Estimation of Materia11Fluxes in an Estuarine Cross Section: A Cr
Analysis of Spatial Measurement Density and Errors
Bjorn Kjerfve; L. Harold Stevenson; Jeffrey A. Proehl; Thomas H. Chrzanowski; Wiley
M. Kitchens
Limnology and Oceanography, Vol. 26, No. 2. (Mar., 1981), pp. 325-335.
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Limnol. Oceanogr., 26(2), 1981, 325335
Estimation of material fluxes in an estuarine cross section:
A critical analysis of spatial measurement
density and errors1
Bjorn K j e r f ~ eL.
, ~Harold S t e v e n ~ o nJeffrey
,~
A. Proehl,
Thomas H . C h r ~ a n o w s k iand
, ~ Wiley M . Kitchens
Belle W . Bamch Institute for Marine Biology and Coastal Research,
University of South Carolina, Columbia 29208
Abstract
Estuarine budget studies often suffer from uncertainties of net flux estimates in view of
large temporal and spatial variabilities. Optimum spatial measurement density and material
flux errors for a reasonably well mixed estuary were estimated by sampling 10 stations from
surface to bottom simultaneously every hour for two tidal cycles in a 320-m-wide cross section
in North Inlet, South Carolina. Discharge and ATP and NH,+-N fluxes were computed. The
analysis method was to form a number of cases, each based on a different number and combination of stations and compare these fluxes to the ideal case using all 10 stations. A percentage error, y, (rms derivation of a given case from the ideal case compared to the tidal
prism) was <15% with only three lateral stations, each located to cover a separate bathymetric
regime. In estuaries with dimensions similar to North Inlet, these results should prove useful
in selecting an optimum (or minimum) number of required stations.
Estuarine investigations typically focus on longitudinal and vertical circulation features and parameter distributions.
Only recently has it become apparent
that cross-sectional variabilities may be
equally significant. For example, lateral
density gradients may maintain net crosschannel flows, and cross-channel differences in velocity and dissolved/suspended material concentrations may have a
large effect on the longitudinal material
flux. In computing longitudinal material
fluxes through an estuarine cross section,
it is reasonable to ask how many lateral
stations must be occupied to estimate the
total flux with less than the maximum acceptable error.
Such an error analysis is lacking in
most estuarine budget studies. The work
of Boon (e.g. 1978) is a notable exception.
He considered suspended solids transport in a salt marsh creek and concluded
that inferences about net transports
This paper is Contribution 337 from the Belle
W. Baruch Institute for Marine Biology and Coastal
Research. The project was supported by National
Science Foundation grant DEB-76-83010.
Also Marine Science Program and Department
of Geology.
Also Department of Biology.
should probably be avoided in the absence of sufficiently detailed spatial (and
temporal) measurements. The question
of temporal sampling rate has received
considerablv more attention than that of
spatial sampling rate. As most of the estuarine flux variability occurs in response
to tidal currents, it is necessary to sample
a number of times over a tidal cycle. The
standard is typically once every hour (or
lunar hour), but sampling is often more
frequent. It may sometimes be acceptable to sample less frequently over the
tidal cycle, e.g. 8 or 6 times. However,
the practice of sampling only twice, during peak ebb and flood or during high
and low water, and of basing flux estimates on such data, is unsound (Boon
1980).
We made a calibration study, preliminary to a study of the annual material
budgets of the North Inlet (South Carolina) marsh-estuary system, to determine
the minimum number of stations in an
estuarine cross section needed to calculate material fluxes within acceptable
error limits. We are basing our flux estimates on simultaneous measurements of
longitudinal velocity and collection of
water samples from which we determined adenosine 5'-triphosphate (ATP)
326
Kjerfve e t al.
whether water, ATP, or NH,+-N is considered most important.
We thank the faculty, technicians, and
students of the University of South Carolina Marine Science Program for collecting the field data. J. D. Spurrier, R.
T. Edwards, L. N. Henry, and M. Marozas also provided assistance. The manuscript was written while B.K. was on sabbatical leave with the Coastal Studies
Unit, Department of Geography, University of Sydney.
