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Fig. 1.41 The temperature gradient (°C/100m) of the main thermocline in the Pacific and Atlantic Oceans. See color plate section.

The vertical temperature gradient of the main thermocline also varies greatly over the world's oceans. Within the subtropical gyres, the gradient is on the order of 2-4°C/100 m; however, it is much larger in the equatorial ocean, varying from 10°C/100 m in the western part to 20°C/100m in the eastern part of the equatorial Pacific Ocean (Fig. 1.41).

1.4.3 Early theories for the wind-driven circulation

Ekman theory

In the early stages, our knowledge of oceanic general circulation was mostly observational. The first milestone was a paper by Ekman (1905), in which he discussed the structure of the wind-driven circulation in the surface boundary layer. According to his theory, the velocity in the boundary layer should have a spiral structure, the vertically integrated volume transport is t/fp0, and it points 90° to the right of the wind stress (in the Northern Hemisphere). This layer is now called the Ekman layer; the flux in this layer is called the Ekman flux, forming the theoretical foundation of modern wind-driven circulation theories.

The Ekman layer and its associated spiral velocity profile in the atmospheric boundary layer can be readily observed. In fact, the beautiful Ekman spiral can be demonstrated by releasing a line with many balloons attached. It took a long time before the Ekman theory could be verified in the ocean. The major difficulty in the ocean is the presence of strong surface waves in the upper ocean. However, after a long time delay, the Ekman spiral in the upper ocean was finally confirmed through in situ measurements (Price et al., 1987). The structure of the stratified Ekman layer is much more complicated. For the most up-to-date information, the reader is referred to Price and Sundermeyer (1999).

Before the 1940s our knowledge of oceanic circulation theory was confined to simple dynamical calculations of currents based on a level of no motion, the Ekman layer, waves, and tides. For example, The Oceans, written by Sverdrup et al. (1942), is an amazing summary of the state of the art of oceanography in the early 1940s. At the time The Oceans was published, it seems that a lot was known about the oceanic circulation. Thus, the book was a rather intimidating collection of knowledge, as Stommel (1984b) recalled in his autobiography.

Sverdrup's theory

Over the past 60 years there have been many major breakthroughs in our understanding of the oceanic general circulation. The second milestone in wind-driven circulation theory is the work by Sverdrup (1947), in which he established the simple relationship between the wind stress curl and the circulation in the basin interior.

In the atmosphere, both westerlies and easterlies are the necessary components of the atmospheric circulation. The westerlies at mid latitudes and the easterlies at low latitudes give rise to a negative wind stress curl in the subtropics, which drives an equatorward flow in the ocean interior. In order to find the circulation in the basin, Sverdrup integrated the wind stress curl westward, starting with a no-zonal flux condition at the eastern boundary. His solution did not include the flow near the western boundary. Apparently, at that time it was unclear how to deal with the western boundary and why the integration should be started from the eastern boundary.

Theories of western intensification

The existence of swift currents, such as the Gulf Stream, near the western boundary of the basin was recognized through frequent trade activity between Europe and the American continent several hundred years ago. However, a dynamical explanation for such swift currents was first established only in the late 1940s, when Stommel (1948) studied an idealized model for the North Atlantic Ocean, including bottom friction and the latitudinal change of the Coriolis parameter, which is now called the j effect. In fact, Stommel was not aware of Sverdrup's theory because it had been published in a journal (Proceedings of the National Academy of Sciences of the USA) rarely used by oceanographers.

Other dynamical explanations of the western boundary current appeared shortly afterward, including a theory of the western boundary layer with lateral friction by Munk (1950), and the inertial western boundary layer theory by Charney (1955) and Morgan (1956).

Homogeneous model or reduced-gravity model The common feature of these models is that they all treat the wind-driven oceanic circulation as a single moving layer. Since seawater is nearly constant, the simplest way to simulate the ocean circulation is to assume that the ocean is homogeneous in density. Such an ocean would have no vertical structure. The model used by Stommel (1948) belongs to this category.

