# Modelling Study of Transport Time Scales for a Hyper-Tidal Estuary

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Transport Time Scale Modelling

_{0}, and M(t) is the amount of tracer remaining in the domain after time t; H(x, y, t) = the water depth at location (x, y), time t; C(x, y, t) = the tracer concentration at location (x, y) and time t. The residence time, or exposure time, can then be defined as:

_{0}) to infinity (t

_{0+∞}), which is impractical for real estuaries. In practice, various studies [3,4] have suggested integrating the remnant function over a finite time period, with this time being sufficiently long enough for most of the tracer to have left the domain of interest.

_{r}is the return coefficient quantifying the contribution of returning water to the exposure time.

#### 2.2. Hydrodynamic and Dispersion Model

_{x}and m

_{y}are the square roots of the diagonal components of the metric tensor, and m = m

_{x}m

_{y}is the Jacobian or square root of the metric tensor determinant. The total depth H = h + ξ, consists of the depth below and the free surface displacement above the undisturbed physical vertical coordinate origin, i.e., z* = 0. In the momentum Equations (6) and (7), f is the Coriolis parameter, A

_{v}is the vertical turbulent or eddy viscosity and Q

_{u}and Q

_{v}are the momentum source-sink terms, which were used to model the subgrid scale horizontal diffusion. The pressure p is the relative hydrostatic pressure in the water column, where ρ and ρ

_{0}are the actual and reference water densities. S = conservative tracer concentration in the transport equation. The numerical scheme adopted in the EFDC model is based on a combination of the finite volume and finite difference spatial discretisation methods on a C grid staggering of the discrete variables. Full details of the EFDC model are given in the corresponding documents [20].

^{2}, which covered the whole of the Bristol Channel and the Severn Estuary. The bathymetry used in this model was obtained by interpolation using a digital bathymetric chart of the area downstream of the second Severn Bridge and observed cross-sectional profiles upstream of the bridge and up through the River Severn [12,23]. The model extended a distance of 180 km in the east–west direction and 72 km in south-north direction (Figure 1). The model was driven by different tidal conditions, including spring and neap tides, at the seaward boundary. The seaward boundary was set between Hartland Point in South West England and Stackpole Head in West Wales. Time varying water levels were specified along this boundary. The upstream landward boundary was set at the tidal limit of the Severn Estuary, located close to Gloucester, to account for the possible impact of the River Severn on both the residence and exposure times in the Severn Estuary. The corresponding water levels at the open boundary were specified using the predicted elevation data from POLPRED [11]. The simulation duration for calibration was 300 h, starting on 20th July and ending on 2nd August 2001.