Fig. 1. Location of experimental cross section in
North Inlet and Belle W. Bamch Coastal Field Laboratory.
and ammonium-nitrogen (NH,+-N) concentrations. The calculated fluxes were
chosen to assure the widest ecological
ramifications. Obviously, t h e flux of
water (discharge) represents a conservative physical quantity controlling the hydrodynamics of the system. The flux of
NH,+-N reflects the biological utilization
and production of a nonconservative nutrient of major importance in marsh and
ocean productivity. The flux of ATP represents a measure of the total microbial
biomass and movement of viable microorganisms across the experimental cross
section. Using a number of different
tests, we intend to outline a way to determine an "optimum" number of stations. Various tests are of different sensitivity and their usefulness depends on
how the hypothesis is formulated. Also,
the conclusion may depend somewhat on
Field and laboratory methods
The North Inlet system (Fig. 1)is a 34k m 9 a l t marsh dominated by Spartina
alterniflora and intersected by winding
tidal creeks. The estuary is characterized
by 30-34%0 salinity, 1.0-2.5-m semidiurnal tidal range, peak tidal currents as
great as 2.3 m . s-l, typical channel depths
of 5 m, and negligible freshwater runoff.
Hydrographic features of the system have
been described elsewhere (Kjerfve 1978;
Kjerfve et al. 1978).
The experimental cross section (Figs.
1 and 2) is 320 m wide and represents the
major water exchange route between the
marsh-estuary and the coastal ocean. Velocity was measured at ten stations every
30 min over three consecutive tidal
cycles 11-12 November 1977. The longitudinal current normal to the cross section was simultaneously measured from
10 four-point-moored boats. Biplane current crosses (Pritchard and Burt 1951)
were used to determine the velocity at
1-m intervals from surface to bottom. The
results of the velocity measurements
have b e e n reported by Kjerfve a n d
Proehl (1979). The choices of ten stations, three tidal cycles, and a 30-min
sampling rate have no significance other
than that they represent the most stations, the longest duration, and the fastest
sampling rate logistically feasible. Each
current cross was calibrated against a
Bendix B-10 current meter to equate the
accuracy of our method with that of the
B-10; the precision was + 2 cm.s-l with
a threshold of 7 cm.s-'.
Material flux
east
A10
.3.01m
/
Vertical exaggeration 20x
NORTH INLET
EXPERIMENTAL CROSS SECTION
-4
-5
-6
-7
-
-8
Fig. 2. Experimental cross section at mean tide showing station notation, distance of stations (m) from
western bank, distance (m) of each subsection over which measurements at a given station are considered
representative, and time-averaged water depths (m) at each station (below bottom trace).
Because of logistic constraints, water
samples were collected at each of the ten
stations from surface, middepth, and bottom once every hour for two of the tidal
cycles, after the hourly velocity measurements. The samples were immediately
transported to the nearby field lab (Fig.
1) and subjected to a variety of chemical
and biological analyses. We will limit our
discussion h e r e to water, ATP, a n d
NH,+-N fluxes.
The ATP data have been reported by
Chrzanowski et al. (1979). Aliquots (10
ml) of the 500-ml water sample were filtered through Whatman G F glass-fiber
filters. The organisms retained on the filter were extracted in 5 ml of boiling buffer (tris-hydroxymethyl amiomethane, 0.02
M, pH 7.75). Extractions were performed
in duplicate. The ATP extracted was determined with the crude luciferin-luciferase enzyme system from Sigma Chemical Co. (FLE-250) and a photometer
(SAI model 3000) operated in the peak
height mode (Stevenson et al. 1979).
Ammonium was determined colorimetrically by the Berthelot reaction (Technicon Ind. Systems 1973). A blue complex, closely related to indophenol,
occurs when a solution of an ammonium
salt is added to sodium phenoxide, fol-
lowed by the addition of sodium hypochlorite. Coprecipitation of the hydroxides of potassium a n d sodium w e r e
prevented by introducing a solution of
potassium sodium tartrate and sodium citrate. A Technicon autoanalyzer allowed
processing of about 40 samples per hour;
it was calibrated frequently. Its sensitivity at 10 pg-atoms N.liter-' was estimate d to correspond to 0.15 absorbance
units. The coefficient of variation at 8
pg-atoms N.liter-l was 0.31%. The sensitivity of this procedure required strict precautions against accidental contamination of the samples. Air introduced into
the sample stream was scrubbed with
concentrated sulfuric acid to remove ammonia. The laboratory area must be free
of agents high in ammonia and cyanides
(e.g. tobacco smoke, floor wax, and detergent).
