Thermophysical property anomalies of Baltic seawater

While the thermodynamic properties of Standard Seawater are very well known, the quantitative effect of sea salt composition anomalies on various properties is difficult to estimate since comprehensive lab experiments with the various natural waters are scarce. Coastal and estuarine waters exhibit significant anomalies which also influence to an unknown amount the routine salinity calculation from conductivity measurements. Recent numerical models of multi-component aqueous electrolytes permit the simulation of physical chemical properties of seawater with variable solute composition. In this paper, the FREZCHEM model is used to derive a Gibbs function for Baltic seawater, and the LSEA DELS model to provide estimates for the conductivity anomaly relative to Standard Seawater. From additional information such as direct density measurements or empirical salinity anomaly parameterisation, the quantitative deviations of properties between Baltic and Standard Seawater are calculated as functions of salinity and temperature. While several quantities show anomalies that are comparable with their measurement uncertainties and do not demand special improvement, others exhibit more significant deviations from Standard Seawater properties. In particular density and sound speed turn out to be significantly sensitive to the presence of anomalous solute. Suitable general correction methods are suggested to be applied to Baltic Sea samples with known Practical Salinity and, optionally, directly determined density. Correspondence to: R. Feistel (rainer.feistel@io-warnemuende.de)


Introduction
From Knudsen's "Normalwasser VI" (Knudsen, 1903) to the current IAPSO 1 service, Standard Seawater (SSW) collected from the North Atlantic and processed into sealed bottles has served for the calibration of oceanographic measuring devices for more than a century.This water has also been used to characterise the properties of seawater (Millero et al., 2008).However, the chemical composition of seawater is not exactly constant.Regional deviations of seawater composition and properties were occasionally investigated, in particular in the 1970s (Rohde, 1966;Cox et al., 1967;Kremling, 1969Kremling, , 1970Kremling, , 1972;;Connors and Kester, 1974;Brewer and Bradshaw, 1975;Millero et al., 1978;Poisson et al., 1981;Millero, 2000), but were generally considered of minor relevance and ignored by previous international oceanographic standards (Forch et al., 1902;Jacobsen and Knudsen, 1940;Lewis, 1981;Millero, 2010).However, the effects of these compositional variations are measureable, and are easily the largest single factor currently limiting the accuracy of empirical formulas for the thermodynamic properties of seawater.It is therefore desirable to investigate the effects of these regional deviations, and to determine how these deviations can be incorporated into routine procedures for obtaining numerical estimates of different seawater properties (Lewis, 1981).
The new TEOS-10 2 formulation of seawater properties (Feistel, 2008;IAPWS, 2008;IOC et al., 2010) supports the analysis of anomalous seawater properties in a first approximation even though methods and knowledge available for the description of the related effects are still immature.An important step in this direction was the definition of the Reference Composition (RC) as a standard composition model for sea salt (Millero et al., 2008).The RC can be used to define a Reference Salinity, which represents the actual mass fraction of solute in seawater of Reference Composition.It also defines a baseline relative to which anomalies can be properly quantified in detail.The RC is defined in the form of exact molar fractions, x RC a > 0, for 15 major sea salt constituents, a. Deviations of molar fractions, x a = x RC a , from the RC found in samples of natural or artificial seawater are regarded as composition anomalies.A second step towards an analysis procedure for anomalous seawater has been to define a parameter, the Absolute Salinity, which will provide the best estimate of the density of a particular seawater sample whose composition is different than the Reference Composition, when used as a numerical input into the TEOS-10 Gibbs function (Wright et al., 2010b).Under this definition, the Absolute Salinity represents the mass fraction of solute in a seawater of Reference Composition with the same density as that of the sample, and can also be called the Density Salinity.It may therefore be different than the actual mass fraction of solute in the sample, which is termed the Solution Absolute Salinity.
In the past, the thermodynamic properties of freshwater and estuarine systems have been found to be approximately described by a heuristic, referred to as "Millero's Rule" here, that states that these properties depend primarily on the mass of solute, and only secondarily on the composition of the solute (Millero, 1975;Chen and Millero, 1984).If this is true for density, then the Density Salinity is a good approximation for Solution Absolute Salinity, even in the presence of composition anomalies.However, recent analysis (Pawlowicz et al., 2010) suggests that this approximation might have a much narrower range of validity than was previously believed.
The Baltic Sea is an obvious place to study the effects of composition anomalies since the existence of composition anomalies in Baltic seawater has been known since the formulation of Knudsen's equation of state (Knudsen, 1901;Forch et al., 1902) in the form of its salinity intercept at zero Chlorinity.The details of these anomalies were determined by chemical analysis beginning in the 1960s (Rohde, 1965;Kremling, 1969Kremling, , 1970Kremling, , 1972;;Feistel et al., 2010a), and some empirical evidence has been gathered on the effects on density (Kremling, 1971;Millero and Kremling, 1976).
The electrical conductivity of anomalous solute in Baltic seawater is not negligible and has led in the past to various mutually inconsistent empirical relations between Practical Salinity and Chlorinity (Kwiecinski, 1965;Kremling 1969Kremling , 1970Kremling , 1972) ) and to an experimental study of whether Practical Salinity is conservative within its measurement uncertainty (Feistel and Weinreben, 2008).Here, conservative means that the salinity value remains the same when temperature or pressure of the sample are changing.However, there is little theoretical knowledge of the reasons for the mag-nitude of the resulting density and conductivity anomalies, and very little is known at all about the quantitative effect of anomalous solutes on the sound speed, the heat capacities, the freezing point, or many other thermodynamic properties (Feistel, 1998).
One drawback of using the Baltic Sea as a test region is that the relative composition of the water is likely not constant with position or depth.The composition variations derive from the inflow of many rivers, which themselves have a wide range of compositions, and these are not well mixed within the Baltic Sea.In addition, these riverine additions are not constant in time and are involved in complex biogeochemical processes during the water residence time of 20-30 years (Feistel et al., 2008b;Reissmann et al., 2009); significant variations apparently occur on at least decadal time scales (Feistel et al., 2010a).Acknowledging this uncertainty, we shall use a highly simplified model of the composition anomaly that represents only the effects arising from the addition of calcium and bicarbonate ions which dominate observed anomalies.
In parallel with the development of TEOS-10, numerical models that can be used to investigate the thermodynamic and transport properties of seawaters from a theoretical basis have been developed and tested (Feistel and Marion, 2007;Pawlowicz, 2010).Known as FREZCHEM (Marion and Kargel, 2008) and LSEA DELS (Pawlowicz, 2009) respectively, these models have been used to extend the range of validity of the thermodynamic Gibbs function into salinities larger and smaller than have been studied experimentally (IAPWS, 2007;Feistel, 2010), and to investigate the effects of composition anomalies resulting from biogeochemical processes on the conductivity and density of seawater (Pawlowicz et al., 2010).In this paper we combine these numerical approaches to study the properties of Baltic Sea water.We create a correction to the TEOS-10 Gibbs function that can be used to determine all the thermodynamic properties of Baltic Sea water, and a correction to the PSS-78 Practical Salinity Scale that can be used to estimate the conductivity of this water.These analytical models are used to study whether the Density Salinity (i.e. the Absolute Salinity as defined by TEOS-10) is in fact a good estimate of the Solution Absolute Salinity (actual mass fraction of solute), and whether or not the Density Salinity can be used in conjunction with the Gibbs function for SSW to determine other thermodynamic parameters.
The composition anomaly of the Baltic Sea, Fig. 1, is dominated by riverine calcium excess (Rohde, 1965;Millero and Kremling, 1976;Feistel et al., 2010a).The dissolved positive Ca ++ ions are charge-balanced mainly by dissolved carbon dioxide, CO 2 , e.g., in the form of two negative bicarbonate HCO − 3 ions.Baltic carbonate concentrations depend in a complex way on exchange with the atmosphere, seasonal solubility, biological activity as well as various chemical reactions with the sediment under occasionally anoxic conditions (Thomas and Schneider, 1999;Nausch et al., 2008;Omstedt Ocean Sci., 6, 949-981, 2010 www.ocean-sci.net/6/949/2010/approach taking into account the experimental uncertainty.In section 6, as functions of the two salinity variables, correlation formulas for the conductivity, Practical Salinity and Reference Salinity of Baltic seawater are derived from results based on the LSEA_DELS model.Combining the previous results, section 7 discusses the errors implied by computing seawater properties directly from Practical Salinity readings, and suggests general correction algorithms for error reduction. Fig. 1: The Baltic Sea is a semi-enclosed estuary with a volume of about 20 000 km 3 and an annual freshwater surplus of about 500 km 3 /a; direct precipitation excess accounts for only 10 % of the latter value (Feistel et al., 2008b).Baltic seawater The Baltic Sea is a semi-enclosed estuary with a volume of about 20 000 km 3 and an annual freshwater surplus of about 500 km 3 a −1 ; direct precipitation excess accounts for only 10% of the latter value (Feistel et al., 2008b).Baltic seawater (BSW) is a mixture of ocean water (OW) from the North Atlantic with river water (RW) discharged from the large surrounding drainage area.Regionally and temporally, mixing ratio and RW solute are highly variable.Collected BSW samples consist of Standard Seawater (SSW) with Reference Composition (RC) plus a small amount of anomalous freshwater solute (FW), which we approximate here to be calcium bicarbonate, Ca(HCO 3 ) 2 .In dissolved form, depending on ambient temperature and pH, Ca(HCO 3 ) 2 is decomposed into the various compounds of the aqueous carbonate system with mutual equilibrium ratios (Cockell, 2008).et al., 2009;Schneider et al., 2010).Additions of solute can cause changes in the equilibrium chemistry (e.g., in pH), and hence can lead to particles of, say, HCO − 3 , being converted into particles of CO 2− 3 by solute-solvent reactions.Such reactions convert H 2 O molecules from being part of the solvent to being part of the solute, or vice versa, such as in the case of Eq. (1.1).A full numerical simulation must model these changes as well, and this requires additional assumptions.
In FREZCHEM an "open system" approach is used.Lime (CaCO 3 ) is added, and then the chemical composition is allowed to evolve to an equilibrium state under the restriction that the partial pressure of carbon dioxide (pCO 2 ) and the total alkalinity (TA) are fixed.This is a reasonable approach for laboratory studies in which waters at 25 • C are stirred in contact with air after the addition of a salt, or for wind-mixed river plumes in equilibrium with the atmosphere.In the additions modelled here, a substantial inflow of CO 2 gas occurs and increases the mass of anomalous solute, so that the final composition is approximately modelled as an addition of Ca ++ and 2 HCO − 3 , i.e. a reaction of the form (Cockell, 2008): (1.1) In LSEA DELS a "closed system" approach is used.In this case a salt is added, and the chemical composition is allowed to evolve to an equilibrium state under the restriction that the total dissolved inorganic carbon (DIC) is fixed.This is a reasonable approach in situations where a TA and DIC anomaly are known.In the Baltic, these anomalies in TA and DIC are almost equal (Feistel et al., 2010), which indicates that the composition change is approximately modelled as an increase in Ca(HCO 3 ) 2 .This again is consistent with a reaction of the form (1.1).
Although the different assumptions in the two models are potentially a source of discrepancy between the results of our investigation into thermodynamic properties, which requires FREZCHEM, and investigation into conductivity properties, which required LSEA DELS, there is little difference between the final compositions obtained using the two approaches in this particular case.From another numerical model referred to as LIMBETA (Pawlowicz et al., 2010), an equilibrium model consistent with LSEA DELS, density is computed for comparison with FREZCHEM in order to quantify the effect of the different boundary conditions.The difference in the predicted density anomalies for a given Ca anomaly is less than 6 g m −3 , as discussed in Sect.6.
The FREZCHEM model results are used here to develop a Gibbs function for Baltic seawater in the form of a small correction to TEOS-10.A Gibbs function is a thermodynamic potential in terms of temperature, pressure and particle numbers and is therefore consistent with "closed system" conditions.The proper thermodynamic potential for FREZCHEM is a function which takes chemical potentials rather than particle numbers as independent variables, such as the Landau potential, = pV , where p and V are pressure and volume (Landau and Lifschitz, 1987;Goodstein, 1975).The Landau potential is related to the Gibbs potential by a Legendre transform (Alberty, 2001;Feistel et al., 2010c).The chemical potential of water in seawater expressed in terms of the Gibbs function is an example for such a Legendre transform.Since the differences between the open and the closed models are small, we refrain from the relatively complicated conversion procedure between Gibbs and Landau potentials in our generalization of the TEOS-10 Gibbs function with respect to an additional salinity variable.The gain expected from this significantly more demanding model will very likely be minor and at this stage does not warrant the additional effort.
Thermodynamic potentials describe unique equilibrium states at given conditions, e.g., in terms of numbers of atoms of the elements present in the system.These atoms may or may not form mutual bound states, and chemical reactions may occur between those compounds, between the solutes www.ocean-sci.net/6/949/2010/Ocean Sci., 6, 949-981, 2010 or the solvent, without affecting the validity of the thermodynamic potential expressed in terms of the system's elementary composition.This very convenient property is evident from the representation of thermodynamic potentials in statistical mechanics such as the canonical or the grand canonical ensemble.Formally, the atom numbers can also be replaced by suitable fixed stoichiometric combinations, i.e. by numbers of certain molecules as independent variables.Hence, the concentrations of Ca ++ and HCO − 3 ions are sufficient to correctly formulate the Gibbs function for Baltic seawater, regardless of any chemical reactions that in reality occur in the marine carbonate system, and which are modelled correspondingly by FREZCHEM and LIMBETA to determine the particular equilibrium states.
The paper is organised as follows.In Sect.2, several required composition variables and basic thermodynamic terms are introduced.In Sect.3, a formal expression for the Gibbs function of Baltic seawater is derived.This expression is used in Sect. 4 to obtain a formulation for the Baltic Sea Gibbs function through an empirical correlation of a specified functional form against results estimated using of the FREZCHEM model.This Gibbs function depends on two salinities, the Absolute Salinity of the SSW part, and a correction proportional to the anomalous calcium excess.In Sect. 5 selected property anomalies are computed from the Gibbs function for Baltic seawater and compared with a density-salinity approach taking into account the experimental uncertainty.In Sect.6, as functions of the two salinity variables, correlation formulas for the conductivity, Practical Salinity and Reference Salinity of Baltic seawater are derived from results based on the LSEA DELS model.Combining the previous results, Sect.7 discusses the errors implied by computing seawater properties directly from Practical Salinity readings, and suggests general correction algorithms for error reduction.

