In typical global balance models, the energy exchange at the ocean-atmosphere interface includes quantitative estimates of five major contributions, namely, solar downward irradiation (≈ 165 W m^-2), thermal upward radiation from the sea surface (≈ – 400 W m^-2), thermal downward radiation from clouds and greenhouse gases (≈ 343 W m^-2), latent heat of evaporated water (≈ – 95 W m^-2), and sensible heat exchange (≈ – 12 W m^-2) by molecular conduction across the interface (Budyko 1963; Josey et al. 2013; Macdonald and Baringer 2013; Rapp 2014). The figures given in brackets are respective rough estimates for the enthalpy fluxes per ocean surface area, which result in a net imbalance of about 1 W m^-2 that is warming up the ocean, an amount 25 times as large as the world’s total energy consumption by humans in 2022 (You 2024). Similarly, the long-term global energy balance at the top of the atmosphere also amounts to about 1 W m^-2 (Meehl et al. 2011). This way, to the globally accumulated energy of the warming climate, the ocean contributes about 89 % (von Schuckmann et al. 2023) or 90 % (Abraham et al. 2013; Cheng et al. 2024) while the atmosphere takes only a minor share of just 1 %. With the advent of the Thermodynamic Equation of State of Seawater – 2010 (TEOS-10) as the current comprehensive international geophysical thermodynamic standard, previous methods for estimating the ocean heat content (Abraham et al. 2013) could be replaced by more accurate modern formulas (McDougall 2003; McDougall et al. 2021), see Section 3.

Each of the energy fluxes mentioned above may, to some extent, be responsible for the observed ocean warming. However, identifying the causes and processes of the enhanced ocean heat imbalance is scientifically challenging. The magnitude of the air-sea flux imbalance is well below the typical uncertainty threshold of at least 10 W m^-2 of climate models (Josey et al. 2013). “The drivers of a larger [Earth energy imbalance] EEI in the 2000s than in the long-term period since 1971 are still unclear, and several mechanisms are discussed in literature. For example, Loeb et al. (2021) argue for a decreased reflection of energy back into space by clouds (including aerosol cloud interactions) and sea ice and increases in well-mixed greenhouse gases (GHG) and water vapor to account for this increase in EEI. Kramer et al. (2021) refer to a combination of rising concentrations of well-mixed GHG and recent reductions in aerosol emissions to be accounting for the increase, and Liu et al. (2020) address changes in surface heat flux together with planetary heat redistribution and changes in ocean heat storage. Future studies are needed to further explain the drivers of this change, together with its implications for changes in the Earth system” (von Schuckmann et al. 2023).

“Low-altitude clouds with cloud top temperatures close to the ground environment generally lead to a weak greenhouse effect. In addition, the radiative cooling at the cloud top and the warming at the cloud base may enlarge the vertical temperature gradient and consequently intensify the turbulence” (Yang et al. 2021: p. 1). However, in some climate models “the low cloud response to [sea-surface temperature] SST change may be too weak” (Loeb et al. 2022: p. 14). As a contributor to the marine heat balance, low-level clouds affect the upwelling thermal radiation by the surface area fraction they cover, as well as by their altitude and the related cloud-base temperature driving their downward longwave radiation. As a function of the surface temperature and relative humidity (RH), an adiabatically ascending air parcel reaches its dew-point at the so-called lifted condensation level (LCL). In this article, equations for the LCL are derived from the international standard TEOS-10, the Thermodynamic Equation of Seawater – 2010 (IOC et al. 2008). As a longwave radiation feedback effect, deviations of the downward black-body radiation of those clouds can this way be estimated from observed trends related to global warming of the ocean surface.

In Section 2 of this paper, selected details of ocean warming are briefly reviewed and discussed in relation to other climatic trends, such as cloudiness reduction, SST increase, or apparently constant marine RH and wind speeds. Section 3 provides a short introduction into the thermodynamic framework of TEOS-10. In Section 4, TEOS-10 equations are formulated for the LCL pressure and temperature as functions of surface pressure, temperature and relative humidity. Rigorous results from those international standard equations may serve as reference values for other estimates. In addition to selected numerical results obtained from an iteration scheme, analytical approximations are suggested for the almost saturated marine troposphere as well as for the tutorial case of simple crude Gibbs function, as explicitly defined in Appendix C. Section 5 investigates the climatic sensitivity of the LCL temperature on the SST at constant RH, with selected numerical solutions of TEOS-10 equations and approximate formulas for crude Gibbs functions. Section 6 concludes the paper with a discussion of the climatological context and derived solutions. Appendix A introduces the Jacobi method for a convenient manipulation of partial derivatives used in TEOS-10. Appendix B calculates from TEOS-10 an equation for the adiabatic lapse rate of the dew point, as required in Section 4. Appendix D mathematically explains Lambert’s W function that is frequently encountered in atmospheric balance equations. Finally, Appendix E provides a list of symbols used in this paper.

