## 1. Introduction

The wind generates near-inertial internal waves by exciting currents in the surface mixed layer. Convergences and divergences in these motions then “pump” propagating near-inertial waves. Some energy is lost to shear-driven turbulence at the base of the mixed layer, but the remainder is free to propagate downward and equatorward as internal “swell,” until it decays by interacting with the rest of the internal wave continuum (Henyey et al. 1986) or the mesoscale flow field (Kunze et al. 1995), or by refracting and undergoing shear instability (Alford and Gregg 2000). In a steady-state ocean, the energy input must ultimately be dissipated. Therefore, knowledge of the magnitude and spatiotemporal patterns of energy input provides useful constraints on internal-wave-induced mixing. Recent focus on tidal sources of internal wave mixing (Sjöberg and Stigebrandt 1992; Morozov 1995; Polzin et al. 1997; Munk and Wunsch 1998; Kantha and Tierney 1997; Egbert 1997; Egbert and Ray 2000) makes a comparison of tidal and wind energy inputs particularly appropriate.

D'Asaro (1985) pioneered the study of the wind–internal wave energy transfer by driving a slab mixed layer model with wind data from a dozen NOAA National Data Buoy Center buoys in the eastern North Pacific and the western North Atlantic. He showed that the forcing is highly intermittent, with winter storms dominating the energy input. Though the spatial coverage of the study was limited, he observed a peak near 50°N, with wintertime fluxes of 2–3 mW m^{−2}.

Since that study, global atmospheric models have improved greatly. In particular, the 2.5° resolution NCEP–NCAR global reanalysis (Kalnay et al. 1996) yields four-times-daily wind estimates that are coherent with buoys at frequencies up to 1.5 cpd. This is sufficient to resolve near-inertial fluctuations at latitudes up to 50°. (At higher latitudes, the inertial frequency is higher than this, and the model fails to resolve the associated motions.)

Here the spatial extent and resolution of D'Asaro's (1985) work is extended using the NCEP–NCAR reanalysis winds from 1996 to 1999. After demonstrating the adequacy of the NCEP winds, I present spatial maps of the flux from 50°S to 50°N. I find magnitudes and seasonal cycles that agree well with his results. The global mean energy flux over all seasons and all oceans from 50°S to 50°N is 0.98 ± 0.08 mW m^{−2}, the same order of magnitude as estimates of tidal input.

In the next section I describe the data, the slab model and the techniques for computing the flux. Then, I compare wind and statistics of the resultant flux from the reanalysis and buoys to demonstrate the validity of the reanalysis winds. I present the results in section 4 and conclude with a discussion.

## 2. Data and methods

### a. Wind data

Wind stress data are from the NCEP–NCAR global reanalysis surface wind fields (Kalnay et al. 1996). These are output four times daily, with 2.5° resolution, from a 17-layer sigma-surface atmospheric model with a consistent data assimilation scheme. The surface winds are those from the lowest *σ* surface. This analysis has recently been completed for the past 40 years. Present data and forecasts are available in near-real time. Data from 1996 to 1999 are used here.

The reanalysis wind fields are validated against measurements from NOAA Data Buoy Center (NDBC) buoys at several locations (section 3). The buoy data are sampled at either ten minute or hourly intervals and adjusted to 10-m height assuming a logarithmic profile. Wind stress from the reanalysis and buoy winds was computed using the Large and Pond (1981) parameterization.

### b. Computing the energy flux: Slab model

*τ***u**

**is the wind stress and**

*τ***u**is the velocity at the surface. Positive fluxes (energy transferred from the atmosphere to the ocean) result when the wind is aligned with the current, tending to accelerate it further.

This flux would be most directly computed using observed surface wind stress and mixed layer velocity data. In reality, only the former is usually available. Fortunately, a simple model (Pollard and Millard 1970) that treats the surface mixed layer as a slab adequately describes the mixed layer response to wind stress (Pollard 1980; Thomson and Huggett 1981; Paduan et al. 1989). Therefore, the approach taken here (as in D'Asaro 1985) is to use wind stress data to drive the mixed layer model and then to compute the flux using the observed wind stress and the model currents in Eq. (1).

