Convergent estuaries have been shortened by dam‐like structures worldwide. Here, we evaluate 31 long‐term water level stations and use a semi‐analytical tide model to investigate how landward‐funneling and a dam influence tidal and storm surge propagation in the greater Charleston Harbor region, South Carolina, where three rivers meet: the Ashley, Cooper, and Wando. Results show that the phase speed and amplification of the principal tidal harmonic (M2) is larger than other long waves such as storm surge (∼1–4days) and setup‐setdown (∼4–10days). Further landward, all waves attenuate, but, as they approach the dam on the Cooper River, a frequency dependent response in amplitude and phase progression occurs. A semi‐analytical tidal model shows that funneling and the presence of a dam amplify tidal waves through wave interference from partial and full reflection, respectively. The different phase progressions of the reflected waves interact with the incident wave to increase or decrease the summed overall wave amplitude. Using a friction‐convergence parameter space, we demonstrate that dominant tides in 23 estuaries and the tidal, storm surge, and setup‐setdown waves in the Cooper River can be delineated into three regimes that describe landward amplification or attenuation associated with funneling, a dam, or both. The regime of each tidal constituent is consistent but can change with the duration and height of each storm surge event; dam associated wave interference can attenuate long‐duration events, while the most intense events (short duration, high water) are amplified by dams more than funneling and greatly increase flood exposure.
]]>Tidal amplitude and flood risk are changing in estuaries and deltas, in part because water levels are being altered by dredging, infrastructure development, land reclamation, climate change, and other local geomorphic processes (e.g., Bosserelle et al., 2022;de Leo et al., 2022;Dijkstra et al., 2019;Orton et al., 2020;Ralston, 2022;Ralston & Geyer, 2019;Talke et al., 2014;van Veen, 1962;Winterwerp & Wang, 2013). Changes in estuarine geometry also alter currents and mixing with shifting locations of erosion and deposition. The consequent change in bathymetry may alter frictional damping, resonance, reflection of tides, as well as the interaction between tides and river discharge (e.g., Buschman et al., 2009;Chernetsky et al., 2010;Familkhalili et al., 2020;Godin, 1999;Kukulka & Jay, 2003;Matte et al., 2014;Talke & Jay, 2020;Talke et al., 2021). Bathymetric change also influences the dynamics of other types of waves, including storm surge and river floods, seiches, meteotsunamis and edge waves
RESEARCH ARTICLE 10.1029/2023JC020498 Special Collection: Physical processes, sediment transport and morphodynamics of estuaries and coastal seas Key Points:
• Estuarine geomorphic funneling and dams produce partial and full reflections, respectively, with magnitudes that are frequency dependent • Three convergence regimes emerge: dominant tides have near peak amplification, overtides attenuate, and long duration surges mildly amplify
• Dams reflect and amplify long waves -increasing flood exposure-the most in weakly convergent estuaries with low friction
Supporting Information may be found in the online version of this article. (Dykstra & Dzwonkowski, 2020, 2021;Familkhalili et al., 2022;Ke & Yankovsky, 2021), and can modify tidal datums, for example, Mean High Water and Mean Water Level (MWL) (e.g., Helaire et al., 2019;Jay et al., 2011;Ralston & Geyer, 2019;Talke et al., 2021). Finally, rising sea-level increases estuarine depth and may further increase longwave magnitudes and high water levels (Lee et al., 2017). Some of the most dramatic shifts in tidal properties over the past 150 years have occurred due to tidal reflection off a dam (Talke & Jay, 2020;Winterwerp & Wang, 2013). For example, reflection from dams on the Hudson River (New York) and Tombigbee River (Alabama), both ∼250 km inland, result in tidal amplification so that the tidal range at the dam is the same as the estuary mouth (Dykstra, Dzwonkowski & Torres, 2022;Georgas, 2012;Ralston & Geyer, 2019). The Ems Estuary, Netherlands/Germany, was shortened with a dam in 1899 and then channelized. These changes increased the tidal range in the estuary up to five-fold and the resulting net landward sediment transport fills the tidal river with fluid mud (Chernetsky et al., 2010;Dijkstra et al., 2019;Talke & Jay, 2013;Talke et al., 2009;Winterwerp & Wang, 2013). Noting that estuaries with natural reflection points can be relatively stable (e.g., James, Potomac, Connecticut Rivers), the closure of a dam may force estuaries toward a new morphodynamic equilibrium (Figueroa et al., 2022).
Dams are built in estuaries throughout the world, from tropical to polar estuaries, and have different names depending on the main purpose, for example, river dams, weirs, locks, sluice gates (Arunpandi et al., 2022;Díez-Minguito et al., 2012;Kuzyk et al., 2008;Webster, 2010). China alone has over 320 salt barrages to reduce salt intrusion (Tilai et al., 2019). However, in situ observations of how dams may interact with tides, storm surge and extreme water levels are limited (Orton et al., 2023). On the Tombigbee River, Alabama, for example, daminduced tidal amplification is damped at high river discharge (Dykstra, Dzwonkowski & Torres, 2022). However, because storm surge can still reach the dam, water levels are increased and flood duration is longer than in the case of a river or coastal flood alone (Dykstra & Dzwonkowski, 2021).
Here we discuss how dam construction influences storm surge, tides, and other long period waves (period of hours to days) dynamics in naturally convergent channels using data collected in Charleston estuary, South Carolina, over time scales ranging from overtides (3 hr) to setup-setdown (∼4-10 days). We analyze long term water level observations from 31 stations and we compare long wave propagation on the dammed Cooper River and undammed Ashley and Wando rivers. We generalize the analysis with the aid of a semi-analytical, onedimensional model and we summarize the results in a friction-convergence parameter space. The primary scientific contributions of this study are developing mechanistic understandings of long waves-beyond tides-in (a) convergent estuary channels and (b) estuaries with dams.
Long waves in estuaries can be generated by astronomical forcing (tides), storms (e.g., surge events) or other oceanographic processes. Each type of long wave has a typical frequency band (or bands) at which they travel. Tides and overtides have astronomically forced frequencies (e.g., M2, M4, M6). Long waves from coastal setupsetdown are atmospherically forced (e.g., wind) and commonly have longer periods with larger ranges than tides and overtides, forming the so-called weather frequency band (e.g., 1-10 days; Yankovsky & Iyer, 2015). Setupsetdown events exceeding an amplitude threshold are classified as storm surges (e.g., Famikhalili et al., 2020). This threshold can be site specific or determined statistically (see Section 4).
On continental shelves, tides are inertial-gravity waves and setup-setdown are vorticity waves. In confined channels, however, both propagate as remotely forced long gravity waves (or diffusive waves; Yankovsky & Iyer, 2015), suggesting that theories and methods developed to study tidal dynamics can be extended to model long waves in the weather frequency band (see also Famikhalili et al., 2020;Proudman, 1955;Spicer et al., 2019).
Idealized, naturally convergent alluvial estuaries reach hydrodynamic equilibrium in the presence of tidal forcing through the balance between landward funneling and friction (Lanzoni & Seminara, 2002). The former generates partial reflections and tidal amplification, the latter attenuates tides (Friedrichs & Aubrey, 1994;Friedrichs et al., 1998;Jay, 1991;Schuttelaars & Swart, 2000;Wright et al., 1973). Partial reflection occurs to enforce continuity in tidal discharge and amplitude when phase speed c and/or cross-sectional area changes, which can occur due to changes in width, depth, or frictional effects (e.g., bottom roughness; Battjes & Labuer, 2014;Dronkers, 1964).
The general solution describing long wave propagation in an estuary contains both an incident and a reflected wave (Dronkers, 1964):
where x is a landward oriented spatial coordinate with origin at the coast,
T is angular frequency (T is wave period), A and B are amplitudes at x = 0 and Re denotes real term. The combined incident and reflected wave is termed "total wave" hereafter. The terms ωt and kx constitute the phase φ (i.e., φ = ωtkx), which decreases landward for the incident wave and decreases seaward for the reflected wave. Because the effect of Earth rotation is negligible in narrow channels, in the absence of friction ω and k are consistently related for all wave periods through the inviscid dispersion relation (ω = k ̅̅̅̅̅ gh √ ) and appear explicitly in Equation 1.
