J.S. ENVIRONMENTAL PROTECTION AGENCY
NATIONAL EUTROPHICATION SURVEY
WORKING PAPER SERIES
LAKE PHOSPHORUS LOADING GRAPHS:
AN ALTERNATIVE
by
D. P. Larsen and H. T. Mercier
Eutrophication & Lake Restoration Branch
Working Paper No. 174
PACIFIC NORTHWEST ENVIRONMENTAL RESEARCH LABORATORY
An Associate Laboratory of the
NATIONAL ENVIRONMENTAL RESEARCH CENTER • CORVALLIS, OREGON
and
NATIONAL ENVIRONMENTAL RESEARCH CENTER - LAS VEGAS, NEVADA
•&GPO 697.032
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LAKE PHOSPHORUS LOADING GRAPHS:
AN ALTERNATIVE
by
D. P. Larsen and H. T. Mercier
Eutrophication & Lake Restoration Branch
Working Paper No. 174
CORVALLIS ENVIRONMENTAL RESEARCH LABORATORY
U.S. ENVIRONMENTAL PROTECTION AGENCY
200 S.W. 35th Street
CORVALLIS, OREGON 97330
July, 1975
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ABSTRACT
As an alternative to loading graphs, a graph of mean influent phosphorus
concentration versus phosphorus retention capacity is proposed to
express the relationship between phosphorus supply and hydraulic flow
to, and resultant trophic state of, lakes. Lines of constant lake
phosphorus concentration drawn on the proposed graph delineate predicted
trophic states and changes caused by altering influent phosphorus concen-
trations and/or lake phosphorus retention capacity. The graph, derived
from the steady state solution of a phosphorus mass balance model,
expresses lake mean phosphorus concentration as a function of mean
influent phosphorus concentration and phosphorus retention capacity.
Because the mean influent phosphorus concentration is a potential lake
concentration reduced by the lake phosphorus retention capacity, a
determination of the latter is critical for predicting lake phosphorus
concentrations. Several empirical expressions are developed for oligo-
trophic lakes relating phosphorus retention capacity to areal hydraulic
load or hydraulic washout coefficient. These expressions can be used to
predict average lake phosphorus concentrations from mean influent
concentrations.
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INTRODUCTION
Mass balance models have been developed to describe the relationship
between phosphorus (P) concentrations in lakes and the supply of P to
lakes (see review by Dillon, 1974; also, Imboden, 1973, 1974; Lerman,
1974; Snodgrass and O’Melia, 1974; Sonzogni, et al., 1974; Vollenweider,
1969, 1975). Since lake P concentrations often indicate trophic state
(Sawyer, 1947; Vollenweider, 1968; Dillon, 1975), these models can be
useful for predicting water quality and changes in water quality associated
with changes in P supply. P loading graphs, which may or may not be
developed from mass balance models, have also been used to summarize the
effects of P supply on the trophic state of lakes and to provide an
assessment of lake changes with changes in P supply (Vollenweider, 1968,
1975; Dillon, 1975). The loading graphs are based on the premise that
increased P supply to lakes increases the productivity of the lakes and
that the effects of supply can be modified by lake characteristics, such
as mean depth and hydraulic retention time. Loading relationships
suggest the supply of P (as g P/m 2 /yr) as the critical factor in deter-
mining a lake’s potential trophic state; however, the average influent P
concentration may be more appropriate as a measure of this potential
trophic state.
An alternative is proposed to the above class of P loading graphs. The
average P concentration in a lake can be described as a relationship
between the mean influent P concentration and a lake’s ability to assim-
ilate P. A graph using this expression can be constructed to predict
trophic state and changes in trophic state caused by manipulating influent
P concentrations and/or manipulating a lake’s P assimilative capacity.
This relationship derives directly from the steady state solution of a
simple P mass balance model.
1
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2
O’Melia (personal cornmunication*) suggested that the important feature
about the P supply to lakes is the P concentration of the influent, not
the loading rate itself. Thus, the greater the influent P concentration,
the greater the average concentration of P in the lake is likely to be-—
hence the higher the trophic state--regardless of the loading rates. In
a well mixed system the steady state lake P concentration will be equi-
valent to the inflowing P concentration in the absence of P transport to
the sediments. The lake P retention capacity--possibly a function of a
particular lake’s physical, chemical and biological properties--reduces
the steady state P concentration from that of the influent.
Since the P retention capacity of lakes is critical in determining the
steady state lake concentrations, lake features which control its mag-
nitude were examined. Kirchner and Dillon (1975) suggested that the
retention capacity was related to areal hydraulic flow. The equation
which they developed was valid for additional lakes examined, however,
several alternative expressions can be generated which are equally
valid. These relate the P retention capacity to either areal hydraulic
flow or to the hydraulic washout coefficient.
