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

1826 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35

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

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. t x y x h Ϫ1 In this study, U0 is chosen to be a constant (0.83 m s ). Ѩ Ѩ␷ Ѩ␷ Ѩ␨ ␷͌ 2 ϩ ␷2 In (5) the quadratic friction has been decomposed into u CD u ϩ u ϩ ␷ ϭϪg Ϫ , and a linear part and a higher-order term. The advantage of Ѩt Ѩx Ѩy Ѩy h ϩ ␨ (5) over the original shallow-water equations is that a Ѩ␨ Ѩ͑h ϩ ␨͒u Ѩ͑h ϩ ␨͒␷ formal perturbation method can be readily applied for ϩ ϩ ϭ 0, ͑2͒ Ѩt Ѩx Ѩy an analytical solution. To be specific, let the solution be a superposition of where u, ␷, ␨, h, x, y, t, C , and g are longitudinal ve- D the first-order, second-order, and higher-order con- locity, lateral velocity, elevation, undisturbed water stituents; that is, depth, longitudinal coordinate, lateral coordinate, time, ϭ ϩ ϩ drag coefficient, and the gravitational acceleration, re- u u1 u2 ···, spectively. ␷ ϭ ␷ ϩ ␷ ϩ ···, and The longitudinal velocity at the head and the lateral 1 2 ␨ ϭ ␨ ϩ ␨ ϩ ͑ ͒ velocity at the side boundaries are zero. The boundary 1 2 ···. 7 conditions for the model can be expressed as It is assumed here that the first-order tide, denoted by | ϭ ␷| ϭ ␨| ϭ ͑␨ i␴t͒ u xϭL 0, yϭ0,D 0, and xϭ0 Re 0e , the subscript 1, has a magnitude proportional to ␧ϭ ␨ ͑3͒ 0/h and the second-order quantity, denoted by the subscript 2, has a magnitude proportional to ␧2. In many ␴ ␨ ␨ where the constants D, , and 0 are the channel width, estuaries and tidal channels, 0 is small relative to the tidal frequency, and amplitude at the mouth, respec- mean water depth. A typical ␧ is 0.1. After substituting tively, and i ϭ ͌Ϫ1. Here, Re is an operator that takes the above equations into the momentum and continuity the real part of the function within the parentheses. equations, one can obtain the first-order equations by Mass balance requires that the cross-channel integra- neglecting terms that are of second order or smaller. tion of the tidally averaged transport over a tidal cycle The second-order equations can then be obtained by must be zero under idealized conditions (i.e., when neglecting terms of third order or smaller. If the first- there is only one tidal frequency and there is no change order linear equations can be solved, the second-order in mean sea level from low-frequency motions such as equations may also be tractable because they are linear remote wind effect), and we have equations driven by products of the known first-order

D solution. A strict and standard approach is to nondi- ͵ ͑h ϩ ␨͒udyϭ 0. ͑4͒ mensionalize the equations and separate equations of 0 different orders by neglecting higher-order terms. The It can be seen later that this condition is only useful for procedure is described in the appendix. Here we will the second-order [O(␧2)] tidally averaged residual cir- use the dimensional equations solely for the purpose of culation. presentation. The dimensional equations are the exact As shown in Li and O’Donnell (1997), a Fourier de- conversions of their nondimensional counterparts. composition (Proudman 1953; Parker 1984) of the qua- The first-order linearized equations of (5) are (see dratic friction leads to the following equations, correct the appendix for the nondimensional version) to the second order: Ѩ Ѩ␨ ␤ u1 1 ϭϪg Ϫ u , Ѩu Ѩu Ѩu Ѩ␨ u ␤ Ѩt Ѩx h 1 ϩ u ϩ ␷ ϭϪg Ϫ ␤ ϩ u␨, Ѩ Ѩ Ѩ Ѩ 2 t x y x h h Ѩ␷ Ѩ␨ ␤ 1 1 ϭϪg Ϫ ␷ , and Ѩ␷ Ѩ␷ Ѩ␷ Ѩ␨ ␷ ␤ Ѩt Ѩy h 1 ϩ u ϩ ␷ ϭϪg Ϫ ␤ ϩ ␷␨, and Ѩt Ѩx Ѩy Ѩy h h2 Ѩ␨ Ѩ Ѩ ␷ 1 u1 h 1 ϩ h ϩ ϭ 0. ͑8͒ Ѩ␨ Ѩ͑h ϩ ␨͒u Ѩ͑h ϩ ␨͒␷ Ѩt Ѩx Ѩy ϩ ϩ ϭ ͑ ͒ Ѩ Ѩ Ѩ 0, 5 t x y The depth is assumed to be a sufficiently smooth (dif- in which ferentiable) but otherwise arbitrary function of the cross-channel position: 8CDU0 ␤ ϭ , ͑6͒ 3␲ h ϭ h͑y͒ ͑0 Ͻ y Ͻ D; D is the channel width͒. ͑9͒

Unauthenticated | Downloaded 09/25/21 10:09 PM UTC OCTOBER 2005 L I A N D O ’ DONNELL 1829

␤ For a single-frequency tide, the solution can be ex- ever, we will assume that CD and are constant for pressed as convenience. Similar to Li and O’Donnell (1997), the tidally aver- ϭ ͑ i␴t͒ ␷ ϭ ͑ i␴t͒ and u1 Re Ue , 1 Re Ve , aged second-order equations can be obtained as (see ␨ ϭ ͑ i␴t͒ ͑ ͒ the appendix for the nondimensional version) 1 Re Ae , 10 where ␴, i, U, V, and A are the angular frequency of the Ѩ Ѩ Ѩ␨ ␤ ␤ ␨ u1 u1 2 u2 u1 1 tide, the unit imaginary number ͌Ϫ1, the complex u ϩ ␷ ϭϪg Ϫ ϩ , 1 Ѩx 1 Ѩy Ѩx h h2 amplitude of the longitudinal velocity, the complex amplitude of the lateral velocity, and the complex ampli- Ѩ␷ Ѩ␷ Ѩ␨ ␤␷ ␤ 1 ϩ ␷ 1 ϭϪ 2 Ϫ 2 ϩ ␷ ␨ tude of the tidal elevation, respectively. The boundary u1 1 g 2 1 1, and Ѩx Ѩy Ѩy h h conditions are Ѩ Ѩ␨ Ѩ ␷ Ѩ␨ ␷ u2 1u1 h 2 1 1 U| ϭ ϭ 0, V| ϭ ϭ 0, and A| ϭ ϭ ␨ . ͑11͒ h ϩ ϩ ϩ ϭ 0, ͑16͒ x L y 0,D x 0 0 Ѩx Ѩx Ѩy Ѩy A solution of (8) for the velocity and tidal elevation has been obtained by Li and Valle-Levinson (1999) with an in which the overbars indicate temporal averages over a assumption that the cross-channel variation of surface complete tidal cycle. elevation is much smaller than that of the tidal ampli- To solve the tidally averaged second-order equations tude. In complex form, the elevation is in (16), we define the transport velocity as in Robinson (1983) and Li (1996): cos͓␻͑x Ϫ L͔͒ ϭ ␨ ͑ ͒ A 0 ͑␻ ͒ , 12 cos L ␨ 1v1 v ϭ ͑1 ϩ ␨րh͒v ϭ v ϩ ϩ higher-order terms, in which T 2 h

