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Conceptual Model of Salinity Intrusion by Tidal Trapping
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R. C. Berger, Ph.D., P.E., D.CE, F.ASCE 1; and Jens Kiesel, Ph.D. 2
Abstract: Shallow bays along the fringe of an estuarine channel impact salinity distribution. The primary mechanism of impact has been
termed “tidal trapping,” and is a result of the phase difference between the filling and emptying of the bay and the flow in the channel. This
mechanism has been proposed as a major contributor to salinity intrusion in some locations. This paper creates a conceptual model to
explain the increase or decrease in salinity intrusion based upon the channel tidal waveform and the character of the adjacent bay. Two
idealized scenarios are used for this explanation: (1) Standing wave in the channel with a small phase lag in the bay tide; and (2) Progressive
wave in the channel again with a small phase lag between the channel tide and the bay. This conceptual approach is useful for inspecting
and understanding salinity intrusion processes in a complex estuary bay setting since the form of the wave can often be determined by
generally available water surface gages along the estuary. A constructed numerical model of an estuary supports the conceptual model.
DOI: 10.1061/(ASCE)HY.1943-7900.0001627. © 2019 American Society of Civil Engineers.
Author keywords: Estuary; Tidal trapping; Added bay; Salinity intrusion; Two-dimensional hydraulics; Adaptive hydraulics model (AdH).
Introduction
The salinity field in an estuary is critical for the ecology of the
waterway, and is also a concern for freshwater supply in industry,
agriculture, and populations. However, anthropogenic modifications to the estuary or the river hydrograph impact the salinity environment. In some cases, these modifications will lead to greater
salinity intrusion. Salinity penetrates within the estuary as a result
of several mechanisms. Among others, these mechanisms include
the tidal filling volume (tidal prism), different flood and ebb paths
(tidal pumping), density-driven currents, and the influence of surrounding tidal bays (tidal trapping).
A typical tide in an estuarine channel is a combination of
two distinct waveforms: a progressive wave and a standing wave
(e.g., Ippen and Harleman 1966). By assessing the apparent phase
lag between successive tide gages or the amplification of the tidal
range along an estuary, an investigator can determine the likely
waveform at a location. The waveform can be derived from relatively inexpensive tide measurements, which are typically plentiful
in estuaries. The relationship between velocity and tide can be
attributed to these two primary waveforms. For the progressive tide,
the velocity and tide are in phase with peak flood velocity occurring
at high water elevation and peak ebb velocity at low water elevation. The standing wave peak flood velocity occurs at the midpoint
in time as the water rises from low water elevation to high water
elevation. Similarly, the peak ebb velocity happens at the midpoint
in time as the water level falls. Salinity is a conservative constituent
1
Retired; formerly Research Hydraulic Engineer, Waterways Experiment Station, US Army Engineer Research and Development Center,
3343 Tucker Rd., Vicksburg, MS 39180 (corresponding author). Email:
Berger.charlie@gmail.com
2
Research Scientist, Dept. of Ecosystem Research, Leibniz-Institute
of Freshwater Ecology and Inland Fisheries (IGB), Berlin 12587, Germany;
Dept. of Hydrology and Water Resources Management, Institute for Natural Resource Conservation, Christian-Albrechts-Univ. Kiel, Kiel 24118,
Germany. Email: kiesel@igb-berlin.de
Note. This manuscript was submitted on April 22, 2018; approved on
February 15, 2019; published online on July 16, 2019. Discussion period
open until December 16, 2019; separate discussions must be submitted for
individual papers. This paper is part of the Journal of Hydraulic Engineering, © ASCE, ISSN 0733-9429.
© ASCE
that is transported with the velocity. The interrelationship with the
waveform is important.