## 3. Results and Discussion

## 4. Conclusions

- (1)
- The average residence and exposure times for a hyper-tidal estuary, such as the Severn Estuary, are not significantly affected by the river flow from the River Severn. Higher river flows give only slightly smaller average residence and exposure times for all modelling scenarios, which suggests that both the exposure and residence times do not show significant seasonal variations for different river flow conditions, as compared with similar results in micro- and macro-tidal water systems.
- (2)
- The effects of river flows from the River Severn on the residence and exposure times in the Severn Estuary are regional in the upstream part of the estuary, for both spring and neap tidal conditions, with the effects for high flow conditions extending slightly further downstream.
- (3)
- The Severn Estuary is a hyper-tidal estuary with the second highest tidal range in the world, and the corresponding impact of this high tidal range on the degree of mixing and water exchange processes is, as expected, found to be significant. A previous study on micro-tidal estuaries has shown that both the exposure and residence times were lower if the tracers were released at higher water levels, regardless of the tide ranges [4]. However, the findings from this study have shown that the tidal effects in the Severn Estuary are quite different. Both the residence and exposure times followed the order of NL (neap low) > NH (neap high) > SL (spring low) > SH (spring high), which means that the tidal range plays a dominant role in the transport time scale, with the higher transport time scales being observed for neap tide conditions and particularly at low water level.
- (4)
- The return coefficient for the Severn Estuary does not vary significantly, with values ranging from 0.75 for the NL scenario to 0.88 for the SH scenario, while the NH scenario gave slighter higher return coefficients of 0.79 and a lower value of 0.81 for the SL scenario. The relatively high return coefficients for both spring and neap tide conditions confirmed that there were significant differences between the exposure and residence times for all scenarios modelled.
- (5)
- For the same tidal range conditions, releasing tracers at higher water levels gave lower residence and exposure times. For macro-tidal coastal waters, such as Dublin Bay, the effects of different return coefficients, under high tidal range conditions, meant that lower exposure times were not guaranteed, such as observed with SH > NH. However, in the hyper-tidal Severn Estuary the higher tidal ranges resulted in lower exposure and residence times. For the same tidal range, then releasing a tracer at a higher water level gave higher return coefficients in the estuary, with SH > SL and NH > NL. This result has a significant impact on designing wastewater treated discharges, particularly under extreme flood conditions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Gao, X.; Zhao, G.; Zhang, C.; Wang, Y. Modeling the exposure time in a tidal system: The impacts of external domain, tidal range, and inflows. Environ. Sci. Pollut. Res. Int.
**2018**, 25, 11128–11142. [Google Scholar] [CrossRef] - Monsen, N.E.; Cloern, J.E.; Lucas, L.V.; Monismith, S.G. A comment on the use of flushing time, residence time, and age as transport time scales. Limnol. Oceanogr.
**2002**, 47, 1545–1553. [Google Scholar] [CrossRef] [Green Version] - de Brauwere, A.; de Brye, B.; Blaise, S.; Deleersnijder, E. Residence time, exposure time and connectivity in the Scheldt Estuary. J. Mar. Syst.
**2011**, 84, 85–95. [Google Scholar] [CrossRef] - Sun, J.; Lin, B.; Li, K.; Jiang, G. A modelling study of residence time and exposure time in the Pearl River Estuary, China. J. Hydro-Environ. Res.
**2014**, 8, 281–291. [Google Scholar] [CrossRef] - Rayson, M.D.; Gross, E.S.; Hetland, R.D.; Fringer, O.B. Time scales in Galveston Bay: An unsteady estuary. J. Geophys. Res Ocean.
**2016**, 121, 2268–2285. [Google Scholar] [CrossRef] [Green Version] - Fischer, H.B.; List, E.J.; Koh, R.C.Y.; Imberger, J.; Brooks, N.H. Mixing in Inland and Coastal Waters; Academic: San Diego, CA, USA, 1979. [Google Scholar]
- Delhez, É.J.M.; de Brye, B.; de Brauwere, A.; Deleersnijder, É. Residence time vs influence time. J. Mar. Syst.
**2014**, 132, 185–195. [Google Scholar] [CrossRef] - Shang, J.; Sun, J.; Tao, L.; Li, Y.; Nie, Z.; Liu, H.; Chen, R.; Yuan, D. Combined effect of tides and wind on water exchange in a semi-enclosed Shallow Sea. Water
**2019**, 11, 1762. [Google Scholar] [CrossRef] [Green Version] - Gao, X.; Chen, Y.; Zhang, C. Water renewal timescales in an ecological reconstructed lagoon in China. J. Hydroinform.
**2013**, 15, 991–1001. [Google Scholar] [CrossRef] [Green Version] - Lyddon, C.; Brown, J.M.; Leonardi, N.; Plater, A.J. Flood hazard assessment for a hyper-tidal estuary as a function of tide-surge-morphology interaction. Estuaries Coasts
**2018**, 41, 1565–1586. [Google Scholar] [CrossRef] [Green Version] - Zhou, J.; Falconer, R.A.; Lin, B. Refinements to the EFDC model for predicting the hydro-environmental impacts of a barrage across the Severn Estuary. Renew. Energy
**2014**, 62, 490–505. [Google Scholar] [CrossRef] - Gao, G.; Falconer, R.A.; Lin, B. Modelling importance of sediment effects on fate and transport of enterococci in the Severn Estuary, UK. Mar. Pollut. Bull.
**2013**, 67, 45–54. [Google Scholar] [CrossRef] [PubMed] - Uncles, R.J. Physical properties and processes in the Bristol Channel and Severn Estuary. Mar. Pollut. Bull.
**2010**, 61, 5–20. [Google Scholar] [CrossRef] [PubMed] - Uncles, R.J.; Stephensa, J.A.; Smithb, R.E. The dependence of estuarine turbidity on tidal intrusion length, tidal range and residence time. Cont. Shelf Res.
**2002**, 22, 1835–1856. [Google Scholar] [CrossRef] - Viero, D.P.; Defina, A. Water age, exposure time, and local flushing time in semi-enclosed, tidal basins with negligible freshwater inflow. J. Mar. Syst.
**2016**, 156, 16–29. [Google Scholar] [CrossRef] - Zimmerman, J.T.F. Mixing and flushing of tidal embayments in the Western Dutch Wadden Sea, Part I: Distribution of salinity and calculation of mixing timescales. Neth. J. Sea Res.
**1976**, 10, 149–191. [Google Scholar] [CrossRef] - Andutta, F.P.; Helfer, F.; de Miranda, L.B.; Deleersnijder, E.; Thomas, C.; Lemckert, C. An assessment of transport timescales and return coefficient in adjacent tropical estuaries. Cont. Shelf Res.
**2016**, 124, 49–62. [Google Scholar] [CrossRef] [Green Version] - Delhez, É.J.M. On the concept of exposure time. Cont. Shelf Res.
**2013**, 71, 27–36. [Google Scholar] [CrossRef] - Delhez, É.J.M.; Heemink, A.W.; Deleersnijder, É. Residence time in a semi-enclosed domain from the solution of an adjoint problem. Estuar. Coast. Shelf Sci.
**2004**, 61, 691–702. [Google Scholar] [CrossRef] - Hamrick, J.M. A Three-Dimensional Environmental Fluid Dynamics Computer Code: Theoretical and Computational Aspect; Hamrick, J.M., Ed.; Special Report in Applied Marine Science and Ocean Engineering; No. 317; Virginia Institute of Marine Science, School of Marine Science, The College of William and Mary: Gloucester Point, VA, USA, 1992; p. 63. [Google Scholar]
- Du, J.; Shen, J. Water residence time in Chesapeake Bay for 1980–2012. J. Mar. Syst.
**2016**, 164, 101–111. [Google Scholar] [CrossRef] [Green Version] - Gao, X.; Xu, L.; Zhang, C. Estimating renewal timescales with residence time and connectivity in an urban man-made lake in China. Environ. Sci. Pollut. Res. Int.
**2016**, 23, 13973–13983. [Google Scholar] [CrossRef] - Xia, J.; Falconer, R.A.; Lin, B. Hydrodynamic impact of a tidal barrage in the Severn Estuary, UK. Renew. Energy
**2010**, 35, 1455–1468. [Google Scholar] [CrossRef] - Stapleton, C.M.; Wyer, M.D.; Kay, D.; Bradford, M.; Humphrey, N.; Wilkinson, J.; Lin, B.; Yang, Y.; Falconer, R.A.; Watkins, J.; et al. Fate and Transport of Particles in Estuaries; Environment Agency Science Report; Environmental Agency: Bristol, UK, 2007; SC000002/SR1-4. [Google Scholar]

