0.31\pm0.14$                                 & $1.10\pm0.14$ & $39\pm1$ &   $11.0\pm1.0$ & $\;\;9.5\pm2.7$  \\
NAv1.56         &  2.301                 &  $481\pm9$                 & $484\pm13$                 & $344\pm11$                 & $73.1\pm2.4$                  & $1.50$     & $0.14\pm0.08$                                 & $1.12\pm0.27$ & $38\pm1$ &   $11.7\pm0.9$ &    $11.3\pm3.1$  \\
\hline
\end{tabular}
\tablefoot{
        $z$ is the redshift inferred from previous CO detection quoted by
        \citetalias{2013ApJ...779...25B} (see the references therein); $F_{250}$,
        $F_{350}$ and $F_{500}$ are the SPIRE flux densities at 250, 350 and 500{\hbox {\,$\mu m$}},
        respectively \citep{2011MNRAS.415..911P}; $F_{880}$ is the SMA flux density
		at 880{\hbox {\,$\mu m$}}; $r_\mathrm{half}$ and $\Sigma_{\rm{SFR}}$ are the intrinsic
		half-light radius at 880\,$\mu$m and the lensing-corrected surface
		$\mathit{SFR}$ (star formation rate) density (Section\,\ref{hto});
		$f_{1.4GHz}$ is the 1.4\,GHz band flux densities from the VLA FIRST survey;
		{\hbox {$T_{\mathrm{d}}$}}\ is the cold-dust temperature taken from \citetalias{2013ApJ...779...25B}
		(note that the errors quoted for {\hbox {$T_{\mathrm{d}}$}}\ are significantly underestimated
		since the uncertainties from differential lensing and single-temperature
		dust SED assumption are not fully considered); $\mu$ is the the lensing
		magnification factor from \citetalias{2013ApJ...779...25B}, except for
		G09v1.124 which is adopted from \cite{2013ApJ...772..137I};
		$\mu$\hbox {$L_{\mathrm{IR}}$}\ is the apparent total \hbox {infrared}\ luminosity mostly inferred from
		\citetalias{2013ApJ...779...25B}. The $\mu$\hbox {$L_{\mathrm{IR}}$}\ in brackets are not listed in
		\citetalias{2013ApJ...779...25B}, thus we infer them from single modified black
		body dust SED fitting using the submm photometry data listed in this table. \\
		a: The cluster source G09v1.124 includes two main components: G09v1.124-W to
		the east and G09v1.124-T to the west (Fig.\,\ref{fig:map-3}) and the values of
		these two rows are quoted from \cite{2013ApJ...772..137I};
		b: Our {\hbox {H$_{2}$O}}\ observation gives $z=2.093$ for G09v1.40. This value is slightly
		different from the value of 2.089 quoted by \citetalias{2013ApJ...779...25B}
		from Lupu et al. (in prep.) obtained by CSO/Z-Spec, but consistent with \co32
		observation by Riechers et al. (in prep.).
}
\label{table:previous_obs_properties}
\end{table*}
\normalsize

In addition, we also observed five sources mostly at lower redshifts
in para-{\hbox {H$_{2}$O}}\ lines \t202111 or \t211202 (Tables\,\ref{table:obs_log}
and \ref{table:previous_obs_properties}) to complete the sample of
our {\hbox {H$_{2}$O}}\ low-excitation study. They are three strongly lensed sources,
G09v1.40, NAv1.56 and SDP11, a hyper-luminous cluster source G09v1.124
\citep{2013ApJ...772..137I}, and a $z \sim 3.7$ source, NCv1.268 for
which we did not propose a $J=3$ {\hbox {H$_{2}$O}}\ observation, considering its
large linewidth which could bring difficulties in line detection.

As our primary goal is to obtain a detection of the submm {\hbox {H$_{2}$O}}\ lines,
we carried out the observations in the compact, D configuration of NOEMA.
The baselines extended from 24 to 176\,m, resulting in a synthesised
beam with modest/low resolution of $\sim$\,$1.0$\,$''\times0.9$\,$''$ to
$\sim$\,$5.6$\,$''\times3.3$\,$''$ as shown in Table \ref{table:obs_log}. The
{\hbox {H$_{2}$O}}\ observations were conducted from January 2012 to December 2013 in
good atmospheric conditions (seeing of $0.3$\,$''$--$1.5$\,$''$) stability and
reasonable transparency (PWV $\leq\,1\,\mathrm{mm}$). The total on source
time was $\sim$\,1.5--8 hours per source. 2\,mm, 1.3\,mm and 0.8\,mm bands
covering 129--174, 201--267 and 277--371\,GHz, respectively, were used.
All the central observation frequencies were chosen based on previous
redshifts given by \citetalias{2013ApJ...779...25B} according to the
previous CO detections (Table\,\ref{table:previous_obs_properties}).
In all cases but one, the frequencies of our detections of {\hbox {H$_{2}$O}}\
lines are consistent with these CO redshifts. The only exception is
G09v1.40 where our {\hbox {H$_{2}$O}}\ redshift disagrees with the redshift of
$z=2.0894\pm0.0009$ given by Lupu et al. (in prep.), which is quoted
by \citetalias{2013ApJ...779...25B}. We find $z=2.0925\pm0.0001$ in
agreement with previous \co32 observations (Riechers et al., in prep.).
We used the WideX correlator which provided a contiguous frequency
coverage of 3.6\,GHz in dual polarisation with a fixed channel
spacing of 1.95\,MHz.

The phase and bandpass were calibrated by measuring standard calibrators
that are regularly monitored at the IRAM/NOEMA, including 3C279, 3C273,
MWC349 and 0923+392. The accuracy of the flux calibration is estimated
to range from $\sim$10\% in the 2\,mm band to $\sim$20\% in the 0.8mm
band. Calibration, imaging, cleaning and spectra extraction were
performed within the
\texttt{GILDAS}\footnote{See \url{http://www.iram.fr/IRAMFR/GILDAS}
for more information about the GILDAS softwares.} packages
\texttt{CLIC} and \texttt{MAPPING}.

\begin{table}[htbp]
\small
\setlength{\tabcolsep}{1.45em}
\caption{Observed CO line properties using the IRAM 30m/EMIR.}
\centering
\begin{tabular}{lcccc}
\hline
\hline
Source     & CO line  & \hbox {$I_{\mathrm{CO}}$}               & \hbox {$\Delta V_\mathrm{CO}$}           \\
           &          & (Jy\,km\,s$^{-1}$) & (km\,s$^{-1}$) \\
\hline
G09v1.97   & \tco54   & $\;\;9.5\pm1.2$    & $224\pm 32$    \\
           & \tco65   & $10.4\pm2.3$       & $292\pm 86$    \\
NCv1.143   & \tco54   & $13.1\pm1.0$       & $273\pm 27$    \\
           & \tco65   & $11.0\pm1.0$       & $284\pm 27$    \\
NAv1.195   & \tco54   & $11.0\pm0.6$       & $281\pm 16$    \\
NAv1.177   & \tco32   & $\;\;6.8\pm0.4$    & $231\pm 15$    \\
           & \tco54   & $11.0\pm0.6$       & $230\pm 16$    \\
NBv1.78    & \tco54   & $10.3\pm0.8$       & $614\pm 53$    \\
           & \tco65   & $\;\;9.7\pm1.0$    & $734\pm 85$    \\
G09v1.40   & \tco43   & $\;\;7.5\pm2.1$    & $198\pm 51$    \\
NAv1.56    & \tco54   & $17.7\pm6.6$       & $\;\;432\pm 182$ \\
\hline
\end{tabular}
\tablefoot{
	       \hbox {$I_{\mathrm{CO}}$}\ is the velocity-integrated flux density of CO; \hbox {$\Delta V_\mathrm{CO}$}\
		   is the linewidth (FWHM) derived from fitting a
		   single Gaussian to the line profile.
           }
\label{table:co_properties}
\end{table}
\normalsize

To compare the {\hbox {H$_{2}$O}}\ emission with the typical molecular gas tracer, CO, we
also observed the sources for CO lines using the EMIR receiver at the IRAM
30m telescope. The CO data will be part of a systematic study of molecular
gas excitation in {\it H}-ATLAS lensed Hy/ULIRGs, and a full description of
the data and the scientific results will be given in a following paper
(Yang et al., in prep.). The global CO emission properties of the sources
are listed in Table\,\ref{table:co_properties} where we list the CO fluxes
and linewidths. A brief comparison of the emission between {\hbox {H$_{2}$O}}\ and CO
lines will be given in Section \ref{CO lines}.

\section{Results}\label{Results}

A detailed discussion of the observation results for each source is given
in Appendix\,\ref{Individual sources}, including the strength of the {\hbox {H$_{2}$O}}\
emission, the image extension of {\hbox {H$_{2}$O}}\ lines and the continuum
(Fig.\,\ref{fig:map-all}), the {\hbox {H$_{2}$O}}\ spectra and linewidths
(Fig.\,\ref{fig:spectra-all}) and their comparison with CO
(Table\,\ref{table:co_properties}). We give a synthesis of these results
in this section.

\subsection{General properties of the {\hbox {H$_{2}$O}}\ emissions}
\label{General properties}

 \begin{subfigures}
 \begin{figure*}[htbp]
	 \begin{center}
 \includegraphics[scale=0.85]{spec-1}
 \caption{
		  Spatially integrated spectra of {\hbox {H$_{2}$O}}\ in the six sources
		  with both $J=2$ para-{\hbox {H$_{2}$O}}\ and $J=3$ ortho-{\hbox {H$_{2}$O}}\ lines observed.
		  The red lines represent the Gaussian fitting to the emission
		  lines. The \htot202111 spectrum of NBv1.78 is taken from
		  \citetalias{2013A&A...551A.115O}. Except for \htot321312 in NAv1.195,
		  all the $J=2$ and $J=3$ {\hbox {H$_{2}$O}}\ lines are well detected, with a high
		  S/N ratio and similar profiles in both lines for the same source.}
 \label{fig:spectra-1}
     \end{center}
 \end{figure*}

 \begin{figure*}[htbp]
	 \begin{center}
 \includegraphics[scale=0.97]{spec-2}
 \caption{
 		  Spatially integrated spectra of {\hbox {H$_{2}$O}}\ of the five sources
 		  with only one $J=2$ para-{\hbox {H$_{2}$O}}\ line observed. The red
 		  lines represent the Gaussian fitting to the emission lines.
 		  Except for the {\hbox {H$_{2}$O}}\ line in G09v1.124,
 		  all the $J=2$ {\hbox {H$_{2}$O}}\ lines are well detected.
		 }
 \label{fig:spectra-2}
     \end{center}
 \end{figure*}
 \label{fig:spectra-all}
\end{subfigures}