*u*and

*υ,*of a mixed layer of depth

*H*and density

*ρ*are

*r*models the loss of mixed layer energy to downward-propagating motions and

*f*is the inertial frequency.

*Z*

_{I}

*Z*

*Z*

_{E}

*T*/

*H*excite inertial oscillations. Mixed layer depth changes are typically negligible relative to wind-stress fluctuations.

^{1}In this case, we take

*H*= H

_{ref}, and

*Z*

_{I}obeys

*Z*

_{I}responds primarily to inertial-frequency wind motions that rotate inertially, that is, clockwise (counterclockwise) in the Northern (Southern) Hemisphere.

To solve Eq. (10), wind stress data from each spatial cell in the NCEP reanalysis are interpolated onto a half-hour grid, using a filter that minimizes the mean-square error between the original and resampled series. Equation (10) is then integrated in time over a given year to produce time series of the inertial mixed layer currents.

*H*

_{ref}

*t*

*τ***u**

_{I}

_{I}is the vector expression of the inertial current,

*Z*

_{I}.

*H.*Following D'Asaro (1985), I take

*H*

_{ref}= 50 m and compute Π(50 m), the flux which would result if the mixed layer depth were 50 m. This enables computation of the flux in “typical” mixed layer conditions from a time series of wind stress alone. With knowledge of the mixed layer depth at a given location and time, the actual flux can be computed, via

In this manner, flux contributions from variations in wind stress and mixed layer depth may be assessed separately. Operationally, Π(50 m) is computed, and then a mixed layer depth climatology (Levitus and Boyer 1994), which yields 1°-resolution maps of monthly mean mixed layer depth,^{2} is interpolated onto the NCEP grid and used in Equation (12). Incorporation of mixed layer depth affects the latitudinal and seasonal dependence of the flux signal, but has little effect on the mean values.

The one free parameter in the model is the damping parameter *r.* For stability, the model requires *r* ≪ *f*. Since *f* varies with latitude, I choose a constant *r*/*f* ratio for the entire domain. This has the advantage of maintaining a constant bandwidth for the calculation. In section 3c it is shown that varying *r* affects the mean flux only little. I therefore take advantage of an opportunity to “tune” the model, and choose the value of *r* that maximizes the coherence between the model inertial current and acoustic Doppler current profiler (ADCP) records from a buoy at 34°N. Extrapolating the results of this analysis, I choose *r* = 0.15*f*.

## 3. Verification

Before presenting the results, the reliability of the NCEP reanalysis winds will be demonstrated in several ways. First, correspondence between the reanalysis and buoy winds will be shown via direct comparison of the time series and by spectral analysis. In addition, it is shown that the current from the slab model driven with the reanalysis winds is coherent with the inertial-band-observed current. Then, the resultant fluxes and their statistics will be compared. Finally, the results' insensitivity to the damping parameter *r* will be examined.

### a. Comparison of reanalysis and buoy winds

Since the model responds primarily to inertial-frequency wind fluctuations, the quality of the inertial-band reanalysis wind fields must be demonstrated. This requirement becomes more stringent as the poles are approached and the inertial frequency increases. The Nyquist frequency of the four-times-daily reanalysis is 2 cpd; however, the model winds begin to fail at about 1.5 cpd, or at 50°N.

This is demonstrated in Fig. 1 using zonal wind data over most of 1997 from three latitudes: 17°, 38°, and 47°N. (Identical conclusions result using the meridional winds.) General coherence is always seen between the reanalysis and buoy time series (left). The uncertainty in the magnitude of the two time series is shown in the center panels, where the slope of a best-fit line differs slightly from unity.

