1. Introduction

Mechanical energy input through wind stress is the most important component of energetics of the global oceans. Faller (1966) made an attempt to estimate mechanical energy sources of the World Ocean, including wind stress, tidal dissipation, and other terms. Lueck and Reid (1984) estimated that the total amount of downward mechanical energy flux in the atmospheric boundary layer is about 510 TW, and “about 2–10% of this downward energy flux eventually enters the ocean.” Thus far, there has been no reliable estimate of the total energy input from wind stress.

Using satellite data and a numerical model, wind energy input through the surface geostrophic current is estimated at 1.3 TW (Wunsch 1998). Wind stress energy input through the surface ageostrophic current consists of two parts: 1) Near-inertial motions are approximately 0.5–0.7 TW as estimated by Watanabe and Hibiya (2002) and Alford (2003) (although these authors seem to be in dispute, the difference in the energy estimate is small). 2) The contribution for subinertial motions is estimated at 2.4 TW (Wang and Huang 2004). Thus, the total amount of wind energy input through the surface Ekman drift is about 3 TW.

The most important term is through surface waves. Many papers and books have been published about surface waves (e.g., Phillips 1977), including in situ and laboratory observations, theoretical studies, and numerical simulations. Energetics of surface waves is one of the most important aspects of surface waves. Nevertheless, there has been no estimate of the total energy input for global oceans. Our goal in this study is to estimate wind stress energy input to surface waves in the World Ocean.

2. Energy input to the surface waves

Using data collected through the Water–Air Vertical Exchange Studies (WAVES) experiment, Terray et al. (1996; their Table 1) calculated the corresponding value of c. This set of data corresponds to waves of a relatively young age, 4.3 < c_p∗ < 7.4, where c_p∗ = c_p/u_∗a is the wave age and c_p is the phase velocity at the wind-sea peak frequency. Drennan et al. (1996; their Table 1) used data collected through the Surface Waves Dynamics Experiment (SWADE) to obtain another set of c. This set of data corresponds to nearly mature waves, 13.5 < c_p∗ < 28.6. Thus, these two sets of data represent good coverage of wave age, with a gap at 7.4 < c_p∗ < 13.5.

Because u_∗a depends not only on wind but also on sea state (Terray 1996), the influence of waves on W_waves is implicitly implied in (2). On the other hand, many existent empirical formulas of u_∗a depend on wind speed only and can give rise to acceptable results (Wu 1982; Watanabe and Hibiya 2002); thus, according to (2), the effect of sea state on W_waves may be relatively weak. The insensitivity of energy flux to sea state was also found by Craig and Banner (1994), when dealing with sea surface dynamical processes, using a “level 2½” turbulence closure scheme.

In addition, the energy flux factor c/u_∗a depends on wave ages. Scaling values from Fig. 6 of Terray et al. (1996), which shows the relation of c/c_p and u_∗a/c_p, produced a relation between cu³_∗a/c³_p and u_∗a/c_p, as shown in Fig. 2. Because cu²_∗a/c³_p = W_waves/ρ_ac³_p, this figure can be used to infer the relation between W_waves and c_p∗. The mean relative errors between the observation data and (2) (indicated by dashed line) are 20%– 65% for the first five datasets listed in the figure and 120% for the last one.

To get the best fit of the experimental data in Fig. 2, we suggest an empirical formula

W_wavesAρ_au³_∗a(3)

where A is the empirical coefficient representing the energy flux factor c/u_∗a:

indicated in Fig. 2 by the bent solid line.

It is clear that (3) and (4) fit the experimental data better than (2), and the mean relative errors are reduced to about 15%–24% for the first five datasets and 60% for the last one listed in Fig. 2.

However, it is difficult to measure the value of c_p∗. Thus, two other parameters, the nondimensional spectral peak wave period T_p∗ = T_pg/u_∗a and the nondimensional significant wave height H∗ = H_sg/u²_∗a, can be used, where T_p is spectral peak wave period, H_s is significant wave height, and g is the gravitational acceleration.