\begin{table*}[htbp]
\small
\setlength{\tabcolsep}{4.5pt}
\caption{The observed properties of {\hbox {H$_{2}$O}}\ emission lines.}
\centering
\begin{tabular}{cccrrrrrrrr}
\hline
\hline
Source     & {\hbox {H$_{2}$O}}\ line & $\nu_\mathrm{H_2O}$ & $S_{\nu}(\mathrm{ct})^{\mathrm{pk}}\:\:\:$ & $S_{\nu}(\mathrm{ct})\:\:\:\:\:$ & $S_\mathrm{H_2O}^{\mathrm{pk}}\:\:\:\:$ & $S_\mathrm{H_2O}\:\:\:\:$& \hbox {$I_{\mathrm{H_2O}}$}$^{\mathrm{pk}}\:\:\:$           & \hbox {$I_{\mathrm{H_2O}}$}\:\:\:\:\:\:    & \hbox {$\Delta V_\mathrm{H_2O}$}\:        & $\mu$\hbox {$L_{\mathrm{H_2O}}$}\:\:\:\: \\
           &            & (GHz)               & ($\mathrm{mJy} \over \mathrm{beam}$)\:\:\:\:\:                   &  (mJy)\:\:\:\:\:               & ($\mathrm{mJy} \over \mathrm{beam}$)\:\:\:\:              & (mJy)\:\:\:\:  & ($\mathrm{Jy\,km\,s}^{-1} \over \mathrm{beam}$) & (Jy\,km\,s$^{-1}$) & (km\,s$^{-1}$) & (10$^8$\,{\hbox {$L_\odot$}})\:\:\:\\
\hline
G09v1.97   & \t211202   & 162.255             & $ 8.9\pm0.2$                       & $ 9.4\pm0.2$         & $14.9\pm2.2$                    &  $15.0\pm2.1$    & $ 3.8\pm0.4$                  & $ 4.1\pm0.4$     & $257\pm27$     & $ 7.4\pm0.7$  \\
           & \t321312   & 250.947             & $21.7\pm0.3$                       & $36.1\pm0.3$         & $ 7.8\pm1.9$                    &  $15.0\pm2.6$    & $ 2.4\pm0.4$                  & $ 3.7\pm0.4$     & $234\pm34$     & $10.4\pm1.0$  \\
G12v2.43   & \t202111   & 239.388             & $16.0\pm0.3$                       & $22.5\pm0.4$         & $10.8\pm2.1$                    &  $17.3\pm3.1$    & $ 3.2\pm0.5$                  & $ 4.8\pm0.6$     & $262\pm35$     & $ 8.8\pm1.0$  \\
           & \t321312   & 281.784             & $31.5\pm0.3$                       & $36.4\pm0.3$         & $25.6\pm3.3$                    &  $25.0\pm3.0$    & $ 4.9\pm0.4$                  & $ 5.9\pm0.5$     & $221\pm20$     & $12.7\pm1.0$  \\
NCv1.143   & \t211202   & 164.741             & $11.2\pm0.1$                       & $13.3\pm0.2$         & $17.4\pm1.3$                    &  $18.7\pm1.3$    & $ 5.6\pm0.3$                  & $ 5.8\pm0.3$     & $293\pm15$     & $10.1\pm0.5$  \\
           & \t321312   & 254.739             & $34.8\pm0.5$                       & $63.5\pm0.5$         & $23.9\pm4.3$                    &  $32.1\pm4.1$    & $ 5.2\pm0.6$                  & $ 8.0\pm0.7$     & $233\pm22$     & $21.3\pm1.8$  \\
NAv1.195   & \t202111   & 250.034             & $14.0\pm0.4$                       & $25.8\pm0.4$         & $ 6.6\pm2.5$                    &  $11.6\pm2.5$    & $ 2.1\pm0.6$                  & $ 4.0\pm0.6$     & $328\pm51$     & $ 6.7\pm1.0$  \\
           & \t321312   & (293.334)           & $17.2\pm0.5$                       & $41.2\pm0.5$         & $      <4.2$                    &  $      <7.3$    & $       <1.5$                 & $      <2.6$     & $ 330^{a}$     & $      <5.0$  \\
NAv1.177   & \t202111   & 261.489             & $26.5\pm0.6$                       & $35.5\pm0.6$         & $16.8\pm4.9$                    &  $21.2\pm4.9$    & $ 4.4\pm0.9$                  & $ 5.4\pm0.9$     & $241\pm41$     & $ 8.2\pm1.2$  \\
           & \t321312   & 307.856             & $38.2\pm0.4$                       & $62.0\pm0.4$         & $14.8\pm2.6$                    &  $25.2\pm3.1$    & $ 4.6\pm0.5$                  & $ 7.3\pm0.6$     & $272\pm24$     & $12.9\pm1.1$  \\
NBv1.78    & \;\,\t202111$^{b}$  & 240.290    & $15.4\pm0.3$                   & $36.9\pm0.4$     & $ 5.0\pm1.0$                    &  $12.3\pm3.2$    & $2.7\pm0.3$               & $6.7\pm1.3$  & $510\pm90$     & $12.2\pm2.4$\\
           & \t321312   & 282.863             & $29.2\pm0.2$                       & $42.6\pm0.2$         & $ 8.8\pm1.0$                    &  $10.6\pm1.0$    & $ 4.8\pm0.4$                  & $ 6.7\pm0.5$     & $607\pm43$     & $14.3\pm1.0$  \\
\hline
G09v1.124-W & \multirow{2}{*}{\t211202} & \multirow{2}{*}{(220.537)} & $6.42\pm0.15$& $7.6\pm0.2$         & $      <1.4$                    &  $      <1.6$    & $   <1.2^{c}$                 & $   <1.4^{c}$     & $850^{c}$      & $<1.3^{c}$    \\
G09v1.124-T &                           &                            & $4.08\pm0.15$& $4.9\pm0.2$         & $      <1.7$                    &  $      <2.0$    & $   <1.0^{c}$                 & $   <1.2^{c}$     & $550^{c}$      & $<1.0^{c}$    \\
G09v1.40   & \t211202   & 243.182             & $16.9\pm0.2$                       & $30.6\pm0.3$         & $17.5\pm2.0$                    &  $27.7\pm1.9$    & $ 4.9\pm0.4$                  & $ 8.2\pm0.4$      & $277\pm14$     & $ 5.7\pm0.3$  \\
SDP11      & \t202111   & 354.930             & $29.2\pm1.3$                       & $52.1\pm1.3$         & $14.8\pm8.4$                    &  $40.3\pm11.7$   & $ 5.2\pm2.0$                  & $ 9.2\pm2.0$      & $214\pm41$     & $ 6.3\pm1.1$  \\
NCv1.268   & \t211202   & 161.013             & $ 6.6\pm0.1$                       & $10.0\pm0.1$         & $ 5.2\pm1.1$                    &  $ 9.0\pm1.2$    & $ 3.7\pm0.4$                  & $ 7.0\pm0.7$      & $731\pm75$     & $12.8\pm1.2$  \\
NAv1.56    & \t211202   & 227.822             & $14.0\pm0.6$                       & $22.7\pm0.6$         & $15.8\pm3.3$                    &  $23.2\pm3.0$    & $ 7.8\pm1.1$                  & $14.6\pm1.3$      & $593\pm56$     & $12.0\pm1.1$  \\

\hline
\end{tabular}
\tablefoot{
		 $\nu_\mathrm{H_2O}$ is the observed central frequency of {\hbox {H$_{2}$O}}\
		 lines, and the values in brackets are the {\hbox {H$_{2}$O}}\ line frequencies
		 inferred from the CO redshifts for the undetected sources;
		 $S_{\nu}(ct)^{\mathrm{pk}}$ and $S_{\nu}(ct)$ are the peak and
		 spatially integrated continuum flux density, respectively;
		 $S_\mathrm{H_2O}^{\mathrm{pk}}$ is the peak {\hbox {H$_{2}$O}}\ line flux and
		 $S_\mathrm{H_2O}$ is the total line flux; \hbox {$I_{\mathrm{H_2O}}$}$^{\mathrm{pk}}$
		 and \hbox {$I_{\mathrm{H_2O}}$}\ are the peak and spatially integrated velocity-integrated
		 flux density of the {\hbox {H$_{2}$O}}\ lines; \hbox {$\Delta V_\mathrm{H_2O}$}\ is the {\hbox {H$_{2}$O}}\ linewidth;
		 $\mu$\hbox {$L_{\mathrm{H_2O}}$}\ is the apparent luminosity of the observed {\hbox {H$_{2}$O}}\ line. \\
		 a: The linewidth of the undetected \htot321312 in NAv1.195 has
		    been set to 330\,km\,s$^{-1}$ by assuming that the widths of
		    the \htot321312 and \htot202111 lines are roughly the same;
		 b: The data of para-\htot202111 in NBv1.78 is taken from
		    \citetalias{2013A&A...551A.115O};
		 c: the 2\,$\sigma$ upper limits of \hbox {$I_{\mathrm{H_2O}}$}\ are derived by
		    assuming that the {\hbox {H$_{2}$O}}\ linewidths are similar to those
		    of the CO lines \citep{2013ApJ...772..137I}.\\
 }
\label{table:h2o_properties}
\end{table*}
\normalsize

To measure the linewidth, velocity-integrated flux density and the
continuum level of the spectra from the source peak and from the
entire source, we extract each spectrum from the CLEANed image at the
position of the source peak in a single synthesis beam and the
spectrum integrated over the entire source. Then we fit them with
Gaussian profiles using \texttt{MPFIT} \citep{2009ASPC..411..251M}.

We detect the high-excitation ortho-\htot321312 in five out of six observed
sources, with high signal to noise ratios ($S/N\;>9$) and velocity-integrated
flux densities comparable to those of the low-excitation $J=2$ para-{\hbox {H$_{2}$O}}\ lines
(Table \ref{table:h2o_properties} and Figs.\,\ref{fig:spectra-all} \& \ref{fig:map-all}).
We also detect nine out of eleven $J=2$ para-{\hbox {H$_{2}$O}}\ lines, either \t202111 or
\t211202, with $S/N\,\ge\,6$ in terms of their velocity-integrated flux density,
plus one
tentative detection of \htot202111 in SDP11. We present the values of
velocity-integrated {\hbox {H$_{2}$O}}\ flux density detected at the source peak in a single synthesised beam,
\hbox {$I_{\mathrm{H_2O}}$}$^{\mathrm{pk}}$, and the velocity-integrated {\hbox {H$_{2}$O}}\ flux density over the entire
source, $I_{\mathrm{H_2O}}$ (Table\,\ref{table:h2o_properties}). The detected
{\hbox {H$_{2}$O}}\ lines are strong, with \hbox {$I_{\mathrm{H_2O}}$}\;$= 3.7\text{--}14.6$\,Jy\,km\,s$^{-1}$.
Even considering gravitational lensing correction, this is consistent with our
previous finding that \hbox {high-redshift}\ Hy/ULIRGs are very strong {\hbox {H$_{2}$O}}\ emitters, with {\hbox {H$_{2}$O}}\
flux density approaching that of CO (Tables\,\ref{table:co_properties} \&
\ref{table:h2o_properties} and Section\,\ref{CO lines}). The majority of the
images (7/11 for $J=2$ lines and 3/4 for $J=3$) are marginally resolved with
\hbox {$I_{\mathrm{H_2O}}$}$^{\mathrm{pk}}$/$I_{\mathrm{H_2O}} \sim 0.4\text{--}0.7$. They show
somewhat lensed structures. The others are unresolved with
\hbox {$I_{\mathrm{H_2O}}$}$^{\mathrm{pk}}$/$I_{\mathrm{H_2O}} > 0.8$. All continuum emission
flux densities ($S_{\nu}(\mathrm{ct})^{\mathrm{pk}}$ for the emission peak
and $S_{\nu}(\mathrm{ct})$ for the entire source) are very well detected
($S/N \ge 30$), with a range of total flux density of 9--64\,mJy for
$S_{\nu}(\mathrm{ct})$.
Fig.\,\ref{fig:map-all} shows the low-resolution images of {\hbox {H$_{2}$O}}\ and
the corresponding dust continuum emission at the observing frequencies.
Because the positions of the sources were derived from {\it Herschel}
observation, which has a large beamsize ($>17$\,$''$) comparing to the
source size, the position of most of the sources are not perfectly centred
at these {\it Herschel} positions as seen in the maps. The offsets are
all within the position error of the {\it Herschel} measurement
(Fig.\,\ref{fig:map-all}). G09v1.124 is a complex HyLIRG system
including two main components eastern G09v1.124-W and western
G09v1.124-T as described in \cite{2013ApJ...772..137I}.
In Fig.\,\ref{fig:map-3}, we identified the two strong components
separated about 10$''$, in agreement with \citet{2013ApJ...772..137I}.
The $J=2$ {\hbox {H$_{2}$O}}\ and dust continuum emissions in NBv1.78, NCv1.195,
G09v1.40, SDP\,11 and NAv1.56, as well as the $J=3$ ortho-{\hbox {H$_{2}$O}}\ and
the corresponding dust continuum emissions in G09v1.97, NCv1.143 and
NAv1.177, are marginally resolved as shown in Fig.\,\ref{fig:map-all}.
Their images are consistent with the corresponding SMA images
\citepalias{2013ApJ...779...25B} in terms of their spatial distribution.
The rest of the sources are not resolved by the low-resolution
synthesised beams. The morphological structure of the {\hbox {H$_{2}$O}}\ emission
is similar to the continuum for most sources as shown in
Fig.\,\ref{fig:map-all}.
The ratio $S_{\nu}(\mathrm{ct})^{\mathrm{pk}}$/$S_{\nu}(\mathrm{ct})$
and $S_{\nu}(\mathrm{H_2O})^{\mathrm{pk}}$/$S_{\nu}(\mathrm{H_2O})$
are in good agreement within the error. However, for NCv1.143 in which
$S_{\nu}(\mathrm{ct})^{\mathrm{pk}}$/$S_{\nu}(\mathrm{ct})=0.55\pm0.01$
and $S_{\nu}(\mathrm{H_2O})^{\mathrm{pk}}$/$S_{\nu}(\mathrm{H_2O})=0.74\pm0.16$,
the $J=3$ ortho-{\hbox {H$_{2}$O}}\ emission appears more compact than the dust continuum.
Generally it seems unlikely that we have a significant fraction of missing
flux for our sources. Nevertheless, the low angular resolution ($\sim$\,$1''$ at best)
limits the study of spatial distribution of the gas and dust in our
sources. A detailed analysis of the images for each source is given
in Appendix\,\ref{Individual sources}.

The majority of the sources have {\hbox {H$_{2}$O}}\ (and CO) linewidths between 210
and 330\,km\,s$^{-1}$, while the four others range between 500 and
700\,km\,s$^{-1}$ (Table\,\ref{table:h2o_properties}). Except NCv1.268,
which shows a double-peaked line profile, all {\hbox {H$_{2}$O}}\ lines are well fit
by a single Gaussian profile (Fig.\,\ref{fig:spectra-all}). The line
profiles between the $J=2$ and $J=3$ {\hbox {H$_{2}$O}}\ lines do not seem to be
significantly different, as shown from the linewidth ratios ranging
from $1.26\pm0.14$ to $0.84\pm0.16$. The magnification from strong
lensing is very sensitive to the spatial configuration, in other words,
differential lensing, which could lead to different line profiles if the different
velocity components of the line are emitted at different spatial positions.
Since there is no visible differential effect between their profiles,
it is possible that the $J=2$ and $J=3$ {\hbox {H$_{2}$O}}\ lines are from similar
spatial regions.

In addition to {\hbox {H$_{2}$O}}, within the 3.6\,GHz WideX band, we have also
tentatively detected {\hbox {H$_2$O$^+$}}\ emission in 3 sources: NCv1.143, G09v1.97
and G15v2.779 (see Section \ref{htop}).

\subsection{Lensing properties}
\label{Lensing properties}
All our sources are strongly gravitationally lensed (except G09v1.124,
see Appendix\,\ref{g09v1.124}), which increases the line flux densities and
allows us to study the {\hbox {H$_{2}$O}}\ emission in an affordable amount of observation
time. However, the complexity of the lensed images complicates the analysis.
As mentioned above, most of our lensed images are either unresolved or
marginally resolved. Thus, we will not discuss here the spatial distribution
of the {\hbox {H$_{2}$O}}\ and dust emissions through gravitational lensing modelling.
However, we should keep in mind that the correction of the magnification
is a crucial part of our study. In addition, differential lensing could have
a significant influence when comparing {\hbox {H$_{2}$O}}\ emission with dust and even
comparing different transitions of same molecular species
\citep{2012MNRAS.424.2429S}, especially for the emission from close
to the caustics.

