Model-predicted distribution of wind-induced internal wave energy in the world s oceans

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2008jc004768, 2008 Model-predicted distribution of wind-induced internal wave energy in the world s oceans Naoki Furuichi, 1 Toshiyuki Hibiya, 1 and Yoshihiro Niwa 1 Received 12 February 2008; revised 17 June 2008; accepted 9 July 2008; published 30 September [1] The distribution of wind-induced internal wave energy in the world s oceans is investigated using a full three-dimensional primitive equation model. Special attention is directed to the global energy input to the surface near-inertial motions and the subsequent downward energy propagation into the deep ocean. We find that the model results for near-inertial energy in the oceanic mixed layer, depth-integrated horizontal energy fluxes, and vertical structures of WKB-scaled kinetic energy are all consistent with the available observations in the regions of significant wind energy input and that the annual mean of the global wind energy input becomes 0.4 TW. It is also found that most of the wind-induced energy resides in high vertical modes, 75 85% of which is dissipated in the surface 150 m. The present study therefore predicts that the total wind-induced near-inertial energy available for deep-ocean mixing is limited to, at most, 0.1 TW, which is an order of magnitude smaller than previously estimated. Adding the energy flux from tide-topography interactions of 0.9 TW, we can conclude that the total energy available for deep-ocean mixing is 1.0 TW, obviously falling short of the required power to sustain the global overturning circulation. This might suggest the existence of other important energy sources, such as the one through geostrophic adjustment processes, and/or additional mechanisms sustaining the global overturning circulation, such as effects of Ekman upwelling in the Southern Ocean. Another possibility is that previous estimates of the volume transport of the global overturning circulation might be too large. Citation: Furuichi, N., T. Hibiya, and Y. Niwa (2008), Model-predicted distribution of wind-induced internal wave energy in the world s oceans, J. Geophys. Res., 113,, doi: /2008jc Introduction [2] Diapycnal mixing processes in the thermocline in the ocean interior are believed to play an important role in sustaining the global thermohaline circulation that controls the climate over much of the Earth [Gargett, 1984; Bryan, 1987]. Munk [1966] and Munk and Wunsch [1998] called for the average value of diapycnal diffusivity of 10 4 m 2 s 1 to sustain the global thermohaline circulation with the volume transport of Sv (1 Sv = 10 6 m 3 s 1 ). Furthermore, Munk and Wunsch [1998] found that the required power to create diapycnal diffusivity of 10 4 m 2 s 1 was 2.1 TW (1 TW = W), which might be available from tidetopography interactions and wind stress fluctuations. [3] The global energy conversion rate from the surface to internal tide was estimated to be 0.9 TW on the basis of the altimetric observations [Egbert and Ray, 2000] and numerical experiments [Niwa and Hibiya, 2001; Simmons et al., 2004]. Furthermore, the pathway of the generated internal tide energy down to dissipation scales was examined [Hibiya et al., 1996, 1998, 2002; MacKinnon and Winters, 2005; Furuichi et al., 2005] to clarify the global distribution of 1 Department of Earth and Planetary Science, Graduate School of Science, University of Tokyo, Tokyo, Japan. Copyright 2008 by the American Geophysical Union /08/2008JC tide-induced diapycnal diffusivity [St. Laurent et al., 2002; Hibiya and Nagasawa, 2004; Hibiya et al., 2006, 2007]. In contrast, while wind energy input has been expected as another major energy source [Munk and Wunsch, 1998], the global energy input from wind stress fluctuations, as well as the resulting distribution of wind-induced diapycnal diffusivity, remains to be clarified. [4] It is well known that traveling storms can excite nearinertial motions in the oceanic mixed layer, whose detailed processes and the associated upper ocean responses, such as mixed layer deepening and sea surface temperature cooling, have been investigated by many authors [Price et al., 1978; Price, 1983; Greatbatch, 1984; Shay et al., 1992; Large and Crawford, 1995]. The generated near-inertial energy can propagate down into the deep ocean, so that it is thought to be a potentially important source for the deep-ocean mixing, rather than other wind energy inputs through subinertial motions [Wunsch, 1998; Wang and Huang, 2004; Huang et al., 2006]. [5] Although theslab modelsestimatetheglobalwindenergy input into surface near-inertial motions to be TW [Pollard and Millard, 1970; D Asaro, 1985; Watanabe and Hibiya, 2002; Alford, 2003a; Watanabe et al., 2005], they are thought to be too simplified to yield accurate estimates [Plueddemann and Farrar, 2006]. Furthermore, the slab models do not explicitly take into account the energy used for the mixed layer deepening (a sink of energy 1of13