THE MASS BALANCE MODEL
A mass balance model for P in lakes can be written as (Vollenweider,
1969, 1975):
d p E V. [ p.]
[ ] — . :i ____ ) _ [ P1 [ P] (1)
dt - V w °p
where [ P] = concentration of P in the lake
*personal communication with Dr. Charles R. O’Melia. University of
North Carolina, Chapel Hill, North Carolina.
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v = flow rate of the th tributary
[ ps] = i concentration in th tributary
V = lake volume
= theoretical hydraulic washout coefficient
= P sedimentation coefficient
t = time.
Assumptions and rationale for adoption of such a mass balance model
discussed by Vollenweider (1969, 1975), Dillon (1974), and Sonzogni, et
al., (1974) will not be repeated here.
The steady state solution is:
[ P] = P (2)
p +o
wp
v. [ p.]
where 9 = and is defined as the volumnar loading.
p
In order to predict [ P] , an estimate of o is necessary. Experimental
determination of o is difficult; therefore Dillon and Rigler (1974a)
and Vollenweider (1975) defined a retention coefficient, which
derives from the steady state solution as:
p 0
R=l- W = (3)
p p+o
wp pw
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4
Solving equation (3) for and substituting for a. . in equation (2),
(l-R )
j = p (4)
Conceptually, the supply of P to lakes can be considered as originating
through a single tributary with a flow weighted average P concentration
v. [ p.] Ev. [ p.] Ev.
equivalent to . Since = ‘ and = — -, the average
concentration of P in the incoming tributaries, [ ], is equivalent to
under the steady state assumption. Thus equation (4) can be rewritten
pw
as:
[ P] = [ ] (1 - Rn). (5)
The retention coefficient can be experimentally determined as (Dillon and
Rigler, 1974a):
v [ p ]
R =1- 00 (6)
exp v [ p ]
where and [ p 0 ] respectively are hydraulic outflow and P concentration
in the outflow. Substituting Rexp for equation (5) becomes
[ P] = [ ] (l_Rexp) (7)
= [ P] 00
(l_Rexp)
Equation 7 describes the relationship between steady state lake and mean
influent P concentrations. This relationship can be summarized by
graphing [ ] vs. Rexp showing the modifying effect of a lake’s P retention
capacity upon the influent P concentration. Lines of constant lake P
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concentration can be drawn as solutions to the mass balance model for
values of [ ] and R . If particular lake mean P concentrations can be
used to designate the trophic state of a lake, curves delineating
trophic states can be drawn analogous to the method in which they have
been drawn on loading graphs. This graph can be useful to determine the
reduction in inflowing P concentration necessary to attain a desired
trophic state, or the influent P concentration above which the receiving
lake can be expected to exhibit noxious conditions.
[ ] VS. Rexp GRAPH
Three sets of data have been used to map the position of lakes on [ ]
vs. Rexp graphs (Figure 1). The first two sets were those summarized by
Dillon and Rigler (1974a) and Dillon (1975) and included estimates of
L (areal P loading, g P/m 2 /yr), (mean depth, m), and Rexp for
various lakes in North America and Europe. The quotient, L /(P )
was used to obtain values for [ ]. Gakstatter (personal communication*)
provided the third set which includes and , as well as inflow and
outflow P concentrations and loadings, for many lakes in northeastern
and midwestern United States studied by the National Eutrophication
Survey. The above loading estimates apparently included rainfall and
direct discharges to the lakes but did not include internal loadings.
Tables 1 and 2 summarize the data used.
The scientific community has not yet agreed upon objective criteria for
identifying the trophic state of lakes, therefore the designations 0
(oligotrophic), M (mesotrophic), E (eutrophic) and HE (hypereutrophic)
are qualitative, based on the assessments of limnologists who have
investigated these lakes. In some cases, the identity of the trophic
state was not available.
*personal communication with Dr. Jack Gakstatter, Pacific Northwest
Environmental Research Laboratory, Corvallis, Oregon.
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6
The selection of lake P concentrations delineating trophic state is
difficult. Sawyer (1947) and Vollenweider (1968) indicate that springtime
concentrations of available P in excess of 10 g/l are likely to produce
noxious blooms of algae. Interpretation of Sakamoto’s (1966) and Dillon
and Rigler’s (1974b) average summer chlorophyll vs. springtime total P
concentration suggest that concentrations of total P in excess of 20
ig/l are likely to produce average summer chlorophyll a concentrations
of about 10 ig/1 or greater. Therefore, values of 10 and 20 iig P/1 were
selected to delineate oligotrophic, mesotrophic and eutrophic states.
These values are used primarily for illustrative purposes.