At a barrier, a fully reflected wave with a phase shift of zero degrees originates and constructively interferes with the incoming wave (i.e., 0°∼ φ rφ i , where the subscripts r and i represent reflected and incident waves, respectively). The reflected wave attenuates seaward due to friction and (potentially) area divergence and becomes progressively less in-phase with the incoming wave (i.e., constructive when 0°< |φ rφ i | < 90°). In the case of wave reflection on a barrier, constructive inference is typically greatest near the reflected boundary (e.g., Chernetsky et al., 2010).
At a location of partial reflection, the phase difference between the incident and reflected wave is non-zero and the net phase difference depends on the sum of all reflections throughout the estuary, which produces a reflected wave with a phase shift relative to the incident wave from -180°to 180°. The reflected wave amplitude is often small compared to the incident wave that continues (i.e., transmitted wave), provided the geometric or wavenumber change is small (Battjes & Labuer, 2014). Because partial reflections are distributed throughout a convergent estuary, their net effects are cumulative and contribute to the resulting wave amplitude and net phase progression. Dam effects, in contrast, are not cumulative but induce full reflection (i.e., no transmission) at a single location, generating a fully reflected wave that propagates seaward along with the partially reflected waves generated by funneling.
Convergence ∆ is a non-dimensional measure of funneling for an idealized estuary of mean depth h and exponentially decreasing area a in the landward direction (Jay, 1991):
where g is gravitational acceleration and λ is tidal wavelength. Τhe e-folding length of cross-sectional area L a determines funneling rate 1/L a and is found by fitting the equation a(x) = a 0 e -x/L a to measurements, with a being cross-sectional area and a 0 being cross-sectional area at the mouth. The parameter ∆ captures the relationship between tidal wavelength (or wavenumber; i.e., ∆ = 1 2 k -1 L -1 a ) and geomorphic funneling. It indicates that convergence scales with wave period.
Frictional effects r, which depend on the square of tidal velocity U t , can be linearized as:
where C d is the drag coefficient. The parameter r is often made non-dimensional with angular frequency ω (i.e., r ω ) and it is used to classify estuaries as strongly dissipative if r ω >> 1 and weakly dissipative if r ω << 1 with respect to long waves (e.g., Lanzoni & Seminara, 1998).
When funneling and frictional effects balance, tidal amplitude is nearly uniform in the landward direction and in a condition called critical convergence ∆ C . The condition occurs when Δ = ∆ C ≈ 1. Convergence is also used to classify estuaries. Estuaries are said to be weakly convergent when 2∆ << 1 and strongly convergent when 2∆ is close to 1. Funneling effects become negligible when 2∆ >> 1 (Friedrichs, 2010;Lanzoni & Seminara, 1998). Estuaries are often considered to be at or below critical convergence for semidiurnal and diurnal tides. Weakly convergent, strongly dissipative behavior is commonly assumed and are conditions for which tidal models are simplified to a single incident wave with the rate of frictional decay offset by a constant funneling rate (see Friedrichs, 2010;Lanzoni & Seminara, 1998;Savenije et al., 2008;van Rijn, 2011;cf. Jay, 1991).
The classification of estuaries based on convergence and friction can be extended to other long waves, including setup-setdown and storm surge waves. Because the period T ∼ 1 ω of these waves is potentially longer (and more variable) than that of semidiurnal tides, the classification of estuaries for surge and setup-setdown waves in terms of r ω and ∆ may differ significantly from that based on tides (see results). Additionally, non-linear interactions between tides, river flow and surge waves may alter frictional resistance, further affecting estuarine classification (see e.g. Familkhalili et al., 2022).
In this section we broaden the classification of estuaries of Section 2.2 to other long waves. This is done by defining convergence and dissipation based on the estuarine geometry and wave properties. The wavenumber of a frictionless long wave k 0 = 2π λ is modified to account for acceleration and friction (k f ; Dronkers, 1964) or acceleration, friction, and convergence (k j ; Jay, 1991) as:
The wavenumber in the presence of friction is complex. When friction is important in a constant width channel, wavelength and wave celerity are smaller (Equation 4b) than in a frictionless estuary (Equation 4a). Convergence is a real term that influences acceleration and further modifies the wavenumber (Equation 4c).
As the wavenumber approaches zero, wavelength becomes infinitely long, the phase stops progressing landward, and a standing wave forms. Under these conditions, the amplitude of the incident and reflected wave are equal (i.e., A = B in Equation 1). While a standing wave wavenumber may not affect wave amplitude in time or space, Jay (1991) shows that additional funneling effects are independent of wavenumber and scale with convergence. Thus, for given r ω , peak amplification from funneling occurs when k → 0 (see Supporting Information S1) and wave amplitude is independent of the dynamics associated with incident and reflected waves (Equation 1), that is η(x) = η 0 e 0.5xL -1 a .
(5)
For each wavenumber, peak amplification occurs when k → 0, that is
Journal of Geophysical Research: Oceans
Equations 6a-6c shows that for small convergence, the solution for k j becomes equivalent to k f . For small friction, ∆ tends toward 1-in agreement with previous results balancing convergence and acceleration through scaling assumptions (e.g., LeBlond, 1978;van Rijn, 2011;Winterwerp & Wang, 2013). In the absence of convergence (i.e., k f ), friction attenuates tides. However, when convergence is modeled in terms of a reflected wave (e.g., Equation 1) and r ω > 1, k f approaches the same results as k j (e.g., amplitudes, see Text S1 and Figure S1 in Supporting Information S1, and Section 5.3).
When convergence is balanced by acceleration and friction in Equation 6c, a complete solution of critical convergence and peak amplification is found. The complex form of Equation 6c is (using a phasor, see Supporting Information S1) lim
where the imaginary component ∆ + is the convergence at which maximum wave amplification occurs,
and the real component ∆ C is critical convergence,
Thus, the convergence parameter ∆ at which the maximum wave amplification occurs (Equation 7b) is offset from the critical convergence condition (constant amplitude condition) by the friction factor r ω . For large r ω , the offset becomes negligible. The importance of ∆ + and ∆ C to model amplitude variation of long-waves in a funnel shaped estuary is explored in Sections 5 and 6. Details on the derivation of Equation 7 are presented in Supporting Information S1.
The Ashley, Cooper, and Wando rivers flow into Charleston Harbor, a partially mixed estuary on the South Atlantic Bight (Figure 1). The city of Charleston lies at the confluence of the Ashley and Cooper Rivers. Since the mid 20th century, flood frequency in Charleston increased from 2 to 25 events per year and is predicted to more than double in the next half century (Morris & Renken, 2020;Sweet & Park, 2014). Despite large scale dredging and harbor development (U.S. Department of Commerce, 1989), observations of mean tidal range (1.6 m) in Charleston Harbor show only minor changes since the 1850s (Talke & Jay, 2020). Subtidal water levels are strongly influenced by coastal setup-setdown, with amplitudes of ∼10-20 cm, much greater than fortnightly tides (e.g., Msf ≲ 5 cm at all stations).
The Charleston Harbor watershed is small, not extending beyond the coastal plain (2,860 km 2 ), and historically (pre-1942) had a combined mean river discharge of less than 10 m 3 s -1 (Kjerfve & Magill, 1990). Landward of the harbor, the watershed areas of the Ashley, Cooper, and Wando rivers are respectively equal to 915, 1,550, and 293 km 2 , and the main channels have relatively similar lengths (∼90, 80, and 50 km, respectively). Each river is shallow and convergent in depth, width, and area (Figure 1, see cross-section DEM Figure S4 in Supporting Journal of Geophysical Research: Oceans 10.1029/2023JC020498 Information S1). Historical maps suggest that, prior to the 19th century, river channels had similar geometries and extensive Tupelo-Cypress swamp headwaters (e.g., Faden et al., 1780).
A river to the north of the Charleston watershed, the Santee River, continues to influence freshwater flows and sediment transport in the harbor due to human interventions. An early 19th century canal connecting the Santee River to Charleston through the Cooper River swamp was replaced with a larger channel that diverted most of the Santee River flow down the Cooper River through a series of dams, starting in 1942. The Santee River diversion resulted in rapid siltation of Charleston Harbor and-in 1985-most of the river flow was diverted back to the Santee River via Lake Moultrie Reservoir (Althausen & Kjerfve, 1992;Kjerfve & Magill, 1990), with a consequent decrease of Cooper River discharge (for further details, see U.S. Department of Commerce, 1989). The post rediversion 1985-2022 mean discharge of 144 m 3 s -1 is meant to stabilize saltwater intrusion and has very limited variability (beyond daily, or twice daily, water releases, called power peaking; e.g., Jay et al., 2016). The Ashley and Wando river discharges remain small. The analysis presented below pertains to the 1985-2022 rediversion period when changes in river discharge and bathymetry were relatively small (Army Corp of Engineers, 2015).
