2.1 Box model and budget equations

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

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 δD.

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 (q₀, R₀). The humidity and isotopic properties of the mass flux export N are thus also (q₀, R₀). 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=q₀), assuming that the main effect of horizontal advection on δD₀ arises from horizontal gradients in δD. Appendix C explains how R_adv can be calculated. At steady state, the water budget of the SCL is written

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 q₀:

The SCL humidity q₀ 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

where R_E is the isotopic composition of the surface evaporation. It is assumed to follow the Craig and Gordon (1965) equation:

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 h₀ 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 P₀ is the surface pressure).

We write the isotopic composition of the rain evaporation, R_evap, as

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 h₀ as a function of q₀ as in B15 or the q profile as a function of q₀. Our ultimate goal is to assess the added value of δD assuming that meteorological measurements are already routinely performed.

By combining all these equations, we get

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 q₀, R₀ 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 R₀ to M is explained physically by the fact that for a given q₀ 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 R₀ is sensitive, remains constant. Rather, since q and R vary with altitude, R₀ is sensitive to the altitude from which the air originates.

Figure 2Schematics showing the simple box model on which the theoretical framework is based, and illustrating the main notations.

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2.2 Closure if δD profile follows a Rayleigh distillation line

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 R₀ from z_orig, we can apply Eq. (6) if we know the q and δD vertical profiles. Conversely, if the goal is to predict z_orig from R₀, we can numerically solve Eq. (6) if we know the q and δD 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 δD are usually bounded by two curves when plotted in a (q, δD) 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):

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

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

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:

Equation (8) tells us that whenever α_eff>1, R₀ decreases as r_orig increases (Fig. 3 red), i.e., as q_orig is moister. Therefore, R₀ 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 δD values is restricted to −70 ‰ to −85 ‰. This explains why in quiescent conditions near the sea level in tropical ocean locations, the water vapor δD 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 R₀ (Appendix B).

Figure 3 (green) shows that the sensitivity to α_eff is relatively small but cannot be neglected. Therefore, predicting δD₀ 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 δD₀ when mixed with the surface evaporation, as illustrated in Fig. 4b and analytically demonstrated in Appendix A. Our main results (more depleted δD₀ as r_orig increases, restricted range of δD₀ variations, relationship with z_orig) hold only for δD 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).

Figure 3δD₀ as a function of r_orig according to Eq. (9), with α_eff=α_eq as an example (red). For this illustrative purpose, we assume SST = 30 ^∘C, h₀=0.8, δD_oce=0 ‰, and $\mathit{\varphi }=\mathit{\eta }=\mathrm{0}$. The sensitivity to the effective fractionation factor α_eff (green) is shown. If rain evaporation is 25 % of surface evaporation (η=0.25), the solid pink and blue curves show the sensitivity to the effective fractionation factor α_evap. If the incoming water vapor by horizontal advection is 25 % of surface evaporation (ϕ=0.25), the dashed pink and blue curves show the sensitivity to the isotopic gradient quantified by β.

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Figure 4Idealized q−δD diagrams showing how the SCL water vapor δD is set, for the case where the tropospheric δD profile follows Rayleigh distillation (a) or a mixing line (b). For this illustrative purpose, we assume SST = 30 ^∘C, h₀=0.8, δD_oce=0 ‰, and $\mathit{\varphi }=\mathit{\eta }=\mathrm{0}$. In (a), the red curve shows the Rayleigh profile starting from the SCL and the purple curve shows the mixing line connecting the air coming from above to the surface evaporation, in the case $r_{\mathrm{orig}}=q_{\mathrm{orig}}/q_{\mathrm{0}}=\mathrm{0.7}$. The green curve shows the Rayleigh profile starting from the SCL and the blue curve shows the mixing line connecting the air coming from above to the surface evaporation, in the case $r_{\mathrm{orig}}=q_{\mathrm{orig}}/q_{\mathrm{0}}=\mathrm{0.5}$. One can see that when r_orig is lower, the mixing line is more curved, leading to more enriched values. In (b), the purple curve joins the SCL air and the air at all altitudes above the SCL. One can see that different values for r_orig can lead to the same value of δD in the SCL.

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3.1 LMDZ simulations

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 δD_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.

3.2 STRASSE observations

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 δD₀ 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 h₀, (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 δD_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 δD₀ 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).