i␴D D gh ͑17͒ ␻2 ϭ ϭϪ ͑ ͒ , with F ͵ ␴ ϩ ␤ր dy, 13 F 0 i h in which the boldface vs are velocity vectors—for ex- and the along-channel velocity is ϭ ␷ ample, v1 (u1, 1). Because the total volume flux of water across unit width at a given time is (h ϩ ␨)v, the g a␻ ϭ ͓␻͑ Ϫ ͔͒ ͑ ͒ transport velocity defined by (17) is a quantity that has U ␴ ϩ ␤ր ͑␻ ͒ sin x L . 14 i h cos L the dimension of velocity and yields the net flux across Using the continuity equation, the lateral velocity can unit width when multiplied by the undisturbed water be obtained as depth h. With this definition, the equations in (16) can be rearranged to 1 D gh␻2 V ϭ ͩi␴y ϩ ͵ dyͪA. ͑15͒ ␴ ϩ ␤ր ␨ Ѩ Ѩ Ѩ␨ h 0 i h 1u1 h u1 u1 gh 2 u ϭ 2 Ϫ ͩu ϩ ␷ ͪ Ϫ , T h ␤ 1 Ѩx 1 Ѩy ␤ Ѩx Equations (10) and (12)–(15) constitute the solution for ␨ ␷ Ѩ␷ Ѩ␷ Ѩ␨ the first-order equations in (8), satisfying the boundary 1 1 h 1 1 gh 2 ␷ ϭ 2 Ϫ ͩu ϩ ␷ ͪ Ϫ , and conditions in (11). The difference between the results T h ␤ 1 Ѩx 1 Ѩy ␤ Ѩy of (12)–(15) and those of Li and O’Donnell (1997) is Ѩ Ѩ ␷ uT h T that the latter use a solution of a flat-bottom channel h ϩ ϭ 0. ͑18͒ for the first-order tide while the former allows an arbi- Ѩx Ѩy trary cross-channel variation in water depth—that is, h ϭ h(y). The condition for this new solution is that the Note that 1) for a narrow channel the surface elevation ⌬␨ cross-channel difference in tidal amplitude ( 0)is of the water is almost uniform along the same cross- much smaller than the cross-channel mean of the tidal channel transect (Friedrichs and Hamrick 1996; Li ץ ␨ץ ␨ ⌬␨ ␨ Ӷ ⌬␨ ␨ amplitude ( 0); that is, 0/ 0 1. The quantity 0/ 0 is 1996; Li and O’Donnell 1997) such that 2/ x is inde- also an estimate of the error of the first-order model. In pendent of cross-channel coordinate y, 2) no-normal- the James River estuary, for example, the maximum flow condition applies on the lateral boundaries (i.e., ⌬␨ ␨ ␷ ϭ ϭ 0/ 0 is about 5% (Li and Valle-Levinson 1999). The 2 0aty 0 and D), and 3) the integrated total first-order solution can also allow a variable drag coef- transport due to tide is zero under idealized conditions ficient as shown by Li et al. (2004). In this paper, how- (i.e., when there is no change in mean sea level from

Unauthenticated | Downloaded 09/25/21 10:09 PM UTC 1830 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35

low-frequency motions such as remote wind effect); thus

D ͵ ϭ ͑ ͒ huT dy 0, 19 0

which is a different format of (4). Using these conditions and by integrating the third equation of (18) across the channel we can solve the tidally averaged pressure gradient to be proportional to

Ѩ␨ 1 2 ϭ Ѩx D g ͵ h2 dy 0

D ϫ ␤ ͵ ␨ ͫ2 u dy IG 1 1 F . 1. (a) The tidally averaged flow at (b) the locations across 0 the channel at the mouth indicated by the dots. The results are obtained by varying the length of the model up to 150 km (or ␦ ϭ D Ѩ Ѩ u1 u1 Ϫ ͵ h2ͩu ϩ ␷ ͪ dyͬ. ͑20͒ 1.56, Table 1, with the given depth function). In (a) is shown the 1 Ѩ 1 Ѩ 0 x y residual velocity from the center (labeled with “a”), shoal (“b”), and slope (“c”) by the thick solid (for a), dashed (for b), and thin From (20) the tidally averaged pressure gradient can be solid (for c) lines, respectively. ␷ ␨ solved with the first-order solution u1, 1, and 1 and an arbitrary depth function h(y). The restriction to h(y)is that it must be smooth and nonzero at all positions. By urnal tide), with a constant width of 2 km. The change substituting the tidally averaged surface slope into the of width has been found to have little effect on the first equation of (18), the along-channel component of structure of the tidally induced mean flow (Li 1996; Li the transport velocity can be calculated. The lateral and O’Donnell 1997) unless the width is too large such component of the transport velocity cannot be resolved that the lateral variation of the surface elevation is com- from the second equation of (18) because the mean parable to the tidal amplitude. We will therefore focus lateral pressure gradient is unknown. The cross-channel on the effects of channel-length variations only. We component of the transport velocity, however, can be have experimented with several different depth func- resolved from the mass-conservation requirement by a tions including those used in Li and Valle-Levinson lateral integration of the third equation of (18) as dem- (1999). For the presentation here we will only discuss onstrated in Li and O’Donnell (1997): the results from one depth function representative of the characteristics of the solution. The selected cross- 1 Ѩ D channel depth function (Fig. 1b) is defined by the fol- ␷ ϭϪ ͩ͵ ͪ ͑ ͒ T Ѩ huT dy . 21 lowing equation (all parameters are in meters): h x 0 ͑ ͒ ϭ ϩ Ϫ͑yϪD/2ր350͒2 ͑ ͒ In the following section, we will analyze and discuss the h y h1 h2e . 22 above solution with different channel lengths and a In the above equations, h ϭ 5 m and h ϭ 10 m. This given depth function. 1 2 depth function is symmetric about the axis of the channel. Such a depth function, though idealized, is repre- 3. Computations and results sentative of the main features of many coastal plain estuaries: a deep channel and shallow shoals, all on the In the calculations of the above solution, we use a order of a few to tens of meters. ϭ semidiurnal tidal period (T 12 h) with a standard To calculate the mean surface slope from (20), we ϭ bottom drag coefficient CD 0.0025 (Proudman 1953), need to calculate the tidally averaged function of the ␤ ϭ Ϫ1 ϭ Ϫ1 which yields 0.001 76 m s given U0 0.83 m s . ␨ product of two functions, for example, 1u1. Assume To examine the effect of channel length on the tidally that a ϭ Re(Aei␻t) and b ϭ Re(Bei␻t), and if we use induced mean flow field and resulting exchange rate subscripts R and I to represent the real and imaginary under different bathymetry conditions, we choose the parts, respectively, that is, length ratio of the model, ␦ as defined by (1), to vary ϭ ϩ ϭ ϩ ͑ ͒ between 0.052 and 1.56 (or 5 and 150 km for a semidi- A AR iAI and B BR iBI, 23

Unauthenticated | Downloaded 09/25/21 10:09 PM UTC OCTOBER 2005 L I A N D O ’ DONNELL 1831

TABLE 1. Mean depth (h0) and length scales.

