## U

If the outflow is in a channel, P can be specified a priori; however, it can be calculated in the general case by

where K = tb/^H is the Ekman number; the ratio of bottom drag to the Coriolis force.

Given the initial values of U, ©, S, H, and W, Eqns. (5.22), (5.24a), (5.24b), (5.26), and (5.27) can be used to calculate the evolution of the streamtube downstream.

Entraining density currents with rotation: an end-point model

We now discuss the simplified relations between the initial conditions and the final product of the outflow. In order to do so, we make three assumptions:

1. The currents are in geostrophy, so Ugeo = g'a/f, where g' = gSp/p is the reduced gravity and a is the topographic slope.

2. The effect of bottom stress on width is retained through

using the continuity equation and neglecting entrainment,

where the subscript src indicates the properties of source water.

3. Mixing is treated as an 'entrainment event' where the Froude number is greater than one. Under these assumptions, the product water density of an outflow is

where Fgeo = , Ugeo / is the geostrophic Froude number.

\Hgeog src

Based on these relations, we can calculate the product density from the initial density and environmental density. Note that the larger the difference is, the smaller the product density will be. This seemingly strange result is due to the strong mixing induced by the large density difference that drives the Froude number supercritical.

Application of the streamtube model to the major sources of deep water in the world's oceans, as discussed above, gives rise to rough estimates for the amount of mixing during the process of descent and entrainment, listed as the last two rows in Table 5.1. From this table, it is clear that mixing for the Mediterranean outflow is much stronger than for the other three outflows. This is one of the major factors that control the depth where the spreading of the Mediterranean outflow is observed in the Atlantic Ocean under the current climate.

In conclusion, the selection of bottom water that can reach and fill up the bottom layer of the world's oceans is controlled by two competing processes: the thermobaric effect and entrainment during descent after overflow over the sill. Calculating the relative contributions due to mixing and entrainment remains a great challenge for theory and numerical modeling.

5.1.4 Mode water formation/erosion

Deepwater formation and bottom water formation provide the sources for the deep ocean, and thus establish the upstream conditions for the deep circulation. However, a complete picture of the meridional overturning circulation in the oceans also involves formation of water masses that sink to a relatively shallower depth. These water masses are also very important parts of the global thermohaline circulation.

Water masses with lighter density are formed primarily in the interiors of subpolar and subtropical basins, and are called mode water. The name "mode water" reflects the fact that these sources of water mass are not uniformly distributed in the temperature-salinity space; instead, owing to specific sea surface conditions favorable for the formation of these water masses, they appear in clusters in the parameter space.

Mode waters in the world's oceans The historical development of the theories about mode water started with the identification of Eighteen Degree Water associated with the Gulf Stream recirculation in the North Atlantic Ocean (Worthington, 1959). The term "Mode Water" was first introduced by Masuzawa (1969) in the description of Subtropical Mode Water in the Kuroshio Extension in the North Pacific Ocean. McCartney (1977) extended the definition of mode water to the region north of the Subantarctic Front in the Southern Ocean and introduced the term "Subantarctic Mode Water." The term "Subpolar Mode Water" was introduced by McCartney and Talley (1982).

Subtropical mode waters exist at the equator side of the strong separated western boundary currents in the Northern Hemisphere and the strong ACC in the Southern Hemisphere. Three types of mode water formation and their primary sites have been discussed by Hanawa and Talley (2001), including the Subtropical Mode Water (and the Eastern Subtropical Mode Water), and the Subpolar Mode Water.

Fig. 5.20 Mode water formation sites in the world's oceans, with the numbers indicating the nominal density of the mode water oceans (Hanawa and Talley, 2001).

In general, mode water is used to describe some special types of water masses in the world's oceans which appear as local maxima of distribution density in the (T, S) space. Water identified as mode water shares similar temperature and salinity properties, and often appears as the local minimum in potential vorticity. Some of the most well-known types of mode water in the world's oceans are included in Figure 5.20, including Subtropical Mode Water, Eastern Subtropical Mode Water, Subpolar Mode Water, and Subantarctic Mode Water (in the Southern Ocean).

Mode water formation commonly occurs through subduction taking place in the upper ocean. Subduction of mode water from the late-winter mixed layer into the permanent thermocline of the subtropical basin interior is realized through the combined effects of vertical pumping and lateral induction. Vertical pumping is related to the Ekman pumping produced by the surface wind stress, and lateral induction is due to the horizontal advection of the wind-driven gyre and the horizontal gradient of the late-winter mixed layer depth. In fact, lateral induction is a dominant player in subtropical mode water formation.

