# Application of Idealised Modelling and Data Analysis for Assessing the Compounding Effects of Sea Level Rise and Altered Riverine Inflows on Estuarine Tidal Dynamics

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- Are data analysis techniques able to provide broad insights into the effects of SLR and varying river inflows on estuarine tidal dynamics?
- What are the dominant effects of SLR and altered riverine inflows on estuarine tidal properties?
- Which estuary types and locations are most vulnerable to changes in mean sea level and river inflows?

## 2. Methods

#### 2.1. Numerical Modelling

_{L}= 160 km (weaker convergence) and C

_{L}= 80 km (stronger convergence) (Figure 1). Combinations of a wide range of estuarine parameters were investigated comprising estuary length (L = 40, 80, and 160 km), uniformly distributed Manning’s roughness (n = 0.015 and 0.03 s/m

^{1/3}), tidal range at the mouth (TR

_{0}= 0.5, 1, and 4 m, representing microtidal, mesotidal, and lower boundaries of macrotidal coastal conditions, respectively), river inflow (Q) over tidal prism (TP) ratio (Q/TP = 0%, 1%, 5%, and 10%), and SLR = 0 and 1 m. The width at the mouth (B

_{0}= 1000 m) and water depth (h = 5 m) were kept constant, and for all cases, sinusoidal M2 tides (as the dominant semi-diurnal tidal constituent along the majority of coastlines and a proxy for tidal range) were applied at the mouth with a period of T = 12.42 h. Initially, no river inflow was considered (Q/TP = 0%) to identify the tidal prism TP (i.e., the volume of water entering an estuary over a flood cycle). Using the identified TP, constant river inflows (Q/TP = 1%, 5%, and 10%) were adopted. The range of considered river inflow represents low to high fluvial input conditions. The combination of all variables resulted in 432 simulation cases (144 simulations for each estuary geometry).

#### 2.2. Tidal Properties

_{t}is the length of equidistant time series data [45]. A positive value of TS denotes a longer rising tide duration and an ebb-dominated system, whereas a negative value suggests a longer falling tide duration and a flood-dominated system [45].

#### 2.3. Data Analysis

## 3. Results

#### 3.1. Effects of SLR and/or Altered Riverine Inflows on Estuarine Tidal Range

- (i)
- SLR led to minor (weak) increases in tidal range along the estuaries (dark green coloured cells);
- (ii)
- SLR substantially increased the tidal range at the mouth and then minimally strengthened it in a landward direction (light green coloured cells).

_{0}= 0.5 and 1 m), while the second pattern was often observed in estuaries with macrotidal conditions (TR

_{0}= 4 m). Where SLR increased the tidal range, this was consistent with reported research of real-world estuaries, such as the Elbe River [47], York River [20], Rappahannock River [20], Hudson–Raritan Estuary [48], and Minnamurra River [49]. In these cases, amplified tides may exceed the crest of protective structures, exacerbate flood events [50], and lead to the failure of surface drainage infrastructures; shoreline erosion; and loss of wetlands, intertidal areas, and their associated ecosystems [7,12,51].

_{L}= 160 km largely varied along the first half of the estuary length (i.e., 0 $\le $ x $\le $ 0.5 L), whereas areas of strong tidal variations were located around the mouth and head (i.e., x = 0 and L) of converging estuaries with C

_{L}= 80 km. This finding aligned with predictions in real-world estuaries, such as the moderately converging Patuxent Estuary (L = ~40 km), which may experience larger variations in tidal range in its first 20 km [20], and the strongly converging Delaware Bay estuary, which may undergo a substantial tidal amplification in its upstream zones [52].

_{0}= 4 m) (Table 1). In these estuaries, the dampening effect of the bottom friction became important (particularly in the upstream part of the estuary), and a larger Manning’s n could lead to a weaker tidal range amplification. For instance, in a converging estuary with C

_{L}= 80 km, TR

_{0}= 4 m, L = 40 km, and Q/TP = 10%, a substantial increase in tidal range with an SLR of 1 m was observed at x = 0, 0.5 L, and L for n = 0.015 s/m

^{1/3}, whereas for n = 0.03 s/m

^{1/3}, the tidal range only increased at x = 0 (cells a-36 and b-36 in Table 1). In this case, it appeared that the convergence effect outweighed the reduced frictional effects under SLR when n = 0.015 s/m

^{1/3}, but friction dominated when Manning’s n increased to n = 0.03 s/m

^{1/3}. There were only four cases with C

_{L}= 80 km, L = 40 and 80 km, n = 0.015 and 0.03 s/m

^{1/3}, and Q/TP = 5% and 10% (cells a-35, b-35, c-35, and a-36 in Table 1) that experienced a substantial increase in tidal range at three or more nodes along the estuary when SLR = 1 m. Such an increase in tidal range due to SLR was observed in several estuaries around the British coast such as the Severn Estuary, which is a converging macrotidal site [53].

