HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus GmbHGöttingen, Germany10.5194/hess-19-2791-2015Revised predictive equations for salt intrusion modelling in estuariesGisenJ. I. A.j.isabellaanakgisen@tudeflt.nlisabella@ump.edu.mySavenijeH. H. G.https://orcid.org/0000-0002-2234-7203NijzinkR. C.https://orcid.org/0000-0002-9999-9883Water Management, Civil Engineering and Geosciences, Deflt University of Technology, Stevinweg 1, 2628CN Delft, the NetherlandsCivil Engineering and Earth Resources, Universiti Malaysia Pahang, Lebuhraya Tun Razak, 26300 Gambang, Kuantan, MalaysiaJ. I. A. Gisen (j.isabellaanakgisen@tudeflt.nl, isabella@ump.edu.my)18June20151962791280317December201416January20152June2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://hess.copernicus.org/articles/19/2791/2015/hess-19-2791-2015.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/19/2791/2015/hess-19-2791-2015.pdf
For one-dimensional salt intrusion models to be predictive, we need
predictive equations to link model parameters to observable
hydraulic and geometric variables. The one-dimensional model of
made use of predictive equations for the Van
der Burgh coefficient K and the dispersion at the seaward boundary
D0. Here we have improved these equations by using an expanded
database, including new previously un-surveyed
estuaries. Furthermore, we derived a revised predictive equation for
the dispersion at tidal average condition and with the boundary
situated at the well identifiable inflection point where the estuary
changes from wave-dominated to tide-dominated geometry. We used 89
salinity profiles in 30 estuaries (including seven recently studied
estuaries in Malaysia), and empirically derived a range of equations
using various combinations of dimensionless parameters. We split our
data in two separated data sets: (1) with more reliable data for
calibration, and (2) with less reliable data for validation. The
dimensionless parameters that gave the best performance depended on
the geometry, tidal strength, friction and the Richardson
number. The limitation of the equations is that the friction is
generally unknown. In order to overcome this problem, a coupling has
been made with the analytical hydraulic model of
, which makes use of observed tidal damping and
by which the friction can be determined.
Introduction
Predictive methods to determine salinity profiles in estuaries can be
very useful to water resources managers, particularly when applied to
ungauged estuaries where only a minimal amount of data are
available. Before any decision is made on collecting detailed field
observations, it is useful to obtain a first estimate of the strength
and range of the salt intrusion in the area of interest. Such estimate
can be made if there are predictive equations available to compute the
longitudinal salinity profile along the estuary. With reliable
predictive equations, water managers are able to estimate how far salt
water intrudes into the river system under different circumstances,
and more importantly, how interventions may change this situation.
The one-dimensional salt intrusion model of
makes use of the Van der Burgh and dispersion equations to represent
the longitudinal variation of the salinity. The Van der Burgh and
dispersion coefficient at the ocean boundary are obtained by
calibration of the simulated salinity curve to
observations. established a predictive equation
for each of these parameters, so that the longitudinal salinity
distribution could be estimated when data were lacking or to monitor
the impact of interventions, such as dredging or fresh water
withdrawal. The predictive equations have subsequently been modified
and tested by several researchers including
, , and
.
In this paper, we shall revisit the predictive equations in the light
of new insights on how friction and estuary shape affect tidal mixing by deriving
a relationship between several governing parameters, making use of the salinity
measurements from 30 estuaries including seven new field observations in previously
ungauged estuaries in Malaysia that were sampled through a consistent approach.
As a result, we present the fully revised and more accurate predictive equations
for the Van der Burgh coefficient and for the boundary value of the dispersion at
a well identifiable location, based on tidal average (TA) condition.
One-dimensional analytical salt intrusion model
The analytical one-dimensional salinity model developed by
, presented below, is
used to simulate the salinity profile in the estuaries studied. In
a steady-state situation, the partial temporal derivative in the salt
balance equation is zero. Considering a constant fresh water discharge
Qf [L3T-1] and tidally averaged cross-sectional area
A [L3], the salt balance equation for tidal average (TA) condition
can then be written as
S-Sf=-A|Qf|DdSdx,
where S=S(x) [ML-3] and D=D(x) [L2T-1]
are the salinity and dispersion at TA condition. Since
discharge has a negative value, the absolute value of Qf
is taken in Eq. (). Sf [ML-3]
represents the fresh water salinity. In 1972, Van der Burgh derived an empirical
equation for the TA dispersion making use of a large amount of salinity
measurements in the Rotterdam Waterway. The equation is then revisited by
who described the relation between dispersion
and salinity to be
dDdx=-K|Qf|A,
in which K [–] is defined as the Van der Burgh coefficient (shape
factor). Substituting Eq. () into Eq. (), the
differential equation for the tidally averaged longitudinal salinity
distribution is expressed as
dSS-Sf=1KdDD.
