Density is one of main physical quantities governing the hydrodynamics, stratification, and mixing in lakes and reservoirs. Water quality in lakes is controlled by biological and biogeochemical processes which depend on the availability of oxygen in deep waters and nutrients in surface waters. Both phenomena are controlled by the duration and extension of the turnover period, which is dependent on density gradients. Therefore, density is a very important variable in numerical models for the simulation of the behaviour of lakes under changing conditions, e.g. due to management measures or phenomena related to global climate change.

The density of lake water (at atmospheric pressure) depends on temperature and dissolved water constituents. Since temperature, chemical composition and concentrations may vary over time, from lake to lake or even within a particular lake due to seasonal stratification or meromixis, numerical models of lakes calculate the density internally. There are several approaches to calculate water density in lakes. Most of them are general equations that do not always reflect specific properties of lakes. If enough measurements of density for the relevant temperature range are available and composition and concentrations of the main constituents are constant, regressions can be used to generate a mathematical formula for density in a specific lake (e.g. Jellison et al., 1999; Vollmer et al., 2002; Karakas et al., 2003). If the composition is constant and the main constituents are ions, electrical conductivity or salinity may be used as an easy-to-measure proxy for concentrations (Bührer and Ambühl, 1975; Chen and Millero, 1986; Pawlowicz and Feistel, 2012).

Imboden and Wüest (1995) discussed the influence of dissolved substances on (potential) density because both the concentration and chemical composition of the total dissolved solids change considerably from lake to lake (see e.g. Boehrer and Schultze, 2008). The effects of dissolved solids on density stratification have been studied in lake-specific investigations in Lake Malawi (Wüest et al., 1996) and in Lake Matano (Katsev et al., 2010). In some cases, the specific contribution of ions such as calcium, carbonate or dissolved iron can control the permanent stratification in lakes such as in La Cruz (Spain) (Rodrigo et al., 2001), Cueva de la Mora (Spain) (Sanchez-España et al., 2009) or Waldsee (Germany) (Dietz et al., 2012).

Density of pure water can be calculated using mathematical expressions such as those in Kell (1975) or Tanaka et al. (2001). Density calculations of natural waters require additional terms to include the contributions of dissolved substances. Specific formulas have been developed for ocean conditions. The UNESCO equations developed by Fofonoff and Millard (1983) have been the standard for a long period. They used temperature and practical salinity based on electrical conductivity measurements. Because seawater conditions are a known reference and the approaches provide stable results over a wide range of temperatures and electrical conductivity, these have been applied in limnic systems and implemented in numerical models such as DYRESM (Imberger and Patterson, 1981; Gal et al., 2009), ELCOM (Hodges and Dallimore, 2007), GOTM (Burchard et al., 1999; Umlauf et al., 2005) or CE-QUAL-W2 (Cole and Buchak, 1995). Recently the ocean standard was replaced by the new Thermodynamic Equation of Seawater 2010 (TEOS-10; IOC et al., 2010). However, as the composition of solutes differs greatly from the ocean, density calculation based on ocean conditions can only be of limited accuracy. Pawlowicz and Feistel (2012) have considered the application of TEOS-10 (IOC et al., 2010) in several cases different from seawater, correcting the salinity values depending on the composition before applying TEOS-10 (IOC et al., 2010). Bührer and Ambühl (1975) developed an equation to calculate density based on temperature and specific conductance at 20 ∘C for alpine lakes. In addition, a popular approach was formulated by Chen and Millero (1986) for tuning ocean approaches to freshwater conditions (salinity < 0.6 psu).

Higher accuracy can be achieved when site-specific density equations are produced. Jellison et al. (1999) developed a density equation for Mono Lake from water samples which have been measured at different temperatures and dilutions. In the case of meromictic lakes, strong differences in the composition of the mixolimnion and monimolimnion must be reflected in the density equations in order to sustain the permanent stratification in the density calculations. Boehrer et al. (2009) and von Rohden et al. (2010) used an equation based on density measurements of the monimolimnion and mixolimnion of the Waldsee.

