Influence of Back‐Barrier Basin Geometry on Multiple Tidal Inlet Systems: The Roles of Resonance and Bottom Friction

Observations of barrier coasts around the world suggest that some systems do not conform to the O'Brien‐Jarret law. Here we explain this by investigating how resonance and bottom friction affect the response of tidal inlets to variations in basin geometry. Therefore, we develop a morphodynamic barrier coast model that is based on the stability concept of Escoffier for the morphological evolution of the inlets, coupled with an idealized hydrodynamic model that describes the water motion in the outer sea, inlets, and arbitrarily shaped back‐barrier basin. We find that the total tidal prism through all inlets is predominantly determined by the (cross‐shore) width of the basin and identify three regimes for this. First, a linear regime for narrow basins (i.e., basin width ≪ tidal wavelength) where a larger basin leads to a linear increase in total tidal prism. Second, a resonant regime for basins with a width around the resonant condition in which the total tidal prism reaches a peak. This resonance condition is a quarter tidal wavelength for basins without friction, which shifts to narrower basins as friction becomes stronger, down to 0.15 tidal wavelength. Third, a dissipative regime for wide basins (i.e., the cross‐shore basin dimension or basin width ≫ resonant condition) with sufficiently strong bottom friction in which the total tidal prism does not change for wider basins, because the tidal wave completely dissipates in the basin.


Introduction
Barrier coasts, covering around 10% of the worldwide coastline (Glaeser, 1978;Stutz & Pilkey, 2011), are often found near densely populated areas. They are important for their economic, ecologic, touristic value and as protection of the mainland coast (e.g., Oost et al., 2012;Wang et al., 2012). The morphology of barrier coasts changes continuously under the influence of tides, waves, and storms (e.g., Escoffier, 1940;de Swart & Zimmerman, 2009), making them dynamic systems. Following Davis and Hayes (1984), barrier coasts are typically categorized according to the relative dominance of either tides (macro-tidal) or waves (micro-tidal), or equal importance of tides and waves (meso-tidal); this study focuses on the latter two.
An important aspect in the natural evolution of barrier coasts is the evolution of tidal inlets since they are the link between the back-barrier basin and the outer sea. Already in the early 20th century the tidal prism of an inlet was linked to the size of tidal basins, where an almost linear response was observed to differences in basin size between tidal inlets in the United States (O'Brien, 1931;1969;Jarret, 1976). While these observations showed that a larger tidal basin corresponds to a larger tidal prism (for the same tidal range), the opposite was observed in the Western Dutch Wadden Sea (see Figure 1) after a large bay was closed off, the Zuiderzee in 1932 (e.g., Kragtwijk et al., 2004;Elias et al., 2012). After this closure, the basin width (i.e., cross-shore dimension) was reduced by approximately 60 km, which led to an increase in inlet cross section for the inlets closest to the intervention (e.g., Elias et al., 2012). One suggested explanation (Elias et al., 2012) for this is the shift from a (cross-shore) basin width close to half the tidal wavelength to a basin width closer to quarter wavelength at which resonance occurs (e.g., in the Bay of Fundy). Another example is the Pamlico Sound (NC, USA) where a large back-barrier basin is connected to the outer sea by just three tidal inlets.
The above illustrates that while we have some understanding of how the equilibrium configuration (i.e., size and spacing) of tidal inlets is affected by spatial variations in cross-shore basin geometry, the underlying mechanisms are still not fully understood. The goal of this study is to better explain how tidal inlets are affected by variations in cross-shore basin geometry and specifically how this is affected by resonance and bottom friction. In particular we aim to answer the following questions: • How does the cross-shore geometry of the back-barrier basin affect the equilibrium configuration of multiple inlets connected to one back-barrier basin? • What is the effect of a reduction of the (cross-shore) basin width (e.g., due to human interventions such as damming or land reclamation) on the equilibrium configuration of multiple inlets connected to one back-barrier basin?
To answer our research questions, we extend the idealized barrier coast model of Roos et al. (2013) to allow back-barrier basins of arbitrary shape. We choose an idealized model because it allows us to study the processes affecting multiple inlet systems in isolation and enables us to perform extensive sensitivity analyses. This extension allows us to simulate barrier coast systems with a back-barrier basin of arbitrary plan-view shape, instead of only a single rectangular basin as the model of Roos et al. (2013). Furthermore, the solution method employed results in low computational costs, making this model well suited to cope with the large spatiotemporal scales involved. This choice also allows us to examine a large set of initial inlet conditions, which is essential because initial conditions have been shown to have a large impact on the equilibrium configuration of tidal inlets in process-based models (Dastgheib et al., 2008).
This paper is organized as follows: The model is described in section 2.1 and the solution method in section 2.2. The results are presented in section 3, and the discussion in section 4, followed by the conclusions in section 5.