Another possible approach is to assume that the wind-driven circulation is confined to the upper layer of the ocean; thus, the circulation can be treated in terms of a single moving layer model, such as those of Sverdrup (1947) and Munk (1950). This is called the reduced-gravity model. The essence of a reduced-gravity model is to treat the main thermocline (or the pycnocline) in the oceans as a step function in density, so that density in the upper layer equals a constant p and density in the lower layer is p + Ap. Furthermore, the lower layer is assumed to be infinitely deep, so the pressure gradient in the lower layer is infinitely small and the corresponding volume transport negligible. As a good approximation, we can assume that the lower layer is motionless; thus we actually deal with a single moving layer. The pressure gradient in the upper layer is described by Vp/p = g'Vh, where h is the upper layer depth and g' = g Ap /p is called a reduced gravity, which is on the order of 0.01-0.02 m/s2.

The basic idea is demonstrated in Figure 1.42, where the structure of the water column is depicted. The seasonal thermocline (pycnocline) near the upper surface can clearly be seen. The main thermocline and the main pycnocline are located at about the same depth (800 m); this is due to the fact that the contribution of salinity to the density stratification is relatively small at this location (and many other locations). The density structure is now represented in terms of two layers of constant density, as shown by the heavy lines in Figure 1.42b. The lower layer extends all the way to the seafloor. Since this layer is very thick, the horizontal pressure gradient and velocity in this layer are very small and negligible. Thus, the reduced-gravity model can be used to capture the first baroclinic mode of the circulation and the depth of the main thermocline.

1.4.4 Theoretical framework for the barotropic circulation

The backbone of theories of the barotropic circulation is the barotropic potential vorticity constraint. Hough (1897) devoted a minor portion of his tidal study to the currents produced by a zonally distributed evaporation and precipitation, ignoring friction. He found that a uniformly accelerated system of purely east-west geostrophic currents would exist. His model had several limitations. First, there was no friction in the model, so that he was unable to obtain a steady solution, nor was he able to discern the long-term effect of evaporation and precipitation. Second, his model had no meridional boundary, which is a crucial constraint

Fig. 1.42 Temperature and stratification at a station (70.5° W,30.5° N) from annual mean climatology: a Temperature (T, °C); b Stratification (oq, kg/m3 ), the heavy lines indicate the equivalent stratification used in a reduced-gravity model; c dT/dz (0.01°C/m); d -do0/dz (0.01 kg/m4).

on the oceanic circulation. Hough published his results without even a figure to show the structure of the solution; thus, his solution remained unnoticed until Stommel (1957) publicized this solution with a beautiful illustration.

Goldsbrough (1933) discussed an ocean model forced by evaporation and precipitation. By choosing a rather special form of precipitation and evaporation pattern (where the zonally integrated evaporation and precipitation along each latitude vanishes), he was able to obtain a steady circulation, even though his model also had no friction. Because Goldsbrough's solution required a special form of precipitation and evaporation, his model was quite unrealistic. The major reason why Goldsbrough's theory has been largely ignored is due to the small size of the barotropic current predicted by his theory. As we discuss later, another major shortcoming in his theory was the lack of salinity in the model. If salt and mixing were added into his model, the baroclinic velocity induced by the freshwater flux could be a hundred times stronger than the barotropic velocity predicted by his original theory.

Nevertheless, the theories of Hough and Goldsbrough are more general than they first appear. The general nature of their theories can be explained more clearly with the aid of two basic mechanisms.

First, Ekman (1905) showed that the frictional stress of the wind is confined to a thin layer on the upper surface, so the motions below this thin surface layer can be treated as frictionless. Thus, for the ocean below this thin layer, the only role that wind stress plays is to create the horizontal Ekman transport in the upper ocean. The horizontal convergence of the Ekman transport gives rise to the Ekman pumping, which drives the interior flow equatorward in the subtropical basin.

In terms of potential vorticity dynamics, the lowest-order balance for the oceanic interior is as follows. In the oceanic interior, the relative vorticity is negligible, so potential vorticity is reduced to f /h, where f is the Coriolis parameter and h is the layer thickness. In order to balance the compression due to the large-scale Ekman pumping associated with the negative wind stress curl, water columns move toward lower latitudes, where f is smaller. Similarly, precipitation plays a role similar to Ekman pumping because it compresses water columns in the ocean, and thus reduces h. As a result, precipitation drives an equatorward flow in the ocean interior, very much like the effect of Ekman pumping.