Data manipulations
Instantaneous vertical profiles of velocity, ATP, and NH,+-N concentrations
were spline-fitted between water surface
and bottom (Kjerfve 1979; Chrzanowski
e t al. 1979). Following Kjerfve (1975,
1979), w e calculated water flux (discharge) from
Kjerfve et al.
depth{+b
(i-I)
ith
ijth segment
(i+l)
Fig. 3. Schematic diagram of a subsection indicating location of interpolated values used in computations (.) and notation.
where Q is the instantaneous cross-sectional discharge; j is a station counter and
i a depth counter; hj is water depth at
station j ; Wij is width of element j at
depth i; n is number of stations in the
cross section used in computation; and Vij
is velocity at depth i and station j , interpolated from the vertical profile at 11
equispaced points between surface (Voj)
and bottom (V,,,), where the velocity vanishes. The velocity, V,,, is assumed constant for the entire i j segment (Fig. 3).
The width, Wii, is measured from the
midpoint between stations ( j - 1) and
(j)to the midpoint between stations ( j )
and ( j + 1)at the depth i. At each side of
the channel, a constant slope of the marsh
(beach) side was introduced whenever
the tide rose above the bankfull stage
(Kjerfve and Proehl 1979).
In an analogous manner, the ATP and
NH4+-N fluxes (both denoted F) were
computed as a concentration-velocity
cross-product, again with interpolated
values at 11 depths. The instantaneous
cross-sectional flux (Kjerfve 1975, 1979)
is
where Cij is ATP or NH4+-N concentration at the depth i and station j; p is water
density, taken to b e constant, 1.02
g.~m-~
and
; other notation is consistent
with that of Eq. 1.
Figure 4 illustrates the variation of discharge, ATP flux, and NH4+-N flux over
the study period, computed according to
Eq. 1 and 2 and with the ten stations in
the experimental cross section. For comparison, the tidal water level is also plotted in the same figure. Although the discharge data set consists of 75 half-hourly
estimates covering three tidal cycles, we
will use here only the 25 discharge estimates that coincide with estimates of
ATP and NH4+-N concentrations. We
thus have three data sets, each consisting
of 25 hourly estimates covering two tidal
cycles (cf. Fig. 4), beginning at 1500
hours on 11 November 1977 and ending
at 1500 hours the next day.
The use of all ten stations in the cross
section for discharge and material flux
computations yields the best possible estimate of the various fluxes. These are our
ideal cases, representing maximum sampling density, as it was not logistically
possible to make more measurements,
more frequently, or for a longer duration.
Adjusting the Wij values appropriately,
we recomputed discharge as well as ATP
and NH,+-N fluxes many times with a
smaller number of stations or a different
combination of stations. To represent
these combinations, we computed each
flux 30 different times. The essence of
the analysis is a comparison by various
statistical methods of the outcome of the
ideal or maximum density case compared
to the other 29 cases.
The arithmetic time means (or net values) for the two tidal cycles and standard
deviations from these net values of discharge, ATP flux, and NH,+-N flux data
sets for the 30 cases are shown in Table
1. Considering the ideal case, 2,750 individual velocity points (10 stations, 11
depths, and 25 time steps) were used to
compute the net discharge. The ATP and
NH4+-N fluxes each required cross-correlation of 2,750 concentration values
with equally many velocity values. The
other 29 cases required fewer data points,
the actual number depending on the
Material jlux
1
5
18
21
00
I I NOV
03
06
09
12 NOV
12
I
Fig. 4. Variability over 1977 two-tidal-cycle study of discharge, ATP flux, NH,+-N flux, and tidal water
elevation. Values represent result of ideal case, using all stations.
number of stations used in the computation. Data manipulations were thus extensive and cumbersome in spite of computer use. Most cases were consistent
with respect to direction of flux. The net
water flow was ebb-directed, NH,+-N
was exported, while ATP was imported.
Over the two tidal cycles, the ideal case
yields an export of 8 x 10" m3 of water
and 0.151 kg of NH,+-N with 3.54 kg of
ATP being imported.
Tests, results, and discussion
Long term flux reliability-A time series of a material flux through an estuarine cross section over a tidal cycle usually displays a large periodic component
superimposed on a small time-averaged
(net) value. The net flux can be either
ebb- or flood-directed for a given tidal
cycle and constituent. Whether a system
in the long term imports or exports a particular constituent is a difficult, costly,
and time-consuming question to answer.
