AQUATIC WATER QUALITY MODELLING

AQUATIC WATER QUALITY MODELLING August 8, 2007 Research Professor Tom Frisk Pirkanmaa Regional Environment Centre P.O.Box 297, FIN-33101 Tampere, Finland E-mail tom.frisk@ymparisto.fi Phone +358 500 739 991

4. BIOLOGICAL AND CHEMICAL PROCESSES, PART I 4.1 Phytoplankton Phytoplankton can be described as community, phytoplankton groups or in principle on species level. In the following, phytoplankton is considered as a community. As a relative measure of phytoplankton, chlorophyll a concentration is often used. It can be converted to phytoplankton biomass by mans of statistical models.

The rate of change of phytoplankton biomass (due to non-hydraulic processes) can be described as follows: dA ---- = μA - ρA - σA - G (4.1) dt where: A = phytoplankton biomass (M L-3) μ = growth rate coefficient of phytoplankton (T-1) ρ = respiration coefficient of phytoplankton (T-1) σ = sedimentation coefficient of phytoplankton (T-1) G = grazing (M L-3 T-1)

The growth rate of phytoplankton is dependent on temperature, light and concentrations of dissolved nutrients. Respiration coefficient is generally described as dependent on temperature. Sedimentation coefficient is also dependent on temperature to some extent, and it is often described as a function of depth. Grazing is a function of phytoplankton biomass.

The dependence of growth rate coefficient on different factors can be described in many ways. • In general it can be described wit the following equation: • μ = μs f(T) f(L, P, N, ...) (4.2) • where: • μs = standard value of growth rate coefficient (T-1) • f(T) = temperature correction function • f(L, P, N, ...) = limitation function of light, phosphorus, • nitrogen and other factors affecting growth

The following form is often used: • μ = μs f(T) f(L) f(P,N) (4.3) • where: • f(L) = limitation function of light • f(P,N) = limitation function of phosphorus and nitrogen In addition to phosphorus and nitrogen, also carbon and silicon are often included in the model. In this lecture they are not treated.

The standard value of growth rate coefficients represents the situation in which light and nutrients are so much available that they do not limit the growth and temperature is the same as the selected standard temperature (generally 20°C). As a temperature correction function, Eq. (3.11) according to which Θ is constant is often used. However, growth rate coefficient does not always increase with increasing temperature but growth has an optimum temperature at which Θ=1. The optimum temperature can be described using e.g. the models of Lassiter and Kearns (1973) or Frisk and Nyholm (1980).

Lassiter-Kearns equation is the following: a(T - To) Tm - T a(Tm - To) μ(T) = μ (To) e ( -------- ) (4.4) Tm - To where: To = optimum temperature of growth Tm = maximum temperature of growth a = empirical constant

Frisk-Nyholm temperature correction is based on the • observation that Θ can be described as a linear function • of temperature: • Θ = a + b T (4.5) • where a ja b = empirical constants Θ is the greatest at temperature 0 and it decreases when temperature grows and so the value of b is negative. • When Θ reaches the value of 1, the temperature is at optimum. Optimum temperature can be calculated as: • To = (1 - a)/b (4.6)

Growth rate coefficient at temperature T can be calculated using the following equation: T ∫ ln Θ dT Ts μ(T) = μ(Ts) e (4.7) where: Ts = standard temperature (generally 20°C) If Eq. (4.5) is valid the exponent of Eq. (4.5) can be calculated in the following way:

T a • ∫ ln Θ dT = (--- + T)(ln(a + b T) -1) • Ts b • a • - (--- + Ts)(ln(a + b Ts) -1) (4.8) • b The Lassiter-Kearns equation is applicable for describing for phytoplankton growth or other processes for which optimum and maximum temperatures can be defined. The Frisk-Nyholm temperature correction function is general and it can be applied to different processes of water quality models.

