# Thermodynamics of Evaporation from the Ocean Surface

^{1}

^{2}

^{*}

## Abstract

**:**

^{−2}per 1 %rh. Within a maximum error of only 0.04 %rh, sea-surface RF may be approximated in terms of dew-point or frost-point temperatures using a simple formula.

## 1. Introduction

^{−2}for the net global and the net atmospheric, oceanic, and land energy balances. … These imbalances are partly due to imperfect closure of the energy cycle in the fluid components.” To demonstrate the significance of that uncertainty, a minor heating flux of just 0.005 W m

^{−2}is sufficient to raise the atmospheric temperature at an observed rate of 2 °C per century [3,4,5,6,7]. “The oceans have a heat capacity about 1000 times greater than the atmosphere and land surface” ([8] p. 420). If also expressed per global surface unit area, even an increase in the large oceanic heat content by as much as 0.5 W m

^{−2}([8] p. 427), [9,10] would remain well below the model uncertainty range. However, “the climate of the Earth is ultimately determined by the temperatures of the oceans” ([8] p. 420).

^{−1}, or, equivalently, 104 W m

^{−2}of latent heat, occurs in the trade-wind belt between 15 and 25° N. An evaporation of 1540 mm yr

^{−1}, or 122 W m

^{−2}, occurs between 10 and 20° S, being particularly intense over the subtropical Indian Ocean.

^{−2}[14,15] with respect to energy fluxes across the air–sea interface is inadequate because “the by far largest part of heat is transferred to the air in the form of latent heat during subsequent condensation along with cloud formation. The heat budget over the sea is mainly controlled by the latent heat released to the air … The heat released to the air in latent form is larger by a multiple than the [sensible] heat transferred immediately to the air” (Original German text as quoted from Albrecht [16]: “Der weitaus größte Teil der Wärme wird der Luft in Form von latenter Wärme und nachfolgender Kondensation bei der Wolkenbildung zugeführt. … Der Wärmehaushalt der Luft über dem Meere wird … hauptsächlich durch die bei der Verdunstung an die Luft abgegebene latente Wärme bestimmt … Die an die Luft in latenter Form abgegebene Wärme ist dabei um ein Vielfaches größer als die durch den Austausch unmittelbar”). A typical evaporation of 1000 mm yr

^{−1}supplies the atmosphere (and cools down the ocean) with the latent water vapour heat at a rate of 79 W m

^{−2}per ocean surface area.

^{−2}, or 6% [17]. Reducing the systematic observational errors and the random uncertainties of ocean–atmosphere fluxes to less than 5 and 15 W m

^{−2}[15], respectively, is an ambitious target: “We need an accuracy of approximately ±15 W m

^{−2}” ([18] p. 59).

## 2. Dalton Equation: Climatological Bias?

_{A}to clearly distinguish it from several alternative, mostly also unitless, oceanographic salinity scales [19]). In Section 4, expressions will theoretically be derived for the computation of the latent heat $L$ in the TEOS-10 framework. In Section 3, the relation between specific humidity and the theoretical thermodynamic driving force of evaporation will be analysed in more detail.

^{−1}, along with global warming, at climatological time scales, and this may substantially strengthen the predicted global evaporation. Consequently, assuming unchanged wind conditions, the use of Equations (4) and (5) in such a numerical climate model will simulate an amplification of oceanic evaporation as a direct consequence of globally rising temperatures. An intensified hydrological cycle is discussed in the climatological literature [7,39,48,58,59,60,61,62,63,64], but the implied putative cooling effect of stronger evaporation from the global ocean would be in contrast to measured data of ocean warming [9,10]. According to recent model comparison studies, “most CMIP6 [»Coupled Model Intercomparison Project Phase 6« [65]] models fail to provide as much heat into the ocean as observed” ([66] p. E1968). The simple mathematical example provided by Equations (4) and (5) may demonstrate how easily minor changes or inappropriate approximations in the parameterisation of marine evaporation could result in systemically biased climate trend projections.

