Scientific background

Thermocline and 10 degrees Celsius isotherme

The thermocline in the plots is calculated using the R package rLakeAnalyzer. Different definitions and calculation methods exist for mixing depth (\(z_{mix}\)) and thermocline, see for example the references in the rLakeAnalyzer documentation (Winslow et al. 2019) As a simple rule of thumb, mixing depth can be estimated as the first depth where the temperature gradient is more than one degree per meter (1K/m). The app uses the rLakeAnalyzer algorithm, to understand what it does is left as an exercise (see below).

Both mixing depth and thermocline are somewhat complex. In reality, hydrophysical gradients are continuous and change dynamically. “Fixed boundaries” do not really exist, but they are good indicators and helpful for further computations. Another practical indicator is the “10 degrees Celsius isotherme”, that is even simpler and less influenced by complex interactions.

Oxygen saturation

Oxygen saturation is given by the ratio of the actual oxygen concentration and the oxygen saturation concentration at the given temperature. The saturation concentration can be approximated as a function of water temperature and barometric pressure using the equation of Mortimer (1981):

\[ C_{O2, sat} = \exp\left( 7.7117 − 1.31403 \cdot \log\left(T + 45.93\right)\right) \cdot \frac{p}{1013.25} \]

where \(T\) is the temperature in degrees Celsius, and \(p\) is the air pressure in hectopascal. The resulting saturation concentration is in units of gO/m\(^3\) or mgO/l.

Underwater light profile and euphotic zone

The underwater light \(I_z\) (\(I=\) irradiation) in a particular depth \(z\) can be estimated from the light intensity immediately below the water surface \(I_0\) using Lambert-Beer's law:

\[ I_z = I_0 \cdot e^{-\varepsilon \cdot z} \]

where it is assumed that the light extinction coefficient \(\varepsilon\) (in some books also named \(k_d\)) is constant over depth. This is of course an approximation for mainly two reasons:

  • light extinction depends on the light wavelength
  • color and particles are evenly distributed over depth

The \(\varepsilon\)-value measured with an underwater light sensor is a mean value over depth and over a certain spectral range (the visible light or the photosynthetic active part), so it can be called the “mean vertical and mean spectral extinction coefficient”.

It can be directly calculated from Lambert-Beer's law by linear regresion of the log-transformed equation

\[ \ln(I_z) = \ln(I_0) -\varepsilon \cdot z \]

that is equivalent to a linear regression, where the coefficient \(b = \varepsilon\):

\[ y = a - b \cdot x \]

Limnological interactions between light, temperature, oxygen pH and conductivity

In aquatic ecosystems, hydrophysical, chemical and biological variables are influenced by hydrology, meteorology and seasonal forcing and influence each other. As a comprehensive description would exceed the space here, we refer to the hydrobiology lecture and the textbooks.

You may consider the following keywords and questions:

  • seasonality and stratification patterns, e.g. dimictic, monomictic, polymictic
  • influence of stratification on oxygen availability, oxygen consumption and production
  • influence of trophic state on the shape of oxygen profile
  • influence of phytosynthetic activity on pH and conductivity (refers to the calcium-carbonate balance)
  • influence of climate warming on stratification duration
  • influence of stratification duration on oxygen in the hypolimnion
  • and more …

Exercises

  • Download data from the course home page and compare temperature and light profiles. Search the internet for background information about the particular Lakes and Reservoirs and discuss how the profile characteristics are related to the lakes. Can you find agreements and disagreements? Are they plausible or surprising?
  • Compare the rLakeAnalyzer thermocline with the 1K/m rule.
  • Try to reproduce the results with Libreoffice, Excel or an own R script.
  • Compare the extinction coefficients with data from a limnology textbook e.g., Lampert and Sommer (2007), Figure 3.4

(Bonus) Recreate the plots in R

Below, the source code to calculate the 10°C isotherme, thermocline depth and 1% light depth, and create the plots shown in this application is provided. It is not required to understand this or to be able to recreate the plots in R. Nevertheless, some skills using the statistical programming language R can be advantageous in your future (academic) career and there are lots of good resources to learn it online.

