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As one of few who have been involved in the equation of state of seawater over the last 40 years, I was invited to review some of the history behind its early development and also the more recent thermodynamic equation of state. The article first reviews early (late 1800s) work by Knudsen and others in defining the concept of salinity. This summary leads into the development of the practical salinity scale. Our studies at the University of Miami Rosenstiel School, along with the work of Alain Poisson's group at Laboratoire de Physique et Chimie, Université Pierre et Marie Curie, and that of Alvin Bradshaw and Karl Schleicher at Woods Hole Oceanographic Institution, were instrumental in deriving the 1980 equation of state (EOS-80) that has been used for 30 years. The fundamental work of Ranier Feistel at Leibniz Institute for Baltic Sea Research led to the development of a Gibbs free energy function that is the backbone of the new thermodynamic equation of state (TEOS-10). It can be used to determine all of the thermodynamic properties of seawater. The salinity input to the TEOS-10 Gibbs function requires knowledge of the absolute salinity of seawater (SA), which is based upon the reference salinity of seawater (SR). The reference salinity is our best estimate of the absolute salinity of the seawater that was used to develop the practical salinity scale (SP), the equation of state, and the other thermodynamic properties of seawater. Reference salinity is related to practical salinity by SR= SP(35.16504/35.000) g kg⁻¹ and absolute salinity is related to reference salinity by SA= SR+ δSA, where δSAis due to the added solutes in seawater in deep waters resulting from the dissolution of CaCO₃(s) and SiO₂(s), CO₂, and nutrients like NO₃ and PO₄ from the oxidation of plant material. The δSAvalues due to the added solutes are estimated from the differences between the measured densities of seawater samples compared with the densities calculated from the TEOS-10 equation of state (Δρ) at the same reference salinity, temperature, and pressure, using δSA= Δρ/0.75179 g kg⁻¹. The values of δSAin the ocean can be estimated for waters at given longitude, latitude, and depth using correlations of δSAand the concentration of Si(OH)₄ in the waters. The SAvalues can then be used to calculate all the thermodynamic properties of seawater in the major oceans using the new TEOS-10. It will be very useful to modelers examining the entropy and enthalpy of seawater.

The determination of salinity by means of electrical conductivity relies on stable salt proportions in the North Atlantic Ocean, because standard seawater, which is required for salinometer calibration, is produced from water of the North Atlantic. To verify the long-term stability of the standard seawater composition, it was proposed to perform measurements of the standard seawater density. Since the density is sensitive to all salt components, a density measurement can detect any change in the composition. A conversion of the density values to salinity can be performed by means of a density–salinity relation. To use such a relation with a target uncertainty in salinity comparable to that in salinity obtained from conductivity measurements, a density measurement with an uncertainty of 2 g m−3 is mandatory. We present a new density–salinity relation based on such accurate density measurements. The substitution measurement method used is described and density corrections for uniform isotopic and chemical compositions are reported. The comparison of densities calculated using the new relation with those calculated using the present reference equations of state TEOS-10 suggests that the density accuracy of TEOS-10 (as well as that of EOS-80) has been overestimated, as the accuracy of some of its underlying density measurements had been overestimated. The new density–salinity relation may be used to verify the stable composition of standard seawater by means of routine density measurements.

Lee, W.D.; Jeong, Y.H., and Jeon, H.S., 2019. Groundwater flow analysis in a coastal aquifer with the coexistence of seawater and freshwater by using a non-hydrostatic pressure model. In: Lee, J.L.; Yoon, J.-S.; Cho, W.C.; Muin, M., and Lee, J. (eds.), The 3rd International Water Safety Symposium. Journal of Coastal Research, Special Issue No. 91, pp. 121-125. Coconut Creek (Florida), ISSN 0749-0208. The study proposes a new hydrostatic pressure model for flow analysis of a coastal aquifer wherein seawater and freshwater coexist. The numerical model introduces an advection–diffusion equation for salinity and temperature to trace the behaviors of seawater and freshwater in the Navier–Stokes (N-S) solver based on the porous body model (PBM) that considers the characteristics of a coastal aquifer (e.g., particle size, porosity, and shape). Furthermore, the density current is analyzed based on a state equation that determines the properties of water (density and viscosity) based on salinity and temperature. This enables the flow analysis of a coastal aquifer based on non-linear interferences between seawater and freshwater. In order to validate the proposed numerical model, a numerical constant water head permeability test device is set up, and the calculated permeability coefficient of the porous media exhibits a high degree of agreement with the value measured in the experiment. Flow simulations of a coastal aquifer are conducted based on the water level difference and salinity difference between seawater and groundwater. Additionally, hydrodynamic characteristics are analyzed from the water table, flow field, salinity distribution, and maximum penetration distance of the coastal aquifer. Specifically, an in-depth discussion of the diffusion of salinity near the shoreline and the movements of the seawater and freshwater interface is presented, which is largely absent in extant models.