Model Formulation 2.1.1. Model Description and Geometry
In our idealized model the morphological evolution of the tidal inlets is governed by the stability concept of Escoffier, and the water motion is described by the linearized shallow water equations, forced by a tidal wave In the left column a length scale and north arrow are provided for reference. In the right column, shown in blue from left to right, are the tidal basin (with a wide and narrow part), the tidal inlets, and the outer sea. Shown in green is the mainland and the barrier islands. Key dimensions such as the (long-shore) basin length L and (cross-shore) basin width B are provided for the entire basin and for different parts of the Wadden Sea basin. The positive x-direction and -direction for both the basin (x b & ) and the outer sea (x o & ) are indicated as well.
on the outer sea. As model domain, we consider a simplified barrier coast consisting of a semi-infinite outer sea bordering a straight coast, interrupted by a set of J tidal inlets that connect the outer sea to an arbitrarily shaped tidal basin ( Figure 2) with surface area A b . Each inlet is assumed to have a rectangular cross section, with width b and depth h . The shape of the inlet is fixed using a constant shape factor 2 = h ∕b (e.g., de Swart & Zimmerman, 2009). This implies The domains used in this study are schematizations of two real-world barrier coast systems. The first is a schematization of the Western Dutch Wadden Sea (see Figure 2 top row) consisting of a narrow and wide part (i.e., the top and bottom part in panel b of Figure 2, respectively). The width of this wide part will be varied later on in our analysis. The second is a schematization of the Pamlico Sound, North Carolina (see Figure 2 bottom row), ranging from Cape Lookout to Cape Hatteras and including parts of the Neuss and Pamlico estuaries. The parameter values corresponding to these domains, one set for the Western Dutch Wadden Sea and one for the Pamlico Sound, are given in Table 1.

Morphodynamics
For meso-tidal barrier coasts, both tides and waves are important for the morphological evolution of the tidal inlets (Davis & Hayes, 1984). The tides export sediment from inlet channels leading to erosion, whereas wave-induced littoral drift imports sediment into the inlet channel leading to accretion. Escoffier (1940) argued that an inlet is in equilibrium if the erosion due to the tides and the accretion due to waves balance each other. Long-shore tidal wave number k t (rad/m) 0 0 Basin length a L (km) 75 135 Wide part basin length a L w (km) 30 100 Wide part basin width b B w (km) 20-120 - Inlet length l (km) 5 5 No. of simulations in one ensemble n (-) 100 a We define basin length as the long-shore dimension of the basin (see Figure 2) . b We define basin width as the cross-shore dimension of the basin (see Figure 2). c The actual initial width of an inlet is randomized using a uniform distribution between 0.9 and 1.1 km.
In the approach of Escoffier (1940), the sediment export, denoted by X, is modeled as a function of the tidal velocity amplitude U in the inlet. The sediment import denoted by M is parameterized as a function of the equilibrium velocity U eq . If U < U eq the inlet accretes, if U = U eq the inlet is in equilibrium, and if U > U eq it erodes. Using this concept and a simple hydrodynamic model with a spatially uniform water level in the basin, Escoffier (1940) found so-called closure curves describing the potential evolution of a single inlet system ( Figure 3). Using these closure curves it is found that either two equilibria (one stable and one unstable), one equilibrium, or no equilibrium at all exists. An equilibrium is stable if after a perturbation of the inlet cross section, it returns to its equilibrium size. More recently, process-based support for this stability concept was provided using complex numerical models (Tran et al., 2012;Nahon et al., 2012).
Escoffier's concept forms the basis of our morphodynamic model governing the evolution of the inlet cross sections. The time evolution of the cross-sectional area A of each inlet is controlled by the volumetric import M (assumed equal for all inlets) and export X of sediment that is assumed to be spatially uniform with l the length of inlet . The tide-driven export X is taken to be proportional to the tidal velocity amplitude in the inlet U cubed: X = U 3 , with a constant. The wave-driven import M is externally imposed and similar to previous studies (van de Kreeke, 2004;van de Kreeke et al., 2008;Roos et al., 2013) assumed identical for all inlets, implying that all necessary sediment can be supplied by the wave-induced long-shore drift. This results in an effective equilibrium velocity U eq by defining M = U 3 eq . Equation (2) is then rewritten as From this equation we can compute the change in cross-sectional area, given the velocity amplitude U in the inlet. To find U , we use a hydrodynamic model that is in part solved analytically and in part numerically. This hydrodynamic model is presented in the next subsection. . Three closure curves for a single inlet, as presented by Escoffier (1940). Each closure curve shows the inlet's velocity amplitude U as a function of inlet area A. Intersection with an equilibrium velocity U eq (i.e., at U = U eq ) yields zero, one, or two equilibria. Two equilibria are found if U > U eq for some A, one is found if both coincide, and none are found if U < U eq for all A. In the case of two equilibria one is stable and one is unstable.

