Phosphorus
Dieter
model of lake eutrophication
M. Imboden
Swiss Federal
Swiss Fcdcral
Institute for Water Resources and Water Pollution
Institutes
of Technology,
Diibcndorf
Control
(EAWAG),
Abstract
A two-box lake model is presented with the subsystems epilimnion
(E) and hypolimnion
rllic mean O2 consmilption
in II as
(I-1) and phosphorus as the limiting nutrient factor.
a function of P loading is calculated and critical P-loading figures above which the lake
turns toward eutrophy are given for varying lake mean depth x and hydraulic
loading facThe results agree with an empirical
relation between x and P loacling given by
tor
q*.
Vollcnweidcr.
In addition,
P rctcntion
factors arc calculated
using the model and compared to measured values..
Eutrophication
of lakes represents one
of the main problems of water pollution
control. In most lakes nitrogen or phosphorus is the limiting nutrient factor. Increasing population density, the use of fcrtilizer in intensive agriculture, and the phosphorus content of detergents have raised
the N and P concentrations in lakes and
thereby their biological
productivity.
A
quantitative study of this cffcct has to take
into account not only biological and chcmical but also physical processes (transport
phcnomcna) within the water system.
Box models are useful and still mathematically simple tools for this kind of investigation. The choice of the subsystems
( boxes) should correspond to regions of
homogeneous chemical and biological conditions. The model then describes the temporal variation of the relevant concentrations (model fullctions)
within each box
depending on a set of model parameters
which themsclvcs can be different in each
box. The external parameters are the flushing coefficient and the loading of the relevant substances; the internal parameters
are the biochemical reaction rates (especially rate of photosynthesis),
the mixing
ratio between the different boxes, the sedimentation rate, and the exchange of matter
at the water-sediment interface. Of course,
these factors undergo seasonal variation.
However, as an approximation
it is possible to choose constant mean values for
them for certain periods such as summer
stagnation and winter circulation (method
of temporally constant parameters).
LIMNOLOGY
AND
OCEANOGHRPIIY
In principle, all thcsc chemical concentrations are interdependent
through various biochemical
reactions, making the
mathematical treatment of the model highly
However, in the case of a
complicated.
dominant controlling nutrient factor it is
possible to decompose the system of cquations. Furthermore, with a stoichiometric
formula of t:he Rcdfield type (Rcdfield et
al. 1963; Stumm 1964), the variation of
other substances (e.g. 0,) can be calculated as a consequence of the P values. As
Thiencmann
(1928) has pointed out, the
analysis of 10, concentration gives important information about the trophic condition of lakes.
Various workers have treated the nutrient
balance using a one-box model for the lake.
Vollenwcidcr
( 1969, in press) has summarized the results, which in their most advanced form lead to an equation connecting mean nutrient concentrations to a few
relevant lake parameters. I-Iowever, disregrading the vertical variation of nutrient
concentrations, as is done in the one-box
model, may produce oversimplified and misleading results. For instance, within the
frame of the one-box model the mean residence time of a substance has to be smaller
than that of the water if the sediments
serve as a sink for this substance (Vollenweider 1969). In fact, as pointed out by
O’Melia (19’72) the real situation is more
complicated, depending on the grade of
vertical inhomogeneity
and the exchange
rate between different depths.
The first step behind the one-box model
297
MARC11
1974,
V.
19(2)
298
Imboden
Q,=O,,
(X,7
1
I
pretation would include all P which is
incorporated in or adsorbed to particulate
matter.
For the rate of mineralization
(or respiration) a first order equation is chosen
(dn/dt),i,
Fig. 1. Nutrient
model for a lake with phosphorus as limiting
nutrient factor.
A and rr represent mean concentrations
of phosphate-P
and
particulate-P.
The indices E, H, and “in” represent epilimnion,
hypolimnion,
and inflow.
Qr,,
and QW are inflow
and outflow
of water per
unit time; g is velocity of sedimentation
of particulate-P; Q is exchange of water between epilimnion
( E ) and hypolimnion
( H ) per unit time; S is cxchange of phosphate at sediment-water
interface
per unit area and unit time; Ph is photosynthesis;
and M is mineralization
or respiration.
is made by the distinction between zones
of photosynthesis (trophogenic zone) and
zones of mineralization
( tropholytic zone),
Since during stratification these zones coincide fairly well with the epilimnion ( E ) and
hypolimnion ( H ) , the expressions E and II
are used for the two subsystems.
