Response of salt intrusion in a tidal estuary to regional climatic forcing

The DBE, which contains one of the longest undammed rivers in the eastern US, is a funnel-shaped estuary that extends from the narrow Delaware River channel at Trenton, NJ (River Kilometer 216) to the large lower bay connected to the Atlantic Ocean. The mean water depth in the DBE is about 7 m, with the deepest depth exceeding 45 m (figure 1). The tides in the DBE are predominantly semi-diurnal with up to 96% of the tidal variance contributed from M2 and S2 tides (Aristizábal and Chant 2013). The tidal amplitude and velocity increase upstream, with the amplitude approximately 1.5 m at the estuary entrance and the maximum tidal current approximately 1 m s⁻¹ (Wong and Sommerfield 2009). The DBE has two main tributaries, the Delaware River and the Schuylkill River, with the former contributing over 50% of the freshwater inflow to the DBE (Whitney and Garvine 2006). The DBE is also manually connected to Chesapeake Bay through the Chesapeake & Delaware Canal (C & D Canal) at River Kilometer 94.

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We constructed the flow conditions for idealized experiments to approximate a rapid transition from a high flow event to an extended, historically worst drought with persistent low flow conditions. Our analysis of streamflow drought was conducted on the two main tributaries that contribute freshwater to DBE. Drought events for each tributary were identified from multi-decadal daily streamflow simulations (1982–2020; more details in the supplemental material) generated from a process-based Distributed Hydrology Soil Vegetation Model (DHSVM; Wigmosta et al 1994, Sun et al 2015).

The drought condition was analyzed by the standardized streamflow index (SSI; Nalbantis and Tsakiris 2009). We calculated SSIs on both 3-month and 6-month timescales, with very similar results for the worst drought case in both tributaries. Here we present the results based on the 3-month SSI. The worst drought event was identified as the one with the lowest SSI and the longest duration (figure 2). For the two tributaries, the most severe drought since 1980 has an SSI value of −2.2, classified as an exceptional drought according to Hao et al (2014, see table 2). The low flow level was then determined by averaging flow during the worst drought period.

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The distribution of salinity in a tidal estuary is mainly a result of competition between the seaward river flow that moves the salt out of the estuary and a upstream salt transport associated with the baroclinic exchange flow and tidal motions (Hansen and Rattray Jr 1966, Monismith et al 2002, Geyer and MacCready 2014). At steady state, there is a balance between the two processes. Suppose in a tidal estuary, the tidally and cross-sectionally averaged salt balance equation can be written as follows (e.g. Monismith et al 2002)

$$\begin{equation} \mathcal{A}\left(x\right)\frac{\partial S}{\partial t} + \frac{\partial}{\partial x}\left(Q_r S\right) = \frac{\partial}{\partial x}\left(\kappa_x\left(x\right)\mathcal{A}\left(x\right)\frac{\partial S}{\partial x}\right),\end{equation} \tag{ 1 }$$

where κ_x is the dispersion coefficient in the along-channel direction (x), $\mathcal{A}$ is the area of local cross-section, Q_r is the river discharge, S is the salinity. At steady state and within the limit of salt flux solely provided by mechanism of gravitational circulation, the salt intrusion length X_S (the distance of the location of isohaline of salinity S measured from a reference point downstream) is proportional to $Q_r^m$ (see Monismith et al (2002) for more detailed derivation) where the scaling with constant cross-section area gives $m = -1/3$ (see equation (6) in Monismith et al 2002). In particular, X₂, defined as the location where the 2 g kg⁻¹ isohaline intersects the estuary bottom as measured from the bay entrance, will be used to indicate the salt intrusion length thereafter due to its significant correlation with the abundance across all trophic levels as reported by Jassby et al (1995). Empirical correlations with observational datasets suggest that this relationship could relax depending on the characteristic of the estuary. In Hudson River for instance, m is about −0.38 and −1.19 for low and high river discharge, respectively (Abood 1974). A weaker relationship was found in San Francisco Bay with $m\sim -1/7$ (Monismith et al 2002). In the DBE, however, Aristizábal and Chant (2013) found that $m\sim -0.103$ based on the results from numerical simulations with a series of constant river discharge. Cook et al (2023) reported that m is close to −1/5 in the DBE.