Integration of Eq. () leads to
S-SfS0-Sf=DD01/K.
The symbols S [ML-3] and D [L2T-1] are
the steady-state salinity and dispersion coefficient at location x,
while S0 [ML-3] and D0 [L2T-1] are the
salinity and dispersion at the estuary mouth. In alluvial estuaries,
the variation of the estuaries shape over the distance upstream can be
expressed in an exponential function as
A=A0exp-xaB=B0exp-xb,
where a [L] and b [L] representing the cross-sectional area and
width convergence length, A0 [L2] and B0 [L] are the
cross-sectional area and width at the mouth, B [L] is the width of
estuary at distance x [L] (towards upstream). Substituting the
exponential relation of Eq. () into Eq. ()
and the integration gives
DD0=1-βexpxa-1withβ=Ka|Qf|D0A0.
Here, β [–] is the dispersion reduction rate. At the salt
intrusion limit (upstream) where only fresh water discharge exists, the
dispersion coefficient becomes zero and x is equal to the salt
intrusion length L [L]. Hence, the intrusion length is expressed by
L=aln1β+1.
Equations ()–() are the general equations used to
compute the longitudinal salinity distribution.
Existing predictive equationsVan der Burgh's coefficient
Van der Burgh's coefficient K is also known as the “shape factor”
in the salinity curve . Based on salinity
measurements of 15 estuaries, Savenije found that K is strongly
related to the geometry (the convergence length a or b and the
width B [L]) and its influence is more significant at the tail of
the salinity curve (upstream). Moreover, observed that every estuary had its own characteristic
value of K, ranging from zero to one. Assuming that the Van der
Burgh coefficient is not time dependent,
established an empirical predictive equation for K as
K=0.16×10-6h00.69g1.12T2.24H00.59b1.10B00.13,
where h0 [L], H0 [L] and B0 [L] are the depth, tidal range
and width at the estuary mouth, respectively. The symbol T [T]
represents the tidal period, while b [L] is the width convergence
length, and g [L T-2] is the gravity acceleration. More than
10 years later, and
made used of an expanded database, modified
the predictive equation involving more parameters:
K=0.3×10-3EH0.65EC20.391-δHb-2.0ba0.85EaA′0.14.
The symbols E [L], H [L] and A′ [L2] refer to the
tidal excursion, tidal range and a boundary value for the
cross-sectional area, respectively. This relation had a correlation of
0.93 and seemed very promising. However, as can be seen from the
equation, the Chezy roughness C [L0.5T-1] and damping
δH [L-1] had to be computed from tidal dynamics
analysis.
Dispersion coefficient
Dispersion is not a physical parameter; it is rather the product of
averaging, representing the mixing of saline and fresh water in an
estuary as a result of residual circulation induced by density
gradients (gravitational circulation) and tidal movement. In salt
intrusion modelling, the definition of dispersion is often unclear as
it is scale dependent and not directly measurable. The role of
dispersion is only meaningful if it is related to the appropriate
temporal and spatial scale of mixing, which here we identify as the
tidal period (timescale), tidal excursion (longitudinal mixing
length), estuary width (lateral mixing length) and depth (vertical
mixing length). A physically based description of the dispersion would
allow the analytical solution of the salt intrusion profile.
Dispersion due to gravitational circulation has been studied since
1957, as summarized by . This type of dispersion is
also known as density-driven dispersion between the two main sources:
sea water and fresh river water. were some
of the first to relate buoyancy to mixing in estuaries, whereby they
introduced the ratio between fresh water discharge and tidal volume to
represent the degree of stratification. This ratio is also known as
the Canter–Cremers number N [–] as defined by
. The buoyancy effect or stratification in
an estuary can also be represented by the estuarine Richardson number
Nr [–] which is the ratio of potential energy of the buoyant fresh
water to the kinetic energy of the tide:
Nr=Δρρghυ2QfTAE,
where ρ [ML-3] is the water density, Δρ
[ML-3] is the density difference over the salt intrusion
length, and υ [L T-1] is the tidal velocity amplitude. The
difference between N and Nr lies in the densimetric
Froude number Fd [–] which is expressed as
Fd=ρΔρ.υ2gh.
