2.1. The standard TEOS10 present version

The aim of this section is to recall how the seawater entropy is calculated in the current standard version of TEOS10, and to show where the reference entropy values for liquid water and sea salts appear.

The specific entropy⁵ 𝜂 called “entropy of seawater” is computed in McDougall, Feistel, et al. (2010, Eq. (2.10.1), p. 20)⁶ as a derivative of the specific seawater Gibbs’ function g:

The specific free enthalpy g = G/m (with m = 1 kg) of a multi-component dilute aqueous electrolyte solution was previously derived according to Feistel and Hagen (1995, Eq. (4.9), p. 266) from the practical osmotic factor considered in Lewis and Randall (1961, Eq. (23-4), p. 334; hereafter LR61) and Falkenhagen and Ebeling (1971, Eq. (116), p. 40), leading to a formulation that can be rewritten as

According to McDougall, Feistel, et al. (2010, Eq. (2.6.1), p. 15) the TEOS10 Gibbs function (2) is split in two and written as the sum of the “pure water” (W) and “salinity” (S) parts

The two Gibbs functions $g^{W}_{\mathrm{Fei03}}(y,z)$ and $g^{S}_{\mathrm{Fei08}}(x,y,z)$ were previously computed by Feistel (2003, Table 10-27, p. 93) and Feistel (2008, Table 17, p. 1666), respectively, as series expansions of xⁱ y^j z^k:

According to (1) applied to (3), and thus to (5) and (6), the entropy coefficients corresponding to (4) are 𝜂_ijk = −(j + 1)g_i,j+1,k/40 and correspond to the relationships shown in Tables 1 and 2, which enable us to compute directly the TEOS10 seawater entropy without using the TEOS10 software⁷ .

The way in which the two parts of g are defined in (3) can be understood by rewriting the first line of (2) using the definition $\mu ^0_{w} = h_{w}-T\, \eta _{w}$ and $\mu ^0_{s} = h_{s}-T\, \eta _{s}$ for the pure-liquid water and sea-salts chemical potentials, with the concentration C = S_A/1000 in kg⋅kg⁻¹ depending on the absolute salinity S_A in g⋅kg⁻¹. Therefore, the first-order constant terms 𝜂_w0 and 𝜂_s0 (independent of T and P) of the entropy 𝜂 impact the Gibbs function via the terms (1 − S_A/1000) 𝜂_w0 + (S_A/1000) 𝜂_s0, which can be rewritten as [𝜂_w0] + [(𝜂_s0 −𝜂_w0)(S_A/1000)]. This explains the natural splitting of this sum in the two TEOS10’s Tables 1 and 2 (for $\eta ^{W}_{\mathrm{Fei03}}$ and $\eta ^{S}_{\mathrm{Fei08}}$) into the two bracketed “pure-water” [𝜂_w0] and “salinity” [(𝜂_s0 −𝜂_w0)(S_A/1000)] parts, respectively. Other linear and non-linear variables terms generated by the three lines of (2) are similarly cast into the “pure-water” and “salinity” parts of g, and thus of 𝜂. In particular, the second line of (2) explains the term x² ln(x) with x² = S_A/40.188617 = 24.88267C.

Therefore, the reference values of the entropies do not impact the pure-water part of the seawater entropy, since [𝜂_w0] is a mere constant with no physical meaning (null differential). Differently, the linear first-order salinity term must have a physical impact on the computation of the seawater entropy. Indeed, its differential (𝜂_s0 −𝜂_w0)(dS_A/1000) is a priori different from zero and depends on the difference of the two reference values. Thus, it depends on both 𝜂_s0 and 𝜂_w0, which must be determined in order to compute the absolute value of the seawater entropy 𝜂 in (4) from Tables 1 and 2.

The first choice retained in pre-TEOS10 formulations was to arbitrarily set 𝜂_w0 = 0 in order to cancel the pure-water entropy at the triple-point conditions: $\eta ^{W}_{\mathrm{Fei03}}(T_{t},p_{t})=0$ (ibid., Eq. (G.2), p. 155). In fact, in the present version of TEOS10 software, the term 𝜂_w0 is set to − g₀₁₀/40 ≈−5.9056/40 ≈ 0.14764 instead, and (𝜂_s0 −𝜂_w0) is set to − g₂₁₀/40 ≈−168.0724/40 ≈ 4.2018 in order to arrive at the cancellation of the whole seawater entropy 𝜂(S_SO,t_SO,p_SO) for the arbitrary standard conditions S_SO = 35.16504 g⋅kg⁻¹, t_SO = 0 °C, and p_SO = 0 dbar (see the explanations p. 155 and Eq. (2.6.6), p. 17 of ibid.).

2.2. A need to update the seawater entropy

The seawater entropy can be written as 𝜂(x,y,z) =𝜂 _w0 + (𝜂_s0 −𝜂_w0)(S_A/1000) + (⋯ ), where (⋯ ) represents the non-linear higher-order terms in x², y or z. Therefore, the first-order terms of the vertical gradient ∂𝜂/∂z and the turbulent flux $\overline{w'\eta '}$ are the product of the measurable quantities ∂S_A/∂z and $\overline{w' S'_{\mathrm{A}}}$ by the difference in reference entropy (𝜂_s0 −𝜂_w0). Then, since the entropy is a thermodynamic state function, the difference of 𝜂 between two points, and thus its gradients and turbulent fluxes, cannot be arbitrary and cannot be either positive, null, or negative, depending on the values and signs of 𝜂_s0 −𝜂_w0. This means that it is not possible to modify or arbitrarily set 𝜂_s0 and/or 𝜂_w0 to this or that value unless modifying the entropy budget (d𝜂/dt = ⋯ ), and thus the second law of thermodynamics.