^{−2}, that is, 7.2% higher than the observations. Remarkable regional overestimations have been reported for latitudes from 10 to 20 °N, related to the excessive seasonal variations in LHF in the MME for the southwest branch of the Kuroshio Current, including the Kuroshio intrusion to the north of the South China Sea. Quite noticeable root–mean–square LHF errors appeared in coastal regions, such as at the west coast of Africa, the northwest coast of the Arabian Sea, the seas to the northeast of Japan, the equatorial eastern Pacific, and northeastern North America. Simulated long-term trends were analysed to be too weak compared to the observed ones, which were hypothesised to originate from uncertainties in both the thermodynamic driving force and the aerodynamic pre-factor, including wind velocity, bulk transfer coefficient, and the mass density of humid air. The most striking biases, however, were recognised for the specific humidity, q, and the wind velocity, u; the simulated rate of the increase in q was a factor of six larger than the observed one, while the simulated rate of increase in u was only half the observed one. Both biases in these trends tend to underestimate the LHF. The reasons for the poor MME performance in the simulations of q and u, again, remain hidden; Zhang et al. [67] concluded that “accordingly, additional exploration is required to enhance our knowledge of the biases in q and u and to find ways to improve this problem in the models as much as possible.” This includes enhanced attention to the spread in observational data, especially surface humidity, one of the most important factors influencing the biases in LHF.

## 3. Evaporation: Thermodynamic Driving Force and Approximations

## 4. Climatological Hydrosphere: Thermodynamics of »Sea Air«

#### 4.1. Equation of State of »Sea Air«

**q**of the $N={N}^{{\mathrm{H}}_{2}\mathrm{O}}+{N}^{\mathrm{A}}+{N}^{\mathrm{S}}$ particles of water, air and salt, respectively, inside the volume $V$, evaluating the potential energy $U\left(\mathit{q}\right)$ of each particular spatial arrangement of the molecules. These configurations include homogeneous states as well as states consisting of separate phases, of which the thermodynamically stable states provide the dominating contributions to the integral (41). For this reason, from the potential function ${F}^{\mathrm{SA}}$, the latent heats of these phase transitions can be mathematically derived by varying the temperature $T$ and analysing the related mass transfer from one phase to another.

#### 4.2. Latent Heats of Phase Transitions

## 5. Relative Fugacity Approximation

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Surface-Pressure Gibbs Functions of Liquid Water and Seawater

$\mathit{j}$ | ${\mathit{g}}_{\mathit{j}0}$ of Equation (A3) | ${\mathit{g}}_{\mathit{j}1}$ of Equation (A4) |
---|---|---|

0 | 0.101 342 743 139 672 × 10^{3} | 0.100 015 695 367 145 × 10^{6} |

1 | 0.590 578 348 518 236 × 10 | –0.270 983 805 184 062 × 10^{3} |

2 | –0.123 577 859 330 390 × 10^{5} | 0.145 503 645 404 680 × 10^{4} |

3 | 0.736 741 204 151 612 × 10^{3} | –0.672 507 783 145 070 × 10^{3} |

4 | –0.148 185 936 433 658 × 10^{3} | 0.397 968 445 406 972 × 10^{3} |

5 | 0.580 259 125 842 571 × 10^{2} | –0.194 618 310 617 595 × 10^{3} |

6 | –0.189 843 846 514 172 × 10^{2} | 0.635 113 936 641 785 × 10^{2} |

7 | 0.305 081 646 487 967 × 10 | –0.963 108 119 393 062 × 10 |

_{P}, is computed from measured electrical conductivity, as defined by the 1978 Practical Salinity Scale, PSS-78 [115,116]. Absolute Salinity, S

_{A}, is computed from measured density, as defined by the 2008 Reference-Composition Salinity Scale [19]. Common oceanographic instruments return S

_{P}values, which are stored in marine databases. In contrast, the equations of TEOS-10 are expressed in terms of S

_{A}, which also considers non-dissociated solutes such as silicate.

_{P}and S

_{A}. For this reason, a single salinity variable S will be used here, which may use either practical salinity unit (psu) or mass fraction units (${\mathrm{g}\mathrm{kg}}^{-1}$, ${\mathrm{kg}\mathrm{kg}}^{-1}$ or unitless). The conversion formula is [19]

$\mathit{i}$ | $\mathit{j}$ | ${\mathit{g}}_{\mathit{i}\mathit{j}0}$ of Equation (A8) | ${\mathit{g}}_{\mathit{i}\mathit{j}1}$ of Equation (A9) |
---|---|---|---|