In order to infer the intrinsic properties of our sample, especially
\hbox {$L_{\mathrm{H_2O}}$}\ as in our first paper \citetalias{2013A&A...551A.115O}, we adopted
the lensing magnification factors $\mu$
(Table\,\ref{table:previous_obs_properties}) computed from the modelling
of the 880\,$\mu$m SMA images \citepalias{2013ApJ...779...25B}. As shown
in the Appendix, the ratio of
$S_{\nu}(\mathrm{ct})^{\mathrm{pk}}$/$S_{\nu}(\mathrm{ct})$
and $S_{\nu}(\mathrm{H_2O})^{\mathrm{pk}}$/$S_{\nu}(\mathrm{H_2O})$
are in good agreement within the uncertainties. Therefore, it is unlikely
that the magnification of the 880\,$\mu$m continuum image and {\hbox {H$_{2}$O}}\
can be significantly different. However, \citetalias{2013ApJ...779...25B}
were unable to provide a lensing model for two of our sources, G12v2.43
and NAv1.177, because their lens deflector is unidentified.
This does not affect the modelling of {\hbox {H$_{2}$O}}\ excitation and the
comparison of {\hbox {H$_{2}$O}}\ and \hbox {infrared}\ luminosities since the differential
lensing effect seems to be insignificant as discussed in
Sections\,\ref{Discussion} and Appendix\,\ref{Individual sources}.

\begin{table*}[!htbp]
\small
\setlength{\tabcolsep}{1.5em}
\caption{IR luminosity, {\hbox {H$_{2}$O}}\ line luminosity and global dust temperature of the entire sample.}
\centering
\begin{tabular}{cllccc}
\hline
\hline
Source     & {\hbox {H$_{2}$O}}\ Transition   & \hbox {$L_{\mathrm{IR}}$}\               & \lhtot211202       & \lhtot202111        & \lhtot321312         \\
           &                    & ($10^{12}$\,{\hbox {$L_\odot$}})  & ($10^{7}$\,{\hbox {$L_\odot$}})  & ($10^{7}$\,{\hbox {$L_\odot$}})   & ($10^{7}$\,{\hbox {$L_\odot$}})    \\
\hline
G09v1.97   & \t211202, \t321312 & $22.1\pm5.9$      & $10.7\pm1.4$     & --                  & $ 15.0\pm1.9$      \\
G12v2.43   & \t202111, \t321312 & $83.2\pm16.6/\mu$ & --                 & $88.4\pm10.7/\mu$ & $143.2\pm11.5/\mu$ \\
NCv1.143   & \t211202, \t321312 & $11.4\pm3.1$      & $ 9.0\pm1.4$     & --                  & $ 18.9\pm3.3$      \\
NAv1.195   & \t202111, \t321312 & $18.0\pm4.6$      & --                 & $16.4\pm3.0$      & $       <12.3$      \\
NAv1.177   & \t202111, \t321312 & $55.0\pm11.0/\mu$ & --                 & $82.0\pm12.8/\mu$ & $129.1\pm10.8/\mu$ \\
NBv1.78    & \t202111, \t321312 & $\;\;8.2\pm2.2$   & --                 & $ 9.4\pm2.1$      & $11.0\pm1.5$       \\
\hline
G09v1.124-W & \t211202          & $33.1\pm3.2$      & $<12.9$           & --                  & --                   \\
G09v1.124-T & \t211202          & $14.5\pm1.8$      & $<6.9$            & --                  & --                   \\
G09v1.40   & \t211202           & $\;\;4.2\pm1.3$   & $ 3.7\pm0.9$      & --                  & --                   \\
SDP11      & \t202111           & $\;\;5.7\pm1.6$   & --                 & $ 5.8\pm1.4$      & --                   \\
NCv1.268   & \t211202           & $\;\;8.6\pm2.3$   & $11.5\pm1.5$     & --                  & --                   \\
NAv1.56    & \t211202           & $\;\;9.7\pm2.6$   & $10.3\pm1.2$     & --                  & --                   \\
\hline
SDP81      & \t202111           & \;\;6.1             & --                 & 3.3                 & --                   \\
NAv1.144   & \t211202           & 11                  &  9.7               & --                  & --                   \\
SDP9       & \t211202           & \;\;5.2             &  7.0               & --                  & --                   \\
G12v2.30   & \t202111           & 16                  & --                 & 13                  & --                   \\
SDP17b     & \t202111           & 16                  & --                 & 20                  & --                   \\
G15v2.779  & \t211202           & 21                  &  26.6              & --                  & --                   \\
\hline
\end{tabular}
\tablefoot{
        $L_\mathrm{IR}$ is the intrinsic total \hbox {infrared}\ luminosity (8-1000{\hbox {\,$\mu m$}}) taken from
		\citetalias{2013ApJ...779...25B}. The intrinsic {\hbox {H$_{2}$O}}\ luminosities are
		inferred from $\mu$\hbox {$L_{\mathrm{H_2O}}$}\ using $\mu$ in \citetalias{2013ApJ...779...25B}. The first
		group of the sources are the ones with both $J=2$ and $J=3$ {\hbox {H$_{2}$O}}\ lines observed, the
		next group are the sources with only $J=2$ {\hbox {H$_{2}$O}}\ observed, and the last group are the
		previous published sources in \citetalias{2013A&A...551A.115O}.
		}
\label{table:Lir_Lh2o}
\end{table*}
\normalsize

\section{Discussion}
\label{Discussion}

\subsection{\hbox {$L_{\mathrm{H_2O}}$}-\hbox {$L_{\mathrm{IR}}$}\ correlation and \hbox {$L_{\mathrm{H_2O}}$}/\hbox {$L_{\mathrm{IR}}$}\ ratio}
\label{correlation}

Using the formula given by \cite{1992ApJ...387L..55S}, we derive the
apparent {\hbox {H$_{2}$O}}\ luminosities of the sources, $\mu$\hbox {$L_{\mathrm{H_2O}}$}\ (Table\,
\ref{table:h2o_properties}), from \hbox {$I_{\mathrm{H_2O}}$}. For the ortho-\htot321312
lines, $\mu$\hbox {$L_{\mathrm{H_2O}}$}\ varies in the range of $6\text{--}22\times10^{8}$\,{\hbox {$L_\odot$}},
while the $\mu$\hbox {$L_{\mathrm{H_2O}}$}\ of the $J=2$ lines are a factor $\sim1.2\text{--}2$
weaker (Table\,\ref{table:h2o_properties}) as discussed in Section\,\ref{hto}.

Using the lensing magnification correction (taking the values of $\mu$
from \citetalias{2013ApJ...779...25B}), we have derived the intrinsic {\hbox {H$_{2}$O}}\
luminosities (Table\,\ref{table:Lir_Lh2o}). The error of each luminosity
consists of the uncertainty from both observation and the gravitational
lensing modelling. After correcting for lensing, the {\hbox {H$_{2}$O}}\ luminosities
of our \hbox {high-redshift}\ galaxies appear to be one order of magnitude higher than those of
local ULIRGs, as well as their \hbox {infrared}\ luminosities (Table\,\ref{table:Lir_Lh2o}),
so that many of them should rather be considered as HyLIRGs than ULIRGs.
Though the ratio of \hbox {$L_{\mathrm{H_2O}}$}/\hbox {$L_{\mathrm{IR}}$}\ in our \hbox {high-redshift}\ sample is close to that of
local ULIRGs \citepalias{2013ApJ...771L..24Y}, with somewhat a statistical
increase in the extreme high \hbox {$L_{\mathrm{IR}}$}\ end (Fig.\,\ref{fig:h2o-ir}).

As displayed in Fig.\,\ref{fig:h2o-ir} for {\hbox {H$_{2}$O}}\ of the three observed
lines, because we have extended the number of detections to 21 {\hbox {H$_{2}$O}}\
lines, distributed in 16 sources and 3 transitions, we may independently
study the correlation of \lhtot202111 and \lhtot211202 with \hbox {$L_{\mathrm{IR}}$}, while we
had approximately combined the two lines in \citetalias{2013A&A...551A.115O}.

 \begin{figure*}[htbp]
	 \begin{center}
 \includegraphics[scale=0.62]{L_L_all}
 \caption{
		  Correlation between \hbox {$L_{\mathrm{IR}}$}\ and \hbox {$L_{\mathrm{H_2O}}$}\ in local ULIRGs and \hbox {high-redshift}\ Hy/ULIRGs.
		  The black points represent local ULIRGs from \citetalias{2013ApJ...771L..24Y}.
		  The blue points with solid error bars are the {\it H}-ATLAS source in
		  this work together with some previously published sources. Red points
		  with dashed error bars are excluded from the fit as described in the text.
		  Upper limits are shown in arrows. The light blue lines show the results
		  of the fitting. The insets are the probability density distributions of
		  the fitted slopes $\alpha$. We find tight correlations between the
		  luminosity of the three {\hbox {H$_{2}$O}}\ lines and \hbox {$L_{\mathrm{IR}}$}, namely \hbox {$L_{\mathrm{H_2O}}$}\;$\propto$\;\hbox {$L_{\mathrm{IR}}$}$^{1.1-1.2}$.
         }
 \label{fig:h2o-ir}
     \end{center}
 \end{figure*}

As found in \citetalias{2013A&A...551A.115O}, the correlation is slightly
steeper than linear (\hbox {$L_{\mathrm{H_2O}}$}\;$\sim$\;\hbox {$L_{\mathrm{IR}}$}$^{1.2}$). To broaden the dynamical
range of this comparison, we also included the local ULIRGs from
\citetalias{2013ApJ...771L..24Y}, together with a few other {\hbox {H$_{2}$O}}\ detections
in \hbox {high-redshift}\ Hy/ULIRGs, for example, HLSJ\,0918 (HLSJ\,091828.6+514223)
\citep{2012A&A...538L...4C, 2014ApJ...783...59R}, APM\,08279
\citep{2011ApJ...741L..38V}, SPT\,0538 (SPT-S\,J053816−5030.8)
\citep{2013ApJ...779...67B} and HFLS3 (\citealt{2013Natur.496..329R},
with the magnification factor from \citealt{2014ApJ...790...40C})
(Fig.\,\ref{fig:h2o-ir}). In the fitting,
however, we excluded the sources with heavy AGN contamination (Mrk\,231 and
APM\,08279) or missing flux resolved out by the interferometry (SDP\,81). We
also excluded the \htot321312 line of HFLS3 considering its unusual high
\lhtot321312/\hbox {$L_{\mathrm{IR}}$}\ ratio as discussed above, that could bias our fitting.
We have performed a linear regression in log-log space using the
Metropolis-Hastings Markov Chain Monte Carlo (MCMC) algorithm sampler
through {\texttt{linmix\_err}} \citep{2007ApJ...665.1489K} to derived
the $\alpha$ in
\begin{equation}
	\label{eq:L_L}
L_\mathrm{H_{2}O} \propto L_\mathrm{IR}^\alpha.
\end{equation}
The fitted parameters are $\alpha=1.06\pm0.19$, $1.16\pm0.13$ and
$1.06\pm0.22$ for {\hbox {H$_{2}$O}}\ line \t202111, \t211202 and \t321312, respectively.
Comparing with the local ULIRGs, the \hbox {high-redshift}\ lensed ones have higher \hbox {$L_{\mathrm{H_2O}}$}/\hbox {$L_{\mathrm{IR}}$}\
ratios (Table\,\ref{table:Lir_Lh2o_ratios}). These slopes confirm our first
result derived from 7 {\hbox {H$_{2}$O}}\ detections in \citepalias{2013A&A...551A.115O}.
The slight super-linear correlations seem to indicate that \hbox {far-infrared}\ pumping
play an important role in the excitation of the submm {\hbox {H$_{2}$O}}\ emission. This
is unlike the high-$J$ CO lines, which are determined by collisional
excitation and follow the linear correlation between the CO line luminosity
and \hbox {$L_{\mathrm{IR}}$}\ from the local to the \hbox {high-redshift}\ Universe \citep{2015ApJ...810L..14L}.
As demonstrated in \citetalias{2014A&A...567A..91G}, using the \hbox {far-infrared}\ pumping
model, the steeper than linear growth of \hbox {$L_{\mathrm{H_2O}}$}\ with \hbox {$L_{\mathrm{IR}}$}\ can be the result
of an increasing optical depth at 100\,$\mu$m ($\tau_{100}$) with increasing
\hbox {$L_{\mathrm{IR}}$}. In local ULIRGs, the ratio of \hbox {$L_{\mathrm{H_2O}}$}/\hbox {$L_{\mathrm{IR}}$}\ is relatively low while most
of them are likely to be optically thin
\citepalias[$\tau_{100}\sim0.1$,][]{2014A&A...567A..91G}. On the other hand,
for the \hbox {high-redshift}\ lensed Hy/ULIRGs with high values of \hbox {$L_{\mathrm{IR}}$}, the continuum optical
depth at \hbox {far-infrared}\ wavelengths is expected to be high (see Section\,\ref{hto}),
indicating that the {\hbox {H$_{2}$O}}\ emission comes from very dense regions of molecular
gas that are heavily obscured.

