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Lecture 5. Mechanical energy balance in the ocean 3/18/2006 10:19 PM

1. Mechanical energy sources/sinks in the world oceans Mechanical energy sources and sinks play vitally important roles in controlling the oceanic circulation, their distribution in the oceans is illustrated in Fig. 1. Wind stress and tidal dissipation are the most important sources of mechanical energy that drive the oceanic general circulation, and they will be discussed in details.

Fig. 1. Mechanical energy diagram for the ocean circulation, (Huang, 2004). A) Wind energy input Wind stress applies to the surface of the ocean and drives both surface currents and waves. From Eq. (23) (Lecture 4) the mechanical energy from wind into the ocean

windW is defined as ( )wind ijW u nσ= ⋅ ⋅ , where the stress tensor includes both the viscosity

stress tensor and the pressure: ij ij ij pσ π δ= + . Thus, energy input due to wind includes contributions from the wind stress and the pressure term. Although wind stress is more frequently discussed in large-scale circulation problems, sea level pressure and its perturbations are important contributor as well. It is hard to separate wind stress from sea level atmospheric pressure because they are closely related to each other, so separation of wind energy input into different categories is hard, and the discussion here is debatable. Wind stress energy input can be roughly separated into different components

( ) 0, 0, 0 0 0) ' ' ' 'wind ij geo ageoW u n U U u p w pwσ τ τ= ⋅ ⋅ = ⋅ ( + + ⋅ + + (1)

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where the overbear indicates average in space and time; the perturbations are defined in terms of the spatial and temporal scales of the surface wavelength; τ , 0,geoU , 0,ageoU are the spatial and temporal averaged tangential stress and surface geostrophic and ageostrophic velocity respectively; 'τ and 0'u are the perturbations; 'p and 0 'w are perturbations of the surface pressure and velocity component normal to the surface; p and 0w 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 is the wind stress work on the quasi-steady currents on the surface, and quasi-steadiness is defined in comparison to the time scale of the typical surface waves. The third term is wind stress work on surface waves, and the last two terms are the work of atmospheric pressure. Energy contributions from these terms will be discussed as follows. a) Geostrophic currents Wind stress energy input through the geostrophic current on the sea surface is

,wind geocurr geoW uτ= ⋅ (2) The surface geostrophic currents can be inferred from the satellite altimeter data,

/u gk fη= ×∇ (η is the sea surface height), except near the equator. Wunsch (1998) estimated that the total amount of energy input is 0.88TW. In order to include the equatorial regime, Wunsch also calculated the wind energy input from a numerical model, and the total amount of energy is about 1.3TW. The distribution is shown in Fig. 2. Although the wind energy input is positive around 040 N , the major energy input is through the Southern Oceans and the equatorial band. In addition, the North Equatorial Counter Current is a place of energy sink because the eastward current is against the easterlies there.

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Fig. 2. Wind energy input through the surface geostrophic current (Wunsch, 1998). Energy input through the surface geostrophic current can be directly fed into the large-scale current, so it can be efficiently turned into gravitational potential energy through the vertical velocity conversion term. b) Surface drift There is a frictional boundary layer on the top of the ocean, in which wind stress is balanced by frictional force. The surface drift is called the Ekman drift. The horizontal momentum equations for the general case including the geostrophic flow are

* * * * * *, ,/ ( ) , / ( )t s x w z z t s y w z zu fv p Au v fu p Avρ ρ− = − + + = − + , (3)

where * *,g e g eu u u v v v= + = + are the sum of geostrophic velocity and ageostrophic velocity in the Ekman layer, and ( , )s sp p x y= is the surface pressure associated with

large-scale circulation. The geostrophic velocity satisfies /g s wu k p f ρ= ×∇ . The corresponding boundary conditions are

, 0 , 0| / , | / , , 0,x ye z z w e z z w e eAu Av u v at zτ ρ τ ρ= == = → → −∞ (4)

where 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.

