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Setup and Setdown in Tidal Bays and Wetlands
Estuarine, Coastal and Shelf Science (2002) 55, 789–794 doi:10.1006/ecss.2001.0940, available online at http://www.idealibrary.com on
Setup and Setdown in Tidal Bays and Wetlands T. L. Walton Jr. U.S. Army Engineer Research and Development Center, Coastal and Hydraulics Laboratory, 3909 Halls Ferry Road, Vicksburg, MS 39180-6199, U.S.A. Received 30 April 2001 and accepted in revised form 20 November 2001 Mean water level elevation change in an idealized inlet and connecting bay is addressed, and analytical results of setup and setdown solutions are discussed. Results of setup and setdown computations are provided for a number of non-dimensionalized inlet-bay system parameters, and parameter sensitivity issues are noted. 2002 Elsevier Science Ltd. All rights reserved.
Keywords: wetlands; tides; inlets; bays; setup; estuary
Introduction Tidal bays and wetlands are critical to the nations ecological balance as they provide both protective spawning grounds for various species of marine life and as protective nurseries for the growth of juvenile marine life. As such, mean water level changes to these systems should be studied intensively when man-made alterations to the systems are planned. Another reason for studying mean water level change in bays is to evaluate property boundary changes that may occur when modifying the connecting channel between a tidally driven water basin and an isolated secondary water basin. In mild slope bays, minor elevation changes brought about by non-linear system water mass exchange may change property boundaries (measured in the horizontal) by significant amounts. The present paper addresses mean water level elevation change which occurs when a channel is cut (or modified) between an ocean (or primary water body having a forcing tide) and a bay (or water basin secondary to the main water body) that has no tidal action initially. This mean water level elevation change due to the introduction of the connecting channel (or modification thereof) may be either a setup or setdown (where setup/setdown refers to a mean ‘ bay ’ elevation higher/lower than the ‘ ocean ’). It is noted again that the term ‘ ocean ’ is used in this paper to describe the primary water body where the forcing tide is known (i.e. measured) and ‘ bay ’ as the secondary water body which has no inflow or outflow other than that due to the tidal driven water levels of the primary water body via the connecting channel. As such, the terminology ‘ ocean ’ and ‘ bay ’ is one of 0272–7714/02/011789+06 $35.00/0
L
ηb
Q
ηo
Cross section area = Ac Area = Ab
F 1. Inlet-Bay System Notation.
convenience and does not necessarily limit solutions provided herein to only ocean/bay systems. An additional requirement of the ocean/bay system is that the ‘ bay ’ be deep such that the tide propagates across the bay instantaneously (see for example Keulegan, 1967). Although this requirement is rarely met, the tide in many bays propagates sufficiently fast to make this assumption a reasonable one. The fluctuating water level response in the ‘ bay ’ due to the mass flow of water through the connecting channel is initially unknown. 2002 Elsevier Science Ltd. All rights reserved.
790 T. L. Walton Jr.
FAoAb /2LAc = 20
0.06 γ=
ξ = 0.2 π
3 π/4
0.04
π/2 π/4
Setup/Ao
0.02 0 0
–0.02
–0.04
–0.06
–0.08
0.5
1
1.5 2 π/(T√gAc /LAb)
2
2.5
F 2. Setup/setdown for positive phasing (Bay coefficient=0·5).
Background Numerous analytical models (Chapman, 1923; Brown, 1928; Keulegan, 1967; Ozsoy, 1978; Escoffier & Walton, 1979; Walton & Escoffier, 1981; DiLorenzo, 1988) have addressed the simplified inletbay 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 and, consequently do not allow for setup or setdown in the bay-tide response when there is no tributary inflow to the bay. An exception to the above is DiLorenzo (1988) which utilizes a ‘ simplified ’ nonlinear solution approach 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, thus providing for a simplified estimate of the setup/ setdown (for the case of constant connecting channel cross sectional area, bay surface area, channel length, channel depth, and friction coefficient). Numerical inlet-bay models (i.e. Van de Kreeke, 1967; Seelig et al., 1977) must be used to provide more realistic answers to the setup/setdown problem
where the ocean tides are complex and where inlet cross sections or bay surface areas are varying with time. The present paper provides a set of setup/setdown estimates in the form of non-dimensional graphs by utilizing DiLorenzo’s (1988) simplified non-linear analytical solution to the inlet/bay system equations with constant channel cross section and bay surface area. It should be noted that this approach deals only with setup/setdown due to influence of an ‘ overtide ’ higher (second) harmonic in the ocean forcing tide, and does not address the issue of non-linear effects due to changing cross sectional area or changing bay surface area. Non-linear effects other than the simple one considered herein may need to be addressed by computational approaches beyond an analytical model. Governing equations For completeness, the governing non-linear inlet/bay equation is developed as per DiLorenzo (1988) as follows: The one dimensional momentum equation for a shallow water wave in a channel has been given (Dronkers, 1964) as:
Setup and setdown in tidal bays and wetlands 791
FAoAb /2LAc = 20
0.06
γ=
ξ = 0.2
–3 π/4
–π/2
–π
0.04
Setup/Ao
0.02
0
–0.02
–0.04
–π/4
0 –0.06
–0.08
0
1
2
3
4
5 6 2 π/(T√gAc /LAb)
7
8
9
10
9
10
F 3. Setup/setdown for negative phasing (Bay coefficient=0·5).
