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

Transcription

1 2104 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y 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-verticalwavenumber 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 30 as well as equatorward of 25. A new finding is that, although PSI is possible equatorward of 30, the efficiency drops sharply as the latitude falls below 25. 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 Diapycnal mixing in the thermocline in the ocean interior is believed to play an important role in maintaining meridional overturning circulation (Bryan 1987). The energy to drive the ocean interior mixing is originally supplied at low vertical wavenumbers by tide topography interactions (Bell 1975; Hibiya 1986, 1988, 1990; Matsuura and Hibiya 1990; Morozov 1995; Merrifield and Holloway 2002; Niwa and Hibiya 2001, 2004; Ray and Cartwright 2001; St. Laurent and Garrett 2002) as well as wind stress fluctuations (Gill 1984; Greatbatch 1984; D Asaro 1985, 1995; Kundu 1993; D Asaro et al. 1995; Nilsson 1995; Niwa and Hibiya 1997, 1999; Nagasawa et al. 2000; Watanabe and Hibiya 2002) that are then transferred across the deep ocean internal wave spectrum to dissipation scales by nonlinear wave interactions. Corresponding author address: Naoki Furuichi, Department of Earth and Planetary Science, Graduate School of Science, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo , Japan. furu1@eps.s.u-tokyo.ac.jp Hibiya et al. (1996, 1998, 2002) carried out a series of numerical experiments to investigate the responses of the vertically two-dimensional oceanic internal wave spectrum to the forcing applied at low-verticalwavenumber M 2 tidal frequency. Their numerical experiments showed that the cascade of low-mode M 2 internal tide energy to small scales is strongly linked with the enhancement of high-vertical-wavenumber current shear. Furthermore, it was shown that, even if similar amount of low-mode M 2 internal tide energy is supplied, the enhancement of high-vertical-wavenumber current shear occurs only at latitudes equatorward of 30. The latitudinal dependence of the intensity of highvertical-wavenumber current shear was confirmed in the real ocean by expendable current profilers (XCP) deployed over a wide area in the North Pacific (Nagasawa et al. 2002; Hibiya and Nagasawa 2004). These authors concluded that the calculated and observed results can be reasonably explained if the energy cascade is dominated by parametric subharmonic instability (PSI) (McComas 1977; McComas and Bretherton 1977; McComas and Müller 1981) that transfers energy from the low-vertical-wavenumber M 2 tidal frequency to high-vertical-wavenumber near-inertial frequency American Meteorological Society

2 NOVEMBER 2005 F URUICHI ET AL In the present study, using the two-dimensional spectral model, we first demonstrate the responses of the vertically two-dimensional oceanic internal wave field to the incidence of the low-mode M 2 internal tide at various latitudes from 13 to 49 (Table 1). Based on the bispectral analysis of the calculated results, we next examine the triad interaction responsible for the latitude-dependent energy cascade of the low-mode M 2 internal tide. 2. Numerical model and bispectral analysis To model the disparate scale wave interactions under the constraint of computer capacity, we restrict our attention to wave motions in a vertically two-dimensional plane by requiring the variability to be independent of one horizontal direction. We use the Navier Stokes equations under the Boussinesq approximation, continuity equation and mass conservation equation. To maximize the range of scales free from numerical diffusive dissipative effects while maintaining the numerical stability, we employ hyperviscosity schemes for subgrid-scale parameterization (Shen and Holloway 1986; Winters and D Asaro 1997). The model employs cyclic boundary conditions at the sidewalls and reflecting boundary conditions at the surface and bottom. A constant background buoyancy frequency N s 1 (buoyancy period is 30 min) is assumed. Basic equations are time advanced using a leapfrog scheme with a time step of 30 s. The vertical size of the numerical model is assumed to be 2.6 km for all of the experiments, whereas the horizontal size of the model is determined so as to correspond to one horizontal wavelength of the lowestvertical-wavenumber M 2 ( s 1 ) internal tide at each latitude between 13 and 49 (Table 1); 4096 and 512 grid points are used in the horizontal and vertical directions, respectively. First, we reproduce the quasi-stationary background Latitude TABLE 1. List of experiments. Inertial frequency (inertial period) Horizontal size of the model (km) 49 N s 1 (15.9 h) N s 1 (18.7 h) N s 1 (22.0 h) N s 1 (24.0 h) N s 1 (25.6 h) N s 1 (27.4 h) N s 1 (29.5 h) N s 1 (33.5 h) N s 1 (38.3 h) N s 1 (53.3 h) 131 internal wave field at each latitude by calculating nonlinear interactions over 10 days among randomly phased linear internal waves of horizontal modes m and vertical modes n with the amplitudes prescribed using the Garrett Munk empirical model (Garrett and Munk 1972, 1975; Munk 1981). Then, the kinetic and potential energy at m n 1is increased instantaneously so as to introduce a 15-m amplitude lowest-vertical-wavenumber M 2 internal tide. Thereafter, while maintaining the levels of the kinetic and potential energy at m n 1, we run the numerical model for another 20 days. In this case, the nonlinear interaction between the lowest-vertical-wavenumber M 2 internal tide (wavenumber k IT ) and the background internal waves (wavenumber k and k ) is accompanied by a kinetic energy exchange 1 LD 0 L 0 D v IT v v dx dz k k X k IT, k, k k IT ± k ± k and a potential energy exchange g 2 L N 2 LD 0 0 D IT v dx dz k k Y k IT, k, k k IT ± k ± k, where x and z are the horizontal and vertical coordinates; L and D are the horizontal and vertical sizes of the numerical model; g is the acceleration due to gravity, 0 is a reference density, N is the buoyancy frequency; v and are the perturbations of velocity and density, respectively; v IT and IT are the corresponding values for the M 2 internal tide; is the Kronecker delta; and X(k IT, k, k ) and Y(k IT, k, k ) are bispectra of kinetic energy and potential energy, respectively, representing the rate at which the M 2 internal tide gains or loses energy through the nonlinear interactions with the background internal waves [for the details about the derivation of bispectra, see Furue (1998)]. Bispectral analysis was used in earlier studies to examine nonlinear energy transfers within the oceanic internal wave field (McComas and Briscoe 1980; Lin et al. 1995; Niwa and Hibiya1997; Furue 1998, 2003). It should be noted that not only resonant interactions but all kinds of triad interactions are taken into account in Eqs. (1) and (2). 3. Results and discussion Figure 1 shows time variations of the energy spectrum at 49, 28, and 18 in the two-dimensional wave- 1 2