Multiplying these two equations by u* and v* and integrating the result over the depth of the Ekman layer leads to

tE S P D= − − , (5) where

0 *2 *20.5( )wE u v dzρ−∞

= +∫

, , (0) (0)x y x yg e g g g e e eS S S S u v S u vτ τ τ τ= + = + = +

0 2 2, ,/ , ( )e s w w e z e zP U p D A u v dzρ ρ

−∞= ∇ = +∫i

where E is the total kinetic energy 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, respectively. Note that /eU k fτ= − × , so

/e s w gP U p uρ τ= ∇ =i i ; (6) 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

/ / se s w e s w e

wS S S

pPdxdy U p dxdy w p dxdy U ndlρ ρρ

= ∇ = − +∫ ∫ ∫ ∫ ∫ ∫ ∫i i , (7)

where S is the area of the ocean. Thus, Eq. (5) can be further reduced to t eE S D= − (8)

Therefore, wind stress energy input to the surface geostrophic currents is equal to the increase of GPE in the world ocean via Ekman pumping, including coastal upwelling/downwelling. On the other hand, wind stress energy input to the surface

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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 2.4TW (Wang and Huang, 2004a) for the frequency higher than 1/(two days). In addition, there is a large amount of energy input through the near-inertial waves, which is due to the resonance at a frequency of fω = − ; i.e., for inertial frequency of the clockwise (anticlockwise) wind stress in the Northern (Southern) Hemisphere. Estimation by Watanaba and Hibiya (2002) is about 0.7TW, but Alford (2003) put it to about 0.5TW. Note that most energy input is through the Southern Ocean, around the latitudes of the Southern Jet Stream. The convergence of the Ekman flux gives rise to a pumping velocity at the base of the Ekman layer ew , which is responsible for pushing the warm water into the subsurface ocean and thus forming the main thermocline in the subtropical ocean. Ekman pumping sets up the wind-driven circulation and the associated bow-shaped main thermocline in the world oceans. In this process gravitational potential energy is increased, which is a very efficient way of converting kinetic energy into gravitational potential energy.

Figure 3. Distribution of the total wind energy input to the surface Ekman layer, cut off at a frequency of 0.5cycle/day, in unit mW/m2. c) Surface waves Another major source of energy induced by wind and sea level atmospheric pressure is fed through the surface waves. Wind stress drives surface waves in the oceans, and this energy input can be treated as the form drag for the atmospheric boundary layer. Wind energy input to surface waves can be estimated as

2 2* *waves a a w wW c u c u cτ ρ ρ= = ≈ (8)

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where c is the effective phase speed, * /a au τ ρ= and * /w wu τ ρ= , aρ and wρ , are density of air and water, τ is stress at the air-sea interface. The corresponding value of c can be determined from field experiments. Observation data fit the following formula

3*3.5waves a aW uρ= . Since *au depends not only on wind but also on sea state, the influence

of waves on wavesW is implicitly implied here. To get the best fit of the experimental data, we used an empirical formula

3*waves a aW A uρ= (9)

where A is the empirical coefficient representing the energy flux factor, */ ac u

* *1/ 3* *

0.5 , 1112 , 11

p p

p p

c as cA

c as c−

≤⎧= ⎨ >⎩

(10)

where * */p p ac c u= is the wave age, and pc is the phase velocity at the wind sea peak frequency. The total contribution to the global oceans is estimated as 60TW (Wang and Huang, 2004b). Although this seems a large amount of energy, it is believed that most of it is transformed into long waves and propagated away from the source area. It is likely that this part of energy is dissipated locally through wave breaking and white capping, but the rest of this incoming energy may be redistributed to remote places in forms of swell. These long waves have a rather low dissipation rate, so their energy is likely to be dissipated, cascading into other forms of energy or through wave breaking along the beaches in the world's oceans; however, the details of this dissipation mechanism remain unclear.

Fig. 4. Distribution of wind stress work on the surface waves (right panel), in units of 2/mW m , calculated from formulas (8, 9), and its latitudinal distribution (left panel).

