Fluctuating water levels in tidal bays and adjacent wetlands are important to the health of the ecological systems that are supported by bay and wetland environs. Water level changes that will effect these systems should be studied intensively when man-made alterations to the systems are planned. Inlets provide the pathway to pump water into the bay-wetland systems and therefore act as a critical control for the ecological health of the system. The ability to assess water level change in the bay-wetland response as a result of modifications to an inlet either man-made (such as channel deepening or channel widening), or natural (due to an extreme open coast storm surge) is a desired engineering capability. One difficulty in assessing bay-wetland changes due to an inlet change stems from the non-linear nature of the governing equations for inlet-bay response. As the governing equations for an inlet bay system are non-linear (i.e., see for example Keulegan (1967)), the typical approach to solving the system response (i.e. the bay water level) involves a non-linear numerical scheme. The present paper addresses the possibility of an alternate approach to assessment of water level elevation changes in the bay through a linear systems approach (i.e. see for example Mayhan (1984)). In the linear systems approach reported herein a linearized friction factor is utilized to assess the viability of modeling the inlet-bay system with the driving force ocean tide consisting of a number of harmonics. Figure 1 provides a definition sketch of the type system considered.

Numerous analytical models (Chapman (1923), Brown (1928), Keulegan (1967), Ozsoy (1978), Escoffier and Walton (1979), Walton and Escoffier (1981), and DiLorenzo (1988) ) have addressed the simplified inlet-bay system mass and momentum equations with various assumptions. Typical approaches have assumed either a simple sinusoidal ocean tide and linearized the momentum equation either explicitly or implicitly. Additionally, simplified solutions to the non-linear inlet-bay system problem exist but again consider only elementary types of ocean forcing functions rather than a realistic suite of astronomical harmonic forcing functions. An example of a non-linear solution is DiLorenzo (1988) which utilizes a "simplified" non-linear solution approach (for the case of constant connecting channel cross sectional area, bay surface area, channel length, channel depth, and friction coefficient) to the complete non-linear inlet-bay system equations for the case of an ocean tide consisting of a primary sinusoidal component and its damped and lagged second harmonic overtide.

Numerical inlet-bay models (i.e. Van de Kreeke(1967), Seelig et al.(1977) ) must be used to provide more realistic answers to the bay tide for the inlet-bay system problem where the ocean tides consist of a number of ocean sinusoidal harmonic components with frequencies that are astronomically controlled. This note discusses utilizing a linear systems approach to modeling the bay tide via computing the linearized friction factor for the complete non-linear inlet-bay system equation, then computing the response gain and phase changes for the linear system response (i.e. see Mayhan(1984)) via the corresponding linear inlet-bay system equation. Comparison of the complete non-linear bay tide and the corresponding linearized solution bay tide are shown in Figures 2 through 5. For purposes of the present discussion, an inlet-bay system such as studied by Keulegan (1967) will be utilized where the channel cross sectional area and bay surface area are considered constant. A requirement of this ocean/bay system is that the bay be sufficiently deep that the tide propagates across the bay instantaneously (i.e.see Keulegan (1967)). Although this requirement may not always be met, the tide in many bays propagates sufficiently fast to make this assumption reasonable and useful from an engineering standpoint. The fluctuating water level response in the bay (i.e. the bay tide) due to the mass flow of water thru the connecting channel is initially unknown and must be solved for via the governing mass and momentum equations.

For completeness, the governing non-linear inlet/bay equation is developed similar to that as per DiLorenzo (1988). The one dimensional momentum equation for a shallow water wave in a channel has been given (Dronkers (1964)) as:

(1)

where g = gravitational acceleration; u = channel velocity; η = elevation of surface wave above still water level; ƒ = dimensionless Darcy-Weisbach friction factor; and h = inlet channel flow depth. Upon integrating the momentum equation along the entire length of the channel and rearranging terms, the resulting "head loss" equation is found:

(2)

where L = length of channel; η₀ = water level at ocean end of inlet channel; η_b = water level at bay end of inlet channel; and where entrance k_en and exit k_ex head losses have also been assigned as per Keulegan(1967).

The conservation of mass (continuity) equation can be written as:

(3)

where A_b bay surface area; and A_c channel cross section area;

The integrated momentum equation and the continuity equation can be combined into one differential equation of form:

(4)

where .
DiLorenzo's(1988) solution assumes that the forcing "ocean" tide is characterized by a harmonic at the primary tidal frequency ( where T₁ the fundamental tidal period), and a damped,lagged second harmonic overtide at twice the frequency of the primary tidal constituent. In the present paper the ocean tide is considered to be a summation of three (astronomically controlled frequency) sinusoidal terms, thus similar to a realistic ocean tide (although with less harmonics than would typically be necessary to provide a major portion of the ocean tidevariance). Thus, the forcing ocean tide here is given as:

(5)

where with being the forcing period of the k^th harmonic constituent of the ocean forcing tide; = the phase lag of the k^th harmonic constituent of the ocean forcing tide; and, A_k = the amplitude of the k^th harmonic constituent of the ocean forcing tide.

A non-dimensionalized form of the complete non-linear inlet-bay equation can then be found as:

(6)

where = non-dimensional bay tide; = dimensionless time; = Helmholtz frequency; ; and = a damping coefficient of the inlet-bay system.

