Rain Evaporation, Snow Melt, and Entrainment at the Heart of Water Vapor Isotopic Variations in the Tropical Troposphere, According to Large‐Eddy Simulations and a Two‐Column Model

Abstract We aim at developing a simple model as an interpretative framework for the water vapor isotopic variations in the tropical troposphere over the ocean. We use large‐eddy simulations of disorganized convection in radiative‐convective equilibrium to justify the underlying assumptions of this simple model, to constrain its input parameters and to evaluate its results. We also aim at interpreting the depletion of the water vapor isotopic composition in the lower and midtroposphere as precipitation increases, which is a salient feature in tropical oceanic observations. This feature constitutes a stringent test on the relevance of our interpretative framework. Previous studies, based on observations or on models with parameterized convection, have highlighted the roles of deep convective and mesoscale downdrafts, rain evaporation, rain‐vapor diffusive exchanges, and mixing processes. The interpretative framework that we develop, valid in case of disorganized convection, is a two‐column model representing the net ascent in clouds and the net descent in the environment. We show that the mechanisms for depleting the troposphere as the precipitation rate increases all stem from the higher tropospheric relative humidity. First, when the relative humidity is larger, less snow sublimates before melting and a smaller fraction of rain evaporates. Both effects lead to more depleted rain evaporation and eventually more depleted water vapor. This mechanism dominates in regimes of large‐scale ascent. Second, the entrainment of dry air into clouds reduces the vertical isotopic gradient and limits the depletion of tropospheric water vapor. This mechanism dominates in regimes of large‐scale descent.


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This supporting information illustrates the convective organization of the large-eddy simulations ( Figure S1), assesses the robustness of the results with respect to the definition for clouds and the environment (Text S1) derives the simple equation for rain evaporation (Text S2), documents how the deep and shallow overturning circulations contribute to set the domain-mean humidity and isotopic profiles ((Text S3) and derives an equation relating the q − δD v steepness coefficients in updrafts and downdrafts to vertical profiles (Text S4).