Boehrer et al. (2010) evaluated the contribution of the different cations and anions separately in terms of partial molal volumes and implemented an algorithm, RHOMV (http://www.ufz.de/webax), to calculate density with a second-order approximation for temperature dependence and ionic strength dependence. Pawlowicz et al. (2011) implemented the LIMBETA method that calculates density from composition. Another approach comes from Pawlowicz et al. (2012) where the authors propose to use TEOS-10 but replace seawater salinity with specific salinities obtained and corrected for freshwaters. This limnic salinity can be calculated using the chemical composition by summing up all the dissolved solutes (Sasoln) or by summing up only the dissolved ions (Saionic) and correcting this value with the dissolved Si(OH)4, Sadens = Saionic + 50.6 × [Si(OH)4] (mol kg-1). Based on partial molal volumes (RHOMV), Dietz et al. (2012) separated the contributions of solutes for freshwater lakes. Moreira et al. (2011) based density on the composition of solutes in their model to reproduce the permanent stratification of the Waldsee numerically using RHOMV to include the reactivity of substances in the density (see also Nixdorf and Boehrer, 2015).

These prior studies therefore highlight the necessity of including the chemical composition to obtain an accurate calculation of density. However, we accept the need for a practical density approach, which can easily be implemented, such as a mathematical formula that relates density to temperature and electrical conductivity. In this manuscript, we develop coefficients for such a formula from the chemical composition. We provide an algorithm – RHO_LAMBDA (from ρλ) – to obtain such coefficients and demonstrate the applicability of the approach with water from Rappbode Reservoir. We also provide an appropriate assessment for the Rappbode Reservoir case and compare the accuracy with other approaches currently in use for limnic waters. For a quantitative judgement of the general applicability of our approach, we also evaluate coefficients for two further freshwater bodies (Lake Geneva, Lake Constance), an extremely saline lake (Mono Lake), a meromictic open pit lake in the mixolimnion and the monimolimnion (Waldsee), and finally seawater as a globally known example and well-defined standard.

We propose a simple equation for density as a numerical approximation of the (potential) density of lake water: ρ≈ρλT,κ25=ρwT+κ25λ0+λ1×T-25∘C, where the first term of the right side ρw is the density of pure water, which can be calculated in a very accurate way using Kell (1975) or Tanaka (2001). Our approach (Eq. 1) correlates density with temperature (T) and electrical conductivity at 25 ∘C (κ25) of a water sample using coefficients λ0 and λ1. The introduction of λ1 reflects temperature dependence of the density contribution of the solutes, which is required for a shifting temperature of maximum density. Only two coefficients need to be determined, and thus this equation is easy to implement. Coefficients λ0 and λ1 can be obtained as follows.

At T=25 ∘C, the λ1 term in Eq. (1) vanishes and λ0 can be calculated using Eq. (2) provided that the water density at 25 ∘C is known from other sources: λ0=ρT=25∘C,κ25-ρwT=25∘Cκ25. If the density is also known for a temperature T≠25 ∘C, λ1 can be calculated in a second step: λ1=ρT,κ25-ρwT/κ25-λ0T-25∘C. Necessary data for Eqs. (2) and (3) can be derived from measurements or from calculations.

In the remaining part of this manuscript, Eq. (1), complemented by Eq. (2) and Eq. (3), will be referenced as the RHO_LAMBDA approach.

In our RHO_LAMBDA approach, we use the Tanaka (2001) equation for pure water density, ρw. If the composition of solutes in the water is known, the density of water is calculated by using RHOMV (Boehrer et al., 2010) and finally κ25 is provided by the algorithm implemented in the PHREEQC code (Parkhust and Appelo, 1999), whose description can be found in Atkins and de Paula (2009) and on Appelo's webpage of PHREEQC (Appelo, 2016). This method (re-implemented in Python from the original code) calculates the specific conductance of a solution from the concentration, the activity coefficient and the diffusion coefficient of all the charged species. The diffusion coefficients can be found in Millero (2001).

The practicability of this approach depends on its accuracy. This will first be assessed for Rappbode Reservoir water and its above-evaluated coefficients. However, for limnologists working on other limnic water bodies, an assessment of accuracy in the general range of limnic water composition is of fundamental interest. In conclusion, we chose a collection of lake waters of different chemical compositions and a wide range of concentrations. We included all lakes we knew of, where a reliable reference density could be provided and the chemical composition was known.

