## 1. Introduction

To maintain the quasi-steady circulation in the ocean, a source of mechanical energy is required to balance the loss of mechanical energy from dissipation. Thus, the distribution of energy sources and sinks dictates the strength of circulation and its variability (Munk and Wunsch 1998; Huang 1998, 1999). This fundamental rule has been overlooked in the past. In fact, a common practice in the study of oceanic general circulation, either theoretically or numerically, has been based on certain arbitrary choices of vertical (diapycnal) mixing parameterization, without checking whether the external mechanical energy required to sustain such parameterization exists in the ocean.

Energetics of ocean circulation were discussed in pioneering works by Faller (1966) and Lueck and Reid (1984). However, many important terms of global energetics remain unknown. Faller (1966) estimated that wind stress contribution to local mixing is about 8 TW and to large-scale circulation is about 1 TW. Using a single sentence, Lueck and Reid (1984) stated that the total amount of energy input to the ocean by wind stress is 2%–10% of the downward flux of energy (510 TW) in the atmospheric planetary boundary layer.

It is only within the past few years that the oceanography community has come to realize the vital importance of external sources of mechanical energy of the ocean circulation. The contribution from tidal dissipation is estimated at 0.9 TW for the deep ocean by Munk and Wunsch (1998). The contribution from wind stress to geostrophic current is estimated as 1.3 TW by Wunsch (1998). The contribution to gravitational potential energy (GPE hereinafter) from geothermal heating is small, about 0.05 TW (Huang 1999, 2002). The contribution from wind stress to the near-inertial motions in the upper ocean has been recently estimated at 0.5 TW (Alford 2003) or 0.7 TW (Watanaba and Hibiya 2002). (These authors disagree on details of the method; however, the slab models used in their studies are crude approximations for the complex processes in the ocean, and the difference in the energy estimate is too slight to be of concern.)

We recently extended the Sandstrom theorem (Sandstrom 1916) in two ways. First, we argue that convective adjustment induced by cooling (buoyancy loss) can lead to a substantial amount of GPE loss in the mixed layer, which is about 0.24 TW (Huang and Wang 2003, unpublished manuscript). Second, a large amount of GPE, estimated at 1.1 TW, is lost by conversion of eddy energy through baroclinic instability (Huang and Wang 2003, unpublished manuscript).

**and**

*τ***U**

_{0}are spatially averaged tangential stress and surface velocity for frequency much lower than the surface waves,

**′ and**

*τ***u**

^{′}

_{0}

*p*′ and

*w*

^{′}

_{0}

**U**

_{0}

**U**

_{0,G}

**U**

_{0,AG}

## 2. Energetics of the Ekman layer

### a. Energy balance

*u*

_{t}

*fυ*

*Au*

_{z}

_{z}

*υ*

_{t}

*fu*

*Aυ*

_{z}

_{z}

*A*is the vertical momentum diffusivity.

*u*and

*υ*and then integrating the result over the depth of the ocean, we obtain the energy balance

*E*

_{t}

*S*

*D,*

*u** =

*u*

_{g}+

*u*

_{e}and

*υ** =

*υ*

_{g}+

*υ*

_{e}are the sum of geostrophic velocity and ageostrophic velocity in the Ekman layer, and

*p*

_{s}=

*p*

_{s}(

*x,*

*y*) is the surface pressure associated with large-scale circulation. The geostrophic velocity satisfies

**u**

_{g}= −

**k**× ∇

*p*

_{s}/

*ρ*

_{w}, where

**k**is the unit vector in the vertical direction. The corresponding boundary conditions arewhere the low limit should be interpreted as the base of the Ekman layer, and within the Ekman layer the vertical shear of the geostrophic velocity is negligible.

*u** and

*υ** and integrating the result over the depth of the Ekman layer leads to

*E*

_{t}

*S*

*P*

*D,*

*P*

**U**

_{e}

*p*

_{s}

*ρ*

_{w}

*τ***u**

_{g}

*S*is the area of the ocean and

**n**is the unit vector normal to the direction of integration

*l.*

Equations (11) and (10) state that wind stress works on the surface geostrophic currents and is equal to the increase of GPE in the World Ocean from Ekman pumping, including coastal upwelling/downwelling (Gill et al. 1974; Fofonoff 1980). On the other hand, wind stress energy input to the surface ageostrophic current is used to maintain the Ekman spiral through the vertical turbulent dissipation in the Ekman layer.