1 | 0 | 0.581 281 456 626 732 × 10^{4} | - |

2 | 0 | 0.141 627 648 484 197 × 10^{4} | −0.331 049 154 044 839 × 10^{4} |

3 | 0 | −0.243 214 662 381 794 × 10^{4} | 0.199 459 603 073 901 × 10^{3} |

4 | 0 | 0.202 580 115 603 697 × 10^{4} | −0.547 919 133 532 887 × 10^{2} |

5 | 0 | −0.109 166 841 042 967 × 10^{4} | 0.360 284 195 611 086 × 10^{2} |

6 | 0 | 0.374 601 237 877 840 × 10^{3} | - |

7 | 0 | −0.485 891 069 025 409 × 10^{2} | - |

1 | 1 | 0.851 226 734 946 706 × 10^{3} | - |

2 | 1 | 0.168 072 408 311 545 × 10^{3} | 0.729 116 529 735 046 × 10^{3} |

3 | 1 | −0.493 407 510 141 682 × 10^{3} | −0.175 292 041 186 547 × 10^{3} |

4 | 1 | 0.543 835 333 000 098 × 10^{3} | −0.226 683 558 512 829 × 10^{2} |

5 | 1 | −0.196 028 306 689 776 × 10^{3} | - |

6 | 1 | 0.367 571 622 995 805 × 10^{2} | - |

2 | 2 | 0.880 031 352 997 204 × 10^{3} | −0.860 764 303 783 977 × 10^{3} |

3 | 2 | −0.430 664 675 978 042 × 10^{2} | 0.383 058 066 002 476 × 10^{3} |

4 | 2 | −0.685 572 509 204 491 × 10^{2} | - |

2 | 3 | −0.225 267 649 263 401 × 10^{3} | 0.694 244 814 133 268 × 10^{3} |

3 | 3 | −0.100 227 370 861 875 × 10^{2} | −0.460 319 931 801 257 × 10^{3} |

4 | 3 | 0.493 667 694 856 254 × 10^{2} | - |

2 | 4 | 0.914 260 447 751 259 × 10^{2} | −0.297 728 741 987 187 × 10^{3} |

3 | 4 | 0.875 600 661 808 945 | 0.234 565 187 611 355 × 10^{3} |

4 | 4 | −0.171 397 577 419 788 × 10^{2} | - |

2 | 5 | −0.216 603 240 875 311 × 10^{2} | - |

4 | 5 | 0.249 697 009 569 508 × 10 | - |

2 | 6 | 0.213 016 970 847 183 × 10 | - |

## Appendix B. Virial Gibbs Function of Humid Air

^{10}= 8.314 472 J mol

^{−1}K

^{−1}. The molar masses of dry air and pure water, respectively, are M

_{A}= 0.028 965 46 kg mol

^{−1}and M

_{W}= 0.018 015 268 kg mol

^{−1}. The new functions that were used are defined below.

**Table A3.**Coefficients of Equations (A16) and (A17). Coefficients ${n}_{4}^{\mathrm{A}}$ and ${n}_{3}^{\mathrm{A}}$ differ from those of Lemmon et al. [118], as they are adjusted to the geophysical reference state at 0 °C and 101,325 Pa.

$\mathit{i}$ | ${\mathit{n}}_{\mathit{i}}^{{A}}$ of Equation (A16) | ${\mathit{n}}_{\mathit{i}}^{{V}}$ of Equation (A17) |
---|---|---|

1 | 0.605 7194 × 10^{−7} | −0.832 044 648 374 969 × 10 |

2 | −0.210 274 769 × 10^{−4} | 0.668 321 052 759 323 × 10 |

3 | −0.158 860 716 × 10^{−3} | 0.300 632 × 10 |

4 | 0.974 502 517 439 48 × 10 | 0.124 36 × 10^{−1} |

5 | 0.100 986 147 428 912 × 10^{2} | 0.973 15 |

6 | −0.195 363 42 × 10^{−3} | 0.127 95 × 10 |

7 | 0.249 088 8032 × 10 | 0.969 56 |

8 | 0.791 309 509 | 0.248 73 |

9 | 0.212 236 768 | 0.128 728 967 × 10 |

10 | −0.197 938 904 | 0.353 734 222 × 10 |

11 | 0.253 6365 × 10^{2} | 0.774 073 708 × 10 |

12 | 0.169 0741 × 10^{2} | 0.924 437 796 × 10 |

13 | 0.873 1279 × 10^{2} | 0.275 075 105 × 10^{2} |

i | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{b}}_{\mathit{i}}$ |
---|---|---|

1 | 0.118 160 747 229 | 0 |

2 | 0.713 116 392 079 | 0.33 |

3 | −0.161 824 192 067 × 10 | 1.01 |

4 | −0.101 365 037 912 | 1.6 |

5 | −0.146 629 609 713 | 3.6 |

6 | 0.148 287 891 978 × 10^{−1} | 3.5 |

i | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{b}}_{\mathit{i}}$ | ${\mathit{c}}_{\mathit{i}}$ | ${\mathit{d}}_{\mathit{i}}$ |
---|---|---|---|---|