2 that is no longer available for the deep-ocean mixing) or the energy radiating downward into the deep ocean. [6] For local regions and/or limited periods, the subsequent radiation of near-inertial energy from the surface to the deep ocean has been investigated by many authors through field observations as well as theoretical and numerical approaches (e.g., the Ocean Storms Experiment [see D Asaro et al., 1995]). The method of expansion into vertical normal modes promoted a physical understanding of the propagation of near-inertial internal waves generated by traveling storms [Gill, 1984; Eriksen, 1988; D Asaro et al., 1995; Nilsson, 1995; Zervakis and Levine, 1995]. It was shown that low-mode internal waves dominated both the vertical and horizontal propagation with their energy transferred to small dissipation scales via the nonlinear wavewave interactions [McComas and Bretherton, 1977; Müller et al., 1986; Niwa and Hibiya, 1997], wave-topography interactions [Rubenstein, 1988], and wave mean flow interactions [Kunze, 1985; Zhai et al., 2005, 2007]. In particular, recent numerical study suggested that these low-mode internal waves propagated equatorward to break down via parametric subharmonic instability (PSI) at latitudes where their frequency is twice the local inertial frequency [Nagasawa et al., 2000]. [7] However, it should be noted that most of the previous numerical studies that examined the basin-scale distribution of wind-induced internal waves [e.g., Nagasawa et al., 2000; Watanabe and Hibiya, 2008; Komori et al., 2008] paid little attention to the dynamical processes in the upper ocean, although it is the crucial energy pathway from the wind forcing to the deep ocean. Furthermore, previous observational studies of wind-induced internal waves in the deep ocean [e.g., Niwa and Hibiya, 1999; Alford, 2003b] only referred to simplified slab models to discuss the relationship between near-inertial responses in surface and deep oceans. As a result, the quantitative aspect of the downward propagation of wind-induced internal waves has not been clarified yet. [8] In order to clarify the global distribution of windinduced internal waves and evaluate the resulting diapycnal diffusivity, it is essential to examine not only the global energy input at the sea surface but also the rate at which the energy penetrates down into the deeper ocean. For this purpose, we conduct here a numerical experiment of basinscale ocean responses to the realistic wind stress fluctuations using a full three-dimensional primitive equation model. 2. Model [9] The numerical model used in the present study is the Princeton Ocean Model (POM) [Mellor, 2003]. We can find recent numerical studies using POM that investigated upper ocean responses to high-frequency (>1 cycle per day) wind stress fluctuations and cyclones [e.g., Ezer, 2000; Chu et al., 2000]. For example, Ezer [2000] demonstrated that high-frequency wind stress components along with a more precise turbulence scheme and effects of short-wave penetration should be taken into account in the basin-scale three-dimensional ocean models to improve their ability to predict the upper ocean processes. [10] The present model uses a terrain-following, sigma coordinate system defined by s =(z h)/d, where z is a vertical Cartesian coordinate positive upward and D is the total water depth (D H + h, where H is the time mean water depth and h is the perturbation sea surface elevation). Thus, the sigma coordinate ranges from s = 0 at the ocean surface down to s = 1 at the ocean bottom. The governing equations are then þ þ @x ¼ fvd þ * * @x ¼ fud þ þ þ F þ F V r d ud; ð1þ r d ¼ gd ð r r 0Þ; ¼ ¼ F T rt ð T ; ð4þ ¼ F S rs ð S c ÞD; ð6þ where t is time; x and y are defined as positive eastward and northward; U and V are the velocity components in the x and y directions; z is the velocity component normal to the s constant surfaces; u and v are the baroclinic components of U and V; f is the Coriolis frequency; T and S are the potential temperature and salinity; T c and S c denote climatological values of temperature and salinity; r is the water density determined from T and S using the equation of state; r 0 is the three-dimensional background density field determined from the annual mean climatological temperature and salinity field of the World Ocean Atlas (WOA98 [Levitus and Boyer, 1994; Levitus et al., 1994]); r * is the reference water density; P is the pressure perturbation associated with the density deviation from the three-dimensional background stratification; g is the acceleration due to gravity; R is the short-wave penetration; F U, F V, F T, and F S represent the viscosity and diffusivity terms; and r d (r) is the damping (restoring) coefficient. It should be noted that, as a result of equation (3), most of the mean flow components are inevitably filtered out from the beginning. 2of13