The positions of the lakes on Figure 1 agree reasonably with their
trophic states. Some lakes violate the pattern. Several reasons can be
suggested for their departure from the model and a more detailed investi-
gation would be necessary to determine the cause of this discrepancy.
Reasons for the discrepancies might be:
1. The observed lakes might not be in or near a steady state and,
consequently, recent changes in { ] may not yet be reflected by changes
in lake characteristics.
2. The data base for the calculations might be inadequate for the
present analysis which requires an adequate estimate of the mean con-
centration of P in iriflowing and outflowing waters. The optimum method
for obtaining P flux into or out of lakes has only recently been cri-
tically evaluated (Treunert, et al., 1974). Thus, the data reflect the
notions of the principal investigators so sampling frequency may be
neither consistent nor optimal.
3. The characteristics of some lakes may sufficiently disagree
with model assumptions to invalidate a comparison. For example, vertical
and horizontal inhomogeneities may be large enough to seriously affect
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the model assumption of homogeneously and instantaneously distributed
properties, or factors other than phosphorus may limit primary production.
4. The identity of trophic state is subjective so that designations
are not likely to be universally acceptable and may not be specifically
related to lake P content.
PHOSPHORUS RETENTION CAPACITY
Determining the P retention capacity is critical to the successful
prediction of lake P concentrations. Empirically, R can be determined
from the ratio of outflow loads or concentrations to inflow loads or
concentrations. Therefore, experimental determination of R is only as
good as measurements of imported and exported P. Furthermore, steady
state values are required if Rexp is to be representative of a particular
lake. Here again no attempts have been made to quantify the effect of
deviation from steady state. Valid questions are: Can Rexp be related
statistically to any lake properties such as mean depth and hydraulic
retention time? Do these statistically derived relationships have
theoretical foundation?
Kirchner and Dillon (1975) obtained a good relationship between Rexp and
the areal hydraulic loading (q 5 = but did not obtain an expected
good relationship between Rexp and the hydraulic washout coefficient.
Voilenweider (1975) obtained a statistical correlation between ln and
in and suggested that could be estimated by 10. Substituting 10
for a in ÷ provides:
GpPw
R= 10 (8)
p
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8
a relationship that also suggests a correlation between Rexp and (is.
The data summarized in Table 2 were analyzed to determine whether similar
relationships existed and whether alternative relationships were equally
valid, particularly whether Rexp was related to Only lower mesotrophic
or oligotrophic lakes were examined because increasing levels of produc-
tivity may decrease a lake’s P retention capacity (Vollenweider, 1975)
thus obscuring relationships between Rexp and other lake properties.
Initial examination revealed a high correlation between Rexp and ln ci
as summarized in Figure 2. However, considerable variability is evident.
To try to reduce this variability by including only those lakes likely
to be oligotrophic and to which P loadings probably had not undergone
recent changes, the data from those lakes whose infiowing P concen-
tration was < 25 pg/i were analyzed further. Table 3 summarizes the
correlations and the correlation coefficients for the 20 lakes selected.
For these lakes, the highest correlation existed between Rexp and ln q 5
(Figure 3, r = -0.92). Note that slightly more than half the lakes used
for these correlations were used by Kirchner and Dillon (1975). A least
squares linear fit to the data provided the relationship:
Rexp = 0.854 - 0.142 in q 5 (9)
The observation that Rexp correlated better with areal hydraulic load
than with volumetric hydraulic load (Kirchner and Dillon, 1975) is also
supported by the results presented in Table 3. However, if values for
Lakes Raven and Talbot are excluded as outliers, a high correlation is
obtained between Rexp and in (Figure 4, r = -0.91). The best fit
linear regression equation was
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Rexp = 0.482 - 0.112 in w (10)
Raven and Talbot are the shallowest lakes ( = 0.73 m and 0.85 m, respec-
tively) in the data set so perhaps this relationship between Rexp and in is
not valid for extremely shallow lakes. Dillon and Rigler (l974a), testing
a simple phosphorus mass balance model, reported that predicted and
observed P values differed markedly for Lakes Raven and Taibot.
Although equations (9) and (10) provide a good fit to observations,
they predict R greater than 1 for low values of q (<0.36 m/yr) or p
1 p S W
(<0.01 yr ) and R less than 0 for high values of q (>395 m/yr) and p
(>74 yr ). For example, equation (10) predicts R = 1.22 for Lake
Tahoe. Predictions of R < 0 or > 1 are contrary to limnological exper-
ience except in unusual circumstances, thus equations (9) and (10) are
not entirely adequate.
Figure 4 also includes a graph of R = for = 1.0 yr
to compare the data spread with the shape of the theoretical R vs ln p curve
(at = 1.0 yr ). For ln p > 0 the curve evidently underestimates
observed values of Rexp and overestimates observed values for in c < 0.