␭ ϫ ␭ ϭ ϭ ϭ ϭ ϭ h0 (m) /4 (km) 0.6 /4 (km) L 30 km L 50 km L 100 km L 105 km L 150 km 8.1 96 58 ␦ ϭ 0.31 ␦ ϭ 0.52 ␦ ϭ 1.04 ␦ ϭ 1.09 ␦ ϭ 1.56 then it can be proven that in this paper show that, as the length of the channel increases, the magnitude of the exchange flow reaches ϭ ͑ i␻t͒ ͑ i␻t͒ ϭ ͑ ϩ ͒ր ͑ ͒ ab Re Ae Re Be ARBR AIBI 2. 24 its maximum when ␦ is at about 0.42 and then decreases ␷ and reaches its minimum (ϳ0) when the length ratio ␦ For convenience, the first-order solution—that is, u1, 1, ␨ is about 0.64 for the selected depth function. For the and 1—is calculated on equally spaced nodes. Because the solution is analytical, the grid size dx and dy can be selected depth function, the corresponding length arbitrary. The calculation of the first-order spatial de- scales are listed in Table 1. When the channel length is rivatives required for the solution of the subtidal fields further increased, the exchange flow at the open end uses the sixth-order compact scheme of Lele (1992) for reverses directions: the flow in the deep center is out- the interior nodes and the sixth-order scheme of Car- ward while the flow in the shoal is inward, which is penter et al. (1993) for the near boundary nodes. The consistent with the model result of Li and O’Donnell integrations such as those in (13) and (20) are calcu- (1997). When the channel length ratio is about 0.92, the lated using the Simpson method. mean flow reaches its maximum again at the open end. In the calculations, we use a 5-km step to vary the All of the experiments have shown that, regardless of length of the channel; that is, the length of the channel the actual depth distributions across the channel, the is chosen to be 5, 10, 15, ...,150km,respectively. In exchange flows are consistent qualitatively among all doing so, for the given depth function, we calculate the short channels and among all long channels. Here the tidal and mean flow fields with 30 different channel division between short and long channels is made by ␦ ϭ lengths. For each of these calculations, we choose 51 0.6–0.7 (0.64 for the example presented). The maxi- cross-channel positions and 61 along-channel positions mum exchange flow for a short channel occurs when for the flow fields and all related variables. By exam- the length ratio is about 0.35–0.5 (0.42 for the example ining the results, the change of the flow patterns with presented). The maximum exchange flow for a long the channel length can be assessed. In the following, we channel occurs when the length ratio is about 0.85–1.0 will examine the exchange flow patterns from two per- (0.92 for the example presented). The variability of spectives: 1) selected locations and 2) plane view of the these length scales is due to the fact that overall fric- entire flow field. tional effect can be different for models with different depth functions even for the same drag coefficient. The a. Residual circulation experiments also demonstrate that the mean flow pattern can be altered if the water-depth distribution 1) EXCHANGE FLOW AT SELECTED LOCATIONS across the channel is altered. An example of this is a By selecting different positions across the channel, dredging activity or storm event that can deepen a the response of the subtidal flow in the along-channel channel, modifying and possibly reversing the exchange direction at the open end as a function of the channel flow patterns if the change in depth causes sufficient ␦ length is shown in Fig. 1a. For all of the depth functions variations in . with which we experimented [results from only one 2) RESIDUAL FLOW FIELD depth function defined by (22) are discussed for brevity], the tidally induced mean flow for “short” channels As a representative example, the plane view of the in the deep water is positive (into the channel or “land- residual flow field in a long channel with the depth ward”) while the mean flow in the shoals is negative function defined in (22) is shown in Fig. 2. The length of (out of the channel or “seaward”). This flow pattern is the channel in this case is 105 km, and the correspond- opposite of that of the model results of Li and ing length ratio ␦ is about 1.09 (Table 1). For brevity, O’Donnell (1997), which are applicable only to a “long we have omitted the same plot for a short channel be- channel” with a progressive tidal wave. In the model of cause the flow pattern in a short channel is the same as Li and O’Donnell (1997), the effect of the lateral vari- that in the closed end of a long channel. This fact is ability of the first-order tidal flow due to the variation verified by all of the experiments with different depth in depth was neglected for convenience, which makes functions that we have used. The residual flow field in the model inappropriate for short channels. The results this case (Fig. 2) is symmetric about the axis of the

Unauthenticated | Downloaded 09/25/21 10:09 PM UTC 1832 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35

channel and the exchange rate. In the present study, for given depth function and parameters, the length of the channel determines the exchange rate and residence time. Figure 3 shows the response of the exchange rate (Fig. 3a) and residence time (Fig. 3b) to the change of channel length for the selected depth function (22). For all depth functions and parameters with which we have experimented, the exchange rate peaks at ␦ ϳ 0.35–0.5 (for short channels) and ␦ ϳ 0.85–1.0 (for long channels). The minimum exchange rate occurs at ␦ ϳ 0.6– 0.7, corresponding to the minimum residual flow at the open end (Figs. 1a and 2). The residence time remains small and relatively constant for the short channels until ␦ ϳ 0.6–0.7 when it reaches maximum and as the exchange rate reaches its maximum. After this peak, the residence time decreases rapidly and then increases again gradually with the channel length.

4. Discussion FIG. 2. Residual flow vectors in a channel of length 105 km. a. Momentum and mass balance The solutions presented here require an assumption channel because the cross-channel depth distribution is that the lateral variation of the surface elevation is symmetric and Coriolis force is not considered in this small in a narrow channel. Although this assumption study. The most interesting feature of this figure is that has been supported by observations using pressure sen- the tidally induced residual flow changes its pattern sors across the James River estuary (Li and Valle- from a short channel to a long channel or from the Levinson 1999), a check on the momentum and mass closed end to the open end. Near the closed end (␦ balances may quantify the errors and verify that the between 0.5 and 1.0 or x between 50 and 100 km), the results do satisfy the momentum and continuity equa- flow pattern is such that the mean inward flow occurs in tions with reasonably small residuals or errors. For this the deep water and the mean outward flow occurs in purpose, we have calculated the errors of the momen- the shallow water. After a short transition zone (x tum and continuity equations for all of the experiments. around 40 km), the flow pattern switches to an opposite These errors are examined at both the first order for sense such that the mean inward flow occurs in the the tidal motion and higher order with the full nonlin- shallow water and the mean outward flow occurs in the ear equations for the residual solution. deep water. The present result for short channels is different from that of Li and O’Donnell (1997), which uses a crude approximation for the first-order tide. In section 4, we will further discuss the change of flow regimes from short to long channels and from the closed end to the open end in a long channel. b. Exchange rate and residence time The along-channel component of the mean flow can be either positive (inward) or negative (outward). By integrating all positive volume flux per unit time across a given section of the channel, the magnitude of the rate of exchange at that section can be examined. If we integrate all negative volume flux per unit time instead, we will get the same exchange rate with a negative sign.

The exchange rate can then be used to calculate the FIG. 3. (a) Exchange rate and (b) residence time as functions of residence time as the ratio between the volume of the the channel length.