Essential ingredients of subtropical mode water formation

As illustrated in Figure 5.21, the basic elements of subtropical mode water formation include the following.

• Background circulation. This transports newly formed mode water away from the formation site, and brings in new water from upstream, preparing the formation site for the next cycle of formation (lower panel of Fig. 5.21).

• Strong seasonal cycle of the mixed layer depth. This is induced by strong cooling due to cold and dry continental air blowing over the relatively warm water in the recirculation regime on the equator side of the separated western boundary current (upper left corner in lower panel of Figure 5.21).

Winter-time: heat loss induces nearly homogenized water properties (low PV)

b Summer-time: surface heating and horiozontal advection rebuild the surface stratification

Winter-time: heat loss induces nearly homogenized water properties (low PV)

Water mass formation through subduction b Summer-time: surface heating and horiozontal advection rebuild the surface stratification

Winter cooling produces low PV mode water

Water mass formation through subduction

Winter cooling produces low PV mode water

Mode water formation site

Fig. 5.21 Sketch of subtropical mode water mass formation through subduction: a winter-time, b summer-time.

Mode water formation site

Fig. 5.21 Sketch of subtropical mode water mass formation through subduction: a winter-time, b summer-time.

During late winter, a large volume of mode water with nearly homogeneous properties, such as temperature and salinity (implying low potential vorticity) is formed (Fig. 5.21a). The rapid retreat of mixed layer depth in early spring leaves the nearly homogenized mode water behind and seals it with a shallow, strongly stratified layer on top, thus completing the formation phase of mode water

• Large horizontal gradient of winter mixed layer depth. This combines with strong horizontal advection of the wind-driven gyre, giving rise to a strong lateral induction (Fig. 5.21a). This is explained in detail shortly.

As an example, the climatological mean temperature structure in the subtropical North Atlantic Ocean is shown in Figure 5.22. In winter-time, cold and dry continental air flowing over the warm water carried by the Gulf Stream forms a major site of heat loss in the ocean, as shown in the annual mean heat flux from oceans to atmosphere discussed in Section 1.1.1.

The strong cooling produces a large drop in near-surface temperature and thus a nearly homogeneous pool of water in the upper ocean, as indicated by the temperature structure in Figure 5.22a. Along the meridional section, late-winter cooling produces nearly vertical isothermals in the upper ocean (Fig. 5.23a). Since the contribution of salinity to density structure is relatively small, this implies a nearly homogenized density structure in the upper ocean, i.e., a very deep mixed layer and a pool of low potential vorticity (f Ap/Ah, where Ap is the density jump across the layer interface and Ah is the layer thickness; thus a thick layer means low potential vorticity).

Fig. 5.22 Climatological temperature structure in the upper ocean along 38.5° N; a March, b September.

When spring comes, this pool of nearly homogenized water is covered by the strong stratification in the upper ocean built up by a rapid shoaling of the mixed layer. In addition, the horizontal advection associated with the subtropical gyre plays two important roles simultaneously. First, it transports the newly formed mode water into the permanent thermocline in the subtropical gyre interior. Second, it brings in new water from upstream, thus preparing the site for the next cycle of mode water formation, as shown in Figures 5.22b and 5.23b.

Mode water formation through subduction is a crucial component of the water mass balance in the world's oceans. In fact, sites of mode water formation are critical windows for the communication of atmospheric signals and tracer input into the oceans. The rate of mode water formation is a good index for climate variability in the oceans. In addition, mode water formation sites play a crucial role in resetting the potential vorticity of water masses in the oceans.

5.1.5 Subduction and obduction

### Introduction

The basic ideas of mode water formation were discussed in the previous section. We now focus on the complex dynamical details of mode water formation. In particular, the rate of mode water formation is defined as the annual mean subduction rate, which is a vital index for climate study.

For the balance of water masses in the world's oceans, if there is water mass formation, there should be a process going on in the other direction; this is called water mass erosion. This process may also be called water mass transformation; however, the most suitable terminology has not yet been generally accepted. The rate of water mass erosion through processes in the upper ocean is called obduction, which will also be discussed in this section.