In particular, we included two further typical freshwater lakes, Lake Geneva and Lake Constance, which are well known in the limnological literature. As an example of saline lakes, we chose Mono Lake (e.g. Jellison et al., 1999). In order to include also water with a rather unusual composition, we chose two water samples from a meromictic open pit lake, the Waldsee, which contains large amounts of sulfate and dissolved iron (e.g. Dietz et al., 2008, 2012; Boehrer et al., 2009; von Rohden et al., 2010; Moreira et al., 2011). Finally, we used seawater, of which the composition is known at high accuracy, as a reference water for a standard comparison. Table 1 presents the original chemical compositions of the different lakes considered in the testing of the RHO_LAMBDA expression. Data were derived from chemical analysis (for experimental details, see Appendix A) or literature (for references, see Table 1). We complemented the set using synthetically produced lake water of differing compositions and concentrations from the work by Gomell and Boehrer (2015) in order to test systematically the influence of composition and concentration on the values of the coefficients λ0 and λ1 (for experimental details, see Appendix A).

For critical comparison with other density equations, this assessment section (Sect. 3) consists of two major parts: firstly we check the accuracy for different lakes and water samples and secondly we provide the lambda coefficients of several aquatic systems where we have direct measurements or a specifically obtained approach to density (e.g. Mono Lake or seawater) to check the accuracy of ρλ in general. Table 2 presents the results of the intermediate step calculations to obtain λ0 and λ1. As references for the assessment, we used the measured (for details, see Appendix A) or published data (Fig. 1a and b, Table 2).

The quantitative comparison between the different methods (including the method presented here, RHO_LAMBDA) and the reference values is shown in Fig. 1. Our approach mainly aimed at representing the density contribution of solutes. Hence, we related the difference to our reference with the contribution of the solutes Rel.Error=ρλ-ρref/ρref-ρw. For temperatures in the range of 1–30 ∘C, usually found in typical limnic conditions, the values of the relative error defined by Eq. (4) are displayed in the right column of Fig. 1. On purpose, we obtained the chemical composition from a different source (sample) than the density measurement. In this way, the variability of the water composition within one lake was included in the error determination in our assessment.

To judge the accuracy of our approach, we also inserted results from other formulas in common use for transferring CTD data into density: we included UNESCO (Fofonoff and Millard, 1983), TEOS-10 (IOC et al., 2010), Chen and Millero (1986) and Bührer and Ambühl (1975) (Fig. 1) as far as possible according to the defined range of applicability of the single formula.

Rappbode Reservoir. The measured conductance (κ25) of our Rappbode Reservoir sample was 0.1579 mS cm-1, which differed only by 4 % from the value 0.1635 mS cm-1 calculated using the PHREEQC electrical conductivity algorithm at 25 ∘C (described in Atkins and De Paula, 2009). This is within the measurement accuracy of the chemical analysis. Reference density was produced by measuring in a PAAR DSA 5000 densitometer from 1 to 30 ∘C.

We can see that the RHO_LAMBDA method reproduced the reference values of the water sample from Rappbode Reservoir with a relative error ranging from -12.7 to 4.3 %. The deviation from the reference was lower than 5 % in the range 10 to 27 ∘C. Among the other compared approaches, TEOS-10 showed the best results, with relative error ranging from -15.7 to 0.1 %. The Bührer and Ambühl (1975) approach resulted in a relative error ranging from -4.0 to 99.3 % and strongly rising with temperatures increasing above 20 ∘C. Results according to Chen and Millero (1986) ranged between -37.8 and -25.3 %.

Lake Geneva. Calculated and measured electrical conductivity (κ25) of a water sample from 7 November 2013 differed by less than 1 % for 25 ∘C (Table 2). Reference density was produced from this sample in a PAAR DSA 5000 densitometer. The relative error ranged from -11.5 to -4.6 % for our RHO_LAMBDA approach. Bührer and Ambühl (1975) (relative error -15.3 to 21.7 %), TEOS-10 (relative error -12.9 to -7.8 %) and Chen and Millero (1986) (relative error -50.9 to -47.6 %) showed larger deviations from the reference.