3 Figure 1. The model domain and bathymetry. Contour interval is 1000 m, and the area where the depth is >2600 m is light gray. Note that the whole model domain is divided into three regions, and a numerical experiment is carried out for each one. The areas outlined by blue and red boxes are used for detailed analysis in sections 3 and 4. [11] The computational domain nearly covers the global region extending from 72 S to 72 N, with horizontal grid spacing of 0.15 in longitude and in latitude. The whole model domain is divided into three subregions (Figure 1), and a numerical simulation is carried out separately for each subregion. In order to reproduce the internal wave pattern smoothly connected with that in the neighboring region, the effect of the artificial model boundary is excluded by assuming a 3 wide buffer zone in each overlapped region. The model topography is constructed from the 1/12 bathymetry data set ETOPO5 [National Geophysical Data Center, 1988] with some modifications; it is first smoothed with a 100 km running mean and then interpolated to the model grid. The regions shallower than 200 m are filled in. The vertical sigma grid has 70 levels, with high resolutions for the upper mixed layer (30 levels in the upper 200 m) and lower resolutions as the depth increases. The maximum depth of the model is set to 4500 m. [12] The vertical mixing coefficients are derived from the recently modified version of the Mellor-Yamada turbulence model where the effects of surface wave breaking are taken into account [Craig and Banner, 1994; Mellor and Blumberg, 2004]. The horizontal mixing coefficients are calculated by the formulation of Smagorinsky [1963]. [13] Following Garratt [1977], the surface wind stress is calculated using the bulk transfer formula which incorporates the wind data at 10 m height for September 1990 to November 1991, which is available from the Japanese 25-year reanalysis data (JRA-25) at a time interval of 6 h with a grid resolution of 1.25 in longitude and latitude [Onogi et al., 2007]. It should be noted that the use of linear interpolation between each 6 h forcing time inevitably leads to underestimates of the near-inertial current amplitude [D Asaro, 1985; Niwa and Hibiya, 1999; Nagasawa et al., 2000; Watanabe and Hibiya, 2002; Alford, 2003a; Watanabe et al., 2005]. In the present study, therefore, we tentatively prepare the data set with a time interval of 90 min by applying the Fourier transform to the original data set, though this does not completely solve the problem associated with coarse temporal resolutions of wind data. Since low-frequency responses in the ocean are outside the scope of the present study, wind stress components with time scales longer than 8 days are filtered out. The calculated wind stress data are linearly interpolated onto the model grid in time and space. [14] The heat flux from the atmosphere to the ocean is calculated by applying the method by Ezer [2000] to 6-hourly data from National Centers for Environmental Prediction National Center for Atmospheric Research reanalysis outputs [Kalnay et al., 1996]. In addition to this heat flux, we employ an additional surface-restoring flux for temperature to the monthly mean sea surface temperature of WOA98. The restoring coefficient for the surface heat flux is set to 50 W m 2 K 1, corresponding to the restoring time of 30 days for the surface 30 m [Ezer, 2000; Chu et al., 2000]. For salinity, we apply a simple surface-restoring flux to the monthly mean sea surface salinity of WOA98 using the same restoring time as for the temperature. [15] In the 3 wide buffer zone near each model boundary, velocity components are damped at all depths so that waves are not allowed to reflect back into the model ocean interior. The damping time is set to 1 day (i.e., r d = s 1 ) at the model boundaries and is increased linearly to infinity (i.e., r d! 0) in the model ocean interior. In a similar way, temperature and salinity are restored to the monthly mean climatological values (WOA98) near each model boundary with a time scale of 1 day. [16] The model integration is divided into preexperimental and experimental stages. The preexperimental stage is assumed to start from an initial state at rest with threedimensional climatological June temperature and salinity fields. Then, the model is integrated for 3 months under no surface forcing with the temperature and salinity fields 3of13

