TEOS-10 Equations for Determining the Lifted Condensation Level (LCL) and Climatic Feedback of Marine Clouds

Abstract

At an energy flux imbalance of about 1 W m⁻², the ocean stores 90% of the heat accumulating by global warming. However, neither the causes of this nor the responsible geophysical processes are sufficiently well understood. More detailed investigations of the different phenomena contributing to the oceanic energy balance are warranted. Here, the role of low-level marine clouds in the air–sea interaction is analysed. TEOS-10, the International Thermodynamic Equation of State of Seawater—2010, is exploited for a rigorous thermodynamic description of the climatic trends in the lifted condensation level (LCL) of the marine troposphere. Rising sea surface temperature (SST) at a constant relative humidity (RH) is elevating marine clouds, cooling the cloud base, and reducing downward thermal radiation. This LCL feedback effect is negative and counteracts ocean warming. At the current global mean SST of about 292 K, the net radiative heat flux from the ocean surface to the LCL cloud base is estimated to be 24 W m⁻². Per degree of SST increase, this net flux is expected to be enhanced by almost 0.5 W m⁻². The climatic LCL feedback effect is relevant for the ocean’s energy balance and may be rigorously thermodynamically modelled in terms of TEOS-10 equations. LCL height may serve as a remotely measured, sensitive estimate for the sea surface’s relative fugacity, or conventional relative humidity.

1. Introduction

“Of the air, the part receiving heat is rising higher. So, evaporated water is lifted above the lower air”. (Leonardo da Vinci, ca. 1508. Paper sheets from the Arundel Codex, after 1508, in handwriting of the old Leonardo, discovered by Anna Maria Brizio in 1951 [1]). Translated from the Italian original by Marianne Schneider (1996) [2], p. 79. Published German text: “Der Teil der Luft steigt höher, der an der Wärme teilhat. Also dringt die Ausdünstung des Wassers hinauf über die niedrige Region der Luft, die sie berührt”.

In the title of this paper, emphasis is placed in particular on “TEOS-10 Equations”. TEOS-10, the Thermodynamic Equation of Seawater—2010 [3], was adopted in 2009 by UNESCO/IOC (IOC: Intergovernmental Oceanographic Commission, https://www.ioc.unesco.org/en, accessed on 10 March 2024) and in 2011 by IUGG (IUGG: International Union of Geodesy and Geophysics, https://iugg.org/, accessed on 10 March 2024) as an international geophysical standard providing the thermodynamic properties of seawater, ice, and humid air. TEOS-10 is designed to cover, with significantly improved quality and range of validity, not only the ocean’s interior [4], but also the properties of the ocean–atmosphere interface. In this paper, the related capabilities and rigor of TEOS-10 equations are demonstrated mathematically by developing and discussing conceptual models in the context of selected current problems from climate research. As with any other model, these are also only imperfect and simplified descriptions of a complex reality.

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), the latent heat of evaporated water (≈−95 W m⁻²), and sensible heat exchange (≈−12 W m⁻²) by molecular conduction across the interface [5,6,7,8]. 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. This is an amount 25 times as large as the world’s total energy consumption by humans in 2022 [9]. Similarly, the long-term global energy balance at the top of the atmosphere also amounts to about 1 W m⁻² [10]. This way, the ocean contributes about 89% [11] or 90% [12,13] of the total to the globally accumulated energy of the warming climate, 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 [12] can be replaced by more accurate modern formulas [14,15], as shown in Section 3.

Each of the energy fluxes mentioned above may, to some extent, be responsible for the observed ocean warming. However, identifying the causes of and processes behind the enhanced ocean heat imbalance is scientifically challenging. The magnitude of the air–sea flux imbalance is well below the typical uncertainty threshold of climate models of at least 10 W m⁻² [6]. “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. [16] 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) [17] 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) [18] 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” [11].

“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” [19]. However, in some climate models “the low cloud response to [sea-surface temperature] SST change may be too weak” [20]. Developing climate models of marine low clouds and their sensitivity to SST variation is still a challenging task [21]. These phenomena are worth modelling using TEOS-10 equations from the Thermodynamic Equation of Seawater—2010 [3]. As a contributor to the marine heat balance, low-level clouds affect the upwelling thermal radiation of the surface area fraction they cover. By their altitude and due to their related cloud-base temperature, low-level clouds drive downward longwave radiation (Figure 1). 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, reference equations for the LCL are derived from the international standard TEOS-10. As a longwave radiation feedback effect, deviations in the downward black-body radiation of those clouds can be estimated this way from the observed trends related to the 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 reductions, SST increases, or apparently constant marine RH and wind speeds. Section 3 provides a short introduction to 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. The rigorous results obtained using those equations may serve as reference values for other non-standard 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 the simple crude Gibbs function, as explicitly defined in Appendix C. Section 5 investigates the climatic sensitivity of the LCL temperature to the SST at constant RH, with selected numerical solutions of TEOS-10 equations and approximate formulas for crude Gibbs functions. Section 6 concludes this paper with a discussion of the climatological context and derived solutions. Appendix A introduces the Jacobi method for a convenient manipulation of the partial derivatives used in TEOS-10. Appendix B calculates an equation from TEOS-10 for assessing the adiabatic lapse rate of the dew point, as required in Section 4. Appendix D mathematically explains Lambert’s W function, which is frequently encountered in atmospheric balance equations. Finally, Appendix E provides a list of symbols used in this paper.

Finally, it should explicitly be underlined at this point again that the very aim of this paper is an introduction to the mathematical apparatus of TEOS-10. For this purpose, selected current problems of climate research at the air–sea interface are briefly reviewed and exploited as tutorial examples, for which simple conceptual models are developed. Familiar such examples may serve to reduce the inevitable learning curve associated with actively making use of TEOS-10. For those models, TEOS-10 equations are presented and solved either numerically or via an approximate analytical method. Before TEOS-10, to the knowledge of the authors, geophysical and climate problems had never been described axiomatically in terms of non-trivial thermodynamic potentials which, however, are well known to be the most elegant, complete, consistent, and accurate tools available for this task. Moreover, TEOS-10 is the current international standard adopted in order to harmonise and overcome the widespread use of various collections of empirical property equations whose levels of mutual consistency are often unclear. An extended list of references regarding previous TEOS-10 work is given for readers who may be interested in the more detailed context of the particular approaches and equations presented here.

2. Application Context: Ocean Warming and Cloudiness

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 [11], see the long-term trends graphically displayed later in this section, with an estimated linear trend of

$$t/°\mathrm{C}\approx 18.4+0.015\times \left(yr-2000\right).$$

(1)

Here, $yr$ is the number of the Gregorian 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 [9]. These values are consistent with a total Earth energy imbalance exceeding 1 W m⁻² [20] and the fact that the ocean alone stores 90% of the globally accumulated heat [13].

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

On the input side of the ocean’s energy imbalance, the systematic global reduction in cloudiness is estimated by the linear trend [23],

$$C\approx 0.664-0.006\times \left(yr-2000\right),$$

(2)

Here, $C$ is defined by the fraction of the surface area covered by clouds. Evidently, this trend is negatively correlated with the tropospheric increase in temperature and specific humidity. Consistent with reduced cloudiness, between 1998 and 2017, Earth’s albedo declined by 0.5 W m⁻² [24,25].

Decreased cloudiness increased the mean solar irradiation, $∆J$, passing through clear-sky regions,

$$∆J/\left(\mathrm{W} {\mathrm{m}}^{-2}\right)\approx -0.41+0.037\times \left(yr-2000\right),$$

(3)

by about 0.74 W m⁻² in the past 20 years [26], see Figure 4 below. This phenomenon is known as the warming “shortwave cloud radiative effect” (SW CRE). This value may possibly 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 [6]. 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 larger that of the ocean. Sensible heat flux may be decreasing due to the faster rise in 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 in the strength of the hydrological cycle. Cheng et al. [13] 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 [27] have 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 [28], there is no evidence yet that mean global marine evaporation has intensified, which would have cooled the ocean in contrast to the warming observed in reality [23,29,30].

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 dominant means of oceanic heat loss and atmospheric energy supply [8,25,31]. According to the Dalton equation, the evaporation flux depends mainly on the RH and the wind speed [29,32,33]. Trends in marine surface RH are weak and are, if measurable at all, well below the level of measurement uncertainty [34,35,36]. “There is a tendency for the RH of the air to remain approximately constant as the climate changes” [31]. Similarly, there is no significant trend in marine wind speeds [37] so that relevant trends in global mean marine evaporation rates are neither expected nor observed [27]. However, the high sensitivity of latent heat flux with respect to RH may be estimated to be about 5 W m⁻² per %rh [25,38,39], so that a small increase of 0.2 %rh may already suffice to explain the oceanic warming rate. Such a minor change in RH, however, could be disguised by the much larger observation uncertainty of 1–5 %rh [34]. “Relative humidity over oceans remained highly uncertain” [36]. 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 through the troposphere via the so-called atmospheric window at about 8–11 µm wavelength, as shown in Figure 5; most energy is blocked by the infrared opacity of clouds and greenhouse gases and sent back down to the ocean via long-wave radiation. This occurs 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 into space and emitting less downward radiation, as shown in Figure 6. It is estimated [26] that this cooling LW CRE almost completely cancels out the warming SW CRE of cloudiness. A consequence of this analysis is that ocean warming may only be attributed to decreasing cloudiness to a minor extent.

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

(i)

In contrast to the SW CRE, the LW CRE strongly depends on the concentration of greenhouse gases present in the cloudless sky. If the clear sky absorbs infrared radiation to a similar extent as clouds do, the greenhouse effect will not decrease significantly due to the substitution of a cloud fraction by a cloudless fraction. The increasing concentration of water vapour in the marine troposphere [36] results in a stronger absorption of longwave radiation [25,40,41,42,43]. The vertically distributed opacity of the clear-sky troposphere results in an effective radiating height of roughly 5000 m at 500 hPa, where the temperature is around 255 K [44]. According to Figure 7, the infrared opacity of the troposphere is between 70% and 85%. A rule-of-thumb estimate [25] for the tropical marine infrared absorption coefficient of 71% is consistent with that range. These values are relatively close to those of the opacity of clouds. “The clear-sky infrared absorption/emission is very important, so ideally the assumed value for the clear sky is calculated using a radiative transfer model driven by reanalysis fields. Cloud radiative effect (as shown in Figure 6) is the difference between the all-sky observed and this modeled cloudless atmosphere” (Coda Phillips, priv. comm.).

