1. Introduction
In the coastal ocean nonlinear internal solitary waves (ISWs) are widely recognized to play an important role in generating turbulent mixing, modulating short-term variability of nearshore ecosystem, and transporting sediment and biochemical materials (Lamb 2014; Woodson 2018; Boegman and Stastna 2019). Their effects on shallow and stratified estuaries are poorly known and have been rarely studied, however. Since the buoyancy frequency in estuaries is typically one to two orders of magnitude larger than that in the coastal ocean, the ISWs in estuaries have a short wavelength of O(10) m and a short period of O(1) min. They are also highly intermittent in time and only emerge during certain periods in a tidal cycle (Dyer 1982; New et al. 1986; Sarabun and Dubbel 1990; Xie et al. 2017a,b). These ISWs are hard to capture in field surveys and thus are often overlooked. In addition, tidal advection of background fields makes the ISWs difficult to trace (Martin et al. 2005). Nevertheless, recent mooring observations in Chesapeake Bay showed that the ISWs had a large vertical displacement, leading to overturning and a dissipation rate of ∼1 × 10−4 m2 s−3, which is three orders of magnitude larger than the background value (Xie et al. 2017a).
Several mechanisms have been shown to be responsible for the ISWs generation in the coastal ocean. In the classic “lee wave” mechanism (Maxworthy 1979), a supercritical tidal flow generates a lee-wave depression on the lee side of a sill which subsequently propagates upstream and evolves into a train of ISWs as the tidal flow slackens. The disintegration of an upstream-propagating internal bore can also lead to ISWs. For example, at the entrance to the Knight Inlet, an undular bore or internal hydraulic jump is generated upstream of the sill crest when the tidal flow becomes supercritical, and this bore subsequently propagates upstream and disperses into a packet of ISWs (Farmer and Armi 1999; Cummins et al. 2003, 2006). ISWs can also be generated by the intrusion of a frontally forced interfacial gravity currents into a two-layer background flow (Bourgault et al. 2016). Similarly, ISWs can be generated from a river plume that flows as a gravity current into the coastal ocean: the horizontal flow convergence at the front of the river plume and downward vertical displacement of the underlying water can develop into ISWs (Nash and Moum 2005).
Once ISWs are generated, they experience nonlinear evolution and eventually dissipate. The shape of an ISW is preserved due to the balance between nonlinear steepening and dispersion. Both processes are sensitive to changes in the stratification and vertical shear as well as variations in the bottom topography (Helfrich and Melville 2006). When the pycnocline is located close to the sea surface, an ISW usually propagates as a wave of depression. In contrast, it propagates as a wave of elevation when the pycnocline sits near the bottom (Scotti et al. 2008). Observations in the coastal ocean found ISWs tend to transform from a depression wave to an elevation wave as they enter the shallower region and experience shoaling as the pycnocline approaches the bottom (Klymak and Moum 2003; Scotti and Pineda 2004; Bourgault et al. 2007).
The breaking and dissipation of shoaling ISWs can take different pathways, depending on the initial wave amplitude, stratification, and the slope of the bottom topography (Lamb 2014). A large-amplitude ISW of depression on a steep slope usually breaks by overturning with energy loss to turbulent dissipation as the trailing edge overtakes the wave trough (Vlasenko and Hutter 2002; Aghsaee et al. 2010; Masunaga et al. 2016). On the other hand, an ISW on a gentle slope tends to fission into a train of shorter waves or boluses, which propagate along the slope with elevated dissipation levels (Bai et al. 2019; Davis et al. 2020; Sinnett et al. 2022). For example, field observations in the Saint Lawrence estuary and idealized 2D numerical modeling showed that ISWs originated in the deep channel of the estuary propagated transversely to the channel axis and moved toward the shallow shoals (Bourgault and Kelley 2003). Some of the ISWs broke into a packet of turbulent boluses that propagated upslope of the breaking zone (Bourgault et al. 2005). Direct turbulence measurements showed a general correlation between the passage of ISWs and elevated turbulence (Richards et al. 2013), but the boluses may not be a primary source of turbulence and could suppress turbulence due to the increased stratification associated with their arrival (Bourgault et al. 2008). It is hard to quantify the contribution of ISWs to estuarine mixing, but the idealized 2D modeling studies by Bourgault and Kelley (2003) suggested that 6% of the wave energy may be converted into potential energy through vertical mixing when the ISWs break on the sloping bottom.