4 Figure 2. (a) A snapshot of the calculated vertical velocity perturbations at 1000 m depth on 18 January Superimposed are contours of wind stress magnitude with the interval of 0.2 N m 2. (b) A snapshot of the calculated meridional velocity perturbations along 177 W on 18 January restored to the climatological September field at all locations and depths in the model ocean interior with a time scale of 30 days (note that the use of the restoring in the ocean interior is limited to the preexperimental run). The purpose of the preexperimental stage is to remove the effect of initial adjustment processes in the model ocean. In fact, we find that the magnitude of the internal wavefield at the end of this run is negligible compared to that in the experimental stage. The final state of the preexperimental run is employed as the initial condition for the experimental stage starting from 1 September During the experimental stage, the model is time advanced for 15 months. [17] For the analysis in sections 3 and 4, we use the calculated results during the final 12 months. The model results, u =(u, v), w, and p, are saved every 1 h, where u is the baroclinic horizontal velocity vector, w is the baroclinic vertical velocity obtained from u and v using the equation of continuity, and p is the baroclinic pressure perturbation obtained from P by subtracting the depth average. These variables are then filtered out to retain frequencies, w, >0.8f 0, where f 0 is the local inertial frequency. Unless otherwise stated, internal wave energy and/or near-inertial energy in sections 3 and 4 are calculated using the highpass-filtered components. However, responses near the equator are outside the scope of the present study because internal wave motions are indistinguishable from other lowfrequency motions. 3. Results 3.1. Brief Overview of Ocean Response [18] A sample snapshot of the calculated vertical velocity perturbations at 1000 m depth and a snapshot of the calculated meridional velocity perturbations along 177 W are shown in Figures 2a and 2b, respectively. Superimposed on Figure 2a are contours of wind stress magnitude with the interval of 0.2 N m 2. Figures 2a and 2b demonstrate that the internal waves excited behind traveling midlatitude storms propagate equatorward and downward while creating vertical mode structures. [19] Extended sample time series of the calculated wind energy input and meridional velocity perturbations at 34 N, 4of13

5 Figure 3. Sample time series of the calculated wind energy input and meridional velocity perturbations at 34 N, 180 E. Contours of potential temperature are superimposed to demonstrate the time variations of mixed layer depth. 180 E are shown in Figure 3, where contours of the potential temperature are superimposed to demonstrate the time variations of mixed layer depth. Since it has been documented that surface near-inertial energy is most efficiently enhanced when wind rotation is resonant at inertial frequency [Large and Crawford, 1995; Crawford and Large, 1996], we calculate the wind energy input to near-inertial motions using W I = t I u surf, where wind stress t I and sea surface velocity u surf are both near-inertial components. [20] Figure 3 shows that the amplitude of near-inertial motion is nearly uniform within the mixed layer. Also confirmed is the deepening of the mixed layer in response to the rapid enhancement of wind stress forcing on 16 January. After the wind energy input, the near-inertial motions spread slowly downward into the thermocline with the surface near-inertial energy gradually decaying. All the calculated results noted above are already well-known features of the upper and deep ocean responses to traveling storms [Gill, 1984; D Asaro et al., 1995; Large and Crawford, 1995; Nagasawa et al., 2000; Garrett, 2001] Global Patterns of Upper Ocean Response [21] We next examine the seasonal variations of stratification and mixed layer depth in the regions of significant wind energy input, both of which are among the important parameters for determining the wind energy input and its partition into internal waves with different spatial scales [Pollard and Millard, 1970; Gill, 1984; Eriksen, 1988; D Asaro et al., 1995]. Figure 4a shows seasonal variations of the thermal structure in the upper 300 m obtained from WOA98 together with the corresponding ones obtained from the present numerical experiment, both of which are averaged within the areas in the North Pacific (Figure 4a, top), the North Atlantic (Figure 4a, middle), and the Indian Ocean (Figure 4a, bottom) outlined by the blue boxes in Figure 1. Figure 4b shows the seasonal variations of areaaveraged and monthly averaged mixed layer depth defined by the density criterion dependent on surface temperature [Kara et al., 2000]. Except for the thermocline during summer in the Indian Ocean, the model results agree fairly well with the climatological data, although the calculated mixed layer depths are found to be different from the climatological data by ±10 30 m at several locations during winter (not shown). [22] Figure 5 shows the global distribution of the wind energy input to the surface near-inertial motions, t I