1. Introduction
Lateral circulation is the flow that is normal to the streamwise direction in a channel. Although lateral circulation is typically one order of magnitude smaller than streamwise flow, it plays an important role in streamwise dynamics (Lerczak and Geyer 2004; Cheng et al. 2017), streamwise dispersion (Fischer 1969; Smith 1976, 1977; Geyer et al. 2008), and morphological development (Huijts et al. 2006; Fugate et al. 2007). A comprehensive review of estuarine lateral circulation has been presented by Chant (2010). As river bends and coastal headlands widely exist in nature, flow curvature has been recognized as a major driving mechanism of lateral circulation. Thomson (1877) investigated the origin of the windings of rivers and showed that the dynamics driving helical circulation in a vertically sheared, curving flow results from a local imbalance between the curvature centrifugal force and lateral water surface slope. The former is proportional to the streamwise velocity and is typically larger in the upper layer such that it drives a vertically sheared crosschannel flow toward the outside bank, whereas the latter, generated by centrifugal motion, is vertically uniform and directed toward the inside bank. Thus, the combined effect of the two forces drives the lateral circulation with outside flow in the upper layer and inside flow near the bottom.
Onedimensional analytical models for curvatureinduced lateral circulation have been developed by Rozovskii (1957) and Kalkwijk and Booij (1986, hereafter KB) for an open channel flow with a logarithmic velocity profile and a parabolic vertical eddy viscosity profile. These models revealed the vertical structure of the lateral circulation and were used to estimate the frictional relaxation length that describes the gradual adaptation of the lateral circulation downstream of the region of flow curvature. The model results were shown to agree well with laboratory measurements and have been widely used to interpret observations. However, there have been discrepancies between the theory and observations when applying the KB model to a coastal region where tides dominate. For example, Geyer (1993) showed that the observed transverse flow around a coastal headland was roughly 3 times stronger than the prediction of the KB model, and Vennell and Old (2007) showed that the observed curvatureinduced lateral flow in a curved tidal channel was 50% more intense than that predicted by the KB model using the drag coefficient that best fitted the streamwise velocity profile. Although those discrepancies were attributed to stratification that increased the lateral flow by enhancing the shear in the streamwise flow and by suppressing vertical eddy viscosity (Geyer 1993; Chant 2002; Vennell and Old 2007), it should be noticed that the use of the KB model is restricted to steady unidirectional flows. An analytical model that describes curvatureinduced lateral circulation in oscillatory flows remains to be explored.
The dynamic theory of tidal currents has been established for rivers, estuaries, and coastal basins. Ianniello (1977) presented a twodimensional analytical model that first described tidal currents and residual currents in a frictional channel, although it ignored lateral processes. Li and O’Donnell (2005) developed a twodimensional analytical model that described depthaveraged tidal motion in a narrow channel without the effects of Earth’s rotation. Their results showed that the channel length affected the spatial pattern of tidally induced subtidal exchange flow due to a change in tidal wave characteristics when shifting between short and long channels. Winant (2007) provided a threedimensional analytical model that described tidal flows in an elongated rotating basin. This model overcame the difficulty of solving the depthintegrated continuity equation and was able to obtain the crossbasin water surface elevation such that it could elaborate the lateral circulation driven by the Coriolis force under the influence of friction for the first time. The strategy of Winant (2007) to solve the depthintegrated continuity equation has been adopted to develop an analytical solution of lateral circulation driven by differential advection in a tidal channel (Cheng et al. 2017). However, all of these previous studies focused on straight tidal channels. Still missing is a model for threedimensional tidal flows in a curved channel. The main objective of the present study is to establish such a model and to explore the essentials of curvatureinduced lateral circulation in a tidal channel on the basis of the work of KB and Winant (2007).
