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Brief Communication: Modulation Instability of Internal Waves in a Smoothly Stratified Shallow Fluid with a Constant Buoyancy Fr

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Brief Communication: Modulation Instability of Internal Waves in a Smoothly Stratified Shallow Fluid with a Constant Buoyancy Fr Nat. Hazards Earth Syst. Sci., 19, 583–587, 2019 https://doi.org/10.5194/nhess-19-583-2019 © Author(s) 2019. This work is distributed under the Creative Commons Attribution 4.0 License. Brief communication: Modulation instability of internal waves in a smoothly stratified shallow fluid with a constant buoyancy frequency Kwok Wing Chow1, Hiu Ning Chan2, and Roger H. J. Grimshaw3 1Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong 2Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong 3Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UK Correspondence: Kwok Wing Chow ([email protected]) Received: 9 August 2018 – Discussion started: 14 August 2018 Revised: 28 February 2019 – Accepted: 5 March 2019 – Published: 19 March 2019 Abstract. Unexpectedly large displacements in the interior breaking of such waves may have an impact on circulation of the oceans are studied through the dynamics of packets of (Pedlosky, 1987). There is a substantial literature on the ob- internal waves, where the evolution of these displacements servations and theories of large-amplitude internal waves in is governed by the nonlinear Schrödinger equation. In cases shallow water (Stanton and Ostrovsky, 1998). Many studies with a constant buoyancy frequency, analytical treatment can concentrate on solitary waves in long-wave situations em- be performed. While modulation instability in surface wave ploying the Korteweg–de Vries equation (Holloway et al., packets only arises for sufficiently deep water, “rogue” in- 1997) but not on the highly transient modes with the poten- ternal waves may occur in shallow water and intermediate tial for abrupt growth. In terms of relevance in other fields depth regimes. A dependence on the stratification parameter of physics and engineering, the actual derivation of the gov- and the choice of internal modes can be demonstrated ex- erning equations may dictate the regime of input parameter plicitly. The spontaneous generation of rogue waves is tested values necessary for rogue waves to occur. here via numerical simulation. 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 1 Introduction and cubic nonlinearity (Zakharov, 1968; Ablowitz and Se- gur, 1979). Modulation instability of plane waves and rogue Rogue waves are unexpectedly large displacements from waves can then occur only if dispersion and cubic nonlin- equilibrium positions or otherwise tranquil configurations. earity are of the same sign. For surface wave packets on a Oceanic rogue waves on the sea surface obviously pose im- fluid of finite depth, rogue modes can emerge for kh > 1:363, mense risk to marine vessels and offshore structures (Dysthe where k is the wavenumber of the carrier wave packet and et al., 2008). As these waves were observed in optical waveg- h is the water depth. Hence, conventional understanding is uides, studies of such extreme and rare events have been that such rogue waves can only occur if the water depth is actively pursued in many fields of science and engineering sufficiently large. (Onorato et al., 2013). Within the realm of oceanic hydro- Other fluid physics phenomena have also been consid- dynamics, observation of rogue waves in coastal regions has ered, such as the effects of rotation (Whitfield and John- been recorded (Nikolkina and Didenkulova, 2011; O’Brien son, 2015) or the presence of a shear current, an opposing et al., 2018). Nearly all experimental and theoretical stud- current (Onorato et al., 2011; Toffoli et al., 2013a; Liao et ies in the literature of rogue waves in fluids focus on sur- al., 2017) or an oblique perturbation (Toffoli et al., 2013b). face waves. Our aim here is to investigate a similar scenario While such considerations may change the numerical value for internal waves. Internal waves play critical roles in the of the threshold (1.363) and extend the instability region, the transport of heat, momentum and energy in the oceans, and Published by Copernicus Publications on behalf of the European Geosciences Union. 584 K. W. Chow et al.: Brief communication: Modulation instability of internal waves requirement of water of sufficiently large depth is probably Table 1. Critical wavelength λc as a function of various internal unaffected. For wave packets of large wavelengths, dynam- mode numbers n (with h D 500 m). ical models associated with the shallow water regime have been employed (Didenkulova and Pelinovsky, 2011, 2016), n (mode number of Rogue waves and such as the well-known Korteweg–de Vries and Kadomtsev– internal waves, with instability can occur Petviashvili types of equations (Grimshaw et al., 2010, 2015; each n representing for wavelengths longer than Pelinovsky et al., 2000; Talipova et al., 2011), which may a different vertical λc given by the following structure) numerical values (in m) also lead to modulation instability under several special cir- cumstances. 