\begin{table*}[!htbp]
\small
\setlength{\tabcolsep}{0.88em}
\caption{Ratio between \hbox {infrared}\ and {\hbox {H$_{2}$O}}\ luminosity, and the velocity-integrated flux density ratio between different {\hbox {H$_{2}$O}}\ transitions.}
\centering
\begin{tabular}{clcccccc}
\hline
\hline
Source     & {\hbox {H$_{2}$O}}\ Transition   & {\hbox {$T_{\mathrm{d}}$}}       & \lhtotlir211202    & \lhtotlir202111    & \lhtotlir321312    & \ihtotihto211202 & \ihtotihto202111 \\
           &                    & (K)       & ($\times 10^{-6}$) & ($\times 10^{-6}$) & ($\times 10^{-6}$) &                  &                  \\
\hline
G09v1.97   & \t211202, \t321312 & $44\pm1$  & $ 4.8\pm1.4$       & --                 & $ 6.8\pm2.0$       & $0.9\pm0.1$      & ($0.8\pm0.2$)    \\
G12v2.43   & \t202111, \t321312 &($39\pm2$) & --                 & $10.6\pm2.5$       & $15.3\pm3.3$       & --               & $1.2\pm0.2$      \\
NCv1.143   & \t211202, \t321312 & $40\pm1$  & $ 7.9\pm2.5$       & --                 & $16.6\pm5.4$       & $1.4\pm0.1$      & ($1.1\pm0.4$)    \\
NAv1.195   & \t202111, \t321312 & $36\pm1$  & --                 & $ 9.1\pm2.9$       & $       <6.8$      & --               & $      <0.7$     \\
NAv1.177   & \t202111, \t321312 &($32\pm1$) & --                 & $14.9\pm3.8$       & $23.5\pm5.1$       & --               & $1.3\pm0.2$      \\
NBv1.78    & \t202111, \t321312 & $43\pm1$  & --                 & $11.4\pm4.7$       & $13.4\pm4.9$       & --               & $1.0\pm0.2$      \\
G09v1.124-W & \t211202          & $40\pm1$  & $<3.9$             & --                 & --                 & --               & --               \\
G09v1.124-T & \t211202          & $36\pm1$  & $<4.8$             & --                 & --                 & --               & --               \\
G09v1.40   & \t211202           & $36\pm1$  & $ 8.8\pm3.5$       & --                 & --                 & --               & --               \\
SDP11      & \t202111           & $41\pm1$  & --                 & $10.2\pm3.8$       & --                 & --               & --               \\
NCv1.268   & \t211202           & $39\pm1$  & $13.4\pm3.9$       & --                 & --                 & --               & --               \\
NAv1.56    & \t211202           & $38\pm1$  & $10.7\pm3.1$       & --                 & --                 & --               &                  \\
\hline
SDP81      & \t202111           & $34\pm1$  & --                 & $ 5.4$             & --                 & --               & --               \\
NAv1.144   & \t211202           & $39\pm1$  & $ 9.7$             & --                 & --                 & --               & --               \\
SDP9       & \t211202           & $43\pm1$  & $13.5$             & --                 & --                 & --               & --               \\
G12v2.30   & \t202111           & $41\pm1$  & --                 & $ 8.1$             & --                 & --               & --               \\
SDP17b     & \t202111           & $38\pm1$  & --                 & $12.5$             & --                 & --               & --               \\
G15v2.779  & \t211202           & $41\pm1$  &  $7.7$             & --                 & --                 & --               & --               \\
\hline
HFLS3                      & \t202111, \t211202, \t321312 & $56^{+9}_{-12}$ & $20.3$  & $22.2$   & $57.3$             & $1.8\pm0.6$      & $2.2\pm0.5$      \\
APM\,08279                 & \t202111, \t211202, \t321312 &$220\pm30$ & $ 2.2$  & $ 6.0$   & $6.4$              & $1.9\pm0.3$      & $0.9\pm0.1$      \\
HLSJ\,0918                 & \t202111                     & $38\pm3$ & $11.4$  &  --      & --                 & --               & --               \\
SPT\,0538                  & \t202111                     & $39\pm2$ & --      & $40.3$   & --                 & --               & --               \\
\hline
local strong-AGN           & \t202111, \t211202, \t321312 & --   &  $3.8$  &  $6.4$   & $6.7$              & $1.1\pm0.4$      & $0.9\pm0.3$      \\
local \ion{H}{ii}+mild-AGN & \t202111, \t211202, \t321312 & --   &  $5.8$  &  $9.2$   & $10.8$             & $1.4\pm0.4$      & $1.1\pm0.3$      \\
\hline
\end{tabular}
\tablefoot{
		The luminosity ratios between each {\hbox {H$_{2}$O}}\ line and their total \hbox {infrared},
		and the velocity-integrated flux density ratio of different {\hbox {H$_{2}$O}}\ transitions. {\hbox {$T_{\mathrm{d}}$}}\ is the cold-dust
		temperature taken from \citetalias{2013ApJ...779...25B}, except for the
		ones in brackets which are not listed \citetalias{2013ApJ...779...25B},
		that we infer them from single modified black-body dust SED fitting using
		the submm/mm photometry data listed in Table\,\ref{table:previous_obs_properties}.
		All the errors quoted for {\hbox {$T_{\mathrm{d}}$}}\ are significantly underestimated especially
		because they do not include possible effects of differential lensing and make
		the assumption of a single-temperature. Line ratios in brackets are derived
		based on the average velocity-integrated flux density ratios between \t211202 and \t202111 lines
		in local infrared galaxies. The local strong-AGN sources are the optically
		classified AGN-dominated galaxies and the local \ion{H}{ii}+mild-AGN sources
		are star-forming-dominated galaxies with possible mild AGN contribution
		\citepalias{2013ApJ...771L..24Y}. The first group of the sources are from
		this work; and the sources in the second group are the previously published
		sources in \citetalias{2013A&A...551A.115O}; the third group contains the
		previously published \hbox {high-redshift}\ detections from other works:
		HFLS3 \citep{2013Natur.496..329R}, APM\,08279 \citep{2011ApJ...741L..38V},
		HLSJ\,0918 \citep{2012A&A...538L...4C, 2014ApJ...783...59R} and SPT\,0538 \citep{2013ApJ...779...67B};
		the last group shows the local averaged values from \citetalias{2013ApJ...771L..24Y}.

 }
\label{table:Lir_Lh2o_ratios}
\end{table*}
\normalsize

Similar to what we found in the local ULIRGs \citepalias{2013ApJ...771L..24Y}, we find
again an anti-correlation between {\hbox {$T_{\mathrm{d}}$}}\ and \lhtot321312/\hbox {$L_{\mathrm{IR}}$}. The Spearman$'$s rank
correlation coefficient for the five \htot321312 detected {\it H}-ATLAS sources is
$\rho = -0.9$ with a two-sided significance of its deviation from zero, $p=0.04$.
However, after including the non-detection of \htot321312 in NAv1.195, the correlation
is much weaker, that is to say, $\rho \lesssim -0.5$ and $p \sim 0.32$. No significant
correlation has been found between {\hbox {$T_{\mathrm{d}}$}}\ and \lhtot202111/\hbox {$L_{\mathrm{IR}}$}\ ($\rho = -0.1$
and $p=0.87$) nor \lhtot211202/\hbox {$L_{\mathrm{IR}}$}\ ($\rho = -0.3$ and $p=0.45$). As explained
in \citetalias{2014A&A...567A..91G}, in the optically thick and very warm galaxies,
the ratio of \lhtot321312/\hbox {$L_{\mathrm{IR}}$}\ is expected to decrease with increasing {\hbox {$T_{\mathrm{d}}$}}. And this
anti-correlation can not be explained by optically thin conditions. However, a larger
sample is needed to increase the statistical significance of this anti-correlation.

Although, it is important to stress that the luminosity of {\hbox {H$_{2}$O}}\ is a
complex result of various physical parameters such as dust temperature,
gas density, {\hbox {H$_{2}$O}}\ abundance and {\hbox {H$_{2}$O}}\ gas distribution relative to the \hbox {infrared}\
radiation field, etc, it is striking that the correlation between \hbox {$L_{\mathrm{H_2O}}$}\ and
\hbox {$L_{\mathrm{IR}}$}\ stays linear from local young stellar objects (YSOs), in which the {\hbox {H$_{2}$O}}\
molecules are mainly excited by shocks and collisions, to local ULIRGs
(\hbox {far-infrared}\ pumping dominated), extending $\sim12$ orders of magnitudes
\citep{2016A&A...585A.103S}, implying that {\hbox {H$_{2}$O}}\ indeed traces the SFR
proportionally, similarly to the dense gas \citep{2004ApJ...606..271G}
in the local infrared bright galaxies. However, for the \hbox {high-redshift}\ sources,
the \hbox {$L_{\mathrm{H_2O}}$}\ emissions are somewhat above the linear correlations which
could be explained by their high $\tau_{100}$ (or large velocity dispersion).
As shown in Table\,\ref{table:Lir_Lh2o_ratios}, HFLS3, with a $\tau_{100}>1$
has extremely large ratios of \hbox {$L_{\mathrm{H_2O}}$}/\hbox {$L_{\mathrm{IR}}$}\ which are stronger than the
average of our {\it H}-ATLAS sources by factors $\sim$\,2 for the $J=2$
lines and $\sim$\,4 for $J=3$ (see Fig.\,\ref{fig:h2o-ir}). The velocity
dispersions of its {\hbox {H$_{2}$O}}\ lines are $\sim$\,900\,km\,s$^{-1}$ (with
uncertainties from 18\% to 36\%), which is larger than all our sources.
For optically thick systems, larger velocity dispersion will increase
the number of absorbed pumping photons, and boost the ratio of
\hbox {$L_{\mathrm{H_2O}}$}/\hbox {$L_{\mathrm{IR}}$}\ \citepalias{2014A&A...567A..91G}.

For the AGN-dominated sources (i.e. APM\,08279, G09v1.124-W and
Mrk\,231) as shown in Fig.\,\ref{fig:h2o-ir}, most of them (except
for the \htot321312 line of Mrk\,231) are well below the fitted correlation
(see Section\,\ref{AGN}).
This is consistent with the average value of local strong-AGN-dominated
sources. The $J \lesssim 3$ {\hbox {H$_{2}$O}}\ lines are \hbox {far-infrared}\ pumped by the 75
and 101\,$\mu m$ photons, thus the very warm dust in strong-AGN-dominated
sources is likely to contribute more to the \hbox {$L_{\mathrm{IR}}$}\ than the $J \lesssim 3$
{\hbox {H$_{2}$O}}\ excitation (see also \citetalias{2013ApJ...771L..24Y}).

\subsection{{\hbox {H$_{2}$O}}\ excitation}\label{hto}
We have detected both $J=2$ and $J=3$ {\hbox {H$_{2}$O}}\ lines in five sources
out of six observed for $J=3$ ortho-{\hbox {H$_{2}$O}}\ lines. By comparing the
line ratios and their strength relative to \hbox {$L_{\mathrm{IR}}$}, we are able to
constrain the physical conditions of the molecular content
and also the properties of the \hbox {far-infrared}\ radiation field.

 \begin{figure}[htbp]
	 \begin{center}
 \includegraphics[scale=0.561]{h2o_sled}
 \caption{
		  Velocity-integrated flux density distribution of {\hbox {H$_{2}$O}}\ normalised to
		  \ihtot202111 adapted from \citetalias{2013ApJ...771L..24Y}. Local
		  averaged values are shown in black dashed line and marks. Among them,
		  AGN-dominated sources are shown in red and star-forming dominated galaxies
		  are shown in blue. Some individual sources are also shown in this plot as
		  indicated by the legend. Green diamonds are the \hbox {high-redshift}\ lensed
		  Hy/ULIRGs from this work. HFLS3 is a $z=6.3$ \hbox {high-redshift}\ galaxy
		  from \cite{2013Natur.496..329R}.
		  }
 \label{fig:h2o-sled}
     \end{center}
 \end{figure}

To compare the {\hbox {H$_{2}$O}}\ excitation with local galaxies, we plot the
velocity-integrated flux density of ortho-\htot321312 normalised by that
of para-\htot202111 in our source on top of the local and \hbox {high-redshift}\ {\hbox {H$_{2}$O}}\
SLEDs (spectral line energy distributions) in Fig.\,\ref{fig:h2o-sled}.
All the six \hbox {high-redshift}\ sources are located within the range of the local galaxies,
with a 1\,$\sigma$ dispersion of $\sim 0.2$. Yet for the $z=6.34$
extreme starburst HFLS3, the value of this ratio is at least 1.7 times
higher than the average value of local sources \citepalias{2013ApJ...771L..24Y}
and those of our lensed \hbox {high-redshift}\ Hy/ULIRGs at $\gtrapprox3\,\sigma$ confidence
level (Fig.\,\ref{fig:h2o-sled}). This probably traces different
excitation conditions, namely the properties of the dust emission,
as it is suggested in \citetalias{2014A&A...567A..91G} that the flux
ratio of \htot321312 over \htot202111 is the most direct tracer
of the hardness of the \hbox {far-infrared}\ radiation field which powers the submm
{\hbox {H$_{2}$O}}\ excitation. However, the line ratios are still consistent
with the strong saturation limit in the \hbox {far-infrared}\ pumping model with a
$T_\mathrm{warm} \gtrsim 65$\,K. The large scatter of the {\hbox {H$_{2}$O}}\ line
ratio between \t321312 and \t202111 indicates different local {\hbox {H$_{2}$O}}\
excitation conditions. As \hbox {far-infrared}\ pumping is dominating the {\hbox {H$_{2}$O}}\
excitation, the ratio therefore reflects the differences in the \hbox {far-infrared}\
radiation field, for example, the temperature of the warmer dust that excites
the {\hbox {H$_{2}$O}}\ gas, and the submm continuum opacity. It is now clear that
\hbox {far-infrared}\ pumping is the prevailing excitation mechanism for those submm
{\hbox {H$_{2}$O}}\ lines rather than collisional excitation
\citepalias{2014A&A...567A..91G} in \hbox {infrared}\ bright galaxies in both the
local and \hbox {high-redshift}\ Universe. The main path of \hbox {far-infrared}\ pumping related to the
lines we observed here are 75 and 101\,$\mu$m as displayed in
Fig.\,\ref{fig:h2o-e-level}. Therefore, the different line ratios are
highly sensitive to the difference between the monochromatic flux at
75 and 101\,$\mu$m. We may compare the global {\hbox {$T_{\mathrm{d}}$}}\ measured from \hbox {far-infrared}\
and submm bands \citepalias{2013ApJ...779...25B}. It includes both cold
and warm dust contribution to the dust SED in the rest-frame, which is,
however, dominated by cold dust observed in SPIRE bands. It is thus
not surprising that we find no strong correlation between {\hbox {$T_{\mathrm{d}}$}}\ and
\ihtot321312/\ihtot202111 ($r \sim -0.3$). The Rayleigh-Jeans tail of
the dust SED is dominated by cooler dust which is associated with
extended molecular gas and less connected to the submm {\hbox {H$_{2}$O}}\ excitation.
As suggested in \citetalias{2014A&A...567A..91G}, it is indeed the
warmer dust ($T_\mathrm{warm}$, as shown by the colour legend in
Fig.\,\ref{fig:h2o-model}) dominating at the Wien side of the dust
SED that corresponds to the excitation of submm {\hbox {H$_{2}$O}}\ lines.

