In a substitution method, a sample (seawater) with an unknown density and a similar, well-known reference (water) are measured (ideally, at the same time) using the same measurement device (densimeter). Deviations in the measurement results caused, for example, by a drift or a temperature deviation can be corrected, as they cause similar effects on seawater and on water. As a result, the measured densities of seawater and water have similar deviations from their true value. The difference equation for calculation of the corrected density from the measurements obtained from seawater and water is ρsubsSW-ρrefH2O=ρmesSW-ρmesH2O, where ρmesSW and ρmesH2O are the measured seawater and water densities, and ρsubsSW and ρrefH2O are the corrected seawater (substitution) density and the well-known water reference density. If the absolute seawater density is determined from a substitution measurement (by calculating ρmesSW-ρmesH2O+ρrefH2O), the result includes the uncertainty in the water reference density. By contrast, if the seawater density relative to water is determined (by calculating ρmesSW-ρmesH2O), the reference uncertainty is not included.

The water reference density was calculated using the equation of state developed by Wagner and Pruß (2002). A description of the calculation is given in Appendix A. The reference density uncertainty is 1 g m-3 for atmospheric pressure, 10 g m-3 for pressures up to 10 MPa, and 30 g m-3 for up to 100 MPa. The uncertainty in a corrected seawater density resulting from a substitution measurement mainly depends on the uncertainty in the water reference density, but also on the similarity of seawater and water in terms of their relevant thermophysical properties, as well as on the stability and linear characteristics of the densimeter used. It should be noted that the linearity is regularly checked in measurements on reference liquids with densities between 700 kg m-3 and 1600 kg m-3; furthermore, the linearity was particularly validated in the seawater density range by means of comparison measurements against a hydrostatic weighing apparatus for both the densimeters used for atmospheric and high pressure; details have been given by Schmidt et al. (2016).

Vibrating-tube densimeters (VTDs) were used for density measurements performed using the substitution method. The core of such a densimeter is a U-shaped tube that is fixed in place on both ends. This tube is filled with the liquid to be measured and then forced to oscillate; the resulting oscillation period is a measure of the liquid density. Since the vibrating tube can be easily accessed from the outside, liquids can be filled in and changed quickly. This feature, together with short-term stability, is necessary for the application of the substitution method. Since the seawater sample and water reference cannot be measured simultaneously, stability is important for the duration of the alternating measurements. Under these conditions, the drift of the densimeter can be quantified using the deviations from the reference density (of water) to correct the sample density (of seawater).

The set-up used for the density measurements at atmospheric pressure is outlined in Fig. 1a. It comprises a fully automated filling system, a VTD, and a peristaltic pump. The filling system was created specifically for small filling volumes to allow more repetitions in the substitution measurements from a limited sample amount. To this end, a sequence of humid air bubbles is used to rinse the previous liquid out of the measuring cell. The bubbles of humid air are inserted into the sample filling tubes using the V2 and V3 valves in addition to the V1 valve to switch between the seawater sample and the water reference. The VTD used for the measurements is a DMA 5000M (Anton Paar GmbH, Graz, Austria). The peristaltic pump used to move the liquids is installed behind the VTD to avoid any interaction of the peristaltic tube material with the seawater or the water before the measurement.

The set-up used for density measurements at high pressures is illustrated in Fig. 1b. It uses an equal filling system to fill the water and seawater like the set-up used for atmospheric pressure. In addition to the filling system, the VTD, and the peristaltic pump, a pressurization part is installed between the VTD and the peristaltic pump. In this part, wherein the pressure is generated and measured, is a syringe pump filled with oil to prevent corrosion of the pressure sensors. The oil transmits the pressure generated in the syringe pump directly to the water without using a pressure transmitter. A long tube is installed between both parts (VTD and pressurization part) to avoid diffusion of oil into the measurement cell of the VTD. Two pressure sensors (P1 up to 14 MPa and P2 up to 70 MPa) are used to increase the accuracy of the pressure measurement. The offsets of these sensors at atmospheric pressure are corrected by the values gained with the atmospheric pressure manometer before each measurement. The VTD used for the measurements at high pressures is a DMA HP (Anton Paar GmbH, Graz, Austria). Details that have been given by Schmidt et al. (2016) are complemented by the Supplement to this article.

The substitution measurements at atmospheric pressure were performed at a constant temperature. The water and seawater were filled and measured in alternation. The water densities measured were thus compared with the reference density; the deviations found were used to correct the seawater measurements.

The procedure for high pressures is similar to that used for atmospheric pressure; however, the liquid is not replaced during a high-pressure run at a constant temperature. Instead, the liquid is replaced after decreasing the pressure back to atmospheric conditions.

The seawater density was measured in the temperature range of 5 to 35 ∘C. The densities were corrected to integer temperatures in ∘C and either to 101 325 Pa or to integer pressures in bar, if the substitution density was determined for high pressures. The measured absolute seawater densities have uncertainties of 2 g m-3 for atmospheric pressure, 14 g m-3 for pressures up to 10 MPa, and 34 g m-3 for pressures up to 65 MPa. If stated relative to water, the seawater densities for high pressures have significantly smaller uncertainties, as they do not include the water reference uncertainty. The measured relative densities have uncertainties of 6 up to 14 g m-3 mainly depending on salinity.