The velocity field in the upper ocean is complicated. Observations indicate that, for time scales longer than a few weeks, the velocity profile in the upper ocean has structure that is closer to the spiral shape as described by the classical theory of the Ekman layer (Price et al. 1987; Plueddemann and Weller 1999). On the other hand, for time scales of near-inertial motions, the velocity structure is best described by a slab model. Wind energy input through near-inertial motions has been discussed in terms of a slab model by D'Asaro (1985), Watanabe and Hibiya (2002), and Alford (2003).

Thus, we will calculate the wind energy input to the surface current, using a classical 1D model for subinertial periods.

### b. The steady solution

*W*

*τ*

^{2}

*ρ*

_{w}

*fD*

_{E}

### c. Time-dependent wind forcing

*q*=

*u*+

*iυ*and

*τ*=

*τ*

^{x}+

*iτ*

^{y}and use the Fourier expansionwhere

*ω*

_{n}

*πn*

*T.*

*n*th component of the momentum equation isThe total amount of wind energy input isIt is clear that, for high-frequency wind stress, the contribution to the energy is modified by a factor of

*f*/(

*f*+

*ω*

_{n})|

*f*< 0 in the Southern Hemisphere, and so the contribution from anticlockwise (clockwise) wind is enhanced (reduced).

## 3. Wind energy input to the surface Ekman layer in the oceans

### a. Choice of Ekman depth

Although the Ekman theory has been the backbone of modern dynamical oceanography, the Ekman spiral predicted by classical theory has not been exactly verified. For a steady wind stress, the velocity vector on sea surface from classical theory is 45° to the right of wind stress (in the Northern Hemisphere), and the velocity vector rotates in the form of a spiral in a vertical direction. On the other hand, observations indicate that the angle between surface wind stress and surface drift velocity vector is in the range between 5° and 20° (Cushman-Roisin 1994). Recent observations also indicate that the surface velocity lies at more than the predicted 45° to the right of the wind. More important, the observed current amplitude decreases at a faster rate than it turns to the right; that is, the observed velocity profiles in the Ekman layer seem “flat” (Chereskin and Price 2001).

Despite efforts to find solutions that fit the observations, all models within the framework of the classical laminar theory (with a diffusivity independent of time) fail to produce the observed flat spiral. Other important dynamic processes, such as buoyancy flux through the air–sea interface, stratification, diurnal cycle, or even Stokes drift, may have to be included to explain the observed structure of the Ekman layer (Price and Sundermeyer 1999). Such models are clearly beyond the scope of this study.

Because of complicated dynamic processes in the upper ocean, parameterization of turbulent dissipation in the upper ocean remains a great challenge. Observations indicate that, in a thin surface layer immediately below the sea surface, waves and turbulent activities are strong, and so dissipation is near constant or slightly increases with depth; however, the dissipation rate declines with depth below this shallow layer. Direct observations in the California Current indicate that turbulent diffusivity declines exponentially for depth below 20 m (Chereskin 1995). Terray et al. (1996) carried out field observations and found that the dissipation rate is higher and roughly constant in a near-surface layer, but below this layer the dissipation rate decays as *z*^{−2}. This result is further refined as a −2.3 power law by Terray et al. (1999).

A crude model to represent this complexity is a two-layer model with a power law of vertical diffusivity in each layer (see the appendix). A simple linear profile *A* = *α*|*z*| was used in previous studies, for example, Madsen (1977). Such a profile is questionable because it is inconceivable that turbulent diffusivity is zero at the sea surface (Huang 1979). For the present case, we choose a linear profile for the surface layer, *n*_{1} = 1, starting with a finite diffusivity on the sea surface. For the second layer, it is found that an inverse power profile with *n*_{2} = −0.7 has a best fit for the diffusivity diagnosed by Chereskin (1995). The roughness and significant wave height can be estimated through an empirical relation *z*_{01} = 0.85*H*_{s} (Terray et al. 1999), where *H*_{s} is significant wave height. For the case of long fetch, *H*_{s} can be estimated as *H*_{s} = 0.30*U*^{2}/*g* (Wilson 1965), where *U* is wind speed and *g* is the gravitational acceleration; *A*_{1} can be calculated as *A*_{1} = *κu*∗, where *κ* is von Kármán's constant and *u*∗ is the friction velocity in the water (Craig and Banner 1994).

From this solution, the vertical diffusivity average *A*_{m} over a depth *H* = *cD*_{e} (*D*_{e} is the *e*-folding depth of horizontal velocity) can be calculated and used in the classical model with a constant diffusivity. For typical cases, the work calculated from these two models is very close. Thus, in the following analysis, we will assume that vertical diffusivity is constant within the Ekman layer (Fig. 1).