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d) Atmospheric loading When sea level atmospheric pressure varies with time, sea surface moves in the vertical direction in response. As a result, changes in sea level atmospheric pressure can transport mechanical energy into the oceans. The relevant process is called atmospheric loading. The common practice in processing satellite altimetry data is to assume that the ocean responses to changes in sea level instantaneously, and the corresponding changes in sea surface height is removed from the original satellite data through the so-called inverse barometer effect. In reality, the response of sea level is not instantaneous, and there is a certain amount of mechanical energy transported from the atmosphere to the oceans. The total amount of energy due to this source for the world's ocean circulation remains unclear. Based on sea surface height from satellite data and atmospheric pressure data, the preliminary estimate for the open oceans is about 0.04TW, Fig. 5 (Wang et al., 2006).

Fig. 5. Global distribution of mechanical energy input due to atmospheric loading 2( / )mW m , averaged over the period of 1993-2002 (right panel) and the zonally

integrated mechanical energy (in GW per degree) input due to atmospheric loading (left panel). B) Tidal dissipation The primary source of mechanical energy supporting deep mixing likely comes from tidal dissipation in the deep ocean. The total amount of tidal dissipation in the world oceans is 3.5TW (Fig. 6), which is calculated from accurate tracking of the moon's orbit. The spatial distribution of tidal energy dissipation remains inaccurate. Based on barotropic tidal simulation with satellite altimeter data assimilation, Walter Munk and Carl Wunsch estimated that 2.6TW is dissipated within the shallow seas of the world's oceans, and the remainder 0.9TW is believed to be distributed in the deep ocean (Fig. 1). Tidal dissipation in the deep ocean takes place primarily over rough topography, where barotropic tidal energy is converted into energy for internal tides and internal waves that sustain the bottom-intensified diapycnal mixing on the order of

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3 210 /m s− . The conversion of kinetic energy to gravitational potential energy through internal waves and turbulence is of rather low efficiency, typically on the range of 20%, which is sometime referred to as the Osborn (1980) coefficient, who was a pioneer in the study of turbulence in stratified fluid environments; thus, the amount of gravitational potential energy generated by tidal dissipation is about 0.18TW.

Fig. 6a. Global distribution of tidal dissipation (Munk and Wunsch, 1998)

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Fig. 6b. Tidal dissipation diagram, unit in TW (modified from Munk and Wunsch, 1998). C) Source/sink of GPE due to thermohaline forcing at the air-sea interface Thermohaline forcing at the air-sea interface has often been treated in terms of the buoyancy flux. Note that using the concept of buoyancy flux may lead to the ignoring of an important physical fact that what goes through the air-sea interface actually includes a freshwater flux and it is associated with other physical quantities, such as mass, enthalpy, and entropy. a) Source/sink of GPE due to surface buoyancy flux As discussed in Appendix, the total amount of GPE source/sink due to surface

heating and cooling is ( )2 heat heat cool cool

p S

d g q h h dxdydt cχ α α= −∫∫

where q is the local heat flux rate, heath ( coolh ) is the depth of heating (cooling) penetration and heatα ( coolα ) is the thermal expansion coefficient during heating (cooling). Assuming the depth where heating/cooling can penetrate is very thin, then this equation suggests that surface heating/cooling on the surface cannot generate much GPE at all. The same argument applies to the surface buoyancy flux.

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Two physical processes in the oceans should be considered. First, solar radiation can penetrate into the first 10-20 m of water, so the effective penetration depth heath is finite. GPE source due to the penetration of solar radiation is affected by many physical processes, including atmospheric conditions, sea state, and biological activity in the upper ocean. It is estimated that the global contribution is on the order of 0.01TW, so it is negligible for theory of oceanic circulation. However, this may be one of the most important controlling factors in marine ecologic system. Second, due to surface waves and turbulence, the effective depth of heating penetration is the depth of the mixed layer. It is to emphasize that GPE generated in this way is not a direct contribution from heat flux; instead, GPE source here is due to the conversion of kinetic energy of surface waves and turbulence. Another important process taking place in the mixed layer is the convective adjustment due to cooling and salinification. During the cooling process, water at the sea surface becomes heavier than water at the depth, thus a gravitational unstable stratification appears. Due to this unstable stratification, a rapid convective adjustment process takes place and density becomes nearly homogenized in the upper part of the water column. During this process, gravitational potential energy is converted into kinetic energy for turbulence and internal waves (Fig. 7a).