Unauthenticated | Downloaded 09/25/21 10:09 PM UTC OCTOBER 2005 L I A N D O ’ DONNELL 1833

The errors of both momentum and continuity equations are calculated at every computational grid point with velocity fields along and across the channel. It shows that the momentum equation is accurately satisfied within the computational error of the computer. This is because the along-channel velocity is solved exactly using the x momentum equation and the cross- channel velocity is obtained exactly by integrating the continuity equation after the surface elevation is solved. This can also be verified by substituting the solution into the original equations. The calculation of the errors of the equations serves the purpose of veri- fying that the computer programs used in the calculations are correct. The surface elevation, on the other hand, is solved with the approximation that the elevation is y independent. As a result, the largest residual is seen in the continuity equation, which has a maximum relative error of about 2 ϫ 10Ϫ3 or 0.2%, a small value FIG. 4. Mean sea level along the channel for different channel lengths. nevertheless. The above examination is made to the momentum and continuity equations. To test the mass conservation which can also be calculated by the first-order tide further, we have also directly calculated the net mass through (20). By integrating the mean surface slope flux through different cross sections along the channel. along the channel, the mean surface elevation is ob- The net mass flux is obtained by integrating the along- tained. Figure 4 shows the results of this integration for

channel component of the transport velocity (uT) across all model runs with the given depth function and pa- the channel. All experiments show practically zero net rameters. Each curve represents the along-channel dis- mass flux. Therefore, the solution does satisfy the equa- tribution of the mean surface for one model run (i.e., tions, and the mass is conserved accurately. with a certain channel length). The results show that the mean surface always increases toward the head of the b. Residual transport velocity channel for all cases. This produces an outward- directed (negative) pressure gradient force. The surface To help to interpret the switch of exchange flow pat- slope is strongly dependent on the length of the chan- terns from short to long channels, we now analyze the nel. For a very short channel, the slope is very small— residual transport velocity (18). We write the right- almost flat (the short lines of Fig. 4). As the length hand side of (18) as a sum of three terms: increases, the slope increases and then oscillates but ϭ ϩ ϩ ͑ ͒ uT uT1 uT2 uT3, 25 with a decreasing magnitude of oscillation in a manner similar to the response of tidal velocity as the length of in which the channel increases (see figures in Li 2001). ␨ Ѩ Ѩ The increase of mean surface elevation toward the 1u1 h u1 u1 u ϭ 2 , u ϭϪ ͩu ϩ ␷ ͪ, and head of the channel can be explained by the propaga- T1 h T2 ␤ 1 Ѩx 1 Ѩy tion of tidal waves. We know that tide in a semienclosed Ѩ␨ channel can be considered as an incident wave plus a gh 2 u ϭϪ . ͑26͒ T3 ␤ Ѩx reflected wave [which can be seen by expressing (9) and (11) in real instead of complex format]. If there is no

The first term uT1 is a combination of two mechanisms: bottom friction, the incident and reflected waves will the bottom friction and the finite tidal amplitude; each have equal amplitudes and are out of phase (e.g. Defant ␨ is equal to 1u1/h, and thus the total has a factor of 2. 1961). With bottom friction, the wave energy is attenu- The second term uT2 is a result of advection. The third ated as it propagates. For a long channel, the bottom

term uT3 is caused by the mean pressure gradient force friction causes more decrease in the reflected wave than induced by a subtidal surface slope. Among these in the incident wave because at a given location the terms, the first two are directly determined by the first- reflected wave has bounced from the head and thus

order tidal motion. The last term uT3 is proportional to traveled longer distance than the incident wave. The the mean surface slope of the water along the channel, net result is a stronger incident wave and weaker re-

Unauthenticated | Downloaded 09/25/21 10:09 PM UTC 1834 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35 flected wave, especially at the open end of the channel. The stronger the incident wave is, the closer the wave is to a progressive wave. A pure progressive wave only exists in an infinitely long channel (regardless of whether there is friction) in which there is no reflection of wave energy such that the elevation and flow are in phase (with 0° phase difference). Equivalently, the high tide of a progressive wave has a maximum inward flow (or flood). The low tide has a maximum outward flow (or ebb). The maximum flood velocity is approximately the same as the maximum ebb velocity in magnitude for a first-order approximation. Because at high tide the total water depth is larger, this progressive wave causes a net inward transport of water mass, which results in an increased mean sea level toward the head. The longer the channel is, the more pronounced effect the bottom friction will have. For short channels, the phase difference between elevation and flow is close to 90° and the transport due to wave motion is nearly zero, causing a “flat” mean surface. The mean surface slope, whenever it exists, causes a seaward-directed pressure gradient force that drives a net outward flow, which in turn prohibits the mean sea level slope from further increase; a state of balance is thus reached, which is the solution.

1) SHORT CHANNEL In this section, we will examine the contributions of each of the three terms in (25). We will discuss the distributions of each term for a short channel and a long channel, respectively. For a short channel, uT1 is positive (into the channel) in the deeper central part and is negative in the shoal area (Fig. 5a). This is because tide in a short channel is closer to a standing wave. A pure standing wave is such that the elevation ␨ and the along- channel velocity have a 90° phase difference. As a result, the tidally averaged product of ␨ and u—that is, ␨u—is zero. With bottom friction, ␨u is not exactly zero but can go either way (positive or negative) for a short channel. To be more specific, the tidally averaged ␨u or ␴ ␴ ϩ ␣ ␣ uT1 is proportional to sin( t) cos( t ), in which is the phase difference in excess of 90° between the sur- Ϫ1 face elevation ␨ and the along-channel velocity u [note FIG. 5. The transport velocity components (m s ) for a short channel [(a) u , (b) u , and (c) u ], and (d) the transport veloc- that sin(␴t) and cos(␴t)havea90° phase difference]. T1 T2 T3 ity u . This quantity is a function of cross-channel distance or T water depth and is identically zero if there is no friction when the phase difference is exactly 90° (i.e., ␣ ϭ 0). less than 0 and for h less than h˜ ␣ is greater than 0. When there is bottom friction, ␣ is zero only at a certain Because value of h ϭ h˜, which should be around the mean depth. Because bottom friction causes the along-channel ve- sin͑␣͒ locity in shallower water to lead that in deeper water, sin͑␴t͒ cos͑␴t ϩ ␣͒ ϭϪsin͑␣͒ sin2͑␴t͒ ϭϪ , 2 that is, ␣ in shallower water is larger than that in deeper water (Li 2001), it follows that for h greater than h˜ ␣ is ͑27͒

Unauthenticated | Downloaded 09/25/21 10:09 PM UTC OCTOBER 2005 L I A N D O ’ DONNELL 1835 the tidally averaged ␨u is an odd function of ␣, which leads to

Ͼ0 ͑h Ͼ h˜ ͒ u ϰ ␨uͭ . ͑28͒ T1 Ͻ0 ͑h Ͻ h˜ ͒

This qualitatively explains the negative values of uT1 in the shoal area and positive values in the deep water for short channels (Fig. 5a). The mean depth for the given depth function is about 8.1 m. The zero uT1 for this case occurs at y ϭ 0.65 and 1.35 km, corresponding to a depth h˜ of about 8 m. This explains the flow pattern of Fig. 5a.