It may be decided not to bother with entropy calculations, since the reference entropy values have no impact on the other thermodynamic variables (specific volumes, heat capacities, expansion coefficients, sound speeds, osmotic pressure, etc.) and only impact the entropy, Helmholtz free energy, and Gibbs free enthalpy functions. But in this case, we have to go right to the end of the consequences and refrain from calculating values of these three functions, leaving these three quantities undetermined to within a function of salinity.

Differently, one of the aims of TEOS10, as Millero reminded us, is to provide precise values for these three functions. So there is no choice if you want to calculate these three functions: you have to leave no room for arbitrariness and trust the recommendations of general thermodynamics. Here lies the strong motivation for using non-arbitrary values for not only the difference in reference entropies 𝜂_s0 −𝜂_w0, but in fact for both 𝜂_s0 and 𝜂_w0, and thus to use the absolute, third-law values of them defined in thermodynamics.

In addition, although the formulations of ML76 and M83 brought a clear theoretical improvement in taking into account the absolute reference entropies of liquid water and sea salts, the analytical and numerical formulation seems somewhat anomalous (see Section 3.1). As a consequence, Millero’s calculations had to be restated using more recent data and methods.

2.3. The absolute entropy of pure liquid water

I recall in the second line of Table 3 that the value of the absolute entropy of (pure) liquid water is well-known since at least LR61 (Table 12.3, p. 137), who published values corresponding to

I will retain for this study the smaller value

2.4. The absolute entropy of sea-salts at 25 °C

The mean absolute entropy for sea-salt aqueous ions has already been computed by ML76 (Table 32, p. 1071) and M83 (Table X, p. 35) from the individual values S(LR61/M83) previously published by LR61 (Table 25.7, p. 400–401) and recalled in Table 3. The resulting value was

Therefore, the mean sea-salt molar mass previously derived by ML76 (M₂ in Eq. (10) in the Appendix, p. 1074) roughly corresponds to ⟨A⟩ = 62.793/1.99944 ≈ 31.405 g⋅mol⁻¹. More precisely, more relevant definitions were suggested by Millero (1982, Eq. (88), p. 427 and Table IV, p. 428), with the total mole written as n_T = n_B + (1/2)∑_in_i and leading to a sum N_B + (1/2)∑_iN_i = 1.0000005 very close to 1 and to M_T = N_BM_B + (1/2)∑_iN_iM_i ≈ 31.3677 g⋅mol⁻¹ indeed indicating double values for all the ions N_i≠N_B (i.e., except the non-ionic Boric term). However, these more relevant 1982 definitions were not considered in ML76 and will not be retained in M83.

Note that the sea-salt aqueous ions listed in Table 3 are relative to $\overline{S}^{\circ }_{\mathrm{H}^+}=0$, with, however, no impact of a non-zero value for $\overline{S}^{\circ }_{\mathrm{H}^+}$ on the mean value for the sea-salt entropy. Indeed, a non-zero value ΔS₀ for $\overline{S}^{\circ }_{\mathrm{H}^+}$ would lead to a change in the average sea salt entropy corresponding to the weighted sum ∑_j(Z_jΔS₀)X_j = (∑_jZ_jX_j)ΔS₀, which depends on the valence (ionic charges) Z_j and the mole fractions X_j. Fortunately, this sum is zero in the Millero (ML76 and M83) and TEOS10 set-ups thanks to the neutrality property ∑_iZ_jX_j = 0 (see the column for Z_j × X_j in Table D.3 of McDougall, Feistel, et al., 2010, p. 144).

Moreover, Millero’s articles suffer from the fact that they do not take into account the two species OH⁻ and $\mathrm{CO}^{\mathrm{aq.}}_{2}$ considered in the more recent studies and TEOS10.

Due to all these inaccuracies and outdated references in the papers by ML76 and M83, it was desirable to use updated and more recent values for the sea-salts entropies and concentrations, like the TEOS10’s sea-salts values listed in McDougall, Feistel, et al. (ibid.), Table D.3, p. 144. It was also necessary to use updated values for the molar absolute sea-salt entropies $\Delta _r S^{\circ }_m$, like those listed in the “Selected auxiliary data” in the NEA-TDB Tables IV.1 and IV.2 of Grenthe, Fuger, et al. (1992, S(G92), p. 64–83) and retained in Grenthe, Gaona, et al. (2020), including uncertainty intervals.

The corresponding updated mean values for M(TEOS10) and S(G92) shown in Table 3 are

2.5. The absolute entropy of sea salts at 0 °C

In order to compute the mean reference sea-salt entropy at 0 °C by using the relationship $S(273.15)=S(295.15)+\overline{C}^{\circ }_{p} \ln (273.15/298.15)$, it is needed to know the mean specific heats of sea salts at constant pressure $\overline{C}^{\circ }_{p}$, for instance by averaging the individual values for each sea salt recalled in Table 4 from several datasets, where individual values are not available for all species. For this reason, three kinds of averaging are made: first with the two main species 0.5% of Na⁺ and 0.5% of Cl⁻ and the concentrations X(2); then for the four main species Na⁺, Mg²⁺, Cl⁻, and $\mathrm{SO}_4^{2-}$ (for LR61, HH96 and M12) and the concentrations X(4); and then for the whole set of the eight species (for HH96 and M12 only) and the concentrations X(8).