Little is understood about the generation, propagation and transformation, and dissipation of ISWs in coastal plain estuaries. Recent observations in Chesapeake Bay found an internal lee wave over the flank of the deep channel that preceded the appearance of the ISWs at mooring sites over the shallow shoal (Xie et al. 2017b). Ekman forcing on the along-channel tidal currents drove a lateral circulation in the cross-channel section and the lateral currents in turn interacted with the stratified water over the channel–shoal topography to generate a supercritical mode-2 internal lee wave (Xie et al. 2017b). It was hypothesized that this lee wave subsequently propagated onto the shallow shoal and evolved into a group of ISWs due to nonlinear steepening. Xie and Li (2019) used a numerical model of Chesapeake Bay to investigate this new wave generation mechanism in estuaries and confirmed that a model-2 wave is generated on the flank of the deep channel when the lateral flow becomes supercritical. In a recent modeling study of a generic coastal plain estuary, Li and Li (2022) showed that the internal lee waves may be generated over a range of river flow and tidal conditions expected in partially mixed and strongly stratified estuaries. They further showed that the wave amplitude decreases with increasing freshwater Froude number but is a parabolic function of the estuarine mixing parameter. However, it remains unclear how the internal lee wave evolves into the ISWs. The previous modeling studies by Xie and Li (2019) and Li and Li (2022) were based on the 3D hydrostatic model which cannot simulate the nonhydrostatic effects and do not have the fine resolution to resolve the small-scale ISWs. This paper is directed at understanding the nonlinear transformation from the internal lee waves to the ISWs.
A major obstacle to a mechanistic modeling investigation into the ISWs in estuaries is the enormous computer resource required to run a 3D nonhydrostatic model of an estuary at a scale of O(100) km and a resolution of O(1) m. Most of the previous nonhydrostatic modeling studies of flow–topography interactions and ISWs are based on the 2D models (Scotti et al. 2008; Chen et al. 2017; Davis et al. 2020; Urbancic et al. 2022). The Coastal and Regional Ocean Community Model (CROCO) is a new ocean modeling system built upon the Regional Ocean Modeling System (ROMS) and the nonhydrostatic kernel for free surface ocean modeling (Auclair et al. 2018). The combination of domain nesting (Penven et al. 2006; Mason et al. 2010), adaptive grid refinement in Fortran (AGRIF; Debreu et al. 2008, 2012), and nonhydrostatic algorithm makes it feasible to resolve fine-scale processes and their interactions with larger-scale flows while keeping the computational cost reasonable. Recently the 3D CROCO model has been used to simulate surf eddies in Grand Popo Beach off Benin in the Gulf of Guinea (Marchesiello et al. 2021). In this modeling study, we use the 3D CROCO model to investigate the generation, propagation, and transformation of ISWs in a coastal plain estuary featuring the channel–shoal bathymetry.
2. Methods
a. Model configuration
To resolve large-scale estuarine circulation as well as small-scale ISWs, we used CROCO to configure triply nested model domains with increasing grid resolution but decreasing domain size (Fig. 1). Such an approach is widely used in mesoscale atmosphere models such as the Weather Research and Forecasting (WRF) Model (e.g., Zhang et al. 2017).
(a) Schematic of the numerical model domain (outer domain model) consisting of a straight estuarine channel and a shelf. The yellow box denotes a child domain with 10-m resolution (middle domain model). (b) Schematic of the nested model domains. The green rectangle denotes the child domain of 2.5-m resolution (inner domain model). (c) Cross-channel section featuring a channel–shoal bathymetry at mid-Chesapeake Bay.
Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0151.1
The outer domain covers an estuary and its adjacent shelf, with a coarse resolution of 250 m inside the estuary (Fig. 1a). This model configuration follows from previous estuarine modeling studies (e.g., Hetland and Geyer 2004; Cheng et al. 2009, 2011; Li et al. 2014; Li and Li 2022). The continental shelf is 80 km wide and has a fixed cross-shelf slope of 0.05%. The estuarine channel is straight and does not have slope in the along-channel direction. The cross-channel section features a channel–shoal bathymetry representative of a coastal plain estuary such as Chesapeake Bay (Fig. 1c). There are 2100 grid cells in the east–west direction, 79 grids in the south–north direction, and 20 layers in the vertical direction. The along-channel grid size is uniform from the estuary mouth to the midestuary (250 m) and then increases exponentially to its head (26 km), providing a highly resolved estuarine region covering 500 km, following the approach of Li et al. (2014). The estuary length, considered here as the region influenced by salinity, is about 800 km. A longer channel (1000 km) is used to damp out tides before they reach the upstream boundary. The cross-channel grid in the estuary is uniformly distributed and the vertical layers are uniformly discretized.