Composition variables
Baltic seawater, BSW, is a mixture of ocean water, OW, from the Atlantic plus a riverine freshwater contribution, RW, which may contain a small amount of salt, Fig. 1.The composition of OW is very close to the RC, i.e., to the composition of IAPSO Standard Seawater (SSW).RW contains various salts with the composition varying strongly in time depending on the different river sources (Perttilä, 2009).On average, the molar ratio of calcium to chloride for RW is significantly higher that for the RC.When RW and OW are mixed to form BSW, the two different origins of the chloride fraction can no longer be distinguished but a measurable calcium excess remains compared to the concentrations seen in SSW of the same Chlorinity and this represents the primary composition anomaly associated with RW inputs to the Baltic.Thus, samples collected from the Baltic Sea can reasonably be regarded as a parent solution of pure-water diluted Stan-dard Seawater, SSW, with Reference Composition, RC, plus a small amount of anomalous freshwater solute, FW, which originates from river discharge and contains mainly the calcium fraction of RW in excess of the expected value based on the Ca/Cl ratio of the RC.Note that the SSW contribution includes pure water plus RC solute from both OW and RW whereas FW refers only to the anomalous solute derived from riverine inputs.
The SSW and FW fractions of BSW are usually separated by the definition that FW does not contain any halides, i.e., that the Chlorinity of BSW determines the SSW fraction, independent of whether or not some of the river water entering the Baltic carries a relevant halide load.Because the RW component does in fact contain a small fraction of halides, the use of Chlorinity to estimate the SSW fraction will always result in this component including a small contribution from RW of all species in the RC.However, because the halide concentrations in OW are so large, the relative change in their concentration due to RW solute is very small, as is the corresponding error in the concentrations of all species in RC, and thus can be neglected.Anomalies of BSW, i.e., the composition of the FW fraction, in chemical species other than calcium and carbonates are neglected in our models.They are less relevant and were also found to vary significantly from author to author and between the analysed samples (Feistel et al., 2010a).
We emphasize that the models considered in this paper are formulated in terms of two independent salinity variables representing the SSW and FW fractions of BSW.In contrast, it is a common practice to assume that the FW composition equals that of RW (Millero and Kremling, 1976;Feistel et al., 2010a), which is consistent with the fact that the composition anomaly of BSW increases with decreasing brackish salinity.When results from our models are discussed or compared with observations, we will make use of such empirical salinity-anomaly relations between SSW and FW to conveniently display the typical anomalous properties as functions of a single variable that is routinely observed, the brackish salinity.In particular, the SSW and FW variables of the models will be approximately linked to the OW and RW concentrations, Eq. (2.16).However, it should be noted that the thermophysical equations derived from our models do not rely on any empirical and climatologically varying relation between SSW and FW; they depend separately on the two concentration variables.
In the FREZCHEM and LSEA DELS models, the FW composition is simplified to consist only of the carbonate equilibrium components that evolve from the dissolution of Ca(HCO 3 ) 2 in pure water, neglecting any other solutes such as sulfate or magnesium.The Gibbs function derived from FREZCHEM takes only the mass fraction of Ca(HCO 3 ) 2 as the FW input variable, regardless of the chemical equilibrium composition details after its dissolution in water.
To describe the thermodynamic properties of a given BSW sample, we first introduce a number of terms and variables.
Ocean Sci., 6, 949-981, 2010 www.ocean-sci.net/6/949/2010/A set of independent primary variables (considered as known) is required to describe the composition of the solutions corresponding to a particular water sample: the number of water molecules from OW, N OW 0 , and from the local freshwater input RW, N RW 0 , and the number of particles N OW a and N RW a of the related solute species, a. Their molar masses of the solvent and solute species are denoted by A 0 and A a , respectively.The number of particles per mole is Avogadro's number, N A .
When conservative mixing and a neutral precipitationevaporation balance are assumed, the number of water and solute particles in BSW are, (2.2) Here, the total solute particle numbers of the SSW and the FW fraction, N SSW S and N FW S , respectively, are chosen so that > 0 for all species of the RC but N FW a = x FW a N FW S = 0 for most of the RC species in the freshwater fraction.The molar fractions of the Reference Composition, x RC a > 0, are defined by Millero et al. (2008), and the molar fractions of the anomalous solute, x FW a ≥ 0, are inferred from the simplified dissociation reaction Eq.(1.1), as Additional basic quantities are derived from the previous variables to determine the related water properties.These quantities include: -the mass of salt from the SSW part, -the mass of the FW part, which consists of the solute only, -the total mass of solvent, M BSW 0 , which equals the solvent mass of the SSW part, -the total mass of solute, M BSW S , -the total mass of the SSW solution, (2.8) -the total mass of the combined BSW sample, M BSW , (2.9) In terms of those basic particle numbers and masses, several other useful properties are defined, such as the total number of water particles, N SSW 0 , in SSW, and of salt, N BSW S , in BSW, (2.10) and the Absolute Salinity of BSW, (2.11) The latter consists of the sum of the mass fractions of sea salt from the SSW, S BSW SSW , and from the FW, S BSW FW , to the BSW, in the form, ) (2.13) Before mixing, the salinities of the two end members are for the OW part, where M OW S is the mass of salt dissolved in the sample mass M OW , and for the RW part, where M RW S is the mass of salt dissolved in the sample mass M RW .Under the plausible assumption that the SSW solute originates from ocean water OW, M SSW S ≈ M OW S , and the FW solute from river discharge, RW, M FW S ≈ M RW S , the relation between the partial salinities before and after the conservative mixing process is given by the mass balance, S BSW FW /S For the estimation of the riverine salinity S RW A from density measurements of Baltic Sea samples, this equation is commonly used under the additional assumption that the SSW end member, North Atlantic surface water, has exactly standard-ocean salinity, S OW A ≈ S SO (Millero and Kremling, 1976;Feistel et al., 2010a), which is given in Table A1.The value of S BSW SSW can be determined from Chorinity measurements since the amount of halides in FW is zero by definition and the value of S BSW FW can then be determined from Eq. (2.11) with the value of S BSW A , Eq. (2.26), estimated from density measurements.
The mean molar masses of the solutes from the SSW and from the FW, respectively, are defined as (2.17) In the final solution, BSW, the total molality 3 of the solute is (2.18) expressed as the sum of the partial molalities, m BSW SSW and m BSW FW , of sea salt from the SSW and from the FW contributions to BSW, Compared to the molalities, Eqs.(2.19), (2.20), the salinities, Eqs.(2.12), (2.13), have the disadvantage that the salinity measure S BSW SSW of salt present with standard composition is (slightly) changing as soon as some anomalous solutes, M FW S , are added or removed, even if the amount of salt that stems from the SSW, M SSW S , and the mass of solvent, M BSW 0 , remain the same.
In general, a formal solute decomposition in the form of Eq. (2.2) is not self-evident.If a seawater sample of a certain molar solute composition x and molality m is given and its original end members are unknown, the decomposition of the solute into a "preformed" part with Reference Composition x RC and molality m RC , and a residual anomalous "freshwater" part with a resulting composition x FW and molality δm takes the form Here, the molar fractions are normalised, x FW a = 1.These mass-balance equations for the n species 3 Molality = moles of solute per mass of solvent do not possess a unique solution for the (n + 1) unknowns m RC , δm and x FW which fully characterise the end members.Consequently, due to this ambiguity of m RC , the "Preformed Salinity" (Wright et al., 2010a) of an arbitrary seawater sample, (2.22) may take any desired value unless it is subjected to a specified additional condition.One suitable, physically reasonable condition is that δm takes a minimum non-negative value and that m RC and all the freshwater fractions x FW are also non-negative, x FW a ≥ 0. In this case, two chemically well-defined and meaningful end members are associated with the given seawater sample.The molar mass A FW , Eq. (2.17), is positive definite under this condition, and the molality, m BSW FW , Eq. (2.20), the salinity, S BSW FW , Eq. (2.13), the mass, M FW S , and the particle numbers, N FW a , of the anomalous solute are nonnegative.The ideal-solution part of the Gibbs function of any aqueous solution, (2.23) possesses a regular and reasonable series expansion with respect to the anomaly if x FW a ≥ 0 and 0 ≤ x FW a δm x RC a m RC , and the chemical potentials of the RC and the FW solutes are mathematically valid and physically meaningful expressions, Eq. (3.6).Symbols newly introduced in Eq. (2.23) are specified in the glossary, Appendix B.
Alternatively, if for certain reasons the separation (Eq.2.21) is formally specified in such a way that at least one of x FW a ≤ 0, x RC a ≤ 0, m RC ≤ 0 or m < m RC is implied, some of the previous convenient properties may no longer be valid and a mathematically more cautious treatment of the thermodynamic perturbation is required.In this respect we can distinguish at least three qualitatively different situations, here referred to as modified, alien, and deficient seawater.The distinction between these cases is necessary only if the anomaly is preferably described in terms of an anomalous solute with thermodynamically well-defined concentration and composition values, i.e., if non-negative molar fractions x FW a and non-negative molalities m RC and δm are relevant for the equations used, and if each of the anomalous concentrations, x FW a δm, is assumed to be small compared to that of the parent solution, x RC a m RC , as exploited in this paper.These conditions are mostly met in the case (a) but partly violated in the cases (b) and (c).Thus, anomalies of the kinds (b) or (c) may require a different Gibbs function approach than the one developed in this paper.a. Modified seawater is defined by the condition x a > 0 for each dissolved species a in the RC (i.e., for all species with x RC a > 0), and x a = 0 for all species a not included Ocean Sci., 6, 949-981, 2010 www.ocean-sci.net/6/949/2010/ in the RC (i.e., for all species with x RC a = 0).Under these conditions, a nonvanishing anomaly implies that x a = x RC a for at least two of them.This is the simplest case and it is considered exclusively in this paper.It occurs when e.g.riverine freshwater or hydrothermal vents increase the concentration of selected species relative to the parent solution with Reference Composition, or if some species are partially precipitated due to supersaturation at high salinity or high temperature, or biologically depleted.If m is the molality of the given sample, the solute can be uniquely separated into a regular part with Reference Composition and the molality m RC < m, and an anomalous part with the molality δm = m−m RC , subject to the conditions x k m − x RC k m RC = 0 for at least one species k ∈ RC. (2.25) The species k is regarded as the key species which is not present in the anomalous part; its molality specifies the regular part via the RC ratios.In this study of the Baltic Sea, chloride will serve as the key species.Because of the condition (Eq.2.24), the anomalous part does not contain species with formally negative concentrations and can be modelled physically/chemically in the form of added salt.Usually, δm m RC will be assumed.
b. Alien seawater is defined by the condition x a > 0 for at least one dissolved species a, the alien species, that is not part of the RC (i.e., x a > 0 for a species for which x RC a = 0).Two examples of this case are when biologically produced silicate or organic compounds are added to seawater at relevant amounts, and when seawater is acidified to prevent precipitation in technical systems.Compared to the Reference Composition, the responsible physical state space dimension must be expanded to cover the alien species, and the representative point for the RC is then located on the boundary of the positive cone of the expanded space rather that in its interior.On the boundary or in its immediate vicinity, thermodynamic properties possess very special properties such as singularities of chemical potentials or electrolytic limiting laws.Thus, alien species cannot be described theoretically by a small linear deviation from a regular point in the phase space; they require specific nonlinear mathematical expressions such as limiting laws.c.Deficient seawater is defined by the condition x a = 0 for at least one species a, the deficient species, that is part of the RC (i.e., for a species with x RC a > 0).The missing constituent may be a volatile or reactive compound such as CO 2 or OH − that has disappeared in a certain physical, chemical or technical environment.Although the resulting composition may be very similar to the RC, a procedure like in case (a) is impossible here since it would formally lead to a zero-molality regular part and an anomalous part that contains all of the solute.In this case it may be more reasonable to specify the anomalous part as a small deviation from the RC concentrations some of which are negative.It is clear that this anomalous part can no longer be considered as an "added salt".
As suggested by observational evidence (Feistel et al., 2010a), the Baltic seawater is modelled here as modified seawater, as specified under case (a).The related Preformed Salinity, Eq. (2.22), is the Absolute Salinity of the diluted SSW, denoted here by A differs from the OW end-member salinity, S OW A , Eq. (2.14), at least due to the dilution with the pure water part of the riverine input and possibly, depending on where and when the BSW sample was collected, due to the riverine contributions to the key species, chloride.We will assume that the dilution effect strongly dominates.The resulting brackish SSW part, the parent solution, can properly be described by the TEOS-10 Gibbs function in terms of S SSW A , T and P .An expression for the correction to this Gibbs function, proportional to the anomalous solute molality, δm, is derived from thermodynamic considerations in the following section.