Twenty-Five water level records spanning from between 1 and 104 years of length are available from the United States Geological Survey (USGS), the National Oceanic and Atmospheric Administration (NOAA), and the National Estuary Research Reserve, and were used in this study (Figure 1, Table S1 in Supporting Information S1). Previously unpublished USGS records were obtained and made available to the public (Dykstra et al., 2022a). To supplement spatial gaps between water level stations, we installed seven additional stations in 2021 using HOBO data loggers (Dykstra et al., 2022a(Dykstra et al., , 2022b)). In total, 32 stations are used in this study and are labeled with the first letter representing the body of water (i.e., H: harbor, A: Ashley River, C: Cooper River, W: Wando River) followed by the along channel distance inland from the estuary mouth in kilometers (e.g., H0 is Fort Sumpter at the harbor mouth). Nearby non-coincident stations within 500 m of each other were assigned the same name (e.g., H0). The longest record, at Charleston Harbor Customs House (H06), has a backup gauge, which we used to supplement gaps and identify them as one station.
All water levels were referenced to the North American Vertical Datum of 1988 and were subsampled to a common hourly interval. Additional discharge data were accessed from USGS 02172002 (station C80). While some data do not temporally overlap, all measurements were concurrent to site H06. Specific details on data sources, access, and length of records at each station is in Supporting Information S1 (Table S1).
Local ground surface and bathymetric elevations were taken from a merged digital elevation model (DEM; CIRES, 2014). Poor data were removed (e.g., water surface elevations in inland reaches). The 3 m resolution Cartesian coordinate DEM was subsampled to analyze the geometry of each river along the longitudinal axis. Following the centerline, at 10 m intervals, perpendicular cross-sections (2 m width resolution) were interpolated to create a longitudinal DEM (Figure S4 in Supporting Information S1; Szot & Dykstra, 2022). To characterize channel geometry (e.g., area, width, and mean depth), the water surface elevation of Low Lower Water was approximated at each station (3.5 percentile based on 1/24.8 hr) and linearly interpolated.
The Ashley, Cooper, and Wando river channel geometries show consistent landward funneling. Landward decrease in cross sectional area is presented in Figures 2a and 2b shows the contributions of width and depth to area for the Cooper River. For each of these parameters, the landward decay is described in terms of a constant efolding length (Figure 2, Table S2 in Supporting Information S1). Overall area funneling 1/L a of the Ashley and Wando are nearly identical (∼1/9 km -1 ) and more than twice the Cooper (1/25 km -1 ; Figure 2a). While estuarine Journal of Geophysical Research: Oceans 10.1029/2023JC020498 funneling is traditionally observed and parameterized using width convergence only, assuming a flat horizontal bed (e.g., Friedrichs & Aubrey, 1994;Prandle & Rahman, 1980;Winterwerp & Wang, 2013), here we consider funneling in terms of area to account for both contributions of width and depth to channel convergence (Figure 2b). Interestingly, in the Wando, depth convergence exceeds width convergence (Table S2 in Supporting Information S1), suggesting that frictional effects may increase in the landward direction.
Observed water levels were decomposed into subtidal and tidal components using a Lanczos filter with a cutoff period equal to 1.7 times the window size (e.g., Dzwonkowski et al., 2015). For the signal in the tidal band, a sensitivity test revealed that the least spectral leaking occurred with window sizes of 35 hr and >45 hr. For water levels in the weather frequency band, the magnitude of setup-setdown was often similar to the ∼24 cm intraannual variability of the South Atlantic Bight (Parker, 2007). To differentiate setup-setdown from intra annual variability, a 35-hour-2-month Lanczos band-pass was utilized, a process that also removed sea-level rise, which is required to make data stationary (Ghanbari et al., 2019). Thus, our approach captures setup-setdown events that have a period of ∼1.5-60 days.
Setup-setdown and storm surge were next defined by first rank ordering the peak heights of all subtidal events in the H06 104-year record. For peak heights less than 3 day apart, only the largest event was retained. Storm surge was then defined from the tail of the probability distribution of events, which typically exhibits a distinct shape that is differentiated from non-extreme events by a threshold (Ghanbari et al., 2019). The probability distribution of setup-setdown heights were fit with a Generalized Pareto Distribution (GPD) using a peak-over-threshold method (i.e., partial duration series). The minimum event threshold was identified by gradually increasing the threshold value (starting from the median) until the shape and scale parameters converge (e.g., Ralston & Geyer, 2019), and was found to be 43 cm (Figure S5 in Supporting Information S1). A total of 32 storm surge events were above this threshold, with 14 occurring post rediversion (after 1985). Due to the peak being resolved from low-passed data, the method could underestimate the height of short duration events and, for duration, does not differentiate the surge from the period of concurrent setup. Our approach may preclude some storm surge events, such as fast-moving tropical storms with a time scale of 12 or 24 hr (see e.g. Familkhalili et al., 2020Familkhalili et al., , 2022 for their assessment of storm surge time scales). Our approach is required to clearly distinguish storm surge and setup/setdown from tides and diurnal power-peaking in frequency space. An assessment of the non-tidal residual suggests that the consequences are minimal.
Observations at H06 were used to evaluate the propagation of storm surge and its magnitude transformation along each of the rivers. For setup-setdown, the lag time tx at each station to the Charleston Custom House (i.e., H06) was determined by finding the temporal offset with the highest cross-correlation between H06 and the measurement. A ratio of the measurement at time lag tx to H06 at time t was determined:
Harmonic analysis (Pawlowicz et al., 2002) was used to evaluate tide amplitude variability and phase progression in each river and investigate reflection effects. To minimize the effects of different lengths of record, only the periods with good spatial coverage and consistent diurnal power peaking were used. This includes 30 days in 2021 (26 May-1 June, 4-14 June, and 4-18 August), a period long enough to minimize potential contamination from fortnightly tidal constituents, which were estimated to be ≲5 cm during 2021. In addition, only constituents with signal to noise ratios >2 were retained (Pawlowicz et al., 2002). Tidal reflection at the Cooper River dam was quantified by separating incident and reflected wave trains for M2 and M2 overtides at C73, C80, and C84 (Díez-Minguito et al., 2012), covering an 11-km region. Because the real channel does not have an ideal frictionless geometry assumed by the wave train analysis (Díez-Minguito et al., 2012), results are used only to support other theory and modeling.
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For a more general investigation of reflected wave dynamics, observational results were compared to results of the 1-dimensional semi-analytical tide model from Talke et al. (2021), based on Dronkers (1964) presented in Section 2 (i.e., Equation 1). The model was chosen because the solution explicitly identifies contributions from reflected and incident waves. The model solves the long wave equation (Equation 1) with the frictional wavenumber k f (Equation 4b). Friction is further linearized to account for convergence (Equation 3). A single sinusoidal wave is prescribed at the open boundary, and a reflective (no flow) boundary condition is applied at the head.
The computational domain is divided into 1 km-long segments, each with segment-averaged width, depth, and drag coefficient. In each segment, Equation 1 is solved iteratively to determine A and B. The simulations presented below ended when the difference between successive approximations of A and B values was less than 0.1%. Model cross sections are rectangular and assume all flow is in the channel, river velocity U r < U t , and η << h. Alternative approaches using numerical models (e.g., Figueroa et al., 2022;Ralston et al., 2018) and many exponential analytical models (e.g., Friedrichs & Aubrey, 1994;van Rijn, 2011) cannot (or do not) explicitly separate incident and reflected waves (but see Jay, 1991 and Supporting Information S1).
We consider non-convergent (uniform) and convergent estuaries as schematically illustrated in Figure 3 in terms of streamwise changes of channel width, depth and drag coefficient. Characteristic dimensions, such as the efolding lengths L w = 35 km and L h = 91 km, are based on Cooper River data. A 300 km long domain was used to simulate pre-dam condition for both constant-width channel and convergent geometries. The long domain enables tides to completely damp out and removes the effect of a complete reflection. To simulate the presence of a dam, the model estuary domain was shortened to 84 km, with no change of e-folding length and geometry at the estuary mouth.