1 | 0.125 335 479 355 23 × 10^{−1} | −0.5 | - | - |

2 | 0.789 576 347 228 28 × 10 | 0.875 | - | - |

3 | −0.878 032 033 035 61 × 10 | 1 | - | - |

4 | −0.668 565 723 079 65 | 4 | - | - |

5 | 0.204 338 109 509 65 | 6 | - | - |

6 | −0.662 126 050 396 87 × 10^{−4} | 12 | - | - |

7 | −0.107 936 009 089 32 | 7 | - | - |

8 | −0.148 746 408 567 24 | 0.85 | 28 | 700 |

9 | 0.318 061 108 784 44 | 0.95 | 32 | 800 |

i | ${\mathit{a}}_{\mathit{i}}$ | ${\mathit{b}}_{\mathit{i}}$ |
---|---|---|

1 | 0.665 687 × 10^{2} | −0.237 |

2 | −0.238 834 × 10^{3} | −1.048 |

3 | −0.176 755 × 10^{3} | −3.183 |

## Appendix C. Surface-Pressure Gibbs Function of Ice Ih

**Table A7.**Coefficients of the TEOS-10 Gibbs function of ice Ih in surface pressure approximation; Equations (A37) and (A38).

Coefficient | Real Part | Imaginary Part | Unit |
---|---|---|---|

${g}_{00}$ | −0.632 020 233 449 497 × 10^{6} | - | ${\mathrm{J}\mathrm{kg}}^{-1}$ |

${g}_{01}$ | 0.655 022 213 658 955 | - | ${\mathrm{J}\mathrm{kg}}^{-1}$ |

${\eta}_{0}$ | −0.332 733 756 492 168 × 10^{4} | - | ${\mathrm{J}\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$ |

${t}_{1}$ | 0.368 017 112 855 051 × 10^{−1} | 0.510 878 114 959 572 × 10^{−1} | |

${t}_{2}$ | 0.337 315 741 065 416 | 0.335 449 415 919 309 | |

${r}_{10}$ | 0.447 050 716 285 388 × 10^{2} | 0.656 876 847 463 481 × 10^{2} | ${\mathrm{J}\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$ |

${r}_{20}$ | −0.725 974 574 329 220 × 10^{2} | −0.781 008 427 112 870 × 10^{2} | ${\mathrm{J}\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$ |

${r}_{21}$ | −0.557 107 698 030 123 × 10^{−4} | 0.464 578 634 580 806 × 10^{−4} | ${\mathrm{J}\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$ |

## Appendix D. Nomenclature

Symbol | Quantity | SI Unit | Equation |

A | Mass fraction of dry air in humid air, $A=1-q$ | kg kg^{−1} | (10) |

${a}_{i}$ | Coefficients of virial coefficients | 1 | (A18) |

$B$ | Mixture virial coefficient | m^{3} mol^{−1} | (A23) |

${B}^{\mathrm{AA}}$ | 2nd virial coefficient of air–air interaction | m^{3} mol^{−1} | (A13) |

${B}^{\mathrm{AW}}$ | 2nd virial coefficient of air–water interaction | m^{3} mol^{−1} | (A15) |

${B}^{\mathrm{WW}}$ | 2nd virial coefficient of water–water interaction | m^{3} mol^{−1} | (A14) |

${b}_{i}$ | Coefficients of virial coefficients | 1 | (A18) |

${C}_{L}$ | Latent heat transfer coefficient [18], ${C}_{L}=1.2\times {10}^{-3}$ | 1 | (35) |

${C}_{p}^{\mathrm{SA}}$ | Isobaric heat capacity of sea air | J K^{−1} | (57) |

${c}_{p}^{\mathrm{Ih}}$ | Specific isobaric heat capacity of ice Ih | J kg^{−1} K^{−1} | (57) |

${c}_{p}^{\mathrm{AV}}$ | Specific isobaric heat capacity of humid air | J kg^{−1} K^{−1} | (57) |

${c}_{p}^{\mathrm{SW}}$ | Specific isobaric heat capacity of seawater | J kg^{−1} K^{−1} | (57) |

${c}_{P}^{\mathrm{W}}$ | Specific isobaric heat capacity of liquid water | J kg^{−1} K^{−1} | (A5) |

