We investigate here the effects of geometric properties (channel depth and cross-sectional convergence length), storm surge characteristics, friction, and river flow on the spatial and temporal variability of compound flooding along an idealized, meso-tidal coastal-plain estuary. An analytical model is developed that includes exponentially convergent geometry, tidal forcing, constant river flow, and a representation of storm surge as a combination of two sinusoidal waves. Nonlinear bed friction is treated using Chebyshev polynomials and trigonometric functions, and a multi-segment approach is used to increase accuracy. Model results show that river discharge increases the damping of surge amplitudes in an estuary, while increasing channel depth has the opposite effect. Sensitivity studies indicate that the impact of river flow on peak water level decreases as channel depth increases, while the influence of tide and surge increases in the landward portion of an estuary. Moreover, model results show less surge damping in deeper configurations and even amplification in some cases, while increased convergence length scale increases damping of surge waves with periods of 12–72 h. For every modeled scenario, there is a point where river discharge effects on water level outweigh tide/surge effects. As a channel is deepened, this cross-over point moves progressively upstream. Thus, channel deepening may alter flood risk spatially along an estuary and reduce the length of a river estuary, within which fluvial flooding is dominant.

An idealized analytical model shows that deepening an estuarine channel reduces the impacts of river flow on peak water level but increases the effects of storm tide.

A friction number shows the competing effects of surge timescale, depth, and convergence on water level amplitudes.

Channel deepening changes the balance of fluvial and coastal flood risks and moves the crossover between storm tide vs. fluvial-dominated flooding landward.

Understanding tidal, surge, and river flow dynamics and how they combine and interact to produce the maximum or total water level (TWL) is important for emergency planning and as an aspect of wave dynamics. It is also a problem that is changing rapidly, as sea level rises and systems are altered by engineering. This contribution therefore analyzes the relative influence of river flow and storm surge effects along the river–estuary continuum from a dynamical perspective that enables us to assess the effects of nonlinear interactions, geometry, and changing (time varying) conditions.

Many low-lying coastal and riverine areas have been affected by combined coastal and riverine floods over the last few decades (e.g., Jongman et al., 2012; Nicholls et al., 2007). In cases such as Hurricane Harvey (Gulf of Mexico, August 2017), flooding was driven primarily by precipitation and runoff (van Oldenborgh et al., 2017; Wang et al., 2018). Other flood events, such as Hurricane Sandy, were forced by the combined effects of tide and storm surge, i.e., by “storm tides” the sum of storm surge and tidal water level (Orton et al., 2016). Some storm events, like Hurricanes Irene and Irma, produce both coastal and inland flooding because both storm surge and river flow produce elevated coastal water levels in a spatially varying pattern (e.g., Orton et al., 2012; Ralston et al., 2013; Talke et al., 2021). Accordingly, a flood influenced by both storm tide and precipitation run-off is a “compound flood” (Zscheischler et al., 2018; Wahl et al., 2015). The relative timing of the coastal and fluvial forcing, and the timescale over which water levels are elevated, matters in terms of impact (e.g., Zheng et al., 2014). Storm surge flooding generally occurs first and for a shorter period (timescales of hours to a day or two) than river flooding, which may last for weeks or even months, particularly in regions with a large watershed and flat topography (e.g., Johnson et al., 2016; IPCC, 2014). The timing of storm surge relative to tidal high-water (Familkhalili and Talke, 2016) or the spring–neap tidal cycle also influences flood heights, even upstream of tidal influence (Helaire et al., 2020).

The spatial variability of compound flooding is influenced by the geometry
of an estuary and may change over time due to system alterations, including
channel deepening, sea-level rise, and wetland reclamation (Ralston et al.,
2019; Helaire et al., 2019, 2020). Recent studies have shown that
human-caused changes to the geometry of estuaries affect the dynamics of
long waves (see reviews by Talke and Jay, 2020; Jay et al., 2022), with
tidal range in some regions more than doubling (e.g., Winterwerp et al.,
2013). Similar effects are observed with storm surge; for example, doubling
the depth of the shipping channel in the Cape Fear Estuary was modeled to
increase the magnitude of a worst-case scenario storm surge in Wilmington
(NC) from 3.8

Here, an idealized approach is used, which enables a large parameter space
to be assessed and the following two dynamical questions to be investigated.