To consider friction caused by river flow, the drag coefficient in each segment was expressed as follows (Figure 3f):
where C dw is the long wave drag coefficient, C dr∞ is the river drag coefficient inland of tides, and a ∞ is the area inland of tides. The second term in Equation 9 causes the river contribution to friction to scale inversely with area, as commonly occurs in tidal environments (see e.g. Buschman et al., 2009;Kästner et al., 2019), but does not directly model river inflow.
Surge dynamics are explored by approximating amplitude and primary wave period of observed events. A similar approach was used by Famikhalili et al. (2020Famikhalili et al. ( , 2022)), supported by observations of estuarine long waves in the weather frequency band that have waveforms like tides (e.g., Proudman, 1955;Schumann & Brink, 2009;Yankovsky & Iyer, 2015). While useful for describing and understanding wave dynamics, our analytical approach is idealized and may be inappropriate for answering site-specific questions.
The model was validated on the Cooper River and was then utilized to study tidal dynamics across the friction and convergence ranges of commonly observed estuaries (i.e., r ω ∼ 0.3-10, ∆ ∼ 0-3; Talke & Jay, 2020). Friction was simplified with no river effects (C d = C dw ) or height changes, representing funneling in only the width. To capture the range of r ω and ∆, we made systematic changes in C d (0.0002-0.007) and L w (8-160 km), respectively, while other variables did not change from one simulation to the other (h:5 m, η 0 :1 m, U t at mouth:1 ms -1 , and ω:M2 frequency).
This section is organized as follows. Field data collected in the Charleston estuary are first presented to discuss how long-waves dynamics vary in presence of a dam. Results of the one-dimensional, analytical tidal model described in Section 2.3 are then analyzed to explain the observed behaviors of tidal constituents (e.g., M2 and overtides), storm surge, and setup-setdown waves. Finally, we generalize results using the model to investigate how wave dynamics vary in the friction-convergence parameter space. This exercise provides new insights into storm surge and setup-setdown wave dynamics in estuaries, and how emplacement of a dam may influence them.
Atlantic Ocean water levels at Charleston Harbor in the weather frequency band (setup-setdown) commonly exhibit longer periods (∼4-10 days) and smaller amplitudes (∼10-20 cm) than the semidiurnal tide. These dynamics are here described using data of a 2009 December winter storm (Figure 4), corresponding to the third largest (positive) storm surge event in the Cooper River recorded since re-diversion. We do not consider the two largest events because they are marked by sizable negative surges and power peaking anomalies.
Water level elevations measured in the Cooper River during the December 2009 event (Figure 4a) show semidiurnal (approximately tidal) fluctuations (Figure 4b) and a storm surge (Figure 4c). In Charleston, the storm surge peaked at 54 cm, with an annual exceedance probability of ∼0.12 corresponding to an ∼8 year return period (see GPD in Figure S5 in Supporting Information S1). The square-wave pattern of daily peak river discharges were similar during the storm, allowing most of the signal to be filtered (Figure 4d). Thus, power-peaking signal is not evident in the band-passed water levels (Figure 4c). Tides are strongly semidiurnal and became smaller through time, capturing a transition to the smallest semi-diurnal tidal amplitudes of the month (monthly beat from M2-N2-S2 harmonics; Figures 4a and 4b). Tidal low water elevations become higher in the landward direction (Figure 4a), indicating a decrease in tidal range and an increase in MWL. The tidal signal was also delayed, more at low water than high water, suggesting asymmetry and overtide generation.
The phase propagation of storm surge followed a similar pattern to the tidal waves, but with less attenuation (Figure 4). As the winter storm approached, Charleston subtidal water levels at H06 were setdown (<0 cm) and slowly increased 39 cm in 2 days (16-18 December; Figure 4c). Subtidal water levels then rapidly increased 38 cm to a peak on 18 December (19:00 GMT). In the landward direction, stations peaked one after another like Journal of Geophysical Research: Oceans 10.1029/2023JC020498 the tide (Figure 4c). Approaching the dam (C84), the peak was delayed 8 hr relative to H06 and occurred nearly simultaneously at the three closest stations (C63, C80, and C84). This standing wave-like pattern suggests that the wave stopped progressing due to reflection at the dam. Dam subtidal water elevations (C84) raised 37 cm in the 22 hr proceeding the peak surge height, indicating that unlike the tidal wave, the surge wave exhibited almost no attenuation. Journal of Geophysical Research: Oceans 10.1029/2023JC020498
Tidal constituent amplitudes in the Ashley, Cooper, and Wando rivers all increased from the mouth to a maximum, then decreased upstream (Figure 5). The amplitude of the largest tidal harmonic (M2) was 0.74 m at the mouth (H0) and amplified landward by 9% on the Cooper River and 19% on the Wando River (Figure 5a). The location of peak amplitude was further seaward in the Cooper River (rkm 22) compared to the more convergent Ashley and Wando Rivers (rkm 38). Overtides M4 and M6 exhibited a 3-8 fold amplification, much more than M2; likely due to shallow water frictional interactions (Figures 5c-5e). Interestingly, the M2, M4, and M6 peak amplitudes were nearly co-located in the Wando an Ashley Rivers, but in the Cooper River M4 and M6 peaked landward compared to M2, suggesting a possible frequency-dependent response associated with reflection from the dam.
All tidal harmonics decayed from peak to zero in a ∼10-15 km reach along the Ashley and Wando (A43-58 and W38-W49, respectively), but, on the Cooper River, they decayed more gradually and stopped decaying around rkm 60. Landward of rkm 60, tidal waves amplified, suggesting an influence of the dam. Harmonics with higher frequencies amplified more (Figure 5). As shown in details in the Supporting Information, M2, M4, M6, and M8 wave trains exhibit near linear wave frequency-amplification (η r /η i = 2,100ω + 0.08, Figure S7 in Supporting Information S1). The timing of tidal constituents relative to values at H06 show a general landward delay (Figures 5b, 5d, and 5f). In each river, the lag times of M4 and M6 were similar to M2, as expected, because they are primarily produced by nonlinear M2 self-interaction (e.g., frictional interaction; see Godin, 1999). Lag times in the Ashley increased landward, indicating a slowing wave celerity that is influenced by shallowing depths. Cooper River lag-times were somewhat shorter, indicating faster wave celerity, maybe due to the smaller convergence. The lag-time curve flattens as the waves approach the dam, indicating little time delay and a standing-wave like behavior. For lower frequencies, the lag-time curve begins to flatten earlier, further seaward of the full reflection point (Figures 5b, 5d, and 5f). The Wando M6 lag time decreased landward, perhaps indicating that the mechanism generating the overtide was changing (Gallo & Vinzon, 2005), because of the strong depth convergence.
An examination of longitudinal subtidal variability from coastal setup-setdown suggests that longer period waves attenuate less and have longer lag times than tidal waves (Figures 6a and 6b). On the three rivers, all observed setup-setdown magnitudes from 2021 were within 12% of H06, indicating small longitudinal variability and little consistent attenuation or amplification (Figure 6a). The setup-setdown also clearly progressed landward as a long wave, with a lag-time that increased monotonically with river kilometer (Figure 6b). Compared to tides, setupsetdown lag times were 2-3 fold larger, indicating a slower longer period wave celerity.
The 2021 setup-setdown had amplitudes and lag time progressions similar to the largest individual setup-setdown events at H06, which is classified as storm surge (setup-setdown heights > 43 cm, maximum = 88 cm; see methods and Figure S5 in Supporting Information S1). An evaluation of 12 storm surge waves since 1985 on the Cooper River shows that at mid-estuary, surge heights decreased by ∼10% and lagged by 4-5 hr (Figures 6c and 6d), similar to the 2021 setup-setdown waves (Figures 6a and 6b). Further landward, of the 10 events observed at Journal of Geophysical Research: Oceans 10.1029/2023JC020498
the dam (C79 or C83), 7 events amplified in the upper estuary and, in 5 cases, surge heights at the dam exceeded heights observed in Charleston (H06). Lag times also stopped increasing as they approached the dam (Figure 6d), with peak surge water levels occurring almost simultaneously, similar to the tidal observations (Figures 4b, 4d, and 4f). Comparing events by storm surge magnitude (ascending from red/small to blue/large in Figures 6c and 6d), the largest events were delayed twice as much as the smallest events (Figure 6d) and amplified more approaching the dam (Figure 6c). The reasons for these observations are explored below. Note that local wind forcing is likely a negligible factor in the evolution of water levels, given the similar evolution of different events and the relatively small fetch the sinuous channels provide for wind, even when blowing in the right general direction (e.g., Jay et al., 2015).