${c}_{i}$ | Coefficients of virial coefficients | 1 | (A19) |

D | Dalton coefficient | m s^{−1} | (1) |

${D}_{e}$ | Vapour-pressure-based transfer coefficient | s m^{−1} | (29) |

${D}_{f}$ | Fugacity-based transfer coefficient | kg m^{−2} s^{−1} | (19) |

${D}_{q}$ | Humidity-based transfer coefficient | kg m^{−2} s^{−1} | (34) |

${d}_{i}$ | Coefficients of virial coefficients | 1 | (A19) |

$e$ | Vapour pressure | Pa | (27) |

${e}^{\mathrm{sat}}$ | Saturation vapour pressure | Pa | (27) |

${F}^{\mathrm{SA}}$ | Helmholtz energy of sea air | J | (40) |

${F}^{\mathrm{SA},\mathrm{id}}$ | Ideal gas Helmholtz energy of sea air | J | (40) |

${f}^{\mathrm{A}}$ | Specific Helmholtz energy of dry air | J kg^{−1} | (A12) |

${f}_{0}^{\mathrm{A}}$ | Thermal ideal-gas Helmholtz energy of dry air | J kg^{−1} | (A13) |

${f}^{\mathrm{AV}}$ | Specific Helmholtz energy of humid air | J kg^{−1} | (42) |

${f}_{0}^{\mathrm{AV}}$ | Thermal ideal-gas Helmholtz energy of humid air | J kg^{−1} | (A21) |

${f}^{\mathrm{Ih}}$ | Specific Helmholtz energy of ice Ih | J kg^{−1} | (42) |

${f}^{\mathrm{mix}}$ | Specific Helmholtz energy of air–water interaction | J kg^{−1} | (A12) |

${f}^{\mathrm{SW}}$ | Specific Helmholtz energy of seawater | J kg^{−1} | (42) |

${f}^{\mathrm{V}}$ | Specific Helmholtz energy of water vapour | J kg^{−1} | (A12) |

${f}_{0}^{\mathrm{V}}$ | Thermal ideal gas Helmholtz energy of water vapour | J kg^{−1} | (A14) |

${f}_{\mathrm{V}}^{\mathrm{AV}}$ | Fugacity of water vapour in humid air | Pa | (13) |

${f}_{\mathrm{V}}^{\mathrm{AV},\mathrm{sat}}$ | Saturation fugacity of water vapour in humid air | Pa | (64) |

${f}_{\mathrm{W}}^{\mathrm{SW}}$ | Fugacity of water in seawater | Pa | (14) |

${G}^{\mathrm{AV}}$ | Gibbs energy of humid air | J | (43) |

${G}^{\mathrm{Ih}}$ | Gibbs energy of ice Ih | J | (43) |

${G}^{\mathrm{SA}}$ | Gibbs energy of sea air | J | (43) |

${G}^{\mathrm{SW}}$ | Gibbs energy of seawater | J | (43) |

${g}^{\mathrm{AV}}$ | Specific Gibbs energy of humid air | J kg^{−1} | (45) |

${g}^{\mathrm{Ih}}$ | Specific Gibbs energy of ice Ih | J kg^{−1} | (45) |

${g}_{jk}$ | Coefficients of the Gibbs function of liquid water | 1 | (A2) |

${g}_{jk}$ | Coefficients of the Gibbs function of ice Ih | J kg^{−1} | (A37) |

${g}_{ijk}$ | Coefficients of the saline Gibbs function | 1 | (A7) |

${g}^{\mathrm{S}}$ | Saline part of the specific Gibbs energy of seawater | J kg^{−1} | (A1) |

${g}^{\mathrm{SW}}$ | Specific Gibbs energy of seawater | J kg^{−1} | (45) |

${g}^{\mathrm{W}}$ | Specific Gibbs energy of liquid water | J kg^{−1} | (A1) |

${g}^{*}$ | Scaling specific Gibbs energy, ${g}^{*}=1{\mathrm{J}\mathrm{kg}}^{-1}$ | J kg^{−1} | (A2) |

${H}^{\mathrm{AV}}$ | Enthalpy of humid air | J | (53) |

${H}^{\mathrm{Ih}}$ | Enthalpy of ice Ih | J | (53) |

${H}^{\mathrm{SA}}$ | Enthalpy of sea air | J | (53) |

${H}^{\mathrm{SW}}$ | Enthalpy of seawater | J | (53) |

${h}^{\mathrm{A}}$ | Specific enthalpy of dry air | J kg^{−1} | (61) |

${h}^{\mathrm{AV}}$ | Specific enthalpy of humid air | J kg^{−1} | (55) |

${h}^{\mathrm{AV},\mathrm{id}}$ | Specific enthalpy of ideal gas humid air | J kg^{−1} | (61) |