The remainder of this paper is structured as follows. Section 2 describes the development of the analytical model and provides solutions of the streamwise tidal flow and lateral circulations driven by the Coriolis and curvature centrifugal forces. Section 3 presents the characteristics of curvatureinduced lateral circulation and the competition between lateral circulations driven by the Coriolis and curvature centrifugal forces. Section 4 discusses the generalized local solution of curvatureinduced lateral circulation, the adaptation time of the lateral circulation, the effects of the lateral circulation on streamwise processes, and the limitations of the model. Section 5 summarizes the main findings of this study.
2. The model
a. Governing equations
The model channel is semienclosed with a varying crosschannel bottom bathymetry and a constant width

The channel is elongated, i.e., the width of the channel is much smaller than the length (λ > 1 or α < ε). A typical range of α is between 0.1 and 0.01. The competition between nonlinear advection and Coriolis force in the streamwise momentum equation [Eq. (4a)] is determined by λ, a large λ indicates that advection is more important than the Coriolis force.

The flow curvatures are mild, i.e., the curvature radius is larger than the channel width (R > 1). Mild bends have less significant lateral redistribution of streamwise momentum along the bend (Blanckaert and de Vriend 2010). Here, we specify that R ranges from 4 to 6, which is similar to the value adopted in the KB model (i.e., 1/R = 0.2). The competition between nonlinear advection and curvature centrifugal force in the streamwise momentum equation [Eq. (4a)] is determined by R, a large R indicates that advection is more important than the curvature centrifugal force.

The width of the channel is constant and the transverse bathymetry is a function of y alone, invariant along the streamwise direction.

The lateral flow is much weaker than the streamwise flow, and the eddy viscosity is thus completely determined by the main flow.
The first two preconditions assume that if the tidal dynamics is linear in the streamwise direction, the Coriolis force and curvature centrifugal force terms in the streamwise momentum equation are negligible. The third precondition is needed to obtain the tidal elevation (Winant 2007) as shown by Eq. (A14) in appendix A. The fourth precondition is needed to separate the lateral momentum equation as shown by Eq. (11). The above assumptions restrict the tidal flow to a certain category. In reality, sinuous rivers and wellmixed estuaries, and headlands in narrow channels might be included in this category.
b. Perturbation and Taylor series expansion
c. Solutions of tidal currents
3. Results
The streamwise characteristics of the tidal motion in an elongated straight channel have been well elaborated by Winant (2007). The existence of meanders in a tidal channel tends to reduce the tidal range and tidal currents and increase the ebb dominance through the enhanced friction of form drag around bends (Pein et al. 2018; Bo and Ralston 2020). Because this study concentrated on lateral circulations, the analysis of streamwise processes was neglected. The reference case chosen for the analysis below has a channel length of approximately a quarter of the tidal wavelength (i.e., κ = 1.5) and a small aspect ratio (i.e., α = 0.05).
a. Characteristics of curvatureinduced lateral circulation
Corresponding to the centrifugal force which has two components [Eq. (10)], the solution of curvatureinduced lateral circulation also has two components with the same amplitude: a periodic component having an overtide frequency and a steady component. The combined effect of the two components on the water surface elevation is illustrated in Fig. 2. The periodic component (i.e., η_{02}) of the water surface elevation shows two cycles during a tidal period, and the lateral water surface slope changes sign during each cycle, whereas the steady component (i.e., η_{00}) has a constant lateral gradient. Because the two components have the same amplitude, the periodic component is compensated during one phase and amplified during the opposite phase over a cycle, resulting in a water surface slope that has a constant direction (positive when the curvature radius is positive) but varies with an overtide frequency during a tidal period.
Associated with the water surface elevation, the velocity of curvatureinduced lateral circulation has the same combination effect. The periodic component of the lateral circulation switches the rotation direction during a cycle. The steady component compensates the periodic component when they have opposite rotation directions and reinforces the periodic component when they have the same rotation direction. As the two components have the same amplitude, the resulting lateral circulation maintains the same rotation direction but changes strength from zero to twice the amplitude of the individual component during a tidal period. In the case shown in Fig. 3, the curvatureinduced lateral circulation exhibits a cycle during the flood and ebb tides respectively. The lateral circulation starts from the beginning of flood or ebb, reaches a maximum near the phase of π/2 after the beginning, and gradually ceases at the end of the flood or ebb. The rotation direction is always clockwise (looking seaward).