Because and seemed related, ln was regressed on in p . Direct
measures of c are not available for these lakes. Therefore, was
pR
determined as l !ReXP (rearranging equation (3)). Data for Lakes Tahoe
exp
and Superior were included because they provided data points at extremely
low values of but Lakes Raven and Talbot were excluded. The relationship
ln = in 0.761 + 0.472 ln (r=084) (11)
= 0.761
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resulted. Substituting into (3) and simplifying provided an equation of
the form
R = (13)
l+cLP 8
where a = 1.3 and 8 = 0.4.
The coefficients a and were also estimated using a Gauss-Newton non-
linear least squares algorithm available from the Oregon State University
Computer Library. The best estimate for a was 1.12 and for 8, 0.49.
Thus, equation (13) can be approximated by the expression:
R = l+p ’ 2 (14)
The correlation between R determined by equation (14) and Rexp was
0.94. Further, the solutions of this expression are not theoretically
impossible at extremely high and extremely low values of w• This
expression is similar to that which develops from a relationship between and
(equation (8)). The correlation between R determined from equation (8)
and Rexp was 0.93.
Table 4 summarizes the various expressions which relate the P
retention capacity to lake properties. The expressions presented are
all based upon hydraulic flow normalized to lake size as either surface
area or volume, the lake attribute which seems to control the P retention
coefficient to the greatest degree in oligotrophic lakes.
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DISCUSSION
Vollenweider (1968) first summarized the notion that the trophic
state of lakes could be related to the supply of P. He proposed a P
loading graph relating the trophic state of lakes to the areal supply of
P (g P/m 2 /yr) modified by lake mean depth (L vs graph). Although an
important first step in stimulating limnologists to consider supplies of
critical nutrients to lakes, the graph has been criticized because it
does not distinguish between conditions of low flow—high influent P
concentration or high flow-low influent P concentration (see review in
Dillon, 1975). Vollenweider (1975) modified this P loading graph,
dividing by T (L vs. 2 /T graph). Since z/Tw is the areal
hydraulic load, this eliminates the effect of on the resultant trophic
state. The latter graph expresses roughly the same concepts as a graph
of [ ] vs. R if L and [ ] are directly correlated and if R is related
to q 5 (Table 4).
Dillon (1975) proposed a graph of L (l_R )/P vs. modifying L to
include the effects of hydraulic washout and P retention. This formu-
lation can be derived directly from equation (4) since
= zv [ p ] a
p V 2
Lines of constant lake P concentration can be drawn on the graph as the
steady state solution to equation (1) for values of R 1 pp,, and 2.
Dillon chose lines of constant [ P] representing the division between
oligotrophic and mesotrophic (10 pg/i) and between mesotrophic and
z v. [ p.]
eutrophic (20 pg/l) conditions. Since L = (S = lake surface
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area, m ) and w —• --- ‘ —n- = . Thus this graph of p p
vs. is equivalent to a graph of [ ] • (l-R ) vs.
A graph of [ ] vs. R better represents the concepts expressed by the
steady state equation. Importantly, [ ] combines both P supply and
hydraulic flow. It is the concentration attainable in a lake in the
absence of P retention, i.e., a potential lake P concentration. (This
differs from the potential concentration defined by Edmondson (1961)
which is , the phosphorus volumnar loading.) Thus the steady state
lake P concentration is a function of the magnitude of [ ] and the
ability of a particular lake to assimilate P, expressed as its P retention
capacity. This assimilation occurs as P is incorporated into the lake
sediments. The graph of [ ] vs. expresses these concepts and conveniently
separates the potential lake P concentration from the lake assimilative
capacity. It can be derived directly from the steady state solution of
a simple mass balance model for phosphorus (equation (7)) and does not
produce results contrary to theoretical expectation. If trophic state
can be related to lake P, then this graph provides a method for quickly
assessing the trophic state of a lake.
The importance of as a factor in determining the steady state P
concentration in lakes is evident from Figure 1 and equation (7). as
defined by equation (3), is sensitive to the relative values of and
If p is >> such as may exist for lakes with high flushing
rates, approaches 0 and is nearly independent of . These lakes are
particularly sensitive to [ ] and changes in [ ]; steady state P concen-
trations would be only slightly lower than [ ]. Thus, in order to
prevent these lakes from becoming eutrophic or to reduce the trophic
state of these lakes, emphasis must be placed upon manipulating [ p].
When is << c , R is nearly independent of the value of either variable
and approaches 1. Large lakes such as Lakes Tahoe and Superior with
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slow water renewal may exhibit these characteristics. These lakes can
absorb high [ b], yet maintain good quality. However, these lakes are
extremely sensitive to changes in and require large reductions in
[ ] to significantly reduce lake P concentrations.