_{0}= 4 m) or a strongly converging shape (C

_{L}= 80 km) (Table 1). For example, in a converging estuary with C

_{L}= 160 km, TR

_{0}= 4 m, L = 160 km, and n = 0.015 s/m

^{1/3}, an SLR of 1 m caused a tidal range amplification at all transects when Q/TP = 1%, with the strongest increases observed at the mouth (cell e-22 in Table 1). However, when Q/TP = 5%, SLR only strongly intensified the tidal range at the mouth, with tidal attenuation observed at x = 0.75 L (cell e-23 in Table 1). Increasing inflows appeared to act against the tidal propagation leading to tidal attenuation in the upstream part of the estuary [54]. A similar trend under rising river inflows has been reported in real-world estuaries, including the Ganges–Brahmaputra–Meghna Delta [55], the Scheldt Estuary [56], and the Pearl River [57].

_{L}= 160 km estuaries) where SLR could reduce the tidal range. These mainly included transects (nodes) at x = 0.25 L for prismatic cases with TR

_{0}= 0.5 and 1 m, L = 160 km, n = 0.015 and 0.03 s/m

^{1/3}, and Q/TP = 0% and 1% (cells e-1, f-1, e-2, f-2, e-5, f-5, e-6, and f-6 in Table 1); transects at 0.25 L ≤ x ≤ 0.5 L for prismatic cases with TR

_{0}= 4 m, L = 40 and 80 km, n = 0.015 s/m

^{1/3}, and Q/TP = 5% (cells a-11 and c-11 in Table 1); and transects at x = 0.75 L for converging estuaries with C

_{L}= 160 km, TR

_{0}= 4 m, L = 40, 80, and 160 km, n = 0.015 and 0.03 s/m

^{1/3}, and Q/TP = 1% and 5% (cells a-22 to e-22 and a-23 to f-23 in Table 1).

**Table 1.**Compounding effects of SLR and varying river inflows on tidal range at five nodes along the estuaries (x = 0, 0.25 L, 0.5 L, 0.75 L, and L), which are represented by five consecutive triangles in each cell. Upward and downward triangles indicate an increase or a decrease in tidal range induced by 1 m of SLR and for four varying river inflows (Q/TP = 0%, 1%, 5%, and 10%). Hollow and solid triangles imply weak (0 $\le \left|\alpha \right|\le $ 0.5) and strong (0.5 $<\left|\alpha \right|\le $ 1) Pearson correlation coefficients, respectively. The symbol “–” denotes the absence of a correlation (for details, see [46]). Dominant patterns of change in tidal properties under compounding effects of SLR and river inflows are highlighted with similar colours across Table 1, Table 2 and Table 3.

a | b | c | d | e | f | |||

L | 40 km | 80 km | 160 km | |||||

n | 0.015 s/m^{1/3} | 0.03 s/m^{1/3} | 0.015 s/m^{1/3} | 0.03 s/m^{1/3} | 0.015 s/m^{1/3} | 0.03 s/m^{1/3} | ||

Prismatic | TR_{0} = 0.5 m | No inflow (Q/TP = 0%) | ||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △▽△△△ | △▽△△△ | 1 | ||

Low inflow (Q/TP = 1%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △▽△△△ | △▽△△△ | 2 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 3 | ||

High inflow (Q/TP = 10%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 4 | ||

TR_{0} = 1 m | No inflow (Q/TP = 0%) | |||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △▽△△△ | △▽△△△ | 5 | ||

Low inflow (Q/TP = 1%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △▽△△△ | △▽△△△ | 6 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 7 | ||

High inflow (Q/TP = 10%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 8 | ||

TR_{0} = 4 m | No inflow (Q/TP = 0%) | |||||||

△△△△△ | △△△△△ | △▲△△△ | △△–△△ | △△△△△ | △△△△△ | 9 | ||

Low inflow (Q/TP = 1%) | ||||||||

▲△△△△ | ▲△△△△ | ▲△△△△ | ▲△–△△ | ▲△△△△ | ▲△△△△ | 10 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