To assess with reliability the magnitude
and direction of the long term flux of a
constituent, we would have to determine
the flux for a long period. Net values for
each tidal cycle and constituent must be
computed. The central limit theorem
shows that these net flux estimates are
normally distributed. By applying the
Student's t-test to the overall mean of the
net values (testing the hypothesis that the
overall mean equals zero), we could determine the long term flux magnitude and
direction with a high degree of confidence. The reliability of this estimate
typically increases with the number of
tidal cycles. Although we have not applied such a test, this is the major objective of the North Inlet study to which this
is preliminary.
Paired t-test-The paired t-test is an
appropriate test for distinguishing between sample means if paired observations are positively correlated (Steel and
Torrie 1960). Each ideal dischargelflux
time series compared to the corresponding series for cases 2-30 yields a legitimate sequence of observations from
which to form pairs. As each case estimates the same dischargelflux value at a
given time, the correlation between the
series can be expected to be large and
positive. Table 2 lists the means (d,,,)and
Kjerfve et al.
330
Table 1. Data summary listing stations used to compute dischargeifluxes for the 30 cases along with
net discharge (m3.s-I) i SD (with respect to time), ATP flux (mgs-I), and NH,+-N flux (mg.s-I). Positive
flux implies export and negative flux import.
Case No.
No of sta
Sta. No.
Discharge
ATP flux
NH,+-N flux
1 (Ideal)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
null hypothesis (type 1 error), but more
concerned with minimizing the probability of accepting a false null hypothesis
(type 2 error).
Using the paired t-test, we can reject
a number of the cases because a computed t-statistic for at least one of the
fluxes is in excess of the tabulated critical
t-value. Thus, cases 9, 12, 13, 17, 21, 22,
23, 24, 27, and 29 are rejected. The rejected cases contain from one to six staZ,,,ISD,,,
(3) tions, which implies that the particular
t =
location of sampling stations in a cross
with 24 df. The computed t-statistics are section is highly critical. It is not just the
also listed in Table 2. The tabulated crit- number of stations that must be considical t-value is 1.32 in a two-tailed test ered but also their particular location on
with 24 df at the 0.20 significance level. the cross section. From the results of this
The unusually large significance level, particular test, this is obvious. For ex0.20, was selected because we were not ample, case 26 with three stations-in
overly concerned about rejecting a true each channel and in between-results in
standard deviations (SD,,,) of the new
data series formed by subtracting each of
the n = 25 observations from case m
from the corresponding observation for
the ideal case, i.e. a sample of a population of paired differences very likely to
be normally distributed.
The null hypotheses to be tested are,
h,:z = 0 for m = 2 (case 1 - case 2),
. . . , m = 30 (case 1 - case 30). The tstatistic is
Material flux
33 1
Table 2. Means and standard deviations of data sets formed from differences between ideal and m
cases along with results of paired t-tests. Discharge differences expressed in mLs-' and ATP and NH,'N flux differences in m g s - l . Null hypothesis was rejected whenever one of the three computed statistics
for a case exceeded tabulated critical t-value of 1.32 with 24 df and a 0.2 significance level.
-
d+SD
NH,+-N flux
ATP flux
D~scharge
Case 1 vs.
case
-
t
di5D
t
d?SD
t
" Rejected on b a s ~ sof test
a better representation of the ideal case
than, for example, case 9 where six stations were used.
Linear regression-It is often desirable
to estimate the constituent flux from a
single or maybe a couple of measurements in the cross section. In view of
this, it is pertinent to evaluate how well
each case ~ r e d i c t sthe ideal case. For this
purpose we used a linear regression
(Steel and Torrie 1960). By regressing the
hourlv discharge and flux values for the
ideal'case (x;) on the corresponding
hourly values (X,,,) for cases m = 2, 3,
. . . , 30, it was possible to evaluate the
intercepts (a,,,)and the regression coefficients (b,,,).The model is
where
E
is a random error term and X , is
predicted flux for the ideal case. The
coefficient of determination r2,,,was used
as a measure of how well XI,,predicts X,.
The results of the regression analyses
are listed in Table 3. For a case to be a
good predictor over the entire range of
dischargelflux, a slope equal to 1 and an
intercept equal to zero are expected without forcing the fit through the origin. In
addition, r' must be close to 1. The r2
value for almost all cases is quite close to
1.0, implying a highly linear relationship
between X, and X,,,. However, if we simultaneously require three somewhat arbitrary criteria to be met for each of the
discharge and fluxes, a pattern emerges.