Theoretical formulation of the Gibbs function for Baltic seawater
In the Baltic Sea, small amounts of anomalous solutes, N FW a , are added to the brackish water body of dilute standard ocean water which consists of N BSW 0 water molecules and N SSW a solute particles.The Gibbs energy of the diluted, anomalyfree parent solution is the sum of the chemical potentials (Feistel and Marion, 2007), (3.6) The chemical potentials, µ a , required here depend only on the properties of the parent solution, µ a = µ 0 a (T ,P ) + kT ln(m a γ a ). (3.7) Here, γ a (m,T ,P ) is the practical activity coefficient of the species a, which depends on the set m = {m a } of all molalities of the parent solution, (3.8) Symbols newly introduced in Eq. (3.7) are specified in the glossary.The particle numbers of the anomalous solutes can be expressed in terms of their mole fractions and their total molalities, (3.9) In these terms, the Gibbs energy anomaly, Eq. (3.6), reads (3.10) Here, R = N A k is the molar gas constant, and γ id FW , γ id a , related by (3.11) are the limiting values of the activity coefficients at infinite dilution.
Note that the Eq.(3.10) is applicable only to anomalous species, x FW a > 0, that are already present in the parent solution, x RC a > 0. Otherwise, in the limit x FW a > 0, x RC a = 0, Eq. (3.10) possesses a logarithmic singularity for "alien" species a that do not belong to the RC but appear in the anomaly.
Dividing the Gibbs energy by the related mass of the solution, we obtain the expressions for the Gibbs functions of the (diluted) parent solution, and of Baltic seawater, (3.13) Here, g SW S SSW A ,T ,P is the TEOS-10 Gibbs function of seawater as a function of Absolute Salinity, S SSW A , Eq. (2.26), of the "preformed" parent solution with Reference Composition (RC) (Millero et al., 2008;Pawlowicz et al., 2010), Newly introduced symbols are explained in the glossary.From Eqs. (3.12) and (3.13), in linear approximation with respect to the anomalous solute concentration, the Gibbs function anomaly is (3.15) The partial specific Gibbs energy, g FW , of the very dilute anomalous solute in the parent solution is inferred from Eqs. (3.10) and (3.15) to depend only on the parent solution properties, in the form where R FW = R/A FW (Table A1) is the specific gas constant of the anomalous solute.The constant γ id FW is the limiting value of γ FW at infinite dilution and is formally introduced here to keep the arguments of the two logarithmic terms dimensionless after their separation; its numerical value is chosen such that the second term disappears at low concentrations.Note that γ FW is defined only up to an arbitrary constant factor which enters the reference state condition, Eq. (4.12), in combination with µ 0 FW .The partial Absolute Salinity, S SSW A , of the salt fraction with Reference Composition in BSW is related to the given molality, m BSW SSW , by means of Eq. (3.14).The chemical potential, µ 0 FW , of the anomalous solute in pure water at infinite dilution is FW , the salinities associated with the salts from the North Atlantic and from the local riverine inputs.The function g BSW depends on the known Gibbs function of SSW, g SW , and an unknown function, g FW , that represents the FW properties in the compact form of Eq. (3.16), and will be determined empirically from simulated data in the next section.
The partial Absolute Salinity, S BSW FW , Eq. (2.13), of the anomalous solute is related to its molality in BSW, m BSW FW , by In terms of the partial salinities S SSW A and S BSW FW , the Absolute Salinity of BSW, S BSW A , Eq. (2.11), is given by the formula (3.21) The salinity variable S BSW A is computed from the molar masses of all the dissolved species and is denoted by S soln A (the mass fraction of dissolved material in solution) in the nomenclature of Wright et al. (2010a).The function g FW depends on the concentration of the SSW part, S SSW A , and the anomalous composition of the FW part but according to Eq. (3.16) it is independent of the concentration, S BSW FW , of the FW part which is assumed to be very dilute.In the next section, an empirical correlation equation for g FW will be derived from model data computed using FREZCHEM (Marion and Kargel, 2008).

Fitting the Baltic Gibbs function to FREZCHEM simulation data
For arbitrary aqueous electrolyte solutions, the related Gibbs function in the form (Feistel and Marion, 2007) g(S A ,T ,P ) = g W (T ,P ) + S A (T ,P ) can be estimated from available Pitzer equations for the constituents using the FREZCHEM model.Here, S A is the Absolute Salinity (mass fraction of dissolved material) of the particular solution, g W is the Gibbs function of pure water, is the partial specific Gibbs energy at infinite dilution, R S is the specific gas constant of the particular solute, and is the activity potential, expressed in terms of the osmotic coefficient, φ, and the mean activity coefficient, γ , of the solution.Infinite dilution is the theoretical asymptotic state of a solution at which the mutual interaction between the solute particles is negligible as the result of their large pairwise separations.Activity coefficients γ are defined only up to an arbitrary constant factor; here, γ id is the limiting value to which the particular γ is normalized at infinite dilution, commonly, γ id = 1 kg mol −1 .Any change of this constant is compensated by the conditions, Eq. ( 4.12), imposed on the freely adjustable coefficients of seawater at the specified reference state (Feistel et al., 2008a).
Using the FREZCHEM model, the absolute salinity, S A = S BSW A , the activity potential, ψ, the specific volume, v = (∂g/∂P ) S A ,T , and the heat capacity, c P = −T (∂ 2 g/∂T 2 ) S A ,P , of Baltic seawater were computed for a number of grid points at given values of T , P , the chloride molality, m Cl (which determines the SSW contribution), and the Calcium molality anomaly, δm Ca (which determines the FW contribution).From these data and Eq.(4.1), an empirical correlation for the partial specific Gibbs energy, g FW , Eq. (3.16), was determined numerically by regression with respect to the anomalies relative to SSW, i.e., relative to δm Ca = 0.
To relate the given molalities, m Cl and δm Ca , to the arguments, m BSW SSW and m BSW FW , of the Gibbs function (3.19), suitable composition models must be specified.For SSW, the Reference Composition model gives Therefore, the SSW composition variable in Eq. (3.19) is obtained from m Cl by Eq. (3.13), In terms of constituents of the RC, the mole fractions of lime dissolved in FW are assumed here to be given by Eq. ( 2.3).
The only purpose of this reaction scheme is its use as a proxy to represent the complex marine carbonate chemistry simulated by FREZCHEM, in order to provide the theoretical Gibbs function model with reasonable molar fractions, Eq. (2.3), and molar masses, Eq. (4.6), of the anomalous solute.The related calcium anomaly of BSW is given by The total calcium molality in BSW is the sum of the SSW and the FW parts, Derived from the structure of the target function of the regression, Eq. (3.16), we use the polynomial expression (Feistel and Marion, 2007), where the dimensionless reduced variables are defined by (Feistel, 2008;IAPWS, 2008), The standard-ocean parameters S SO , T SO and P SO are given in Table A1.Comparing equal powers of T and P of the logarithmic term in Eqs.(3.16) and (4.8) in the limit x → 0, the coefficients r j k are analytically available from the relation to be The coefficients c 000 and c 010 are arbitrary and chosen to satisfy reference state conditions which determine the absolute energy and the absolute entropy of the anomalous solute.
Here we employ the reference state conditions From the Gibbs function (3.19) in conjunction with the functional form (4.8) we derive expressions for the available properties v, c P and ψ in terms of the remaining unknown coefficients, c = c ij k .These coefficients are then determined numerically by the requirement to minimise the penalty function, in which δv i , δc P i and δψ i are property anomalies of Baltic seawater relative to the parent solution at the grid points chemistry implemented in FREZCHEM.If, for example, results were calculated without allowing for the contribution from atmospheric CO2 in the reaction (1.1), then a mismatch between Millero's Rule and FREZCHEM of approximately 30% occurs in the modified results corresponding to Fig. 2; this difference results from the smaller molar mass of the solute, FW A , eq. ( 2.17), and hence the smaller contribution to salinity from the FW source eq.i of the FREZCHEM simulation results, weighted by estimated uncertainties ω.Selected examples of the data for δv i , δc P i and δψ i are displayed in Figs. 2, 3 and 4. In our Gibbs function, the original complex chemistry implemented in FREZCHEM is represented in the simplified form represented by the reaction (1.1) in conjunction with the analytical expression (4.8).Since Eq. (4.13) measures the deviation between the two numerical models, the uncertainties ω cover their numerical round-off and mutual misfit rather than any experimental accuracy.In practice, the ω values were suitably chosen to allow a reasonably smooth fit.Experimental uncertainties are irrelevant for the regression considered in this section and will be discussed in the subsequent section where the properties of the resulting Gibbs function (4.8) are analysed.The scatter of the FREZCHEM points relative to the fitted Gibbs function are shown in Figs. 5, 6 and 7.