Since then, researchers have tried to look for a relation between
dispersion and estuarine numbers. Laboratory results of WES flume
, Delft flume
and
indicated an agreement with the result of in
computing the salt intrusion length, using shear velocity instead of
mean velocity in the estuarine Richardson number. Subsequently, the
relationship between the dispersion and modified Nr
also gave good correlation for all the other cases (mostly flume
experiments). suggested that the longitudinal
dispersion is proportional to the salinity gradient and included this
in his one dimensional analytical salt intrusion model, which later
was used by to model the vertical salinity and
velocity distribution. A disadvantage of all these methods was that
they did not account for convergence (implicitly assuming an
infinitely large convergence length) and that the tidal excursion, as
the most important mixing length scale, was missing in the
derivations.
Deriving the dimensionless dispersion coefficient from scaling the
steady-state salt balance equation, developed the
following empirical predictive relation for the longitudinal
dispersion at the estuary mouth for high water slack (HWS):
D0HWS=1400h‾aNr0.5υE.
The estuary shape was represented by the ratio of the averaged depth
h‾ [L] to the convergence length a, while the dispersion
was made dimensionless by the tidal velocity amplitude and tidal
excursion which was not considered in any of the earlier studies. The
applicability of these predictive equations has been widely tested in
many estuaries including multi-channel estuaries.
later modified the empirical equation using
salinity measurements from 13 estuaries, in which they introduced the
inclusion of the dimensionless friction (C2/g). The predictive
equation was divided into two depending on the types of channel – prismatic
and convergent:
convergent channel:
D0=60αcπΔρgh0/ρυC2g|u|υ0.5Eaυh0,
prismatic channel:
D0=6αcπΔρgh0/ρυC2g|u|υ0.5υh0,
where u [L T-1] is the fresh water velocity. These equations can
be used to calculate dispersion locally at any location. The coefficient αc
is an additional calibration coefficient with the range of 0.7 to 1.3.
From the result of , it is observed that the αc
coefficients for prismatic channels have values that are closer to 1.0,
whereas for convergent channels, the coefficients are scattered within
the range.
Salt intrusion length
Several researchers have tried to develop a general relation for the salt
intrusion length. The development of such predictive equations was done
empirically based on a reasonable amount of data. A pioneer effort was made
by , making use of prototype information from the Dutch and
German estuaries. His equation for the salt intrusion length as summarized by
is as follows:
LTA=26πh0KF-1.0N-0.5with F=υgh0and N=|Qf|TPt=A|u|TAυT⋅π=|u|υ⋅π.
In this equation, LTA [L] is the salt intrusion length at
TA situation, F [–] is the Froude number, and Pt [L3] is the
tidal flood volume. re-analysed
the data by Rigter (1973) and included the Darcy–Weisbach roughness and the
densimetric Froude number, resulting in
LLWS=17.7h0fD0.625Fd-0.75N-0.25,
where LWS denotes low water slack.
It is important to note here that Van der Burgh's coefficient K is replaced
by the Darcy–Weisbach roughness fD=8g/C2 [–] and F
is represented by the densimetric Froude number Fd [–].
About 20 years later, established
a similar equation with a slightly different
coefficient:
LLWS=4.4h0fDFd-1N-1.
All these methods were based on flume data with prismatic
geometry. who
explicitly accounted for channel convergence and the tidal excursion,
developed a predictive equation for the salt intrusion length at
HWS. The reasoning was that the maximum salt intrusion length occurs
during HWS, which is most important for water resources management.
Based on Eq. (), the equation reads as
LHWS=aln1400h‾E0υ0Ka2u0Nr0.5+1,
where υ0 [L T-1] is the tidal velocity amplitude at the
mouth. It is worth noting that Savenije follows Van der Burgh's
equation, with an additional shape indicator referring to the area
convergence length a.
Most of the empirical equations discussed above are based on LWS
except for Van der Burgh's and Savenije's methods which are based on
TA and HWS, respectively. However, they can easily be brought in
agreement with each other by adding E/2 or E to LHWS,
respectively. Here, we aim to develop a universal predictive equation
for estimating the Van der Burgh and dispersion coefficient for TA
condition, which can be applied in the salt intrusion model to predict
the salinity profile for any estuary worldwide under different tidal
and flood conditions.
Methods
In this paper, the main focus is on the mixing mechanisms which lead
to longitudinal dispersion in estuaries: the tide- and density-driven
dispersion. Key parameters are developed based on measurable
parameters of geometry, tidal hydraulics and fresh water discharge. In
total 89 measurements data of 30 estuaries worldwide have been used to
develop the predictive equations. Measurements in seven newly surveyed
estuaries were collected from 2011 to 2013 in Malaysia
, whereas the remainder were compiled by
revisiting existing data available in the database of
and from professional reports. The locations of
the estuaries studied are displayed in Fig. .
Global map showing the locations of the estuaries studied.