Since the salinity uncertainty, which is 0.5 g m-3 in terms of density, is significant compared to the density measurement uncertainty for atmospheric pressure, it has to be considered in the development of the density–salinity relation. This had already been done at this point by adding the salinity uncertainty to the density measurement uncertainty.

To determine salinity by means of conductivity, the PSS-78 relations were developed based on five data sets

All publications cited here were also reprinted together (JPOTS, 1981b).

Both the salinities of the samples used to develop the conductivity–salinity relation PSS-78 and the salinities of the samples used to develop the density–salinity relation were thus determined by weighing measurements. If a relation between density and conductivity is set using both relations, then both (relation) uncertainties have to be taken into account. It should be noted that the density–conductivity relation is only valid if standard seawater is consistent in its composition. Conversely, this relation can therefore be used to check the standard seawater composition.

The uncertainty in a salinity determined by means of conductivity measurement that is supported by PSS-78 is (in a best-case scenario) 0.0022 using a laboratory salinometer and 0.0034 using a conductivity–temperature–depth device (Le Menn, 2011). These uncertainties are 2 and 3 g m-3 in terms of density. The accuracy of the seawater densities for atmospheric pressure fulfils these criteria, both in absolute terms and relative to the water reference. In the high-pressure range, it is currently not possible to achieve a comparable accuracy in absolute density using the substitution method and a water reference, as here, the uncertainty in the water reference density is too high. This can be circumvented by stating the seawater density relative to water.

Since the aim of developing the density–salinity relation was to determine the salinity by measuring density with higher accuracy than by measuring conductivity, a relative density–salinity relation was developed instead of an absolute density–salinity relation. The accuracy of a salinity that is determined by measuring density at high pressure and subsequent calculation using the (relative) density–salinity relation is thus comparable to the salinity accuracy of conductivity–temperature–depth devices.

Water shows a variation in its isotopic composition. The natural variation comprises the H–D relation and the 16O–17O–18O relation. The isotopic abundance of a water sample is usually stated relative to that of the reference material VSMOW, whose isotopic composition is based on a mixture of ocean waters (IAEA, 2006). The D isotopic abundance (as well as the 18O abundance) is thus expressed as the ratio of the amount-of-substance ratio of D and H in the sample to the respective ratio in VSMOW, δD: δD=DHSampleDHSampleDHVSMOWDHVSMOW-1. The 17O abundance is usually not monitored, as it is very small compared to the 18O abundance. In Earth's deep ocean layers, the isotopic composition varies by up to 4 ‰ in D and 0.3 ‰ in 18O, whereas in the surface ocean layers, these variations are up to 35 and 3 ‰ (Ferronsky and Polyakov, 2012) due to precipitation. A variation in the isotopic abundance affects the density directly: The corresponding variations are 0.1 g m-3 for the deep ocean and 1.3 g m-3 for the surface ocean if calculated using Eq. (A2) given in Appendix A. Isotopic composition variations in the water of the pedosphere are even more significant.

The D and 18O isotopic abundances δD and δ18 in the natural seawater that was used as the raw material for the diluted seawater preparation at the area of sampling were measured in 1972 and made available by Ostlund et al. (1987). The water which is deionized and used for dilution of the natural seawater (N. Higgs, personal communication, 2011) is tap water from Havant, UK, where the supplier of the IAPSO SSW is located. Darling et al. (2003) analysed the isotopic composition of freshwaters in the British Isles. They used isotope measurement data collected from around 1978 to 2003, including in the region from which the water for dilution was taken. The relevant values and uncertainties given by Ostlund et al. and Darling et al. are given in Table 2. The equations used to calculate the isotopic abundances of the diluted seawater after mixing standard seawater with water can be derived from the amount-of-substance balance of the isotope considered. For D and 18O, the equations derived are δDDSW=δDH2O+1⋅mH2O+δDSSW+1⋅mSSW⋅1-SAmH2O+mSSW⋅1-SA and δ18DSW=δ18H2O+1⋅mH2O+δ18SSW+1⋅mSSW⋅1-SAmH2O+mSSW⋅1-SA, where “DSW” refers to diluted seawater (after mixing) and SA=SR=35.1650435.1650435⋅SP⋅(gkg-1)35⋅SP⋅(gkg-1) is the absolute salinity of standard seawater, which is assumed to be equal to the reference salinity of IAPSO SSW according to the recommendation of Millero et al. (2008). The calculated isotopic abundance values and corresponding uncertainties of the seawater samples used for the density measurements are given in Table 2. For calculation of the uncertainty, only the isotopic abundances of the water and seawater were taken into account, as the other contributions are insignificant (for example, the salinity of the natural seawater, which is diluted, may differ by multiple g kg-1 without affecting δDDSW and δ18DSW significantly).

The density difference due to the isotopic abundance change during preparation, ΔρprepSW, is estimated using Eq. (A2), where ΔδD=δDDSW-δDSSW and Δδ18=δ18DSW-δ18SSW are inserted for this purpose. Following this procedure, the isotopic abundance effect on density is assumed to be the same for seawater as for water at Tρmax= 3.98 ∘C and p0= 101 325 Pa and is calculated relative to the isotopic composition of IAPSO SSW. ΔρprepSW is approximated by ΔρprepSWS,Tρmax,p0gm-3=-0.0700⋅S+2.4577, where ΔρprepSWS,T,p≈ΔρprepSWS,Tρmax,p0, and S, T, and p are the salinity, temperature, and (absolute) pressure, respectively. The uncertainty in ΔρprepSW is estimated to be 0.3 g m-3; uncertainties in the isotopic abundances are insignificant.