(ii)

The SW CRE is relevant at daytime only, while the LW CRE acts all day and night. It is unclear whether the reported reduction in global cloudiness differs between day and night [45]. In that case, it may have distinct impacts on SW and LW CRE.

(iii)

Cloudiness is most pronounced in the tropics and the west-wind belts (Figure 3). Moreover, the SW CRE is most relevant at low latitudes, while the LW CRE acts all over the globe. It is unclear how the reported reduction in global cloudiness is correlated with latitude and it may have different impacts on SW and LW CRE.

(iv)

Through the LCL, the increasing ocean SST has an effect on the altitude of cloud formations. This changes the cloud base’s temperature and, in turn, its downward thermal radiation. This feedback effect is analysed thermodynamically in Section 4 and Section 5 of this paper. In addition, low-level cumulus cloud formation is highly correlated with the diurnal cycle of solar irradiation, latitude, and land–ocean distribution.

(v)

Through the LCL, a so far unnoticed minor increase in ocean surface RH may have an effect on the altitude of cloud formation. This could change the cloud base temperature and, in turn, cause its downward thermal radiation to be in the opposite direction compared to the SST trend. This negative feedback effect is briefly quantified thermodynamically in Section 4.2.

3. Mathematical Method: TEOS-10 Equations of State

The world’s ocean is a vast and dynamically changing store of heat. However, the theoretical description of its heat transport and conversion processes is thermodynamically challenging [12,14,15,46,47,48,49]. 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 [50,51]. 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 has been developed by the SCOR/IAPSO Working Group 127 in cooperation with the International Association for the Properties of Water and Steam [52].

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 [53] 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 to the mathematics used throughout TEOS-10. However, extensive software libraries are available from the web (www.teos-10.org, accessed on 10 March 2024), as outlined in the digital supplement of Wright et al. [54]. These conveniently provide direct access to numerical properties without detailed expertise being required regarding the sometimes-demanding algorithms behind these calculations [54,55].

More than 150 years ago, in 1873, Willard Gibbs 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 [3] in terms of empirical correlation equations for just three thermodynamic potentials:

(i)

The specific Gibbs energy of seawater, $g^{\mathrm{S}\mathrm{W}}\left(S,T,p\right)$, which is a function of absolute salinity, $S$, absolute temperature, $T$, and absolute pressure, $p$ [56,57,58];

(ii)

The specific Gibbs energy of ambient hexagonal ice Ih, $g^{\mathrm{I}\mathrm{h}}\left(T,p\right)$, which is a function of temperature and pressure [59,60];

(iii)

The specific Helmholtz energy of humid air, $f^{\mathrm{A}\mathrm{V}}\left(A,T,\rho \right)$, which is a function of the dry-air mass fraction, $A$, and the mass density of humid air, $\rho$ [61,62]. From these potential functions, all thermodynamic properties of seawater, ice, and humid air, as well as their mutual equilibria, can be derived mathematically in a perfectly consistent way via analytical or numerical means [39,63,64].

“Helmholtz energy” is sometimes also known as “free energy”, and “Gibbs energy” is sometimes referred to as “free enthalpy” in the literature. In TEOS-10, temperature is expressed on the 1990 International Temperature Scale [65]. In the vicinity of the triple point of water, ITS-90 deviates only insignificantly from the current Thermodynamic Temperature Scale [66] and will remain in practical use for the foreseeable future [67]. The isotopic composition of water is assumed to be that of Vienna Standard Mean Ocean Water (VSMOW) [68]. Salinity, $S$ in IAPSO Standard Seawater, is defined by the Reference-Composition Salinity Scale [57] 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

$$\frac{1}{r}=\frac{1}{q}-1={\left(\frac{1}{A}-1\right)}^{-1}=\frac{M_{\mathrm{A}}}{M_{\mathrm{W}}}\left(\frac{1}{x}-1\right).$$

(4)

Here, $M_{\mathrm{W}}=18.015268 \mathrm{g}{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$ and $M_{\mathrm{A}}=28.96546 \mathrm{g}{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$, respectively, are the molar masses of water and of dry air (IAPWS G08-10 2010) [69]. The freely adjustable constants for the absolute energies and entropies of water, sea salt, and dry air are consistently defined at reference states, for which the triple point of water and the standard ocean state are chosen [39,55,56,58,62,64,70].

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 [56]. This evaluated at high densities of liquid water via

$$g^{\mathrm{S}\mathrm{W}}\left(0,T,p\right)\equiv g^{\mathrm{W}}\left(T,p\right)=f^{\mathrm{F}}\left(T,\rho \right)+\frac{p}{\rho },$$

(5)

where pressure is given by

$${p={\rho }^2\left(\frac{{\partial f}^{\mathrm{F}}}{\partial \rho }\right)}_T.$$

(6)

3.1. Ocean Heat Content

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

$${\eta }^{\mathrm{S}\mathrm{W}}=-{\left(\frac{{\partial g}^{\mathrm{S}\mathrm{W}}\left(S,T,p\right)}{\partial T}\right)}_{S,p},$$

(7)

and the specific enthalpy of seawater, $h^{\mathrm{S}\mathrm{W}}$, is available by

$$h^{\mathrm{S}\mathrm{W}}\left(S,{\eta }^{\mathrm{S}\mathrm{W}},p\right)=g^{\mathrm{S}\mathrm{W}}\left(S,T,p\right)+{\eta }^{\mathrm{S}\mathrm{W}}T.$$

(8)

This way, the ocean heat content, $OHC$, is estimated from the integral [15] via

$$OHC=\int \left[h^{\mathrm{S}\mathrm{W}}\left(S,{\eta }^{\mathrm{S}\mathrm{W}},p_0\right)-h_{\mathrm{S}\mathrm{O}}^{\mathrm{S}\mathrm{W}}\right]{\rho }^{\mathrm{S}\mathrm{W}}\left(S,{\eta }^{\mathrm{S}\mathrm{W}},p\right)\mathrm{d}V.$$

(9)

This is carried out using the ocean’s volume and its local in situ properties $\left[S,{\eta }^{\mathrm{S}\mathrm{W}}\left(S,T,p\right),p\right]$. By definition [3,58,64,70] 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

$$\frac{1}{{\rho }^{\mathrm{S}\mathrm{W}}}={\left(\frac{\partial h^{\mathrm{S}\mathrm{W}}}{\partial p}\right)}_{S,{\eta }^{\mathrm{S}\mathrm{W}}}.$$

(10)

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, $h^{\mathrm{S}\mathrm{W}}\left(S,{\eta }^{\mathrm{S}\mathrm{W}},p_0\right)-h_{\mathrm{S}\mathrm{O}}^{\mathrm{S}\mathrm{W}}$, to an external measurement device via sensible heat flux. 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 [39].

The approach of Equation (9) is therefore consistent with the fact that thermodynamically, heat is an exchange quantity rather than a state quantity, so that, rigorously speaking, any kind of stored “heat content” does not unambiguously exist, but is definite only as an amount of heat which is exchanged between a given system and its surroundings [71]. According to Clausius [72] and Gibbs [73],

$$\mathrm{d}{h}^{\mathrm{S}\mathrm{W}}=T^{*}\mathrm{d}{\eta }^{\mathrm{S}\mathrm{W}}=c_p^{\mathrm{S}\mathrm{W}}\mathrm{d}{T}^{*}$$

(11)

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. While the enthalpy difference $\int \mathrm{d}{h}^{\mathrm{S}\mathrm{W}}$ is always an unambiguous state quantity, the total heat $\int T^{*}\mathrm{d}{\eta }^{\mathrm{S}\mathrm{W}}$ is ambiguous and depends on the particular heat exchange process performed along the integration path. Heat output, for example, may result from work input, or vice versa. “The obsolete hypothesis of heat being a substance is excluded” [74].

3.2. Humid Air

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) [61],

$$f^{\mathrm{A}\mathrm{V}}\left(1,T,\rho \right)\equiv f^{\mathrm{A}}\left(T,\rho \right).$$

(12)

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 [56], evaluated at low densities of water vapour

$$f^{\mathrm{A}\mathrm{V}}\left(0,T,\rho \right)\equiv f^{\mathrm{F}}\left(T,\rho \right).$$

(13)

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

$$g^{\mathrm{A}\mathrm{V}}\left(A,T,p\right)=f^{\mathrm{A}\mathrm{V}}\left(A,T,\rho \right)+{\rho \left(\frac{{\partial f}^{\mathrm{A}\mathrm{V}}}{\partial \rho }\right)}_{A,T}$$

(14)

by

$$\eta =-{\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T,p\right)}{\partial T}\right)}_{A,p}.$$

(15)

Note that, derived from Equation (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 the work of Feistel and Hellmuth (2023) [29].

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

$$h^{\mathrm{A}\mathrm{V}}\left(A,\eta ,p\right)=g^{\mathrm{A}\mathrm{V}}\left(A,T,p\right)+\eta T.$$

(16)

3.3. Relative Fugacity

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

$${\mu }_{\mathrm{V}}^{\mathrm{A}\mathrm{V}}=g^{\mathrm{A}\mathrm{V}}-A{\left(\frac{g^{\mathrm{A}\mathrm{V}}}{\partial A}\right)}_{T,p},$$

(17)

or equivalently from the enthalpy by the following (also see Equation (A26))

$${\mu }_{\mathrm{V}}^{\mathrm{A}\mathrm{V}}=h^{\mathrm{A}\mathrm{V}}-\eta {\left(\frac{{\partial h}^{\mathrm{A}\mathrm{V}}}{\partial \eta }\right)}_{A,p}-A{\left(\frac{{\partial h}^{\mathrm{A}\mathrm{V}}}{\partial A}\right)}_{\eta ,p}.$$

(18)

The chemical potential of liquid water equals its specific Gibbs energy,

$${\mu }_{\mathrm{W}}^{\mathrm{W}}=g^{\mathrm{W}}\left(T,p\right),$$

(19)

and, similarly, the chemical potential of ice is,

$${\mu }_{\mathrm{W}}^{\mathrm{I}\mathrm{h}}=g^{\mathrm{I}\mathrm{h}}\left(T,p\right).$$

(20)

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

$${\mu }_{\mathrm{V}}^{\mathrm{A}\mathrm{V}}\left(A^{\mathrm{s}\mathrm{a}\mathrm{t}},T,p\right)={\mu }_{\mathrm{W}}^{\mathrm{W}}\left(T,p\right),$$

(21)

and is defined with respect to ice by

$${\mu }_{\mathrm{V}}^{\mathrm{A}\mathrm{V}}\left(A^{\mathrm{s}\mathrm{a}\mathrm{t}},T,p\right)={\mu }_{\mathrm{W}}^{\mathrm{I}\mathrm{h}}\left(T,p\right).$$

(22)

To make the use of TEOS-10 equations more convenient, the numerical solutions of Equations (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 sea–ice–air (SIA) library of TEOS-10, freely available from the open source code and accessible at www.teos-10.org, accessed on 10 March 2024.