*A*or the Ekman layer depth

*D*

_{E}. In this study, we choose an empirical formula:where

*γ*≈ 0.25–0.4 is an empirical constant, determined from observations (Coleman et al. 1990; Price and Sundermeyer 1999). This relation implies that the equivalent vertical diffusivity at a given location is not a constant; instead, it depends on the wind stress and is inversely proportional to the Coriolis parameter. Using six sets of observations, including LOTUS-3 and LOTUS-4 (Long-Term Upper Ocean Study; Table 1 in Schudlich and Price 1998), SWAPP (Surface Waves Processes Program; Fig. 4a in Plueddemann and Weller 1999), Krauss (1993), Eastern Boundary Current (EBC), and 10°N (Fig. 11b in Price and Sundermeyer 1999), we found the best-fit value is

*γ*≈ 0.5 (Fig. 2). Thus, energy input to the Ekman layer iswhere

*u*

_{∗w}is defined by the time-mean wind stress. From this formula, the energy input to the surface Ekman layer is roughly proportional to the third power of the frictional velocity, with a relatively minor dependency on the Coriolis parameter and frequency of the wind stress. It is also clear that, when

*ω*→ −

*f,*resonance of the near-inertial motions can substantially boost the energy input, however, our focus in this study does not really cover this regime, as discussed above.

Note that Eq. (16) is based on the classical Ekman spiral, which predicts an angle of 45° to the right of the wind stress (Northern Hemisphere). As discussed above, field measurements indicate a much smaller angle in deviation; thus, using this formula we may underestimate the energy input.

### b. Application to the World Ocean

The 1D model has been applied to the daily daily-mean wind stress data from the National Oceanic and Atmospheric Administration/National Centers for Environmental Prediction (NCEP)–National Center for Atmospheric Research (NCAR) from 1948 to 2002. These wind stress data have a zonally uniform spacing of 1.875° and a meridionally nonuniform spacing that varies from 1.89° at the Poles to 2.1° near the equator.

Using the complex-variable fast Fourier transform to the time series at each grid point, the cutoff frequency is *ω* = 0.5 cycle day^{−1}. Thus, our calculation basically does not cover the energy input to near-inertial motions. Recent studies on wind energy input to near-inertial motions in the oceans give an estimate of 0.5–0.7 TW (Watanabe and Hibiya 2002; Alford 2003), and the calculation discussed here covers the contribution to a steady-state and subinertial frequency.

Wind energy input to the surface Ekman layer is very strong over three regimes (Fig. 3): 1) the Antarctic Circumpolar Current; 2) the South Indian Ocean, in particular; and 3) subpolar basins in the North Pacific and North Atlantic Oceans, which are coincident with the location of the storm track. On the other hand, this energy input is relatively weak along the equatorial band.

The total amount of energy from our calculation is 2.3 TW for the global oceans; see Table 1. Note that energy input to the Southern Hemisphere is 80% larger than to the Northern Hemisphere, probably because of strong currents driven by wind stress over the Southern Ocean, especially the Antarctic Circumpolar Current. The input through the steady component of wind stress in the Southern Hemisphere (0.4 TW) is much larger than in the Northern Hemisphere (0.14 TW), which is probably due to the strong steady wind stress over the Southern Ocean.

As a comparison, we also performed a calculation based on the assumption of a constant depth of the Ekman spiral, *D*_{E} = 50 m. Note that under such an assumption energy input would be proportional to the fourth power of friction velocity instead of the third power implied in Eq. (16). The total amount of energy and its distribution in the space and frequency domains change accordingly. The global sum of energy input coincidently remains almost the same, 2.27 TW. However, the spatial distribution is different; there would be a peak of energy input near the equator, which may not be realistic (figure not shown).

In the Northern Hemisphere, energy input by the clockwise rotating wind stress is 0.39 TW, which is slightly larger than that from the anticlockwise rotating wind stress (0.3 TW); see Table 1. In the Southern Hemisphere, energy input by the anticlockwise rotating wind stress is 0.60 TW, which is much larger than that from the clockwise rotating wind stress (0.44 TW). The difference in energy inputs from clockwise and anticlockwise wind stress is, at least partially, due to the factor of 1 + *ω*_{n}/*f* in the denominator of Eq. (16).

