We present high-resolution, three-dimensional simulations of rotation-modified mode-2 internal solitary waves at various rotation rates and Schmidt numbers. Rotation is seen to change the internal solitary-like waves observed in the absence of rotation into a leading Kelvin wave followed by Poincaré waves. Mass and energy is found to be advected towards the right-most side wall (for a Northern Hemisphere rotation), leading to increased amplitude of the leading Kelvin wave and the formation of Kelvin–Helmholtz (K–H) instabilities on the upper and lower edges of the deformed pycnocline. These fundamentally three-dimensional instabilities are localized within a region near the side wall and intensify in vigour with increasing rotation rate. Secondary Kelvin waves form further behind the wave from either resonance with radiating Poincaré waves or the remnants of the K–H instability. The first of these mechanisms is in accord with published work on mode-1 Kelvin waves; the second is, to the best of our knowledge, novel to the present study. Both types of secondary Kelvin waves form on the same side of the channel as the leading Kelvin wave. Comparisons of equivalent cases with different Schmidt numbers indicate that while adopting a numerically advantageous low Schmidt number results in the correct general characteristics of the Kelvin waves, excessive diffusion of the pycnocline and various density features precludes accurate representation of both the trailing Poincaré wave field and the intensity and duration of the Kelvin–Helmholtz instabilities.

Over recent decades non-linear internal solitary waves (ISWs) have been the
subject of continuing research due, in part, to their common presence in
coastal waters (

The dominant laboratory insights on Kelvin waves in channel geometry come
from the experimental work of

In

In the simpler case of a rotating fluid adjusting without the presence of
side walls, it has been observed that if dispersive effects are accounted
for, a leading solitary wave is created, which then breaks down into a
non-linear wave packet as it propagates (

The work by

There have been a number of studies on Kelvin waves from a model equation
approach.

Rotation-modified mode-1 ISWs within a cylindrical geometry have been investigated by

Results on non-rotating mode-2 ISWs, especially with regards to their mass
transport capabilities
(

Our primary qualitative results that could be confirmed in the laboratory
concern the fundamentally three-dimensional nature of the shear instability
at the edges of the mode-2 wave's core and the details of the spatial
structure of the span-wise kinetic energy flux. The former could be
visualized by a particle image velocimetry (PIV) system with a light sheet oriented in the span-wise
direction or along-tank light sheets at varying distances from the focusing
wall. The latter could be characterized by the more usual along-tank PIV set-up. Moreover, the quantitative results of the Kelvin wave–Poincaré wave
resonance and the formation of secondary Kelvin waves in our simulation
should provide an easier comparison than field-oriented simulations such as
those of

The remainder of the paper is structured as follows: the set-up of the
numerical experiments and numerical methods are outlined first, with the
results that follow structured to identify fundamental differences between
rotating and non-rotating evolution; the three-dimensional
structure of near wall overturning is characterized; and the importance of using a
realistic Schmidt number other than

We have run a series of direct numerical simulations (DNSs) in a set-up similar
to that of

We form the gravity intrusion by releasing a large density perturbation into
a quiescent, quasi-two-layer background stratification (Fig.

Stratification parameters.

Schematic of the numerical domain. The blue region, centred at the mid-depth,
corresponds to

We have completed a suite of numerical simulations at various rotation rates
and Schmidt numbers. The rotation rate has been specified using the Coriolis
parameter,

Following the work of

The internal Rossby radius of deformation is defined as

Case parameters and characterizations.

We briefly contrast our set-up to that of

Our numerical model solves the so-called Boussinesq equations of motion on an

The equations used differ from the oceanic situation in that we take the
density as a variable to be evolved, whereas in the ocean it is the salinity
and temperature that evolve, with density recovered from an equation of
state. The non-linearity of the equation of state leads to a variety of
complex phenomena (e.g. salt fingering, cabbeling, the fact that pure water
has a density maximum at 4

Numerical simulations were completed using the Spectral Parallel
Incompressible Navier–Stokes Solver (SPINS;

The size of the channel and the grid resolution are listed in Table

Tank dimensions and numerical resolution.