The channel is often lined with bays along its length. The channel
and bay flow and salinity interact. Tidal trapping is an important component of this interaction. Tidal trapping has been described as an
exchange between bays and an adjacent estuarine channel in which
a constituent (e.g., salinity) fills and empties out of phase with the
main channel (Fischer et al. 1979). This exchange can be described
by two processes, one driven by the channel currents and the other by
the filling and emptying of the bay caused by water surface difference. The first process explains tidal trapping as a dispersion process
induced by lateral velocity gradients within the main channel (Fischer
1972; Okubo 1973; Smith 1976; Simpson et al. 2001; Bowen and
Geyer 2003). The currents along the channel have lateral gradients
due to the friction of the channel and the irregularity of the channel
edge. These lateral current gradients result in an exchange with the
surrounding bays that is characterized by turbulence and by larger
scale current eddies at the bay entrance. Usually, this process is modeled using one-dimensional cross sections in which the result of this
tidal trapping is a smearing or dispersion process.
The second process is exemplified by MacVean and Stacey
(2011) in which the bay is long and its entrance narrow. They
describe the process as advective rather than dispersive. As a tide
progresses up an estuary, it will travel somewhat faster along the
deeper portion of the channel. There is a tendency for flow to
spread from the channel laterally into the shallower areas and into
the bays along the fringe. This will cause the bays to fill and empty
due to the water surface gradients. The filling and emptying will
tend to be out of phase with the channel currents and salinity
and modify the channel salinity.
The bay’s geometry influences the impact of these two processes, wherein the first process becomes more important if the
surrounding bay is small and its opening to the channel is large.
That is, the bay is large compared to its opening into the channel.
The second process is for bays that are long or large compared to
their openings into the channel. Here, the exchange due to the water
surface slope between the channel and the bay is much more
important than the velocity gradient exchange at the entrance. This
second process is considered in this paper.
The previous MacVean and Stacey (2011) study is valuable in
describing tidal trapping in complex estuary bay settings using
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Conceptual Model of Tidal Trapping and Salinity
Intrusion
Normalized Tide, Velocity, or Salinity
A conceptual model is a tool used primarily for developing understanding of a process. The tool may also be helpful in making
informed decisions for screening purposes. To develop this conceptual model, the authors utilize the waveform in the channel and link
this waveform to tidal trapping through water-surface-driven flow
exchange with the bay. As with the prior studies, the authors take a
one-dimensional or cross-section average approach for the conceptual model. The channel is considered to be one-dimensional with a
branch to represent the bay. The filling and emptying of the bay are
a result of the rising and falling of the tide in the channel. The salinity in the channel is directly linked to the velocity there. The salinity
reaches its maximum value at a channel location at the time of slack
after flood. The channel salinity falls to its minimum at slack after
ebb. With the bay filling in association with the tide and the salinity
values associated with the currents, then it is reasonable to consider
the waveform in the channel as a critical principle in developing a
conceptual model.
A progressive tide, as the name implies, progresses along the
estuary and will have later and later times of high water at gages
moving upland along the waterway [Fig. 1(a)]. The phase lag between gages will be close to that calculated by the wave celerity.
With the progressive waveform, the velocity is synced with the
wave. That is, high water elevation and the maximum upland
(flood) velocity occur simultaneously and at low water elevation,
the maximum oceanward (ebb) velocity occurs. Slack velocity then
is at midtide elevation. This is the typical waveform for straight
1
0.5
0
-0.5
-1
-0.5
-0.25
0
(a)
Normalized Tide, Velocity, or Salinity
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detailed observations. A direct transfer of these results to other,
even seemingly similar settings is, however, not possible, leaving
the practical engineer only with the option to carry out detailed
measurements or modeling just to understand the impact on salinity
intrusion in their own region or estuary of interest. Therefore, in
order to make qualitative predictions, the identification and generalization of processes impacting tidal trapping in a complex estuary
bay setting is needed.
Expanding on the work of MacVean and Stacey (2011), the authors develop a conceptual model to describe a simplified tidal trapping water surface slope-based mechanism. The conceptual model
describes the important parameters and predicts the likely direction
of the change (increase or decrease) to salinity intrusion caused by
tidal trapping. The model is explained using two canonical cases of
tidal trapping linked to the waveform in the channel and is confirmed by using a two-dimensional numerical model to evaluate
the impact of an added bay with changing depths in an estuarine
example.
