A Hydraulic Model for Multiple-Bay-Inlet Systems on Barrier Islands

Abstract

In this study, we apply the traditional hydraulic engineering approach to model an inter-connected multiple-bay-inlet system that can represent the Great South Bay-Moriches Bay system on Long Island, New York. We show that the hydraulic model captures the essential physics of the system, despite its apparent simplicity in mathematical expressions. The model gives good estimates of bay tidal transmissions, including the tidal ranges, phase lags, and the flood-ebb asymmetry behavior in Moriches Bay. The hydraulic modeling results compare well with the simulations from a 3D coastal ocean circulation model, in particular the changes in bay tides due to the breach of Old Inlet by Hurricane Sandy. The modeled inlet discharge rates are in good agreement with the observations.

Appendix: Appendix A: Dimensionless equations and dynamical parameters

Denoting the dimensionless variables by primes, we choose the following normalization.

$$ t^{\prime}\,=\,t\omega, \quad \left( \eta^{\prime}_{oi}, \eta^{\prime}_a, \eta^{\prime}_b\right) \,=\, \left( \eta_{oi},\eta_a,\eta_b\right)/a, \quad u^{\prime}_i\,=\,u_i /(a \omega), $$

(A.7)

where ω = 2π/T is the angular frequency, T and a are the period and amplitude of the dominant constituent of ocean tidal forcing. Substituting into (2.1)–(2.6) and dropping the primes for brevity, the dimensionless equations are written, identifying the parameters that control the dynamics of the system.

$$ \frac{\mathrm{d}\eta_a}{\mathrm{d} t} = b_1 u_1+b_2 u_2-b_4 u_4, $$

(A.8)

$$ b_0\frac{\mathrm{d}\eta_b}{\mathrm{d} t} = b_4 u_4+b_3 u_3, $$

(A.9)

$$ c_i\frac{\mathrm{d} u_i}{\mathrm{d} t} =\left( \eta_{oi}-\eta_a\right) - r_i |u_i|u_i, \quad i=1,2, $$

(A.10)

$$ c_3\frac{\mathrm{d} u_3}{\mathrm{d} t} = \left( \eta_{o3}-\eta_b\right) -r_3 |u_3|u_3, $$

(A.11)

$$ c_4\frac{\mathrm{d} u_4}{\mathrm{d} t} = \left( \eta_a-\eta_b\right) -r_4 |u_4|u_4, $$

(A.12)

where

$$ b_0=A_b/A_a, \quad b_i = A_i/A_a $$

(A.13)

are the sizing parameters, and

$$ c_i =\ell_i\omega^2/g, \quad r_i = \frac{a\omega^2}{2g}\left( \frac{f_i\ell_i}{4 R_{h,i}} + \kappa_{en,i}+\kappa_{ex,i}\right) $$

(A.14)

are the parameters determining the dynamics of inlet flows. Rewriting c _i = (ℓ _i /a)(a ω ²)/g, it is seen that c _i compares the flow acceleration in an inlet of friction length ℓ _i to gravity, under the excitation of ocean tide. Thus, c _i provides a measure of the inertia of an inlet (i.e., its resistance to be accelerated), hence characterizing the transient time scale of an inlet in response to changes in ocean forcing. The longer an inlet is, the stronger inertia it has, hence the longer transient time it takes to reach a new equilibrium state of motion. Rewriting $r_{i} = {b_{i}^{2}}/{K_{i}^{2}}$, where

$$ K_i = \frac{A_i}{A_a} \frac{T}{2\mathrm{\pi} a}\sqrt{\frac{2ga}{{f_i\ell_i}/{(4R_{h,i})}+\kappa_{en,i}+\kappa_{ex,i}}} $$

(A.15)

is the coefficient of filling or repletion (Keulegan 1967) for inlet i based on the bay area A _a . It summarizes the effects of hydraulic resistance in an inlet, the basin size and ocean forcing. The repletion coefficient K _i is a measure of the ability of an inlet to fill a bay that is characterized by the surface area A _a : For a given water level difference between the bay and ocean, the larger values of K _i are, the faster the bay water level rises. It should be mentioned, however, Keulegan’s definition of K _i may not be properly interpreted for inlet 4, since it does not connect to the ocean.