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**Atmospheric Chemistry and Physics**
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- Abstract
- Introduction
- Theoretical framework
- Model simulations, observations, and methods
- Results from LMDZ
- Results from observations
- Discussion: what can we learn from water isotopes on mixing processes?
- Conclusions
- Code and data availability
- Appendix A: Closure if the tropospheric profile follows a mixing line
- Appendix B: Diagnostics for rain evaporation in LMDZ
- Appendix C: Diagnostics for horizontal advection in LMDZ
- Appendix D: LMDZ free-tropospheric profiles
- Author contributions
- Competing interests
- Acknowledgements
- Review statement
- References

**Research article**
02 Oct 2019

**Research article** | 02 Oct 2019

Controls on the water vapor isotopic composition near the surface of tropical oceans and role of boundary layer mixing processes

^{1}Laboratoire de Météorologie Dynamique, IPSL, CNRS, Sorbonne Université, Paris, France^{2}Department of Earth and Planetary Sciences, University of New Mexico, Albuquerque, USA^{3}Sorbonne Université, CNRD/IRD/MNHN, LOCEAN, IPSL, Paris, France^{4}CNRM, Université de Toulouse, Météo-France, CNRS, Toulouse, France

^{1}Laboratoire de Météorologie Dynamique, IPSL, CNRS, Sorbonne Université, Paris, France^{2}Department of Earth and Planetary Sciences, University of New Mexico, Albuquerque, USA^{3}Sorbonne Université, CNRD/IRD/MNHN, LOCEAN, IPSL, Paris, France^{4}CNRM, Université de Toulouse, Météo-France, CNRS, Toulouse, France

**Correspondence**: Camille Risi (camille.risi@lmd.jussieu.fr)

**Correspondence**: Camille Risi (camille.risi@lmd.jussieu.fr)

Abstract

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Understanding what controls the water vapor isotopic composition of
the sub-cloud layer (SCL) over tropical oceans (*δ**D*_{0}) is
a first step towards understanding the water vapor isotopic composition
everywhere in the troposphere. We propose an analytical model to predict
*δ**D*_{0} motivated by the hypothesis that the altitude from
which the free tropospheric air originates (*z*_{orig}) is an important
factor: when the air mixing into the SCL is lower in altitude, it
is generally moister, and thus it depletes the SCL more efficiently.
We extend previous simple box models of the SCL by prescribing the
shape of *δ**D* vertical profiles as a function of humidity profiles
and by accounting for rain evaporation and horizontal advection effects.
The model relies on the assumption that *δ**D* profiles are steeper
than mixing lines, and that the SCL is at steady state, restricting
its applications to timescales longer than daily. In the model, *δ**D*_{0}
is expressed as a function of *z*_{orig}, humidity and temperature
profiles, surface conditions, a parameter describing the steepness
of the *δ**D* vertical gradient, and a few parameters describing
rain evaporation and horizontal advection effects. We show that *δ**D*_{0}
does not depend on the intensity of entrainment, in contrast to several
previous studies that had hoped that *δ**D*_{0} measurements
could help estimate this quantity.

Based on an isotope-enabled general circulation model simulation,
we show that *δ**D*_{0} variations are mainly controlled by mid-tropospheric
depletion and rain evaporation in ascending regions and by sea surface
temperature and *z*_{orig} in subsiding regions. In turn, could *δ**D*_{0}
measurements help estimate *z*_{orig} and thus discriminate between
different mixing processes? For such isotope-based estimates of *z*_{orig}
to be useful, we would need a precision of a few hundred meters in
deep convective regions and smaller than 20 m in stratocumulus regions.
To reach this target, we would need daily measurements of *δ**D*
in the mid-troposphere and accurate measurements of *δ**D*_{0}
(accuracy down to 0.1 ‰ in the case of stratocumulus clouds,
which is currently difficult to obtain). We would also need information
on the horizontal distribution of *δ**D* to account for horizontal
advection effects, and full *δ**D* profiles to quantify the uncertainty
associated with the assumed shape for *δ**D* profiles. Finally,
rain evaporation is an issue in all regimes, even in stratocumulus
clouds. Innovative techniques would need to be developed to quantify
this effect from observations.

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How to cite.

Risi, C., Galewsky, J., Reverdin, G., and Brient, F.: Controls on the water vapor isotopic composition near the surface of tropical oceans and role of boundary layer mixing processes, Atmos. Chem. Phys., 19, 12235–12260, https://doi.org/10.5194/acp-19-12235-2019, 2019.

1 Introduction

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The water vapor isotopic composition (e.g., $\mathit{\delta}D=\left(R/{R}_{\mathrm{SMOW}}-\mathrm{1}\right)\times \mathrm{1000}$
expressed in per mill, where *R* is the *D*∕*H* ratio and SMOW is the
Standard Mean Ocean Water reference), has been shown to be sensitive
to a wide range of atmospheric processes (Galewsky et al., 2016),
such as continental recycling (Salati et al., 1979; Risi et al., 2013);
unsaturated downdrafts (Risi et al., 2008, 2010a);
rain evaporation (Worden et al., 2007; Field et al., 2010);
the degree of organization of convection (Lawrence et al., 2004; Tremoy et al., 2014);
the convective depth (Lacour et al., 2017b); the proportion of
precipitation that occurs as convective or large-scale precipitation
(Lee et al., 2009; Kurita, 2013; Aggarwal et al., 2016);
vertical mixing in the lower troposphere (Benetti et al., 2015; Galewsky, 2018a, b),
mid-troposphere (Risi et al., 2012b) or upper-troposphere
(Galewsky and Samuels-Crow, 2014); convective detrainment (Moyer et al., 1996; Webster and Heymsfield, 2003); and
ice microphysics (Bolot et al., 2013). It is therefore
very challenging to quantitatively understand what controls the isotopic
composition of water vapor.

A first step towards this goal is to understand what controls the
water vapor isotopic composition in the sub-cloud layer (SCL) of tropical
(30^{∘} S–30^{∘} N) oceans. Indeed, this water vapor is
an important source moistening air masses traveling to land regions
(Gimeno et al., 2010; Ent and Savenije, 2013)
and towards higher latitudes (Ciais et al., 1995; Delaygue et al., 2000).
It is also ultimately the only source of water vapor in the tropical
free troposphere since water vapor in the free troposphere ultimately
originates from convective detrainment (Sherwood, 1996), and
convection ultimately feeds from the SCL air (Bony et al., 2008).
Therefore, the water vapor isotopic composition in the SCL of tropical
oceans serves as initial conditions to understand the isotopic composition
in land waters and in the tropospheric water vapor everywhere on Earth.
We focus here on the SCL because, by definition, there is no complication
by cloud condensation processes.

The goal of this paper is thus to propose a simple analytical equation
that allows us to understand and quantify the factors controlling
the *δ**D* in the water vapor in the SCL of tropical oceans.
So far, the most famous analytical equation for this purpose has been
the closure equation developed by Merlivat and Jouzel (1979) (MJ79).
This closure equation can be derived by assuming that all the water
vapor in the SCL air originates from surface evaporation. The water
balance of the SCL can be closed by assuming a mass export at the
SCL top (e.g., by convective mass fluxes) and a totally dry entrainment
into the SCL to compensate for this mass export. The MJ79 equation has
proven very useful to capture the sensitivity of *δ**D* and second-order
parameter d-excess to sea surface conditions (Merlivat and Jouzel, 1979; Ciais et al., 1995; Risi et al., 2010d).
However, the *δ**D* calculated from this equation suffers from
a high bias in tropical regions (Jouzel and Koster, 1996). This bias
can be explained by the neglect of vertical mixing between the SCL
and air entrained from the free troposphere (FT). The MJ79 equation
can better reproduce surface water vapor observation when extended
to take into account this mixing (Benetti et al., 2015, hereafter
B15). This extension requires us to know the specific humidity (*q*)
and water vapor *δ**D* of the entrained air. To get these values,
they assume that the air entrained into the boundary layer comes from
a constant altitude. However, this does not reflect the complexity
of entrainment and mixing processes in marine boundary layers.

Figure 1 summarizes our knowledge about these entrainment and mixing processes. In stratocumulus regions, clouds are thin and the inversion is just above the lifting condensation level (LCL). Air is entrained from the FT by cloud-top entrainment driven by radiative cooling or wind shear instabilities (Mellado, 2017), possibly amplified by evaporative cooling of droplets (Lozar and Mellado, 2015). Both direct numerical simulations (Mellado, 2017) and observations of tracers (Faloona et al., 2005) and cloud holes (Gerber et al., 2005) show that air is entrained from a thin layer above the inversion, thinner than 80 m and as small as 5 m. The boundary layer itself is animated by updrafts, downdrafts, and associated turbulent shells that bring air from the cloud layer downward (Brient et al., 2019; Davini et al., 2017).

In trade-wind cumulus regions, the cloudy layer is a bit deeper. Observational studies and large-eddy simulations have pointed out the important role of thin subsiding shells around cumulus clouds, driven by cloud-top radiative cooling, mixing, and evaporative cooling of droplets (Jonas, 1990; Rodts et al., 2003; Heus and Jonker, 2008; Heus et al., 2009; Park et al., 2016). This brings air from the cloudy layer to the SCL. Subsiding shells may also cover overshooting plumes of the cumulus clouds, entraining FT air into the cloud layer (Heus and Jonker, 2008).

In deep convective regions, unsaturated downdrafts driven by rain evaporation (Zipser, 1977) are known to contribute significantly to the energy budget of the SCL (Emanuel et al., 1994). Large-eddy simulations show that subsiding shells, similar to those documented in shallow convection, also exist around deep convective clouds (Glenn and Krueger, 2014). In the clear-sky environment between clouds, turbulent entrainment into the SCL may also play a significant role (Thayer-Calder and Randall, 2015).

Therefore, whatever the cloud regime, air entering the SCL from above
may originate from either the cloud layer or the free troposphere,
depending on the mixing mechanism. Therefore, in this paper in contrast
with B15, we let the altitude from which the air originates, *z*_{orig},
be variable. We do not call it “entrained” air because entrainment
sometimes refers to mixing processes through an interface (e.g., De Rooy et al., 2013; Davini et al., 2017),
whereas air in the SCL may also enter through deep, coherent, and penetrative
structures such as unsaturated downdrafts. We do not call it FT air
either since it may originate from the cloudy layer.

To acknowledge the diversity and complexity of mixing mechanisms,
we extend the B15 framework in several ways. First, we assume that
we know the shape of *δ**D* profiles as a function of *q*. Second,
we write the specific humidity of the air originating from above the
SCL as a function of *z*_{orig}. Third, we account for rain evaporation
and horizontal effects.

While B15 focused on observations during a field campaign, we also
apply the extended equation to global outputs of an isotope-enabled
general circulation model, with the aim to quantify the different
factors controlling the *δ**D* variability in the tropics. The
variable *z*_{orig} will emerge as an important factor. Therefore,
we discuss the possibility that *δ**D* measurements at the near
surface and through the lower FT could help estimate *z*_{orig}
and thus the mixing processes between the SCL and the air above.