The modeled channel is designed to have two bends with opposite curvature in order to explore the spatial distribution of curvatureinduced lateral circulation. The steady component of the curvatureinduced lateral circulation is used to show the differences between the two bends (Fig. 4). As the water surface piles up at the outside bank of a bend, the water surface slope has the same sign as the curvature such that it is positive at the first bend and negative at the second bend. Correspondingly, the lateral circulation, which is represented using the streamwise vorticity (i.e., ∂υ_{00}/∂z), also changes the rotation direction at the two bends. It is noticed that the amplitudes of the water surface slope and lateral flow velocity are larger at the first bend than at the second bend, showing asymmetries in curvatureinduced lateral circulation. Due to friction, the streamwise tidal current velocity decreases landward, and the tidal current velocity is thus larger at the first bend and leads to stronger lateral circulation there. The phase of the periodic component of the curvatureinduced lateral flow (i.e., V_{2}) has an offset of 180° between the two bends, consistent with the opposite rotation directions of the lateral circulation. At each bend, the phase is leading at the middle of the channel because the change in the streamwise tidal current starts from the thalweg of the channel.
b. Effects of friction
When friction is low (δ = 0.1), the curvatureinduced lateral circulation is confined within the bottom boundary layer where the shear of the streamwise tidal flow is large (Fig. 5). When friction is intermediate (δ = 0.5), the lateral circulation extends toward the surface and is evident throughout the transverse section. This pattern is similar to that in the high friction case (δ = 1.0) shown in Fig. 3. Because the lateral circulation is forced by the streamwise tidal flow that is concentrated within the bottom boundary layer, increasing friction leads to a thicker bottom boundary layer and a larger vertical extent of the lateral circulation. The effect of friction on the pattern of curvatureinduced lateral circulation is similar to that of lateral circulation driven by the Coriolis force (Winant 2007), as both lateral circulations are associated with the streamwise tidal flow.
The strength of curvatureinduced lateral circulation measured using the sectionaveraged amplitudes (i.e., 2V_{2} because of the two components) at the apexes of the two bends as a function of friction (δ) and curvature radius (R) is shown in Fig. 6a. A larger curvature radius leads to a weaker lateral flow, consistent with the analytical solution. The strength of curvatureinduced flow is not simply inversely related to friction but initially increases with δ, reaching a maximum value near 0.1, and then decreases toward zero. If the friction is extremely small, the bottom boundary layer is thin and restricts the development of lateral circulation. A relatively strong mixing produces a thicker bottom boundary layer that allows the lateral circulation to be developed. Therefore, the strength of the lateral circulation increases with δ when friction is weak.
To examine the relation between the curvatureinduced lateral circulation and streamwise tidal flow, the ratio V_{2}/U^{2} is calculated and averaged over the transverse section at the apexes of the two bends because V_{2} and U^{2} have the same frequency such that the phase between them can be evaluated. According to Eq. (23), this ratio is a function of the curvature radius and friction (Fig. 7). The amplitude of the ratio is inversely proportional to R and initially increases with friction, reaching a maximum near 1/3, and then decreases toward zero. The phases of this ratio at the two bends have a similar trend but with a difference of π. The phase of this ratio generally decreases with increasing friction and tends to be in phase when friction is very high. In particular, when the amplitude of the ratio is the largest, the corresponding phase is approximately −100°, indicating that the strongest lateral circulation appears near the slack water.
