HESSHydrology and Earth System SciencesHESSHydrol. Earth Syst. Sci.1607-7938Copernicus PublicationsGöttingen, Germany10.5194/hess-21-3287-2017The physics behind Van der Burgh's empirical equation, providing a new predictive equation for salinity intrusion in estuariesZhangZhilinz.zhang-5@tudelft.nlSavenijeHubert H. G.https://orcid.org/0000-0002-2234-7203Department of Water Management, Faculty of Civil Engineering and
Geosciences, Delft University of Technology, Delft, the
NetherlandsZhilin Zhang (z.zhang-5@tudelft.nl)4July20172173287330520October201615November201612May20176June2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://hess.copernicus.org/articles/21/3287/2017/hess-21-3287-2017.htmlThe full text article is available as a PDF file from https://hess.copernicus.org/articles/21/3287/2017/hess-21-3287-2017.pdf
The practical value of the surprisingly simple Van der Burgh equation in
predicting saline water intrusion in alluvial estuaries is well documented,
but the physical foundation of the equation is still weak. In this paper we
provide a connection between the empirical equation and the theoretical
literature, leading to a theoretical range of Van der Burgh's coefficient of
1/2<K<2/3 for density-driven mixing which falls within the feasible
range of 0<K<1. In addition, we developed a one-dimensional predictive
equation for the dispersion of salinity as a function of local hydraulic
parameters that can vary along the estuary axis, including mixing due to
tide-driven residual circulation. This type of mixing is relevant in the
wider part of alluvial estuaries where preferential ebb and flood channels
appear. Subsequently, this dispersion equation is combined with the salt
balance equation to obtain a new predictive analytical equation for the
longitudinal salinity distribution. Finally, the new equation was tested and
applied to a large database of observations in alluvial estuaries, whereby
the calibrated K values appeared to correspond well to the theoretical
range.
Introduction
Estuaries play an essential role in the human–earth system, affecting fresh
water resources, the mixing between ocean and river water, and the health of
aquatic ecosystems. This makes the functioning of estuarine systems an
important field of research. A crucial element of estuarine dynamics is the
interaction between saline and fresh water. The river discharges fresh water
into estuaries, flushing out the salt, while saline water penetrates landward
as a result of density gradients. The temporal and spatial distribution of
salinity in an estuary is determined by the competition between fresh water
flushing and penetration of saline water by gravity.
Dispersion is the mathematical reflection of the spreading of a substance
(e.g., salinity s) through a fluid as a function of a gradient in the
concentration of the substance (e.g., the salinity gradient ds/ dx).
Hence, dispersion is the mathematical description of mixing. The physical
process driving dispersion differs at different scales, depending on the
dominant mechanism. For instance, at the molecular scale, the dominant
mechanism is the Brownian movement of water molecules. At the scale of river
flow, the process is driven by the transfer of friction from the riverbed
into the cross section through turbulence. At this scale, the dispersion
coefficient is called hydraulic eddy viscosity (KE) . The
most important type of mixing in estuaries is the result of salinity
gradients and the non-concurrence of the velocity and salinity field
(u′s′‾) , which is the result of
gravitational and tidal mixing processes. Finally, there is mixing by
residual circulation, driven by the tide, where ebb and flood flows of
different densities mix e.g.,.
The dispersion resulting from density gradients is closely connected to the
stratification number NR, which is the balance between the
potential energy resulting from the buoyancy of fresh water flowing into the
estuary and the kinetic energy of the tide that provides the energy of
mixing. This stratification number, also known as the estuarine Richardson
number, is widely used in theoretical and practical studies
e.g.,. If NR
is large, potential energy of river discharge dominates and stratification
occurs; if NR is small, the estuary is well mixed due to
sufficient kinetic energy to reduce the density gradient.
developed a purely empirical method with excellent
practical performance e.g.,, combining into one
equation the effects of all mixing mechanisms. However, the physical meaning
of Van der Burgh's coefficient K is still unknown. Starting from this
equation, the dispersion coefficient D can be shown to be proportional to
the salinity gradient to the power of K/(1-K). The
literature presents different values for this power. Transferring these back
with this relationship to Van der Burgh's coefficient, we found a theoretical
value of 1/2, of 1
, a series of 0, 1/2 and 2/3 or an empirical range of 0.20–0.75
. This article aims to provide a theoretical
background for this coefficient.
Traditionally, researchers focused on vertical/longitudinal dispersion in
prismatic estuaries or cross-sectional varying estuaries
. concluded that the
lateral gravitational circulation is dominant over the sum of vertical
oscillatory shear, net vertical circulation and lateral oscillatory shear.
also stated the importance of lateral circulation to the
momentum budget in estuaries, but they used straight and prismatic channels,
whereas the fact that the cross sections of natural alluvial estuaries obey
an exponential function is relevant. In addition, almost all researchers
split up dispersion into its components by decomposed salinity and velocity
e.g.,.