The second term uT2 is all positive, which indicates that advection induces inward transport of water mass. The deeper the water is, the larger this inward transport is (Fig. 5b). The third term uT3 is all negative (Fig. 5c), which is consistent with the positive mean surface slope as discussed earlier (Fig. 4). A larger outward transport by uT3 is through the deep water. This is because the pressure gradient is laterally independent and thus the vertically integrated transport due to this mechanism is proportional to water depth, as is evident from the definition of uT3 [(26)]. It turns out that although uT1 has a smaller magnitude than either uT2 or uT3, the inward advective transport and the outward transport due to the mean pressure gradient force are similar in magnitude. The net effect, or the sum of all the three terms uT

(Fig. 5d), is therefore similar to uT1, which is a result of the finite amplitude of the tide and the phase-difference distribution in different water depth due to bottom friction. Therefore, the net transport in a short channel is such that the inward flow occurs through the deep water while the outward flow occurs through the shallow water.

2) LONG CHANNEL For a long channel, the tidally averaged product of surface elevation and velocity—that is, ␨u—can be expressed as being proportional to sin(␴t) sin(␴t ϩ ␣). The reason for this expression, just as that used earlier Ϫ for a short channel, that is, sin(␴t) cos(␴t ϩ ␣),isto FIG. 6. The transport velocity components (m s 1) for a long channel [(a) u , (b) u , and (c) u ], and (d) the transport veloc- have a small value for ␣ for the sake of discussion. For T1 T2 T3 ity u . an infinite tidal channel without friction, tide is purely T progressive and ␣ in sin(␴t) sin(␴t ϩ ␣) is identically zero. When the channel is long, the bottom friction wave becomes more important relative to the incident damps the reflected wave significantly and the wave at wave and uT1 is still positive but its magnitude de- the open end is close to a pure progressive wave—that is, creases. Farther inside the channel where the wave is ␣ is close to zero. This ensures that sin(␴t) sin(␴t ϩ ␣) close to a standing wave, the situation is similar to a ϳ 2 ␴ ϭ Ͼ Ͼ sin ( t) 1/2 0, and thus uT1 0 across the short channel (Fig. 5a), where the positive uT1 is in the ϭ channel (Fig. 6a; ocean side of x 50 km). As the deep water and the negative uT1 is in the shallow water position goes farther inside the channel, the reflected (Fig. 6a).

Unauthenticated | Downloaded 09/25/21 10:09 PM UTC 1836 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35

ץ ␷ץ ␷ Similar to short channels, the second term uT2 is all cross-channel direction [ 1( 1/ y)] and 2) the earlier positive and thus the advection induces inward trans- model has a y-independent ␨ЈuЈ term. Just as in the

port of water mass (Fig. 6b). The third term uT3 is again present model, all the three terms (uT1, uT2, and uT3) all negative (Fig. 6c), which is consistent with the posi- contribute to the residual circulation. The force terms tive mean surface slope (Fig. 4) into the channel as in the momentum equations are force per unit mass.

discussed earlier. Now the magnitudes of uT2 and uT3 The total force in the water column is integrated over are not similar in pattern any more. A larger outward the water column, which has less mass in shallower wa- transport by uT3 is through the deeper water, because of ter than in deeper water. The total force in the water an almost laterally independent along-channel pressure column is thus depth dependent. The residual circula- gradient. It turns out that, for long channels, the first tion is generated because of this nonuniform depth dis- term uT1 has larger values at shallow waters near the tribution. This is made clearer when the vorticity is open end (Fig. 6a). This is apparently caused by a larger examined as in the earlier work: all three terms con- nonlinear effect due to finite tidal amplitude in shallow tribute to the production of residual vorticity or circu- water than is present in deep water of a progressive lation. The current work is not fundamentally different wave. The second term, or the advective transport, uT2 in that aspect, although we have demonstrated new re- is relatively small relative to either the first or the third sults when the model is refined to solve the first-order term. This is in contrast to the short-channel cases. The equations without the approximation employed in the third term has a negative value with a larger magnitude earlier study. than that of uT1 in the deep water. The net effect at the The new result of this model is that the flow regimes open end is thus an outward flow in the deep water and can change along a long channel or they can be differ- an inward flow over the shoals (Fig. 6d). As the position ent between short and long channels at the open end. goes farther into the channel, the progressive nature of This explains why the residual flow patterns in the the tide decreases until at a point where the flow re- James River estuary (Li et al. 1998) are different from sembles that of a short channel again: with an inward that of a short channel (Winant and Gutiérrez de Ve- flow in the deep water and an outward flow in the lasco 2003). It also demonstrates that the tidally in- shallow water. duced flow can reinforce the density-driven flow (inward flow through the deep water but outward flow c. Comparison with previous studies through the shallow water; Hamrick 1979; Wong 1994) The difference between the earlier work (Li and in a short channel or at the closed end of a long channel O’Donnell 1997) and this work is that the former is while competing with the density-driven flow at the overly simplified such that the effect of the depth de- open end of a long channel. This may be used to explain pendence of the first-order tidal velocity is neglected, the flow regime change in Winyah Bay (Kim and Voul- and that results in an underestimate (not an exclusion) garis 2005) between the upper bay and lower bay. of the advective effect, particularly in a short channel in Note also that Zimmerman (1981) has predicted a which the advection is relatively more important than residual flow pattern consistent with that of a short that in a long channel. The earlier model also shows channel using vorticity dynamics in a flat-bottom dependence of the residual flow on the length ratio, but semienclosed basin with only lateral friction. In that the exchange flow does not reverse direction between model, the only nonlinearity considered is the advec- short and long channels. The depth-averaged along- tion. By neglecting other nonlinear terms, that model channel component of the residual transport velocity in effectively excludes long channels. The physical expla- the earlier model is as follows: nation of the residual flow pattern of Zimmerman’s ␨ Ѩ Ѩ␨Ј model is as follows: the current in the center of the pup h up u ϭ 2 Ϫ ͩu ϩ g ͪ, ͑29͒ channel is larger than that close to the lateral boundary T h ␤ p Ѩx Ѩx because of less side-wall friction away from the bound- ␨ in which p and up are the “prediction” solution for tide ary. This produces a vorticity in opposite senses on the or the approximate solution of the first-order equation two sides of the axis of the channel, which also alter- that ignores cross-channel depth variation and uЈ and ␨Ј nates the sense of rotation between flood and ebb tides. are the second-order residual velocity and water eleva- Because the water particles advecting through the basin tion, respectively. The differences between the present affected by the lateral friction will have reduced veloc- result—for example, the first equation in (18)—and the ity when it returns to the ocean during ebb tide, the net earlier result in (29) include the facts that 1) the earlier effect will be an inward flow in the center of the chan- results do not have the advective force term in the nel and a return flow near the side walls across the