Hydrodynamics
The hydrodynamic model simulates the water motion in the outer sea, the tidal inlets, and the tidal basin. It is forced by an M 2 -tidal wave in the outer sea resulting in a flow of water through the inlets, which in turn triggers oscillations in the back-barrier basin and in the outer sea. On the coastline of the outer sea and on the basin boundaries, a no-flow boundary condition is prescribed.
To describe the water motion in each inlet , we consider the linearized cross-sectionally averaged momentum equation, with u the cross-sectionally averaged flow velocity that is uniform over the channel length. Furthermore, assuming a constant surface gradient in the inlet, the momentum equation reduces to with h the inlet depth, o the water level in the outer sea (denoted by subscript o), and b the water level in the tidal basin (denoted by subscript b). The angle brackets denote averaging over the width of a tidal inlet , located at either x o = 0 or x b = 0 and defined as with the centre location of inlet . Here x b is the x-direction in the basin (with x b = 0 at the basin side of the inlet) and x o the x-direction in the outer sea (with x o = 0 at the outer sea side of the inlet); both are shown in Figure 2, together with the -direction that is the same for the basin and outer sea. Furthermore, in equation (4) the coefficient r is a linearized bottom friction coefficient following from Lorentz's linearization (Lorentz, 1922): with drag coefficient c d and velocity scale U (see section 2.2.2). The pressure gradient over the inlet channel ( times the r.h.s. of equation (4)) is due to the difference in elevation between the outer sea ⟨ o ⟩ and the basin ⟨ b ⟩ , both averaged over the inlet width, with g = 9.81 m s −2 the gravitational acceleration.
Since our model equations are linearized, the elevation at the open sea o (t, x o , ) can be decomposed in an elevation related to the incoming tidal wave (i.e., Z cos( t + k t ), the tidal elevation that forces the entire system) and contributions from each inlet o, (t, x o , ) due to radiating waves Since the water depth in the outer sea is large and the influence of Coriolis acceleration on the cross-sectional stability of tidal inlets is small , we neglect bottom friction and Coriolis acceleration. The resulting linearized depth-averaged shallow water equations read with h o the outer sea depth (assumed to be spatially uniform and constant in time), = ( ∕ x o , ∕ ), and u o = (u o , v o ) the depth-averaged flow velocity with components in the x o -direction and -direction, which are indicated in Figure 2. At the outer sea's closed boundaries (i.e., where no inlets are present), we require the normal velocity to vanish, that is Furthermore, we only allow radiating waves propagating away from the inlets.
In the basin we include the effect of bottom friction, as the basin is much shallower than the outer sea, but we still neglect Coriolis acceleration. The resulting linearized system of equations reads: with h b the basin depth (assumed to be spatially uniform and constant in time), = ( ∕ x b , ∕ ), and The linearized friction coefficient r b according to Lorentz's linearization (Lorentz, 1922) is with U b the velocity scale in the basin (see section 2.2.2). At the closed boundaries of the tidal basin (i.e., where no inlets are present), we require the normal velocity to vanish with n the normal vector at the coastline Ω where no inlets are present. Finally, at the tidal inlets we require the transport of water through each inlet to match the transport of water in the adjacent sea and basin, that is where the angle brackets again denote averaging over the width of a tidal inlet .

Outline of Solution Method
In this section we present an outline of the solution method for both the morphodynamic and the hydrodynamic part of the model. The former is solved using a forward Euler discretization of equation (3), with timestep Δt (given in Table 1). The latter is solved analytically-except for the eigenfunctions in the basin that are found numerically (see Appendices A and B)-and yields flow velocities and water levels in the outer sea, tidal inlets, and basin.
The hydrodynamic part of the model is solved as follows. We first express the variables as a product of a complex amplitude and a time-periodic factor. Next, we express both the water motions in the basin and those in the outer sea in terms of the flow velocities in all inlets (see Appendix A). Combining this with the momentum equation for each inlet yields a system of linear equations for the velocity amplitudes in the inlets. This system of equations is solved numerically using standard techniques.