3 2106 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y 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 M 2 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 M 2 internal tide. Numerals on the solid lines denote the wave period. number space after the injection of the M 2 internal tide energy. Each spectrum is scaled by the unforced, freely decaying, reference spectrum. At 28, we can find spectral energy density gradually increasing with time at vertical wavelengths m and a period of about 24 h. In contrast, no significant enhancement of spectral energy density takes place at 49 and 18. Figures 2, 3, and 4 show the bispectra at 28, 49, and 18, respectively, in the two-dimensional wavenumber space which is averaged over t days. These correspond to the sum of X and Y [Eqs. (1) and (2)], namely, the rate of nonlinear energy transfer from the lowest-vertical-wavenumber M 2 tidal frequency to each spectral location. The red circles in these figures denote the spectral locations of internal waves satisfying the condition for resonant interaction with the M 2 internal tide; namely, k ± k k IT 3 k ± k k IT, where (k) is a frequency for each wavenumber k (Phillips 1977; McComas 1977; McComas and Müller 1981).

4 NOVEMBER 2005 F U R U I C H I E T A L FIG. 2. The nonlinear energy transfer rate at 28 N. The energy in the cold-colored area is transferred from the lowest-verticalwavenumber M 2 internal tide, whereas the energy in the warmcolored area is transferred to the lowest-vertical-wavenumber M 2 internal tide. The red circles denote the spectral locations of internal waves satisfying the condition for resonant interaction with the lowest-vertical-wavenumber M 2 internal tide. Numerals on the solid lines denote the wave period. Of special importance is that the efficient energy cascade to high vertical wavenumbers at 28 is associated with the triad interaction satisfying k k k IT 4 k k k IT 2, which is the resonant condition for PSI. It is interesting to note that spectral enhancement also occurs at small horizontal ( 500 m) and vertical wavelengths ( 50 m) FIG. 3. As in Fig. 2 but for the case at 49 N. FIG. 4. As in Fig. 2 but for the case at 18 N. (the upper-right-hand corner of the 20-day panel of Fig. 1b) where direct energy transfer from the M 2 internal tide cannot be recognized in Fig. 2. This is consistent with the eikonal calculation by Watanabe and Hibiya (2005) showing that the development of high-verticalwavenumber current shear is prerequisite to enhanced turbulent dissipation. Both at 49 and 18, in contrast, no noticeable energy cascade to high vertical wavenumbers is recognized in the corresponding bispectra (Figs. 3 and 4). It is interesting to note that, although the resonant condition (4) can be satisfied equatorward of 30, PSI is not operating at 18. Actually, calculated time development of the squared 30-m vertical shear at various latitudes (Fig. 5) indicates that the efficiency of PSI in transferring the low-vertical-wavenumber M 2 internal tide energy to high-vertical-wavenumbers rapidly drops as the latitude falls below 25, 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 M 2 internal tide and two nearly identical internal waves with horizontal wavelengths of km and periods 4 h (Figs. 2, 3, and 4). That the frequency and wavenumber of the M 2 internal tide are both lowest among the triad members [ (k ) (k ) (k IT ), k k k IT ] 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 M 2 internal tide is an order of magnitude lower than that by PSI, indicating that this interaction plays only a minor role in cascading the low-mode M 2 internal tide energy.