A comparison of Lakes Superior and Zurichsee illustrates the relative
importance of [ ] and R . Lake Superior (Rexp = 0.9 [ ] 40 pg/l,
mean lake P = 3-5 pg/i) is oligotrophic (Beeton and Chandler, 1966;
Patalas, 1972; Dobson, et al., 1974), while Zurichsee exp = 0.25, [ ]
= 40 pg/i, [ P] = 32 pg/i at vernal circulation) is eutrophic (Vollen-
welder, 1968). Assuming that both lakes approximate a steady state,
Lake Superior’s ability to assimilate incoming P allows it to manifest
oligotrophic characteristics while Zurichsee’s much lower assimilatory
ability dictates that the same [ ] results in eutrophic conditions.
Lake Superior probably could absorb a 40-50 pg/l increase in [ ] and
remain oligotrophic, while a similar increase in [ for Zurichsee
probably would aggravate an already eutrophic problem. Slight changes
in Zurichsee’s assimilatory capacity would only slightly affect the
steady state P concentrations, while a slight reduction in Superior’s
retention capacity--for example, from 0.9 to 0.8--could double the
steady state lake concentration. This sensitivity of lakes with high R
to changes in R suggests that a careful determination of is necessary
if the results of influent P changes are to be reasonably predicted for
these lakes.
The P retention capacity of oligotrophic lakes may be primarily controlled
by p or q 5 ,--that is, hydraulic flow normalized to a measure of lake
size (Table 4). The relationships summarized in Table 4, if generally
valid, may provide an upper limit for because it apparently decreases
as productivity increases (Volienweider, 1975). These relationships may
be useful for predicting lake P content for known or hypothesized influent
P levels. No attempt has been made to quantify this reduction in R due
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to increased productivity. Other factors such as lake morphometry,
climate, or lake chemistry may influence but a more thorough investi-
gation is necessary to refine these predictive estimates.
If loading and flow data are available, the expression of trophic charac-
teristics in terms of [ ] and not only allows a rapid assessment of
trophic state, but also provides a preliminary guide useful for management
decisions. Effects of alternatives such as loading reduction or nutrient
dilution can be expressed as changes in [ ]. Furthermore, the effect of
manipulations affecting if quantifiable in terms of can be
assessed. Thus the results of nutrient inactivation or hypolimnetic
aeration might be determined as increases in R and, therefore, as
reductions in steady state P levels. In addition, the effect of combined
treatments can be predicted through their changes in [ ] and R .
The [ ] vs. R graph provides an alternative to the P loading graphs.
It develops from a simple mass balance equation for P (equation (1)) and,
therefore, is only as good as the mass balance model itself. Mass
balance models of this nature are intended to describe general charac-
teristics of phosphorus dynamics in lakes. Thus annual averages of the
input variables and rate coefficients are used and results are compared
with average conditions in lakes. This type of model has been used to
explain existing conditions (Vollenweider 1969, 1975) or to predict
changes as a result of reducing P supplies to lakes (Sonzogni, et a].,
1974; Sonzogni and Lee, 1974; Megard, 1970). Slightly more complex mass
balance models developed by Imboden (1973, 1974) and Snodgrass and
O’Melia (1974) may provide a closer fit to average conditions and,
more importantly, to seasonal changes in lake P. However, these models
require further testing to ascertain their generality.
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SUMMARY
An expression derived from the steady state solution of a phosphorus mass
balance model was developed relating mean lake phosphorus concentration
to mean influent phosphorus concentration and lake retention capacity.
A graph of mean influent phosphorus concentration versus lake phosphorus
retention capacity was proposed to summarize this relationship. Lines
of constant lake phosphorus concentration drawn on the graph to delineate
trophic states help predict changes in trophic state occurring when mean
influent phosphorus concentration and/or phosphorus retention capacity
are changed.
The mean phosphorus influent concentration is a potential lake phosphorus
concentration. The assimilative capacity of lakes acts to reduce this
potential to produce mean lake concentrations. Empirical expressions
were developed to relate phosphorus assimilative capacity in oligotrophic
lakes to either areal hydraulic load or hydraulic washout coefficient.
These expressions can be used to predict the resultant mean lake phosphorus
concentrations if mean influent concentrations are known.
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LIST OF SYMBOLS
Symbol Definition
L Areal phosphorus loading (g P/m 2 /yr).
9.. p Volumnar phosphorus loading (g P/m 3 /yr).
[ P1 Phosphorus concentration in lake (pg/i).
[ ] Mean inflowing phosphorus concentration (pg/i).