The adjustment timescale, the time it takes for salinity distribution in the estuary to adjust to a new condition, can be written in an analytical format by assuming there is an abrupt change in the freshwater flux in a partially-mixed estuary with length $\mathcal{L}$ (Kranenburg 1986, Hetland and Geyer 2004):

$$\begin{equation} T_\textrm{adj} = \frac{1}{S_0}\int^{\mathcal{L}}_{0}\mathcal{A}\left(x\right)\left|\frac{\partial S}{\partial \mathcal{Q}_r}\right|\,\textrm{d}x,\end{equation} \tag{ 2 }$$

where ${S_0}$ is the salinity at the estuary entrance, and $S(\mathcal{Q}_f, x)$ is the cross-sectionally averaged salinity in a steady state at a given $\mathcal{Q}_r$. MacCready (1999) reexamined the time-dependent response of a rectangular estuary to both changes in riverine flows and mixing using a two-layer model, and reported longer adjustment timescale in intermediate depth estuaries compared to shallow and deep estuaries. Using a three-dimensional numerical model in a configuration of an idealized prismatic estuary with constant vertical mixing and steady river flow, Hetland and Geyer (2004) revisited the classic theories of estuarine circulation, and found asymmetric responses of adjustment timescale to increasing/decreasing river flows, which cannot be answered by the linear theory of Kranenburg (1986). Instead, this asymmetry was attributed to quadratic bottom drag.

These theories, based on both theoretical and empirical correlations, provide a useful framework to estimate the salt intrusion length and the associated adjustment timescale. However, these theories do not answer how salt intrusion length and adjustment timescale would change under a combination of regional climatic forcing such as long-term drought and SLR, partly due to the lacking of observations on such future projections. And we try to examine this using a three-dimensional high-resolution model that will be described in section 2.4.

The DBE model was based on the Finite Volume Community Ocean Model (FVCOM; Chen et al 2003), which has been extensively used to study the nonlinear interactions between river flow, storm surge, and tides (Xiao et al 2021), and has also been instrumental in investigating compound flooding (Deb et al 2023) and enhancing storm surge forecasts during extreme events (Liu et al 2020). The model grid used for this study is modified from those in the earlier applications to include the broad shelf outside the DBE (figure 3). It consists of 126 722 triangle elements and 68 416 nodes with 30 uniform sigma layers in the vertical direction. The resolution of the model grid ranges from 4 km on the shelf to 100 m in the DBE with the finest resolution of O(30)m in the Delaware River channel. The sea surface wind and air pressure fields are obtained from NCEP Climate Forecast System Version 2 (CFSv2) Selected Hourly Time-Series Products (Saha et al 2014). Freshwater discharge for the model validation period is prescribed at two main tributaries using the USGS stream gauge data from Delaware River at Trenton, NJ (USGS 1463500) and Schuylkill River at Philadelphia, PA (USGS 01474500). The salinity on the open ocean boundary is extracted from the HYCOM database of GOFS 3.1: 41-layer HYCOM + NCODA Global 1/12^∘ Analysis (Metzger et al 2017). The tidal open boundary condition is specified with time series of surface elevations obtained from the Oregon State University TPXO global ocean tide database v9 (Egbert and Erofeeva 2002) and includes 21 tidal constituents: M2, S2, N2, K2, K1, O1, P1, Q1, MM, M4, MN4, MS4, 2N2, S1, 2Q1, J1, L2, M3, MU2, NU2, and OO1.

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The model is evaluated based on a three year-long simulation forced by the historical meteorological forcing from year 2010 to 2012. The salinity field is initialized from the climatological salinity distribution along the river kilometer system based on the Delaware River Basin Commission monthly salinity survey from 1979 to 2016, of which the values in general increase from the bay entrance to the upper bay. Then the model is validated against multiple data sets. An example of the model comparison with the observed water level at 12 National Oceanic and Atmospheric Administration (NOAA) tidal gauges and the observed salinity transect along the River Kilometer system (black line in figure 1) during the Rutgers research voyages in June 2010 are shown in figures 4 and 5, respectively. The results show that the DBE model well reproduces the hydrodynamics, in particular, the location of the salt front (e.g. the isohaline of 2 g kg⁻¹) which is crucial to understand the response of salt intrusion to regional climatic forcing.