3.3 Estimating the altitude from which the air originates

Here we explain how z_orig is estimated based on LMDZ outputs. First, we assume that the q and δD at 500 hPa (q_f, δD_f) belong to a Rayleigh distillation line starting from the surface with effective fractionation α_eff:

In a real field campaign, this assumption means that we do not need to measure the full vertical profile of δD, but only δD_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>q₀) due to horizontal advection or convective detrainment from nearby moister regions; meanwhile, δD 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 , δD_oce, h₀, and δD₀ 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 δD_f we assume a constant α_eff=1.07 based on LMDZ simulation and that α_eq, α_K , δD_oce, h₀, and δD₀ 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 δD₀. 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, q₀, or δD₀ or vertical variations in q₀ or δD₀ within the SCL.

Figure 5Schematics illustrating the typical structure of tropical marine boundary layers. The sub-cloud layer (SCL) extends from the surface to the lifting condensation level (LCL) and the cloud layer extends from the LCL to the inversion (z_i). EIS stands for the estimated inversion strength. Left: shape of the vertical profile in q (black) and q_s (green). Right: shape of the vertical profile in potential temperature θ, inspired by Wood and Bretherton (2006). The LCL, z_orig , $z_{\mathrm{orig},r_{\mathrm{orig}}=\mathrm{0.6}}$, and z_i altitudes defined in Sect. 3.4 are indicated.

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3.4 Boundary layer structure diagnostics

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).

3.5 Averages and composites

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 (ω₅₀₀, 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 ω₅₀₀ (Bony et al., 2004). To make such plots, we divide the ω₅₀₀ 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 ω₅₀₀ belongs to this interval (e.g., Fig. 8a will be an example). Note that such composites are carried out on seasonal-mean ω₅₀₀ 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 ω₅₀₀ 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 δD₀.

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.

Figure 6Maps of winter-mean ω₅₀₀ (a) and EIS (b) simulated by LMDZ.

3.6 Decomposition method for δD₀

To understand what controls the δD₀ spatiotemporal variations, δD₀ 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 δD₀ as a function of r_orig, α_eff, SST, h₀, 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 δD₀ is estimated as δD_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, h₀, ϕ, and η to δD₀ are estimated as detailed in Table 1. All the contributions have the same units as δD₀ (‰). The sum of these components yields a quantity that is very close to the simulated δD₀, 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 δD variability are quantified by performing a linear regression of each of the components as a function of δD₀. If the correlation coefficient is significant for a given factor, then the slope quantifies the contribution of this factor to the variability of δD₀. 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 δD₀ can be quantified by performing the regression among all locations and seasons. The contributions to the daily variability of δD₀ can be quantified by performing the regression among all days of a given season at a given location.

Table 1Equations to calculate the relative contributions of r_orig, α_eff, SST, h₀, ϕ, and η to δD₀ and the physical meaning of these contributions.

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3.7 Decomposition method for r_orig

To understand what controls r_orig, a similar method as for the decomposition of δD₀ can be applied. We can write r_orig as

where $\stackrel{\mathrm{‾}}{T}\left(z_{\mathrm{orig}}\right)+\mathit{\delta }T\left(z_{\mathrm{orig}}\right)=T\left(z_{\mathrm{orig}}\right)$ is the temperature at altitude z_orig, $\stackrel{\mathrm{‾}}{T}$ is the tropical-ocean-mean temperature profiles, h(z_orig) and P(z_orig) are the relative humidity and pressure at z_orig, and δT(z_orig) is the temperature perturbation compared to $\stackrel{\mathrm{‾}}{T}$. Therefore, the variability of r_orig is decomposed into the effect of four factors: q₀, z_orig, h(z_orig), and δT(z_orig). In practice, r_orig and z_orig are calculated following Sect. 3.3, and then Eq. (11) is applied.

4.1 Decomposition of δD₀ variability

The spatial variations in δD₀ 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, δD₀ values exhibit a maximum for weakly ascending or subsiding regions: δD₀ decreases with increasing vertical velocity of both signs (Fig. 8a black); δD₀ 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, h₀, rain evaporation, and horizontal advection effects (Sect. 3.6). We check that the reconstructed δD₀ from the sum of its four contributions is very similar to the simulated δD₀ (Figs. 7b, 8 dashed black).