u surf, averaged over 3 months for each season. Note that the equatorial region within ±3 is excluded from the calculation. [23] We can recognize that a large amount of near-inertial energy is excited poleward of 30 in both hemispheres in response to the intermittent passage of the midlatitude storms. The annual mean of the global wind energy input is estimated to be 0.4 TW (395 GW; 1 GW = 10 9 W). Figures 5 confirms that the calculated spatial pattern of the wind-induced energy for each season is in good agreement with that of the slab models [Watanabe and Hibiya, 2002; Alford, 2003a], while the total estimate of the wind energy input is somewhat smaller than predicted by the slab models ( TW). This is consistent with the work by Plueddemann and Farrar [2006], who pointed out that slab models using oversimplified empirical decay constants cannot reproduce the observed quick damping of surface near-inertial energy, thus tending to overestimate the wind work during strong wind forcing events. [24] Park et al. [2005] examined the global statistics of near-inertial energy in the mixed layer, which is defined as the product of inertial energy density at the sea surface and mixed layer depth. Using the Argo float data, namely, surface trajectories of Argo floats and temperature and salinity profiles in the upper ocean, they showed that the near-inertial energy in the ocean mixed layer is globally O(10 3 Jm 2 ) and reaches its maximum value during winter in both hemispheres, with the range of seasonal variation largest in the North Atlantic and smallest in the Southern Ocean [see Park et al., 2005, Figure 4]. [25] Figure 6 shows the seasonal variations of nearinertial (0.7f 0 < w < 1.3f 0 ) energy in the mixed layer obtained by taking the product of the near-inertial energy density at the sea surface and the mixed layer depth. Note that the energy values are monthly averaged and area averaged within the same regions adopted by Park et al. [2005] (outlined by the red boxes in Figure 1). The pattern 5of13

6 Figure 4. (a) Seasonal variations of thermal structure of the upper 300 m obtained from (left) WOA98 monthly climatological data and (right) the present numerical experiment, which are area averaged and monthly averaged within the areas outlined by the blue boxes in Figure 1 for (top) the North Pacific, (middle) the North Atlantic, and (bottom) the Indian Ocean. Contour interval is 1 C; thick lines follow 12,15,18,21, and 24 C contours. (b) Seasonal variations of the mixed layer depth area averaged and monthly averaged within the areas outlined by the blue boxes in Figure 1 from WOA98 (thin lines) and from the present numerical experiment (thick lines). Following the work by Kara et al. [2000], the density criterion dependent on surface temperature is used to define the mixed layer depth. 6of13

7 Figure 5. Global distribution of the wind energy input into the surface near-inertial motions (t I u surf ) averaged over 3 months for each season. Note that the equatorial region within ±3 is excluded from the calculation. of seasonal variation in each area is very similar to that obtained by Park et al. [2005], although the amplitude is somewhat smaller (say, by 10 30%) Global Patterns of Deep-Ocean Response [26] Figure 7 shows the global distribution of the horizontal kinetic energy (r * (u 2 + v 2 )/2, yellow to red shading) and that of the horizontal energy flux (pu, arrows), both integrated from 150 m depth down to the bottom. The notable feature is that the horizontal energy fluxes are directed equatorward away from the regions of large wind energy input at midlatitudes and high latitudes. Presumably reflecting the seasonal variation of surface near-inertial energy (Figure 6), the winter enhancement of horizontal kinetic energy is seen to be more significant in the Northern Hemisphere than in the Southern Hemisphere. It is interesting to note that the horizontal energy fluxes in the North Pacific are somewhat larger than those in the other oceans. These features are all in general agreement with the estimates from historical mooring records in the North Pacific, the North Atlantic, and part of the Southern Hemisphere [Alford, 2003b; Alford and Zhao, 2007; Alford and Whitmont, 2007]. [27] Alford [2003b] evaluated the annual mean equatorward energy flux across 30 N in the North Pacific to be 12 GW. The corresponding value obtained from the present numerical experiment is 7.6 GW, about two thirds of Alford s estimate. Nevertheless, this estimate seems to be consistent with the field observation, considering that (1) Alford s estimates are based on the data from only five moorings adjacent to 30 N, (2) the data sampling periods of Alford [2003b] and the present study are different, and (3) near-inertial motions at this latitude are susceptible to direct diurnal tidal forcing and subharmonic resonance in the real