When available, the order of magnitude and the signs of $\overline{C}^{\circ }_p$ are similar for the 6 species Na⁺, K⁺, Cl⁻, $\mathrm{SO}_4^{2-}$, $\mathrm{HCO}_3^{-}$ and Br⁻. The discrepancies are larger for Ca²⁺ and Mg²⁺, with even not the same signs for Mg²⁺. However, the value for Mg²⁺ computed in a recent paper by Caro et al. (2020, Table 7, p. H) is negative (−155) and close to the value −16 − 2 × 71 = −158 published in both HH96 and M12. For this reason the old positive values of LR61 for Mg²⁺ might be inaccurate.

The mean values computed with 0.5% of Na⁺ and 0.5% of Cl⁻ are close to −42 to −46 units. The mean values computed with the 4 major species Na⁺, Mg²⁺, Cl⁻ and $\mathrm{SO}_4^{2-}$ are more negative (by about 10 units) and remain close to each other (about −53 ± 2 units), which is close to the mean values computed with the more complete set of 8 species (about −52 ± 1 units).

Therefore, the mean specific heat at constant pressure for sea salts may be set to

2.6. The TEOS10 absolute seawater entropy

Therefore, it is possible to compute the absolute seawater entropy by considering the absolute values for the reference entropies 𝜂_s0 and 𝜂_w0, to be taken into account in the standard TEOS10 value $\eta (x,y,z) = \eta ^{W}_{\mathrm{Fei03}}(y,z) + \eta ^{S}_{\mathrm{Fei08}}(x,y,z)$ recalled in (4), with the pure-water and salinity parts $\eta ^{W}_{\mathrm{Fei03}}(y,z)$ and $\eta ^{S}_{\mathrm{Fei08}}(x,y,z)$ shown in Tables 1 and 2, respectively.

Both values of the constant terms 𝜂_w0 and − g₀₁₀/40 ≈ 0.14764 in $\eta ^{W}_{\mathrm{Fei03}}(y,z)$ have no physical impact (zero differential, and thus zero time derivatives, gradients, and turbulent fluxes). Differently, the impact of the first-order salinity term in $\eta ^{S}_{\mathrm{Fei08}}(x,y,z)$, recalled in the first line of Table 2, can be summarised by the increment term

The present values for 𝜂_s0 and 𝜂_w0 due to the arbitrary hypothesis 𝜂(S_SO,t_SO,p_SO) = 0 made in TEOS10 can be replaced by the third-law absolute values at 0 °C given in (17) and (9):

The absolute-entropy increment (20) is large in comparison with the present (arbitrary) entropy value recalled in the first line of $\eta ^{S}_{\mathrm{Fei08}}(x,y,z)$ shown in Table 2, with x² = S_A/40.188617 and g₂₁₀ ≈ 168.072408311545 leading to − g₂₁₀ × (x²/40) ≈−104.6 × (S_A/1000). This is about 6% of the value of Δ𝜂_s given by (20) and only 6 times the uncertainty in Δ𝜂_s.

3.1. Comparisons with the Millero and others entropies

It became apparent during the review process that it was necessary to describe in some detail the comparisons between the more recent Feistel’s and TEOS10 formulations and the older papers of ML76 and M83, where for the first time an attempt was published to take into account the absolute values of the entropies of pure water and ocean salts to calculate, in some ways, the absolute seawater entropy. In particular, it was necessary to explain how entropy is calculated in Millero’s papers in a somewhat atypical way, in a so-called “relative” way, which explains the significant differences compared to more recent formulations.

Figure 1 show the salinity parts at constant pressure departure (p = 0) of the seawater entropies 𝜂_sw/sal.(t,S_A) =𝜂 _sw(t,S_A) −𝜂_w(t) defined as the increment from the liquid-water part 𝜂_w(t). For the specific seawater entropy written like in (4), the salinity part 𝜂_sw/sal.(t,S_A) can be computed as follows:

The three Feistel’s and TEOS10 curves labeled FH95, Fei05, and TEOS10 are clearly very close to each other in the Figure 1(a). The older formulation of Feistel (1993, in blue, labelled Fei93) differs from these three almost superimposed curves, with a positive bias (vertical blue thin lines) but with similar negative and decreasing values for S_A > 10 g⋅kg⁻¹.

This positive bias can be explained by the different arbitrary definitions made for the zero of enthalpies and entropies in both Fei93 and Feistel and Hagen (1994, hereafter FH94). This was recalled in FH95, where it was explicitly stated (p. 269) that the arbitrary choice made in Fei93 (and still retained in FH94) was to “define the enthalpy per salt particle to vanish at infinite dilution” (i.e., S = 0 PSU), and to arbitrarily decide from FH95 onwards “to set entropy and enthalpy to zero for the standard ocean statet = 0 °C, p = 0 MPa andS = 35 PSU” because “entropy per salt particle diverges when salinity goes to zero” in Fei93 and FH94.