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Following model validation, a series of simulations (table 1) are conducted in an idealized setting to investigate the effect of various regional climatic forcing factors, including SLR, shelf processes, and estuarine wind on salt intrusion. Because this study is mainly focused on salt intrusion into the main stem of Delaware River from the ocean boundary, in the baseline case (R1), the C & D Canal is assumed to be closed as a simplification to remove its effect. In each simulation, the model is initialized with the same low river flow condition extracted from the previously validated simulation at 28 September 2010 12:00:00 UTC (0 Day in model time). This initial condition was chosen because the salinity distribution in the river channel was in a quasi-steady state due to the low flow condition in September. A few days after this time, there was a high flow event that pushed the salt front downstream and reshuffled the estuary. Therefore it allows for the investigation on the internal characteristic of the estuarine adjustment in drought condition. The idealized river discharge is constructed to represent a quick shift from flood to drought conditions, starting with the peak flow of a high flow event observed in October 2010 (Xiao et al 2021), followed by a prolonged five-month drought condition with a persistent low river flow (figure 6), resembling the most extreme drought condition identified in section 2.2. Cases R2-R6 represent the scenarios with SLR ranging from 0.5 m to 4.0 m, in which the water elevation is uniformly augmented across the entire domain including the open boundary by assuming no shoreline retreat, i.e. keeping the channel geometry unchanged. The choice of SLR is inspired by the latest projections for relative SLR along the U.S. coastlines by Sweet et al (2022)(see their table 2.3). The salinity values on the shelf ($S_\textrm{obc}$) outside the DBE are reduced by 5% in Case R7 and they are increased by 5% in Case R8 compared to Case R1, which is motivated by the relative change (∼5%) in surface salinity on the Northeast Shelf over the past few decades (National Oceanic and Atmospheric Administration 2020). In particular, Case R7 represents the salinity decrease on the shelf potentially caused by southward transport of less saltier water arising from accelerated ice melting in the arctic in a warmer future, and R8 represents the salinity increase on the shelf potentially due to excess evaporation or other broader circulation shifts. The estuarine wind forcing is turned off in Case R9 to examine the effect of meteorological forcing on the salt intrusion.

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Table 1. Summary of simulations.

Cases	Sea level rise (m)	$S_\textrm{obc}$	Wind
R1	0.0	100%	On
R2	0.5	100%	On
R3	1.0	100%	On
R4	2.0	100%	On
R5	3.0	100%	On
R6	4.0	100%	On
R7	0.0	95%	On
R8	0.0	105%	On
R9	0.0	100%	Off

We started the analysis by examining how SLR affects salt intrusion length X₂. The temporal variations of tidally averaged X₂ with different SLR, shown in figure 6(b), indicate that X₂ remains relatively stable for all cases with respect to the initial value ($X_{2,0} \sim$ River Kilometer 115) in the first few days of the simulation, before the discharge pulses. The river discharge at Schuylkill River increases until reaching the peak within a day, followed with a ∼12 h delay by an increase in the Delaware River discharge. The summed discharge pulses from both rivers push the salt front location downstream until it reaches a minimum $X_\textrm{2, min}$. The minimum salt intrusion length ($X_\textrm{2, min}$) increases with SLR, ranging from River Kilometer 83 in the baseline case (R1) to River Kilometer 100 in the case with the highest SLR (R6). After the transient adjustment associated with the freshwater pulses, since day 17, the salt front starts to move upstream, as river flow drops to a base level during the simulated drought condition. As a result, the salt intrusion length increases until it reaches a new steady state, defined by the maximum value of X₂. The maximum salt intrusion length, $X_\textrm{2,max}$, increases with SLR, ranging from River Kilometer 127 in the baseline case (R1) to River Kilometer 152 in the highest SLR scenario (R6).