# required libraries
library(readxl)
library(reshape2)
library(ggplot2)
library(rLakeAnalyzer)

# an example data set can be downloaded from:
# https://github.com/tpetzoldt/hydrobio/blob/master/data/lake_profile.xlsx

# read in the data
DF <- read_excel("lake_profile.xlsx")

# reshape the data to "long" format
df_long <- melt(DF, id.vars = "Depth")

# add a column to create the different subplots
df_long <- merge(df_long,
    data.frame(variable = c("Temp", "Oxygen", "pH", "Cond", "chla", "Turb", "Light"),
               plot = c("Temp & O2", "Temp & O2", "pH", "Conductivity",
                        "Chlorophyl-a", "Turbidity", "Light")))

# first remove NAs (Not Available)from the data
DF_valid <- na.omit(DF[c("Depth", "Temp")])

# calculate 10°C isotherme
z_iso10 <- approx(DF_valid$Temp, DF_valid$Depth, 10, ties=mean)$y

## calculate thermocline using rLakeAnalyzer
# calculate thermocline depth
z_thermo <- thermo.depth(DF_valid$Temp, DF_valid$Depth)

## calculate 1% light depth
# first fit a liner model to the logarithmic light data
m <- lm(log(DF$Light) ~ DF$Depth)
# then calculate the depth where 1% light is left
z_light <- log(0.01) / m$coefficients[2]

## first plots (Temp, O2, ph, and cond)
# create subset of the data containing only Temp, O2, ph, and cond
dat_p1 <- subset(df_long, df_long$variable %in% c("Temp", "Oxygen", "pH", "Cond"))
# plot the data using ggplot
p1 <- ggplot(dat_p1, aes(x = Depth, y = value, col = variable)) + geom_line() +
  geom_point() + coord_flip()  +
  theme(legend.position="bottom") + xlab("Depth (m)")  +
  scale_x_continuous(trans = "reverse") +
  facet_grid(.~plot, scales = "free")

# add the thermocline depth, 10°C isotherme and 1% light depth to the plot

p1 <- p1 + geom_vline(data = data.frame(x = z_iso10, variable = "10 °C isotherme"),
                      aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_light, variable = "1% light depth"),
             aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_thermo, variable = "thermocline"),
             aes(xintercept = x, col = variable), linetype = "dashed")


# show the plot
p1

## create chla and Trub plot
# subset of data containing chla and Turb
dat_p2 <-  subset(df_long, df_long$variable %in% c("chla", "Turb"))

# plot the data
p2 <- ggplot(dat_p2, aes(x = Depth, y = value, col = variable)) + geom_line() +
  geom_point() + coord_flip()  +
  theme(legend.position="bottom") + xlab("Depth (m)")  +
  scale_x_continuous(trans = "reverse") +
  facet_grid(.~plot, scales = "free")

# add the thermocline depth, 10°C isotherme and 1% light depth to the plot

p2 <- p2 + geom_vline(data = data.frame(x = z_iso10, variable = "10 °C isotherme"),
                      aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_light, variable = "1% light depth"),
             aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_thermo, variable = "thermocline"),
             aes(xintercept = x, col = variable), linetype = "dashed")

# show the plot
p2


## create the light plots
# create subset with light data
dat_p3 <- subset(df_long, df_long$variable %in% c("Light"))

# plot the data (allready with the thermocline depth, 10°C isotherme and 1% light depth)
p3 <- ggplot(dat_p3, aes(x = Depth, y = value, col = variable)) +
  geom_line() + geom_point() + coord_flip() +
  theme(legend.position="bottom") + xlab("Depth (m)")  +
  scale_x_continuous(trans = "reverse") +
  ggtitle("Light") +
  geom_vline(data = data.frame(x = z_iso10, variable = "10 °C isotherme"),
             aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_light, variable = "1% light depth"),
             aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_thermo, variable = "thermocline"),
             aes(xintercept = x, col = variable), linetype = "dashed")

# show the plot
p3
## now plot log light and add the linear fit
# get the coefficients from the linear model to show the equation in the plot
eqt <- paste0("y = ", round(coef(m)[1], 2), " ",
              round(coef(m)[2], 2), " * x")
# plot log light (allready with the thermocline depth, 10°C isotherme and 1% light depth)
p4 <- ggplot(dat_p3, aes(x = Depth, y = value, col = variable)) +
  geom_point() + coord_flip() + scale_y_log10() +
  theme(legend.position="bottom") + xlab("Depth (m)")  +
  scale_x_continuous(trans = "reverse") +
  geom_smooth(method = "lm", aes(col = "linear fit"))  + ggtitle("log(light) with linear fit") +
  geom_vline(data = data.frame(x = z_iso10, variable = "10 °C isotherme"),
             aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_light, variable = "1% light depth"),
             aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_thermo, variable = "thermocline"),
             aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_text(data = data.frame(Depth = 1, value = 1, variable = "linear fit"),
            parse = FALSE, label = eqt, color = "black")

# show the plot
p4

References

Lampert, Winfried, and Ulrich Sommer. 2007. Limnoecology: The Ecology of Lakes and Streams. Oxford university press.