ΔρprepSW is illustrated in Fig. 2. The more water is used for dilution, the more the density decreases, as the water is depleted in heavy isotopes compared to seawater. The density difference caused by the difference between the isotopic composition of VSMOW and that of IAPSO SSW (which is given in Table 2), ΔρisoSW, is 0.3 g m-3.

The seawater used for the measurements was stored in 230 mL borosilicate glass vessels (Bacon et al., 2007) from the time of preparation at OSIL to the time of measurement. During this time, glass material that dissolved into the seawater has significantly altered the chemical composition, and thus the density.

Usually, seawater samples used in highly accurate density measurements in laboratories are air-saturated, as any degassing procedure may change the salt composition. For water, the effect of air solubility on density has been measured directly, e.g. by Bignell (1983), by comparing the densities of saturated and desaturated water.

For our density measurements, the seawater samples were taken directly from the vessels delivered by OSIL as shown in Fig. 1a. The vessels were stored in our laboratory at a temperature of approximately 20 ∘C, at which the seawater equilibrated with the air inside the (closed) vessels. Since the seawater was also pumped into the VTD at this temperature, the air saturation was 100 % at 20 ∘C. After filling the VTD, the seawater temperature was altered to the measurement temperature. During this time, the saturation changed to undersaturation at temperatures lower than 20 ∘C and to oversaturation at temperatures higher than 20 ∘C, as there was no contact to air during the time of temperature equilibration, which is approximately 15 min. This temperature-dependent aeration is significant compared to the density measurement uncertainty. For consistency of the air saturation, the measured densities have to be corrected to a saturation of either 0 or 100 %. Because the density corrections to 100 % are significantly smaller than those to 0 %, and because any degassing procedure is problematic, the density values were corrected to 100 % air saturation. Following this procedure, the density–salinity relation was developed with the least loss in accuracy.

The density correction is estimated taking into account the fact that the amount of air molecules remains constant while the liquid temperature changes from 20 ∘C to measurement temperature before density measurement. To quantify the density change of seawater by saturation with nitrogen, oxygen and argon in an atmosphere with 100 % humidity, a complex calculation similar to that for water of Harvey et al. (2005) was carried out. For this calculation, the partial molar volumes of nitrogen, oxygen and argon in water were assumed to be equal in seawater. Salinity-dependent solubility data of nitrogen and argon were taken from Hamme and Emmerson (2004) and of oxygen from Garcia and Gordon (1992).

Carbon dioxide exists in three different significant forms in seawater, i.e. as free aqueous molecule, CO2, as bicarbonate ion, HCO3-, and as carbonate ion, CO32-, the sum of all being called dissolved inorganic carbon (DIC). The DIC concentration as well as that of each species depends on salinity, temperature as well as CO2 partial pressure, if the seawater is in contact with the atmosphere (as described by Henry's law for CO2). If the air is humid, the CO2 partial pressure changes with temperature due to the water vapour pressure, whereby CO2 is released from (for temperature increase) or absorbed into seawater (for temperature decrease). Since this affects the HCO3- and CO32- concentration, the absolute salinity and thus the density are affected by temperature changes. For standard seawater with a salinity of 35 exposed to air with 100 % humidity, DIC is ≈ 2190 µmol kg-1 for 0 ∘C, ≈ 2050 µmol kg-1 for 20 ∘C, and ≈ 1870 µmol kg-1 for 40 ∘C 

DIC was calculated using the CO2calc software developed by Robbins et al. (2010) with the carbonate constants given by Millero (2010) that are also valid for low salinities, the acidity constant of hydrogen sulfate given by Dickson (1990), the boron to chlorinity ratio given by Lee et al. (2010) and the total pH scale. In the calculations, the total alkalinity was 2300 µmol kg-1 (IOC et al., 2010), which is constant for CO2 exchange (Zeebe and Wolf-Gladrow, 2001), and the CO2 molar fraction was 400 ppm.

The complex calculation showed that the different gas solubilities in water and seawater are negligible in terms of density, as the deviation between the calculated density change of seawater and that of water (of Harvey et al., 2005) is around 0.1 g m-3. Furthermore, it was found that it is sufficient to consider only the nitrogen solubility to calculate the density correction that is approximated by ΔρaerSW=1-nN2100%,20∘CnN2100%,T⋅ΔρaH2O100%,T, where nN2 (100 %, 20 ∘C) and nN2 (100 %, T) are the dissolved nitrogen amounts of a substance at 100 % saturation at 20 ∘C and at measurement temperature as well as ΔρaH2O (100 %, T) being the corresponding density effect, whose calculation is described in Appendix A.

At measurement temperatures higher than 20 ∘C, the seawater is oversaturated during density measurement, as it was saturated at 20 ∘C before filling. It is assumed that the nucleation of microbubbles due to the oversaturation takes significantly longer than the time of temperature stabilization and density measurement, which is always less than 30 min. The density effect caused by oversaturation is therefore assumed to be proportionally equal to that up to saturation. The calculated density correction and the corresponding estimated uncertainty, which is 0.4 g m-3, are illustrated in Fig. 4. The density correction is significant at temperatures less than 15 ∘C compared to the measurement uncertainty of 2 g m-3, as the gas solubility is significantly higher at low temperatures.