${h}^{\mathrm{Ih}}$ | Specific enthalpy of ice Ih | J kg^{−1} | (56) |

${h}^{\mathrm{SW}}$ | Specific enthalpy of seawater | J kg^{−1} | (54) |

${h}^{\mathrm{V}}$ | Specific enthalpy of water vapour | J kg^{−1} | (61) |

${h}^{\mathrm{W}}$ | Specific enthalpy of liquid water | J kg^{−1} | (70) |

${J}_{k}$ | Irreversible Onsager flux | various | (6) |

${J}_{\mathrm{W}}$ | Evaporation mass-flux density | kg m^{−2} s^{−1} | (1) |

${k}_{\mathrm{B}}$ | Boltzmann constant | J K^{−1} | (40) |

L | Specific evaporation enthalpy | J kg^{−1} | (1) |

${L}^{\mathrm{evap}}$ | Specific evaporation enthalpy of liquid water | J kg^{−1} | (57) |

${L}^{\mathrm{subl}}$ | Specific sublimation enthalpy of ice Ih | J kg^{−1} | (57) |

${M}_{\mathrm{A}}$ | Molar mass of dry air, M_{A} = 0.028 965 46 kg mol^{−1} | kg mol^{−1} | (15) |

${M}_{\mathrm{AW}}$ | Molar mass of humid air | kg mol^{−1} | (A22) |

${M}_{\mathrm{S}}$ | Molar mass of dissolved sea salt, M_{S} = 0.031 403 822 kg mol^{−1} | kg mol^{−1} | (16) |

${M}_{\mathrm{W}}$ | Molar mass of water, M_{W} = 0.018 015 268 kg mol^{−1} | kg mol^{−1} | (13) |

${m}^{\mathrm{A}}$ | Mass of dry air | kg | (40) |

${m}^{\mathrm{Ih}}$ | Mass of ice Ih | kg | (42) |

${m}^{{\mathrm{H}}_{2}\mathrm{O}}$ | Mass of water | kg | (40) |

${m}^{\mathrm{S}}$ | Mass of sea salt | kg | (40) |

${m}^{\mathrm{V}}$ | Mass of water vapour | kg | (42) |

${m}^{\mathrm{W}}$ | Mass of liquid water | kg | (42) |

$N$ | Number of particles | 1 | (41) |

${N}^{\mathrm{A}}$ | Number of dry-air particles | 1 | (41) |

${N}^{\mathrm{S}}$ | Number of sea salt particles | 1 | (41) |

${N}^{{\mathrm{H}}_{2}\mathrm{O}}$ | Number of water molecules | 1 | (41) |

${n}_{i}^{\mathrm{A}}$ | Coefficients oft he Helmholtz function of dry air | 1 | (A16) |

${n}_{i}^{\mathrm{V}}$ | Coefficients oft he Helmholtz function of water vapour | 1 | (A17) |

p | Pressure | Pa | (13) |

${p}_{\mathrm{SO}}$ | Standard ocean surface pressure, ${p}_{\mathrm{SO}}=\mathrm{101,325}\mathrm{Pa}$ | Pa | (65) |

${p}_{\mathrm{t}}$ | Triple-point pressure of water, ${p}_{\mathrm{t}}=611.657\mathrm{Pa}$ | Pa | (A38) |

${p}^{*}$ | Scaling pressure, ${p}^{*}={10}^{8}\mathrm{Pa}$ | Pa | (A2) |

${Q}_{L}$ | Latent heat flux density | W m^{−2} | (1) |

${Q}^{\mathrm{SA}}$ | Statistical configuration integral of sea air | 1 | (40) |

${Q}_{V}$ | Liquid water evaporation velocity | m s^{−1} | (1) |

q | Specific (or absolute) humidity | kg kg^{−1} | (1) |

$\mathit{q}$ | Vector of positions and orientations | (41) | |

${q}^{\mathrm{sat}}$ | Saturation specific humidity | kg kg^{−1} | (32) |

q_{eq} | Equilibrium specific humidity | kg kg^{−1} | (1) |

$R$ | Molar gas constant,$R=8.31446261815324{\mathrm{J}\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$ | J (mol K)^{−1} | (13) |

R^{10} | 2010 molar gas constant, R^{10} = 8.314 472 J mol^{−1} K^{−1} | J (mol K)^{−1} | (A13) |