ocean. [28] The calculated horizontal kinetic energy at depth can also be compared with the historical mooring data [Alford and Whitmont, 2007]. Figure 8 shows the vertical structure of WKB-scaled horizontal kinetic energy, [(u 2 + v 2 )/2][N 0 /N(z)], for each season averaged over the two areas in the Northern Hemisphere outlined by red boxes in Figure 1, where N(z) and N 0 denote the buoyancy frequency and its depthaveraged value at each location, respectively. We find that the WKB-scaled energy is limited to (1 10) 10 4 m 2 s 2, with a seasonal modulation by a factor of 3 4 at all depths. These features are again in general agreement with the historical mooring data in the Northern Hemisphere [see Alford and Whitmont, 2007, Figures 2 and 6]. [29] Another interesting feature found in Figure 7 is energy enhancement near the equator. Figure 9 shows the latitudinal distribution of the annual mean wind energy input (Figure 9, top) and that of the depth-integrated horizontal kinetic energy of each frequency band (Figure 9, bottom). We can see that energy with w >1.3f 0 (the difference Figure 6. Seasonal variations of near-inertial energy in the oceanic mixed layer obtained by calculating the product of surface near-inertial energy and mixed layer depth. Note that the energy values are area averaged and monthly averaged within the areas outlined by the red boxes in Figure 1. 7of13

8 Figure 7. Global distribution of the horizontal kinetic energy (r * (u 2 + v 2 )/2, yellow to red shading) and that of the horizontal energy flux (pu, arrows), both integrated from 150 m depth to the bottom. The equatorial region within ±3 is excluded from the calculation. Energy fluxes with magnitudes <0.2 kw m 1 are not shown. between the thick and thin solid lines) gradually increases approaching the equator, although no significant wind energy input takes place. This suggests that the energy enhancement near the equator is caused not only by the locally generated near-inertial internal waves but also by internal waves propagating from higher latitudes. 4. Discussions 4.1. Distribution of Energy Dissipation Rate in the World s Oceans [30] Since the validity of the present numerical experiment has been confirmed via the comparison with the available observations in section 3, we examine here the spatial distribution of energy dissipation rate in the world s oceans. Instead of the viscosity terms (F U and F V, equations (1) and (2)), which include not only the effect of energy dissipation but also that of downward momentum transport from the wind stress field, we use the wind energy input to surface near-inertial motions and horizontal and vertical components of internal wave energy flux to estimate energy dissipation rate at each depth range. [31] Figure 10 shows the global distribution of the seasonally averaged vertical energy flux, pw, across 150 m depth, which is found to be much less than the wind energy input (Figure 5). This suggests that most of the windinduced near-inertial energy in the world s oceans is confined within the upper ocean. Figure 11 shows the annual mean energy balance in each area outlined by the red boxes in Figure 1, namely, 140 E 120 W, N (the North Pacific); W, N (the North Atlantic); and W, S (the Southern Ocean). [32] In the North Pacific, the annual mean wind energy input to surface inertial motions is estimated to be 58 GW. Out of this, 43.8 GW (75% of the wind energy input) is dissipated within the surface 150 m, 6.7 GW is dissipated from 150 m depth to the bottom, and the remaining 7.6 GW radiates horizontally away from this area. Nearly the same energy balance can be found in the North Atlantic and the Southern Ocean, where the annual mean wind energy inputs, 26.2 GW and 151 GW, respectively, are dissipated within the surface 150 m by more than 80%. In the global energy budget, 75 85% of the wind energy input is found to be dissipated within the surface 150 m (Figure 11). [33] A recent numerical study suggested that wind-induced internal waves propagated equatorward to break down via Figure 8. Vertical structure of WKB-scaled horizontal kinetic energy [(u 2 + v 2 )/2][N 0 /N(z)] for each season averaged over the two areas in the Northern Hemisphere outlined by red boxes in Figure 1. 8of13