Based on a measured substitution density, ρsubsSW, that has been corrected to the uniform isotopic water and chemical compositions and to 100 % air saturation, a seawater density, ρSW, was calculated by ρSW=ρsubsSW-ΔρprepSW+ΔρisoSW-ΔρstorSW+ΔρaerSW+ΔρtarSW, where ΔρtarSW is a density correction to integer salinities introduced for practicability.

The density of air-saturated seawater is modelled based on degassed water, whose density is given by ρ0H2O. Salt that has a relative composition similar to that dissolved in standard seawater is added to the degassed water. The salt content is given implicitly by the salinity. The density of the degassed water changes after the salt is added by Δρ0SW. In addition, air with a defined composition is absorbed, as a result of which the density changes by ΔρaSW. The density of air-saturated seawater, ρSW, is thus given by ρSW=ρ0H2O+Δρ0SW+ΔρaSW, where ρ0H2O is the density of degassed water, Δρ0SW is the density change due to dissolved salt, and ΔρaSW is the density change due to absorbed air. The density change due to dissolved salt and absorbed air may be summarized by ΔρSW and may also be called relative density of air-saturated seawater, as the seawater density was measured relative to water in the substitution measurements.

If the salt is added at the atmospheric pressure p0, the water density changes by Δρ0SWp0. If the salt is added at the pressure p≠p0, the water density changes by Δρ0SWp. If the difference between the two changes is ΔΔρ0SWp-p0, then the density change due to dissolved salt at any pressure is given by Δρ0SW=Δρ0SWp0+ΔΔρ0SWp-p0, where Δρ0SWp0 is the density change due to dissolved salt at the atmospheric pressure p0 and ΔΔρ0SWp-p0 is the difference between the density changes at the pressure p and at the atmospheric pressure p0.

The solubility of gases in liquids is described well at infinite dilution and low pressure by means of Henry's law, according to which the number of absorbed gas molecules is proportional to the gas pressure above the liquid. However, since there is no reservoir for additional gas at high pressure, only the air absorption at the gas pressure p0 is taken into account for modelling. In addition, it is assumed that the absorbed air is incompressible. In the model, the density change due to absorbed air is therefore not treated as a function of pressure, i.e. ΔρaSW≠fp.

The solubility of air in seawater depends on salinity (Hamme and Emmerson, 2004; Garcia and Gordon, 1992). The comparison of the N2, O2, and Ar solubilities in water and seawater showed that the resulting density change in both liquids is approximately equal. In the model, the density change due to absorbed air is therefore not a function of salinity, i.e. ΔρaSW=ΔρaH2O≠fS.

The values of the seawater density for atmospheric pressure, ρSW, which were obtained from the measurements and corrected to the uniform conditions, were broken down according to Eq. (10) into the corresponding values of the water density, ρ0H2O, and the values yielded by the density change due to dissolved salt and absorbed air (or relative density of air-saturated seawater), ΔρSW. For this purpose, the water density was calculated using the equation of state developed by Wagner and Pruß (2002), by means of which the water reference density for the substitution measurements was calculated as well. Therefore, the uncertainty in the relative density is up to 20 % lower than that in the absolute density.

The values of the relative density of air-saturated seawater ΔρSW were broken down into the resulting values of the density change due to dissolved salt, Δρ0SWp0, and the corresponding values of density change due to absorbed air, ΔρaSW. For this purpose, the values of ΔρaSW were calculated using the equation of Harvey et al. (2005), which is valid for the absorption of air into water at p0= 101 325 Pa, but were adopted for the absorption of air into seawater according to the physical model described above: ΔρaSWTgm-3=0.103-2.371×105⋅T∘C+75-2.5+1.82×10-7⋅T∘C+753, where the model air composition is 78.1 % N2, 20.9 % O2, 0.9 % Ar, and 0.4 ‰ CO2. This equation is also given in Appendix A, but is repeated here for clarity. Free aqueous CO2 contributes less than 0.2 g m-3 to ΔρaSW and is therefore negligible.

The values of the relative density of degassed seawater, Δρ0SWp0, were used to fit the coefficients ai,j of the following empirical equation: Δρ0p0=Δρ0o⋅σ⋅∑i=05∑j=05-iai,j⋅τi⋅σj, where Δρ0o=30kgm-3, τ=TTToTo is the reduced temperature with T being the temperature in K and To=288.15K, and σ=SSSoSo is the reduced salinity with S being the salinity and So=35. The values of Δρ0o, To, and So (as well as ΔΔρ0o and πo below) were chosen for practical handling of the fit coefficient values and do not have a physical meaning.

The linear fit coefficients ai,j were determined by uncertainty-weighted least squares fitting within the Monte Carlo based approach described in Appendix B. The fit coefficients were initially averaged from up to n=1500 runs, where no longer significant effects on calculated values or uncertainties thereof were found. Finally, the coefficients were averaged from n=15000 runs to be certain. The fitting yielded the values of ai,j given in Table 4, which were reduced to the significant number of digits.

The residuals of the fit using the coefficients given in Table 4 are illustrated in Fig. 5, where they are compared with the density–salinity relation uncertainty, U, whose determination is described below. The residual standard deviation is 1.1 g m-3. No systematic deviation of the residuals depending on salinity or temperature was found.