Moreover, several researchers determined the dispersion based on a downstream
boundary , instead of calculating
local dispersion on the basis of local hydraulic variables, as done in this
research.
Although the processes of mixing and saline water intrusion are clearly
complex and three-dimensional, it is remarkable that a very simple, empirical
and one-dimensional approach, such as Van der Burgh's relationship, has
yielded such surprisingly good results. This paper tries to bridge the gap
between the theoretical approaches developed in the literature and the
empirical results obtained with Van der Burgh's relationship, considering the
complex interaction between tide, geometry, salinity and fresh water that
govern dispersion in alluvial estuaries. In addition, we present a
one-dimensional general dispersion equation for convergent estuaries that
includes lateral exchange through preferential ebb and flood channels, using
local tidal and geometrical parameters. This equation was validated on a
broad database of salinity distributions in alluvial estuaries.
Linking Van der Burgh to the traditional literature
The one-dimensional mass-conservation equation averaged over the cross
section and over a tidal cycle can be written as
e.g.,A∂s∂t-|Qf|∂s∂x-∂∂xDA∂s∂x=0
where A=Bh is the cross-sectional area, B is the width, h is the depth,
s is the cross-sectional average salinity, t is time, Qf is the fresh
water discharge, x is the distance from the estuary mouth and D is the
effective longitudinal dispersion coefficient. The positive direction of flow
is in the upstream direction.
At steady state, where ∂s/∂t=0, using the boundary
condition at x→∞, s=sf and ∂s/∂x=0,
integration yields
-|Qf|As-sf=Ddsdx
where sf is the fresh water salinity, usually close to zero.
found an empirical equation describing the tidal
average longitudinal variation of the effective dispersion:
dDdx=-K|Qf|A,
where the dimensionless coefficient K∈(0,1) according to
.
Combining Eqs. () and () yields
DD1=ss1K
where D1 and s1 are the dispersion coefficient and salinity at a given
point x1, generally taken at the inflection point in the exponential
estuary geometry. This equation is special in that it links the dispersion to
the salinity instead of the salinity gradient, as most other researchers do
e.g.,.
Interestingly, using Eqs. () and () we
can derive the dispersion as a function of the salinity gradient
:
DD1=-AD1|Qf|s1dsdxK1-K,
which connects the dispersion coefficient to local variables (A,
ds/ dx), boundary conditions (D1, s1) and K.
derived an equation for the exchange
term theoretically:
u′s′‾-KHdsdx=m1h2uf2KS+KH-dsdx+m2gcsh5ufKSKE-dsdx2+m3g2cs2h8KSKE2-dsdx3=-Ddsdx,
where u′s′‾ is the tidal average and width average exchange flow
salt flux, u′ is the depth-varying velocity, s′ is the depth-varying
salinity, KH is the along-channel diffusion coefficient,
m1=2105, m2=19420×48 and
m3=19630×482 are constant values following MacCready's
vertical integration, uf=|Qf|/A is the depth-averaged
velocity of fresh water, KS is the effective vertical eddy diffusivity,
g is the gravity acceleration, cs is the saline expansivity
equal to 7.7×10-4, and KE is the effective hydraulic
eddy viscosity. For the latter, we use the equation KE=0.12πu∗h, with u∗=gCυ as
the shear velocity in relation to the tidal velocity amplitude υ (=
πET; E is tidal excursion length; T is tidal period), where
C=Kmh1/6 is the coefficient of Chézy, and Km
is Manning's coefficient. Comparing the salt balance equation of MacCready to
Eq. () implies that Eq. () is identical
to -Ddsdx. MacCready assumed the estuary to
be narrow and rectangular, in the sense that cross-sectional shape does not
basically modify the width-averaged dynamics. In the derivation, he also
assumed the effective vertical eddy viscosity to be constant with depth,
following , and that the salinity gradient of the
depth-varying part is much smaller than the depth-averaged part, following
. Additionally, other effects like salt storage,
internal hydraulics and the Coriolis force were considered negligible.
After division of all terms by the salinity gradient, it becomes an equation
for the dispersion coefficient D:
D=m1h2uf2KS+KH+m2gcsh5ufKSKE-dsdx+m3g2cs2h8KSKE2-dsdx2,
whereby the first term is not dependent on the salinity gradient, the second
is directly proportional to it, and the third term depends on the square of
the salinity gradient.
Based on Eq. () we can also derive an expression for the
dispersion:
D=D1A1D1l|Qf|K1-K-AA1ls1dsdxK1-K,
where A1 is the cross-sectional area at the inflection point (at x=x1),
l=L-x1 is the distance from the inflection point to where salinity becomes
the same as the fresh water salinity, and L is the total intrusion length.
