Iff v iff pUnontidalVv jjj putidai vrdv 3105

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The sign of the first term on the right-hand side of Eqn. (3.105) is unclear; however, tidal flows are driven by tidal force, so that the last term in Eqn. (3.105) is positive. Thus, the last two terms in Eqn. (3.101) represent a positive contribution to the non-tidal motions, and they correspond to similar terms associated with the barotropic tides discussed in Section 3.5.3.

3.5.5 Interpretation of energy integral equations

Wind energy sources for the kinetic energy

In addition to the contribution from tides as discussed above, the KE of oceanic motions comes primarily from surface forcing. Averaged over a long period, the surface integral in Eqn. (3.89) is reduced to jj (-pu + fi,uU) ■ nds & jj [—pw + (utx + vTy)]zz=-H dxdy (3.106)

Thus, the surface integral consists of a kinetic energy source from the upper surface and a kinetic energy sink from the seafloor.

The kinetic energy source on the upper surface is further separated into contributions from two categories: large spatial scale and low frequency versus small spatial scale and high frequency.

The energy contribution due to large-scale and low-frequency atmospheric pressure perturbations is the work done associated with the so-called atmospheric loading. The global sum of this term is small, estimated as 0.04 TW by Wang et al. (2006). Wind energy input to the surface currents can be classified as: (1) wind stress work on the surface geostrophic currents, estimated as 0.88 TW (Wunsch, 1998); and (2) wind stress work on the surface ageostrophic currents (the Ekman layer), estimated as 3 TW (Wang and Huang, 2004a).

In the meantime the contribution due to small-scale and high-frequency atmospheric pressure perturbations and wind stress is the energy input into the surface waves, estimated as 60 TW (Wang and Huang, 2004b).

The kinetic energy sink from the seafloor boundary represents energy dissipation due to bottom friction and form drag, whose value remains uncertain.

Sources of internal energy

In the internal energy balance equation, the first term on the right-hand side of Eqns. (3.9o) or (3.1o2) constitutes only a minor source of internal energy due to evaporation and precipitation. The major source/sink of internal energy comes from the exchange with the atmosphere above and geothermal heat from the seafloor below, as expressed in the surface integral —(q + js-^) ■ Jds, where the fluxes can be parameterized as h h q + ^ F = Frad — pCpKj V T — — pks V S (3.1o7)

where the first term on the right-hand side is the solar radiation; the second term is the turbulent flux, which is normally parameterized as a downward temperature gradient flux. Similarly, one can parameterize the diffusive salt flux as a downward salinity gradient. Strictly speaking, the diffusive salt flux used in the formulae should be defined in the coordinates moving with the center of mass, and this can be in a form slightly different from the traditional definition of salt flux used in numerical models. In addition, a salinity (temperature) gradient can induce temperature (salinity) diffusion. However, the diffusion parameterization in currently used numerical models is too crude to take into account such physics.

Although salt diffusion may contribute to internal energy for each individual grid cell in the ocean interior, there is no salt flux across the air-sea interface. Thus, the corresponding flux term is zero at the sea surface, assuming that freshwater flux across the air-sea interface is balanced and has the same temperature. In this discussion, the salt flux across the water-sea ice interface is excluded.

Of note is the fact that the sum of the internal energy fluxes through the system is not self-balanced, even though the system reaches a quasi-steady state. If all these equations are added up, the exchange terms in Eqns. (3.1o1, 3.1o2, 3.1o3) cancel out exactly. Thus, for a quasi-steady state, the energy balance is to be set up among the inputs, including internal energy from the sea surface and seafloor, plus wind and tidal energy inputs. As required by the first law of thermodynamics, energy should be conserved, i.e., the total amount of energy input should be equal to the amount of energy output. The only way for the ocean to lose energy is in the form of heat loss to the atmosphere; therefore the total heat loss to the atmosphere is the sum of the following terms: the heat absorbed by the ocean, the heat flux from geothermal heating (32 TW), and the net mechanical energy received by the oceans, including the mechanical energy source due to wind (64 TW) and tidal dissipation (3.5 TW). The net heat loss to the atmosphere, i.e., the difference between the outgoing heat flux and the incoming heat flux, is estimated as 99.5 TW.

The balance of GPE

The GPE plays an important role in the balance of mechanical energy in the ocean. From the balance of GPE (see Eqn. (3.1o3)), the only direct source term is that due to mass exchange through the air-sea interface, namely that from precipitation and evaporation;

however, this term turns out to be extremely small and negligible, as will be shown later. The contribution from wind stress manifests itself in the conversion term Po, which represents the transfer between the potential and kinetic energy. There are two pathways for the ocean to receive GPE.

1. Surface wind stress. As explained in detail in the first subsection of Section 3.6.1, wind stress energy input to the surface geostrophic current is converted into GPE in the world's oceans through Ekman pumping. This is a major source of GPE in the ocean. However, strong baroclinic instability associated with strong zonally oriented currents, such as the ACC, is a major sink of GPE.

