The total energy of the saturated parcel is limited by the sensible enthalpy difference of the phase change from vapor to liquid, δQ=dH, and can be written in finite forms as follows (hereafter the signs in the energy balance equations describe the direction of energy transfer): 2δQ=dH∴Lv⋅ql=-cp,cloud⋅ΔT=-cp,cloud⋅(T0-T). Where ql is the liquid water mixing ratio (kgkg-1), cp,cloud is the specific heat capacity of cloudy air (Jkg-1K-1), and Lv is the latent heating of vaporization (Jkg-1). The ql and cp,cloud are defined as the following: 3ql=LWCρair and, 4cp,cloud=cpd+cpv⋅rs+cpl⋅ql. Where, ρair is the dry air density (kgm-3), LWC is the liquid water content (kgm-3), rs is the saturation water vapor mixing ratio (kgkg-1), cpd is the specific heat of dry air at constant pressure (1005 Jkg-1K-1), cpv is the specific heat of water vapor at constant pressure (1850 Jkg-1K-1), cpl is the specific heat of liquid water at constant pressure (4218 Jkg-1K-1). We assume the warmer temperature (T0) in the cp.cloud and Lv calculations, as it represents the initial condition of the saturated parcel. For small temperature differences at cloud base (<∼1K), cp.cloud varies only weakly with rs and ql. Since T and T0 represent the temperatures of a saturated parcel under isobaric conditions, they are equivalent to the dew-point temperature at the corresponding pressure. Furthermore, rs is calculated as follows: 5rs=Rd/Rv⋅es(T)p-es(T) Where Rv is the gas constant for water vapor (461.5 Jkg-1K-1), Rd is the gas constant for dry air (287.05 Jkg-1K-1), and es(T) is the saturation water vapor pressure (Pa) (Bolton, 1980; Rogers and Yau, 1989). The values of es can be calculated for a wide range of temperatures in Celsius (-30°C≤Tc≤35°C) with minor uncertainties (∼0.1%) as follows: 6es=611.2⋅exp⁡17.67⋅TcTc+243.5

During the phase change, the internal energy decreases (T0>T) as ql and Q increase. The phase change process leads to an increase in the air temperature, represented here by the latent-heat-equivalent temperature (TQ), which is approximately conserved during the vapor-to-liquid conversion, i.e.: 7TQ=T+ΔTQ≈T0 Where, 8ΔTQ=Lv⋅qlcp,cloud

The formulation of RH describes the amount of water vapor in the air relative to the maximum amount of water vapor the air can hold for a given temperature, i.e., the saturation ratio, and is defined as follows: 9RH=100⋅e(T)es(T)[%]

Where e(T) is the ambient water vapor pressure (Pa). For a saturated parcel or cloud undergoing cooling (from T0 to T) the reference-state saturation ratio (RHc) is calculated as follows: 10RHc=100⋅es(T)es(T0)[%] and the cloud supersaturation (Sc) is formulated as follows: 11Sc=RHs-RHc=100-RHc[%] where RHs=100% is the saturation ratio at the reference state (T0), and Sc expresses the thermodynamic excess of water vapor relative to equilibrium with respect to liquid water at the reference state. Since the cooling process is related to the increase of ql (shown in Eq. 2), the increase in Sc is associated with vapor-to-liquid conversion and Q.

