The Effect of Channel Length on the Residual Circulation in Tidally Dominated Channels

1826 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35 The Effect of Channel Length on the Residual Circulation in Tidally Dominated Channels CHUNYAN LI Coastal Studies Institute, Department of Oceanography and Coastal Sciences, Louisiana State University, Baton Rouge, Louisiana JAMES O’DONNELL Department of Marine Sciences, University of Connecticut, Groton, Connecticut (Manuscript received 25 March 2004, in final form 21 March 2005) ABSTRACT With an analytic model, this paper describes the subtidal circulation in tidally dominated channels of different lengths, with arbitrary lateral depth variations. The focus is on an important parameter associated with the reversal of the exchange flows. This parameter (␦) is defined as the ratio between the channel length and one-quarter of the tidal wavelength, which is determined by water depth and tidal frequency. In ϭ ␦ this study, a standard bottom drag coefficient, CD 0.0025, is used. For a channel with smaller than 0.6–0.7 (short channels), the exchange flow at the open end has an inward transport in deep water and an outward transport in shallow water. This situation is just the opposite of channels with a ␦ value larger than 0.6–0.7 (long channels). For a channel with a ␦ value of about 0.35–0.5, the exchange flow at the open end reaches the maximum of a short channel. For a channel with a ␦ value of about 0.85–1.0, the exchange flow at the open end reaches the maximum of a long channel, with the inward flux of water occurring over the shoal area and the outward flow in the deep-water area. However, near the closed end of a long channel, the exchange flow appears as that in a short channel—that is, the exchange flow changes direction along the channel from the head to the open end of the channel. For a channel with a ␦ value of about 0.6–0.7, the tidally induced subtidal exchange flow at the open end reaches its minimum when there is little flow across the open end and the water residence time reaches its maximum. The mean sea level increases toward the closed end for all ␦ values. However, the spatial gradient of the mean sea level in a short channel is much smaller than that of a long channel. The differences between short and long channels are caused by a shift in dynamical balance of momentum or, equivalently, a change in tidal wave characteristics from a progressive wave to a standing wave. 1. Introduction along the channel. For convenience of discussion, we will use a nondimensional length parameter ␦ defined Tides behave differently in long and short channels. by the ratio between the geometric length of the chan- In a long channel, the tidal wave is progressive because nel and one-quarter of the wavelength: of a significant damping effect of bottom friction; in a short channel, it is a standing wave because of a rela- 4L ͌gh tively smaller damping effect. Note that the length of ␦ ϭ ͩ␭ ϭ ͪ, ͑1͒ ␭ f the channel is not measured by the geometric length alone but by a ratio between the geometric length and where g, h, and f are the gravitational acceleration, the tidal wavelength. With a given bottom drag coeffi- depth, and tidal frequency, respectively. cient, this ratio determines the overall damping effect The rationale for the selection of this parameter is based on the following considerations: 1) For a friction- less tidal wave propagating in a channel, resonance oc- Corresponding author address: Chunyan Li, Coastal Studies In- stitute, Howe-Russell Geoscience Complex, Louisiana State Uni- curs when the channel length equals one-quarter of the versity, Baton Rouge, LA 70803. wavelength, or when ␦ ϭ 1. With friction, the magni- E-mail: [email protected] tude of the response is finite, and the peak is shifted © 2005 American Meteorological Society Unauthenticated | Downloaded 09/25/21 10:09 PM UTC JPO2804 OCTOBER 2005 L I A N D O ’ DONNELL 1827 slightly toward a smaller ␦ value. Therefore, one should simplification in producing the second-order residual expect ␦ to be a useful parameter for tidal-wave char- circulation? The work presented in this paper will re- acteristics for problems with friction. 