Unexpectedly large displacements in the interior of the oceans are studied through the dynamics of packets of internal waves, where the evolution of these displacements is governed by the nonlinear Schrödinger equation. In cases with a constant buoyancy frequency, analytical treatment can be performed. While modulation instability in surface wave packets only arises for sufficiently deep water, “rogue” internal waves may occur in shallow water and intermediate depth regimes. A dependence on the stratification parameter and the choice of internal modes can be demonstrated explicitly. The spontaneous generation of rogue waves is tested here via numerical simulation.

Rogue waves are unexpectedly large displacements from equilibrium positions or otherwise tranquil configurations. Oceanic rogue waves on the sea surface obviously pose immense risk to marine vessels and offshore structures (Dysthe et al., 2008). As these waves were observed in optical waveguides, studies of such extreme and rare events have been actively pursued in many fields of science and engineering (Onorato et al., 2013). Within the realm of oceanic hydrodynamics, observation of rogue waves in coastal regions has been recorded (Nikolkina and Didenkulova, 2011; O'Brien et al., 2018). Nearly all experimental and theoretical studies in the literature of rogue waves in fluids focus on surface waves. Our aim here is to investigate a similar scenario for internal waves. Internal waves play critical roles in the transport of heat, momentum and energy in the oceans, and breaking of such waves may have an impact on circulation (Pedlosky, 1987). There is a substantial literature on the observations and theories of large-amplitude internal waves in shallow water (Stanton and Ostrovsky, 1998). Many studies concentrate on solitary waves in long-wave situations employing the Korteweg–de Vries equation (Holloway et al., 1997) but not on the highly transient modes with the potential for abrupt growth. In terms of relevance in other fields of physics and engineering, the actual derivation of the governing equations may dictate the regime of input parameter values necessary for rogue waves to occur.

Theoretically, the propagation of weakly nonlinear, weakly dispersive
narrowband wave packets is governed by the nonlinear Schrödinger
equation, where the dynamics are dictated by the competing effects of second-order dispersion and cubic nonlinearity (Zakharov, 1968; Ablowitz and Segur,
1979). Modulation instability of plane waves and rogue waves can then occur
only if dispersion and cubic nonlinearity are of the same sign. For surface
wave packets on a fluid of finite depth, rogue modes can emerge for

Other fluid physics phenomena have also been considered, such as the effects of rotation (Whitfield and Johnson, 2015) or the presence of a shear current, an opposing current (Onorato et al., 2011; Toffoli et al., 2013a; Liao et al., 2017) or an oblique perturbation (Toffoli et al., 2013b). While such considerations may change the numerical value of the threshold (1.363) and extend the instability region, the requirement of water of sufficiently large depth is probably unaffected. For wave packets of large wavelengths, dynamical models associated with the shallow water regime have been employed (Didenkulova and Pelinovsky, 2011, 2016), such as the well-known Korteweg–de Vries and Kadomtsev–Petviashvili types of equations (Grimshaw et al., 2010, 2015; Pelinovsky et al., 2000; Talipova et al., 2011), which may also lead to modulation instability under several special circumstances.

The goal here is to establish another class of rogue-wave occurrence through
the effects of density stratification, namely, internal waves in the
interior of the oceans. Internal waves in general display more complex
dynamical features than their surface counterparts. As an illustrative
example, a given density profile may allow many internal modes characterized
by the number of nodes in the vertical structures. This family of allowed
states will be generically represented in this paper by an integer

For a basin depth (

Critical wavelength

The important point is not just a difference in the numerical value of the cutoff but that rogue waves now occur for water depths lower than a certain threshold. Our contribution is to extend this result. The nonlinear focusing mechanism of internal rogue waves is (i) determined by an estimation of the growth rate of modulation instability and (ii) elucidated by a numerical simulation of the emergence of rogue modes with the optimal modulation instability growth rate as the initial condition.

The dynamics of small-amplitude (linear) waves in a stratified shear flow
with the Boussinesq approximation is governed by the Taylor–Goldstein
equation (

The emergence of rogue-wave modes from a background continuous wave
perturbed by a long-wavelength unstable mode. Larger baseband gain implies a
smaller time is required for the rogue-wave modes to emerge.

For the simple case of constant buoyancy frequency

the maximum growth rate is (imaginary part of

the growth rate for long-wavelength disturbance is

In terms of significance in oceanography, the constraint

An intensively debated issue in the study of rogue waves through a
deterministic approach is the proper initial condition that may generate or
favor the occurrence of such large-amplitude disturbances. Modulation
instability refers to the growth of a small disturbance in a system due to the
interplay between dispersive and nonlinear effects (Craik, 1984), and here
we examine this instability by solving the nonlinear Schrödinger equation (Eq. 4)
numerically. One suggestion is the role played by long-wavelength modes
associated with modulation instability (also known as “baseband instability”; Baronio
et al., 2015). To examine this effect and to clarify the role of
stratification as well as the choice of internal wave modes, numerical
simulations are performed where baseband modes with the scaled modulation
instability growth rate on a plane wave background and, for example, 5% amplitude
are selected as the initial condition (Chan and Chow, 2017; Chan et al.,
2018):

This choice of a preferred modulation instability mode as the initial
condition is different from other approaches in the literature, such as one
using random noise. A pseudospectral method with a fourth-order Runge–Kutta
scheme for progressing through time is applied to solve the nonlinear
Schrödinger equation (Eq. 4) numerically. When the wavenumber

The growth rate of the baseband mode is a crucial factor of rogue-wave generation. A stronger baseband growth rate will trigger a rogue wave within
a shorter period of time. From Eqs. (6) and (7), the baseband growth rate

The baseband growth rate increases as the fluid depth

Figure 1 shows that rogue waves can emerge sooner when the fluid is deeper. Remarkably, this implies that baseband instability is stronger when the system is closer to the singular limit where the cubic nonlinearity changes sign. On the other hand, the degree of the background density stratification creates only a minor effect on the baseband mode. Apart from choosing a preferred baseband mode, another perspective taken in the literature is to select a random field as the initial condition. For the present nonlinear Schrödinger equation, “rogue-wave-like” entities would then emerge as well (Akhmediev et al., 2009).

An analytically tractable model for packets of internal waves was studied
here through four input parameters,

No data sets were used in this article.

KWC initiated the idea of studying and comparing the connections between modulation instability and rogue waves for internal versus surface modes. RHJG advised on the literature on internal waves and refined the formulation as well as the writing style. HNC performed the numerical simulations.

The authors declare that they have no conflict of interest.

Partial financial support has been provided by the Research Grants Council (contracts HKU17200815 and HKU17200718).

This paper was edited by Ira Didenkulova and reviewed by Efim Pelinovsky, Yury Stepanyants and one anonymous referee.