Since the ISWs were observed to develop in the middle part of Chesapeake Bay where stratification was strongest (Xie et al. 2017b), we configured the middle domain to cover a short section (length of 1.6 km and width of 10 km) in the middle of the estuary at a resolution of 10 m (Fig. 1a). Based on the results from the middle domain model, we refined the resolution on the shallow part of the estuarine cross section using the adaptive grid refinement approach, such that the inner domain has a grid size of 2.5 m (Fig. 1b). Results from the parent model domain are automatically used as the boundary forcing to drive the child model domain through one-way nesting (Penven et al. 2006) such that the outer, middle, and inner domain models are run simultaneously.
The outer domain model is forced by semidiurnal tides at the offshore (eastern) open boundary and by river flow at the upstream (western) end of the estuarine channel. In this study, the tidal amplitude ηt was set to be 1.0 m and the river flow Ur was set to be 0.01 m s−1 (equivalent to a river discharge of 1400 m3 s−1 over a cross sectional area of 1.4 × 105 m2). The inflowing river water is prescribed to have zero salinity while the salinity of the coastal ocean is set at 35 psu. To simplify, temperature is uniform everywhere and does not change with time. A quadratic stress is exerted at the seabed, assuming that the bottom boundary layer is logarithmic with a roughness height of 0.5 mm. The Coriolis parameter f is set at 1.0 × 10−4 rad s−1. The vertical eddy viscosity and diffusivity are computed using the k–ε turbulence closure scheme (Warner et al. 2005) with the background diffusivity and viscosity set at 1 × 10−5 m2 s−1. The horizontal eddy viscosity and diffusivity are set to zero. To reduce numerical diffusion, the total variation diminishing (TVD) scheme with the Van Leer limiter (Sweby 1984) is employed for momentum advection and the fifth-order weighted essentially nonoscillatory (WENO5) scheme (Jiang and Shu 1996) is employed for advection of the scalar fields.
The outer domain model was initialized with no flow, a flat sea surface, and salinity distribution from 0 psu at the river end to 35 psu at the estuary mouth, linearly increasing along the estuarine channel. It was first run for 360 days to reach a quasi-steady state when the tidally averaged circulation and stratification in the estuary did not change with time. The salinity distribution from this model run was then used as the initial salinity condition for the nested domain model simulations. Results obtained after 10 tidal cycles were used for analysis.
b. Diagnostics of model results
The variability of the coefficients α and β has been studied to understand the nonlinear transformation of ISWs in the coastal ocean (Holloway et al. 1999; Small 2001a,b). The sign of the quadratic nonlinear coefficient α determines the polarity (elevation or depression) of an ISW and mostly depends on the location of pycnocline (maximum N2), according to the solution of the KdV equation for a two-layer fluid (Grimshaw et al. 1997). If the pycnocline lies closer to the seabed, α > 0, favoring an elevation wave. If the pycnocline lies closer to the sea surface, α < 0, favoring a depression wave. The dispersion parameter β quantifies the strength of wave dispersion such that a large value of β usually signals the dispersion of an ISW into a train of waves. When analyzed in combination with wave amplitude and wavelength, α and β yield insights into the nonlinear transformation processes of ISWs.
3. Results
The distribution of tidally averaged salinity along the channel thalweg shows that salinity increases from the head to mouth of the estuary and water is stratified above a well-mixed bottom boundary layer (Fig. 2a). A midestuary cross section was selected to present a detailed investigation into the generation, evolution, and dissipation of the nonlinear internal waves. The results from the middle model domain (10-m resolution) are used to analyze the generation and evolution of an internal lee wave in sections 3a and 3b and the results from the inner domain (2.5-m resolution) are used to analyze the ISWs in sections 3c and 3d.