The best indicators of the adequacy of the reanalysis data are the spectra and squared coherence to the right of the figure (panels c,f,i). At all three locations, the buoy and the reanalysis winds are coherent and have equal spectra at the local inertial frequency (dotted line). North of 50°N, the inertial-band coherence between model winds and buoys begins to fall (not shown). Based on these analyses, I have chosen 50° N/S as the maximum resolvable latitude, and conclude that the reanalysis winds are a good proxy for the buoy winds closer to the equator.^{3}

### b. Comparison of reanalysis and buoy fluxes

In this section, examples of time series and statistics of energy flux from the reanalysis and buoy wind fields are compared. Close qualitative agreement is found with the findings of D'Asaro (1985), who provides a detailed discussion. For the present purposes, the primary conclusion is that the reanalysis and buoy wind fields yield very similar fluxes.

For the comparison, data from the reanalysis winds at 35.0°N, 120°W are compared to those computed from wind data from NDBC buoy 46054, at latitude 34.2°N. This buoy was selected because it has a downlooking ADCP, which allows comparison of the slab-model currents to bandpassed observed currents. (The only other available buoys with ADCPs are part of the Tropical Atmosphere–Ocean array. Situated on the equator, they are not useful for study of inertial fluxes.) At this location, the reanalysis winds are a factor of 0.8 less than the buoy winds. For the purposes of this discussion, they have been multiplied by 1/0.8 to facilitate direct comparison with the buoy winds.

#### 1) A typical simulation

Figure 2a shows the zonal (black) and meridional (gray) components of the wind stress over a one-month period in 1997. In Figs. 2a–d, heavy lines correspond to reanalysis results, and thin lines to buoy results. The zonal mixed layer velocity is plotted in Fig. 1b (the buoy trace is offset by 0.2 m s^{−1}). The flux, Π(50 m), is plotted in Fig. 1c, and its time integral, the cumulative energy input, is shown in Fig. 1d.

The results are qualitatively similar to those presented by D'Asaro (1985). As in that study, positive and negative fluxes result. Positive values are more likely, resulting in a positive transfer of energy from the wind to mixed layer motions. Nearly half of the month's flux results from the single event at yearday 48.

#### 2) Comparison of model and observed currents

Though the validity of the Pollard and Millard (1970) slab model has been demonstrated before (Pollard and Millard 1970; Pollard 1980; Thomson and Huggett 1981; Paduan et al. 1989), it is worth demonstrating that the mixed layer currents resulting from the reanalysis winds are coherent with observed currents. To perform the comparison, a time series of zonal velocity from a single ADCP bin spanning 25–41 m (the shallowest one) was high-pass filtered with a cutoff of two cpd, and overplotted in Fig. 2b. The comparison was performed during winter to maximize the likelihood that the (unknown) mixed layer depth was always > 41 m. The model and observed mixed layer currents are visually correlated and have a correlation coefficient of 0.58.

#### 3) Probability distribution

D'Asaro (1985) found that a small number of very energetic forcing events produced most of the flux, indicating a highly skewed probability distribution. Do the reanalysis wind fields yield fluxes with the same statistics as those computed using buoy data?

The probability distribution of Π(50 m) is shown in Fig. 3. The entire year of 1997 is used in the comparison. Recall that the reanalysis winds are multiplied by 1/0.8 prior to running the model, for this section only. The resultant distributions are plotted on a logarithmic *x* axis, in variance-preserving form, so that the area under each curve between any two bounding values is proportional to the likelihood of Π(50 m) lying between them. Positive values are shown with solid lines, and negative values with dashed lines. As seen in the last section, positive values are more likely than negative ones, resulting in a net flux of energy into the ocean. Importantly, the distribution of the reanalysis- and buoy-derived flux is nearly the same.

As another test, the flux is computed form 10-min buoy data. Since the distribution does not differ between the 10-min and 6-h buoy data, I conclude that the interpolation scheme used does not affect the fluxes.

### c. Dependence upon damping

The damping parameter *r* is an adjustable parameter in the slab model. Its inverse, *r*^{−1}, represents the *e*-folding decay time for impulsively forced oscillations. D'Asaro (1985) noted that varying *r* affects the variability of Π(50 m), but only weakly influences its mean. The same is true in the present study, as shown below. For validity, the model requires *r* ≪ *f*. For stability, it requires *r* > 0. I have taken *r* = *cf*, where *c* is determined below. Taking *r* as a constant fraction of the local inertial frequency maintains a constant forcing bandwidth.