Adjustments have been made to the geometry (see Fig. S1 in the Supplement)
and salinity analysis for some of the estuaries to ensure consistency in the
input data used. The entire data set was split into two: reliable and less
reliable data. The reliable data set have been used to develop the predictive
equations, whereas the less reliable ones have been used for verification
purposes. The study was performed based on Savenije's (1993b, 2005, 2012)
method for predicting K and D0 with some modifications. The
modifications include:
All geometry and tide information used refers to the well
identifiable inflection point x1 as the boundary condition.
Analyses were performed on TA condition instead of HWS, which is
consistent with the geometry information.
Estuary roughness and the ratio of estuary width to river width
have been added in the predictive equations.
The parameters chosen are mostly independent and easy to observe
without the need for prior calibration.
Although the predictive equations are based on the TA
situation, one can still compute the salinity distribution for both LWS and HWS by shifting the curve over E/2 in
seaward and landward direction.
Selecting the dimensionless ratios
Revising the parameters selected by , we
found that the latter contained some parameters that required tidal dynamics
analysis while one of the ratios was not dimensionless. The following are
the dimensionless ratios selected for the revised predictive equation for the
Van der Burgh coefficient:
K=fBfB1,gC2,E1H1,h‾1b2,h‾1H1,λ1E1,
where Bf [L] is the river regime width (located upstream of
the tidal limit where the convergence of the river width is modest and near
constant), and λ1 = Tgh‾1/rs [L] is
the wave length at the inflection point with rs [–] being the
storage width ratio (defined as the ratio between storage width and stream
width). The symbols B1 [L], E1 [L], h1 [L], H1 [L] and
b2 [L] represent the estuary width, tidal excursion, averaged
estuary depth, tidal range and width convergence length at the
inflection point x1. It is worth noting that the roughness C=Kmh‾11/6 was obtained through calibration
using the tidal dynamics solution of which makes
use of observed tidal damping. In the above equation, it can be seen
that all parameters used have been defined at the inflection point
x1. It is also important to note that the convergence length
adopted is of the second reach and not of the first part of the
estuary. Generally the tidal indicators E and H are defined at
the mouth. In order to obtain the tidal excursion and tidal damping at
the inflection point, a projection can be made considering tidal
damping as follows :
H1=H0⋅expδHx1E1=E0⋅expδHx1,
where the damping factor δH also follows from the tidal
dynamics simulation of . The values of H1 and
E1 used in the dimensionless ratios represent the condition of
spring tide, where υ is considered to be close to
1 ms-1. This is to ensure that K is time independent
representing a general characteristic of an estuary. As a result, E
essentially reflects the tidal period as described in
(see also Table 1)
E=υTπ.
Data used to develop the predictive equation for the Van der Burgh
coefficient K.
NoEstuaryA1a2B1Bfb2h‾1x1H0E0TKmδHKK[103](km)(m)(m)(km)(m)(km)(m)(km)(h)(10-6m-1)CalPre(m2)Reliable sets for calibration 1Kurau0.74613020286.23.62.3141230-6.300.400.352Perak9.2372070130216.34.02.81412653.000.200.243Bernam4.525127045175.34.32.91412701.700.200.224Selangor1.01327035133.72.84.0141240-3.700.340.425Muar1.610028055318.23.92.0141245-2.680.250.326Endau2.04431072446.54.81.9141245-1.300.400.337Maputo4.7161150100164.15.13.31412582.000.380.328Thames10.92378050408.231.05.31412451.100.200.249Corantijn26.8645000400486.718.03.1141240-1.700.210.2710Sinnamary1.13947095123.92.73.3141240-5.000.450.4611MaeKlong1.11502401501504.63.23.6141240-4.200.300.4812Lalang2.91673601309410.30.02.6282484-0.540.650.5713Limpopo1.1115180901156.320.01.91412431.700.500.3814Tha Chin1.48726045875.65.02.6141250-5.500.350.3115ChaoPhya3.11304702001306.512.03.4282465-2.200.750.7116Edisto5.215125060154.12.03.2141230-8.800.350.3117Elbe_Flanders27.3703040350808.533.04.71412322.000.300.2717aElbe_Kuijper46.06645003506610.20.04.71412322.000.300.2517bElbe_Savenije43.06628803505011.70.04.61412322.000.300.2818Pangani0.91527035153.23.14.214124210.000.600.4119Linggi1.5832025133.20.52.0141230-14.000.300.3620Landak2.060230100608.70.01.6282445-6.700.600.69Less reliable sets for verification 213,4Delaware255.04137 655120426.40.01.81412550.650.220.09222,3Westerschelde150.02716 00050279.40.04.01412462.800.250.10231,2,4Pungue14.519520050192.80.06.7141231-8.500.300.22242Incomati1.14038022402.815.03.3141256-19.900.150.34252,4Solo2.1226225952269.20.01.82824313.000.600.64262,4Eems120.01931 62355193.80.03.6141231-0.700.300.11272,3Tejo100.01320 000180135.00.03.61412562.200.900.16282,4Rompin0.8110140501106.119.02.5141215-33.400.300.64292,4Ulu Sedili Besar0.73814035494.14.32.5141230-25.500.300.45301,3Gambia35.79637001101008.833.01.83141235-1.000.600.16
Note: 1 Non-steady state (NSS); 2 uncertain discharge
(UQ); 3 non-alluvial (NA); 4 information lacking (IL).