What factors determine the region in which river flow effects or tide and surge effects dominate the total water level?

How does the transition from coastal to fluvial dominance shift as geometry changes or as properties of storm surge (e.g., timescale and magnitude) and river flow (magnitude) change?

We combine a three-sinusoidal-wave analytical model based on Jay (1991) with the multi-wave and multi-segment approach of Giese and Jay (1989) (see Familkhalili et al., 2020 for details) to quickly query a parameter space or relevant factors and provide insight into how factors such as storm timescale and the relative magnitudes of different forcing factors influence the dynamics of compound flooding.

Both analytical solutions and numerical models are regularly used to explore the mechanism of surge and tidal waves propagation along an estuary (see Talke and Jay, 2020). While numerical models can simulate tidal wave propagation more accurately than analytical models considering the measurements in a real system, numerical models are typically calibrated for an existing bathymetric, meteorological, and boundary forcing configurations (e.g., Brandon et al., 2014; Bertin et al., 2012; Orton et al., 2012). On the other hand, idealized numerical models with simplified configurations can be used to develop sensitivity studies to investigate the effects of changing hydrodynamic variables on surge and tidal wave interactions in a system (e.g., Shen and Gong, 2009; Familkhalili and Talke, 2016), but a downside of these numerical approaches is that studying an entire parameter space is computationally expensive. In contrast, analytical models rely on fundamental underlying physics and are transparent. Thus, they are good tools to explain some of the factors (e.g., channel depth, convergence length, river discharge, and surge amplitude and timescale changes) that alter flood levels in an estuary.

We apply an analytical approach to investigate the TWL caused by river discharge, tides, and surge in an idealized estuary. Various forms of one-dimensional analytical solutions of tidal wave propagation have long been used for idealized and real estuaries (e.g., Dronkers, 1964; Prandle and Rahman, 1980; Jay, 1991; Friedrichs and Aubrey, 1994; Savenije, 1998; Lanzoni and Seminara, 1998; Godin, 1999). More complex idealized tidal models investigate overtide generation and evolution (e.g., Chernetsky et al., 2010), the effects of variable cross section and bottom slope (e.g., Savenije et al., 2008; Kästner et al., 2019), and the effects of multiple tidal constituents and river discharge (Giese and Jay, 1989; Buschman et al., 2009). Other studies have used a tidal model combined with regression analysis (e.g., Godin, 1999; Kukulka and Jay, 2003a) to investigate river discharge effects. Such idealized models, by the parameter space analyzed, can be used to obtain fundamental insights into how long waves in estuaries are affected by depth, convergence, friction, and boundary forcing.

In our approach, we develop an analytical model that is driven by three sinusoidal constituents and a constant river discharge. Our approach idealizes storm surge as the sum of two sinusoids and neglects factors such as the potential role of wetlands and the floodplain in order to gain insight into some of the important, along-channel factors that govern the system response to a compound event. Similarly, we neglect processes such as Coriolis acceleration, wind waves, and gravity waves, and focus on the specific case of an incident long wave that propagates from the coast in the landward direction and is eventually completely damped out. Though a reflected wave is produced by convergent geometry in analytical models (Jay, 1991), we neglect the partial reflections caused by depth and width changes and do not consider the case of a reflective upstream boundary. Such factors are important for tidal changes in many estuaries, particularly for locations that are near resonance such as the Ems (see Ensing et al., 2015) or near where total reflections occur (see Ralston et al., 2019). Moreover, we simplify our approach by considering only constant river flow conditions, a valid approximation for situations in which the timescale of a river flood event is much longer than a storm surge. These simplifications enable a solution that is much faster than numerical models and enables a tractable sensitivity study of storm surge and river flow effects on water levels for different depths, convergence, and boundary conditions.