[ ] Phosphorus concentration of th tributary (pg/l).
[ p 0 ] Phosphorus concentration of outlet (pg/i).
q 5 Areal hydraulic load (m 3 /m 2 .yr = m/yr).
Rexp Experimentally determined phosphorus retention coefficient
(dimensionless).
R Theoretical phosphorus retention coefficient (dimensionless).
S Lake surface area (m 2 )
V Lake volume (m 3 ).
v Flow rate of th tributary (m 3 /yr).
Flow rate of outlet (m 3 /yr).
Mean depth (m).
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Hydraulic washout coefficient (yr ).
Phosphorus deposition coefficient (yr ).
Tw Hydraulic retention time = (yr).
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References
Beeton, A. M., and D. C. Chandler. 1966. The St. Lawrence Great Lakes.
In: D. G. Frey (ed.). Limnology in North America. pp. 535-558.
Univ. of Wisc. Press.
Edmondson, W. T. 1961. Changes in Lake Washington following an
increase in the nutrient income. Verh. mt. Verein. Limnol.
14: 167-1 75.
Dillon, P. J. 1974. A critical review of Vollenweider’s nutrient
budget model and other related models. Water Resources Bull.
10:969-989.
Dillon, P. J. and F. H. Rigler. 1974a. A test of a simple nutrient
budget model predicting the phosphorus concentration in lake water.
J. Fish. Res. Bd. Can. 31 :1771-1778.
Dillon, P. J. and F. H. Rigler. 1974b. The phosphorus—chlorophyll
relationship in lakes. Lirnnol. Oceanogr. 19:767-773.
Dillon, P. J. 1975. The phosphorus budget of Cameron Lake, Ontario:
The importance of flushing rate to the degree of eutrophy of lakes.
Lirnnol. Oceanogr. 20 (l):28-39.
Dobson, H. F. H., M. Gilbertson, and P. G. Sly. 1974. A summary and
comparison of nutrients and related water quality in Lakes Erie,
Ontario, Huron, Superior. J. Fish. Res. Bd. Can. 31:731-738.
Imboden, D. M. 1973. Limnologische Transport- und N hrstoffmode1le.
Schweiz. Z. Hydrol. 35:29-68.
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19
Imboden, D. M. 1974. Phosphorus model of lake eutrophication. Limnol.
Oceanogr. 19:297-305.
Kirchner, W. B. and P. J. Dillon. 1975. An empirical method of
estimating the retention of phosphorus in lakes. Water Resources
Res. 11:182-183.
Lerman, A. 1974. Eutrophication and water quality of lakes: control
by water residence time and transport to sediments. Hydrological
Sciences Bulletin. 19:25-34.
Megard, R. 0. 1970. Lake Minnetonka: nutrients, nutrient abatement,
and the photosynthetic system of the phytoplankton. Interim Report
No. 7. Limnological Research Center, University of Minnesota. 214
p.
Patalas, K. 1972. Crustacean plankton and the eutrophication of the
St. Lawrence Great Lakes. J. Fish. Res. Bd. Can. 29:1451-1462.
Sakamoto, M. 1966. Primary production by phytoplankton community in
some Japanese lakes and its dependence on lake depth. Arch.
•Hydrobiol. 62:1-28.
Sawyer, C. N. 1947. Fertilization of lakes by agricultural and urban
drainage. J. N. Engl. Water Works Ass. 61:109-127.
Snodgrass, W. J. and C. R. O’Melia. 1974. A predictive model for
phosphorus in lakes. University of North Carolina. Unpubi. Manu-
script. 28 p.
Sonzogni, W. C., P. D. Uttormark, and G. F. Lee. 1974. The phosphorus
residence time model: theory and application. Submitted to Water
Res.
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Sonzogni, W. C. and G. F. Lee. 1974. Diversion of wastewater from
Madison Lakes. J. of the Environmental Engineering Division, ASCE.
100:153-170.
Treunert, E., Wilhelms, A., and Bernhardt, H. 1974. Einflu der
Probenentnahme-Häufigkeit auf die Ermittlung der Jahres-Phosphor-
Frachtwerte mittlerer B che. Hydrochem. Hydrogeol. Mitt. 1:175-
198.
Vollenweider, R. A. 1968. Scientific fundamentals of the
eutrophication of lakes and flowing waters, with particular
reference to nitrogen and phosphorus as factors in eutrophication.
Tech. Rep. OECD. Paris. DAS/CSI/68.27. 159 p.
Vollenweider, R. A. 1969. Möglichkeiten und Grenzen elementarer
Modelle der Stoffbilanz von Seen. Arch. Hydrobiol. 66:1-36.
Vollenweider, R. A. 1975. Input-output models. Schweiz. Z. Hydrol. In
press.