Hence D∝γK1-K with γ=-AA1ls1dsdx. Given the
function Fγ=γK1-K, a Taylor series expansion near γ=1 can be derived as
Fγ=(2K-1)(3K-2)2(1-K)2+K(2-3K)1-K2AA1ls1-dsdx+K2K-121-K2AA1ls12-dsdx2+R2x,
where R2(x) is the residual term, considered to be small. To
analyze the
importance of the different terms of Eq. (),
Fig. presents the factors g1=(2K-1)(3K-2)2(1-K)2,
g2=K(2-3K)1-K2 and
g3=K2K-121-K2.
g1 is the closure term which compensates for g2 and g3 so as to make
∑gi=1 (i=1,2,3). It is clear that the absolute value of the first
term is much smaller than the density-driven terms. Also, the larger the
value of K, the more important the third term is. This is in accordance
with traditional literature. If K=1/2, F(γ)=AA1ls1-dsdx, and dispersion is
proportional to the salinity gradient. If K=2/3, F(γ)=AA1ls12-dsdx2, then dispersion is proportional to the square of the salinity
gradient, which means that the dispersion is more sensitive to the salinity
gradient.
Comparison between the factors in the Taylor series expansion of F(γ) as a function of the Van der Burgh coefficient K.
Considering only the density-dependent terms in Eqs. ()
and (), the proportionality results in
2-3K2K-1=36KE|Qf|gcsh3A1ls1=7.2E|Qf|gcsh2A1CTls1=w,
leading to an analytical expression for K:
K=2+w3+2w.
According to Eq. (10), K is not time independent, as was previously assumed
by ; rather, it is determined by the tidal excursion and
the fresh water discharge. In the case of a relatively constant discharge, a
larger tidal excursion implies less stratification, a larger value of w and
K approaching the lower limit (1/2). On the other hand, a smaller tidal
excursion implies more stratification, a smaller value of w and K
approaching the higher limit (2/3), which corresponds to the situation
where the dispersion is more sensitive to the salinity gradient. We have used
this expression to compute K values in 18 real estuaries using the database
of . These K values are in a range of 0.51–0.64 (see
Sect. 4.3).
Overall, there are three results for the estimation of Van der Burgh's
coefficient: (1) by comparison with traditional studies (K=1/2 or K=2/3),
(2) by comparison with MacCready considering the salinity relevant terms
(1/2<K<2/3), and (3) based on empirical calibration (see Sect. 4). These
results are surprisingly close, even though the theoretical comparison is
limited to density-driven mixing.
Including residual circulation in wide estuaries
In the theory about mixing in estuaries, several authors have distinguished
between tide-driven and density-driven dispersion
e.g.,. The tide is an active hydraulic
driver that creates shear stresses in the flow as momentum, resulting from
friction along the boundaries, transferred to the heart of the channel by
turbulence. Generally these shear stresses reduce stratification and hence
reduce dispersion. However, at a large scale, the tide facilitates mixing by
tidal trapping and residual circulation, which enhances dispersion. Tidal
trapping results from irregularities of the channel, leading to pockets of
relatively high or low salt concentrations that later reunite with the
stream. The mixing length scale of tidal trapping is the tidal excursion. By
using the tidal excursion as the mixing length, tidal trapping can be
incorporated into a predictive equation. Residual circulation is more
complicated. It can be a very powerful tide-driven mechanism in the wider
parts of estuaries where the tide causes mixing by the cross-over of
preferential ebb and flood channels that develop in wide estuaries, such as
the Schelde, described by . But how can we parameterize
residual circulation? Here a different approach is followed from
, trying to combine this effect in the regular
one-dimensional advection–dispersion equation.
Model including residual circulation
Figure presents the sketch of a box model used to
include lateral exchange in longitudinal dispersion. Water particles in the
middle can mix longitudinally and laterally within their respective mixing
lengths. For the longitudinal mixing length we consider the tidal excursion
and for the lateral exchange half of the estuary width. The balance of the
mass can then be described as
VΔs2Δt=|Qf|(s2-s1)+d(s1-2s2+s3)+r(sL-2s2+sR)
where V=AE is the water volume, si is the salinity at different
locations i, and d and r are longitudinal and lateral exchange flows.
Conceptual sketch for lateral and longitudinal mixing.
Longitudinal and lateral mixing lengths are Δx and Δy, respectively.
The balance equation then becomes
V∂s∂t-|Qf|∂s∂xΔx-d∂2s∂x2(Δx)2-r∂2s∂y2(Δy)2=0
where Δx and Δy are the mixing lengths, which are taken as
Δx=E and Δy=B/2.