0.25
0.5
0.75
1
0.5
0.75
1
Fraction of Cycle
1
0.5
0
-0.5
-1
-0.5
-0.25
0
0.25
Fraction of Cycle
(b)
Tide
Velocity
Salinity
Bay Tide
Fig. 1. Channel and bay tide phase lag, channel velocity, and salinity for (a) a progressive wave; and (b) for a standing wave. Please note the
indication of the filling and emptying process of the bay.
© ASCE
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channels or ones in which there is not a strong convergence of the
cross section in the upland direction. A standing wave, on the other
hand, is a sloshing mode. A pure standing wave (for a quarter wavelength) will find all the channel stations experiencing high water at
the same time. Peak maximum upland (flood) velocity occurs at the
midtide elevation as the tide rises from low water to high water
elevation [Fig. 1(b)]. Peak maximum oceanward (ebb) velocity occurs at midtide for falling tidal water elevation. The standing wave
is the result of a wave reflection due to a closed upland end or a
rapid cross-section convergence, which will cause an amplification
of the tidal range.
In a single channel with no accompanying bays, regardless of
the waveform, the salinity is transported upstream during the time
of flood velocity, normally reaching its maximum salinity at a location at slack water after flood. Alternatively, the location will normally find its minimum salinity at slack after ebb. This is when the
salt water has been transported as far out of the estuary as possible.
If a narrow-throated bay is added along the channel, it fills and
empties with the tide. There can be a phase lag between the tide
in the bay and an adjacent channel location. The lag may be
due to the depth of the bay or a tenuous connection to the channel.
These dependencies are described mathematically in the following formulation. The description involves the channel waveform
and the phase lag between the bay tide and the channel tide. These
conceptual model findings are then synthesized into a salinity
intrusion index.
Model Definition
A simple mathematical description of the tides and velocity at a
channel location is now given for a standing wave and a progressive
wave. All explanations of the symbols utilized are given in the
Notation section at the end of the paper. The water surface for
the channel is defined as
2πt
ηc ¼ ac Sin
ð1Þ
T
and for the added side embayment as
2πt
ηb ¼ ab Sin
−δ
T
ð2Þ
This lag is relative to the channel tide. The authors will consider
the phase lag of the channel itself to be zero; the lag then is between
the bay and the main channel.
The flow into and out of the bay (from the channel) is assumed
to be solely driven by the water surface differential. This means that
the bay amplitude will be given by
π
ab ¼ ac Sin
−δ
ð3Þ
2
The maximum water surface elevation in the bay is determined
by the elevation of the channel and the phase lag. The peak bay
water surface elevation will match the channel water surface elevation at that time. A larger phase lag would then mean a smaller
amplitude in the bay. If the bay planar area remains constant, then
the reduction in the bay tide range results in a similar drop in the
volumetric exchange with the channel.
The authors will furthermore explain the phased filling and
emptying of this bay and how that impacts the salinity of the channel. Bay filling and emptying is influenced by the channel tidal
wave type. The tidal wave is likely a combination of a progressive
wave and a standing wave. The authors separate these two wave
© ASCE
types and consider a purely progressive wave or a purely standing
wave. The phase of the current velocity determines the wave type.
The channel current velocity is given as follows:
2πt
uc ¼ ûSin
þ∝
ð4Þ
T
where û = maximum velocity in the channel adjacent to the bay. For
a purely progressive tide, the value of α is zero. The velocity is in
phase with the tide. For a standing wave, caused by a closed end of
a channel, the value of α is π=2 radians (or a quarter tidal cycle).
Herein, α will be termed the “velocity phase.”
The following assumptions are used to facilitate an understanding and applicability of the model:
• The bay tidal prism is sufficiently small so as not to materially
impact the tidal prism of the estuary channel nearby. The impacts of the added bay are then a result of tidal trapping and
not caused by increasing the estuarine channel tidal prism.
• The analysis assumes a one-dimensional system in which the
channel is considered to be cross-sectionally uniform.
• The authors will further assume that the exchange between the
bay and the channel is dominated by flow due to the lateral surface gradient and not by a turbulent or eddy-driven exchange at
the entrance to the bay. This suggests that the connection between the bay and adjacent channel is narrow compared to
the size of the bay.