Unauthenticated | Downloaded 09/25/21 10:09 PM UTC OCTOBER 2005 L I A N D O ’ DONNELL 1837 channel. This explanation holds if one substitutes the ematics that the integral can be expressed as the func- ϭ lateral friction with bottom friction and considers a tion at an intermediate point [h h0(y0)] multiplied by depth function that varies across the channel, in which the interval; that is, case the net inward flow will be through the deep water i␴D i␴D (not necessarily in the center of the channel) and out- ␻ ϭ Ϸ L L D ward flow will be through the shallow water. This can gh gh0D ΊϪ͵ dy ΊϪ be seen from Fig. 5b, which shows that the net advec- ␴ ϩ ␤ր ␴ ϩ ␤ր 0 i h i h0 tive transport is larger in the deep water than in the shallow water, although here the advective term alone L i␤ ϭ ␲ ͱ Ϫ ͑ ͒ does not meet the mass-balance requirement. 2 ␭ 1 ␴ , 31 h0 d. Multiple-frequency problems where h0 is a depth value between the minimum and ␭ ϭ ͌ maximum depths and gh0/f is the wavelength. The solution given here only contains a single tidal The reason that the above equation is approximate is frequency and the calculations have only been applied that the function being integrated is a complex one and to a semidiurnal tidal frequency. In many coastal wa- the intermediate points for the real and imaginary parts ters, semidiurnal tide is indeed the main constituent. In of the integration can be different. other coastal waters, diurnal tide is dominant or tide is Using the above result, (12) can be expressed as a mixture of different constituents of comparable magnitudes. For these situations, the present results do not L ␤ x ͫ ␲ ͱ Ϫ ͩ Ϫ ͪͬ apply. The solution, however, can be readily modified ␨ cos 2 ␭ 1 i ␴ 1 A 0 h0 L for these problems because the solution for the second- Ϸ . ͑32͒ h0 h0 L ␤ order equations is not done in the frequency domain. It ͩ ␲ ͱ Ϫ ͪ cos 2 ␭ 1 i ␴ is an explicit form that only depends on the first-order h0 tide, regardless of whether or not it is a solution of a The intermediate depth h is roughly equal to the mean single frequency. Because the first-order tide is solved 0 depth. Our calculations show that it is a very good ap- from a set of linear equations in frequency domain, we proximation. This equation indicates several nondi- can add all the major tidal constituents of the first-order mensional parameters: 1) the nonlinear parameter ␧ϭ tide for different frequencies together before solving ␨ /h , 2) the length ratio ␦/4 ϭ L/␭, and 3) the time-scale the second-order subtidal flow field. However, for a 0 0 ratio ␶ ϭ ␤/␴h (the ratio between the tidal time scale multiple-frequency problem in which different frequen- 0 1/␴ and the frictional decay time scale h /␤). All of cies may have comparable magnitudes in the energy 0 these parameters are also present in the earlier model spectrum, the definition of residual flow can be prob- of Li and O’Donnell (1997). In fact, the right-hand side lematic because there is no obvious choice for the time of (32) is the same as the solution of Li and O’Donnell period over which the average can be made to yield a (1997). It can be shown that a similar operation to the “mean” quantity. first-order tidal velocity does not indicate any new parameter except a velocity scale that is a combination of e. Other parameters the above parameters. The second-order tidally in- In this study, we have focused on the effect of a single duced flow is dependent on the first-order tide, and parameter, ␦ ϭ 4L/␭. We now take a closer look at the therefore all of the above parameters affect the second- solution for other parameters that may also affect the order solution. solution. The first-order tide, (12), can be expressed in In our presentation, we have only used results from a ␻ ϭ a nondimensional form. First notice that L in (12) can standard bottom drag coefficient CD 0.0025. By vary- be written as ing depth function (and h0) and CD (equivalently vary- ␤ ␴ ing the third parameter: the time-scale ratio / h0), the i␴D i␴D length scales will be modified (e.g., the length scale that ␻L ϭ Lͱ ϭ L . ͑30͒ F D gh divides between short and long channels). For given ΊϪ͵ dy i␴ ϩ ␤րh bottom drag coefficient (CD) and velocity magnitude 0 ␤ ␶ (U0) or friction coefficient ( ), the time-scale ratio is Because both the real and imaginary parts of the func- inversely proportional to mean water depth h0. For the tion being integrated in the above equation {that is, ␤ value used in this work (0.001 76 m sϪ1), ␶ varies from gh/[i␴ ϩ (␤/h)]} are continuous and can be integrated as 2.42 to 0.6 when the mean depth changes from 5 to 20 a function of h(y), we can apply a theorem of the math- m. In particular, when the mean depth is 8.1 m as is the

Unauthenticated | Downloaded 09/25/21 10:09 PM UTC 1838 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35

case in this paper (Table 1), ␶ is 1.99, indicating a strong open end. At the open end of a long channel, an inward bottom friction effect. The qualitative result (i.e., the flow occurs in shallow water and outward flow occurs in contrast between short and long channels) is, however, deep water, just the opposite of that of a short channel. not sensitive to the change of these parameters. In ad- The interior of the long channel close to the head, how- dition, the cross-channel depth gradient also affects the ever, has a flow pattern consistent with that of a short solution. In the earlier model, the depth gradient is channel. When the channel is intermediate in length, or expressed by a ratio between the minimum and maxi- ␦ is about 0.6–0.7, the residual flow at the open end

mum depths hmin/hmax. In the present work, it is difficult reaches its minimum, causing a maximum water resi- to isolate a similar nondimensional parameter for the dence time. When a channel’s depth distribution is al- depth gradient because of the format of the depth func- tered (such as by a dredging activity or a storm event

tion. By experimenting with different h1 and h2 values that involves a large scale transport of sediment), the in (22), however, we have examined the effect of bot- length ratio may change even if the geometric length of tom slope. We have also experimented with different ␧ the channel remains the same. The exclusion of density- values. All of the results of these parameter space gradient-related mechanism allows a thorough analysis analyses are qualitatively similar to what have been of the response of tidally induced flows to different presented and do not change the conclusions. There- channel lengths. As a result, this work only applies to fore, we have omitted detailed discussion of the effect well-mixed tidal channels. It may, however, provide of all the other parameters and have only focused on useful reference and methods to study those more com- the effect of ␦. Only the effect of ␦ can cause the re- plicated flow conditions, such as when the density gra- versal of the exchange flows. dient is included.

Acknowledgments. This study was supported for the 5. Conclusions most part by the University of Connecticut through We have solved an analytic model that allows the use several fellowships to Chunyan Li. It was also sup- of arbitrary cross-channel depth functions to resolve ported by the Center for Coastal Physical Oceanogra- the subtidal flow regimes in well-mixed tidally domi- phy at Old Dominion University, Georgia Sea Grant nated channels of different lengths. The model is vali- (NA06RG0029-RR746-007/7512067), South Carolina dated by verifications of momentum balance, continu- Sea Grant (NOAA NA960PO113), and Georgia DNR ity equation, and the overall mass balance across the (RR100-279-9262764). Constructive comments and channel. Different depth functions are used to repre- suggestions from two reviewers are highly appreciated. sent different shapes of the cross-channel bathymetry. George Li provided some useful editorial suggestions. The solutions are relatively insensitive to the choice of depth profile, and therefore we have only discussed the APPENDIX results of a selected depth function in detail. By varying The Perturbation Equations the channel length we are able to discuss the responses This appendix shows derivation from (5) to the first- of the residual flows and their contributing compo- order and second-order (8) and (16). It is done using nents. nondimensional variables as a formal procedure of a The results show that the tidally induced flows in normal perturbation method. The variables are made elongated channels are strongly dependent on the dimensionless using the following transformations: length for a given value of friction. The flow direction reverses between short and long channels as well as x y u ␷ ˆt ϭ ␴t, xˆ ϭ , yˆ ϭ , uˆ ϭ , ␷ˆ ϭ , from the open end to the head in a long channel. The L0 D U0 V0 length of the channel is measured by the ratio between h ␨ g the geometric length of the channel and a quarter of the ˆ ϭ ␨˜ ϭ ϭ ␨ ͱ ϭ ⑀␴ h , ␨ , U0 0 , V0 D, tidal wavelength. When this ratio is smaller than 0.6– h0 0 h0 0.7, the channel is considered to be short, in which case ␨ ␭ T͌gh 2␲ the tide is close to a standing wave. The short channel ⑀ ϭ 0 ϭ ϭ 0 ϭ ͑ ͒ , L0 ␲ ␲ , and T ␴ , A1 exhibits an outward flow over the shallow water and h0 2 2 ␴ ␨ inward flow through the deep water. When this ratio is where , L0, D, U0, V0, 0, and T are the angular fre- larger than 0.6–0.7, the channel is considered to be long, quency of semidiurnal tide, tidal length scale, width of the tide is close to a progressive wave, and the tidally the estuary, scale of the longitudinal velocity, scale of induced flow is dominated by the nonlinear wave the lateral velocity, tidal amplitude at the mouth, and propagation and the mean pressure gradient at the the period of tide. The choice of the scales is the same