Second, mass conservation requires a return flow, which is accomplished by western boundary currents. The frictional or inertial western boundary layers provide a vital dynamical component that helps to close the circulation in terms of the conservation of mass, energy, and potential vorticity. The interior flow field can therefore be combined with some kind of western boundary layer, such as the bottom friction model of Stommel (1948), the lateral friction model of Munk (1950), or the inertial model of Charney (1955) and Morgan (1956). Thus, the boundary layer theory that was developed in traditional fluid dynamics found its use in dynamical oceanography. In mathematical terms, the problem is treated by the perturbation method. In the interior ocean the flow is described by the low-order dynamics, i.e., essentially inviscid and linear. This lower-order dynamics cannot satisfy the boundary conditions at the western wall, so that within the western boundary layer, high-order terms, such as frictional or inertial terms, must be included to provide some corrections in order to match the interior solution and the western boundary conditions.

In this spirit, Stommel proposed to use western boundary currents to close the circulation driven by evaporation at low latitudes and precipitation at high latitudes. The solution he proposed overcomes the strong limitations implied in Goldsbrough's original model. Accordingly, circulation in a closed basin driven by an arbitrary pattern of wind stress, or freshwater flux across the air-sea interface due to evaporation minus precipitation, can be very well described by the theory.

Therefore, these seemingly unrelated theories for the oceanic circulation are essentially linked to each other in terms of potential vorticity balance, as shown in Figure 1.43. The dynamics of the reduced-gravity model will be discussed in detail in connection with the theory of the wind-driven circulation. In addition, the further development of three-dimensional structure of the wind-driven circulation also follows similar approaches based on potential vorticity dynamics.

Baroclinic Circulaiton Theory
Fig. 1.43 Connections between different paths during the early development of the oceanic general circulation.

1.4.5 Theories of the baroclinic wind-driven circulation

Although single-layer models have been useful tools in describing wind-driven circulation, the theories of wind-driven circulation based on such models remained primarily two-dimensional and they could not provide the vertical structure of the gyre. In the 1950s and 1960s, people tried very hard to work out theories about the vertical structure of the oceanic circulation.

As discussed above, one of the most outstanding features in the oceans is the existence of the main thermocline, which is closely related to the velocity structure in the upper ocean. Thus, the baroclinic theory of the wind-driven circulation is also called the thermocline theory. The theory of the thermocline was first proposed in two papers published side by side in Tellus by Welander (1959) and Robinson and Stommel (1959). There have been many attempts to find solutions to the thermocline equations; however, most of these solutions are similarity solutions that cannot satisfy some of the essential boundary conditions. The most serious deficit of these solutions is their inability to satisfy the Sverdrup constraint. Without satisfying this constraint, these solutions are incapable of describing the basin-wide structure of the wind-driven gyre.

During the 1960s and 1970s, the development of oceanic circulation theory was relatively slow, owing to a lack of understanding of the physical processes involved in the circulation. Numerical models had been developed; however, without the physical insights obtained from observations or theoretical studies, results from numerical experiments proved as hard to understand as data from the oceans. The second phase of development of the theory of oceanic circulation began in the 1980s. This new phase is characterized by combining observations, theory, and numerical models.

Three-dimensional structure of the wind-driven circulation

As our understanding of the oceanic circulation deepened, it was realized that the ocean can be described to a very good approximation as an ideal fluid system. Recent field observations have indicated that diapycnal diffusion in the subsurface ocean interior is on the order of 10-5 m2/s. A major theoretical difficulty in the 1970s was the puzzle of how the subsurface layers in an ideal-fluid model ocean could be in motion. In a rotating fluid, large-scale motions should follow the geostrophic contours, f /Ah (where Ah is the layer thickness). Since interfacial friction is assumed to be infinitesimal, simple intuition suggests that geostrophic contours in subsurface layers should be parallel to the latitudinal circles. (However, such a simple intuition turns out to be incorrect, as discussed below.) Because of the no-flux condition at the eastern wall, all geostrophic contours in subsurface layers are blocked, and the subsurface layers should be stagnant.

This puzzle was solved by Rhines and Young (1982). Using a quasi-geostrophic model, they were able to show that closed geostrophic contours can be formed due to large interfacial deformation induced by strong forcing applied to the surface layer. As a result, there could be infinitely many possible non-stagnant solutions to the problem instead of the solution of no motion, as was thought previously. Furthermore, they showed, subject to assumptions about the effects of eddies on the mean flow, that potential vorticity should be homogenized within these closed geostrophic contours; thus the system should possess a unique solution that is stable to small perturbations. Their theory provided a theoretical background for motions in the subsurface layers.