Note that we focus on *δ**D* only. Results for *δ*^{18}O
are similar. We do not aim at capturing the second-order parameter
d-excess because our model requires some knowledge about free-tropospheric
vertical profiles of isotopic composition. While *δ**D* is known
to decrease with altitude (Ehhalt, 1974; Ehhalt et al., 2005; Sodemann et al., 2017),
vertical profiles of d-excess are more diverse and less well understood
(Sodemann et al., 2017). In addition, there is more need for an
extension of MJ79 for *δ**D* than for d-excess
since the effect of convective mixing is larger on *δ**D* than
on d-excess (Risi et al., 2010d; Benetti et al., 2014).

2 Theoretical framework

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Building on Benetti et al. (2014) and B15, we consider a simple
box representing the SCL (Fig. 2). We assume
that the air comes from above (*M*) and from the incoming large-scale
horizontal advection (*F*_{adv}) and is exported through the SCL
top (*N*, e.g., turbulent mixing or convective mass flux) and by outgoing
large-scale horizontal advection (*F*_{adv,out}). We assume that
the SCL is at steady state. For example, its depth is constant. Since
the SCL properties may exhibit a diurnal cycle (Duynkerke et al., 2004),
this hypothesis restricts the application of this model to timescales
longer than daily. The air mass budget of the SCL is thus

$$\begin{array}{}\text{(1)}& M+{F}_{\mathrm{adv}}=N+{F}_{\mathrm{adv},\mathrm{out}}.\end{array}$$

These fluxes also transport water vapor and isotopes. In addition,
surface evaporation *E* and rain evaporation *F*_{evap} import water
vapor and isotopes (Fig. 2).

Hereafter, to simplify equations, we use the isotopic ratio *R* instead
of *δ**D*.

The SCL is usually well mixed (Betts and Ridgway, 1989; Stevens, 2006; De Roode et al., 2016).
We thus assume that the humidity and isotopic properties are constant
vertically and horizontally in the SCL. They are noted (*q*_{0},
*R*_{0}). The humidity and isotopic properties of the mass flux export
*N* are thus also (*q*_{0}, *R*_{0}). The properties of the flux
*M* are noted (*q*_{orig}, *R*_{orig}). The properties of the incoming
air by horizontal advection are noted (*q*_{adv}, *R*_{adv}). For
simplicity here we neglect the effect of horizontal gradients in humidity
(i.e., *q*_{adv}=*q*_{0}), assuming that the main effect of horizontal
advection on *δ**D*_{0} arises from horizontal gradients in *δ**D*.
Appendix C explains how *R*_{adv} can be calculated.
At steady state, the water budget of the SCL is written

$$\begin{array}{}\text{(2)}& M\cdot {q}_{\mathrm{orig}}+E+{F}_{\mathrm{evap}}+{F}_{\mathrm{adv}}\cdot {q}_{\mathrm{0}}=(N+{F}_{\mathrm{adv},\mathrm{out}})\cdot {q}_{\mathrm{0}}.\end{array}$$

This model is consistent with SCL water budgets that have already
been derived in previous studies (Bretherton et al., 1995), except
that we consider steady state. This equation can be solved for *q*_{0}:

$$\begin{array}{}\text{(3)}& {q}_{\mathrm{0}}={q}_{\mathrm{orig}}+{\displaystyle \frac{E+{F}_{\mathrm{evap}}}{M}}.\end{array}$$

The SCL humidity *q*_{0} is thus sensitive to *M*, justifying that
it can be used to estimate the mixing intensity or the “entrainment
velocity” ${w}_{\mathrm{e}}=M/{\mathit{\rho}}_{\mathrm{0}}$ (*ρ* being the air volumic mass)
(Bretherton et al., 1995).

At steady state, the water isotope budget of the SCL is written

$$\begin{array}{}\text{(4)}& \begin{array}{rl}& M\cdot {q}_{\mathrm{orig}}\cdot {R}_{\mathrm{orig}}+E\cdot {R}_{\mathrm{E}}+{F}_{\mathrm{evap}}\cdot {R}_{\mathrm{evap}}+{F}_{\mathrm{adv}}\cdot {q}_{\mathrm{0}}\cdot {R}_{\mathrm{adv}}\\ & =(N+{F}_{\mathrm{adv},\mathrm{out}})\cdot {q}_{\mathrm{0}}\cdot {R}_{\mathrm{0}},\end{array}\end{array}$$

where *R*_{E} is the isotopic composition of the surface evaporation.
It is assumed to follow the Craig and Gordon (1965) equation:

$$\begin{array}{}\text{(5)}& {R}_{\mathrm{E}}={\displaystyle \frac{{R}_{\mathrm{oce}}/{\mathit{\alpha}}_{\mathrm{eq}}-{h}_{\mathrm{0}}\cdot {R}_{\mathrm{0}}}{{\mathit{\alpha}}_{\mathrm{K}}\cdot (\mathrm{1}-{h}_{\mathrm{0}})}},\end{array}$$

where *R*_{oce} is the isotopic ratio in the surface ocean water,
*α*_{eq} is the equilibrium fractionation calculated at the
sea surface temperature (SST) (Majoube, 1971), *α*_{K}
is the kinetic fractionation coefficient (MJ79), and *h*_{0} is the
relative humidity normalized at the SST (${h}_{\mathrm{0}}={q}_{\mathrm{0}}/{q}_{\mathrm{s}}(\mathrm{SST},{P}_{\mathrm{0}})$,
where *q*_{s} is the saturation-specific humidity at SST and *P*_{0}
is the surface pressure).

We write the isotopic composition of the rain evaporation, *R*_{evap},
as

$${R}_{\mathrm{evap}}={\mathit{\alpha}}_{\mathrm{evap}}\cdot {R}_{\mathrm{0}},$$

where *α*_{evap} is an effective fractionation coefficient.
For example, if droplets are formed near the cloud base, some of them
precipitate and evaporate totally into the SCL (e.g., in non-precipitating
shallow cumulus clouds), then *α*_{evap}=*α*(*T*_{cloud base}).
In contrast, if droplets are formed in deep convective updrafts after
total condensation of the SCL vapor, and then a very small fraction
of the rain is evaporated into a very dry SCL, then ${\mathit{\alpha}}_{\mathrm{evap}}=\mathrm{1}/\mathit{\alpha}\left({T}_{\mathrm{SCL}}\right)/{\mathit{\alpha}}_{\mathrm{K}}$
(Stewart, 1975).

We note $\mathit{\eta}={F}_{\mathrm{evap}}/E$ the ratio of water vapor coming from rain evaporation to that of surface evaporation, and $\mathit{\varphi}={F}_{\mathrm{adv}}\cdot {q}_{\mathrm{adv}}/E$ the ratio of water vapor coming from horizontal advection to that coming from surface evaporation. We note $\mathit{\beta}={R}_{\mathrm{adv}}/{R}_{\mathrm{0}}$ the ratio of isotopic ratios of horizontal advection to that of the SCL.

Note that in all our equations, we assume that temperature and humidity
profiles and all basic surface meteorological variables are known.
We do not attempt to express either *h*_{0} as a function of *q*_{0} as
in B15 or the *q* profile as a function of *q*_{0}. Our ultimate
goal is to assess the added value of *δ**D* assuming that meteorological
measurements are already routinely performed.

By combining all these equations, we get

$$\begin{array}{}\text{(6)}& \begin{array}{rl}& {R}_{\mathrm{0}}=\\ & {\displaystyle \frac{(\mathrm{1}-{r}_{\mathrm{orig}})\cdot {R}_{\mathrm{oce}}/{\mathit{\alpha}}_{\mathrm{eq}}+{\mathit{\alpha}}_{\mathrm{K}}\cdot (\mathrm{1}-{h}_{\mathrm{0}})\cdot {r}_{\mathrm{orig}}\cdot \left(\mathrm{1}+\mathit{\eta}\right)\cdot {R}_{\mathrm{orig}}}{(\mathrm{1}-{r}_{\mathrm{orig}})\cdot {h}_{\mathrm{0}}+{\mathit{\alpha}}_{\mathrm{K}}\cdot (\mathrm{1}-{h}_{\mathrm{0}})\cdot \left(\mathrm{1}+\mathit{\eta}+(\mathrm{1}-{r}_{\mathrm{orig}})\cdot (\mathit{\varphi}\cdot (\mathrm{1}-\mathit{\beta})-\mathit{\eta}\cdot {\mathit{\alpha}}_{\mathrm{evap}}\right)}}\end{array},\end{array}$$

where ${r}_{\mathrm{orig}}={q}_{\mathrm{orig}}/{q}_{\mathrm{0}}$ is the proportion of the water vapor in the SCL that originates from above.

An intriguing aspect of this equation is that the sensitivity to *M*
disappears. In contrast to *q*_{0}, *R*_{0} is not sensitive to
*M*. Therefore, it appears illusory to promise that water vapor isotopic
measurements could help constrain the entrainment velocity that many
studies have striven to estimate (Nicholls and Turton, 1986; Khalsa, 1993; Wang and Albrecht, 1994; Bretherton et al., 1995; Faloona et al., 2005; Gerber et al., 2005, 2013).
The lack of sensitivity of *R*_{0} to *M* is explained physically
by the fact that for a given *q*_{0} and *q*_{orig}, if *M* increases,
then *E*+*F*_{evap} increases in the same proportion to maintain the
water balance. Therefore, the relative proportion of the water vapor
originating from surface and rain evaporation to that coming from
above, to which *R*_{0} is sensitive, remains constant. Rather, since
*q* and *R* vary with altitude, *R*_{0} is sensitive to the altitude
from which the air originates.

Equation (6) requires knowing *q*_{orig} and *R*_{orig}.
B15 take these values from general circulation model (GCM) outputs at 700 hPa. In contrast, here
we acknowledge the diversity and complexity of mixing mechanisms by
keeping the possibility to take *q*_{orig} and *R*_{orig} at a variable
altitude *z*_{orig}.

If the goal is to predict *R*_{0} from *z*_{orig}, we can apply
Eq. (6) if we know the *q* and *δ**D* vertical
profiles. Conversely, if the goal is to predict *z*_{orig} from *R*_{0},
we can numerically solve Eq. (6) if we know the *q*
and *δ**D* vertical profiles. No analytical solution exists in
the general case, but a numerical solution can be searched for the *z*_{orig}
based on Eq. (6). However, the existence and unicity
of the solution is not warranted for all kinds of profiles (e.g., Appendix A).

In practice, full isotopic profiles are costly to measure. In addition,
our goal is to develop an analytical model. Therefore, in the following
we simplify the problem by assuming that the vertical profile of *R*
follows a known relationship as a function of *q*. Measured vertical
profiles of *δ**D* are usually bounded by two curves when plotted
in a (*q*, *δ**D*) diagram (Sodemann et al., 2017):
Rayleigh distillation curve and mixing line.