Unauthenticated | Downloaded 09/25/21 10:09 PM UTC OCTOBER 2005 L I A N D O ’ DONNELL 1839

␨ as in Ianniello (1977), which are based on the scales of in which 0 and h0 are the tidal amplitude and mean semidiurnal tide and the geometry of the channel. water depth, respectively. In a typical tidal channel, ␧ is Equations (5) can then be written as on the order of 0.1. We now define a power series ex- pansion for the velocity and elevation in terms of ␧ as Ѩuˆ U Ѩuˆ V Ѩuˆ g␨ Ѩ␨ˆ ␤ uˆ ϩ 0 ϩ 0 ␷ ϭϪ 0 Ϫ ␴ uˆ Ѩ ␴ ˆѨ ␴ Ѩ ␴ uˆ ϭ uˆ ϩ ⑀uˆ ϩ ⑀2uˆ ϩ ···, Ѩˆt L0 xˆ D yˆ U0L0 xˆ h0 hˆ 1 2 3 ␷ˆ ϭ ␷ˆ ϩ ⑀␷ˆ ϩ ⑀2␷ˆ ϩ ···, and ␤␨ ␨ˆ 1 2 3 0 uˆ ϩ , ␴ 2 ˆ 2 ␨ˆ ϭ ␨ˆ ϩ ⑀␨ˆ ϩ ⑀2␨ˆ ϩ ͑ ͒ h0 h 1 2 3 ···, A8 in which Ѩ␷ˆ U Ѩ␷ˆ V Ѩ␷ˆ g␨ Ѩ␨ˆ ␤ ␷ˆ ϩ 0 uˆ ϩ 0 ␷ˆ ϭϪ 0 Ϫ ␴ Ѩ ␴ Ѩ ␴ Ѩ ␴ ϭ ͑ ˆ͒ Ѩˆt L0 xˆ D yˆ V0D yˆ h0 hˆ uˆ i uˆ i xˆ, yˆ, t , ␤␨ ␷␨ˆ ␷ ϭ ␷ ͑ ˆ͒ 0 ˆ ˆi ˆi xˆ, yˆ, t , and ϩ , and ␴ 2 ˆ 2 h0 h ␨ˆ ϭ ␨ˆ ͑ ˆ͒ ͑ ͒ i i xˆ, yˆ, t , A9 Ѩ␨ˆ h U Ѩ͑hˆ ϩ ⑀␨ˆ͒uˆ h V Ѩ͑hˆ ϩ ⑀␨ˆ͒␷ˆ where i ϭ 1,2,3,....Bysubstituting (A8) into (A6), ϩ 0 0 ϩ 0 0 ϭ ͑ ͒ ␴␨ Ѩ ␴␨ Ѩ 0. A2 the O(␧1) and O(␧2) equations are obtained as Ѩˆt 0L0 xˆ 0D yˆ Ѩ Ѩ␨ˆ It is well established that (Ianniello 1977) uˆ 1 1 uˆ 1 ϭϪ Ϫ D , Ѩˆ Ѩxˆ ˆ U V g␨ h U t h 0 ϭ ⑀ 0 ϭ ⑀ 0 ϭ 0 0 ϭ ␴ , ␴ , ␴ 1, ␴␨ 1, and L0 D U0L0 0L0 Ѩ␷ Ѩ␨ˆ ␷ ˆ1 1 ˆ1 ϭϪH Ϫ D , and h V Ѩˆ Ѩyˆ ˆ 0 0 ϭ ͑ ͒ t h ␴␨ 1. A3 0D Ѩ␨ˆ Ѩ Ѩˆ␷ 1 uˆ 1 hˆ1 To reduce the complexity of the notation, we now write ϩ hˆ ϩ ϭ 0 ͑A10͒ Ѩˆ Ѩxˆ Ѩyˆ the coefficient of bottom friction using t ␤ and ϭ ͑ ͒ ␴ D A4 h0 Ѩ Ѩ Ѩ Ѩ␨ˆ ␨ˆ uˆ 2 uˆ 1 uˆ 1 2 uˆ 2 uˆ 1 1 ϩ uˆ ϩ ␷ˆ ϭϪ Ϫ D ϩ D , 1 Ѩ 1 Ѩ Ѩ 2 and the coefficient of the across-channel pressure gra- Ѩˆt xˆ yˆ xˆ hˆ hˆ dient term using Ѩ␷ Ѩ␷ Ѩ␷ Ѩ␨ˆ ␷ ␷ ␨ˆ ˆ2 ˆ1 ˆ1 2 ˆ2 ˆ1 1 g␨ ͌gh 2 ϩ uˆ ϩ ␷ˆ ϭϪH Ϫ D ϩ D , and 0 ϭ ϭ ͩ 0ͪ ͑ ͒ Ѩˆ 1 Ѩxˆ 1 Ѩyˆ Ѩyˆ ˆ ˆ 2 ␴ H ␴ . A5 t h h V0D D Ѩ␨ˆ Ѩuˆ Ѩ␨ˆ uˆ Ѩhˆ␷ˆ Ѩ␨ˆ ␷ˆ Equation (A2) is therefore equivalent to 2 ϩ ˆ 2 ϩ 1 1 ϩ 2 ϩ 1 1 ϭ ͑ ͒ h Ѩ Ѩ Ѩ Ѩ 0. A11 Ѩˆt xˆ xˆ yˆ yˆ Ѩuˆ Ѩuˆ Ѩuˆ Ѩ␨ˆ uˆ uˆ␨ˆ ϩ ⑀uˆ ϩ ⑀␷ˆ ϭϪ Ϫ D ϩ ⑀D , Ѩ Ѩ Ѩ 2 If one applies a temporal average over a tidal cycle, Ѩˆt xˆ yˆ xˆ hˆ hˆ (A11) leads to Ѩ␷ˆ Ѩ␷ˆ Ѩ␷ˆ Ѩ␨ˆ ␷ˆ ␷ˆ␨ˆ ϩ ⑀ ϩ ⑀␷ ϭϪ Ϫ ϩ ⑀ Ѩ Ѩ Ѩ␨ˆ ␨ˆ uˆ ˆ D D , and uˆ 1 uˆ 1 2 uˆ 2 uˆ 1 1 Ѩˆ Ѩxˆ Ѩyˆ Ѩyˆ ˆ ˆ 2 uˆ ϩ ␷ˆ ϭϪ Ϫ D ϩ D , t h h 1 Ѩ 1 Ѩ Ѩ 2 xˆ yˆ xˆ hˆ hˆ Ѩ␨ˆ Ѩ͑hˆ ϩ ⑀␨ˆ͒uˆ Ѩ͑hˆ ϩ ⑀␨ˆ͒␷ˆ ϩ ϩ ϭ ͑ ͒ Ѩ␷ Ѩ␷ Ѩ␨ˆ ␷ ␷ ␨ˆ Ѩ Ѩ 0. A6 ˆ1 ˆ1 2 ˆ2 ˆ1 1 Ѩˆt xˆ yˆ uˆ ϩ ␷ˆ ϭϪH Ϫ D ϩ D , and 1 Ѩ 1 Ѩ Ѩ 2 xˆ yˆ yˆ hˆ hˆ Because we are interested in the nonlinear effect due to Ѩ Ѩ␨ˆ Ѩˆ␷ Ѩ␨ˆ ␷ a finite tide–depth ratio, the assumption is that the pa- uˆ 2 1uˆ 1 hˆ2 1ˆ1 hˆ ϩ ϩ ϩ ϭ 0. ͑A12͒ rameter ␧ is not negligible but is smaller than 1: Ѩxˆ Ѩxˆ Ѩyˆ Ѩyˆ ␨ 0 Equations (A10) and (A12) are the nondimensional ⑀ ϭ Ͻ 1, ͑A7͒ h0 formats of the dimensional equations in (8) and (16),