Expansion of Variables
We first express both the water levels and flow velocities as the product of complex amplitudes (denoted by a hat) and a time-periodic factor Here, ℜ means taking the real part and is the tidal frequency already introduced in section 2.1.3. Next, by substituting equations (14) to (16) in equations (6) and (7), the momentum equation (equation (4)) for an inlet a can be written as with 2 a = 1−ir a ∕( h a ) the frictional correction factor. Equation (17) shows that the flow of water through inlet a , û a is affected by all inlets s . There are three contributions to û a : one is related to the tidal elevation in the outer sea due to the forced tidal wave (Term I), the second contribution is related to waves radiating from the tidal inlets resulting in an additional sea surface elevation at the inlets (Term II), and the third contribution is due to the oscillating water level in the basin (Term III).
To calculate the velocity amplitudes in the inlet, given the tidal wave characteristics, we express the latter two surface elevation contributions (i.e., Terms II and III) in terms of û s . To this end we use the so-called sea impedance coefficients z o, s , a and basin impedance coefficients z b, s , a , which follow from the model equations (i.e., equation (10b)). These coefficients provide a relation between the surface elevation amplitude experienced by inlet a due to a flow of water through inlet s where explicit expressions for the basin impedance coefficients z b, s , a and sea impedance coefficients z o, s , a are given in Appendix A. Substituting this expression in equation (17) resulting in a linear system of equations for the velocity amplitudes û that is solved numerically using standard techniques.

Velocity Scales for the Friction Formulation
The velocity scales U b and U , as used in the friction coefficients r b and r (equations (5) and 12) and defined as will be determined iteratively since they are both model input and output. First, an initial guess is used as input. Next, the velocity scales are updated iteratively by applying an underrelaxation procedure until the relative difference between the two is below an error tolerance threshold of 10 −10 .

Design of Model Experiments 2.3.1. Single Simulation
In each simulation, we start with a so-called oversaturated coast containing J init equidistantly spaced tidal inlets and simulate the morphologic evolution of the inlets. We slightly randomize the initial width (up to ±10% from the value in Table 1) of all tidal inlets in each simulation, allowing for possible different equilibrium configurations to be found. The simulation spans 1,000 years, a period that is large enough for the system to reach an equilibrium (or near-equilibrium) configuration. The model results consist of the inlet cross-sections A ; the tidal elevation amplitude in the basin̂b and outer seâo; and the tidal velocity amplitude in the basin u b , inlet û , and outer sea u o .
To demonstrate that our model is capable of simulating various barrier coast systems, we perform three representative model runs, two for Wadden Sea parameters and one for Pamlico Sound parameters (see Table 1). For the Wadden Sea parameters, we consider both the present-day system in which B w = 30 km and the former system (i.e., pre-closure) in which B w = 120 km. Since the Western Dutch Wadden Sea is a meso-tidal barrier coast with drumstick shape barrier islands and four tidal inlets (although up to 10 have been open in the last two millennia, see Vos & Knol, 2015), we expect our model to simulate multiple inlets and barrier islands with limited lengths (i.e., long-shore direction). Since the Pamlico Sound is a micro-tidal barrier coast with very long and thin barrier islands and three tidal inlets (although multiple inlets have opened and closed of the past two millennia, see Mallinson et al., 2018), we expect our model to simulate a small number of inlets and thus very long barrier islands.