A second way of getting the subsurface water in motion was proposed by Luyten et al. (1983): in their model, the isopycnal outcropping effectively bypasses the blocking due to the eastern boundary. In some sense, their model is a very nice extension of the classical conceptual model of ventilation through outcropping proposed by Iselin (1939) much earlier. Of course, such an extension included many conceptual breakthroughs, such as formulating the model in a solid dynamical framework and introducing the concepts of the ventilated zone, the pool regime, and the shadow zone.

Although Iselin proposed his conceptual model of late winter ventilation, it was not clear why the ocean should pickup only the late winter properties for ventilation. To explain this phenomenon, Stommel (1979) analyzed the physical processes involved and showed that there are indeed some processes in the oceans that select only the late winter properties for ventilation. This mechanism is now called the Stommel demon. Accordingly, in order to study the climatological mean circulation, it is possible to avoid the complexity of the seasonal cycle by simply choosing the late winter properties, such as the mixed layer depth and density. To date, the Stommel demon has remained a main theoretical backbone of the modern theory of wind-driven ocean circulation.

Another classical approach to the thermocline theory is the ideal-fluid thermocline theory proposed by Welander (1959, 1971a). His theory is basically to treat the wind-driven circulation as a perturbation to the background stratification set up by an external thermohaline circulation (which is not explicitly included in the model). Welander (1971a) showed that the ideal-fluid thermocline problem can be reduced to solving a second-order ordinary differential equation. However, his solution can satisfy only two boundary conditions in the vertical direction, and for a long time it was not clear how to improve his theory to accommodate more boundary conditions, as required by the physics.

The connections between these seemingly different approaches were unified into a theory of the three-dimensional structure of the wind-driven circulation in the continuously stratified oceans by Huang (1988a, b). It was demonstrated that the problem can be reduced to solving a free-boundary problem of a second-order ordinary differential equation in density coordinates. This theory was further extended to a model incorporating a mixed layer on top (Huang, 1990a; Williams, 1991). The model with a mixed layer can provide a more realistic description of the three-dimensional structure of the wind-driven circulation in the oceans.

The baroclinic structure of the inertial western boundary currents Although theories of western boundary currents based on a single-moving layer have been simple, elegant, and successful, the corresponding part associated with the multiple moving layers has not. The difficulty associated with multi-layer inertial western boundary currents was first discussed by Blandford (1965). Basically, he searched for solutions with two moving layers, but failed to find any continuous solutions. Instead, he found that the solutions break down before they reach the latitude corresponding to the Gulf Stream's separation from the coast. The difficulties associated with the discontinuity of the inertial western boundary current have been discussed by Luyten and Stommel (1985) in terms of virtual control.

Using a streamfunction coordinate transformation, Huang (1990b) showed that continuous solutions for the two-moving-layer inertial western boundary currents do exist, and that these solutions can be matched to the multi-layer ventilated thermocline solution in the ocean interior (Huang, 1990c). However, the continuity of the inertial western boundary currents does impose certain dynamical constraints on the thermocline structure of the interior oceans.

1.4.6 Theory of thermohaline circulation

The Stommel-Arons theory for the deep circulation In the early theory of the thermohaline circulation, the deep part of the circulation was idealized as a source-sink-driven circulation. In a series of seminal papers, Stommel and his colleagues developed the framework of the deep circulation (e.g., Stommel, 1957; Stommel and Arons, 1960a, b).

The basic assumptions of their models are as follows. Deep circulation in the world's oceans is assumed to be steady and it is driven by idealized point sources prescribed in an ocean with no bottom topography. The source of deep water is balanced by a uniform upwelling prescribed over the whole basin.

The most important dynamical consequence of these assumptions is as follows. First, uniform upwelling specified in the ocean drives poleward flow in the abyssal oceanic interior, as dictated by the linear potential vorticity constraint, similar to the case of wind-driven circulation discussed above. Second, in order to balance mass and potential vorticity in the abyssal ocean, deep western boundary currents are required.

In fact, the most crucial conclusion from such a simple theory is the prediction of the existence of deep western boundary currents. Almost immediately after the theory was postulated by Stommel (1957), a deep western boundary current was found off the eastern coast of the USAby Swallow and Worthington (1957). This combination of theory and in situ observations was hailed as one of the most important discoveries in physical oceanography of the last century.