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 [75] that had not yet or only insufficiently been addressed by TEOS-10. Among those issues was the ambiguity and mutual inconsistency of different definitions of RH in practical use, such as the discrepancy between meteorology and climatology [34]. 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 [62]; Equation (10) in Feistel and Lovell-Smith 2017 [76])

$${\psi }_f\left(A,T,p\right)\equiv \exp \left\{\frac{{\mu }_{\mathrm{V}}^{\mathrm{A}\mathrm{V}}\left(A,T,p\right)-{\mu }_{\mathrm{V}}^{\mathrm{A}\mathrm{V}}\left(A^{\mathrm{s}\mathrm{a}\mathrm{t}}\left(T,p\right),T,p\right)}{R_{\mathrm{W}}T}\right\}.$$

(23)

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 Equation (21) or (22). Note that Equation (23) is only valid for temperatures below the boiling point of water at a given pressure (Feistel and Lovell-Smith 2017) [76], a condition which is naturally fulfilled under ambient geophysical conditions.

In ideal gas approximation, RF is identical to conventional RH in terms of water vapour partial pressures. For air that is close to saturation, such as an amarine surface layer with about 80 %rh, the Clausius–Clapeyron formula is an excellent approximation of TEOS-10 values for RF, even if it refrains from the perfect gas assumption [77]. RF appears naturally in expressions for the Onsager driving force of non-equilibrium fluxes, such as evaporation from the ocean surface [29,33]. For the numerical computation of RF, a source code extension to the SIA library is available from Feistel et al. (2022) [77].

Mathematical transformation between different thermodynamic quantities and partial derivatives with respect to alternative independent variables, as is 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) [78], as briefly described in Appendix A.

4. Model Results: Isentropically Lifted Condensation Level (LCL)

“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” [79]. Subtropical marine clouds show a pronounced “transition between unbroken sheets of stratocumulus and fields of scattered cumulus” [80]. 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 frostpoint, 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

$$z_{\mathrm{L}\mathrm{C}\mathrm{L}}-z={\gamma }_{\mathrm{L}\mathrm{C}\mathrm{L}}\left(T-T_{\mathrm{d}\mathrm{p}}\right),$$

(24)

where $z$, $T$ and $T_{\mathrm{d}\mathrm{p}}$, respectively, are the initial values of the parcel’s height, in situ temperature, and dew-point temperature. Commonly, the latter is related to the saturation vapour pressure and the RH via 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 $165 \mathrm{m}{\mathrm{K}}^{-1}$ [81], with an optimum of $125 \mathrm{m}{\mathrm{K}}^{-1}$ suggested by Lawrence (2005) [82]. Also, nonlinear mathematical expressions, more complicated than Equation (24), were suggested in the literature and 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, the highly accurate TEOS-10 results selected 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 [8,30,31,83,84,85]. Assuming that an air parcel starts ascending at a constant specific humidity, $A=A_0=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$, from the sea surface at a given air pressure, $p_0$, and a certain 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 Equation (23):

$${\mu }_{\mathrm{V}}^{\mathrm{A}\mathrm{V}}\left(A_0,T_0,p_0\right)=g^{\mathrm{W}}\left(T_0,p_0\right)+R_{\mathrm{W}}{T}_0\ln {\psi }_f.$$

(25)

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

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

$$\eta ={\eta }_0=-{\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)}{\partial T}\right)}_{A,p}.$$

(26)

At the LCL, the entropy value,

$$\eta ={\eta }_{\mathrm{L}\mathrm{C}\mathrm{L}}=-{\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial T}\right)}_{A,p},$$

(27)

needs to be the same as at the surface, Equation (26), and may be eliminated from the mathematical problem by equating Equation (26) with (27),

$${\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)}{\partial T}\right)}_{A,p}={\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial T}\right)}_{A,p}.$$

(28)

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 drops to the dew point, $T_{\mathrm{L}\mathrm{C}\mathrm{L}}$. This is determined using Equation (21):

$${\mu }_{\mathrm{V}}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)=g^{\mathrm{W}}\left(T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right).$$

(29)

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

$$g^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)-A{\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial A}\right)}_{T,p}=g^{\mathrm{W}}\left(T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right).$$

(30)

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

$$g^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)-A{\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)}{\partial A}\right)}_{T,p}=g^{\mathrm{W}}\left(T_0,p_0\right)+R_{\mathrm{W}}{T}_0\ln {\psi }_f.$$

(31)

Equations (28), (30) and (31) constitute a closed system of three nonlinear implicit equations for determining 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 performed because the dissolution of air into 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.

4.1. Numerical Iterative Solution

The system of Equations (28) and (30), used for the calculation of $T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}$ from $A$; Equation (31), used for the determination of $A$ from the given surface values of $T_0,p_0;$ and the relative fugacity can be solved iteratively, offering the best accuracy currently available. Expanding Equation (31) into a power series up to linear terms in the increment, $∆A$, of a starting estimate, $A$,

$$\begin{gathered} g^{\mathrm{A}\mathrm{V}}\left(A+∆A,T_0,p_0\right)-\left(A+∆A\right){\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A+∆A,T_0,p_0\right)}{\partial A}\right)}_{T,p} \\ =g^{\mathrm{W}}\left(T_0,p_0\right)+R_{\mathrm{W}}{T}_0\ln {\psi }_f \end{gathered}$$

(32)

leads to the solution for the improvement, $A_{i+1}=A_i+∆A_i$, of iteration step $i$,

$$∆A_i=\frac{g^{\mathrm{A}\mathrm{V}}\left(A_i,T_0,p_0\right)-A_i{\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A_i,T_0,p_0\right)}{\partial A}\right)}_{T,p}-g^{\mathrm{W}}\left(T_0,p_0\right)-R_{\mathrm{W}}{T}_0\ln {\psi }_f}{A_i{\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A_i,T_0,p_0\right)}{\partial A^2}\right)}_{T,p}}.$$

(33)

Practically, this Newton–Cotes iteration may start from almost dry air, such as $A_0=0.999999$.

In a similar manner, the other increments $∆T_{\mathrm{L}\mathrm{C}\mathrm{L}},∆p_{\mathrm{L}\mathrm{C}\mathrm{L}}$ follow on from Equations (28) and (30) and can be given in matrix notation as

$$\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)\left(\begin{array}{c}∆T_{\mathrm{L}\mathrm{C}\mathrm{L}}\\ ∆p_{\mathrm{L}\mathrm{C}\mathrm{L}}\end{array}\right)=\left(\begin{array}{c}{b}_1\\ b_2\end{array}\right).$$

(34)

By Cramer’s rule, the solution is

$$∆T_{\mathrm{L}\mathrm{C}\mathrm{L}}=\frac{\left|\begin{array}{cc}{b}_1& a_{12}\\ b_2& a_{22}\end{array}\right|}{\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|}=\frac{b_1{a}_{22}-b_2{a}_{12}}{a_{11}{a}_{22}-a_{21}{a}_{12}}$$

(35)

and

$$∆p_{\mathrm{L}\mathrm{C}\mathrm{L}}=\frac{\left|\begin{array}{cc}{a}_{11}& b_1\\ a_{21}& b_2\end{array}\right|}{\left|\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right|}=\frac{a_{11}{b}_2-a_{21}{b}_1}{a_{11}{a}_{22}-a_{21}{a}_{12}}.$$

(36)

The coefficients of this system are as follows:

$$a_{11}={\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial T^2}\right)}_{A,p},$$

(37)

$$a_{12}={\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial T\partial p}\right)}_{A,p},$$

(38)

$$a_{21}={\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial T}\right)}_{A,p}-A{\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial A\partial T}\right)}_p-{\left(\frac{{\partial g}^{\mathrm{W}}\left(T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial T}\right)}_p,$$

(39)

$$a_{22}={\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial p}\right)}_{A,T}-A{\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial A\partial p}\right)}_T-{\left(\frac{{\partial g}^{\mathrm{W}}\left(T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial p}\right)}_T,$$

(40)

$$b_1={\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)}{\partial T}\right)}_{A,p}-{\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial T}\right)}_{A,p},$$

(41)

$$b_2=g^{\mathrm{W}}\left(T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)-g^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)+A{\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial A}\right)}_{T,p}.$$

(42)

All derivatives utilized here are numerically available from the TEOS-10 SIA library [54].

Table 1. At constant surface RH of 80 %rh and pressure of 1013.25 hPa, as functions of the surface temperature $T_0$ and the surface dew-point temperature $T_{\mathrm{d}\mathrm{p}}$, dry-air mass fraction $A=1-q$, LCL pressure $p_{\mathrm{L}\mathrm{C}\mathrm{L}}$, the sensitivity of LCL pressure with respect to SST increases, LCL temperature $T_{\mathrm{L}\mathrm{C}\mathrm{L}}$, and the sensitivity of that temperature with respect to increasing SST are reported in the columns. The values result from iteratively solving TEOS-10 Equations (34)–(36). Differences $∆p_{\mathrm{L}\mathrm{C}\mathrm{L}}$,$∆T_0$ and $∆T_{\mathrm{L}\mathrm{C}\mathrm{L}}$ are obtained from the lower minus the upper row of the related values, so that their signs reflect the climatic warming trend. Climatic sensitivities α, γ and $\beta$ of $A$, $p_{\mathrm{L}\mathrm{C}\mathrm{L}}$ and $T_{\mathrm{L}\mathrm{C}\mathrm{L}}$, respectively, are defined in Equations (87) and (92) and computed from Equation (94), as shown in Section 5. All values are calculated with full TEOS-10 accuracy and rounded only upon printing. The row printed in bold approximates the current global mean SST (Figure 7).

