fairly well into the geostrophic and ageostrophic components. The energy input into the geostrophic current calculated from a numerical model for this period is 0.87 TW; almost the same as that deduced from satellite data.

Energy input through the surface geostrophic current can be fed directly into the large-scale current, and can thereby be efficiently turned into GPE through the vertical velocity conversion term.

Surface ageostrophic currents can be described in terms of the Ekman theory. The horizontal momentum equations, including the time-dependent term and the geostrophic flow, are ut - f v = -pSx/P0 + (Auz)z, vt + fu = -ps,y/p0 + (Avz)z (3.112)

where u = ug + ue, v = vg + ve are the sum of geostrophic velocity and ageostrophic velocity in the Ekman layer, p0 is the reference density, and ps = ps(x, y) is the surface pressure associated with large-scale circulation. The geostrophic velocity satisfies ug = k x Vps/f p0. The corresponding boundary conditions are

Aue,z lz=0 = Tx/P0, Ave,z |z=0 = Ty/P0, (ue, ve) ^ 0, at z ^ -TO (3.113)

where the lower limit is to be understood as the base of the Ekman layer, and within the Ekman layer the vertical shear of the geostrophic velocity is negligible.

Multiplying these two equations by u and v, respectively, and integrating the result over the depth of the Ekman layer, we obtain

where r0

S = Sg + Se, Sg = TXUg + Ty Vg, Se = TXUe (0) + Ty ve (0) (3.116)

where E is the total KE of the Ekman layer, S the rate of wind energy input, P the rate of pressure work by the current integrated over the Ekman layer, and D the rate of dissipation integrated over the Ekman layer. Note that Ue = p0f°OQ u^dz = —kx.x/f, so

i.e., pressure work done by the Ekman transport is exactly the same as the wind stress work on the surface geostrophic currents. The pressure work done by the Ekman transport is also related to the GPE generated by Ekman pumping:

ff PdA = H Ue • Vps/P0dA =- ff Weps/p0dA + fi Ue • Udl (3.119)

where A is the area of the ocean. Thus, Eqn. (3.114) can be further reduced to

Therefore, we conclude that wind stress energy input to the surface geostrophic currents is equal to the increase in GPE in the world's oceans via Ekman pumping, including coastal upwelling/downwelling. On the other hand, wind stress energy input to the surface ageostrophic current is used to maintain the Ekman spiral via the vertical turbulent dissipation in the Ekman layer.

The exact amount of wind stress energy input through the Ekman spiral is estimated as 2.4 TW (Wang and Huang, 2004a) for frequencies greater than 1/(2 days). The spatial pattern of this energy source is quite similar to that of the energy input to surface waves, detailed in the next subsection. In addition, there exists a large amount of energy input through the near-inertial waves, which is due to the resonance at a frequency of m = —f. Alford (2003) used a slab model and obtained an estimate of 0.47 TW. However, Plueddemann and Farrar (2006) argued that a slab model may give rise to a near-inertial energy input that is twice as much as that obtained from a more accurate mixed-layer model. Thus, a more realistic value might be 0.23 TW.

The above estimate is based on the smoothed wind stress data, excluding the contribution from strong nonlinear events such as hurricanes and typhoons. Using tropical cyclone data from 1984 to 2003, Liu et al. (2008) estimated that there is an additional contribution due to hurricanes/typhoons of 0.1 TW to the surface currents, including 0.03 TW to the near-inertial waves.

The convergence of the Ekman flux gives rise to a pumping velocity at the base of the Ekman layer we, which is responsible for pushing the warm water into the subsurface ocean and thus forming the main thermocline in subtropical oceans. Ekman pumping sets up wind-driven circulations and an associated bow-shaped main thermocline in the world's oceans. During this process GPE is increased, which is a very efficient way indeed of converting KE into GPE.

Another major channel of energy input from wind and sea-level atmospheric pressure into the ocean is through surface waves. Wind stress drives surface waves in the oceans, and this energy input can be treated by means of the form drag of the atmospheric boundary layer. Wind energy input to surface waves can be estimated as

where C is the effective phase speed, u*a = Vtfpa and u*w = Vrfpw, pa and pw are the density of air and water, and t is the stress at the air-sea interface. The corresponding value of C can be determined from field experiments. Observation data fit the formula Wwaves = 3.5paula. Since u*a depends on both wind and sea state, the effects of waves on Wwaves is implied here. To have the best fitting for the observational data, an empirical formula is proposed:

where A is the empirical coefficient representing the energy flux factor, C /u*a:

where Cp* = Cp/u*a is the wave age, and Cp is the phase velocity at the wind sea peak frequency.

Using the smoothed NCEP (National Centers for Environmental Prediction) wind stress data, the total contribution to the global oceans is estimated as 60 TW (Wang and Huang, 2004b) (Fig. 3.11). How this huge amount of energy is distributed and dissipated in the oceans remains unclear at this time. It is roughly estimated that 36 TW of this energy is dissipated locally through wave breaking, turbulence, and internal wave generation in the upper ocean; about 20 TW is propagated to remote areas in the form of long waves, which gradually dissipate their energy. Eventually, these long waves break and dissipate the rest of their energy in the shallow/marginal seas and along the beaches of the world's oceans.

The strong nonlinear events in the atmosphere are smoothed out in the low-resolution wind stress dataset. The total contribution of subtropical cyclones, hurricanes, and typhoons to surface waves is estimated as 1.6 TW (Liu et al., 2008). Since the total amount of 60 TW mentioned above is the result of rounding up from a number slightly lower than 60 TW, we

1000 500 0 GW/degree

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