A new two-fluid layer model consisting of forced rotation-modified Boussinesq equations is derived for studying tidally generated fully nonlinear, weakly nonhydrostatic dispersive interfacial waves. This set is a generalization of the Choi–Camassa equations, extended here with forcing terms and Coriolis effects. The forcing is represented by a horizontally oscillating sill, mimicking a barotropic tidal flow over topography. Solitons are generated by a disintegration of the interfacial tide. Because of strong nonlinearity, solitons may attain a limiting table-shaped form, in accordance with soliton theory. In addition, we use a quasi-linear version of the model (i.e. including barotropic advection but linear in the baroclinic fields) to investigate the role of the initial stages of the internal tide prior to its nonlinear disintegration. Numerical solutions reveal that the internal tide then reaches a limiting amplitude under increasing barotropic forcing. In the fully nonlinear regime, numerical experiments suggest that this limiting amplitude in the underlying internal tide extends to the nonlinear case in that internal solitons formed by a disintegration of the internal tide may not reach their table-shaped form with increased forcing, but appear limited well below that state.

Tidally generated internal solitons are a widespread phenomenon in the oceans
and have been observed and studied for decades

For internal solitons as such, an archetypal model has been the Korteweg–de
Vries (KdV) equation, which is based on the assumption of weak nonlinearity
and weak nonhydrostaticity. The equation gives prediction for the relation
between amplitude, width and phase speed of the solitons, as well as the
shape itself. In the KdV equation there is, mathematically speaking, no limit
to the amplitude that solitons may reach (although, of course, at some point
the underlying assumption of weak nonlinearity would be violated). This
behaviour changes fundamentally if a higher-order (i.e. cubic) nonlinear term
is included, leading to the so-called extended KdV (eKdV) equation, as
discussed in e.g.

In this paper, we focus on another limiting factor, which comes into play even before solitons arise, namely in the internal tide itself. In a purely linear system, the amplitude of the internal tide increases linearly with the strength of the barotropic tidal flow. Here we study how this changes if one includes quasi-linear terms, i.e. retaining products of barotropic and baroclinic fields in the advective terms while still ignoring interactions of the baroclinic field with itself. We demonstrate that a saturation in the amplitude of the internal tide occurs, and increasing the barotropic flow further does not produce a larger internal tide. As a consequence, when one includes the genuinely nonlinear effects, i.e. products of baroclinic terms, resulting solitons may stay well below their formal limiting amplitude, no matter how strong the forcing.

To study these effects we derived a set of fully nonlinear, weakly
nonhydrostatic model equations, by extending the MCC equations with a
barotropic tidal forcing over topography and with Coriolis effects, which
have previously been shown to play a key role in soliton generation from
internal tides

The presence of a topography greatly complicates the subsequent handling of the equations, but we demonstrate that the set of equations can be obtained and can be cast in a form amenable to numerical solving.

An extension of the MCC theory with Coriolis effects (MCC-

The paper is organized as follows. We derive a new two-fluid layer model
consisting of a set of forced rotation-modified Boussinesq equations in
Sect.

The numerical methods and schemes are described in
Appendix

We start from the continuity and Euler equations and consider a two-fluid
layer system (Fig.

The two-fluid layer system for which the forced-MCC-

Boundaries are defined at the surface, taken to be a rigid lid, which is
located at

The kinematic boundary conditions at the surface and bottom read as

For later convenience, we write pressure as the sum of hydrostatic and
dynamic parts, the latter being denoted by primes:

The second equation in (

The next step is to bring the equations into an appropriate dimensionless
form for which we introduce the following scales. The scale for the
undisturbed water depth is taken to be

In

Since we allow waves to have large amplitudes (i.e. to be strongly
nonlinear), we may take horizontal current velocities to scale with

The typical scale of

In summary, then, we can introduce the following dimensionless variables,
indicated by asterisks,

The goal is now to derive a reduced set of equations from
Eqs. (

We vertically integrate the equations over the upper and lower layers and
expand them to the orders

For this reason we need to apply the Leibniz integral
rule below with respect to

After integration of
Eqs. (

For the lower layer one proceeds similarly, except that now both boundaries
are variable. Applying the boundary conditions
(

The six integrated Eqs. (

At the lowest order (

At lowest order, the vertical momentum equation (

At lowest order, then, the set of integrated equations is closed; together
with the (exact) integrated continuity equations (

Recall that

At order

This allows us to write the horizontal momentum equations as

The remaining problem is to find an expression for

By vertically integrating the continuity equation (

We have thus obtained a closed set of six dimensionless equations, namely the
exact continuity equations (

We combine the continuity equations (

Given that

If we now substitute the time derivative of the oscillating topography
(

Equation (

Far from the sill (i.e.