${\mathit{T}}_{\mathbf{0}}$ K	${\mathit{T}}_{\mathbf{d}\mathbf{p}}$ K	$\mathit{A}$ %	α % K−1	${\mathit{p}}_{\mathbf{L}\mathbf{C}\mathbf{L}}$ hPa	$\frac{\mathbf{∆}{\mathit{p}}_{\mathbf{L}\mathbf{C}\mathbf{L}}}{\mathbf{∆}{\mathit{T}}_{\mathbf{0}}}$	γ hPa K−1	${\mathit{T}}_{\mathbf{L}\mathbf{C}\mathbf{L}}$ K	$\frac{\mathbf{∆}{\mathit{T}}_{\mathbf{L}\mathbf{C}\mathbf{L}}}{\mathbf{∆}{\mathit{T}}_{\mathbf{0}}}$	$\mathit{\beta }$ K K−1
286	282.633	99.2655	−0.0483	963.093	−0.2757	−0.2742	281.883	0.9632	0.9634
288	284.580	99.1631	−0.0542	962.542		−0.2773	283.810		0.9629
290	286.526	99.0482	−0.0608	961.984	−0.2823	−0.2806	285.735	0.9621	0.9624
292	288.471	98.9196	−0.0680	961.419		−0.2841	287.659		0.9619
294	290.416	98.7758	−0.0759	960.847	−0.2897	−0.2878	289.583	0.9611	0.9614
296	292.361	98.6154	−0.0846	960.268		−0.2917	291.505		0.9608
298	294.305	98.4368	−0.0942	959.680	−0.2981	−0.2959	293.426	0.9600	0.9603
300	296.248	98.2381	−0.1047	959.084		−0.3004	295.346		0.9597

reports selected LCL states computed from TEOS-10, iteratively solving Equations (34)–(36) at $p_0=1013.25 \mathrm{h}\mathrm{P}\mathrm{a}$ and ${\psi }_f=80 \%\mathrm{r}\mathrm{h}$. The results are rounded to 6 digits. The climatic warming trend moves LCL to higher altitudes ($p_{\mathrm{L}\mathrm{C}\mathrm{L}}$), where the temperature difference between ocean ($T_0$) and cloud base ($T_{\mathrm{L}\mathrm{C}\mathrm{L}}$) becomes larger and consequently, the downward radiative heat flux becomes reduced compared to (fictitious) lower clouds. This cooling effect is a negative, stabilising feedback of the cloud cover, slowing down ocean warming. The observed shrinking cloudiness, in turn, likely renders the cooling influence of clouds globally smaller.

4.2. Linear Analytical Approximation

If the initial temperature, $T_0$, is only slightly higher than the dew point, $T_{\mathrm{d}\mathrm{p}}$, as is typical at the ocean surface, the LCL pressure may be estimated from TEOS-10 equations that are linearised based on small differences as $\left(T_{\mathrm{L}\mathrm{C}\mathrm{L}}-T_0\right)$ and $\left(p_{\mathrm{L}\mathrm{C}\mathrm{L}}-p_0\right)$. Performing this, the relative fugacity may be very accurately expressed by the Clausius–Clapeyron equation for the dew point (Feistel and Hellmuth 2022: Equation (14) therein) [77]

$$R_{\mathrm{W}}{T}_0\ln {\psi }_f\approx L_{\mathrm{L}}\left(T_{\mathrm{d}\mathrm{p}},p_0\right)\left(1-\frac{T_0}{T_{\mathrm{d}\mathrm{p}}}\right).$$

(43)

Here, $L_{\mathrm{L}}$ is the specific evaporation enthalpy of liquid water at an equilibrium with (saturated) humid air. Its value is available from the thermodynamically rigorous relations (Feistel et al. 2010b; Feistel and Hellmuth 2022: Equations (C.5) and (C.6) therein) [62,77],

$$\frac{L_{\mathrm{L}}\left(T,p\right)}{T}={\eta }^{\mathrm{A}\mathrm{V}}\left(A^{\mathrm{s}\mathrm{a}\mathrm{t}}\left(T,p\right),T,p\right)-A^{\mathrm{s}\mathrm{a}\mathrm{t}}\left(T,p\right){\left(\frac{{\partial \eta }^{\mathrm{A}\mathrm{V}}}{\partial A}\right)}_{T,p}-{\eta }^{\mathrm{W}}\left(T,p\right),$$

(44)

which is equivalent to

$$\frac{L_{\mathrm{L}}\left(T,p\right)}{T}=-A^{\mathrm{s}\mathrm{a}\mathrm{t}}\left(T,p\right){\left(\frac{{\partial A}^{\mathrm{s}\mathrm{a}\mathrm{t}}}{\partial T}\right)}_p{\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}}{\partial A^2}\right)}_{T,p}.$$

(45)

Here, ${\eta }^{\mathrm{A}\mathrm{V}}$ and ${\eta }^{\mathrm{W}}$, respectively, are the specific entropies of humid air and liquid water.

Equation (43) represents the initial linear term of a truncated Taylor series with respect to powers of $\left(T_0-T_{\mathrm{d}\mathrm{p}}\right)$, regardless of whether ideal gas assumptions have been applied or not.

A similar series expansion of the Gibbs function of liquid water in Equation (30) with respect to $∆T\equiv \left(T_{\mathrm{L}\mathrm{C}\mathrm{L}}-T_0\right)$ and $∆p\equiv \left(p_{\mathrm{L}\mathrm{C}\mathrm{L}}-p_0\right)$ gives

$$g^{\mathrm{W}}\left(T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)\approx g^{\mathrm{W}}\left(T_0,p_0\right)+{\left(\frac{{\partial g}^{\mathrm{W}}\left(T_0,p_0\right)}{\partial T}\right)}_p∆T+{\left(\frac{{\partial g}^{\mathrm{W}}\left(T_0,p_0\right)}{\partial p}\right)}_T∆p.$$

(46)

The expansion coefficients are the specific entropy, ${\eta }^{\mathrm{W}}$, and the specific volume, $v^{\mathrm{W}}$, of liquid water at $\left(T_0,p_0\right)$,

$${∆g}^{\mathrm{W}}\equiv g^{\mathrm{W}}\left(T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)-g^{\mathrm{W}}\left(T_0,p_0\right)\approx -{\eta }^{\mathrm{W}}∆T+v^{\mathrm{W}}∆p.$$

(47)

A similar expansion about $\left(T_0,p_0\right)$ can be carried out for the Gibbs function of humid air,

$${∆g}^{\mathrm{A}\mathrm{V}}\equiv g^{\mathrm{A}\mathrm{V}}\left({A,T}_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)-g^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)\approx -{\eta }^{\mathrm{A}\mathrm{V}}∆T+v^{\mathrm{A}\mathrm{V}}∆p.$$

(48)

Subtracting Equation (31) from (30) results in

$${∆g}^{\mathrm{A}\mathrm{V}}-A{\left(\frac{{\partial ∆g}^{\mathrm{A}\mathrm{V}}}{\partial A}\right)}_{T,p}-{∆g}^{\mathrm{W}}=-R_{\mathrm{W}}{T}_0\ln {\psi }_f,$$

(49)

which may by expressed in terms of the linear offsets (47), (48), as

$$\left[{\eta }^{\mathrm{A}\mathrm{V}}-A{\left(\frac{{\partial \eta }^{\mathrm{A}\mathrm{V}}}{\partial A}\right)}_{T,p}-{\eta }^{\mathrm{W}}\right]∆T-\left[v^{\mathrm{A}\mathrm{V}}-A{\left(\frac{{\partial v}^{\mathrm{A}\mathrm{V}}}{\partial A}\right)}_{T,p}-v^{\mathrm{W}}\right]∆p=R_{\mathrm{W}}{T}_0\ln {\psi }_f.$$

(50)

Up to small linear terms in the subsaturation, $A-A^{\mathrm{s}\mathrm{a}\mathrm{t}}\left(T_0,p_0\right)$, at the surface, the coefficient of $∆T$ equals the specific evaporation entropy, Equation (44),

$$\frac{L_{\mathrm{L}}\left(T_{\mathrm{d}\mathrm{p}},p_0\right)}{T_{\mathrm{d}\mathrm{p}}}=∆{\eta }_{\mathrm{L}}\left(T_{\mathrm{d}\mathrm{p}},p_0\right)\approx {\eta }^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)-A{\left(\frac{{\partial \eta }^{\mathrm{A}\mathrm{V}}}{\partial A}\right)}_{T,p}-{\eta }^{\mathrm{W}}\left(T_0,p_0\right)$$

(51)

and the term in front of $∆p$ equals approximately the specific evaporation volume [62],

$$∆v_{\mathrm{L}}\left(T_{\mathrm{d}\mathrm{p}},p_0\right)\approx v^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)-A{\left(\frac{{\partial v}^{\mathrm{A}\mathrm{V}}}{\partial A}\right)}_{T,p}-v^{\mathrm{W}}\left(T_0,p_0\right).$$

(52)

Concerning linear approximation with respect to the LCL pressure and temperature offset, Equation (50) then reads

$$\frac{L_{\mathrm{L}}\left(T_{\mathrm{d}\mathrm{p}},p_0\right)}{T_{\mathrm{d}\mathrm{p}}}∆T-∆v_{\mathrm{L}}\left(T_{\mathrm{d}\mathrm{p}},p_0\right)∆p=R_{\mathrm{W}}{T}_0\ln {\psi }_f.$$

(53)

During adiabatic uplift, temperature and pressure change are mutually related by the moist lapse rate, Equation (A18),

$$∆T=\mathsf{\Gamma }∆p.$$

(54)