Lake Constance. The composition shown in Table 1 mainly coincided with the analysis done by Stabel (1998). The calculated conductivity at 25 ∘C (κ25) of 0.330 mS cm-1 differed from the measured value of 0.322 mS cm-1 by 3 %. The reference density was again from measurements in a PAAR DSA 5000 densitometer. The relative error ranged from -9.7 to -4.2 % for the RHO_LAMBDA approach. TEOS-10 (relative error -12.2 to -8.6 %), Bührer and Ambühl (1975) (relative error -17.4 to 14.6 %) and Chen and Millero (1986) (relative error -46.6 to -44.1 %) again had larger deviations from the reference. The strong increase in the relative error of Bührer and Anbühl (1975) with temperature was smallest for Lake Constance compared to the other freshwater lakes.

Mono Lake. We evaluated density for a water sample of conductivity κ25=85.67 mS cm-1 which was provided by Jellison et al. (1999) and which differed by 12 % from the calculated value 96.61 mS cm-1 using the PHREEQC algorithm. The density formula by Jellison et al. (1999) was used as the reference density. In this case, the relative error using RHO_LAMBDA ranged from -9.5 to -1.5 % even in this lake with such saline waters and unusual composition. Also in this case, TEOS-10 showed larger deviation from the reference (relative error -10.4 to -5.2 %). The largest relative error was found for the UNESCO equation according to Foffonof and Millard (1983) (relative error -39.9 to -36.0 %).

Waldsee mixolimnion/Waldsee monimolimnion. This case presented a meromictic open pit lake (Boehrer et al., 2008; Dietz et al., 2008, 2012; von Rohden et al., 2010; Moreira et al., 2011) of moderate salinity (0.22 psu in the mixolimnion and 0.6 psu in the monimolimnion, Moreira et al., 2011), but its composition differed from the usual carbonate or chloride waters. Composition was obtained from Dietz et al. (2008, 2012). The DOC (dissolved organic carbon) contribution was added according to Dietz et al. (2012). This correction increased density by 0.015 kg m-3 in the mixolimnion and by 0.06 kg m-3 in the monimolimnion.

The calculated κ25 differed by 7.0 % from the reference value in the mixolimnion and by 7.6 % in the monimolimnion (Table 2). This was the highest difference between reference and calculated values of all waters considered in this study. Probably, the very special chemical composition of the waters was the reason. The missing data for ammonia and silicate may also have contributed, in particular in the monimolimnion. Measurements in the work of Boehrer et al. (2009) were used as a density reference.

The relative error of the RHO_LAMBDA approach ranged from -8.4 to -3.9 % in the mixolimnion. In the monimolimnion, the relative error ranged from -11.9 to -9.8 % for the RHO_LAMBDA approach. The deviation from the reference was substantially larger for all other compared approaches (Fig. 1a and b). The averages of the absolute values of the relative error were 22.8, 52.0 and 52.3 % for TEOS-10 (IOC et al., 2010), Chen and Millero (1986) and UNESCO (Foffonof and Millard, 1983) in the mixolimnion, respectively. In the monimolimnion, the values were 35.2 % for TEOS-10, 60.0 % for Chen and Millero (1986) and 60.2 % for UNESCO (Foffonof and Millard, 1983).

Seawater. The seawater composition was obtained from Millero et al. (2008) and we used TEOS-10 (IOC et al., 2010) as our seawater density reference. Electrical conductivity was calculated for this composition and resulted in 53.76 mS cm-1, while the reference value given by Millero et al. (2008) was 53.06 mS cm-1. That meant the deviation was 1.3 %. As expected – both formulas were specifically designed for ocean water –, the relative error of the UNESCO approach according to Foffonof and Millard (1983) was very small, ranging between -0.02 and -0.01 %. This was probably a result of numerical uncertainties of the calculations. The relative error of our RHO_LAMBDA approach ranged between -0.75 and 0.68 %.

In all cases, our density approach reproduced the density contribution of the salts to within 10 %. This is better than most of the other approaches, which differed by up to 60 % from the correct values. Even in the case of very low concentrations (Rappbode Reservoir) and very high concentrations (Mono Lake) as well as in very special water composition (mine lake Waldsee), the 10 % accuracy for the salt contribution was achieved with our RHO_LAMBDA approach. The observed strong increase in the relative error with temperature for Bührer and Ambühl (1975) was caused by its validity limited to 24 ∘C. Spatial and temporal variability of solute composition could contribute to errors in density calculation. However, where we attained chemical composition separately (i.e. from another sample) from the density information, this error is intrinsically included in our assessment and hence in the value that we supply for the RHO_LAMBDA approach.