First, we explore the case of a Rayleigh distillation curve (Dansgaard, 1964), as in Galewsky and Rabanus (2016):

$$\begin{array}{}\text{(7)}& {R}_{\mathrm{orig}}={R}_{\mathrm{0}}\cdot {r}_{\mathrm{orig}}^{{\mathit{\alpha}}_{\mathrm{eff}}-\mathrm{1}},\end{array}$$

where *α*_{eff} is an effective fractionation coefficient. Typically,
*q* decreases with altitude, so *R* also decreases with altitude.
However, in observations and models, vertical profiles of *R* can
be very diverse (Bony et al., 2008; Sodemann et al., 2017).
The water vapor may be more (Worden et al., 2007) or less (Sodemann et al., 2017)
depleted than predicted by a Rayleigh curve using a realistic fractionation
factor that depends on local temperature. Therefore, here we let *α*_{eff}
be a free parameter larger than 1. Rather than assuming a true Rayleigh
curve, we simply assume that *R* and *q* are logarithmically related.
Effects of horizontal advection and rain evaporation on tropospheric
profiles are encapsulated into *α*_{eff}.

Injecting Eq. (7) into Eq. (6), we get

$$\begin{array}{}\text{(8)}& \begin{array}{rl}& {R}_{\mathrm{0}}={\displaystyle \frac{{R}_{\mathrm{oce}}}{{\mathit{\alpha}}_{\mathrm{eq}}}}\\ & \cdot {\displaystyle \frac{\mathrm{1}}{{h}_{\mathrm{0}}+{\mathit{\alpha}}_{\mathrm{K}}\cdot (\mathrm{1}-{h}_{\mathrm{0}})\cdot \left((\mathrm{1}+\mathit{\eta})\cdot \frac{\mathrm{1}-{r}_{\mathrm{orig}}^{{\mathit{\alpha}}_{\mathrm{eff}}}}{\mathrm{1}-{r}_{\mathrm{orig}}}-\mathit{\eta}\cdot {\mathit{\alpha}}_{\mathrm{evap}}+\mathit{\varphi}\cdot (\mathrm{1}-\mathit{\beta})\right)}}\end{array}.\end{array}$$

A simpler form can be found if neglecting horizontal advection and rain evaporation effects ($\mathit{\varphi}=\mathit{\eta}=\mathrm{0}$):

$$\begin{array}{}\text{(9)}& {R}_{\mathrm{0}}={\displaystyle \frac{{R}_{\mathrm{oce}}}{{\mathit{\alpha}}_{\mathrm{eq}}}}\cdot {\displaystyle \frac{\mathrm{1}}{{h}_{\mathrm{0}}+{\mathit{\alpha}}_{\mathrm{K}}\cdot (\mathrm{1}-{h}_{\mathrm{0}})\cdot \frac{\mathrm{1}-{r}_{\mathrm{orig}}^{{\mathit{\alpha}}_{\mathrm{eff}}}}{\mathrm{1}-{r}_{\mathrm{orig}}}}}.\end{array}$$

As a consistency check, in the limit case where the air coming from
above is totally dry (*r*_{orig}=0), Eq. (9) becomes
the MJ79 equation:

$$\begin{array}{}\text{(10)}& {R}_{\mathrm{0}}={\displaystyle \frac{{R}_{\mathrm{oce}}}{{\mathit{\alpha}}_{\mathrm{eq}}}}\cdot {\displaystyle \frac{\mathrm{1}}{{h}_{\mathrm{0}}+{\mathit{\alpha}}_{\mathrm{K}}\cdot (\mathrm{1}-{h}_{\mathrm{0}})}}.\end{array}$$

Equation (8) tells us that whenever *α*_{eff}>1,
*R*_{0} decreases as *r*_{orig} increases (Fig. 3
red), i.e., as *q*_{orig} is moister. Therefore, *R*_{0} decreases
as *z*_{orig} is lower in altitude. This result may be counterintuitive,
but can be physically interpreted as follows. If *z*_{orig} is high,
mixing brings air with very depleted water vapor, but since the air
is dry, the depleting effect is small. In contrast, if *z*_{orig}
is low, mixing brings air with water vapor that is not very depleted,
but since the air is moist, the depleting effect is large (Fig. 4a).

Figure 3 (red) shows that the range of possible
*δ**D* values is restricted to −70 ‰ to −85 ‰.
This explains why in quiescent conditions near the sea level in tropical
ocean locations, the water vapor *δ**D* varies little (Benetti et al. (2014),
Françoise Vimeux, personal communication, 2018). In the limit case where *r*_{orig}→1
(i.e., the air comes from the SCL top), ${R}_{\mathrm{0}}\to \frac{{R}_{\mathrm{oce}}}{{\mathit{\alpha}}_{\mathrm{eq}}}\cdot \frac{\mathrm{1}}{{h}_{\mathrm{0}}+{\mathit{\alpha}}_{\mathrm{K}}\cdot (\mathrm{1}-{h}_{\mathrm{0}})\cdot {\mathit{\alpha}}_{\mathrm{eff}}}$
(L'Hôpital's rule was used to calculate this limit). This lower bound
is not so depleted compared to the more depleted water vapor observed
in regions of deep convection (e.g., Lawrence et al., 2002, 2004; Kurita, 2013).
This is because when *r*_{orig}→1, the water vapor coming
from above has a composition very close to that of the SCL, so the
depleting effect is limited. In addition, surface evaporation strongly
damps the depleting effect of mixing. Only rain evaporation or liquid–vapor
exchanges (Lawrence et al., 2004; Worden et al., 2007) can further
decrease *R*_{0} (Appendix B).

Figure 3 (green) shows that the sensitivity to
*α*_{eff} is relatively small but cannot be neglected. Therefore,
predicting *δ**D*_{0} requires having some knowledge about the
steepness of the isotopic profiles in the FT. Rain evaporation and
horizontal advection can have either an enriching or depleting effect,
but do not qualitatively change the results (Fig. 3
purple and blue).

Now we consider the case of a mixing line. Detailed calculations in
Appendix A show that the sensitivity
to *r*_{orig} is lost. An infinity of FT end members can lead to
the same *δ**D*_{0} when mixed with the surface evaporation,
as illustrated in Fig. 4b and analytically
demonstrated in Appendix A. Our main
results (more depleted *δ**D*_{0} as *r*_{orig} increases, restricted
range of *δ**D*_{0} variations, relationship with *z*_{orig})
hold only for *δ**D* profiles that are steeper than a mixing
line. This is the case for profiles that are intermediate between
a Rayleigh and a mixing line, as is usually the case in nature (Sodemann et al., 2017)
or in a general circulation model (Appendix D1).

3 Model simulations, observations, and methods

Back to toptop
We use an isotope-enabled general circulation model (GCM) as a laboratory
to test our hypotheses and investigate what controls the isotopic
composition. We use the LMDZ5A version of LMDZ (Laboratoire de Météorologie
Dynamique Zoom), which is the atmospheric component of the IPSL–CM5A
coupled model (Dufresne et al., 2012) that took part in CMIP5
(Coupled Model Intercomparison Project; Taylor et al., 2012).
This version is very close to LMDZ4 (Hourdin et al., 2006). Water
isotopes are implemented the same way as in the predecessor LMDZ4
(Risi et al., 2010c). We use 4 years (2009–2012) of a simulation of the AMIP (Atmospheric
Model Intercomparison Project) (Gates, 1992)
that was initialized in 1977. The winds are nudged towards ERA-40
reanalyses (Uppala et al., 2005) to ensure a more realistic simulation.
Such a simulation has already been described and extensively validated
for isotopic variables in both precipitation and water vapor (Risi et al., 2010c, 2012a).
The ocean surface water *δ**D*_{oce} is assumed constant and
set to 4 ‰. The resolution is 2.5^{∘} in latitude
by 3.75^{∘} in longitude, with 39 vertical levels. Over
the ocean, the first layer extends up to 64 m, and a typical SCL
extending up to 600 m is resolved by six layers. Around 2500 m, a
typical altitude for the inversion for trade-wind cumulus clouds,
the resolution is about 500 m.

For our calculations, we only use tropical grid boxes (30^{∘} S–30^{∘} N)
over tropical oceans (>80 % ocean fraction in the grid box). In addition,
to avoid numerical problems when estimating the effect of horizontal advection
and rain evaporation, only grid boxes and days where *E*>0.5 mm d^{−1}
are considered. This represents 99.7 % of all tropical oceanic grid
boxes.

Specific diagnostics for horizontal advection and rain evaporation are detailed in Appendix B and C.

We also apply our theoretical framework to observations during the
STRASSE (Sub-Tropical Atlantic Surface Salinity Experiment) cruise
that took place in the northern subtropical ocean in August and September
2012 (Benetti et al., 2014). This campaign accumulates several
advantages that are important for our analysis: (1) continuous *δ**D*_{0}
measurements in the surface water vapor (17 m) at a high temporal frequency
during 1 month (Benetti et al., 2014, 2015, 2017b),
(2) associated surface meteorological measurements, including SST
and *h*_{0}, (3) 22 radio soundings relatively well distributed over
the campaign period and providing vertical profiles of altitude, temperature,
relative humidity and pressure, (4) ocean surface water *δ**D*_{oce}
measurements (Benetti et al., 2017a), (5) a variety of
conditions ranging from quiescent weather to convective conditions,
(6) on many vertical profiles, a well defined temperature inversion
allows to calculate the inversion altitude.

We use *δ**D*_{0} measurements on a 15 min time step. The
measurements in ocean water were interpolated on the same time steps
using a Gaussian filter with a width of 3 d. The radio-soundings
are used together with all water vapor isotopic measurements that
are within 30 min of the radio-sounding launch. Only profiles
during the ascending phase of the balloon are considered because
the descent phase is often located far away from the initial launch
point (McGrath et al., 2006; Seidel et al., 2011).

Here we explain how *z*_{orig} is estimated based on LMDZ outputs.
First, we assume that the *q* and *δ**D* at 500 hPa (*q*_{f},
*δ**D*_{f}) belong to a Rayleigh distillation line starting from
the surface with effective fractionation *α*_{eff}:

$${\mathit{\alpha}}_{\mathrm{eff}}=\mathrm{1}+{\displaystyle \frac{\mathrm{ln}({R}_{\mathrm{f}}/{R}_{\mathrm{0}})}{\mathrm{ln}({q}_{\mathrm{f}}/{q}_{\mathrm{0}})}}.$$

In a real field campaign, this assumption means that we do not need
to measure the full vertical profile of *δ**D*, but only *δ**D*_{f}
at a given free-tropospheric altitude (e.g., 500 hPa).

We checked that results are similar when defining the end member at
400 hPa rather than 500 hPa. However, the end member should be defined
above 500 hPa to ensure that it is well above boundary layer processes.
If the end member is defined below 500 hPa (e.g., 600 hPa), there
are a few cases where *q* increases with altitude (*q*_{f}>*q*_{0})
due to horizontal advection or convective detrainment from nearby
moister regions; meanwhile, *δ**D* decreases monotonically, leading
to unrealistic values for *α*_{eff}.