(a) Tidally averaged salinity distribution along the channel thalweg with 1-psu contour interval. The white dashed line marks the location of the cross-channel section in (c)–(g). (b) Time series of the depth-averaged along-channel velocity and lateral velocity in the bottom layer at the channel–shoal interface. The dashed vertical lines indicate the timing of the cross-channel snapshots shown in (c)–(g) and in Figs. 3–6 and 8. The gray shaded region marks the flood tide and the yellow shaded region marks the ebb tide. (c)–(g) Distributions of salinity (contours) and the velocity vectors (arrows) at a midestuary cross section. Contour intervals are at 0.5-psu increments. The magenta dashed line in (c)–(f) marks the location of the channel–shoal interface where the mode-2 Froude numbers (Fr2) are calculated.
Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0151.1
a. Lee wave generation and propagation
During the ebb tide, a counterclockwise lateral circulation (looking into the estuary) develops in the cross-channel section and the rightward currents in the bottom layer advect the stratified water across the shoal–channel topography (Fig. 2c). To characterize the flow–topography interaction, the Froude number (Fr) at the channel–shoal interface, defined as Fri = υ/ci where υ is the depth-averaged lateral flow velocity in the lower layer, is calculated, following previous studies on flow–topography interactions (e.g., Lansing and Maxworthy 1984; Hibiya 1986; Vlasenko et al. 2013; da Silva et al. 2015). When the lateral flows become supercritical (Fr2 = 1.2) at the channel–shoal interface, a large-amplitude internal lee wave with a characteristic mode-2 bulge structure appears at the left flank of the deep channel (Fig. 2d). The lee wave is arrested over the deep channel by the supercritical lateral flows for ∼2.5 h, during which its trailing edge (right edge) steepens (Fig. 2e).
Distributions of salinity (contour lines) and the vertical velocity w (color) at the midestuary cross section at T4–T6 marked in Fig. 2b. Contour intervals are at 0.2-psu increments. The purple arrow line marks the phase line of the lee wave, and the magenta arrow line marks the propagation of maximum w from T4 to T6.
Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0151.1
In the long-wave limit (kh ≪ 1), the horizontal group velocity cg,x ≈ c2 = 0.11 m s−1 and the vertical group velocity cg,z = cg,x tan(90° − θ) = 0.001 m s−1. The shoreward translation of the maximum vertical velocity w along the channel slope illustrates the wave propagation (Fig. 3). The maximum w is transported ∼800 m horizontally and ∼7 m vertically over 2 h from T4 to T6, matching the direction of the lee wave phase line and the estimated group speeds.
b. Wave steepening, dispersion, and propagation
The wave modal structure also changes as the internal lee wave propagates. The lee wave evolves from a mode-2 wave at T4 to mode-1 waves at T6 (Figs. 2f,g). On flood tide strong mixing in the bottom boundary layer erases stratification in the deep channel and the lower branch of the mode-2 wave. On the other hand, the upper branch of the mode-2 lee wave, in the form of an elevation wave, propagates into the stratified water in the top 10 m. The upward propagation of the elevation wave perturbs the initially flat isopycnals there, such that the elevated isopycnals are followed by the depressed isopycnals to the right, indicating the formation of a trailing depression wave.
The subsequent evolution of the elevation and depression wave is shown in a sequence of snapshots of isopycnals and the lateral and vertical velocity distributions (Figs. 4a–h). Both the elevation and depression waves flatten and disperse between T6 and T7 (Figs. 4e,f). Subsequently the depression wave disintegrates into a train of small amplitude waves (Figs. 4g,h). On the other hand, the leading edge of the elevation wave steepens, accompanied by an intensification of the upward velocity (Figs. 4f–h). These wave transformation processes can be understood by studying the spatial variation of the nonlinear term c1 + αη and dispersion parameter β in the KdV equation. The term αη determines how nonlinearity changes the wave phase speed at the wave crest/trough and may cause the steepening and flattening of the wave. The dispersion parameter β quantifies the magnitude of wave dispersion which disperses the wave energy and may cause wave fission. The parameter β increases by a factor of 2 (from 3 to 5 m3 s−1) in the deeper water region where the depression wave propagates to the right. Strong dispersion causes the depression wave to flatten and disintegrate into a train of small-amplitude waves. The parameter β is considerably smaller on the left shallow shoal (in the range of 1–2 m3 s−1) where the elevation wave propagates. In the meantime, the linear phase and nonlinear term become more important in driving the transformation of the elevation wave there. The nonlinear parameter α is negative everywhere and would lead to the flattening of the elevation wave. However, the magnitude of α decreases as the left shore is approached (Figs. 4i–k), implying this flattening effect decreases as the elevation wave propagates to the left. Moreover, the spatial variation in the wave phase speed c1 due to changes in the water depth and stratification can lead to either steepening or flattening of a wave. A decrease in the phase speed c1 toward the left shoal will cause the steepening of the elevation wave since the wave crest moves faster than the leading edge. Indeed c1 decreases from 0.25 to 0.3 m s−1 at the channel shoal interface to under 0.2 m s−1 at the left shallow region. Such a large reduction in c1 overwhelms αη, causing the elevation wave to steepen (cf. Figs. 4f–h).
