Knowledge of the processes governing salt intrusion in estuaries is important, since it influences the eco-environment of estuaries as well as its water resource potential in many ways. Analytical models of salinity variation offer a simple and efficient method for studying salt intrusion in estuaries. In this paper, an unsteady analytical solution is presented to predict the spatio-temporal variation in salinity in convergent estuaries. It is derived from a one-dimensional advection–diffusion equation for salinity, adopting a constant mixing coefficient and a single-frequency tidal wave, which can directly reflect the influence of the tidal motion and the interaction between the tide and runoff. The deduced analytical solution is illustrated with an application to the Humen estuary of the Pearl River Delta (PRD) and proves to be an efficient and accurate approach for predicting the salt intrusion in convergent estuaries. The unsteady analytical solution is tested against observations from six study sites to validate its capability to predict intratidal variation in salt intrusion. The results show that the proposed unsteady analytical solution can be successfully used to reproduce the spatial distribution and temporal processes governing salinity dynamics in convergent, well-mixed estuaries. The proposed method provides a quick and convenient approach for deciding on water-fetching methods to make good use of water resources.

Salt intrusion in a river connecting to the sea is largely controlled by the river flow (Keulegan, 1966). The salinity of estuary waters is the result of the balance between river and tidal fluxes and mixing between them. The natural variability in river and tidal inputs to estuaries has been greatly disrupted as a result of the impact of global climate change and sea level rise as well as of local human activities, such as dam construction and channel dredging. These changes cause salt intrusion to become a serious problem in estuaries. It influences water quality, agricultural development in lowland areas, water utilization in upstream catchments, and the aquatic environment in estuaries (Han et al., 2010; Mo et al., 2007; Savenije, 1992). To address this issue worldwide, research efforts devoted to salt intrusion have been conducted in laboratory tanks, with numerical models, and using analytical approaches.

Nowadays, numerical models have become the most popular tool for studying salinity distribution in estuaries because they can provide visible results presenting the spatio-temporal variation in detail (e.g. Gong et al., 2012; Lerczak et al., 2006; MacCready, 2004; Wu and Zhu, 2010). However, the application of a numerical model is not an easy task, since it requires detailed data of the bathymetry and of hydrological boundary conditions, which are not available for all estuaries in the world. Here, a comparatively simple and convenient analytical model is developed as an efficient method for studying the salt intrusion in well-mixed estuaries. Analytical models are widely used because they are simple yet retain the basic physical characteristics involved. In the early 1960s, when systematic methods were developed to explore the factors controlling the instantaneous longitudinal salinity distribution, an expression was developed to compute the salt intrusion length as a function of the estuary length, mean depth, tidal amplitude, tidal period, fresh-water discharge, ocean salinity, and estuary roughness (Ippen and Harleman, 1961). In a subsequent period, analytical models of increasing complexity were developed based on the one-dimensional advection diffusion equation (Cameron and Pritchard, 1963), and on two-dimensional equations (Hansen and Rattray, 1965), capturing the dynamics of buoyancy-driven exchange flow and tidal mixing, satisfying salt conservation. Since the 1970s, numerous empirical and semi-empirical one-dimensional analytical models were put forward that correlated the salt intrusion length to the estuarine dynamical conditions and geomorphology based on the flume experiments and field measurements (e.g. Brockway et al., 2006; Fischer, 1974; Gay and O'Donnell, 2007, 2009; Kuijper and Van Rijn, 2011; Lewis and Uncles, 2003; Prandle, 1981, 1985; Rigter, 1973; Savenije, 1986). Although the literature on salt intrusion is vast, most studies concentrate on salt-water intrusion in a prismatic flume for reasons of convenience. However, the majority of estuaries in the world converge in width. The topography of the estuary is crucial to salt intrusion because the two main drivers (i.e. river flow and the tidal motion) both depend on the topography. The cross-sectional area determines the amount of the salt water entering the estuary and the efficiency of fresh water carrying salt out of the estuary. Savenije (1986) developed a fully analytical and predictive model to predict salt intrusion that applies to the natural topography of alluvial estuaries. It has been validated well in numerous estuaries where the width converges exponentially (e.g. Savenije, 1989; Savenije and Pagès, 1992; Nguyen and Savenije, 2006; Eaton, 2007; Ervine et al., 2007; Nguyen et al., 2008). In the years 2000–2010, another analytical approach (Brockway et al., 2006) was put forward which can be considered to be a modified and simplified version of the method presented in earlier studies (Prandle, 1981; Savenije, 1986). The dispersion coefficient in Brockway's model is assumed to be constant along the estuary, while it is assumed to be proportional to the spatial integral of the subtidal axial velocity in Savenije's model. In the theoretical models described above, the salt intrusion is predicted as a steady-state solution during slack water. Few studies have focussed on analysing the intratidal variation in salinity analytically. Song et al. (2008) proposed an unsteady-state model applicable to laboratory flumes and artificial channels where the cross section is assumed to be constant along the channel. Elaborating on the work of Song et al. (2008), here, an unsteady-state model is developed to predict the intratidal salinity intrusion dynamics in alluvial estuaries where the cross-sectional area typically converges. The aim of this study is to introduce a simple, unsteady analytical solution to the problem of predicting the intratidal variation in salt intrusion in convergent, well-mixed estuaries.