Fig. 7. Gravitational potential energy loss due to convective adjustment and baroclinic instability. As a result, the effective center of cooling is not at the sea surface; instead it is located half way of the well-mixed layer's depth. Because of this asymmetry associated with cooling/salinification, surface buoyancy forcing gives rise to a sink of gravitational potential energy. The total amount of energy loss through this process remains unclear. A preliminary estimate based on the monthly mean climatology for this sink term is about 0.24TW. The corresponding gain of GPE associated with buoyancy gain is estimated as

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0.13 TW (Huang and Wang, 2003). However, calculation in which the diurnal cycle is resolved may give rise to a much larger value. Note that gain of GPE is converted from kinetic energy associated surface waves and turbulence, so this should not be counted again as a separated source of mechanical energy for the world oceans.

Fig. 8. Annual mean GPE loss to convective adjustment, in unit of 2/mW m . b) Source/sink of GPE due to surface mass flux Freshwater flux through the air-sea interface is actually a mass flux. As discussed in Lecture 4, GPE balance of the world ocean is

( ) TsV S V

dv ds dv Pt tδ φρφ ρφ ρδ

∂= − ⋅ + +

∂∫∫∫ ∫∫ ∫∫∫u u n

The second term on the right-hand side is GPE source/sink due to surface mass exchange associated with hydrological cycle. The global sum of this term is small, estimated as

0.007− TW (Huang, 1998). The negative sign is due to the fact that freshwater is transported to atmosphere from low latitudes where sea level is high, and returns to the oceans at high latitudes where sea level is low. D) Loss of GPE through baroclinic instability a) Meso-scale eddies in the oceans Meso-scale eddies are the most important features in the oceans. It is estimated that the amount of eddy kinetic energy is about 100 times larger than the kinetic energy of the time-mean flow. Despite great effort in carrying out field observations and numerical simulation, there is no reliable estimate on the total amount of eddy kinetic energy in the world oceans. However, satellite observations have provided global distribution of both the mean and eddy kinetic energy on the sea surface, as shown in Fig. 9. The ratio of these two forms of kinetic energy is on the order of 100, consistent with theoretical prediction based on scaling.

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Fig. 9. a) Kinetic energy of the four-year time average surface geostrophic ocean circulation as seen in altimeter data relative to a best-estimated geoid. b) Kinetic energy of the time variable part of the ocean circulation as seen in the altimeter. c) Ratio of the kinetic energy of the time variable to the time-mean geostrophic flow (Wunsch, 2002). b) GPE conversion from the mean state to eddies The steep isopycnal surface in the oceans, such as that along the meridional edges of the wind-driven gyre, is unstable due to baroclinic instability. Because of this, a large amount of gravitational potential energy of the mean state can be converted into the eddy kinetic/potential energy through baroclinic instability. Baroclinic instability is one of the most important mechanisms for GPE transform from the mean state to eddies, and parameterization of eddies has been a major focus over the past decade. A parameterization proposed by Gent and McWilliams (1990) has been widely used in non-eddy-resolving numerical simulations with some success. The basic idea is to parameterize the baroclinic instability in terms of isopycnal layer thickness diffusion. Their formulation can be better explained in terms of the energy conversion.