If the density of air-saturated seawater is calculated using the density–salinity relation, the (fitted) relative seawater density plus the (artificially inserted) density change due to absorbed air is used, i.e. ΔρSW=Δρ0SW+ΔρaSW. However, ΔρaSW has practically no statistical influence on the fitting of Δρ0SW, and therefore no statistical influence on ΔρSW. Consequently, if ΔρSW is calculated, its uncertainty is UΔρ0SW, i.e. that of the degassed seawater density, whereas, if Δρ0SW is calculated, its uncertainty is [U(Δρ0SW)2+U(ΔρaSW)2]1/2, i.e. that of the degassed seawater density and that of the density change due to absorbed air. The uncertainty in Δρ0SWp0 was determined and verified using the approach described in Appendix B. The calculated uncertainty is at least (0.7 g m-3) at a salinity of 15 and at 25 ∘C, and increases as expected at higher salinities, as well as at lower and higher temperatures (up to 1.2 g m-3). The subsequent uncertainty verification yielded four inconsistent densities whose residuals were higher than their corresponding uncertainties. The uncertainty was therefore increased to 2 g m-3 in the entire measurement region of salinities up to 35 and temperatures from 5 to 35 ∘C.

Since the density–salinity relation may be used for calculations in a wider region, e.g. salinities up to 40 and temperatures from 0 to 40 ∘C, we also estimated the uncertainty in Δρ0SWp0 for this region in the absence of measurement data. The density uncertainty in the wider (extrapolation) region was also calculated using the approach described in Appendix B, whereby the possible variation of the fit polynomial outside the measured salinity and temperature region is taken into account. The uncertainties resulting from this calculation are shown in Fig. 6a together with the uncertainty of the measurement region. For practicability, the highest uncertainty in a particular region was assigned. The uncertainties in the extrapolation region are at least twice as much as in the measurement region.

To calculate salinity using relative density and temperature values by means of the density–salinity relation, the uncertainty in salinity was also determined in the measurement and extrapolation region. The salinity uncertainty was calculated by multiplying the density uncertainty by the partial derivative of salinity by density, i.e. US=UΔρSW⋅∂S∂S∂ρ∂ρ. The uncertainties resulting from this calculation are shown in Fig. 6b. A salinity determined by means of a calculation using the relation has an uncertainty of 3×10-3. If measurement values are used for calculation, their uncertainty has to be considered.

Since the mathematical formulation of the density-salinity relation is empirical and does not contain any theoretical boundary conditions for infinite dilution, as for example implemented in TEOS-10, the question arises whether the relation correctly predicts the density for very low salinities. Additionally, no uncertainty verification in the extrapolation region is possible using the fitting data set. Therefore, additional substitution density measurements were conducted: The density of diluted standard seawater with salinity 2 was measured at some temperatures and the density of some samples of the seawater used for determination of the density–salinity relation was measured at 1 ∘C. The seawater with salinity 2 was prepared like the seawater with salinities from 5 to 30. Unfortunately, the precision in the salinity-2-calibration was lower, so that the uncertainty in salinity is 0.0028 corresponding to an uncertainty in density of 2.2 g m-3. The density results were corrected to the uniform isotopic water and the chemical salt compositions as well as air saturation as described in Sect. 3. The density deviations of the corrected results from the predicted values of the density–salinity relation are shown in Fig. 7. In both cases, the deviations are well within the uncertainty in the density–salinity relation. For the measurements of seawater with salinity 2, even if the uncertainty in salinity is treated as an offset to all deviations, the deviation is within its uncertainty. No inconsistencies are caused by the non-compliance with theoretical boundary conditions for very low salinities and atmospheric pressure.

The values of the seawater density for high pressures, ρSW, were broken down according to Eq. (10) into corresponding values of the water density, ρ0H2O, and the values yielded by the density change due to dissolved salt and absorbed air (or relative density of air-saturated seawater), ΔρSW. Since the water density was calculated analogously to the atmospheric pressure densities, the uncertainty in the relative density is up to 50 % lower than that in the absolute density for pressures up to 10 MPa and up to 80 % lower for up to 65 MPa.

The values of the relative density of air-saturated seawater, ΔρSW, were broken down into the values yielded by the density change due to dissolved salt, Δρ0SW, and the corresponding values of the density change due to absorbed air, ΔρaSW. For this purpose, the values of ΔρaSW were calculated analogously to the atmospheric pressure densities using Eq. (12).

The values of the density change due to dissolved salt, Δρ0SW, were further broken down according to Eq. (11) into the values of the density change due to dissolved salt at the atmospheric pressure p0= 101 325 Pa, Δρ0SWp0, and the difference between the density change at (high) pressure p and that at the pressure p0, ΔΔρ0SWp-p0. The relative density values for atmospheric pressure that had been used to fit the coefficients of Eq. (13), were used for this purpose.

The resulting values of the density difference ΔΔρ0SWp-p0 were used to fit the coefficients bi,j,k of the following empirical equation: ΔΔρ0SWp-p0=ΔΔρ0o⋅σ⋅π⋅∑i=04∑j=04-i∑k=04-i-jbi,j,k⋅τi⋅σj⋅πk, where ΔΔρ0o=2kgm-3, π=pppopo-1pppopo-1πoπo with p being the pressure in MPa, po=p0= 0.101325 MPa, and πo=1000. Due to the formulation of the dimensionless pressure π, ΔΔρ0 is exactly zero at p0, thereby ensuring the high accuracy of the density at atmospheric pressure Δρ0p0.