The first coefficient λ0 varied by more than a factor of 2 between 0.37 and 0.88 kg cm m-3 mS-1; see Fig. 2. This explained that a density formula with constant coefficients could never be able to mimic density accurately for a larger range of lake waters. Obviously the coefficient λ0 depended on the composition of the solutes. A dominance of double-charged ions – as opposed to single-charged ions – led to higher values of λ0. This effect was clearly visible in the inclusion of calculated values for a NaCl solution of 1 g L-1 and a CaSO4 solution of 1 g L-1 (Fig. 2).

Also, the concentration of solutes had a decisive effect on the coefficients. We used density measurements of a dilution series of synthetic lake waters by Gomell and Boehrer (2015) of 1, 3, 10, 30, or 90 g L-1 of a mixture of KCl, NaHCO3, and Na2SO4. We included lambda coefficients from RHO_LAMBDA approach “Mix” together with regressions of published measured data “Mix-M” (Fig. 2). Both empirical data as well as RHO_LAMBDA results reflected the concentration effect on λ0 of a factor of 1.5. Although not perfect, the agreement between empirical data and RHO_LAMBDA values lay within the 10 % margin we found for lake waters above.

Values for λ1 nearly all lay between -0.001 and -0.002 kg cm m-3 mS-1 K-1. Hence, the λ1 term delivered a small contribution in all cases, i.e. always an order of magnitude smaller compared to the λ0 term. As a consequence, it could be neglected for most limnological applications. Though not really necessary for an absolute density calculation, λ1 was included to also represent the shift of temperature of maximum density for a given lake water composition, which could not be achieved with the λ0 term alone. Negative values of λ1 indicated a shift of the temperature of maximum density to lower temperatures. A closer look at the λ1 values revealed that some empirical values (also the Mono Lake reference derived from empirical measurements) were considerably lower than expected from coefficients of physical chemistry. However, the difference posed the question of how accurately the shift of temperature of maximum density would actually be indicated by coefficients of the physical chemistry literature. The largest discrepancies appeared for freshwater lakes (Rappbode Reservoir, Lake Geneva, Lake Constance) where the shift is small.

The values of λ0 and λ1 have also been calculated using direct measurements of density (starred values λ0∗ and λ1∗). In the case of λ0 only slight differences can be found between the values calculated from chemical composition and from direct measurements of density. However, those differences increase in the case of λ1, as mentioned above.

We showed that the correlation between electrical conductivity and density depends strongly on the composition and concentration of solutes. As a consequence, the limnic range cannot be covered with one formula with constant coefficients. However, a simple mathematical addition of two terms to a pure water formula is able to represent the density contribution of solutes in all our examples with an error of less than 10 %. This is sufficient for most limnological applications and is better than any other density approach based on CTD data, if not specifically designed for a given lake water.

Only two coefficients λ0 and λ1 need to be evaluated: while λ0 varies considerably between lakes, the numerical evaluation of λ1 delivers very similar values of λ1∼-0.0015 kg cm m-3 mS-1 K-1 for any lake water composition. Hence, once λ0 has been evaluated for a lake, a rather accurate and simple density formula can be used for CTD data. The approach uses conductance κ25, which can be measured in limnic waters, and thus avoids salinity, which is badly defined for limnic waters and, hence, is a precarious quantity. The inclusion of this simple and more accurate approach for potential density calculation in numerical lake models is therefore recommended.

For convenient use and implementation, a density calculator tool is provided at https://sourceforge.net/projects/densitycalc.

Webax-Web access to numerical tools of limnology is available at http://www.ufz.de/webax. TEOS-10 original Fortran 90 library and related tools are available at http://www.teos-10.org/software.htm. Density calculator is available at https://sourceforge.net/projects/densitycalc. All the data used in this manuscript to calculate the lambda coefficients and density values are provided in Tables 1 and 2.