We can thus combine the horizontal momentum equations (

All in all, we now have five equations for five unknowns (

Before concluding this section, it is worthwhile noting an alternative
approach. Given the assumption of a rigid lid, one could have also taken

Whilst not designed to represent a specific region of the world oceans, we aim to investigate in a general manner the conditions by which tidally generated solitons may evolve and, eventually, develop limiting amplitudes in ocean-like scenarios. It is then desirable that leading solitons can propagate towards a mature stage before overtaking preceding internal tides; otherwise, although these are form-preserving features, the tracking of their wave properties becomes cumbersome. For this reason, the parameters that we describe in the following were selected to highlight the qualitative features of these nonlinear processes for a broad range of (mimicked) tidal forcing strengths.

Although the model is solved and discussed in nondimensional form, we also present the parameter values in dimensional form to put them in an oceanographic context.

We define the (dimensional) topography analytically following

At this point it is worthwhile recalling that the oscillation of the
topography is introduced in dimensionless form as

The topographic obstacle (ridge, sill, etc.) is always centered on the

To characterize the hydraulic state, we use the Froude number calculated as

Importantly, we also use the Froude number in
Appendix

Results from this model comparison confirm a near equivalence between both
models within the parameter framework of study, which we restrict to

We adopt a two-layer system where the total water depth,

Amplitude of the linear,

Summary of runs. Varying parameters are the reduced gravity,

In Table

In Sect.

For convenience, wave properties are scaled as follows. The interfacial
displacement,

Tide-generated solitons emerge from nonlinear disintegration of the underlying internal tides and may be, therefore, naturally subjected to the properties of the latter. For this reason, we find it insightful to investigate first the properties of the underlying internal tides, prior to their nonlinear disintegration, within the parameter space of this study.

As described in Sect.

Accordingly, Fig.

In the purely linear experiments, the amplitude of the internal tide
increases linearly with the barotropic tidal flow strength. However, the
quasi-linear internal tide exhibits a limiting amplitude in all runs, when
the tidal forcing increases well above

Regarding the comparison between runs with different parameters, we find the
following. In Fig.

Although not shown, it is worth mentioning that the wavelength of the
quasi-linear tides does not deviate from the linear case in any of the
settings of study and is independent of the strength of the tidal forcing
(and hence of the Froude number) and of the height of the topography.
However, as predicted from linear theory for interfacial waves, an increase
in

The amplitude saturation described above is further illustrated in
Fig.

Snapshots of the interfacial displacement of leftward-propagating
quasi-linear internal tides for run A1 (

These findings raise the question as to whether solitons emerging from a
disintegration of the initially quasi-linear internal tides may be subjected
to saturation before they reach a limiting “table-top” shape. We examine
this question in the next section by focussing on runs A1, B1 and C1, varying
the height of the topography and the thickness of the upper layer while
preserving a high stratification. The latter allows us to investigate the
broadest range of interfacial wave amplitudes, as suggested by
Fig.

In this section we investigate the conditions by which tidally generated fully nonlinear solitons may attain a limiting amplitude. Special attention is devoted to factors conditioning the growth of fully nonlinear waves as “table-top” solitons. The main question to address is whether the amplitudes of tidally generated solitons may be subjected to limiting amplitudes of the underlying quasi-linear internal tides, as we hypothesized in the previous section, thus qualifying predictions from classical eKdV and MCC theories.

In Fig.