In a similar manner a linearized inlet-bay equation can be established via linearizing the non-linear friction term in Equation 4 with a linearized friction coefficient (=λ) where

(7)

where = a suitably averaged discharge scale measure averaged over the tidal cycle. In the present approach, this discharge scale measure is equated to (i.e. see Lorentz(1926), Dronkers(1964));

(8)

with T_r = time period of repetition for the ocean forcing tide (i.e. the "beat period") and as before. This form of discharge scale measure is based on the concept of equating the work done by the bottom shear in both the linear and quadratic frictional forms.

Utilizing the linearized friction coefficient, a non-dimensional form of the linearized inlet-bay system equation can be given as:

(9)

where the same ocean tide forcing function (consisting of three harmonics ) as previously given is provided again.

The above equation is the well known second order non-homogeneous constant coefficient differential equation of a damped spring-mass system (i.e. see Ross (1974)) where the forced mode solution is given (for the particular forcing function provided herein) as:

(10)

where

(11)

with

(12)

and

(13)

and where an arbitrary phasing for γ₁ has been chosen as γ₁= 0.

To establish the feasibility of utilizing a linearized friction coefficient along with linear systems theory to solve for the bay tide in an inlet-bay system, a systematic procedure was established to first find the bay tide using the corresponding non-linear inlet-bay system governing equation (Eq. 6). This was accomplished utilizing a 4th order Runge-Kutta numerical integration scheme and dropping the initial transient portion (i.e. approximately the first 2 tidal cycles) of the solution to eliminate transient responses due to the assumed initial conditions in the system dynamics. After solving for the non-linear solution bay tide, the continuity equation (Eq. 3) was used to obtain the non-linear solution discharge in the inlet channel which is proportional to the derivative of the bay tide. Given the non-linear solution discharge, a linearized friction coefficient (=λ) can be calculated as per Eqs. 7 and 8 via simple integration over a repetition period of the tidal cycle (for the case considered, the minimal spanning period of the three harmonic constituant periods provided for the forcing ocean tide). Upon establishment of the linearized friction coefficient the coefficients for the gain (M_k where k = 1.3) and phase lag (Ψ_k where k =1.3) of the linearized solution (Eqs. 12 and 13) are calculated to provide for computation of the linearized bay tide (Eq. 10). In the following case studies, the non-dimensionalized non-linear solution bay tide and the linearized solution bay tide for the inlet-bay system as driven by three harmonics in the ocean tide are presented as a function of dimensionless time (= τ). Each case study is a function of the inlet-bay system dimensionless parameter β (= the system [frictional] damping coefficient), and the dimensionless ocean tide parameters (which for this non-dimensional solution are also functions of the Helmholtz frequency = Ω_H of the inlet-bay system) as follows: the amplitude of the ocean tide k^th harmonic constituant divided by the amplitude of the fundamental ocean tide constituant (for k = 1.3); α_k = the ocean tide radial frequency divided by the Helmholtz frequency (for k = 1.3); and γ_k = ocean tide phase lags (for k = 1.3) where γ₁ has been assumed =0 without loss of significance.

Using a representative range of parameters for ƒ, A₁, A_c, h, L, and A_b from real inlets as calculated from data contained in O'Brien and Clark(1973), it was decided to consider the dimensionless damping parameter β to range over values from 0.5 to 500. Additionally representative parameters of a realistic inlet with a semidiurnal period lead to considering a value of the fundamental forcing frequency α₁ =0.2.
Case 1
; ; ; ; ; ; ;

This case assumes a low frictional damping and harmonics that are reasonably consistant with that of a semi-diurnal ocean tide harmonic, a diurnal ocean tide harmonic of lesser amplitude, and, a low amplitude overtide of the semi-diurnal ocean tide harmonic. Results of this case are shown in Figure 2 where there is shown to be an imperceptable difference between the non-dimensional non-linear solution bay tide and the non-dimensional linearized solution bay tide. Additionally, the ocean tide is very close to the bay tide in amplitude and phasing.
Case 2
; ; ; ; ; ; ;

This is the same as Case 1 with an increased frictional damping coefficient . A comparison of the non-linear bay tide solution with the linearized solution bay tide is shown in Figure 3. The bay tide lags the ocean tide but remains similar in amplitude although a bit larger than the ocean tide due to apparent limited resonance. The non-linear bay tide solution and the linearized bay tide solution appear so close that the difference between the two is imperceptable in the figure.
Case 3
; ; ; ; ; ; ;

This is the same as Case 1 and 2 with an increased frictional damping coefficient . A comparison of the non-linear bay tide solution with the linearized solution bay tide is shown in Figure 4. In this case the bay tide is lagged and reduced from the ocean tide. The non-linear bay tide solution and the linearized bay tide solution appear very close although there is a perceptable difference between the two bay tides in this figure.
Case 4
; ; ; ; ; ; ;

This is the same as Cases 1,2,and 3 with an increased frictional damping coefficient β. A comparison of the non-linear bay tide solution with the linearized solution bay tide is shown in Figure 5. In this case the bay tide is further lagged and further reduced from the ocean tide than was the situation in the previous case. The non-linear bay tide solution and the linearized bay tide solution appear close, although again, there is a very small difference between the two bay tide solutions in this figure.

Results of the case studies shown suggest that linear system analysis can provide a solution for bay tide (and consequent inlet currents) that is not very different (in many cases imperceptible from an engineering standpoint) than the more complicated non-linear numerical solution. The key to successfully addressing the linearized problem lies in the proper characterization of the linearized friction coefficient for the inlet-bay system.

Should you have any questions concerning forecasting or regression for your specific project needs call: Todd Walton, Ph.D., P.E. (850-644-2847).

References

© Copyright 2004 Todd L. Walton Jr.
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