Numerical solutions of the Korteweg–de Vries (KdV) and extended Korteweg–de Vries (eKdV) equations are used to model the transformation of a sinusoidal internal tide as it propagates across the continental shelf. The ocean is idealized as being a two-layer fluid, justified by the fact that most of the oceanic internal wave signal is contained in the gravest mode. The model accounts for nonlinear and dispersive effects but neglects friction, rotation and mean shear. The KdV model is run for a number of idealized stratifications and unique realistic topographies to study the role of the nonlinear and dispersive effects. In all model solutions the internal tide steepens forming a sharp front from which a packet of nonlinear solitary-like waves evolve. Comparisons between KdV and eKdV solutions are made. The model results for realistic topography and stratification are compared with observations made at moorings off Massachusetts in the Middle Atlantic Bight. Some features of the observations compare well with the model. The leading face of the internal tide steepens to form a shock-like front, while nonlinear high-frequency waves evolve shortly after the appearance of the jump. Although not rank ordered, the wave of maximum amplitude is always close to the jump. Some features of the observations are not found in the model. Nonlinear waves can be very widely spaced and persist over a tidal period.

Internal waves (IWs) are present throughout earth's oceans wherever there is stratification, from the shallowest near-shore waters to the deepest seas. IWs are important to physical oceanographers because they transport momentum and energy, horizontally and vertically, through the ocean (e.g. Munk, 1981; Gill, 1982). They provide shear to turbulence, which results in energy dissipation and vertical mixing (e.g. Holloway, 1984; Sandstrom and Elliott, 1984). Biological oceanographers are interested because the IWs carry nutrients onto the continental shelf and into the euphotic zone, (e.g. Shea and Broenkow, 1982; Sandstrom and Elliott, 1984; Holloway et al., 1985). They are of interest to geological oceanographers because the waves produce sediment transport on the shelf (e.g. Cacchione and Drake, 1986). Civil, hydraulic and ocean engineers are also interested because of the IWs tidal and residual currents (e.g. Willmott and Edwards, 1987), which can cause scour on nearshore as well as offshore structures (e.g. Osborne et al., 1978). Large nonlinear IWs are also of interest to the navy because they cause large vertical displacements and large vertical velocities that may affect underwater operations.

This study is focused on the internal tide and subsequent evolution of
nonlinear waves. IWs in the ocean span the frequency spectrum from the
buoyancy frequency,

We are interested in nonlinear IWs because they are a very energetic part of the signal in time series that we have observed on continental shelves and in the shallow ocean. We are guided by numerical solutions of Korteweg–de Vries (KdV) type equations that incorporate both weak nonlinear and weak dispersive effects. The state of the art of the evolution of internal solitary waves (ISWs) across the continental shelf is reviewed in Grimshaw et al. (2010). Grimshaw et al. (2004) simulated the transformation of ISWs across the North West Shelf of Australia, the Malin shelf edge and the Arctic shelf; Holloway (1987) discussed the evolution of the internal tide in a two-layer ocean on the Australian North West Shelf. Our model simulations of the evolution of the internal tide across the Middle Atlantic Bight topography are unique since these waves have never been modelled across such topography and stratifications, and the model results are compared with observations made at moorings off Massachusetts during the Coastal Mixing and Optics (CMO) experiment. Whereas most modelling studies regarding wave propagation over linearly sloping bottom and realistic topography have focused on the behaviour of a single soliton, this work is concerned with the development and evolution of a packet of solitary waves.

The goal of this paper is to study the observed variability in the evolution of the internal tide as it crosses the continental shelf resulting from different stratifications and varying topography. In Sect. 2, the model framework is presented, and model runs and results of simulations are discussed for cases of linearly sloping bottom topography and that at the site of the CMO. Model results are compared with data and observations collected at the CMO site in Sect. 3. A summary and conclusions are presented in Sect. 4.