5 2108 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 35 FIG. 5. Time development of the squared 30-m vertical shear (S 2 30 ) at each latitude after the injection of the lowest-vertical-wavenumber M 2 internal tide energy. Note that each value of S 2 30 is horizontally and vertically averaged. 4. Conclusions We have carried out bispectral analysis of the numerically reproduced spectral responses of the twodimensional oceanic internal wave field to the incidence of the low-mode M 2 internal tide. The most important result of the present study is that the low-mode M 2 internal tide at 28 interacts dominantly with two high-vertical-wavenumber diurnal (near-inertial) internal waves, forming PSI triads. As the energy level at high vertical wavenumbers is increased by this PSI interaction, energy cascade to small horizontal and vertical scales is significantly promoted (see Figs. 1b and 2). This is consistent with the eikonal calculation by Watanabe and Hibiya (2005) showing that the development of high-vertical-wavenumber shear is prerequisite to enhanced turbulent dissipation. In contrast, no noticeable energy cascade to high vertical wavenumbers has been recognized in the bispectra poleward of 30 as well as equatorward of 25. A new finding is that, although PSI is possible equatorward of 30, the efficiency drops quite sharply as the latitude falls below 25. The definite physical explanation for this latitudinal dependence is an intriguing work to be explored in the future. At all latitudes, another resonant interaction suggestive of induced diffusion (ID) has been found between the low-mode M 2 internal tide and high-frequency internal waves. The amount of energy drained from the low-mode M 2 internal tide is, however, an order of magnitude smaller than that by PSI, so this interaction is thought to play only a minor role in cascading the low-mode M 2 internal tide energy. Admittedly, there are some limitations with the present numerical approach. These include the calculation domain being limited to a vertically two-dimensional plane, the background density stratification assumed to be uniform over the full ocean depth, and the highly idealized way M 2 internal tides are incorporated. Because of the dynamical constraint imposed by the two-dimensionality, in particular, the present model inevitably lacks possible triad interactions in the deep ocean. Nevertheless, we believe that studying the twodimensional problem is the useful first step in understanding three-dimensional oceanic internal waves. In concluding the present study, it should be emphasized that we are concerned here only with the cascade of the low-mode M 2 internal tide energy. At low latitudes, in particular, low-mode near-inertial internal waves originating from the winter storm track (Nagasawa et al. 2000) as well as low-mode diurnal internal tides become susceptible to PSI and cause diapycnal mixing in the deep ocean. Acknowledgments. The authors express their gratitude to two anonymous reviewers for their invaluable comments. The numerical experiments were carried out using the Hitachi SR8000/128 and SR8000/MPP supercomputers at the Information Technology Center of the University of Tokyo. REFERENCES Bell, T. H., Jr., 1975: Topographically generated internal waves in open ocean. J. Geophys. Res., 80, Bryan, F., 1987: Parameter sensitivity of primitive equation ocean general circulation models. J. Phys. Oceanogr., 17, D Asaro, E. A., 1985: The energy flux from the wind to near-inertial motions in the surface mixed layer. J. Phys. Oceanogr., 15, , 1995: Upper-ocean inertial currents forced by a strong storm. Part II: Modeling. J. Phys. Oceanogr., 25, , C. C. Eriksen, M. D. Levine, P. P. Niiler, C. A. Paulson, and P. Van Meurs, 1995: Upper-ocean inertial currents forced by a strong storm. Part I: Data and comparisons with linear theory. J. Phys. Oceanogr., 25, Furue, R., 1998: Importance of local interactions within the smallscale oceanic internal wave spectrum for transferring energy to dissipation scales: A three dimensional numerical study. Ph.D. thesis, University of Tokyo, 112 pp.