The applied criteria are that r',,, > 0.98;
that 1.20 > b,,, > 0.80; and that a,,, does
not deviate from zero by more than *50%
of the appropriate net ideal case value
Kjerfve et al.
Table 3. Results of linear analyses regressing ideal case on case m for discharge (Q), ATP flux, and
NH,+-N flux. Values of regression coefficient (slope, b ) and coefficient of determination ( r Y )are nondimensional; intercept ( a ) is in mLs-', mg.ssl, and mg.s-l.
Case 1 vs
case
Slope ( b )
Intercept (a)
Q
ATP
NH,--N
Q
ATP
r2
NH,--N
ATP
Q
NH,+-N
(Table 1). Only nine cases (2, 4, 5, 6, 7,
y = 100 DIP
8 , 9, 20, and 26) are acceptable on the
basis of these constraints. Only three of where the rms deviation for case m is
these are based on data from six or fewer
stations: case 9 is based on six, case 20
!i=l
on four, and case 26 on three stations. In
for discharge
each of these cases, the stations cover the
major bathymetric regions of the cross
section: the deep eastern channel, the
shallow midchannel region, and the secI
for mass transport
ondary channel region on the western
side.
and Qnlkand Fmkare the discharge and
Percentage error and lateral station mass flux for case m and time k. The
density parameter-The percentage error tidal prism for case m is
of a given case relative to the ideal case
was successfully evaluated in this final
Qnlk
for discharge
2r
test. By comparing the rms deviation, D
(m3),of case m from the ideal case to the
tidal prism, P (m3),we computed an error
Fnlk 1 for mass transport
percentage
xI
I
1
Material flux
Table 4. Computed percentage errors (y) for discharge, ATP-flux, and NH,+-N flux for all cases.
Percentage error, y
Case 1 vs.
case
Q
ATP
NH,--N
,
1 :
1
2
,
,
3
4
NUMBER
,
5
,
6
OF
,
,
7
8
,
,
9
1
0
STATIONS
Fig. 5. Average percentage error, y, for the three
flux constituents plotted vs. number of stations in
each instance using case with smallest average y.
percentage errors in the sense that if the
discharge percentage error is smallest for
a case based on X stations, then usually
the ATP and NH,+-N flux percentage
errors are also lower for the same case
than for the other cases based on the
same number of stations. As expected,
the percentage error increases as the
number of stations decreases. It can be
seen from Fig. 5 that information gained
by increasing the number of stations to
four or five from three is minimal. Similarly, very little is gained by increasing
the number of stations from six to seven,
eight, or nine. Thus, depending on the
where k is a time counter; n is number level of percentage error that is acceptof sampling times, here 25; subscript m able, the choice of either three or six starefers to the cases 2, 3, . . . , 30; subscript tions would be optimal in continued ex1 refers to the ideal case; r is number of perimentation in this North Inlet cross
tidal cycles, here 2; and At is sampling section. Because the slope of the curve in
rate, here 3,600 s. Thus, the tidal prism, Fig. 5 becomes increasingly flatter as we
P, is computed as either the average dis- add more stations, it is unlikely that more
charge (m3.s-l) or the average mass trans- than ten stations would change the flux
port (kges-') of a given dissolved or sus- estimates significantly.
pended constituent passing through the
If we assume a constant tidal prism, a
cross section during half a tidal cycle. For deep cross section would be relatively
the ideal case, the computed tidal prism narrow and a shallow cross section quite
is 2.31 x lo7m3 of water, 21.4 kg of ATP, wide. The wider the cross section, the
and 0.86 kg of NH,+-N.
more lateral bathymetric irregularity is
Several conclusions can be drawn from likely to exist. Thus, for a given estuary,
this analysis. From the computed per- the greater the water depth, the fewer latcentage errors listed in Table 4, it is ap- eral hydrographic stations would usually
parent that there typically is good agree- be required for flux computations. It is
ment between the three flux constituent convenient to define a lateral station den-
Kjerfve et al.