Second, *r*_{orig} is estimated based on Eq. (9),
using *α*_{eff}, *α*_{eq}, *α*_{K} , *δ**D*_{oce},
*h*_{0}, and *δ**D*_{0} simulated by LMDZ.

Third, the altitude *z*_{orig} is estimated from *r*_{orig}. Using
the *q* vertical profile, we find *z*_{orig} so that $q\left({z}_{\mathrm{orig}}\right)={r}_{\mathrm{orig}}\cdot {q}_{\mathrm{0}}$
(Fig. 5, red).

When estimating *z*_{orig} from observations, we follow the same
methodology except that in absence of measurements for *q*_{f} and
*δ**D*_{f} we assume a constant *α*_{eff}=1.07 based on
LMDZ simulation and that *α*_{eq}, *α*_{K} , *δ**D*_{oce},
*h*_{0}, and *δ**D*_{0} come from surface observations.

Note that *r*_{orig} and *z*_{orig} are not direct diagnostics from
the simulation, but rather a posteriori estimates to match the simulated
*δ**D*_{0}. Therefore, if assumptions underlying Eq. (9)
are violated, then the estimate of *r*_{orig}, and subsequently *z*_{orig},
will be biased. The estimate of *r*_{orig} encapsulates the effect
of mixing processes, but also all other processes that have been neglected
in our theoretical framework, such as temporal variations in SCL depth,
*q*_{0}, or *δ**D*_{0} or vertical variations in *q*_{0} or
*δ**D*_{0} within the SCL.

Figure 5 illustrates the structure of a typical tropical marine boundary layer covered by stratocumulus or cumulus clouds (Betts and Ridgway, 1989; Wood, 2012; Wood and Bretherton, 2004; Neggers et al., 2006; Stevens, 2006). The cloud base corresponds to the lifting condensation level (LCL). Below is the well mixed SCL. Above is the cloud layer, topped by a temperature inversion. Above the inversion is the FT.

The LCL is calculated as the altitude at which the specific humidity near the surface equals the specific humidity at saturation of a parcel that is lifted following a dry adiabat (Fig. 5).

The temperature inversion is an abrupt increase in temperature that
caps the boundary layer. Therefore, a method to automatically estimate
its altitude *z*_{i} is to detect a maximum in the vertical gradient
of potential temperature (Stull, 1988; Oke, 1988; Sorbjan, 1989; Garratt, 1994; Siebert et al., 2000).
This method is sensitive to the resolution of vertical profiles (Siebert et al., 2000; Seidel et al., 2010).
Therefore, we adapted this method in order to yield *z*_{i} values
that best agree with what we would estimate from visual inspection
of individual temperature profiles. In LMDZ, we calculate *z*_{i}
as the first level at which the vertical potential temperature gradient
exceeds 3 times the moist-adiabatic lapse rate. In observations, we
calculate *z*_{i} as the first level at which the vertical potential
temperature gradient exceeds 5 times the moist-adiabatic lapse rate
because radio-soundings are noisier than simulated profiles.

Finally, we calculate *z*_{orig}(*r*_{orig}=0.6), which is the *z*_{orig}
altitude if *r*_{orig} is set to 0.6. This usually coincides with
the altitude of strong humidity decrease near the inversion (Fig. 5).

All calculations are performed on daily values for LMDZ and on 15 min values for observations.

For LMDZ, when analyzing spatial and seasonal variability, seasonal averages are calculated at each grid box over tropical oceans by averaging all days of all years that belong to each season. Seasons are defined as boreal winter (December–January–February), spring (March–April–May), summer (June–July–August), and fall (September–October–November). For illustration purpose, all maps are plotted for boreal winter. Standard deviations are also calculated among all days of all years for each season.

The type of clouds and mixing processes depends strongly on the large-scale
velocity at 500 hPa (*ω*_{500}, map shown in Fig. 6a),
with shallow clouds in subsiding regions and deeper clouds in ascending
regions (Fig. 1). Therefore, it is convenient
to plot variables as composites as a function of *ω*_{500} (Bony et al., 2004).
To make such plots, we divide the *ω*_{500} range from −30 to
50 hPa d^{−1} into intervals of 5 hPa d^{−1}. In each given interval, we average
all seasonal-mean values at all locations over tropical oceans for
which seasonal-mean *ω*_{500} belongs to this interval (e.g.,
Fig. 8a will be an example). Note
that such composites are carried out on seasonal-mean *ω*_{500} because
cloud processes and their associated diabatic heating are tied to
the large-scale circulation through energetic constraints (Yanai et al., 1973; Emanuel et al., 1994)
that are best valid at longer timescales, otherwise, the energy storage
term may become significant (e.g., Masunaga and Sumi, 2017). This is
why *ω*_{500} is generally averaged over a month or longer
(e.g., Bony et al., 1997; Williams et al., 2003; Bony et al., 2004; Wyant et al., 2006; Bony et al., 2013).
In addition, we primarily focus on understanding the seasonal and
spatial distribution of *δ**D*_{0}.

The cloud cover strongly correlates with the inversion strength, which can be quantified by the estimated inversion strength (EIS; Wood and Bretherton, 2006) (map shown in Fig. 6b) as a measure of inversion strength. We thus also plot variables as composites as a function of EIS. To make such plots, we divide the EIS range from −1 to 9 K into intervals of 0.5 K. In each given interval, we average all seasonal-mean values at all locations over tropical oceans for which seasonal-mean EIS belongs to this interval (e.g., Fig. 8b will be an example). Using seasonal-mean values is consistent with Wood and Bretherton (2006) and with the better link at longer timescales between cloud processes and the large-scale dynamical regime.

To understand what controls the *δ**D*_{0} spatiotemporal variations,
*δ**D*_{0} is decomposed into four contributions based on Eq. (8).
First, we define ${r}_{\mathrm{orig},\mathrm{bas}}=\mathrm{0.3}$, ${\mathit{\alpha}}_{\mathrm{eff},\mathrm{bas}}=\mathrm{1.09}$, SST_{bas}=25 ^{∘}C,
${h}_{\mathrm{0},\mathrm{bas}}=\mathrm{0.7}$, *ϕ*_{bas}=0, *η*_{bas}=0, *β*_{bas}=1,
and ${\mathit{\alpha}}_{\mathrm{evap},\mathrm{bas}}=\mathrm{1}$ as a basic state. We call $\mathit{\delta}{D}_{\mathrm{0},\mathrm{func}}({r}_{\mathrm{orig}},{\mathit{\alpha}}_{\mathrm{eff}},\mathrm{SST},{h}_{\mathrm{0}},\mathit{\varphi},\mathit{\beta},\mathit{\eta},{\mathit{\alpha}}_{\mathrm{evap}})$
the function giving *δ**D*_{0} as a function of *r*_{orig},
*α*_{eff}, SST, *h*_{0}, *ϕ*, *β*, *η*, and *α*_{evap}
following Eq. (8), and $\mathit{\delta}{D}_{\mathrm{0},\mathrm{bas}}=\mathit{\delta}{D}_{\mathrm{0},\mathrm{func}}({r}_{\mathrm{orig},\mathrm{bas}}$, *α*_{eff,bas}, SST_{bas}, *h*_{0,bas}, *ϕ*_{bas}, *β*_{bas}, *η*_{bas}, *α*_{evap,bas}).
The relative contribution of *r*_{orig} to *δ**D*_{0} is estimated
as *δ**D*_{0,func} (*r*_{orig}, *α*_{eff,bas}, SST_{bas}, *h*_{0,bas}, *ϕ*_{bas}, *β*_{bas}, *η*_{bas}, ${\mathit{\alpha}}_{\mathrm{evap},\mathrm{bas}})-\mathit{\delta}{D}_{\mathrm{0},\mathrm{bas}}$.
Similarly, the contributions of *α*_{eff}, SST, *h*_{0}, *ϕ*,
and *η* to *δ**D*_{0} are estimated as detailed in Table 1. All the contributions have the
same units as *δ**D*_{0} (‰). The sum of these components
yields a quantity that is very close to the simulated *δ**D*_{0},
which confirms the validity of this linear decomposition. These components
and their sum can be plotted as maps: Fig. 7
provides an example.

The relative contributions of each of these components to the *δ**D*
variability are quantified by performing a linear regression of each
of the components as a function of *δ**D*_{0}. If the correlation
coefficient is significant for a given factor, then the slope quantifies
the contribution of this factor to the variability of *δ**D*_{0}.
The sum of all contributions may not always be 1 due to nonlinearity.
Such a method has already been applied in previous studies (e.g., Risi et al., 2010b; Oueslati et al., 2016).
The contributions to the seasonal spatial variability of *δ**D*_{0}
can be quantified by performing the regression among all locations
and seasons. The contributions to the daily variability of *δ**D*_{0}
can be quantified by performing the regression among all days of a
given season at a given location.

To understand what controls *r*_{orig}, a similar method as for the
decomposition of *δ**D*_{0} can be applied. We can write *r*_{orig}
as

$$\begin{array}{}\text{(11)}& {r}_{\mathrm{orig}}={\displaystyle \frac{h\left({z}_{\mathrm{orig}}\right)\cdot {q}_{\mathrm{s}}\left(\stackrel{\mathrm{\u203e}}{T}\left({z}_{\mathrm{orig}}\right)+\mathit{\delta}T\left({z}_{\mathrm{orig}}\right),P\left({z}_{\mathrm{orig}}\right)\right)}{{q}_{\mathrm{0}}}},\end{array}$$

where $\stackrel{\mathrm{\u203e}}{T}\left({z}_{\mathrm{orig}}\right)+\mathit{\delta}T\left({z}_{\mathrm{orig}}\right)=T\left({z}_{\mathrm{orig}}\right)$ is the temperature
at altitude *z*_{orig}, $\stackrel{\mathrm{\u203e}}{T}$ is the tropical-ocean-mean temperature
profiles, *h*(*z*_{orig}) and *P*(*z*_{orig}) are the relative humidity
and pressure at *z*_{orig}, and *δ**T*(*z*_{orig}) is the temperature
perturbation compared to $\stackrel{\mathrm{\u203e}}{T}$. Therefore, the variability of
*r*_{orig} is decomposed into the effect of four factors: *q*_{0},
*z*_{orig}, *h*(*z*_{orig}), and *δ**T*(*z*_{orig}). In practice,
*r*_{orig} and *z*_{orig} are calculated following Sect. 3.3,
and then Eq. (11) is applied.

4 Results from LMDZ

Back to toptop
The spatial variations in *δ**D*_{0} simulated by LMDZ (Fig. 7a) are characterized by depleted values
near midlatitudes and in dry subsiding regions (e.g., off the coast
of Peru and over other regions of oceanic upwelling) and regions of
atmospheric deep convection (e.g., Maritime Continent). Consistently,
*δ**D*_{0} values exhibit a maximum for weakly ascending or subsiding
regions: *δ**D*_{0} decreases with increasing vertical velocity
of both signs (Fig. 8a black); *δ**D*_{0}
decreases as EIS increases reflecting more stable, subsiding conditions
(Fig. 8b black). This pattern is
consistent with previous studies (e.g., Good et al., 2015).
For the first time, we propose a theoretical framework to interpret
this pattern, decomposing it into six contributions: *r*_{orig}, *α*_{eff},
SST, *h*_{0}, rain evaporation, and horizontal advection effects (Sect. 3.6). We check that the reconstructed *δ**D*_{0}
from the sum of its four contributions is very similar to the simulated
*δ**D*_{0} (Figs. 7b, 8
dashed black).