To further explore the physical properties of the {\hbox {H$_{2}$O}}\ gas content
and the \hbox {far-infrared}\ dust radiation related to the submm {\hbox {H$_{2}$O}}\ excitation,
we need to model how we can infer key parameters, such as the {\hbox {H$_{2}$O}}\
abundance and those determining the radiation properties, from the
observed {\hbox {H$_{2}$O}}\ lines. For this purpose, we use the \hbox {far-infrared}\ pumping
{\hbox {H$_{2}$O}}\ excitation model described in
\citetalias{2014A&A...567A..91G} to fit the observed \hbox {$L_{\mathrm{H_2O}}$}\ together
with the corresponding \hbox {$L_{\mathrm{IR}}$}, and derive the range of continuum optical
depth at 100\,$\mu$m ($\tau_{100}$), warm dust temperature
($T_\mathrm{warm}$), and {\hbox {H$_{2}$O}}\ column density per unit of velocity
interval (\hbox {$N_{\mathrm{H_2O}}$}/$\Delta V$) in the five sources with both $J=2$ and
$J=3$ {\hbox {H$_{2}$O}}\ emission detections. Due to the insufficient number of
the inputs in the model, which are \hbox {$L_{\mathrm{H_2O}}$}\ of the two {\hbox {H$_{2}$O}}\ lines and \hbox {$L_{\mathrm{IR}}$},
we are only able to perform the modelling by using the pure \hbox {far-infrared}\ pumping
regime. Nevertheless, our observed line ratio between $J=3$ and $J=2$
{\hbox {H$_{2}$O}}\ lines suggests that \hbox {far-infrared}\ pumping is the dominant excitation
mechanism and the contribution from collisional excitation is minor
\citepalias{2014A&A...567A..91G}. The $\pm 1\,\sigma$ contours from
$\chi^2$ fitting are shown in Fig.\,\ref{fig:h2o-model} for each warm
dust temperature component ($T_\mathrm{warm} = 35\text{--}115$\,K)
per source. It is clear that with two {\hbox {H$_{2}$O}}\ lines (one $J=2$ para-{\hbox {H$_{2}$O}}\
and ortho-\htot312312), we will not be able to well constrain $\tau_{100}$
and \hbox {$N_{\mathrm{H_2O}}$}/$\Delta V$. As shown in the figure, for $T_\mathrm{warm} \lesssim 75$\,K,
both very low and very high $\tau_{100}$ could fit the observation data together
with high \hbox {$N_{\mathrm{H_2O}}$}/$\Delta V$, while the dust with $T_\mathrm{warm} \gtrsim 95$\,K
are likely favouring high $\tau_{100}$. In the low continuum optical depth part
in Fig.\,\ref{fig:h2o-model}, as $\tau_{100}$ decreases, the model needs to
increase the value of \hbox {$N_{\mathrm{H_2O}}$}/$\Delta V$ to generate sufficient \hbox {$L_{\mathrm{H_2O}}$}\ to be able
to fit the observed \hbox {$L_{\mathrm{H_2O}}$}/\hbox {$L_{\mathrm{IR}}$}. This has been observed in some local sources
with low $\tau_{100}$, such as in NGC\,1068 and NGC\,6240. There are no absorption
features in the \hbox {far-infrared}\ but submm {\hbox {H$_{2}$O}}\ emission have been detected in these
sources \citepalias{2014A&A...567A..91G}. The important feature of such
sources is the lack of $J\geq4$ {\hbox {H$_{2}$O}}\ emission lines. Thus, the observation
of higher excitation of {\hbox {H$_{2}$O}}\ will discriminate between the low and high
$\tau_{100}$ regimes.

 \begin{figure*}[!htbp]
	 \begin{center}
 \includegraphics[scale=0.82]{model}
 \caption{
		  Parameter space distribution of the H$_2$O \hbox {far-infrared}\ pumping excitation
		  modelling with observed para-{\hbox {H$_{2}$O}}\ \t202111 or \t211202 and
		  ortho-\htot321312 in each panel. $\pm 1\,\sigma$ contours are
		  shown for each plot. Different colours with different line styles
		  represent different temperature components of the warm dust as
		  shown in the legend. The explored warm dust temperature range is
		  from 35\,K to 115\,K. The temperature contours that are unable to
		  fit the data are not shown in this figure. From the figure, we
		  are able to constrain the $\tau_{100}$, $T_\mathrm{warm}$ and
		  \hbox {$N_{\mathrm{H_2O}}$}/$\Delta V$ for the five sources. However, there are strong
		  degeneracies. Thus, we need additional information, such as the
		  velocity-integrated flux densities of $J\geq4$ {\hbox {H$_{2}$O}}\ lines, to
		  better constrain the physical parameters.
		  }
 \label{fig:h2o-model}
     \end{center}
 \end{figure*}

Among these five sources, favoured key parameters are somewhat different
showing the range of properties we can expect for such sources. Compared
with the other four Hy/ULIRGs, G09v1.97 is likely to have the lowest
$T_\mathrm{warm}$ as only dust with $T_\mathrm{warm} \sim 45-55$\,K can
fit well with the data. NCv1.143 and NAv1.177 have slightly different
diagnostic which yields higher dust temperature as $T_\mathrm{warm} \sim 45\text{--}75$\,K,
while NBv1.78 and G12v2.43 tend to have the highest temperature range,
$T_\mathrm{warm} \sim 45\text{--}95$\,K. The values of $T_\mathrm{warm}$
are consistent with the fact that {\hbox {H$_{2}$O}}\ traces warm gas. We did not find
any significant differences between the ranges of \hbox {$N_{\mathrm{H_2O}}$}/$\Delta V$ derived
from the modelling for these five sources, although G09v1.97 tends to have
lower \hbox {$N_{\mathrm{H_2O}}$}/$\Delta V$ (Table\,\ref{table:model_para}). As shown in Section\,\ref{AGN},
there is no evidence of AGN domination in all our sources, the submm {\hbox {H$_{2}$O}}\
lines likely trace the warm dust component that connect to the heavily obscured
active star-forming activity. However, due to the lack of photometry data on the
Wien side of the dust SEDs, we will not be able to compare the observed values
of $T_\mathrm{warm}$ directly with the ones derived from the modelling.

By adopting the 100\,$\mu$m dust mass absorption coefficient from
\cite{2003ARA&A..41..241D} of $\kappa_{100}$\,=\,27.1\,cm$^2$\,g$^{-1}$,
we can derive the dust opacity by
\begin{equation}
	\label{eq:tau}
       \tau_{100}=\kappa_{100} \, \sigma_\mathrm{dust}= \kappa_{100}
       \left( M_\mathrm{dust} \over A \right) = \kappa_{100}
       \left( M_\mathrm{dust} \over 2 \pi r_\mathrm{half}^2 \right)
\end{equation}
where $\sigma_\mathrm{dust}$ is the dust mass column density, $M_\mathrm{dust}$
is the dust mass, $A$ is the projected surface area of the dust continuum
source and $r_\mathrm{half}$ is the half-light radius of the source at submm.
As shown in Table\,\ref{table:previous_obs_properties}, among the five sources
in Fig.\,\ref{fig:h2o-model}, the values of $M_\mathrm{dust}$ and
$r_\mathrm{half}$ in G09v1.97, NCv1.143 and NBv1.78 have been derived
via gravitational lensing \citepalias{2013ApJ...779...25B}. Consequently,
the derived approximate dust optical depth at 100\,$\mu$m in these three
sources is $\tau_{100} \approx\,$\,1.8, 7.2 and 2.5, respectively. One should
note that, the large uncertainty in both the $\kappa_{100}$ and $r_\mathrm{half}$
of these \hbox {high-redshift}\ galaxies can bring a factor of few error budget. Nevertheless,
by adopting a gas-to-dust mass ratio of $X = 100$
\cite[e.g.][]{2011ApJ...740L..15M}, we can derive the gas depletion
time using the following approach,
\begin{equation}
	\label{eq:t_dep}
t_\mathrm{dep} = {M_\mathrm{gas} \over \mathit{SFR}} =
                   {X \tau_{100} \over \Sigma_\mathit{SFR} \kappa_{100}}
                   \approx 1.8 \times 10^{4}  \left( \tau_{100} \over {\Sigma_\mathit{SFR} \over \mathrm{M_\odot\,yr^{-1}\,kpc^{-2}}} \right) \, \mathrm{Myr}
\end{equation}
where $M_\mathrm{gas}$ is the total molecular gas mass and $\Sigma_\mathit{SFR}$
is the surface $\mathit{SFR}$ density derived from \hbox {$L_{\mathrm{IR}}$}\ using \citet{1998ARA&A..36..189K}
calibration by assuming a Salpeter IMF
\citepalias[][and Table\,\ref{table:previous_obs_properties}]{2013ApJ...779...25B}.
The implied depletion time scale is $t_\mathrm{dep} \approx 35\text{--}60$\,Myr
with errors within a factor of two, in which the dominant uncertainties are
from the assumed gas-to-dust mass ratio and the half-light radius. The $t_\mathrm{dep}$
is consistent with the values derived from dense gas tracers, like HCN in local
(U)LIRGs \citep[e.g.][]{2004ApJ...606..271G, 2012A&A...539A...8G}. As suggested
in \citetalias{2014A&A...567A..91G}, the {\hbox {H$_{2}$O}}\ and HCN likely to be located in
the same regions, indicate that the {\hbox {H$_{2}$O}}\ traces the dense gas as well. Thus,
the $\tau_{100}$ derived above is likely also tracing the \hbox {far-infrared}\ radiation source
that powers the submm {\hbox {H$_{2}$O}}\ emissions. \citetalias{2013ApJ...779...25B} also has
found that these {\it H}-ATLAS \hbox {high-redshift}\ Hy/ULIRGs are expected to be optically thick
in the \hbox {far-infrared}. By adding the constrain from $\tau_{100}$ above, we can better derive
the physical conditions in the sources as shown in Table\,\ref{table:model_para}.

\begin{table}[htbp]
\setlength{\tabcolsep}{0.64em}
\small
\caption{Parameters derived from \hbox {far-infrared}\ pumping model of {\hbox {H$_{2}$O}}.}
\centering
\begin{tabular}{lrrrr}
\hline
\hline
Source   & $\tau_{100}$ & $T_\mathrm{warm}$ &     \hbox {$N_{\mathrm{H_2O}}$}/$\Delta V$\;\;\;       & \hbox {$N_{\mathrm{H_2O}}$}\;\;\;\;\;                  \\
         &              &       (K)\;\,     &     (cm$^{-2}$\,km$^{-1}$\,s)    & (cm$^{-2}$)\;\;\;\;              \\
\hline
G09v1.97 &     1.8      &      45--55       &   (0.3--0.6)\,$\times 10^{15}$   & (0.3--1.1)\,$\times 10^{17}$     \\
G12v2.43 &     --       &      45--95       & $\gtrsim$\,0.7\,$\times 10^{15}$ & $\gtrsim$\,0.7\,$\times 10^{17}$ \\
NCv1.143 &     7.2      &      45--55       &   (2.0--20)\,$\times 10^{15}$    & (2.0--60)\,$\times 10^{17}$      \\
NAv1.177 &     --       &      45--75       & $\gtrsim$\,1.0\,$\times 10^{15}$ & $\gtrsim$\,1.0\,$\times 10^{17}$ \\
NBv1.78  &     2.5      &      45--75       & $\gtrsim$\,0.6\,$\times 10^{15}$ & $\gtrsim$\,0.6\,$\times 10^{17}$ \\
\hline
\end{tabular}
\tablefoot{ $\tau_{100}$ is derived from Eq.\,\ref{eq:tau} with errors of a few units (see text), while
			$T_\mathrm{warm}$ and \hbox {$N_{\mathrm{H_2O}}$}/$\Delta V$ are inferred from the {\hbox {H$_{2}$O}}\ excitation model. \hbox {$N_{\mathrm{H_2O}}$}\
			is calculated by taking a typical $\Delta V$ value range of $100\text{--}300$\,km\,s$^{-1}$
			as suggested by \citetalias{2014A&A...567A..91G}.
       }
\label{table:model_para}
\end{table}
\normalsize