Ensemble Simulation
To study the effect of basin geometry on the size and spacing of tidal inlets, we performed a sensitivity analysis by varying the basin width (i.e., the cross-shore dimension) in the Wadden Sea inspired domain (i.e., shown in the top row of Figure 4). For this sensitivity analysis, we use 21 ensembles in which the width of the back-barrier basin in the wide part of the basin B w is varied from 20 km (i.e., B w = B n ) to 120 km (the maximum extent of the Western Dutch Wadden Sea before closure of the Zuiderzee). Each ensemble consists of 100 simulations, using the Wadden Sea parameters and the same set of 100 randomized inlet widths for all ensembles, to ensure that variations in equilibrium configurations are solely due to changes in basin geometry.
Two types of experiments are carried out. The first set of experiments is used to investigate to what extent the geometry in one part of the basin affects the equilibrium configuration of tidal inlets in other parts of the basin. To this end, we kept the width in the narrow part of the basin B n constant, while varying B w . These experiments are referred to as fixed-geometry runs.
The second set of experiments is used to study the effect of basin reduction on the equilibrium configuration of tidal inlets. In these experiments, we first consider a wide tidal basin (B w = 120 km). After 500 years (i.e., halfway the simulation) the basin width is reduced in the wide part of the basin, this reduction varies from 5 to 100 km. With this set of experiments, we simulate how the equilibrium configuration of tidal inlets changes due to an intervention, instead of finding the equilibrium state that an initially oversaturated coast will reach. These experiments are referred to as intervention runs.
We analyze the results of each model run by determining three metrics for the total basin (denoted by a subscript t): the ratio of the inlet cross sections and its initial size A ∕A init, for all open inlets , all in the total basin (i.e., A ,all ∕A init, ,all ), the number of open inlets J t , and the dimensionless total tidal prism P t ∕P ref,t , with Here A ref basin is the basin surface area. Since we vary the basin area of the Wadden Sea basin, we choose one reference basin area A ref basin that is equal to the basin area of the smallest Wadden Sea basin that we consider in this study (i.e., for which B w = B n = 20 km). Similarly, these metrics can be defined for the wide and narrow part of the Wadden Sea basin, by replacing the total quantities by quantities for the wide part of the basin (A ,w ∕A init, ,w , J w , and P w ∕P ref,w with P ref,w = ZL w B n ) and the narrow part of the basin (A ,n ∕A init, ,n , J n , and P n ∕P ref,n with P ref,n = Z(L − L w )B n ). Per ensemble we aggregate the three metrics and determine the median, 50% envelope, and 100% envelope for each basin configuration.

Representative Model Runs
Here we present the results of three representative model runs (using both the Wadden Sea and Pamlico Sound parameters from Table 1). These representative model runs are individual realizations from the ensembles described in section 2.3.2. Figure 4 shows the initial configuration (panels a, d, and g), evolution ) shows that our model correctly simulates a decrease in tidal amplitude as the basin extends further. For the former Wadden Sea domain, our model appears to predict a number of inlets and barrier island lengths that are in the same range as found in reconstructed paleographic maps (Oost et al., 2012;Vos & Knol, 2015;de Haas et al., 2018). The simulated number of inlets is slightly larger than the number of inlets currently present in the Western Dutch Wadden Sea.
Comparison of the example Pamlico Sound model run (Figure 4 bottom row) with the real-world system (Inman & Dolan, 1989; United States Geological Survery, 2019; National Oceanic and Atmospheric Administration, 2020) reveals that our model correctly simulates the tidal amplitude pattern (i.e., lower in the basin and higher on the Pamlico river) and the number of inlets and barrier island lengths (although these results slightly vary for different initial conditions) but overestimates their total cross section. Finally, the time period over which the inlets reach a stable equilibrium differs per example run. For the present-day Wadden Sea, this period ranges from 100 to 500 year, for the former Wadden Sea, this period ranges from 100 to 800 year, and for the Pamlico Sound, this period ranges from 50 to 250 year.

Fixed-Geometry Runs
The experiments for the fixed-geometry runs consist of 21 ensembles of 100 model runs each. For the ensemble corresponding to the basin of the example run where B w = 30 km, the evolution of the ratio A ,all ∕A init, ,all for each open inlet in the total basin (panel a), the number of inlets J t (panel b), and dimensionless tidal prism P t ∕P ref,t (panel c) are shown are shown in Figure 5.
The first panel shows that the inlets that remain open are approximately five times larger than their initial size. The second panel shows that the number of inlets reaches its equilibrium value right before the end of the simulation period. Finally, the third panel shows that the equilibrium tidal prism approaches its equilibrium faster than the inlet cross section and number of inlets. This indicates that it reaches its equilibrium despite the individual inlets not having reached their equilibrium.
The same three metrics are shown for the equilibrium or end states of the fixed-geometry runs in Figure 6: for all inlets , all in the entire basin (panels a-c), the inlets , w in the wide part of the basin (panels d-f), and the inlets , n in the narrow part of the basin (panels g-i). Our results show that the inlets in all parts of the basin are affected by the changes in basin size. For larger basin sizes up to B w ∕ tide = 0.15 (with tide the wavenumber of the tide without friction), both the number of inlets along the back-barrier basin J t and the total tidal prism P t increase , with a maximum for B w ∕ tide %0.15. For even larger basins, both J and P decrease again until both aggregated variables reach a constant value. These results also show a large spread in inlet cross-section A ,all for given basin geometry, which remains more or less constant for different basin geometries. Only the cross sections of inlets in the wide part of the basin (where the width is varied) weakly respond to variations in basin geometry, as they tend to become larger than those in the narrow part of the basin (where the width is not varied).