9 Figure 9. (top) Latitudinal distribution of annual mean wind energy input and (bottom) that of annual mean horizontal kinetic energy integrated from 150 m depth to the bottom. Note that each value is averaged from 160 E to 160 Wforw >0.8f 0 (thick solid line), 0.8f 0 < w <1.3f 0 (thin solid line), and 2f 0 < w < 3f 0 (dashed line, only for horizontal kinetic energy), where f 0 is the local inertial frequency. PSI into turbulence at latitudes where their frequency is twice the local inertial frequency [Nagasawa et al., 2000]. However, the equatorward internal wave energy flux estimated from the present numerical experiment is only 10% of the total wind energy input (Figure 11). This implies that the role of PSI in cascading the wind-induced energy to small dissipation scales is minor, although the existence of PSI-induced mixing itself cannot be denied because the energy with 2f 0 < w <3f 0 is identified to be somewhat enhanced at latitudes <20 (Figure 9) Expansion Into Vertical Normal Modes [34] To understand the underlying dynamics for the dissipation of wind-induced energy in the surface 150 m (Figure 11), we use the method of vertical normal mode expansion by which the horizontal and vertical propagation of the wind-induced internal wave energy can be best discussed [e.g., Gill, 1984; D Asaro et al., 1995; Levine and Zervakis, 1995]. For each location where the total water depth is >2600 m (Figure 1), the vertical normal modes are computed using the seasonally averaged stratification for the entire water column. Then, u and p are expanded into the vertical normal modes by least squares fitting at each time. [35] First, we calculate the wind energy input to each mode, W m = t I u m (0), and the equatorward energy flux of each mode, p m v m, where u m (z) =(u m (z), v m (z)) and p m (z) denote the horizontal velocity and pressure associated with the mth vertical mode, respectively. Figure 12 (top) shows the annual mean area-integrated W m (blue) and annual mean p m v m integrated over the equatorward cross section of each area (purple) (areas are outlined by the red boxes in Figure 1). We can see that although most of the wind-induced energy is fed into higher modes (m 6), the equatorward energy flux is dominated by the two lowest modes in each area. [36] The vertical distribution of energy dissipation for each mode can be best discussed by introducing E m (z) = X m (z) X m 1 (z), where X m 2 X m ðþ z ¼ r u X m 2 * j¼1 jðþ z þ v j¼1 jðþ z =2: We integrate the annual mean, area-averaged E m within each depth range to obtain the results in Figure 12 (bottom). We can see that most of the horizontal kinetic energy of higher modes (m 6) is confined within the surface 150 m in each area. Figure 10. Global distribution of the seasonally averaged vertical energy flux pw across 150 m depth. The equatorial region within ±3 is excluded from the calculation. Note that the cold colors denote the downward energy flux. 9of13

10 Figure 11. Schematic diagram showing the annual mean energy balance for each of the three areas outlined by the red boxes in Figure 1. Labels are as follows: a, annual mean wind energy input to each area; b, annual mean energy dissipation rate within the surface 150 m in each area together with its ratio to the local wind energy input; c, annual mean energy dissipation rate from 150 m depth to the bottom in each area together with its ratio to the local wind energy input; d, annual mean energy dissipation rate from 1000 m depth to the bottom in each area together with its ratio to the local wind energy input; e, annual mean equatorward energy flux integrated over the equatorward cross section of each area together with its ratio to the local wind energy input. Figure 12. (top) Annual mean area-integrated wind energy input to each mode (blue) and annual mean equatorward energy flux of each mode integrated over the equatorward cross section of each area (purple) (areas are outlined by the red boxes in Figure 1). The purple and blue values show the ratio of each modal component to the total wind energy input. (bottom) Annual mean area-averaged horizontal kinetic energy E m integrated within each depth range (for the definition of E m, see text). 10 of 13

11 Figure 13. Sample time series of (a) the cumulative wind energy input and (b d) the horizontal kinetic energy E m integrated within each depth range at 34 N, 180 Eform = 1 2 (purple), m = 3 5 (green), m 6 (red), and the total (black). [37] Figure 13 shows a sample time series of E m for m =1 2, m =3 5,andm 6inresponsetothelargeamountofwind energy input on 16 January. We can see that in contrast to E 1 and E 2 immediately spreading throughout the water column, E 3 E 5 travel slowly down to the deeper ocean and E m (m 6) is quickly dissipated within the surface 150 m. [38] In summary, Figures 12 and 13 suggest that roughly 20% of the wind-induced energy is fed into modes 1 and 2, about half of which is then carried away equatorward; 30% of the wind-induced energy is fed into modes 3 5, which is dissipated both within and below the surface 150 m without being carried away equatorward; and the remaining 50% of the wind-induced energy is fed into higher modes (m 6), nearly all of which is quickly dissipated within the surface 150 m. This is consistent with the result of the present numerical experiment that most (75 85%) of the wind-induced energy is dissipated in the surface 150 m (Figure 11). 