Bispectral Analysis of Energy Transfer Within the Two-Dimensional Oceanic Internal Wave Field

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 interaction, 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 equatorward 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 results 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.

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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 numerical 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-

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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. The red triangle at the lower left in each panel shows the spectral location of the lowest-vertical-wavenumber M2 internal tide. Numerals on the solid lines denote the wave period.

number space after the injection of the M2 internal tide namely, the rate of nonlinear energy transfer from the energy. Each spectrum is scaled by the unforced, freely lowest-vertical-wavenumber M2 tidal frequency to each decaying, reference spectrum. spectral location. The red circles in these figures denote At 28°, we can find spectral energy density gradually the spectral locations of internal waves satisfying the increasing with time at vertical wavelengths 50–1000 m condition for resonant interaction with the M2 internal and a period of about 24 h. In contrast, no significant tide; namely, enhancement of spectral energy density takes place at kЈ ± k؆ ϭ k 49° and 18°. IT ͮ ͑ ͒ ␻͑ Ј͒ ␻͑ Љ͒ ϭ ␻͑ ͒ , 3 Figures 2, 3, and 4 show the bispectra at 28°,49°, and k ± k kIT 18°, respectively, in the two-dimensional wavenumber where ␻(k) is a frequency for each wavenumber k space which is averaged over t ϭ 10–20 days. These (Phillips 1977; McComas 1977; McComas and Müller correspond to the sum of X and Y [Eqs. (1) and (2)], 1981).

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FIG. 2. The nonlinear energy transfer rate at 28°N. The energy FIG. 4. As in Fig. 2 but for the case at 18°N. in the cold-colored area is transferred from the lowest-vertical- wavenumber M2 internal tide, whereas the energy in the warm- colored area is transferred to the lowest-vertical-wavenumber M2 internal tide. The red circles denote the spectral locations of in- (the upper-right-hand corner of the 20-day panel of Fig. ternal waves satisfying the condition for resonant interaction with 1b) where direct energy transfer from the M2 internal the lowest-vertical-wavenumber M2 internal tide. Numerals on tide cannot be recognized in Fig. 2. This is consistent the solid lines denote the wave period. with the eikonal calculation by Watanabe and Hibiya (2005) showing that the development of high-vertical- wavenumber current shear is prerequisite to enhanced Of special importance is that the efficient energy cas- turbulent dissipation. cade to high vertical wavenumbers at 28° is associated Both at 49° and 18°, in contrast, no noticeable energy with the triad interaction satisfying cascade to high vertical wavenumbers is recognized in Խ ЈԽ Ϸ Խ ЉԽӷԽ Խ k k kIT the corresponding bispectra (Figs. 3 and 4). It is inter- ͮ, ͑4͒ ␻͑kЈ͒ Ϸ ␻͑kЉ͒ ≅ ␻͑k ͒ր2 esting to note that, although the resonant condition (4) IT can be satisfied equatorward of ϳ30°, PSI is not oper- which is the resonant condition for PSI. It is interesting ating at 18°. Actually, calculated time development of to note that spectral enhancement also occurs at small the squared 30-m vertical shear at various latitudes horizontal (Ͻ500 m) and vertical wavelengths (Ͻ50 m) (Fig. 5) indicates that the efficiency of PSI in transfer-

ring the low-vertical-wavenumber M2 internal tide energy to high-vertical-wavenumbers rapidly drops as the latitude falls below ϳ25°, although the definite physical explanation for this latitudinal dependence remains to be explored in the future. All the bispectra show that another type of triad interaction exists between the lowest-vertical-wavenumber

M2 internal tide and two nearly identical internal waves with horizontal wavelengths of 0.5–20 km and periods Ͻ4 h (Figs. 2, 3, and 4). That the frequency and wave-

number of the M2 internal tide are both lowest among ␻ Ј Ϸ ␻ Љ Ͼ ␻ | Ј| Ϸ | Љ| ӷ the triad members [ (k ) (k ) (kIT), k k | | kIT ] strongly suggests that this resonant interaction is induced diffusion (ID) (McComas and Bretherton 1977). However, the amount of energy drained from

the lowest-vertical-wavenumber M2 internal tide is an order of magnitude lower than that by PSI, indicating that this interaction plays only a minor role in cascading

FIG. 3. As in Fig. 2 but for the case at 49°N. the low-mode M2 internal tide energy.