Iselin's model

As discussed in Section 4.1.5, one of the major conceptual difficulties in understanding how subsurface layers are set in motion is that they are not in direct contact with the local atmospheric forcing. However, most isopycnals are in contact with the atmosphere, primarily at high latitudes. Iselin (1939) postulated the preliminary framework for water mass formation through a link between the T-S relation found in a vertical section and the winter-time mixed layer at higher latitudes. His schematic picture for this ventilation and water mass formation process is shown in Figure 4.26. The arrows indicate the speculated motions. In modern terminology, the basic idea is that within the subtropical gyre a water mass is formed at the sea surface in late winter, and is pushed downward into the thermocline by Ekman pumping. Afterward, it downwells along isopycnals, continuing its equatorward motion induced by Sverdrup dynamics. The motion of the particles after their ejection from the base of the mixed layer is confined within the corresponding isopycnal surfaces because mixing is relatively weak within the main thermocline.

Iselin's model was the first prototype for water mass formation in the oceans; however, it was incomplete in two major regards. First, Iselin ignored the mixed layer, which plays a vitally important role in water mass formation. Second, since mixed layer depth and density change greatly from season to season, it was not clear how to make the link between water mass properties and winter-time mixed layer properties, as he postulated.

How do we calculate the water mass formation rate?

According to Iselin's model, the Ekman pumping rate might serve as the water mass formation rate. Although this seemingly simple concept had dominated for a long time, it turned out that the Ekman pumping rate is not exactly the rate of water mass formation. A better way is to calculate the mass flux across the base of the mixed layer. Mixed-layer models have been developed, and can provide an accurate description of the seasonal cycle of the entrainment/detrainment rate across the base of the mixed layer. Can we use the annually integrated detrainment rate as the local water mass formation rate? The answer is "No." Water leaving the mixed layer may not enter the permanent pycnocline; instead, some of the water detrained from the mixed layer at one location may be re-entrained into the mixed layer downstream. Similarly, the simple annually integrated rate of entrainment cannot be used as the rate of water mass erosion. This is due to the fact that water entrained into the mixed layer may not come from the permanent pycnocline; instead it may be water temporarily detrained upstream.

A modified conceptual model is shown in Figure 5.24. The upper ocean is divided into four layers: the Ekman layer, the mixed layer, the seasonal pycnocline, and the permanent pycnocline. The Ekman layer plays the role of collecting the horizontal volume transport which is driven by surface wind stress and produces the convergence/divergence. In the subtropical basin, the convergence gives rise to Ekman pumping, and in the subpolar basin the divergence gives rise to Ekman sucking (upwelling). The mass exchange between the mixed layer and the seasonal pycnocline is called entrainment/detrainment, while the mass exchange between the seasonal pycnocline and the permanent pycnocline is called sub-duction/obduction. Accordingly, the annual mean subduction rate is defined as the total

Subduction/obduction rate

Intergyre boundary

Heating subtropical basin

Cooling subpolar basin

Ekman layer

Ekman layer

Permanent pycnocline

Fig. 5.24 Water mass formation and erosion through subduction and obduction processes. The vertical two-way arrows indicate a continuous mass exchange between the mixed layer and the seasonal thermocline.

### Permanent pycnocline

Fig. 5.24 Water mass formation and erosion through subduction and obduction processes. The vertical two-way arrows indicate a continuous mass exchange between the mixed layer and the seasonal thermocline.

amount of water going from the mixed layer, passing through the seasonal pycnocline, to the permanent pycnocline irreversibly in one year. This definition excludes the contribution due to the so-called temporal detrainment, i.e., the detrainment which re-enters the mixed layer downstream. Similarly, the annual mean obduction rate is defined as the total amount of water going from the permanent pycnocline, passing through the seasonal pycnocline, to the mixed layer irreversibly in one year.

### The Stommel demon

A major technical difficulty in calculating the water mass formation rate is the complicated seasonal cycle in the mixed layer. Both water properties and mixed layer depth vary considerably within the seasonal cycle. By carefully analyzing the processes involved, Stommel (1979) was able to show that a process is at work that selects only the late-winter water for actual subduction into the permanent pycnocline (Fig. 5.25). This mechanism is now called the "Stommel demon."