Fig. 4.1 a: Θ as a function of temperature b: K(T)/K(Ts) as a function of temperature (Frisk 1980)

Fig. 4.2 Phytoplankton growth (according to the data of Reynolds and Goldstein 1979) (Frisk & Nyholm 1980)

Fig. 4.3 (Frisk & Nyholm 1980) Sedimentation rate of particles. Data calculated on the basis of Kajosaari (1973) BOD decomposition coefficient according to the data of GOTAAS (1949), -------- = Streeter-Phelps curve

Fig. 4.4 Benthic carbon dioxide production according to the data of Bergström (1979) (Frisk & Nyholm 1980)

As for temperature correction, also for light correction several ways of description have been presented. One simple way is to compare light with nutrients and to describe light correction with the following function: L f(L) = --------- (4.9) KL + L where: L = light intensity (M T-3) KL = half saturation constant of light (M T-3)

Incoming light is absorbed in the lake when light penetrates deeper, and absorption obeys Lambert’s law: dL(z) -------- = - ε L(z) (4.10) dz where: L(z) = light intensity at depth z ε = light absorption coefficient (extinction coefficient)(L-1) The solution of Eq. (4.10) is the following: - ε z L(z) = L(0) e (4.11)

where: L(0) = light intensity on the surface • Light absorption coefficient can be calculated as follows: • ε = ε0 + kh ch + ka A (4.12) where: ch = concentration of dissolved organic matter, particularly humus (M L-3) A = phytoplankton biomass (M L-3) ε0, kh ja ka = constants

In Finland, humus is by far the most significant factor contributing to light absorption compared to other factors. Absorption coefficient can usually be calculated by means of colour of water. The value of the limitation function at certain depth can be calculated using Eqs. (4.9) and (4.11). If light limitation in a certain water layer is considered the average value of the light limitation function (Eq. 4.9) is calculated:

-εz 1 z2 L(0)e f(L) = ------ ∫ --------------------- dz z2-z1z1 -εz KL + L(0)e -εz1 1 KL + L(0)e = ---------- ln( ---------------------) (4.13) ε(z2 -z1) -εz2 KL + L(0)e where: L(0) = light intensity on the surface

However, phytoplankton growth has an optimum light intensity at higher values of which growth is slower. To describe this, Steele’s (1965) is often used: L (1 - L/Lo) f(L) = ---- e (4.14) Lo where: Lo = optimum intensity of light. The value of the limitation function can be calculated applying Eqs. (4.11) and (4.14).

The limitation function between depths z1 and z2 is obtained by calculating the average: -εz -εz 1 z2 L(0)e (1 – L(0)e /Lo) f(L) = -------- ∫ ------------ e dz z2 - z1 z1 Lo L(0) -εz2 L(0) -εz1 (1 - ------ e ) (1 - ----- e ) 1 Lo Lo = ----------(e - e ) (4.15) (z2-z1)ε

Fig. 4.5 = Eq. (4.14) ------------ = Eq. (4.9) (Rossi 1991)

The limitation function of phosphorus in the models is usually the following: P f(P) = --------- (4.16) KP + P where: P = concentration of available phosphorus (M L-3) KP = half saturation constant of phosphorus (M L-3) As available phosphorus, phosphate phosphorus can be used or…

… it can be calculated on the basis of total phosphorus • concentration using a statistical model e.g. in the • following way: • P = -aP + bP TP - αP A (4. 17) • where: • aP ja bP = constants which can be determined on the • basis of total phosphorus and phosphate phosphorus • measurements • A = phytoplankton biomass (M L-3) • αP = phosphorus content of phytoplankton

If it is assumed that the different fractions of nitrogen are as well available to phytoplankton the concentration of available phosphorus can be calculated as the sum of inorganic nitrogen fractions: • N = N1 + N2 + N3 (4.18) • where: • N1 = ammonia nitrogen concentration (M L-3) • N2 = nitrite nitrogen concentration (M L-3) • N3 = nitrate nitrogen concentration (M L-3)

The concentration of available nitrogen can also be • calculated on the basis of total nitrogen: • N = -aN + bN TN - αN A (4.19) • where: • aN ja bN = constants which can be calculated utilizing • measurements of total nitrogen and nitrogen fractions • A = phytoplankton biomass (M L-3) • αP = nitrogen content of phytoplankton

The limitation function of nitrogen is similar to that of phosphorus: N f(N) = --------- (4.20) KN + N where: N = concentration of available nitrogen (M L-3) KN = half saturation constant of nitrogen (M L-3) The quantitative importance of nitrate in lakes is generally so small that it is not included in the models.