For deep-water waves, the dispersion relation is ω = gk, where k is the wavenumber, and the relation between c_p∗ and T_p∗ is c_p∗ = (1/2π)T_p∗; thus, the wind energy flux factor is

The wave age c_p∗ and the nondimensional significant wave height H∗ are related through H∗ = c^2/3_p∗ (Toba 1974; Maat et al. 1991), and so we have the relation between A and H∗:

3. Wind energy input to surface waves for the global oceans

b. Calculations

Wind energy to surface waves is calculated in two ways:

1) Quasi-steady calculation

The annual mean wind energy input to the surface waves is defined as the mean of the energy input based on daily mean wind stress; that is,

where N is the total number of days of wind stress data and w_waves,I, is the wind energy input for the ith day.

2) Spectral distribution of energy

It is useful to explore the spectral distribution of wind energy input to surface waves. First, wind stress can be expanded into the Fourier series

where

Wind energy input into surface waves is

W_wavesτtCt(9)

where

and u∗ is the mean friction velocity at a given station. Wind stress energy input corresponding to the nth component is

where

T²_nT²_x,nT²_y,n

and the rate of energy input is

These formulas were used to estimate the wind energy input to the surface waves in the World Ocean (Table 1). For example, the spatial distribution of energy input calculated from (2) and (7) and the latitudinal distribution are shown in Fig. 3. It is clear that the major site of energy input is within the Antarctic Circumpolar Current, which consists of more than 50% of the total input. In addition, the storm tracks in the Northern Hemisphere are also major sites of wind energy input.

As a next step, the wind energy input over the past 54 yr (1948–2002) is examined. Energy input distribution among different frequencies is calculated from (11) and assuming A = 3.5 (Fig. 4). There are clearly four peaks, corresponding to periods of 1, 1/2, 1/3, and 1/4 yr, with energy input of 1.50, 0.20, 0.03, and 0.01 TW. In comparison, the rate of energy input corresponding to the mean wind stress is 38.4 TW. The total energy input averaged over the past 54 yr is 52.8 TW, which is noticeably smaller than the values obtained for the time period of July 1997–June 2002.

Wind energy input to the surface waves increases about 20% over the past 50 yr (Fig. 5). Although such an increase may be partially due to the systematic errors induced by the introduction of new instruments, the general trend of increase may be related to wind stress, especially after 1976, which is clearly related to the systematic change in the atmospheric circulation.

Most of the energy input is through the Southern Hemisphere, which consists of 2/3 of the total input, while the energy input into the Northern Hemisphere is about 1/3. In addition, the interannual variability of the energy input in the Southern Hemisphere is relatively large.

4. Conclusions

Three estimates of energy input to surface waves, based on an empirical formula and numerical simulations, gave similar results. Thus, a tentative conclusions is that the total amount of wind energy input through surface waves is about 60 TW (for the time period 1997– 2002). These estimates are preliminary in nature; however, this study will serve as a first step toward a more accurate balance of the mechanical energy in the World Ocean.

Energy input through the surface waves is at least 20 times that through the surface Ekman drift and the geostrophic current. However, this result does not necessarily mean energy input through the surface waves is the dominating term of mechanical energy supporting gravitational potential energy in the World Ocean.

Most mechanical energy input in surface waves may be dissipated within the surface layer of the oceans through wave breaking, and only a small fraction may be transferred to other forms and locations by the mechanisms of wave–wave interaction and wave–current interaction. Because total energy flux is huge, the energy transferred to other forms is not negligible when compared with other energy sources like wind stress to geostrophic current and Ekman drift, and so on, which will contribute to mixing of the global oceans. Long surface waves may transfer energy generated by local wind stress into remote sites where dissipation can be much stronger, such as beaches. However, the balance of the surface energy on the local scale remains one of the most challenging questions.