where cg = PR2 is the group velocity of Rossby waves, and Re is the first baroclinic radius of deformation.

Decadal variability of gyres in the North Atlantic Ocean

The maximal strength of meridional volume transport (defined as the meridional circulation rate between 20° N and 40° N) and the corresponding poleward heat flux associated with gyration and throughflow in the Atlantic Ocean vary greatly over a decadal time scale (Fig. 5.184).

The most outstanding feature was the dramatic intensification of gyration in the 1970s, apparently associated with the regime shift in the atmospheric circulation around 1975-6. Although the volume flux associated with throughflow remains virtually constant, the gyration rate went up more than 100% during this time period. In addition, there seems to be a gradual intensification of gyration over the past 50 years.

Poleward heat flux associated with throughflow and gyration has changed greatly over the past 50 years. There is a clear trend of decline in poleward heat flux associated with throughflow. In particular, there is a large amplitude change that took place in the 1970s and 1980s; the contributions due to both gyration and meridional throughflow went up, although the corresponding increase in the meridional throughflow was relatively small. This is may be due to large changes in the thermal structure in the upper ocean associated with the wind-driven circulation in response to the regime shift in the 1970s.

Fig. 5.184 Decadal variability of the maximal poleward fluxes in the Atlantic Ocean, within the latitudinal band of 20-40° N, where ThF indicates throughflow: a volume flux and b heat flux (Jiang etal, 2008).

The poleward heat flux associated with gyration also changed. The regime shift seems to have had a noticeable impact on the poleward heat flux associated with gyration in the 1970s. In addition, the poleward heat flux associated with gyration has increased slightly over the past 50 years, although the overall trend of poleward heat flux declined slightly over the same period owing to the weakening of the contribution associated with the throughflow.

Paleoproxy evidence indicates that the production of North Atlantic Deep Water (NADW) was much reduced or even shut down at times in the past that coincided with rapid changes in global climate (as recorded in ice cores and elsewhere; see, e.g. Broecker, 1998). Although the atmospheric response was mostly confined to the Northern Hemisphere, planetary wave propagation through the world's oceans may have brought about rapid global changes (Doscher et al., 1994). The global adjustment process associated with the planetary waves can be examined by studying an idealized model in which the global circulation is represented in terms of a linear single mode.

One of the major assumptions in the classical theory of deep circulation by Stommel and Arons (1960a) is that the circulation is stationary. For the non-stationary circulation, planetary Kelvin waves along the coasts and in the equatorial waveguide, and Rossby waves, both play a major role in setting up the circulation in a closed basin by transporting water masses (Kawase, 1987). In this section we extend the theory to the multi-basin world oceans.

We begin with a linear shallow water model on an equatorial j plane (Kawase, 1987):

Note that His the equivalent depth of the shallow water equation, and its specification will be discussed later. Thus, the horizontal momentum equations are geostrophic balance plus the time-dependent term and a simple Rayleigh friction, Ku or Kv; the continuity equation includes the time-dependent term, the deepwater source distribution Q, and a simple Rayleigh damping term Xh. The most crucial departure from the Stommel-Arons formulation in this model is the inclusion of the time-dependent terms and replacing the specified upwelling with the term Xh. These equations can be nondimensionalized by introducing

5.5.3 Global adjustment of the thermocline

Adjustment to a quasi-steady state Introduction

Model formulation ut - ßyv = -ghx - Ku vt + ßyu = -ghy - Kv ht + H (ux + vy) = -Xh + Q

the following scales for velocity, length , depth , and time:

c = (KgH/X)1'2, L = (c/P)1/2, H, T = (cP)-1'2 (5.395) For X = K, this equation set is reduced to ut — yv = —hx — ru (5.396)

where r = KT is the new nondimensional friction parameter, and Q = TQ/H.