We begin by looking at how the ISW is affected by rotation through the
Coriolis force. We have chosen the rotation to match that of the Northern
Hemisphere, which causes objects to be deflected towards the right of their
trajectory. In the context of our experiment this leads to span-wise
variation in the developing ISW.

We investigate the location of the ISW crests using the scaled, vertically
integrated kinetic energy,

The time evolution of the scaled, vertically integrated kinetic energy,

The time evolution of

As time progresses, Poincaré waves form behind the Kelvin wave, as
previously described by

Scaled, vertically integrated kinetic energy,

The presence of the Kelvin and Poincaré waves in this context is quite
common and is dependent on the rotation rate (Fig.

It needs to be mentioned that the presence of side walls removes the possibility of a span-wise invariant geostrophic state forming in the collapse region since the presence of walls enforces that flow is in the along-tank direction. This means that the release of mass and energy into the ISWs is greater than when no side wall is present. The detailed dynamics of the near field are interesting but beyond the scope of the present paper.

A secondary boundary trapped wave also forms in the rotation-modified cases
(Fig.

Scaled, vertically integrated kinetic energy,

More importantly, at high rotation rates the second Kelvin wave is clearly
formed by a resonance with the Poincaré waves at the focused wall, which is
directly contrary to how the trailing ISWs form without rotation (they are
the excess mass that is not trapped by the leading wave). This description
appears to be valid for rotation rates greater than

As described by

As energy is drained from the leading Kelvin wave and deposited into the
secondary wave, the secondary wave will eventually become more energetic than
the first, resulting in an eventual overtaking. Our simulations do not show
this feature since our channel is not long enough and we are focused on the
shorter timescales associated with the energetics of the leading Kelvin
wave. See

The differentiation of whether a wave is a Kelvin wave or a Poincaré wave
is made difficult because of the non-linearity associated with the large
amplitude of these waves. The classical linear theory, as presented in
standard textbooks (

We observed that the radiating Poincaré waves resonate to form a secondary Kelvin wave. Since this secondary wave is separated from
the chaotic leading wave, it could possibly fit better in the description of a classical Kelvin wave. Figure

Vertically integrated

The dynamics seen thus far are fundamentally different than in the case without the side walls. As described in

Span-wise kinetic energy flux density for case 10_1 at

The advection of kinetic energy within the Kelvin and Poincaré waves is key
to understanding how the distribution of this energy is influenced by the
side wall. The span-wise flux of KE (Fig.

The secondary Kelvin wave (near

Horizontal kinetic energy flux density for case 10_1 at

For comparison purposes, we have completed the same simulation in two dimensions while allowing transverse flow to be coupled to horizontal motion through the Coriolis force (i.e. a two-and-a-half-dimensional model). Since there is no side wall, the radiated ISWs are smaller because much of the energy remains within the geostrophic state. Regardless of the waves being smaller and thus travelling slower, the KE flux in the leading wave of the 2-D case (not shown) is different from that of the 3-D case with side walls, especially near the wall. Further from the side walls, the two become more similar yet remain distinct in the magnitudes and distributions of the KE flux.

The span-wise variation in the KE flux along the mid-depth
(

Now that the description of the global wave field has been presented, we move
on to the localized behaviour of the leading Kelvin wave at the focusing wall.
As described by

Density anomaly,

In the highest rotation rate (case 10_1) the instability is confined to
within approximately 10 cm (one-quarter of the channel width) of the
focusing wall (Fig.

Length scales and timescales associated with localized shear instabilities.

For all cases, the K–H billows simultaneously form as pairs with a vortex above and below the pycnocline.
Furthermore, except during the early energetic K–H formation, these billows remain synchronous, even during the process of being broken down.
Comparison to the observations of

At the wave crest, the wave has the expected exponential decay (Fig.