From their modelling of local infrared galaxies, \citetalias{2014A&A...567A..91G}
find a range of $T_\mathrm{warm} =45\text{--}75$\,K, $\tau_{100}=0.05\text{--}0.2$
and \hbox {$N_{\mathrm{H_2O}}$}/$\Delta V=(0.5\text{--}2)\times10^{15}$\,\,cm$^{-2}$\,km$^{-1}$\,s. The
modelling results for our \hbox {high-redshift}\ sources are consistent with those in local galaxies
in terms of $T_\mathrm{warm}$ and \hbox {$N_{\mathrm{H_2O}}$}/$\Delta V$. However, the values of $\tau_{100}$
we found at \hbox {high-redshift}\ are higher than those of the local \hbox {infrared}\ galaxies. This is consistent
with the higher ratio between \hbox {$L_{\mathrm{H_2O}}$}\ and \hbox {$L_{\mathrm{IR}}$}\ at \hbox {high-redshift}\ \citepalias{2013ApJ...771L..24Y}
which could be explained by higher $\tau_{100}$ \citepalias{2014A&A...567A..91G}.
However, as demonstrated in an extreme sample, a very large velocity dispersion
will also increase the value of \hbox {$L_{\mathrm{H_2O}}$}/\hbox {$L_{\mathrm{IR}}$}\ within the sources with $\tau_{100} > 1$.
Thus, the higher ratio can also be explained by larger velocity dispersion (not
including systemic rotations) in the \hbox {high-redshift}\ Hy/ULIRGs. Compared with local ULIRGs,
our {\it H}-ATLAS sources are much more powerful in terms of their \hbox {$L_{\mathrm{IR}}$}. The dense
warm gas regions that {\hbox {H$_{2}$O}}\ traces are highly obscured with much more powerful
\hbox {far-infrared}\ radiation fields, which possibly are close to the limit of maximum starbursts.
Given the values of dust temperature and dust opacity, the radiation pressure
$P_\mathrm{rad} \sim \tau_{100} \sigma T_\mathrm{d} / c$ ($\sigma$ is
Stefan-Boltzmann$'$s constant and $c$ the speed of light) of our sources is about
$0.8 \times 10^{-7}$\,erg\,cm$^{-3}$. If we assume a H$_2$ density $n_\mathrm{H_2}$
of $\sim$\,10$^6$\,cm\,$^{-3}$ and take $T_\mathrm{k} \sim$\,150\,K as suggested
in \citetalias{2014A&A...567A..91G}, the thermal pressure
$P_\mathrm{th} \sim n_\mathrm{H_2} k_\mathrm{B} T_\mathrm{k} \sim 2 \times 10^{-8}$\,erg\,cm$^{-3}$
($k_\mathrm{B}$ is the Boltzmann constant and $T_\mathrm{k}$ is the gas temperature).
Assuming a turbulent velocity dispersion of
$\sigma_\mathrm{v} \sim 20\text{--}50$\,km\,s$^{-1}$ \citep{2015A&A...575A..56B}
and taking molecular gas mass density $\rho \sim 2 \mu n_\mathrm{H_2}$ (2$\mu$
is the average molecular mass) would yield for the turbulent pressure
$P_\mathrm{turb} \sim \rho \sigma_\mathrm{v}^2/3 \sim 4 \times 10^{-6}$\,erg\,cm$^{-3}$.
This might be about an order of magnitude larger than $P_\mathrm{rad}$ and two
orders of magnitude larger than $P_\mathrm{th}$, but we should note that all
values are very uncertain, especially $P_\mathrm{turb}$ which could be
uncertain by, at maximum, a factor of a few tens. Therefore, keeping in mind
their large uncertainties, turbulence and/or radiation are likely to play an
important role in limiting the star formation.

\subsection{Comparison between {\hbox {H$_{2}$O}}\ and CO}
\label{CO lines}

The velocity-integrated flux density ratio between submm {\hbox {H$_{2}$O}}\ and submm CO lines
with comparable frequencies is 0.02--0.03 in local PDRs such as Orion
and M\,82 \citep{2010A&A...521L...1W}. But this ratio in local ULIRGs
\citepalias{2013ApJ...771L..24Y} and in {\it H}-ATLAS \hbox {high-redshift}\ Hy/ULIRGs
is much higher, from 0.4 to 1.1 (Table\,\ref{table:co_properties} and
\ref{table:h2o_properties}). The former case is dominated by typical
PDRs, where CO lines are much stronger than {\hbox {H$_{2}$O}}\ lines, while the
latter sources shows clearly a different excitation regime, in which
{\hbox {H$_{2}$O}}\ traces the central core of warm, dense and dusty molecular gas
which is about a few hundred parsec \citep{2010A&A...518L..43G} in
diameter in local ULIRGs and highly obscured even at \hbox {far-infrared}.

Generally, submm {\hbox {H$_{2}$O}}\ lines are dominated by \hbox {far-infrared}\ pumping that
traces strong \hbox {far-infrared}\ dust continuum emission, which is different from
the regime of molecular gas traced by collisional excited CO lines.
In the active star-forming nucleus of the \hbox {infrared}-bright galaxies, the
\hbox {far-infrared}\ pumped {\hbox {H$_{2}$O}}\ is expected to trace directly the \hbox {far-infrared}\ radiation
generated by the intense star formation, which can be well correlated
with the high-$J$ CO lines \citep{2015ApJ...810L..14L}. Thus there
is likely to be a correlation between the submm {\hbox {H$_{2}$O}}\ and CO emission.
From our previous observations, most of the {\hbox {H$_{2}$O}}\ and CO line profiles
are quite similar from the same source in our \hbox {high-redshift}\ lensed Hy/ULIRGs
sample (Fig.\,2 of \citetalias{2013A&A...551A.115O}). In the present
work, we again find similar profiles between {\hbox {H$_{2}$O}}\ and CO in terms
of their FWHM with an extended sample (Table\,\ref{table:co_properties}
and \ref{table:h2o_properties}). In both cases the FWHMs of {\hbox {H$_{2}$O}}\ and
CO are generally equal within typical 1.5\,$\sigma$ errors (see special
discussion for each source in Appendix\,\ref{Individual sources}).

As the gravitational lensing magnification factor is sensitive
to spatial alignment, the similar line profiles could thus suggest
similar spatial distributions of the two gas tracers. However,
there are a few exceptional sources, such as SDP\,81 \citep{2015ApJ...808L...4A}
and HLSJ0918 \citep{2014ApJ...783...59R}. In both cases, the {\hbox {H$_{2}$O}}\
lines are lacking the blue velocity component found in the CO line
profiles. Quite different from the rest sources, in SDP\,81 and HLSJ0918,
the CO line profiles are complicated with multiple velocity components.
Moreover, the velocity-integrated flux density ratios between these CO components may vary
following the excitation level (different $J_\mathrm{up}$). Thus,
it is important to analyse the relation between different CO excitation
components (from low-$J$ to high-$J$) and {\hbox {H$_{2}$O}}. Also, high resolution
observation is needed to resolve the multiple spatial gas components
and compare the CO emission with {\hbox {H$_{2}$O}}\ and dust continuum emission
within each component.

\subsection{AGN content}
\label{AGN}
It is still not clear how a strong AGN could affect the excitation of
submm {\hbox {H$_{2}$O}}\ in both local ULIRGs and \hbox {high-redshift}\ Hy/ULIRGs. Nevertheless, there
are some individual studies addressing this question. For example, in
APM\,08279, \cite{2011ApJ...741L..38V} found that AGN is the main power
source that excites the high-$J$ {\hbox {H$_{2}$O}}\ lines and also enriches the
gas-phase {\hbox {H$_{2}$O}}\ abundance. Similar conclusion has also been drawn by
\cite{2010A&A...518L..43G} that in Mrk\,231 the AGN accounts for at
least 50\,\% contribution to the \hbox {far-infrared}\ radiation that excites {\hbox {H$_{2}$O}}.
From the systematic study of local sources \citepalias{2013ApJ...771L..24Y},
slightly lower values of \hbox {$L_{\mathrm{H_2O}}$}/\hbox {$L_{\mathrm{IR}}$}\ are found in strong-AGN-dominated
sources. In the present work, the decreasing ratio of \hbox {$L_{\mathrm{H_2O}}$}/\hbox {$L_{\mathrm{IR}}$}\ with AGN is
clearly shown in Fig.\,\ref{fig:h2o-ir} where Mrk\,231, G09v1.124-W
and APM\,08279 are below the correlation by factors between 2 and 5 with
less than 30\% uncertainties (except the \htot321123 line of Mrk\,231).

In the \hbox {far-infrared}\ pumping regime, the buried AGN will provide a strong \hbox {far-infrared}\
radiation source that will pump the {\hbox {H$_{2}$O}}\ lines. However, the very warm
dust powered by the AGN will increase the value of \hbox {$L_{\mathrm{IR}}$}\ faster than the
number of $\geq$\,75\,$\mu m$ photons that is dominating the excitation
of $J \leq 3$ {\hbox {H$_{2}$O}}\ lines \citep[e.g.][]{2015ApJ...814....9K}. If we assume
that the strength of the {\hbox {H$_{2}$O}}\ emission is proportional to the number of
pumping photons, then in the strong-AGN-dominated sources, the ratio of
\hbox {$L_{\mathrm{H_2O}}$}/\hbox {$L_{\mathrm{IR}}$}\ will decrease since much warmer dust is present. Moreover, strong
radiation from the AGN could dissociate the {\hbox {H$_{2}$O}}\ molecules.

To evaluate the AGN contribution to the {\it H}-ATLAS sources, we extracted
the 1.4\,GHz radio flux from the FIRST radio survey \citep{1995ApJ...450..559B}
listed in Table\,\ref{table:previous_obs_properties}. By comparing the
\hbox {far-infrared}\ and radio emission using the q parameter
\citep{1992ARA&A..30..575C},
$q \equiv \log ( {L_\mathrm{FIR} / 3.75 \times 10^{12}\,\mathrm{W}} )
 - \log ( {{L_\mathrm{1.4\,GHz}} / {1\,\mathrm{W}\,\mathrm{Hz}^{-1}}} )$,
we derive values of $q$ from 1.9 to 2.5 in our sources. These values
follow the value $2.3\pm0.1$ found by \citet{2001ApJ...554..803Y} for non
strong-radio AGN. This may suggest that there is also no significant indication
of a high radio contribution from AGN. This is also confirmed by the
Wide-field Infrared Survey Explorer \citep[WISE,][]{2010AJ....140.1868W},
which does not detect our sources at 12\,$\mu$m and 22\,$\mu$m.
However, rest-frame optical spectral observations show that
G09v1.124-W is rather a powerful AGN (Oteo et al, in prep.),
which is the only identified AGN-dominated source in our sample.

\section{Detection of {\hbox {H$_{2}$O}}$^+$ emission lines}
\label{htop}

 \begin{figure*}[htbp]
	 \begin{center}
 \includegraphics[scale=0.86]{h2op}
 \caption{
         {\em Left panel}: from top to bottom are the full NOEMA
         spectrum at $\nu_\mathrm{rest} \sim 750$\,GHz of NCv1.143,
         G09v1.97 and G15v2.779, respectively. The reference frequency
         is the redshifted frequency of the line \htot211202. The
         frequencies of the main {\hbox {H$_2$O$^+$}}(\t211202)\,$_{(5/2-5/2)}$ and
         {\hbox {H$_2$O$^+$}}(\t202111)\,$_{(5/2-3/2)}$ lines are indicated by
         grey vertical dashed lines. The three dashed squares in
         the spectrum of NCv1.143 show the position of each zoom-in
         spectrum of the {\hbox {H$_2$O$^+$}}\ (or the H$_2^{18}$O) as displayed
         in the {\it right panel} indicated by the A, B or C. The superposed
         blue dashed histograms represents the spectra of \htot211202
         centred at the frequencies of the {\hbox {H$_2$O$^+$}}\ lines. Note that,
         in many cases, the observed frequency ranges (yellow histograms)
         do not include the full expected profiles for the {\hbox {H$_2$O$^+$}}\ lines.
         The red curve represents the Gaussian fitting to the spectra.
         We have detected both {\hbox {H$_2$O$^+$}}\ lines in NCv1.143, and
         tentatively detected {\hbox {H$_2$O$^+$}}(\t202111)\,$_{(5/2-3/2)}$ in G09v1.97
         and {\hbox {H$_2$O$^+$}}(\t211202)\,$_{(5/2-5/2)}$ in G15v2.779.
         {\em Right panel}: from top to bottom are the spectra dominated by
          lines of {\hbox {H$_2$O$^+$}}(\t211202)\,${_{(5/2-5/2)}}$, {\hbox {H$_2$O$^+$}}(\t202111)\,$_{(3/2-3/2)}$
          and H$_2^{18}$O(\t211202), respectively, displayed as the
          filled yellow histograms. The reference frequency is the frequency
          of each of these lines. Weaker {\hbox {H$_2$O$^+$}}(\t202111)\,$_{(3/2-3/2)}$ and
		  {\hbox {H$_2$O$^+$}}(\t211202)\,$_{(5/2-3/2)}$ components are indicated by additional
		  grey vertical dashed lines. The superposed blue dashed histograms
		  represent the spectra of para-\htot211202 in NCv1.143 centred at each
		  line frequency. The red curve represents the Gaussian fitting
		  to the spectra, and the green dashed curves are the decomposed
		  Gaussian profiles for each fine structure line. The violet error
		  bar indicates the $\pm$\,1\,$\sigma$ uncertainties of the spectrum.
         }
 \label{fig:h2o-ions1}
     \end{center}
 \end{figure*}