As discussed in Section 4.1.7, the Stommel demon has become the backbone of the modern theory of wind-driven circulation. Similarly, the theory of mode water formation/erosion through subduction/obduction is also based on the Stommel demon. As discussed later, the effective detrainment period is marked by the Lagrangian trajectories of water particles released from the base of the mixed layer. For simplicity, we assume that the vertical velocity is nearly constant, equal to we, in the upper ocean, and the mixed layer depth is horizontally uniform. Such simplifications will be replaced by more accurate statements in the discussion below.

The basic mechanism is as follows. The mixed layer reaches its annual maximum density and depth in late winter, so there is a very thick layer of almost vertically homogenized water. When spring comes, the mixed layer shoals very quickly (as indicated by the sharp turning of the mixed layer depth in the upper and lower panels of Figure 5.25) and leaves the homogenized water behind, so that the water subducted has properties very close to those of the late-winter mixed layer. It can readily be seen that if the time evolution of the mixed layer depth is approximately a S function, i.e., AT ^ 0, the subducted water would have the properties of late-winter water.

In the present case, both the vertical velocity and the annual maximal depth of the mixed layer are assumed to be constant everywhere. As a result, the annual mean subduction rate is equal to we.

Note that the subduction process in the oceans is a very complicated process involving the seasonal cycle. In fact, now the challenging problem is to calculate the annual mean subduction rate including the seasonal cycle. In one way, this mean can be considered as some kind of weighted average of the instantaneous detrainment rate. Choosing the late-winter properties is equivalent to using a S function as the weight function. Stommel's suggestion has been used extensively in almost all theoretical models of the ventilated pycnocline. What Stommel suggested yields an elegant solution to this rather intricate problem. This can be used as the lowest-order solution. The next step is to find out a weight

5.1 Water mass formation/erosion Spring Summer Fall Winter

Mixed layer depth (m)

Mixed layer density

Lagrangian trajectory

5.1 Water mass formation/erosion Spring Summer Fall Winter

Mixed layer depth (m)

Mixed layer density

Lagrangian trajectory

Fig. 5.25 The Stommel demon: the mixed layer properties at late winter are selected through the subduction process; the horizontal axis represents both time and distance along the 1-year trajectory.

Fig. 5.25 The Stommel demon: the mixed layer properties at late winter are selected through the subduction process; the horizontal axis represents both time and distance along the 1-year trajectory.

function that is better than the 8 function. In other words, we would like to know the next-order correction to the subduction rate calculated according to the Stommel formula. As such a correction must include the seasonal cycle, it is not an easy problem to solve.

### Subduction

We begin with a layered model without a seasonal cycle. In such a model the ventilation/subduction process can be divided into two steps. First, ventilation occurs when water flows downward from the mixed layer into a layer below. Second, water in each density layer follows an equatorward motion induced by the Sverdrup dynamics. Thus, water in a dense layer will move underneath the next layer with lighter density. This process of submersion of a denser layer under a lighter one is called subduction. The term subduction has been used in geology to describe a similar process during the movement of tectonic plates. According

to this strict classification, as the number of layers increases, the first stage (ventilation) becomes shorter and shorter. It can readily be seen that, for a continuously stratified ocean, these two stages will merge; we use the term subduction, reserving the term ventilation for the general case of either subduction or obduction.

The seasonal cycle plays one of the most important roles in the upper ocean dynamics, so we must include the seasonal cycle in our subduction model. The most crucial parameter describing the subduction/ventilation process is the subduction rate. The instantaneous detrainment rate is defined as the volume flux of water leaving the base of the mixed layer per unit horizontal area (Cushman-Roisin, 1987):

where wmb = we - f f-h vdz and Vmb are the vertical and horizontal velocity at the base of the mixed layer, and hm is the mixed layer depth (Fig. 5.26). The first term on the right-hand side is the contribution due to vertical pumping at the base of the mixed layer, which is slightly smaller than the Ekman pumping rate due to the geostrophic flow in the mixed layer. The second term is due to the lateral induction. The third term is due to the temporal change of the mixed layer depth.

If there were no seasonal cycle, the subduction rate should equal the detrainment rate

Thus, this equation can be used for calculating the subduction rate, if there is no seasonal cycle; however, the strong seasonal cycle in the oceans makes the calculation of the subduction rate much more complicated.

Another parameter commonly used in the description of tracer ventilation is the so-called ventilation rate of an individual isopycnal or water mass, defined as

where S is the subduction rate defined above. Physically, the ventilation rate determines the average time (in years) it takes to renew the entire water mass through the ventilation process, or the average time that water particles remain in a water mass category (Jenkins, 1987).