Snapshots of (a)–(d) the lateral velocity (color) and salinity (contour), (e)–(h) the vertical velocity (color) and salinity (contour), (i)–(l) the nonlinearity coefficient α and dispersion coefficient β in the KdV equation, and (m)–(p) the mode-1 phase speed c1 at T6–T9 marked in Fig. 2b. Contour intervals of salinity are at 0.5-psu increments.
Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0151.1
The steepening between T6 and T9 changes the characteristics of the elevation wave. The wavelength decreases from ∼1000 to ∼300 m. Moreover, this short wave is accompanied by large vertical velocities (Fig. 4h), with the upward velocity on the leading edge of the elevation wave reaching 3.0 mm s−1 and the downward velocity on the trailing edge reaching 1.2 mm s−1.
c. Emergence of ISWs
(a)–(d) Snapshots of the vertical velocity (color) and salinity (contour) at T9–T13 marked in Fig. 2b. Contour intervals of salinity are at 0.5 psu increments. The green contour in (a) marks the isopycnal tracked in Fig. 6. The magenta dashed line in (d) marks the location of the time series shown in Fig. 7.
Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0151.1
Figure 6 provides a detailed view of the solitary wave evolution by tracking the isopycnal of 9 psu marked in Fig. 5a. The soliton of elevation has a symmetric shape at T8 but becomes asymmetric as it moves up the shoaling bottom. Its leading edge is rarefied whereas its training edge steepens. From T9 to T11, the isopycnal displacement between the wave crest and the trailing edge increases from 0.4 to 0.8 m. At T12 the isopycnal drops down 1.2 m from the crest to the trailing edge, overshooting its equilibrium position. A small-amplitude depression wave is then formed at the tail of the elevation wave. The amplitude of the depression wave continues to grow and reaches 2 m at T13 while the trailing edge disintegrates into a train of ISWs due to dispersion, such that a large-amplitude leading depression wave is followed by a train of small-amplitude undular waves.
Hovmöller diagram for isopycnal displacement of 9 psu marked in Fig. 5a. In the vertical coordinate, the time interval of each line is 10 min, equivalent to 2.5-m isopycnal displacement. The gray shaded region represents the bottom topography on the left shoal.
Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0151.1
Time series at a site on the shoal (∼2.5 km away from the coastline, marked in Fig. 5d) further illustrate the features of the wave packet (Fig. 7). The wave train contains 4 solitons in rank order, lasting ∼20 min. The period of one solitary wave is only ∼5 min on average. The leading wave in the wave packet has the largest amplitude of 2 m, accompanied by vertical velocity as strong as 0.03 m s−1. The rest of the waves have much smaller amplitudes. The signal of ISWs is also shown in the time series of the horizontal velocities (Fig. 7b). To sustain the conservation of water mass, the internal waves generate additional surface and bottom lateral flows in opposite directions between the upward and downward vertical velocity. Unlike the elevation wave that strengthens the two-layer lateral flows in both the surface and bottom layers, the depression wave weakens and even reverses the lateral flows, leading to flow convergence in the upper layer and divergence in the lower layer. At the leading depression wave, the lateral velocity in the upper layer dramatically drops from 0.13 to −0.02 m s−1 (positive is rightward) while the lateral velocity in the lower layer changes from −0.02 to 0.03 m s−1.
(a) Time–depth distributions of salinity (contours) and vertical velocity (color) at a location on the left shoal marked by the magenta line in Fig. 5d. (b) Time series of the lateral velocity at the depths of 2 and 6 m at the same location on the shoal.
Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0151.1
d. Dissipation of ISWs
The subsequent destruction of the ISWs is shown in a group of snapshots in Fig. 8. At the beginning, the wave packet contains 4 solitary waves (Fig. 8a). As it propagates farther shoreward, the number of solitons in the packet decreases while the amplitudes are dampened (Fig. 8b). When the ISWs arrive at 1.8 km away from the coastline, only the leading depression wave remains (Fig. 8c). The wave trailing edge rarefies, shaping the wave form into a bore front. The wave is finally merged into the salinity front adjacent to the coastline before it runs into the coast (Fig. 8d).
Distributions of salinity (contours) and dissipation rate ε (color) on the left shoal at times T13–T16 marked in Fig. 2b.
Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0151.1
Dissipation rate ε on the shallow shoals is high due to strong turbulence in the tidal bottom boundary layer. The dissipation rate ε below the wave is about O(10−6) m2 s−3, which dissipates the small-amplitude ISWs. The ε at the nearshore salinity front reaches O(10−5) m2 s−3, leading to the final wave destruction. It is interesting to note that there exists a pocket of high ε anomaly at 5–7-m depth below the wave (Figs. 8a–c). Dissipation rate ε reaches 4 × 10−6 m2 s−3 at T14 after the passage of the ISWs (Fig. 8b). The leading large-amplitude depression wave enhances velocity shear below the wave, lowering the local Richardson number (Fig. 9). The correspondence between the enhanced ε and large velocity shear suggests internal solitary wave can also contribute to local turbulent mixing in the stratified pycnocline region.
Distributions of salinity (contours) and (a) the along-channel velocity shear, (b) the lateral velocity shear, and (c) Richardson number at T13 marked in Fig. 2a.
Citation: Journal of Physical Oceanography 54, 2; 10.1175/JPO-D-23-0151.1
4. Discussion and conclusions
Using a 3D nonhydrostatic model with nested domains and adaptive grids, we investigated the generation, propagation, transformation, and dissipation of nonlinear internal waves in a coastal plain estuary. The model results showed that the internal lee wave generated by flow–topography interaction at the channel–shoal interface can evolve into a train of ISWs when propagating over the shoaling bottom. Although the observations in Chesapeake Bay suggested a potential link between the lee wave and ISWs (Xie et al. 2017b), this modeling study provides a mechanistic explanation for the ISWs captured in the bottom-mounted ADCPs at the mooring stations on the western shore. Given the enormous computer resources required to run a 3D nonhydrostatic model of an entire estuary at a 1–2-m resolution, currently we cannot conduct hindcast simulations of a realistic estuary for a direct model–data comparison. Nevertheless, there is a general agreement on the characteristics of the ISWs between this study and the observations of Xie et al. (2017b), such as the timing (from late flood to early ebb) for the initiation of the ISWs during a tidal cycle, the duration of the ISWs (20–40 min), and wave amplitude (1–3 m).
The first step in the wave transformation process involves a change from a mode-2 lee wave into a mode-1 elevation wave. Changes in the wave modal structure have been widely reported and are often triggered by changes in stratification and/or bottom bathymetry. Previous observational and modeling studies of lee waves in Chesapeake Bay showed that the mode-2 wave content decreases and the mode-1 wave content increases as the lee wave propagates from the deep channel to the western shoal (Xie and Li 2019; Li and Li 2022). In the South China Sea, the energy of a mode-2 wave approaching a shallow plateau was scattered into mode-1 waves owing to steep bathymetric changes and wave reflection (Klymak et al. 2011). Similarly, observations on the New Jersey shelf showed the decay time scale of mode-2 wave energy is much shorter than mode 1, leading to a transformation from mode 2 to mode 1 during the wave propagation toward the coast (Shroyer et al. 2010). In this study the modal change appears to be mainly driven by changes in the stratification. With a magnitude of O(0.1) m s−1, the lateral currents in estuaries are usually subcritical with respect to mode 1 but may become supercritical with respect to mode 2. The lee waves generated by the interaction between the lateral flows and the channel–shoal bathymetry are typically of a mode-2 structure (Li and Li 2022). However, turbulent mixing in the bottom boundary layer can undergo large temporal changes over a flood–ebb tidal cycle, destroying stratification and precipitating a wave modal shift as documented in Figs. 2f and 2g.