We next use the model introduced in Section 2 to investigate the influence of convergence and dam closure (i.e., partial and full reflection) on semidiurnal tidal amplitudes (Figures 7a, 7c, and 7e) and 2-day period waves (Figures 7b, 7d, and 7e) on the Cooper River. The model was validated on the Cooper River and closely replicates observed M2 waves in the harbor and Cooper River with RMSE of 0.039 m, n = 12 (Figure 7a). Model validation at higher frequencies is presented in Supporting Information S1 for M2 overtides (Figure S6).
The influence of convergence is studied by comparing results obtained for uniform non-convergent (red lines in Figure 7 and left column in Figure 3) and convergent geometries (blue lines in Figure 7 and right column in Figure 3). Similarly, the influence of dam closure on incident and reflected wave is studied by running the model for a width-convergence estuary with and without a reflective boundary (blue and yellow lines, respectively). For each geometry, incident (solid line) and reflected (dashed line) waveforms are shown in amplitude (Figures 7c and 7d) and phase (Figures 7e and 7f) to interpret the dynamics (note: Figures 5 and 6 show time lag in hours). We test model scenarios using sinusoidal waveforms with the same boundary amplitude (0.74 m) and two different periods, representing the M2 tide and a 2-day storm surge, a midrange period of commonly observed storm surge waves (Familkhalili et al., 2020), to investigate impacts of friction and convergence on wave propagation.
For a channel with uniform width and depth, the model predicts landward attenuation of the total (incident plus reflected) M2 wave across the lower estuary, followed by amplification closer to the dam with wave amplitude reaching values similar to those at the mouth (red line, Figure 7a). Separating incident and reflected waves shows attenuation and phase increase; landward for the incident wave and seaward for the reflected wave (Figures 7c-7e). It is important to note that if the reflected wave reaches the ocean (x = 0) with non-zero amplitude, an artificial reflection may occur (e.g., Liu & Gan, 2020). However, any artificial reflection at the ocean would attenuate and become negligible in the mid and upper estuary, that is, close to the dam. In particular, simulations with a longer estuary (not shown here) indicated that model results did not qualitatively change in the lower estuary results, suggesting negligible artificial reflection.
The 2-day surge wave has similar dynamics but exhibits smaller phase progression and amplitude change than the M2 tide (Figures 7b, 7d, and 7f). The M2 is faster than the 2-day wave as it respectively takes 2.9 and 3.6 hr to travel from the mouth to the dam. This difference is in agreement with the measured lag times of Figures 5b and 6d. The nonconvergent-no dam scenario has a total (reflected + incident) wave that was nearly identical in amplitude and phase to the incident wave of the non-convergent estuary with a dam (Figures 7c-7e), and is therefore not shown.
In the convergent channel, the incident wave amplitude exhibits a maximum ∼35 and 60 km from the mouth for the M2 and 2-day waves, respectively (Figures 7c and 7d). When a dam is present, the reflected wave amplitude exhibits a minimum for both M2 and 2-day waves approximately 20-25 km from the dam, at Rkm 60-65 (see blue dashed lines in Figures 7c and 7d), just downstream of the maximum reflected phase (Figures 7e and 7f). Incident and reflected amplitudes of the 2-day wave, combined with the small phase progression (Figures 7f), leads to the approximately constant wave amplitude observed in Figure 7b-hence, the apparent similarity between wave dynamics in convergent and non-convergent channels for the surge wave hides substantial differences. By contrast, changes of incident and reflected wave amplitudes of the M2 wave occur over a shorter distance; this makes them asymmetric (Figure 7c), resulting in the observed Sshaped blue curve in Figure 7a.
Phase progression for incident and reflected waves in the convergent channel changes rapidly near the dam (∼80 km inland), and suggests strong spatial gradients in wave celerity (Figures 7e and 7f). Moreover, a maximum is formed in the phase progression of the reflected wave near the dam, more prominently at the M2 frequency than at the storm surge frequency (blue dashed line, Figures 7e and 7f). The maximum phase location of the reflected wave leads to the non-intuitive conclusion that the reflected wave reaches the ocean boundary before it starts at the dam (Figures 7e and 7f). This behavior occurs because the phase shift in a partial reflection is nonzero and is strongly phase-locked in our configuration to the incident phase. The phase locking occurs due to partial reflection points that are distributed evenly throughout the estuary (see Sections 2.1 and 6.2, and below) with a slight variability in relative phase of the incoming and outgoing wave occurring for variations in velocity and hence friction (see the convergent, no-dam case).
The maximum reflected phase of Figures 7e and 7f occurs because of the interaction of full reflection (phase shift of zero degrees, progressive phase variation) and locally generated, convergence-driven partial reflection. Near the dam, substantial differences in amplitude and phase are observed between the case with a dam (blue line) and the no-dam case (yellow line), indicating the existence of a ∼20 km long region of dam influence (Figures 7c-7f). As dam influence subsides in the seaward direction, the phase difference in the reflected wave relaxes to the phase shift caused by partial reflection only (yellow dashed line, Figures 7e and 7f).
Reflection dynamics differ strongly by wave period (Figure 7). Phase and amplitude differences between the dam and no-dam cases persist for a longer distance (∼35 km) downstream of the dam for the 2-day wave than the M2 wave and are more pronounced for the reflected than the incident wave (compare blue and yellow lines in Figures 7c and 7d, or Figures 7e and 7f). The reflected M2 wave has a higher velocity than the 2-day wave due to both larger amplitude and higher frequency. The reflected M2 wave is consequently attenuated for frictional effects more rapidly than the reflected 2-day wave. The reflected wave influences the overall flow velocity, and therefore the frictional effects of the reflected wave is felt by both the incoming and reflected waves. For this reason, the incoming wave is slightly different in no-dam and dam cases, with a bigger influence observed at the 2-day frequency, likely because the reflected wave persists longer (yellow vs. blue lines near the dam in Figure 7c vs. Figure 7d).
Phase comparison between convergent and non-convergent geometries shows different interference. The nonconvergent-dam scenarios show landward attenuation of the total M2 wave across the lower estuary, followed by amplification, reaching almost the same amplitude at the dam as the mouth (red line in Figure 7a). Near the dam, phases of incident and reflected wave are similar, indicating constructive interference and thus amplification. Phase difference between incident and reflected waves is greatest in the lower estuary, resulting in destructive interference.
In a convergent estuary, funneling influences the incoming and reflected wave asymmetrically (Jay, 1991), with the reflected wave at the dam attenuating over a shorter distance due to the area divergence. Hence, for the convergent scenarios considered here, the transition to destructive interference (phase difference between incident and reflected waves greater than 90°) almost happens for the 12-hr wave, reaching 88°at rkm 72, and is not close for the 2-day wave (reaching 40°; Figures 7e and 7f). At locations seaward of ∼rkm 60, the phase difference between the incident and reflected wave is dominated by partial reflection and is 40°-47°for the M2 wave and 15°-33°for the surge wave. For the convergent-no dam scenario, similar phase differences occur because partial reflection dominates everywhere (yellow, Figures 7e and 7f). Thus, for the cases considered here, partial reflection always causes constructive interference. The larger M2 phase difference reduces constructive interference and causes the total M2 wave to be smaller than the total 2-day surge wave throughout most of the estuary (Figures 7a and 7b).
Summarizing the trends of reflected waves, the direction in which phases increase and amplitudes decrease are seaward for fully reflected waves and landward for partially reflected waves; a result of being phase locked to the incident wave (Figures 7c and 7e). By only analyzing total wave amplitude, the traditional approach of identifying the seaward extent of dam influence (e.g., Chernetsky et al., 2010;Ensing et al., 2015;Li et al., 2016), suggests that dam effects extend seaward for only 12 km (rkm 72) (Figure 7a). The decomposition into incident and reflected waves, however, suggests that dam effects exceed convergence effects for 21 km (to rkm 63), almost twice the distance seaward. For the Cooper River, the model indicates dam reflection increases the 2-day surge event high water levels in the landward half of the river, reaching up to a ∼35% increase near the dam (blue and red lines, Figure 7b).