Intervention Runs
For a single ensemble experiment from the intervention runs (corresponding to the basin of the example run where, at t = 500 year, the basin width B w is reduced from 120 to 30 km), the temporal evolution is shown in Figure 7. The temporal evolution clearly shows that the system reaches an approximate equilibrium after 500 years; after the intervention, the system is no longer in equilibrium anymore and evolves to a new equilibrium. This is reflected in both the relative inlet size of all open inlets A ,all ∕A init, ,all and the dimensionless tidal prism P t ∕P ref,t . The only metric that appears to be unaffected is the number of inlets J t .
The same three metrics are shown for the end states of the intervention runs in Figure 8; for the total basin (panels a-c), the wide part of the basin (panels d-f), and the narrow part of the basin (panels g-i). Similar to the fixed-geometry runs results we plotted the median, the 50% envelope, and the 100% envelope. The end states of the intervention runs show the same results for the total tidal prism P t as the fixed-geometry runs for the same value of B w ∕ tide , but the results for the inlet cross-section A ,all and number of inlets J t are different. The effect of reducing the basin width after an equilibrium has been reached is that after the intervention the number of inlets hardly varies, primarily the size of the inlets varies to compensate for a different equilibrium tidal prism. These results are discussed further in section 4.3.

The Effect of Basin Area on Tidal Prism
Our results for the total tidal prism P t (panel c in Figures 6 and 8) show that for a multiple inlet system the tidal prism is predominantly determined by the overall basin area A basin and much less by variations in (initial) inlet characteristics, as can be seen by the overlapping of the median and 50% and 100% envelopes. Further evidence for this comes from the fact that the total tidal prism for the entire basin is the same for the fixed-geometry runs and the intervention runs, implying that it is unaffected by the initial conditions of the tidal inlets. This also implies that the variations in the total tidal prism in the wide and narrow part of the basin are mostly due to variations in the equilibrium configuration of the tidal inlets.
Thus, the equilibrium configuration of the tidal inlets adapts to convey the total tidal prism P t . This helps explaining the variations in tidal prism in the (unaltered) narrow part of the basin due to variations in (cross-shore) basin width in the wide part of the basin. The basin area determines the equilibrium tidal prism P t of the system, so variations in basin area will lead to variations in P t . In turn, the tidal inlets in the entire basin will evolve (due to inlet interaction) such that in their equilibrium configuration the entire tidal prism is conveyed by them.
For single inlet systems, the tidal prism has been observed to satisfy a linear relation with basin area (e.g., O'Brien, 1931), that is, a linear regime, sometimes referred to as a pumping mode. This observation was also used for modeling these systems (e.g., van de Kreeke, 1990;1990;Van Goor et al., 2003;D'Alpaos et al., 2010;Brouwer et al., 2012). Our results for multiple inlet systems only support the occurrence of linear regimes in short basins (i.e., B ≪ tide ) but clearly indicate that this observation does not hold in longer basins where resonance can occur. Two additional regimes can be identified besides the classic linear regime, in our results on multiple inlet systems. First, a clear maximum can be seen in the total tidal prism P t , implying a resonant regime. Second, the total tidal prism P t becomes invariant for ever larger basins, implying a dissipative regime. In the next subsection these observations and the processes causing them are further discussed.

The Effect of Resonance and Bottom Friction on Tidal Prism
Our results show that the total tidal prism P t (panel c in Figures 6 and 8) increases for an increase in basin width and then decreases for large enough values of B w ∕ tide eventually reaching a constant value, where tide is the wavelength of the frictionless tidal wave. The relative width of the basin at which this peak happens is B w ∕ tide = 0.15. This suggests that this behavior is due to the well-known quarter wavelength resonance even though the maximum is found for B w ∕ tide < 0.25.
To further investigate the importance of resonance, we used a 1D analytical model to study the effect of basin width B on the equilibrium cross section A ,all of a single tidal inlet connected to a rectangular basin (see Figure 9a and Appendix C). This model is forced by a tidal elevation amplitude at the seaside of the inlet (neglecting other seaside processes as Coriolis and radiative damping) and is again based on the linearized shallow water equations with a linearized friction coefficient. Using this model we compute for which inlet cross-section A the inlet velocity is exactly equal to the equilibrium velocity (i.e., U = U eq ). Our results in Figure 9b show that a clear resonance peak is present at B∕ tide = 0.25 in case no bottom friction is present (i.e., r b = 0). Stronger bottom friction (i.e., a higher value of r b ) leads to a shift of this peak toward narrower basins and a reduction of the peak. For larger basins, a constant value for the inlet cross section A is obtained, if bottom friction is sufficiently strong. In narrow basins (i.e., B ≪ tide ) resonance behavior does not occur, and the increase in inlet cross section depends linearly on basin area, also referred to as linear regime, see Figure 9b. Our results show that this linear regime gives way to a resonant regime around B∕ tide = 0.1 or B ∼ 31 km for the parameters in this study, although it will be lower for higher values of r b .
Thus, the relation between basin size and tidal prism is no longer fully linear (as observed for single inlet systems; e.g., O'Brien, 1931), if the basin is sufficiently wide.
For wider basins with sufficient friction, the resonant regime gives way to a dissipative regime with constant inlet cross section A for increasing basin widths. Strong bottom friction causes the inlet cross-section A to reach a constant value with only a small resonance peak; weak bottom friction leads to a distinct resonance peak, before also reaching a constant value. This implies that the tidal wave completely dissipates in the basin and that increasing the basin width further has no effect on the water motion inside the basin and hence on the inlet cross section. Thus, a very wide basin can be accompanied by few/small tidal inlets, as can be observed in the wide and shallow Pamlico Sound behind the Outer Banks of North Carolina, USA (e.g., Inman & Dolan, 1989;Luettich et al., 2002).