Apart from the western boundary, the solution consists of equatorial Kelvin waves leaving the western boundary to propagate toward the eastern boundary. At the eastern boundary the equatorial Kelvin waves are reflected as westward Rossby waves, plus two poleward Kelvin waves. On their poleward paths, the coastal Kelvin waves send out Rossby waves propagating westward and establishing the circulation in the interior. Along their pathways, wave amplitude gradually declines due to dissipation. As the circulation approaches equilibrium, the time-dependent terms drop off, and the steady-state solution, which is the combination of the Kelvin waves and their reflection in Rossby waves, in an ocean bounded at X = 0 in the west and X = LB in the east, is (Cane, 1989):

h = AF, F = (cosh 2rf):1/2 e— y2/2tanh2r^ (5.399)

where A is a constant for each basin to be determined by volumetric flux balance in the model, F > e—r^y for very small r and F = 1 along the eastern boundary and the equator, and f = (Lb — X) /L is the nondimensional zonal coordinate.

When r is very small, the volumetric communication rate M between two adjacent basins is primarily controlled by the semi-geostrophic current around the southern tip of the continent separating the basins. Denoting its latitude by ys, the volumetric flux from one basin to the next is M = Ah/ys, where Ah is the layer thickness change across the boundary current. For example, the volumetric balance in the Indian Ocean is the influx equal to the upwelling inside the basin plus the outflow

(Aa — AjFjw) /yAj = r jjf hdxdy + (Ai — ApFBw) /ysI P (5.400)

The subscripts A, J, and P indicate the Atlantic, Indian, and Pacific Oceans; Sj is the surface area of the Indian Ocean; FJw ~ e~rfjwy'py'P; fJw is the western boundary of the Indian Ocean; and ysA J (ysj P) represent the southern tip of the continent separating the Atlantic and Indian (Indian and Pacific) Oceans (Fig. 5.185).

For small friction , r //S/ hdxdy — rAJSJ to the first order inr. There is a similar relation for the Pacific Ocean

(Aj — ApFpw) /yj p = r ff hdxdy + Ap/yP a (5.401)

j jsp

Atl Ind Pac

Atl Ind Pac

Fig. 5.185 Circulation in the idealized world's oceans driven by apoint source of deep water Q. Sa, Si , and Sp are the areas of each ocean, and ySj p is the southern tip of the continent separating the Indian and Pacific Oceans (Huang et al., 2000).

In addition, we have the total volumetric balance in the world's oceans:

The solution can be found by combining Eqns. (5.400, 5.401, 5.402). When r is very small, the first term on the right-hand side of Eqns. (5.400, 5.401) can be neglected.

Numerical experiments

The numerical model is integrated using Ax = 1° and Ay = 0.5°. In order to simulate the sea level and main thermocline appropriately, a combination of the first baroclinic and second baroclinic modes is used, and the equivalent depth scale is H = 0.92 m (Zebiak and Cane, 1987). The Rayleigh friction and damping coefficients are K = X = 4 x 10-10/s, which corresponds to a nondimensional friction coefficient r = 4.8 x 105. For a mean thermocline depth of 300 m, this corresponds to a diapycnal diffusivity of 0.36 x 10-4 m2/s, which is larger than the low open-ocean mixing coefficient of 10-5 m2/s, inferred from tracer release experiments; however, it is smaller than the 0(10-4 m2/s) pivotal values inferred from the global tracer budget.

A source of 10 Sv, approximately the rate at which NADW currently crosses the equator from the North Atlantic Ocean, is uniformly distributed within a latitudinal band north of 50° N in the North Atlantic Ocean. After 500 years the solution seems to approach a quasi-steady state. Note that a slowdown in deepwater formation is equivalent to a source of upper layer water and a sink of lower layer water. Thus, the h field discussed below should be interpreted as the downward motion of the thermocline in response to a slowdown of deepwater formation, or its upward motion in response to increased deepwater formation.

