We study the interaction of small-scale internal wave packets with a large-scale internal solitary wave using high-resolution direct numerical simulations in two dimensions. A key finding is that for wave packets whose constituent waves are short in comparison to the solitary wave width, the interaction leads to an almost complete destruction of the short waves. For mode-1 short waves in the packet, as the wavelength increases, a cutoff is reached, and for larger wavelengths the waves in the packet are able to maintain their structure after the interaction. This cutoff corresponds to the wavelength at which the phase speed of the short waves upstream of the solitary wave exceeds the maximum current induced by the solitary wave. For mode-2 waves in the packet, however, no corresponding cutoff is found. Analysis based on linear theory suggests that the destruction of short waves occurs primarily due to the velocity shear induced by the solitary wave, which alters the vertical structure of the waves so that significant wave activity is found only above (below) the deformed pycnocline for overtaking (head-on) collisions. The deformation of vertical structure is more significant for waves with a smaller wavelength. Consequently, it is more difficult for these waves to adjust to the new solitary-wave-induced background environment. These results suggest that through the interaction with relatively smaller length scale waves, internal solitary waves can provide a means to decrease the power observed in the short-wave band in the coastal ocean.

Internal waves are commonly observed in stably stratified fluids such as the Earth's atmosphere and oceans. They exist in
a variety of environmental conditions, including those with background shear currents, and on different length and timescales.
The interaction between internal waves and other physical processes
results in energy exchange between the waves and the background environment

The interaction between internal waves of different length scales also occurs naturally

In this work, we study the interaction of small-scale mode-1 internal waves initialized from linear waves with an ISW
initialized from the exact Dubreil–Jacotin–Long equation, using high-resolution direct numerical simulations in two
dimensions. Internal waves that are short in terms of wavelength compared to the fluid depth are generally less
documented in the nonlinear wave literature. In fact, the derivation of the model equations of most weakly nonlinear
theories, such as the Korteweg–de Vries equation and its variations, assumes large horizontal scales and thus filters out
short waves

The remainder of the paper is organized as follows: theoretical descriptions of internal waves are introduced in
Sect.

Example of

In the classical linear theory, the horizontal structure of internal waves is usually described by the travelling wave
ansatz

Due to the nonlinear nature of fluid flows, purely linear waves are a mathematical idealization. For large-amplitude waves
or on timescales long enough for nonlinear effects to manifest themselves, results predicted by the linear theory do not
agree with measurements. Weakly nonlinear theory attempts to better describe internal wave dynamics by expanding flow
variables asymptotically and retaining corrections that correspond to finite amplitude (nonlinearity) and wavelength
(dispersion). The most famous weakly nonlinear model is probably the Korteweg–de Vries (KdV) equation, given by

While the KdV theory correctly predicts some properties of internal waves, it can only be expected to perform
well within certain asymptotic limits (e.g. the small-amplitude limit and long-wave limit). For large-amplitude waves, solutions of the
KdV equation and its variations have been shown to be different from waveforms predicted by the fully nonlinear theory

Graph of the model setup. Solid curves are isopycnals indicating the ISW and the linear waves in the initial field.

The governing equations for the present work are the incompressible Navier–Stokes equations under the rigid lid and
Boussinesq approximations, given by

A complete description of the numerical model used in this study can be found in

A graph showing the model setup is given in Fig.

We focus on flows in a quasi-two-layer stratification with a dimensionless density difference

Solitary wave parameters. Here, the amplitude is measured by the
maximum isopycnal displacement

Filled contours showing the horizontal velocity induced by the ISW,
with positive current shown in red and negative current shown in blue. The
dashed curve shows the isopycnal displacement along the pycnocline. The black
contours show the gradient Richardson number with

The initial solitary wave is specified by interpolating a solution of the DJL equation onto the ISW subdomain. Parameters
of the particular solitary wave solution considered in this work are given in Table

Linear wave parameters. In the case labels, O indicates an “overtaking” collision and H indicates a “head-on” collision, and the proceeding digits correspond to the wavelength of the linear waves.

Shaded density contours (full range of density shown, green denotes
the pycnocline centre) showing the solitary wave and the linear waves in the
case O2

We perform a suite of simulations in which the solitary wave propagates to the right and interacts with a small-scale wave
packet initialized from linear waves. The linear waves are specified by solving the TG equation numerically using
a pseudo-spectral technique

Same as Fig.

We consider linear waves of wavelengths ranging from 0.2 to 0.6

We would like to mention that these linear waves are in fact not purely linear during the simulations. However, by scaling
the relevant terms (i.e.

An impression of the overall flow behaviour in the case O2 can be gained from Fig.

Detailed density contours of the simulations

Vertical structure profiles of the linear waves with wavelengths of

The density profiles of the case O6 are shown in Fig.

Detailed density contours showing the mode-2 wave packet

In Fig.

As suggested in Fig.

In Fig.