In Fig. 1 a two-box model is presented in
which phosphorus is the only limiting nutrient factor. This assumption itself cannot
be proved within the model, but has to be
elaborated by additional
considerations.
Within certain boundaries the P concentrations then do not depend on other concentrations.
The total P concentration
[P,] is separated into two components
[P,] = x + %-.
(1)
A represents the dissolved P available for
bioproduction
( thus x - [POa-P] ), n the
particulate-P.
The question of the attribution of dissolved organic phosphorus is
left open, as its role in organic production
seems to be still unrevealed and rather of
the order of a correction term (Gachter
1968; Lean 1973). Furthermore, it is assumed that the rr component mainly consists
of organic matter, although an exact inter-
= -( dx/d&,
= -Rrr,
( 4:)
where the rate coefficient R has the dimension time-l. R is a function of temperature
as a consequence of Van’t Hoff’s law. It
changes considerably if the lake becomes
anaerobic. The range of aerobic conditions
can be calculated within the model.
The rate of photosynthesis depends on
both the biomass concentration and the nutrient concentration,
However, if P really
is the limiting nutrient factor, the linear
equation
(dx/d&
= -(drr/dt)ph
= -CXX
(W
is a useful approximation ( Imboden 1973).
It signifies that within the trophogenic
layer practically all the available phosphate
is steadily consumed independently
of the
actual biomass concentration.
This is at
least true for most lakes during summer
and corresponds to a recent investigation
by Lean (1973). The rate coefficient a!
can be determined by various field measurements (e.g. Gachter 1968).
From a biological
point of view the
above assumptions may look rough or even
Of course, one could replace
incorrect.
equations 1 to 3 by more complicated ones;
e.g. splitting [P,] into three components
or choosing a more complex model of photosynthesis, However, such alterations would
result in an increase of variables and system
parameters. The aim of the model presented here is to describe a large class of
lakes with a few common parameters in
order to compare them in particular wa:ys,
and its assumptions therefore represent a
compromise between simplicity and reality.
Of course, the modeling of a specific lake
would justify the introduction
of more :refined equations.
The model results in a system of four
coupled linear differential
equations for
the two epilimnetic and hypolimnetic
concentrations :
of
P model
Table
Model
1.
parameters
and their
299
lake eutrophication
range
of variation.
Winter/
Circulation
Summer/
Stagnation
External
parameters
qs = Qin/A:
Hydraulic
loading
(Qin: Water inflow per unit time; A:
Lake Surface)
Extreme cases:
Lake Tahoe
Zirichsee
(Lake Zurich)
0.001
0.1
= L,, + L, : P-loading
per lake surface
and unit time
and par(LA Lx : Loading of dissolved
ticulate-p,
respectively)
Extreme cases: Lake Tahoe
Bodensee (Lake Constance)
Lt
Internal
-1
mg P m-* day-'
parameters
0.001-0.01
0.1-0.4
9:
Velocity
a:
Rate coefficient
of photosynthesis
R:
Rate coefficient
Hypolimnion RH
of mineralization
of sedimentation
Epilimnion
*
t
9
I
m day
0.1
11
I = Q/V,: Exchange coefficient
(Q: Flux of water between E and H per
unit time; VE: Volume of E)
S:
Unit
l-2
.
0.003-0.02
RE*
at sediment-
day-'
0.1-0.4
m day -1
0.06-0.3
day-'
0.003-0.02
3 RH
Exchange of phosphate
water interface
+
0.05-0.5
day -1
RH
0.271:
up to 500 §
mg P m-* day-'
If the temperature
difference
between E and H is 12°C in summer and 0 in winter.
S is positive
if the flux goes out of the sediments into the water.
Theoretical
estimate by Stumm and Leckie (1971).