The Effect of Basin Reduction
If a basin reduction is implemented, the geometry of the basin changes, and thus, the total tidal prism P t will change as well. The results from our intervention runs (see Figure 8) show that the equilibrium configuration of the inlets indeed adapts to the new basin geometry, and a new equilibrium total tidal prism is reached that is not noticeable different from that reached in the fixed-geometry runs for the same basin geometry. However, because only a limited number of inlets are open when the basin geometry is changedf the change in tidal prism has to be accommodated by changes in the inlet cross sections. This closely mimics the situation in most barrier coast systems that are actively managed. In most of these systems new inlets are not allowed to open.
This implies that a significant reduction of basin width could lead to a regime shift from a constant to resonant regime or from a resonant to a linear regime. Thus, a decrease in basin width could lead to an increase in tidal prism and inlet cross sections. This has indeed been observed in the Western Dutch Wadden Sea, where a large bay (the former Zuiderzee) was separated from the main basin by a dam, resulting in a basin width reduction from ∼115 to ∼30 km. The tidal inlets in the wide part of the basin grew in size because the basin width moved closer to a resonant state (e.g., Elias et al., 2012).

Model Validity and Limitations
To assess the performance of our model we compare the model results with the real-word systems that inspired our domain and parameter choices: the Wadden Sea and Pamlico Sound. Comparison of the example runs with the real-world systems (see section 3.1) revealed that our model is capable of qualitatively reproducing observed phenomena. Our fixed-geometry runs (section 3.2) show that for basins narrower than the resonant condition (i.e., B w ≪ tide ) a larger basin generally corresponds to more and larger inlets (per km barrier coast), agreeing with observations (Davis & Hayes, 1984;Stutz & Pilkey, 2011). Furthermore, our intervention runs (section 3.3) agree with observations from the Western Dutch Wadden Sea. Closure of a large bay significantly reduced the basin width from B w ≈ 0.4 tide to B w ≈ 0.1 tide , which brings the system closer to resonance. This has led to larger inlets and an increased tidal prism (Elias et al., 2003), as shown by our model results.
By basing the evolution of tidal inlets on the stability concept of Escoffier (1940) and coupling that with an idealized hydrodynamic model accounting for tidal oscillations under the influence of bottom friction and resonance, we were able to get a good qualitative comparison between our model results and the observations. To improve this comparison, one has to extend the model by including morphological features and processes neglected in this study. Previous studies have noted the importance of morphological features such as tidal divides (van de Kreeke et al., 2008;de Swart & Volp, 2012), channel networks (Kragtwijk et al., 2004;Reef et al., 2018), ebb and flood deltas (Gerritsen et al., 2013;Elias et al., 2012), and the formation of new tidal inlets through storm-induced breaching of barrier islands (Sallenger, 2000;Kraus et al., 2002). It has also been suggested that processes such as the morphological evolution in the basin and outer sea (Kragtwijk et al., 2004;Elias et al., 2012), nonlinear hydrodynamics (Salles et al., 2005), residual transport (Duran-Matute et al., 2014;Sassi et al., 2015), and changes in boundary conditions due to climate change such as sea level rise and changes in long-shore drift (Glaeser, 1978;Stutz & Pilkey, 2011) affect the long-term evolution of barrier coast systems as well.