The variability of tidal amplitude with friction and convergence is investigated by further simplifying the model: no full (dam) reflection, funneling is from width changes only (details in Section 4.2). Amplitudes are determined at a distance from the sea x equal to the e-folding length of cross-sectional area L a . Results are presented in Figure 8 in terms of relative amplitude defined as the difference between tidal amplitude at x = L a , η La , and tidal amplitude at the sea η o , normalized with η o .
In agreement with previous studies (Jay, 1991;Lanzoni & Seminara, 1998), tides amplify or decay depending on friction and convergence ( r ω , ∆). Generally, large values of convergence ∆ lead to tides that are amplified by 0%-39% (bluish coloring in Figure 8a). Nonetheless, the region of tidal amplification or parity is strongly modulated Journal of Geophysical Research: Oceans 10.1029/2023JC020498 by the friction parameter r ω . As friction r ω increases, the region of convergence ∆ with decaying tidal amplitude increases greatly (brown-yellow coloring in Figure 8a). The isoline of 1-corresponding to no amplitude change -has convergence varying over one order of magnitude within the parameter space (∆ ∼ 0.2-2) and corresponds to the analytically defined critical convergence ∆ C (continuous line in Figure 8). The line of maximum relative amplitude ∆ + in Figure 8 is also similar to the analytical prediction (dash line in Figure 8).
As expected, ∆ C and ∆ + increase with friction due to modeling friction with a complex wavenumber (i.e., k f , k j , Equations 4a-4c). For a given value of friction, however, amplitude does not necessarily increase with convergence. As convergence increases (a shift up in Figure 8), amplitude reaches a maximum value and then decreases until it becomes negligible (relative amplitude ≈1). This counter-intuitive result can be more clearly understood in the context of tidal waves amplifying in long funneling estuaries and not short coastal bays.
An important note here is that delineating frictional regimes in reference to dissipation alone is inadequate (i.e., dotted line at log 10 r ω = 0). In Figure 8, critical convergence divides the parameter space in different regimes: the lower weakly convergent regime (∆ < ∆ C ), where funneling effects are weaker than friction, and the strongly convergent regime (∆ C < ∆ < ∆ + ), where funneling effects are stronger than friction. Our regimes provide new delineations to the descriptions of Lanzoni and Seminara (1998).
As funneling becomes strong enough (high convergence), amplification reaches a maximum and then declines. This suggests a new regime needs to be defined for ∆ > ∆ + , which we call the hyper-convergent regime (∆ > ∆ + ).
Data pertaining to dominant tides in 23 estuaries are presented in the parameter space of Figure 8a (Figure 8b). Most estuaries are strongly dissipative (r > ω, i.e., log 10 r ω > 0) and many are near critical convergence (e.g., ∆ C ± ∆/4), such as the Cooper River, Mobile Bay, and Ems Estuary. Approximately half (11) the estuaries are in the weakly convergent parameter space. Consistent with results of Figure 5, the dominant semidiurnal tides in Charleston estuaries appear to be strongly dissipative and in the strongly convergent or hyper-convergent regimes (Figure 8b). In particular, dominant tides of the Ashely and Wando Rivers are more convergent and more dissipative than for the Cooper River (up and to the right in Figure 8b); however, because these factors balance, the overall funneling related dynamics are similar and show the distance across parameter space in this direction may have limited effects. Even though the Ashley River is more convergent, the Wando River is closer to ∆ + , where peak tidal amplification from funneling is greater and is consistent with observations (see Figure 5a). (Talke & Jay, 2020). An amplification value of -1 in panel (a) occurs when wave amplitude is completely attenuated (red coloring); a value >0 denotes amplification (green and blue coloring). Flow in estuaries in panel (b) is classified as weakly and strongly frictional (r = ω, vertical dotted line) and relative to critical convergence (Δ C ; solid line) as relative amplitude amplifies above or attenuates below the Δ C line. For a given friction, maximum amplification is reached at Δ + (dashed line). Field data are available from: Ashley, Cooper, Wando (this study); Mobile and Tombigbee (Dykstra, Dzwonkowski & Torres, 2022); Guadalquivir (Díez-Minguito et al., 2012); Ems (Talke & Jay, 2020); Cook Inlet (Danielson et al., 2016); gray squares (Lanzoni & Seminara, 1998, updated by Toffolon et al., 2006).
Research: Oceans 10.1029/2023JC020498 6. Discussion 6.1. Expanding the Friction-Convergence Parameter Space for All Long Waves 6.1.1
To gain further insight into different wave dynamics, we use the parameter space of Figure 8a to study the propagation of different wave types (e.g., overtide, tide, storm surge) in the Cooper River (Figure 9). To reduce the effects of dam-reflection, we limit the analysis to the mid-estuary region, and use mean channel geometry and observed wave characteristics at H08 to represent waves. In Figure 9, individual tidal constituents are shown as single points, the purple box represents typical storm surge and the orange box setup-setdown waves. For the idealized storm surge parameters (i.e., Equations 2 and 3), we use a period of 12-72 hr, following Familkhalili et al. (2020). It is interesting to note that in Figure 9 storm surge purple box is shifted rightwards of the smaller amplitude, generally longer period setup-setdown waves (orange box) indicating higher friction for large wave amplitudes. Further, long period waves, such as setup-setdown waves and some storm surge waves, have a large convergence ∆ (Figure 9). Wave period minimally effects the r ω parameter because the scaling of friction r with wave current U t is normalized by wave frequency ω.
Because different waves interact nonlinearly (e.g., Jay et al., 2016;Parker, 2007), some variability between model results (e.g., Figure 7b) and observed waves (e.g., Figure 6c) is expected, and the wave classification in parameter Figure 9. Long waves in friction-convergence parameter space for the Cooper River, evaluated at H08 (rkm 8) using the landward funneling rate of L a -1 = 25.1 km -1 . Layout follows Figure 8. The parameter space of storm surge and setupsetdown is only partially shown and extends from ∆ = 1.1 to 7 and ∆ = 2.3 to 16, respectively. Major tidal harmonics are shown with friction estimated from the amplitude of each harmonic observed at H08 (filled circles) and the total friction of the estuary (based on the mean semidiurnal current amplitude; open squares). The friction of M3, M6, and M8 are log10(r/ ω) < -0.5 and not shown. M2 modeled in a convergent geometry with a dam (i.e., Figure 7a) shows the longitudinal variability of M2 (blue line) at 10 km intervals (dots). Rkm 70 is near the minimum total amplitude. All other parameters were determined using observations except storm surge periods, which are 12-72 hr following (Familkhalili et al., 2020), and approximating wave currents as U t ≈ ηωL a h (Friedrichs, 2010).
10.1029/2023JC020498 space therefore depicts general tendencies (Figure 9). Enhanced frictional loss and nonlinear interaction of real estuaries usually only modify rather than completely change wave amplitude and phase progression (Familkhalili et al., 2020;Godin, 1999;Yankovsky & Iyer, 2015).
We next investigate how the area occupied by long-waves in the parameter space governs the observed response, depending on whether an individual wave is in the hyper-convergent, strongly convergent, or weakly convergent regime of Figure 9. The semidiurnal M2 tide is in the strongly convergent regime marked by amplification as observed (Figure 5a). Diurnal tides such as K1 are found in the hyper-convergent regime marked by amplification or near parity (circles). Setup-setdown waves, with relatively small amplitudes and large periods, are in the hyper-convergent regime of small or negligible amplification, in agreement with observed setup-setdown waves with nearly constant amplitude throughout the estuary (Figure 6a). Surge waves (purple box, Figure 9) exhibit more frictional effects (larger r ω ) than setup-setdown. Moreover, because surge waves have periods (12-72 hr) that are often smaller than most setup-setdown events, the convergence value ∆ can be smaller. Hence, storm surge is within a regime marked either by amplification (low end of its ∆ range) or a nearly constant amplitude, as with setup-setdown waves (compare Figure 8a, which shows relative amplification, with Figure 9).