Similar to Equation (24), the final formula for the LCL pressure, Formula (53), therefore also makes use of Equation (43),

$$p_{\mathrm{L}\mathrm{C}\mathrm{L}}-p_0=\frac{L_{\mathrm{L}}\left(T_{\mathrm{d}\mathrm{p}},p_0\right)\left(1-\frac{T_0}{T_{\mathrm{d}\mathrm{p}}}\right)}{\frac{L_{\mathrm{L}}\left(T_{\mathrm{d}\mathrm{p}},p_0\right)}{T_{\mathrm{d}\mathrm{p}}}\mathsf{\Gamma }-∆v_{\mathrm{L}}\left(T_{\mathrm{d}\mathrm{p}},p_0\right)}=-\frac{{T_0-T}_{\mathrm{d}\mathrm{p}}}{\mathsf{\Gamma }\left(T_{\mathrm{d}\mathrm{p}},p_0\right)+{\mathsf{\Gamma }}_{\mathrm{d}\mathrm{p}}\left(T_{\mathrm{d}\mathrm{p}},p_0\right)}.$$

(55)

Here, the lapse rate of the dew point is negative, as shown in Appendix B, Equation (A31),

$${\mathsf{\Gamma }}_{\mathrm{d}\mathrm{p}}={\left(\frac{\partial T_{\mathrm{d}\mathrm{p}}}{\partial p}\right)}_A=-\frac{T_{\mathrm{d}\mathrm{p}}∆v_{\mathrm{L}}}{L_{\mathrm{L}}\left(T_{\mathrm{d}\mathrm{p}},p\right)}.$$

(56)

The hydrostatic balance equation (height $z$, gravity constant $g_{\mathrm{E}}=9.81 \mathrm{m} {\mathrm{s}}^{-2}$),

$$v^{\mathrm{A}\mathrm{V}}\mathrm{d}p=-g_{\mathrm{E}}\mathrm{d}z,$$

(57)

can be integrated along an isentropic vertical profile, due to the thermodynamic relation,

$$v^{\mathrm{A}\mathrm{V}}={\left(\frac{\partial h^{\mathrm{A}\mathrm{V}}}{\partial p}\right)}_{A,\eta },$$

(58)

to give

$$z_{\mathrm{L}\mathrm{C}\mathrm{L}}-z_0=-\frac{1}{g_{\mathrm{E}}}\left[h^{\mathrm{A}\mathrm{V}}\left(A,\eta ,p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)-h^{\mathrm{A}\mathrm{V}}\left(A,\eta ,p_0\right)\right].$$

(59)

This enthalpy difference may be used to estimate the coefficient of the LCL height, Equation (24), from the LCL pressure, through

$${\gamma }_{\mathrm{L}\mathrm{C}\mathrm{L}}\equiv \frac{z_{\mathrm{L}\mathrm{C}\mathrm{L}}-z_0}{T_0-T_{\mathrm{d}\mathrm{p}}}.$$

(60)

Rigorous numerical TEOS-10 results for $\left(z_{\mathrm{L}\mathrm{C}\mathrm{L}}-z_0\right)$ and ${\gamma }_{\mathrm{L}\mathrm{C}\mathrm{L}}$ from Equations (59) and (60) at selected typical marine conditions, obtained through iteratively solving Equations (34)–(36), are reported in

Table 2. Iterative solutions for the rigorous Equations (34)–(36), TEOS-10 values of LCL height, $z_{\mathrm{L}\mathrm{C}\mathrm{L}}-z_0$, computed from Equation (59), and the LCL coefficient, ${\gamma }_{\mathrm{L}\mathrm{C}\mathrm{L}}=\left(z_{\mathrm{L}\mathrm{C}\mathrm{L}}-z_0\right)/\left(T_0-T_{\mathrm{d}\mathrm{p}}\right)$, computed from Equation (60) at a surface pressure of 1013.25 hPa and marine relative fugacity of ${\psi }_f=80\%\mathrm{r}\mathrm{h}$ for selected SST values, $T_0$. The row printed in bold approximates current global mean SST.

${\mathit{T}}_{\mathbf{0}}$ K	${\mathit{\psi }}_{\mathit{f}}$ %rh	${\mathit{T}}_{\mathbf{d}\mathbf{p}}$ K	$\mathit{A}$ %	${\mathit{z}}_{\mathbf{L}\mathbf{C}\mathbf{L}}\mathbf{-}{\mathit{z}}_{\mathbf{0}}$ m	${\mathit{\gamma }}_{\mathbf{L}\mathbf{C}\mathbf{L}}$ m K−1
286	80	282.633	99.2655	423.468	125.778
288	80	284.580	99.1631	431.481	126.157
290	80	286.526	99.0482	439.660	126.553
292	80	288.471	98.9196	448.017	126.967
294	80	290.416	98.7758	456.561	127.403
296	80	292.361	98.6154	465.305	127.862
298	80	294.305	98.4368	474.263	128.345
300	80	296.248	98.2381	483.449	128.857

and

Table 3. Iteratively solving the rigorous Equations (34)–(36), TEOS-10 values of LCL height, $z_{\mathrm{L}\mathrm{C}\mathrm{L}}-z_0$, computed from Equation (59), and LCL coefficient, ${\gamma }_{\mathrm{L}\mathrm{C}\mathrm{L}}=\left(z_{\mathrm{L}\mathrm{C}\mathrm{L}}-z_0\right)/\left(T_0-T_{\mathrm{d}\mathrm{p}}\right)$ from Equation (60) at surface pressure of 1013.25 hPa and SST of $T_0=292 \mathrm{K}$ for selected relative fugacities ${\psi }_f$. The row printed in bold approximates current global mean SST. Note the significant sensitivity of the LCL height on RH, affecting the LCL temperature, $T_{\mathrm{L}\mathrm{C}\mathrm{L}}$, as well as the related downward thermal radiation flux, ${\sigma }_{\mathrm{S}\mathrm{B}}{T}_{\mathrm{L}\mathrm{C}\mathrm{L}}^4$.

${\mathit{T}}_{\mathbf{0}}$ K	${\mathit{\psi }}_{\mathit{f}}$ %rh	${\mathit{T}}_{\mathbf{d}\mathbf{p}}$ K	$\mathit{A}$ %	${\mathit{z}}_{\mathbf{L}\mathbf{C}\mathbf{L}}\mathbf{-}{\mathit{z}}_{\mathbf{0}}$ m	${\mathit{\gamma }}_{\mathbf{L}\mathbf{C}\mathbf{L}}$ m K−1	${\mathit{T}}_{\mathbf{L}\mathbf{C}\mathbf{L}}$ K	${\mathit{\sigma }}_{\mathbf{S}\mathbf{B}}{\mathit{T}}_{\mathbf{L}\mathbf{C}\mathbf{L}}^4$ W m−2
292	74	287.262	99.0012	600.040	126.632	286.182	380.348
292	76	287.674	98.9740	548.289	126.745	286.685	383.030
292	78	288.077	98.9468	497.632	126.857	287.177	385.668
292	80	288.471	98.9196	448.017	126.967	287.659	388.263
292	82	288.857	98.8923	399.396	127.076	288.131	390.818
292	84	289.235	98.8650	351.724	127.184	288.594	393.334
292	86	289.604	98.8378	304.959	127.291	289.048	395.812
292	88	289.966	98.8105	259.061	127.396	289.493	398.255

.

A linear approximation based on the LCL pressure, Equation (55), is performed as follows:

$${\gamma }_{\mathrm{L}\mathrm{C}\mathrm{L}}\approx -\frac{v^{\mathrm{A}\mathrm{V}}}{g_{\mathrm{E}}}\frac{\left(p_{\mathrm{L}\mathrm{C}\mathrm{L}}-p\right)}{\left(T-T_{\mathrm{d}\mathrm{p}}\right)}\approx \frac{v^{\mathrm{A}\mathrm{V}}}{g_{\mathrm{E}}\left(\mathsf{\Gamma }+{\mathsf{\Gamma }}_{\mathrm{d}\mathrm{p}}\right)}.$$

(61)

In Table 3, in addition to the calculated LCL coefficients, the climatic feedback to possible RH trends is also reported as the thermal radiation flux, ${\sigma }_{\mathrm{S}\mathrm{B}}{T}_{\mathrm{L}\mathrm{C}\mathrm{L}}^4$, downward from the cloud base, which tends to warm the ocean if RH is rising.

If the specific volume of humid air in Equation (52) is much larger than the remaining terms of the evaporation volume, $∆v_{\mathrm{L}}\approx v^{\mathrm{A}\mathrm{V}}$, the simplified LCL coefficient given by Equation (61), expressing the lapse rate by Equation (A18), is $\mathsf{\Gamma }={\alpha }^{\mathrm{A}\mathrm{V}}{v}^{\mathrm{A}\mathrm{V}}T/c_p^{\mathrm{A}\mathrm{V}}$. In terms of the thermal expansion coefficient, ${\alpha }^{\mathrm{A}\mathrm{V}}$, and the specific isobaric heat capacity, $c_p^{\mathrm{A}\mathrm{V}}$, of humid air, by virtue of Equation (56) and $T\approx T_{\mathrm{d}\mathrm{p}}$, we state that

$${\gamma }_{\mathrm{L}\mathrm{C}\mathrm{L}}\approx {\left[g_{\mathrm{E}}{T}_{\mathrm{d}\mathrm{p}}\left(\frac{{\alpha }^{\mathrm{A}\mathrm{V}}}{c_p^{\mathrm{A}\mathrm{V}}}-\frac{1}{L_{\mathrm{L}}}\right)\right]}^{-1}.$$

(62)

In ideal gas approximation, ${\alpha }^{\mathrm{A}\mathrm{V}}\approx 1/T_{\mathrm{d}\mathrm{p}}$, this becomes

$${\gamma }_{\mathrm{L}\mathrm{C}\mathrm{L}}\approx {\left[g_{\mathrm{E}}\left(\frac{1}{c_p^{\mathrm{A}\mathrm{V}}}-\frac{1}{∆{\eta }_{\mathrm{L}}}\right)\right]}^{-1}.$$

(63)

Here, $∆{\eta }_{\mathrm{L}}=L_{\mathrm{L}}/T_{\mathrm{d}\mathrm{p}}$ is the evaporation entropy of liquid water, as shown in Equation (51).

Analytical low-pressure approximations for the TEOS-10 Gibbs functions of liquid water and humid air are provided by Feistel and Hellmuth (2023: Appendix A therein) [29] and are given in an extremely simplified, crude form in Appendix C of this paper. These methods are successfully used in the following section.

4.3. Clausius–Clapeyron Expansion

In the context of global warming, the Clausius–Clapeyron equation for the saturated vapour pressure of water is a widely exploited and sufficiently accurate approximation method for estimating the increasing humidity of the troposphere. Here, an expression for the LCL is derived with a similar accuracy from the TEOS-10 equations.