2) Because tide visit the model and solve the problem without imple- can be considered as the driving force of residual cir- menting the crude approximation used in the earlier culation in tidally dominated channels, it is reasonable model. The first-order solution for tide is then used to to suggest that the length ratio ␦ also have some impact solve the second-order mean circulation by a normal on the residual flow patterns when bottom friction is perturbation method. This approach does not pose any given, based on the fact that tidal characteristics are technical challenge for the first-order tide by using the different in long and short channels. solution of Li and Valle-Levinson (1999), nor does it Some observations support the idea about the pos- significantly complicate the solution of the second or- sible importance of this parameter. Observations in the der. The improvement of the present model is made in James River estuary show that tidal wave is partially the first-order tidal flow, and it is dependent on the progressive (e.g., Browne and Fisher 1988). By separat- depth variations. The results provide some insight to ing the density-driven flows and the tidally induced sub- the variation of flow regimes among channels of differ- tidal flows obtained from a vessel-towed ADCP in the ent lengths and within different segments of the same James River estuary, Li et al. (1998) show that the tid- channel for long channels. The results at the mouth of ally induced subtidal flow has a net inward transport the channel, where the exchange flow pattern is of most through the shallow water but a net outward transport interest, are also consistent with the earlier model for through the deep channel, a pattern consistent with an long channels in which the tidal wave is progressive. analytic model that considers the lateral variation of The results for short channels or near the closed end depth (Li and O’Donnell 1997). This result is also con- of long channels are different from the earlier model. sistent with some other observations (e.g. Zimmerman This result indicates that the residual flow regime 1974; Charlton et al. 1975; Robinson 1960, 1965). In changes with channel length or varies within a long contrast, more recent observations of Winant and channel. Gutiérrez de Velasco (2003) in a well-mixed short channel (Laguna San Ignacio) with standing wave characteristics demonstrate that the residual flow pattern 2. The model and solutions is just the opposite of that of the tidally induced flow observed in the James River estuary. In a more recent The basic idea of the analytic model that we are using study of hydrodynamics within Winyah Bay (Kim and is to solve the first-order [O(␧)] equations of the tide Voulgaris 2005), it is found that the different segments first and then to use it as the forcing for the second- of the same estuary (the upper, middle, and lower order tidally averaged motion. By varying relevant pa- bay) have different regimes of subtidal flows even dur- rameters, the response of the latter can be analyzed. In ing low discharge periods: the lower bay (near the mathematical terms, this is a normal perturbation mouth) behaves more like a tidally driven flow (Li and method. O’Donnell 1997) while the upper bay shows a flow pat- We are only interested in narrow channels (a few tern consistent with that of a density dominated situa- kilometers) open at one end (where it is connected to tion, or the opposite of the tidally driven flows. the ocean) and with no density gradients, in which Co- A question thus arises: does the theory of Li and riolis effect can be neglected. A practical definition of a O’Donnell (1997) apply to both long and short chan- narrow channel is one in which the lateral variation of ⌬␨ nels? In this paper we will address this issue by using a the tidal amplitude ( 0) is much smaller than the ␨ general analytic model that allows an almost arbitrary cross-channel mean of the tidal amplitude ( 0)—that is, ⌬␨ Ӷ ␨ cross-channel depth variation and by an examination of 0 0 (Li and Valle-Levinson 1999). the solution within a large range of channel length val- In the model, the x axis is along the boundary and ues. The earlier model uses an approximate first-order points toward the closed end of the channel. The y axis solution obtained from a flat-bottom channel to drive is along the open boundary at x ϭ 0. A single- the second-order residual circulation in a channel with frequency, semidiurnal tide is imposed at the open end y-dependent depth. Here “first order” is O(␧), in which of the channel, and both the amplitude and phase of the ␧ϭtidal amplitude/mean depth. The mechanism of sea level variation at the open end are assumed to be lateral depth variation in driving the residual circula- uniform across the channel. tion is demonstrated at the second order. The question The depth-averaged, shallow-water momentum and is then, is the first-order tide in a flat bottom an over- continuity equations are used: Unauthenticated | Downloaded 09/25/21 10:09 PM UTC 1828 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35 Ѩu Ѩu Ѩu Ѩ␨ C u͌u2 ϩ ␷2 where CD and U0 are the bottom drag coefficient and ϩ ϩ ␷ ϭϪ Ϫ D Ѩ u Ѩ Ѩ g Ѩ ϩ ␨ , the magnitude of the longitudinal velocity, respectively.

Load more