▲▽▽△△ | ▲△–△△ | ▲▽▽△△ | ▲△△△△ | ▲△△△△ | ▲△△△△ | 11 | ||

High inflow (Q/TP = 10%) | ||||||||

▲△△△△ | ▲△△△△ | ▲△△△△ | ▲△△△△ | ▲△▲△△ | ▲△△△△ | 12 | ||

Converging with C_{L} = 160 km | TR_{0} = 0.5 m | No inflow (Q/TP = 0%) | ||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 13 | ||

Low inflow (Q/TP = 1%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 14 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 15 | ||

High inflow (Q/TP = 10%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 16 | ||

TR_{0} = 1 m | No inflow (Q/TP = 0%) | |||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 17 | ||

Low inflow (Q/TP = 1%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 18 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 19 | ||

High inflow (Q/TP = 10%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 20 | ||

TR_{0} = 4 m | No inflow (Q/TP = 0%) | |||||||

▲△△△△ | ▲△△△△ | ▲△△△△ | ▲△△△△ | ▲△▲△△ | ▲△△△△ | 21 | ||

Low inflow (Q/TP = 1%) | ||||||||

▲△△▽△ | ▲△△▽△ | ▲△△▽△ | ▲△-▽△ | ▲△△▽▲ | ▲△△△▲ | 22 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

▲–▲▽△ | ▲△–▽△ | ▲▽▲▽△ | ▲△△▽△ | ▲△△▽△ | ▲△△▽△ | 23 | ||

High inflow (Q/TP = 10%) | ||||||||

▲▲△△△ | ▲▲△△△ | ▲▲△△△ | ▲▲△△△ | ▲▲△△△ | ▲▲△△△ | 24 | ||

Converging with C_{L} = 80 km | TR_{0} = 0.5 m | No inflow (Q/TP = 0%) | ||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 25 | ||

Low inflow (Q/TP = 1%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 26 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△–△△ | △△△△△ | 27 | ||

High inflow (Q/TP = 10%) | ||||||||

△△△△△ | △△△△△ | △△△△▲ | △△△△▲ | △△△△▲ | △△△△▲ | 28 | ||

TR_{0} = 1 m | No inflow (Q/TP = 0%) | |||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 29 | ||

Low inflow (Q/TP = 1%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△▲△ | 30 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△–△ | △△△△△ | 31 | ||

High inflow (Q/TP = 10%) | ||||||||

△△△△▲ | △△△△– | △△△△▲ | △△△△▲ | △△△–▲ | △△△△▲ | 32 | ||

TR_{0} = 4 m | No inflow (Q/TP = 0%) | |||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△▲△△ | △△△△△ | 33 | ||

Low inflow (Q/TP = 1%) | ||||||||

▲△△△△ | ▲△△△△ | ▲△△△△ | ▲△△△△ | ▲△△△△ | ▲△△▲△ | 34 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

▲▲▲▲– | ▲△▲▲▽ | ▲▲▲▲△ | ▲△△▲▽ | ▲△△▲△ | ▲△△▲△ | 35 | ||

High inflow (Q/TP = 10%) | ||||||||

▲△▲△▲ | ▲△△△△ | ▲△▲△△ | ▲△△△△ | ▲△△△▲ | ▲△△△▲ | 36 |

#### 3.2. Effects of SLR and/or Altered Riverine Inflows on Estuarine Maximum Current Velocity

- (i)
- Compounding effects of SLR and moderate to high inflows (Q/TP = 5–10%) increased this parameter particularly in the first half of estuaries (0 $\le $ x $\le $ 0.5 L) (e.g., dark and light shades of red in Table 2);
- (ii)
- SLR minimally reduced this parameter when L = 40 and 80 km, TR
_{0}= 0.5 and 1 m, and Q/TP = 0–1% (e.g., very light shades of red in Table 2).

_{0}= 0.5 and 1 m and Q/TP = 0%, but for long estuaries (L = 160 km), both small increases and decreases were observed (Table 2). This observation of a mixed pattern was consistent with findings in the Hudson River, a long, mesotidal, moderately converging estuary (L $\cong $ 245 km, TR

_{0}$\cong $ 1.3 m, C

_{L}$\cong $ 140 km) [58,59], where SLR would lead to minor increases in the residual current speed mid-estuary and minor decreases in the upper bay [48]. Although the maximum velocity does not change considerably under SLR, the current velocity distribution (e.g., ebb or flood velocity values) may vary throughout the system [22].