We are interested in modelling the evolution of the internal tide as it
propagates shoreward from the shelf break. Since the greatest oceanic signal
is the first internal mode, the stratification of the continental
shelf/slope region is modelled as a two-layer fluid. This approximation
greatly simplifies the problem; the numerical scheme is much less complex
for the two-layer case than the continuously stratified case, and the
results are easier to interpret. Using this model configuration, we study
the propagation of the internal tide over linear sloping and CMO topography.
All cases have been run within the quadratic nonlinear framework of the KdV
equation, and the results are compared with an extended form of it, the eKdV
Eq. (1) model, written as

Note, the KdV equation is well known to be a suitable physical model for
describing weakly nonlinear advective effects and linear dispersion in IWs.
It was originally developed by Benney (1966) and extended to second order by
Lee and Beardsley (1974). The KdV and eKdV Eq. (1) equations are derived
following the procedure of Lee and Beardsley (1974) and the discussion by
Lamb and Yan (1996). The two-layer KdV model approximation is discussed in
Grimshaw et al. (2002), and justified since most of the energy in the ocean
appears to be contained in the first mode anyway (see e.g. Alford and Zhao,
2007). The problem has been investigated for slowly varying topography and
stratification by Grimshaw (1979) and Pelinovsky et al. (1977). An
interesting reference is Lamb and Xiao (2014), who took a similar approach
to ours, comparing predictions of the KdV and eKdV models, and
also the regularized long-wave (RLW) equation, with fully nonlinear numerical simulations for two-layer
stratification over selected topographies. See O'Driscoll (1999) for a full
discussion of our experiments. For all simulations the density difference
between the two layers is chosen to be a constant:

Using the KdV equation, we investigate two cases with a constant-sloping bottom, one with a horizontal interface and one with a sloping interface, and finally make model runs with realistic topography at the CMO site.

For convenience in solving the equation, we avail of a transformation,
utilized by Pelinovsky and Shavratsky (1976), of the space and time
variables

Case A (constant-sloping bottom with level interface,

The propagation of the internal tide
along constant-sloping topography was studied for cases of constant upper-layer thicknesses (Case A) and sloping interface (Case B), both of which are
possible on continental shelves. We chose the starting layer thicknesses at

We first investigate the case of constant-sloping bottom with fixed upper-layer thickness (

Figure 1b and c show the internal tide signal for
Case A at increasing

As the internal tide propagates into shallow water the front of the wave
trough steepens but the decreasing magnitude of

The leading waves are slightly more nonlinear than dispersive when

Case A, leading waves of elevation (black line) at various distances

For Case B with constant-sloping bottom and sloping upper layer, we also
begin in 200 m water with

The CMO site was located in the Middle Atlantic Bight. Conductivity, temperature, and depth (CTD) profiles were made across the continental shelf from shallow water to beyond the continental slope. Boyd et al. (1997) have concluded that the internal tide at the site is primarily a first mode IW, further justifying our choice of a two-layer model. An upper-layer thickness of 25 m is a representative average value for the duration of the experiment (July and August 1996).

Figure 4a shows KdV parameter values as a function of

The model runs discussed in Sect. 2.1 were also made with the eKdV
equation. The ratio of the nonlinear parameters

Same as Fig. 1 but for Case B (constant-sloping bottom with sloping interface ).

Same as Fig. 1 but for CMO experiment site (with flat interface,

Quadratic nonlinear parameter,

For the CMO case, a comparison of KdV and eKdV results shows a more
significant difference than for Case A. Figure 7a–c show the KdV and eKdV
model results for a 4 m internal tide having propagated 60 km to a water
depth of 69 m. The leading KdV model solitary wave (solid line) arrives at
the CMO central mooring

To learn more about the evolution of a sine wave to waves with sech

Another investigation to explore the evolution in the eKdV model (constant
parameters) was made using an initial condition of a sech

The data to be presented and discussed were collected during the CMO; for location see Fig. 9. The CMO experimental field program was conducted to increase our understanding of the role of vertical mixing processes in determining the mid-shelf vertical structure of hydrographic and optical properties. The field program was conducted on a wide shelf so as to reduce the influences of shelf break and nearshore processes. The data we discuss was collected from the CMO central mooring in July and August 1996, a time when a strong thermocline is present as a result of large-scale surface heating (Boyd et al., 1997).