Equations (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 relations may be formulated in terms of “crude” approximations, taking all heat capacities as constant, all liquids as incompressible, and all gases as perfect gases. The related crude Gibbs functions, ${\tilde{g}}^{\mathrm{W}}\left(T,p\right)$ and ${\tilde{g}}^{\mathrm{A}\mathrm{V}}\left(A,T,p\right)$, are indicated here by tildes and are provided in Appendix C.

Using those Gibbs functions, the conservation of entropy during adiabatic ascent, shown in Equation (28), is given by

$${\left(\frac{{\partial \tilde{g}}^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)}{\partial T}\right)}_{A,p}={\left(\frac{{\partial \tilde{g}}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial T}\right)}_{A,p}.$$

(64)

This takes the form of the conventional adiabatic equation of state,

$${\tilde{c}}_p^{\mathrm{A}\mathrm{V}}\ln \frac{T_0}{T_{\mathrm{L}\mathrm{C}\mathrm{L}}}=R_{\mathrm{A}\mathrm{V}}\ln \frac{p_0}{p_{\mathrm{L}\mathrm{C}\mathrm{L}}}.$$

(65)

Here, ${\tilde{c}}_p^{\mathrm{A}\mathrm{V}}\left(A\right)\equiv \left(1-A\right){\tilde{c}}_p^{\mathrm{V}}+A{\tilde{c}}_p^{\mathrm{A}}$ and $R_{\mathrm{A}\mathrm{V}}\left(A\right)\equiv \left(1-A\right)R_{\mathrm{W}}+A{R}_{\mathrm{A}}$, respectively, are the heat capacity and the specific gas constant of humid air.

The saturation of air at the LCL, Equation (30), expresses the equation chemical potentials of water in the gas and the liquid phase,

$${\tilde{g}}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)-A{\left(\frac{{\partial \tilde{g}}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial A}\right)}_{T,p}={\tilde{g}}^{\mathrm{W}}\left(T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right).$$

(66)

In crude approximation, this condition reads

$$\left({\tilde{c}}_p^{\mathrm{W}}-{\tilde{c}}_p^{\mathrm{V}}\right)T_{\mathrm{L}\mathrm{C}\mathrm{L}}\ln \frac{T_{\mathrm{L}\mathrm{C}\mathrm{L}}}{T_{\mathrm{t}}}+R_{\mathrm{W}}{T}_{\mathrm{L}\mathrm{C}\mathrm{L}}\ln \frac{x{p}_{\mathrm{L}\mathrm{C}\mathrm{L}}}{p_{\mathrm{t}}}={\tilde{g}}_0^{\mathrm{W}}+{\tilde{g}}_1^{\mathrm{W}}{T}_{\mathrm{L}\mathrm{C}\mathrm{L}}+\frac{p_{\mathrm{L}\mathrm{C}\mathrm{L}}}{{\tilde{\rho }}^{\mathrm{W}}}.$$

(67)

Here, $T_{\mathrm{t}}$ and $p_{\mathrm{t}}$, respectively, are temperature and pressure at the triple point of water, and ${\tilde{\rho }}^{\mathrm{W}}$ is the density of liquid water. They appear in this context because the triple point is used as the reference state at which the arbitrary constants ${\tilde{g}}_0^{\mathrm{W}}$ and ${\tilde{g}}_1^{\mathrm{W}}$ are adjusted to the phase equilibrium properties of fluid water, computed from TEOS-10, as shown in Appendix C.

Finally, the relative fugacity (or RH) at the surface is specified by Equation (31) in terms of crude Gibbs functions

$${\tilde{g}}^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)-A{\left(\frac{{\partial \tilde{g}}^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)}{\partial A}\right)}_{T,p}={\tilde{g}}^{\mathrm{W}}\left(T_0,p_0\right)+R_{\mathrm{W}}{T}_0\ln {\psi }_f.$$

(68)

Inserting the crude equations given in Appendix C, results in

$$\left({\tilde{c}}_p^{\mathrm{W}}-{\tilde{c}}_p^{\mathrm{V}}\right)T_0\ln \frac{T_0}{T_{\mathrm{t}}}+R_{\mathrm{W}}{T}_0\ln \frac{{xp}_0}{{{\psi }_{f}p}_{\mathrm{t}}}={\tilde{g}}_0^{\mathrm{W}}+{\tilde{g}}_1^{\mathrm{W}}{T}_0+\frac{p_0}{{\tilde{\rho }}^{\mathrm{W}}}.$$

(69)

From the latter equation, the vapour mole fraction at the surface, $x\left({\psi }_f,T_0,p_0\right)$, is obtained from the relative fugacity, ${\psi }_f$, by

$$x\left({\psi }_f,T_0,p_0\right)={\psi }_f\frac{p_{\mathrm{t}}}{p_0}\exp \left\{\frac{{\tilde{g}}_0^{\mathrm{W}}+{\tilde{g}}_1^{\mathrm{W}}{T}_0}{R_{\mathrm{W}}{T}_0}+\frac{p_0}{R_{\mathrm{W}}{T}_0{\tilde{\rho }}^{\mathrm{W}}}-\frac{\left({\tilde{c}}_p^{\mathrm{W}}-{\tilde{c}}_p^{\mathrm{V}}\right)}{R_{\mathrm{W}}}\ln \frac{T_0}{T_{\mathrm{t}}}\right\}.$$

(70)

From $x$, the dry-air mass fraction $A$ is available through the use of Equation (4).

Subtracting Equation (67) from (69) and inserting entropy conservation (65) as

$$\ln \frac{p_{\mathrm{L}\mathrm{C}\mathrm{L}}}{p_{\mathrm{t}}}=\ln \frac{p_0}{p_{\mathrm{t}}}+\frac{{\tilde{c}}_p^{\mathrm{A}\mathrm{V}}}{R_{\mathrm{A}\mathrm{V}}}\ln \frac{T_{\mathrm{L}\mathrm{C}\mathrm{L}}}{T_0},$$

(71)

the saturation Equation (68) can be written in the form of

$$Q\ln \frac{T_0}{T_{\mathrm{L}\mathrm{C}\mathrm{L}}}-\ln {\psi }_f=\frac{{\tilde{g}}_0^{\mathrm{W}}}{R_{\mathrm{W}}}\left(\frac{1}{T_0}-\frac{1}{T_{\mathrm{L}\mathrm{C}\mathrm{L}}}\right)+\frac{1}{{\tilde{\rho }}^{\mathrm{W}}{R}_{\mathrm{W}}}\left(\frac{p_0}{T_0}-\frac{p_{\mathrm{L}\mathrm{C}\mathrm{L}}}{T_{\mathrm{L}\mathrm{C}\mathrm{L}}}\right).$$

(72)

Here, the abbreviation

$$Q\equiv \frac{{\tilde{c}}_p^{\mathrm{W}}-{\tilde{c}}_p^{\mathrm{V}}}{R_{\mathrm{W}}}+\frac{{\tilde{c}}_p^{\mathrm{A}\mathrm{V}}}{R_{\mathrm{A}\mathrm{V}}}$$

(73)

is introduced. The commonly used Clausius–Clapeyron equation for the saturation vapour pressure makes use of the fact that far from the critical point of water, the liquid density is much higher than the vapour density. This may be quantified by saying that the dimensionless parameter

$$\epsilon =\frac{p_0}{R_{\mathrm{W}}{T}_0{\tilde{\rho }}^{\mathrm{W}}}\ll 1$$

(74)

is a small number. The leading term of the perturbation expansion

$$T_{\mathrm{L}\mathrm{C}\mathrm{L}}=T_{\mathrm{L}\mathrm{C}\mathrm{L}}^{\left(0\right)}+\epsilon T_{\mathrm{L}\mathrm{C}\mathrm{L}}^{\left(1\right)}+\cdots ,$$

(75)

applied to Equation (72) may be regarded as an Clausius–Clapeyron approximation,

$$Q\ln \frac{T_0}{T_{\mathrm{L}\mathrm{C}\mathrm{L}}^{\left(0\right)}}-\ln {\psi }_f=\frac{{\tilde{g}}_0^{\mathrm{W}}}{R_{\mathrm{W}}}\left(\frac{1}{T_0}-\frac{1}{T_{\mathrm{L}\mathrm{C}\mathrm{L}}^{\left(0\right)}}\right).$$

(76)

Equation (76) may be rearranged in the form

$$\frac{B{T}_0}{T_{\mathrm{L}\mathrm{C}\mathrm{L}}^{\left(0\right)}}\exp \left(-\frac{{BT}_0}{T_{\mathrm{L}\mathrm{C}\mathrm{L}}^{\left(0\right)}}\right)=E$$

(77)

with the abbreviations,

$$B\equiv -\frac{{\tilde{g}}_0^{\mathrm{W}}}{Q{R}_{\mathrm{W}}{T}_0}$$

(78)

$$E\equiv B\exp \left\{-B+\frac{\ln {\psi }_f}{Q}\right\}.$$

(79)

The solution of the nonlinear Equation (77) is

$$T_{\mathrm{L}\mathrm{C}\mathrm{L}}^{\left(0\right)}=-\frac{B}{w\left(-E\right)}{T}_0.$$

(80)

Here, the function $w\left(y\right)$ is the well-defined Lambert W function, as shown in Appendix D, which is one of the two solutions of the equation

$$w\exp \left(w\right)=y.$$

(81)

Because $y=-E$ is negative, such as $E=0.174718$ at $T_0=292 \mathrm{K}$, solutions for the lower branch, $w\left(y\right)<-1$, are chosen here,

$$T_{\mathrm{L}\mathrm{C}\mathrm{L}}^{\left(0\right)}=287.658 \mathrm{K}.$$

(82)

Eventually, by means of Equation (71), the Clausius–Clapeyron approximation for the LCL pressure is

$$p_{\mathrm{L}\mathrm{C}\mathrm{L}}^{\left(0\right)}=p_0\exp \left\{\frac{{\tilde{c}}_p^{\mathrm{A}\mathrm{V}}}{R_{\mathrm{A}\mathrm{V}}}\ln \left[-\frac{B}{w\left(-E\right)}\right]\right\}.$$

(83)

In fact, these “crude” Clausius–Clapeyron approximations (80) and (83) already agree quite well with the rigorous iterative TEOS-10 results of Equation (34), see

Table 4. Comparison of LCL temperatures and pressures of the rigorous TEOS-10 solutions of Equation (34) with the Clausius–Clapeyron approximations, Equations (80) and (83). The row printed in bold approximates the current global mean SST.