In addition, it is consistent with the physics of the damping, since the decay of mixed layer energy is effected through propagation of near-inertial waves. D'Asaro (1989) showed that this occurs after several inertial periods, as the meridional wavenumber decreases via *l* = −*βt,* where *β* = ∂*f*/∂*y* (the *β* effect). A latitude-dependent damping parameter ensures that the decay occurs on the correct timescale over the entire domain.

The inertial currents, and flux, Π(50 m), are computed at 34°N, 120°W from reanalysis data for *r* values ranging from 0.05*f* to 0.3*f*. The correlation coefficient is computed between the resultant currents and the observed inertial-band ADCP currents for yeardays 1–90, and plotted in Fig. 4. Both components are most correlated with the observed currents in a range *r* = (0.14–0.19)*f*. The flux is maximum for *r* = 0.17*f*, but varies by only 20% over the range *r* = (0.1–0.3)*f*—smaller than the wind stress error (next section). Concluding that the particular value of *r* is not critical, I have taken *r* = 0.15*f*, attempting to tune the model for optimum correlation with observations.

### d. Errors

It has been shown that the reanalysis winds show high coherence with buoy observations up to frequencies of about 1.5 cpd, allowing resolution of the inertial-band winds as far poleward as 50°. The magnitude at a given location, however, can be off by as much as 30%. Since the flux depends on the fourth power of the wind speed, flux errors of 280% result for individual measurements. Wind stress uncertainty is the largest source of error in the flux calculation. Assuming a (conservative) 1000-km decorrelation scale for the reanalysis winds, these figures result in a 10% standard error for the global mean.

Bias is a more serious issue. To ensure that the wind stress errors are random and not systematic, a comparison was performed between the surface wind field and another NCEP wind product, the 10-m wind. (The surface winds are used in favor of the 10-m winds due to their much higher coherence with the buoys.) The magnitudes of the two wind fields differ by less than 2%, implying less than 10% bias in the flux estimates. Furthermore, the monthly mean NCEP winds show no significant biases relative to either the European Centre for Medium-Range Weather Forecasts or the U.S. Navy model wind fields (McDermott et al. 1997).

A potential source of low bias results from the reanalysis winds' underresolution of tropical storms (Swail and Cox 2000), whose scales are comparable to the reanalysis wind 2.5° grid spacing. Tropical storms are very efficient generators of near-inertial fluxes (D'Asaro 1985). Nilsson (1995) roughly estimates the global power input to near-inertial waves due to hurricanes at *O*(0.01 TW). K. Emanuel (2000, personal communication) using a 1D coupled hurricane-resolving model, estimates a global near-inertial power input of 0.05 TW due to tropical storms (however, the 1D model may underestimate the true flux somewhat). This value would augment the present study's global total of 0.29 TW (section 4d) by about 16%—an upper bound since *some* of the tropical storm fluxes are in fact resolved by the reanalysis winds.

## 4. Results

### a. Spatial distribution

Spatial maps of Π(〈*H*〉) during a typical year (1997) are presented by season in Fig. 5. Spatial patterns in the mixed layer depth are included through use of the Levitus mixed layer depth climatology in Eq. (12), averaged over the appropriate season. Examination of Fig. 5 shows that the flux varies by orders of magnitude, while the mixed layer depth varies by only a factor of 2 or 3, during any given season. Therefore, incorporation of the mixed layer depth affects the magnitude and spatial patterns of Fig. 5 only weakly. (It has a greater effect on the seasonal dependence of the signals; section 4b.)

The plotting scale is logarithmic, with red values indicating 30 mW m^{−2}. Values smaller than 0.1 mW m^{−2} appear as blue. The equatorial band yields zero flux since *f* = 0 there and the model gives zero currents. A few (<1%) seasonally averaged values are < 0 (loss of energy to the atmosphere), but all of these have magnitudes less than 0.1 mW m^{−2}, and so are also plotted in blue. The rough land edges reflect the 2.5° grid spacing.