Through the last 70 years, the global mean sea surface temperature (SST) has increased from about 17.8 °C in 1955 to about 18.8 °C in 2023 (Cheng et al. 2024), see Figure 5 below, with an estimated linear trend of

Here, $yr$ is the number of the calendar year. Accordingly, depending on the dataset used for the analysis, the heat content of the upper 2000 m of the world ocean is currently increasing at a rate of either about 0.9 W m^-2 or 1.3 W m^-2 per ocean surface area (You 2024). These values are consistent with the total Earth energy imbalance exceeding 1 W m^-2 (Loeb et al. 2022) and the fact that the ocean alone stores 90 % of the globally accumulated heat (Cheng et al. 2024).

Recent ocean warming is strongest along the global west-wind belts, Figure 2, and is spatially correlated with regions of high mean cloud coverage, Figure 3.

On the input side of the ocean’s energy imbalance, the systematic global reduction of cloudiness is estimated by the linear trend (Foster et al. 2023),

Here, $C$ is defined by the fraction of the surface area covered by clouds. Evidently, this trend is negatively correlated with the tropospheric increase of temperature and specific humidity. Consistent with reduced cloudiness, between 1998 and 2017 Earth’s albedo declined by 0.5 W m^-2 (Goode et al. 2021; Feistel and Hellmuth 2021).

Decreased cloudiness has increased the mean solar irradiation, $∆J$, passing through clear-sky regions, by about 0.74 W m^-2 in the past 20 years (Phillips and Foster 2023), see Figure 4 below. This phenomenon is known as the warming “shortwave cloud radiative effect” (SW CRE). Possibly, this value may explain the observed ocean warming if the ocean’s heat loss remains sufficiently below the SW CRE heat gain.

On the output side of the ocean’s energy imbalance, the sensible heat exchange of roughly 10 W m^-2 between ocean and atmosphere is relatively small and does not exceed the uncertainty of flux estimates (Josey et al. 2013). The ocean’s heat capacity exceeds that of the atmosphere by a factor of 1000, so that 1 % of added ocean’s heat content corresponds to an atmospheric temperature increase 10 times as large as that of the ocean. Sensible heat flux may be decreasing due to the faster rise of tropospheric temperatures compared to those of the sea. Estimates for such a trend are not available, though, and the quantitative contribution of sensible heat flow to ocean warming remains a pending problem.

On the output side of the ocean’s energy imbalance, the problem of ocean warming is closely related to a suspected change of the strength of the hydrological cycle. Cheng et al. (2024: p. 6) argue that the observed increase in salinity contrast in the upper 2000 m “is generally consistent with many atmosphere-based estimates and strengthens the evidence that the global water cycle has been amplified with global warming”. On the other hand, Held and Soden (2006: p. 5687-5689) had already emphasised that “it is important that the global-mean precipitation or evaporation, commonly referred to as the strength of the hydrological cycle, does not scale with Clausius–Clapeyron. … We can, alternatively, speak of the mean residence time of water vapor in the troposphere as increasing with increasing temperature”. In contrast to some model predictions (Weller et al. 2022), there is no evidence yet that mean global marine evaporation has intensified which in that case would have cooled the ocean in contrast to the observed warming in reality (Feistel and Hellmuth 2021, 2023; Blunden et al. 2023).