The assumption used is that the lateral exchange is proportional to the
longitudinal :
r∂2s∂y2∝d∂2s∂x2.
As a result, longitudinal and lateral processes can be combined into one
single one-dimensional equation:
∂s∂t-|Qf|A∂s∂x-dEA1+C2BE2∂2s∂x2=0
Comparing Eq. () with the traditional salt balance equation,
the effective longitudinal dispersion is
D=dEA1+C2BE2.
Subsequently, the longitudinal exchange flow d is assumed to be
proportional to the amplitude of the tidal flow (driving the circulation)
(Qt^=Aυ), and to the stratification number to the power of
K:
d=C1(NR)KQt^,
with NR defined as the ratio of potential energy of the river discharge to
the kinetic energy of the tide over a tidal period:
NR=Δρρghυ2|Qf|TAE,
where Δρ/ρ=css is the relative density difference between
river water and saline water.
The reason why the exchange flow is a function of the stratification number
to the power of K is because it is in agreement with Eq. (), Δρ/ρ being directly proportional to s.
We then obtain a simple dimensionless expression for the dispersion
coefficient, simulating to the one by but incorporating
lateral exchange flow:
DυE=C1(NR)K1+C2BE2,
where C1 and C2 are constants.
Analytical solution
In almost all estuaries, the ratio of width to excursion length is quite
small, particularly upstream where salinity intrusion happens. So for further
analytical solutions we can focus on the first part of Eq. ():
D=C1NRKυE.
The traditional approach by merely uses this equation
as the boundary condition at x=x1, after which D(x) values are obtained
by integration of Van der Burgh's equation along the estuary axis. But, in
principle, with this equation the dispersion can be calculated at any point
along the estuary, provided local hydraulic and geometric variables are
known. Equation (20) can be elaborated into
D(x)=C1csgπKs|Qf|υ3BKυE
where all local variables are now a function of x.
The following equations are used for the tidal velocity amplitude, width and
tidal excursion:
υ(x)=υ1eδυx-x1,B(x)=B1e-x-x1b,E(x)=E1eδHx-x1,
where δυ≈δH are the damping/amplifying rate of
tidal velocity amplitude and tidal range, and b is the width convergence
length (b1 downstream of the inflection point and b2 upstream).
At the inflection point, the predicted equation is given by
D1=C1csgπKs1|Qf|υ13B1Kυ1E1,
where the subscript “1” means parameters are evaluated at the inflection
point (x=x1).
Substitution of Eqs. ()–() into Eq. () gives
D(x)=D1ss1KeΩ(x-x1)
with Ω=2δH-3KδH+K/b.
Differentiating D with respect to x and using Eq. () results
in
dDdx=KDsdsdx+ΩD.
Combining the result with the time-averaged salt balance, Eq. ()
results in
dDdx=ΩD-K|Qf|A.
For a prismatic channel (b→∞) with constant width and little tidal
damping, Ω=0 and Eq. () becomes Van der Burgh's equation.
As a result, the exponent of NR in this model represents Van der
Burgh's coefficient.
The cross-sectional area A is given by
A(x)=A1e-x-x1a,
where a is the cross-sectional convergence length (a1 downstream of the
inflection point and a2 upstream).
Substitution of Eq. () into Eq. () gives
dDdx=ΩD-K|Qf|A1ex-x1a.
In analogy with , the solution of this linear
differential equation is
DD1=eΩ(x-x1)+K|Qf|A1D1ζeΩ(x-x1)-e(x-x1)/a
with ζ=a1-Ωa.
The maximum salinity intrusion length is obtained from Eq. ()
after substitution of D→0 at x=L:
L=ζlnA1D1K|Qf|ζ+1+x1.
This is the same equation as in if ζ=a.
Using Eq. (), the longitudinal salt distribution becomes
ss1=1+K|Qf|A1D1ζ1-e(x-x1)/ζ1/K.
This solution is similar to the solution by , with the
difference that Kuijper and Van Rijn used a constant value of K=0.5 and
that their value of Ω depended on the bottom slope.
So with these new analytical equations, the local dispersion and salinity can
be obtained, using the boundary condition at the inflection point. This
method is limited since it only works when B/E<1. If we want to account for
residual circulation using Eq. (), then we have to use
numerical integration of Eq. () using
Eq. () for D.
Empirical validationSummary information
Eighteen estuaries with quite
different characteristics, covering a diversity of sizes, shapes and
locations, have been selected from the database of . It
appears that all these alluvial estuaries can be schematized in one or two
segments separated by a well-defined inflection point .
As an example, Fig. shows the geometry of two
estuaries: Maputo with an inflection point and the Thames without an
inflection point. It can be seen that the natural geometry fits well on
semilogarithmic paper, indicating an exponential variation of the cross
section and width. Geometric data of all 18 estuaries are presented in
Appendix A.