Unauthenticated | Downloaded 09/25/21 10:09 PM UTC 1840 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35 respectively, which can be verified by changing the vari- tion in partially stratified estuaries. Ph.D. thesis, University ables back to dimensional. of California at Berkeley, 451 pp. Referring to (3) and (4), the boundary conditions for Ianniello, J. P., 1977: Non-linear induced residual currents in tidally dominated estuaries. Ph.D. dissertation, University of the nondimensional equations are Connecticut, 250 pp. Kim, Y. H., and G. Voulgaris, 2005: Effect of channel bifurcation | ϭ ␷| ϭ ␨ˆ| ϭ ͑ itˆ͒ ͑ ͒ uˆ xˆϭ1 0, ˆ yˆϭ0,1 0, xˆϭ0 Re e , and A13 on residual estuarine circulation: Winyah Bay, South Caro- lina. Estuarine, Coastal Shelf Sci., in press. 1 ͵ ͑ ˆ ϩ ␨ˆ͒ ϭ ͑ ͒ Lele, S. K., 1992: Compact finite difference schemes with spectral- h uˆ dyˆ 0. A14 like resolution. J. Comput. Phys., 103, 16–42. 0 Li, C., 1996: Tidally induced residual circulation in estuaries with The first-order boundary conditions are cross channel bathymetry. Ph.D. dissertation, University of Connecticut, 242 pp. itˆ ——, 2001: 3D analytic model for testing numerical tidal models. uˆ | ϭ ϭ 0, ␷ˆ | ϭ ϭ 0, ␨ˆ | ϭ ͑ ͒ and 1 xˆ 1 1 yˆ 0,1 1 xˆϭ0 Re e , J. Hydraul. Eng., 127, 709–717. ͑A15͒ ——, and D. O’Donnell, 1997: Tidally driven residual circulation in shallow estuaries with lateral depth variation. J. Geophys. 1 Res., 102, 27 915–27 929. ͵ ˆ ϭ ͑ ͒ ——, and A. Valle-Levinson, 1999: A two-dimensional analytic huˆ 1 dyˆ 0. A16 0 tidal model for a narrow estuary of arbitrary lateral depth variation: The intratidal motion. J. Geophys. Res., 104, The last condition (A16) is automatically satisfied be- 23 525–23 543. cause the first-order tide is oscillatory (a sine function) ——, ——, K.-C. Wong, and K. M. M. Lwiza, 1998: Separating ϭ baroclinic flow from tidally induced flow in estuaries. J. Geo- and its temporal average is always zero; that is, uˆ 1 0. The second-order boundary conditions are phys. Res., 103, 10 405–10 417. ——, ——, L. P. Atkinson, K.-C. Wong, and K. M. M. Lwiza, 2004: Estimation of drag coefficient in James River estuary uˆ | ϭ ϭ 0, ␷ˆ | ϭ ϭ 0, ␨ˆ | ϭ and ͑ ͒ 2 xˆ 1 2 yˆ 0,1 2 xˆϭ0 0, A17 using tidal velocity data from a vessel-towed ADCP. J. Geo- phys. Res., 109, C03034, doi:10.1029/2003JC001991. 1 ͵ ͑hˆ uˆ ϩ ␨ uˆ ͒ dyˆ ϭ 0. ͑A18͒ Parker, B. B., 1984: Frictional effects on the tidal dynamics of 2 1 1 shallow estuary. Ph.D. dissertation, The Johns Hopkins Uni- 0 versity, 291 pp. Proudman, J., 1953: Dynamical Oceanography. Methuen, 409 pp. REFERENCES Robinson, A. H. W., 1960: Ebb-flood channel systems in sandy bays and estuaries. Geography, 45, 183–199. Browne, D. R., and C. W. Fisher, 1988: Tide and tidal currents in ——, 1965: The use of the sea bed drifter in coastal studies with the Chesapeake Bay. NOAA Tech. Rep. NOS OMA3, 84 pp. particular reference to the Humber. Z. Geograph. Suppl., 7, Carpenter, M. H., D. Gottlieb, and S. Abarbanel, 1993: The sta- 1–22. bility of numerical boundary treatments for compact high- Robinson, I. S., 1983: Tidally induced residual flow. Physical order finite-difference schemes. J. Comput. Phys., 108, 272– Oceanography of Coastal and Shelf Seas, B. Johns, Ed., 295. Elsevier, 321–356. Charlton, J. A., W. McNicoll, and J. R. West, 1975: Tidal and Winant, C. D., and G. Gutiérrez de Velasco, 2003: Tidal dynamics freshwater induced circulation in the Tay estuary. Proc. Roy. and residual circulation in a well-mixed inverse estuary. J. Soc. Edinburgh B, 75, 11–27. Phys. Oceanogr., 33, 1365–1379. Defant, A., 1961: Physical Oceanography. Vol. 2, Pergamon Press, Wong, K.-C., 1994: On the nature of transverse variability in a 598 pp. coastal plain estuary. J. Geophys. Res., 99, 14 209–14 222. Friedrichs, C. T., and J. M. Hamrick, 1996: Effects of channel ge- Zimmerman, J. T. F., 1974: Circulation and water exchange near ometry on cross sectional variations in along channel velocity a tidal watershed in the Dutch Wadden Sea. Netherlands J. in partially stratified estuaries: Buoyancy effects on coastal Sea Res., 8, 126–138. and estuarine dynamics. Coastal Estuarine Stud., 53, 283–300. ——, 1981: Dynamics, diffusion and geomorphological signifi- Hamrick, J. M., 1979: Salinity intrusion and gravitational circula- cance of tidal eddies. Nature, 290, 549–555.

Unauthenticated | Downloaded 09/25/21 10:09 PM UTC