2104 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35 Bispectral Analysis of Energy Transfer within the Two-Dimensional Oceanic Internal Wave Field NAOKI FURUICHI,TOSHIYUKI HIBIYA, AND YOSHIHIRO NIWA Department of Earth and Planetary Science, Graduate School of Science, The University of Tokyo, Tokyo, Japan (Manuscript received 23 November 2004, in final form 4 May 2005) ABSTRACT Bispectral analysis of the numerically reproduced spectral responses of the two-dimensional oceanic internal wave field to the incidence of the low-mode semidiurnal internal tide is performed. At latitudes just equatorward of 30°, the low-mode semidiurnal internal tide dominantly interacts with two high-vertical- wavenumber diurnal (near inertial) internal waves, forming resonant triads of parametric subharmonic instability (PSI) type. As the high-vertical-wavenumber near-inertial energy level is raised by this interac- tion, the energy cascade to small horizontal and vertical scales is enhanced. Bispectral analysis thus indicates that energy in the low-mode semidiurnal internal tide is not directly transferred to small scales but via the development of high-vertical-wavenumber near-inertial current shear. In contrast, no noticeable energy cascade to high vertical wavenumbers is recognized in the bispectra poleward of ϳ30° as well as equator- ward of ϳ25°. A new finding is that, although PSI is possible equatorward of ϳ30°, the efficiency drops sharply as the latitude falls below ϳ25°. At all latitudes, another resonant interaction suggestive of induced diffusion is found to occur between the low-mode semidiurnal internal tide and two high-frequency internal waves, although bispectral analysis shows that this interaction plays only a minor role in cascading the low-mode semidiurnal internal tide energy. 1. Introduction Hibiya et al. (1996, 1998, 2002) carried out a series of numerical experiments to investigate the responses Diapycnal mixing in the thermocline in the ocean of the vertically two-dimensional oceanic internal wave interior is believed to play an important role in main- spectrum to the forcing applied at low-vertical- taining meridional overturning circulation (Bryan wavenumber M2 tidal frequency. Their numerical ex- 1987). The energy to drive the ocean interior mixing is periments showed that the cascade of low-mode M2 originally supplied at low vertical wavenumbers by internal tide energy to small scales is strongly linked tide–topography interactions (Bell 1975; Hibiya 1986, with the enhancement of high-vertical-wavenumber 1988, 1990; Matsuura and Hibiya 1990; Morozov 1995; current shear. Furthermore, it was shown that, even if Merrifield and Holloway 2002; Niwa and Hibiya 2001, similar amount of low-mode M internal tide energy is 2004; Ray and Cartwright 2001; St. Laurent and Garrett 2 supplied, the enhancement of high-vertical-wavenumber 2002) as well as wind stress fluctuations (Gill 1984; current shear occurs only at latitudes equatorward of Greatbatch 1984; D’Asaro 1985, 1995; Kundu 1993; 30°. The latitudinal dependence of the intensity of high- D’Asaro et al. 1995; Nilsson 1995; Niwa and Hibiya vertical-wavenumber current shear was confirmed in 1997, 1999; Nagasawa et al. 2000; Watanabe and Hibiya the real ocean by expendable current profilers (XCP) 2002) that are then transferred across the deep ocean deployed over a wide area in the North Pacific (Na- internal wave spectrum to dissipation scales by nonlin- gasawa et al. 2002; Hibiya and Nagasawa 2004). These ear wave interactions. authors concluded that the calculated and observed re- sults can be reasonably explained if the energy cascade is dominated by parametric subharmonic instability Corresponding author address: Naoki Furuichi, Department of (PSI) (McComas 1977; McComas and Bretherton 1977; Earth and Planetary Science, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, McComas and Müller 1981) that transfers energy from Japan. the low-vertical-wavenumber M2 tidal frequency to E-mail: [email protected] high-vertical-wavenumber near-inertial frequency. © 2005 American Meteorological Society Unauthenticated | Downloaded 09/29/21 02:48 PM UTC JPO2816 NOVEMBER 2005 F URUICHI ET AL. 2105 In the present study, using the two-dimensional spec- internal wave field at each latitude by calculating non- tral model, we first demonstrate the responses of the linear interactions over 10 days among randomly vertically two-dimensional oceanic internal wave field phased linear internal waves of horizontal modes m ϭ ϭ to the incidence of the low-mode M2 internal tide at 0–512 and vertical modes n 1–128 