Abyssal circulation beyond the Stommel-Arons theory

The Stommel-Arons theory of the abyssal circulation was very simple and successful, so it dominated the theoretical field of deep circulation for more than 20 years. However, their theory is essentially limited by several of the assumptions it makes. Although their theory has been very successful in predicting the deep western boundary currents, the uniform poleward flow in the abyssal ocean could not be verified. It was realized in the 1980s that in order to describe the deep circulation accurately, many of the simplifications made in their theory must be replaced with more realistic assumptions.

Departing from the steady-state assumption, Kawase (1987) studied the spin-up process of an inverse reduced-gravity model (the meaning of this term will be explained in later chapters), in which he assumed that the interfacial upwelling is linearly proportional to the interfacial displacement from the mean. His solution clearly demonstrated the critical role of the coastal Kelvin waves in setting up the deep circulation, especially the deep western boundary currents. Rhines and MacCready (1989) noted that the bottom of the oceans is far from being flat. In fact, the oceans' bottom has the shape of a bowl, like a Chinese wok. Since the horizontal area of the deep oceans increases upward, the deep circulation in the interior oceans may be clockwise, instead of counterclockwise as suggested by the classical Stommel-Arons theory.

Stommel and Arons (1960a) assumed that the upwelling velocity was uniform basin-wide; such an assumption was a way to simplify the model, but it is not necessarily true. There is much evidence to suggest that upwelling is not uniform. Using an idealized two-level model, Huang (1993a) was able to show that upwelling is very strong along the equator and the eastern boundary.

The baroclinic structure of the abyssal circulation with continuous stratification has been discussed by Pedlosky and his co-workers in a series of papers (Pedlosky, 1992; Christopher and Pedlosky, 1995). For a model with a flat bottom and given stratification, it was shown that the vertical and meridional velocity can have alternate signs because the basic equations have eigenfunctions that oscillate in both vertical and horizontal directions.

These theories are based on simple assumptions about the bottom topography and mixing. However, the situation in the oceans is much more complicated. As recent field experiments have indicated, mixing is highly inhomogeneous in both space and time. As a result of the nonlinear interaction between stratification, flow over topography, and mixing, the abyssal circulation is very complicated; it is therefore one of the most exciting research frontiers.

Conceptual models for the thermohaline circulation The seemingly similar situation that both the ocean and the atmosphere are under strong thermal forcing has convinced many people to draw parallel comparisons between the thermal circulation in the atmosphere and in the oceans. Since the discovery of cool and dense water overlying the bottom of the world's oceans, the concept of thermally driven circulation in the ocean has been gradually formed. Because salinity should also contribute to the density distribution, and thus the oceanic general circulation, the circulation related to density differences in the ocean is called the thermohaline circulation. The advances in atmospheric science in understanding the thermally driven circulation, plus our experience from thermal engines in daily life, would have been the major impetus for the formation of early theories of thermohaline circulation.

The essential element of the early theories is that cold and dense deep water formed at high latitudes plays the role of the driving force for the thermohaline circulation in the world's oceans. A typical example is the two-dimensional conceptual model proposed by Wyrtki (1961), in which high-latitude cooling produces dense water that sinks to the seafloor. The equatorward spreading of deep water drives the meridional circulation, including the poleward branch of the return flow. The same idea was also used in the classical box model of Stommel (1961), in which he assumed that the overturning rate is linearly proportional to the meridional difference of density in the upper ocean.

However, there is an essential difference in terms of how heating/cooling applies to atmosphere and ocean. In fact, at the beginning of the twentieth century Sandstrom (1908,1916) had postulated that thermal forcing in the ocean is incapable of driving any strong circulation. Although his postulation was also cited in the classic textbook Physical Oceanography by Defant (1961), it was largely ignored. Beginning in the late 1990s, the energetics of thermohaline circulation became a research frontier. According to the new paradigm, the thermohaline circulation is not driven by surface heating/cooling; instead, it is driven by external sources of mechanical energy from wind stress and tidal dissipation, as briefly discussed below.

The hydrological cycle and the haline circulation It is common knowledge that salinity is a key factor in controlling the thermohaline circulation. Thus, freshwater flux across the air-sea interface should be one of the primary driving forces of the oceanic general circulation. Although salinity distribution in the ocean has been simulated in numerical models based on the surface salinity relaxation condition, the dynamical role of freshwater flux was not explored thoroughly in most early studies.