Thermodynamic property anomalies
Various salinity measures such as Reference Salinity SR, Absolute Salinity, SA, Density Salinity, SD, or Chlorinity Salinity, SCl, have the same values for SSW but differ from each other for BSW.The estimate of Density Salinity based on inversion of the expression for density in terms of the Gibbs function for SSW at arbitrary values of temperature and pressure is represented by D S , and referred to as "measured" Density Salinity since it is based on whatever the conditions of the direct density measurement are.It is the Absolute Salinity of SSW (here assumed to have Reference Composition) that has the same density as BSW at given temperature and pressure, i.e., ( 5 . 1 ) Fig. 7. Scatter of the activity potential anomalies computed from FREZCHEM, δψ i , relative to the activity potential anomalies computed from the Gibbs function, δψ (c), Eq. (4.33), at 1260 given data points.The rms deviation of the fit is 3.1×10 −5 .Symbols 0-5 indicate the pressures of 0.1 MPa, 1 MPa, 2 MPa, 3 MPa, 4 MPa and 5 MPa, respectively.These residual anomalies should be compared with the total anomalies δψ i shown in Fig. 4.
-Absolute Salinity of anomalous seawater can be computed from its density using the TEOS-10 equation of state, and results in the same value at any temperature or pressure at which the density was measured, as well as that -the properties of anomalous seawater can be computed from the TEOS-10 Gibbs function if Absolute Salinity is used as the composition variable, and finally, the first two rules combined, that -the properties of anomalous seawater can be estimated by the TEOS-10 functions in terms of SSW properties evaluated at the same density, temperature and pressure.
In this section, we discuss the validity of Millero's Rule and compare the results derived from the FREZCHEM model with those from the TEOS-10 Gibbs function evaluated at the same Absolute Salinity.In the next section, we again discuss the validity of Millero's Rule and compare the results derived from the fitted Gibbs function of Baltic seawater with those from the TEOS-10 Gibbs function evaluated at the same Absolute Salinity or at the same density.
In Figs. 2, 3 and 4, the simulated FREZCHEM data are compared with those estimated from Millero's Rule, i.e., property differences computed from the already available TEOS-10 Gibbs function at the Absolute Salinities S BSW A and S SSW A .The very good agreement visible in Fig. 2 between the simulated density anomalies and those estimated from Millero's Rule depends on two factors.The first factor is how well the rule estimates the results of the FREZCHEM simulation.In other words, how consistent the rule is with the Pitzer equations for the specific volume in the special case of the Baltic seawater composition.The second factor is how well the simple static composition model of the anomaly, Eq. (1.1), used here for the construction of the Gibbs function with intentionally only two representative conservative composition variables, is capable of approximately covering the underlying complicated dynamic solute chemistry implemented in FREZCHEM.If, for example, results were calculated without allowing for the contribution from atmospheric CO 2 in the reaction (1.1), then a mismatch between Millero's Rule and FREZCHEM of approximately 30% occurs in the modified results corresponding to Fig. 2; this difference results from the smaller molar mass of the solute, A FW , Eq. (2.17), and hence the smaller contribution to salinity from the FW source Eq. (4.6), which changes the value of S BSW A used for Millero's Rule at a specified value of the Calcium molality anomaly.
The analytical expressions required in Eq. (4.13) for the fit of the anomalous properties are derived from Eqs. (3.15) and (4.8), in the form and (4.15) The required analytical formula for the activity potential anomaly δψ (c) expressed explicitly in terms of the TEOS-10 Gibbs function g SW and the Gibbs function correction, g FW , which depends on the unknown coefficients c, is more complicated to obtain.From the Gibbs function for BSW, g BSW , Eq. (4.1), the activity potential is derived, and similarly that of SSW, (4.17) Ocean Sci., 6, 949-981, 2010 www.ocean-sci.net/6/949/2010/After some algebraic manipulation of the difference between Eqs. (4.16) and (4.17), the activity potential anomaly, δψ (c), takes the form Here, A BSW is the molar mass of Baltic sea salt, the Gibbs function of pure water is and the partial specific Gibbs energy at infinite dilution is computed from Eq. (4.1) in the mathematical zero-salinity limit, (4.21) Since the TEOS-10 Gibbs function is defined as a series expansion in salinity, in the form (Feistel et al., 2010b), it follows immediately from Eq. (4.21) that SSW is given by SSW (T ,P ) ≡ g 2 (T ,P ). (4.23) The function BSW (T ,P ) in Eq. (4.18) is the coefficient of the linear salinity term of the Gibbs function g BSW and can be determined by comparison of the two different expressions available for g BSW , on the one hand, Eq. (4.1), in terms of Pitzer equations, and on the other hand, Eq. (3.19), in the form of a linear correction to TEOS-10, Note that g BSW in Eqs.(4.24) and (4.25) represent different approximations of the Gibbs function that we want to determine.The Gibbs function given by Eq. (4.24) is nonlinear in the anomaly.For the composition model given, its activity potential ψ BSW can be computed from complicated systems of Pitzer equations.To derive a simpler correlation function, we estimate ψ BSW here by means of the Gibbs function, Eq. (4.25), which is linear in the anomaly, S BSW FW .We consider the series expansions of Eqs.(4.24), (4.25) with respect to salinity s and require that the coefficients of the terms s 0 , s ln s and s 1 are identical in the two equations.As the small expansion parameter we choose s ≡ S BSW A under the condition that the composition ratio r ≡ S BSW FW /S BSW A remains constant in the mathematical limit s → 0.
In terms of s and r, the salinity variables are The truncated series expansions are for Eq.(4.24), for Eq.(4.25), for Eq.(4.22), and for Eq.(3.16), Note that the limiting laws of ψ BSW and ln γ /γ id are of the order O s 1/2 .The combination of Eqs.(4.28), (4.29), (4.30) gives Here we used the specific gas "constant" Note that BSW (T ,P ) depends on the composition of BSW, in particular on the ratio r = S BSW FW /S BSW A of the two independent salinity variables.
In Eq. (4.18), we replace BSW by Eq. (4.32) and get the final formula for the required activity potential anomaly, δψ (c), Here, the saline part of the Gibbs function of SSW is g S S SSW A ,T ,P =g SW S SSW A ,T ,P −g SW (0,T ,P ), (4.34) or, using Eq.(4.22), Similarly, the saline part of the partial Gibbs function of freshwater solute is defined by g F S SSW A ,T ,P = g FW S SSW A ,T ,P − µ 0 FW (T ,P ) or, using Eq.(3.16), Note that in the zero-salinity limit of Eq. (4.33), the singularity lim g F S SSW A ,T ,P of Eq. (4.37) cancels exactly with the corresponding singularity of g S /S SSW A , Eq. (4.35).In Eq. (4.33), all terms are known at the FREZCHEM data points except for g F which depends on the set of coefficients c = c ij k to be adjusted by the regression, Eq. (4.13).After this compilation, the reference state conditions, Eq. (4.12), must be satisfied.After setting c 000 = 0 and c 010 = 0 in g FW , the final values are computed from the equations c 000 = − g FW (S SO ,T SO ,P SO ) The results for the coefficients are given in (5.1) In contrast, the true Density Salinity is defined to be strictly conservative and represented by S dens A in the nomenclature of Wright et al. (2010a).To ensure that it is independent of temperature and pressure, it is computed using Eq. ( 5.1) evaluated at T = 298.15K and P = 101325 Pa, and is by definition the same for the given sample at any other T or P .
Chlorinity Salinity, S Cl , is the Absolute Salinity of SSW that has the same Chlorinity as BSW, Density Salinity and Chlorinity Salinity can be measured in the Baltic Sea; readings are currently related by the approximate empirical relation (Feistel et al., 2010a) in the form of Eq. (2.16),    S , computed from eq. (5.1) for Baltic seawater at the standard ocean surface pressure and temperatures between 0 and 25 °C.The uncertainty of Density Salinity measurements is 2 g m -3 / (βρ) = 2.5 mg kg -1 (Feistel et al., 2010a), indicated by the solid horizontal lines.
The density anomaly of the Baltic Sea is shown in Fig. 9 as the difference between the densities with and without the freshwater solute, i.e., of SSW and BSW with the equal Chloride molalities (roughly, equal chlorinities),     5.6), between the densities with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure and temperatures between 0 and 25 °C.The uncertainty of density measurements is 2 g m -3 (Feistel et al., 2010a), indicated by the solid horizontal line.
The Baltic Sea anomaly of the thermal expansion coefficient is shown in Fig. 10 as the difference between the coefficients with and without the freshwater solute, i.e., of SSW and BSW with equal Chloride molalities (roughly, equal chlorinities), Using βρ ≈ 0.8 × (10 6 g m −3 )/(10 6 mg kg −1 ), it is seen that division of the numerical values of δρ/(g m −3 ) in Fig. 9 by 0.8 provides an approximate conversion to the units used in Fig. 8 so that comparison of the results in these two figures reveals that the relative errors associated with using S D in place of S BSW A to estimate salinity anomalies due to the addition of calcium carbonate is at most 25%, and only about 2.5% for a typical brackish salinity value of S SSW A ≈ 8 g kg −1 .Note that the salinity change associated with the added calcium carbonate solute (S D − S SSW A ) is itself a small fraction of the salinity change associated with the addition of fresh water (S SO − S SSW A ). Using Eq. (5.4), the ratio is approximated by (S D − S SSW A )/(S SO − S SSW A ) ≈ (130 mg kg −1 )/S SO ≈ 0.4%.
The Baltic Sea anomaly of the thermal expansion coefficient is shown in Fig. 10 as the difference between the coefficients with and without the freshwater solute, i.e., of SSW and BSW with equal chloride molalities (roughly, equal Chlorinities), The uncertainty of the TEOS-10 thermal expansion coefficient is estimated as 0.6 ppm K -1 , so the Baltic anomalies are within the uncertainty and can in practice be neglected.(5.7), between the thermal expansion coefficients (solid lines) with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure and temperatures between 0 and 25 °C, in comparison to estimates from Millero's Rule, D δα , based on Density Salinity (dashed lines), eq. (5.8), and A δα , based on Absolute Salinity (dotted lines, temperatures not labelled), eq.(5.9).For the latter two, the responsible difference between BSW A S and D S is shown in Fig. 8.The estimated experimental uncertainty of the thermal expansion coefficient is 0.6 Fig. 10.Difference δα, Eq. (5.7), between the thermal expansion coefficients (solid lines) with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure and temperatures between 0 and 25 • C, in comparison to estimates from Millero's Rule, δα D , based on Density Salinity (dashed lines), Eq. (5.8), and δα A , based on Absolute Salinity (dotted lines, temperatures not labelled), Eq. (5.9).For the latter two, the responsible difference between S BSW A and S D is shown in Fig. 8.The estimated experimental uncertainty of the thermal expansion coefficient is 0.6 ppm K −1 (Feistel and Hagen, 1995;IAPWS, 2008)   S SSW A ,0,T ,P SO g BSW P S SSW A ,0,T ,P SO .(5.9) The uncertainty of the TEOS-10 thermal expansion coefficient is estimated as 0.6 ppm K −1 , so the Baltic anomalies are within the uncertainty and can in practice be neglected.
For seawater with varying composition, there are several ways to define the haline contraction coefficient, depending on the particular thermodynamic process by which the composition is changing with salinity.Here we consider the anomalous contraction coefficient which provides the density change with respect to the addition of freshwater solute  5.12), between the haline contraction coefficients (solid lines) of the parent solution with respect to the addition of FW solute and of SSW solute for Baltic seawater.Values are determined at the standard ocean surface pressure and temperatures between 0 and 25 °C.The standard-ocean value of the haline contraction coefficient is 0.781 = 781 ppm g -1 kg.The haline contraction coefficient associated with the addition of calcium carbonate is within 20% of the haline contraction coefficient for Standard Seawater.
The Baltic Sea anomaly of the isobaric specific heat is shown in Fig. 12 as the difference between the values with and without the freshwater solute, i.e., of SSW and BSW with the equal chloride molality (roughly, equal Chlorinity), The Baltic Sea anomaly of the isobaric specific heat is shown in Fig. 12 as the difference between the values with and without the freshwater solute, i.e., of SSW and BSW with the equal chloride molality (roughly, equal Chlorinity),  5.13), between the specific isobaric heat capacity (solid lines) with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure and temperatures between 0 and 25 °C, in comparison to estimates from Millero's Rule, D δ P c , based on Density Salinity (dashed lines), eq.(5.14), and A δ P c , based on Absolute Salinity (dotted lines, temperatures not labelled), eq.(5.15).For the latter two, the responsible difference between BSW A S and D S is shown in Fig. 8.The experimental uncertainty of cP relative to pure water is 0.5 J kg -1 K -1 , as indicated by the solid horizontal line.A typical value for the heat capacity of water or seawater is 4000 J kg -1 K -1 .The changing curvature of the solid curves below 5 g kg -1 is probably a numerical edge effect of the regression.
The sound speed c is computed from the Gibbs function g using the formula, ( 5 . 1 6 ) The Baltic Sea anomaly of the speed of sound is shown in Fig. 13 as the difference between the values with and without the freshwater solute, i.e., of SSW and BSW with equal chloride molalities (roughly, equal Chlorinities), Fig. 12. Difference δc P , Eq. ( 5.13), between the specific isobaric heat capacity (solid lines) with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure and temperatures between 0 and 25 • C, in comparison to estimates from Millero's Rule, δc D P , based on Density Salinity (dashed lines), Eq. (5.14), and δc A P , based on Absolute Salinity (dotted lines, temperatures not labelled), Eq. (5.15).For the latter two, the responsible difference between S BSW A and S D is shown in Fig. 8.The experimental uncertainty of c P relative to pure water is 0.5 J kg −1 K −1 , as indicated by the solid horizontal line.A typical value for the heat capacity of water or seawater is 4000 J kg −1 K −1 .The changing curvature of the solid curves below 5 g kg −1 is probably a numerical edge effect of the regression.
The anomalies of c P remain with the experimental uncertainty of 0.5 J kg −1 K −1 , Fig. 12.The errors associated with using Millero's Rule are similar to those associated with simply neglecting the FW solute and are again negligible.
The sound speed c is computed from the Gibbs function g using the formula, c = g P g T T g 2 T P − g T T g P P . (5.16) The Baltic Sea anomaly of the speed of sound is shown in Fig. 13  Rule is only slightly better than totally neglecting the influence of the FW solute on sound speed estimates.In eq. ( 5.16), the largest contribution to the sound speed anomaly comes from the anomaly of the compressibility, gpp, which is of order of magnitude up to 0.07%.Compressibility estimates from FREZCHEM have larger uncertainties than e.g.those of the density or the heat capacity (Feistel and Marion, 2007).5.17), between the sound speed (solid lines) with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure Fig. 13.Difference δc, Eq. (5.17), between the sound speed (solid lines) with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure and temperatures between 0 and 25 • C, in comparison to estimates from Millero's Rule, δc D , based on Density Salinity (dashed lines), Eq. (5.18), and δc A , based on Absolute Salinity (dotted lines, temperatures not labelled), Eq. (5.19).For the latter two, the responsible difference between S BSW A and S D is shown in Fig. 8.The experimental uncertainty of c is 0.05 m s −1 , indicated by the solid horizontal line.
The anomalies of c are much larger than the experimental uncertainty of 0.05 m s −1 , Fig. 13 and poorly approximated by Millero's Rule.Except at very low salinities, use of Millero's Rule is only slightly better than totally neglecting the influence of the FW solute on sound speed estimates.In Eq. (5.16), the largest contribution to the sound speed anomaly comes from the anomaly of the compressibility, g pp , which is of order of magnitude up to 0.07%.Compressibility estimates from FREZCHEM have larger uncertainties than e.g.those of the density or the heat capacity (Feistel and Marion, 2007).
Because of the freely adjustable constants, only relative enthalpies can reasonably be compared between samples that have different compositions.The Baltic Sea anomaly of the relative specific enthalpy is shown in Fig. 14 as the difference of relative enthalpies between the values with and without the freshwater solute, i.e., of SSW and BSW with the equal chloride molalities (roughly, equal Chlorinities),  5.20), between the relative specific enthalpies (solid lines) with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure and temperatures between 5 and 25 °C, in comparison to estimates from Millero's rule, D δh , based on Density Salinity (dashed lines, only the 15 -25 °C results are labelled), eq. ( 5.21), and A δh , based on Absolute Salinity (dotted lines, temperatures not labelled), eq.(5.22).For the latter two, the responsible difference between BSW A S and D S is shown in Fig. 8.The experimental uncertainty of the relative enthalpies is 0.5 J kg -1 × t /°C.
For the computation of the freezing temperature of Baltic seawater we need a formula for the chemical potential, µW, of water in Baltic seawater similar to µ0 in eq.(3.1), but on a mass rather than on a particle number basis: ( 5 . 2 3 ) Here, µW is defined by (5.22) For the computation of the freezing temperature of Baltic seawater we need a formula for the chemical potential, µ W , of water in Baltic seawater similar to µ 0 in Eq. (3.1), but on a mass rather than on a particle number basis: Here, µ W is defined by and apply the chain rule, (5.26) to obtain the result (5.27) This general formula is simplified in our case using the linear expression Eq. (3.19), to give: (5.28) At the freezing point, T f S SSW A ,S BSW FW ,P , the chemical potential µ W equals that of ice, µ Ih (IAPWS, 2009b): (5.29) The Baltic Sea anomaly of the freezing temperature is shown in Fig. 15 as the difference of freezing points between the values with and without the freshwater solute, i.e., of SSW and BSW with the equal chloride molalities (roughly, equal Chlorinities), as a function of Density Salinity, computed from Eqs. (5.4) and (5.5).For comparison, the anomaly is estimated by Millero's Rule using Density Salinity S D , Eq. ( 5.1), from The experimental uncertainty of the freezing temperature of seawater is 2 mK.The anomaly is of the same order of magnitude and can normally be ignored.Millero's Rule does not provide much improvement over neglecting the anomalies.( ) , is computed from the condition that the chemical potential of water in seawater, W μ , eq. ( 5.28), equals that of vapour, g V (IAPWS, 2009a, Feistel et al., 2010b): (5.33) The Baltic Sea anomaly of the vapour pressure is shown in Fig. 16 as the difference of pressures between the values with and without the freshwater solute, i.e., of SSW and BSW with the equal Chloride molalities (roughly, equal chlorinities), The experimental uncertainty of the freezing temperature of seawater is 2 mK.The anomaly is of the same order of magnitude and can normally be ignored.Millero's Rule does not provide much improvement over neglecting the anomalies.The vapour pressure of Baltic seawater, P vap S SSW A ,S BSW FW ,T , is computed from the condition that the chemical potential of water in seawater, µ W , eq. (5.28), equals that of vapour, g V (IAPWS, 2009a, Feistel et al., 2010b): A ,S BSW FW ,T ,P vap = g V T ,P vap . (5.33) The Baltic Sea anomaly of the vapour pressure is shown in Fig. 16 as the difference of pressures between the values with and without the freshwater solute, i.e., of SSW and BSW with the equal chloride molalities (roughly, equal Chlorinities), (5.36) The anomalies shown in Fig. 16 are a factor of 10 smaller than the uncertainty of the most accurate experimental data (Robinson, 1954;Feistel, 2008).(5.36) The anomalies shown in Fig. 16 are a factor of 10 smaller than the uncertainty of the most accurate experimental data (Robinson, 1954;Feistel, 2008).
The "measured" Density Salinity S D is given by Eq. ( 5.1) as a function of S SSW A , S BSW FW , T and P .When a sample's temperature is changing, its molalities m Cl and δm Ca are conservative, and so are the salinities S SSW A and S BSW FW computed from Eqs. (4.4) and (4.6).On the contrary, Density Salinity, Eq. (5.1), is not strictly conservative unless the thermal expansion coefficient and compressibility of BSW happen to be exactly the same as those for SSW. Figure 17   FW is conservative with respect to the temperature.Density Salinities are less sensitive to temperature changes than density measurements but may need to be stored together with the temperature at which they were determined.Note that the mass fraction of anomalous solute in Baltic seawater is larger than that present anywhere in the deep ocean.For a typical Baltic Sea salinity of 8 g kg −1 the mass temperature changes than density measurements but may need to be stored together with the temperature at which they were determined.Note that the mass fraction of anomalous solute in Baltic seawater is larger than that present anywhere in the deep ocean.For a typical Baltic Sea salinity of 8 g kg -1 the mass fraction of anomalous solute is approximately 0.004 × (35 -8) g kg -1 = 0.108 g kg -1 , about 7 times as large as the maximum mass fraction of anomalous solute in the deep North Pacific where composition anomalies are largest in the open ocean. is not necessarily zero for temperatures different from 25 • C; typical results are shown in Fig. 18.These density errors are relatively small in comparison to the typical Baltic density anomalies of 50-100 g m −3 that are associated with fresh water solute (Fig. 9).The anomalies discussed in this section describe the differences between thermodynamic properties of BSW and of SSW if both have the same Absolute Salinity of the SSW part, S SSW A .For a given sample of BSW, S SSW A can for instance be determined from a Chlorinity measurement.This is expensive and time-consuming, cannot be carried out in Ocean Sci., 6, 949-981, 2010 www.ocean-sci.net/6/949/2010/situ and usually requires skilled personnel, in contrast to routine CTD casts that automatically produce in-situ readings of Practical Salinity, S P .Due to the electrolytic conductivity of the freshwater solute, the relation between S SSW A and S P of BSW is influenced by a significant anomaly that cannot be estimated from the Gibbs function g BSW .This problem is addressed in the following section.