I show in Table 5 a comparison of the lower-order terms (g[1;0;0] + g[1;1;0] y) x² ln(x) + (g[2;0;0] + g[2;1;0] y) x² for the Gibbs function g_sw, which enter the entropy formula via 𝜂_sw = −∂g_sw/∂t =  − (1/40)∂g_sw/∂y, and thus via − (1/40)[g[1;1;0] x² ln(x) + g[2;1;0] x²]. If the varying coefficient g[2;0;0] only concerns the enthalpy, the other varying coefficient g[2;1;0] impacts the value of the seawater entropy, and Fei93’s value is indeed very different from those for FH95, Fei05, and TEOS10, with a large impact on the seawater entropy via the term − [g[2;1;0]/1.6](S_A/1000). This means from (23) that the changes in −g[2;1;0]/1.6 correspond to those in 𝜂_s0 −𝜂_w0 computed at 0 °C. The difference of about 160.9 + 615.7 = 776.6 between Fei93’s and FH95’s formulations corresponds to a change in 𝜂_s0 −𝜂_w0 of about −485.7 J⋅K⁻¹⋅kg⁻¹, and from (23) to a change in 𝜂_sw/sal.(t,S_A) of about −19.4 J⋅K⁻¹⋅kg⁻¹ at S_A = 40 g⋅kg⁻¹, which is in agreement with Figure 1(a) and with the same vertical shift of about −19.4 J⋅K⁻¹⋅kg⁻¹ (from −12.164 to −31.561) between Fei93’s and FH95’s curves.

The Millero curves (ML76 or M83) plotted in orange on Figure 1(a) show an even greater vertical shift, with values made positive and increasing for all values of S_A. We could try to interpret this increased shift relative to the curves of Fei93, FH95, Fei05, and TEOS10 as a result of the absolute definitions of reference entropies a priori included in ML76 and M83. However, the impact of these absolute definitions is negative when using the formula (20) established above for Δ𝜂_s, as shown by the differences between the black curve for TEOS10 (standard) and the purple curve for TEOS10-abs. It therefore becomes important to be able to explain the reasons for such unusual features for the impacts in ML76 and M83 of the absolute definitions of reference entropies.

It can first be noted that the orange and increasing curve for ML76 or M83 has already been plotted by Sharqawy et al. (2010, Eq. (44), p. 373, Figure 16, p. 374) and Qasem et al. (2023, Figure 5.1, p. 125), who already showed that all other “specific salinity” curves plotted from Chou (1968), Pitzer et al. (1984), Levelt Sengers and Dooley (1992, IAPWS), Sun et al. (2008), Cooper and Dooley (2008, IAPWS), and Nayar et al. (2016) were decreasing with the salinity S_A. This means that the increasing orange curve for ML76 + M83 in Figure 1(a) is truly atypical, as it differs from all the others (Chou, 1968; Pitzer et al., 1984; Levelt Sengers and Dooley, 1992; Feistel, 1993; Feistel, 2003; Feistel, 2005; Sun et al., 2008; Cooper and Dooley, 2008; McDougall, Feistel, et al., 2010; Nayar et al., 2016). Conversely, the decreasing black curve for TEOS10 (standard), which I relied on in my paper, is very similar to all the others except those by Millero.

In fact, it can be shown that the formulations proposed by ML76 (Eq. (136), p. 1072) and M83 (Eq. (147), p. 36) corresponding to the curves in orange (ML76 or M83) do not include the absolute values of the reference entropies. Indeed, they do not correspond to the definitions (21) and (22) leading to the saline parts (23) for the entropy of seawater. In order to prove this, it can be noted that the plot of the curve M83a (in bold dashed red) for the quantity (h − g)/T is superimposed on that of the entropy 𝜂 (ML76 + M83), with h the enthalpy and g = h − T 𝜂 the Gibbs function previously defined both in ML76 (Eq. (67), p. 1050 and Eq. (118), p. 1068) and M83 (Eqs. (54)–(57), p. 15–16 and Eqs. (126)–(129), p. 32). These definitions were derived long before the absolute liquid-water and sea-salt entropies $\overline{\eta }_1^{\circ }$ and $\overline{\eta }_2^{\circ }$ were respectively defined in ML76 (p. 1070–1072) and M83 (p. 34–35)⁹ . This confirms that neither ML76, M83, or M83a can represent the absolute version of the saline part of seawater entropy.

In order to continue the investigation, it is possible to start from the definition of the specific heat of seawater expressed in ML76 (Eqs. (52), p. 1044) and M83 (Eqs. (34)–(37), p. 11) as

The next step is to define a value of the seawater entropy by integrating c_psw(t)/(t + 273.15) up to an arbitrary function of salinity 𝜂₀ only, and independent of the temperature. This leads, with 𝜂₀(0 °C) = 0, to the red dotted curve M83b, with a small negative bias with respect to the FH95, Fei05, and TEOS10 curves (thin vertical red lines).