${\mathit{T}}_{\mathbf{0}}$ K	${\mathit{T}}_{\mathbf{L}\mathbf{C}\mathbf{L}}$ K	${\mathit{T}}_{\mathbf{L}\mathbf{C}\mathbf{L}}^{\mathbf{\left(}\mathbf{0}\mathbf{\right)}}$ K	${\mathit{p}}_{\mathbf{L}\mathbf{C}\mathbf{L}}$ hPa	${\mathit{p}}_{\mathbf{L}\mathbf{C}\mathbf{L}}^{\mathbf{\left(}\mathbf{0}\mathbf{\right)}}$ hPa
286	281.883	281.883	963.093	963.066
288	283.810	283.810	962.542	962.525
290	285.735	285.735	961.984	961.972
292	287.659	287.658	961.419	961.395
294	289.583	289.584	960.847	960.858
296	291.505	291.506	960.268	960.276
298	293.426	293.428	959.680	959.712
300	295.346	295.348	959.084	959.118

.

5. Model Results: Marine Climatic LCL Feedback

Refraining from discussing the complexity of spectrally resolved greenhouse gas attenuation, as shown in Figure 5 and in proper 3D radiation propagation models [42,86,87], low-level marine clouds as well as the ocean surface may approximately be described as planar black bodies, mutually interacting according to the Stefan–Boltzmann law of thermal radiation, as shown in Figure 8.

Denoting the Stefan–Boltzmann constant as ${\sigma }_{\mathrm{S}\mathrm{B}}=5.670374419\times {10}^{-8} \mathrm{W} {\mathrm{m}}^{-2} {\mathrm{K}}^{-4}$, the net radiative heat loss, $J_{\mathrm{E}}$, of the cloud-covered ocean is given in a 1D model:

$$J_{\mathrm{E}}={\sigma }_{\mathrm{S}\mathrm{B}}\left(T_0^4-T_{\mathrm{L}\mathrm{C}\mathrm{L}}^4\right).$$

(84)

Its sensitivity with respect to ocean warming is determined as follows:

$${\left(\frac{\partial J_{\mathrm{E}}}{\partial T_0}\right)}_{p_0,{\psi }_f}=4{\sigma }_{\mathrm{S}\mathrm{B}}\left[T_0^3-T_{\mathrm{L}\mathrm{C}\mathrm{L}}^3{\left(\frac{\partial T_{\mathrm{L}\mathrm{C}\mathrm{L}}}{\partial T_0}\right)}_{p_0,{\psi }_f}\right],$$

(85)

assuming that the marine surface RH remains unaffected by the SST change. For the LCL values listed in Table 1, the upward and downward radiative fluxes, $J_{\uparrow }={\sigma }_{\mathrm{S}\mathrm{B}}{T}_0^4$ and $J_{\downarrow }={\sigma }_{\mathrm{S}\mathrm{B}}{T}_{\mathrm{L}\mathrm{C}\mathrm{L}}^4$, respectively, as well as estimated sensitivities, as in Equation (85), were calculated and are shown in

Table 5. At a given surface RH of ${\psi }_f=80\%\mathrm{r}\mathrm{h}$ and pressure of $p_0=101325 \mathrm{P}\mathrm{a}$, as functions of the SST and $T_0$, the columns report upward thermal radiation flux, $J_{\uparrow }={\sigma }_{\mathrm{S}\mathrm{B}}{T}_0^4$; LCL temperature, $T_{\mathrm{L}\mathrm{C}\mathrm{L}}$; downward thermal radiation flux, $J_{\downarrow }={\sigma }_{\mathrm{S}\mathrm{B}}{T}_{\mathrm{L}\mathrm{C}\mathrm{L}}^4$; upward net flux, ${J_{\mathrm{E}}=J}_{\uparrow }-J_{\downarrow }$; and the estimated sensitivity of the net flux with respect to increasing SST. The row printed in bold approximates the current global mean SST (Figure 7).

${\mathit{T}}_{\mathbf{0}}$ K	${\mathit{J}}_{\mathbf{\uparrow }}$ W m−2	${\mathit{T}}_{\mathbf{L}\mathbf{C}\mathbf{L}}$ K	${\mathit{J}}_{\mathbf{\downarrow }}$ W m−2	${{\mathit{J}}_{\mathbf{E}}=\mathit{J}}_{\mathbf{\uparrow }}\mathbf{-}{\mathit{J}}_{\mathbf{\downarrow }}$ W m−2	$\frac{\mathbf{∆}{\mathit{J}}_{\mathbf{E}}}{\mathbf{∆}{\mathit{T}}_{\mathbf{0}}}$
286	379.381	281.883	358.006	21.375	0.419
288	390.105	283.810	367.893	22.213
290	401.055	285.735	377.977	23.078	0.447
292	412.233	287.659	388.263	23.971
294	423.644	289.583	398.751	24.893	0.476
296	435.290	291.505	409.444	25.846
298	447.174	293.426	420.345	26.829	0.508
300	459.300	295.346	431.455	27.845

.

The mathematical dependence of $T_{\mathrm{L}\mathrm{C}\mathrm{L}}$ on $T_0$ is complicated, even in the case of crude approximations, as shown in Equation (80). In this section, a rigorous theoretical expression for this dependence is derived systematically from the TEOS-10 approach. As an aside, it should be noted that this longwave radiative interaction is an irreversible process with an entropy production rate [88,89,90] of

$$P=J_{\mathrm{E}}\left(\frac{1}{T_{\mathrm{L}\mathrm{C}\mathrm{L}}}-\frac{1}{T_0}\right).$$

(86)

The latest assessments [30] have shown that at the ocean surface, current climate change is characterised by an increase in SST along with a constant surface RH, or similarly, it is characterised in the TEOS-10 formalism by a constant relative fugacity, ${\psi }_f$. As a dimensionless quantity, a suitable thermodynamic expression for describing a related climatic trend of the LCL greenhouse effect is the sensitivity of the cloud base temperature, $T_{\mathrm{L}\mathrm{C}\mathrm{L}}\left(T_0,p_0,{\psi }_f\right)$, with respect to an increase in the surface temperature, $T_0$,

$$\beta \equiv {\left(\frac{\partial T_{\mathrm{L}\mathrm{C}\mathrm{L}}}{\partial T_0}\right)}_{p_0,{\psi }_f}.$$

(87)

This LCL temperature sensitivity controls the sensitivity of the ocean–cloud heat flux balance, Equation (85), with respect to rising SST. A rigorous formula for this sensitivity may be derived from the set of LCL equations that describe entropy conservation during uplift, as in Equation (28), where

$$f_1\equiv {\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)}{\partial T}\right)}_{A,p}-{\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial T}\right)}_{A,p}=0.$$

(88)

Air saturation at the cloud base, as in Equation (30), is determined by

$$f_2\equiv g^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)-A{\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial A}\right)}_{T,p}-g^{\mathrm{W}}\left(T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)=0,$$

(89)

and the given relative humidity at the sea surface, as in Equation (31), is determined by

$$f_3\equiv g^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)-A{\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)}{\partial A}\right)}_{T,p}-g^{\mathrm{W}}\left(T_0,p_0\right)-R_{\mathrm{W}}{T}_0\ln {\psi }_f=0.$$

(90)

During the climatic increase in the SST, $T_0$, at a constant RH, ${\psi }_f$, and surface pressure, $p_0$, these equations always remain fulfilled for $i=1,2,3$,

$$f_i\equiv 0,\hspace{1em}\hspace{1em}\hspace{1em}\frac{\mathrm{d}{f}_i}{\mathrm{d}{T}_0}\equiv 0,$$

(91)

so that other properties, namely, $A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}$, may vary along with $T_0$. Taking the partial derivatives of the functions $f_1$, $f_2$ and $f_3$ with respect to $T_0$, while considering the dependence of $A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}$ on $T_0$, a set of three equations is obtained for their three unknown sensitivities with respect to ocean warming:

$$\alpha \equiv {\left(\frac{\partial A}{\partial T_0}\right)}_{p_0,{\psi }_f}, \beta , \mathrm{and} \gamma \equiv {\left(\frac{\partial p_{\mathrm{L}\mathrm{C}\mathrm{L}}}{\partial T_0}\right)}_{p_0,{\psi }_f}.$$

(92)

These derivatives are mutually related by Equations (91),

$${\left(\frac{\partial f_i}{\partial A}\right)}_{T_0,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}}\alpha +{\left(\frac{\partial f_i}{\partial T_{\mathrm{L}\mathrm{C}\mathrm{L}}}\right)}_{A,T_0,p_{\mathrm{L}\mathrm{C}\mathrm{L}}}\beta +{\left(\frac{\partial f_i}{\partial p_{\mathrm{L}\mathrm{C}\mathrm{L}}}\right)}_{{A,T}_0,T_{\mathrm{L}\mathrm{C}\mathrm{L}}}\gamma +{\left(\frac{\partial f_i}{\partial T_0}\right)}_{A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}}=0.$$

(93)

In matrix notation, this is

$$\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& 0& 0\end{array}\right)\left(\begin{array}{c}\alpha \\ \beta \\ \gamma \end{array}\right)=\left(\begin{array}{c}{b}_1\\ 0\\ b_3\end{array}\right).$$

(94)

By Cramer’s rule, the solution of this system for the LCL sensitivity $\beta$ is straight forward:

$$\beta =\frac{\left|\begin{array}{ccc}{a}_{11}& b_1& a_{13}\\ a_{21}& 0& a_{23}\\ a_{31}& b_3& 0\end{array}\right|}{\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& 0& 0\end{array}\right|}=\frac{b_1{a}_{31}{a}_{23}-b_3\left(a_{11}{a}_{23}-a_{21}{a}_{13}\right)}{a_{31}\left(a_{12}{a}_{23}-a_{22}{a}_{13}\right)}.$$

(95)

The coefficients of this system of linear equations are as follows

$$a_{11}={\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)}{\partial A\partial T}\right)}_p-{\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial A\partial T}\right)}_p,$$