with the ampli- various latitudes from 13° to 49° (Table 1). Based on tudes prescribed using the Garrett–Munk empirical the bispectral analysis of the calculated results, we next model (Garrett and Munk 1972, 1975; Munk 1981). examine the triad interaction responsible for the lati- Then, the kinetic and potential energy at m ϭ n ϭ 1is tude-dependent energy cascade of the low-mode M2 increased instantaneously so as to introduce a 15-m am- internal tide. plitude lowest-vertical-wavenumber M2 internal tide. Thereafter, while maintaining the levels of the kinetic ϭ ϭ 2. Numerical model and bispectral analysis and potential energy at m n 1, we run the numeri- cal model for another 20 days. To model the disparate scale wave interactions under In this case, the nonlinear interaction between the the constraint of computer capacity, we restrict our at- lowest-vertical-wavenumber M2 internal tide (wave- tention to wave motions in a vertically two-dimensional number kIT) and the background internal waves (wave- plane by requiring the variability to be independent of number kЈ and kЉ) is accompanied by a kinetic energy one horizontal direction. We use the Navier–Stokes exchange equations under the Boussinesq approximation, conti- 1 L 0 ͔ ١͒ nuity equation and mass conservation equation. To ͵ ͵ и ͓͑ и vIT v v dx dz maximize the range of scales free from numerical dif- LD 0 ϪD fusive–dissipative effects while maintaining the numeri- ϭ ͑ Ј Љ͒␦͑ Ј Љ͒ ͑ ͒ cal stability, we employ hyperviscosity schemes for sub- ͚͚X kIT, k , k kIT ± k ± k 1 kЈ kЉ grid-scale parameterization (Shen and Holloway 1986; Winters and D’Asaro 1997). and a potential energy exchange The model employs cyclic boundary conditions at the g2 1 L 0 ١͒␳͔ ␳ и ͓͑ и ͵ ͵ sidewalls and reflecting boundary conditions at the sur- 2 2 IT v dx dz ␳ N LD Ϫ face and bottom. A constant background buoyancy fre- 0 0 D Ϫ3 Ϫ1 quency N ϭ 3.49 ϫ 10 s (buoyancy period is 30 ϭ ͑ Ј Љ͒␦͑ Ј Љ͒ ͑ ͒ ͚͚Y kIT, k , k kIT ± k ± k , 2 min) is assumed. Basic equations are time advanced kЈ kЉ using a leapfrog scheme with a time step of 30 s. where x and z are the horizontal and vertical coordi- The vertical size of the numerical model is assumed nates; L and D are the horizontal and vertical sizes of to be 2.6 km for all of the experiments, whereas the the numerical model; g is the acceleration due to grav- horizontal size of the model is determined so as to cor- ␳ ity, 0 is a reference density, N is the buoyancy fre- respond to one horizontal wavelength of the lowest- quency; v and ␳ are the perturbations of velocity and ␻ ϭ ϫ Ϫ4 Ϫ1 vertical-wavenumber M2 ( 1.41 10 s ) internal ␳ density, respectively; vIT and IT are the corresponding tide at each latitude between 13° and 49° (Table 1); ␦ values for the M2 internal tide; is the Kronecker delta; 4096 and 512 grid points are used in the horizontal and Ј Љ Ј Љ and X(kIT, k , k ) and Y(kIT, k , k ) are bispectra of vertical directions, respectively. kinetic energy and potential energy, respectively, rep- First, we reproduce the quasi-stationary background resenting the rate at which the M2 internal tide gains or loses energy through the nonlinear interactions with the TABLE 1. List of experiments. background internal waves [for the details about the derivation of bispectra, see Furue (1998)]. Bispectral Inertial frequency Horizontal size of analysis was used in earlier studies to examine nonlin- Latitude (inertial period) the model (km) ear energy transfers within the oceanic internal wave Ϫ Ϫ 49°N 1.09 ϫ 10 4 s 1 (15.9 h) 203 field (McComas and Briscoe 1980; Lin et al. 1995; Niwa ϫ Ϫ5 Ϫ1 40°N 9.35 10 s (18.7 h) 170 and Hibiya1997; Furue 1998, 2003). It should be noted 33°N 7.92 ϫ 10Ϫ5 s Ϫ1 (22.0 h) 154 30°N 7.27 ϫ 10Ϫ5 s Ϫ1 (24.0 h) 149 that not only resonant interactions but all kinds of triad 28°N 6.83 ϫ 10Ϫ5 s Ϫ1 (25.6 h) 145 interactions are taken into account in Eqs. (1) and (2). 26°N 6.38 ϫ 10Ϫ5 s Ϫ1 (27.4 h) 142 24°N 5.91 ϫ 10Ϫ5 s Ϫ1 (29.5 h) 140 Ϫ Ϫ 3. Results and discussion 21°N 5.21 ϫ 10 5 s 1 (33.5 h) 136 Ϫ5 Ϫ1 18°N 4.49 ϫ 10 s (38.3 h) 134 Figure 1 shows time variations of the energy spec- ϫ Ϫ5 Ϫ1 13°N 3.27 10 s (53.3 h) 131 trum at 49°,28°, and 18° in the two-dimensional wave- Unauthenticated | Downloaded 09/29/21 02:48 PM UTC 2106 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35 FIG. 1. Time variations of the energy spectrum at (a) 49°, (b) 28°, and (c) 18°N in the two-dimensional wavenumber space after the injection of the lowest-vertical-wavenumber M2 internal tide energy. Note that each spectrum is scaled by the unforced, freely decaying, reference spectrum.