c. Interaction of lateral circulations driven by Coriolis and curvature centrifugal forces
The relative importance of the terms in the lateral momentum equation [i.e., Eq. (4b)] can be evaluated using classic nondimensional parameters. For example, based on the KB model, Alaee et al. (2004) categorized lateral flow into four regimes using Reynolds and Rossby numbers. The competition between curvature centrifugal and Coriolis forces in driving lateral circulation is measured by a Rossby number,
Winant (2007) has shown that the strength of the lateral circulation driven by the Coriolis force is a function of the Coriolis parameter (f) and friction (δ). The sectionaveraged amplitudes of the Coriolis force–driven lateral circulation (V_{1}) at the apexes of the two bends are illustrated in Fig. 6b. The strength of the Coriolis force–driven lateral circulation is proportional to f, and initially increases with friction, reaching a maximum, and then decreases toward zero; a similar trend was found by Winant (2007). The value of δ at which the amplitude is the maximum increases approximately from 0.05 to 1.5 as f increases. The strength of both the lateral circulations driven by the curvature centrifugal and Coriolis forces varying with friction has a similar trend but with different values of δ at which the amplitude is a maximum.
The ratio V_{1}/2V_{2} is taken as a measure of the relative strength of the two lateral circulations and shows a nonlinear relation with the Rossby number (Fig. 8a). Since the two circulations are affected by friction, the ratio is also plotted as a function of both the Rossby number and friction (Fig. 8b). The critical Rossby number is discerned from the contour to have a value of 1.0. The actual critical Rossby number is less than 1 and generally decreases with increasing friction, indicating that the damping effect of friction on the Coriolis force–driven flow is more effective than on curvatureinduced lateral flow.
The Coriolis force–driven lateral circulation is proportional to the streamwise tidal flow and it thus changes the rotation direction between flood and ebb tides. When the rotation driven and curvatureinduced lateral circulations are comparable, the two lateral circulations augment each other during one phase and compete during the opposite phase, producing asymmetries (Fig. 9). At a positive bend, the curvatureinduced lateral circulation is reduced by the Coriolis force–driven lateral circulation during the flood and amplified during the ebb. At a negative bend, the trend is reversed.
4. Discussion
a. Local solution of curvatureinduced lateral flow
The local solution helps interpret the discrepancies between the KB model and observations in curved tidal flows. First,
b. Adaptation time of curvatureinduced lateral circulation
Two types of adaptation of lateral circulation are considered here: spindown of the fully developed lateral circulation after the streamwise flow leaves a meandering region when the driving force vanishes (e.g., a bend followed by a straight reach) and the spinup of lateral circulation after the streamwise flow enters a meandering region where the driving force works (e.g., a straight reach followed by a bend). Similar to the relaxation length defined in the KB model, the spindown time is defined as the time at which the lateral circulation has reduced to e^{−1} of its original value, and the spinup time is defined as the time at which the lateral circulation has increased to 1 − e^{−1} of its ultimate value.
As the transition solution comprises a series of cosine modes, the adaptation time of the lateral circulation can be simplified as the mean of adaptation times of the leading modes. The percentage of each mode for averaging is determined by the weight of the mode which can be calculated using Eqs. (30b) and (32b). The minimum number of leading modes is two because the first mode represents a unidirectional flow and the sum of the first two modes represents a twolayer circulation. The actual number of leading modes needed for the estimation of adaptation time is determined by examining the weights. Nidzieko et al. (2009) conducted principal component analysis for lateral velocity profiles at a mooring and found that the first three modes accounted for 92% of the variance. As the first mode in their results is equivalent to the sum of the first two cosine modes of the solution, four cosine modes are needed for the estimation of adaptation time in this case.
It has been observed that the relaxation time is relatively short. The relaxation time has been estimated as approximately 400 s from a relaxation length of 150 m during the ebb tide in a narrow channel (Fong et al. 2009) and approximately 300 s from a relaxation length of 300 m in Otago Harbor (Vennell and Old 2007). To allow the lateral circulation to fully develop in a bend, the spinup time must be shorter than the streamwise advection time, which is the bend length divided by the streamwise velocity (Nidzieko et al. 2009). The observed short adaptation time supports the fast development of lateral circulation.
c. Effects of curvatureinduced lateral circulation on streamwise processes
d. Limitations of the model
The preconditions of the analytical model simplify the model but also limit the model validity. The streamwise velocity is assumed to be unaffected by the lateral circulation. While it has been found that advection of streamwise momentum by lateral circulation can modify the streamwise velocity profiles (Blanckaert and Graf 2004) and results in accelerated flows near the inner bend (Johannesson and Parker 1989).