Case A (constant-sloping bottom with flat interface,

CMO experiment site (realistic topography with level interface,

Width vs. amplitude of the leading waves of the KdV and eKdV solutions
at the CMO mooring site (

The central mooring of the CMO experiment was located at
40

Site of the Coastal Mixing & Optics (CMO) experiment located in the Middle Atlantic Bight to the south of Massachusetts. The data discussed was collected at the mooring marked “CTD #6”.

Pressure (tidal) record at the CMO mooring site including for the period day 210–245 of 1996 (from Boyd et al., 1997).

Energy spectra of the first internal mode at the CMO mooring for the
period day 210–245

Observations at the CMO mooring site over a semi-diurnal period.

Wave amplitude vs. wave width at the CMO mooring for waves from all events during the period day 210–245. Also plotted are the two leading waves from six of the nine model runs shown in Fig. 10 (diamonds).

This same pattern can often be seen in the observed time series of the first internal mode. Figure 13 shows several individual jumps at the CMO mooring. Figure 13i (a) shows first modes which best match the model results of Fig. 10iii. Some features of the observations compare well with the model. The slackened leading face of the tide is always followed by a steep – almost shock like – front followed by several highly-nonlinear short-period waves. Although not rank ordered, the largest amplitude wave in the observed packet is always at or near the jump. The model results show that the amplitude of the jump is greater for larger initial conditions, and decreases with distance from the point of generation. Although nonlinear waves continue to evolve, their amplitudes decrease as they propagate shoreward from their generation point, and they become “thicker”, i.e. they become more table-top like. Although the modelled waves have amplitudes less than the theoretical limit for local eKdV parameters, they nonetheless fit the shape of several observed waves at the CMO site.

There are also features of the observations that are not found in the model. Figure 13i (f and g) differ in that the packet that follows the shock-like front persists until the end of the tidal period, and the waves are spread apart from each other. Figure 13i (c) shows two packets of solitary-like waves propagating past the mooring site over a tidal period. The leading slackened face is followed by a shock-like front and a packet of solitary waves. The trailing face then slackens to assume a slope similar to the leading face but a second shock-like front, followed by a packet of solitary waves, passes before the end of the tidal period. This could be from a second internal tide front coming from another generation site, there can be overlapping semi-circles of IW fronts from multiple generation sites, see for example discussion in Apel et al. (1988). Another possible generation mechanism is the nonlinear evolution of inertia-gravity waves forming behind ISWs due to rotation, see further details in Grimshaw et al. (2014), and Lamb and Warn-Varnas (2015) who have shown that rotation effects can become important after one or more inertial periods. However, in this model-to-observation comparison, the waves have travelled for not much longer than one inertial period, and rotation has been ignored in the model runs. It is also possible that multiple packets form each tidal period, due to different generation mechanisms such as multiple tidal constituents or harmonics of tidal components as found, for example, at the site of the Littoral Optics Experiment, where the 4th harmonic of the semi-diurnal tide was used to successfully simulate the evolution of the internal tide (O'Driscoll, 1999).

Another common observation that is not found in the model results is a “drop” in amplitude before the jump that occurs at the beginning of the wave packet. Figure 13i (h) shows that the first internal mode drops between day 243.5 and 243.6 but the slackening slope is restored before the arrival of the jump and packet of solitary waves. Similar “drops” also occur in Fig. 13i (b and e) and Fig. 13ii (i). Another phenomenon observed is that the slope of the leading face of the tide changes sign before the packet in several of the examples in Fig. 13ii. In Fig. 13ii (h) the low-frequency slope changes sign at day 236, and the solitary waves appear as usual ahead of the trailing, low-frequency signal. The signal becomes even more complicated when both a “drop” and low-frequency slope change are present, e.g. Fig. 13ii (d). In this case, the slope of the leading slackening low frequency signal changes sign at day 242.5 and is followed by a packet of four solitary waves. The low-frequency signal is restored before the passage of a jump followed by a packet of five large solitary waves. The trailing face retains the slope of the low-frequency signal. Figure 13iii shows a series of jumps which are more complex than those in the top panels, though they retain the basic structure of the model results over the tidal period.