Measurements of Mauensee, a very small and highly eutrophic
Swiss lake (Internal
Report
08 Swiss Federal Institute
for Water Resources and Water Pollution
Control,
Dibendorf,
Zurich, 1971).
dx,/dt
h-ddt
h&t
= -hn(qsh
+ 6+ 4
+ XI& + &IT”
-I- Lx/xa;
= X&/E) - h(m)
+ m&
+ s/Gr;
=
mz(
t/e
+
g/~,)
-7C,-I(W + g/h
+ &I).
(4)
The system 4 can be solved numerically
or analytically,
although the latter leads
to very complicated expressions.
The definition of the model parameters
q,, & LX,RI<, Err, LX, L,, and g is given in
Table 1. X, xx,, ~~1are the total, epilimnetic,
and hypolimnetic
mean depths, V = VE
+ VII the corresponding vohrmes, and A
is the lake surface: V = Ax. Since the aim
of this work is mainly to compare different
lakes, the number of morphometric parameters is reduced by introducing
an idealized form of the lake which results in the
equations x =: xE + xII and zrr/zn = Vr,/Vn
( = E), i.e. ths surface of the epilimnion is
assumed to be equal to the “surface” of the
hypolimnion.
Consequently, there is no di-
300
Imboch
rect epilimnion-sediment
interface, and direct phosphate exchange between E and
the sediments is neglected in the first equation of 4. The hydraulic characterization of
the lake is uniquely given by the loading
qs, and the total and the cpilimnctic mean
depth, x and xr+ In modcling a single lake,
equation 4 could easily be rewritten for a
general lake morphology.
A detailed mathematical analysis of the
model (steady state concentrations, relaxation time, determination and sensitivity of
parameters) is presented elsewhere (Imboden 1973). In connection with the eutrophication problem we are interested in
the following
questions.
(a) How long does the system take to
reach the steady state during the summer?
(b) What is the hypolimnetic
oxygen
consumption corresponding to this steady
state?
( c) To what degree does the circulation
period destroy the lake’s “memory” of the
summer period?
As will be shown, the model allows us
to answer the first two questions, but the
third one requires a more refined knowledge of mixing processes within the lake.
Concerning (a), I have shown (Imboden
1973) that in summer the concentrations
move to their
rrrc and rrIr approximately
steady state, 7~~~and 7&r, in the same way
as the fmlction
-x0} + x0
(5)
x(t) = ,-(w/~{X(tO)
moves from its initial state x( to) at time to
to its steady state x0 for t-+m. The exponential relaxation is characterized by the relaxation time r after which the function
reaches its steady state x0 within 37% of
the original deviation {x(to) -x0>.
The relaxation times, r( nE) and r( rrrr),
for 7rr;: and rII are given approximately by
r(m)
= (q&E + g/+ky = Gc/( 4s + g)
7( n1) = (%I + g/GI)-l.
(6)
For most lakes qs << g, so that the upper
limit of 7 (nm ) - &c/g - 50 days is valid.
For deep lakes r ( nrr) is approximated by
l/RI1 = 50 to 300 days; for shallower lakes
is smaller. Since in summer a is
much larger than the other parameters such
as t, @h, and RE, the steady state value
of x1<:is approached together with z-~ and
7~~1.Note that the value of a does not have a
significant influence on the concentrations
as long as a is great. Finally, hrI only approaches its steady state very slowly, depending on the exchange rate t between E
and II. In the extreme case e = 0, hII does
not have a finite upper limit.
(b) As first pointed out by I-Iutchinson
( 1938) and Deevey ( 1940)) a relation
should exist between hypolimne tic oxygen
consumption and epilimnetic processes (bioproduction).