H$_2$O can be formed through both solid-state and gas-phase chemical
reactions \citep{2013ChRv..113.9043V}. On dust-grain mantles, surface
chemistry dominates the formation of {\hbox {H$_{2}$O}}\ molecules. Then they can be
released into the interstellar medium (ISM) gas through sublimation.
In the gas phase, {\hbox {H$_{2}$O}}\ can be produced through two routes: the
neutral-neutral reaction,
usually related to shocks, creates {\hbox {H$_{2}$O}}\ via \ce{O + H2 -> OH + H};
\ce{OH + H2 -> H2O + H} at high temperature ($\gtrsim$\,300\,K). At
lower temperature ($\lesssim$\,100\,K), the ion-neutral reactions in
photon-dominated regions (PDRs), cosmic-ray-dominated regions and
X-ray-dominated regions \citep[e.g.][]{2005A&A...436..397M} generate
{\hbox {H$_{2}$O}}\ from O, H$^+$, H$_3^+$ and H$_2$, with intermediates such as
O$^+$, OH$^+$, {\hbox {H$_2$O$^+$}}\ and H$_3$O$^+$, and finally \ce {H3O^+ + e -> H2O + H}.
However, classical PDRs are not likely linked to these highly excited
submm {\hbox {H$_{2}$O}}\ emissions \citepalias{2013ApJ...771L..24Y}. Therefore,
{\hbox {H$_2$O$^+$}}\ lines are important for distinguishing between shock- or
ion-chemistry origin for {\hbox {H$_{2}$O}}\ in the early Universe, indicating
the type of physical regions in these galaxies: shock-dominated
regions, cosmic-ray-dominated regions or X-ray-dominated regions.
Indeed, they can be among the most direct tracers of the cosmic-ray
or/and X-ray ionization rate
\citep[e.g.][]{2010A&A...518L.110G, 2010A&A...521L..10N, 2013A&A...550A..25G}
of the ISM, which regulates the chemistry and influences many key
parameters, for example, X-factor \citep{2007MNRAS.378..983B} that connects
the CO luminosity to the H$_2$ mass. Moreover, the significant detections
of {\hbox {H$_2$O$^+$}}\ emission in \hbox {high-redshift}\ Hy/ULIRGs could help us understanding {\hbox {H$_{2}$O}}\
formation in the early Universe.

When observing our sources with redshift $z \gtrsim 3.3$, it is possible
to cover all the following lines with the NOEMA WideX bandwidth: para-\htot211202
at 752\,GHz and four ortho-{\hbox {H$_2$O$^+$}}\ lines (two intertwined fine structure doublets
of two different lines whose frequencies almost coincide by chance):
\t202111\,$_{(5/2-3/2)}$ at 742.1\,GHz, \t211202\,$_{(5/2-3/2)}$ at 742.3\,GHz,
\t202111\,$_{(3/2-3/2)}$ at 746.3\,GHz and \t211202\,$_{(5/2-5/2)}$ at 746.5\,GHz,
in the 3.6\,GHz band simultaneously (the rest-frame frequencies are taken from the
CDMS catalogue: \url{http://www.astro.uni-koeln.de/cdms}, see energy level
diagram of {\hbox {H$_2$O$^+$}}\ in Fig.\,\ref{fig:h2o-e-level} and the
full spectra in Fig.\,\ref{fig:h2o-ions1}).
Additionally, within this range, we can also cover the H$_2^{18}$O(\t211202)
line at 745.3\,GHz. There are three sources of our sample
that have been observed in such a frequency setup: NCv1.143, NCv1.268 and
G09v1.97. We have also included the source G15v2.779 from our previous
observation \citepalias{2013A&A...551A.115O}, in which we have covered both
\htot211202 at 752\,GHz and {\hbox {H$_2$O$^+$}}\ lines around 746\,GHz.
We have detected both main lines of {\hbox {H$_2$O$^+$}}\ in NCv1.143, and tentatively
detected one line in G09v1.97 and G15v2.779 (Fig.\,\ref{fig:h2o-ions1}).
For NCv1.268, due to the large noise level and the complex line
profile, we were not able to really identify any {\hbox {H$_2$O$^+$}}\ line detection.

\begin{table}[htbp]
\setlength{\tabcolsep}{0.65em}
\small
\caption{Observed ortho-{\hbox {H$_2$O$^+$}}\ fine structure line parameters of the \hbox {high-redshift}\ {\it H}-ATLAS lensed HyLIRGs.}
\centering
\begin{tabular}{lcrrr}
\hline
\hline
Source             &  {\hbox {H$_2$O$^+$}}\ transition        & $\nu_{\rm rest}$ & $\nu_{\rm line}$  & $I_{\rm H_{2}O^+}$ \\
                   &                           &    (GHz)         &    (GHz)          & (Jy\,km\,s$^{-1}$) \\
\hline
NCv1.143           &  \t211202\,$_{(5/2-5/2)}$ &   746.5          &   163.53          & $1.6\pm0.5$        \\
                   &  \t202111\,$_{(3/2-3/2)}$ &   746.3          &   163.48          & $0.2\pm0.5$        \\
				   &  \t211202\,$_{(5/2-3/2)}$ &   742.3          &   162.61          & $0.3\pm0.4$        \\
				   &  \t202111\,$_{(5/2-3/2)}$ &   742.1          &   162.56          & $1.6\pm0.4$        \\
G09v1.97           &  \t202111\,$_{(5/2-3/2)}$ &   742.1          &   160.14          & $1.4\pm0.4$        \\
G15v2.779          &  \t211202\,$_{(5/2-5/2)}$ &   746.5          &   142.35          & $1.2\pm0.3$        \\
\hline
\end{tabular}
\tablefoot{
	    The {\hbox {H$_2$O$^+$}}\,(\t202111)\,$_{(5/2-3/2)}$ line in G09v1.97 is
	    blended by (\t211202)\,$_{(5/2-3/2)}$, and
	    {\hbox {H$_2$O$^+$}}\,(\t211202)\,$_{(5/2-5/2)}$ line in G15v2.779 is
	    blended by (\t202111)\,$_{(3/2-3/2)}$. However, the contribution
	    from the latter in each case is small, likely less than
	    20\,\% as shown in the case of the {\hbox {H$_2$O$^+$}}\ lines in NCv1.143.
	    Note that the quoted uncertainties do not include the missing
	    parts of the spectra cut by the limited observed bandwidth
	    (Fig.\,\ref{fig:h2o-ions1}).
       }
\label{table:htop_intensity}
\end{table}
\normalsize

As shown in Fig.\,\ref{fig:h2o-ions1}, in NCv1.143, the dominant
{\hbox {H$_2$O$^+$}}\ fine structure lines \t211202\,$_{(5/2-5/2)}$ at 746.5\,GHz
and \t202111\,$_{(5/2-3/2)}$ at 742.1\,GHz are well detected.
The velocity-integrated flux densities of the two lines from a
two-Gaussian fit are $1.9\pm0.3$ and $1.6\pm0.2$\,Jy\,km\,s$^{-1}$,
respectively. These are the approximate
velocity-integrated flux densities of the dominant {\hbox {H$_2$O$^+$}}\ lines \t211202\,$_{(5/2-5/2)}$
and \t202111\,$_{(5/2-3/2)}$ if neglecting the minor
contributions from {\hbox {H$_2$O$^+$}}\ lines \t202111\,$_{(3/2-3/2)}$ at 746.2\,GHz
and \t211202\,$_{(5/2-3/2)}$ at 742.3\,GHz. However, the {\hbox {H$_2$O$^+$}}\ line
profile at 746.5\,GHz is slightly wider than the {\hbox {H$_{2}$O}}\ line
(Fig.\,\ref{fig:h2o-ions1}), probably due to a contribution from the
fairly weak fine structure line {\hbox {H$_2$O$^+$}}(\t202111)\,$_{(3/2-3/2)}$ at
746.3\,GHz. The ratio between total velocity-integrated flux density of the {\hbox {H$_2$O$^+$}}\
lines and \htot211202 is $0.60\pm0.07$ (roughly 0.3 for each dominant
{\hbox {H$_2$O$^+$}}\ line), being consistent with the average value from the local
\hbox {infrared}\ galaxies \citepalias{2013ApJ...771L..24Y}\footnote{As suggested by
\cite{2013A&A...550A..25G}, due to the very limited spectral resolution
of {\it Herschel}/SPIRE FTS, the ortho-{\hbox {H$_2$O$^+$}}(\t202111)\,$_{(3/2-3/2)}$
line at 746.5\,GHz quoted in \citetalias{2013ApJ...771L..24Y} is
actually dominated by ortho-{\hbox {H$_2$O$^+$}}(\t211202)\,$_{(5/2-5/2)}$,
considering their likely excitation and relative strength.}. In order
to derive the velocity-integrated flux density of each fine structure doublets around 742
and 746\,GHz, we have also performed a four-Gaussian fit with fixed
line positions (equal to $\nu_\mathrm{rest}/(1+z)$) and linewidth
(equals to that of \htot211202). We find the velocity-integrated flux densities
of the two fine structure lines of {\hbox {H$_2$O$^+$}}(\t211202) are $1.6\pm0.5$
and $0.3\pm0.4$\,Jy/km\,s$^{-1}$, while they are $1.6\pm0.4$ and
$0.2\pm0.5$\,Jy/km\,s$^{-1}$ for the two fine structure lines of
{\hbox {H$_2$O$^+$}}(\t202111) (Table\,\ref{table:htop_intensity}).
We should note that these fitting results have much larger
uncertainties due to the blending. Nevertheless, they are consistent
with the earlier fitting results without de-blending. The similarity
of the velocity-integrated flux densities between the {\hbox {H$_2$O$^+$}}(\t202111) and {\hbox {H$_2$O$^+$}}(\t211202)
lines is in good agreement with the regime of \hbox {far-infrared}\ pumping
as submm {\hbox {H$_{2}$O}}\ \citep{2013A&A...550A..25G}. As a first approximation,
if these {\hbox {H$_2$O$^+$}}\ lines are optically thin and we ignore the additional
pumping from ortho-{\hbox {H$_2$O$^+$}}\ 2$_{02}$ to ortho-{\hbox {H$_2$O$^+$}}\ $J=3$ energy levels,
the statistical equilibrium applied to energy level 2$_{02\,5/2}$
implies that all population arriving per second at 2$_{02\,5/2}$
should be equal to all population leaving the level per second.

After subtracting the Gaussian profiles of all the {\hbox {H$_2$O$^+$}}\ lines
in the spectrum, we find a $3$\,$\sigma$ residual in terms
of the velocity-integrated flux density around 745.3\,GHz
($I = 0.6\pm0.2$\,Jy\,km\,s$^{-1}$, see Fig.\ref{fig:h2o-ions1}).
This could be a tentative detection of the H$_2^{18}$O(\t211202)
line at 745.320\,GHz. The velocity-integrated flux density ratio
of H$_2^{18}$O(\t211202) over \htot211202 in NCv1.143 would hence
be $\sim0.1$. If this tentative detection was confirmed, it would
show that ALMA could easily study such lines. But sophisticated
models will be needed to infer isotope ratios.

The spectrum of the \htot211202 line in G09v1.97 covers both
the two main {\hbox {H$_2$O$^+$}}\ fine structure lines (Fig\,\ref{fig:h2o-ions1}).
However, due to the
limited sensitivity, we have only tentatively detected the
{\hbox {H$_2$O$^+$}}(\t202111)\,$_{(5/2-3/2)}$ line just above 3\,$\sigma$
(neglecting the minor contribution from {\hbox {H$_2$O$^+$}}(\t211202)\,$_{(5/2-3/2)}$),
and the velocity-integrated flux density is $1.4\pm0.4$\,Jy\,km\,s$^{-1}$ using a
single Gaussian fit. We did not perform any line de-blending for
this source considering the data quality. The {\hbox {H$_2$O$^+$}}\ line
profile is in good agreement with that of the {\hbox {H$_{2}$O}}\ (blue dashed
histogram in Fig.\,\ref{fig:h2op-h2o}). The velocity-integrated flux density
of the undetected {\hbox {H$_2$O$^+$}}(\t211202)\,$_{(5/2-5/2)}$ line could also be
close to this value as discussed in the case of NCv1.143, yet somewhat lower
and not detected in this source. More sensitive observation should
be conducted to further derive robust line parameters.