For the dispersion coefficient, eight dimensionless ratios have been
selected with 18 different types of equations including the one of
as benchmark. The dispersion
coefficient is represented in dimensionless form as
D1υ1E1=fNr1,h1a2,gC2,H1E1,h1E1,λ1E1,λ1a2,B1h1with:Nr1=Δρρgh1υ12QfTA1E1,
where Nr1 [–] is the estuarine Richardson number with
υ1 [L T-1] being the tidal velocity amplitude, both at the
inflection point. It is important to note that the values taken for E1 and
H1 in the dispersion analysis are based on the real-time data captured
during measurements and the depth is referring to the depth at the inflection
point. In general, the density difference between the saline and fresh water
is taken as (25/35)S0kgm-3 and the fresh water density as
1000 kgm-3. The fresh water discharge data were adjusted for the
seven newly surveyed estuaries so that the runoff contribution downstream of the
gauging station was also considered in the analysis. Stepwise multiple
regression analysis has been used to identify the best combination of the
dimensionless ratios in predicting K and D1. The efficiency of the
established equations was examined by comparing the correlation coefficient
R2 and the standard error SE. The predicted results calculated by the most
suitable equations were plotted against the calibrated values to evaluate
their predictive performance (see Fig. S2 in the Supplement).
Substitution of predictive equations in the salt intrusion model
Since the predictive dispersion is computed at the inflection point
x1, reverse calculation has to be done to obtain the dispersion at
the mouth. This is necessary to enable the simulation of the
longitudinal salinity distribution starting from the mouth to the salt
intrusion limit. Inverse integration from x1 to x=0 of
Eq. () yields for the dispersion at the estuary mouth:
D0TA=D1TA1+βrevTA1-exp-x1a1withβrevTA=Ka1α1TAA1andα1TA=D1TAQf,
where βrev [–] is the reversed dispersion reduction rate,
whereas A1 [L2], D1 [L2T-1] and α1
[L-1] are the cross-sectional area, dispersion coefficient and
mixing number at the inflection point, respectively. It is important to note
that the convergence length a1 [L] applied in Eqs. () and
() is of the first section of the estuary. The relation between
dispersion and salinity is then expressed by
STA-SfTAS0TA-SfTA=DTAD0TA1K0≤x≤x1STA-SfTAS1TA-SfTA=DTAD1TA1Kx>x1,
where S0 [ML-3] and S1 [ML-3] refer to the
salinity at the estuary mouth and the inflection point, respectively.
Substituting the tidally average dispersion coefficient into the
general form of the salt intrusion length of yields
LTA=x1+a2ln1β1TA+1withβ1TA=Ka2α1TAA1.
Note that all parameters used in these equations refer to the
inflection point. We obtain the salinity profile at HWS and LWS by
moving the salinity curve over E/2 in the upstream and downstream
direction. Similarly, the maximum salt intrusion length can be
obtained by shifting the intrusion length at TA in the landward direction
by half of the tidal excursion at the mouth as
LHWS=LTA+E02
and the LWS intrusion length by moving the tidal excursion seaward:
LLWS=LTA-E02.
Data
Data were divided into two categories: reliable and less reliable. There are
47 measurements grouped under the reliable data set, and 38 measurements under
the less reliable data set (see Table S2 and S3 in the Supplement). This
distinction was made based on the following criteria.
Criteria for classifying estuaries as reliable:
the estuary is generally in steady-state condition;
the fresh water discharge is estimated, observed or measured
correctly;
the estuary is alluvial and undisturbed;
complete measurement data for tidal dynamics and salinity
analysis are available.
Criteria for classifying estuaries as less reliable:
The estuary is not in steady state particularly during low river
discharge. This depends on the ratio of the timescale of system response to
the timescale of discharge reduction see (NSS).
The estimation of the fresh water discharge is uncertain (UQ).
The estuary may not be alluvial (e.g. dredged, modified or
constricted by rocky banks) (NA).