The largest inputs occur from 30°–50°N during Northern Hemispheric fall/winter, in bands across the western North Atlantic and Pacific, associated with northern midlatitude storms. Large fluxes also occur at 30°–50°S during southern winter, particularly in the Indian Ocean.

Broad minima span the central and eastern portions of each basin. Interestingly, D'Asaro's (1985) eastern Pacific analysis was conducted in a relatively low flux region. He obtained higher values in the eastern Atlantic, consistent with this general picture.

Spatial maps of the inertial component of the wind would produce identical patterns. These differ substantially from maps of wind speed (or, for example, maps of *ρu*^{3}_{*}

### b. Seasonal cycle

The seasonal cycle is examined by computing the zonal average across the bands 37.5°N and 37.5°S. Examining the cycle of Π(50 m) (Fig. 6a), a strong maximum is seen near December/January in the Northern Hemisphere [in agreement with D'Asaro (1985)], and near June in the Southern Hemisphere. This pattern is consistent with primary forcing by winter storms.

The zonally averaged mixed layer depth (Fig. 6b) is deepest during winter. Its incorporation into the flux correspondingly weakens the seasonal cycle of Π(〈*H*〉) relative to Π(50 m), and shifts the peaks towards October in the Northern Hemisphere, also as seen by D'Asaro (1985). Seasonal cycles are still present in both hemispheres, however, with maxima resulting as a combination of large fluxes and a thin mixed layer.

### c. Zonally averaged flux

The latitudinal dependence of the flux is examined by averaging the global flux zonally and in time. Figure 7a shows the result for Π(50 m). Thin traces represent annual averages for 1996–99, and the 4-yr mean is plotted in heavy black. Incorporating mixed layer depth variability, whose latitude dependence is shown in Fig. 7b, yields Π(〈*H*〉), plotted in Fig. 7c. The two differ little since the zonal average of *H* varies by only about a factor of 2.

Very little interannual variability is seen from 1996 to 1999, especially in the Southern Hemisphere. A north–south asymmetry is evident, especially when mixed layer depth dependence is included (Fig. 7c). The large value near 5°N results from the active region near the coast of Venezuela (Fig. 5), where large Π(50 m) values conspire with shallow mixed layer depth to produce large fluxes, that dominate the zonal band.

The zonal averages show a peak near 40°N, roughly comparable with the peak near 50°N found by D'Asaro (1985) for the wintertime, eastern Pacific. An analysis of 14 NDBC buoys (Fig. 8) was conducted for comparison and to attempt to extend the results northward of 50°N. For each buoy that had at least 90 days of wintertime data in a given year, the mean flux, 〈Π(50 m)〉, was computed.

The 4-yr mean flux as a function of latitude, and the values found by D'Asaro (1985) (spanning various 2–4-yr intervals over 1974–79), are overplotted in Fig. 7a. The magnitudes of these buoy fluxes are not directly comparable to the zonally averaged model flux, since they are computed over wintertime only, and represent the low eastern Pacific fluxes. The buoy fluxes show considerable scatter, since they result from a 2–4-yr average of highly variable single-point values. Still, they show an overall pattern of a peak near 40°–50°N, which is reproduced in the zonally averaged model fluxes.

### d. Mean values

The mean fluxes, and the total power dissipated, from 1996 to 1999 are summarized in Table 1. The mean flux is computed over each basin, taking into account the reduction of area of latitude/longitude bins as the poles are approached on the globe. In this manner, it is seen that the north/south flux asymmetry occurs predominantly in the Pacific; elsewhere, north/south contributions are more even.