Several numerical experiments were carried out, including cases in which the world's oceans are represented by the highly idealized rectangular basins (Fig. 5.185). All results from these numerical experiments fit the simple formulae very closely. As an example, experiments with a realistic coastline are shown in Figure 5.186.

The first case is the control case: the Indonesian Passage is closed, the corresponding southern tips of the African, Australian and American continents are 34° S, 44° S, and 55° S, respectively, and the amplitude of the thermocline depth perturbation in the world's oceans is AA : Aj : AP = 105 : 87 : 46 (m) (Fig. 5.186a). Thus, the shallow-water model predicts that a deepwater formation rate of 10 Sv induces an upward motion of the thermocline on the order of 50-100 meters. Assuming r > 0, Eqns. (5.400,5.401,5.402) are in error by less than 10%. Errors are due to the approximation in neglecting friction, additional dissipation in the numerical model for a realistic, jagged coastline, and the fact that the solution has not completely reached the steady state. It is interesting to note that there is a current of 3 Sv going through the Drake Passage. From Eqn. (5.402), it can readily be seen that if the friction/damping coefficient is reduced 10 times, this recirculation will increase 10 times because the amplitude of the perturbation is inverse to r.

In the second case, the Indonesian Passage is open, and the thermocline perturbation ratio is AA : Aj : AP = 106 : 67 : 54 (Fig. 5.186b). The effective southern tip of the boundary between the Indian and Pacific Oceans is moved up to 9.1° S, still far enough away from the equator so that the boundary current is geostrophically controlled. The deepwater flow takes a short-cut through the Indonesian Passage. This direct route gives rise to a slightly larger thermocline depth perturbation in the Pacific Ocean. The volumetric flux through the Indonesian Passage is 3.3 Sv. Here again, reducing r 10 times can give rise to a flux 10 times larger.

In the third case, both the Indonesian Passage and Drake Passage are closed, and the thermocline perturbation ratio is AA : Aj : AP = 98 : 81 : 43 (Fig. 5.186c). With the Drake Passage closed, Eqns. (5.400) and (5.401) need modification: the last term on the right-hand side of Eqn. (5.401) is eliminated and the area of the Pacific Basin is expanded to the Antarctic coast. It is interesting that in this simple model the thermocline adjustment in the global oceans is not sensitive to whether the Drake Passage is open or closed. However, the Drake Passage plays such a vitally important role in controlling the thermohaline and thermocline circulation in the world's oceans that the meaning of this result should be interpreted with caution.

The role of Kelvin/Rossby waves

Both Kelvin and Rossby waves establish the steady solutions discussed above. When NADW is formed, Kelvin waves are generated at the site of deepwater formation and move to the other parts of world's oceans, as shown in Figure 5.187.

The pathways of Kelvin waves are as follows. First, they move southward along the eastern coastline of North America in the form of coastal Kelvin waves (Leg 1). At the equator, these waves turn eastward and propagate along the equatorial waveguide (Leg 2). At the eastern boundary they bifurcate and become the poleward Kelvin waves (Legs 3 and 11). After turning the corner of Cape of Hope, they continue to move equatorward along the eastern coast of Africa (Leg 4). The rest of the passage (Legs 5, 6,7,8, 9, and 10) is similar.

Closed Indonesian Passage

Closed Indonesian Passage

a 80°W 80°W 0° 40°E 80°E 120°E 160°E 160°W 120°W 80°W 40°W 0°

Open Indonesian Passage

Open Indonesian Passage

b 80°W 80°W 0° 40°E 80°E 120°E 160°E 160°W 120°W 80°W 40°W 0°

Closed Indonesian Passage, Closed Drake Passage

Closed Indonesian Passage, Closed Drake Passage

Fig. 5.186 The displacement of the thermocline (m) induced by NADW source of 10 Sv: a closed Indonesian Passage; b open Indonesian Passage; c closed Indonesian Passage, closed Drake Passage (Huang et al, 2000).
Thermocline Simple Diagram

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