Anomalies of Conductivity, Practical Salinity and Reference Salinity
Conductivity is a non-equilibrium, transport property of seawater and is not available either from the TEOS-10 Gibbs function, or from the FREZCHEM model, which provides only equilibrium thermodynamic properties.Since Practical Salinity, the currently most important solute concentration measure in oceanography, is determined from conductivity measurements, it is important to estimate the effects of the Baltic composition anomaly on measured conductivities.This conductivity effect could reduce or increase the difference between the actual thermodynamic properties of Baltic water and those determined for Standard Seawater diluted to the same conductivity, relative to the differences between the actual thermodynamic properties of Baltic water and those determined for Standard Seawater diluted to the same chloride molality which were discussed previously in Sect. 5.These property differences for waters of the same conductivity will be discussed in Sect.7, once we have determined how conductivity is affected by the composition changes present in the Baltic.In addition, predictions of conductivity also allow us to validate at least some of the model calculations against actual observations.At present, theoretical models of aqueous solution conductivity, based on arbitrary chemical composition, are not accurate enough to study the Baltic (or any other) anomalous seawater directly.However, the composition/conductivity theory of Pawlowicz (2008), which is valid for conductivities in limnological low salinity situations, has been adapted (Pawlowicz, 2009;Pawlowicz et al., 2010) using a linearization about the known characteristics of Standard Seawater to study changes in composition/conductivity/density relationships in seawater, arising from small composition perturbations that originate from biogeochemical processes.This linearization approach, implemented in the numerical model LSEA DELS, is now used to investigate changes in the relationship between Chlorinity and conductivity-based Reference Salinity, using our idealized model of the Baltic composition anomaly, Eq. (1.1).All considerations in this section refer to conditions at an arbitrary temperature, set to 25 • C unless otherwise specified, and atmospheric pressure, P = 101325 Pa.However, these parameters are omitted from the formulas for notational simplicity. .Deviation (5.37) between the density of Baltic seawater and the density computed from conservative Density Salinity, S dens A , Eq. (5.38).The experimental uncertainty of density measurements is 2 ppm (Feistel et al., 2010a), indicated by the solid lines.

Definitions
The starting point of simulations is a composition vector C SSW , specifying the molar composition of all constituents in a base seawater.In contrast to the development in Sect.2, but more straightforwardly linked to the structure of the Gibbs function (3.19), this base seawater is not an "ocean end member" with S P = 35.Instead, it is SSW diluted by the addition of pure water so that chloride molality will remain unchanged as the calcium carbonate solute is "added" to create Baltic water.The conductivity κ SSW = κ C SSW and density ρ SSW = ρ C SSW of this water depend on the composition, and the true mass fraction of dissolved material (Solution Salinity) will be S SSW A .Since this water is just a dilution of SSW, the Reference Salinity: based on using the observed conductivity in the algorithm S P (.) specified by the Practical Salinity Scale 1978, is scaled by an appropriate choice of the constant u P to give the Solution Salinity S SSW A .The factor u P is not exactly the same as u PS when anomalies are being calculated because LSEA DELS calculations are based on a SSW composition model that slightly differs from the RC (Wright et al., 2010a).
The composition of Baltic seawater is described by the composition vector C BSW .Exact details of the way in which C BSW is related to C SSW are discussed in Section 6.2, but both compositions have the same chloride molality.The composition C BSW has a Solution Salinity S BSW A , a conductivity κ BSW = κ C BSW and a density ρ BSW = ρ C BSW that will differ from that of the base seawater.All of these parameters can be estimated using LSEA DELS once the as computed from the model results is then directly comparable to that calculated using Eq.(5.6).This parameter can therefore be used to validate the densities calculated by LSEA DELS against the Gibbs function (itself based on FREZCHEM model calculations).In addition, the change in Solution Salinity between the original base seawater and the Baltic water is, Eq. (3.21): The approximation is valid when the amount of solute added is small, as it is in this case.
Typically, conductivity measurements in the ocean are used with SSW parameterizations for different properties under the assumption that the properties of the measured water are well-modelled by the properties of SSW diluted to the same conductivity.Thus we infer a third "reference" water type, described by a composition vector C BSW R , with Solution Salinity S BSW R , whose composition is that of SSW diluted by pure water, but whose conductivity matches that of BSW: The Solution Salinity of the reference water is then the Reference Salinity of the Baltic Sea water.The ultimate purpose of the modelling in this section is then to compare the change in the Reference Salinity between Baltic Sea water and diluted Standard Seawater of the same conductivity with the actual Solution Salinity change S BSW FW from Eq. (6.3).If the added solute has the same conductivity as that of sea salt, then S R = S BSW FW .If the added solute is not conductive, then S R = 0, irrespective of the value of S BSW FW .In addition, the density of this reference water, denoted as the reference density ρ BSW R = ρ C BSW R , will differ from the true density of Baltic water ρ BSW , and the change between the true and reference densities can then be directly compared with measurements of the density anomaly in the Baltic.Previous investigations have suggested that LSEA DELS calculations for S R have an error of between 1 and 10%, depending on the details of the composition anomaly.This uncertainty ultimately arises from uncertainties in the basic chemical data for binary electrolytes from which model parameters for the conductivity algorithm were extracted, as well as inadequacies in the theoretical basis of the model at higher salinities.Errors in the LSEA DELS density algorithms are themselves much smaller than those for conductivities, but since the Reference Salinity calculation implicitly involves conductivity changes, errors in conductivity will carry over into the density anomaly calculation.
The calculations described above can be carried out at any desired temperature.However, the temperature-dependence of the conductivity and density of seawaters may also vary with the composition anomaly.This implies that the value of S BSW R as calculated above may have a slight temperature dependence.For Baltic seawater, this non-conservative effect was shown experimentally to remain within the measurement uncertainty (Feistel and Weinreben, 2008), and neglect of this effect is also supported by numerical experimentation with LSEA DELS, which suggest the maximum error is less than 0.001 g kg −1 .