Phytoplankton has been found to prefer ammonia over nitrate, even though different results have been obtained in some studies. The phenomenon can be taken into account using the so called preference coefficient. The limitation function of phosphorus can then be written as follows: p N1 + N3 f(N) = ---------------------- (4.21) KN + p N1 + N3 where p = preference coefficient of ammonia nitrogen

If the value of the preference coefficient is = 1, ammonium and nitrate are as well available. If p>1, ammonia is preferred and if p<1, nitrate is preferred. How to combine the limitation functions of phosphorus and nitrogen is a questions to which many answers have been given. • The solution according to Liebig’s law of minimum is the • following: • f(P,N) = min{f(P), f(N)} (4.22) • where: • f(P,N) = combined limitation function of P and N • f(P) = limitation function of P (Eq. 4.16) • f(N) = limitation function of N (Eq. 4.20 or 4.21)

According to Eq. (4.22) the limitation function has the value of that limitation function which has lower value. • Another way is to multiply the limitation functions: • f(P,N) = f(P) · f(N) (4.23) A third generally used way is to calculate the harmonic average of the limitation functions: 2 f(P,N) = ------------------ (4.24) 1 1 ------ + ----- f(P) f(N)

Also other ways of combining have been presented. Let’s have a look at an example in which the limitation function of phosphorus f(P) has the value of 0.5 and the limitation function of nitrogen f(N) the value of 0.7. According to Eq. (4.22) the combined limitation function f(P,N) has the value of 0.5. According to Eq. (4.23) f(P,N) has the value of 0.35 and according to Eq. (4.24) the value of 0.58. We can find that the values differ from each other quite much. Thus the values of the half saturation constants in the limitation functions are not independent of the way of describing the combined effect.

If phytoplankton growth is described, as presented before, as a function of nutrient concentrations in water and nutrient uptake is considered as the same process, we talk about Michaelis-Menten-Monod kinetics (MMM). Growth is often described as dependent on intracellular nutrient concentration. This dependence is called (e.g.) Droop kinetics and if nutrient uptake obeys Michaelis- Menten kinetics, we can tak about Michaelis-Menten- Droop kinetics (MMD).

The loss terms of phytoplankton (the right hand side of Eq. 4.1) are respiration, sedimentation ad grazing. If zooplankton is not a state variable in the model grazing is usually included in the respiration term. Other mortality than in the connection of grazing is also included in the respiration term, except in the models in which detritus is a state variable. Phytoplankton is then assumed to become detritus. Thus it is important to keep in mind the meaning of “respiration” in different model versions.

The temperature dependence of respiration coefficient can be described by the temperature correction functions presented above. In lakes, the optimum temperature is not reached. Sedimentation of phytoplankton is often described as an areal reaction instead of voluminal reaction. Then in Eq. (4.1) there is the term (σs/Δz)A (where Δz is the thickness of the water layer)instead of the term σA. Temperature dependence of sedimentation can also be described using the same correction functions as with other processes, but as already mentioned, the dependence is weaker then for growth, respiration or decomposition of organic matter.

Grazing is dependent on biomass of herbivorous zoo- • plankton. It can be described by a simple equation: • G = g Z (4.29) • where: • G = grazing (M L-3 T-1) • g = grazing coefficient (T-1) • Z = zooplankton biomass (M L-3) In a more sophisticated description the phenomena that different phytoplankton groups can be utilized at different intensities and that different zooplankton groups have different diets.

4.2 Zooplankton The rate of change of zooplankton biomass (due to non-hydraulic processes) can be calculated as follows: dZ ---- = μz Z - ρz Z - Pz - Mz (4.30) dt where: μz = growth rate coefficient of zooplankton (T-1) ρz = respiration coefficient of zooplankton (T-1) Pz = predation to zooplankton (M L-3 T-1) Mz = mortality due to other reasons than predation (M L-3 T-1)

If predators eating zooplankton (planktivorous zooplankton or fish) are not separately simulated, predation term is not usually included in the model but predation is taken into account in the respiration term. Also the mortality term is Mz is often omitted and mortality is included in the respiration term. The growth rate coefficient of zooplankton is usually described as a function of phytoplankton biomass in the form of Michaelis-Menten function. However, no growth of zooplankton is assumed if phytoplankton biomass is smaller than a certain a limit value:

A - AL μz =μzs--------------- (4.31) KA + A - AL where: μzs = constant KA = half saturation constant of phytoplankton (in the growth of zooplankton) (M L-3) A = phytoplankton biomass (M L-3) AL = the lowest value of phytoplankton biomass at which zooplankton can grow (M L-3)

The processes of zooplankton are also dependent on temperature. However, in building a model is important to be able to separate the primary and secondary effects so that the impact of temperature on phytoplankton is not taken into account twice in the model.

AQUATIC WATER QUALITY MODELLING