_{0}= 4 m and Q/TP = 0%, SLR generally increased the velocity up to three quarters of the prismatic and converging cases with C

_{L}= 160 km (e.g., cells a-21 to f-21 in Table 2) but decreased at the mouth and head of converging estuaries with C

_{L}= 80 km (e.g., cells e-33 and f-33 in Table 2). For low inflows (Q/TP = 1%) and TR

_{0}= 0.5 and 1 m, SLR often decreased the current velocity along prismatic and converging with C

_{L}= 160 km estuaries with L = 40 or 80 km (e.g., cells a-6 to d-6 in Table 2), whereas it increased in estuaries with L = 160 km (e.g., cells e-6 and f-6 in Table 2). This finding was consistent with the microtidal, ~100 km long Barataria Bay, where SLR decreased the longitudinal velocity by nearly 58% in the mid-estuary region [60]. When Q/TP = 1% and TR

_{0}= 4 m, nearly all prismatic and converging with C

_{L}= 160 km estuaries (cells a-10 to f-10 and a-22 to f-22 in Table 2) and almost all converging cases with C

_{L}= 80 km experienced an increase in their maximum current velocity due to SLR.

_{0}= 4 m). For instance, for a converging case with C

_{L}= 80 km, TR

_{0}= 4 m, L = 80 km, and Q/TP = 10%, SLR strongly increased the maximum current velocity from the mouth up to the mid-estuary (x = 0.5 L) when n = 0.015 s/m

^{1/3}(cell c-36 in Table 2) and up to x = 0.25 L when n = 0.03 s/m

^{1/3}(cell d-36 in Table 2). In converging estuaries, the funnelling effects outweighed the frictional effects from the entrance to a distance along the estuary before reaching a balance between these two forces [29,62].

**Table 2.**Compounding effects of SLR and varying river inflows on maximum current velocity at five nodes along the estuaries (x = 0, 0.25 L, 0.5 L, 0.75 L, and L), which are represented by five consecutive triangles in each cell. Upward and downward triangles indicate an increase or a decrease in maximum current velocity induced by 1 m of SLR and for four varying river inflows (Q/TP = 0%, 1%, 5%, and 10%). Hollow and solid triangles imply weak (0 $\le \left|\alpha \right|\le $ 0.5) and strong (0.5 $<\left|\alpha \right|\le $ 1) Pearson correlation coefficients, respectively. The symbol “–” denotes the absence of a correlation (for details, see [46]). Dominant patterns of change in tidal properties under compounding effects of SLR and river inflows are highlighted with similar colours across Table 1, Table 2 and Table 3.

a | b | c | d | e | f | |||

L | 40 km | 80 km | 160 km | |||||

n | 0.015 s/m^{1/3} | 0.03 s/m^{1/3} | 0.015 s/m^{1/3} | 0.03 s/m^{1/3} | 0.015 s/m^{1/3} | 0.03 s/m^{1/3} | ||

Prismatic | TR_{0} = 0.5 m | No inflow (Q/TP = 0%) | ||||||

▽▽▽▽▽ | ▽▽▽▽▽ | △▽▽▽▽ | △▽▽▽▽ | △△△▽▽ | △△△▽▽ | 1 | ||

Low inflow (Q/TP = 1%) | ||||||||

▽▽▽▽△ | ▽▽▽▽△ | △▽▽▽△ | △▽▽▽△ | △△△▽△ | △△△▽△ | 2 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 3 | ||

High inflow (Q/TP = 10%) | ||||||||

▲▲▲△△ | ▲▲▲△△ | ▲▲▲△△ | ▲▲▲△△ | ▲▲▲△△ | ▲▲▲△△ | 4 | ||

TR_{0} = 1 m | No inflow (Q/TP = 0%) | |||||||

▽▽▽▽▽ | ▽▽▽▽▽ | △▽▽▽▽ | △▽▽▽▽ | △△△▽▽ | △△△▽▽ | 5 | ||

Low inflow (Q/TP = 1%) | ||||||||

▽▽▽▽△ | ▽▽▽▽△ | △▽▽▽△ | △▽▽▽△ | △△△▽△ | △△△▽△ | 6 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 7 | ||

High inflow (Q/TP = 10%) | ||||||||

▲▲▲△△ | ▲▲▲△△ | ▲▲▲△△ | ▲▲▲△△ | ▲▲▲△△ | ▲▲▲△△ | 8 | ||

TR_{0} = 4 m | No inflow (Q/TP = 0%) | |||||||

△▽△△▽ | △▽△△▽ | △▲△△▽ | △△–△▽ | △△△△▽ | △△△△▽ | 9 | ||

Low inflow (Q/TP = 1%) | ||||||||

△▽△△△ | △▽△△△ | △△△△△ | △△-△△ | △△△△△ | △△△△△ | 10 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△▲△△▲ | △△–▲▲ | △▲△△▲ | △△△▲▲ | △△△△▲ | △△△▲▲ | 11 | ||