Composition anomalies
Although the Baltic Sea composition anomaly is idealized in this paper as arising from the addition of calcium carbonate, calcium itself is not directly measured in the Baltic.However, anomalies in the Total Alkalinity (TA), defined in LSEA DELS as and Dissolved Inorganic Carbon (DIC), defined as are known to be approximately equal.In this section, the usual chemical notation of total stoichiometric molalities by brackets [..] is preferred for convenience.Thus we assume for the anomalies δTA = δDIC.(6.8) The addition of Ca 2+ is then inferred from mass and charge balance considerations: δm Ca ≡ Ca 2+ = δTA/2.(6.9)Using Eqs.(6.6)-(6.9), the complete composition at any particular chloride molality can be determined as a function of the molality of the calcium anomaly.This will provide a direct comparison with the Gibbs function described in Sect. 4. In order to apply these calculations specifically to the Baltic (i.e. as in Sect.5), we relate some parameter to a function of the chloride salinity S Cl (or, alternatively, any other salinity measure) in the Baltic.The value of δTA at a chloride salinity of zero, which is taken as an endpoint of linear correlations in mixing diagrams, is estimated from observations to be 1470 µmol kg −1 (Feistel et al., 2010a).The TA anomaly in Baltic waters is then δTA = 1470 µmol kg −1 × 1 − S Cl S SO (6.10) Eqs. (6.8)-(6.10),hereafter denoted as "model-1", then specify the composition C BSW of Baltic water at all chloride Ocean Sci., 6, 949-981, 2010 www.ocean-sci.net/6/949/2010/ The LSEA_DELS model calculations for R δρ , eq. ( 6.2), using model-1 anomalies can also be compared directly (Fig 20) against calculations from the Gibbs function, eq.(5.6), with the Baltic anomaly being modelled using eq.(5.4).This is a complete intercomparison of not only the density algorithms but also different approaches for specifying the composition anomalies.The two independent calculations agree quite well, with values being within 6 3 m g − of each other at all temperatures.(Feistel et al., 2010a) as well as Eq. ( 5.3), and model predictions.
molalities.However, the composition is only specified in terms of aggregate variables TA and DIC.A carbonate chemistry model within LSEA DELS, based on equations for the equilibrium chemistry, is used to calculate the complete ionic chemical composition in a new chemical equilibrium.This involves changes to CO 2 , HCO − 3 , CO 2− 3 , B(OH) 3 and B(OH) − 4 , as well as to pH and pCO 2 .Although the actual compositional perturbation is now somewhat more complex than indicated by Eq. (1.1) almost all of the change that occurs at the pH of seawater is described by an increase in HCO − 3 , similar in LSEA DEL and in FREZCHEM.From Eqn. (1.1), the change in Solution Salinity due to the added mass of dissolved solute is S BSW FW ≈ 162.1 g mol −1 × δm Ca (i.e., the molar mass of Ca(HCO 3 ) 2 times the change in calcium molality, neglecting the change in the mass of solution).The change in Solution Salinity calculated directly from the full chemical compositions used by LSEA DELS is less than 3% larger than this value, which is insignificant here in comparison with other uncertainties.This procedure allows us to determine the conductivity and density anomalies at a particular S Cl within the Baltic.
Later we will discuss whether disagreements between the model predictions and observations of density anomalies arise from inadequacies in LSEA DELS, or whether they are inherent to the idealized composition anomaly used to model Baltic seawater.For this purpose we introduce a second model for composition anomalies in the Baltic that is slightly more complex.Sulfate is the next largest component of the actual Baltic composition anomaly after calcium carbonate.The sulfate anomaly is estimated (Feistel et al., 2010a) to have a zero-Chlorinity limit of about 166 µmol kg −1 (with a considerable uncertainty), With anomalies in both Ca 2+ and SO 2− 4 , charge balance considerations now require a modification to Eq. (6.9) to balance the charge associated with the sulfate anomaly, δ Ca 2+ − δ SO 2− 4 = δTA/2, (6.12) which will increase the size of the calcium anomaly.

Model validation
Although the model/data predictions will be shown to be in rough agreement, it is useful at this stage to enumerate possible sources of disagreement.The first potential source of disagreement is the error in density anomaly predictions from the conductivity model, which can themselves be in error by as much as 10% for a given composition anomaly.The second potential source is the idealization of the composition anomaly, which is only a simplified version of the true Baltic composition anomaly.This error can be investigated by comparing model-1 and model-2 predictions.A third potential source of disagreement is inhomogeneities in the chemical composition of the Baltic, which will tend to scatter results at a particular Chlorinity over a wider range than predicted by measurement uncertainty alone.A final potential source of disagreement is measurement uncertainty in the data itself.Feistel et al. (2010a) report 437 observations of the density anomaly δρ R in the Baltic Sea over the years 2006-2008, mostly at salinities of 10-20 g kg −1 .66 of these replicate measurements on water were obtained from 11 stations.The observations (Fig. 19a) show a large scatter.Part of this scatter arises from observational error in the density measurements, which can be estimated at about ±9 g m −3 (coverage factor 2) from replicate values about the means.However, scatter in excess of this value is present.The additional scatter likely derives from spatial variations in the magnitude and composition of the anomaly.The concentrations of TA in different rivers inflowing into the Baltic can vary by an order of magnitude, and these effects are not always well-mixed within the Baltic.In addition, the solute is subject to various complex chemical processes and interaction with the sediment over the residence time of 20-30 years.
In general, model calculations of δρ R using either model-1 or model-2 are quite consistent with the observations (Fig. 19a), within the limits of observational uncertainty and presumed spatial inhomogeneity.LSEA DELS predicts an anomaly of zero at S R = 35.16504g kg −1 , rising to 48 and 58 g m −3 for model-1 and model-2 anomalies respectively, www.ocean-sci.net/6/949/2010/Ocean Sci., 6, 949-981, 2010 at S R = 5 g kg −1 .The scatter in the observations is large enough that it is not clear which of the two models better describes the data.The model-2 results fall somewhat closer to the raw data at salinities of 15-20 g kg −1 .On the other hand, although both models predict much larger density differences than are observed at salinities <5 g kg −1 , the comparison is better for model-1.It should be noted that the small number of observations in this low-salinity range are from the Gulfs of Bothnia and Finland (Feistel et al., 2010a), Fig. 1, which are not representative of the freshwater inflows as a whole.Hence, complete agreement is not expected.We conclude that spatial inhomogeneities in the composition anomalies are likely the limiting factor in the present model/data comparison, rather than the accuracy of LSEA DEL itself.
The LSEA DELS calculations for both model-1 and model-2 anomalies suggest that δρ R is not a linear function of the salinity, but rather one with a pronounced downward curvature, especially at low salinities.The curvature is large enough that there is little change in predicted anomalies at salinities less than 5 g kg −1 .This downward curvature is somewhat consistent with the low density anomalies observed for S R < 5 g kg −1 , although as just discussed the lack of data makes it unlikely that the observed values are completely representative of mean Baltic values.The curvature in the model results arises because conductivity changes will account for an increasingly large proportion of the total salinity change at low salinities, although this will not become clear until Sect.6.4.
The δρ R observations are derived from measurements of density and conductivity.A small number of measurements were also made of density and Chlorinity in 2008 (Feistel et al., 2010a).Comparison of differences between Density Salinity and Chlorinity Salinity from these observations (Fig. 19b) against predictions using model-1 and model-2 anomalies again shows reasonably good agreement, with predictions using model-1 anomalies closer to the approximate empirical parameterization, Eq. (5.3).In this case, conductivity effects are not involved and the model curves are nearly straight lines, deriving from the straight lines in Eqs.(6.10) and (6.11).Although the expanded uncertainty (coverage factor 2) of the Chlorinity measurements is about 0.5% (Feistel et al., 2010a), the relationships, Eqs.(6.10), (6.11) are themselves fits to scattered data (again probably reflecting inhomogeneities in the Baltic's chemical composition), so better agreement is not expected.
The LSEA DELS model calculations for δρ R , Eq. (6.2), using model-1 anomalies can also be compared directly (Fig. 20) against calculations from the Gibbs function, Eq. (5.6), with the Baltic anomaly being modelled using Eq.(5.4).This is a complete intercomparison of not only the density algorithms but also different approaches for specifying the composition anomalies.The two independent calculations agree quite well, with values being within 6 g m −3 of each other at all temperatures.(Feistel et al., 2010a), with LSEA_DELS model predictions.b) Comparison of model results with 3 observational estimates of the anomalies between Density Salinity SD and the Chlorinity Salinity SCl (Feistel et al., 2010a) as well as eq.( 5.3), and model predictions  .In this section we determine a correction factor for conductivity effects as a function of the same parameters using LSEA_DELS with the model-1 parameterization.
First, calculating R S Δ , eq. ( 6.13), for a grid of points in the range 0 < SCl < 35 g/kg and 0 < δmCa < 800 µmol/kg, we find that the calculated change in conductivity-based Reference Salinity, decreases significantly for a fixed δmCa as the salinity increases (Fig. 21).This reflects a commonly observed phenomenon that the conductivity per mole of charges (the equivalent conductivity), decreases as concentrations increase in solutions where the amount of solute is much less than the amount of solvent (Pawlowicz, 2008).The physical effects

Corrections to Practical Salinity required for Gibb function calculations
The Gibbs function determined in Sect. 4 is a function of chloride molality and the calcium anomaly, or equivalently S SSW A and S BSW FW .In this section we determine a correction factor for conductivity effects as a function of the same parameters using LSEA DELS with the model-1 parameterization.
First, calculating S R , Eq. (6.13), for a grid of points in the range 0 < S Cl < 35 g kg −1 and 0 < δm Ca < 800 µmol kg −1 , we find that the calculated change in conductivity-based Reference Salinity, decreases significantly for a fixed δm Ca as the salinity increases (Fig. 21).This reflects a commonly observed phenomenon that the conductivity per mole of charges (the equivalent conductivity), decreases as concentrations increase in solutions where the amount of solute is much less than the amount of solvent (Pawlowicz, 2008).The physical effects which reduce electrolytic conductivity are the relaxation force, electrophoresis and ion association; each of them tends to strengthen with increasing ion concentration (Ebeling et al., 1977(Ebeling et al., , 1979)).This change is largest at the lowest concentrations, with the decreases from its infinite dilution endpoint being proportional to √ S Cl in this limit, in accordance with limiting laws.At lower temperatures, S R for a given addition δm Ca is slightly larger than at higher temperatures.However, at all temperatures the changes S R are almost perfectly proportional to the magnitude of the composition anomaly.As expected, the ratio of S R to S BSW FW still depends significantly on S SSW A = S Cl / 1 − S BSW FW ≈ S Cl , Eq. ( 5.2), and also shows a slight temperature dependence.The results can be fit to an equation of the form, (6.14)where the reduced variables are τ = (T − 298.15 K)/(1 K) and ξ = S SSW A / 1 g kg −1 , and the coefficients a ij are given in Table 3. Numerical check values are available from Table A2.
The root-mean-square error of this fit is 5.3 × 10 −4 , but note that the model results themselves may be biased by as much as 0.05 (i.e., 10%).In Sect.7, Eqs.(6.13) and (6.14) will be used in conjunction with Eq. (3.19) to determine thermodynamic anomalies for waters of a measured conductivity.
Overall, conductivity changes will account for about 30-50% of the total change in salinity resulting from the presence of the anomaly, with the lower percentages occurring at highest salinities.
It had been shown experimentally that estimates of the Practical Salinity of Baltic seawater are independent of the sample temperature, within reasonable uncertainty (Feistel and Weinreben, 2008).From Eq. (6.13) and Fig. 21 we infer a weak temperature dependence of the Reference Salinity S R at constant S SSW A and S BSW FW if S R = u PS × S P is computed from Practical Salinity S P of Baltic seawater.Figure 23 shows the deviation from Practical Salinity conservation, FW /u PS , (6.15) as a function of salinity S SSW A and temperature T , where S BSW FW is estimated from the empirical relations (5.4), (5.5), and the abscissa value from Eq. (6.13), S R = S SSW A + f S SSW A ,T S BSW FW .The model results suggest that the measured salinity will vary by no more than 0.001 over a 15 degree temperature change at Practical Salinities of 5 to 10. Experimental evidence (Feistel and Weinreben, 2008) finds that any changes are smaller than this value, i.e., the violation of conservation does not exceed the measurement uncertainty of salinity.