• Furthermore, the authors develop this conceptual model considering only the velocity, tide, and salinity in the channel adjacent
to the bay.
• The salinity is assumed to be a step function that has a value of ŝ
for ½ tidal cycle. The salinity value in the channel adjacent to
the added embayment is zero for the rest of the cycle. This
means that the assumed salinity is a constant and during a flood,
it doesn’t reach the bay location for ¼ T after the time of slack
after ebb. The water has a value of salinity of ŝ for a ¼ T while
the channel has a flood direction, and also for a ¼ T after the
current has turned and is ebbing. The salt water reaches the bay
position halfway through the flood flow and completely leaves
this position by halfway through the ebb flow. This is true no
matter what the waveform happens to be at this location. The
assumption that the salinity reaches the bay location ¼ T before
the slack flow after flood and recedes back to zero at a ¼ T
afterwards is arbitrary. Any choice that results in salinity being
present at the location that is symmetric in the time about slack
after flood would suffice. All choices within this limitation
would produce a conceptual model with qualitatively similar
results.
• The authors reason that an increase in the salinity at this station
would result in an increase in salinity intrusion since the salt
water is being transported past this location to the upstream regions. Similar reasoning is applied for a decrease in salinity at
this location due to the presence of the bay.
Conceptual Model Application
To enable a clearer understanding of the illustrated conceptual
model, the cases of two primary waveforms are described. The
two forms are: (1) a standing wave in combination with a bay with
a specified phase lag relative to the channel; and (2) a progressive
wave in combination with a bay with a phase lag relative to the
channel.
1. For the purely standing waveform, the velocity leads the water
surface by π=2. Slack after flood occurs at high water elevation.
From the assumptions on the salinity time history at the channel
station, there is a half cycle with salinity (centered around high
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water and flood slack) and the other half is totally fresh. With a
phase lag between the channel tide and the bay tide, the velocity
and the tide are shown in Fig. 1(b). The amplitude of the bay tide
diminishes as the phase lag increases, as shown by Eq. (3). If
there is no phase lag, then the channel and bay tide are identical.
Whenever the bay tide is rising, the bay is filling with channel
water and salinity. Channel salinity is also shown in Fig. 1(b).
The salinity that enters the bay is that which is found in the
channel adjacent to the bay. When the bay water surface is falling, the salinity concentration is leaving the bay and recedes
back into the channel. Under the assumption of no mixing within the bay, the salt water will reenter the channel in the reverse
order that it previously entered the bay. At first, the bay begins
filling with freshwater and then later with salt water. Initially,
consider the case in which there is no phase lag between the
bay and channel tides. In this case, the salt water that enters
the bay then exits back to the channel while the channel is still
salty. Therefore, the bay has no impact on the channel salinity.
If, however, there is a phase lag, then the filling of the bay is
pushed later in time so that more time is spent filling while there
is salt water in the channel. The bay will then return this saline
water back to the channel partly while the channel is also saline
but will also, for the time length of the bay tide phase lag, return
saline water into a freshwater channel. This means that the tide
phase lag from the channel to the bay will result in an increase in
the salinity of the channel. And, in turn, the increased local salinity will create an increase in salinity intrusion. The larger the
tide phase lag, the longer the fraction of the bay fill time is when
there is salinity in the channel. This means that the bay salinity
will be higher. The saline return flow back to the channel will
also tend to occur during a time with low channel salinity. The
phase lag of the bay tide from that of the channel means that for
a standing waveform in the channel, the bay will increase salinity intrusion. With no phase lag, the bay will have no impact on
channel salinity.
2. Now consider the filling and emptying of a small embayment
along the channel, in which the channel tide is characterized
as a progressive wave [Fig. 1(a)]. First, consider the case that
the bay’s geometry causes no phase lag between the bay and
the channel. The bay fills with rising bay tide and empties with
falling bay tide in exactly the same manner as the standing wave
case. For a fully progressive wave, the channel flow is in phase
with the tide. This means that maximum flood velocity occurs at
channel high water elevation and channel low water elevation is
simultaneous with peak ebb velocity. In this scenario, the salinity reaches the channel location adjacent to the bay at high
water and is completely swept out of the channel at low water.