High inflow (Q/TP = 10%) | ||||||||

▲▲▲▲▲ | ▲▲▲▲△ | ▲▲▲▲△ | ▲▲▲▲△ | ▲▲▲▲△ | ▲▲▲△△ | 12 | ||

Converging with C_{L} = 160 km | TR_{0} = 0.5 m | No inflow (Q/TP = 0%) | ||||||

▽△▽▽▽ | ▽△▽▽▽ | ▽△▽▽▽ | ▽△▽▽▽ | ▽△△▽▽ | ▽△△▽▽ | 13 | ||

Low inflow (Q/TP = 1%) | ||||||||

▽△▽△△ | ▽△▽△△ | ▽△▽△△ | ▽△▽△△ | ▽△△△△ | ▽△△△△ | 14 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△▽△△△ | △▽△△△ | △△△△△ | △▲△△△ | △▲△△△ | △▲△△△ | 15 | ||

High inflow (Q/TP = 10%) | ||||||||

▲▲△△△ | ▲▲△△△ | ▲▲△△△ | ▲▲▲△△ | ▲▲▲△△ | ▲▲▲△△ | 16 | ||

TR_{0} = 1 m | No inflow (Q/TP = 0%) | |||||||

▽△▽▽▽ | ▽△▽▽▽ | ▽△▽▽▽ | ▽△▽▽▽ | ▽△△▽▽ | ▽△△▽▽ | 17 | ||

Low inflow (Q/TP = 1%) | ||||||||

▽△▽△△ | ▽△▽△△ | ▽△▽△△ | ▽△▽△△ | ▽△△△△ | ▽△△△△ | 18 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△▽△△△ | △▽△△△ | △△△△△ | △▲△△△ | △▲△△△ | △▲△△△ | 19 | ||

High inflow (Q/TP = 10%) | ||||||||

▲▲△△△ | ▲▲△△△ | ▲▲△△△ | ▲▲▲△△ | ▲▲▲△△ | ▲▲▲△△ | 20 | ||

TR_{0} = 4 m | No inflow (Q/TP = 0%) | |||||||

△△△△▽ | △△△△▽ | △△△△▽ | △△△△▽ | △△▲△▽ | △△△△▽ | 21 | ||

Low inflow (Q/TP = 1%) | ||||||||

△△▽△△ | △△▽△△ | △△▽△△ | △△–△△ | △△△△▲ | △△△▽▲ | 22 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△–△△△ | △▽–△△ | △▲△△△ | △△△△△ | △▽△△△ | △△△△△ | 23 | ||

High inflow (Q/TP = 10%) | ||||||||

▲▲△△△ | ▲▲△△△ | ▲▲△△△ | ▲▲△△△ | ▲▲△△△ | ▲▲△△△ | 24 | ||

Converging with C_{L} = 80 km | TR_{0} = 0.5 m | No inflow (Q/TP = 0%) | ||||||

▽▽▽▽▽ | ▽▽▽▽▽ | ▽▽▽▽▽ | ▽▽▽▽▽ | ▽▽△▽▽ | ▽▽△▽▽ | 25 | ||

Low inflow (Q/TP = 1%) | ||||||||

▽△△△△ | ▽△△△△ | ▽△△△△ | ▽△△△△ | ▽△△△△ | ▽△▽△△ | 26 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△–△△ | △△△△▽ | 27 | ||

High inflow (Q/TP = 10%) | ||||||||

▲▲△△▲ | ▲▲△△▲ | ▲▲△△△ | ▲▲△△△ | ▲▲△△△ | ▲▲△△△ | 28 | ||

TR_{0} = 1 m | No inflow (Q/TP = 0%) | |||||||

▽▽▽▽▽ | ▽▽▽▽▽ | ▽▽▽▽▽ | ▽▽▽▽▽ | ▽▽△▽▽ | ▽▽△▽▽ | 29 | ||

Low inflow (Q/TP = 1%) | ||||||||

▽△△△△ | ▽△△△△ | ▽△△△△ | ▽△△△△ | ▽△▽△△ | ▽△▽△△ | 30 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△▽ | △△△–▽ | △△△△▽ | 31 | ||

