A Nonlinear Model of the Shoaling and Refraction of Interfacial Solitary Waves in the Ocean
Total Page：16
File Type：pdf, Size：1020Kb
3184 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 A Nonlinear Model of the Shoaling and Refraction of Interfacial Solitary Waves in the Ocean. Part II: Oblique Refraction across a Continental Slope and Propagation over a Seamount JUSTIN SMALL Defence Evaluation Research Agency, Winfrith Technology Centre, Dorchester, United Kingdom (Manuscript received 28 August 2000, in ®nal form 21 March 2001) ABSTRACT The refraction of internal solitary waves can be due to variations in water depth, nonlinearity, strati®cation, or shear. This paper addresses the effects of the ®rst two of these effects. Part I of this paper investigated how the amplitude and phase speed of an interfacial solitary wave varied as the wave propagated across a one- dimensional slope. Important differences between the predictions of Korteweg±de Vries (KdV) and extended KdV (EKdV) theory were found. In particular the EKdV limited the amplitude growth seen in KdV as the wave passed into shallow water, leading to an associated strong retardation in speed. In this second part of the paper these results are applied to two-dimensional bathymetry. The propagation of an oblique planar wave across a wedge-shaped slope is ®rst studied. Whereas the linear internal-wave theory predicts small refraction of the wave front, the nonlinear EKdV model shows that the refraction of large amplitude internal waves is signi®cantly greater. This is explained by the stronger retardation seen in the shoaling experiments. Second, the propagation of a planar internal wave across a seamount is investigated. Here signi®cant differences are found between the KdV and EKdV theory. Linear theory and nonlinear EKdV theory predict a focusing of rays past the seamount, and an eventual caustic. The EKdV predicts that the focusing occurs at shorter ranges than in the linear case, with the range dependent on the wave amplitude. In contrast KdV theory predicts that the focusing and consequent amplitude increase leads to such an increase in speed that the rays start to spread again and the caustic is avoided. It is concluded that nonlinearity has a signi®cant effect on the refraction of interfacial solitary waves. Further, as the strongest refraction occurs in water depths where the EKdV appears to be the more appropriate theory, the EKdV results are considered to be more reliable. 1. Introduction: The refraction of linear and (where H is the water depth for the surface wave case nonlinear waves and h1 and h2 are the upper and lower layer thicknesses in the two-layer case, Dr is the density difference across The oblique incidence of internal wave rays onto the interface, r is the reference density, and g the ac- varying bathymetry may cause refraction of the wave 0 celeration due to gravity). The Boussinesq approxima- fronts, due to the variation of phase speed along the tion, of differences between the layer densities being only wave front. As with surface waves, the internal wave important when multiplied by g, has been assumed here phase speed is dependent on depth (as well as strati®- and will be used because in many open ocean situations cation and current shear) as shown below for the linear the density difference between layers is, in fact, quite case. For both surface waves, and internal waves on an small [O(0.001) of the density value]. The assumption interface whose mean depth does not change with range, of long waves is equivalent to that of shallow water phase speed decreases with decreasing depth. This can be seen simply from the linear phase speed expression waves: that is, the wavelength is signi®cantly longer than the water depth. Clearly c decreases with decreasing H for long surface waves (c ), and long two-layer internal s s while c decreases with decreasing h (assuming h re- waves (c ) (assuming a rigid lid): 0 2 1 0 mains fairly constant), and hence with water depth. g9h Dr For long linear waves the refraction of the wave obeys c 5 ÏgH c 5 1 g95g (1) Snell's law: s 0 1 h /h ! 