The cross-sectional area in this paper is described as an exponential
function:

The tidal velocity amplitude

Since the tidal flow is assumed to vary as a simple harmonic wave, the
unsteady salinity model is here presented in its simplest form, with a
single frequency. As the tidal propagation celerity in the estuary is
assumed to be constant, the tidal phase at each site can be made up of an
initial phase

We note that in the approach presented above, the tidal excursion at the
mouth is inferred from salinity data, whereas an alternative theoretical
approach may be applicable that is less dependent on in situ data. Tidal-wave propagation can be described analytically by a set of four implicit
equations (Cai et al., 2012), namely the phase lag equation

The Pearl River estuary (PRE) is located midway along the northern boundary of the South China Sea. It receives a large amount of fresh water from the Pearl River, which has three major branches (i.e. the West River, the North River, and the East River) in the upper drainage basin. The annual river discharge, with 80 % occurring in the wet season, empties into the South China Sea via eight outlets (Zhao, 1990). The Lingding bay is created by the inflows of fresh water from the Pearl River through four major discharge outlets, namely Humen, Jiaomen, Hongqimen, and Hengmen. Historically, about 50 %–55 % of the river flow enters the Lingding bay, while the remaining fresh water directly flows into the South China Sea through the four southwestern outlets (i.e. Modaomen, Jitimen, Hutiaomen, and Yamen).

The Humen is the largest river outlet in the Lingding bay and contributes
34.6 % of the water discharge, i.e. about

The tide in the Pearl River estuary has a mixed semidiurnal–diurnal character (Zhang et al., 2012). Among the eight outlets of the Pearl River estuary, Humen is most strongly dominated by the tide, with an annual average and maximum tidal range of 1.63 and 2.59 m, respectively, at the mouth of the estuary (Li and Lei, 1998).

As a major tributary of the Pearl River, the Humen estuary can be divided into two waterways: the Guangzhou channel (the upper reach), with an average width of 431 m, and the Shiziyang channel (the lower reach), which is about 2200 m wide (Mai et al., 2001). It is a NW–SE branch of the Pearl River estuary, with a width of about 4 km at the mouth, resembling an inverted funnel with a narrow neck in the north and a wide mouth opening to the south. The Humen outlet has the highest tidal prism in the Pearl River estuary due to the large width of the mouth, resulting in strong tidal motion. Especially during spring tide in the dry season, when the river discharge is lowest, the downstream area becomes saline.

The information available for the model application in this study includes data on topography, salinity, river discharge, and the tidal flow. A field survey for salt intrusion was conducted during the dry season in 2005. It was a project carried out by Guangdong Province Hydrology Bureau and the Pearl Hydrology Bureau from the River Conservancy Commission. In this paper, the field data from 29 January to 3 February were used, which were measured at six gauge stations along the channel (Fig. 1). Considering the impact of shipping, the measuring positions were near the banks, with certain distances ranging from 605 to 70 m. A Global Positioning System was used to confirm the exact measuring locations (Table 1). The Humen estuary is well-mixed under normal flow conditions during the dry season (Ou, 2009; Luo et al., 2010). Due to 3 years of drought, the river discharge decreased by 50 % during the study period in 2005 compared to a normal year (Liao et al., 2008). Thus, there is no doubt that well-mixed conditions prevailed during the calibration and validation. The average salinity of vertical profiles was calculated based on the hourly water samples. At each location, the saline water was sampled at two different elevations: at one-fifth and four-fifths of the depth of channel from the bed, and salinity was obtained using a salimeter. The water discharge at stations was provided by the hydrology bureaus during the field survey. The cross section was measured at mean sea level, with the help of an ultrasonic echo sounder.

General information of hydrological stations in the Humen waterway.

Map of the Humen estuary, showing the gauging stations where salinity concentration was measured during the field survey from 29 January to 3 February 2005.