In addition to SLR, we also examined how changes in salinity on the shelf affect the salt intrusion in the DBE. The minimum and maximum of X₂ is about River Kilometer 82 and River Kilometer 125 in Case R7, about River Kilometer 85 and River Kilometer 129 in Case R8, respectively, indicating only a weak effect on salinity intrusion into the Delaware Bay. The effect of estuarine wind forcing also seems to not be significant as the maximum X₂ is around River Kilometer 127 when wind forcing is excluded, which is similar to that in R1 in which the wind forcing is included.

Figure 6(b) shows that different SLR cases have different adjustment timescales to equilibrium under drought conditions. To compare its variation with different scenarios, we define the adjustment timescale ($T_\textrm{adj}$) as an asymptotic exponential function between $X_\textrm{2, min}$ and $X_\textrm{2, max}$,

$$\begin{equation} X_\textrm{2,max} - X_2\left(t\right) = \left(X_\textrm{2,max}-X_\textrm{2,min}\right) e^{-\frac{t}{T_\textrm{adj}}},\end{equation} \tag{ 3 }$$

where $X_2(t)$ is the X₂ at time t with $X_\textrm{2,min}$ and $X_\textrm{2,max}$ for the minimum and maximum value in X₂, and equation (3) can be written as

$$\begin{equation} \ln\left({\frac{X_\textrm{2,max}-X_2\left(t\right)}{X_\textrm{2,max}-X_\textrm{2,min}}}\right) = -\frac{t}{T_\textrm{adj}} \quad ,\end{equation} \tag{ 4 }$$

which can be rewritten as

$$\begin{equation} T_\textrm{adj} = -\alpha/\beta,\end{equation} \tag{ 5 }$$

where $\beta = \ln({\frac{X_\textrm{2,max}-X_2(\alpha)}{X_\textrm{2,max}-X_\textrm{2,min}}})$ and α is time in days. This indicates that the steeper the slope ($\beta/\alpha$) is, the shorter the adjustment timescale.

The adjustment timescale ($T_\textrm{adj}$) for different scenarios are summarized in figure 7. Note that $T_\textrm{adj}$ varies nonlinearly with respect to SLR (table 2). Adjustment timescale was primarily influenced by SLR. The changes in the shelf salinity did not significantly affect the adjustment timescale, whereas $T_\textrm{adj}$ was 40 and 39 day when salinity reduced/increased by 5%, respectively. Also, without estuarine wind forcing, $T_\textrm{adj}$ in R9 decreased by 5 day compared to the baseline case.

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Table 2. Summary of adjustment timescale $T_\textrm{adj}$ (day).

	R1	R2	R3	R4	R5	R6	R7	R8	R9
$T_\textrm{adj}$ (day)	39	40	48	44	35	26	40	39	34

Figure 7 indicates that $T_\textrm{adj}$ tends to first increase with SLR, then after 1 m SLR it starts to decrease. This suggests that there is a regime shift in the estuarine flow with increasing SLR. To address this question, we compared the adjustment timescale, $T_\textrm{adj}$ to the Simpson number,

$$\begin{equation} Si \equiv \frac{N_{x}^{2}H^{2}}{C_{d}U_{t}^{2}} \equiv -\frac{g \gamma H^2}{\rho_0 C_dU_t^2}\frac{\partial S}{\partial x},\end{equation} \tag{ 6 }$$

a ratio of potential energy increases due to gravitational straining, and decreases due to tidal mixing (Stacey and Ralston 2005, Burchard and Hetland 2010, Geyer and MacCready 2014), where $N_{x}^{2}$ is along-channel buoyancy gradient, C_d is bottom drag coefficient, U_t is tidal velocity, and H is water depth, g is gravitation acceleration, ρ₀ is reference density, γ is haline expansion coefficient, and ${\partial S}/{\partial x}$ is along-channel salinity gradient. Here Si is averaged over a control volume with lateral boundary bounded by River Kilometer 65 and X₂. Si > 1 indicates gravitational circulation dominates and the estuary tends toward more stratified, and values much smaller than one indicate mixing dominates and the estuary tends toward well-mixed.