At a first stage, Fig. 4b, the internal tide splits up into two different groups of rank-ordered solitons: a train of depressions on the leading edge, and a train of elevations, after the former packet, with initially smaller amplitudes. At a later stage, Fig. 4c, the largest elevations have reached the smaller depressions in the train, and three leading solitons at the front present almost equal amplitudes. Previous solitary wave packets, already propagating away from the generation area, are shown in Fig. 4d and e and correspond to preceding disintegrated internal tides. The “table-top” soliton observed at the leading edge of every preceding internal tide emerged in all cases from the first of the three solitons described previously in Fig. 4c.

As the leading soliton evolves and reaches its maximum amplitude, it also
broadens, as predicted by soliton wave theory

Snapshots of the interfacial displacement of nonlinear internal
tides and solitons in run A1 for a supercritical regime
(

Wave evolution of leftward-propagating leading solitons in run A1
under different forcing strengths (see legend). In all panels the horizontal
axis indicates the run time and soliton age (in brackets) in tidal periods.
The (dimensionless) wave properties are

Because tidally generated solitons are part of the evolving internal tides,

Using the above criteria, Fig.

The soliton reaches its maximum amplitude slightly before the flow becomes
critical (

Generally speaking, we distinguish between two types of solitons regarding
their timescales of growth (see Fig.

In agreement with the above description, the phase speed graphs also reveal a
clear distinction between the subcritical and critical/supercritical regimes
(Fig.

Solitary wave solutions for mature leading solitons in run A1 from
the KdV (grey line), eKdV (black line) and MCC (red line) theories compared
to numerical solutions from the forced-MCC equations (coloured dots refer to
the Froude number and strength of the tidal flow; see legend).

Finally, we compare in Fig.

These wave properties correspond to solitons of State II (mature solitons) after time averaging over a tidal cycle.

with KdV-type and MCC soliton solutionsAs expected, small tide-generated solitons approach the linear long-wave
phase speed for interfacial waves (

As regards the relationship between the soliton width and amplitude,
tide-generated solitons follow a similar behaviour to that predicted by eKdV
and MCC theory, broadening as they approach their maximum amplitude. By this
broadening, strongly nonlinear solitons develop the “table-top” shape,
although forced-MCC equations generate some larger and narrower solitons than
their eKdV and MCC counterparts (Fig.

Wave evolution of leftward-propagating leading solitons in run B1
under different forcing strengths (see legend). In all panels the

We use for runs B1 and C1 a similar range of Froude numbers as for run A1;
however, they present a more weakly nonlinear regime where a striking feature
emerges. Leading solitons exhibit a maximum amplitude which is not related to
a “table-top” form and which cannot be exceeded by further increasing the
tidal forcing (see Figs.

The above results support the idea that tidally generated solitons might be
subject to a limited growth which is beyond the classical KdV and MCC-type
models, being due to the saturation of the underlying quasi-linear internal
tide as the tidal forcing increases (see Sect.

According to their phase speed, and in agreement with findings from run A1,
two types of leading solitons also emerge in runs B1 and C1. The larger
nonlinear solitons (critical and supercritical regimes) exhibit an
oscillating speed, in phase with the tidal flow, which increases over time.
The smaller nonlinear solitons (subcritical regime) exhibit a nearly constant
phase speed (Figs.

From Figs.

Same as Fig.

On the one hand, the smaller topography generates quasi-linear internal tides
which are smaller than those in run A1 (see Fig.

When compared with solitary wave solutions from eKdV and MCC theories, the
growth-limiting effect of the tidal forcing becomes a remarkable feature of
forced-MCC solitons generated in runs B1 and C1, since they reach a limiting
amplitude but do not attain a “table-top” form
(Fig.

Snapshots of the interfacial displacement of nonlinear internal
tides and solitons in run B1 for a supercritical regime
(

Snapshots of the interfacial displacement of nonlinear internal
tides and solitons in run C1 for a supercritical regime
(

Solitary wave solutions for mature leading solitons in run B1 (top
row) and run C1 (bottom row) from KdV (grey line), eKdV (black line) and MCC
(red line) theories compared to numerical solutions from the forced-MCC
equations (coloured dots refer to the Froude number and strength of the tidal
flow; see legend).

Regarding the relationship between the soliton phase speed and amplitude,
both runs B1 and C1 follow a similar curve to that predicted by the eKdV and
MCC theories (Fig.

In Fig.