Because of the complex river network upstream of the Humen area in the Pearl
River estuary, the river discharge is difficult to determine. The total flux
through the Humen outlet is composed of three parts which come from three
main sources: the East River, the North River, and the Liuxihe River. The river
discharge used in this paper was measured at upstream stations (Sanshui for
the North River, Boluo for the East River, and Laoyagang for the Liuxihe River)
from 29 January to 3 February. These data were collected from the official
databases of the Hydrology Bureaus mentioned above. In the lower reach of
the East River and the Liuxihe River, respectively, the Boluo station and
Laoyagang station are located about 80 km upstream from the Humen outlet.
The daily discharge measured at the Boluo station varied from 260 to 400 m

To demonstrate the practical application of the proposed analytical solution, the model has been used to simulate and analyse the spatio-temporal variation in salt intrusion in the Humen estuary. In the following, the parameters of the analytical solution are obtained from calibration.

The spatial decay of the cross-sectional area of the Humen estuary can be
described by the exponential function expressed in Eq. (1). The field data
(triangles) and the best-fit line are shown in Fig. 2. The cross-sectional
area at the mouth at mean tide,

Shape of the Humen estuary, showing the correlation between the
cross-sectional area

The relative salinity is plotted as

Relative salinity concentration along the Humen estuary. The
circlers represent observations, and the lines represent the fit to Eq. (12).

Dispersion coefficient of salt intrusion in Humen estuary.

The tidal excursion at the mouth of the estuary is obtained through Eq. (18). For each tidal excursion at each day, the period-averaged value,
maximum value, and minimum value of salinity at the mouth are obtained by
statistical analysis, and the longitudinal dispersion coefficient

The six calibration parameters (i.e. the convergence length of cross
section

Calibrated values of parameters.

Comparison between calibration results and measured salinity concentration along the river on 29 January 2005, showing values of measured salinity at high water slack (circles) and low water slack (inverted triangles) and the calibrated salinity curves at high water slack (red curve) and low water slack (blue curve).

A validation of the unsteady model is offered in two separate parts, i.e. the longitudinal distribution of salinity along the channel and the temporal variation in salinity during the tidal period. In the first part, observations during two characteristic conditions (i.e. HWS and LWS) are chosen for comparison against the calculated results of the salinity distribution. In the second part, a model for expressing the change process of salinity during tidal periods is established, according to the measurement on 31 January.

Based on the field measurements from 30 January to 3 February, Eqs. (15), (16), and (18) are used to calculate the longitudinal variation in salinity. Conditions of neap tide are considered to last from 31 January to 2 February. The calibration results are presented in Fig. 5. The good agreement between the computation and the measured data indicates that the performance of the unsteady analytical model is to a certain extent satisfactory in the Humen estuary. The analytical model is found to better reproduce the distribution of salinity at high water (HW) than at low water (LW). This can be attributed to different degrees of mixing, which is stronger at HW. As the estuary is assumed to be well-mixed, the analytical model undoubtedly will perform better when mixing is higher. Fluctuations around the theoretical curve may partly be caused by the unequal distribution of salinity over the cross section or by the indirect derivation of the salinity at HWS and LWS, which is replaced with the daily maximum and minimum values, respectively.

Comparison between validation result and measured salinity concentration along the river from 30 January to 3 February 2005.

It can be seen that the analytical model substantially overestimates the
salinity in the downstream part of the estuary partly because of the
special locations of the stations (some are located at the confluence of
rivers). The expression for the distribution analysis of salinity, Eq. (11),
is multiplied by the tidal average salinity with an extra component that
reflects the effect of the tide and the interaction of the tide and river
flow. This time-dependent component is a sine function, namely

The observations of salinity at hourly intervals along the Humen estuary are
used to calibrate the dispersion coefficient in the model and to analyse
the change of salinity with time. The results indicate that the calibrated
unsteady analytical model fits the observations well. Figure 6, where the
analytical solution is compared with observation, demonstrates that the
proposed unsteady analytical solution is able to reflect the change process
of salinity over a tidal cycle. Additionally, the simplification and
assumption of the tidal celerity (

Comparison between the predicted and measured salinity
concentration over time on 31 January (neap tide) at each study site,
showing that the analytical model captures the temporal variation in
salinity. The hourly salinity measurements are represented by rectangles,
while the simulated salinity varying with time is represented by the red
solid line. In the figure,