In ascending regions, the main contribution explaining the more depleted
*δ**D*_{0} in deep convective regions is that of *α*_{eff}
(Figs. 7d, 8a
red). *α*_{eff} is higher in more ascending regions (Fig. D1d).
This means that the main factor depleting *δ**D*_{0} in deep
convective regions is the fact that the mid-troposphere is more depleted.
This leads to a steeper gradient (higher *α*_{eff}), and thus
a more efficient depletion by vertical mixing. This is consistent
with deep convection depleting the water vapor most efficiently in
the mid-troposphere (Bony et al., 2008). The second main contribution
is that associated with *r*_{orig} (Figs. 7c,
8a green). *r*_{orig} is larger
in deep convective regions (as explained in Sect. 4.2).

In subsidence regions, SST is the main factor controlling *δ**D*_{0} (Figs. 7e, 8a
pink): as subsidence is stronger, or as EIS increases, SST is colder, leading to larger *α*_{eq} and thus more depleted *δ**D*_{0}. Another important factor is *h*_{0} (Figs. 7f, 8a
purple): as subsidence is stronger, *h*_{0} is drier, leading to more depleted *δ**D*_{0}. The contribution of *r*_{orig} is also a significant contribution
to the depletion of *δ**D*_{0} in the cold upwelling regions,
for example off Peru or Namibia (Fig. 7c).
The shallower boundary layer there is associated with higher *r*_{orig}.

The contribution of rain evaporation on *δ**D*_{0} is minor compared
to other contributions, except in the deepest convective regions (Fig. 7g). Rain evaporation is a slightly depleting
effect in regions of strong deep convection and a slightly enriching
effect in regions of moderate deep convection. When the fraction of
raindrops that evaporate is small, isotopic fractionation favors evaporation
of the lighter isotopologues. Therefore in convective, moist regions,
rain evaporation has a depleting effect on the SCL (Worden et al., 2007).
In contrast, in drier regions, rain evaporates almost totally. The
evaporation flux thus has almost the same composition as the initial
rain, which is more enriched than the water vapor.

The contribution of horizontal advection to *δ**D*_{0} is significant
only where isotopic gradients are the largest (Fig. C1h).
Horizontal advection has slightly enriching in deep convective regions
and depleting in coastal regions (e.g., off the coasts of California,
Peru, Mauritania, Namibia, India, and Australia). For example, the
Saharan layer off the northwestern African coast leads to
a strong effect of horizontal advection (Lacour et al., 2017a).

From a quantitative point of view, we can decompose the *δ**D*_{0}
seasonal spatial variations into these different effects (Sect. 3.6). In regions of large-scale ascent, *α*_{eff}
is the main factor explaining the *δ**D*_{0} seasonal spatial
variations (33 %), followed by rain evaporation (20 %) and *r*_{orig}
(19 %; Table 2). In regions of
large-scale descent, SST is the main factor explaining the seasonal spatial
variations (54 %), followed by *r*_{orig} (29 %), *h*_{0} (13 %),
and *α*_{eff} (10 %) (Table 2).
Note that the contribution of *r*_{orig} would be similar if we neglect
rain evaporation and horizontal advection effects (Table 2).

The decomposition method can also be applied to decompose the *δ**D*_{0}
variability at the daily timescale at each location and for each
season (Table 3). On average,
in ascending regions, *r*_{orig} is the main factor (52 %), followed
by rain evaporation (48 %) and *α*_{eff} (35 %). In subsiding
regions, the effect of SST is muted due to its slow variability, and
*r*_{orig} (82 %) becomes the main factor.

Overall, the results highlight the importance of *r*_{orig} as one
of the main factors controlling the spatiotemporal variability of
*δ**D*_{0}.

Given the importance of *r*_{orig} in controlling the *δ**D*_{0}
variations, we now decompose *r*_{orig} into its four contributions:
*q*_{0}, *z*_{orig}, *h*_{orig}, and *δ**T*_{orig} (Sect. 3.6). Spatially, *r*_{orig} is maximum in regions
of strong large-scale ascent (Fig. 10a)
such as the Maritime Continent (Fig. 9a) and
in very stable regions (Fig. 10b) such
as upwelling regions (Fig. 10a). We check
that the reconstructed *r*_{orig} from the sum of its four contributions
is very similar to the simulated *r*_{orig} (Figs. 9b,
10 dashed black).

In regions of strong large-scale ascent, *r*_{orig} is larger mainly
because *h*_{orig} is larger (Figs. 9e, 10a
pink). This is because the moister the FT, the higher the contribution
of vapor coming from above to the vapor of the SCL, and thus the higher
*r*_{orig} and the more depleted *δ**D*_{0}. This mechanism
through which a moister FT leads to a more depleted *δ**D*_{0}
is consistent with that argued in B15. *z*_{orig} damps this effect:
when convection is stronger and the FT moister, convection is also
deeper, so the air originates from higher altitudes where the air
is drier.

In very stable regions, *r*_{orig} is larger because *q*_{0} is
larger (Figs. 9c, 10b
green), consistent with the drier conditions in these regions of large-scale
descent. Note that this effect can be seen only in the most stable regions,
but when considering all subsiding regions, the contribution is small
(Table 2). *r*_{orig} is also larger because *z*_{orig} is lower
in altitude (Figs. 9d, 10b
red). As EIS increases, the boundary layers are shallower, the air
comes from lower in altitude, *r*_{orig} is higher, and thus *δ**D*_{0}
is more depleted. This mechanism was not considered in B15 but our
decomposition shows that it is a key mechanism driving *r*_{orig}
and thus *δ**D*_{0} variations in stable regions.

Quantitatively, in ascending regions, the main factor controlling
the seasonal spatial variations in *r*_{orig} is *h*_{orig} (182 %),
dampened by *z*_{orig} (−67 %) (Table 4).
In descending regions, the main factor is also *h*_{orig} (96 %),
followed by *z*_{orig} (41 %) (Table 4).
At the daily scale, the same two factors dominate the variability
of *r*_{orig}: *h*_{orig} and *z*_{orig} contribute to 78 % and
39 % of *r*_{orig} variations on average over ascending regions
and to 118 % and 39 % on average over descending regions (Table 5).

Estimated altitude *z*_{orig} is at a minimum in dry subsiding regions,
especially in upwelling regions (Figs. 11a, and 12),
corresponding to regions with the strongest inversion (Fig. 11).
This contributes to the depleted *δ**D*_{0} in these regions.

As explained in Sect. 3.3, our estimate of *z*_{orig}
may be artificially biased due to the neglect of some processes in
our theoretical framework. Ideally, to check whether *z*_{orig} really
physically represents the altitude from which the air originates,
additional model experiments where water vapor from different levels
are tagged (Risi et al., 2010b) would be needed. While
we leave this for future work, we check whether *z*_{orig} estimates
are consistent with what we expect based on what we know about mixing
processes in the marine boundary layers. We expect that in stratocumulus
regions, air originates from a very shallow (a few tens of meters)
layer above the inversion, whereas the mixing processes may be more
diverse, and possibly deeper in the FT, as the boundary layer deepens
(Fig. 1).

To check whether estimated *z*_{orig} is consistent with this picture,
we compare *z*_{orig} to *z*_{orig}(*r*_{orig}=0.6) (*z*_{orig} that
we would estimate is *r*_{orig} was set constant to 0.6) and *z*_{i}
(Sect. 3.4), which are measures of the altitude of
the humidity drop and temperature inversion, respectively. As expected
from Fig. 1, they are minimum in dry upwelling
regions, intermediate in trade-wind regions, and maximum values in
convective regions (Figs. 11c–d, 12
green, blue). Therefore, the low *z*_{orig} in upwelling regions
reflects the low *z*_{i}. Consistently, in subsiding regions, *z*_{orig}
correlates well with ${z}_{\mathrm{orig},{r}_{\mathrm{orig}}=\mathrm{0.6}}$ (correlation coefficient
of 0.52, statistically significant beyond 99 %). If we focus on
very stable regions only (EIS >7 K), *z*_{orig} correlates well with
both *z*_{orig}(*r*_{orig}=0.6) and *z*_{i} (correlation coefficient
of 0.58 and 0.52, respectively, statistically significant beyond 99 %).
The altitude *z*_{orig} is a few meters above the inversion in stratocumulus
regions, and up to 1 km above the inversion in cumulus and deep convective
regions (Fig. 12), consistent with our expectations
from Fig. 1. This lends support to the
fact that at least in subsiding regions, our isotope-based *z*_{orig}
estimate effectively reflects the origin of air coming from above.

In ascending regions, in contrast, *z*_{orig} does not correlate
significantly with *z*_{orig}(*r*_{orig}=0.6) or *z*_{i}. This may
indicate either that our *z*_{orig} estimate is biased by neglected
processes such as rain evaporation or that in deep convective regions
the origin of FT air into the SCL is very diverse due to the variety
of mixing processes (Fig. 1).

5 Results from observations

Back to toptop
To check whether our results obtained with LMDZ are realistic, we apply our methods to the measurements gathered during the STRASSE campaign. For simplicity and in absence of all necessary measurements, here we neglect the effects of rain evaporation and horizontal advection.

Throughout the cruise, *δ**D*_{0} shows a large variability,
ranging from around −75 ‰ in quiescent conditions to −120 ‰
during the two convective conditions (Benetti et al., 2014)
(Fig. 13a red). Variability in *r*_{orig}
is the major factor contributing to this variability (58 %) (Fig. 13a green, Table 6).
This crucial importance of mixing processes is consistent with B15.

During the two convective events, the estimated *r*_{orig} saturates
at 1 (Fig. 13b). This proves that *r*_{orig}
estimated in these conditions is biased high because it encapsulates
the effect of neglected processes, i.e., depletion by rain evaporation.
Equation (9) is not valid in this case. In addition,
at the scale of a few hours, the steady-state assumptions may be violated.
Rain evaporation may strongly deplete the SCL before surface evaporation
has the time to play its dampening role, hence the possibility of
reaching very low *δ**D*_{0} that cannot be predicted even when
considering rain evaporation (Appendix B).

During the rest of the cruise, the main factors controlling the *r*_{orig}
variability are *z*_{orig} (90 %) and *h*_{orig} (70 %). The
importance of FT humidity in controlling *r*_{orig} was already highlighted
in B15. However, in their paper, the variability in *z*_{orig} was
neglected, whereas it appears here as the main factor.