In ascending regions, the main contribution explaining the more depleted δD₀ 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 δD₀ 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 δD₀ (Figs. 7e, 8a pink): as subsidence is stronger, or as EIS increases, SST is colder, leading to larger α_eq and thus more depleted δD₀. Another important factor is h₀ (Figs. 7f, 8a purple): as subsidence is stronger, h₀ is drier, leading to more depleted δD₀. The contribution of r_orig is also a significant contribution to the depletion of δD₀ 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 δD₀ 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 δD₀ 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 δD₀ seasonal spatial variations into these different effects (Sect. 3.6). In regions of large-scale ascent, α_eff is the main factor explaining the δD₀ 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 %), h₀ (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 δD₀ 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 δD₀.

Figure 7(a) Map of winter-mean δD₀ simulated by LMDZ. (b) Map of winter-mean δD₀ reconstructed as the sum of the four contributions. Tropical-mean δD₀ was added to compare with (a) on the same color scale. (c) Map of the contribution of r_orig on winter-mean δD₀ calculated from Eq. (9) (see Sect. 3.6). (d) Same as (b) but varies for α_eff. (e) Same as (b) but for SST. (f) Same as (b) but for h₀.

Figure 8Composites as a function of ω₅₀₀ (a) and of EIS (b) of the seasonal averages of δD₀ simulated by LMDZ over all tropical ocean locations (black). Same for the sum of the contributions (black dashed and for each individual contribution to δD₀: r_orig varies (green), α_eff (dark red), SST (pink), h₀ (purple), rain evaporation (dashed brown), and horizontal advection (dashed blue). The tropical mean δD₀ was added to each contribution to plot on the same scale as simulated δD₀. The number of samples in each bin is indicated on a logarithmic scale on the right-hand side (dotted black line).

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Table 2Decomposition of the spatial and seasonal variation in δD₀ into its six contributions: effect of r_orig, α_eff, SST, h₀, rain evaporation, and horizontal advection. For each contribution, we show the correlation coefficient of the linear regression of the contribution as a function of δD₀. The analysis is performed separately for ascending and subsiding regimes. All seasons and locations over tropical oceans (30^∘ N–30^∘ S, ocean fraction >80 %, surface evaporation >0.5 mm d⁻¹) are considered. The threshold for the correlation coefficient to be statistically significant at 99 % is 0.15 or lower in all cases. We write correlation coefficient and slope values between brackets when they are not significant at 99 %.

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Table 3As in Table 4 but at the daily scale. The correlation coefficients and slopes are averaged over all seasons and locations over tropical oceans (30^∘ N–30^∘ S, ocean fraction >80 %), separately for ascending and subsiding regimes.

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4.2 Decomposition of r_orig variability

Given the importance of r_orig in controlling the δD₀ variations, we now decompose r_orig into its four contributions: q₀, z_orig, h_orig, and δT_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 δD₀. This mechanism through which a moister FT leads to a more depleted δD₀ 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 q₀ 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 δD₀ 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 δD₀ 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).

Figure 9(a) Map of winter-mean r_orig simulated by LMDZ. (b) Map of winter-mean r_orig reconstructed as the sum of the four contributions. Tropical-mean r_orig was added to compare with (a) with the same color scale. (c) Map of winter-mean r_orig calculated from Eq. (11) if only q₀ varies (see Sect. 3.6). (d) Same as (b) but if only z_orig varies. (e) Same as (b) but if only h(z_orig) varies. (f) Same as (b) but if only δT_orig varies.

Figure 10Composites as a function of ω₅₀₀ (a) and of EIS (b) of the seasonal averages of r_orig simulated by LMDZ over all tropical ocean locations (black). Same for the sum of the contributions (black dashed) and for each individual contribution to r_orig:q₀ varies (green), z_orig (dark red), h(z_orig) (pink), and δT_orig (purple). The number of samples in each bin is indicated on a logarithmic scale on the right-hand side as bars.

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Table 4Decomposition of the spatial-seasonal variation in r_orig into its four contributions: effect of q₀, z_orig, h_orig, and δT_orig variations. For each contribution, we show the correlation coefficient of the linear regression of the contribution as a function of r_orig. The analysis is performed separately for ascending and subsiding regimes. All seasons and locations over tropical oceans (30^∘ N–30^∘ S, ocean fraction >80 %) are considered. The threshold for the correlation coefficient to be statistically significant is 0.15 or lower in all cases. We write correlation coefficient and slope values between brackets when they are not significant at 99 %.

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Table 5As in Table 4 but at the daily scale. The correlation coefficients and slopes are averaged over all seasons and locations over tropical oceans (30^∘ N–30^∘ S, ocean fraction >80 %), separately for ascending and subsiding regimes.

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4.3 Estimating altitude z_orig

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 δD₀ 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).