The theoretical results of the periodic variation in salinity are not always consistent with the observations. As can be seen in Fig. 6, the analytical model for simulating the temporal process of salinity has a relatively poor performance at the sites near the mouth of estuary, such as the Dahu station. By comparing the variation in salinity at different sites (Fig. 6), it shows that salinity variation is more symmetrical further away from the study site. The discrepancies near the mouth may have three reasons. Firstly, lateral residual circulation usually exists at the mouth of an estuary, where the cross section is widest. Secondly, the mouth of estuary is close to Lingding bay, where the salt dynamics are influenced by coastal and ocean currents. Thirdly, near the outlet, comprehensive salinity measurements are much more difficult to take due to the impact of tidal flats and complex hydrodynamics, influenced by the Coriolis force and wind effects. All the influences above are related to the width of the channel, which gradually decreases in the landward direction.

The observations at the Machong station show some non-periodic variation, which may relate to the proximity of the confluence of the East River and the Shiziyang channel. At the Dasheng station, about 2.6 km upstream from the Machong station and near another confluence, the simulated temporal process of salinity shows fairly good agreement with the observations. To understand the irregular changes of salinity at the Machong station, the daily averaged discharges at the Machong and Dasheng stations are analysed by integrating over the tidal period. The results are presented in Fig. 7, where the positive values represent the mean discharge transporting in the seaward direction. At the Machong station, the mean discharge is directed inland, which can be attributed to Stokes transport (Buschman et al., 2010; Hoitink and Jay, 2016). At the Dasheng station, only a few kilometres upstream, the mean discharge is seaward, as expected. The tide-averaged discharge thus converges in the estuary during low river flow, which will increase the total water volume in the estuary, and create a mean water level rise. We expect that this process has an impact on the mean salt balance, which explains part of the observed discrepancies.

Subtidal discharge measured at Machong station and Dasheng station from 29 January to 3 February. Positive values indicate discharge in the seaward direction.

For comparison, the result obtained by Song's model is also presented here.
The unsteady analytical model developed by Song et al. (2008) can reproduce
the salinity process in an idealized estuary with constant depth and
constant width, which is expressed as

Comparison between observed and computed salinity concentration over time on 31 January (neap tide) at study sites along the Humen estuary.

Calibration results of Song's model.

The amplitude of salinity can be described by

Three constant discharge values of 200, 600, and 1800 m

The tidal effect is studied using three different tidal excursions. The
tidal excursion values result in the plots that are shown in Fig. 10. The
longitudinal salinity distribution at tidal average conditions is
independent of the tidal excursion, as can be seen in Fig. 10a. From Eq. (23), since the salinity amplitude coefficient

In estuaries, it is noticed that the maximum salinity appears after HWS and
the minimum salinity appears before LWS. However, often, the salinity at HWS
and LWS corresponds approximately to the maximum and minimum salinity,
respectively. The accuracy of this approximation cannot be inferred from
existing steady-state models for salt intrusion, as time variation is
neglected. As shown in Fig. 11, the unsteady analytical solution proposed
in this paper demonstrates that the phase lag between tidal velocity and
salinity transportation is

Salinity and tidal flow velocity over a tidal cycle at Huangpuyou station. The measured salinity is represented by triangles, and the measured flow velocity is indicated by circles (on 31 January 2005). The dashed line is the calculated tidal velocity, while the dashed–dotted line is the total velocity of tidal flow and river flow. The red solid curve represents salinity simulated by the unsteady analytical solution, which reproduces the time lag HWS and maximum salinity.

More generally, Eq. (14) offers a simple expression yielding qualitative insight into the role of the river discharge in the spatio-temporal variation in salinity in a well-mixed estuary. The time lag between salinity extremes and slack water is determined by the strength of the river flow in a way that is consistent with the previous observations in which the maximum salinity appears after HWS and the minimum salinity appears before LWS. The estimated river flow velocity at the Huangpuyou station is about one-sixth of the tidal flow amplitude, resulting in a time lag between HW (at maximum salinity) and HWS (when total velocity is zero) of less than 30 min. At this station, it is acceptable to assume that the salinity reaches the maximum value at HWS and the minimum value at LWS.

Estuaries are crucial feeding and breeding grounds for many life forms and are a source of drinking water. Intrusion of salt water can temporarily halt the production of drinking water and put stress on plant and animal species that have adapted to the typical salt concentrations along the estuary. In China, a value of 0.5 ‰ salinity is considered to be the upper limit of drinking water (SWEQ PRC, 2002), while turbot farmed in man-made ponds needs to live in water with no less than 12 ‰ salinity. The unsteady solution proposed in this paper shows reproducing the intratidal variation in salt intrusion, which allows estimating the window of opportunity for drinking-water intake and has the potential of application in aquaculture and water-fetching methods in estuaries.