This means that the embayment fills with freshwater and then
releases this freshwater back into the channel while the channel
has a salinity of ŝ. The added embayment then is reducing the
concentration in the channel and likely reducing the distance of
salinity intrusion. Second, consider the case that there is a phase
lag between the channel and bay tide, as shown in Fig. 1(a). In
this case, if the bay fills later during the tidal cycle, it would tend
to fill part of the time when the channel has salt water. This is in
contrast to the no phase lag condition, which only filled with
freshwater. The conceptual approach shows that this length
of salt water filling time is the same length as the phase lag
of the bay tide. A progressive tide with a bay tidal phase lag
will also reduce the salinity intrusion, but less than the no phase
lag condition. This is consistent with the site-specific findings of
MacVean and Stacey (2011) in which the continued flood condition in the channel after high water elevation resulted in the
maximum salinity at the channel station occurring before slack
© ASCE
water after flood. The fraction of the tide form that was progressive meant that the bay was emptying freshwater into the channel while the channel was still flooding.
Here is a short summary of the findings. For the no bay phase
lag, no channel waveform increases salinity intrusion. In fact,
the progressive waveform would reduce salinity intrusion. If there
is a bay phase lag, the standing wave tide will increase salinity
intrusion. The progressive waveform would still reduce salinity
intrusion, but not as much as the no phase lag condition.
A typical tide in an estuarine channel is not purely a standing
wave or a progressive wave. Instead, it is somewhere in between.
This discussion now will be about the generalization of the conceptual model from these two canonical forms. The examples show
that if the phase lag of the bay places the time of bay high water
elevation at the same time as slack after flood in the channel, then
the bay is neutral in terms of salinity intrusion. That is, it shouldn’t
raise or lower the channel salinity because the peak flood velocity is
midway through the time of the saline period in the channel. The
phase lag for neutral conditions can then be calculated
δn ¼
π
−α
2
ð5Þ
If the phase lag in the bay is greater than δ n then the bay tends to
increase salinity intrusion. If the phase lag in the bay is less than δn
then the bay will likely reduce salinity intrusion. This relationship
between phase lag in the bay and the phase between the channel
velocity and tide holds over all bay conditions for this conceptual model.
Under certain limiting conditions, the relative impact of a bay on
salinity intrusion can be developed for a range of α and δ conditions. The authors create a salinity intrusion index, SI , that can give
some guidance on the relative salinity contribution to intrusion using the conceptual model. The index is positive for increased
intrusion and negative for a reduction in intrusion. The magnitude
of the index gives a suggestion of the strength of the impact. In
order to produce this index, a significant additional constraint
(assumption) is required. This assumption is that the bay tidal prism
is directly proportional to the bay tide amplitude. This means that
the bay plan area remains constant.
This salinity index is defined in Eq. (6):
R ð14þ2πδ ÞT
SI ≡
δ
ÞT
ð−14þ2π
SðtÞ dηdtb dt − ab ŝ
ac ŝ
ð6Þ
Fig. 2 visualizes the salinity index, SI , for five velocity phases.
The index synthesizes the conceptual model to a single graph. The
index plots extend over the range of potential bay phase lags (from
0 to 0.25 cycle). This extends the two examples of a progressive
wave and a standing wave. If the Eq. (6) numerator and denominator are each multiplied by the plan area, then the index can be
explained in terms of volumes and salinity mass. The integral then
would represent the salinity mass gained by the bay during filling.
The terms that are subtracted in the numerator represent the mean
salinity mass that the bay could potentially gain during filling at the
given bay amplitude. The denominator then is used to normalize
the index by using the average channel salinity times the maximum
volume the bay could gain. This maximum volume is a result of the
bay tide being identical to the channel tide. All this results in a
normalized index with potential values between −1 and 1. The
standing wave in Fig. 2 (∝ ¼ π2) is neutral at zero phase lag, but
it shows an increase in salinity intrusion for all other phase lags.