In the analytical model, the streamwise tidal current velocity is symmetric between flood and ebb tides, having a pronounced nearsurface maximum velocity profile that always predicts typical curvatureinduced lateral circulations over a tidal cycle. However, it has been observed that the lateral circulation has opposite rotation directions between flood and ebb tides in a curved tidal channel. The reversed lateral circulation is often found during a flood tide and is associated with a vertical profile of the streamwise tidal current that has a nearbed maximum. This type of vertical velocity distribution changes the direction of the velocity shear and the rotation direction of lateral circulation. Such enhanced bottom current during flood may result from tidal straining (Winterwerp et al. 2006) or seawater inflows that are strong near the bottom or at middepth (Nakayama et al. 2016). In addition, seawater intrusion during flood can generate a lateral density gradient that might be strong enough to reverse the typical curvatureinduced lateral circulation (Winterwerp et al. 2006; Kranenburg et al. 2019).
In natural channels, the transverse section is typically asymmetric at bends, with a thalweg near the outside bank and shoals near the inside bank, resulting from feedback between the lateral circulation and bottom bathymetry. Due to the assumption that the lateral bathymetry is invariant along the tidal channel, the analytical model neglected this feedback process. Those limitations discussed above might be overcome in future work using a threedimensional numerical model that includes the inhomogeneous density field, nonlinear effects of tidal currents, and sediment transport and morphodynamics.
5. Conclusions
A linear threedimensional analytical model was developed to examine the lateral circulation in an elongated tidal channel with mildly curved bends for which the radius of curvature is larger than the width. The curvatureinduced lateral circulation was found to have a steady component and a periodic component with a frequency of overtide. As the two components have the same amplitude, the periodic component is compensated by the steady component during one phase and is amplified during the opposite phase over a cycle, resulting in a lateral circulation of which the strength varies periodically and the rotation direction is unchanged during a tidal period.
The general local solution of curvatureinduced lateral flow, as a combination of the two components, has an amplitude of the steady component that can be determined using the streamwise tidal current velocity, and a phase of the periodic component that is a function of friction. The existence of the phase indicates that the lateral circulation is not necessarily in phase with the streamwise tidal flow.
Friction modifies the structure, strength, and phase of curvatureinduced lateral circulation. The lateral circulation is confined within the bottom boundary layer when friction is low and is evident throughout the transverse section when friction is high. The strength of the lateral flow initially increases with friction, reaching a maximum value, and then decreases toward zero. The phase between the lateral circulation and squared streamwise tidal flow velocity decreases with increasing friction and tends to be in phase when friction is very high.
As the Coriolis force–driven lateral circulation changes the rotation direction between flood and ebb tides, it augments the curvatureinduced lateral circulation during one phase and competes during the opposite phase, producing asymmetric lateral circulation. The relative importance of Coriolis and curvature centrifugal forces is measured by the Rossby number and friction that dampens the Coriolis force–driven lateral flow more effectively.
Curvatureinduced lateral circulation spins down when the streamwise tidal flow leaves a bend and spins up when the tidal flow enters. The adaptation times of spinup and spindown are the same and proportional to the square of the water depth and inversely proportional to the eddy viscosity, reflecting the fact that the adaptation of lateral circulation is independent of driving forces. Because the transition solution of lateral circulation consists of a series of cosine modes, the adaptation time is determined by the average of leading modes.
Acknowledgments.
This work was supported by National Natural Science Foundation of China (42076008, 41476004, and 92058205). We are grateful to the two anonymous reviewers for their insightful comments that greatly improved this work.
Data availability statement.
