Density calculations are essential to study stratification, circulation patterns, internal wave formation and other aspects of hydrodynamics in lakes and reservoirs. Currently, the most common procedure is the use of CTD (conductivity, temperature and depth) profilers and the conversion of measurements of temperature and electrical conductivity into density. In limnic waters, such approaches are of limited accuracy if they do not consider lake-specific composition of solutes, as we show. A new approach is presented to correlate density and electrical conductivity, using only two specific coefficients based on the composition of solutes. First, it is necessary to evaluate the lake-specific coefficients connecting electrical conductivity with density. Once these coefficients have been obtained, density can easily be calculated based on CTD data. The new method has been tested against measured values and the most common equations used in the calculation of density in limnic and ocean conditions. The results show that our new approach can reproduce the density contribution of solutes with a relative error of less than 10 % in lake waters from very low to very high concentrations as well as in lakes of very particular water chemistry, which is better than all commonly implemented density calculations in lakes. Finally, a web link is provided for downloading the corresponding density calculator.

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

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 (

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

We propose a simple equation for density as a numerical approximation of the
(potential) density of lake water:

At

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,

We demonstrate our density approach with the example of Rappbode Reservoir
(Germany; for details on this reservoir, see Rinke et al., 2013, and
references therein); its low electrical conductivity indicates low
concentrations of solutes. From chemical analysis of a surface sample from 19
November 2010, we knew the major cations were calcium (13.8 mg L

For this sample, an electrical conductance

According to RHOMV, the density of this sample at 25

Putting these numbers into Eq. (2) delivered

Similarly, we evaluated

Finally, inserting the lambdas as coefficients into Eq. (1) delivered a density formula for Rappbode Reservoir.

Test cases. The left panel presents the density curves of the
different methods and the right panel shows the relative error of the density
contribution of the solutes with respect to the reference. In all the cases,
measured values have been used as a reference, except in the cases of Mono
Lake (Jellison et al., 1999) and seawater (TEOS-10, IOC et al., 2010), which
use specific density equations.

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

Original chemical composition of the water in different test cases
presented in Sect. 3. All values except pH are presented in mg L

* Density modified by addition of this quantity expressed in
kg m

Summary of the calculated values for obtaining RHOMV_LAMBDA
coefficients. Lambdas and references:

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

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

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.

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

The calculated

The relative error of the RHO_LAMBDA approach ranged from

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

The first coefficient

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

Distribution of the values of

Values for

The values of

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

For convenient use and implementation, a density calculator tool is provided
at

Webax-Web access to numerical tools of limnology is available at

All samples were taken as surface samples and stored cooled and without bubbles in polyethylene bottles until measurements and analysis in the lab.

Density measurements were done in 1

pH was measured using a HQ11d pH meter (Hach-Lange, Germany) in the lab.
Sulfate (SO

Nitrate (NO

Fluoride (F

If the charge balance between cations and anions was higher than 5 % or
below

The initial algorithm of TEOS10 according to IOC et al. (2010) was applied only for seawater serving as a reference. In all other cases, the adaptation for limnic systems proposed by Pawlowicz and Feistel (2012) was used since all other systems are limnic. Because the only difference between both algorithms is the calculation of the so-called absolute salinity and the equation for density is the same, “TEOS-” was used in the legends of all diagrams in Fig. 1a and b.

For systematic investigation of dependencies of coefficients,

All the density methods have been implemented in Python 2.7 except the
TEOS-10 (IOC et al., 2010). For TEOS-10 the original Fortran 90 library has
been downloaded from

We thank Ulrich Lemmin for taking and sending a water sample from Lake Geneva, and Karsten Rinke for a water sample from Lake Constance.The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association. Edited by: M. Hipsey Reviewed by: three anonymous referees