Quantitatively, corresponding to about 1200 mm of annual evaporation, the latent heat exchange of roughly 100 W m^-2 between ocean and atmosphere is the dominating way of oceanic heat loss and atmospheric energy supply (Randall 2012; Rapp 2014; Feistel and Hellmuth 2021). According to the Dalton equation, the evaporation flux depends mainly on the RH and the wind speed (Stewart 2008; Feistel and Hellmuth 2023, 2024). Trends in marine surface RH are weak and are, if at all, well below measurement uncertainty (Lovell-Smith et al. 2016; Dunn et al. 2023; Willett et al. 2023). “There is a tendency for the RH of the air to remain approximately constant as the climate changes” (Randall 2012: p. 147). Similarly, there is no significant trend in marine wind speeds (Azorin-Molina et al. 2023), so that relevant trends in global mean marine evaporation rates are neither expected nor observed (Held and Soden 2006). However, the high sensitivity of latent heat flux with respect to RH may be estimated to about 5 W m^-2 per %rh (Feistel and Hellmuth 2021), so that a small increase by 0.2 %rh may already suffice to explain the oceanic warming rate. Such a minor change of RH, however, could be disguised by the much larger observation uncertainty of 1 – 5 %rh (Lovell-Smith et al. 2016). “Relative humidity over oceans remained highly uncertain” (Willett et al. 2023: p. S49). Therefore, the relevance of latent heat with respect to ocean warming remains unclear.

On the output side of the ocean’s energy imbalance, the thermal radiation of the sea surface may easily be estimated from the Stefan-Boltzmann law of black-body radiation. Only a small fraction of this radiation may pass the troposphere in the so-called atmospheric window at about 8-11 µm wavelength, see Figure 6; most energy is blocked by the infrared opacity of clouds and greenhouse gases which send back down to the ocean long-wave radiation, according to their particular altitudes and temperatures. This “longwave cloud radiative effect” (LW CRE) of reduced cloudiness is cooling the ocean by letting more thermal radiation pass to space and emitting less downward radiation, Figure 7. It is estimated (Phillips and Foster 2023) that this cooling LW CRE almost completely cancels out the warming SW CRE of cloudiness, so that, following that analysis, ocean warming may be attributed to decreasing cloudiness only to a minor extent.

In addition to merely the lateral area covered by clouds, the mutual energy balance between SW and LW CRE is influenced also by other processes.

The world ocean is a vast and dynamically changing storage of heat, however, the theoretical description of its heat transport and conversion processes is thermodynamically challenging (McDougall 2003; Abraham et al. 2013; McDougall et al. 2013, 2021, 2023; Graham and McDougall 2013; Tailleux and Dubos 2024). For example, the former 1980 Equation of State of Seawater (EOS-80) did not provide any equations for the calculation of seawater entropy or enthalpy (Unesco 1981; Fofonoff and Millard 1983). In order to meet the growing demands of climate research and to support numerical circulation models of the ocean, starting in 2005 an improved comprehensive standard description of seawater thermodynamics had been developed by the SCOR/IAPSO Working Group 127 in cooperation with the International Association for the Properties of Water and Steam (Pawlowicz et al. 2012).

Proper axiomatic systems of mathematical statements possess the internal properties of independence, consistency and completeness. In contrast to any earlier collections of empirical seawater property equations, the International Thermodynamic Equation of Seawater – 2010 (TEOS-10) is designed in such an axiomatic manner and offers a comprehensive description of thermodynamic properties of water in its geophysical mixtures, phases and mutual equilibria. Thermodynamics has a reputation of being mathematically difficult. For beginners, though, an encouraging anecdote quoted by Fink (2009: p.1) is Arnold Sommerfeld’s experience with the use of the thermodynamic formalism: “Thermodynamics is a funny subject. The first time you go through it, you don’t understand it at all. The second time you go through it, you think you understand it, except for one or two points. The third time you go through it, you know you don’t understand it, but by that time you are so used to that subject, it doesn’t bother you anymore.” Appendix A provides a brief tutorial introduction into the mathematics used throughout TEOS-10. However, extensive software libraries are available from the web (www.teos-10.org), as outlined in the digital supplement of Wright et al. (2010), which conveniently provide direct access to numerical properties without detailed expertise to be required regarding sometimes demanding algorithms behind (Feistel et al. 2010a; Wright et al. 2010).