Computation of properties from Practical Salinity readings
Regular oceanographic practice in Baltic Sea observation (Feistel et al., 2008b) ignores composition anomalies; readings of Practical Salinity are commonly inserted directly into SSW formulas to compute seawater properties.For conductive anomalies such as in the Baltic Sea, using Practical Salinity (or Reference Salinity S R ) rather than Chlorinity Salinity S Cl as the input of the Gibbs function can be expected to result in a better approximation of the anomalous property (Lewis, 1981).Nevertheless, the related error in density is known from direct density measurements (Millero and Kremling, 1976;Feistel et al., 2010a).The corresponding errors of other computed properties such as sound speed, freezing point or enthalpy are simply unknown even though they may be relevant for, say, echo sounding or submarine navigation.In this section we first estimate typical errors related to this practice and eventually provide algorithms for their reduction, based on the results of the previous sections.In Sect.5, the deviations from SSW properties are discussed for given Density Salinities S D which are not available from regular CTD measurements.However, our models directly estimate S BSW FW and S R as functions of S SSW A , so we can easily compute and display pairs (δq R , S R ) using S SSW A as a running dummy variable, where δq R is the error of a property computed from the Gibbs function g BSW between the salinity pairs (S SSW A , S BSW FW ), the "true salinity", and (S R , 0), the "conductivity salinity".At the end of this section we shall invert the relations used in this procedure in order to estimate S SSW A and S BSW FW from practically measured values of S R and eventually compute more accurate property estimates from the Gibbs function g BSW , but first we consider a more theoretical approach in which S SSW A is treated as if it were measured.
For a given point S SSW A ,T ,P , we compute the empirical Baltic Density Salinity anomaly from Eq. (5.4),   3. Numerical check values are available from Table A2.
Table 3: Coefficients of the correlation function f, eq.(6.14) i j aij i j aij 0 0 +0.578390505245625 0 1 -0.000180931852871 1 0 -0.089779871747927 1 1 -0.000294811756809 2 0 -0.001654733793251 2 1 -0.000012798749635 3 0 +0.012951706126954 3 1 +0.000079702941453Reference Salinity is then available from Eqs. (6.4), (6.13) and ( 6 The anomaly-related error of any considered property q available from the Gibbs function g BSW S SSW A ,S BSW FW ,T ,P , Eq. (3.19), is calculated as the difference between the best model estimate, q BSW , and the result q SW obtained using Reference Salinity, S R = u PS × S P , in the TEOS-10 Gibbs function: δq R = q BSW S SSW A ,S BSW FW ,T ,P − q SW (S R ,T ,P ). .The model results suggest that the measured salinity will vary by no more than 0.001 over a 15 degree temperature change at Practical Salinities of 5 to 10. Experimental evidence (Feistel and Weinreben, 2008) finds that any changes are smaller than this value, i.e., the violation of conservation does not exceed the measurement uncertainty of salinity.54 model estimate, q BSW , and the result q SW obtained using Reference Salinity, , in the TEOS-10 Gibbs function: The density deviation of the form (7.4), ( ) is displayed in Fig. 24.Comparison with experimental data (Feistel et al., 2010a) and with LSEA_DELS results shows reasonable agreement with each, with slightly better agreement with the experimental data.Compared to Figs. 9 or 20, the density anomaly is reduced by almost 50% as a result of the conductivity of the anomalous salt influencing SR and representing part of the associated density changes through the second term on the right side of eq.(7.5).Similarly, the conductivity effect changes the sign of the curvature and significantly reduces the temperature dependence of the density anomaly.  is displayed in Fig. 25.The sound speed formula is given by Eq. (5.16).This figure is very similar to Fig. 13, i.e., the conductivity effect on the sound speed anomaly is only minor.
Consequently, CTD sound speed sensors with a resolution of 1 mm s −1 (Valeport, 2010) that are carefully calibrated with respect to SSW can be expected to be capable of measuring Baltic anomalies in situ and to observationally confirm the numerical model results shown here.
is displayed in Fig. 25.The sound speed formula is given by eq.(5.16).This figure is very similar to Fig. 13, i.e., the conductivity effect on the sound speed anomaly is only minor.
Consequently, CTD sound speed sensors with a resolution of 1 mm/s (Valeport, 2010) that are carefully calibrated with respect to SSW can be expected to be capable of measuring Baltic anomalies in situ and to observationally confirm the numerical model results shown here.  .Since h depends on an arbitrary constant, only differences of enthalpies belonging to the same salinities are reasonable to be considered here.Compared to Fig. 14, the enthalpy changes are  The relative enthalpy deviation of the form (7.4), is displayed in Fig. 27.Freezing temperature is computed from Eq. (5.29).Compared to Fig. 15, the error is reduced by about 80% due to the conductivity effect and is well below the experimental uncertainty of freezing point measurements.
The above examples show that in some cases it may be desirable to correct for the anomaly or at least to check its significance in the particular case of interest.Even though this may be unnecessary in some situations, we note that there is now a general method for the calculation of the Baltic property anomaly based on the empirical Gibbs and Practical Salinity functions developed in this paper.Two practical situations are considered, (i) only Practical Salinity (plus T and P ) is known for a given sample, and, (ii) a direct density measurement is also available for the sample.
(i) Practical Salinity S P is known: Since no direct information is available on the magnitude of the anomaly, an empirical relation is used for its estimate.The Eqs. (6.4), (6.13), (5.4) and ( 5 Here, the functions g and f are evaluated at salinity S R = u PS × S P .The constant u PS is given in Table A1.The Gibbs function (3.19) with the arguments S SSW A and S BSW FW can now be used to compute the corrected property.The above examples show that in some cases it may be desirable to correct for the anomaly or at least to check its significance in the particular case of interest.Even though this may be unnecessary in some situations, we note that there is now a general method for the calculation of the Baltic property anomaly based on the empirical Gibbs and Practical Salinity functions developed in this paper.Two practical situations are considered, (i) only Practical Salinity (plus T and P) is known for a given sample, and, (ii) a direct density measurement is also available for the sample.

(i) Practical Salinity SP is known
Since no direct information is available on the magnitude of the anomaly, an empirical relation is used for its estimate.The equations (6.4), (6.13), (5.4) and (5.5), (ii) Both Practical Salinity S P and density ρ are known: Since density ρ is known, the estimate, Eq. (7.10), is not required here and is replaced by a more reliable value.The remaining equations (7.17) The functions g and f are again evaluated at salinity S R = u PS × S P .(8.2) Note that a single salinity variable such as Eq.(8.2) is insufficient for the description of Baltic seawater properties.Rather, the Gibbs function (8.1) takes two separate salinity variables, one for the SSW part and one for the additional anomalous (freshwater-related) part.The anomalous part of the Gibbs function, g FW , is available from the correlation expression (4.8) with regression coefficients reported in Table 1 and numerical check values in Table A2.
Computed from the Baltic Gibbs function, g BSW , various property anomalies are quantitatively displayed in Figs.8-18 and discussed in relation to Millero's Rule which provides generally reasonable, and sometimes very good estimates although it cannot be assumed a priori to be valid in general.Density Salinity is a good proxy for the actual Absolute Salinity of the Baltic Sea when the composition anomaly is represented by Ca 2+ and 2HCO    shows that these results are somewhat sensitive to the particular composition of the anomaly.The influence of dissolved calcium that is in charge balance and in chemical equilibrium with the marine carbonate system is estimated from LSEA DELS simulation results and is effectively represented by the conductivity factor f S SSW A ,T which correlates the anomalous mass-fraction salinity, S BSW FW , with Practical Salinity, S P , in the form, Eq. (6.13), The salinity conversion factor u PS is given in Table A1.The correlation function f S SSW A ,T has the mathematical form (6.14) with coefficients given in Table 3 and numerical check values in Table A2.The pressure dependence of f is unknown but is assumed to be of minor relevance for the relatively shallow Baltic Sea compared to the general uncertainties of the models and the scatter of the data employed here.
The above discussion regards the influence of anomalous solute as an addition to the preformed SSW part of the Absolute Salinity.When dealing with field measurements, it is often more convenient to consider anomalies from the Reference-Composition Salinity S R = u PS × S P .In this case, the conductivity effect of the anomalous solute influences the value of S R and reduces the anomalies in comparison to those computed with respect to estimates based on the preformed Absolute Salinity, S SSW A , as shown in Figs.24-27.This conclusion is similar to earlier studies on regional ocean waters (Cox et al., 1967;Lewis, 1981).
For some properties the use of S R = S P × u PS as the salinity argument of the TEOS-10 Gibbs function (IOC et al., 2010) proves sufficiently accurate for Baltic seawater but may be insufficient in cases such as for density or sound speed, depending on the actual application purposes.In these cases, estimates of S SSW A and S BSW FW are required for use in the Gibbs function, Eq. (8.1).Two alternative methods, Eqs.(7.12), (7.13) or (7.16), (7.17), are suggested to www.ocean-sci.net/6/949/2010/Ocean Sci., 6, 949-981, 2010 Fig.1.The Baltic Sea is a semi-enclosed estuary with a volume of about 20 000 km 3 and an annual freshwater surplus of about 500 km 3 a −1 ; direct precipitation excess accounts for only 10% of the latter value(Feistel et al., 2008b).Baltic seawater (BSW) is a mixture of ocean water (OW) from the North Atlantic with river water (RW) discharged from the large surrounding drainage area.Regionally and temporally, mixing ratio and RW solute are highly variable.Collected BSW samples consist of Standard Seawater (SSW) with Reference Composition (RC) plus a small amount of anomalous freshwater solute (FW), which we approximate here to be calcium bicarbonate, Ca(HCO 3 ) 2 .In dissolved form, depending on ambient temperature and pH, Ca(HCO 3 ) 2 is decomposed into the various compounds of the aqueous carbonate system with mutual equilibrium ratios(Cockell, 2008).
2.1) respectively.Regardless of the -usually unknown -precise origin in terms of N OW a and N RW a of the particle numbers finally found in the mixture, N BSW 0 , N BSW a , they actually define the composition of a given Baltic seawater sample and represent the starting point of our model.The aim of this paper is to estimate the deviation of thermophysical properties of BSW from those of SSW due to the excess of calcium ions in BSW.For this reason we formally divide the BSW particle numbers N BSW 0 , N BSW a into a major SSW fraction with Reference Composition, and a minor fraction of FW solute,

Fig. 2 .
Fig. 2. Specific volume anomaly of Baltic seawater at the standard ocean surface pressure and a typical salinity of S SSW A = 10.306gkg −1 for six different temperatures 0-25 • C as indicated by the curves, computed by the FREZCHEM model and by Millero's Rule (dashed lines, without temperatures indicated).The latter curves are the differences between the specific volumes computed from the TEOS-10 Gibbs function at salinities S BSW A , Eq. (3.21), and S SSW A , Eq. (3.19).Experimental uncertainties are considered in the following section.
Fig. 3: Heat capacity anomaly of Baltic seawater at the standard ocean surface pressure and a typical salinity of g/kg 10.306 SSW A = S for six different temperatures 0 -25 °C as indicated by the curves, computed by the FREZCHEM model and by Fig. 3. Heat capacity anomaly of Baltic seawater at the standard ocean surface pressure and a typical salinity of S SSW A = 10.306gkg −1 for six different temperatures 0-25 • C as indicated by the curves, computed by the FREZCHEM model and by Millero's Rule (dashed lines).The latter curves are the differences between the heat capacities computed from the TEOS-10 Gibbs function at salinities S BSW A , Eq. (3.21), and S SSW A , Eq. (3.14).Experimental uncertainties are considered in the following section.
Rule (dashed lines).The latter curves are the differences between the heat capacities computed from the TEOS-10 Gibbs function at salinitiesBSW   A   .(3.14).Experimental uncertainties are considered in the following section.

Fig. 4 :Fig. 4 .
Fig. 4: Activity potential anomaly of Baltic seawater at the standard ocean surface pressure and a typical salinity of g/kg 10.306 SSW A = S for six different temperatures 0 -Fig.4. Activity potential anomaly of Baltic seawater at the standard ocean surface pressure and a typical salinity of S SSW A = 10.306gkg −1 for six different temperatures 0-25 • C as indicated by the curves, computed by the FREZCHEM model and by Millero's Rule (dashed lines, different temperatures graphically indistinguishable).The latter curves are the differences between the activity potentials computed from the TEOS-10 Gibbs function at salinities S BSW A , Eq. (3.21), and S SSW A , Eq. (3.14).

5 Fig. 5 :
Fig. 5: Scatter of specific volume anomalies computed from FREZCHEM, i v δ , relative to the specific volume anomalies computed from the Gibbs function, ( ) c v δ , eq. (4.14), at 1260 given data points.The rms deviation of the fit is 1.5 mm³/kg.Symbols 0 -5 indicate the pressures of 0.1 MPa, 1 MPa, 2 MPa, 3 MPa, 4 MPa and 5 MPa, respectively.These residual anomalies should be compared with the total anomalies i v δ shown in Fig. 2.

Fig. 6 .
Fig.6.Scatter of heat capacity anomalies computed from FREZCHEM, δc P i , relative to the heat capacity anomalies computed from the Gibbs function, δc P (c), Eq. (4.15), at 210 given data points at atmospheric pressure.The rms deviation of the fit is 3.4 mJ/(kg K).Symbols 0-5 indicate the temperatures of 0-25 • C, respectively.These residual anomalies should be compared with the total anomalies δc P i shown in Fig.3.

Fig. 7 :
Fig. 7: Scatter of the activity potential anomalies computed from FREZCHEM, i ψ δ , relative to the activity potential anomalies computed from the Gibbs function, ( ) c ψ δ , eq. (4.33), at 1260 given data points.The rms deviation of the fit is 3.1E-5.Symbols 0 -5 indicate the pressures of 0.1 MPa, MPa, 2 MPa, 3 MPa, 4 MPa and 5 MPa, respectively.These residual anomalies should be compared with the total anomalies i ψ δ shown in Fig. 4.

c
000 = 0,c 010 = 0 and c 010 = −(40 K) × g FW T (S SO ,T SO ,P SO ) c 000 =0,c 010 =0.(4.38) .3) which is based on density measurements made at 20 • C and Chlorinity determinations at 3 different stations.Using Eq. (5.2) in the form S Cl ≈ S SSW A in Eq. (5.2), we have S D approximately given as a function of S SSW A for typical Baltic seawater conditions, S D = S SSW A + 130 mg kg −1 × 1 − S SSW A S SO .(5.4)This empirical relation is used here to conveniently present the comparisons for typical Baltic conditions as a function of Ocean Sci., 6, 949-981, 2010 www.ocean-sci.net/6/949/2010/ Fig. 8: Difference 0.8 provides an approximate conversion to the units used in Fig.8 so
Fig.9: Difference ρ δ , eq. (5.6), between the densities with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure and temperatures between 0 and 25 °C.The uncertainty of density measurements is 2 g m -3(Feistel et  al., 2010a), indicated by the solid horizontal line.