334
sity parameter, p, as the tidal prism divided by the net cross-sectional depth,
h,, and the optimum number of stations
(N = 3 or 6). In the case of North Inlet
P
.=
2x
x
lo6m2.sta.-' for y = 15%
lo6m2.sta.-l for y = 5%
(8)
(cf. Table 4). ,8 is computed based on
water flux only, as relative agreement between water and constituent fluxes is
good. P and h, can easily be estimated
from a hydrographic chart and tide table,
in the absence of detailed velocity measurements. Equation 8 could presumably
be useful in determining N, the optimum
number of lateral stations, for other estuarine studies. Whether the c o m ~ u t e d
relationship between p and N for a given
y holds true for estuaries that are an order
of magnitude larger or smaller than North
Inlet has obviously not been shown.
However, the present results provide a
rational method for determining t h e
number of lateral hvdrogra~hicstations
n e e d e d in estuari*e b i d i e t studies.
More independent checks of the general
validity- of
- Eq. 8 for given y levels would
be useful.
However, it is not only the number of
stations that must be optimized; the particular location of stations in the cross
section must also be considered. This is
less obvious from the previous tests. But
comparison of Fig. 2 and Table 1 (with
the optimum cases from Table 4) indicates the need to sample the major lateral
bathymetric regimes. Thus, with two
channels and a separating bar (Fig. 2),
both cases 9 and 26 point to the necessity
of locating stations in each channel as
well as above the separating bar. Comparison of cases 26 and 28 (Table 4) clearly indicates the need to sample the shallow bar as well as the channels. T h e
detailed velocity results from this cross
section (Kjerfve and Proehl 1979) also
bear this out. The imdications are that if
more distinct channels and shallow areas
exist in a cross section, each bathymetric
region requires a lateral station for accurate flux measurements.
-
-
-
Kjerfve and Proehl (1979) implied that
for well mixed estuaries detailed knowledge of the cross-sectional velocity structure is more important than the same detailed spatial knowledge of t h e
concentration of a dissolved constituent.
Table 4 does indeed support this statement. The percentage errors for ATP flux
are for the most part only slightly greater
than the discharge percentage errors.
And, still, the ATP concentration was
shown to vary significantly in the cross
section (Chrzanowski et al. 1979). The
ATP flux, of course, includes ATP concentration as well as velocity variability.
The percentage errors for the NH,+-N
fluxes are systematically less than both
discharge a n d ATP flux percentage
errors. This is curious and implies a fortuitous covariance of velocity and NH,+-N
concentration.
Conclusions
Based on the preceding analysis, we
decided to use three cross-sectional stations for further flux computations in this
North Inlet cross section. Case 26, based
on stations 2, 6, and 9, was selected as
being optimum, considering both a small
relative flux error and logistic feasibility.
Other conclusions include the following: for a percentage error <15%, the lateral station density parameter of 2 x lo6
m'. sta.-l could probably be used in other
estuarine environments to select the
needed minimum number of lateral stations; distinct cross-sectional bathymetric regimes probably need to be sampled
if the along-channel material flux is the
end goal; cross-sectional flux calibration
based on discharge is also an accurate indicator of dissolved material fluxes-at
least in a reasonably well mixed estuary;
in a well mixed estuary, a single lateral
station (sampled from surface to bottom)
can effectively be calibrated to ~ i e l da
good estimate of the cross-sectional discharge or material flux; and finally, the
number of cross-sectional stations and
their lateral positioning are equally important in measuring discharge and material fluxes accurately.
To answer the long term net transport
Material .flux
question for a n y c o n s t i t u e n t , it is essential to make measurements for many consecutive tidal cycles-two c y c l e s are obviously too few; 30 or more may be
appropriate.
References
BOON,J. D., 111. 1978. Suspended solids transport
in a salt marsh creek-an analysis of errors, p.
147-159. In B. Kjerfve [ed.], Estuarine transport processes. Univ. South Carolina.
. 1980. Comment on "Nutrient and particulate fluxes in a salt march ecosystem: Tidal exchanges and inputs by precipitation and
groundwater" by Valiela e t al. Limnol. Oceanogr. 25: 182-183.
CHRZANOWSKI,
T. H., L. H. STEVENSON,
AND B.
KJERFVE. 1979. Adenosine 5'-triphosphate
flux through the North Inlet marsh system.
Appl. Environ. Microbiol. 37: 841-848.
KJERFVE,B. 1975. Velocity averaging in estuaries
characterized by a large tidal range to depth
ratio. Estuarine Coastal Mar. Sci. 3: 311-323.
. 1978. Bathymetry as an indicator of net circulation in well mixed estuaries. Limnol.
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. 1979. Measurement and analysis of water
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Submitted: 24 January 1980
Accepted: 11 September 1980