1 1305 The goal here is to establish another class of rogue- 2 652 wave occurrence through the effects of density stratification, 3 435 namely, internal waves in the interior of the oceans. Internal 4 326 waves in general display more complex dynamical features 5 261 than their surface counterparts. As an illustrative example, a given density profile may allow many internal modes char- acterized by the number of nodes in the vertical structures. the optimal modulation instability growth rate as the initial This family of allowed states will be generically represented condition. in this paper by an integer n (referred to here as the mode number). There is an extensive literature on large-amplitude internal solitary waves that are spatially localized pulses es- 2 Formulation sentially propagating without change of form (Grimshaw et al., 2004; Osborne, 2010). Our focus here is on internal rogue 2.1 Nonlinear Schrödinger theory for stratified shear waves that are modeled as wave pulses localized in both flows space and time. The asymptotic multiple scale expansions for internal wave packets under the Boussinesq approximation The dynamics of small-amplitude (linear) waves in a strat- also yield the nonlinear Schrödinger equation (Grimshaw, ified shear flow with the Boussinesq approximation is gov- 1977, 1981; Liu and Benney, 1981). When the buoyancy fre- erned by the Taylor–Goldstein equation (φ.y/ is the vertical quency is constant, modulation instability in one horizontal structure, k is the wavenumber, c is the phase speed and U.y/ is the shear current): space dimension will only occur for kh < kch D 0:766nπ, where the fluid is confined between rigid walls that are dis- 2 Uyy N φ tance h apart, n is the vertical mode number of the internal φ − k2 C φ C D 0; (2) yy U − c .U − c/2 wave and the critical wavenumber kc is given by the follow- ing equation (Liu et al., 2018): where N is the Brunt–Väisälä frequency (or the “buoyancy nπ 1=2 frequency”; ρ is the background density profile) by k D 41=3 − 1 : (1) c h g dρ N 2 D − (3) For a wave packet associated with the first internal mode ρ dy (n D 1), modulation instability or a rogue wave can occur with the carrier wavenumber k and shallow fluid of depth h The evolution of weakly nonlinear, weakly dispersive wave in the range of kh < 0:766π or 2.406. packets is described by the nonlinear Schrödinger equation For a basin depth (h) of, for example, 500 m, the critical for the complex-valued wave envelope S, obtained through wavelength (λc) is as follows: a multi-scale asymptotic expansion, which involves calcu- lating the induced mean flow and second harmonic (β and 2π 2h λ D D ; γ being parameters determined from the density and current c 1=2 kc n41=3 − 1 profiles): and the ranges of “shallow” and “intermediate” depths are 2 2 iSτ − βSξξ − γ jSj S D 0; τ D " t; ξ D " x − cgt ; (4) covered (Table 1). The important point is not just a difference in the numer- where τ is the slow timescale, ξ is the group velocity (cg) ical value of the cutoff but that rogue waves now occur for coordinate and " is a small amplitude parameter. water depths lower than a certain threshold. Our contribution is to extend this result. The nonlinear focusing mechanism of 2.2 Constant buoyancy frequency internal rogue waves is (i) determined by an estimation of the growth rate of modulation instability and (ii) elucidated by a For the simple case of constant buoyancy frequency N0, the numerical simulation of the emergence of rogue modes with formulations simplify considerably in the absence of shear Nat. Hazards Earth Syst. Sci., 19, 583–587, 2019 www.nat-hazards-earth-syst-sci.net/19/583/2019/ K. W. Chow et al.: Brief communication: Modulation instability of internal waves 585 Figure 1. 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. (a) For N0 D 2, h D 4, k D 0:5, n D 1, r D 0:2 and baseband instability growth rate D 0.868, a rogue wave emerges at τ ≈ 17. (b) For N0 D 2, h D 1, k D 0:5, n D 1, r D 0:2 and baseband instability growth rate D 0.193, a longer time is required for the emergence of a rogue wave in a shallower fluid (τ ≈ 55). (c) For N0 D 1, h D 4, k D 0:5, n D 1, r D 0:2 and baseband instability growth rate D 0.868, a rogue wave emerges at about the same time (τ ≈ 14) as compared to the case with a higher buoyancy frequency N0 D 2.
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