Information on tidal dynamics and salinity is lacking or unclear
(IL).
The estuaries that fall under category NSS, UQ, NA and IL are listed
in Table . It is worth noting that only the reliable
set is used in regression analysis. The less reliable ones are merely
plotted for verification purpose.
Performance of the predictive equation for the Van der Burgh
coefficient against the calibrated values.
Results and analysisPredictive equation for the Van der Burgh coefficient K
Results from the stepwise multiple regression analyses show that the
best combinations of the dimensionless ratios to represent the Van der
Burgh predictive equation are
K=8.03×10-6BfB10.30gC20.09E1H10.97h‾1b20.11H1h‾11.10λ1E11.68
or
K=151.35×10-6Bf0.30H10.13T0.97B10.30C0.18υ10.71b20.11h‾10.15rs0.84,
where Eq. () is the simplified form. The correlation
coefficient R2 and the standard error SE obtained for predictive
K equation is 0.72 and 0.11, respectively. If we had used the
cross-sectional area convergence a2 instead of b2, then the
correlation would be slightly poorer. Hence the width convergence is
a better indicator, which is fortunate because it is easier to
determine. From the equation, we can see that the parameters that have
the most influence on the Van der Burgh coefficient is the tidal
period, storage width ratio and tidal velocity amplitude which have the
power of 0.93, 0.84 and 0.71, respectively. The importance of the friction
appears to be minor, which is also fortunate as C is not directly
observable. The estuary to river width ratio shows higher power than the
convergence length, which indicates that the width is a better shape
indicator. Result from the regression is shown in Table S4 (supplementary
material). Finally, we should realize that 0≤K≤1 according
to . For prismatic channels where b2 becomes
infinity, K approaches zero, implying constant dispersion.
Figure shows the plot of the predicted K against the
calibrated value. All the reliable data points appear to fall close to
the perfect agreement line. About half the unreliable data points were
outliers particularly the Gambia (30) and Tejo (27) estuaries which lie
much further away from the perfect agreement line. This is not strange
in the sense that the Tejo Estuary is not entirely alluvial, and its
narrow and deep mouth caused by a rock outcrop formation turns it into
a fjord type estuary. As for the Gambia, it is an unsteady-state
estuary. Nevertheless, for the rest of the outliers we believe that
they would fit better if good data had been available. The results are
summarized in Table .
Predictive equation for the dispersion coefficient D
In this study, 18 combinations of the dimensionless ratios were established
by a multiple regression method of which the results are displayed in Table S1
(equations) and Fig. S2 in the Supplement (correlations and standard error).
By observing the exponent, it can be seen that the power of the estuarine
Richardson number Nr varies little, indicating the clear
correlation with Nr compared to the other parameters. The next
parameter that has a high exponent is the dimensionless roughness, of which
the inclusion improves the correlation. As for the rest of the dimensionless
ratios, it appears that the contribution is minimal. Hence, the best
equations chosen for further analysis are
D1TAυ1E1=0.1167Nr0.57D1TAυ1E1=0.3958Nr0.57gC20.21D1TAυ1E1=1.9474Nr⋅gC20.51.
More information about the equations tested is provided in the Supplement
(Table S1). Equations (), () and ()
correspond with Eqs. (R2), (R4) and (R9) in Table S1.
It is interesting to note that the performance of the benchmark equation of
(Eq. R1) is rather poor, with R2 and
SE of 0.67 and 0.33. These significant differences may be caused by the
homogenization of the input information (e.g. geometry), and the use of
selective data for calibration. With more or less equal performance, it is
decided that the simplest equation with the best performance is the most
attractive one. Therefore, we conclude that Eq. () is the best to
predict the tidal average dispersion coefficient at x1. This is also
theoretically the most attractive, since laboratory experiments of the WES
flume , Delft flume
and have
demonstrated that both Nr and the roughness are key parameters.
Nevertheless, if the Chezy roughness is unknown, then Eq. () can
be applied. Equations (), () and () have
an R2 of 0.84, 0.86 and 0.80 with SE of 0.14, 0.13 and 0.15, respectively.
We can also conclude that although estuary shape is the key in defining K,
the dispersion boundary condition D1 appears to be determined by hydraulic
parameters. The detailed results obtained from the regressions are shown in
Table S5 (Supplement).
Performance of the predictive equations for the dispersion
coefficient (left panel) and mixing number (right panel) against calibrated
values.
Comparison between predicted and calibrated maximum salt
intrusion LHWS for Eqs. (43), (44) and (45).