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Table 1
Summary of lakes and characteristiscS used in Figure
1. Data for lakes 1-17 were obtained from Dillon
(1975) and references therein and for lakes 18-37
from Gakstatter (personal communication).
Trophic R
No. Lake Name State mg/i ex
1 Baldeggersee E 230 0.61
2 Bodensee-Obersee E 200 0.65
3 Erie M-E 150 0.84
4 Greifensee E 170 0.62
5 Haliwilersee 0 75 0.36
6 Kalamalka E 610 0.90
7 Kegonsa E 505 0.09
8 Menona E 330 0.71
9 Michigan 0-M 85 0.9
10 Norrviken (1961-1962) E 425 0.49
11 Okanagan 0 305 0.95
12 Pfaffikersee E 195 0.77
13 Sebasticook E 90 0.48
14 Skaha M—E 95 0.65
15 Waubesa E 620 0
16 Wood E 2590 0.90
17 227 (ELA) E 325 0.89
Lake/State
18 Beaverdam (Dodge County)/WI E 265 0.0
19 Bemidji/MN E 35 0.36
20 Blackduck/MN E 135 0.55
21 Chemung/MI E 105 0.63
22 Clearwater/MN E 180 0.76
23 Cokato/MN E 385 0.44
24 Delavan/WI E 410 0.55
25 Fremont/MI HE 550 0.47
26 Green/MN M-E 55 0.67
27 Jordan/MI E 130 0.17
28 Le Homme Dieu/MN E 135 0.78
29 Nagawicka/WI E 205 0.57
30 Nest/MN E 90 0.56
31 Sebasticook/ME E 60 0.29
32 Shawano/WI E 35 0.23
33 Swan/WI E 140 0.18
34 Townline/WI E 255 0.52
35 Trout/MN E 375 0.92
36 Wagonga/MN E 4300 0.57
37 Wapogasset/WI E 70 0.43
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TABLE 2
Summary of lakes and characteristics used in Figure 1 and for an analysis of relationships between Rexp and
lake characteristics. Data for lakes 38-45 were obtained from Dillon (1975) and references therein, for lakes
46—57 from Dillon and Rigler (1974a), and for lakes 58-73 from Gakstatter (personal communication). Asterisks (*)
indicate lakes used for generating best empirical relationships between Rexp and lake characteristics.
Trophic [ ] R w - °p
No. Lake Name State mg/i exp ( yr 1 ) (j ( yr )
38 Aegerisee 0 30 0.68 0.115 49 0.24
39 0 25 0.80 0.13 12.5 0.52
40 Leman M 55 0.20 0.083 155 0.02
41 Ontario M 50 0.78 0.152 84 0.54
42 *Superior 0 40 0.90 0.0053 148 0.05
43 *Tahoe 0 100 0.93 0.0014 303 0.02
44 Turlersee M 45 0.80 0.465 14 1.86
45 Zurichsee E 40 0.25 0.680 50 0.23
46 *Beech 8 0.07 22.7 9.8 1.71
47 *Bob 24 0.71 0.37 18.0 0.91
48 *Cameron 16 0.30 18.9 7.1 8.1
49 *Eagje...Moose 9 0.36 2.03 12.8 1.14
50 Four Mile 45 0.82 0.26 9.3 1.18
51 *Oblong_HaIiburton 20 0.72 0.32 17.7 0.82
52 *Hails 8 0.53 0.96 27.2 1.08
53 9 0.26 8.0 11.6 2.81
54 *pine 8 0.01 18.5 7.4 0.19
55 Raven 20 0.55 14.9 0.73 18.2
56 Talbot 24 0.69 4.9 0.85 10.9
57 Mile--Boshkung 8 0.33 2.38 18.1 1.18
Lake/State
58 *Bay of Naples/ME 0 9 0.19 14.0 4.3 3.28
59 Canadaigua/NY 0 55 0.65 0.07 39.0 0.13
60 Carlos/MN M 40 0.56 0.27 13.1 0.34
61 *Carry Falls/NY 0 14 0.28 9.61 5.4 3.74
62 Cass/MN M 40 0.63 1.17 7.6 1.99
63 *Charlevojx/MI 0 24 0.63 0.31 16.8 0.53
64 Higgins/MI 0 28 0.70 .06 14.9 0.14
65 Houghton/MI M 32 0.47 0.76 2.3 0.68
66 Long (Aroostook County)/ME M 30 0.56 0.31 13.4 0.39
-------
TABLE 2 (Continued)
67
68
69
70
71
72
73
*Long (Cumberland County)/ME
*Mattawamkegg/ME
*Moosehead/ME
Pelican/ME
*Rangeley/ME
*Sebago/ME
Winnipesaukee/NH
M
M
0
M
0
0
0
16
20
15
77
18
14
40
0.50
0.31
0.46
0.46
0.49
0.57
0.77
0.83
8.69
0.33
0.30
0.36
0.19
0.25
10.4
3.7
16.4
2.7
14.3
30.8
13.1
0.83
3.90
0.28
0.25
0.35
0.25
0.84
N)
-------
24
Table 3. Correlations between Rexp and various transformations
and combinations of and for 20 lakes whose [ ]
< 25 pg/i (Table 2).