This small negative bias of the curve M83b is clearly reduced with the red curve M83c plotted with the following alternative choice for 𝜂₀(0 °C), obtained by adding the impact of the Gibbs ideal-solution term g₁x² ln(x)y with the value g₁ = g[1;1;0] ≈ 851.3 typical of FH95 + Fei05 + TEOS10, for which the impact on entropy is − [(851.3/1600)ln(S/40)]S > 0 for 0 < S < 40, and null for S = 0 and S = 40. The same method can be used by integrating the values of c_psw(t)/(t + 273.15) published in the earlier articles of Fofonoff (1962, Eqs. (29)–(30)–(31), p. 13–14)) and Fofonoff (1985, Table-7, p. 3339), with the blue curves Fof62c and Fof85c obtained by adding the same term − (g₁/40)x² ln(x) with g₁ ≈ 851.3 as for M83c.

As a result, all curves for Fof62c, M83c, Fof85c, FH95, Fei05, and TEOS10 almost overlap in Figure 1(a). This means that the ML76 + M83 formulation of c_psw(t,S) recalled in (24) is relevant, and that the atypical behavior of the M83 and M83b curves must correspond to a deeper difference in the functions h, g = h − T 𝜂 and 𝜂 defined in these papers. Furthermore, Feistel (1993, p. 102–103) indicates that several problems exist in the previous Millero’s formulations (forming parts of what he called the “EOS80” system of equations): “…there is a numerical mismatch in the paper of ML76, in transforming heat capacity to apparent molal heat capacity in the form of a generalized Debye–Hückel formula. Because this formula is used later to compute enthalpy and free enthalpy (and thus entropy), the temperature dependence of both is inconsistent with heat capacity. …theory predicts the power series in salinity would begin with$S\, \sqrt{S}$ in the case of relative enthalpy and withS ln(S) in the case of relative free enthalpy (and thus of entropy). The corresponding functions reported are smoothed out at small salinities to begin simply with S. Therefore they cannot be used correctly to compute, say, the chemical potential (and thus the entropy). …while density etc. are defined in absolute terms, thermal properties are only given relative to an infinitely diluted reference state. It is not obvious how these quantities have to be combined correctly if necessary, for example when computing the chemical potential (and thus the entropy). ‘‘Relative” means thatG(S,t,p) is zero at the infinite dilution reference state, where all ion-ion interactions are neglected at temperaturet = 0 °C (T = 273.15 K) and pressurep = 0 bar (P = 1 atm).”

Indeed, it turns out that the ML76 and M83 definitions of “relative” and “relative apparent” parts of any extensive quantity S (like the heat content or enthalpy, the volume, the available enthalpy or Gibbs function, and the entropy noted S) are the same as those published in the previous paper by Millero, Hansen, et al. (1973), and in fact go back to the older books by LR61 and Harned and Owen (1958; 1950; 1943). The “relative” feature means considering only the deviations S_rel. = S − S^° from the molar or specific actual value S = n₁S₁ + n₂S₂ and the “infinite-solution” value $S^{\circ } = n_1 S^{\circ }_1 + n_2 S^{\circ }_2$. They are both computed for the actual n₁ moles of liquid water and n₂ moles of sea salts, and with the “infinite-solution” values $S^{\circ }_1$ and $S^{\circ }_2$ corresponding to n₂→0 and thus S_A→0. It is thus clear that the omission of a function S^° for entropy—which varies with salinity—renders this “relative” view of entropy, defined by S(t,S_A,p) − S^°(t,S_A,p), somewhat meaningless for the study of the entropy function S(t,S_A,p). This is why Chou (1968, Eq. (5)–(9), p. 96) defined the specific entropy of a solution consisting of n₁ mole of water and n₂ moles of salt as $s = n_1 \, \overline{s}^{\circ }_1 + n_2\, \overline{s}^{\circ }_2 + (\cdots )$, therefore including the term $n_1 \, \overline{s}^{\circ }_1 + n_2\, \overline{s}^{\circ }_2$ not included in Millero’s “relative” formulations. Similarly, Pitzer et al. (1984, Eq. (1), p. 3) defined the Gibbs function of the real system as $G = n_1 G^{\circ }_1 + n_2\, \overline{G}^{\circ }_2 + (\cdots )$, again including the terms not included in “relative” Millero’s formulations.

Starting from (21) the “infinite-solution” and “relative” value of the seawater entropy (noted 𝜂) can be written as

It therefore appears that the so-called “relative” entropy of ML76 and M83 is very different from the salinity parts studied in Fei93, FH95, Fei05, and TEOS10, with the liquid-water entropy 𝜂_w(t) in the last term of the salinity-part formula (23) replaced by the infinite-solution sea-salts entropy 𝜂_s(t,S_A = 0) in the “relative-entropy” formula (30), together with the additional first term 𝜂_w(t).

According to ML76 (p. 1072) and M83 (p. 36), “Since future workers may wish to look at entropy surfaces in the oceans,” the “relative-entropy” formula was fitted to the function [h(t,S) − g(t,S)]/(t + 273.15) to give

To do so, the ML76 + M83 specific absolute reference sea-salt entropy is first recalled in (11). Then, the entropy of liquid water 𝜂_w(t) was computed by ML76 (p. 1072) and M83 (p. 36) by first fitting the specific heat capacity c_pw(t) given by (27) in terms of the absolute temperature T (K). This leads to