Through September, the cruise goes from a shallow boundary layer in
early September to deeper boundary layers with higher inversions,
before reaching the convective conditions (Fig. 13c).
Consistently with this deepening boundary layer, the air originates
from increasingly higher altitudes. Remarkably, there are 6 d when
*z*_{orig} coincides with *z*_{i} with a root-mean-square error
of 31 ‰ and correlation coefficient of 0.996 (Fig. 13c).
This indicates that the air comes exactly from the inversion layer.
When recalling that *z*_{orig} and *z*_{i} are estimated from completely
independent observations, the coincidence is remarkable and lends
support to the fact that on these days, our *z*_{orig} estimate is
physical. However, there remain 9 d when *z*_{orig} is much higher
than *z*_{i}. This may reflect more penetrative downdrafts as we
approach deeper convective regimes. But it may also be an artifact
of our neglect of horizontal advection. For example, on these days
which are characterized by lower *h*_{0}, neglecting the advection
of enriched water vapor from nearby regions with higher *h*_{0} could
be misinterpreted as lower *r*_{orig} and thus higher *z*_{orig}.

6 Discussion: what can we learn from water isotopes on mixing processes?

Back to toptop
We have shown in the previous section that one of the main factors
controlling *δ**D*_{0} at the seasonal spatial and daily scales
is the proportion of the water vapor in the SCL that originates from
above (*r*_{orig}) and that one of the main factors controlling *r*_{orig}
is the altitude from which the air originates (*z*_{orig}). In turn,
could we use water vapor isotopic measurements to constrain *z*_{orig}?
This would open the door to discriminating between different mixing
processes at play (Fig. 1). Since mixing
processes are crucial to determine the sensitivity of cloud fraction
to SST (Sherwood et al., 2014; Bretherton, 2015; Vial et al., 2016),
such a prospect would allow us to improve our knowledge of cloud feedbacks,
and hence of climate sensitivity.

With this in mind, we assess the errors associated with *z*_{orig}
estimates from *δ**D*_{0} measurements, and discuss whether they
are small enough for *z*_{orig} estimates to be useful. In stratocumulus
clouds where the air is believed to originate from the first few tens
of meters above cloud top (Faloona et al., 2005; Mellado, 2017),
*z*_{orig} estimates are not useful if the errors are larger than
a few tens of meters, e.g., 20 m. In cumulus clouds where mixing processes
are more diverse and possibly deeper (Fig. 1),
*z*_{orig} estimates may be useful if errors are of the order of
80 m.

Let us assume that we have a field campaign where we measure *δ**D*_{0},
surface meteorological variables, temperature and humidity profiles
(e.g., radio soundings), and a few *δ**D* profiles (e.g., by aircraft).
This is what we can expect for example from the future EUREC4A (Elucidating
the role of clouds-circulation coupling in climate) campaign to study
trade-wind cumulus clouds (Bony et al., 2017). Below we quantify
the effects of five sources of uncertainty on *z*_{orig} estimates.

The first source of uncertainty is measurement errors. We recalculate
*z*_{orig} assuming an error of 0.4 ‰ on *δ**D*_{0}
(typical of what we can measure with in-situ laser instruments; Aemisegger et al., 2012; Benetti et al., 2014)
and 1 ‰ on *δ**D*_{f} (larger errors due to lower humidity
and the increased complexity of measurements in altitude). The averaged
errors on *z*_{orig} and their standard deviations are plotted as
a function of EIS in Fig. 14a. Whereas errors on
*δ**D*_{f} lead to errors on *z*_{orig} of the order of 20 m
(Fig. 14a, green), errors on *δ**D*_{0} lead
to errors on *z*_{orig} of the order of 80 m (Fig. 14a,
red). Yet in stratocumulus, no one expects the air to originate from
a higher altitude than 80 m above the inversion. Therefore, *δ**D*_{0}
measurements would need to be more accurate than usual to be useful
in stratocumulus regions, i.e., 0.1 ‰ to yield a 20 m precision
on *z*_{orig}. In trade-wind cumulus regions, the precision of 0.4 ‰
is enough for *z*_{orig} to be useful.

The second source of uncertainty is associated with neglecting rain
evaporation. This effect can be quantified in a model, but it is very
difficult to quantify in nature because it is complicated and uncertain
to measure *η* (Rosenfeld and Mintz, 1988), and it is even
more complicated to measure or predict *α*_{evap}. Rain evaporation
can have a depleting or enriching effect depending on microphysical
details that are too complex to be addressed here (Graf et al., 2019).
Neglecting rain evaporation leads to an error of the order of 500 m
in regions of low EIS and 250 m in regions of strong EIS (Fig. 14b,
brown). In regions of stratocumulus regions, rain evaporation is
a significant source of error in spite of the relatively small amount
of precipitation available to evaporate. This is because total evaporation
of the rain efficiently enriches the SCL and easily modifies *δ**D*_{0}
by more than the 0.1 ‰ targeted precision explained above.
However, it is possible that LMDZ overestimates this source of error
in trade-wind cumulus and stratocumulus regions. LMDZ is one of the
GCMs producing the strongest rain in stratocumulus regions (Zhang et al., 2013),
and GCMs are known to trigger convection too often in trade-wind cumulus
regions (Nuijens et al., 2015a, b).

The third source of uncertainty is associated with horizontal advection.
In nature, *ϕ* can be estimated from meteorological analyses and
*β* can be estimated from near-surface isotopic measurements
at several locations (e.g., sounding arrays during typical field campaigns).
In absence of these additional measurements, neglecting this effect
leads to an error of the order of 800 m (Fig. 14b,
purple). This limits the usefulness of *z*_{orig} estimates for all
cloud regimes.

The fourth source of uncertainty arises from the daily variability
in *α*_{eff} (Appendix D2). Estimating *α*_{eff}
requires us to measure *δ**D*_{f} at 500 hPa. Satellite measurements
are available but are affected by random errors that are too large
for our application (Worden et al., 2011, 2012; Lacour et al., 2015).
Precise in situ measurements of water vapor *δ**D* in altitude
are costly and difficult (Sodemann et al., 2017).

Let us assume that we have only one *δ**D*_{f} value that represents
the seasonal average at a given location. To estimate the resulting
error on *z*_{orig}, we re-estimate *z*_{orig} every day and at
each location using $\stackrel{\mathrm{\u203e}}{{\mathit{\alpha}}_{\mathrm{eff}}}+{\mathit{\sigma}}_{{\mathit{\alpha}}_{\mathrm{eff}}}$ and
$\stackrel{\mathrm{\u203e}}{{\mathit{\alpha}}_{\mathrm{eff}}}-{\mathit{\sigma}}_{{\mathit{\alpha}}_{\mathrm{eff}}}$. The error on *z*_{orig}
is calculated as $\left({z}_{\mathrm{orig}}(\stackrel{\mathrm{\u203e}}{{\mathit{\alpha}}_{\mathrm{eff}}}-{\mathit{\sigma}}_{{\mathit{\alpha}}_{\mathrm{eff}}})-{z}_{\mathrm{orig}}(\stackrel{\mathrm{\u203e}}{{\mathit{\alpha}}_{\mathrm{eff}}}+{\mathit{\sigma}}_{{\mathit{\alpha}}_{\mathrm{eff}}})\right)/\mathrm{2}$.
The averaged error and its standard deviation is plotted as a function
of EIS in Fig. 14c (black). It is of the order of
400 m and rarely below 200 m. If we attempt to estimate *α*_{eff}
as the fractionation coefficient as a function of local temperature,
errors would be even more dissuasive (Fig. 14c,
blue).

Therefore, estimating *z*_{orig} from daily *δ**D*_{0} measurements
cannot be useful unless we measure *δ**D*_{f} on a daily basis
as well. Practically, we could imagine measuring FT properties (*δ**D*_{f})
at the top of a mountain while we measure *δ**D*_{0} at the sea
level (e.g., on islands such as Hawaii or Réunion: Galewsky et al., 2007; Bailey et al., 2013; Guilpart et al., 2017).

Finally, as a fifth source of uncertainty comes the assumption that
the *δ**D* profile follows a Rayleigh distillation line (Sect. 2.2). However, in both LMDZ (Appendix D1)
and nature (Sodemann et al., 2017), *δ**D* profiles
are usually intermediate between Rayleigh and mixing lines. The precision
of our *z*_{orig} estimate is at a maximum in the Rayleigh distillation
case.

When trying to find a numerical solution for *z*_{orig} directly
from Eq. (6), a solution can be found only in 0.1 %
of cases. This is because simulated *δ**D* profiles are often
close to a mixing line in the lower troposphere (Appendix D1).
Whatever *z*_{orig} in the lower troposphere, the *δ**D*_{0}
calculated from Eq. (6) is nearly constant because
the *δ**D* profile is close to a mixing line (Appendix A,
Fig. 4b). Whatever *z*_{orig} in the
middle troposphere, the *δ**D*_{0} calculated from Eq. (6)
is also nearly constant because *r*_{orig} there is very small. So
whatever *z*_{orig}, the *δ**D* calculated from Eq. (6)
is nearly constant, and the numerical solution fails.

However, it is possible that *δ**D* profiles simulated by LMDZ
are closer to mixing lines than real profiles since GCMs are known
to overestimate vertical mixing through the troposphere (Risi et al., 2012b)
and to mix the lower free troposphere too frequently by deep convection
in trade-wind regions (Nuijens et al., 2015a, b).
Therefore, the shape of *δ**D* profiles simulated by LMDZ is
not a sufficient reason to reject the Rayleigh assumption. The uncertainty
associated with this assumption is very difficult to quantify in LMDZ.
More measurements of full *δ**D* profiles are very welcome to
help quantify it.

To summarize, *δ**D*_{0} measurements could potentially be useful
to estimate *z*_{orig} with a useful precision, but only if we measure
daily *δ**D*_{f} in the mid-troposphere, if the shape of *δ**D*
profiles can be better documented, if we measure *δ**D*_{0} at
different places to quantify the effect of horizontal advection, and
if we can invent innovative techniques to better quantify the effect
of rain evaporation. In addition, in stratocumulus clouds, we need
to measure *δ**D*_{0} with an accuracy of 0.1 ‰.

7 Conclusions

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We propose an analytical model to predict the water vapor isotopic
composition *δ**D*_{0} of the sub-cloud layer (SCL) over tropical
oceans. This model relies on the hypothesis that the altitude from
which the air originates, *z*_{orig}, is an important factor. We
build on B15, who extended the Merlivat and Jouzel (1979) closure equation
to make the explicit link between *δ**D*_{0} and mixing processes.
We further extend their equation: we assume a shape for the *δ**D*
vertical profiles as a function of *q*, and we account for horizontal
advection and rain evaporation effects.