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Accordingly, the conversion rate of mean GPE to eddy GPE is governed by the following equation

( ) ( )*

th thz

D g z g L g w gDt

ρ ρρ ρκ ρ κρ

∇ ⋅∇= ∇⋅ + + (11)

where * /D Dt is the substantial derivative that advects with the effective transport velocity, including the Eulerian mean velocity and the eddy transport velocity; thκ is the thickness diffusivity; / zL ρ ρ= −∇ , ρ is the potential density. The first term on the right-hand side integrates to zero, so it does not contribute. The second term is the conversion from the kinetic energy to the GPE. The third term is a sink because zρ is negative, and this is the term associated with the eddies. Accordingly, the total conversion rate in the world oceans can be calculated by integrating this term over the total volume

,e bi thz

P g dxdydzρ ρκρ

∇ ⋅∇= ∫ ∫ ∫ (12)

Potential density gradient is calculated using the center of each layer as the reference level for the potential density, and a central difference scheme in space. As suggested by Gent et al. (1995), we have chosen 21000 /th m sκ = , and the density field is calculated from Levitus 1994 annual mean climatology. GM90 is a good approximation that applies for the case of relatively small isopycnal slope. The common practice is to set a limit for the isopycnal slope. For the case with the isopycnal slope limit | / | 0.01zρ ρ∇ ≤ , the total rate of GPE transform is , 1.1e biP TW= .

Fig. 10. Conversion rate of mean GPE to eddy GPE through baroclinic instability based on the empirical eddy parameterization of Gent and McWilliams (1990), in units of

2/mW m .

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The horizontal distribution of this GPE conversion rate is shown in Fig. 10. It is clear that most energy transform takes place in the Antarctic Circumpolar Current, the Gulf Stream, and to a much smaller degree in the Kuroshio. E) Diapycnal and along isopycnal mixing Tracers, including temperature and salinity, are mixed in the oceans through internal wave breaking and turbulence. Mixing can be classified into diapycnal mixing and along isopycnal mixing. Since along isopycnal mixing involves the least amount of gravitational potential energy, it is the dominating form of mixing on large scales. On the other hand, in stratified fluid vertical mixing increases gravitational potential energy because light fluid is pushed downward and heavy fluid is pushed upward. Although molecular diffusion can play the role of mixing, the corresponding rate is too small and thus the result is irrelevant to the oceans. In the upper ocean, below the mixed layer, diapycnal mixing rate is on the order of 5 210 /m s− , which is much larger than the mixing rate due to molecular mixing. In other places of the oceans, mixing rate can be much higher than this background rate. The strong diapycnal (or vertical) mixing in the oceans is driven by strong internal wave and turbulence.

Fig. 11. Bottom intensified mixing in the Brazil Basin (Ledwell et al., 2000) The energy sources supporting diapycnal mixing in the ocean include: a) Wind stress input through the sea surface and tidal dissipation, which will be discussed in the next section. b) Turbulent mixing associated with fast current, especially that associated with flow over sill and the down-slope flow afterward can provide a strong source of energy supporting mixing. There is much observational evidence indicating that mixing can be on the order of 3 1 210 10 /m s− −− . The mixing rate can be as large as 20.1 /m s within the