To examine the details of the wave packets themselves, the width vs. amplitude was estimated for each wave from all events during the period day 210–245 (Fig. 14). These waves are plotted along with the leading two waves from six of the nine model runs shown in Fig. 10iii. Also shown are the theoretical relations for solitary waves for the eKdV and KdV equations using CMO site parameters. The observed nonlinear waves vary greatly in amplitude and width (s). Larger amplitude observed waves are well approximated by model runs with large initial amplitude, particularly the 4 m model. The 6 m model run from 24 km seaward of the CMO site is also a very good match for several of the observed waves. A large fraction of observed waves with amplitude less than 15 m, and particularly less than 10 m, are much “thinner” than model waves with similar amplitudes. However, it seems reasonable to say that the observed waves are a good fit to the model waves.

While some features of the observations are reproduced in the model, there are many differences. The eKdV model used here is highly idealized. There are many effects that have not been considered, including bottom and internal friction, earth's rotation and mean shear. Given these limitations, we conclude that the observations are reasonably well matched by our model.

Observations of highly-nonlinear IWs contained in the first mode time series on the mid-continental shelf and in current metre records in shallow water have led us to investigate the transformation of the shoaling internal tide. Observations were made in the mid-continental shelf at the site of the CMO experiment. An existing model based on generalized KdV and eKdV equations has been simplified for use in a two-layer ocean, which is representative of realistic stratification. The model accounts for weakly-nonlinear and dispersive properties of the internal tide. Earth's rotation, internal dissipation, bottom friction and internal shear are not included. The internal tide was forced with a periodic sinusoidal boundary condition and allowed to propagate shoreward.

The model was first run within a KdV framework with realistic continental
shelf, constant-sloping bottom with flat and sloping interface, and CMO
shelf parameters. The internal tide steepens on the back face of its crest
as it propagates shoreward, a direct result of the much greater magnitude of
the nonlinear term in comparison with the dispersive term. Nonlinear waves
evolve from the internal tide after the back face forms a shock-like front.
The waves can appear as a rank-ordered packet with the leading waves
travelling fastest, since they are the most nonlinear. The leading waves of
depression usually travel faster than the linear wave speed,

The model runs made within the KdV framework were also made within the eKdV
framework, which includes a cubic nonlinearity term scaled by

To better understand the evolution of waves toward table-top form in an eKdV
framework, without the complications of varying parameters, model runs were
made using constant eKdV parameters representative of the CMO site. Upon
formation, the leading waves of the packet are similar to sech

Model runs with varying initial amplitudes and generation regions were made to help interpret the observations made at the CMO site. Some features of the observations compare well with the model. The leading face of the trough of the internal tide steepens to form a shock-like front. Nonlinear high-frequency waves evolve shortly after the appearance of the jump. Although not rank ordered, the wave of maximum amplitude is always close to the jump. Some features of the observations are not found in the model. Nonlinear waves can be very widely spaced and persist over a tidal period. The amplitude of the observed waves often decreases before the arrival of the jump, while the leading face may change slope before the jump arrives.

Individual observed waves were examined and the details compared to model results. The observed nonlinear waves vary greatly in amplitude and width, generally having amplitudes of between 5 and 25 m, and widths of between 200 and 600 s. Larger amplitude waves are well approximated by waves evolving from large amplitude model waves. A large fraction of smaller amplitude observed waves, particularly less than 10 m, are thinner than model waves of similar amplitude. We conclude that the observed waves are a good match to modelled waves given the highly idealized eKdV model used, and the fact that we have neglected friction, rotation and mean shear.

Model result data is available from Kieran O'Driscoll. Observation data is available from Oregon State University, see reference Boyd et al. (1997) above.

KO conducted this work while a graduate student at the College of Oceanic & Atmospheric Sciences, Oregon State University, in partial fulfillment of the degree of Master of Science. ML was the student's advisor.

The authors declare that they have no conflict of interest.

Kieran O'Driscoll would like to thank Jack Barth for a substantial review of a previous version of this manuscript, and two referees for considerable reviews and helpful comments. This work was supported by funding from the Office of Naval Research and Oregon State University. Edited by: John M. Huthnance Reviewed by: two anonymous referees