Multiplying each mean flux value by the area of the basin, we obtain the total power dissipated, *D* = Π*A.* The global total from 50°S–50°N is 0.29 TW. This value is considered in the context of the global ocean energy budget in the discussion. Since the area equatorward of 50° is 92% of the total area of the oceans and since the latitudinal peak fluxes near 40°–50°N appear to have been resolved, it is not likely that the global total from 90°S–90°N is much higher, especially given the increasing ice coverage proceeding poleward. Contributions from the high winds in the Southern Ocean [which Wunsch (1998) found dominated the energy input into lower-frequency motions; see discussion] may be significant; however, their resolution must await improved atmospheric models.

## 5. Discussion

Munk and Wunsch (1998) provide a useful framework for discussion of these results in terms of the global energy budget. Extending the 1D analysis of Munk (1966), they compute that an average diapycnal diffusivity *κ*_{AS} = 10^{−4} m^{2} s^{−1}, corresponding to a total dissipation of *D*_{AS} = 2 TW, is necessary to prevent the deep oceans from filling up with the cold water sinking at the poles. The power input is needed to stir the cold water upward. They note that *κ*_{AS} is likely a proxy representing the average of the oft-observed “pelagic” *κ*_{pel} = 10^{−5} m^{2} s^{−1} (*D*_{pel} = 0.2 TW) and localized hot spots of much higher mixing.

The wind and the tides are identified as power supplies. Since the *M*_{2} tidal dissipation is well established at 2.5 ± 0.1 TW from astronomical measurements, plenty of tidal power is available, but past calculations have indicated that nearly all of it is dissipated in shallow seas and thus unavailable for abyssal mixing. However, launch of the TOPEX/Poseidon satellite and the improvement of tidal models have enabled calculations of *M*_{2} tidal flux divergence, which suggest that 0.36 TW (Kantha and Tierney 1997) to 0.7 TW (Egbert and Ray 2000) occurs in the open ocean. Theoretical estimates of barotropic to baroclinic conversion are the same order of magnitude, ranging from 0.29 TW (Munk 1966), to 0.5 TW (Bell 1975), to 1.1 TW (Morozov 1995), to 1.3 TW (Sjöberg and Stigebrandt 1992).^{4} These results suggest that internal tides, generated by barotropic tidal flow past topography, break and provide a portion of *D*_{AS}.

The balance, according to a recent study by Wunsch (1998), may be provided by wind work on the general circulation. From TOPEX/Poseidon data, confirmed with a model, he computed ** τ** ·

**u**

_{g}, where

**u**

_{g}is the geostrophic circulation, for 10-day timescales and longer. The resultant 1 TW, with the largest contribution in the Southern Ocean, enters the flow at synoptic scales. Its pathway to dissipation scales is subject to speculation.

The present value of 0.29 TW is non-negligible relative to these numbers, and is not included in either the tidal or the geostrophic wind work budgets. It is suggested that near-inertial wind work may be yet another player in the global mixing scenario.

For this to be true, a substantial fraction of the near-inertial energy flux must penetrate to deep isopycnals before mixing. If, for example, all of the mixing occurred due to shear instability at the base of the mixed layer, none would remain for deep propagation and abyssal mixing. D'Asaro et al. (1995) found that the depth-integrated energy input by a storm decreased by (36 ± 10)% after three weeks, consistent with horizontal propagation out of the experiment region. Therefore, it appears that a large fraction of the near-inertial energy is, in fact, available to propagate.

Many observations of deep near-inertial oscillations support this interpretation. For example, Alford and Gregg (2000) found that a surface-generated near-inertial wave penetrated at least to 300 m, the maximum depth of their observations. The wave underwent active mixing at the stratification peak near 100 m. But once below, mixing was weak enough for the wave to continue to propagate for months, potentially reaching great depth. It is significant that abyssal isopycnals are relatively shallow at high latitudes: since most near-inertial energy flux is at these high latitudes, propagating waves can reach and mix these isopycnals with relative ease.

Once generated, near-inertial waves are constrained to propagate equatorward. (It is therefore significant that most generation occurs at high latitudes; if most occurred near the equator, it would be difficult to explain near-inertial waves at high latitudes.) Knowledge of the propagation characteristics of the generated waves would enable construction of maps of dissipation, rather than of their generation.