According to the first law of thermodynamics, the change in internal energy of a closed system is determined by the balance between heat exchange and pressure-volume work. For an isobaric process involving only pressure-volume work, the change in sensible enthalpy (dH) can be interpreted as the sum of the change in internal energy and the expansion work performed by the system. Under adiabatic conditions, the dH equals the heat exchanged at constant pressure by a saturated parcel. Thus, for a saturated parcel undergoing condensational growth from the reference state, the latent heat released (δQ=dH) is the source of heat in the thermodynamic system, and the energy budget can be calculated following the first law as follows: 12Q=ΔU+Ws where Q represents the latent heat released by condensation, ΔU is the change in internal energy of the parcel associated with vapor–liquid conversion, and Ws is the specific work done by water vapor during expansion. The terms of ΔU and Ws are calculated for a saturated parcel under isobaric cooling as follows: 13ΔU=-cp,cloud⋅(T0-Tws) and, 14Ws=rs⋅∫αvs0αvsesdαv=rs⋅∫αvs0αvsRv⋅Tαvdαv≈rs0⋅Rv⋅T0⋅ln⁡αvsαvs0=rs0⋅Rv⋅T0⋅ln⁡TT0⋅es(T0)es(T) Where αv is the specific volume of water vapor for saturated conditions (m3kg-1). The Ws term represents the effective thermodynamic work associated with changes in the specific volume of saturated water vapor during cloud-adiabatic cooling. Under quasi-isobaric conditions and small cloud base cooling intervals, the integration is referenced to the initial thermodynamic state of the saturated parcel (T0), and Ws depends primarily on T0 and T.

Analogous to Eq. (2), the energy budget from Eq. (12) can be calculated as follows: 15Lv⋅ql=-cp,cloud⋅ΔTws+rs0⋅Rv⋅T0⋅ln⁡αvsαvs0 or, 16Lv⋅ql-rs0⋅Rv⋅T0⋅ln⁡αvsαvs0=-cp,cloud⋅(T0-Tws)

From this budget, Tws, the saturated temperature, is slightly warmer than T calculated in Eq. (2) due to the Ws term. Similarly, the resulting air temperature (Tair) will be slightly cooler than TQ due to the Ws term. Thus, Tair is calculated as follows: 17Tair=TQ-Wscp,cloud The differences of the saturated temperatures (Tws-T) and air temperature (T0-Tair) are the same.

The reversibility of the energy budget for isobaric conditions relies on the fact that the magnitude of the total latent energy (δQ=dH) can be expressed equivalently in terms of either LWC or the temperature difference (ΔT) between two saturated states. In Sect. 3.1, we show that the relationship between LWC and ΔT is one-to-one since the parameters are calculated as a function of the initial thermodynamic properties of the saturated parcel. In this framework, this dependence is treated explicitly through an iterative solution, ensuring consistency between condensate and heat capacity.

When the total energy is partitioned as Q=ΔU+Ws, the internal-energy component represents a reduced cooling of the saturated parcel compared to the case without accounting for the work term Ws. Despite this partitioning, the functional dependence between ql and ΔTws remains unchanged, so the relationship between thermodynamic variables remains one-to-one. Since the parameters at Ws, cp,cloud and Lv are calculated for the initial conditions of the parcel (T0), the reversibility of the heat budget is preserved, ensuring a consistent and reversible relationship between condensate and temperature within the proposed framework. The numerical differences while applying Eqs. (2) and (12) to calculate Q are negligible (<∼0.01%). The differences between the thermodynamic and energetic components of the cloud parcel for simulations with a range of initial temperatures T0 (2–30 °C) and adiabatic cooling magnitudes (ΔT) are presented in Sect. S1 in the Supplement. The calculations are performed for the energy and vapor budgets at an atmospheric pressure of 950 hPa, similar as shown in the Sect. 4.1.

The calculation of liquid supersaturation (Sl) and water vapor supersaturation (Sv) from the cloud supersaturation (Sc) is performed as a function of the energy budget formulation described in Sect. 3.3, where Sl and Sv are the energy contributions of ΔU and Ws in the total energy of the saturated parcel (Q), respectively.

The signs in the energy balance equations shown above describe the direction of energy transfer. However, Sl and Sv are not directional variables. They represent the partition of Sc considering the energy budget, which is defined from the absolute magnitudes of the energetic contributions, and thus, for a given latent heat released, due to condensation Sl and Sv are calculated as follows: 18Sl=Sc⋅|ΔU|Q[%],19Sv=Sc⋅WsQ[%], Where, 20Sc=Sl+Sv[%]