Due to the serious increase in salt intrusion in recent years, the water intake from Humen estuary is more suitable for saline-water aquaculture than residential use. However, the salinity along the estuarine channel is changing all the time according to the variations in the tides as well as the fresh-water discharge. This makes it important to capture the temporal variation in salinity for optimizing the water intake of the man-made ponds around the estuary. The analytical model proposed in this study provides a simple and efficient approach for predicting the variation in salinity, which is economical and practical, with the limited amount of data available.

Close to the Sishengwei station, there is an aquaculture area with many man-made
ponds of different sizes. Optimizing water intake is a key issue here. The
applicability of the analytical model is illustrated by focussing on turbot
farming, which requires salinity of no less than 12 ‰.
The observed salinity data on 29 January are used to calibrate the model,
where the determination of three parameters is needed, i.e. tide-averaged
salinity at mouth

Time for water intake of given salinity that is higher than
12 ‰.

Since the fresh-water discharge influences the slope (Brockway et al., 2012), it is reasonable to assume that the slope remains constant in a short timescale, since the fresh-water discharge variation has a timescale of days to months (Fig. 12a). The tidal excursion is the integral over time of the tidal velocity between the low water slack and high water slack. It varies from day to day as the tidal wave changes from neap tide to spring tide (Savenije, 2005). Therefore, the tidal excursion is assumed to be independent of time in the neap cycle from 29 January to 3 February. Besides, Eq. (18) is demonstrated to be a useful equation for the calculation of the tidal excursion, which offers an approach for estimating the tidal excursion with salinity data. The predicted salinity fits well with observed values, indicating that the estimation of the tide-averaged salinity during the neap tide is acceptable. However, the prediction accuracy of the model can be higher if more observed tide-averaged salinity data are available.

An unsteady-state analytical solution of salt intrusion is proposed based on the one-dimensional advection–diffusion equation for salinity, assuming a harmonic tidal wave with a single-frequency and a constant mixing coefficient. The predictive skill of the model has been illustrated from an application to the Humen estuary, which shows that it can offer an efficient approach for calculating the variation in salinity in a well-mixed estuary where the channel area is convergent. The results show that the analytical model is able to reproduce the intratidal variation in salt intrusion and can be a useful tool for computing the time windows in which salinity remains below a critical threshold in an estuary.

All the research data have been deposited in the public data repository “4TU.Centre for Research Data” (

YX and WZ formulated the overarching research goals and aims. YX, AJFH, and WZ contributed to the development of the methodology. YX, AJFH, JZ, and WZ discussed and interpreted the results. YX created the figures and wrote the original draft. AJFH, JZ, KK, and WZ reviewed and edited the draft.

The authors declare that they have no conflict of interest.

This work was supported by the National Key R&D Program of China (grant nos. 2017YFC0405900), the Fundamental Research Funds for the Central Universities (grant nos. 2018B56214, 2017B21514, and 2018B13114), the National Natural Science Foundation of China (grant nos. 41676078 and 41506100), the Open Foundation of Key Laboratory of Coastal Disasters and Defense of the Ministry of Education (grant nos. 201704), the China Postdoctoral Science Foundation (grant nos. 2017M621611), the Open Research Foundation of Key Laboratory of the Pearl River Estuarine Dynamics and Associated Process Regulation of the Ministry of Water Resources (grant nos. 2018KJ05 and 2017KJ04), and the Six Talent Peaks Project of Jiangsu Province (grant no. XXRJ-008). We thank the editor, Hubert H. G. Savenije, as well as Huayang Cai and an anonymous reviewer for constructive comments on the initial draft of this paper.

This research has been supported by the National Key R&D Program of China (grant no. 2017YFC0405900), the Fundamental Research Funds for the Central Universities (grant nos. 2018B56214, 2017B21514 and 2018B13114), the National Natural Science Foundation of China (grant nos. 41676078 and 41506100), the Open Foundation of Key Laboratory of Coastal Disasters and Defense of the Ministry of Education (grant no. 201704), the China Postdoctoral Science Foundation (grant no. 2017M621611), the Open Research Foundation of Key Laboratory of the Pearl River Estuarine Dynamics and Associated Process Regulation of the Ministry of Water Resources (grant nos. 2018KJ05 and 2017KJ04), and the Six Talent Peaks Project of Jiangsu Province (grant no. XXRJ-008).

This paper was edited by Hubert H. G. Savenije and reviewed by Huayang Cai and one anonymous referee.