5. Conclusions [39] Munk and Wunsch [1998] claimed that 2.1 TW of energy was required to create the deep-ocean mixing so that the global thermohaline circulation with the volume transport of Sv can be sustained and that wind stress fluctuations might be one of the major sources to supply nearly half of the required energy. Near-inertial motions in the oceanic mixed layer excited by traveling storms have been thought to be a potentially important source for the deep-ocean mixing since they can propagate into the deep ocean as internal waves. [40] In order to examine the global distribution of windinduced internal wave energy and to evaluate the resulting diapycnal diffusivity in the deep ocean, we have carried out numerical experiments of wind-induced internal waves in the world s oceans using a full three-dimensional primitive equation model. Special attention has been directed to the global wind energy input into the surface near-inertial motions and the downward energy propagation into the deep ocean. [41] The annual mean of the global wind energy input into surface near-inertial motions has been estimated to be 0.4 TW. The global pattern of the wind energy input has been confirmed to be in agreement with that from the slab models for each season [Watanabe and Hibiya, 2002; Alford, 2003a], while the total estimate of the wind energy input has been somewhat smaller than predicted by the slab models. The near-inertial energy in the ocean mixed layer, the depth-integrated equatorward energy flux, and the vertical structure of WKB-scaled horizontal kinetic energy have been shown to be consistent with the available field observations in the regions of significant wind energy input. [42] The most important result obtained in the present study is that in total, 75 85% of the global wind energy 11 of 13

12 input to surface near-inertial motions is dissipated in the surface 150 m (Figure 11). The analysis using vertical normal mode expansion has shown that a large part of wind energy input resides in high modes that are subject to local dissipation in the upper ocean. As a result, even if the remaining 20% of energy is assumed to contribute to the deep-ocean mixing quite efficiently, the total available energy is limited to, at most, 0.1 TW (0.4 TW 0.2). We believe that the results of the present numerical experiment are robust and not numerical artifacts because additional numerical experiments for a limited region of the North Pacific have shown that the vertical energy flux across the 150 m depth increases only by 3% even when the vertical resolution is doubled. [43] The present study has thus predicted that the total wind-induced, near-inertial energy available for deep-ocean mixing is an order of magnitude smaller than previously estimated by Munk and Wunsch [1998]. Adding the energy flux from tide-topography interactions, estimated to be 0.9 TW, we can conclude that the total turbulent energy available for deep-ocean mixing is, at this time, 1.0 TW. This might suggest the existence of other important energy sources, such as the one through geostrophic adjustment processes [Wunsch and Ferrari, 2004], and/or some additional mechanisms sustaining the global thermohaline circulation, such as effects of Ekman upwelling in the Southern Ocean [Hasumi and Suginohara, 1999; Webb and Suginohara, 2001]. Another possibility is that the volume transport of the global thermohaline circulation, Sv, might be overestimated. Actually, Schmitz [1995] suggested a much lower value of Sv. More detailed studies are absolutely necessary for the energetics and volume transport of the global thermohaline circulation. [44] There might be some ambiguities associated with the wind stress data in the present numerical experiment. For example, estimates of the wind energy input to surface nearinertial motions are shown to be quite sensitive to the temporal and spatial resolutions of wind stress data [D Asaro, 1985; Watanabe et al., 2005; Jiang et al., 2005]. The interannual variability of the observed wind stress ignored in the present study should also be taken into account. Other ambiguities might occur because of the exclusion of the background mean flows and the background internal waves and rough bathymetry [Garrett and Munk, 1972, 1975; Müller et al., 1986; Rubenstein, 1988]. The presence of background mean flows, for example, might enable the penetration of high-mode, near-inertial internal waves into the deep ocean [Kunze, 1985; Zhai et al., 2005, 2007]. How much the results of the present study would be modified by these processes remains to be clarified in the future. [45] Acknowledgments. The authors express their gratitude to two anonymous reviewers for their invaluable comments. 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Niwa, Department of Earth and Planetary Science, Graduate School of Science, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo , Japan. (furu1@eps.s.u-tokyo.ac.jp) 13 of 13