The calculations of RHc and cloud supersaturations (Sc, Sl, and Sv) at cloud bases are performed from measurements collected during the ACRIDICON-CHUVA campaign using HALO. Cloud probes and meteorological sensors on board HALO have been used to collect the necessary data. Table 1 summarizes the general thermodynamic conditions and time intervals of cloud properties measured at cloud bases. The cloud measurements were conducted with the Cloud Combination Probe-Cloud Droplet Probe (CCP-CDP) (Lance, 2012). In this study, we focus on the initial stage of cloud formation. Therefore, only 1 Hz cloud passes (individual in-cloud sampling intervals) with Nd greater than 20 cm-3, developing under updraft conditions, are considered. This threshold of Nd=20cm-3 is used to avoid haze and well-mixed cloud elements in the analysis. Furthermore, cloud passes with measured raindrops are removed from our analysis. In the supplementary material (Sect. S2), we present the sensitivity analysis assuming the proposed threshold for cloud identification and varying the values of Nd and LWC for the derived supersaturations and their relationships. These analysis shows that the derived supersaturations and their relationships assuming the proposed threshold (Nd>20cm-3) are robust with respect to variations in cloud identification criteria. The remaining variability is primarily driven by environmental heterogeneity and turbulence rather than by the threshold definition itself.

The closure analysis relies on the reversibility of the cloud adiabatic parcel as described in Sect. 3.4. The calculations of RHc, Sc, Sl, and Sv require the information of p, LWC, T, and T0 near cloud bases. From the airborne data at cloud bases, p, T and LWC are measured. The distinction between the measured cloud-base quantities and the diagnostically reconstructed thermodynamic state is illustrated in Fig. 1b. Therefore, we estimate T0 and Tair for each cloud pass. Below are the sequential steps used in our analysis:

T0 (or TQ) is iteratively estimated for each cloud pass from Q (calculated from LWC, T and p) using Eq. (2). RHc and Sc are subsequently calculated from T and estimated T0.

During flights AC11, AC14 and AC17, the NCCN(Sv) spectra from two CCN counters (CCNa and CCNb) were measured below cloud base prior to cloud base measurements with the CCP-CDP probe. This comparison provides an important evaluation of the derived water vapor component of cloud supersaturation (Sv), since CCN chamber measurements are independent of in-cloud condensational growth processes. The Nd(Sv) spectra were converted to standard temperature and pressure (STP) conditions before comparison. The comparison between Nd(Sv) and NCCN(Sv) aimed to evaluate the physical consistency of the proposed thermodynamical framework. In the CCN counters, Sv is determined from the Köhler theory using particles with a known physicochemical composition, typically ammonium sulfate, and is therefore independent of cloud processes.

To illustrate the thermodynamic basis of the energetic budget analysis at cloud base, we first describe the fundamental relationships between the initial temperature of the saturated parcel (T0) and adiabatic cooling ΔT for T0 ranging from 2–30 °C. Figure 2 shows the simulated values of LWC, Q, ΔU, and Ws resulting from the energy balance at a pressure level of 950 hPa. All quantities from the energy balance are expressed as positive values to represent the partition of the total energy (Q) during the condensational growth. In the case in which water vapor is completely condensed (Q=dH) the LWC (solid lines) increase almost linearly with cooling rate for different T0. For the same cooling rate, a slightly larger LWC is calculated for lower temperatures owing to the higher dry air density of the saturated parcel. When accounting for Ws in the energy budget, the resulting values of LWC (dotted lines) are relatively smaller. At 950 hPa, the fractional reduction in LWC due to Ws increases approximately linearly with temperature, ranging from about 4 %–6 % at low temperatures (∼2–6°C) to 15 %–20 % at higher temperatures (∼25-30°C) (see Fig. S1 in the Supplement). The values of ΔU are slightly smaller than Q since the total energy is partitioned with Ws, which increases approximately linearly with temperature and logarithmically with the cooling rate.