0.08 γ= π
0.06
FAoAb /2LAc = 20
3 π/4
ξ = 0.2
π/2 0.04
π/4 0
Setup/Ao
0.02
0
–0.02
–0.04
–0.06
–0.08
0
1
2
3
4
5
6
7
8
2 π/(T√gAc /LAb)
F 4. Setup/setdown for positive phasing (Bay coefficient=2·0).
792 T. L. Walton Jr. 0.08 γ= –π
FAoAb /2LAc = 20
0.06
ξ = 0.2 0.04 0
Setup/Ao
0.02
0
–0.02 –3 π/4 –0.04 –π/2 –0.06
–0.08
–π/4
0
1
2
3
4
5 6 2 π/(T√gAc /LAb)
7
8
9
10
F 5. Setup/setdown for negative phasing (Bay coefficient=2·0).
where g=gravitational acceleration; u=channel velocity; =elevation of surface wave above still water level; f=dimensionless Darcy–Weisbach friction factor; and h=inlet channel flow depth. Upon integrating the momentum equation above along the entire length of the channel and rearranging terms, the resulting ‘ head loss ’ equation is found:
where L=length of channel; 0 =water level at ocean end of inlet channel; b =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). The conservation of mass (continuity) equation can be written as:
where Ab =bay surface area; and Ac =channel cross section area.
Now, the integrated momentum equation and the continuity equation can be combined into one differential equation of form:
where DiLorenzo’s (1988) solution assumes that the forcing ‘ ocean ’ tide is characterized by a harmonic at the primary tidal frequency (frequency= 1T where T=the tidal period), and a damped, lagged second harmonic at twice the frequency of the primary tidal constituent. The ‘ ocean ’ tide is then given as: o =Aosin(t)+Aosin(2t+)
(5)
where =2/T with T being the fundamental forcing period of the ‘ ocean ’ tide; =the dimensionless damping coefficient of the second harmonic; and, =the phase lag of the second harmonic. An overview of the simplified inlet-bay system is provided in Figure 1.
Setup and setdown in tidal bays and wetlands 793 0.08 π/2
FAoAb /2LAc = 20
0.06
ξ = 0.2
π/4 0.04
Setup/Ao
0.02 –3 π/4 0
–0.02 π –0.04 0
–0.06
–0.08
0
1
2
3
4
5 6 2 π/(T√gAc /LAb)
7
8
9
10
9
10
F 6. Setup/setdown for positive phasing (Bay coefficient=20·0). 0.08 FAoAb /2LAc = 20
γ= –π
0.06
ξ = 0.2
0.04 0
Setup/Ao
0.02
0
–0.02
–0.04
–3 π/4
–0.06 –π/2 –π/4 –0.08
0
1
2
3
4
5
6
7
8
2 π/(T√gAc /LAb)
F 7. Setup/setdown for negative phasing (Bay coefficient=20·0).