Hibiya et al. (1999) made the first such attempt at mapping wind-induced mixing. They assumed that near-inertial waves propagate equatorward until their frequency becomes twice inertial, at which point they undergo parametric subharmonic instability (PSI) and break. Mapping the locations of this occurrence, they concluded that wind-induced mixing should be greatest between 20°–30°N, where few microstructure measurements have been taken.

Observational evidence for the occurrence of PSI is scarce, and Garrett (2001) points out that the waves would reflect off the bottom many times before they underwent PSI. Still, the basic finding that dissipation should occur equatorward of the generation sites is reasonable. A better understanding of near-inertial propagation is necessary for further progress.

As a final note, even if Fig. 5 is not representative of the spatial patterns of the actual mixing, it does reflect the times and locations of large near-inertial energy. Since the NCEP reanalysis data is available in near-real time, and the processing time for the flux calculation is low, these techniques may be of use in planning cruise locations and times. Using the NCEP forecast wind fields, “internal swell forecasts” may even be possible.

## Acknowledgments

This work was supported by M. A.'s startup funding at the Applied Physics Laboratory, and by a SECNAV/CNO grant. I am indebted to Mike Gregg for this support. NCEP 4× Daily Global Reanalyses data were provided by the NOAA–CIRES Climate Diagnostics Center, Boulder, Colorado, from their Web site at http://www.cdc.noaa.gov/. The Levitus mixed layer climatology was obtained from the Lamont-Doherty Web site. Discussions with Eric D'Asaro were very helpful.

## REFERENCES

Alford, M., and M. Gregg, 2000: Near-inertial mixing: Modulation of shear, strain and microstructure at low latitude.

*J. Geophys. Res.,*in press.Bell, T. H. Jr, . 1975: Topographically generated internal waves in the open ocean.

,*J. Geophys. Res***80****,**320–327.D'Asaro, E. A., 1985: The energy flux from the wind to near-inertial motions in the mixed layer.

,*J. Phys. Oceanogr***15****,**943–959.D'Asaro, E. A., 1989: The decay of wind-forced mixed layer inertial oscillations due to the

*β*effect.,*J. Geophys. Res***94****,**2045–2056.D'Asaro, E. A., C. E. Eriksen, M. D. Levine, P. Niiler, C. A. Paulson, and P. V. Meurs, 1995: Upper-ocean inertial currents forced by a strong storm. Part I: Data and comparisons with linear theory.

,*J. Phys. Oceanogr***25****,**2909–2936.Egbert, G. D., 1997: Tidal data inversion: Interpolation and inference.

*Progress in Oceanography,*Vol. 40, Pergamon, 53–80.Egbert, G. D., and R. D. Ray, 2000: Significant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data.

,*Nature***45**(6,) 775–778.Garrett, C., 2001: What is the “near-inertial” band and why is it different from the rest of the internal wave spectrum?

,*J. Phys. Oceanogr***31****,**962–971.Henyey, F. S., J. Wright, and S. M. Flatté, 1986: Energy and action flow through the internal wave field.

,*J. Geophys. Res***91****,**8487–8495.Hibiya, T., M. Nagasawa, and Y. Niwa, 1999: Model predicted distribution of internal wave energy for diapycnal mixing processes in the deep waters of the North Pacific.

*Dynamics of Oceanic Internal Gravity Waves: Proc.'Aha Huliko'a Hawaiian Winter Workshop,*P. Müller and D. Henderson, Eds., Hawaii Institute of Geophysics, 205–215.Kalnay, E. M., and Coauthors,. . 1996: The NCEP/NCAR 40-Year Reanalysis Project.

,*Bull. Amer. Meteor. Soc***77****,**437–471.Kantha, L., and C. Tierney, 1997: Global baroclinic tides.

*Progress in Oceanography,*Vol. 40, Pergamon, 163–178.Kunze, E., R. W. Schmitt, and J. M. Toole, 1995: The energy balance in a warm-core ring's near-inertial critical layer.

,*J. Phys. Oceanogr***25****,**942–957.Large, W. G., and S. Pond, 1981: Open ocean momentum flux measurements in moderate to strong winds.

