These authors contributed equally to this work.

Nonlinear oceanic internal solitary waves are considered under the influence of the combined effects of saturating nonlinearity, Earth's rotation, and horizontal depth inhomogeneity. Here the basic model is the extended Korteweg–de Vries equation that includes both quadratic and cubic nonlinearity (the Gardner equation) with additional terms incorporating slowly varying depth and weak rotation. The complicated interplay between these different factors is explored using an approximate adiabatic approach and then through numerical solutions of the governing variable depth, i.e., the rotating Gardner model. These results are also compared to analysis in the Korteweg–de Vries limit to highlight the effect of the cubic nonlinearity. The study explores several particular cases considered in the literature that included some of these factors to illustrate limitations. Solutions are made to illustrate the relevance of this extended Gardner model for realistic oceanic conditions.

Oceanic internal waves are an important class of nonlinear wave processes. In particular, the internal solitary waves (ISWs) are the most ubiquitous type of solitons in the natural environment. These waves often propagate for long distances over several inertial periods, and the effect of Earth's background rotation is potentially significant

The rotation-induced solitary wave decay can be suppressed in certain ambient shear flows wherein the sign of the rotation coefficient is changed

Another important feature of oceanic solitons is that, in many cases, they are strongly nonlinear, so that the Korteweg–de Vries approximation (KdV) involving only the quadratic nonlinearity is inapplicable

In this paper, we make the next step by adding a sloping bottom effect to the Gardner model with rotation. Correspondingly, the results of

The work by

A standard model for the evolution of large-amplitude oceanic internal solitary waves is the rotating Gardner (rG), or extended KdV equation with rotation and variable depth

Additionally, any initial condition to Eq. (

In the absence of rotation,

There are three families of steady solitary wave solutions given by Eqs. (

Assuming that

The result, Eq. (

In the KdV equation,

For later reference, we first consider the adiabatic theory for the rotating KdV equation (

In the absence of rotation,

When

Comparison of the rG adiabatic radiation decay theory for homogeneous conditions (black lines) and rG numerical solutions for

While it is not necessary to make the sign of

As discussed above, in a non-rotating system, the right-hand side of Eq. (

Radiation decay in a homogeneous environments (where

In a homogeneous environment, the first term on the right-hand side of Eq. (

Figure

Figure

Numerical solution of Eq. (

In this section, the adiabatic theory (Eq.

In the homogeneous case where the coefficients

Figure

The complicated evolution of the decaying solitary wave and the trailing radiation is illustrated in Fig.

The example in Fig.

Adiabatic theory for wave propagation from deep to shallow in a two-layer system, with

To illustrate the combined effects of inhomogeneity and rotation, a two-layer Boussinesq stratification with upper layer depth

rG numerical solutions for

The bottom slope will be taken constant, and the lower layer depth is given by the following:

Note that the linear bottom slope in Eq. (

Adiabatic theory for wave propagation from deep to shallow in a two-layer system with

The evolution of the wave amplitude

Figure

The variation in wave mass,

Also shown in Fig.

There are two examples of the full rG solutions for

In the examples above, the cubic nonlinearity was not an essential feature of the evolution. Indeed, the wave evolution is qualitatively similar to the rotating KdV solutions in Fig. 2 of

The transmitted signals at

In this paper we have continued a series of studies of nonlinear internal waves of a moderate amplitude in the shallow, stratified areas of the ocean (see the references in the Introduction). Based on the classical Korteweg–de Vries equation, we added the main factors, making the analysis closer to the physical reality, i.e., cubic nonlinearity, Earth's rotation, and sloping bottom. The interplay of these factors makes the problem rather complicated, both physically and mathematically. To better explain the qualitative effect of each of them, we first briefly reproduce the effect of rotation in the medium with quadratic nonlinearity (KdV with rotation), then that with both quadratic and cubic nonlinearities (Gardner with rotation), and, as the main content of this paper, the joint effect of rotation and inhomogeneity in the Gardner equation. The specific qualitative effect of the latter is the limiting soliton amplitude, and the corresponding increase in the wavelength so that the topography effect becomes especially important. Along with the approximate adiabatic approach, a direct numerical study of the rG equation was performed and confirmed the adiabatic theory and highlighted its limitations. This combined approach allowed us to demonstrate a rather complicated behavior of shoaling internal solitons. For example, whereas the soliton energy always decreases due to the radiation losses, the displacement amplitude and mass can increase in a shoaling wave at a finite distance due to the decrease in total depth. In turn, the oscillating tail reveals a complicated behavior that includes, in particular, the formation of nonlinear wave packets as in

Future research should include further comparisons of the theory with observational results in different oceanic environments and extend the above results to the strongly nonlinear waves with rotation.

The eigenvalue problem for the linear long wave phase speed

In a two-layered stratification, with depths

This work does not include any externally supplied code, data, or other material. All material in the text and figures was produced by the authors using standard mathematical and numerical analysis by the authors.

KRH and LO equally contributed to formulating the problem, obtaining analytical solutions, and discussing the results. Numerical modeling was performed by KRH.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors thank Yury Stepanyants, for the helpful discussions.

This research has been supported by the Office of Naval Research (grant no. N00014-18-1-2542) and National Science Foundation (grant no. OCE-1736698).

This paper was edited by Victor Shrira and reviewed by two anonymous referees.