Mortimer, C. H. 1981. The oxygen content of air-saturated fresh waters over ranges of temperature and atmospheric presure of limnological interest. Mitteilungen Internationale Vereinigung für theoretische und angewandte Limnologie, 22:1–23.

Winslow, Luke, Jordan Read, Richard Woolway, Jennifer Brentrup, Taylor Leach, Jake Zwart, Sam Albers, and Doug Collinge. 2019. RLakeAnalyzer: Lake Physics Tools. https://CRAN.R-project.org/package=rLakeAnalyzer.

Wissenschaftliche Grundlagen

Thermokline und 10 Grad Celsius-Isotherme

Für die Plots wird die Thermokline mit Hilfe des R-Pakets rLakeAnalyzer berechnet.

Generell existieren für die Berechnung der Sprungsschicht (\(z_{mix}\)) bzw. Thermokline unterschiedliche Definitionen und Berechnungsverfahren, siehe z.B. Winslow et al. 2019.

Als einfache Faustregel wird oft die Tiefe angegeben, bei der der Temperaturgradient mehr als 1 Grad pro Meter (1K/m) beträgt. Die vorliegende App nutzt den rLakeAnalyzer-Algorithmus, das Verständnis der Funktiosweise belassen wir als Aufgabe (siehe unten).

In der Realität sind Durchmischungstiefe bzw. Thermokline physikalisch komplex gesteuert. Außerdem sind die physikalischen Gradienten (z.B. Temperatur, Dichte, Turbulenz) in der Realität kontinuierlich und ändern sich dynamisch. Feste Grenzen gibt es eigentlich nicht, als Indikatoren und als Werkzueg für weitere Berechnungen sind sie jedoch sehr nützlich. Ein weiterer noch einfacherer pragmatischer Indikator ist die 10 Grad Celsius-Isotherme.

Sauerstoffsättigungskonzentration

Die Sauerstoffsättigung ist das Verhältlis zwischen der gemessenen Sauerstoffkonzentration (in mg/L bzw. mmol/L) und der theoretischen Konzentration von sauerstoffgesättigtem Wasser bei einem bestimmten Luftdruck und einer bestimmten Temperatur. Zur Abschätzung existieren verschiedene empirische Formeln, z.B. die recht einfache Formel von Mortimer (1981):

\[ C_{O2, sat} = \exp\left( 7.7117 − 1.31403 \cdot \log\left(T + 45.93\right)\right) \cdot \frac{p}{1013.25} \]

mit Temperatur \(T\) in Grad Celsius und Luftdruck \(p\) in Hektopascal. Die Sättigungskonzentration nach dieser Formel hat die Maßeinheit g O/m\(^3\) bzw. mg O/L. Weitere Formeln und entsprechende Literaturangaben finden sich im R-Paket marelac.

Unterwasserlichtprofil und euphotische Zone

Die Unterwasser-Lichtintensität \(I_z\) (\(I=\) irradiation) in einer bestimmten Tiefe \(z\) ergibt sich aus der Lichtintensität unmittelbar unter der Wasseroberfläche \(I_0\) über das Lambert-Beer'sche Gesetz:

\[ I_z = I_0 \cdot e^{-\varepsilon \cdot z} \]

Hierbei wird angenommen, dass der Lichtextinktionskoeffizient \(\varepsilon\) (in manchen Büchern \(k_d\) genannt) über die Tiefe konstant ist. Das ist aus mehreren Gründen eine Vereinfachung, weil:

  • die Lichtextinktion von der Wellenlänge abhängig ist und
  • Färbung und Partikel im Wasser nicht gleichmäßig verteilt sind.

Aus diesem Grund ist der mit Hilfe eines Unterwasserlichtsensors gemessene \(\varepsilon\)-Wert ein Mittelwert über die Tiefe und über einen bestimmten Spektralbereich, z.B. das sichtbare Licht oder den photosynthetisch aktiven Bereich (photosynthetisch aktive Strahlung PAR). Man spricht deshalb vom “mittleren vertikalen und mittleren spektralen Lichtextinktionskoeffizient”.