Figure S1: Snapshots of the large-eddy simulations
To assess the convective organization in our simulations, we plot snapshots of maps of the precipitation rate and of near-surface water vapor δD v . We can see that convection is disagregated, with isolated cumulonimbi (figure S1).
Text S1: Robustness of the results with respect to the definition for clouds and the environment In our simple two-column framework, we decide to separate cloudy regions from their environment based on a threshold on cloud water content (e.g. Thayer-Calder and Randall (2015)): we define parcels as "cloudy" when the cloud water content exceeds 10 −6 g/kg.
In the previous studies, alternative definitions have been based on vertical velocity (e.g. Hohenegger and Bretherton (2011)) and/or buoyancy (e.g. Siebesma and Cuijpers (1995)), or position in altitude-equivalent potential temperature diagrams (Pauluis & Mrowiec, 2013). We thus test here the robustness of our results to different definitions, by defining "very cloudy regions" with cloud water content larger than 10 −3 g/kg, "cloudy updrafts" with cloud water content larger than 10 −6 g/kg and ascending vertical velocity, "saturated drafts" with relative humidity larger than 99%, "nearly-saturated drafts" with relative February 8, 2021, 1:42pm : X -3 humidity larger than 95%, and (7) "moist static energy updrafts" including all parcels falling into bins of frozen moist static energy in which the vertical velocity is positive (Pauluis & Mrowiec, 2013). "Cloudy updrafts" and "nearly-saturated regions" are the most and least restrictive definitions respectively ( Figure S2a,f). In all definitions, the cloudy region fraction remains below 10% except in the free lower and middle troposphere. In stricter definitions, the cloudy regions are characterized by a larger vertical velocity ( Figure S2b) and a larger cloud water content (not shown). The entrainment is not strongly sensitive to the definition in the free troposphere ( Figure S2c).
The ratio of the isotopic ratio in the rain evaporation over that in the environment vapor (φ = R ev /R v ) is not very sensitive to the definition for the ctrl ( Figure S2e), but its value near the melting level is quite sensitive ( Figure S2g). In all definitions, we can see the negative anomaly near the melting level, but it is much more negative in the loosest definitions. This is because in stricter definitions, the non-fractionating evaporation of cloud water droplets takes place in the environment. Since droplet evaporation takes place in shells around convective updrafts, and does not directly affect the environment, we chose a loose definition for the "cloudy regions".
The ratio of the large-scale mass flux over the cloudy mass flux, η, for HighPrec is larger in loose definitions ( Figure S2h). This is because the cloudy regions incorporates cloudy downdrafts that compensate for the upward mass flux in cloudy updrafts. This large η in the loose definition may contribute to the overestimate of the direct effect of large-scale forcing on δD by the two-column model, and ultimately to the underestimate of the "vapor amount effect". The quick equilibration between the rain and vapor motivates us to use a simple equation in which some mass q l0 of rain, with isotopic ratio R l0 , partially evaporates and isotopically equilibrates with some mass q e0 of vapor (subscript e for environment), with isotopic ratio R e0 . After the evaporation and equilibration process, the masses of rain and vapor are noted q l and q v : where q ev is the mass of evaporated rain water. The corresponding isotopic budget writes: where R l , R e and R ev are isotopic ratios in the final rain, final vapor and evaporation flux. Isotopic equilibrium writes: where α eq is the equilibrium fractionation coefficient.
We define: February 8, 2021, 1:42pm Re-arranging these equations, we get: If the mass of rain is much greater than than of vapor, i.e. g ≫ 1, the equation becomes: Therefore, φ scales with λ. In addition, φ increases with f ev from φ = λ/α eq for f ev = 0 (first order approximation) to φ = λ pour f ev = 1 (total evaporation).
Text S3: How do the deep and shallow overturning circulations contribute to set the domain-mean humidity and isotopic profiles?
In our two-column model, we calculate bulk profiles for our input parameters γ (saturation specific humidity lapse rate), η (ratio of the large-scale vertical velocity relative to the cloudy region velocity), f ev (evaporated fraction) and φ e (relative isotopic ratio of rain evaporation), ǫ (entrainment rate) and δ (detrainment rate). These bulk profiles hide large horizontal disparities. In particular, we expect a deep overturning circulation in high-cloud parts of the domain and a shallow overturning circulation in low-cloud or clear-sky parts of the domain. How do the deep and shallow overturning circulations contribute to set the domain-mean humidity and isotopic profiles?
To address this question, we repeated all our diagnostics for each simulations on two subdomains: the high-cloud columns (columns for which the cloud condensate mixing ratio February 8, 2021, 1:42pm X -6 : exceeds 10 −6 g/kg in at least one level between 7 and 11 km) and the low-cloud/clear-sky columns (remaining columns).  Figure S3g-h). In particular, φ e is negative in highcloud columns, due to the evaporation of rain arising from the melting of snow formed in the anvil clouds. Therefore, if we applied our two-column model separately for the two sub-domains, we would get very different relative humidity and δD v profiles.
Yet, the relative humidity, δD v and α z profiles simulated by the LES are remarkably similar between the two sub-domains ( Figure S3i-k). This means that the relative humidity and δD v profiles are quickly homogenized between the sub-domains. This is likely due to the disorganized state of convection in our simulations. Isolated cumulonimbi develop randomly in the domain and decay within a few hours, so that each location of the domain regularly undergo the influence of deep convective processes (figure S1). This prevents the building of strong horizontal gradients between high-cloud and low-cloud/clear-sky sub-domains. As a consequence, in our simulations, both deep and shallow overturning circulations simultaneously act on the domain-mean relative humidity and δD v profiles. This justifies mixing them together in our two-column framework.
In case of organized convection however, stronger humidity and isotopic horizontal variations are expected to build at the meso-scale. Thus, the two sub-domains may show more contrasted humidity and δD v profiles. Our two-column framework applied on domainmean profiles may thus not apply so well.
Text S4: How do the q − δD v steepness coefficients in updrafts and downdrafts relate to vertical profiles?
In our simple SCL water budget, the efficiency of updrafts and downdrafts to deplete the SCL is quantified by the q − δD v steepness coefficients in updrafts and downdrafts ,α u and α d (Risi et al., 2020). In our LES, α u and α d scale with the domain-mean q − δD v steepness in the vertical, α z (Risi et al., 2020). Is this scaling a universal property for all convective conditions, or is it specific to some conditions that our LES satisfy? To address this question, we theoretically calculate α u as a function of α z , and highlight the underlying hypotheses. In brief, we will show that α u and α d scale with α z only if the convection is disorganized.