We use an idealized one-dimensional analytical model developed by
Familkhalili et al. (2020) to investigate how combinations of tides, storm
surge, and river flow affect water levels in an estuary. In this model,
storm surge is approximated as the sum of a primary and a secondary
sinusoidal wave. A third sinusoidal frequency is reserved for the

One-dimensional long-wave propagation along an idealized, funnel-shaped
estuary is described by the cross-sectionally integrated equations of mass
and momentum conservation (e.g., Jay, 1991; Kukulka and Jay, 2003a;
Familkhalili et al., 2020):

The sectionally and vertically averaged velocity term in Eq. (3) (

We use a multi-segment approach (Dronkers, 1964) to divide the model domain
into

An example fit using two sinusoidal waves to a surge caused by Hurricane
Irene (August 2011) is shown in Fig. 2. The surge signal is calculated by
subtracting predicted tide from observed water level at Lewes, DE (NOAA
Station ID: 8557380). Fitting two sinusoidal waves approximates the surge
signal with correlation of

An example of decomposing surge into two sinusoidal waves. The red
circles represent surge and are calculated by subtracting predicted tide
from measured water level during Hurricane Irene (2011) at Lewes, DE (NOAA
Station ID: 8557380). The blue line is the model fit that is the sum of

Typical amplitudes, frequencies, and phases of the two component surge waves
are determined by fitting two sinusoids to 354 storm surge events from
Lewes, DE. These results are used to define the parameter space that we
investigate (Sect. 4) and are typical of coastal storm surge characteristics
on the Mid-Atlantic Bight. Only significant events with surges larger than
0.5 m are fit. The largest resulting primary surge wave amplitude was about
1.1 m, larger than but of the same order as the main tidal constituent
(

The presence of river discharge (

River flow alters the water surface slope, and this behavior influences the
spatial distribution of total water level (e.g., Fig. 1b). Here, we use the
tidally averaged one-dimensional equation of motion to investigate water
level gradients, following Kukulka and Jay (2003b) and Godin (1999). For
simplicity, the component of mean water level caused by the tidal Stokes
drift is neglected. The parameter

Previous studies (e.g., Ralston et al., 2019; Helaire et al., 2019; Talke et
al., 2021) showed that reduced friction due to increased channel depth can
alter the tidally averaged water level gradient (

The above tide–surge analytical model has previously been compared against two one-constituent analytical models (the Toffolon and Savenije, 2011, and Jay, 1991, tidal solutions) and idealized Delft-3D numerical model results for situations without river flow (Familkhalili et al., 2020). Results showed that our analytical model is capable of capturing tidal wave amplitudes that are in good agreement with numerical model results. In this section, we update the validation to include the effects of river flow and compare our results against idealized Delft-3D numerical model results using the same bathymetry and forcing (Type I). Following this, we compare our analytical model results against an idealized numerical model developed for the Cape Fear Estuary, North Carolina (Familkhalili and Talke, 2016). This numerical model simulates storm surge from tropical storms by using a parametric model of hurricane wind and pressure forcing that is applied over the continental shelf (Type II). Table 1 shows the model parameters that were used to compare analytical model results with numerical models.

Analytical model parameters used in this study. See Appendix A for a
description of the parameters. Nondimensional river discharge (

Analytical/numerical comparisons were made for a weakly convergent and
strongly dissipative estuary with constant depth of 5m and a width profile
defined by Type I (Table 1, see Fig. 1). The
estuary section of the model domain (

Figure 3 shows the spatial pattern of the dominant tidal constituent
(

Dominant tidal constituent (

In addition, results for the tidally averaged water levels (i.e.,

We further validate our analytical model results (Type II) with the
idealized numerical modeling of Familkhalili and Talke (2016). This model
includes a storm surge produced at the continental shelf and six semidiurnal
and diurnal tidal constituents. Upstream of river kilometer 12, the estuary
is convergent with an

Comparison of normalized surge amplitude as a function of depth
for an estuary resembling the Cape Fear Estuary at an inland location at the
approximate location of Wilmington, North Carolina. The dashed line is the
analytical model result, and the solid line is the numerical result. The
idealized numerical model uses a surge event with a mean amplitude of 0.6 m
at the ocean boundary (data from Familkhalili and Talke, 2016). The fill area
is the range of results due to different relative phase of the storm surge
and tide wave. The “analytical model” results are for a 12 h surge that had
an amplitude of 0.5 m and is evaluated at