(96)

$$a_{12}=-{\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial T^2}\right)}_{A,p}=\frac{c_p^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{T_{\mathrm{L}\mathrm{C}\mathrm{L}}},$$

(97)

$$a_{13}=-{\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial T\partial p}\right)}_A,$$

(98)

$$a_{21}=-A{\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial A^2}\right)}_{T,p},$$

(99)

$$a_{22}={\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial T}\right)}_{A,p}-A{\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial A\partial T}\right)}_p-{\left(\frac{{\partial g}^{\mathrm{W}}\left(T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial T}\right)}_p=\frac{L_{\mathrm{L}}\left(T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{T_{\mathrm{L}\mathrm{C}\mathrm{L}}},$$

(100)

$$a_{23}={\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial p}\right)}_{A,T}-A{\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A,T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial A\partial p}\right)}_T-{\left(\frac{{\partial g}^{\mathrm{W}}\left(T_{\mathrm{L}\mathrm{C}\mathrm{L}},p_{\mathrm{L}\mathrm{C}\mathrm{L}}\right)}{\partial p}\right)}_T=∆v_{\mathrm{L}},$$

(101)

$$a_{31}=-A{\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)}{\partial A^2}\right)}_{T,p},$$

(102)

$$b_1=-{\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)}{\partial T^2}\right)}_{A,p}=\frac{c_p^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)}{T_0},$$

(103)

$$b_3=-{\left(\frac{{\partial g}^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)}{\partial T}\right)}_{A,p}+A{\left(\frac{{{\partial }^{2}g}^{\mathrm{A}\mathrm{V}}\left(A,T_0,p_0\right)}{\partial A\partial T}\right)}_p+{\left(\frac{{\partial g}^{\mathrm{W}}\left(T_0,p_0\right)}{\partial T}\right)}_p+R_{\mathrm{W}}\ln {\psi }_f.$$

(104)

All partial derivatives required here, in Equation (96) through to Equation (104), are explicitly numerically available from the TEOS-10 SIA library. Reported in Table 1, the LCL sensitivities computed from Equation (94) are the rigorous figures that are close to the previous estimates shown in Section 4. A value of $\beta <1$ means that $T_{\mathrm{L}\mathrm{C}\mathrm{L}}$ is rising slower than $T_0$, so that the net radiation balance between sea surface and cloud base is distorted in favour of the upward heat flux, reducing the ocean’s warming rate.

The cloud base is cooling relative to global SST warming, as shown in Figure 7, and the net upward radiative heat flux of about 24 W m⁻² is intensifying by 0.45 W m⁻² per °C of SST as negative feedback to the warming ocean by currently about 1 W m⁻² [9]. This feedback does not provide a possible explanation for the warming; on the contrary, it worsens the problem. While LCL feedback does not dominate the sea–air energy transfer, it is large enough anyway to not be ignored. The effect is about 100 times as large as the rate of atmospheric warming.

For tutorial reasons or first estimates, the coefficients of Equation (94) may be expressed as crude approximations of the Gibbs functions, shown in Appendix C. These are as follows:

$${\tilde{a}}_{11}=\left({\tilde{c}}_p^{\mathrm{V}}-{\tilde{c}}_p^{\mathrm{A}}\right)\ln \frac{T_0}{T_{\mathrm{L}\mathrm{C}\mathrm{L}}}+\left(R_{\mathrm{A}}-R_{\mathrm{W}}\right)\ln \frac{p_0}{p_{\mathrm{L}\mathrm{C}\mathrm{L}}},$$

(105)

$${\tilde{a}}_{12}=\frac{{\tilde{c}}_p^{\mathrm{A}\mathrm{V}}}{T_{\mathrm{L}\mathrm{C}\mathrm{L}}},$$

(106)

$${\tilde{a}}_{13}=-\frac{R_{\mathrm{A}\mathrm{V}}}{p_{\mathrm{L}\mathrm{C}\mathrm{L}}},$$

(107)

$${\tilde{a}}_{21}=-\frac{R_{\mathrm{A}}{R}_{\mathrm{W}}}{\left(1-A\right)R_{\mathrm{A}\mathrm{V}}}{T}_{\mathrm{L}\mathrm{C}\mathrm{L}},$$

(108)

$${\tilde{a}}_{22}=\frac{L_{\mathrm{L}}}{T_{\mathrm{L}\mathrm{C}\mathrm{L}}},$$

(109)

$${\tilde{a}}_{23}=\frac{R_{\mathrm{W}}{T}_{\mathrm{L}\mathrm{C}\mathrm{L}}}{p_{\mathrm{L}\mathrm{C}\mathrm{L}}}-\frac{1}{{\tilde{\rho }}^{\mathrm{W}}},$$

(110)

$${\tilde{a}}_{31}=-\frac{R_{\mathrm{A}}{R}_{\mathrm{W}}}{\left(1-A\right)R_{\mathrm{A}\mathrm{V}}}{T}_0,$$

(111)

$${\tilde{b}}_1=\frac{{\tilde{c}}_p^{\mathrm{A}\mathrm{V}}}{T_0},$$

(112)

$${\tilde{b}}_3={\tilde{g}}_1^{\mathrm{W}}+\left({\tilde{c}}_p^{\mathrm{V}}-{\tilde{c}}_p^{\mathrm{W}}\right)\ln \frac{T_0}{{eT}_{\mathrm{t}}}-R_{\mathrm{W}}\ln \frac{{xp}_0}{{\psi }_f{p}_{\mathrm{t}}}.$$

(113)

With either the TEOS-10 SIA library functions, or even with the simplified equations of Appendix C, the calculation of the LCL sensitivity, shown in Equation (95), is straightforward, but does likely need to be carried out numerically.

6. Discussion

Climate research relies on observation and causal prediction models. Causality itself cannot be observed, but is the most reliable prediction tool [91,92,93,94]. The comparison of predicted phenomena with future observations serves as a validity criterion for causal models [95,96]. In 2023, the global ocean heat content (OHC) increased by $1.5\times {10}^{22} \mathrm{J}$, a value 25 times as large as the world’s total human energy consumption [9], and even exceeded the atmospheric heat gain by at least 100 times [11]. This currently observed excessive OHC increment [13], rising at a rate of 1.3 W m⁻², remains largely unexplained so far [16,30] and raises questions regarding the correctness, completeness, and internal consistency of climate models.

The dominating input–output processes of the ocean heat balance include solar shortwave and tropospheric longwave irradiation; thermal radiation from the ocean surface; latent heat exchange by phase transitions, including water vapour and ice; and sensible heat exchange with the atmosphere and lithosphere. The input and output processes each amount to a total of about 500 W m⁻², and the observed imbalance arises from a small mutual mismatch by just 0.2% between those. Evidently, to reasonably explain a small difference of 1 W m⁻² between two big numbers of about 500 W m⁻², each of those must be known to within an uncertainty of below 1 W m⁻². A key question is how this small amount may selectively be attributed to the various individual physical contributions involved. Unfortunately, at the ocean–atmosphere interface, current climate models exhibit uncertainties larger than 10 W m⁻² and cannot reliably resolve the requisite heat flux differences.

TEOS-10, the Thermodynamic Equation of Seawater—2010, provides the most accurate and mutually consistent equations currently available for assessing the thermodynamic properties of seawater, humid air, and ice. It was adopted as an international geophysical standard by IOC/UNESCO in 2009 and by IUGG in 2011 to support climate research.

Current numerical climate models tend to underestimate ocean warming [28]. They typically implement Dalton equations to estimate evaporation rates [32] in a historic form, which understands increasing evaporation as a consequence of rising temperature and, in turn, of the vapour pressure of seawater. Quantitatively underpinned by TEOS-10, irreversible thermodynamics, however, suggests that constant marine relative humidity (RH) is the driving force of evaporation, rather than increasing vapour pressure [29,33,39]. A putative tiny increase in global mean RH, even below instrumental resolution, may suffice to explain the rate of ocean warming; this renders latent heat as the preferred candidate responsible for the observed OHC imbalance [25].

A second candidate possibly contributing to the OHC increase is the radiation balance, strongly affected by clouds and greenhouse gases. Global cloudiness has systematically been decreasing in the past decades, enhancing oceanic shortwave irradiation by decreasing shadowed areas. At the same time, the downward longwave irradiation from those clouds is also decreasing, with an estimated result that these two effects almost completely cancel one another [26]. However, global warming does not only reduce cloudiness; the trend of higher sea surface temperature (SST) also elevates marine clouds by an increasing the lifted condensation level (LCL), in turn cooling the cloud base and reducing their downward thermal radiation. In order to include such a cooling effect in the OHC balance investigations, in this paper the sensitivity of LCL and of the related ocean–cloud radiation balance is analysed with the TEOS-10 thermodynamic formalism. Rigorous equations are derived for estimating LCL effects in numerical climate models.

The 2023 global mean SST is about 292 K. At this temperature, the upward thermal radiation flux from the sea surface is 412 W m⁻² (Table 5), while the downward flux from the LCL cloud base is 388 W m⁻². Under LCL cloud cover, the remaining net upward radiation of 24 W m⁻² grows by 0.45 W m⁻² per one degree of further ocean warming. This LCL feedback effect is relevant in comparison to the 2023 OHC gain of 1.3 W m⁻² and may not be ignored in balance investigations. As negative cooling feedback, this LCL effect does not help to explain observed ocean warming; rather, it enables the problematic heating rate to be explained by other contributing effects of the OHC balance, such as a putative uncertain minor climatic change in evaporation.

However, atmospheric RH can be observed to within an uncertainty in the range 1 %rh–5 %rh [34]. An increase in ocean surface RH of 1 %rh is estimated to reduce the latent heat flux of evaporation by about 5 W m⁻² [25,29,38,39], thus warming up the ocean at this rate. In addition, a 1 %rh increase may lower the LCL by about 25 m, in turn intensifying the thermal downward radiation flux from the cloud bottom by more than 1 W m⁻² (see Table 3). LCL height may serve as a remotely measured, sensitive estimate for sea surface relative fugacity or conventional RH. It appears to be a plausible working hypothesis that the excessive ocean warming of 2023 by 1.3 W m⁻² may be caused to a large extent by slightly rising ocean surface RH, not exceeding the uncertainty of observation.