If we ignore the mixed layer, i.e., set its thickness to zero, then the only term contributing to subduction is the vertical pumping, which is the same as the Ekman pumping, since the mixed layer thickness is zero. Such an oversimplified model for the subduction calculation can lead to the misconception that the subduction rate is the same as the Ekman pumping rate. Since the mixed layer depth is non-zero and it varies with time and location, each term on the right-hand side of Eqn. (5.32) contributes differently.

First, because the mixed layer has a finite thickness, the vertical velocity at the base of the mixed layer is slightly smaller than the Ekman pumping velocity. Second, the lateral induction term actually contributes to subduction substantially. In the North Atlantic Ocean, the winter-time mixed layer depth varies greatly. Within 3,000 km it increases northward from 100 m to about 400 m, so that the slope of mixed layer depth is about 0.0001. The meridional velocity in the mixed layer is about 0.01 m/s, so the lateral induction term is 10-6 m/s, which is of the same order as the vertical pumping term. According to a more accurate calculation for the North Atlantic Ocean, the contribution from the vertical pumping amounts to 12.1 Sv, and that from the lateral induction is about 12.7 Sv (Huang, 1990a).

When the time-dependent term is non-zero, the situation becomes even more complicated. There are two prominent cycles in the mixed layer, i.e., the diurnal cycle and the annual cycle. For simplicity, here we discuss the seasonal cycle only. First of all, there is the seasonal pycnocline between the mixed layer and the permanent pycnocline. Thus, a complete picture must consist of four layers, as shown in Figure 5.24. The seasonal pycnocline plays the role of a buffer, i.e., the mass exchange between the mixed layer and the permanent pycnocline must go through the seasonal pycnocline.

As stated above, the mass flux from the seasonal pycnocline to the permanent pycnocline is called subduction. Since we have assumed that the flow in the permanent pycnocline is time-independent, the subduction across the base of the seasonal pycnocline does not vary with time. The mass exchange between the mixed layer and the seasonal pycnocline has a prominent seasonal cycle, and the corresponding exchange rate is called the detrain-ment/entrainment rate. The subduction rate is, therefore, different from the detrainment rate because they represent different processes. Owing to the existence of the annual cycle, the commonly used subduction (obduction) rate discussed in this section is defined as the annual mean of the corresponding rates.

Between late winter and early fall, detrainment is activated owing to the Ekman pumping and mixed layer shoaling. This period can be further divided into two sub-phases. From late winter to early spring, water entering the seasonal pycnocline from the mixed layer will eventually reach the permanent pycnocline; this process is called the effective detrainment. From early spring to early fall, water entering the seasonal pycnocline will be re-taken by the rapid mixed layer deepening during the winter season, resulting in temporary (ineffective) detrainment (Fig. 5.27). From early fall to late winter, the mixed layer deepens rapidly - the entrainment phase. It appears that the temporal and spatial inhomogeneity of motions in the

Effective Ineffective detrainment detrainment

Effective Ineffective detrainment detrainment

Fig. 5.27 The annual mean subduction rate defined in Lagrangian coordinates; the horizontal axis represents both spatial and temporal coordinates along the 1-year trajectory.

mixed layer can create a fairly complicated detrainment/entrainment process, and a comprehensive understanding of the intermittent and sporadic nature of detrainment/entrainment and its contribution to subduction is yet to come through observations and theoretical investigations.

Subduction rate defined as an integral property Were the mixed layer to overlay a stagnant ocean, the subduction rate would be a purely local property. In the oceans, the mixed layer overrides currents in the seasonal/permanent pycnocline. As soon as water particles are left behind the mixed layer, they are carried downstream by currents. There is no chance that water particles could be overtaken at the same location where they first left the mixed layer. This situation is very similar to a human breathing air. A person confined in a small box may breathe the same air again and again; but a jogger can never inhale the same air that he exhales.

In the oceans, the mixed layer can overtake only the water pumped down from upstream, but not the water pumped down at the same location. Although a person sitting at a recording station may know the local rate of mixed layer entrainment/detrainment as a function of time, that person cannot be sure how much of this water actually reaches the permanent pycnocline. To obtain the correct answer, one has to check at stations downstream, because subduction is a non-local process.

The annual mean subduction rate can be defined in different ways depending on the coordinates used. First, it can be defined in Lagrangian coordinates (Woods and Barkmann, 1986):

0 0