There are several peaks in the energy power spectrum, which correspond to periods of 1, 1/2, and 1/3 yr (Fig. 4). The most interesting phenomenon is the maximum in energy flux at the annual frequency along the coast of Somalia, which is closely related to the reverse of the Somali Jet, and in the western boundary of the subtropical gyre of the North Pacific. Such spatial patterns of wind energy input may have important implications for mixed layer dynamics at these locations (Fig. 5).

Wind energy input to the surface Ekman layer varies greatly from year to year (Fig. 6). Over the past 54 yr, the total energy input has increased from 2.1 to 2.4 TW, and this change is clearly related to the increase of wind stress over the same period of time. The increase of wind stress over the past 50 yr may be partially related to the improvement of instrumentation. In addition, climate change may also contribute. However, the exact nature of such a long-term trend in wind stress is beyond the scope of this study. For the period of 1997–2002, the mean energy input rate is 2.43 TW.

## 4. Conclusions

Using a combination of a simple analytical formula for the Ekman spiral and an empirical formula for the Ekman depth, we have calculated the energy input through the surface ageostrophic current from the Ekman spiral. For the period of 1997–2002, the rate of energy input for the subinertial frequency range is estimated as 2.4 TW; thus, in addition to the near-inertial wave contribution of 0.5–0.7 TW calculated by Alford (2003) and Watanabe and Hibiya (2002), the total energy input to the Ekman layer is estimated as 3 TW.

Energy input through the subinertial range is likely to be spent on supporting turbulence and mixing in the Ekman layer and thus maintains the velocity and stratification field in the upper ocean. This is an important part of the oceanic general circulation, although this energy input does not seem to contribute to the large-scale circulation directly. However, there might be interaction between the Ekman layer and fluids below, so that some part of this energy may be used to support turbulence and mixing below the Ekman layer; however, the details of these interactions remain unclear at this time.

On the other hand, wind stress energy input to the surface geostrophic current, estimated as 1.3 TW by Wunsch (1998), is mostly converted into the GPE through Ekman pumping. However, there is an intimate interplay between the geostrophic and ageostrophic components of the velocity in the upper ocean. Thus, energy input through the geostrophic currents and ageostrophic currents shows the currents are closely related to each other.

Our estimate should be interpreted with caution. First, our calculation is based on the classical spiral, assuming a constant viscosity. It is well known that circulation and water mass properties in the upper ocean can also be described in terms of a slab model, especially for a time scale comparable with the inertial period. The choice of a spiral model used in our calculation is, of course, an idealization. Further study is needed to explore this separation in the frequency domain more carefully.

Second, choice of the Ekman layer depth is primarily based on an empirical formula. Although the case with a constant Ekman depth was also briefly discussed, a close examination on the effect of Ekman layer depth is needed.

Third, many complicated processes have been omitted in this discussion, such as the effect of stratification, surface waves, and heat and freshwater fluxes through the air–sea interface. Including these processes will make the calculations more accurate and meaningful.

Last, we emphasize the approximate nature of our calculation. All numbers presented in this study are offered with some uncertainty. It is anticipated that these calculations will serve as a first step toward understanding the energy input through the Ekman layer and its dynamical consequence.

## Acknowledgments

Author WW was supported by The National Natural Science Foundation of China through Grant 49976003. RXH was supported by the National Science Foundation through Grant OCE-0094807 to the Woods Hole Oceanographic Institution. Reviewers' comments helped to clean up the presentation.

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## APPENDIX

### Ekman Spiral of a Two-Layer Model

*A*

_{2}=

*A*

_{1}(

*z*

_{01}+

*h*)

^{n1}

*n*

_{1}and

*n*

_{2}are arbitrary real numbers;

*z*

_{01}and

*z*

_{02}are roughnesses at the surface and interface, respectively. Through straightforward but tedious algebraic manipulation, the horizontal velocity of the Ekman spiral is as follows:where subscripts 1 and 2 represent the upper and lower layers,

*J*

_{b}(

*s*) and

*Y*

_{b}(

*s*) are the

*b*th-order Bessel functions of the first and second kind,with

*s*

_{jh}=

*s*

_{j}(

*h*),

*j*= 1, 2. The wind energy input to the Ekman spiral isAs compared with the energy flux based on a classical Ekman solution with a constant diffusivity

*A*

_{m}, the ratio is

Distribution of the wind energy input to the surface Ekman layer (TW), with cutoff frequency at ω = 0.5 cycle day^{−1}

^{*}

Woods Hole Oceanographic Institution Contribution Number 10868.