At constant pressure, a mixture of water parcels with equal density but differing salinity and temperature will be denser than the parent water parcels. This is known as cabbeling and is a consequence of the nonlinear equation of state for seawater density. With a source of turbulent vertical mixing, cabbeling has the potential to trigger and drive convection in gravitationally stable water columns and there is observational evidence that this process shapes the thermohaline structure of high-latitude oceans. However, the evolution and maintenance of turbulent mixing due to cabbeling has not been fully explored. Here, we use turbulence-resolving direct numerical simulations to investigate cabbeling’s impact on vertical mixing and pathways of energy in closed systems. We find that cabbeling can sustain convection in an initially gravitationally stable two-layer configuration where relatively cold/fresh water sits atop warm/salty water. We show the mixture of the cold/fresh and warm/salty water is constrained by a density maximum and that cabbeling enhances mixing rates by four orders of magnitude. Cabbeling’s effect is amplified as the static stability limit is approached, leading to convection being sustained for longer. We find that available potential energy, which is classically thought to only decrease with mixing, can increase with mixing due to cabbeling’s densification of the mixed water. Our direct numerical dimulations support the notion that cabbeling could be a source of enhanced ocean mixing and that conventional definitions of energetic pathways may need to be reconsidered to take into account densification under mixing.

A reduced-order model (ROM) of the global oceans is developed by projecting the hydrostatic Boussinesq equations of motion onto a proper orthogonal decomposition (POD) basis. Three-dimensional POD modes are calculated from the ocean fields of an ensemble climate reanalysis dataset. The coefficients in the POD ROM are calculated using a regression approach. The performance of various POD ROM configurations are assessed. Each configuration is derived from an alternate sea-water equation of state, linking the density and temperature fields. POD ROM variants incorporating an equation of state in which density is a quadratic function of temperature, are able to reproduce the statistics of the large-scale structures at a fraction of the computational cost required to numerically simulate this flow. Due to the speed and efficiency of calculation, such reduced-order models of the global geophysical system will enable researchers and policy makers to assess the physical risk for a broader range of potential future climate scenarios.

Knowledge of thermodynamic properties of aqueous solutions of CO 2 is crucial for various applications including climate science, carbon capture, utilisation and storage (CCUS), and seawater desalination. However, there is a lack of reliable experimental data, and the equation of state (EOS) predictions are not reliable, particularly for sound speeds in low CO 2 concentrations typical of water resources. For this reason, we have measured speeds of sound in three different aqueous solutions containing CO 2 . We report speeds of sound in the single-phase liquid region for binary mixtures of water and CO 2 for mole fractions of CO 2 of 0.0118, 0.0066 and 0.0015 at temperatures from 273.15 K to 313.15 K and at pressures up to 50 MPa, measured using a dual-path pulse-echo apparatus. The relative standard uncertainties of the sound speeds are 0.05 %, 0.03 % and 0.01 % at 0.0118, 0.0066 and 0.0015 CO 2 mole fractions, respectively. The change in sound speeds as functions of composition, pressure and temperature are analysed in this study. We find that dissolution of CO 2 in water increases its sound speeds at all conditions, with the greatest increase occurring at the highest mole fractions of CO 2 . Our sound speed data agree well with the limited available experimental data in the literature but deviate from the EOS-CG of Gernert and Span by up to 7 % at the lowest temperatures, highest pressures, and highest CO 2 mole fraction. The new low-uncertainty sound speed data presented in this work could provide a basis for development of an improved EOS and in establishing reliable predictions of the change in thermodynamic properties of seawater-like mixtures due to absorption of CO 2 gas.