In the steady state, the rate
of mineralization
in II per unit volume is
given by the input of particulate-P from E
into II, namely (no means steady state)
(dw’dt)arin = r”rc -ghr
+ (7.r”Ii:-O&/E,
(7)
if we assume that the organic matter which
is lost to the sediments is completely mineralized there as well. The last equation of
4 yields the following
relation between
+& and non:
Together with the averaged stoichiometric
relation ( m and O2 in mg n+ )
(d[0~1/~~)mn
=
140( dn/dt)m
w
we finally get the oxygen consumption in
II per unit tolumc as a function Of-TOE:
Assuming that S = 0 (which means that
phosphate is neither adsorbed nor redissolved at the sediment-water
interface),
the only inhomogeneous terms in equation
4 are proportional to the specific phosphorus
input LX and L,. Therefore, the stea.dy
state concentrations also have to bc proportional to Lx and L,. In fact, as can be
shown, these values mainly depend on Lt
and are insensitive to the partition of the
P input between phosphate and particulate
form. One can write
P <model of lake eutrophication
MEAN
DEPTH
zc (m>
I
I 0.04
Fig. 2. Tolerable
annual phosphorus
loading per lake arca Li,lrlnx as a function
of mean depth
x for different hydraulic
loading factors Q,~. Comparison
of theoretic
Lt,,,l,,s for oligotrophy
with the
empirical
curve by Vollenwcider
( 1968 ) . Choice of model parameters:
a = 1 day-‘; .g = 0.2 m day-‘;
XFI= 10 m; RII = 0.02 day-l; RI,: = 3 RH; .$= 0.001 day”. In addition, two cxtremc casts with qS =
0.02 m day-’ are plotted to demonstrate the sensitivity
due to tho variation
of H and E. Curve A:
Rrr = 0.02 day-l, 4 = 0.01 day-‘; Curve 13: RII = 0.003 day”, .$:= 0.001 day-‘. The points show the
present condition
of the lake, the end points of the arrow the theoretic Lt,,,,ns. Abbreviation
for the
lakes as in Table 2.
7& = fLt,
(11)
where non can bc calculated from an analytical expression (Imbodcn 1973) containing all the model parameters. Thus, we get
f by normalizing noIi: by the corresponding
loading Lt. As can be shown, the mineralization rates Rrr and RI<;are the only internal parameters to which f is really sensitive :
by shifting Rrr from 0.02 day-l to 0.003
(and doing the same for RE using Rri: = 3&)
we decrease f by 30 to 50% depending on
ys and x.
Replacing the time dependent rrE by its
steady state value no F2and assuming a mean
duration of the stagnation period Tst (180 days ) , WC get an average oxygen depletion in II at the end of the stagnation
period:
which is proportional to the total specific
P-loading LtI The various model parameters
enter into 12 both directly and through the
factor f. Given an upper limit for the tolerable O2 consumption A[O~]~,,~~for an oligotrophic lake, we get the corresponding up-
302
Imboden
per limit L t,Inar above which a lake starts
to show eutrophic characteristics :
~~1~:
and 7.r”E) with increasing mean depth z;,
but is independent of the total P-loadmg
Lt, as hole and voE themselves are proporL t,mnx = * [%I mnx X
tional to Lt. From a mathematical point of
view this result is an obvious consequence
of the linear system 4, just as the steady
state concentrations are proportional to Lit
In Fig. 2 Lt,max is plotted as a function of as mentioned in equation 11.
x and q, for fixed internal parameters and
Of course, equation 14 is only valid for
steady state conditions and for S = 0. With
m-3. The resulting
*[GImnx
= 1,000 mg
curves show surprisingly
good agreement
increasing Lt phosphate redissolution at the
with Vollenweider’s
( 1968) empirical rela- sediment-water
interface may produce a
tion between mean depth and tolerable P breakdown of the retention capacity of the
loading. Although the choice of A [021max lake; %p may even become negative. A.s
and with it the absoute values of Lt,mnx may an important conclusion it follows that the
constancy of .$Zp with varying Lt reflects a
seem arbitrary, the theoretical and empirnormal adaptation of the lake to changing
ical variations of L t,mnxwith x show a similarity which cannot be reached with a external conditions, while a change in gp
simple one-box model. Regarding the rela- signalizes a basic alteration in the nutrient
tively low value of A[02]mnx compared to balance of the lake.
In Table 2, calculated and measured rethe 02 saturation, we have to take into
account the effectively inhomogeneous OtL tention factors of a few lakes are compared.
Bodensee (Lake Constance) is the only
depletion within the hypolimnion,
which
normally has its maximum at the bottom of lake with a theoretical value significantly
the lake and may produce anaerobic zones lower than the measured one. Since th.e
corresponding theoretical 7roE is very high
even with a low mean O2 deficit.