High inflow (Q/TP = 10%) | ||||||||

▲▲△△△ | ▲▲▽▽– | ▲▲△△△ | ▲▲△△△ | ▲▲△–△ | ▲▲△△△ | 32 | ||

TR_{0} = 4 m | No inflow (Q/TP = 0%) | |||||||

▽△▽△▽ | ▽△▽△▽ | ▽△▽△▽ | ▽△▽△▽ | ▽△▲△▽ | ▽△△△▽ | 33 | ||

Low inflow (Q/TP = 1%) | ||||||||

△△▽△△ | △△▽△△ | △△▽△△ | △△▽△△ | △△▽△▲ | △△▽△▲ | 34 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△▲△▲- | △▽△▲△ | △▲△▲△ | △△△△△ | △△△△△ | △△△△△ | 35 | ||

High inflow (Q/TP = 10%) | ||||||||

▲▲▲△▽ | ▲▲△△△ | ▲▲▲△△ | ▲▲△△△ | ▲▲△△△ | ▲▲△△△ | 36 |

#### 3.3. Effects of SLR and/or Altered Riverine Inflows on Estuarine Asymmetry

_{0}= 4 m). These increased transformed skewness values do not necessarily imply ebb tide domination but can also represent a reduced flood tide domination. This is important as increasing transformed skewness may reinforce seaward sediment export/flushing and jeopardise shoreline stability and marsh accretion. This finding was consistent with the analytical predictions of [67,68], indicating that flood tide domination reduces with increasing water depth (e.g., under SLR) in estuaries with no (or small) tidal flats or with considerable overland inundation.

_{L}= 80 km and TR

_{0}= 0.5 and 1 m, an SLR of 1 m generally increased the transformed skewness along most estuaries, leading to less flood tide domination or more ebb tide domination where Q/TP ≤ 5% (rows 25–27 and 29–31 in Table 3). Where Q/TP = 10% and TR

_{0}= 0.5 and 1 m (rows 28 and 32 in Table 3), SLR enhanced ebb tide dominance along all transects (nodes) except at x = 0.75 L, where SLR brought about flood tide dominance. When C

_{L}= 80 km and TR

_{0}= 4 m, SLR considerably reduced the flood tide dominance in all cases with Q/TP = ≤ 1 % (rows 33 and 34 in Table 3). As a converging estuary with minor river inflows, the Tamar River estuary in Australia is a real-world example that will likely experience a reduction of up to 40% in its flood tide dominance under SLR, enhancing flushing and altering the geomorphic trajectory of the system [69]. Converging cases with C

_{L}= 160 km largely followed similar trends as cases with stronger convergence (C

_{L}= 80 km), except for cases with L = 40 and 80 km, TR

_{0}= 0.5 and 1 m, and Q/TP = 0%, where SLR reduced flood tide dominance at all cross-sections except the mid-estuary (cells a-13 to d-13 and a-17 to d-17 in Table 3). Prismatic estuaries with TR

_{0}= 0.5 and 1 m, L = 40 km, and Q/TP = ≤ 1% were the only cases that underwent an increased flood tide dominance at their mouths (cells a-1, b-1, a-2, b-2, a-5, b-5, a-6, and b-6 in Table 3), boosting sediment import and basin infilling.

_{L}= 160 km, TR

_{0}= 4 m, L = 80 km, and Q/TP = 5%, the transformed skewness strongly increased along the estuary from the oceanic entrance to the head when n = 0.015 s/m

^{1/3}(cell c-23 in Table 3) but only strongly increased the transformed skewness at the entrance and head and decreased at other transects (nodes) when n = 0.03 s/m

^{1/3}(cell d-23 in Table 3). As such, a system may shift from ebb tide dominant (or less flood dominant) to flood tide dominant (or less ebb tide dominant) by increasing friction, leading to an increasingly turbid estuary, as observed in the Ems River [71].