Conclusions
We studied the effect of basin geometry and (cross-shore) basin width reduction on the long-term evolution and equilibrium configuration of tidal inlets in a meso-tidal barrier coast. Our newly developed model allowed us to simulate the long-term morphological evolution, from an oversaturated state to an equilibrium state, of multiple tidal inlets connected to a back-barrier basin with a non-uniform basin width. Simulations of the Western Dutch Wadden Sea and the Pamlico Sound showed that our model was able to get results that have a good agreement with observations.
Our results display a strong relation between the basin size and the total tidal prism in the basin. However, this relation is not always linear (i.e., a linear regime) as has been observed for single inlet systems with a small tidal basin. For shorter basins, the tidal prism increases approximately linearly for an increase in basin size linear regime, but after a resonance peak at B w ∕ tide = 0.15 (resonant regime) the tidal prism decreases for an increase in basin size and reaches a constant value (dissipative regime), given that bottom friction is strong enough.
Furthermore, the equilibrium configuration of tidal inlets in the entire basin is affected by the local basin geometry (and human interventions). That is for instance the case in the Wadden Sea, where the construction of the Afsluitdijk affected all inlets in the basin, also further away from the closure dam. This is caused by changes in flow patterns in the basin and outer sea, in turn affecting inlets in the entire basin. Therefore, it is necessary to consider entire multiple inlet systems, when evaluating the effects of basin geometry or land reclamation on barrier coast systems.

A1. Basin Impedance Coefficients
In this subsection, we seek an expression for the basin impedance coefficients z b, s , a for inlet a due to a flow of water in inlet s (see equation (18b)). First, we formulate the model equation 10b in terms of complex amplitudeŝb(x b , ), the surface amplitude in the basin, and û b , the velocity amplitude in the basin defined in equation (15) 2̂b 10.1029/2019JF005261 is a frictional correction factor. The basin boundary conditions that describe the exchange of water with the tidal inlets and no exchange along the other parts of the basin (see equation (12)), read̂b Here û b is prescribed by using equation (13) for the complex amplitudes of the velocity (i.e., h û = h b ⟨û b ⟩ ) and assuming that exchange of water between the inlet and the basin is uniformly distributed over the inlet cross-section (i.e., h û = h b û b ). Next, an explicit solution for̂b(x b , ) can be found by using the Green's function G b (see, e.g., Sommerfeld, 1949) that describes the impact of a Dirac-type of flow over the basin boundary on the basin hydrodynamics and that solves equation (A1a) with x s and s being the coordinates where the boundary condition in equation (A2) is applied. The influence of one inlet on̂b( where Green's function G b associated with equation (A1a) and the boundary conditions in equations (A2) and (A3) is given by (e.g., Sommerfeld, 1949;Polyanin, 2002) with m the eigenfunctions for a closed basin, ||...|| 2 the L 2 -norm, = 2 b k 2 b , and m the eigenvalue associated with m . Here we assume ≠ m . For cases with friction, this is always the case since has an imaginary part and m does not. By averaginĝb , s over action inlet a and applying the matching condition in equation (13), we obtain an expression for the basin impedance z b, s , a as defined in equation (18b) by More information about the numerical procedure that is used to find the eigenfunctions m and eigenvalues m is given in Appendix B.

A2. Sea Impedance Coefficients
We seek an expression for the sea impedance coefficients z o, s , a for inlet a due to a flow of water in inlet s (see equation (18a)). First, we formulate the model equation (8b) in terms of complex amplitudeŝo(x o , ), the surface amplitude in the outer sea, and û o , the velocity amplitude in the outer sea defined in equation (14) 2̂o + k 2 ôo = 0, where m are the corresponding eigenvalues. As such, there is a distinct set of eigenfunctions and corresponding eigenvalues that depends on the geometry of the basin.
For simple geometries, such as rectangular and conical basins, the eigenfunctions are easily found to be cosines and (a linear combination of) Bessel functions, respectively. For more complex geometries, as used in this study, finding the eigenfunctions and eigenvalues is not as straightforward. For a basin of arbitrarily shape, we can solve a discretized EVP of the form 10.1029/2019JF005261 2 û b By applying the boundary conditions (equations (C1) and (C3)), the solution in the basin is found to be ) .
By combining equation (C5) and the solution for̂b at x = 0, we find the following expression for A Here is a frictional correction factor that depends on the depth of the inlet, and thus on the inlet cross section (using equation ((1)). Since both sides of equation (C9) thus depend on A we use an iterative underrelaxation procedure to find the inlet cross section for given parameters, basin length L = 5 km), and requiring the inlet velocity amplitude to equal 1 m s −1 . We use the Wadden Sea parameters (see Table 1) and a basin of length L = 5 km (i.e., along-shore) and a varying width B from 0 to 155 km (i.e., cross-shore).