The resulting equation highlights the fact that *δ**D*_{0} is
not sensitive to the intensity of mixing processes. Therefore, it
is unlikely that water vapor isotopic measurements could help estimate
the entrainment velocity that many studies have striven to estimate
(Bretherton et al., 1995). In contrast, *δ**D*_{0} is sensitive
to the altitude from which the air originates. Based on a simulation
with LMDZ and observations during the STRASSE cruise, we show that
*z*_{orig} is an important factor explaining the seasonal spatial
and daily variations in *δ**D*_{0}, especially in subsidence
regions. In turn, could *δ**D*_{0} measurements, combined with
vertical profiles of humidity, temperature, and *δ**D*, help estimate
*z*_{orig} and thus discriminate between different mixing processes?
For such isotope-based estimates of *z*_{orig} to be useful, we would
need a precision of a few hundreds of meters in deep convective regions
and smaller than 20 m in stratocumulus regions. To reach this target,
we would need daily measurements of *δ**D* in the mid-troposphere
and very accurate measurements of *δ**D*_{0}, which are currently
difficult to obtain. We would also need information on the horizontal
distribution of *δ**D* to account for horizontal advection effects,
and full *δ**D* profiles to quantify the uncertainty associated
with the assumed shape for *δ**D* profiles. Finally, rain evaporation
is an issue in all regimes, even for stratocumulus clouds. Innovative
techniques would need to be developed to quantify this effect from
observations.

This study is preliminary in many respects. First, it would be safe
to check using water tagging experiments in LMDZ that *z*_{orig}
estimates really represent the altitude from which the air originates
and are not biased by our simplifying assumptions. Second, the coarse
vertical resolution of LMDZ and the simplicity of mixing parameterizations
(e.g., cloud top entrainment is not represented) are a limitation of
this study. Ideally, the relationship between *δ**D*_{0}, *z*_{orig},
and the type of mixing processes should be investigated in isotope-enabled
large-eddy simulations (LESs) (Blossey et al., 2010; Moore et al., 2014).
Artificial tracers and structure detection methods (Park et al., 2016; Brient et al., 2019),
combined with conditional sampling methods (Couvreux et al., 2010),
could help detect the different kinds of mixing structures, estimate
their contributions to vertical transport, and describe their isotopic
signature. This would allow us to confirm, or disprove, many of the
hypotheses and conclusions in this paper. Finally, if the sensitivity
of *δ**D*_{0} to the type of mixing processes is confirmed, paired
isotopic simulations of single-column model (SCM) versions of general
circulation models (GCMs) and LES, forced by the same forcing, could
be very useful to help evaluate and improve the representation of
mixing and entrainment processes in GCMs, as is routinely the case
for non-isotopic variables (Randall et al., 2003; Hourdin et al., 2013; Zhang et al., 2013).

Code and data availability

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Code and data availability.

LMDZ can be downloaded from http://lmdz.lmd.jussieu.fr/ (last access: 26 September 2019). Program codes used for the analysis are available on https://prodn.idris.fr/thredds/catalog/ipsl_public/rlmd698/article_mixing_processes/d_pgmf/catalog.html (last access: 26 September 2019).

Isotopic measurements from STRASSE can be downloaded from http://cds-espri.ipsl.fr/isowvdataatlantic/ (last access: 26 September 2019). All other datasets and processed files are available on https://prodn.idris.fr/thredds/catalog/ipsl_public/rlmd698/article_mixing_processes/catalog.html (last access: 26 September 2019).

Appendix A: Closure if the tropospheric profile follows a mixing line

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For simplicity, we neglect here horizontal advection and rain evaporation
effects, but results would be similar otherwise. If we assume that
*R*_{orig} is uniquely related to *q*_{orig} through a mixing line
between the SCL air and a dry end member (*q*_{f}, *R*_{f}),

$$\begin{array}{}\text{(A1)}& {q}_{\mathrm{orig}}=a\cdot {q}_{\mathrm{0}}+(\mathrm{1}-a)\cdot {q}_{\mathrm{f}}\end{array}$$

and

$$\begin{array}{}\text{(A2)}& {R}_{\mathrm{orig}}=a\cdot {q}_{\mathrm{0}}\cdot {R}_{\mathrm{0}}+(\mathrm{1}-a)\cdot {q}_{\mathrm{f}}\cdot {R}_{\mathrm{f}}.\end{array}$$

Reorganizing Eq. (A1), we get $a=\frac{r-p}{\mathrm{1}-p}$ with
$p={q}_{\mathrm{f}}/{q}_{\mathrm{0}}$. Since ${q}_{\mathrm{f}}\le {q}_{\mathrm{orig}}\le {q}_{\mathrm{0}}$, *p*≤*r*_{orig}.
Injecting Eq. (A2) into Eq. (6), we get

$$\begin{array}{}\text{(A3)}& {R}_{\mathrm{0}}={\displaystyle \frac{{R}_{\mathrm{oce}}/{\mathit{\alpha}}_{\mathrm{eq}}+p/(\mathrm{1}-p)\cdot {R}_{\mathrm{f}}\cdot {\mathit{\alpha}}_{\mathrm{K}}\cdot (\mathrm{1}-{h}_{\mathrm{0}})}{{h}_{\mathrm{0}}+{\mathit{\alpha}}_{\mathrm{K}}\cdot (\mathrm{1}-{h}_{\mathrm{0}})/(\mathrm{1}-p)}}.\end{array}$$

As a consistency check, in the limit case where the end member is
totally dry (*p*=0), we find the MJ79 equation, i.e., Eq. (10).

It is intriguing to realize that *r*_{orig} has disappeared from
Eq. (A3). This can be understood physically: if the
vertical profile follows a mixing line, it does not matter from which
altitude the air comes: ultimately, what matters is how much dry air
has been mixed directly or indirectly into the SCL (Fig. 4b).
Therefore, if *R*_{orig} follows a mixing line, we lose the sensitivity
to *z*_{orig}.

Appendix B: Diagnostics for rain evaporation in LMDZ

Back to toptop
Rain evaporation can be accounted for in Eq. (8)
if we can quantify *η*, the ratio of water vapor originating from
rain evaporation to that originating from surface evaporation, and
*α*_{evap}, the ratio of isotopic ratio in the rain evaporation
flux to *R*_{0}.

In LMDZ, two parameterization schemes can produce rain evaporation:
the convective scheme and the large-scale condensation scheme. Their
respective precipitation evaporation tendencies, ${\left(\mathrm{d}q/\mathrm{d}t\right)}_{\mathrm{evap},\mathrm{conv}}$
and ${\left(\mathrm{d}q/\mathrm{d}t\right)}_{\mathrm{evap},\mathrm{lsc}}$, are given in ${\mathrm{kg}}_{\mathrm{water}}\cdot {{\mathrm{kg}}_{\mathrm{air}}}^{-\mathrm{1}}\cdot {\mathrm{s}}^{-\mathrm{1}}$
and are used to calculate *F*_{evap} in ${\mathrm{kg}}_{\mathrm{water}}\cdot {\mathrm{m}}^{-\mathrm{2}}\cdot {\mathrm{s}}^{-\mathrm{1}}$:

$${F}_{\mathrm{evap}}=\sum _{k=\mathrm{1}}^{{k}_{\mathrm{LCL}}}\left({\left(\mathrm{d}q/\mathrm{d}t\right)}_{\mathrm{evap},\mathrm{conv}}+{\left(\mathrm{d}q/\mathrm{d}t\right)}_{\mathrm{evap},\mathrm{lsc}}\right)\cdot {\displaystyle \frac{\mathrm{\Delta}{P}_{k}}{g}},$$

where *k*_{LCL} is the last layer below the LCL, Δ*P*_{k}
is the depth of layer *k* in pressure coordinate, and *g* is gravity.

The isotopic equivalent of this flux, *F*_{evap,iso}, is used to
calculate ${R}_{\mathrm{evap}}={F}_{\mathrm{evap},\mathrm{iso}}/{F}_{\mathrm{evap}}$.

Only grid boxes and days where *F*_{evap}>0.05 mm d^{−1} are considered
to calculate *α*_{evap}. This represents 94.0 % of all tropical
oceanic grid boxes.

Consistent with the larger amount of precipitation available for evaporation,
*η* is at a maximum in regions of deep convection, reaching 30 % around
the Maritime Continent (Fig. C1a). It is
minimal over the dry descending regions, reaching 5 % off the coasts
of Mauritania, Peru, and Namibia. The rain evaporation is more depleted
than the SCL in regions of strong deep convection, by as much as 70 ‰
around the Maritime Continent (Fig. C1b).
When the fraction of raindrops that evaporate is small, as is the
case in such moist regions, isotopic fractionation favors evaporation
of the lighter isotopologues. In these regions, rain evaporation has
a depleting effect on the SCL, consistent with Worden et al. (2007).
In contrast, in other regions, rain evaporation has an enriching effect
on the SCL, up to 70 ‰ in dry regions. This is because in
dry regions, rain evaporates almost totally, so that the evaporation
flux has almost the same composition as the initial rain, which is
more enriched than the water vapor.

Appendix C: Diagnostics for horizontal advection in LMDZ

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We can account for horizontal advection in Eq. (8) if we can quantify parameters $\mathit{\varphi}=\frac{{F}_{\mathrm{adv}}\cdot {q}_{\mathrm{adv}}}{E}$, the ratio of water vapor coming from horizontal advection to that coming from surface evaporation, and $\mathit{\beta}={R}_{\mathrm{adv}}/{R}_{\mathrm{0}}$, the ratio of isotopic ratios of horizontal advection to those of the SCL.

Let us assume that the box representing the SCL has a zonal extent
Δ*y* and a meridional extent Δ*x* and is composed of *k*_{LCL}
layers of vertical extent Δ*z*_{k}. The quantity *F*_{adv}⋅*q*_{adv}
represents the mass flux of water entering the grid box by horizontal
advection per surface area, expressed in ${\mathrm{kg}}_{\mathrm{water}}\cdot {\mathrm{s}}^{-\mathrm{1}}\cdot {\mathrm{m}}^{-\mathrm{2}}$.
Assuming an upstream advection scheme, it can be expressed as

$$\begin{array}{}\text{(C1)}& \begin{array}{rl}& {F}_{\mathrm{adv}}\cdot {q}_{\mathrm{adv}}=\\ & {\displaystyle \frac{{\sum}_{k=\mathrm{1}}^{{k}_{\mathrm{LCL}}}\left({\mathit{\rho}}_{k}\cdot \left|{u}_{k}\right|\cdot {q}_{uk}\cdot \mathrm{\Delta}y\cdot \mathrm{\Delta}{z}_{k}+{\mathit{\rho}}_{k}\cdot \left|{v}_{v}\right|\cdot {q}_{vk}\cdot \mathrm{\Delta}x\cdot \mathrm{\Delta}{z}_{k}\right)}{\mathrm{\Delta}x\cdot \mathrm{\Delta}y}}\end{array},\end{array}$$

where *u*_{k} and *v*_{k} are the zonal and meridional wind components
at layer *k*, *ρ*_{k} is the volumic mass of air at layer *k*,
and *q*_{uk} and *q*_{vk} are the humidities of the incoming air
from zonal and meridional advection at layer *k*. When *u*_{k}>0,
*q*_{uk} is the humidity in the grid box to the west. When *u*_{k}<0,
*q*_{uk} is the humidity in the grid box to the east. When *v*_{k}>0,
*q*_{vk} is the humidity in the grid box to the south. When *v*_{k}<0,
*q*_{vk} is the humidity in the grid box to the north.