Semi-logarithmic presentation of estuary geometry, comparing
simulated (lines) to observations (symbols), including cross-sectional area
(green), width (red) and depth (blue).
∗ Note (a): the estuaries with an asterisk-marked label used Km from ,
and others .
Note (b): data about Shatt al-Arab Estuary come from .
In Table the general geometry of estuaries is summarized,
where Bf is the bankfull stream width. It is obvious that these
estuaries cover a wide range of sizes. An estuary with x1=0 means there is
no inflection point. In addition, the larger the convergence length a2
(b2), the slower the cross section (width) declines upstream. With a large
b2, a relatively small value of b1 suggests the channel with a
pronounced funnel shape with fast decrease in width near the mouth. In
contrast, a relatively large value of b2 indicates estuaries with
near-prismatic shape. The same values of a and b indicate that the depth
is constant.
Tables and
contain summary information of estuaries on different measurement dates,
where H1 is the tidal range at x1, η is tidal amplitude, α=D1|Qf| is the mixing coefficient and β=Ka2|Qf|A1D1 is the dispersion reduction ratio. Tidal
excursion and tidal period are more or less the same in all estuaries, except
for Lalang and Chao Phraya with a diurnal tide. Most estuaries damp upstream,
with negative values of δH. In addition, most estuaries have a small
tidal amplitude to depth ratio, which means relatively simple solutions of
hydraulic equations are possible . K values have been
obtained by calibration of simulated salinities to observations in 18
estuaries. According to Eq. (4), the K value affects the salinity mostly in
the upstream reach, where D/D1 is small. Using an automatic solver, the
best result was obtained with C1=0.10, C2=12 and K=0.58. For
individual estuaries, K values were obtained ranging between 0.45 and 0.78.
The dispersion at the inflection point has a range of
50–600 m2 s-1 in a variety of estuaries, which is consistent with
. The mixing coefficient demonstrates to what extent the
dispersion overcomes the flushing by river flow. The larger the river
discharge, the smaller the α, meaning it is difficult for the salinity
to penetrate into the estuary. The dispersion reduction ratio determines the
longitudinal variation of dispersion. suggested that the
transition from a well-mixed to a strongly stratified estuary occurs when the
values of stratification number NR are in the range of 0.08–0.8.
With a ratio of π between Fischer's and our expressions for the
stratification number, the range becomes 0.25–2.51. It is obvious that all
estuaries are partially to well mixed, with NR below 2.51.
Comparison between simulated and observed salinity at
high water slack (thin lines) and low water slack (thick lines),
scaled by the salinity s1 at the inflection point x1 for
different C2 values. Observations at high water slack are represented
by triangles and low water slack by circles. Observe that the
Thames only has low water slack observations.
Sensitivity to C2
Through the use of C2 we can use a single dispersion equation accounting
for two-dimensional effects in a one-dimensional model. The assumption that
lateral exchange is proportional to longitudinal dispersion suggests C2 to
be independent of x. Figure and Appendix B demonstrate how
salinity changes with varying C2. Salinities were simulated by numerical
solution of Eq. () with Eq. () based
on the boundary condition at x=x1. Typically, C2 matters mainly near
the mouth, but there is almost no effect on narrow estuaries like Lalang,
Limpopo, Tha Chin and Chao Phraya. Hence, the inclusion of the residual
circulation improves the accuracy of salinity simulation in wide estuaries
and more particularly near the mouth of the estuaries where the ratio of
width to tidal excursion is relatively large.
To check the sensitivity to C2, values of 1, 10 and 50 have been used to
calculate salinity curves. It is demonstrated that the larger the value of
C2, the smaller the salinity gradient and the flatter the salinity curve
near the estuary mouth. However, because of the interdependence of D, s
and ds/dx through Eq. () in the
upstream part, a larger value of C2 can lead to larger salinities (e.g.,
Thames, Elbe, Edisto, Maputo and Corantijn). Basically, C2=10 (green
lines) can perform perfectly in 14 out of 18 estuaries (e.g., Maputo and the
Thames). We can see that larger values than C2=10 cause exaggerated
salinity in the downstream part of these estuaries, which is why a general
value of C2=10 is recommended. The poorer results occur in estuaries that
have peculiar shapes near the mouth. A larger value of C2 applies to the
Kurau. This may be because the width is underestimated in the wide estuary
mouth, due to misinterpretation of the direction of the streamline (the width
is determined according to a line perpendicular to the streamline). As a
matter of fact, the width should be larger and dispersion should be larger
with smaller salinity gradients, which would then result in a lower value of
C2. The same applies to Endau. By contrast, a smaller value of C2 in
Perak fits better, because of overestimation of the width. Here the
topographical map suggests a wider estuary mouth, whereas the tidal flow is
concentrated in a much narrower main channel due to the north bank protruding
into the estuary and a spur from the south projecting into the mouth. The
Selangor has a similar situation. It shows that the configuration of the
mouth is important for the correct simulation of the salinity near the
estuary mouth. But, fortunately, a relatively poor performance near the mouth
of these estuaries does not affect the salinity distribution upstream as long
as C2 is not too large. In conclusion, C2=10 appears to be a suitable
default value as long as the trajectory of the tidal currents can be
considered properly.