Before the 1990s, the role of freshwater-driven circulation was discussed in very few papers, such as those by Hough (1897), Goldsbrough (1933), and Stommel (1957, 1984a). These studies focused on the barotropic circulation driven by freshwater flux.

In a departure from the traditional salinity boundary conditions, Huang (1993b) postulated that the suitable condition for salinity balance should be the net freshwater flux through the air-sea interface. Assuming strong diapycnal mixing, numerical simulations showed that freshwater flux can drive a haline circulation. Although the barotropic component of such a haline circulation is relatively weak, the baroclinic component has a strength that is comparable to the circulation driven by heat flux or wind stress. The unique role of surface freshwater flux in maintaining and regulating the circulation is also a vital research frontier related to the thermohaline circulation, as discussed below.

The multiple equilibria and variability of the thermohaline circulation

The multiple equilibria for the thermohaline circulation were first discussed in a seminal paper by Stommel (1961). Based on a two-box model, he predicted that there should be two steady states (one stable and one unstable) in the so-called thermal mode, and a stable steady state in the haline mode. In common with much of his work, his model was thought to be too simple, and people did not appreciate its physical meaning for two decades. However, this changed rapidly in the 1980s. Due to a very strong need to understand the climate system, people started to consider possible multiple equilibrium states of the climate, including multiple solutions to the thermohaline circulation.

Major contributions include F. Bryan's (1986) work on the multiple states of the model Atlantic Ocean and the associated change in the poleward heat flux. He also introduced the use of the so-called mixed boundary conditions, which have since gained wide acceptance by numerical modelers. Manabe and Stouffer (1988) found multiple states in an air-sea coupled general circulation model, which shares some similarities to the solutions predicted by the box model of Stommel. Marotzke (1990) found many more multiple states of the thermohaline circulation, including the so-called flushing phenomenon associated with the halocline catastrophe.

The critical role of virtual salt flux (or freshwater flux) in controlling the thermohaline catastrophe and its variability on the decadal time scale was explored in numerical models. One of the most interesting topics is the thermohaline variability on decadal or longer time scales (e.g., Weaver and Sarachik, 1991).

The flux condition for the salinity balance seems to be the essential ingredient for the thermohaline variability, as shown by many studies (e.g., Weaver et al., 1991). In fact, freshwater flux alone can give rise to haline oscillation on a decadal time scale (e.g., Huang and Chou, 1994). Thus, freshwater flux due to evaporation and precipitation may be the essential ingredient for climate variability.

1.4.7 Mixing and energetics of the oceanic circulation

Since the early days of oceanic general circulation modeling it was realized that the choice of sub-grid parameterizations was rather subjective and may involve great uncertainty. In order to assess the vertical diffusivity in the oceans, Munk analyzed the balance of tracers in the world's oceans; his paper "Abyssal recipes" (Munk, 1966) remains a classic. Using a one-dimensional model, he concluded that the global mean vertical diffusivity is approximately 10-4 m2/s.

Mixing in the oceans is non-uniform Many in situ observations carried out afterward, however, indicated that diffusivity in the oceans is highly inhomogeneous. In particular, diffusion is very weak (on the order of 2 x 10-5 m2/s; Ledwell et al., 1993) in the subsurface ocean, which is much smaller than the global mean value of 10-4 m2/s. To narrow the gap between Munk's global mean diffusivity and the low diffusivity observed in the subsurface ocean, it was postulated that strong diffusion near the bottom or side boundaries of the ocean may contribute to the seemingly high global mean value of Munk. In fact, recent in situ observations indicate that diapycnal diffusivity can be very strong near the bottom and close to the mid-ocean ridge (on the order of 10-3 m2/s; Ledwell et al, 2000).

Highly non-uniform mixing in the deep ocean posed serious questions about the validity of the classical Stommel-Arons theory. It is obvious that abyssal circulation in the presence of complicated bottom topography and driven by such non-uniform mixing can be drastically different from the classical theory of Stommel and Arons (1960a). Flow induced by bottom-intensified mixing over sloping boundaries has been studied extensively. Phillips (1970) and Wunsch (1970) pointed out that a thermal insulation boundary condition applied to a sloping bottom requires that isotherms must be perpendicular to the local slope. In a rotating fluid such a density gradient near the bottom boundary thus induces an along-slope flow (the primary circulation) and an uphill flow (the secondary circulation) in the bottom boundary layer. Phillips et al. (1986) also showed that a tertiary flow perpendicular to the slope might exist, due to the convergence and divergence of the secondary circulation. Mixing over a sloping bottom boundary has been studied extensively (Garrett et al., 1993). Recent numerical studies of abyssal circulation induced by bottom-intensified mixing demonstrate rather complicated flow patterns (e.g., Cummins and Foreman, 1998).