The index drops to zero at the maximum phase lag since the tide
range in the bay is zero. For the purely progressive wave (∝ ¼ 0),
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0.6
0.4
Salinity Indicator
0.2
0
-0.2
-0.4
-0.6
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-0.8
-1
0
0.05
0.1
0.15
0.2
0.25
Bay Tide Phase Lag (fraction of a cycle)
Fig. 2. Salinity intrusion index showing the salinity indicator versus
bay tide phase lag for five velocity phases (α).
the index is negative throughout (except for the maximum phase lag
point). This indicates that the progressive wave would tend to reduce salinity intrusion, but intrusion would be reduced less with
greater bay phase lag. Other velocity phases are in between these
extremes. The index could likely be scaled by the range of salinity
at the location, as well as the planar area of the bay. This would
perhaps make the index applicable between different sites along
the estuary.
In summary, the conceptual model shows:
• The smaller the channel velocity phase is, the greater the reduction in salinity intrusion by a bay at that location is. A bay associated with a progressive tide form reduces salinity intrusion
more than a standing wave.
• As the phase lag is increased between the bay tide and the
channel tide, an increase in channel salinity intrusion can be
expected.
This conceptual model and salinity intrusion index provide insight into the potential salinity change in the channel due to the
addition or modification of a small bay. This also provides practical
guidance for planning potential mitigation sites for restoration projects using wetlands.
Numerical Example
A numerical example is presented to test the conclusions developed
with the conceptual approach. The numerical example is based on
the Delaware estuary, but is invalidated and should be considered a
representative flume for the purposes of this study. The bathymetry
of the bay is artificially managed by the authors to illustrate points
for the article and is not that found in the actual estuary. While the
model is used as an example flume, the roughness coefficients are
reasonable (Manning’s n of 0.025 for most of the estuary and 0.030
in the very shallow regions). The numerical code used is the twodimensional module of the adaptive hydraulics model (AdH)
(Berger and Lee 2005; Savant et al. 2011) and is a shallow water
model with baroclinic pressure terms. The AdH calculates water
surface, velocity, and salinity. Since this module of the AdH is
two-dimensional, complete vertical mixing is assumed. The standard procedure was followed to demonstrate spatial and temporal
computational convergence. The details are not necessary for the
purposes of this article. Fig. 3 shows a representation of the model
domain. There is also an ocean area that is not shown in Fig. 3.
© ASCE
This domain includes a bay that can be added or removed to assess
its impact. The tidal phase lag between the bay and channel is produced by varying the depth of the added bay. The model is run with
a tide that is a pure semi-diurnal tide period of 12.42 h (the principal
lunar semi-diurnal component, M2) with a range of 0.924 m. While
this is the only driving tidal component, the nonlinear nature of the
system also generates overtides (of higher frequencies). The river
inflow is a constant 50 m3 =s. After running for a long enough time
so that the salinity field reaches an equilibrium, every tidal cycle
thereafter has the identical water surface, velocity, and salinity at a
specific location and phase of the cycle. Results are reported for the
final complete cycle of a 2,000-h simulation. This simulation length
is over 160 tidal cycles.
The conceptual model uses inputs from the numerical model in
terms of the hydrodynamic descriptions. These inputs include the
velocity phase in the channel and the phase lag between the water
surface in the channel and the bay tides. The velocity phase is from
the base condition in which there is no bay. Data throughout this
comparison will be drawn from two stations shown in Fig. 3 (inset).
The stations are labeled TC and TB for stations in the channel
adjacent to the bay and within the bay, respectively. The base condition is provided with no added bay, that is, the bay is removed.
The two plans are a constant bay elevation of 0 m (Plan 1) and a
constant bay elevation of 0.5 m (Plan 2). Fig. 3 shows the elevations
for the channel bed.
Fig. 4 shows the simulated tide and velocity at station TC. The
velocity phase is about 1.8 h, which in terms of radians is 0.28π.