The next problem is that the entropy of sea salts 𝜂_s(t,S_A), depending on the temperature and salinity, is not calculated explicitly by ML76 or M83, like that 𝜂_w(t) of liquid water in (39) and that relative 𝜂_sw/rel.(t,S) for seawater in (31). However, the absolute value of the reference sea-salt entropy at infinite dilution 𝜂_s0(25 °C) ≈ 1513.1 J⋅K⁻¹⋅kg⁻¹ recalled in (11) was explicitly calculated by ML76 and M83 at 25 °C. We can therefore imagine a first-order formulation to be used in (36) and based on (16), but valid for variable absolute temperatures T and based on the specific heat $\overline{C}_{p{s}} (25~ \mbox{{\textdegree }C}) \approx {-}1656~ \mathrm{J}{\cdot }\mathrm{K}^{-1}{\cdot }\mathrm{kg}^{-1}$ calculated in (15), leading to

This M83d-abs curve corresponds to the expected and hoped-for result, as it validates, in a way, the absolute definition of seawater entropy as I propose and have plotted with TEOS10-abs. The M83d-abs curve corrects the “relative” ML76 + M83 + M83a aspect that was too atypical and was not included in the subsequent IAPWS and TEOS10 projects, in which Millero has nevertheless participated since 1983. The difference between the M83d-abs and TEOS10-abs curves can probably be explained by the remaining approximations and imperfections in the ML76 and M83 formulations, as listed by Feistel (1993, p. 102–103) about the “EOS80” system of equations.

It should be noted that the decrease in entropy with salinity is more pronounced for absolute versions and is not contrary to the fact that entropy must increase during the process of dissolving salts in pure water. As an example, the process of dissolution of crystals of NaCl corresponds to a positive change in seawater entropy for the reaction NaCl_cr⟶(Na⁺)_aq + (Cl⁻)_aq. Indeed, the entropy for a crystal of NaCl is 72.14 J⋅K⁻¹⋅mol⁻¹, whereas the entropy for the separate ions (Na⁺)_aq and (Cl⁻)_aq is 59.00 + 56.36 = 115.36 J⋅K⁻¹⋅mol⁻¹. This leads to an increase of entropy of 115.36 − 72.14 = +43.22 J⋅K⁻¹⋅mol⁻¹, representing the impact on the entropy of the dissolution of the NaCl crystal. In fact, the decrease in absolute entropy with salinity must be interpreted differently: it is as a decrease in the average entropy per unit mass when a mass of water with a higher salinity is added to another with a lower salinity, with the salts already dissolved in both masses of water. For such a process the entropy decreases because the reference entropy of ocean salts (1513.1 J⋅K⁻¹⋅kg⁻¹) is smaller than that of liquid water (3885.65 J⋅K⁻¹⋅kg⁻¹). In this sense, the errors in the tuning leading to the ML76 and M83 “relative” formulation had to be corrected in the present formulation based on the amended version of TEOS10, with, moreover, the use of the absolute values of the reference entropies.

The second plot in Figure 1(b) shows the joint variations with the salinity of the curves 𝜂_sw(t,S_A) for a selection of three temperatures: t = 0 °C, 20 °C, and 40 °C. We can see that the behaviors observed at 40 °C remain unchanged at other temperatures, with a more noticeable decrease in all cases in Figure 1(b–d) for the absolute versions of the entropies.

Finally, the impact of the term x² ln(x) in the TEOS10’s entropy formulation corresponding to the term S_A ln(S_A) introduced by Onsager and Fuoss (1932, Eq. (1.1.2), p. 2689) is to create a vertical tangent at the origin. This is particularly visible in Figure 1(c) and (d), where I show that this behaviour starts to exist only for S_A much smaller than 1 g⋅kg⁻¹. Differently, a general decrease of the absolute entropy exists for the whole oceanic range of salinity from 5 to 40 g⋅kg⁻¹.

3.2. The temperature-salinity (t − S_A) diagram

The same results are obtained with the study of the other temperature-salinity (t − S_A) diagram plotted in the Figure 2. The green lines are for constant values of the TEOS10 potential-density anomaly¹⁰ (𝜎^𝜃 =𝜌 ^𝜃 − 1000, in g⋅m⁻³). Using potential density instead of local density will make it easier to compare the density of points of vertical profile at different depths in the second part of the paper, always using the same family of green lines (surface, p_r = 0).

In this t − S_A diagram, the new absolute seawater entropy red lines are different from the standard TEOS10 black lines: the new absolute values increase with the salinity, whereas the standard TEOS10 values are almost constant (especially for the smaller values of salinity and temperature). Similar differences (not shown) exist between the standard and absolute versions of the seawater entropy plotted with the entropy-temperature diagram published in Feistel, Wright, Kretzschmar, et al. (2010, Figure 4, p. 103).

Three black and red circles are plotted to show the differences in temperature associated with an isentropic process starting at about 6.6 °C and with a change in salinity from 10 to 35 g⋅kg⁻¹: the change in temperature is about 4 °C greater for the absolute version than for the standard TEOS10 version (Δt ≈ +0.3 °C compared with +4.2 °C). This difference is significant, as shown by the applications to real cases described in Part II.

It should be noted here (like in Part II) that the term “isentropic” (or same entropy) includes possible joint variations in pressure, temperature, and salinity, without prejudging any adiabatic (no exchange of heat) or closed (same salinity) aspect of the evolution of ocean parcels. This is analogous to the definition of “isopycnal” processes, where density (or potential density) remains constant regardless of joint variations in pressure, temperature, and salinity, with variations in temperature and salinity along isopycnals described with spiciness.