The dataset of the analytical model can be made available after request to the corresponding author.
APPENDIX A
Solution of Tidal Elevation N_{1}
APPENDIX B
Solution of Elevation of Overtide N_{2}
REFERENCES
Alaee, M. J., G. Ivey, and C. Pattiaratchi, 2004: Secondary circulation induced by flow curvature and Coriolis effects around headlands and islands. Ocean Dyn., 54, 27–38, https://doi.org/10.1007/s1023600300583.
Blanckaert, K., and W. H. Graf, 2004: Momentum transport in sharp open channel bends. J. Hydraul. Eng., 130, 186–198, https://doi.org/10.1061/(ASCE)07339429(2004)130:3(186).
Blanckaert, K., and H. J. de Vriend, 2010: Meander dynamics: A nonlinear model without curvature restrictions for flow in openchannel bends. J. Geophys. Res., 115, F04011, https://doi.org/10.1029/2009JF001301.
Bo, T., and D. K. Ralston, 2020: Flow separation and increased drag coefficient in estuarine channels with curvature. J. Geophys. Res. Oceans, 125, e2020JC016267, https://doi.org/10.1029/2020JC016267.
Chant, R. J., 2002: Secondary circulation in a region of flow curvature: Relationship with tidal forcing and river discharge. J. Geophys. Res., 107, 3131, https://doi.org/10.1029/2001JC001082.
Chant, R. J., 2010: Estuarine secondary circulation. Contemporary Issues in Estuarine Physics, A. ValleLevinson, Ed., Cambridge University Press, 100–124.
Cheng, P., A. Wang, and J. Jia, 2017: Analytical study of lateralcirculationinducedexchange flow in tidally dominated wellmixed estuaries. Cont. Shelf Res., 140, 1–10, https://doi.org/10.1016/j.csr.2017.03.013.
Farlow, S., 1993: Partial Differential Equations for Scientists and Engineers. Dover Publications, 414 pp.
Fischer, H. B., 1969: Effect of bends on dispersion in streams. Water Resour. Res., 5, 496–506, https://doi.org/10.1029/WR005i002p00496.
Fischer, H. B., J. E. List, R. C. Y. Koh, J. Imberger, and N. H. Brooks, 1979: Mixing in Inland and Coastal Waters. Academic Press, 483 pp.
Fong, D. A., S. G. Monismith, M. T. Stacey, and J. R. Burau, 2009: Turbulent stresses and secondary currents in a tidalforced channel with significant curvature and asymmetric bed forms. J. Hydrol. Eng., 135, 198–208, https://doi.org/10.1061/(ASCE)07339429(2009)135:3(198).
Fugate, D. C., C. T. Friedrichs, and L. P. Sanford, 2007: Lateral dynamics and associated transport of sediment in the upper reaches of a partially mixed estuary, Chesapeake Bay, USA. Cont. Shelf Res., 27, 679–698, https://doi.org/10.1016/j.csr.2006.11.012.
Geyer, W. R., 1993: Threedimensional tidal flow around headlands. J. Geophys. Res., 98, 955–966, https://doi.org/10.1029/92JC02270.
Geyer, W. R., R. Chant, and R. Houghton, 2008: Tidal and springneap variations in horizontal dispersion in a partially mixed estuary. J. Geophys. Res., 113, C07023, https://doi.org/10.1029/2007JC004644.
Huijts, K. M. H., H. M. Schuttelaars, H. E. de Swart, and A. ValleLevinson, 2006: Lateral entrapment of sediment in tidal estuaries: An idealized model study. J. Geophys. Res., 111, C12016, https://doi.org/10.1029/2006JC003615.
Ianniello, J. P., 1977: Tidally induced residual currents in estuaries of constant breadth and depth. J. Mar. Res., 35, 755–786.
Johannesson, H., and G. Parker, 1989: Velocity redistribution in meandering rivers. J. Hydrol. Eng., 115, 1019–1039, https://doi.org/10.1061/(ASCE)07339429(1989)115:8(1019).
