### Calculating density

Temperature variations and concentration differences are responsible for density differences in lake water. These density differences lie in the range of permille relative to avarage density. However, these small density differences control the circulation of a lakes. In conclusion, high acccuracy approaches are required to reflect the processes connected to density differences in lakes. The density stratification controls the transport of matter, such as nutrients and oxygen, and hence is a central attribute to the ecology, water quality and the foodweb that can establish in the lake.

As density in lakes cannot be measured at the required accuracy in-situ, indirect approaches have been developed, which base density on easier and more accurately measurable quantities. Usually measurements of temperature and electrical conductivity are implemented. Elecrical conductivity itself shows a temperature sensitivity and hence needs to be processed from in-situ values to a reference temperature usually 25°C (but also other temperatures have been used), before it can be used as a quantitative measure for dissolved substances.

The contribution of solutes to electrical conductivity and to density are specific. As a consequence, the high accuracy required for limnological purposes cannot cover the range of lake waters found on Earth. As a consequence, several approaches have been developed to approach density numerically. These approaches can be grouped into four classes. We only list potential density values here, which in limnological sciences is simply referred to as "density". If you are not familiar with the difference between in-situ density and potential density, or neither with in-situ temperature and potential temperature read in the appropriate literature:

#### Ocean conditions and brackish water, UNESCO formula

Density of ocean water can be calculated floowing the UNESCO formula (Foffonoff an Millard 1983, 1986). This approach uses in-situ electrical conductivity and temperature measuremnts to calculate salinity, which for ocean conditions and brackish water is a very good approximation of the dissolved substnaces in the water in grams of solutes in one kilogram of ocean water (permille and someimes referred to as psu = practical salinity units). The ocean has a salinity of about 35. The formula is valid for waters from salinity 2 to 42. Despite its limited formal applicability to lake water, most numerical models use this approach to calculate density for its large range of reliable behaviour.

#### Numerical approaches used in more than 1 lake

In the (European) alpine lakes, scientists have implemented the approach by Bührer and Ambühl (1975) in their calculations. The formula uses a third order polynomial for temperature and a linear term for electrical conductivity. Only a slight variation was found for Lake Constance (Obersee)

Being aware of the short comings of the UNESCO equation at low salinity, Chen and Millero (1986) published a special aproach for lakes with a low concentration of solutes. the composition of the considered waters is not explicitely mentioned, but the calculated values indicate a dominance of NaCl. The formula has been used in various lakes on Earth, but the formula has been critizied, when applied to lakes, where NaCl was not dominating the solutes. (no accessible internet source known to me - let me know)

#### lake specific approach

If a lake does not follow the cmposition of solutes closely enough to one of the above mentioned lakes, the formulation of a lake-specific numerical approach may be required.

This can be done with only one sample: 1) Take a sample of water, 2) measure its electrical conductivity at 25°C, 3) measure its density in a densitometer over the interval that is required for your considerations, 4) make a fourth order fit for the density against temperature 5) use a density formula for pure water (e.g. Kell 1975 or Chen and Millero 1986) and add a term for the conductivity.
Lake Constance: Heinz (1990) or Bäuerle et al. (1998) with a constant term
Mine Lake 111: Karakas et al. (2003), conductivity term includes a third order polynomial in temperature, to also reflect the effect on temperature of maximum density
One short coming of this approach is the fact that the composition of the salt is assumed to be constant for any concentration.

In lakes that show chemical gradients, the chemistry can be included much better, when a second water sample is taken. Especially meromictic lakes are candidates for such an approach. Best collect the water samples from the low end of conductivity and at the high end. Proceed like above, but replace the pure water curve by the lower conductivity sample.
Lake Waldsee: Boehrer et al. (2009)
Though the composition is not assumed to be the same, the samples are asumed to lie on the mixing line between both samples, which can be a good approach for many cases.

It is also possible to take more samples to include non-linearities.
Lake Mono: Jellison and Melack (1999)
Lake Goitsche (Niemegk): Gräfe and Boehrer (2001)
In lake, the heterogeneity however may overcome the non-linearities.

A last lake specific approach includes the use of coefficients from physical chemistry for both electrical conductivity and density. For a given water composition, both values can be calculated. Finally the density effect is correlated to the electrical conductivity.
Lake Malawi: Wüest et al. (1996)
For a lake specific approach, you may also use the density calculator RHOMV, and use those densities.

#### calculating density from chemical composition of lake water

RHOMV is a numerical code that calculates (potential) density of limnic waters based on the chemical composition of its solutes and its temperature. The scientifc background of this approach has been published in the paper Boehrer B., Herzsprung P., Schultze M., Millero F.J. (2010) Limnol. Oceanogr. (accepted). The numerical code was written by Moreira S. and Boehrer B. (2010). This approach is particularly suited for geochemical stratification models od lakes, where the density is based on the local composition of solutes.