**Figure 6.**Exposure time (

**a**–

**c**) and residence time (

**d**–

**f**) distribution at spring tide at high water level (SH).

**Figure 7.**Exposure time (

**a**–

**c**) and residence time (

**d**–

**f**) distribution at spring tide at low water level (SL).

**Figure 8.**Exposure time (

**a**–

**c**) and residence time (

**d**–

**f**) distribution at neap tide at high water level (NH).

**Figure 9.**Exposure time (

**a**–

**c**) and residence time (

**d**–

**f**) distribution at neap tide at low water level (NL).

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**MDPI and ACS Style**

Gao, G.; Xia, J.; Falconer, R.A.; Wang, Y.
Modelling Study of Transport Time Scales for a Hyper-Tidal Estuary. *Water* **2020**, *12*, 2434.
https://doi.org/10.3390/w12092434

**AMA Style**

Gao G, Xia J, Falconer RA, Wang Y.
Modelling Study of Transport Time Scales for a Hyper-Tidal Estuary. *Water*. 2020; 12(9):2434.
https://doi.org/10.3390/w12092434

**Chicago/Turabian Style**

Gao, Guanghai, Junqiang Xia, Roger A. Falconer, and Yingying Wang.
2020. "Modelling Study of Transport Time Scales for a Hyper-Tidal Estuary" *Water* 12, no. 9: 2434.
https://doi.org/10.3390/w12092434