Der Koeffizient kann mit Hilfe der logarithmisch-transformierten Form des Lambert-Beerschen Gesetzes:

\[ \ln(I_z) = \ln(I_0) -\varepsilon \cdot z \]

bestimmt werden, analog einer linearen Regression. Die Steigung \(b\) der Gerade entspricht dem Extinktionskoeffizienten \(\varepsilon\):

\[ y = a - b \cdot x \]

Limnologische Wechselwirkungen zwischen Licht, Temperatur, Sauerstoff und Leitfähigkeit

Die hydrophysikalischen, chemischen und biologischen Variablen aquatischer Ökosysteme werden durch meteorologische, hydrologische und andere saisonale Faktoren gesteuert und beeinflussen sich gegenseitig. Eine umfassende Beschreibung würde den Rahmen dieses Textes überschreiten, deshalb wird auf die Vorlesung und die Lehrbücher verwiesen.

Als Anregung für Wiederholung und Selbststudium dienen die folgenden Stichworte und Fragen:

  • Saisonalität und Schichtungsmuster, z.B. dimiktisch, monomiktisch, polymiktisch
  • Einfluss der Schichtung auf die Sauerstoffverfügbarkeit und die Form des Sauerstoffprofils
  • Einfluss des Gewässertrophie auf die Sauerstoffkurve
  • Einfluss der photosynthetischen Aktivität auf pH-Wert und Leitfähigkeit (siehe Kalk-Kohlensäure-Gleichgewicht)
  • Einfluss der Klimaerwärmnug auf die Dauer der Sommerstratifikation
  • Einfluss der Schichtungsdauer auf den Sauerstoffhaushalt im Hypolimnion
  • usw.

Übungsaufgaben

  • Laden sie sich die Daten von der Kurs-Homepage herunter und vergleichen Sie die Temperatur- und Lichtprofile. Suchen Sie im Internet nach den Charakteristika der jeweiligen Seen bzw. Talsperren und diskutieren Sie den Zusammenhang zwischen beobachteten Profilen und den Seeneigenschaften. Gibt es Übereinstimmungen oder Widersprüche? Sind diese plausibel oder überraschend?
  • Vergleichen Sie die vom rLakeAnalyzer abgeschätzten Werte der Thermokline mit der 1K/m-Regel
  • Versuchen Sie, die Ergebnisse mit LibreOffice, Microsoft Excel oder eigenen R-Skripten nachzuvollziehen
  • Vergleichen sie die Extinktionskoeffizienten mit Werten aus Lehrbüchern, z.B. Lampert and Sommer (2007), Abbildung 3.4

(Bonus) R-Scripte für die Vertikalprofile

Anbei finden Sie ein R-Script für die Berechnung der 10°C-Isotherme, der Thermokline und der 1%-Lichttiefe. Ein volles Verständnis von R oder einer anderen Skriptsprache ist kein Bestandteil dieser Übung. Andererseits können Kenntnisse in einer Datenanalysesprache wie R für Ihre zukünftige (akademische oder praktische) Karriere nützlich sein. Im Internet finden Sie zahlreiche gute Quellen für das Selbststudium.

# required libraries
library(readxl)
library(reshape2)
library(ggplot2)
library(rLakeAnalyzer)

# an example data set can be downloaded from:
# https://github.com/tpetzoldt/hydrobio/blob/master/data/lake_profile.xlsx

# read in the data
DF <- read_excel("lake_profile.xlsx")

# reshape the data to "long" format
df_long <- melt(DF, id.vars = "Depth")

# add a column to create the different subplots
df_long <- merge(df_long,
    data.frame(variable = c("Temp", "Oxygen", "pH", "Cond", "chla", "Turb", "Light"),
               plot = c("Temp & O2", "Temp & O2", "pH", "Conductivity",
                        "Chlorophyl-a", "Turbidity", "Light")))

# first remove NAs (Not Available)from the data
DF_valid <- na.omit(DF[c("Depth", "Temp")])

# calculate 10°C isotherme
z_iso10 <- approx(DF_valid$Temp, DF_valid$Depth, 10, ties=mean)$y

## calculate thermocline using rLakeAnalyzer
# calculate thermocline depth
z_thermo <- thermo.depth(DF_valid$Temp, DF_valid$Depth)

## calculate 1% light depth
# first fit a liner model to the logarithmic light data
m <- lm(log(DF$Light) ~ DF$Depth)
# then calculate the depth where 1% light is left
z_light <- log(0.01) / m$coefficients[2]

## first plots (Temp, O2, ph, and cond)
# create subset of the data containing only Temp, O2, ph, and cond
dat_p1 <- subset(df_long, df_long$variable %in% c("Temp", "Oxygen", "pH", "Cond"))
# plot the data using ggplot
p1 <- ggplot(dat_p1, aes(x = Depth, y = value, col = variable)) + geom_line() +
  geom_point() + coord_flip()  +
  theme(legend.position="bottom") + xlab("Depth (m)")  +
  scale_x_continuous(trans = "reverse") +
  facet_grid(.~plot, scales = "free")