${R}^{95}$ | Molar gas constant of [109], ${R}^{95}={R}_{\mathrm{W}}^{95}\times {M}_{\mathrm{W}}$ | J (mol K)^{−1} | (A14) |

$\mathrm{Re}$ | Real part of a complex number | (A37) | |

${r}_{jk}$ | Coefficients of the specific Gibbs energy of ice Ih | J (kg K)^{−1} | (A37) |

${R}^{\mathrm{L}}$ | Molar gas constant of [117], ${R}^{\mathrm{L}}=8.31451{\mathrm{J}\mathrm{mol}}^{-1}{\mathrm{K}}^{-1}$ | J (mol K)^{−1} | (A13) |

${R}_{\mathrm{W}}$ | Specific gas constant of water, ${R}_{\mathrm{W}}=R/{M}_{\mathrm{W}}$ | J (kg K)^{−1} | (13) |

${R}_{\mathrm{W}}^{95}$ | Specific gas constant of water [109], ${R}_{\mathrm{W}}^{95}=461.51805{\mathrm{J}\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$ | J (kg K)^{−1} | (A14) |

S | Specific (or absolute) salinity | kg kg^{−1} | (1) |

${S}^{*}$ | Scaling salinity, ${S}^{*}=40\mathrm{psu}=0.040188617$ | kg kg^{−1} | (A7) |

T | (Absolute) temperature, ITS-90 | K | (1) |

t | Celsius temperature, $t=T-273.15\mathrm{K}$ | °C | |

${t}_{\mathrm{dp}}$ | Dew-point Celsius temperature | °C | (71) |

${T}_{\mathrm{AV}}$ | Temperature of humid air | K | (10) |

${T}_{\mathrm{dp}}$ | Dew-point temperature | K | (66) |

${T}_{\mathrm{fp}}$ | Frost-point temperature | K | (67) |

${t}_{k}$ | Coefficients of the specific Gibbs energy of ice Ih | 1 | (A37) |

${T}_{\mathrm{mp}}$ | Melting-point temperature | K | (68) |

${T}_{\mathrm{ref}}$ | Reference temperature | K | |

${T}_{\mathrm{SO}}$ | Standard ocean temperature, ${T}_{\mathrm{SO}}=273.15\mathrm{K}$ | K | (A2) |

${T}_{\mathrm{SW}}$ | Temperature of seawater | K | (10) |

${T}_{\mathrm{t}}$ | Triple-point temperature of water, ${T}_{\mathrm{t}}=273.16\mathrm{K}$ | K | (A37) |

${T}^{*}$ | Scaling temperature, ${T}^{*}=40\mathrm{K}$ | K | (A2) |

$U$ | N-particle interaction potential | J | |

u | Wind speed | m s^{−1} | (1) |

V | Volume | m^{3} | (40) |

${v}^{\mathrm{W}}$ | Specific volume of liquid water | m^{3} kg^{−1} | (A4) |

${X}_{k}$ | Onsager force | various | (6) |

${X}_{\mathrm{Q}}$ | Onsager force of sensible heat flux | (K m)^{−1} | (9) |

${X}_{\mathrm{W}}$ | Onsager force of water diffusion flux | J (kg K m)^{−1} | (8) |

${x}_{\mathrm{V}}$ | Mole fraction of water vapour in humid air | mol mol^{−1} | (13) |

${x}_{\mathrm{V}}^{\mathrm{sat}}$ | Saturation mole fraction of water vapour in humid air | mol mol^{−1} | (21) |

${x}_{\mathrm{W}}$ | Mole fraction of water in seawater | mol mol^{−1} | (14) |

z | Vertical coordinate | m | (8) |

${\epsilon}_{\mathrm{V}}$ | Numerically negligible historical deviation | 1 | (A30) |

${\epsilon}_{f}$ | Numerically negligible historical deviation | 1 | (A31) |

${\eta}_{0}$ | TEOS-10 specific residual entropy of ice Ih | J (kg K)^{−1} | (A37) |

${\eta}^{\mathrm{SW}}$ | Specific entropy of seawater | J (kg K)^{−1} | (A10) |

${\eta}^{\mathrm{W}}$ | Specific entropy of liquid water | J (kg K)^{−1} | (A5) |

$\lambda $ | 2-box water–air interface thickness | m | (10) |

${\mu}_{\mathrm{W}}$ | Specific chemical potential of water | J kg^{−1} | (8) |

${\mu}_{\mathrm{W}}^{\mathrm{id}}$ | Ideal gas part of the chemical potential of water | J kg^{−1} | (13) |