Applying the hydrostatic equation at each layer ($\mathrm{\Delta}{P}_{k}={\mathit{\rho}}_{k}\cdot g\cdot \mathrm{\Delta}{z}_{k}$,
where *g* is gravity and Δ*P*_{k} is the vertical extent of
the layer *k* in pressure coordinate), we get

$${F}_{\mathrm{adv}}\cdot {q}_{\mathrm{adv}}=\sum _{k=\mathrm{1}}^{{k}_{\mathrm{LCL}}}{\displaystyle \frac{\mathrm{\Delta}{P}_{k}}{g}}\cdot \left({\displaystyle \frac{\left|{u}_{k}\right|\cdot {q}_{uk}}{\mathrm{\Delta}x}}+{\displaystyle \frac{\left|{v}_{k}\right|\cdot {q}_{vk}}{\mathrm{\Delta}y}}\right).$$

The quantity *F*_{adv} represents the incoming air mass flux by horizontal
advection, and *q*_{adv} represents the humidity of the incoming
air. We can thus write them as

$${F}_{\mathrm{adv}}=\sum _{k=\mathrm{1}}^{{k}_{\mathrm{LCL}}}\left({\displaystyle \frac{\left|{u}_{k}\right|}{\mathrm{\Delta}x}}+{\displaystyle \frac{\left|{v}_{k}\right|}{\mathrm{\Delta}y}}\right)\cdot {\displaystyle \frac{\mathrm{\Delta}{P}_{k}}{g}}$$

and

$${q}_{\mathrm{adv}}={\displaystyle \frac{{\sum}_{k=\mathrm{1}}^{{k}_{\mathrm{LCL}}}\left(\frac{\left|{u}_{k}\right|}{\mathrm{\Delta}x}\cdot {q}_{uk}+\frac{\left|{v}_{k}\right|}{\mathrm{\Delta}y}\cdot {q}_{vk}\right)\cdot \frac{\mathrm{\Delta}{P}_{k}}{g}}{{\sum}_{k=\mathrm{1}}^{{k}_{\mathrm{LCL}}}\left(\frac{\left|{u}_{k}\right|}{\mathrm{\Delta}x}+\frac{\left|{v}_{k}\right|}{\mathrm{\Delta}y}\right)\cdot \frac{\mathrm{\Delta}{P}_{k}}{g}}}.$$

The same budget as in Eq. (C1) can be written for water isotopes:

$$\begin{array}{rl}& {F}_{\mathrm{adv}}\cdot {q}_{\mathrm{adv}}\cdot {R}_{\mathrm{adv}}=\\ & {\displaystyle \frac{{\sum}_{k=\mathrm{1}}^{{k}_{\mathrm{LCL}}}\left({\mathit{\rho}}_{k}\cdot \left|{u}_{k}\right|\cdot {q}_{uk}\cdot {R}_{uk}\cdot \mathrm{\Delta}y\cdot \mathrm{\Delta}{z}_{k}+{\mathit{\rho}}_{k}\cdot \left|{v}_{v}\right|\cdot {q}_{vk}\cdot {R}_{vk}\cdot \mathrm{\Delta}x\cdot \mathrm{\Delta}{z}_{k}\right)}{\mathrm{\Delta}x\cdot \mathrm{\Delta}y}}\end{array},$$

where *R*_{adv} represents the isotopic ratio of the incoming water
vapor:

$${R}_{\mathrm{adv}}={\displaystyle \frac{{\sum}_{k=\mathrm{1}}^{{k}_{\mathrm{LCL}}}\left(\frac{\left|{u}_{k}\right|}{\mathrm{\Delta}x}\cdot {q}_{uk}\cdot {R}_{uk}+\frac{\left|{v}_{k}\right|}{\mathrm{\Delta}y}\cdot {q}_{vk}\cdot {R}_{vk}\right)\cdot \frac{\mathrm{\Delta}{P}_{k}}{g}}{{\sum}_{k=\mathrm{1}}^{{k}_{\mathrm{LCL}}}\left(\frac{\left|{u}_{k}\right|}{\mathrm{\Delta}x}\cdot {q}_{uk}+\frac{\left|{v}_{k}\right|}{\mathrm{\Delta}y}\cdot {q}_{vk}\right)\cdot \frac{\mathrm{\Delta}{P}_{k}}{g}}}.$$

Note that the upstream advection scheme assumed here overestimates the effect of advection compared to the Van Leer (1977) advection scheme used in LMDZ. We thus estimate an upper bound for the advection effect here.

In practice, rather than calculating $\mathit{\beta}={R}_{\mathrm{adv}}/{R}_{\mathrm{0}}$ , we calculate
$\mathit{\beta}={R}_{\mathrm{adv}}/{R}_{\mathrm{SCL}}$, where *R*_{SCL} is the isotopic ratio on
average through the SCL:

$${R}_{\mathrm{SCL}}={\displaystyle \frac{{\sum}_{k=\mathrm{1}}^{{k}_{\mathrm{LCL}}}{q}_{k}\cdot {R}_{k}\frac{\mathrm{\Delta}{P}_{k}}{g}}{{\sum}_{k=\mathrm{1}}^{{k}_{\mathrm{LCL}}}{q}_{k}\frac{\mathrm{\Delta}{P}_{k}}{g}}}.$$

This prevents the advected water vapor to be systematically more depleted when the mixed-layer hypothesis is not exactly verified.

Parameter *ϕ* is at a maximum where winds are maximum, such as near
the extra-tropics or in the North Atlantic (Fig. C1a).
Horizontal advection has an enriching effect in deep convective regions
(probably because water vapor comes from nearby drier regions that
have been less depleted by deep convection) and a depleting effect
near the coasts (probably because of winds bringing vapor from the
nearby land that is depleted by the continental effect) (Fig. C1b).

Note that in this formulation, parameters *ϕ* and *β* are
resolution-dependent. For example, in a finer resolution, *ϕ*
would be larger and *β* would be closer to 1, but ${F}_{\mathrm{adv}}\cdot {q}_{\mathrm{adv}}\cdot {R}_{\mathrm{adv}}$
and thus the contribution of horizontal advection in Eq. (8)
would remain the same.

Appendix D: LMDZ free-tropospheric profiles

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The goal of this appendix is to document the spatiotemporal variability
in the shape (Sect. D1) and steepness (Sect. D2) of simulated free-tropospheric *δ**D* profiles.
Note that a detailed interpretation of these profiles is beyond the
scope of this paper. This paper aims at understanding *δ**D*_{0},
which is the first step towards understanding full tropospheric profiles.
In turn, understanding full tropospheric profiles in future studies
will help refine our model for *δ**D*_{0}.

First, we test whether the *δ**D* vertical profiles simulated
by LMDZ follow a Rayleigh or mixing line as a function of *q*. For
the Rayleigh curve, *α*_{eff} is estimated as explained in Sect. 3.3. For the mixing line (Appendix A),
the end member (*q*_{f}, *R*_{f}) is also taken at 500 hPa.

The tropical-mean vertical *δ**D* profiles simulated by LMDZ
are bounded by Rayleigh and mixing lines (Fig. D1a).
To better document the spatial variability in the shape of *δ**D*
profiles, we plot parameter $f=\frac{\mathit{\delta}{D}_{\mathrm{LMDZ}}-\mathit{\delta}{D}_{\mathrm{Rayleigh}}}{\mathit{\delta}{D}_{\mathrm{mix}}-\mathit{\delta}{D}_{\mathrm{Rayleigh}}}$,
describing how close the simulated *δ**D* (*δ**D*_{LMDZ}) is
to the Rayleigh (*δ**D*_{Rayleigh}) and mixing (*δ**D*_{mix})
lines. We have *f*=0 in the case of a Raleigh line, *f*=1 in the case of
a mixing line, and *f*>1 if *δ**D* is more enriched than a mixing
line. In the lower troposphere, *δ**D*_{LMDZ} is close to a mixing
line (and sometimes even more enriched) in deep convective regions
(Indian Ocean, South Pacific Convergence Zone, Atlantic ITCZ), probably
because deep convection efficiently mixes the lower troposphere. Elsewhere,
*δ**D*_{LMDZ} is intermediate between the two lines (Fig. D1e).
In the middle troposphere, *δ**D*_{LMDZ} is relatively closer
to Rayleigh everywhere (Fig. D1b).

The daily variability of *f* is large everywhere and at all levels
(Fig. D1c, e), with standard deviation of 0.23 and
0.44 on tropical average at 1000 and 4000 m, respectively. A large
daily variability in the shape of profiles is also observed in nature
(Sodemann et al., 2017).

The steepness of the *δ**D* gradient from the surface to the
middle troposphere is described by the parameter *α*_{eff}.
It is at a maximum in regions of deep convection, for example around the
Maritime Continent (Fig. D2a). This is consistent
with the maximum depletion simulated in deep convective regions in
the mid-troposphere simulated by models (Bony et al., 2008),
leading to steeper *δ**D* profiles. The pattern of *α*_{eff}
may also reflect horizontal advection effects, where strong isotopic
gradients align with winds
(e.g., from the eastern to the western Pacific; Dee et al., 2018).

Values of *α*_{eff} are of the same order of magnitude as real
fractionation factors, but the spatial variations do not reflect those
predicted if using a fractionation coefficient *α*_{eq} as a
function of temperature *T* (Fig. D2b).

The daily standard deviation of *α*_{eff} (${\mathit{\sigma}}_{{\mathit{\alpha}}_{\mathrm{eff}}}$)
for a given season ranges from 5 ‰ in the central Atlantic
to 40‰ near the Maritime Continent (Fig. D2c).
On average over all seasons and locations, daily *α*_{eff}−1
at a given location varies within ±25 % of its seasonal-mean
mean value.

Author contributions

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Author contributions.

CR thought about the equations, ran the LMDZ simulations, performed the analysis, and wrote the paper. JG initiated the discussion on the subject and discussed regularly about the results. GR provided the STRASSE radiosoundings. FB provided insight and references about cloud processes. JG, GR, and FB all gave comments on the paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This work was granted access to the HPC resources of IDRIS under the allocation 2092 made by GENCI. Joseph Galewsky was supported by the LABEX-IPSL visitor program, the Franco-American Fulbright Foundation, and NSF AGS grant 1738075. We thank Marion Benetti and Sandrine Bony for her previous studies on this subject and useful discussions. We thank the two anonymous reviewers for their comments.

Review statement

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Review statement.

This paper was edited by Farahnaz Khosrawi and reviewed by two anonymous referees.

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Short summary

Water molecules can be light (one oxygen atom and two hydrogen atoms) or heavy (one hydrogen atom is replaced by a deuterium atom). These different molecules are called water isotopes. The isotopic composition of water vapor can potentially provide information about physical processes along the water cycle, but the factors controlling it are complex. As a first step, we propose an equation to predict the water vapor isotopic composition near the surface of tropical oceans.

Water molecules can be light (one oxygen atom and two hydrogen atoms) or heavy (one hydrogen...

Atmospheric Chemistry and Physics

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