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Linear systems analysis approach to inlet–bay systems
Ocean Engineering 31 (2004) 513–522 www.elsevier.com/locate/oceaneng
Technical note
Linear systems analysis approach to inlet–bay systems Todd L. Walton Jr. Beaches and Shores Resource Center, Florida State University, 2035E Paul Dirac Drive, Morgan Bldg., Tallahassee, FL 32310, USA Received 22 March 2003; accepted 9 July 2003
Abstract Fluctuating water levels in tidal inlet-bay systems are investigated by comparison of bay water levels resulting from a linearization of the combined continuity and integrated equation of motion to bay water levels from a numerical model with the complete non-linear equation terms. In this approach a linearized friction factor is computed using results from the non-linear model. The present paper addresses the possibility of utilizing this alternate approach to assessment of water level elevation changes in the bay for realistic tides consisting of a number of harmonic components. # 2003 Elsevier Ltd. All rights reserved. Keywords: Tides; Tidal hydraulics; Inlets; Bays; Water levels; Friction
1. Introduction 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 affect 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 asses
Tel. +1-850-644-2847; fax: +1-850-644-6293. E-mail address: [email protected] (T.L. Walton).
0029-8018/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2003.07.002
514
T.L. Walton / Ocean Engineering 31 (2004) 513–522
Fig. 1. Inlet–bay system notation.
sing 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 nonlinear numerical scheme. The present paper addresses the possibility of an alternate approach for the assessment of water level elevation changes in the bay through a linear system approach (i.e. see for example Mayhan, 1984). In the linear system 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. An overview of the simplified inlet–bay system considered is provided in Fig. 1. 2. Background Numerous analytical models (Chapman, 1923; Brown, 1928; Keulegan, 1967; Ozsoy, 1978; Escoffier and Walton, 1979; Walton and Escoffier, 1981; 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 or 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 in DiLorenzo (1988), which utilizes a ‘‘simplified’’ non-linear solution approach (for the case of constant connecting channel
T.L. Walton / Ocean Engineering 31 (2004) 513–522
515
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. The present paper investigates the possibility of utilizing a linear system 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 nonlinear bay tide and the corresponding linearized solution bay tide will be shown. For purposes of the present paper, 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 enough 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 through the connecting channel is initially unknown and must be solved for via the governing mass and momentum equations.
3. Governing 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 @u 1 @u2 @g fujuj þ ¼ g @t 2 @x @x 8h
ð1Þ
where g is the gravitational acceleration; u the channel velocity; g the elevation of surface wave above still water level; f the dimensionless Darcy–Weisbach friction factor; and h is the inlet channel flow depth. Upon integrating the momentum equation along the entire length of the channel and rearranging the terms, the resulting ‘‘head loss’’ equation is found: L du fL ujuj þ ken þ kex þ þ gb ¼ go ð2Þ g dt 4h 2g where L is the length of the channel; go the water level at ocean end of inlet channel; gb the water level at bay end of inlet channel; and where entrance ken and exit kex head losses have also been assigned as per Keulegan (1967).