The linear fit coefficients bi,j,k were determined analogously to the fit coefficients ai,j using the approach described in Appendix B. The fitting yielded the values of bi,j,k given in Table 5, which were reduced to the significant number of digits. The residuals of the fit using the coefficients given in Table 5 are illustrated in Fig. 8. The uncertainty of the measured relative densities underlying the fit range from 6 up to 14 g m-3 for salinities from 5 up to 35 and were estimated conservatively. The fit standard deviation of 2.3 g m-3, which is 5 g m-3 for a probability of 95.45 %, and the fact that no systematic deviation of the residuals depending on salinity, temperature, or pressure is found suggest that the uncertainty in the measured relative densities may have been overestimated, i.e. their accuracy may have been underestimated. The density uncertainty in the measurement region was determined using the approach described in Appendix B and yielded an uncertainty of 6 g m-3, thereby supporting the suggestion. The uncertainty of 6 g m-3 was therefore adopted for the measurement region. One residual for 41.5 MPa, one for 52 MPa and two for 65 MPa, of 49, respectively, exceed this uncertainty significantly.

The data set for fitting ΔΔρ0SWp-p0 comprises pressures up to 65 MPa. Since the density–salinity relation may be used for calculations over a wider range, e.g. pressures up to 100 MPa, the uncertainty in this range was estimated in the absence of measurement data. Summarized results of this calculation are shown in Fig. 9a and c together with the results of the measurement region. For practicability, the highest uncertainty in a particular region was assigned. The uncertainties in the extrapolation region are at least twice as much as in the measurement region. For calculating salinity using relative density, temperature, and pressure values by means of the density–salinity relation, the uncertainty in salinity was also determined in the measurement and extrapolation region. The salinity uncertainty was calculated by multiplying the density uncertainty by the partial derivative of salinity by density, i.e. US=UΔρSW⋅∂S∂S∂ρ∂ρ. The uncertainties yielded by this calculation are shown in Fig. 9b and d. A salinity determined by means of a calculation using the relation in the measurement region has an uncertainty of 8×10-3. If measurement values are used for calculation, their uncertainty has to be included.

As pointed out above, the mathematical formulation of the density–salinity relation is empirical and does not contain any theoretical boundary conditions for infinite dilution. This is also an issue for the density at high pressures, as here the measurement uncertainty in density is higher, thereby causing more variability in the shape of the relation for very low salinities. Therefore, additional measurements were conducted on diluted standard seawater with salinity 2 for some temperatures. The samples used were obtained from the same seawater as described above in Sect. 4.2; the corrections were similar. The density deviations of the corrected values from predicted values of the density–salinity relation are shown in Fig. 10. The deviations are well within the uncertainty in the relation. No inconsistencies are caused by the non-compliance with theoretical boundary conditions for very low salinities and high pressures.

For atmospheric pressure, the density deviation of TEOS-10 from the density–salinity relation is shown in Fig. 11a. TEOS-10 density values are always higher than those of the density–salinity relation. The increase in the deviation with salinity is approximately linear. At salinities higher than 25, the deviation exceeds the estimated uncertainty of 8 g m-3 significantly. At salinities smaller than 5, the deviation, although consistent, is unexpectedly high. Salinity 0, which is pure water, defines the zero-line of TEOS-10 and of the density–salinity relation.

To leave the linear increase in the deviation with salinity seen in Fig. 11a out of consideration, a reduced form is shown in Fig. 11b. Here, Δρ-Δρ(S=35)⋅SΔρ(S=35)⋅S3535 is visualized. It is found that the reduced deviation is always less than 5 g m-3.

To find possible causes of the unexpectedly high density deviation, the density data on which TEOS-10 is based were examined, where the uncertainty in salinity was considered negligible. In the S,T,p0 region of interest, according to Feistel (2003, 2008), TEOS-10 is based on a data set (JPOTS, 1981c, pp. 36–56) that consists of normalized density data of Millero et al. (1976) and of Poisson et al. (1980), where the density data of Millero et al. have a significantly higher precision. For atmospheric pressure, this data set was also used to fit the previous reference equation of state EOS-80 (JPOTS, 1981c), and, therefore, no comparison with EOS-80 was carried out.

Millero et al. (1976) measured the density of diluted and standard seawater of batch P63 using a magnetic float densimeter. The comparison between the normalized densities measured by Millero et al. and TEOS-10 shown in Fig. 12a suggests that TEOS-10 is well fitted to these densities. Furthermore, for salinities less than 30 (compared to for salinities greater than or equal to 30), the deviation is strongly scattered and the salinity value of each deviation value is different. This may be explained by the fact that not all density measurements were carried out in a closed measuring vessel (JPOTS, 1981c, p. 35), thereby avoiding evaporation, which would increase the salinity and density during a measurement. To exclude the impact of the data normalization, a comparison of the original densities of Millero et al. (1976) and the density–salinity relation is shown in Fig. 12b, where the measurements that were carried out in a closed measuring vessel are separated from those that were putatively carried out in an open measuring vessel. The deviations of the closed-vessel measurements (for salinity 30 and 35) are the smallest and scatter the least, whereas the deviations of the open-vessel measurements (for all other salinities) scatter highly. If it is assumed that evaporation occurred during the open-vessel measurements, then the measured densities can be systematically too high (or the assigned salinities too small), which would cause the open-vessel deviations to be too high. Furthermore, a linear fit curve that was developed using the closed-vessel deviations is shown, as it is possible that there is a systematic deviation increasing linearly with salinity besides the evaporation. The smallest open-vessel deviations, which are most likely not significantly affected by evaporation, correlate conspicuously with this fit curve, thereby supporting the possibility of systematic deviation. The open-vessel densities for salinity 40, which are visible as the highest deviations in Fig. 12b, were corrected (using the closed-vessel densities) when the density data of Millero et al. and Poisson et al. were normalized (JPOTS, 1981c, pp. 35 and 58), which is why there are no significant deviations for salinity 40 in Fig. 12a. It should be noted that for calculation of the density deviations given in Fig. 12a and b, the temperatures at which Millero et al. made their measurements were converted from the International Practical Temperature Scale 1968 to the International Temperature Scale 1990 (CCT, 1997). To identify plausible causes of the systematic deviation, we thoroughly examined the magnetic flotation method used by Millero et al. for possible issues.