The poor fit in the downstream parts of the Lalang and Chao Phraya, in which
measured salinities are lower than simulated, can be explained by a complex
downstream boundary. The Lalang estuary has a pronounced riverine character
and is a tributary to the complex estuary system of the Banyuasin, sharing
its outfall with the large Musi River. So the salinity near its mouth is
largely affected by the Musi. Also, pockets of fresh water can decrease the
salinity near the confluence. The Chao Phraya opens to the Gulf of Thailand
where the salinity is influenced by historical discharges rather than ocean
salinity, remaining relatively fresh. Other measurement uncertainties may
cause outliers as well.
A possible solution for K
The physical meaning of Van der Burgh's coefficient has been analyzed,
linking it to traditional theoretical research. Equation ()
shows a direct relation between this coefficient and MacCready's parameters,
which are measurable quantities. Hence, the coefficient is affected by tide,
geometry and fresh water discharge. also found K values
to depend on river discharge and considered its value to increase upstream in
a range of 0–1 due to different mechanisms along the estuary.
A 1:1 plot is presented in Fig. , relating the empirical K
values to the predicted values using Eq. (). The predicted K
values have a smaller range (0.51–0.64) than the calibrated ones
(0.45–0.78). Moreover, it can be seen that there is a steep linear relation
between predictive and calibrated K values, which reveals that the
predictive method overestimates the low calibrated K values and
underestimates the high values. Fully tide-driven dispersion would correspond
to K=0. But the predictive method does not consider tidal mixing and, as a
result, the predicted K values are too high in the lower region. Smaller
calibrated values imply that the tide plays a prominent role in the estuary.
For the higher calibrated values, another explanation applies. The
theoretical approach follows width-averaged dynamics, whereas the empirical
approach relies on natural estuaries with cross-sectional variations. A K
value larger than the predicted value could result from a strong lateral
salinity gradient due to shearing in a complex geometry, which strengthens
the sensitivity to the salinity gradient. In addition, there is quite some
uncertainty in calibrating a partly empirical analytical model to data in
real estuaries, as a result of a whole range of uncertain factors related
with observational errors, data problems, the assumption of steady state and
other factors. Some estuaries may be in non-steady state (e.g., the Thames).
However, considering the K values have been obtained from different
approaches, they are still quite similar. As a result, this correspondence
forms, at least partly, a physical basis for the Van der Burgh coefficient.
All K values are very close to 0.58, which may be a good starting value
in estuaries where information on geometry and channel roughness is lacking.
Comparison between predicted and calibrated K
values. Labels are used to distinguish estuaries. The blue marks
used Km from and the red ones from
; 25 % sensitivity of fresh water discharge
is indicated by the whiskers.
Discussion and conclusion
Overall, the single one-dimensional salinity intrusion model including
residual circulation appears to work well in natural estuaries with a
diversity of geometric and tidal characteristics, by both analytical and
numerical computation. The new equation is a simple and useful tool for
analyzing local dispersion and salinity directly on the basis of local
hydraulic variables. In a calibration mode, K is the only parameter to be
calibrated using C1=0.10 and C2=10. In a predictive mode, a value of
K=0.58 can be used as a first estimate. If information on river discharge,
roughness and geometry is available, K can be determined iteratively by
taking K=0.58 as the predictor and subsequently substituting s1 and l
from the first iteration by Eqs. () and () and
repeating the procedure until the process converges.
The addition of the factor (1+C2(B/E)2) in the dispersion equation
proved valuable near the mouth of estuaries where residual circulation due to
interacting ebb and flood channels dominates dispersion. The value C2=10
was found to perform best in most estuaries, indicating that residual
circulation is dominant in wide estuaries where ebb and flood currents
prevail.
Van der Burgh's coefficient determines the way dispersion relates to the
stratification number by a power function. Two approaches, theoretical
derivation from the traditional literature and empirical validation based on
observations in a large set of estuaries, provided similar estimates of Van
der Burgh's coefficient. Under MacCready's assumptions, there are three ways
to estimate K: 0.51<K<0.64 from empirical application of
Eqs. () and (); 1/2<K<2/3 as the physical boundaries
of Eq. (); and the comparison with traditional approximations
(K=1/2 or K=2/3). After calibration of the new analytical model to the
database of field observations, the values of K were in a range of
0.45–0.78 for a wide range of conditions, with an average of 0.58, close to
the predicted values. MacCready's equation determines dispersion via a
decomposition method, using depth-varying velocity and salinity. Although
these 1-D expressions of velocity and salinity may be simplifications of
reality, the good correspondence between Van der Burgh's equation and
MacCready's theory provides a promising theoretical basis for Van der Burgh's
equation.