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T.L. Walton / Ocean Engineering 31 (2004) 513–522
The conservation of mass (continuity) equation can be written as Ab
dg ¼ Ac u dt
ð3Þ
where Ab is the bay surface area; and Ac , channel cross-section area. The integrated momentum equation and the continuity equation can be combined into one differential equation of the form d2 g FAb dg dg gAc gAc þ þ g¼ g ð4Þ dt2 2LAc dt dt LAb LAb o where F ¼ ðken þ kex þ fL=4hÞ. DiLorenzo’s (1988) solution assumes that the forcing ‘‘ocean’’ tide is characterized by a harmonic at the primary tidal frequency (frequency ¼ 1=T1 where T1 is 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 tide variance). Thus, the forcing ocean tide here is given as go ¼ A1 cosðx1 tÞ þ A2 cosðx2 t þ c2 Þ þ A3 cosðx3 t þ c3 Þ
ð5Þ
where xk ¼ 2p=Tk with Tk being the forcing period of the kth harmonic constituent of the ocean forcing tide; ck , the phase lag of the kth harmonic constituent of the ocean forcing tide; and, Ak , the amplitude of the kth harmonic constituent of the ocean forcing tide. A non-dimensionalized form of the complete non-linear inlet–bay equation can then be found as d2 g dg dg A2 A3 þ g ¼ cosða1 sÞ þ þb cosða2 s þ c2 Þ þ cosða3 s þ c3 Þ ð6Þ 2 ds ds ds A1 A1 where g ¼ g=A1 is the non-dimensional bay tide; s ¼ XH t, dimensionless time; XH ¼ ðgAc =LAb Þ1=2 , Helmholtz frequency; ak ¼ xk =XH ; and b ¼ FA1 Ab =2LAc , a damping coefficent 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 Eq. (4) with a linearized friction coefficient (=k) where k¼
FQs 2LAc
ð7Þ
where Qs is 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
T.L. Walton / Ocean Engineering 31 (2004) 513–522
Lorentz, 1926; Dronkers, 1964): ð Tr jQ3 j dt 0 Qs ¼ ð Tr Q2 dt
517
ð8Þ
0
with Tr the time period of repetition for the ocean forcing tide (i.e. the ‘‘beat period’’) and Q ¼ Ac u ¼ Ab dg=dt 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 d2 g k dg A2 A3 þ g ¼ cosða1 sÞ þ þ cosða2 s þ c2 Þ þ cosða3 s þ c3 Þ XH ds ds2 A1 A1
ð9Þ
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 g ¼
3 X gk
ð10Þ
k¼1
where gk ¼ Mk Ak cosðak þ ck wk Þ
ð11Þ
with " Mk ¼
1
x2k X2H
2 #1=2 kxk 2 þ X2H
ð12Þ
and wk ¼ arctan
kxk =X2H 1 x2k =X2H
ð13Þ
and where an arbitrary phasing for c1 has been chosen as c1 ¼ 0.
4. Methodology and case results To establish the feasibility of utilizing a linearized friction coefficient along with linear system theory to solve for the bay tide in an inlet–bay system, a systematic
518
T.L. Walton / Ocean Engineering 31 (2004) 513–522
procedure was established to first find the bay tide using the corresponding nonlinear inlet–bay system governing equation (Eq. (6)). This was accomplished utilizing a fourth order Runge–Kutta numerical integration scheme and dropping the initial transient portion (i.e. approximately the first two 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 (=k) 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 constituent periods provided for the forcing ocean tide). Upon establishment of the linearized friction coefficient, the coefficients for the gain (Mk where k ¼ 1; . . . ; 3) and phase lag (wk where k ¼ 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 (s). Each case study is a function of the inlet–bay system dimensionless parameter b (the system [frictional] damping coefficient), and the dimensionless ocean tide parameters (which for this non-dimensional solution are also functions of the Helmholtz frequency, XH of the inlet–bay system) as follows: Ak =A1 is the amplitude of the ocean tide kth harmonic constituent divided by the amplitude of the fundamental ocean tide constituent (for k ¼ 1; . . . ; 3); ak , the ocean tide radial frequency divided by the Helmholtz frequency (for k ¼ 1; . . . ; 3); and ck , the ocean tide phase lags (for k ¼ 1; . . . ; 3) where c1 has been assumed to be zero without loss of significance. Using a representative range of parameters for f, A1, Ac, h, L, and Ab from real inlets as calculated from data contained in O’Brien and Clark (1973), it was decided to consider the dimensionless damping parameter b to range over values from 0.5 to 500. Additionally, representative parameters of a realistic inlet with a semi-diurnal period lead to considering a value of the fundamental forcing frequency a1 ¼ 0:2. Case 1. b ¼ 0:5;
A2 A3 p p ¼ 0:3; ¼ 0:1; a1 ¼ 0:2; a2 ¼ 0:1; a3 ¼ 0:4; c2 ¼ ; c3 ¼ 4 6 A1 A1
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 Fig. 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.