The time series of Si, shown in figure 6(c), shows that the magnitude of Si increases with SLR, particularly between 20 and 50 day in the simulation time when the salt front starts to move upstream to its new equilibrium position. The variation of $T_\textrm{adj}$ with respect to the time averaged Simpson number $\overline{Si}$ and SLR, presented in figure 8, shows that $T_\textrm{adj}$ tends to be separated into two regimes where $T_\textrm{adj}$ grows until capped by the limit under low SLR and starts to fall under high SLR, while $\overline{Si}$ keeps increasing with SLR, consistent with trend of the freshwater residence time which increases with SLR as well (not shown). Since the larger $\overline{Si}$ indicates that the advection of horizontal salinity gradient becomes dominant over the tidal mixing through the water column, the result suggests that for SLR greater than 1 m, it enters a different regime where the estuary flow gets more stratified and less well mixed under drought conditions.

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In fact, the role of SLR on $\overline{Si}$ and salt intrusion can be seen by comparing the evolution of each component in $\overline{Si}$ (figures 6(d) and (e)) as H increases with increasing SLR. Numerator $N_x^2H$ increases with higher SLR (e.g. between 20 and 50 simulation day) whereas denominator $C_dU_t^2/H$ decreases with higher SLR. Both contribute to a larger $\overline{Si}$. In other words, there will be positive feedback from both horizontal advection and tidal mixing on $\overline{Si}$.

The adjustment timescale, $T_\textrm{adj}$, is inversely proportional to the estuarine circulation, $T_\textrm{adj} \sim u_{e}^{-1}$. The estuarine circulation, u_e , can be scaled as

$$\begin{equation} u_e \sim -\frac{H^3 N_x^2}{\nu} \sim - \frac{H^2 N_x^2}{C_d U_t}\end{equation} \tag{ 7 }$$

where ν is (constant) viscosity (Hansen and Rattray Jr 1966, Hetland and Geyer 2004). That is, the estuarine circulation increases with increasing baroclinicity (${H N_x^2}$), and decreases with increasing drag (${C_d U_t}$), the same tendencies represented in the Simpson number (Si).

The two dynamical regimes can be explained by understanding the shift in the tidal dynamics, which do not have a linear response to SLR throughout the Bay. For SLR up to 1 m, increasing sea level increases the magnitude of the tidal currents linearly with SLR. In this case, the drag associated with the increasing tidal currents increases faster than the increase in baroclinicity caused by the increased water depth, with the net effect of reducing the estuarine circulation. Therefore $T_\textrm{adj}$ increases with SLR in this case. However, for SLR of 1 m and greater, the magnitude of the tidal currents, and hence also the associated drag, is roughly constant. This is most likely because the location of the primary integrated drag on the tidal wave propagating up the estuary has shifted upstream to shallower areas (figure 9, note that the rapid changes from positive to negative values and vice versa between 90 and 150 km occur where there are river islands, meander, or confluence in the channel, highlighting the role of local topography and bathymetry in altering the circulation), upstream of the area with strong salinity gradients. Now, increasing water depth increases the baroclinicity of the circulation with no compensating increase in drag, so the net response is that the estuarine circulation increases. Therefore $T_\textrm{adj}$ decreases with SLR in this case.

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One weakness of this approach is that we do not consider that the inundated area associated with SLR will increase. In other words, we assume that the human response to SLR is to build coastal walls to maintain the position of the present day coastline. While this assumption is reasonable for populated regions, it may not be true for coastal marsh regions. Different mitigation strategies to coastal defense may alter tides in different ways than our model predicts, thus change the point at which SLR alters the fundamental balance of the Bay and the response to drought. However, we believe that, generally, our result that it is possible for SLR to alter Bay response to drought in ways that are not just extrapolations of present day dynamical balances—or just generally alter the response timescale of the Bay to changing freshwater forcing—is robust, even though the details will heavily depend on local Bay geometry and human response to SLR.