Kalkwijk, J. P. T., and H. J. De Vriend, 1980: Computation of the flow in Shallow River bends. J. Hydraul. Res., 18, 327–342, https://doi.org/10.1080/00221688009499539.
Kalkwijk, J. P. T., and R. Booij, 1986: Adaptation of secondary flow in nearlyhorizontal flow. J. Hydraul. Res., 24, 19–37, https://doi.org/10.1080/00221688609499330.
Kranenburg, W. M., W. R. Geyer, A. M. P. Garcia, and D. K. Ralston, 2019: Reversed lateral circulation in a sharp estuarine bend with weak stratification. J. Phys. Oceanogr., 49, 1619–1637, https://doi.org/10.1175/JPOD180175.1.
Lerczak, J. A., and W. R. Geyer, 2004: Modeling the lateral circulation in straight, stratified estuaries. J. Phys. Oceanogr., 34, 1410–1428, https://doi.org/10.1175/15200485(2004)034<1410:MTLCIS>2.0.CO;2.
Li, C., and J. O’Donnell, 2005: The effect of channel length on the residual circulation in tidally dominated channels. J. Phys. Oceanogr., 35, 1826–1840, https://doi.org/10.1175/JPO2804.1.
Marani, M., S. Lanzoni, D. Zandolin, G. Seminara, and A. Rinaldo, 2002: Tidal meanders. Water Resour. Res., 38, 1225, https://doi.org/10.1029/2001WR000404.
Nakayama, K., D. H. Nguyen, T. Shintani, and K. Komai, 2016: Reversal of secondary flows in a sharp channel bend. Coast. Eng. J., 58 (2), 1–23, https://doi.org/10.1142/S0578563416500029.
Nidzieko, N. J., J. L. Hench, and S. G. Monismith, 2009: Lateral circulation in wellmixed and stratified estuarine flows with curvature. J. Phys. Oceanogr., 39, 831–851, https://doi.org/10.1175/2008JPO4017.1.
Pein, J., A. ValleLevinson, and E. V. Stanev, 2018: Secondary circulation asymmetry in a meandering, partially stratified estuary. J. Geophys. Res. Oceans, 123, 1670–1683, https://doi.org/10.1002/2016JC012623.
Rozovskii, I. L., 1957: Flow of Water in Bends of Open Channels. Academy of Sciences of the Ukrainian SSR, 233 pp.
Smith, R., 1976: Longitudinal dispersion of a buoyant contaminant in a shallow channel. J. Fluid Mech., 78, 677–688, https://doi.org/10.1017/S0022112076002681.
Smith, R., 1977: Long term dispersion of contaminants in small estuaries. J. Fluid Mech., 82, 129–146, https://doi.org/10.1017/S0022112077000561.
Thomson, J., 1877: V. On the origin of windings of rivers in alluvial plains, with remarks on the flow of water round bends in pipes. Proc. Roy. Soc. London, 25, 5–8, https://doi.org/10.1098/rspl.1876.0004.
Vennell, R., and C. Old, 2007: Highresolution observations of the intensity of secondary circulation along a curved tidal channel. J. Geophys. Res., 112, C11008, https://doi.org/10.1029/2006JC003764.
Winant, C. D., 2007: Threedimensional tidal flow in an elongated, rotating basin. J. Phys. Oceanogr., 37, 2345–2362, https://doi.org/10.1175/JPO3122.1.
Winant, C. D., 2008: Threedimensional residual tidal circulation in an elongated, rotating basin. J. Phys. Oceanogr., 38, 1278–1295, https://doi.org/10.1175/2007JPO3819.1.
Winterwerp, J. C., Z. B. Wang, T. van der Kaaij, K. Verelst, A. Bijlsma, Y. Meersschaut, and M. Sas, 2006: Flow velocity profiles in the lower Scheldt estuary. Ocean Dyn., 56, 284–294, https://doi.org/10.1007/s1023600600634.