This is a fairly even mix of a progressive and a standing wave. Fig. 5
shows the water surface elevation for station TC for the base condition and the water surface time history in the bay at station TB for
the two plans. The phase lag for Plan 1 is about 0.75 h and for Plan
2 it is about 1.0 h. These lags are 0.06 and 0.08 of a tidal cycle,
respectively. In terms of radians, this is a phase lag of 0.12π for Plan
1 and 0.16π for Plan 2. There is a significant shallow water effect in
Plan 2, given by the longer phase lag for low water than for high
water. The neutral phase lag [from Eq. (5)] is found to be 0.22π.
Since both values are less than δ n , the conceptual model would
suggest that each value would reduce salinity intrusion. Plan 1
would be expected to reduce salinity intrusion more than Plan 2.
The same information can be determined using the salinity
intrusion index from Fig. 2. The salinity intrusion index for the
two plans would be −0.37 for Plan 1 and −0.24 for Plan 2.
The salinity intrusion index is a rough prediction of the salinity
of the bay scaled by the salinity range and the ratio of the bay tide to
the channel tide range. The salinity intrusion index of zero would
imply the bay salinity to be about that of the channel. Fig. 6 shows
the salinity at TC for the base condition and the bay salinity at TB
for the two plans. The average salinity for the base at the channel
station is 20.4 ppt, while the bay average salinity at TB is 19.0 for
Plan 1 and 19.8 for Plan 2. The greater phase lag in Plan 2 leads to
higher bay salinity. The channel salinity profile for the base and
plans is shown in Fig. 7. This is a profile along the channel thalweg.
The profile begins at Station TC and progresses upstream (landward). The base shown is the salinity at the time of maximum
intrusion. The two plans are shown as differences from the base at
that same time. Both plans lower the salinity in the upper reach,
reducing intrusion. Plan 2, however, has a higher salinity than
Plan 1. Plan 2 has a greater salinity intrusion than Plan 1, but less
salinity intrusion than the base. This finding supports the conceptual model conclusions.
Fig. 8 shows the 25 ppt isohaline for the base and Plan 2 conditions at a time of minimum salinity. This is for a region oceanward of the fringe bay. The difference between the base and Plan 2
shows that with the added bay, the preferred path for freshwater is
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Fig. 3. Spatial overview of the Delaware River Estuary and its depth distribution with small bay denoted and inset showing the detailed location of the
gage sites (TB in the bay and TC in the channel).
much more on the west side of the estuary (away from the bay). The
bay fills from much of the channel cross section, but empties along
the channel eastern shore. This is a subtlety that a conceptual model
cannot represent and serves as a cautionary note regarding the limitations of a simple representation like this one. The conceptual
model assumes channel cross-section uniformity and the model results show that is a weak assumption here. Nevertheless, the conceptual model provides meaningful insight.
© ASCE
Conclusions
A conceptual model should be simple enough that the behavior is
represented and the results are understandable. Its goal is to provide
insight into the direction of change (increasing, decreasing, or no
change) resulting from a feature such as an added bay. The authors
have developed a conceptual model to describe the impact on salinity intrusion caused by tidal trapping in an estuary. The description
04019033-6
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J. Hydraul. Eng.
2
1.5
1
0.5
0
-0.5
0
20
-0.1
-0.2
15
-0.3
10
-0.4
5
0
0
4
8
12
Tide [m]
-0.5
0
20
Base
Velocity [m/s]
Fig. 4. Channel station (TC) water surface elevation and velocity, base
condition.
40
60
-0.6
Distance Upstream of Bay [km]
16
Time [hrs]
Plan1-Base
Plan2-Base
Fig. 7. Longitudinal salinity profile at maximum intrusion for the base
condition and the difference between each plan and base; absolute
base salinity is shown on the left y-axis and the differences (Δ) between
base conditions and the plans are shown on the right y-axis.
Water Surface Elevation [m]
3
2.5
2
1.5
1
0.5
0
4
8
12
16
Time [hrs]
TC Base
TB Plan1
TB Plan2
Fig. 5. Water surface elevation at station TC for base condition and at
bay station TB for Plans 1 and 2.