3.3. General impacts of the third law in physics

It may be important to recall the first validations for H₂, Ar and Hg made close to 0 K and recalled by Nernst (1926, p. 185–186) for the Sackur–Tetrode–Planck absolute entropy constants defined and studied by Planck (1917), Nernst (1918; 1921) and Planck (1921). These authors showed that the theoretical and statistical-quantum absolute value C₀ ≈−1.61 was in agreement with the experimental and calorimetric third-law value C₀ ≈−1.62 ± 0.03. This justified and allowed the computation of the saturation pressures from the third-law absolute entropies if the latent heats are known (see Section 12.1 in the SM).

It is possible to show another modern validation of the theoretical and calorimetric third-law absolute values computed at 0 °C for the water-vapour and ice-Ih entropies. I have first computed (see Section 4.6 and Eq. (126) in the SM) the third-law statistical-quantum water-vapour entropy

As a consequence, a new validation of the third law of thermodynamics can be obtained by computing (see Section 3.5 and p. 19 in the SM) the calorimetric water-vapour entropy at T₀ = 273.15 K and p₀ = 10⁵ Pa via the relationship

Conversely, one can use (41) and (42) to calculate either the latent heat of sublimation L_sub(T₀) or the saturation pressure p_sat(T₀) variable from the other variable and from the difference in third-law entropies

These calculations show the predictive power of the statistical (for water vapour) and calorimetric (for Ice-Ih) third-law independent computations leading to the blue and red disks, respectively. These calculations form a validation of the physical significance of the third-law values of the entropy by forming a little-known link between the numerical values of L_sub(T₀) and p_sat(T₀), which are generally considered to be independent experimental constants. This creates an unexpected third-law implicit impact on the thermodynamic conditions at the surface of the oceans and thus impacts the measurable evaporation processes.

Incidentally, these calculations may also provide a justification for and a physical interpretation of the residual entropy of 189 J⋅K⁻¹⋅kg⁻¹. Indeed, if we subtract this value from the absolute entropy of ice, then according to (44), the quantity 𝛿𝜂(T₀,p₀) increases from about 8023 to about 8212 J⋅K⁻¹⋅kg⁻¹, resulting in, according to (45) and (46), the unrealistically high values of L_sub(T₀) ≈ 2885.7 kJ⋅kg⁻¹ and p_sat(T₀) ≈ 917.6 Pa in comparison with the experimental values of L_sub(T₀) ≈ 2834.5 ± 0.5 kJ⋅kg⁻¹ and p_sat(T₀) ≈ 611.15 ± 0.10 Pa. The bias of 189 × 273.15 ≈ 51.6 kJ⋅kg⁻¹ for L_sub(T₀) and the multiplicative factor of exp(189/461.65) ≈ 1.506 for p_sat(T₀) are indeed unrealistic values. This means that it was not possible to separate the residual value from the absolute calorific entropy of ice-Ih, as suggested by Feistel (2019) who tried “Distinguishing between Clausius, Boltzmann and Pauling entropies of frozen non-equilibrium states.” This also means that this quantity, 189 J⋅K⁻¹⋅kg⁻¹, is likely hidden somewhere in the quantum-statistical calculations carried out by Sackur, Tetrode, Planck, and others for the translational, rotational, and vibrational degrees of freedom of atoms and molecules.

Moreover, I recall in Section 12.2 of the SM that the equilibrium constants of chemical reactions K(T) ultimately depend on the absolute entropies S⁰ of the reactants and products, via the well-known stability relationship ΔG_r = ΔH_r − TΔS⁰ = −RT ln(K). General forms like ln(K) = ΔS⁰/R −ΔH_r/(RT) = A + B/T + C ln(T) are considered both in ozone chemistry (NASA-JPL publication by Burkholder et al., 2020) and seawater chemistry, where it was calculated in Weiss (1974) by adjustments to the observed data. They are also considered, for instance, by Millero (1995) for the study of carbon dioxide concentration in the oceans and more generally described in the DOE (1994) publication. In the general form above, the third-law absolute entropies of reactants and products must impact the constant term A. I show in Section 12.2 of the SM such an impact for the atmospheric reactions Cl + O₂ ⟷ ClO₂ and NO + NO₂ ⟷ N₂O₃ and the seawater reaction $\mathrm{HCO}_3^- \longleftrightarrow \mathrm{H}^+ + \mathrm{CO}_3^{2-}$ (ionization of the bicarbonate anion). The constant terms A can be computed from the theoretical or experimental values of the third-law absolute entropies of molecules, anions, and cations acting as reactants and products. Therefore, the concentrations of ozone in the atmosphere and of sea salts in the oceans both depend on the third-law absolute reference entropies as those considered in the present paper, in Millero’s papers, and in some few others in atmospheric science (like Hauf and Höller, 1987; Marquet, 2011, …), via the photochemistry of the stratosphere and the electrolytic chemistry of the oceans. This is how the absolute values of the entropies influence the observed values of the concentrations of ozone and sea salts, where the concentrations vary with the temperature and pressure. This influences, in return and via the interactions with the radiations, the vertical profiles of the temperature, which is an obvious observable quantity.