2. Surface thermohaline forcing has to undertake a zigzag route to be converted into GPE. First it is converted into internal energy of water parcels, then it is converted into kinetic energy through expansion/contraction, and finally it can be converted to the gravitational potential energy through the Po term.

For a steady state, the total vertical mass transport averaged over a period of tides should be zero at any fixed level, so that Po vanishes, i.e., there is no net exchange between KE and GPE. However, for an ocean with a seasonal cycle, the GPE balance also goes through a similar cycle, in which the late winter cooling-induced convection appears as a major sink of GPE and the diapycnal mixing and other dynamical processes constitute the sources of GPE. As an example, a detailed analysis of GPE balance for the case with a regular seasonal cycle is discussed in Section 3.7.3.

If we neglect any contribution due to evaporation and precipitation, integration of Eqn. (3.1o3) leads to a statement of o = o. Thus, one seems to be led to the conclusion that the balance of GPE is a trivial problem, and the study of GPE for the large-scale circulation is neither interesting nor useful.

However, such a seemingly trivial relation may be one of the most important equations for the study of oceanic energetics. The term Po reflects synthetically quite a few physical processes. Averaging over a period of tides, this term can be rewritten as gp w = gpw + g p 'w' (3.1o8)

This equation can be interpreted in at least two possible ways, as follows.

First, the overbars are interpreted as the ensemble mean of the product of density and vertical velocity at each point, and the primes indicate the deviation from the ensemble mean due to turbulence and internal waves. Then the first term on the right-hand side of Eqn. (3.1o8) is the energy transfer between the GPE and KE of the mean state, while the second term is due to the turbulence and internal waves. When the second term is examined in further detail, we find two cases to be differentiated. Within the deep ocean interior with stable stratification, vertical mixing pushes heavy water upward (downward), so that p' and w' are both positive (negative); therefore this term is positive. This means that mixing due to turbulence and internal waves in the stably stratified ocean increases the GPE of the mean state. On the other hand, within the convective overturning zone, instantaneous stratification is unstable before the convective overturning takes place. During convective adjustment, dense and heavy water parcels (p' > 0) move downward (w' < 0); thus, p' and w' are of opposite signs, and convective mixing reduces the GPE of the mean state.

Second, the overbars are interpreted as the time and horizontal mean, and the primes stand for the deviation from such a mean. As shown in Section 3.7.3, the perturbation term can be interpreted as the meridional transport of GPE associated with advection.

In the above argument, we exclude the time-varying component of gravity. In fact, in a slightly different formulation, we can include the time-dependent gravity, and the relation corresponding to Eqn. (3.108) is gpw = gopw + gop 'w'+ gT p w (3.109)

The last term is an additional term due to tidal force, and here it appears as a net source of energy conversion.

3.5.6 An energy diagram for the world's oceans

World ocean circulation encompasses an immense range of spatial and time scales. Many of the basic elements of the energetics of the circulation remain unclear owing to the severe technical challenges of collecting data over such a huge dimension and under the extreme conditions in the open ocean.

Energy in the ocean can be roughly classified into four categories: GPE, KE, internal energy, and chemical potential (Fig. 3.8). In the study of the oceanic general circulation we further separate the GPE and KE into two parts: the energy contained in the mean state and that in meso-scale eddies, turbulence, and internal waves.

The oceans receive a huge amount of thermal energy through solar insolation, and exchange heat with the atmosphere in the forms of short-wave radiation, latent heat flux, sensible heat flux, and long-wave radiation. The total amount of internal energy in the ocean is immense, estimated as 20 YJ (2 x 1025 J). Internal energy can be converted into KE through expansion/compression, pV ■ v. However, the ocean's capability of converting internal energy into KE is very limited, as discussed in previous sections; thus, we will not discuss the balance of internal energy in this book. Readers who are interested in this subject can find much useful information in the book Physics of Climate by Peixoto and Oort (1992).

We include chemical potential as a form of energy, in addition to mechanical energy and internal energy. However, chemical potential in the ocean is directly linked to evaporation induced by the solar radiation. The ocean has chemical potential because seawater is a multi-component mixture. It is the existing concentration difference of salt in seawater that gives rise to the chemical potential. The total amount of chemical potential in the oceans is estimated as 3.6 x 1024 J. However, we should be aware of the fact that only the difference in chemical potential between different water parcels is dynamically active. The hydrological cycle in the form of evaporation and precipitation gives rise to salinity difference in the oceans. In fact, evaporation (precipitation) extracts (inputs) pure water from

Short-wave Latent

Short-wave Latent

Fig. 3.8 Energy diagram for the world's oceans.