# add the thermocline depth, 10°C isotherme and 1% light depth to the plot

p1 <- p1 + geom_vline(data = data.frame(x = z_iso10, variable = "10 °C isotherme"),
                      aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_light, variable = "1% light depth"),
             aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_thermo, variable = "thermocline"),
             aes(xintercept = x, col = variable), linetype = "dashed")


# show the plot
p1

## create chla and Trub plot
# subset of data containing chla and Turb
dat_p2 <-  subset(df_long, df_long$variable %in% c("chla", "Turb"))

# plot the data
p2 <- ggplot(dat_p2, aes(x = Depth, y = value, col = variable)) + geom_line() +
  geom_point() + coord_flip()  +
  theme(legend.position="bottom") + xlab("Depth (m)")  +
  scale_x_continuous(trans = "reverse") +
  facet_grid(.~plot, scales = "free")

# add the thermocline depth, 10°C isotherme and 1% light depth to the plot

p2 <- p2 + geom_vline(data = data.frame(x = z_iso10, variable = "10 °C isotherme"),
                      aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_light, variable = "1% light depth"),
             aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_thermo, variable = "thermocline"),
             aes(xintercept = x, col = variable), linetype = "dashed")

# show the plot
p2


## create the light plots
# create subset with light data
dat_p3 <- subset(df_long, df_long$variable %in% c("Light"))

# plot the data (allready with the thermocline depth, 10°C isotherme and 1% light depth)
p3 <- ggplot(dat_p3, aes(x = Depth, y = value, col = variable)) +
  geom_line() + geom_point() + coord_flip() +
  theme(legend.position="bottom") + xlab("Depth (m)")  +
  scale_x_continuous(trans = "reverse") +
  ggtitle("Light") +
  geom_vline(data = data.frame(x = z_iso10, variable = "10 °C isotherme"),
             aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_light, variable = "1% light depth"),
             aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_thermo, variable = "thermocline"),
             aes(xintercept = x, col = variable), linetype = "dashed")

# show the plot
p3
## now plot log light and add the linear fit
# get the coefficients from the linear model to show the equation in the plot
eqt <- paste0("y = ", round(coef(m)[1], 2), " ",
              round(coef(m)[2], 2), " * x")
# plot log light (allready with the thermocline depth, 10°C isotherme and 1% light depth)
p4 <- ggplot(dat_p3, aes(x = Depth, y = value, col = variable)) +
  geom_point() + coord_flip() + scale_y_log10() +
  theme(legend.position="bottom") + xlab("Depth (m)")  +
  scale_x_continuous(trans = "reverse") +
  geom_smooth(method = "lm", aes(col = "linear fit"))  + ggtitle("log(light) with linear fit") +
  geom_vline(data = data.frame(x = z_iso10, variable = "10 °C isotherme"),
             aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_light, variable = "1% light depth"),
             aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_vline(data = data.frame(x = z_thermo, variable = "thermocline"),
             aes(xintercept = x, col = variable), linetype = "dashed") +
  geom_text(data = data.frame(Depth = 1, value = 1, variable = "linear fit"),
            parse = FALSE, label = eqt, color = "black")

# show the plot
p4

Literaturverzeichis

Lampert, Winfried, and Ulrich Sommer. 2007. Limnoecology: The Ecology of Lakes and Streams. Oxford university press.

Mortimer, C. H. 1981. The oxygen content of air-saturated fresh waters over ranges of temperature and atmospheric presure of limnological interest. Mitteilungen Internationale Vereinigung für theoretische und angewandte Limnologie, 22:1–23.

Winslow, Luke, Jordan Read, Richard Woolway, Jennifer Brentrup, Taylor Leach, Jake Zwart, Sam Albers, and Doug Collinge. 2019. RLakeAnalyzer: Lake Physics Tools. https://CRAN.R-project.org/package=rLakeAnalyzer.

Disclaimer

This app is set up for teaching purposes. Occasional use by guests is allowed as long as server load remains within reasonable limits. The service will be available for limited time.

Source code is licensed free of charge under the GNU General Public License 2.0 available from https://github.com/tpetzoldt/hydrobio

In case of questions, please consult the authors T. Petzoldt and J. Feldbauer.

Links

  • R: A free software environment for statistical computing and graphics
  • Shiny by Rstudio: Web application framework