( 128 mg m-3) and above measured conIn addition to the mean depth x, I have
introduced the hydraulic loading q8 as a centrations, the steady state represents a
poor approximation in this case. Three lakes
second factor determining the P tolerance
( Baldeggersee, I-Iallwilersee,
Greifensee )
of a lake. The result seems to agree with
experimental knowledge. For instance, Lake show lower measured than theoretical values. For Baldeggersee, Vollenweider ( 1968,
Tahoe with a mean depth of 300 m, which
p. 79 and table 4.7) has computed from exwould have a high Lt,,nax in the pure Lt/x
diagram, in this model has a very low P perimental data that redissolution of P from
tolerance ( in accordance with observations ) the sediments, which may even exceed the
P input from the river, may occur during
due to its extremely low q8 = 0.001 m day-l.
Vollenweider
( in press) has come to a periods on the order of several months.
The breakdown of the P retention capacity
similar conclusion by introducing a modified diagram in which he plots Lt,mnx as a seems to be even more pronounced for
Hallwilersee.
function of q8 instead of X.
The theory of maximum tolerable depleThe P-retention factor gr is defined by
the relative amount of P which is lost to tion of 02 is only reasonable if the hypelimnetic O2 reserves are completely renewed
the sediments. Under steady state conditions the P outflow
of the lake per unit time during the circulation
period, i.e. if the
is given by Qout ( hoE + moE) with Qout = annual O2 deficits do not accumulate from
Qin; the P input is A X Lt (A: lake surface).
one year to the next. Thus, we are left with
Thus,
question (c) about the “memory” of the
lake.
L7?p= l- (q8/Lt)(X0n+rrolp).
(14)
Obviously, a detailed answer is not possible within the frame of this two-box model
& decreases with increasing hydraulic
loading qs and (through the variation of for the simple reason that the hypolimnion
303
P model of lake eutrophication
Table
Fig. 2).
2. Phosphorus retention factors
Data from Vollenweider
(1968).
Lake and
abbreviation
Aegerisee A
Baldeggersee B
Lake
Constance Bo
Greifensee G
Hallwilersee
H
Lake Geneva L
Pfaffikersee
P
Sempachersee
Tirlersee
T
Zlrichsee
Z
Lake Tahoe Ta
Vinern V
Zellersee
Ze
*
O-oligotrophic,
Z
Retention factor
(%)
theory
measured
qs
(m day-')
(m)
0.019
0.015
50
34
;;
0.037
0.058
0.019
90
19
21
::
83
0.036
0.033
0.0074
0.018
0.098
154
fi"6
1":
0.001
0.001
0.038
300
:;
it
M-mesotrophic,
and P loading
iif;
31
86
99
65
68
61
z;
36
:;
-80
25
93
--
of some lakes (abbreviations
P loading
(mgm -2 y ear -' )
tolerable
actual
are for use with
L act /Lto,
State*
0
E
160
1,800
170
130
0.94
13.8
4,000
1,600
550
700
1,360
770
300
1,320
480
100
186-i
6:0
40
150
1,200
160
4;:
100
250
3::
1::
i
4.3
3.6
M:E
E
0.25
1.7
6.7
O/oE
E
E-eutrophic.
is not in fact homogeneous. Therefore the
circulation of a lake cannot be simply described by increasing the exchange parameter 6, as the exchange may for example
be great at the epilimnion-hypolimnion
interface (m 10 m) but practically zero in
a depth of 100 m in a meromictic lake.
However, as the aim of this model is only
to synthesize a general rough behavior for
lakes characterized by a few parameters
such as Lt, x, and q8, it cannot be expected
to describe single events within a lake (e.g.
temporal algal blooms) nor all kinds of
lakes (e.g. amictic lakes where transport
of dissolved substances only occurs by diffusion processes).
The model may be extended by replacing the hypolimnetic box by a onc-dimensional diffusion model for O2 consumption
as a function of time and depth during
stratification (Imboden 1973). In the same
way the renewal of 02 during circulation
could be calculated, to give a more detailed answer about the maximum O2 depletion capacity of the lake. IIowever, we
cannot expect this kind of extended model
to bc applicable in a general way, with a
few fixed parameters, to a group of different lakes. Instead, it could describe indi-
vidual properties due to specific morphometric and meteorological conditions, such
things as the eddy diffusion coefficient as
a function of depth and season that has
recently been calculated for the Ziirichsee
by Li (1973), using monthly temperatures
measured over 10 years.