More than 150 years ago, in 1873, Willard Gibbs had discovered that a single mathematical function is sufficient to describe all thermodynamic properties of a given substance in its equilibrium state. Such functions are known as “thermodynamic potentials”. At its very core, TEOS-10 is mathematically defined (IOC et al. 2010) in terms of empirical correlation equations for just three thermodynamic potentials, namely,

“Helmholtz energy” is sometimes also known as “free energy”, and “Gibbs energy “ as “free enthalpy” in the literature. In TEOS-10, temperature is expressed on the 1990 International Temperature Scale (ITS-90, Preston-Thomas 1990). In the vicinity of the triple point of water, ITS-90 deviates only insignificantly from the current Thermodynamic Temperature Scale (BIPM 2019) and will remain in practical use for the foreseeable future (Harvey et al. 2023). The isotopic composition of water is assumed to be that of Vienna Standard Mean Ocean Water (VSMOW, IAPWS G5-01 2016). Salinity $S$ of IAPSO Standard Seawater is defined by the Reference-Composition Salinity Scale (Millero et al. 2008) as the mass fraction of dissolved salt in seawater. The dry-air mass fraction $A$ of humid air is related to the specific humidity $q=1-A$, the mixing ratio $r$ and the water-vapour mole fraction $x$ by

Here, $M_{\mathrm{W}}=\mathrm{18.015\; 268}\mathrm{g}{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$ and $M_{\mathrm{A}}=\mathrm{28.965\; 46}\mathrm{g}{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$ , respectively, are the molar masses of water and of dry air (IAPWS G08-10 2010). The freely adjustable constants for the absolute energies and entropies of water, sea salt and dry air are consistently defined at reference states chosen to be the triple point of water and the standard ocean state (Wagner and Pruß 2002; Feistel et al. 2008, 2010a,b; Feistel 2018).

The Gibbs function of freshwater, $g^{\mathrm{S}\mathrm{W}}\left(0,T,p\right)$, equals the Gibbs function of liquid water derived from the specific Helmholtz energy,$f^{\mathrm{F}}\left(T,\rho \right)$, of fluid water (Wagner and Pruß 2002) evaluated at high densities of liquid water, where the pressure is given by

The specific entropy of seawater, ${\eta }^{\mathrm{S}\mathrm{W}}$, is available from the Gibbs function by and the specific enthalpy of seawater, $h^{\mathrm{S}\mathrm{W}}$, by

This way, the ocean heat content, $OHC$, is estimated from the integral (McDougall et al. 2021), carried out over the ocean’s volume with its local in-situ properties $\left[S,{\eta }^{\mathrm{S}\mathrm{W}}\left(S,T,p\right),p\right]$. By definition (Feistel et al. 2008; IOC et al. 2010; Feistel 2018), the TEOS-10 enthalpy of the standard-ocean reference state is $h_{\mathrm{S}\mathrm{O}}^{\mathrm{S}\mathrm{W}}\equiv 0$. The local in-situ density ${\rho }^{\mathrm{S}\mathrm{W}}\left(S,{\eta }^{\mathrm{S}\mathrm{W}},p\right)$ is available from the relation

Formula (9) describes a fictitious process by which each mass element, $\mathrm{d}m={\rho }^{\mathrm{S}\mathrm{W}}\mathrm{d}V$, in the volume $V$ is isentropically lifted to the surface pressure,$p_0$, where it transfers all its excess enthalpy, $h^{\mathrm{S}\mathrm{W}}\left(S,{\eta }^{\mathrm{S}\mathrm{W}},p_0\right)-h_{\mathrm{S}\mathrm{O}}^{\mathrm{S}\mathrm{W}}$, via sensible heat flux to an external measurement device. Subsequently, the same amount of heat is reversibly conducted back, after which the parcel returns to its original spatial position. Then, $OHC$ is the total amount of heat reversibly removed from the ocean across its surface by this process.

The approach of eq. (9) is therefore consistent with the fact that thermodynamically, heat is an exchange quantity rather than a state quantity, so that, rigorously spoken, any kind of stored “heat content” does not unambiguously exist but only an amount of heat which is exchanged between a given system and its surrounding (Landau and Lifschitz 1966: §13). According to Clausius (1876) and Gibbs (1873), is the proper thermodynamic expression for heat exchange at constant pressure and constant amounts of water and salt. Here, $T^{*}$ is the parcel’s changing surface temperature during the fictitious heat transfer process, and $c_p^{\mathrm{S}\mathrm{W}}$ is its isobaric heat capacity.