Energetics of oceanic circulation Oceanic circulation requires a source of mechanical energy to overcome friction and dissipation. For a long time, the common wisdom has been that thermal forcing can provide the mechanical energy sustaining the thermohaline circulation. However, a close examination reveals the following fundamental difference between the atmospheric circulation and the oceanic circulation: the atmosphere is heated from below and cooled from the middle and upper levels, so the atmosphere can work as a thermal engine. However, the ocean is heated and cooled from the upper surface, at roughly the same level; such a forcing is now called horizontal differential heating. About 100 years ago, Sandstrom (1908, 1916) had postulated that thermal forcing in the ocean is incapable of driving a strong circulation. Thus, the oceanic circulation is not a heat engine; instead, it is a conveyor belt for heat and freshwater driven by external sources of mechanical energy.

In order to maintain the stratification observed in the ocean, the cold and dense deep water must be warmed up and returned to the surface. This process implies mass flux across isopycnal surfaces, and is called diapycnal mixing. Since isopycnals in the ocean are nearly horizontal, diapycnal mixing is often referred to as vertical mixing. Vertical mixing in a stratified fluid pushes light water downward and heavy water upward. As a result, the center of mass is moved upward owing to vertical mixing, and the total amount of gravitational potential energy is increased. Therefore, mixing requires external sources of mechanical energy. The sources of external mechanical energy and their distribution within the ocean are critically important for oceanic circulation and climate.

The mechanical energy required to sustain the oceanic circulation was first explored by Faller (1966); however, the connection between sources of external mechanical energy and oceanic general circulation remained unexplored. Although many scientists working on small-scale mixing problems knew that mixing in a stratified fluid requires mechanical energy, most numerical modelers and theoreticians for large-scale oceanic circulation completely ignored the possible link between diapycnal mixing and the external mechanical energy required to sustain the mixing. Vertical diffusivity in numerical models remains as an adjustable parameter in the models, which were tuned to fit the meridional overturning circulation to observations. For example, in the book Physics of Climate (Peixoto and Oort, 1992), the energetics for the oceanic circulation was mostly confined to balance of thermal energy.

The thermohaline circulation in the oceans is strongly influenced by diffusion; thus, the traditional tools based on ideal-fluid assumptions, such as conservation of tracers or potential vorticity, cannot provide complete dynamical pictures. Studying the thermohaline circulation from other viewpoints, such as the balance of mechanical energy, is necessary; the major breakthrough in the energetics of oceanic circulation came only in the late 1990s.

The dominating role of wind stress over the Southern Ocean in providing mechanical energy and sustaining the thermohaline circulation was demonstrated through numerical experiments by Toggweiler and Samuels (1993,1998). Wunsch (1998) made the first reliable estimate of wind energy input to geostrophic currents in the world's oceans.

A somewhat surprising conceptual breakthrough was postulated by Munk. After studying tidal problems for many decades, he finally came up with the crucial idea that tidal dissipation can provide the mechanical energy sustaining diapycnal mixing in the ocean. As a result, tidal dissipation can play a key role in regulating oceanic circulation, and thus the climate, on a planet. Munk and Wunsch (1998) postulated that tidal dissipation and energy due to wind work on the surface geostrophic current are the vital sources of mechanical energy sustaining diapycnal mixing in the world's oceans. An attempt was made to establish the balance of different kinds of energy for the oceanic general circulation by Huang (1998a). In particular, the importance of external sources of mechanical energy required for sustaining diapycnal mixing and the balance of available potential energy were examined (e.g., Huang, 1998b, 1999). Many studies regarding the energetics of the oceanic circulation have appeared since then. Progress along these lines has been summarized in review papers by Wunsch and Ferrari (2004) and Ferrari and Wunsch (2009). I discuss some fundamental issues related to the energetics of the oceanic general circulation in Chapters 3, 4 and 5.

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