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

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

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.

The analytical one-dimensional salinity model developed by

Van der Burgh's 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

Deriving the dimensionless dispersion coefficient from scaling the
steady-state salt balance equation,

convergent channel:

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

About 20

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

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

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

All geometry and tide information used refers to the well
identifiable inflection point

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.

Revising the parameters selected by

Data used to develop the predictive equation for the Van der Burgh
coefficient

Note:

For the dispersion coefficient, eight dimensionless ratios have been
selected with 18 different types of equations including the one of

Since the predictive dispersion is computed at the inflection point

Substituting the tidally average dispersion coefficient into the
general form of the salt intrusion length of

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

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).

Performance of the predictive equation for the Van der Burgh coefficient against the calibrated values.

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

Figure

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

It is interesting to note that the performance of the benchmark equation of

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

Figure

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.

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. (

The salinity curve can be computed by applying Eqs. (

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

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

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

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

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.

The new set of dimensionless ratios proposed in this study to
establish the predictive equation for

For the predictive dispersion equation, the ratio of the depth to the
convergence length is no longer important, but the longitudinal length
scale

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

Calibrating

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

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.

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