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Romanche Fracture Zone, where water drops more than 500 meters within 100 km of a downward flow. c) Internal lee waves generated by flow over topography, such as the Antarctic Circumpolar Current, are probably one of the most important contributors. The total amount of energy supporting diapycnal mixing in the oceans remains unclear because mixing is highly non-uniform in space and time. According to the theory of thermohaline circulation, the meridional overturning rate is directly controlled by the strength and distribution of the external mechanical energy available for supporting mixing in the ocean, including mixing in the mixed layer and mixing in the subsurface layers. Therefore, understanding the physics related to the spatial and temporal distribution of mixing is one of the most important research frontiers in physical oceanography. Note that our understanding of the role of mixing in the thermohaline circulation remains rudimentary, and it is fair to say that there is no simple relation between them. F) Geothermal heating In addition, geothermal heat and hot plumes provide a total heat flux of 32TW. Although this is much smaller than the heat flux across the air-sea interface, it may be a significant component of the driving force for the abyssal circulation. Since the geothermal heating is applied at great depth, and the corresponding cooling takes place at the sea surface, geothermal heat can be more efficiently converted into gravitational potential energy. The source of GPE due to geothermal heat can be calculated using the formula discussed in Appendix. The global rate of this conversion is about 0.05TW, which is a small term compared to other major terms; nevertheless is not negligible, especially for the abyssal circulation and temperature distribution in the abyss. G) Bottom drag Oceanic currents moving over rough bottom topography must overcome bottom drag or form drag, as shown in Fig. 1. The total amount of bottom drag for the world's ocean circulation remains unclear; however, the preliminary estimate for the open oceans is about 0.4TW (Wunsch and Farrari, 2004). The bottom drag remains one of the major unknown in oceanic circulation theory. There is no reliable observation data about bottom drag, and most numerical models cannot resolve the deep circulation accurately. H) Double diffusion: Seawater contains salt, thus it is a two-component chemical mixture. On the level of molecular mixing, the heat diffusion is 100 times faster than the salt diffusion. The difference in heat and salt diffusion for laminar fluid and turbulent fluid environment is one of the most important aspects of the thermohaline circulation in the oceans. Double diffusion in the oceans primarily manifests itself in two forms: salt fingers and diffusive convection. Salt fingers appear when warm and salty water lies over cold and fresh water. The mechanism that drives the instability is illustrated in Fig. 12a, Assume a vertical pipe which connects the cold and fresh water in the lower part of the water column with the warm and salty water in the upper part of the water column. The wall of the pipe is very thin, so it allows a rather efficient heat flux into the pipe, thus

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warming up the cold water. At the same time, the low salt diffusivity preserves the freshness of the ascending water parcel. Therefore, the buoyancy difference drives the upward motion of water inside the pipe, and a self-propelled fountain can be built in the ocean.

Fig. 12. Two possible cases for double diffusion in the oceans. The subtropical gyre interior is a salt-finger's favorite place, where strong solar insolation and excessive evaporation leads to warm and salty water above the main thermocline. In this salt-finger favorite regime, warm and salty fingers move downward and cold and fresh plumes move upward. Since heat is 100 times more easily diffused between the salt fingers and the environment, salt fingers lose their buoyancy and continue their movement downward. In this way gravitational potential energy released is used to drive the mixing. According to the new in-site observations in the upper thermocline of the subtropical North Atlantic, the mixing rate for the temperature (salinity) is on the order of 5 24 10 /m s−× ( 5 28 10 /m s−× ), and the equivalent density mixing rate is negative. The other possible double diffusive process is the diffusive convection that takes place, if cold and fresh water overlies warm and salty water. The system is relatively stable and allows an oscillatory instability, Fig. 12b. GPE release due to salt fingering in the world oceans Based on Levitus (1998), GPE release due to salt fingers can be calculated, using the double diffusive flux associated with salt fingering parameterization Zhang, Schmitt, and Huang (1998), Fig. 13. The global sum of GPE release is small, in the order of 0.02TW. Since GPE is increased due to thermal diffusion associated with salt fingers, with a global total of 0.0145TW; thus, the net is about 0.006TW. Note that: a) This energy should not be considered as external mechanical energy because salt fingers can only release GPE from the stratification which is set up by the large-scale circulation, and the circulation itself is driven by external mechanical energy. b) Although GPE release from salt fingers seems small compared with other sources of energy, it may play an important role in some parts of the world oceans, such as the subtropical North Atlantic.

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Fig. 13. Global distribution of GPE release due to salt diffusion associated with salt fingering, in unit of mW/m2 (the background diffusivity is set to 0K ∞ = ). I) Cabbeling: Due to the nonlinearity of the equation of state, water density is increased during both diapycnal and isopycnal mixing. The newly formed water with higher density sinks, and it is called cabbeling, as shown in Fig. 1. As a result, gravitational potential energy is converted into energy for internal waves and turbulence. The total amount of gravitational potential energy loss associated with cabbeling in the oceans remains unknown. 2. Attempt at balancing the mechanical energy in the ocean The balance of mechanical energy, including both the kinetic energy and gravitational potential energy, is shown in Fig. 14. It is clear that at this time we do not know even the lowest-order balance of mechanical energy. There is much kinetic energy input into the ocean, but it is not clear how such energy is distributed and eventually dissipated in the oceans. On the other hand, there are two major sinks of gravitational potential energy, but it is not clear how energy is transported and supplied to such energy sinks. It is fair to say that most of the energy fluxes listed in Fig. 14 is accurate to the factor of two only. More accurate energy pathways and estimates require further study.