A previous analytical salinity intrusion model was developed by
, from which the K values resulted in a range of
0.20–0.75 by calibration and 0.22–0.71 by prediction. These solutions cover a
wider range than our estimates because of Gisen's assumption that K does
not depend on river discharge and because of three improvements made in this
paper. Firstly, we used the local hydraulic parameters to simulate the
salinities, while Gisen used a constant depth and no damping (Ω=0).
In addition, by using an uncertainty bound of 25 % on fresh water discharge
we could reduce the inaccuracy of the tail of the salinity curve and obtain a
better fit (where K matters most). And finally, all geometric analyses were
improved by revisiting the fit to observations.
An important consequence of this research is that K depends on time and
space. Where Gisen assumed K to be constant for each estuary, we find
substantial variability for estuaries where a larger range of discharges is
available: e.g. in the Maputo 0.57<K<0.70; in the Limpopo 0.61<K<0.72;
and in the Edisto 0.48<K<0.58. The implication of discharge dependence
needs to be tested further for predictive purposes.
In some particular cases, the simulated salinity with C2=10 does not fit
the observations near the estuary mouth. So one should be aware of peculiar
configurations of streamlines and geometries near the estuary mouth when
using this model. also pointed out that the effect of
irregular channel shape is important. However, a poor fit near the estuary
mouth has almost no effect on the total salinity intrusion length. It is
suggested that in future research, the assumption that lateral exchange is
proportional to longitudinal exchange needs to be tested further. Finally,
this predictive one-dimensional salinity intrusion model, having a stronger
theoretical basis, may be a useful tool in ungauged estuaries.
About the data, all observations are available on the
website at https://salinityandtides.com/.
Notation
SymbolMeaningDimensionSymbolMeaningDimensionacross-sectional convergence length(L)Qtamplitude of tidal flow(L3/T)Across-sectional area(L2)rlateral exchange flow(L3/T)bwidth convergence length(L)R2(x)residual term(–)Bwidth(L)ssalinity(M/L3)Bfbankfull stream width(L)sffresh water salinity(M/L3)Ccoefficient of Chézy(L1/2/T)s′depth-varying salinity(M/L3)Ciconstant(–)ttime(T)cssaline expansivity(–)Ttidal period(T)dlongitudinal exchange flow(L3/T)uflow velocity(L/T)Ddispersion coefficient(L2/T)u′depth varying flow velocity(L/T)Etidal excursion length(L)ufvelocity of fresh water(L/T)ggravity acceleration(L/T2)u∗shear velocity(L/T)gifactor(–)Vwater volume(L3)hdepth(L)xdistance(L)Htidal range(L)αmixing coefficient(L-1)KVan der Burgh's coefficient(–)βdispersion reduction ratio(–)KHdiffusion coefficient(L2/T)γdimensionless argument(–)KEhydraulic eddy viscosity(L2/T)δdamping/amplifying rate(L-1)KmManning's coefficient(L1/3/T)Δx,Δymixing lengths(L)KSvertical eddy diffusivity(L2/T)υtidal velocity amplitude(L/T)lintrusion length from inflection point(L)ζadjusted convergence length(L)Lintrusion length(L)ηtidal amplitude(L)miconstant(–)ρdensity of water(ML-3)NRstratification number(–)Ωadjustment parameter(L-1)Qffresh water discharge(L3/T)
Compilation of the geometry
The same as Fig. 3.
Sensitivity to C2
The same as Fig. 4.
The authors declare that they have no conflict of
interest.
Acknowledgements
The first author is financially supported for her PhD research by the China
Scholarship Council. Edited by: Insa
Neuweiler Reviewed by: two anonymous referees
ReferencesAbdullah, A. D., Gisen, J. I. A., van der Zaag, P., Savenije, H. H. G.,
Karim, U. F. A., Masih, I., and Popescu, I.: Predicting the salt water
intrusion in the Shatt al-Arab estuary using an analytical approach, Hydrol.
Earth Syst. Sci., 20, 4031–4042, 10.5194/hess-20-4031-2016, 2016.
Banas, N. S., Hickey, B. M., and MacCready, P.: Dynamics of Willapa
Bay, Washington: A highly unsteady, partially mixed estuary,
J. Phys. Oceanogr., 34, 2413–2427, 2004.Cai, H., Savenije, H. H. G., and Toffolon, M.: A new analytical
framework for assessing the effect of sea-level rise and dredging
on tidal damping in estuaries, J. Geophys. Res.-Oceans, 117, C09023 10.1029/2012JC008000, 2012.