Density anomaly,

We find that changes in the rotation rate (i.e. the strength of the Coriolis
force) influence the intensity of the K–H billows (Fig.

As time progresses, the wave sheds energy and mass in the shear instabilities
until a critical amplitude is reached, after which the wave remains stable but
continues to decay because of the lossy behaviour of the core region of large
amplitude mode-2 ISWs (

The emergence of shear instabilities is correlated with the rotation rate; that is, a higher rotation rate is associated with an
earlier shear instability. The higher rotation causes greater fluid to be directed towards the focusing wall, which leads to a
greater initial amplitude, which creates favourable conditions for shear instabilities. Though the initial amplitude is correlated
with the rotation rate (Fig.

Wave amplitude as a function of time for different rotation rates (

The wave amplitude decay rate,

Kinetic energy density at

Figure

Over time, due to the radiation of energy into trailing waves, the leading
Kelvin wave reduces in amplitude, span-wise width, and KE. The time-dependent
nature of the width, and thus the exponential decay, results in an increased
localization of KE along the focusing wall (seen very clearly in the second
row of Fig.

Kinetic energy density at the location of maximum amplitude for

The shear instabilities and associated dynamics are fundamentally small-scale behaviour which are damped by viscosity and smeared by diffusion, both of which are determined by the properties of the fluid, namely the molecular diffusivity and the viscosity. Experimentally, the diffusivity is fixed by the choice of stratifying solute. Physical values of a salt stratified experiment, which are typical for experiments of this type, give a Schmidt number of approximately 700. Since DNSs at these values are unattainable due to the resolution required, here we provide a short description of the impact that various Schmidt numbers have on the results presented thus far. Of importance is measuring the change in the shear instabilities by varying the Schmidt number.

For longer simulations, such as the ones conducted here, the pycnocline will
diffuse, causing the waves to propagate in a slightly different stratification
near the end of the simulation compared to the beginning. Smaller Schmidt
numbers have greater diffusion, causing a greater impact (Fig.

We find that the wave amplitude
is unaffected when compared at various Schmidt numbers. Thus, the general
features characterizing the wave are fairly similar at

Density anomaly,

The span-wise profile of the wave also shows differences (right column of
Fig.

In the

We have performed a series of numerical experiments of a lock-release
configuration, exploring the effects that rotation and side walls have on the
evolution of mode-2 ISWs. When the stratification has a single pycnocline
form, naturally occurring mode-2 waves are quite likely to exhibit regions of
overturning, and hence our configuration is ideal for exploring the combined
effects of rotation and instability. Matching with the results of

These instabilities cause the leading wave to lose energy and thus higher rotation rates, resulting in a faster decay in wave amplitude than cases with a lower rotation rate. The increased instability generation along the side wall also serves to create an asymmetry in the extent of mixing across the width of the tank. This effect could be observed in fjords or narrow lakes by making a careful comparison of mixing levels and wave amplitudes across the channel.

The high level of mass and kinetic energy along the focusing wall also
resulted in the radiation of Poincaré waves, as previously described by

The results presented above suggest two clear avenues for future work. One avenue would focus on the rotation-modified instability region. While in the above, clear evidence of transitional behaviour was presented, it is unclear to what extent a truly turbulent state was achieved. This is because trapping by the leading Kelvin wave is incomplete and turbulence may lose energy due to a spreading in space. Moreover, while the resolution of the numerical simulations was excellent for the full domain, a study that is focused on turbulent transition could optimize the domain and stratification parameters. For example, the domain could be shortened and the initial perturbation increased in size to increase the wave amplitude. While we have speculated that no-slip side walls would not fundamentally alter the shear instability, this remains to be confirmed by using a clustered Chebyshev grid in the span-wise direction. A more likely mechanism that would alter the shear instability would be to raise the pycnocline such that the layers are of unequal depths to better approximate an oceanographic stratification.

A second possible avenue for future work would explore the effect of the
span-wise extent. Figure 15 of

Data are 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,