Moreover, I show in Section 12.9 of the SM that similar impacts exist for the turbulence acting on the observable temperature-like variables. This, according to Richardson (1919; 1922), should be calculated via a turbulent mixing acting on the absolute entropy variable and not on the temperature variable (see the Section “A need to update the seawater entropy”). I show in Section 12.9 of the SM that the Lewis number (ratio of the thermal and water exchange coefficients) is different from unity for the moist-air absolute entropy variable 𝜃_s (Marquet, 2011), for which K_s > 0. This generates a counter-gradient term for the usually used liquid-water potential temperature 𝜃_l variable (Betts, 1973), for which K_h does not even have a definite sign. This means that the vertical and horizontal structures of temperature must ultimately depend on the absolute definition of the difference in entropy (like s_vr − s_dr for the atmosphere and 𝜂_s0 −𝜂_w0 in the ocean) and the corresponding absolute-entropy exchange coefficients. Accordingly, it is possible to make a link between these atmospheric turbulent phenomena and those in the oceans. Indeed, several of the parameterizations are based on the principle of “double diffusion,” first studied by Stern (1960) for the “Salt-Fountain” and still considered in Ma and Peltier (2024) for the polar oceans. Here, the exchange coefficients are different for the large-scale velocity (momentum), heat (temperature), and salinity, respectively (see, for instance, Canuto et al., 2002).

Moreover, in order to comply with Richardson’s requirements, an equivalent of the atmospheric variable 𝜃_s = T₀ exp[(s − s_d0)/c_pd] (Marquet, 2011; Marquet and Stevens, 2022) to be used in turbulent schemes of oceanic simulations could be defined as the seawater absolute-entropy (potential?) temperature 𝜃_𝜂 derived from the value of the absolute entropy 𝜂_abs given by (18), (19), and (20), but rewritten as 𝜂_abs = c_w ln(𝜃_𝜂/T₀), and thus with 𝜃_𝜂 = T₀ exp(𝜂_abs/c_w)written as

I show in Figures 42 to 44 in Section 12.9 of the SM that 𝜃 and Θ remain very close to each other for several arctic and tropical CTD profiles (up to ± 0.03 °C). The differences between the actual temperature T and both 𝜃 and Θ are also small close to the surface (up to ± 0.03 °C for depths smaller than 250 dbar), where the impact of salinity on 𝜃_𝜂 is the largest and can reach ± 1.5 °C. Note the interesting feature that 𝜃_𝜂 becomes similar to both 𝜃 and Θ for layers deeper than 3000 dbar (up to ± 0.04 °C), where the differences with T can reach −0.6 °C. These numerical evaluations confirm the potential of the 𝜃_𝜂 variable to become a kind of thermal variable on which ocean turbulence is acting and becoming a kind of “absolute seawater potential temperature”. This variable 𝜃_𝜂 is numerically similar to 𝜃 and Θ in deeper layers, but different from them in the mid and surface layers. This simply means that the seawater entropy (and thus 𝜃_𝜂) is very likely the more natural conservative variable.

In addition to the examples in Section 3.1 concerning the unexpected links between latent heat and the saturation pressure of water vapour, the examples described in this section show that there are observable impacts in geophysics of the absolute reference entropy values, thereby providing a physical meaning for calculations and studies of absolute seawater entropy. However, the same doubts concerning the physical relevance of absolute entropy will continue to exist in atmospheric and oceanic sciences, much like doubts can still exist for the ultimate realism of the principles of many general principles of physics, such as the existence of the Michelson–Morley–Lorentz–Einstein’s limiting velocity (c) of physical phenomena in a vacuum or the existence of the Planck’s quantum of action (h). Indeed, there is no demonstration in the logical, mathematical, or physical senses of these facts, which are simply observed, never disproved, and therefore elevated to the status of general principles. The same applies to the third law and the principle of unattainability of the absolute zero of temperature, subject to amending Planck (1911; 1917) formulation by adding possible residual entropies at 0 K in the calorimetric computations for some species like H₂O, as logically included in 𝜂_abs and 𝜃_𝜂 for seawater.

Accordingly, other contributions than TEOS10 have admitted the possibility of at least considering as an option the absolute entropies for liquid water and dry air (ML76, M83, Feistel and Hagen, 1995; Lemmon et al., 2000; Feistel and Wagner, 2006). Nonetheless, it is often still considered that the TEOS10’s choice is consistent with the first conclusion of the 5th International Conference on the Properties of Steam in London (5th-ICPS et al., 1956). Indeed, this conference recommended (p. 1/30–1/32 and 3/34) that the specific internal energy and the specific entropy of the liquid water should be set equal to zero at the triple point temperature of 0.01 °C instead of 0.00 °C, without affecting any measurable thermodynamic properties of the climate system. However, I show in Section 12.5 and Figure 32 of the SM that this first 1956 recommendation (p. 3/34) was only valid for the case of a pure liquid-water steam system. It was not valid for a mixture involving other species, like in the moist-air atmosphere and the seawater. Moreover, it was decided at the same time (ibid., p. 3/35–3/37) that the absolute values for the entropy of liquid water (computed from hypotheses made at 0 K) should also be mentioned in all subsequent studies. This is precisely what I have achieved in the present study by defining both 𝜂_abs and 𝜃_𝜂 for seawater, including the impacts of absolute and non-arbitrary definitions for both pure liquid-water and sea-salt reference entropies.