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Fig. 14. Mechanical energy balance for the world oceans, unit in TW. 3. Remaining challenges in the energetics of the world oceans A. Energy from wind stress should be the focus of oceanic circulation and climate change: 1. Wind stress energy input is the dominating player in controlling the meridional overturning circulation in the Atlantic Ocean under current climate condition. 2. Wind stress energy input changes greatly over a broad range of time scales, from inter-annual, decadal, centennial, to other longer time scales. 3. Wind stress energy input affects the upper ocean dynamics directly, and this is the most important dynamical zone for many applications, including weather, fishery, transportation, and environments. B. Changes in wind stress energy input The most important point is that: wind stress energy input to surface geostrophic current varies greatly from year to year, and it has been increasing over the past 50 years!

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Fig. 15. Annual-mean wind energy input to the surface geostrophic current diagnosed from the numerical model (circle), and the results from altimetry data (triangle), vertically displaced 0.3TW upward, in units of TW, (Huang et al., 2006).

Figure 16. The time evolution of the normalized wind energy input through geostrophic current (circle), surface waves (triangle), and Ekman layer (star) , (Huang et al., 2006).

Wind energy input in all three major forms increase greatly over the past 50 years. Thus, the oceanic general circulation should adjust in response to such changes in energy input.

Trend of increasing energy input after 1965, and why?

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Fig. 17. Decadal-mean zonal wind stress in the world’s oceans, and the deviation from the 50-year mean (in Pa), based on NCEP wind stress, (Huang et al., 2006). C. Change in tidal dissipation (1) Global scale changes in land-sea distribution due to continental drift, which has time scale of millennial or longer. Tidal dissipation has varied greatly over the geological past, thus the energy supporting diapycnal mixing may vary as well, Fig. 18. Note that this result was based on a numerical model with a flat bottom. It is well-known, bottom topography plays vitally important roles in controlling the barotropic tidal flow and dissipation; however, reconstruction of the paleo topography may be very challenging. Thus, the results from such simple tidal model are quoted here to demonstrate the basic idea that tidal dissipation may have changed tremendously over the geological past. If we go back to much earlier history in the geological past, the change in the gravitational field of the Earth-Moon system and the rotation rate of the Earth may have to be taken into consideration. Paleo records indicate that the Earth’s rotation has slowed down over the past 900 million years, and this should affect the tidal flow and dissipation rate.

Westerly intensified greatly over the past 50yr.

Equatorial Easterly diminished over the past 50

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Fig. 18. Changes of the M2 tide dissipation over the past 550 millions years (after Kagan and Sundermann, 1996) (2) Global sea level change. For example, during last glacial maximum great changes took place. The next figure shows changes in the M2 tides. Great changes in tidal dissipation during last glaciations maximum:

1 Sea level was more than 100 m lower than that of present day. 2 Tidal dissipation in the shallow seas was much reduced. As a result, tidal flow in

the deep oceans was much fast. 3 Global tidal dissipation during LGM was 50% higher than present day (Egbet et

al., 2003). 4 The meridional temperature in the atmosphere was larger, so wind was much

stronger; thus, wind energy input into the ocean was much stronger. 5 It would be a great project to simulate the oceanic circulation and climate during

LGM with a new parameterization of diapycnal mixing which changes with climate.

These peaks may correspond to relatively warm period in the past.

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Fig. 19. Changes in M2 tide amplitude, taken from Egbet et al. (2003) Thus, for climate variability on decadal or centennial time scales, tidal dissipation can be treated as fixed, and our discussion can be focused on energy input from wind stress.