q and R as a function of w
In each grid point, at a given altitude, we assume that the specific humidity q and water vapor isotopic ratio R are controlled by vertical advection, a source term and a sink term.
The source term may be turbulent mixing, advection or condensate evaporation The sink can be either mixing or advection without fractionation or condensation: February 8, 2021, 1:42pm where w is the vertical velocity, S is the source term, p is the sink rate, R S is the effective isotopic ratio of the source and α p is the effective isotopic fractionation coefficient of the sink (e.g. α p = 1 for mixing or advection and α p = α eq for condensation) For simplicity and consistent with the approximations in our two-column model, we assume: At permanent state, we find: If we assume that horizontal variations of q and R associated with w are small, we can linearize q and R as a function of w: We can check that q increases with w (Risi et al., 2020). In general, we expect α z > α p , and thus R also increases with w (Risi et al., 2020). 2. Domain-mean q and R:q andR If S, p and γ do not vary too much with w, then we have: If in addition, if α p and α z do not vary too much with w, then we have: We can thus rewrite q and R as: 3. Effective q and R in updrafts: q u and R u Following (Risi et al., 2020), M u , q u and R u are defined as: February 8, 2021, 1:42pm where U is the ensemble of grid points where w > 0 and n is the total number of grid points. Thus, Using equation 1 , we get: This corresponds to the ratio of w variance over the w mean among ascending grid points. We thus have: Similarly, 4. q − δD v steepness coefficient in updrafts: α u α u is defined as: Using equations 2 and 3, we get: February 8, 2021, 1:42pm : X -11 α u = 1 + ln 1 + γ·αz p·αp σ − ln 1 + γ p · σ ln 1 + γ p · σ Since we assumed that horizontal variations of q and R associated with w were small, we can linearize the logarithms: Therefore, α u scales with α z , consistent with our LES simulations (Risi et al., 2020).
Similarly, we can show that α d scales with α z . We expect α u and α d to equal α z (if α p = 1) or to be slightly smaller (if α p > 1). The slightly larger values of α u and α d in our simulations may be due to slightly violated assumptions, or to the fact that we use α z 1 km above the SCL top.

Discussion of the underlying hypotheses
We assumed two main hypotheses: 1. q and R variations associated with w are small 2. S, p, γ, α p and α z do not vary too much with w.
Together, these assumptions mean that q and R variations with w are small.
We can check that this is the case in our simulations: q and R anomalies relative to the domain-mean remain are smaller than 10% and 2% of the domain-mean respectively at the SCL top (Risi et al., 2020). This is because the simulated convection is disorganized: isolated cumulonimbi develop randomly in the domain and decay within a few hours, so that each location of the domain regularly undergo the influence of deep convective processes ( figure S1). This is why in our LES simulations, α u and α d scale with α z .

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In contrast, in a case of organized convection, we expect q and R to vary much more strongly at the meso-scale. First, meso-scale convective systems live longer, so some regions of the domain can remain several days without undergoing any effect of deep convective processes. This may contribute to the drier troposphere observed when convection is more organized (Bretherton & Khairoutdinov, 2015;Tobin et al., 2012). Second, mesoscale convective systems are larger, reducing the effect of horizontal advection and mixing, and thus allowing strong horizontal gradients to build. In other words, meso-scale convective systems are better protected from the environment, and vice versa (Tobin et al., 2012). The observation of strongly depleted water vapor in tropical cyclones (Lawrence et al., 2004), squall lines (Tremoy et al., 2014) and mature meso-scale convective systems in general (Kurita, 2013) support our expectation that larger horizontal variations in R are expected in case of organized convection.
Therefore, in case of organized convection, S, p, γ, α p and α z may strongly vary with w. Consequently, α u and α d may not scale with α z . Rather, horizontal variations in R at the meso-scale may control α u and α d .