**Table 3.**Compounding effects of SLR and varying river inflows on transformed skewness at five nodes along the estuaries (x = 0, 0.25 L, 0.5 L, 0.75 L, and L), which are represented by five consecutive triangles in each cell. Upward and downward triangles indicate an increase or a decrease in transformed skewness induced by 1 m of SLR and for four varying river inflows (Q/TP = 0%, 1%, 5%, and 10%). Hollow and solid triangles imply weak (0 $\le \left|\alpha \right|\le $ 0.5) and strong (0.5 $<\left|\alpha \right|\le $ 1) Pearson correlation coefficients, respectively. The symbol “–” denotes the absence of a correlation (for details, see [46]). Dominant patterns of change in tidal properties under compounding effects of SLR and river inflows are highlighted with similar colours across Table 1, Table 2 and Table 3.

a | b | c | d | e | f | |||

L | 40 km | 80 km | 160 km | |||||

n | 0.015 s/m^{1/3} | 0.03 s/m^{1/3} | 0.015 s/m^{1/3} | 0.03 s/m^{1/3} | 0.015 s/m^{1/3} | 0.03 s/m^{1/3} | ||

Prismatic | TR_{0} = 0.5 m | No inflow (Q/TP = 0%) | ||||||

▽△▽△△ | ▽△▽△△ | △△▽△△ | △△▽△△ | △▲▽△△ | △▲▽△△ | 1 | ||

Low inflow (Q/TP = 1%) | ||||||||

▽△△△△ | ▽△△△△ | △△△△△ | △△△△△ | △▲▽△△ | △▲▽△△ | 2 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△▽△△ | △△▽△△ | 3 | ||

High inflow (Q/TP = 10%) | ||||||||

△△△▽△ | △△△▽△ | △△△▽△ | △△△▽△ | △△▽▽△ | △△▽▽△ | 4 | ||

TR_{0} = 1 m | No inflow (Q/TP = 0%) | |||||||

▽△▽△△ | ▽△▽△△ | △△▽△△ | △△▽△△ | △▲▽△△ | △▲▽△△ | 5 | ||

Low inflow (Q/TP = 1%) | ||||||||

▽△△△△ | ▽△△△△ | △△△△△ | △△△△△ | △▲▽△△ | △▲▽△△ | 6 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△▽△△ | △△▽△△ | 7 | ||

High inflow (Q/TP = 10%) | ||||||||

△△△▽△ | △△△▽△ | △△△▽△ | △△△▽△ | △△▽▽△ | △△▽▽△ | 8 | ||

TR_{0} = 4 m | No inflow (Q/TP = 0%) | |||||||

△△▲△△ | △△▲△△ | △▲▲△△ | △△-△△ | △△▽△△ | △△▽△△ | 9 | ||

Low inflow (Q/TP = 1%) | ||||||||

▲△▲▲▲ | ▲△▲▲▲ | ▲△▲▲▲ | ▲△–▲▲ | ▲△▽▲▲ | ▲△▽▲▲ | 10 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

▲△△△▲ | ▲△–▼▲ | ▲△△△▲ | ▲△▽▼▲ | ▲△▽△▲ | ▲△▽▼▲ | 11 | ||

High inflow (Q/TP = 10%) | ||||||||

▲△▼▽▲ | ▲△△▽▲ | ▲△△▽▲ | ▲△△▽▲ | ▲△▽▽▲ | ▲△△▽▲ | 12 | ||

Converging with C_{L} = 160 km | TR_{0} = 0.5 m | No inflow (Q/TP = 0%) | ||||||

△△▽△△ | △△▽△△ | △△▽△△ | △△▽△△ | △△△△△ | △△△△△ | 13 | ||

Low inflow (Q/TP = 1%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 14 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △▽△△△ | △△△△△ | △▲▽△△ | △△▽△△ | 15 | ||

High inflow (Q/TP = 10%) | ||||||||

△△△▽△ | △△△▽△ | △△△▽△ | △△▽▽△ | △△▽▽△ | △△▽▽△ | 16 | ||

TR_{0} = 1 m | No inflow (Q/TP = 0%) | |||||||

△△▽△△ | △△▽△△ | △△▽△△ | △△▽△△ | △△△△△ | △△△△△ | 17 | ||

Low inflow (Q/TP = 1%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 18 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △▽△△△ | △△△△△ | △▲▽△△ | △△▽△△ | 19 | ||

High inflow (Q/TP = 10%) | ||||||||

△△△▽△ | △△△▽△ | △△△▽△ | △△▽▽△ | △△▽▽△ | △△▽▽△ | 20 | ||

TR_{0} = 4 m | No inflow (Q/TP = 0%) | |||||||

▲△▲▲▲ | ▲△▲▲▲ | ▲△▲▲▲ | ▲△△▲▲ | ▲△▼▲▲ | ▲△△▲△ | 21 | ||

Low inflow (Q/TP = 1%) | ||||||||

▲△▲▲▲ | ▲△▲▲▲ | ▲△▲▲▲ | ▲△-▲▲ | ▲△△▲▲ | ▲△△▲▲ | 22 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