This kind of investigation
lies beyond
the aim of the general P model presented
here. However, from our box-model calculation we are at least able to draw the following qualitative
conclusions concerning
the memory and the rate of response of a
lake to changes in nutrient input:
1. A lake, oscillating between two (or
more) annual states, each described by a
different set of model parameters, reaches
the corresponding steady states faster if its
mean depth is small and its hydraulic loading factor is large. In this case its memory
does not persist over the seasonal changes.
The response to changed input factors
(mainly P loading) is felt within 1 year in
its P concentration values.
2. A deep lake may have a memory in its
hypolimnion
consisting of both slow approach to steady state and incomplete renewal of its oxygen reserves. Its persistence against increase of nutrient input
304
Imboden
may keep the lake oligotrophic for a certain
time. The inverse effect would slow down
a desired response to reduction of nutrient
input.
3. Both the shallow (quickly responding) and deep (inert) lake may in addition
have an important and long lasting memory
in the sediments. Even after lowering the
P input, once the lake is anaerobic for part
of the year, P could be returned to the water as a consequence of redissolution out
of the sediments until the sediment surface
is depleted allowing a new (aerobic) state.
References
DEEVEY,
E.
S.
Connecticut.
741.
1940. Limnological
studies in
Part 5. Am. J. Sci. 238: 717-
R.
19168. Phosphorhaushalt
und
planktische
Primarproduktion
im Vierwaldsttittersee (Holwcr
Bucht).
Schweiz. Z. IIydrol. 30: l-66.
HUTCIIINSON,
G. E. 1938. On the relation between oxygen deficit
and the productivity
and typology
of lakes.
Int. Rev. Gcsamten
Hydrobiol.
36: 336-355.
IMBODEN,
D. M.
1973. Limnologische
Transport-und
Nahrstoff-Modclle.
Schweiz
Z.
Hydrol. 35( 1) : in press.
LEAN,
D. R. S. 1973. Phosphorus dynamics in
lake water.
Science 1’79:
678-680.
LI, Y. H.
1973. Diffusion
in lakes.
Schweiz.
Z. Hydrol.
35( 1) : in press.
GXCHTER,
C. R. 1972. An approach
to the
modeling of lakes.
Schweiz. Z. TIydrol. 3~&:
l-34.
REDFIELD,
A. C., B. I-I. KETCI-IUM,
AND F. A.
RICHAIXDS . 1963.
The influence
of organisms on the composition
0E seawater, p. 2%
77. In M. N. Hill fed.], The sea, v. 2. Interscience.
STUMM, W.
1964. Discussion (Methods for the
removal of phosphorus and nitrogen from sewage plant effluents by G. A. Rohlich), p. 21%
229. In W. W. Eckenfelder
[ea.], Advancaes
in water pollution
research. Proc. 1st Int.
Conf., London,
1962, v. 2. Pergamon.
-,
AND J. 0. LECKIE.
19’71. Phosphate exchange with sediments:
its role in the productivity
of surface waters, p. III-26/l-1.6
In S. II. Jenkins [ed.], Advances in water
pollution
research. Proc. 5th Int. Conf., San
Francisco and Hawaii, 1970, v. 2. Pergamon.
TIIIENEMANN,
A. 1928. Der Sauerstoff
im cutrophen
und oligotrophen
See. Binncngcwaesser 4.
VOLLENWEIDER,
R. A. 1968. Water
management research.
OECD-IRep.
68.27. Paris.
159 p.
-.
1969. Moglichkeiten
und Grenzen elementarer Modellc
der Stoffbilanz
von Seen.
Arch. Hydrobiol.
66:
l-36.
-.
In press.
Input-output
models of lak,es.
In W. Stumm and K. Wuhrmann
[eds.], Proc.
Kastanicnbaum
Conf ,, April 1972.
O’MELIA,
Submitted: 21 June 1973
Accepted: 30 November 1973