The stratification is primarily controlled by temperature in subtropical regions (alpha ocean) and by salinity in subpolar regions (beta ocean). Between these two regions lies a transition zone, often characterized by deep mixed layers in winter and responsible for the ventilation of intermediate or deep layers. While of primary interest, no consensus on what controls its position exists yet. Among the potential candidates, we find the wind distribution, air–sea fluxes, or the nonlinear cabbeling effect. Using an ocean general circulation model in an idealized basin configuration, a sensitivity analysis is performed testing different equations of state. More precisely, the thermal expansion coefficient (TEC) temperature dependence is explored, changing the impact of heat fluxes on buoyancy fluxes in a series of experiments. The polar transition zone is found to be located at the position where the sign of the surface buoyancy flux reverses to become positive, in the subpolar region, while wind or cabbeling are likely of secondary importance. This inversion becomes possible because the TEC is reducing at low temperature, enhancing in return the relative impact of freshwater fluxes on the buoyancy forcing at high latitudes. When the TEC is made artificially larger at low temperature, the freshwater flux required to produce a positive buoyancy flux increases and the polar transition moves poleward. These experimets demonstrate the important role of competing heat and freshwater fluxes in setting the position of the transition zone. This competition is primarily influenced by the spatial variations of the TEC linked to meridional variations of the surface temperature.

We consider the hydrostatic Boussinesq equations of global ocean dynamics, also known as the “primitive equations”, coupled to advection–diffusion equations for temperature and salt. The system of equations is closed by an equation of state that expresses density as a function of temperature, salinity and pressure. The equation of state TEOS-10, the official description of seawater and ice properties in marine science of the Intergovernmental Oceanographic Commission, is the most accurate equations of state with respect to ocean observation and rests on the firm theoretical foundation of the Gibbs formalism of thermodynamics. We study several specifications of the TEOS-10 equation of state that comply with the assumption underlying the primitive equations. These equations of state take the form of high-order polynomials or rational functions of temperature, salinity and pressure. The ocean primitive equations with a nonlinear equation of state describe richer dynamical phenomena than the system with a linear equation of state. We prove well-posedness for the ocean primitive equations with nonlinear thermodynamics in the Sobolev space H 1 . The proof rests upon the fundamental work of Cao and Titi (Ann. Math. 166:245–267, 2007) and also on the results of Kukavica and Ziane (Nonlinearity 20:2739–2753, 2007). Alternative and older nonlinear equations of state are also considered. Our results narrow the gap between the mathematical analysis of the ocean primitive equations and the equations underlying numerical ocean models used in ocean and climate science.

Turbulent entrainment properties of thermohaline internal hydraulic jumps in quiescent ambient water are investigated. Underflow and overflow jumps are considered. The study is mainly concerned with thermohaline buoyancy effects on flow development. The thermohaline buoyancy is determined from a seawater equation of state, which is nonlinear in temperature and linear in salinity, and is accurate throughout the range of temperature and salinity of interest (T = 0 to 40 °C and S = 0 to 40 ppt). The results indicate that thermohaline buoyancy produces smaller less diluted underflow jumps and larger more diluted overflow jumps. These nonlinear buoyancy effects are particularly significant when buoyancy arising from temperature and salinity act in opposite directions. Hydraulic controls by vertical and/or horizontal constrictions downstream reveal, using a matching technique, unique stationary miscible jump solutions.

Salinity distribution depends strongly on estuary geometry, tidal amplitude, and freshwater discharge, which counteract salt intrusion. As an important part of the “New Western Land and Sea Corridor” in China, the Pinglu Canal Project (PCP) plan has been proposed to dredge a navigational channel from the Inland Qinjiang River to the Maowei Sea in South China Sea. Understanding the salinity transport process in estuary region is important for both water resource management and ecosystem health in the Qinjiang River Estuary (QRE). In this study, a predictive model of high-water slack was constructed based on the steady-state equation of salt intrusion, and the related measurable or quantifiable variables were considered through empirical relations. The developed model herein was applied to the QRE, and the predicted salt distribution appeared to agree well with the observations. Then, the response of fresh water discharge and tide on the process of salt intrusion in the QRE before and after the PCP construction was analyzed. The results indicate that when the water flow discharge is Q<100 m 3 /s, the saltwater intrusion length in the QRE during the high tide period can attach to 10–27 km after the channel deepening, increased by 4.5 km compared to the natural condition in a whole year, and the seawater will intrude to the Youth Ship Lock when the fresh water is Q<20 m 3 /s and tidal range is H 0 >2 m. During the dry period, the Youth Hydro-hub can alleviate the saltwater intrusion by increasing the water discharge when the tidal range is H 0 >2.0 m, and then decreasing the water discharge when the tidal range is H 0 <1 m. The predictive model herein can be used to estimate the salt intrusion length under variable fresh water and tide level in the QRE region and provide efficient suggestions for quantifying upstream water discharge to mitigate salt intrusion after PCP construction.