We have also tentatively detected the {\hbox {H$_2$O$^+$}}(\t211202)\,$_{(5/2-5/2)}$
line in G15v2.779 ($S/N\;\sim 4$ by neglecting the minor contribution
from the {\hbox {H$_2$O$^+$}}(\t202111)\,$_{(3/2-3/2)}$ line). The line
profile is in good agreement with that of \htot211202 (blue dashed
histogram in Fig.\,\ref{fig:h2o-ions1}). The velocity-integrated flux density
derived from a double-peak Gaussian fit is $1.2\pm0.3$\,Jy\,km\,s$^{-1}$ (we did
not perform any line de-blending for the {\hbox {H$_2$O$^+$}}\ doublet considering
the spectral noise level). There could be a minor contribution from
the {\hbox {H$_2$O$^+$}}(\t202111)\,$_{(3/2-3/2)}$ line to the velocity-integrated
flux density. However, such a contribution is likely to be negligible
as in the case of NCv1.143. The contribution is also within the
uncertainty of the velocity-integrated flux density. Nevertheless,
the position of {\hbox {H$_2$O$^+$}}\ has a small blue-shift compared with {\hbox {H$_{2}$O}},
but note that the blue part of the line is cut by the limited
observed bandwidth (yellow histogram).

 \begin{figure}[htbp]
	 \begin{center}
 \includegraphics[scale=0.441]{L_h2op_h2o}
 \caption{
		  Correlation between the luminosity of $J=2$ ortho-{\hbox {H$_2$O$^+$}}\
		  and para-\htot211202. The fitted function is
		  \hbox {$L_{\mathrm{H_2O^+}}$}\;$\propto$\;\hbox {$L_{\mathrm{H_2O}}$}$^{\alpha}$. We found a very good
		  correlation between \hbox {$L_{\mathrm{H_2O^+}}$}\ and \hbox {$L_{\mathrm{H_2O}}$}\ with a slope close
		  to one. Black points are from the local
		  ULIRGs as listed in Table\,\ref{table:htop}. Dark blue
		  ones are \hbox {high-redshift}\ starbursts from this work. Black solid
		  lines indicate the $\chi^2$ fitting results while the
		  grey dashed lines and the grey annotations represent
		  the average ratio between the \hbox {$L_{\mathrm{H_2O^+}}$}\ and \hbox {$L_{\mathrm{H_2O}}$}.
         }
 \label{fig:h2op-h2o}
     \end{center}
 \end{figure}

After including the local detections of {\hbox {H$_2$O$^+$}}\ lines from
\citetalias{2013ApJ...771L..24Y} (Table\,\ref{table:htop}),
we find a tight linear correlation between the luminosity of {\hbox {H$_{2}$O}}\
and the two main lines of {\hbox {H$_2$O$^+$}}\ (slopes equal to $1.03\pm0.06$ and
$0.91\pm0.07$, see Fig.\,\ref{fig:h2op-h2o}). However, one should keep
in mind that, because the local measurement done by {\it Herschel}
SPIRE/FTS \citep{2010SPIE.7731E..16N} has rather low spectral resolution,
neither {\hbox {H$_2$O$^+$}}(\t211202)\,$_{(5/2-3/2)}$ and {\hbox {H$_2$O$^+$}}(\t202111)\,$_{(5/2-3/2)}$,
nor {\hbox {H$_2$O$^+$}}(\t211202)\,$_{(5/2-5/2)}$ and {\hbox {H$_2$O$^+$}}(\t202111)\,$_{(3/2-3/2)}$ can be
spectroscopically resolved. In the correlation plot (Fig.\,\ref{fig:h2op-h2o})
and Table\,\ref{table:htop}, we use the total luminosity from the 742\,GHz and
746\,GHz lines, by assuming the contribution from {\hbox {H$_2$O$^+$}}(\t211202)\,$_{(5/2-3/2)}$
and {\hbox {H$_2$O$^+$}}(\t202111)\,$_{(3/2-3/2)}$ to the velocity-integrated flux density
of the line at 742\,GHz and 746\,GHz is small ($\sim18$\,\%) and does not
vary significantly between different sources. Hence, the velocity-integrated
flux density ratio between each of the two dominant {\hbox {H$_2$O$^+$}}\ fine structure
lines and {\hbox {H$_{2}$O}}\ in NCv1.143, G15v2.779 and G09v1.97 is $\sim 0.3$
(uncertainties are less than 30\%), which is consistent with local
galaxies as shown in the figure. This ratio is much larger than the abundance
ratio of {\hbox {H$_2$O$^+$}}/{\hbox {H$_{2}$O}}\,$\sim$\,0.05 found in Arp\,220, an analogue of \hbox {high-redshift}\
ULIRGs \citep{2011ApJ...743...94R}.

As discussed above, the AGN contribution to the excitation of the submm
lines of most of our sources appears to be minor. Thus, the formation
of {\hbox {H$_2$O$^+$}}\ is likely dominated by cosmic-ray ionization, rather than
X-ray ionization. Given the average luminosity ratio of
{\hbox {H$_2$O$^+$}}/{\hbox {H$_{2}$O}}\;$\sim 0.3\pm0.1$ shown in Fig.\,\ref{fig:h2op-h2o}, \citet{2011A&A...525A.119M}
suggest a cosmic-ray ionization rate of 10$^{-14}$--10$^{-13}$\,s$^{-1}$.
Such high cosmic-ray ionization rates drive the ambient ionization
degree of the ISM to 10$^{-3}$--10$^{-2}$, rather than the canonical
10$^{-4}$. Therefore, in the gas phase, an ion-neutral route likely
dominates the formation of {\hbox {H$_{2}$O}}. However, {\hbox {H$_{2}$O}}\ can also be enriched
through the water-ice sublimation that releases {\hbox {H$_{2}$O}}\ into the gas-phase
ISM. As the upper part, $\sim 90$\,K, of the possible range for
$T_\mathrm{warm}$ is close to the sublimation
temperature of water ice. Hence, the high {\hbox {H$_{2}$O}}\ abundance
(\hbox {$N_{\mathrm{H_2O}}$}\;$\gtrapprox 0.3 \times 10^{17}$\,cm$^{-2}$, see Section\,\ref{hto})
observed is likely to be the result of ion chemistry dominated by
high cosmic-ray ionization and/or perhaps water ice desorption. However,
further observation of {\hbox {H$_2$O$^+$}}\ lines of different transitions and a
larger sample is needed to constrain the contribution to {\hbox {H$_{2}$O}}\
formation from neutral-neutral reactions dominated by shocks.

\section{Conclusions}
\normalsize
\label{Conclusions}
In this paper, we report a survey of submm {\hbox {H$_{2}$O}}\ emission at redshift
$z \sim 2\text{--}4$, by observing a higher excited ortho-\htot321312
in 6 sources and several complementary $J=2$ para-{\hbox {H$_{2}$O}}\ emission lines
in the warm dense cores of 11 \hbox {high-redshift}\ lensed extreme starburst galaxies
(Hy/ULIRGs) discovered by {\it H}-ATLAS. So far, we have detected an
{\hbox {H$_{2}$O}}\ line in most of our observations of a total sample of 17 \hbox {high-redshift}\
lensed galaxies, in other words, we have detected both $J=2$ para-{\hbox {H$_{2}$O}}\ and $J=3$
ortho-{\hbox {H$_{2}$O}}\ lines in five, and in ten other sources only one $J=2$
para-{\hbox {H$_{2}$O}}\ line. In these \hbox {high-redshift}\ Hy/ULIRGs, {\hbox {H$_{2}$O}}\ is the second strongest
molecular emitter after CO within the submm band, as in local ULIRGs.
The spatially integrated {\hbox {H$_{2}$O}}\ emission lines have a velocity-integrated
flux density ranging from 4 to 15\,Jy\,km\,s$^{-1}$, which yields the
apparent {\hbox {H$_{2}$O}}\ emission luminosity, $\mu$\hbox {$L_{\mathrm{H_2O}}$}\, ranging from
$\sim 6\text{--}22 \times 10^{8}$\,{\hbox {$L_\odot$}}. After correction for gravitation
lensing magnification, we obtained the intrinsic \hbox {$L_{\mathrm{H_2O}}$}\ for para-{\hbox {H$_{2}$O}}\
lines \t202111, \t211202 and ortho-\htot321312. The luminosities of the
three {\hbox {H$_{2}$O}}\ lines increase with \hbox {$L_{\mathrm{IR}}$}\ as
\hbox {$L_{\mathrm{H_2O}}$}\;$\propto$\;\hbox {$L_{\mathrm{IR}}$}$^{1.1\text{--}1.2}$. This correlation indicates
the importance of \hbox {far-infrared}\ pumping as a dominant mechanism of {\hbox {H$_{2}$O}}\
excitation. Comparing with $J=3$ to $J=6$ CO lines, the linewidths
between {\hbox {H$_{2}$O}}\ and CO are similar, and the velocity-integrated flux
densities of {\hbox {H$_{2}$O}}\ and CO are comparable. The similarity in line profiles
suggests that these two molecular species possibly trace similar intense
star-forming regions.

Using the \hbox {far-infrared}\ pumping model, we have analysed the ratios between $J=2$ and
$J=3$ {\hbox {H$_{2}$O}}\ lines and \hbox {$L_{\mathrm{H_2O}}$}/\hbox {$L_{\mathrm{IR}}$}\ in 5 sources with both $J$ {\hbox {H$_{2}$O}}\ lines detected.
We have derived the ranges of the warm dust temperature ($T_\mathrm{warm}$), the {\hbox {H$_{2}$O}}\
column density per unit velocity interval (\hbox {$N_{\mathrm{H_2O}}$}/$\Delta V$) and the optical depth
at 100\,$\mu$m ($\tau_{100}$). Although there are strong degeneracies, these
modelling efforts confirm that, similar to
those of local ULIRGs, these submm {\hbox {H$_{2}$O}}\ emissions in \hbox {high-redshift}\ Hy/ULIRGs trace the
warm dense gas that is tightly correlated with the massive star forming activity.
While the values of $T_\mathrm{warm}$ and \hbox {$N_{\mathrm{H_2O}}$}\ (by assuming that they have
similar velocity dispersion $\Delta V$) are similar to the local ones, $\tau_{100}$
in the \hbox {high-redshift}\ Hy/ULIRGs is likely to be greater than 1 (optically thick), which
is larger than $\tau_{100}=0.05\text{--}0.2$ found in the local \hbox {infrared}\ galaxies.
However, we notice that the parameter space is still not well constrained in
our sources through {\hbox {H$_{2}$O}}\ excitation modelling. Due to the limited excitation
levels of the detected {\hbox {H$_{2}$O}}\ lines, we are only able to perform the modelling
with pure \hbox {far-infrared}\ pumping.

The detection of relatively strong {\hbox {H$_2$O$^+$}}\ lines opens the possibility to
help understanding the formation of such large amount of {\hbox {H$_{2}$O}}. In these
\hbox {high-redshift}\ Hy/ULIRGs, the {\hbox {H$_{2}$O}}\ formation is likely to be dominated by ion-neutral
reactions powered by cosmic-ray-dominated regions.
The velocity-integrated flux density ratio between {\hbox {H$_2$O$^+$}}\ and {\hbox {H$_{2}$O}}\
($I_{\mathrm{H_2O^+}}/\hbox {$I_{\mathrm{H_2O}}$} \sim 0.3$), is remarkably constant from low to
high-redshift, reflecting similar conditions in Hy/ULIRGs. However,
more observations of {\hbox {H$_2$O$^+$}}\ emission/absorption and also OH$^{+}$
lines are needed to further constrain the physical parameters of the
cosmic-ray-dominated regions and the ionization rate in those regions.

We have demonstrated that the submm {\hbox {H$_{2}$O}}\ emission lines are strong and
easily detectable with NOEMA. Being a unique diagnostic, the {\hbox {H$_{2}$O}}\
emission offers us a new approach to constrain the physical conditions
in the intense and heavily obscured star-forming regions dominated by
\hbox {far-infrared}\ radiation at \hbox {high-redshift}. Follow-up observations of other gas tracers,
for instance, CO, HCN, {\hbox {H$_2$O$^+$}}\ and OH$^+$ using the NOEMA, IRAM 30m and JVLA
will complement the {\hbox {H$_{2}$O}}\ diagnostic of the structure of different
components, dominant physical processes, star formation and chemistry
in \hbox {high-redshift}\ Hy/ULIRGs.

With unprecedented spatial resolution and sensitivity, the image from
the ALMA long baseline campaign observation of SDP\,81 \citep[also known
as {\it H}-ATLAS\,J090311.6+003906, ][]{2015ApJ...808L...4A, 2015MNRAS.452.2258D,
2015MNRAS.451L..40R}, shows the resolved structure of the dust, CO and
{\hbox {H$_{2}$O}}\ emission in the $z=3$ ULIRG. With careful reconstruction of the
source plane images, ALMA will help to resolve the submm {\hbox {H$_{2}$O}}\ emission
in \hbox {high-redshift}\ galaxies into the scale of giant molecular clouds, and
provide a fresh view of detailed physics and chemistry in the early
Universe.

\begin{acknowledgement}
We thank our referee for the very detail comments and suggestions
which have improved the paper. This work was based on observations
carried out with the IRAM Interferometer
NOEMA, supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain).
The authors are grateful to the IRAM staff for their support. CY thanks
Claudia Marka and Nicolas Billot for their help of the IRAM 30m/EMIR
observation. CY also thanks Zhi-Yu Zhang and Iv\'an Oteo for insightful
discussions. CY, AO and YG acknowledge support by NSFC grants \#11311130491,
\#11420101002 and CAS Pilot B program \#XDB09000000. CY and YG also
acknowledge support by NSFC grants \#11173059. CY, AO, AB
and YG acknowledge support from the Sino-French LIA-Origin joint exchange
program. E.G-A is a Research Associate at the Harvard-Smithsonian
Center for Astrophysics, and thanks the Spanish Ministerio de
Econom\'{\i}a y Competitividad for support under projects
FIS2012-39162-C06-01 and  ESP2015-65597-C4-1-R, and NASA grant ADAP
NNX15AE56G. RJI acknowledges support from ERC in the form of the Advanced
Investigator Programme, 321302, COSMICISM. US participants in {\it H}-ATLAS
acknowledge support from NASA through a contract from JPL. Italian
participants in {\it H}-ATLAS acknowledge a financial contribution
from the agreement ASI-INAF I/009/10/0. SPIRE has been developed by
a consortium of institutes led by Cardiff Univ. (UK) and including:
Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI,
Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden);
Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK);
and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has
been supported by national funding agencies: CSA (Canada); NAOC (China);
CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden);
STFC, UKSA (UK); and NASA (USA). CY is supported by the China
Scholarship Council grant (CSC No.201404910443).
\end{acknowledgement}

\footnotesize
\bibliographystyle{aa}