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2 FIG. 5. Time development of the squared 30-m vertical shear (S30) at each latitude after the injection of the lowest-vertical-wavenumber M2 internal tide energy. Note that each value of 2 S30 is horizontally and vertically averaged.

4. Conclusions highly idealized way M2 internal tides are incorporated. Because of the dynamical constraint imposed by the We have carried out bispectral analysis of the nu- two-dimensionality, in particular, the present model in- merically reproduced spectral responses of the two- evitably lacks possible triad interactions in the deep dimensional oceanic internal wave field to the inci- ocean. Nevertheless, we believe that studying the two- dence of the low-mode M2 internal tide. dimensional problem is the useful first step in under- The most important result of the present study is that standing three-dimensional oceanic internal waves. the low-mode M2 internal tide at 28° interacts domi- In concluding the present study, it should be empha- nantly with two high-vertical-wavenumber diurnal sized that we are concerned here only with the cascade (near-inertial) internal waves, forming PSI triads. As of the low-mode M2 internal tide energy. At low lati- the energy level at high vertical wavenumbers is in- tudes, in particular, low-mode near-inertial internal creased by this PSI interaction, energy cascade to small waves originating from the winter storm track (Na- horizontal and vertical scales is significantly promoted gasawa et al. 2000) as well as low-mode diurnal internal (see Figs. 1b and 2). This is consistent with the eikonal tides become susceptible to PSI and cause diapycnal calculation by Watanabe and Hibiya (2005) showing mixing in the deep ocean. that the development of high-vertical-wavenumber shear is prerequisite to enhanced turbulent dissipa- Acknowledgments. The authors express their grati- tion. tude to two anonymous reviewers for their invaluable In contrast, no noticeable energy cascade to high ver- comments. The numerical experiments were carried tical wavenumbers has been recognized in the bispectra out using the Hitachi SR8000/128 and SR8000/MPP su- ϳ ϳ poleward of 30° as well as equatorward of 25°.A percomputers at the Information Technology Center of new finding is that, although PSI is possible equator- the University of Tokyo. ward of ϳ30°, the efficiency drops quite sharply as the ϳ latitude falls below 25°. The definite physical expla- REFERENCES nation for this latitudinal dependence is an intriguing work to be explored in the future. Bell, T. H., Jr., 1975: Topographically generated internal waves in open ocean. J. Geophys. Res., 80, 320–327. At all latitudes, another resonant interaction sugges- Bryan, F., 1987: Parameter sensitivity of primitive equation ocean tive of induced diffusion (ID) has been found between general circulation models. J. Phys. Oceanogr., 17, 970–985. the low-mode M2 internal tide and high-frequency in- D’Asaro, E. A., 1985: The energy flux from the wind to near-iner- ternal waves. The amount of energy drained from the tial motions in the surface mixed layer. J. Phys. Oceanogr., low-mode M internal tide is, however, an order of 15, 1043–1059. 2 ——, 1995: Upper-ocean inertial currents forced by a strong magnitude smaller than that by PSI, so this interaction storm. Part II: Modeling. J. Phys. Oceanogr., 25, 2937–2952. is thought to play only a minor role in cascading the ——, C. C. Eriksen, M. D. Levine, P. P. Niiler, C. A. Paulson, and low-mode M2 internal tide energy. P. Van Meurs, 1995: Upper-ocean inertial currents forced by Admittedly, there are some limitations with the a strong storm. Part I: Data and comparisons with linear present numerical approach. These include the calcula- theory. J. Phys. Oceanogr., 25, 2909–2936. Furue, R., 1998: Importance of local interactions within the small- tion domain being limited to a vertically two-dimen- scale oceanic internal wave spectrum for transferring energy sional plane, the background density stratification as- to dissipation scales: A three dimensional numerical study. sumed to be uniform over the full ocean depth, and the Ph.D. thesis, University of Tokyo, 112 pp.

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