1 12r 0 sinfrefc ref 5 0 , (2) in in Corresponding author address: Dr. Justin Small, International Pa- sinf c0 ci®c Research Center, School of Ocean and Earth Science and Tech- in ref nology, University of Hawaii at Manoa, 2525 Correa Rd., Honolulu, where f and f refer to the incident and refracted HI 96822. angles, respectively, between the wave direction and the E-mail: [email protected] gradient of bathymetry (Pelinovsky et al. 1994), and Unauthenticated | Downloaded 10/06/21 01:56 PM UTC NOVEMBER 2001 SMALL 3185 TABLE 1. De®nition of acronyms. As the phase speed is dependent on amplitude for DERA Defence Evaluation Research Agency both ®rst- and second order theory, it is clear that the DR Djordevic and Redekopp (1978) change of amplitude across varying bathymetry, the cKdV Cylindrical KdV shoaling effect, is important to the refraction of nonlin- EKdV Extended Korteweg±de Vries ear waves. In Part I the shoaling of solitary waves as JKKD Intermediate long-wave theory of Joseph (1977) they passed on/offshelf over one-dimensional bathym- and Kubota et al. (1978) KdV Korteweg±de Vries etry was examined. Comparisons were made with an- MB Michallat and Bartholemey (1998) alytical results for adiabatic propagation over a slowly NRL Naval Research Laboratory changing slope, and with ®ssion laws for rapidly chang- SAR Synthetic aperture radar ing slopes. The agreement was good. Both ®rst-order SES Shelf Edge Study, a study of oceanographic pro- (vKdV) and extended (vEKdV) nonlinear evolution cesses at the Malin shelf SESAME SES Acoustic Measurement Experiment, a DERA/ models were tested. In the adiabatic KdV case, the sol- NRL study in conjunction with SES itary wave slowly increases in amplitude but retains the TVB Turner and Vanden-Broek (1988) solitary waveform (except for the formation of a long, vEKdV Variable coef®cient EKdV small amplitude tail or ``shelf'' behind it). The phase speed slowly reduces with depth. In contrast, when the bathymetry is rapidly changing, the soliton is seen to in ref split into a number of waves, a process known as soliton cc00and are the corresponding phase speeds of the incident and refracted wave, dependent only on water ®ssion, and the lead wave amplitude is reduced relative depth (assuming the top layer thickness does not to the adiabatic case. change). EKdV dynamics predicted rather different results: un- In this paper the extent by which the linear refraction der adiabatic conditions the solitary wave initially grows effects are modi®ed by the introduction of nonlinear but then in shallow water the amplitude is reduced. This terms is investigated. For instance, as the wave approach- is due to a theoretical upper limit on EKdV amplitude, es shallow water it may shoal and grow in amplitude. given by hc[(27) of Part I], namely that the wave cannot According to ®rst-order nonlinear theory the phase speed depress the interface beyond middepth. This also leads of a wave should increase with increasing amplitude, as to a rapid reduction in wave phase speed. In water of rapidly changing depth, the ®ssion effect of KdV is not follows [Ostrovsky and Stepanyants (1989); see appendix seen, as the shape of the EKdV wave does not drastically B of Part I (Small 2001) of this paper]: change between deep and shallow water (in contrast to ha the signi®cant narrowing of the KdV wave). Offshelf c 5 c 1 0 , (3) 0 3 propagation was also examined. In most cases this lead to the reverse of the onshelf effect, but in cases where where a is a function of the density distribution and h 0 the initial wave was close to the theoretical limit hc, is the amplitude, which leads to the question of whether the wave was so large that the bathymetric slope was this amplitude-dependent effect will cancel the depth- often steeper than the wave, and hence ®ssion of the dependent effect. KdV theory is believed to be relevant wave occurred. for small amplitude waves in many two-layer scenarios From these one-dimensional results it can be seen that (see, e.g., Michallet and Bartholemey 1998, hereafter nonlinearity not only increases the phase speed of the MB); see Table 1 for de®nition of acronyms. internal waves relative to the linear case, but it can also When the wave amplitudes become so large that sec- lead to stronger retardation of the wave (i.e., the ratio ond-order effects become important, a new relationship of phase speed in shallow water to that in deep water between phase speed and amplitude occurs, which was is reduced). This is likely to lead to changes in the two- dealt with explicitly in Part I (appendix B) and which dimensional refractive properties of the waves, as will may be summarized as be discussed in the remainder of this paper, organized as follows. Section 2 recaps on the model developed in c 5 c001 f(h ), (4) Part I and presents the ray spreading algorithm. Section where f is a function of amplitude and density distri- 3 investigates the oblique incidence of internal waves bution and increases with increasing absolute amplitude. on a wedge-shaped continental slope. Section 4 then applies the model to simulate refraction of a planar wave Fully second-order theories exist (see, e.g., Koop and over a seamount. A detailed discussion of the merits of Butler 1981) but in this paper the partially second-order the theoretical basis of the model is presented in the extended KdV (EKdV theory) is used (Holloway et al. discussion, section 5. Section 6 then presents the sum- 1999).