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 [email protected]
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
dt
(4.1)
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
( -------- )
Tm - To
where:
To = optimum temperature of growth
Tm = maximum temperature of growth
a = empirical constant
(4.4)
Frisk-Nyholm temperature correction is based on the
observation that Θ can be described as a linear function
of temperature:
Θ=a+bT
(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)
b
Ts
a
- (--- + Ts)(ln(a + b Ts) -1)
b
(4.8)
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
(Frisk 1980)
b: K(T)/K(Ts) as a function of temperature
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) = --------KL + L
(4.9)
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)
-------dz
= - ε L(z)
(4.10)
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:
-εz
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:
1
f(L) = -----z2-z1
=
-εz
z
L(0)e
2
∫ --------------------z
-εz
1
KL + L(0)e
dz
-εz1
1
KL + L(0)e
---------- ln( ---------------------)
ε(z2 -z1)
-εz2
KL + L(0)e
where:
L(0) = light intensity on the surface
(4.13)
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
Lo
(4.14)
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:
1
f(L) = -------z2 - z1
1
= ----------(e
(z2-z1)ε
-εz
-εz
z
L(0)e
(1 – L(0)e
/Lo)
2
∫ ------------ e
dz
z
Lo
1
L(0) -εz2
(1 - ------ e
)
Lo
L(0) -εz1
(1 - ----- e
)
Lo
-e
)
(4.15)
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) = --------KP + P
(4.16)
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) = --------KN + N
(4.20)
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) = ---------------------KN + p N1 + N3
(4.21)
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) = -----------------1
1
------ + ----f(P)
f(N)
(4.24)
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 MichaelisMenten kinetics, we can tak about Michaelis-MentenDroop 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 zooplankton. It can be described by a simple equation:
G=gZ
(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
dt
(4.30)
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 --------------KA + A - AL
(4.31)
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.