(to) the seawater, acting as the source of chemical potential energy. A difference in salinity induces a difference in chemical potential, which is the driving force of salt mixing at the molecular level. As a matter of fact, however, salt mixing in the oceans is primarily regulated by turbulence and internal waves, which are much stronger than the molecular mixing.

The most important forms of energy of the oceanic circulation are kinetic and potential energy. The sources of KE of the mean state include tidal force and wind stress. However, wind stress also makes direct contributions to meso-scale eddies, turbulence, and internal waves, mostly through surface waves and other dynamical processes in the upper ocean.

We note that oceanic circulation takes place in the environment of a gravity field, and gravitational force is one of the major forces regulating the stratification and circulation. The total amount of GPE is immense; however, only a very small proportion of this energy is dynamically active. We will delay this discussion until the end of this chapter. GPE and KE can be converted into each other through vertical motions, which can be expressed as p gw.

A major part of the mechanical energy in the oceans is contained in meso-scale eddies, turbulence, and internal waves. In fact, it is estimated that the total energy associated with meso-scale eddies is two orders of magnitude larger than that associated with the mean currents. Unfortunately, there is no reliable estimate of the energy stored in oceanic motions in these forms.

At the smallest spatial and temporal scale, mixing plays a vital role in maintaining the stratification due to temperature and salinity. One of the major characteristics of stratified fluid is the following: vertical mixing in a stratified ocean pushes light water downward and heavy water upward, resulting in an increase in GPE of the mean state. Therefore, turbulent mixing in the oceans (due to internal wave breaking and small-scale turbulence) does not turn all mechanical energy into the so-called dissipation heat; instead, a small fraction of the turbulent KE, estimated as approximately 2o%, is fed back to the mean state on a global scale. It is imaginable that the efficiency of mixing varies greatly, depending upon the environmental conditions; however, its dynamical specification remains at the frontiers of research. This energy transform is indicated by the arrows at the bottom of Figure 3.8.

3.6 Mechanical energy balance in the ocean

3.6.1 Mechanical energy sources/sinks in the world's oceans

It has been shown in the previous section that mechanical energy sources and sinks play vital roles in maintaining/regulating the oceanic circulation; in this section we focus on the mechanical energy balance in the world's oceans. A sketch of mechanical energy distribution in the oceans (Fig. 3.9) shows that wind stress and tidal dissipation are the most important sources of mechanical energy driving the oceanic general circulation.

Fig. 3.9 Mechanical energy diagram for the ocean circulation.

Wind energy input

Wind stress on the surface of the ocean drives both surface currents and waves. The mechanical energy transferred from wind into the ocean, Wwind, is defined as Wwind = (aij ■ U) ■ u, where the stress tensor includes both the viscosity stress tensor and the pressure: aij = nij + Sijp. Thus, energy input from wind includes contributions from wind stress and the pressure. Although wind stress is more frequently discussed in relation to large-scale circulation, sea-level atmospheric pressure and its perturbations also affect the large-scale circulation.

It is difficult to separate wind stress from sea-level atmospheric pressure because they are closely related to each other; however, wind stress energy input can be roughly divided into the following components

Wwind = {oij ■ u) ■ n=TU ■ (Uo,g + Uo,ag) + TU' ■ U'0 + p'w'0 + pW0 (3.110)

where the overbar indicates average in space and time; the perturbations are defined in terms of the spatial and temporal scales of the surface waves; U, U0,g, U0, ag are the spatially and temporally averaged tangential stress, surface geostrophic and ageostrophic velocity, respectively; t' and U0 are the perturbations; p' and w'0 are perturbations of the surface pressure and velocity component normal to the surface; and p and W0 are the sea surface pressure and vertical velocity averaged over time scales much longer than the typical wave periods.

The first and second terms on the right-hand side of Eqn. (3.110) are the wind stress work on the quasi-steady currents on the surface, and quasi-steadiness is defined in terms of the time scale of the typical surface waves. The third term is work done by wind stress on surface waves, and the last two terms are the work done by atmospheric pressure. Energy contributions from these terms are discussed separately as follows.

Work input to surface geostrophic currents

Wind stress energy input through the surface geostrophic current is

where surface geostrophic currents can be calculated from U0,g = gk x Vn/f (where n is the sea surface height inferred from either satellite altimeter data or numerical models), except near the equator. The total amount of energy input is estimated as 0.88 TW (Wunsch, 1998). Although the wind energy input is positive around 40° N, the major energy input is through the Southern Ocean and the equatorial band (Fig. 3.10). In addition, the North Equatorial Counter Current (NECC) is a place of energy sink because the eastward current is against the easterlies there.

By means of numerical modeling, energy input to surface currents can be calculated, and this energy input for the period 1993-2003 is estimated as 1.16 TW (Huang et al., 2006). Away from a narrow band (within ±3° of the equator), the surface currents can be separated

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Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable.

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