794 T. L. Walton Jr.
Results The resulting expression for the setup/setdown (DiLorenzo, 1988) is provided here as:
where b =setup/setdown from the original ‘ no tide ’ bay mean water level; and
setdown does not exceed approximately 8% of the fundamental harmonic amplitude Ao. For an ‘ ocean ’ tide fundamental harmonic constituent having an amplitude of approximately 1 m, this signifies that setup/setdown could be on the order of several centimetres. The complexity of the setup/setdown variability suggest that close attention to details must be exercised when making such calculations as minor changes in parameter values could make significant changes in the setup/setdown estimated values. Acknowledgment Permission was granted by the Office of the Chief of Engineers, U.S. Army Corps of Engineers, to publish this information. References
and where Calculations of the setup/setdown are provided herein for an assumed second harmonic damping coefficient of =0·2 which may be considered a reasonable upper limit to the damping of the second harmonic constituent of the forcing tide. Figures 2 to 7 provide a (non-dimensionalized by A0) setup/ setdown for three different values of the quantity FAoAb/2LAc for varying values of ranging from 0 to 2. Results show that the setup/setdown changes slowly/rapidly with the quantity
for larger/
smaller values of FAoAb/2LAc. In the cases calculated (i.e. =0·2 and FAoAb/2LAc varying from 0·5 to 20), it is noted that the maximum absolute value of setup/
Brown, E. I. 1928 Inlets on sandy coasts. Proceedings of the American Society of Civil Engineers LIV, 505–553. Chapman, S. 1923 A note on the fluctuation of water level in a tidal power reservoir. Philosophical Magazine and Journal of Science XLVI, 101–108. DiLorenzo, J. L. 1988 The overtide and filtering response of small inlet-bay systems. In Hydrodynamics and Sediment Dynamics of Tidal Inlets (Aubrey, D. G. & Weishar, L., eds). Springer-Verlag Publishing, New York, N.Y., pp. 24–53. Dronkers, J. J. 1964 Tidal Computations. North Holland Publishing Company, Amsterdam, 518 pp. Escoffier, F. F. & Walton, T. L. Jr. 1979 Inlet stability solutions for tributary inflow. Journal of Waterways and Harbors Division, WW4, 105, ASCE, 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, U.S. Army Engineers Waterways Experiment Station, Vicksburg, MS, 89 pp. Ozsoy, E. 1978 Notes published in Stability of Tidal Inlets by P. Bruun. Elsevier Science Publishing, New York, N.Y, 510 pp. Seelig, W. N., Harris, D. L. & Herchenroder, B. E. 1977 A Spatially integrated numerical model of inlet hydraulics. GITI Report No. 14, U.S. Army Coastal Engineering Research Center, Fort Belvoir, VA, 59 pp. Van de Kreeke, J. 1967 Water level fluctuations and flow in tidal inlets. Journal of Waterways and Harbors Division, WW4, ASCE 93, 97–106. Walton, T. L. Jr. & Escoffier, F. F. 1981 Linearized solution to the inlet equation with inertia. Journal of Waterways and Harbors Division, WW3, ASCE 107(3), 191–195.
Setup and Setdown in Tidal Bays and Wetlands T. L. Walton Jr. U.S. Army Engineer Research and Development Center, Coastal and Hydraulics Laboratory, 3909 Halls Ferry Road, Vicksburg, MS 39180-6199, U.S.A. Received 30 April 2001 and accepted in revised form 20 November 2001 Mean water level elevation change in an idealized inlet and connecting bay is addressed, and analytical results of setup and setdown solutions are discussed. Results of setup and setdown computations are provided for a number of non-dimensionalized inlet-bay system parameters, and parameter sensitivity issues are noted. 2002 Elsevier Science Ltd. All rights reserved.
Keywords: wetlands; tides; inlets; bays; setup; estuary
Introduction Tidal bays and wetlands are critical to the nations ecological balance as they provide both protective spawning grounds for various species of marine life and as protective nurseries for the growth of juvenile marine life. As such, mean water level changes to these systems should be studied intensively when man-made alterations to the systems are planned. Another reason for studying mean water level change in bays is to evaluate property boundary changes that may occur when modifying the connecting channel between a tidally driven water basin and an isolated secondary water basin. In mild slope bays, minor elevation changes brought about by non-linear system water mass exchange may change property boundaries (measured in the horizontal) by significant amounts. The present paper addresses mean water level elevation change which occurs when a channel is cut (or modified) between an ocean (or primary water body having a forcing tide) and a bay (or water basin secondary to the main water body) that has no tidal action initially. This mean water level elevation change due to the introduction of the connecting channel (or modification thereof) may be either a setup or setdown (where setup/setdown refers to a mean ‘ bay ’ elevation higher/lower than the ‘ ocean ’). It is noted again that the term ‘ ocean ’ is used in this paper to describe the primary water body where the forcing tide is known (i.e. measured) and ‘ bay ’ as the secondary water body which has no inflow or outflow other than that due to the tidal driven water levels of the primary water body via the connecting channel. As such, the terminology ‘ ocean ’ and ‘ bay ’ is one of 0272–7714/02/011789+06 $35.00/0
L
ηb
Q
ηo
Cross section area = Ac Area = Ab
F 1. Inlet-Bay System Notation.
convenience and does not necessarily limit solutions provided herein to only ocean/bay systems. An additional requirement of the ocean/bay system is that the ‘ bay ’ be deep such that the tide propagates across the bay instantaneously (see for example Keulegan, 1967). Although this requirement is rarely met, the tide in many bays propagates sufficiently fast to make this assumption a reasonable one. The fluctuating water level response in the ‘ bay ’ due to the mass flow of water through the connecting channel is initially unknown. 2002 Elsevier Science Ltd. All rights reserved.