Dyer, K. R.: (New revised issue) Estuaries: a Physical Introduction,
John Wiley & Sons Aberdeen, UK, 1997.
Fischer, H. B.: Mass transport mechanisms in partially stratified
estuaries, J. Fluid Mech., 53, 671–687, 1972.
Fischer, H. B.: Mixing and dispersion in estuaries,
Annu. Rev. Fluid Mech., 8, 107–133, 1976.
Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J.,
and Brooks, N. H.: Mixing in inland and coastal waters, Academic Press, New York, 1979.
Gisen, J. I. A.: Prediction in ungauged estuaries, Delft
University of Technology, Delft, 2015a.
Gisen, J. I. A., Savenije, H. H. G., Nijzink, R. C., and Abd Wahab, A. K.:
Testing a 1-D analytical salt intrusion model and its predictive equations
in Malaysian estuaries, Hydrolog. Sci. J., 60, 156–172, 2015b.
Hansen, D. V. and Rattray, M.: Gravitational circulation in straits and
estuaries, J. Mar. Res., 23, 104–122, 1965.
Kuijper, K. and Van Rijn, L. C.: Analytical and numerical analysis of
tides and salinities in estuaries; Part II: Salinity distributions
in prismatic and convergent tidal channels, Ocean Dynam., 61, 1743–1765, 2011.Lerczak, J. A. and Geyer, W. R.: Modeling the lateral circulation in
straight, stratified estuaries, J. Phys. Oceanogr., 34, 1410–1428, 2004.
MacCready, P.: Toward a unified theory of tidally-averaged estuarine
salinity structure, Estuaries, 2, 561–570, 2004.
MacCready, P.: Estuarine adjustment, J. Phys. Oceanogr., 37, 2133–2145, 2007.
MacCready, P. and Geyer, W. R.: Advances in estuarine physics,
Annu. Rev. Mar. Sci., 2, 35–58, 2010.
MacCready, P.: Calculating estuarine exchange flow using isohaline
coordinates, J. Phys. Oceanogr., 41, 1116–1124, 2011.
Nguyen, A. D., Savenije, H. H. G., Van der Vegen, M., and Roelvink, D.:
New analytical equation for dispersion in estuaries with a
distinct ebb-flood channel system, Estuar. Coast. Shelf S., 79, 7–16, 2008.
Prandle, D.: Salinity intrusion in estuaries, J. Phys.
Oceanogr., 11, 1311–1324, 1981.
Pritchard, D. W.: Salinity distribution and circulation in the Chesapeake
Bay Estuarine system, J. Mar. Res., 11, 106–123, 1952.Ralston, D. K. and Stacey, M. T.: Longitudinal dispersion and lateral
circulation in the intertidal Zone, J. Geophys. Res.-Oceans, 110, C07015, 10.1029/2005JC002888, 1983.
Savenije, H. H. G.: A one-dimensional model for salinity intrusion in
alluvial estuaries, J. Hydrol., 85, 87–109, 1986.
Savenije, H. H. G.: Salt intrusion model for high-water slack, low-water slack,
and mean tide on spread sheet, J. Hydrol., 107, 9–18, 1989.
Savenije, H. H. G.: Composition and driving mechanisms of longitudinal
tidal average salinity dispersion in estuaries, J. Hydrol., 144, 127–141, 1993a.
Savenije, H. H. G.: Determination of estuary parameters on basis
of Lagrangian analysis, J. Hydraul.Eng.-Asce, 119, 628–642, 1993b.
Savenije, H. H. G.: Salinity and tides in alluvial estuaries,
Elsevier, New York, 2005.Savenije, H. H. G.: Salinity and tides in alluvial estuaries,
2nd Edn., available at: http://salinityandtides.com/ (last access: 19 October 2016), 2012.
Savenije, H. H. G.: Prediction in ungauged estuaries: An integrated
theory, Water Resour. Res., 51, 2464–2476, 2015.Shaha, D. C. and Cho, Y.-K.: Determination of spatially varying Van der
Burgh's coefficient from estuarine parameter to describe salt transport in an
estuary, Hydrol. Earth Syst. Sci., 15, 1369–1377,
10.5194/hess-15-1369-2011, 2011.
Thatcher, M. L. and Najarian, T. O.: Transient hydrodynamic and salinity
simulations in the Chesapeake Bay Network, Estuaries, 6, 356–363, 1983.
Van der Burgh, P.: Ontwikkeling van een methode voor het voorspellen van
zoutverdelingen in estuaria, kanalen en zeeen, Rijkswaterstaat Rapport, 10–72, 1972.