T.L. Walton / Ocean Engineering 31 (2004) 513–522
519
Fig. 2. Ocean and bay tides (non-linear and linearized); Case 1.
Case 2. b ¼ 5:0;
A2 A3 p p ¼ 0:3; ¼ 0:1; a1 ¼ 0:2; a2 ¼ 0:1; a3 ¼ 0:4; c2 ¼ ; c3 ¼ 4 6 A1 A1
This is the same as Case 1 with an increased frictional damping coefficient b. A comparison of the non-linear bay tide solution with the linearized solution bay tide is shown in Fig. 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. b ¼ 50:0;
A2 A3 p p ¼ 0:3; ¼ 0:1; a1 ¼ 0:2; a2 ¼ 0:1; a3 ¼ 0:4; c2 ¼ ; c3 ¼ 4 6 A1 A1
This is the same as Cases 1 and 2 with an increased frictional damping coefficient b. A comparison of the non-linear bay tide solution with the linearized solution
Fig. 3. Ocean and bay tides (non-linear and linearized); Case 2.
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T.L. Walton / Ocean Engineering 31 (2004) 513–522
bay tide is shown in Fig. 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. b ¼ 500:0;
A2 A3 p p ¼ 0:3; ¼ 0:1; a1 ¼ 0:2; a2 ¼ 0:1; a3 ¼ 0:4; c2 ¼ ; c3 ¼ 4 6 A1 A1
This is the same as Cases 1–3 with an increased frictional damping coefficient b. A comparison of the non-linear bay tide solution with the linearized solution bay tide is shown in Fig. 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.
Fig. 4. Ocean and bay tides (non-linear and linearized); Case 3.
Fig. 5. Ocean and bay tides (non-linear and linearized); Case 4.
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5. Conclusions An approach to assess the viability of utilizing linearized system analysis for prediction of bay tides in inlet–bay systems of the type studied by Keulegan (1967) and others have been discussed and specific case studies have been investigated where an ocean tide consisting of three harmonic components was utilized as the driving force for the problem. Results of the case studies suggest that linear system analysis can provide a solution for bay tide (and consequent inlet currents) that is not very different (in many cases imperceptable 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. In the present paper, this characterization was via means of the approach utilized by Lorentz (1926) that equates work done on the boundary by the linear and non-linear approaches and requires the discharge to be known (i.e. measured). Possible other concepts of assessing the linearized friction factor through measured amplitude responses of the system are yet to be investigated.
Acknowledgements A special thanks to Dr. Robert Dean and Dr. Rod Sobey for knowledge gained from their instructional courses in the hydraulics of inlet systems.
References Brown, E.I., 1928. Inlets on sandy coasts. Proc. Am. Soc. Civ. Eng. LIV, 505–553. Chapman, S., 1923. A note on the fluctuation of water level in a tidal power reservoir. Phil. Mag. J. Sci. XLVI, 101–108. DiLorenzo, J.L., 1988. The overtide and filtering response of small inlet–bay systems. In: Aubrey, D.G., Weishar, L. (Eds.), Hydrodynamics and Sediment Dynamics of Tidal Inlets. Springer-Verlag Publishing, New York, NY, pp. 24–53. Dronkers, J.J., 1964. Tidal Computations. North Holland Publishing Company, Amsterdam. Escoffier, F.F., Walton, Jr., T.L., 1979. Inlet stability solutions for tributary inflow. J. Waterways Harbors Div., ASCE WW4 (105), 341–355. Keulegan, G.H., 1967. Tidal flow in entrances: water level fluctuations of basins in communication with the seas. Committee on Tidal Hydraulics Technical Bulletin No.14. US Army Engineers Waterways Experiment Station, Vicksburg, MS. Lorentz, H.A., 1926. Report of the Government Zuiderzee Commission. The Hague, Netherlands. Mayhan, R.J., 1984. Discrete-Time and Continuous-Time Linear Systems. Addison Wesley, Reading, MA. O’Brien, M.P., Clark, R.R., 1973. Hydraulic Constants of Tidal Entrances I: Data from NOS Tide Tables, Current Tables, and Navigation Charts, UFL/COEL/TR-021. Coastal and Oceanographic Engineering Laboratory, University of Florida, Gainesville, FL. Ozsoy, E., 1978. Notes Published in Stability of Tidal Inlets by P. Bruun. Elsevier Science Publishing, New York, NY. Ross, S.L., 1974. Differential Equations. John Wiley and Sons, New York, NY.