Figure displays the plots of the predicted D1 and α1
against the calibrated values for both the reliable and less reliable
data sets using Eqs. (), () and (). Here,
it is shown that all the reliable data points fall nicely within the range of
a factor 1.5. Some of the less reliable data points are also within or near
the range except for several obvious outliers such as the Delaware, Schelde,
Pungue and Tejo. This is because the Pungue is often in an unsteady-state
condition, while the Schelde is dredged, and the Tejo and Delaware are not
completely alluvial. In addition, the doubt on the accuracy of the discharge
data is also one of the factors contributing to poor results. It can be seen
that all the predictive equations selected have underestimated the values of
the dispersion coefficient for the outlying data points, indicating
a possible underestimation of the river discharge.
Comparing the outliers in both plots, it appears that the unreliable data are
distributed closer to the reference lines if the dispersion is represented in
term of the mixing number. This implies that the fresh water discharge is
partly to blame for the discrepancy. The data used for the regression and
results of the predicted dispersion are tabulated in Table S2 in the
Supplement.
Calibrated (solid lines) and predicted (dashed lines) salinity
curves compared to observations (symbols) for HWS, TA and LWS in the seven newly
surveyed Malaysian estuaries.
Modified predictive equation for maximum salt intrusion length LHWS
Comparison between the predicted and calibrated salt intrusion length has
been done for HWS condition instead of TA. This is because the salt intrudes
furthest into the river system at HWS, and the maximum intrusion is the
information water managers are most interested in. Substituting the
predictive dispersion Eqs. (), () and ()
into the general form for salt intrusion length yields
LHWS=x1+a2ln0.1167E1υ1Ka2u1Nr0.57+1+E02LHWS=x1+a2ln0.3958E1υ1g0.21Ka2u1C0.42Nr0.57+1+E02LHWS=x1+a2ln1.9474E1υ1g0.51Ka2u1C1.02Nr0.51+1+E02,
where u1 represents the driver flow velocity at the inflection point.
Figure shows the performance of these equations in predicting
the maximum salt intrusion length. In the plots using Eqs. (),
(), and (), all data points fall within the range of
factor 1.5 except the Solo Estuary. The list of data and the results are
summarized in Table S3 in the Supplement. It appears that the predictive
equations overestimated the intrusion length in the Solo Estuary. This may be
due to the nearly prismatic shape of the channel which has a very long
convergence length of 226 km.
Longitudinal salinity profiles
The salinity curve can be computed by applying Eqs. () and
() with the different dispersion calculated by each of the
predictive measures developed. Considering the substantial amount of salinity
measurements available, only the salinity profiles of the seven newly surveyed
estuaries are discussed. Plots of all salinity profiles are available
as electronic material at the website http://salinityandtides.com.
Figure demonstrates the performance of the simulated
longitudinal salinity distribution calibrated against the measurement data,
and the salinity profile obtained from the predictive equations of K and
D1.
From the salinity curve comparison, it appears that all the predictive
equations did not perform very well for Kurau and Bernam estuaries. This may
be caused by the uncertainty in discharge data. The Kurau and Bernam
discharge calculations were based on the discharge observed in a small part of
the catchments of about 12 and 20 % of the total area, respectively
. Thus, it is possible that we may have
underestimated the discharge draining into the Kurau Estuary, and
overestimated the one for Bernam Estuary. It is also interesting to note that
Eq. () works better in predicting the salinity distribution for
some of the estuaries such as the Perak, Linggi and Endau. As for
most of the cases, Eq. () appears to give the best fit. The
difference in the performance of these equations suggests that there is
a possibility that the equations are subject to improvement if more reliable
measurements are available. Thus, it is appropriate to retain the three
Eqs. (), (), and () for consideration.
Discussion
Before Savenije's (1993a) effort to develop predictive equations for the Van
der Burgh and dispersion coefficient, these parameters could only be obtained
by calibration. Without site measurements, it was impossible to make any
estimate of the salinity distribution along an estuary. The predictive
equations of were able to estimate the
value of K and D reasonably well in reference to the calibration data.
However, after re-evaluating and re-analysing the available data, we found
that the equations do not work as well for all estuaries.
In this study, we have collected an additional 32 salinity profiles
from 16 new estuaries for consideration in the analysis. Moreover, the
measurements were split into two data sets to make sure that only the
reliable data were used for establishing the revised equations. In
previous work, the data were not split. The selection process is
important so that the results are not influenced by incomplete or
uncertain data. Re-examining the available measurements from the old
database ensures that all data used are accessible and consistent. The
new compilation also provides a section containing important
information about each measurement (see electronic additional material
– salinity worksheet at http://salintyandtides.com).