Figure 3 shows the corresponding values of RHc, Sc, Sl, and Sv derived from the energy budget framework. Figure 3a and b show RHc decreasing below 100 %, with a corresponding increase of Sc above 0 % as a function of the prescribed adiabatic cooling. For a given cooling magnitude, the increase in Sc becomes slightly smaller at warmer temperatures because the fractional reduction in RHc[es(T)/es(T0)] for a fixed ΔT decreases slightly with increasing T0, consistent with the temperature dependence of the Clausius–Clapeyron relation. Figure 3c and d show the partitioning of Sc into Sl and Sv resulting from the energy budget analysis. The values of Sl are about one order of magnitude larger than Sv, indicating that most of the enthalpy change associated with adiabatic cooling and condensation contributes to vapor-to-liquid conversion and condensational growth, while Ws represents a smaller fraction of the total energy budget. Since Ws increases with T0, Sv becomes relatively larger under warmer cloud-base conditions for the same cooling magnitude, whereas the opposite behaviour is observed for Sl. This emphasizes the increasing contribution of saturation work to the total energy budget at warmer temperatures, which enhances Sv while reducing the condensational growth of cloud particles.

These relationships between energetic partitioning and the vapor budget remain qualitatively similar under lower-pressure conditions. However, as pressure decreases, the LWC is reduced because the density of dry air decreases. Furthermore, for a given T0 and ΔT, the magnitude of Ws is slightly enhanced at lower pressures due to the larger saturation mixing ratio (see Eq. 5), thereby increasing the contribution of water vapor expansion to the total energy budget. Consequently, a smaller fraction of the enthalpy change associated with cooling and condensation contributes to vapor-to-liquid conversion and condensational growth. At 850 hPa, for example, the reduction in LWC associated with Ws ranges from approximately 5 %–7 % at low temperatures (∼2-6°C) to about 18 %–22 % at higher temperatures (∼25-30°C) (Fig. S2 in the Supplement). The larger contribution of Ws to the energy budget leads to slightly larger Sv and smaller Sl relatively to those at 950 hPa.

The data collected during the ACRIDICON-CHUVA campaign over the Amazon Basin provides a particularly useful testbed for the cloud base closure analysis because the same aircraft and instrumentation sampled a wide range of cloud base environments, from relatively clean marine and forested conditions to polluted air masses influenced by biomass burning and the Manaus urban plume (see the locations of the cloud base measurements at Fig. S3 in the Supplement).

Figure 4 illustrates a case study of the dependence of Nd at cloud base on RH, RHc, cloud supersaturations (Sc, Sl, Sv), and the associated energetic terms (Q, ΔU, Ws) derived from measurements collected during Flight AC14. During this flight leg, cloud bases of developing cumuli were sampled under RH conditions ranging from ∼60% to ∼102%, at an altitude of about 1600 m and temperatures near 17.6 °C. Figure 4a shows that RH generally increases as a function of Nd even though in most cloud passes RH is smaller than 100 %. This variability in RH measurements arises from the characteristics of in situ temperature measurements: liquid water impinging on the sensor can induce evaporative cooling, whereas the latent heat released during particle condensational growth and buoyant air volumes may locally warm the sensor. RHc decreases consistently with Nd associated with higher LWC. For cloud passes where RH<RHc the local vapor pressure e(T) is typically smaller than expected from the cooling trajectory of the initially saturated parcel, suggesting enhanced entrainment of unsaturated environmental air and reduced condensational growth (see Fig. S4 in the Supplement). Under such conditions, smaller droplets evaporate preferentially due to the inhomogeneous nature of mixing in convective clouds. This selective evaporation reduces the local LWC, and the remaining droplet population is biased toward droplets formed on CCN with higher hygroscopicity, which are more resistant to evaporation. Conversely, cases in which RH>RHc indicate that the parcel retains a larger vapor content relative to theoretical calculations, favouring continued condensational growth, enhanced Sv, yielding higher Nd for a given CCN population.

Figure 4b shows the dependence of Nd on Sv and Sl at cloud base. Both Sl and Sv increase with Nd, although Sv remains nearly one order of magnitude smaller than Sl. The calculated Sv values exhibit the expected positive relationship between supersaturation and droplet concentration commonly observed in CCN activation spectra. The Nd(Sv) relationship exhibits substantial variability at cloud base owing to the inhomogeneous and turbulent nature of the in situ measurements. The dispersion is particularly enhanced for cases in which relatively large Nd values are associated with small Sv, corresponding to conditions where the local vapor pressure remains below the theoretical simulations (RH<RHc). Nevertheless, most of the measured water vapor pressures remain within the estimated uncertainty range of the theoretical estimates (see Fig. S4), indicating that the overall Nd-Sv relationship remains representative of the cloud base thermodynamic conditions. Figure 4c presents the energy budget components as a function of Sc, while Fig. 4d depicts the cloud supersaturations derived from the measured LWC. These figures show that the total energy (Q) increases strongly with LWC and is dominated by changes in internal energy (ΔU/Q; ∼90% on average), while the contribution from saturation work (Ws) is comparatively minor.