Magnetic float densimeters have the advantage over hydrostatic weighing densimeters that no mechanical coupling by means of a suspension is needed to determine the buoyancy force acting on a float (or sinker). Instead, this is achieved with a magnetic coupling by placing a magnet into a float. The float is brought to mechanical equilibrium, i.e. floats in the liquid, by means of a current-carrying coil; here, the current is a measure of the force, and thus of the liquid density. However, for density measurement the characterization of the magnetic coupling is necessary in addition to the determination of the float volume, as in the case of a hydrostatic weighing densimeter.

The densimeter used by Millero et al. for measuring the seawater density consisted of a hollow float in the measuring liquid of a vessel that had a volume of 250 mL, with the coil mounted underneath. The float was made of Pyrex, contained a permanent magnet that was a stirring bar and was therefore probably made of Alnico, and had a volume of 32 cm-3 (Millero, 1967). The float was weighted with platinum weights to adjust its buoyancy. The current that passed through the coil was used to pull the float to the bottom of the measuring vessel. Subsequently, the current intensity was gradually reduced until the float lifted off the bottom. The equilibrium current determined in this way, which was assumed to define the state of floating, was a measure of the liquid density.

Bignell (2006) discussed various methods for determining the buoyancy force in magnetic float densimeters. For the design of the magnetic coupling system, the magnetic force exerted on a permanent magnet by a current-carrying, circular coil (without a metal core) was given by Fmag=m⋅Gz,R=m⋅-32⋅μ⋅R2⋅zR2+z25⋅I, where m is the magnetic momentum, Gz,R is the magnetic field gradient (along the axis perpendicular to the coil plane through the coil centre point), μ is the permeability of the medium between the permanent magnet and the coil, R is the circular coil radius, z is the distance between the magnet and the coil, and I is the current. In a measurement obtained from seawater, the magnetic force is therefore dependent on the magnetic water properties and on the magnet-coil distance.

Bignell pointed out that the force on the magnet is also dependent on the magnetic field, even for magnetically hard materials. The magnetic force is therefore not linearly (as in Eq. 15) but quadratically dependent on the equilibrium current, i.e. Fmag=f1⋅I+f2⋅I2, where f1 and f2 are magnetic coupling constants. For a magnetically hard material, the force mainly depends on the linear term, whereas the quadratic term is used as a correction.

Millero et al. used a cylindrical (instead of a circular) coil and summarized the magnetic force as Fmag=f⋅I, where the calibration factor f was determined with measurements obtained from air-saturated water by weighing the float with platinum weights. The seawater density was determined relative to water, i.e. relative to the calibration using water: ΔρSW=f⋅ISW-IH2OV+mPtmPtρPtρPt, where ISW and IH2O are the currents resulting from the measurements obtained from air-saturated seawater and water, V is the float volume, which is also determined by the calibration, and mPt and ρPt are the mass and density of the platinum weights, which were identical in a measurement obtained from seawater and water.

Since seawater and water have different magnetic properties, it is possible that the calibration factor f is significantly different, i.e. μSW≉μH2O⟹fSW≉fH2O. To rule out this possibility, we carried out a representative calculation. Since, theoretically, Fmag∝μ⋅I for a cylindring (and circular) ring coil, it follows directly that μSWμSWμH2OμH2O=fSWfSWfH2OfH2O if the permanent magnet is in the same position in both measurements; the calibration factor of seawater is thus calculated from that of water. The permeabilities are calculated by μ=μ0⋅1+χ, where μ0=4⋅π⋅10-7NA-2 is the vacuum permeability, χSW=-8.25×10-6, and χH2O=-9.04×10-6 are the (dimensionless) volume susceptibilities of seawater with a salinity of 29 (Imhmed, 2012) and of water. The relative density deviation due to the different permeabilities being neglected was calculated by ΔρSWfSW,fH2O-ΔρSWf=fH2O using (i) the calibration factor for water f=fH2O=-3.5308gA-1 for 25 ∘C (Millero, 1967), (ii) the currents ISW=0.4A and IH2O=0.15A, (iii) the platinum mass and density mPt=0.7g and ρPt= 21 450 kg m-3, and (iv) the float volume given above. The values (ii) and (iii) were chosen based on a plot of calibration data of the flotation densimeter given by Millero (1967), and correspond to a relative seawater density of 28 kg m-3. The calculation yields a density deviation at the order of 0.01 g m-3; as a result, the differences in the magnetic properties of seawater and water are not problematic.