Another important modification in this work is the change in the
selected boundary condition. In this research, we decided to process
the cross-sectional data in reference to the TA situation, whereas previous methods were based on HWS and LWS, which led to
inconsistencies because the geometry during low and high water can be
different from TA situation. Moreover, in this study we
fixed the location of the downstream boundary at the inflection point
x1 and not at the estuary mouth (adopted by all earlier
researchers). The reasons and advantages of moving the downstream
boundary to the inflection point are:
to eliminate the difficulty of determining the exact location of
the estuary mouth;
to reduce the effect from wind and waves;
to eliminate the dilemma of which geometry parameters to use in
the predictive equation.
In Savenije's (1993a, 2005) and Kuijper and van Rijn's (2011)
predictive model, the cross-sectional area convergence length applied
to calculate the salt intrusion length was the weighted value obtained
from an iteration process. With the change of the downstream
boundary to x1, this process is no longer needed and the predictive
measures are more consistent.
The new set of dimensionless ratios proposed in this study to
establish the predictive equation for K contains mostly measurable
independent parameters. The selection was made based on the existing
equations, considering only the parameters that are easy to obtain. The ratio (1-δHb) has been removed from
the equation because the tidal damping changes from spring to neap
tide. Furthermore, the tidal level tends to increase or decrease when
it moves upstream and the changes are highly influenced by fresh water
discharge. The river to estuary width ratio has been added in the new
equation as an additional geometry indicator besides the depth and
convergence length. This ratio appeared to have great influence on K.
For the predictive dispersion equation, the ratio of the depth to the
convergence length is no longer important, but the longitudinal length
scale E and velocity amplitude υ remain important in the
scaling of D1. The elimination of h‾/a allows the new
equation to be applied also in prismatic channels. In the old
equation, when a2 approaching infinity, the calculation became
invalid. Since suggested that the friction
parameter is related to the vertical mixing, g/C2 has been included
in this new equation and it indeed improved the
correlation. did not consider roughness in his
predictive equation for dispersion.
Although some improvements and simplicity have been introduced in this
study, there are limitations in using the new equations. Until now, we
have only taken into account single-network estuaries. Furthermore, it
has implicitly been assumed that no water is entering or leaving the
tributaries in the estuary region. If there are large tributaries or
large areas draining on the estuary, then these should be accounted
for. From the plot of Van der Burg's coefficient, we found that the
performance in predicting K is rather low. This indicates that the
equation has to be used with caution. Another constraint in using the
developed equations is the friction factor. The Chezy roughness is not
directly measurable and can only be obtained by calibration using
a tidal dynamics model. However, if this information is impossible to
get, it can be neglected (the correlation only decreases to 0.70 for
the predictive equation without roughness). If cross-sectional area
information is lacking, then b1 can be used to replace a1. For
the depth estimates, one can make use of the method presented by
which links h‾1 to the bankfull
discharge.
Conclusions
Calibrating K and D1 is only possible if measurement of the
salinity distribution is available. In a situation where data are
limited, a predictive equation is required to estimate the desired
variables. A good predictive equation should be simple (parameters can
be easily measured) and efficient. The predictive equations
established in this study consist of mostly measurable independent
parameters. Options are suggested for the case in which data are very
limited. The adjustment of the downstream boundary to the inflection
point has clarified the selection of the right geometry parameters to
be used and the position of the downstream boundary.
The analysis based on tidal average conditions enables the entire
process to be carried out consistently, whereby model and data errors
can be reduced. The obtained salt intrusion can easily be converted
from TA to HWS by adding half of the tidal excursion. The performance
of the predictive equation for K is rather weak with a R2 value
of 0.72 but still acceptable. For the dispersion, the correlation of 0.86
seems very promising. All the reliable data points fall within a factor
of 1.5 for both the predicted K and D1 results. Some less reliable
ones are also within this range. This indicates that the predictive
equations developed are appropriate to be applied in getting a first
estimate of K and D1. Subsequently, the longitudinal salinity
distribution in an estuary can be estimated.
Hence, these tools can be very helpful for water managers and
engineering to make preliminary estimates on the salt intrusion in an
estuary of interest and to analyse the impact of
interventions. Finally, it is recommended to collect more reliable
measurements to strengthen the development of the empirical
relationships. New data are also required for validation purposes.
The Supplement related to this article is available online at doi:10.5194/hess-19-2791-2015-supplement.
Acknowledgements
We would like to express our gratitude to: Universiti Teknologi Malaysia
(UTM) and colleagues Huayang Cai for their invaluable support and assistance
in completing the field works in Malaysia; the Department of Irrigation and
Drainage (DID) Malaysia for providing the hydrological data; and Kees Kuijper
(Deltares) for sharing the surveyed data of the Elbe
Estuary.Edited by: A. D. Reeves
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