Figure 11(a) Map of winter-mean z_orig estimated from δD₀ simulated by LMDZ. (b) Same as (a) but z_orig that we would estimate if r_orig were constant set to 0.6, z_orig(r_orig=0.6). (c) Same as (a) but for z_i simulated from LMDZ. Only days when EIS >2 K are considered, otherwise z_i is difficult to estimate. (d) Same as (a) but for LCL simulated by LMDZ.

Figure 12Composites as a function of EIS of seasonal mean of z_orig (black), z_orig(r_orig=0.6) (green), z_i (blue), and LCL (red). The composite profiles of cloud cover are also shown, showing deep clouds when EIS is close to 0 and the shallowest clouds when EIS is largest.

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6.1 Measurement errors

The first source of uncertainty is measurement errors. We recalculate z_orig assuming an error of 0.4 ‰ on δD₀ (typical of what we can measure with in-situ laser instruments;  Aemisegger et al., 2012; Benetti et al., 2014) and 1 ‰ on δD_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 δD_f lead to errors on z_orig of the order of 20 m (Fig. 14a, green), errors on δD₀ 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, δD₀ 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.

6.2 Neglecting rain evaporation

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 δD₀ 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).

6.3 Neglecting horizontal advection

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.

6.4 Daily variability in the steepness of δD profiles

The fourth source of uncertainty arises from the daily variability in α_eff (Appendix D2). Estimating α_eff requires us to measure δD_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 δD in altitude are costly and difficult (Sodemann et al., 2017).

Let us assume that we have only one δD_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 $\stackrel{\mathrm{‾}}{{\mathit{\alpha }}_{\mathrm{eff}}}+{\mathit{\sigma }}_{{\mathit{\alpha }}_{\mathrm{eff}}}$ and $\stackrel{\mathrm{‾}}{{\mathit{\alpha }}_{\mathrm{eff}}}-{\mathit{\sigma }}_{{\mathit{\alpha }}_{\mathrm{eff}}}$. The error on z_orig is calculated as $\left(z_{\mathrm{orig}}(\stackrel{\mathrm{‾}}{{\mathit{\alpha }}_{\mathrm{eff}}}-{\mathit{\sigma }}_{{\mathit{\alpha }}_{\mathrm{eff}}})-z_{\mathrm{orig}}(\stackrel{\mathrm{‾}}{{\mathit{\alpha }}_{\mathrm{eff}}}+{\mathit{\sigma }}_{{\mathit{\alpha }}_{\mathrm{eff}}})\right)/\mathrm{2}$. 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 δD₀ measurements cannot be useful unless we measure δD_f on a daily basis as well. Practically, we could imagine measuring FT properties (δD_f) at the top of a mountain while we measure δD₀ 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).

6.5 Rayleigh assumption for the shape of δD profiles

Finally, as a fifth source of uncertainty comes the assumption that the δD profile follows a Rayleigh distillation line (Sect. 2.2). However, in both LMDZ (Appendix D1) and nature (Sodemann et al., 2017), δD 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 δD profiles are often close to a mixing line in the lower troposphere (Appendix D1). Whatever z_orig in the lower troposphere, the δD₀ calculated from Eq. (6) is nearly constant because the δD profile is close to a mixing line (Appendix A, Fig. 4b). Whatever z_orig in the middle troposphere, the δD₀ calculated from Eq. (6) is also nearly constant because r_orig there is very small. So whatever z_orig, the δD calculated from Eq. (6) is nearly constant, and the numerical solution fails.

However, it is possible that δD 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 δD 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 δD profiles are very welcome to help quantify it.

To summarize, δD₀ measurements could potentially be useful to estimate z_orig with a useful precision, but only if we measure daily δD_f in the mid-troposphere, if the shape of δD profiles can be better documented, if we measure δD₀ 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 δD₀ with an accuracy of 0.1 ‰.

Figure 14Errors when estimating z_orig from δD₀ observations, as a function of EIS, as predicted by LMDZ. (a) Error we would make if δD₀ is measured with a 1 ‰ error (red), and error we would make if δD_f is measured with a 1 ‰ error (green). (b) Error we would make if we neglect horizontal advection effects (purple) and rain evaporation effects (brown). (c) Error if one uses α_eq as a function of local temperature to estimate α_eff (blue), error if one uses the seasonal-mean profile instead of the daily profile to estimate α_eff (black). The standard deviations among all daily errors estimated in each bin of EIS are also shown.

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