${\mu}_{\mathrm{W}}^{\mathrm{AV}}$ | Specific chemical potential of water in humid air | J kg^{−1} | (10) |

${\mu}^{\mathrm{Ih}}$ | Specific chemical potential of ice Ih | J kg^{−1} | (50) |

${\mu}_{\mathrm{W}}^{\mathrm{SW}}$ | Specific chemical potential of water in seawater | J kg^{−1} | (10) |

${\mu}_{\mathrm{W}}^{\mathrm{W}}$ | Specific chemical potential of liquid water | J kg^{−1} | (A5) |

$\pi $ | Number Pi, $\pi =2\mathrm{arcsin}\left(1\right)=3.14\dots $ | 1 | (41) |

ρ | Mass density of humid air | kg m^{−3} | (A12) |

${\rho}^{\mathrm{A}}$ | Partial mass density of dry air | kg m^{−3} | (A13) |

${\rho}^{\mathrm{AV}}$ | Mass density of humid air | kg m^{−3} | (1) |

${\rho}^{\mathrm{Ih}}$ | Mass density of ice Ih | kg m^{−3} | (42) |

${\rho}^{\mathrm{SW}}$ | Mass density of seawater | kg m^{−3} | (42) |

${\rho}^{\mathrm{V}}$ | Partial mass density of water vapour | kg m^{−3} | (A14) |

${\rho}^{\mathrm{W}}$ | Mass density of liquid water | kg m^{−3} | (1) |

${\rho}^{*}$ | Scaling mass density, ${\rho}^{*}=1{\mathrm{kg}\mathrm{m}}^{-3}$ | kg m^{−3} | (A13) |

${\rho}_{\mathrm{A}}^{*}$ | Scaling air density, ${\rho}_{\mathrm{A}}^{*}={\rho}_{\mathrm{AA}}^{*}\times {M}_{\mathrm{A}}$ | kg m^{−3} | (A16) |

${\rho}_{\mathrm{AA}}^{*}$ | Scaling molar density, ${\rho}_{\mathrm{AA}}^{*}=\mathrm{10,447.7}{\mathrm{mol}\mathrm{m}}^{-3}$ | kg m^{−3} | (A18) |

${\rho}_{\mathrm{AW}}^{*}$ | Scaling molar density, ${\rho}_{\mathrm{AW}}^{*}={10}^{6}{\mathrm{mol}\mathrm{m}}^{-3}$ | mol m^{−3} | (A20) |

${\rho}_{\mathrm{WW}}^{*}$ | Scaling molar density, ${\rho}_{\mathrm{WW}}^{*}=\left(322{\mathrm{kg}\mathrm{m}}^{-3}\right)/{M}_{\mathrm{W}}$ | mol m^{−3} | (A19) |

$\sigma $ | Entropy production density | W K^{−1} m^{−3} | (7) |

$\tau $ | Temperature variable, $\tau =\left(132.6312\mathrm{K}\right)/T$ | 1 | (A16),(A18) |

$\tau $ | Temperature variable, $\tau =\left(647.096\mathrm{K}\right)/T$ | 1 | (A17),(A19) |

$\tau $ | Temperature variable, $\tau =T/\left(100\mathrm{K}\right)$ | 1 | (A20) |

${\psi}_{f}$ | Relative fugacity | Pa Pa^{−1} | (22) |

${\psi}_{q}$ | Relative humidity (climatological definition) | kg kg^{−1} | (3) |

${\psi}_{x}$ | Relative humidity (metrological definition) | Pa Pa^{−1} | |

${\mathsf{\Omega}}_{kl}$ | Onsager coefficient | various | (6) |

${\mathsf{\Omega}}_{\mathrm{WW}}$ | Onsager coefficient of irreversible evaporation | J K (m s)^{−1} | (12) |

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**MDPI and ACS Style**

Feistel, R.; Hellmuth, O.
Thermodynamics of Evaporation from the Ocean Surface. *Atmosphere* **2023**, *14*, 560.
https://doi.org/10.3390/atmos14030560

**AMA Style**

Feistel R, Hellmuth O.
Thermodynamics of Evaporation from the Ocean Surface. *Atmosphere*. 2023; 14(3):560.
https://doi.org/10.3390/atmos14030560

**Chicago/Turabian Style**

Feistel, Rainer, and Olaf Hellmuth.
2023. "Thermodynamics of Evaporation from the Ocean Surface" *Atmosphere* 14, no. 3: 560.
https://doi.org/10.3390/atmos14030560