In agreement with previous studies, we observe in all panels that an increase
in the latitude leads to larger dispersive effects due to Coriolis
dispersion, which prevents the nonlinear internal tide from disintegrating
into strongly nonlinear solitons

Effects of the Earth's rotation through a set of snapshots from
runs A1 (

We investigate limiting amplitudes of internal tides and solitons using a
generalization of the fully nonlinear MCC equations

The application of an oscillating topography is not completely equivalent to
the oceanic case of a tidal flow over a topography at rest. For this reason
we have restricted our analyses to a parameter space where a semi-equivalence
between both forcing systems was demonstrated (Appendix

Numerical solutions show that strongly nonlinear tide-generated solitons
attain in some cases a limiting table-shaped form, in agreement with
classical soliton theory. However, results also suggest that tide-generated
solitons may alternatively be limited by saturation of the underlying
quasi-linear internal tide. In the purely linear system the amplitude of the
internal tide increases linearly with the strength of the barotropic tidal
flow. But in the quasi-linear case, as the forcing becomes stronger,
advective terms become stronger too and cannot be neglected. (Again, in the
quasi-linear case, barotropic advection is retained, but interactions of the
baroclinic field with itself are neglected). As a result, a saturation in the
amplitude of the internal tide occurs; a further increase in the tidal flow
does not produce a larger internal tide. This effect seems to have passed
unnoticed in previous studies, but might be a key factor in the subsequent
disintegration of the internal tide into solitons. It implies that when one
includes the genuinely nonlinear effects, i.e. products of baroclinic terms,
resulting solitons may stay well below their formal limiting amplitude, no
matter how strong the forcing. Interestingly, an increase in the tidal
forcing above the value that generates table-shaped solitons, or above the
value that

Motivated by the above finding, we performed analogous runs using the full
set of weakly nonlinear equations derived in

Another departure from classical theories is that strongly nonlinear
tide-generated solitons may exhibit larger maximum amplitudes than predicted
from eKdV and MCC solutions, while soliton phase speeds are always smaller.
We attribute these differences to the fact that tide-generated solitons

In relation to the rotational cases, and in agreement with previous studies

Before concluding we must note, reiterating arguments by

The modeling data used by this study are freely available but not otherwise published in any publicly accessible database. The data can nonetheless be provided on request via e-mail to the first author.

We define a grid in time and space for discretization of the various
derivatives of the system. Then,

The various derivatives in the model are discretized with centered difference
approximations

Initially the system is at rest with horizontal velocities,

The time derivatives of the

However, solving numerically

It is important to recall here that

If we now discretize the time derivative of

The choice of the space–time steps

For the simulations we present, it was not needed to filter out wavenumbers
above a threshold to control Kelvin–Helmholtz instabilities, as we designed
the space–time grid to avoid this problem. However, in some cases,
especially in the simulations where the forcing was fairly strong, an
additional trick was needed to retain stability around the generation area

In Appendix

Firstly, all terms of the

The quasi-linear forced-MCC-

We notice that the linear runs were actually done somewhat indirectly by
taking the quasi-linear version of forced-MCC-

A Galilean transformation involves two frames of reference which move with constant and rectilinear speed with respect to each other. Hence, observations made in one frame can be converted to another, as physical laws are identical. However, our oscillating topography is not an inertial frame since it is accelerated with respect to a situation where the topography is at rest (as in the ocean). It is, therefore, not evident that the results from the two frames are equivalent.

Linear (left panels) and quasi-linear (right panels) interfacial
waves generated via a tidal flow over a sill from the model equations derived
in

We use the generation model of weakly nonlinear, weakly nonhydrostatic
interfacial waves derived in

In Appendix Fig.

Results from Fig.

Financial support was provided by the Spanish government (Ministerio de Ciencia e Innovación) through a PhD grant (FPU) awarded to the first author (AP2007-02307), and through the Netherlands Organization for Scientific Research (NWO), section Earth and Life Sciences (ALW), via ZKO grant no. 839.08.431 awarded to the “INdian-ATlantic EXchange in present and past climate” (INATEX) programme. The authors also wish to thank the reviewers for their helpful remarks and suggestions. Edited by: V. Shrira Reviewed by: V. I. Vlasenko and two anonymous referees