6 NOVEMBER 2005 F U R U I C H I E T A L. 2109, 2003: Energy transfer within the small-scale oceanic internal wave spectrum. J. Phys. Oceanogr., 33, Garrett, C. J. R., and W. H. Munk, 1972: Space time scales of internal waves. Geophys. Fluid Dyn., 2, , and, 1975: Space time scales of internal waves: A progress report. J. Geophys. Res., 80, Gill, A. E., 1984: On the behavior of internal waves in the wakes of storms. J. Phys. Oceanogr., 14, Greatbatch, R. J., 1984: On the response of the ocean to a moving storm: Parameters and scales. J. Phys. Oceanogr., 14, Hibiya, T., 1986: Generation mechanism of internal waves by tidal flow over a sill. J. Geophys. Res., 91, , 1988: The generation of internal waves by tidal flow over Stellwagen Bank. J. Geophys. Res., 93, , 1990: Generation mechanism of internal waves by a vertically sheared tidal flow over a sill. J. Geophys. Res., 95, , and M. Nagasawa, 2004: Latitudinal dependence of diapycnal diffusivity in the thermocline estimated using a finescale parameterization. Geophys. Res. Lett., 31, L01301, doi: /2003gl , Y. Niwa, K. Nakajima, and N. Suginohara, 1996: Direct numerical simulation of the roll-off range of internal wave shear spectra in the ocean. J. Geophys. Res., 101, ,, and K. Fujiwara, 1998: Numerical experiments of nonlinear energy transfer within the oceanic internal wave spectrum. J. Geophys. Res., 103, , M. Nagasawa, and Y. Niwa, 2002: Nonlinear energy transfer within the oceanic internal wave spectrum at mid and high latitudes. J. Geophys. Res., 107, 3207, doi: /2001jc Kundu, P. K., 1993: On internal waves generated by traveling wind. J. Fluid Mech., 254, Lin, C.-L., J. R. Koseff, and J. H. Ferziger, 1995: On triad interactions in a linearly stratified ocean. J. Phys. Oceanogr., 25, Matsuura, T., and T. Hibiya, 1990: An experimental and numerical study of the internal wave generation by tide-topography interaction. J. Phys. Oceanogr., 20, McComas, C. H., 1977: Equilibrium mechanisms within oceanic internal wave field. J. Phys. Oceanogr., 7, , and F. P. Bretherton, 1977: Resonant interaction of oceanic internal waves. J. Geophys. Res., 82, , and M. G. Briscoe, 1980: Bispectra of internal waves. J. Fluid Mech., 97, , and P. Müller, 1981: The dynamic balance of internal waves. J. Phys. Oceanogr., 11, Merrifield, M. A., and P. E. Holloway, 2002: Model estimates of M 2 internal tide energetics at the Hawaiian ridge. J. Geophys. Res., 107, 3179, doi: /2001jc Morozov, E. G., 1995: Semidiurnal internal wave global field. Deep-Sea Res., 42A, Munk, W., 1981: Internal waves and small-scale processes. Evolution of Physical Oceanography, B. A. Warren and C. Wunsch, Eds., The MIT Press, Nagasawa, M., Y. Niwa, and T. Hibiya, 2000: Spatial and temporal distribution of the wind-induced internal wave energy available for deep water mixing in the North Pacific. J. Geophys. Res., 105, , T. Hibiya, Y. Niwa, M. Watanabe, Y. Isoda, S. Takagi, and Y. Kamei, 2002: Distribution of fine-scale shear in the deep waters of the North Pacific obtained using expendable current profilers. J. Geophys. Res., 107, 3221, doi: / 2002JC Nilsson, J., 1995: Energy flux from traveling hurricanes to the oceanic internal wave field. J. Phys. Oceanogr., 25, Niwa, Y., and T. Hibiya, 1997: Nonlinear processes of energy transfer from traveling hurricanes to the deep ocean internal wave field. J. Geophys. Res., 102, , and, 1999: Response of the deep ocean internal wave field to traveling midlatitude storms as observed in long term current measurements. J. Geophys. Res., 104, , and, 2001: Numerical study of the spatial distribution of the M 2 internal tide in the Pacific Ocean. J. Geophys. Res., 106, , and, 2004: Three-dimensional numerical simulation of M 2 internal tides in the East China Sea. J. Geophys. Res., 109, C04027, doi: /2003jc Phillips, O. M., 1977: The Dynamics of the Upper Ocean. Cambridge University Press, 336 pp. Ray, R. D., and D. E. Cartwright, 2001: Estimates of internal tide energy fluxes from Topex/Poseidon altimetry: Central North Pacific. Geophys. Res. Lett., 28, Shen, C. Y., and G. Holloway, 1986: A numerical study of the frequency and the energetics of nonlinear internal gravity waves. J. Geophys. Res., 91, St. Laurent, L., and C. Garrett, 2002: The role of internal tides in mixing the deep ocean. J. Phys. Oceanogr., 32, Watanabe, M., and T. Hibiya, 2002: Global estimates of the windinduced energy flux to inertial motions in the surface mixed layer. Geophys. Res. Lett., 29, 1239, doi: /2001gl , and, 2005: Estimates of energy dissipation rates in the three-dimensional deep ocean internal wave field. J. Oceanogr., 61, Winters, K. B., and E. A. D Asaro, 1997: Direct simulation of internal wave energy transfer. J. Phys. Oceanogr., 27,