790 T. L. Walton Jr.
FAoAb /2LAc = 20
0.06 γ=
ξ = 0.2 π
3 π/4
0.04
π/2 π/4
Setup/Ao
0.02 0 0
–0.02
–0.04
–0.06
–0.08
0.5
1
1.5 2 π/(T√gAc /LAb)
2
2.5
F 2. Setup/setdown for positive phasing (Bay coefficient=0·5).
Background Numerous analytical models (Chapman, 1923; Brown, 1928; Keulegan, 1967; Ozsoy, 1978; Escoffier & Walton, 1979; Walton & Escoffier, 1981; DiLorenzo, 1988) have addressed the simplified inletbay 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 and, consequently do not allow for setup or setdown in the bay-tide response when there is no tributary inflow to the bay. An exception to the above is DiLorenzo (1988) which utilizes a ‘ simplified ’ nonlinear solution approach 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, thus providing for a simplified estimate of the setup/ setdown (for the case of constant connecting channel cross sectional area, bay surface area, channel length, channel depth, and friction coefficient). Numerical inlet-bay models (i.e. Van de Kreeke, 1967; Seelig et al., 1977) must be used to provide more realistic answers to the setup/setdown problem
where the ocean tides are complex and where inlet cross sections or bay surface areas are varying with time. The present paper provides a set of setup/setdown estimates in the form of non-dimensional graphs by utilizing DiLorenzo’s (1988) simplified non-linear analytical solution to the inlet/bay system equations with constant channel cross section and bay surface area. It should be noted that this approach deals only with setup/setdown due to influence of an ‘ overtide ’ higher (second) harmonic in the ocean forcing tide, and does not address the issue of non-linear effects due to changing cross sectional area or changing bay surface area. Non-linear effects other than the simple one considered herein may need to be addressed by computational approaches beyond an analytical model. Governing equations For completeness, the governing non-linear inlet/bay equation is developed as per DiLorenzo (1988) as follows: The one dimensional momentum equation for a shallow water wave in a channel has been given (Dronkers, 1964) as:
Setup and setdown in tidal bays and wetlands 791
FAoAb /2LAc = 20
0.06
γ=
ξ = 0.2
–3 π/4
–π/2
–π
0.04
Setup/Ao
0.02
0
–0.02
–0.04
–π/4
0 –0.06
–0.08
0
1
2
3
4
5 6 2 π/(T√gAc /LAb)
7
8
9
10
9
10
F 3. Setup/setdown for negative phasing (Bay coefficient=0·5).
0.08 γ= π
0.06
FAoAb /2LAc = 20
3 π/4
ξ = 0.2
π/2 0.04
π/4 0
Setup/Ao
0.02
0
–0.02
–0.04
–0.06
–0.08
0
1
2
3
4
5
6
7
8
2 π/(T√gAc /LAb)
F 4. Setup/setdown for positive phasing (Bay coefficient=2·0).
792 T. L. Walton Jr. 0.08 γ= –π
FAoAb /2LAc = 20
0.06
ξ = 0.2 0.04 0
Setup/Ao
0.02
0
–0.02 –3 π/4 –0.04 –π/2 –0.06
–0.08
–π/4
0
1
2
3
4
5 6 2 π/(T√gAc /LAb)
7
8
9
10
F 5. Setup/setdown for negative phasing (Bay coefficient=2·0).
where g=gravitational acceleration; u=channel velocity; =elevation of surface wave above still water level; f=dimensionless Darcy–Weisbach friction factor; and h=inlet channel flow depth. Upon integrating the momentum equation above along the entire length of the channel and rearranging terms, the resulting ‘ head loss ’ equation is found:
where L=length of channel; 0 =water level at ocean end of inlet channel; b =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). The conservation of mass (continuity) equation can be written as:
where Ab =bay surface area; and Ac =channel cross section area.