Fig. 9 .
Fig.9.Difference δρ, Eq. (5.6), between the densities with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure and temperatures between 0 and 25 • C. The uncertainty of density measurements is 2 g m −3(Feistel et al., 2010a), indicated by the solid horizontal line.
Fig. 10: Difference α δ , eq.(5.7), between the thermal expansion coefficients (solid lines) with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure and temperatures between 0 and 25 °C, in comparison to estimates from Millero's Rule, and exceeds the range shown in the figure.
as a function of Density Salinity, computed from Eqs. (5.4) and (5.5).For comparison, the anomaly is estimated by Millero's Rule using Density Salinity S D , Eq. (5.1), from δα D = g SW T P (S D ,T ,P SO ) g SW P (S D ,T ,P SO ) Fig. 11: Difference β δ , eq. (5.12), between the haline contraction coefficients (solid lines) of the parent solution with respect to the addition of FW solute and of SSW solute for Baltic seawater.Values are determined at the standard ocean surface pressure and temperatures between 0 and 25 °C.The standard-ocean value of the haline contraction coefficient is 0.781 = 781 ppm g -1 kg.The haline contraction coefficient associated with the addition of calcium carbonate is within 20% of the haline contraction coefficient for Standard Seawater.

Fig. 11 .
Fig. 11.Difference δβ, Eq. (5.12), between the haline contraction coefficients (solid lines) of the parent solution with respect to the addition of FW solute and of SSW solute for Baltic seawater.Values are determined at the standard ocean surface pressure and temperatures between 0 and 25 • C. The standard-ocean value of the haline contraction coefficient is 0.781 = 781 ppm g −1 kg.The haline contraction coefficient associated with the addition of calcium carbonate is within 20% of the haline contraction coefficient for Standard Seawater.
Fig.12: Difference P c δ , eq. (5.13), between the specific isobaric heat capacity (solid lines) with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure and temperatures between 0 and 25 °C, in comparison to estimates from Millero's Rule, D δ P c , based on Density Salinity (dashed lines), eq.(5.14), and Fig.13: Difference cδ , eq. (5.17), between the sound speed (solid lines) with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure Fig. 14: Difference h δ , eq.(5.20), between the relative specific enthalpies (solid lines) with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure and temperatures between 5 and 25 °C, in comparison to estimates from Millero's rule, D δh , based on Density Salinity (dashed lines, only the 15 -25 °C results are labelled), eq.(5.21), and A δh , based on Absolute Salinity (dotted lines, temperatures not labelled), eq.(5.22).For the latter two, the responsible difference between BSW

Fig. 14 .
Fig.14.Difference δh, Eq. (5.20), between the relative specific enthalpies (solid lines) with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure and temperatures between 5 and 25 • C, in comparison to estimates from Millero's rule, δh D , based on Density Salinity (dashed lines, only the 15-25 • C results are labelled), Eq. (5.21), and δh A , based on Absolute Salinity (dotted lines, temperatures not labelled), Eq. (5.22).For the latter two, the responsible difference between S BSW A and S D is shown in Fig.8.The experimental uncertainty of the relative enthalpies is 0.5 J kg −1 × t/ • C.
Fig. 15: Difference T δ , eq. (5.30), between the freezing temperature (solid line) with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure, in comparison to estimates from Millero's Rule, D δT , based on Density Salinity (dashed line), eq.(5.31), and A δT , based on Absolute Salinity (dotted line), eq.(5.32).For the latter two, the responsible difference between BSW A S and D S is shown in Fig. 8.The experimental uncertainty of the freezing temperature of seawater is 2 mK, indicated by the solid horizontal line.

Fig. 15 .
Fig. 15.Difference δT , Eq.(5.30), between the freezing temperature (solid line) with and without the freshwater solute for Baltic seawater at the standard ocean surface pressure, in comparison to estimates from Millero's Rule, δT D , based on Density Salinity (dashed line), Eq. (5.31), and δT A , based on Absolute Salinity (dotted line), Eq. (5.32).For the latter two, the responsible difference between S BSW A and S D is shown in Fig.8.The experimental uncertainty of the freezing temperature of seawater is 2 mK, indicated by the solid horizontal line.
Fig. 16: Difference P δ , eq. (5.34), between the vapour pressures (solid line) with and without the freshwater solute for Baltic seawater at 20 °C, in comparison to estimates from Millero's Rule, D δP , based on Density Salinity (dashed line), eq.(5.35), and

Fig. 16 .
Fig.16.Difference δP , Eq. (5.34), between the vapour pressures (solid line) with and without the freshwater solute for Baltic seawater at 20 • C, in comparison to estimates from Millero's Rule, δP D , based on Density Salinity (dashed line), Eq. (5.35), and δP A , based on Absolute Salinity (dotted line), Eq. (5.36).For the latter two, the responsible difference between S BSW A and S D is shown in Fig.8.The related experimental uncertainty is 0.02% or 0.4 Pa, well beyond the range of this graph.and using Absolute Salinity, S BSW A , Eq. (3.21), shows the salinity difference S D (t) = S D S SSW A ,S BSW FW ,T SO + t,P SO −S D S SSW A ,S BSW FW ,T SO + 25 • C,P SO (5.37) as a function of the Density Salinity at 25 • C for typical Baltic anomaly pairs of S SSW A and S BSW FW computed from Eqs. (5.4) and (5.5).
Figure 17 is similar to Fig. 8 in which S BSW

Fig. 17 .
Fig. 17: Difference ( ) t S D Δ , eq. (5.37), between the Density Salinities computed at different temperatures from eq. (5.1) at the same mass-fraction salinities SSW A S and Fig.18.Deviation (5.37)  between the density of Baltic seawater and the density computed from conservative Density Salinity, S dens A , Eq. (5.38).The experimental uncertainty of density measurements is 2 ppm(Feistel et al., 2010a), indicated by the solid lines.

Fig. 19 .
Fig. 19.(a) Comparison between 437 measured density anomalies (Feistel et al., 2010a), with LSEA DELS model predictions.(b) Comparison of model results with 3 observational estimates of the anomalies between Density Salinity S D and the Chlorinity Salinity S Cl(Feistel et al., 2010a) as well as Eq.(5.3), and model predictions.

Fig. 19
Fig.19 a) Comparison between 437 measured density anomalies(Feistel et al., 2010a), with LSEA_DELS model predictions.b) Comparison of model results with 3 observational estimates of the anomalies between Density Salinity SD and the Chlorinity Salinity SCl(Feistel et al., 2010a) as well as eq.(5.3), and model predictions

Fig. 20 :
Fig. 20: Comparison of the density anomalies between SSW and Baltic seawater of the same chloride molality, computed by the Gibbs function and by LSEA_DELS.Curves are drawn for temperatures of 0, 5, 10, 15, and 20 °C, with the highest curves corresponding to the lowest temperatures.

6. 4
Corrections to Practical Salinity required for Gibb function calculationsThe Gibbs function determined in Section 4 is a function of chloride molality and the calcium anomaly, or equivalently SSW A

Fig. 20 .
Fig. 20.Comparison of the density anomalies between SSW and Baltic seawater of the same chloride molality, computed by the Gibbs function and by LSEA DELS.Curves are drawn for temperatures of 0, 5, 10, 15, and 20 • C, with the highest curves corresponding to the lowest temperatures.
.012951706126954 3 1 +0.000079702941453 be accurately expressed as the product of a function, f , that depends only on the salinity associated with the base seawater and temperature, and the change in solute mass fraction S f on both T and S Cl is shown in Fig.21but curves corresponding to different values of δm Ca at a fixed temperature are visually indistinguishable at this scale.
Fig. 21: Anomaly of the Reference Salinity

Fig. 21 .
Fig. 21.Anomaly of the Reference Salinity S R , Eq. (6.13), as a function of S Cl at different temperatures and anomalies δm Ca , estimated using LSEA DELS.51
.14) as a function of S SSW A and S BSW FW , S R = S SSW A + f S SSW A ,T S BSW FW .(7.3) Fig.24.Comparison with experimental data(Feistel et al., 2010a) and with LSEA DELS results shows reasonable agreement with each, with slightly better agreement with the experimental data.Compared to Fig.9or 20, the density anomaly is reduced by almost 50% as a result of the conductivity of the anomalous salt influencing S R and representing part of the associated density changes through the second term on the right side of Eq. (7.5).Similarly, the conductivity effect changes the sign of the curvature and significantly reduces the temperature dependence of the density anomaly.The sound speed deviation of the form (7.4),δc R = c BSW S SSW A ,SBSW FW ,T ,P SO − c SW (S R ,T ,P SO ), (7.6) Ocean Sci., 6, 949-981, 2010 www.ocean-sci.net/6/949/2010/

Fig. 23 :Fig. 23 .
Fig. 23: Temperature dependence, eq.(6.15), of Practical Salinity relative to 15 °C of a given sample of Baltic seawater at atmospheric pressure Fig. 23.Temperature dependence, Eq. (6.15), of Practical Salinity relative to 15 • C of a given sample of Baltic seawater at atmospheric pressure.

Fig. 24 :
Fig.24: Error in density, eq.(7.5), if computed from measured Reference Salinity, using the Gibbs function for SSW.Results are shown for temperatures between 0 and 25 °C and at atmospheric pressure.

Fig. 24 .
Fig. 24.Error in density, Eq. (7.5), if computed from measured Reference Salinity, using the Gibbs function for SSW.Results are shown for temperatures between 0 and 25 • C and at atmospheric pressure.

Fig. 25 :
Fig. 25: Error in sound speed, eq.(7.6), if computed from measured Reference Salinity using the Gibbs function for SSW.Results are shown for temperatures between 0 and 25 °C and at atmospheric pressure

Fig. 25 .Fig. 26 :
Fig. 25.Error in sound speed, Eq. (7.6), if computed from measured Reference Salinity using the Gibbs function for SSW.Results are shown for temperatures between 0 and 25 • C and at atmospheric pressure.56 is almost completely captured by the conductivity effect and the enthalpy anomalies are therefore negligible.

Fig. 26 .
Fig. 26.Error in relative enthalpy, Eq. (7.7), if computed from measured Reference Salinity using the Gibbs function for SSW.Results are shown for temperatures between 1 and 25 • C and at atmospheric pressure.
.5), u PS × S P ≡ S R = S SSW A in linear approximation of the anomaly, δS R = S R − S SSW A .The solution reads

Fig. 27 :
Fig. 27: Error in freezing temperature, eq.(7.8), if computed from measured Reference Salinity using the Gibbs function for SSW.Results shown correspond to atmospheric pressure

Fig. 27 .
Fig. 27.Error in freezing temperature, Eq. (7.8), if computed from measured Reference Salinity using the Gibbs function for SSW.Results shown correspond to atmospheric pressure.

For
Baltic seawater with a simplified composition anomaly representing only inputs of calcium carbonate, Eq. (1.1), a Gibbs function is determined based on theoretical considerations and results from FREZCHEM model simulations.The new Gibbs function, Eq. (3.19), combines the TEOS-10 Gibbs function of Standard Seawater (SSW), g SW S SSW A ,T ,P , with an anomalous part, g FW S SSW A ,T ,P , proportional to the Absolute Salinity of the anomalous (freshwater) salt, S BSW FW , resulting in the form g BSW S SSW A ,S BSW FW ,T ,P = 1 − S of the "preformed" SSW part, the parent solution, is denoted by S SSW A , Eq. (2.26).From the mass balance, the Absolute Salinity of Baltic seawater is given by Eq. (3.21), FW is determined, the Gibbs function g BSW of Baltic seawater can be computed from Eq. (3.15), in the form Table 1, and the results of the fit in Table 2.The scatter of the FREZCHEM data points with respect to the resulting partial Gibbs function g FW is shown in Figs. 5, 6 and 7 for δv, δc P and δψ, respectively.Numerical check values are available from TableA2.Various salinity measures such as Reference Salinity S R , Absolute Salinity, S A , Density Salinity, S D , or Chlorinity Salinity, S Cl , have the same values for SSW but differ from each other for BSW.The estimate of Density Salinity based on inversion of the expression for density in terms of the Gibbs function for SSW at arbitrary values of temperature and pressure is represented by S D , and referred to as "measured" Density Salinity since it is based on whatever the conditions of the direct density measurement are.It is the Absolute Salinity of SSW (here assumed to have Reference Composition) that has the same density as BSW at given temperature and pressure, i.e., FW ,T ,P = g SW P (S D ,T ,P ).

Table 2 .
Results of the regression, Eq. (4.8), with respect to properties of Baltic seawater simulated with FREZCHEM.
SW (S R ,T ,P SO ) + h SW (S R ,T SO ,P SO ), by the conductivity effect and the enthalpy anomalies are therefore negligible.The freezing point deviation of the form (7.4), δT R = T BSW S SSW A ,S BSW FW ,T ,P SO − T SW (S R ,T ,P SO ), (7.8) is displayed in Fig.26.Enthalpy is computed from the Gibbs function by h = g − T g T .Since h depends on an arbitrary constant, only differences of enthalpies belonging to the same salinities are reasonable to be considered here.Compared to Fig.14, the enthalpy changes are is almost www.ocean-sci.net/6/949/2010/Ocean Sci., 6, 949-981, 2010 completely captured

Table A2 .
Numerical check values of the Gibbs function anomaly g FW , Eq. (4.8), and of the conductivity function, f , Eq. (6.14).

Table B1 .
Glossary of formula symbols.