The Helmholtz function of dry air, $f^{\mathrm{A}\mathrm{V}}\left(1,T,\rho \right)$, equals (up to modified reference-state conditions) the equation of state of Lemmon et al. (2000),

The Helmholtz function of pure water vapour, $f^{\mathrm{A}\mathrm{V}}\left(0,T,\rho \right)$, equals the specific Helmholtz energy,$f^{\mathrm{F}}\left(T,\rho \right)$, of fluid water (Wagner and Pruß 2002) evaluated at low densities of water vapour

For the following calculations of condensation levels, the specific entropy of humid air, $\eta$, is available from the related Gibbs function by

Note that derived from (14) for atmospheric applications, a simplified analytical low-pressure formulation for the Gibbs function $g^{\mathrm{A}\mathrm{V}}$ is available from Appendix B of Feistel and Hellmuth (2023).

The specific enthalpy of humid air is computed from the Gibbs function as

For the computation of phase equilibria such as dew- or frost-points, TEOS-10 provides the chemical potentials of water in the different phases or mixtures.

The chemical potential of water vapour, ${\mu }_{\mathrm{V}}^{\mathrm{A}\mathrm{V}}$, in humid air is available from the Gibbs function by or equivalently from the enthalpy by, see eq. (A26),

The chemical potential of liquid water equals its specific Gibbs energy, and, similarly, the chemical potential of ice is,

The dry-air mass fraction, $A^{\mathrm{s}\mathrm{a}\mathrm{t}}\left(T,p\right)$, of humid air saturated with respect to liquid water is defined by the equilibrium condition and with respect to ice by

To make the use of TEOS-10 equations more convenient, the numerical solutions of eqs. (21) and (22) for either $A^{\mathrm{s}\mathrm{a}\mathrm{t}}\left(T,p\right)$, $T\left(A^{\mathrm{s}\mathrm{a}\mathrm{t}},p\right)$, or $p\left(A^{\mathrm{s}\mathrm{a}\mathrm{t}},T\right)$, together with numerous other properties (Feistel et al. 2010a; Wright et al. 2010), are implemented in the freely available from the open source code, sea-ice-air (SIA) library of TEOS-10, accessible at www.teos-10.org.

After the official adoption of TEOS-10, the IAPSO/SCOR/IAPWS Joint Committee on the Properties of Seawater (JCS) continued to address pending climate-related problems (Feistel et al. 2016) that could have not yet or only insufficiently been addressed by TEOS-10. Among those is the ambiguity and mutual inconsistency of different definitions of RH in practical use, such as between meteorology and climatology (Lovell-Smith et al. 2016). With respect to deviations from water phase equilibria such as saturation properties, the real-gas equivalent of conventional RH is the relative fugacity (RF) (Feistel et al. 2010b: eq. (10.10) therein, Feistel and Lovell-Smith 2017),

Here, $R_{\mathrm{W}}=R/M_{\mathrm{W}}$ is the specific gas constant of water, and $R$ is the universal molar gas constant. As a function of $\left(T,p\right)$, the chemical potential at saturation, ${\mu }_{\mathrm{V}}^{\mathrm{A}\mathrm{V}}\left(A^{\mathrm{s}\mathrm{a}\mathrm{t}},T,p\right)$, may be expressed by means of either eq. (21) or (22). Note that eq. (23) is valid only for temperatures below the boiling point of water at the given pressure (Feistel and Lovell-Smith 2017), a condition which is naturally fulfilled under ambient geophysical conditions.

In ideal-gas approximation, RF is identical with conventional RH in terms of water-vapour partial pressures. For air close to saturation, such as the marine surface layer with about 80 %rh, the Clausius-Clapeyron formula is an excellent approximation of TEOS-10 values for RF, even if refraining from the perfect-gas assumption (Feistel et al. 2022). RF appears naturally in expressions for the Onsager driving force of non-equilibrium fluxes, such as evaporation from the ocean surface (Feistel and Hellmuth 2023, 2024). For the numerical computation of RF, a source code extension to the SIA library is available from Feistel et al. (2022).

Mathematical transformation between different thermodynamic quantities and partial derivatives with respect to alternative independent variables, such as frequently exploited in TEOS-10 to derive the desired functional dependencies, may conveniently and error-free be executed by means of the formal Jacobi method developed by Norman Shaw (1935), as briefly described in Appendix A.