Now, the integrated momentum equation and the continuity equation can be combined into one differential equation of form:
where DiLorenzo’s (1988) solution assumes that the forcing ‘ ocean ’ tide is characterized by a harmonic at the primary tidal frequency (frequency= 1T where T=the tidal period), and a damped, lagged second harmonic at twice the frequency of the primary tidal constituent. The ‘ ocean ’ tide is then given as: o =Aosin(t)+Aosin(2t+)
(5)
where =2/T with T being the fundamental forcing period of the ‘ ocean ’ tide; =the dimensionless damping coefficient of the second harmonic; and, =the phase lag of the second harmonic. An overview of the simplified inlet-bay system is provided in Figure 1.
Setup and setdown in tidal bays and wetlands 793 0.08 π/2
FAoAb /2LAc = 20
0.06
ξ = 0.2
π/4 0.04
Setup/Ao
0.02 –3 π/4 0
–0.02 π –0.04 0
–0.06
–0.08
0
1
2
3
4
5 6 2 π/(T√gAc /LAb)
7
8
9
10
9
10
F 6. Setup/setdown for positive phasing (Bay coefficient=20·0). 0.08 FAoAb /2LAc = 20
γ= –π
0.06
ξ = 0.2
0.04 0
Setup/Ao
0.02
0
–0.02
–0.04
–3 π/4
–0.06 –π/2 –π/4 –0.08
0
1
2
3
4
5
6
7
8
2 π/(T√gAc /LAb)
F 7. Setup/setdown for negative phasing (Bay coefficient=20·0).
794 T. L. Walton Jr.
Results The resulting expression for the setup/setdown (DiLorenzo, 1988) is provided here as:
where b =setup/setdown from the original ‘ no tide ’ bay mean water level; and
setdown does not exceed approximately 8% of the fundamental harmonic amplitude Ao. For an ‘ ocean ’ tide fundamental harmonic constituent having an amplitude of approximately 1 m, this signifies that setup/setdown could be on the order of several centimetres. The complexity of the setup/setdown variability suggest that close attention to details must be exercised when making such calculations as minor changes in parameter values could make significant changes in the setup/setdown estimated values. Acknowledgment Permission was granted by the Office of the Chief of Engineers, U.S. Army Corps of Engineers, to publish this information. References
and where Calculations of the setup/setdown are provided herein for an assumed second harmonic damping coefficient of =0·2 which may be considered a reasonable upper limit to the damping of the second harmonic constituent of the forcing tide. Figures 2 to 7 provide a (non-dimensionalized by A0) setup/ setdown for three different values of the quantity FAoAb/2LAc for varying values of ranging from 0 to 2. Results show that the setup/setdown changes slowly/rapidly with the quantity
for larger/
smaller values of FAoAb/2LAc. In the cases calculated (i.e. =0·2 and FAoAb/2LAc varying from 0·5 to 20), it is noted that the maximum absolute value of setup/
Brown, E. I. 1928 Inlets on sandy coasts. Proceedings of the American Society of Civil Engineers LIV, 505–553. Chapman, S. 1923 A note on the fluctuation of water level in a tidal power reservoir. Philosophical Magazine and Journal of Science XLVI, 101–108. DiLorenzo, J. L. 1988 The overtide and filtering response of small inlet-bay systems. In Hydrodynamics and Sediment Dynamics of Tidal Inlets (Aubrey, D. G. & Weishar, L., eds). Springer-Verlag Publishing, New York, N.Y., pp. 24–53. Dronkers, J. J. 1964 Tidal Computations. North Holland Publishing Company, Amsterdam, 518 pp. Escoffier, F. F. & Walton, T. L. Jr. 1979 Inlet stability solutions for tributary inflow. Journal of Waterways and Harbors Division, WW4, 105, ASCE, 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, U.S. Army Engineers Waterways Experiment Station, Vicksburg, MS, 89 pp. Ozsoy, E. 1978 Notes published in Stability of Tidal Inlets by P. Bruun. Elsevier Science Publishing, New York, N.Y, 510 pp. Seelig, W. N., Harris, D. L. & Herchenroder, B. E. 1977 A Spatially integrated numerical model of inlet hydraulics. GITI Report No. 14, U.S. Army Coastal Engineering Research Center, Fort Belvoir, VA, 59 pp. Van de Kreeke, J. 1967 Water level fluctuations and flow in tidal inlets. Journal of Waterways and Harbors Division, WW4, ASCE 93, 97–106. Walton, T. L. Jr. & Escoffier, F. F. 1981 Linearized solution to the inlet equation with inertia. Journal of Waterways and Harbors Division, WW3, ASCE 107(3), 191–195.