Since the volume of the float was also determined by means of the calibration measurement using water, it is possible that this resulted in a significant deviation in the relative seawater density. We therefore carried out a further representative calculation using the values (i–iv). Using this calculation, a density deviation of only 3 g m-3 is yielded for a relative volume deviation of 10-4. Although the volume results indirectly from an extrapolation of the linear relation of the magnetic coupling, Fmag=f⋅I, which is quadratic even for magnetically hard materials according to Bignell (2006), it is unlikely that a volume deviation of this magnitude will occur in the calibration measurement; the float volume calibration is therefore not problematic.

We performed a final calculation to estimate how significant the precise height positioning of the permanent magnet is, i.e. the distance from the coil. Two reasons for a change of the distance are conceivable. On the one hand, the position of the magnet (inside the float) or of the coil can change in the time between the calibration measurement obtained from water and the measurement obtained from seawater; the permanent magnet was fixed in the hollow float using wax (Millero, 1967). Density deviations that result from such position changes are minimized if, after each measurement obtained from seawater, a measurement obtained from water had also been carried out (a quasi-substitution measurement). On the other hand, the “lift-off” process, wherein the equilibrium current is determined by sight, is not the same for seawater and water in terms of speed (among other factors). Density deviations that result from such dissimilarities are minimized, if, in additional to the “lift-off” current, the “drop-down” current had been determined in the opposite manner and both currents had been averaged for seawater and water, respectively. Or, if in the measurement obtained from seawater, the float was weighted with the aim to yield the same current as in the calibration measurement using water.

For the calculation, it was assumed that the height dependence of the magnetic force given by z in Eq. (15) for the circular coil is similar for the cylindrical coil used by Millero et al. (1976). If the distance between magnet and coil is z+Δz, then fz+Δzfz+Δzfzfz=z+Δzz+Δzzz⋅R2+z2R2+z2R2+z+Δz2R2+z+Δz25522 holds. The displacement of the coil or of the magnet can be treated mathematically as the same, since fSW=f(z+Δz) applies to the measurement obtained from seawater and fH2O=f(z) applies to the measurement obtained from water in both cases. Using the values (i–iv), the coil radius R=20 mm and the distance z=40 mm for an unconsidered distance increase in Δz=3 µm yield a relative seawater density which is too high by 10 g m-3. R and z were estimated based on a sketch and a dimension of the flotation densimeter used (Millero, 1967). If a temporal or permanent distance increase exists that is not considered, an approximately linear density increase (or decrease) as seen in Fig. 12b results.

The high sensitivity of the measurement density to the magnet height position is one reason why magnetic flotation densimeters that were developed later and that share a similar principle, e.g. that of Bignell (1982), use position sensing systems with accuracies that are at least in the micrometre range to keep the height, z, constant. The actual cause of the significance of the density deviations seen in Fig. 12a and b may therefore be an overestimation of the accuracy and precision of the magnetic flotation method used.

TEOS-10 may be used to calculate densities for pressures up to 100 MPa. In the S,T,p>p0-region of interest, the relative density data given by Chen and Millero (1976), as well as thermal expansion data given by Bradshaw and Schleicher (1970) and speed-of-sound data given by Del Grosso (1974) were used for fitting (Feistel, 2003, 2008). Chen and Millero (1976) directly measured the seawater density, i.e. the specific volume, relative to water using a magnetic float densimeter whose magnetic force on the float was determined as described above. By contrast, the data of Bradshaw and Schleicher (1970), and of Del Grosso (1974) allows only the calculation of density differences using thermodynamic relations, i.e. relative to a reference state of the absolute seawater density with defined salinity, temperature, and pressure.

An overview of the density deviation of TEOS-10 from the density–salinity relation in the entire salinity–temperature region for atmospheric pressure is given in Fig. 13a. The increase in the deviation with salinity seen in Fig. 11a for 5, 20, and 35 ∘C is also present for 0 ∘C. For higher temperatures and salinities of around 20, the deviation increases unexpectedly. A similar overview of the density deviation for 30 MPa is given in Fig. 13b. The density deviation for this pressure is higher than that for atmospheric pressure. In the measurement region, this trend continues globally for up to 65 MPa as seen in Fig. 13c, but, in the extrapolation region, discontinues locally for up to 100 MPa as seen in Fig. 13d. For all pressures, the densities calculated using TEOS-10 are higher than the densities calculated using the density–salinity relation. The uncertainty in the deviations, however, is not exceeded significantly for higher pressures.

Chen and Millero measured the seawater density using a densimeter that is similar to that for atmospheric pressure used by Millero et al. (1976), which is why similar systematic deviations are likely. Both the thermal expansion data of Bradshaw and Schleicher and the speed-of-sound data of Del Grosso can only be compared with the density–salinity relation if the absolute seawater and water density are included in the calculation. The uncertainty in the water density calculated using IAPWS-95 is 10 g m-3 for pressures up to 10 MPa and 30 g m-3 for up to 100 MPa. Since the deviation between TEOS-10 and the density–salinity relation shown in Fig. 13b–d is comparable to this uncertainty, the water density may be considered as a cause. For example, Lin and Trusler (2012) showed by rough calculation of the water density using their measured speed-of-sound data that the IAPWS-95 density for 0 to 40 ∘C and pressures up to 100 MPa is within its uncertainty, but may be too low by a few 10 g m-3. A detailed analysis of this issue was given by Wagner and Thol (2015).