▲–▲△▲ | ▲▲–△▲ | ▲▲▲△▲ | ▲▽▽▽▲ | ▲▲▽▽▲ | ▲▽▽▽▲ | 23 | ||

High inflow (Q/TP = 10%) | ||||||||

▲△△▽▲ | ▲△▽▽▲ | ▲△△▽▲ | ▲△▽▽▲ | ▲△▽▽▲ | ▲△▽▽▲ | 24 | ||

Converging with C_{L} = 80 km | TR_{0} = 0.5 m | No inflow (Q/TP = 0%) | ||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 25 | ||

Low inflow (Q/TP = 1%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 26 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△–△▲ | △△▽△▲ | 27 | ||

High inflow (Q/TP = 10%) | ||||||||

△△△▽▲ | △△△▽▲ | △△△▽▲ | △△△▽▲ | △△△▽△ | △△△▽△ | 28 | ||

TR_{0} = 1 m | No inflow (Q/TP = 0%) | |||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 29 | ||

Low inflow (Q/TP = 1%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | △△△△△ | 30 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

△△△△△ | △△△△△ | △△△△△ | △△△△▲ | △△▽-▲ | △△▽△▲ | 31 | ||

High inflow (Q/TP = 10%) | ||||||||

△△△▽▲ | △△△▽- | △△△▽▲ | △△△▽▲ | △△▽-△ | △△△▽△ | 32 | ||

TR_{0} = 4 m | No inflow (Q/TP = 0%) | |||||||

▲△▲▲▲ | ▲△▲▲▲ | ▲△▲▲▲ | ▲△▲▲▲ | ▲△▲▲▲ | ▲△△▲△ | 33 | ||

Low inflow (Q/TP = 1%) | ||||||||

▲△▲▲▲ | ▲△▲▲▲ | ▲△▲▲▲ | ▲△▲▲▲ | ▲△△▲▲ | ▲△△△▲ | 34 | ||

Moderate inflow (Q/TP = 5%) | ||||||||

▲▲▲▲– | ▲▽▲▲▲ | ▲▲▲▲▲ | ▲▽▽▽▲ | ▲▽▽▽▲ | ▲▽▽▽▲ | 35 | ||

High inflow (Q/TP = 10%) | ||||||||

▲△△△▲ | ▲△▽△▲ | ▲△△△▲ | ▲△▽△▲ | ▲△▽△▲ | ▲△▽△▲ | 36 |

## 4. Discussion

- (i)
- The most common patterns of change in tidal properties along the length of estuaries;
- (ii)
- The estuary types influenced by compounding effects of SLR and varying riverine inflows; and,
- (iii)
- The most vulnerable estuarine cross-sections.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Plan view of different estuarine geometries investigated in this study including prismatic (

**top**panel), converging with C

_{L}= 160 km (

**middle**panel), and converging with C

_{L}= 80 km (

**bottom**panel). All panels depict co-ordinate system, driving forces, and boundary conditions.

**Figure 2.**An example of a gridded prismatic estuary together with five different locations at x = 0, 0.25 L, 0.5 L, 0.75 L, and L, where data of water surface elevations and horizontal velocity components were extracted to calculate tidal range, maximum current velocity, and transformed skewness. Star symbols show the nodes where the flow data were extracted and examined.

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**MDPI and ACS Style**

Khojasteh, D.; Vibhani, T.; Shafiei, H.; Glamore, W.; Felder, S.
Application of Idealised Modelling and Data Analysis for Assessing the Compounding Effects of Sea Level Rise and Altered Riverine Inflows on Estuarine Tidal Dynamics. *J. Mar. Sci. Eng.* **2023**, *11*, 815.
https://doi.org/10.3390/jmse11040815

**AMA Style**

Khojasteh D, Vibhani T, Shafiei H, Glamore W, Felder S.
Application of Idealised Modelling and Data Analysis for Assessing the Compounding Effects of Sea Level Rise and Altered Riverine Inflows on Estuarine Tidal Dynamics. *Journal of Marine Science and Engineering*. 2023; 11(4):815.
https://doi.org/10.3390/jmse11040815

**Chicago/Turabian Style**

Khojasteh, Danial, Tej Vibhani, Hassan Shafiei, William Glamore, and Stefan Felder.
2023. "Application of Idealised Modelling and Data Analysis for Assessing the Compounding Effects of Sea Level Rise and Altered Riverine Inflows on Estuarine Tidal Dynamics" *Journal of Marine Science and Engineering* 11, no. 4: 815.
https://doi.org/10.3390/jmse11040815