The results of the model comparison (Figs. 3, 4 and 5) show that both the analytical and idealized numerical models produce broadly consistent results. Therefore, our neglect of acceleration in the subtidal model (Fig. 4) and the use of linearized friction is justified. Both numerical and analytical models are complementary tools. A 3D model with resolved bathymetry is clearly best used to evaluate the specific effect of bathymetric alterations in a particular estuary (e.g., Pareja-Roman et al., 2020; Helaire et al., 2019) or to run simulations using complex, real valued boundary forcing (river and coastal). However, our analytical model runs substantially more quickly than even the idealized numerical models, facilitating investigation of a larger parameter space. Moreover, numerical models cannot unambiguously separate tide, fluvial, and surge effects. Currently, the best-practice approach is to run the numerical model with and without relevant forcing; for example, by running a surge model with and without tides, one can approximate the effect that tides have on total water level (Shen et al., 2006). When combined, tide and surge waves travel faster (due to deeper water depth; see Horsburgh and Wilson, 2007), and frictional energy loss in each wave component is also larger (Familkhalili et al., 2020). Due to the multiple feedbacks and nonlinear interactions, decomposing numerical results into individual surge and tide wave transformations is inherently ambiguous. The analytical approach, while not including all interactions (such as the phase modulation caused by depth variability), is also able to individually estimate transformations in the primary surge and tide constituent amplitudes under different river discharge conditions. This approach, to our knowledge, has not previously been approached to understanding the fundamental bathymetric and boundary condition factors that influence compound events.

We use our validated analytical model to further investigate the effects of
channel depth, river flow, channel width convergence, and surge timescale
on the spatial evolution of water levels along estuaries. For all
simulations, the primary tidal constituent period and amplitude are fixed to
12 h (i.e., a semidiurnal or

Parameter space used in analytical model.

Nondimensional variables provide insights into which parameters produce the
greatest effect on system response. From the scaling of Eq. (3) (see also
Familkhalili et al., 2020), we derive the three most relevant independent
nondimensional variables.

Parameter (

The friction number

Parameter (

For plotting purposes, we define two additional nondimensional numbers:

In our models, we assume that the two surge waves are symmetric with a phase
lag (

A symmetric surge wave that is the result of two sinusoidal waves
(i.e., surge

We employ the validated model to study how bathymetry, river discharge, and surge characteristics affect water floods in an idealized estuary. First, the effects of surge amplitude and period on water levels are examined. Following this, the effects of river discharge and width convergence on surge amplitude are presented, and finally compound flooding of tide, surge, and river flow is investigated.

The influence of wave characteristics (i.e., period and magnitude) on
tidally averaged water level is tested by modeling a set of waves with
periods of 12 and 24 h and amplitudes of 0.5 and 1 m at the ocean
boundary (i.e.,

The effects of wave period (i.e., 12 and 24 h) and amplitude
(0.5 and 1 m at the ocean boundary

The effects of river flow

The rate at which a surge decays away from the ocean entrance varies with
river flow and surge period. Figure 8 shows the effects of river discharge
and surge period on the

The effects of convergence length scale and river discharge on
primary surge (12 h,

Model results also show that higher river discharge will increase the
damping of surge amplitudes (Fig. 8). When

Surge amplitudes also decay more slowly (larger

Combined contribution of tide, surge, and river flow to water
level for depths of 5 m (left panel subplots) and 10 m (right panel
subplots). Colors and the labeled contours denote water level. The total
water level

Consistent with other studies (e.g., Kukulka and Jay, 2003b; Hoitink and
Jay, 2016), both the analytically and numerically modeled water level slope
(

Long-wave propagation along an estuary is characterized by a balance of
inertial effects, friction, and convergence. Figure 9 shows the normalized
amplitude (

The convergence of an estuary influences surge amplitudes (Fig. 9), similar
to its well-known effects on tidal amplitudes (e.g., Jay, 1991). All surge
amplitudes decrease landward for all depth cases in a weakly convergent
(

We next investigate how variations in river flow influence the total water level (TWL), caused by the combination of tide, storm surge, and river discharge effects. The highest possible total water level (HTWL) during such a compound event occurs when the surge occurs at high water, coincident with peak river flow. Because the timing of a meteorological event is usually random relative to tides, and because peak surge usually precedes peak river discharge, HTWL rarely if ever occurs. However, it is a useful metric of the potential flooding. Such a worst-case scenario could occur, for example, when multiple storms occur in close succession. The HTWL therefore provides a way to compare different parameter regimes and evaluate the effect of long-term changes in the geometry of an individual estuary.