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Seelig, W.N., Harris, D.L., Herchenroder, B.E., 1977. A Spatially Integrated Numerical Model of Inlet Hydraulics, GITI Report No. 14. US Army Coastal Engineering Research Center, Fort Belvoir, VA. Van de Kreeke, J., 1967. Water level fluctuations and flow in tidal inlets. J. Waterways Harbors Div., ASCE WW4, 97–106. Walton, Jr., T.L., Escoffier, F.F., 1981. Linearized solution to the inlet equation with inertia. J. Waterways Harbors Div., ASCE WW3, 191–195.
Technical note
Linear systems analysis approach to inlet–bay systems Todd L. Walton Jr. Beaches and Shores Resource Center, Florida State University, 2035E Paul Dirac Drive, Morgan Bldg., Tallahassee, FL 32310, USA Received 22 March 2003; accepted 9 July 2003
Abstract Fluctuating water levels in tidal inlet-bay systems are investigated by comparison of bay water levels resulting from a linearization of the combined continuity and integrated equation of motion to bay water levels from a numerical model with the complete non-linear equation terms. In this approach a linearized friction factor is computed using results from the non-linear model. The present paper addresses the possibility of utilizing this alternate approach to assessment of water level elevation changes in the bay for realistic tides consisting of a number of harmonic components. # 2003 Elsevier Ltd. All rights reserved. Keywords: Tides; Tidal hydraulics; Inlets; Bays; Water levels; Friction
1. Introduction 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 affect 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 asses
Tel. +1-850-644-2847; fax: +1-850-644-6293. E-mail address: [email protected] (T.L. Walton).
0029-8018/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2003.07.002
514
T.L. Walton / Ocean Engineering 31 (2004) 513–522
Fig. 1. Inlet–bay system notation.
sing 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 nonlinear numerical scheme. The present paper addresses the possibility of an alternate approach for the assessment of water level elevation changes in the bay through a linear system approach (i.e. see for example Mayhan, 1984). In the linear system 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. An overview of the simplified inlet–bay system considered is provided in Fig. 1. 2. Background Numerous analytical models (Chapman, 1923; Brown, 1928; Keulegan, 1967; Ozsoy, 1978; Escoffier and Walton, 1979; Walton and Escoffier, 1981; 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 or 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 in DiLorenzo (1988), which utilizes a ‘‘simplified’’ non-linear solution approach (for the case of constant connecting channel
T.L. Walton / Ocean Engineering 31 (2004) 513–522
515
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. The present paper investigates the possibility of utilizing a linear system 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 nonlinear bay tide and the corresponding linearized solution bay tide will be shown. For purposes of the present paper, 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 enough 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 through the connecting channel is initially unknown and must be solved for via the governing mass and momentum equations.
3. Governing 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 @u 1 @u2 @g fujuj þ ¼ g @t 2 @x @x 8h
ð1Þ
where g is the gravitational acceleration; u the channel velocity; g the elevation of surface wave above still water level; f the dimensionless Darcy–Weisbach friction factor; and h is the inlet channel flow depth. Upon integrating the momentum equation along the entire length of the channel and rearranging the terms, the resulting ‘‘head loss’’ equation is found: L du fL ujuj þ ken þ kex þ þ gb ¼ go ð2Þ g dt 4h 2g where L is the length of the channel; go the water level at ocean end of inlet channel; gb the water level at bay end of inlet channel; and where entrance ken and exit kex head losses have also been assigned as per Keulegan (1967).