As friction and convergence vary as a wave propagates through an estuary, the same wave may be represented in different parts of the ( r ω ,∆) parameter space (Figure 9). To illustrate this we refer to the modeled M2 wave of Figure 7. The M2 wave at the Cooper River mouth has large ∆ and r ω ≈ 1 (blue line, Figure 9) and falls in the hyper-convergent regime with funneling amplification. As the wave travels upstream, the wave transitions first into the highly convergent and then into the weakly convergent, highly frictional regime. The trajectory reflects increases in frictional effects ( r ω ) due to decreased depth and increased river velocity, but decreases in convergence effects ∆ as depth reduces and wavelength shortens. The maximum r ω value is reached at rkm 72. As the wave moves through parameter space, it crosses critical thresholds (compare the blue line from Figure 7a now shown in Figure 9): the location of peak M2 amplitude corresponds with ∆ + (∼rkm 20; Equation 7b), the peak incident wave amplitude corresponds with ∆ C (∼rkm 30; Equation 7c), and the minimum total wave amplitude (incident + reflected) corresponds with peak frictional effects (largest r ω ; ∼rkm 70).
Upstream of rkm 72, dam reflection effects become more prominent (i.e., amplitude of fully reflected wave exceeds the partially reflected wave) and r ω decreases due to decreased tidal velocity. This causes the wave to move up-left in parameter space with increasing convergence and decreasing friction and approach ∆ + in agreement with the observed amplification (Figure 5a). Like the M2 wave, a longitudinal transformation should affect all wave types and shift them in their respective regions of the parameter space.
Peak amplification from landward funneling at ∆ + (Equation 7b), lower boundary of the hyper-convergent regime (i.e., ∆ ≥ ∆ + ), occurs when acceleration is balanced by the forces of convergence and friction. This delineation is also marked by increased celerity ω k , due to the wavenumber approaching zero (see Section 2.2). Insitu results show that M2 amplified more from convergence and exhibited faster wave celerity than storm surge and setup-setdown (Figures 5 and 6), providing observational support that wave proximity in parameter space to ∆ + increases amplification and celerity. Within the hyper-convergent regime, amplification and celerity are inversely related to wave period and wavelength, becoming minimally affected by convergence and friction as ∆ >> ∆ + (Equation 7b). At this extreme convergence, incident and reflected waves have nearly the same magnitude (i.e., A ∼ B) and low spatial variability, as previously noted (see e.g., Figure 6a). This indicates that storm surge events of relatively short duration are more influenced by acceleration effects than long duration events and may amplify when near ∆ + in parameter space. Frictional effects are minimal, though, and additional friction from interaction with tides and river discharge adds temporal variability that may shift events horizontally in parameter space. These complexities partially explain the range of observed Cooper River storm surge dynamics (Figures 6c and 6d). Amplification of the dominant M2 tides in the Ashley and Wando rivers was similar to the Cooper River. Though more convergent, these estuaries are also shallower and more frictional, placing the waves further up and to the right in parameter space. Thus, the Ashley and Wando retain similar proximity to ∆ + as the Cooper (Figure 8b), leading to similar along-channel behavior (Figure 5a). For estuaries with weaker funneling, like the Guadalquivir, Gironde, or Potomac Rivers (L w ≈ 60, 87, and 230 km, respectively; Figure 8b), storm surge events would likely extend to both sides of ∆ + , suggesting medium duration events may amplify the most from funneling and intense short duration-large amplitude events could attenuate landward the most.
We examine the effect of dam closure on wave propagation in funneling estuaries by introducing a reflective boundary at x dam = 3L a in the model configurations of Figure 8a (i.e., constant depth, L a = L w , and no river flow). This length scale is chosen to limit the effects of resonance (see e.g. Cai et al., 2016). Figure 10a shows the maximum amplitude change caused by shortening an estuary with a dam relative to the no-dam scenario in terms of dη d (x) = (η damη)/η, where η dam is the amplitude of the short estuary. The maximum change is always at the dam (Figure 10b), so η and η dam are from x = 3L a . Results show that dam-induced amplification is large when convergence ∆ is small. The solution for ∆ + identifies amplitudes that are mildly reduced in the hyper-convergent regime (∆ > ∆ + ) from those that are substantially increased in the strongly and weakly convergent regimes (e.g., amplifying waves by >50% for ∆ < 0.5). A weak dependence of dam effects on r ω is observed, with slightly more amplification in more frictional configurations (Figure 10a).
The variable response in amplitude when a dam is introduced is due to constructive interference peaking at ∆ + , which occurs when incident and reflected wave phases are the same. The reflected wave, which contains only partial reflection from funneling in the no-dam case, increases in amplitude and phase with the addition of dam reflection (Figure 7). When convergence ∆ < ∆ + , the addition of dam reflection increases constructive interference between incident and reflected waves more than the increased phase difference reduces it. In the case of hyper-convergence ∆ > ∆ + , by contrast, dam amplification increases constructive interference less than the increased phase difference reduces it. Considering different wave types, amplification caused by a dam could simultaneously be large for overtides and irrelevant for setup-setdown.
The influence of a full reflection on wave amplitude decreases seaward of the dam, with spatial extent that reduces as convergence and friction increase (Figure 10b). We quantify the spatial extent of dam influence as the seaward distance in which the full reflection alters amplitudes by 10% or more, compared to the no-dam case (i.e., |dη d | > 0.1 in Figure 10a). We normalize results using the inviscid wavelength (i.e., λ = T√gh). Overall, the reflected wave propagates farthest in weakly convergent estuaries (small ∆ ∼ λ/L a ). Seaward propagation of Journal of Geophysical Research: Oceans 10.1029/2023JC020498 reflection effects is also largest in weakly dissipative estuaries and is reduced by friction. For the strongly convergent and weakly convergent regimes (i.e., ∆ < ∆ + ), the region of dam influence scales with the e-folding length of landward funneling (L a ). In strongly dissipative estuaries (i.e., 1 < r ω < 10), for example, the reflection from the dam significantly influences dynamics ∼1L a -1.5L a (for strong and weak convergence). This identified reflection zone of influence is similar in size to that of the weakly convergent estuary of Friedrichs (2010; ∼L a ) and much larger than that of van Rijn (2011; ∼L a /3).
Due to convergence effects (e.g., proximity to ∆ + ), in the strongly convergent regime (∆ C < ∆ < ∆ + ), the seaward extent of dam effects can counterintuitively increase with friction. While this is an interesting finding, the parameter space where dams most strongly amplify waves and produce the largest zone of influence is in the lower left of the parameter space, where convergence and friction are both weak (Figures 10a and 10b). In the parameter space shaded dark blue (Figure 10a), dams can nearly double long wave amplitudes. Under these conditions, reflection effects extend as far seaward as the wavelength itself, which in most estuaries would be the entire system.
Convergence affects long wave amplitudes in markedly different ways close to a dam than further seaward where dam effects are minimal (Figures 8a and 10a). Landward amplification from convergence occurs when ∆ > ∆ C and from dams occurs when ∆ < ∆ + , allowing amplification to be summarized according to our three regimes (Figure 9): (a) hyper-convergent, where amplification is from only convergence (∆ > ∆ + , white), (b) weakly convergent, where amplification is from only dams (∆ < ∆ C , dark gray), and (c) strongly convergent, where amplification is from convergence and dams (∆ C < ∆ < ∆ + , light gray). Further seaward, amplification from a dam located relatively far inland may be minimal (e.g., x damx > L a ).
From the classifications introduced into Figure 9, general patterns are deduced. For estuaries with dominant tides in the strongly convergent regime, such as the Cooper River (Figure 8b):
• dominant tides amplify from convergence and dams;
• storm surge is likely to be in the hyper-convergent regime-like setup-setdown waves-causing it to amplify with convergence but not dams (Figures 9 and 10); • overtides and other waves with a 4-8-hr period are in the weakly convergent regime, where waves attenuate landward because frictional dissipation exceeds amplification from convergence. Such waves, however, amplify in the landward direction near dams. The overtide effects scale inversely with period and are supported by observations showing that M6 attenuated more strongly landward than M4 where dams were not present (i.e., Ashley and Wando Rivers), but exhibited stronger amplification when a dam was present (i.e., Cooper River; Figure 5, Figure S6 in Supporting Information S1). We note again that overtides are also strongly influenced by M2 overtide production, and the overall frictional environment set by tides and other waves (Parker, 2007). Thus, though our parameter-space analysis appears to capture features of in situ overtides, full consideration of wave-wave interaction may modify conclusions.