Comparison of the contribution of tide, surge, and river flow to
compound flooding between 5 and 10 m depth channel and

The HTWL (Fig. 10a and e) follows a pattern set by the contradictory
effects of river flow and marine forcing (tides and surge). Far upstream
(

Importantly, the HTWL is not merely the superposition of river flow, tide,
and surge effects considered in isolation. Rather, as shown by the
non-vertical contour lines for tides and surge (e.g., Fig. 10f and g),
increases in the relative influence of river flow (larger

Crossover point location for 7–15 m channel depth compared to the 5 m
case, (

Changes in the depth of an estuary, whether by dredging, sea-level rise, or
sedimentation and erosion, also exert a strong, spatially variable influence on
the HTWL (Figs. 10 and 11). When depth is small (5 m; Fig. 10a), the HTWL is
greater in the upstream domain (

The differences in the response of river flow and storm surge to a depth increase lead to a crossover point, which we define as the location in which river flow effects on HTWL are larger than marine effects for a given set of forcing conditions (see the zero-contour line in Fig. 11a). Since the crossover point moves upstream as depth increases (Fig. 12), processes such as dredging, erosion, or sea-level rise that increase depth can alter the relative influence of marine and river effects, for a given storm surge and river flow. Similarly, a decrease in mean river inflow, as has occurred in many river estuaries due to flow regulation, may also cause a landward migration in the crossover point (Fig. 12).

Other factors that influence long wave amplitudes also influence the
crossover point, including the period of the surge (Fig. 8), convergence
length

Other studies, such as Bilskie and Hagen (2018), have defined flood zone transitions between marine and fluvial dominance; close to the coast, tide and surge-based flooding dominates, while river floods dominate far upstream. In between, there is a transition zone with compound flooding in which both coastal and fluvial processes are important. Here, our model also suggests that the transition zone location is sensitive to changes in estuary geometry, such as depth, in addition to being dependent on the relative strength of river flow, tide, and surge amplitudes.

In this study, we have applied a new river–tide–surge analytical model to
investigate the interactions of tide, surge, and river flow along idealized
estuaries. The novelty of our approach is that we develop a quasi-linear
analytical model, previously applied to tides, that considers the nonlinear
interaction between tides, storm surge, and river discharge. To the best of
our knowledge, these processes (river flow

We show that the rate of damping in a storm tide (surge

Globally, natural and local anthropogenic changes in estuaries (e.g., sea-level rise, channel deepening for navigation and landfilling) produce alterations in tidal and surge amplitudes (see review by Talke and Jay, 2020, and references therein). This study shows that river flow and its interaction with tides and surge must also be considered when evaluating changes to water levels. For example, increasing the river discharge relative to tide velocity reduces the amplification of the surge wave. Moreover, channel deepening produces a reduction in the water level caused by river discharge, leading to a domain in which channel deepening produces lower water levels upstream but larger water levels in the estuary (Figs. 10–12; see also Helaire et al., 2019; Ralston et al., 2019). Our findings are consistent with other studies that find that reduced frictional effects (e.g., caused by channel deepening) can cause increases to tides and surge (see, e.g., Ralston et al., 2019; Talke et al., 2021). Overall, anthropogenic changes to estuary geometry and frictional characteristics can cause large changes in the amplitude and spatial distribution of compound flooding.

The data used in this study can be obtained from NOAA's Center for Operational Oceanographic
Products and Services website (

RF developed the model and carried out the model runs and postprocessing. SAT and DAJ designed the research. RF, SAT, and DAJ contributed to interpretation of results and writing.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank the two anonymous reviewers for their helpful comments.

This material is based upon work supported by the National Science Foundation under NSF grant nos. 2013280 and 1455350.

This paper was edited by Xinping Hu and reviewed by two anonymous referees.