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T.L. Walton / Ocean Engineering 31 (2004) 513–522
The conservation of mass (continuity) equation can be written as Ab
dg ¼ Ac u dt
ð3Þ
where Ab is the bay surface area; and Ac , channel cross-section area. The integrated momentum equation and the continuity equation can be combined into one differential equation of the form d2 g FAb dg dg gAc gAc þ þ g¼ g ð4Þ dt2 2LAc dt dt LAb LAb o where F ¼ ðken þ kex þ fL=4hÞ. DiLorenzo’s (1988) solution assumes that the forcing ‘‘ocean’’ tide is characterized by a harmonic at the primary tidal frequency (frequency ¼ 1=T1 where T1 is 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 tide variance). Thus, the forcing ocean tide here is given as go ¼ A1 cosðx1 tÞ þ A2 cosðx2 t þ c2 Þ þ A3 cosðx3 t þ c3 Þ
ð5Þ
where xk ¼ 2p=Tk with Tk being the forcing period of the kth harmonic constituent of the ocean forcing tide; ck , the phase lag of the kth harmonic constituent of the ocean forcing tide; and, Ak , the amplitude of the kth harmonic constituent of the ocean forcing tide. A non-dimensionalized form of the complete non-linear inlet–bay equation can then be found as d2 g dg dg A2 A3 þ g ¼ cosða1 sÞ þ þb cosða2 s þ c2 Þ þ cosða3 s þ c3 Þ ð6Þ 2 ds ds ds A1 A1 where g ¼ g=A1 is the non-dimensional bay tide; s ¼ XH t, dimensionless time; XH ¼ ðgAc =LAb Þ1=2 , Helmholtz frequency; ak ¼ xk =XH ; and b ¼ FA1 Ab =2LAc , a damping coefficent 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 Eq. (4) with a linearized friction coefficient (=k) where k¼
FQs 2LAc
ð7Þ
where Qs is 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
T.L. Walton / Ocean Engineering 31 (2004) 513–522
Lorentz, 1926; Dronkers, 1964): ð Tr jQ3 j dt 0 Qs ¼ ð Tr Q2 dt
517
ð8Þ
0
with Tr the time period of repetition for the ocean forcing tide (i.e. the ‘‘beat period’’) and Q ¼ Ac u ¼ Ab dg=dt 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 d2 g k dg A2 A3 þ g ¼ cosða1 sÞ þ þ cosða2 s þ c2 Þ þ cosða3 s þ c3 Þ XH ds ds2 A1 A1
ð9Þ
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 g ¼
3 X gk
ð10Þ
k¼1
where gk ¼ Mk Ak cosðak þ ck wk Þ
ð11Þ
with " Mk ¼
1
x2k X2H
2 #1=2 kxk 2 þ X2H
ð12Þ
and wk ¼ arctan
kxk =X2H 1 x2k =X2H
ð13Þ
and where an arbitrary phasing for c1 has been chosen as c1 ¼ 0.