Appendix: Source/sink of GPE due to heating/cooling a) For a water column on the upper surface The amount of GPE for a water column with unit horizontal area, located from the

sea surface to a depth of h, is 0 cenmghχ = (A.1)

where g is gravity, cenh is the center of mass relative to the reference level for GPE, and 0

0( )h

m z dz hρ ρ−

= ∫ is the total mass of the water column, ( )0 zρ is the density profile

in the water column, ρ is the mean reference density. Assuming this water column receives an amount of heat Q. After heating, temperature increases

/ pT Q c hδ ρ= , (A.2)

Strong M2 tides during last glacial maximum, which induced strong mixing in the oceans.

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and water column height increases p

Qhc

αδρ

, where α is the thermal expansion

coefficient, pc is the specific heat under constant pressure. Thus, after heating the center of mass moves upward / 2hδ and the total GPE of this water column is

1 ( / 2)cenmg h hχ δ= + . (A.3) The net change of GPE is

0

0( )

2 2h

p p

z dzg Q g hQc c

ρα αχρ

−Δ = ∫ (A.4)

Assuming the rate of heating and cooling is balanced, this leads to the total GPE source/sink due to surface heating/cooling

( )2 heat heat cool cool

p S

d g q h h dxdydt cχ α α= −∫∫ (A.5)

where q is the local heat flux rate and the integration is taken over the global sea surface. Eq. (A.5) can be easily extended to the general case with buoyancy flux

( ) ( )( ). .2 buoy gain buoy lossS

d g bh bh dxdydtχ= −∫∫ (A.6)

Where ( )p

qb P Ecα β= + − is the surface buoyancy flux, P E− is the rate of

precipitation minus evaporation. Since a mass coordinate is used, the results are independent of the choice of the reference level for GPE. b) For a water parcel below the sea surface

Assume a water parcel sitting at a depth h below the sea surface, with initial thickness of hΔ . The amount of GPE for a water column with unit area is again

0 cenmghχ = , where cenh is the center of mass related to the reference level for GPE, and 0

0( )h

m z dz hρ ρ−

= =∫ is the total mass of the water column, ( )0 zρ is the density profile

in the water column. Assuming this water column receives an amount of heat Q. After heating, the temperature increases / pT Q c hδ ρ= , and the water column height increases

/ ph Q cδ α ρ . As a result, the whole water column above is pushed upward for a distance of hδ , and the total GPE of this water column is

1 ( )cenmg h hχ δ= + . (A.5) The net change in GPE for the water column after heating a water parcel is

0

0 ( )h

p p

z dzg Q g hQc c

ρα αχρ

−Δ = ∫ . (A.6)

Note that the thickness hΔ of the water parcel is assumed to be much smaller than h, so changes in GPE of this water parcel is neglected.

From this formula, it is clear that in order to make heating an efficient source of energy, it should be placed at deep levels. Geothermal heating is such an example.

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Note that for a steady state, the total amount of heat through heating and cooling should be balanced. Assuming specific heat is constant, GPE generated from thermal forcing should be

( )h h c cp

gQ h hc

χ α αΔ = − (A.7)

where hα ( cα ) and hh ( ch ) are the thermal expansion coefficient and geometric height for the heating (cooling) source.

c) Cautions: 1) We must use a mass-conserving coordinates in the discussion of GPE. If you

try to use a Boussinesq model in which the mass conservation is replaced by the volume conservation, you may get some incorrect results. In general, models based on Boussinesq approximations may have artificial sources/sinks of mass and GPE.

2) In the analysis above, we assumed that heating energy is entirely converted to internal energy, as 0/ pT Q c hδ ρ= (Eq. A.2). In fact, part of the input thermal energy should be converted into GPE, thus, the corresponding temperature change should include a correction T Tδ εδ− . However, it can be shown that the correction term is much smaller then the first term, | | | |T Tεδ δ , and thus negligible, even for the case of heating and cooling from the sea floor. Reference Alford, M. H., 2003: Improved global maps and 54-year history of wind-work on ocean

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