Similar characteristics of the energy budget were observed for the different flight legs measured in the Amazon Basin and Atlantic Ocean (Figs. S5–S18 in the Supplement). In these flight segments, most of the measured water vapor pressures were within the estimated uncertainty range of the theoretical cloud-adiabatic estimates. The estimated fraction of ΔU/Q accounted for ∼88%-91% in the energy budget, lower values were measured for warmer (marine) clouds during flight AC19. As shown in Sect. 4.1, Sv has a strong linear dependence on temperature, since this supersaturation component is associated with the Ws term. These results show that in warmer environments, a larger amount of energy relative to Q is expended for the air parcel's expansion, leading to a smaller decrease in the cloud parcel's internal energy and a lower cooling rate.

Figure 5 shows the droplet number spectra as a function of cloud supersaturations (Sc, Sl, Sv), Nd(S), derived from cloud base measurements for all flight segments (see Tables 3–5 for the statistics of the fit parameters). The Nd(S) were derived using power-law fits of the form Nd=C⋅Sk, where C is a scaling factor and k is the spectral exponent. Across all three supersaturations derived larger values of C and k parameters are estimated for aerosol-rich conditions (e.g., AC07), whereas smaller C and k values (e.g., AC19) reflect more CCN-limited regimes. The values of C and k have closer values for the power-law fits of Nd-Sc and Nd-Sl due to the similar values of Q and ΔU in the energetic budget partition (ΔU∼0.9⋅Q). The high coefficients of determination (R2>0.87) and low root-mean-square errors (RMSE<0.3) indicate robust agreement between the fitted relationships and the observations under different atmospheric conditions. The tables show that the values of R2 are sensitive to sample size, and larger cloud base datasets typically result in lower R2 values due to the inclusion of a broader range of atmospheric variability. In our observations, larger samples encompass more heterogeneous conditions in updraft velocity, turbulence intensity, and CCN concentrations, which increase scatter in the Nd-S relationships. Nevertheless, the fitted parameters C and k remain relatively stable across the different environmental conditions owing to the underlying physical dependence between Nd and S. A more detailed description of this analysis accounting for different cloud detection thresholds and measurements uncertainties is provided in Sect. S2.3.

The Nd(Sv) spectra calculated from the energetic budget analysis at cloud base are comparable to the NCCN(Sv) spectra observed near the surface level over marine regions and the Amazon Basin during the dry season (Khain and Pinsky, 2018; Pöhlker et al., 2016). Under clean marine conditions (flight AC19), the estimated Nd expected to activate at cloud base is about 740 cm-3 at Sv=1%. The maximum estimated Sv derived from the measurements is 0.27 %, corresponding to an adiabatic cooling (ΔTc) of ∼0.4 °C resulting from the closure analysis. In contrast, measurements obtained under more polluted conditions over forested regions during flights AC09 and AC18 show Nd values roughly twice as large for the same Sv. The maximum estimated Sv values derived from measurements were 0.34 % (ΔTc=0.53°C) and 0.44 % (ΔTc=0.71°C) for flights AC09 and AC18, respectively. Even larger Nd (Sv=1%) values are estimated for the data over the Central Amazon and the deforestation arc, where the droplet concentrations were more strongly influenced by emissions from urban activities and biomass burning. The largest estimated Sv value derived from the measurements (1.15 %) was calculated for flight leg AC17a corresponding to a ΔTc of 1.81 °C.