According to the solution of the KdV equation for a two-layer fluid, nonlinear parameter α is negative if the pycnocline lies closer to the sea surface and a depression wave is favored. The presence of an elevation wave in the upper part of the water column (as shown in Fig. 5a) is thus surprising, despite that a negative value of α is consistent with the KdV theory (Figs. 4i–l). Obviously, the stratification does not fit into a two-layer system since water is nearly uniformly stratified in the top 5 m but has homogeneous salinity (density) below that depth (Fig. 5). It is hard to locate the pycnocline, but the isopycnal displacement is largest just above the top of the bottom boundary layer, which is near the middepth. Another difficulty is the separation of the wave-induced shear from the “background” shear (associated with the lateral circulation), as explained in McSweeney et al. (2020). The α values shown in Figs. 4i–l did not consider the background shear. We recalculated α using the instantaneous velocity shear and still obtained negative values. Nevertheless, the formation of an elevation wave makes sense given its history evolving from the upper branch of the mode-2 lee wave.
The subsequent transformation of the elevation wave into a train of rank-ordered ISWs on the shallowing shoal shares common characteristics as the transformation of a solitary wave that propagates toward the coast on the continental shelf. A solitary wave of depression usually develops in the deeper ocean where the pycnocline is located close to the ocean surface. As this wave propagates onto the shoaling continental shelf, its leading edge rarefies while its trailing edge steepens and evolves into a solitary wave of elevation followed by ranked-ordered ISWs near the seabed (Klymak and Moum 2003; Shroyer et al. 2009). The transition from the depression to elevation wave sometimes occurs at a location where the wave’s polarity changes (Grimshaw et al. 2004; Shroyer et al. 2009). In other situations, sign change in α is not necessary. In Massachusetts Bay, the transition from the depression to elevation wave occurred at a location offshore of where the KdV theory predicts polarity switching should occur (Scotti et al. 2008). Our result is similar since α remains to be negative over the entire left shoal (Figs. 4i–l). The shoaling on the sloping bottom causes the rarefaction of the leading wave edge and the steepening of the trailing wave edge. The effect of dispersion then leads to the disintegration into a train of rank-ordered ISWs (Fig. 6).
The cubic nonlinear coefficient α1 attained from the equations in Grimshaw et al. (1997) ranges from −0.005 to −0.01 across the shoal during the wave propagation. When α1 is negative, the polarity of an ISW only depends on the sign of the quadratic nonlinear coefficient α (Helfrich and Melville 2006). Besides, the amplitude of the elevation wave is only ∼0.5 m, which means quadratic nonlinearity is dominant during this stage. Therefore, the existence of cubic nonlinearity does not affect the wave transformation. After the elevation wave transforms into the large-amplitude depression wave, cubic nonlinearity becomes more important. Contrary to quadratic nonlinearity, the cubic nonlinearity decreases the wave speed. As both α and α1 keep negative on the shoal, the depression wave maintains its soliton-like shape until dissipated, similar to the cases found on the Arctic shelf (Dokken et al. 2001; Grimshaw et al. 2004).
Our model results also highlight the role of the background turbulence (turbulence generated in the tidal boundary layer) in dissipating ISWs as well as the generation of elevated energy dissipation by large-amplitude ISWs themselves. Previous idealized numerical studies found bottom friction may significantly dampen the amplitude of ISWs (Holloway et al. 1997; Liu et al. 1998). Observations of near-bottom elevation waves also showed energy dissipated by bottom friction could be comparable to the loss to internal turbulence production (Scotti and Pineda 2004). In the coastal ocean pycnocline often locates well above the bottom, whereas in shallow coastal plain estuaries bottom friction is expected to be more important in the wave dampening. Furthermore, the ISWs in the coastal plain estuary may enhance vertical shear and generate strong energy dissipation locally (Richards et al. 2013), similar to the observations of enhanced local turbulent mixing via shear instability in the ISWs propagating on the continental shelf (Moum et al. 2003, 2007; Jones et al. 2020). On the other hand, the ISWs can also bring more stratification than shear when they break on the shoals and form boluses that can reduce the turbulent mixing, as shown in the Saint Lawrence estuary (Bourgault et al. 2008).
Acknowledgments.
We are grateful to NSF (OCE-1756155) for the financial support. This is UMCES Contribution Number 6343.
Data availability statement.
The numerical model simulations upon which this study is based are too large to transfer. Instead, we provide all the information needed to replicate the simulations. The model code, compilation script, and configuration files are available at https://doi.org/10.5281/zenodo.8198319.
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