Same as Fig.

The above analysis suggests that as the linear waves enter into the solitary-wave-induced background state, they are
subject to a modified stratification and a velocity shear due to solitary-wave-induced current, and it is this velocity
shear across the deformed pycnocline that leads to the deformation of short waves. This process is in many ways similar to
that found in

In Fig.

Phase speed of mode-1 linear waves in the ISW-induced background
shear current (solid curves) and a hypothetical zero-background current
(dashed curves) for

Froude number of mode-1 linear waves in the ISW-induced background
shear current for

Recall from Fig.

In addition to the stratification, the presence of background current will also modify the phase speed. In
Fig.

Power spectral density (PSD) of the initial horizontal velocity fields, computed from some of the overtaking collision cases.

We would like to note that given the nonlinear nature of the ISW, the interaction is indeed a nonlinear process, and thus
the linear theory can only provide some rough guide for the flow behaviour. To measure the nonlinearity of the fluid flows,
we shall introduce the Froude number which, in the context of internal wave dynamics, is usually defined as

A function that describes a physical process can be represented either in the physical space or in the Fourier space. The
two different representations are connected through the Fourier transform. Suppose

In the remainder of this section, we compute the PSD of horizontal velocity in the layer above the pycnocline and use it
to estimate the amount of wave energy being transferred during the collisions. The location of the horizontal layer chosen for
the analysis is

Scaled PSD of linear waves in the simulations with

Maximum values of the scaled PSD observed in Fig.

In Fig.

Quantitative measurement of the maximum values plotted in Fig.

The maximum value of the scaled PSD as a function of wavelength is plotted in Fig.

Scaled PSD of the cases

Quantitative measurement of the peak values observed in Fig.

For the cases O6 and H6, simulations were performed for an extended period of time in order to allow for repeated
collisions between the solitary wave and the linear wave packet. For each of these cases, four complete collision cycles
were observed, and the scaled PSD has been computed at

Time series of scaled maximum vertically integrated kinetic energy (KE).
The figure shows the difference between simulations with and without the linear
waves. Note the different scales in the

During the collision with the linear wave packet, the solitary wave is also affected by the linear waves that pass through
it. We note, however, that the kinetic energy carried by the linear waves is much smaller than that carried by the
solitary wave, and hence the impact of linear waves on the solitary wave is also small. Here, we define the kinetic
energy (KE) per unit mass following standard convention (which drops the reference density and hence changes the
dimensions of the quantity) by

To analyze changes in the solitary wave and determine if they are results of the collision, we performed an additional
simulation with the same solitary wave but without the linear waves. We estimated the vertically integrated KE at the
crest of the ISW for simulations with and without linear waves, and plotted the difference as time series (i.e. as functions
of scaled time

We have also attempted to detect the phase shift of the solitary waves from the locations of maximum vertically integrated KE. However, we found that such a phase shift, if it exists at all, is on the order of millimetres. In other words, the detected phase shift is on the grid scale and is subject to numerical error. For this reason, the results are not shown here.

In this work we performed two-dimensional direct numerical simulations to study the interaction between a large-scale
fully nonlinear ISW and small-scale linear internal waves. We demonstrated that there was a net energy transfer from the
small-scale linear waves to the large-scale solitary waves. This contrasts the conclusion in

We performed analysis based on linear wave theory and showed that during the nonlinear interaction with the ISW, the destruction of short linear waves occurs primarily due to the presence of ISW-induced velocity shear, which alters the vertical structure of the short waves in a nonlinear manner, leading to significant wave amplitudes on only one side of the deformed pycnocline centre. On the other hand, a shift of the location of the pycnocline plays a secondary role during the collision, as its main effect is to alter the propagation speed of the linear waves, and shift the location of the maximum of the vertical structure downward. However, the vertical structure is unchanged with respect to the pycnocline centre. We also demonstrated that a critical layer is not present during the collision, regardless of the wavelength of the linear waves, since the phase speed approaches the maximum ISW-induced current asymptotically as the wavelength approaches 0.

A clear avenue of future research is to explore the parameter space, in particular the Richardson number effect, of the
solitary wave. In the present work, we studied the ISW whose minimum Richardson number is 0.246. Although none of the
simulations show evidence of the generation of shear instability, this does not necessarily mean that the wave–wave
interaction considered in the present work is Richardson number independent. Moreover,

We note that our findings are in many ways similar to those in

While all of the simulations discussed in this work are performed on the laboratory scale, the scaling-up of the current
experiments to the field scale is left as a topic for future work. When the field scale is considered, waves with a much
larger range of wavelengths can be expected to breakdown, including short waves affected by self-interaction

Data is available upon request by email to the first author.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Extreme internal wave events”. It is a result of the EGU, Vienna, Austria, 23–28 April 2017.

Time-dependent simulations were completed on the high-performance computer cluster Shared Hierarchical Academic Research
Computing Network (SHARCNET,