4. Methodology and case results To establish the feasibility of utilizing a linearized friction coefficient along with linear system theory to solve for the bay tide in an inlet–bay system, a systematic
518
T.L. Walton / Ocean Engineering 31 (2004) 513–522
procedure was established to first find the bay tide using the corresponding nonlinear inlet–bay system governing equation (Eq. (6)). This was accomplished utilizing a fourth order Runge–Kutta numerical integration scheme and dropping the initial transient portion (i.e. approximately the first two 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 (=k) 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 constituent periods provided for the forcing ocean tide). Upon establishment of the linearized friction coefficient, the coefficients for the gain (Mk where k ¼ 1; . . . ; 3) and phase lag (wk where k ¼ 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 (s). Each case study is a function of the inlet–bay system dimensionless parameter b (the system [frictional] damping coefficient), and the dimensionless ocean tide parameters (which for this non-dimensional solution are also functions of the Helmholtz frequency, XH of the inlet–bay system) as follows: Ak =A1 is the amplitude of the ocean tide kth harmonic constituent divided by the amplitude of the fundamental ocean tide constituent (for k ¼ 1; . . . ; 3); ak , the ocean tide radial frequency divided by the Helmholtz frequency (for k ¼ 1; . . . ; 3); and ck , the ocean tide phase lags (for k ¼ 1; . . . ; 3) where c1 has been assumed to be zero without loss of significance. Using a representative range of parameters for f, A1, Ac, h, L, and Ab from real inlets as calculated from data contained in O’Brien and Clark (1973), it was decided to consider the dimensionless damping parameter b to range over values from 0.5 to 500. Additionally, representative parameters of a realistic inlet with a semi-diurnal period lead to considering a value of the fundamental forcing frequency a1 ¼ 0:2. Case 1. b ¼ 0:5;
A2 A3 p p ¼ 0:3; ¼ 0:1; a1 ¼ 0:2; a2 ¼ 0:1; a3 ¼ 0:4; c2 ¼ ; c3 ¼ 4 6 A1 A1
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 Fig. 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.
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Fig. 2. Ocean and bay tides (non-linear and linearized); Case 1.
Case 2. b ¼ 5:0;
A2 A3 p p ¼ 0:3; ¼ 0:1; a1 ¼ 0:2; a2 ¼ 0:1; a3 ¼ 0:4; c2 ¼ ; c3 ¼ 4 6 A1 A1
This is the same as Case 1 with an increased frictional damping coefficient b. A comparison of the non-linear bay tide solution with the linearized solution bay tide is shown in Fig. 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. b ¼ 50:0;
A2 A3 p p ¼ 0:3; ¼ 0:1; a1 ¼ 0:2; a2 ¼ 0:1; a3 ¼ 0:4; c2 ¼ ; c3 ¼ 4 6 A1 A1
This is the same as Cases 1 and 2 with an increased frictional damping coefficient b. A comparison of the non-linear bay tide solution with the linearized solution
Fig. 3. Ocean and bay tides (non-linear and linearized); Case 2.
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bay tide is shown in Fig. 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. b ¼ 500:0;
A2 A3 p p ¼ 0:3; ¼ 0:1; a1 ¼ 0:2; a2 ¼ 0:1; a3 ¼ 0:4; c2 ¼ ; c3 ¼ 4 6 A1 A1
This is the same as Cases 1–3 with an increased frictional damping coefficient b. A comparison of the non-linear bay tide solution with the linearized solution bay tide is shown in Fig. 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.
Fig. 4. Ocean and bay tides (non-linear and linearized); Case 3.
Fig. 5. Ocean and bay tides (non-linear and linearized); Case 4.
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5. Conclusions An approach to assess the viability of utilizing linearized system analysis for prediction of bay tides in inlet–bay systems of the type studied by Keulegan (1967) and others have been discussed and specific case studies have been investigated where an ocean tide consisting of three harmonic components was utilized as the driving force for the problem. Results of the case studies suggest that linear system analysis can provide a solution for bay tide (and consequent inlet currents) that is not very different (in many cases imperceptable 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. In the present paper, this characterization was via means of the approach utilized by Lorentz (1926) that equates work done on the boundary by the linear and non-linear approaches and requires the discharge to be known (i.e. measured). Possible other concepts of assessing the linearized friction factor through measured amplitude responses of the system are yet to be investigated.
Acknowledgements A special thanks to Dr. Robert Dean and Dr. Rod Sobey for knowledge gained from their instructional courses in the hydraulics of inlet systems.
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