,*J. Phys. Oceanogr***11****,**324–336.Levitus, S., and T. Boyer, 1994:

*World Ocean Atlas 1994*. Vol. 4:*Temperature,*NOAA Atlas NESDIS 4, 117 pp.McDermott, D. A., D. E. Harrison, and N. K. Larkin, 1997: An intercomparison of near-surface wind products over the ocean on monthly mean and longer time scales, 1985–1995. NOAA Tech. Memo. ERL PMEL-110, 163 pp.

Morozov, E. G., 1995: Semidiurnal internal wave global field.

,*Deep-Sea Res***42****,**135–148.Munk, W. H., 1966: Abyssal recipes.

,*Deep-Sea Res***13****,**707–730.Munk, W. H., and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing.

,*Deep-Sea Res***45****,**1977–2010.Nilsson, J., 1995: Energy flux from travelling hurricanes to the internal wave field.

,*J. Phys. Oceanogr***25****,**558–573.Paduan, J. D., R. A. de Soerke, and R. A. Weller, 1989: Inertial oscillations in the upper ocean during the Mixed Layer Dynamics Experiment (MILDEX).

,*J. Geophys. Res***94****,**4835–4842.Pollard, R. T., 1980: Properties of near-surface inertial oscillations.

,*J. Phys. Oceanogr***10****,**385–398.Pollard, R. T., and R. C. Millard, 1970: Comparison between observed and simulated wind-generated inertial oscillations.

,*Deep-Sea Res***17****,**153–175.Polzin, K. L., J. M. Toole, J. R. Ledwell, and R. W. Schmitt, 1997: Spatial variability of turbulent mixing in the abyssal ocean.

,*Science***276****,**93–96.Sjöberg, B., and A. Stigebrandt, 1992: Computations of the geographical distribution of the energy flux to mixing processes via internal tides and the associated vertical circulation in the ocean.

,*Deep-Sea Res***39****,**269–291.Swail, V. R., and A. T. Cox, 2000: On the use of NCEP–NCAR reanalysis surface marine wind fields for a long-term North Atlantic wave hindcast.

,*J. Atmos. Oceanic Technol***17****,**532–545.Thomson, R., and W. Huggett, 1981: Wind-driven inertial oscillations of large spatial coherence.

,*Atmos.–Ocean***19****,**281–306.Wunsch, C., 1998: The work done by the wind on the oceanic general circulation.

,*J. Phys. Oceanogr***28****,**2332–2340.

Mean flux, Π(*H*), area, *A,* and total dissipation, *D* = Π*A,* by basin. All means are computed over 50°S–50°N, and 1 TW = 10^{12} W. The total area of the world oceans is 3.2 × 10^{14} m^{2}

^{1}

This approximation cannot always be justified solely on the basis of scale analysis. A nighttime convection cycle, for example, can substantially modulate mixed layer depth. To address the resultant errors, I ran simulations with and without a diurnally cycling mixed layer. When *H* was varied diurnally by a (very large) factor of 5 about its mean value, the flux time series were affected somewhat, but the annual mean flux values were affected by less than 50%: much less than the errors resulting from wind stress uncertainties. The errors were random and without bias. Apparently, the long-tail distribution of wind stress fluctuations [section 3b(3)] causes them to dominate the flux inputs, even when *H* varies appreciably.

^{2}

The mixed layer depth is computed as the shallowest occurrence of *σ*_{θ} > 0.125 kg m^{−3}.

^{3}

The skill of a data-assimilating model should deteriorate in data-poor regions (such as the Southern Ocean). Therefore, this analysis is an optimistic one. A better comparison would result using buoys that are not assimilated into the reanalysis model. On the other hand, the requirement of high coherence is probably overly stringent: the specific timing of wind events may be slightly off in the model, lowering the coherence. But, as long as the spectral levels are similar in the inertial band (a weaker criterion), the mean flux should not be significantly impacted.

^{4}

The former two values are computed by extrapolation from the North Pacific.