Dam impacts may further change through time due to channel modifications and be explainable using the parameter space. Localized changes may increase funneling, such as dredging predominantly in seaward regions or land reclamation of inland regions, shifting waves up in parameter space. These common changes, along with length shortening from dam emplacement, describe the Ems and Guadalquivir River estuaries (Figure 8b; Chernetsky et al., 2010;Díez-Minguito et al., 2012;Ensing et al., 2015;Ruiz et al., 2014;Talke & Jay, 2020). As depth increased over time, the dominant semidiurnal tidal wave in the Ems estuary shifted from the weakly convergent regime to the strongly convergent regime, near ∆ + . This change in the parameter space suggests that modern funneling amplification is much larger while dam amplification has moderately decreased. From a different historical parameter space location, the same shift for longer period waves-such as storm surges-may have decreased flooding risks, while the same shift for short period overtides may have greatly increased the seaward distance of dam amplification, possibly contributing to their sediment transport and fluid mud problems (see also Dijkstra et al., 2019). On the Hudson River, New York, navigational dredging of inland regions approaching a dam in Troy, New York have reduced both convergence and friction, shifting the region down-left in parameter space. The resulting increase in incident magnitude, and the larger zone over which reflection
Journal of Geophysical Research: Oceans 10.1029/2023JC020498 matters, helps explain the doubling of amplitudes in Albany, New York, near the dam (see also Ralston & Geyer, 2019;Talke & Jay, 2020). Similar changes observed on the Cooper River and Tombigbee River, where storm surge is also observed reflecting, suggest dredging navigational channels to dams compounds dam effects, increasing dam amplification and the region affected.
The wave effects on water level are also expressed in long-wave currents, and thus also exert control on friction ( r ω ). Wave reflections, both partial and full, act to reduce the overall currents and friction parameter, allowing the vertical water level waves to amplify. The phase lag between the total water level and the total current (incident + reflected) is expressed in the derivation of k j (see Equation S8 in Supporting Information S1),
where solving the four-quadrant tan -1 reveals a phase range from 0°to -180°(see Figure S2a in Supporting Information S1 plotted in friction-convergence parameter space). Without convergence, friction only has a range of 0°to -90°, but when convergence exceeds 1, the phase lag ranges -90 to -180°. Future work in estuaries with continuous velocity measurements may use this phase lag relationship to determine the convergence regime or gain insights into the properties of the incident and reflected waves.
Dam construction can be motivated for various reasons, such as restricting salt intrusion, altering sediment transport patterns, and reducing ecological impacts (e.g., plankton blooms, fish migrations; Kimmerer et al., 2019), factors that can be driven more by secondary than first-order processes. However, studies of dam effects almost exclusively focus on the first-order dominant tides without quantifying the magnitude of dam reflection or accounting for dam effects on other waves (e.g., Arunpandi et al., 2022;Figueroa et al., 2022;Kidd et al., 2017). Observational results can also appear to conflict; strong dam-associated effects are found in some estuaries and none in others (Prandle & Rahman, 1980). These contradictory observations likely stem from considering different wave types in systems with different parameter spaces. Modeling studies like Figueroa et al. (2022) carefully cover a wide range of estuarine circulation parameter space (i.e., mixing number vs. freshwater Froude number; Geyer & MacCready, 2014) but their geometry and semidiurnal waves follow a Regional Ocean Model default case that place their study in regions beyond our modeled ( r ω , ∆) parameter space (L a > 3, ∆ > 10 or L a ≈ ∞, ∆ ≈ 0). Similarly, Du et al. (2018) call L a = 60 km "strongly convergent" and model a narrow range of low convergence (∆ < 0.45), concluding length is the key factor determining how estuaries respond to sea-level rise. However, we show that length becomes less critical as convergence increases and negligible as ∆ >> ∆ + (e.g., Figure 10b). Future studies of dam effects on secondary processes might be improved by identifying where primary effects from different wave types (not just M2) are in the ( r ω , ∆) parameter space .
As the number of dams in marine environments continues to grow (Figueroa et al., 2022;Tilai et al., 2019), engineers and managers should consider the potential effects associated with all wave forms. Kidd et al. (2017) argue that the regulation of secondary processes such as salinity intrusion by estuarine dams do not outweigh the costs and are often counterproductive. Our results suggest that storm surge amplification can occur from convergence and dams; hence, the influence of a dam on flood hazard should be considered. Estuaries traditionally identified as weakly convergent for dominant tides likely have some storm surge events that strongly amplify due to funneling (i.e., are strongly hyper convergent and near Δ + ); other storm surge events in the same estuary may be in the weakly convergent regime and are at risk for amplification from dams. Because funneling often decreases in tidal rivers toward a constant width channel (e.g., Jay & Flinchem, 1997), the reflection effects in a small ∆ regime may be larger than further seaward. Further, long period waves (such as the 2-day surge considered here) may be most influenced, as the distance waves propagate inland scales with wave period (Jay & Flinchem, 1997). Even beyond the head of tides, longer period waves may reflect off dams and increase flood risk (e.g., Dykstra & Dzwonkowski, 2021). For managing estuaries that are hyper-convergent for dominant tides, dam effects on short period overtides, and the subsequent transport patterns, may be overlooked. New and proposed barriers (e.g., for the Hudson River; Ralston, 2022) need to consider how the distance landward may interact with the local geometry for a range of surge and tidal conditions or compound storm tides. Lastly, management plans may need flexibility as dam amplification could also oscillate with natural variability, such as tidal cycles (e.g., spring-neap, lunar node), river flows, meteorological forcing, and seasonality, such as ice (Georgas, 2012;Wang et al., 2012), or change through time from humans directly impacting the geometry or sea-level rise extending marine processes landward.
The effects of dam closure on tides, storm surge, and other long-waves in naturally convergent and modified estuarine systems are investigated in the Charleston Harbor. Results are then generalized in terms of friction and convergence. Main conclusions of the study are summarized as follows:
1. Convergence ∆-which we define as the ratio of geomorphic funneling to wave frequency-is the most important factor controlling amplification in a dammed frictional estuary, followed by friction. 2. For a given friction r ω , funneling effects maximize wave amplitude and celerity when friction, convergence, and acceleration cause the complex wavenumber to approach zero (wavelength approach infinity), a condition at which we delineate ∆ + . 3. Dam reflection (full reflection) can increase or decrease constructive interference; reflection from a barrier amplifies waves when convergence is weak (∆ < ∆ + ) and attenuates waves when convergence is strong (∆ > ∆ + ). 4. The analysis in ( r ω , ∆) parameter space reveals that convergence and dam closure on long waves have distinctive dynamics which fall into three different regimes: weakly convergent, strongly convergent, and hyper-convergent (see Figures 8 and 9). 5. Storm surge can straddle different regimes in the ( r ω , ∆) parameter space, depending on period and amplitude, as well as local geometric characteristics. Because dominant tides are often near critical convergence (Figure 8b), the most likely location in the parameter space for storm surge is the hyper-convergent regime with minimal amplitude variability. However, for large amplitude, short duration events or weakly funneling estuaries, storm surge may be in the weakly convergent regime and amplify at a reflection point, e.g. dam. Thus, the characteristics of events that cause a flood hazard at a dam should be carefully considered with respect to the parameter space. In some estuaries, the most intense storm surge may occur near dams. 6. While both partial and full reflections propagate seaward, the critical difference is apparent in their opposite longitudinal trends: funneling induced partial reflections are cumulative-with an overall signal that is phaselocked to incident waves-and exhibit a landward phase increase and amplitude decrease, while dam induced full reflections are characterized by a seaward phase increase and amplitude decrease (Figures 7c-7f). 7. For the case of Charleston Harbor-due to the strong landward funneling-flood risks associated with the dam, from tide and storm surge, are shown in observations and modeling to be localized to the upper Cooper River. However, if dredging were extended inland, dam amplification may increase and cause amplified water levels further seaward. 8. The proposed analysis of wave propagation in terms of friction and convergence will help researchers and managers to approximate how funneling and dams affect estuarine processes and flooding risks, both now and under future changes from human development and sea-level rise.
data), CIRES (2014), and Dykstra et al. (2022aDykstra et al. ( , 2022b)). Individual stations are further detailed in Table S1 in Supporting Information S1.