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 603×108 m3 in terms of annual water discharge (Ren et al., 2006). The fresh-water input into Lingding bay through the Humen outlet comes from three sources: the East River, the Liuxihe River, and the North River. The annual river discharge with a peak of 1870 m3 s-1, measured at the Niuxinling station in Liuxihe River, is about 10 times less than that with a flood peak in excess of 12 000 m3 s-1, measured in the other two rivers (Luo et al., 2002).

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.

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 m3 s-1 during the survey period, while it was about 20 m3 s-1 at the Laoyagang station. The discharge of the East River and the Liuxihe River entirely flows toward the South China Sea through the Humen estuary (Ren et al., 2011). The North River is an important source for the river discharge to the Humen outlet. River discharge from the North River reaches the Humen channel through a network of channels which connects to the western part of the Pearl River delta. The Sanshui station is the primary hydrological station in the North River. About 11 % of the measured discharge flowed into the Humen estuary during the survey in 2005. The Sanshui station is located further upstream than the other two stations (Boluo and Laoyagang). The response lag of salinity variation at the estuary mouth to discharge variation at the Sanshui station is about 2 d, while the river flow spends about 1 d to travel from the Boluo and Laoyagang station to the estuary mouth.

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, A0, is calculated as 37 822 m2, and the convergence length of cross section a is obtained by curve fitting as 16.7 km.

The relative salinity is plotted as ln⁡(s/s0) against exp⁡(x/a) in Fig. 3. There is a straight fit between these two variables, confirming the constancy of the dispersion coefficient. The dispersion coefficient D can be computed from the slope of the fitted lines according to Eq. (12), where k is the slope. This approach has been shown previously to be efficient (e.g. Brockway et al., 2006; Fang et al., 2006; Zhang et al., 2010). Table 2 shows the results of the fit for all these surveys carried out between 29 January to 3 February (the slope in column 4 and the dispersion coefficient in column 6). The dispersion coefficient estimates obtained from this fitting procedure can be interpreted as a spatial average, representing the entire reach. The coefficient of determination (R2) lies in the range between 0.85 and 0.92, with a mean value of 0.89. The assumption of the dispersion coefficient independent of distance is demonstrated to be reasonable and acceptable in the present case. The dispersion coefficient from data on 29 January is therefore used as the calibrated value.

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 D is computed by linear fitting, as shown previously. Moreover, the damping length of the tidal excursion e is calibrated using the observed salinity along the estuary. Similar to the tidal excursion, the value of the propagation celerity c is also obtained by calibration based on observations. The initial tidal phase φ0 is calculated via a reverse procedure by calibration on the salinity at the mouth of estuary. Data from 31 January are used to verify the change of salinity over a tidal cycle.

The six calibration parameters (i.e. the convergence length of cross section a, the dispersion coefficient D, the tidal excursion E0, the damping length of the tidal excursion e, the initial phase φ0, and the tidal celerity c) are obtained based on the measurements at the mouth of the estuary, as shown in Table 3. Based on the observed data on 29 January, the results of the model calibration can be seen in Fig. 4.

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.

The amplitude of salinity can be described by 23s^=s‾x⋅Is, where s‾x is the tide-averaged salinity along the estuary and is a function of the river discharge, i.e. Eq. (12). The parameter Is is the salinity amplitude coefficient that is defined as 24Is=-EQf2DA, representing the interaction between tides and the river discharge. To investigate the longitudinal salinity distribution and intratidal salinity variation for different discharge and tidal dynamic conditions in the Humen estuary, Eqs. (12) and (23) are used to plot the longitudinal salinity curve and intratidal variation in salinity, respectively. The implemented parameters are the same as shown in Table 3; only the river discharge and the tidal excursion are variable.

Three constant discharge values of 200, 600, and 1800 m3 s-1 are used to evaluate the impact of the river discharge on the salinity variation. The discharge values are chosen because the minimum discharge in the dry season is around 600 m3 s-1 in the Humen estuary, and low salinity can be measured at the Huangpuyou station when the discharge is larger than 1800 m3 s-1. In addition, the discharge in the extreme dry season is set to be 200 m3 s-1. The longitudinal salinity curve can be seen in Fig. 9a. At tidal average conditions, the salt intrusion length becomes smaller when the discharge increases. The steepest salinity gradient can be found at the highest discharge (Qf=1800 m3 s-1). It is clear from Fig. 9b that the salinity amplitude increases firstly and then decreases as the river discharge increases. This is because during periods of low river discharge (Qf=200 m3 s-1), the tide-averaged salinity is larger but the salinity amplitude coefficient Is is smaller, which indicates the weaker interaction between the river flow and the tides. However, the tide-averaged salinity decreases rapidly with the increasing river discharge, as we can see from Fig. 9a, resulting in a smaller amplitude of salinity during periods of high river discharge (Qf=1800 m3 s-1).

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 Is is in direct proportion to the tidal excursion, the amplitude of the salinity shows a linearly increasing trend with the increased tidal excursion (Fig. 10b).

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 π/2, which means that the extreme values of salinity appear when the tidal velocity is zero. Our unsteady equation for salinity (i.e. Eq. 11) demonstrates the influence of the river discharge on the occurrence of maximum and minimum salinity relative to HWS and LWS, respectively.

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 s‾0, the slope k=Qfa/DA0, and the tidal excursion E. The decreasing trend of subtidal salinity is close to a linear relation from 29 January to 3 February (Fig. 12b). Thus the tide-averaged salinity value of the predicted model is set as 90 % of that on 29 January, considering the slight change of the subtidal salinity in the 5 d after 29 January. Moreover, the slope and the tidal excursion are assumed to be constant during the whole period from 29 January to 3 February. As shown in Fig. 12c, the prediction by the model is in good agreement with the observation in this case. Furthermore, if more observed data are available to calibrate the tide-averaged salinity covering the period from 29 January to 3 February, Eq. (14) performs better, as shown in Fig. 12d. The available time for water intake can be obtained from the prediction, when the salinity concentration reaches a value 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.