24
22
Salinity [ppt]
Downloaded from ascelibrary.org by Universidad de la Sabana on 08/05/19. Copyright ASCE. For personal use only; all rights reserved.
-1
25
Salinty (Plan-Base) [ppt]
2.5
Base Salinity [ppt]
Wat. Surf. Elev. or Velocity
3
Salinity (25ppt)
20
Base
(no Bay)
18
Plan 2
(Bay 0.5m deep)
16
0
4
8
12
16
Time [hrs]
TC Base
TB Plan1
TB Plan2
Fig. 6. Salinity for station TC for the base condition and bay station TB
for Plans 1 and 2.
Fig. 8. Spatial overview of the minimum salinity 25 ppt isohaline for
base and Plan 2 conditions.
is applicable to bays that have sufficiently narrow connections to
the channel so that the exchange is dominated by the flow due to
surface water slope, rather than the turbulent exchange at the entrance to the bay. The bay should not be so large as to significantly
impact the tidal prism of the main channel. The conceptual model
demonstrates that the form of the tide wave in the channel and the
tidal phase lag of the bay are the most important factors. A stronger
component of a progressive wave will tend to suggest a reduction in
© ASCE
04019033-7
J. Hydraul. Eng., 2019, 145(10): 04019033
J. Hydraul. Eng.
Downloaded from ascelibrary.org by Universidad de la Sabana on 08/05/19. Copyright ASCE. For personal use only; all rights reserved.
the channel salinity intrusion with the bay. A stronger standing
wave component has a more neutral effect. In both cases, any geometric change to the bay that increases the water surface lag between the bay and the adjacent channel will favor an increase in
salinity intrusion. The value of this conceptual model is that the
waveform of the tide in most channels is known. The geometry
and bathymetry of the associated bays are also typically known.
Often, the only initial data that a planner or researcher has are
the tidal records along an estuary. These tidal records are often adequate to determine the form of the tide. The conceptual model is
supported by using a numerical model based on an estuary flume
example. The numerical model results are in accord with the expectations of the conceptual model.
The conceptual model illustrates that seemingly small features,
such as these small bays, can play an important role in the salinity
distribution in the estuary. Estuarine modelers validate their models
for hydrodynamics, often by comparisons to field tides and tidal
fluxes. This conceptual model suggests that even though these
small bays may not noticeably contribute to the tidal flux or modify
the tide, they could have a strong influence on the salinity intrusion
length and distribution. This then requires the modelers to accurately represent these small and perhaps shallow features.
The conceptual model is useful in understanding the likely direction of the salinity change as a result of a proposed added feature, such as wetland restoration sites or marinas. The conceptual
model also provides a reasonable understanding of the salinity
intrusion caused by tidal trapping.
Future work can focus on further generalization of the model
and reducing the number of assumptions. This work could include
a consideration of the range of the salinity and planar area of the
bay in generalizing the salinity intrusion index to different sites
along the estuary.
Acknowledgments
While one of the authors (Berger) is a retiree of the U. S. Army
Corps of Engineers Research and Development Center, this paper
was created independent of that organization. J.K. acknowledges
funding obtained through the GLANCE project (Global change effects on river ecosystems; 01LN1320A) supported by the German
Federal Ministry of Education and Research (BMBF). The authors
also acknowledge the valuable contributions of the reviewers and
the associate and chief editors.
Notation
The following symbols are used in this paper:
ab = tidal amplitude of the bay;
© ASCE
ac =
SðtÞ =
SI =
ŝ =
T=
t=
uc =
û =
∝=
δ=
δn =
ηb =
ηc =
tidal amplitude of the channel adjacent to the bay;
channel salinity as a function of time;
salinity intrusion index;
salinity maximum in the channel (a step function);
tidal period;
time;
channel velocity, adjacent to the bay, as a function of
time;
maximum channel velocity, adjacent to the bay;
angle in radians by which the velocity leads the water
surface;
phase lag of the tide in the bay compared to the tide in
the channel;
neutral phase lag;
bay water surface elevation; and
channel water surface elevation.
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