“Synoptic weather observations from ships throughout the World Ocean have been analyzed to produce a climatology of total cloud cover and the amounts of nine cloud types. … Among the cloud types, the most widespread and consistent relationship is found for the extensive marine stratus and stratocumulus clouds” (Eastman et al. 2011: p. 5914). Subtropical marine clouds show a pronounced “transition between unbroken sheets of stratocumulus and fields of scattered cumulus“ (Sandu et al. 2010). The cumulus cloud base is typically observed below 500 m height, as shown in Figure 1. For the climatic feedback effect of marine clouds, the lifted condensation level (LCL) is a key parameter that describes the pressure (or altitude) at which an isentropically ascending parcel reaches its dew- or frost-point, starting from given values of temperature and RH, such as at the sea surface. A typical empirical equation for the LCL height, $z_{\mathrm{L}\mathrm{C}\mathrm{L}}$, is where $z$, $T$ and $T_{\mathrm{d}\mathrm{p}}$, respectively, are the initial values of the parcel’s height, in-situ and dew-point temperature. Commonly, the latter is related to the saturation vapour pressure and the RH by the Clausius-Clapeyron formula. Previous estimates for the coefficient ${\gamma }_{\mathrm{L}\mathrm{C}\mathrm{L}}$ range widely between $123\mathrm{m}{\mathrm{K}}^{-1}$ and $165\mathrm{m}{\mathrm{K}}^{-1}$ (Romps 2017), with an optimum of $125\mathrm{m}{\mathrm{K}}^{-1}$ suggested by Lawrence (2005). Also, nonlinear mathematical expressions, more complicated than eq. (24), had been suggested in the literature, derived from varying approximate thermodynamic relations for humid air, liquid water and ice. Often, such relations are chosen according to the authors’ personal preferences and are scarcely supported by internationally agreed standard formulations. In Section 4.2 below, for typical values of the marine troposphere, selected highly accurate TEOS-10 results for ${\gamma }_{\mathrm{L}\mathrm{C}\mathrm{L}}$ range between 125 and 129 $\mathrm{m}{\mathrm{K}}^{-1}$.

Humid air at the sea surface has a typical climatological RH of about 80 %rh, a value that may vary seasonally or regionally between 70 %rh and 90 %rh, but appears likely to be only very weakly affected by global warming, if at all (Thomas and Stamnes 1999; Liou 2002; Dai 2006; Randall 2012; Rapp 2014; Blunden 2023). Assuming that an air parcel starts ascending at constant specific humidity, $A=A_0=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, from the sea surface at the air pressure, $p_0$, and the sea surface temperature, $T_0$, the value of the initial air fraction $A_0$ can be computed in implicit form from the sea surface relative fugacity, ${\psi }_f={\psi }_f\left(A_0,T_0,p_0\right)$, using eq. (23), by

Here, surface values of independent variables are indicated by the subscript 0 which may be dropped at constant $A$ for simplicity. Similarly, the subscript LCL is used for those properties at the lifted condensation state. Associated function values such as ${\psi }_f$ or $g^{\mathrm{W}}$ are without state-dependent subscript as long as there is no risk of confusion.

The tropospheric entropy, assumed to remain constant during the uplift, can be expressed by eq. (15),

At the LCL, the entropy value, needs to be the same as at the surface, eq. (26), and may be eliminated from the mathematical problem by equating eq. (26) with (27),

The LCL pressure, $p_{\mathrm{L}\mathrm{C}\mathrm{L}}$, is approached when at fixed values of $\left(A,\eta \right)=\left(A_0,{\eta }_0\right)$ the temperature has dropped to the dew-point, $T_{\mathrm{L}\mathrm{C}\mathrm{L}}$, eq. (21), that means,

The chemical potential of water vapour in humid air, ${\mu }_{\mathrm{V}}^{\mathrm{A}\mathrm{V}}$, may be expressed by the related Gibbs function, eq. (17),

If the initial value of RH is given, rather than the initial specific humidity, $q=1-A$, then eq. (25) may be added to the system in the form of

Eqs. (28), (30) and (31) constitute a closed system of three nonlinear implicit equations for the three unknowns $\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)$ as functions of the given input values $\left({\psi }_f,T_0,p_0\right)$. These thermodynamic relations are the exact conditions for the LCL properties with the only minor approximation that the dissolution of air in liquid water, influencing the saturation state, is neglected. This system may be solved iteratively using the TEOS-10 functions implemented in the SIA library, see Appendix B, or it may be exploited analytically after introducing quantitatively reasonable approximations.