The cloud properties observed under CCN-limited regimes were generally associated with regions of weaker buoyant forcing, resulting in lower LWC and weaker w, with maximum values ∼2ms-1 (Fig. S19 in the Supplement). In contrast, aerosol-rich environments were associated with stronger buoyant forcing and enhanced adiabatic cooling at cloud base, leading to larger LWC, higher Sv, and stronger w (up to ∼4ms-1). Enhanced buoyant forcing also increased the variability of cloud-base measurements by strengthening fluctuations in the thermodynamic state of the cloud parcel (LWC, T, and p), which are the primary variables used in the energy budget and supersaturation calculations. However, the uncertainties in Sv do not scale directly with buoyant forcing itself, but rather with the uncertainty in these thermodynamic and microphysical measurements (see Sect. S2.2). Consequently, the largest observed Sv and w values were generally associated with larger cloud base LWC values, consistent with stronger adiabatic cooling and enhanced condensational growth.

The comparison of Nd(Sv) spectra vs. NCCN(Sv) spectra is performed for four flight segments in Central Amazon, AC11, AC14, AC17a, and AC17b. The Nd(Sv) spectra are interpreted as representative of the sub-cloud NCCN(Sv) population that was activated into cloud droplets after buoyant ascent to cloud base. During flights AC14 and AC17, the NCCN(Sv) measurements were performed about 700 m below cloud base altitude, while during flight AC11, they were performed about 1400 m below cloud base (Tables 1 and 2). Therefore, the comparison should be interpreted as a closure between nearby sub-cloud CCN populations and cloud base droplet activation rather than as a strict Lagrangian comparison of identical air parcels. This observational configuration is illustrated schematically in Fig. 1a.

Figure 6 shows NCCN(Sv) spectra measured with CCNa for Sv ranging from ∼0.2% to ∼0.55% (red marks), while NCCN(Sv) from CCNb was measured with a fixed Sv at ∼0.55% (blue marks). The corresponding Nd(Sv) spectra and the power-law fit parameters for each flight segment are shown in black. The fixed-Sv measurements from CCNb illustrate the variability of sub-cloud CCN concentrations for each flight segment. For segments AC11, AC14 and AC17a, the differences between minimum and maximum NCCN(Sv) exceeded 1300 cm-3, whereas during AC17b this range remained below 700 cm-3. The large variability in NCCN(Sv) is associated with heterogeneous air masses influenced by pollution plumes from biomass-burning and urban emissions. Under these conditions, the typical monotonic increase of NCCN(Sv) with increasing Sv is not consistently observed in the CCNa measurements. This effect of pollution plumes in CCNa measurements is clearly shown in the figure where larger concentrations were measured for smaller Sv. In the flight segment AC17b, where a smaller variability of NCCN(Sv) is observed in the CCNb sampling, NCCN(Sv) measurements from CCNa are more consistent with those observed in homogenous air masses.

Figure 7 shows the Nd(Sv) spectra and the NCCN(Sv) measurements for time intervals during which both CCNa and CCNb were operating and the sampling duration exceeded 20 s (∼1km horizontal distance). The NCCN(Sv) measurements are displayed using the same color for each fixed-Sv cycle of the CCNa measurements. The measurements during Flight AC11 exhibit a clear influence of pollution plumes, with CCN concentrations reaching up to ∼2000cm-3 in both counters measuring in different Sv. Large concentration peaks are frequently observed in flight segment AC17a for a wide range of Sv measurements. Overall, most Nd(Sv) and NCCN(Sv) values remain within the combined uncertainty bounds, whereas the largest deviations are associated with variability in polluted air masses and with derived Nd(Sv) values from cloud passes more strongly affected by the entrainment and mixing of dry air. The spatial and temporal heterogeneity of the sampled air masses during AC11, AC14, and AC17a also contributes to deviations in the fitted relationships over specific supersaturation ranges (e.g., near Sv≈0.55% during AC14). Additional discrepancies between Nd(Sv) and NCCN(Sv) concentrations may arise from differences in the particle diameter (d) detection ranges of the CCP-CDP probe (approximately 3–50 µm) and the CCN counters (activated droplets with d≳1µm), since the CCN counters may detect activated particles that have not yet grown to the minimum detectable size of the CCP-CDP probe at cloud base.