Correlation Correlation
Correlation Coefficient Correlation Coefficient
Rexp vs. -0.74 Rexp vs. 0.57
0.33
1 p
— 0.68 — 0.00
pw z
1 0.18
z
in p -0.77 in 0.58
w p
w
in z 0.09 0.34
2 -0.80 -0.69
in p • 2 -0.92 1 0.25
w
z
1 0.75 0.55
IW
-------
25
Table 4. Summary of various relationships between and or for
twenty selected lakes (indicated by (*) in Table 2). Correlation coefficients
in parenthesis are values obtained when Lakes Superior and Tahoe are deleted.
1. R = 1 r 0.94 (0.91)
p 1 + 1/2
w
2. R = 0.482 - 0.112 ln r = 0.93 (0.91)
3. R = q r = 0.93 (0.89)
4. R = O.426e(_0 271 ) + 0.574e °°° 949 1 ) r = 0.94 (0.90)
5. R = 0.854 - 0.142 in q 5 r = 0.94 (0.91)
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26
LIST OF FIGURES
1. Graph of mean tributary phosphorus concentration ( [ ]) vs. phosphorus
retention coefficient (Rexp) for lakes summarized in Tables 1 and 2.
0 = oligotrophic; M = mesotrophic; E = eutrophic; HE = hypereutrophic.
2. Regression of Rexp on ln for lakes summarized in Table 2.
3. Regression of Rexp on in for iakes whose [ ] was < 25 pg/i
(Table 2).
4. Regression of Rexp on in p for lakes whose [ ] was < 25 pg/i
(Table 2). Dashed line is regression equation excluding Raven (#55) and
Talbot (#56). Solid line is a graph of R = for =
l.Oyr . p w
5. Comparison of solution of R = 1 1/2 with Rexp for lakes whose
l+p
w
[ ] was < 25 pg/l. Raven and Talbot have been excluded; Tahoe
= 100 pg/i) and Superior ( [ ]) = 40 pg/l) have been added.
-------
ioo
: :
‘65 M
• •67M
690
‘52
0.2 0.3 0.4 0.5
REXP
1000
• 15E
•7E
4300
t36E
18E
•25 HE
• ‘IOE.24E
23 E
6E
•34E
.33 E
27 E
‘8E
•l
E
17
29E •2E
•4E
‘20 E
, 12 E
22E
‘50
•40M
‘31 E
•13E
•70M
37 E
‘281
•30E
,2 1 E
‘14 M-E
.45 E
32 E
‘19 E
M
68 M
•66M
‘50
730
‘6 1M
420
•390
‘72 0
.49
5
0 0.1
0.7
0.8 0.9
1.0
FIGURE 1
-------
1.0
REXP 078—0 121 In q
- ‘42
r -0.80
0.8 •44
•4I
.73
.51
.47
.56
•38
.59
‘63 ‘62
0.6
66 \.55 ‘52
a-
‘71 •67
•70 ‘65 .69
0.4
.49
.57
‘68\ .48
‘61
.45
0.2 - ‘40 ‘58
‘46
0 I I I
-I 0 I 2 3 4 5 6
Inq
‘50
.39
- .64
.53
FIGURE 2
-------
I I I I
REXP 0.86-0.14 3 In q
.39
.72
r -0.92
0.8
.55
.52
‘67
.49
\.57
•68\
•6 1
0.2
0. I I I I
•7I
‘69
0 I 2 3
lnq
•48
I 54 .i
4 5 6
1.0
51
.47
•56
0.6-
a-
w
04-
.53
•58
‘46
FIGURE 3
-------
1.0
0.8
0.6
a-
x
U i
0.4
02
Pw
REXP 0.482-0.112 In
Excluding
r=
#55
-0.91
and
#56,
.5’
.47
•56
•72
52
\
.
•7 1
•69
7
.55
\
.49
.57
•48
•68
6 1
‘58
\
0
-3 -2 -I 0 I 2 3 4
In
FIGURE 4
-------
x
U i
o
04
0.2
o
4
.
.39
51
..47
•63
•72
\‘ 52
67
•7 1
69
.49
057 68
\ . •48
53•\\
•58
-4
-2
Pw
0
2
FIGURE 5
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