W Afil if L

dTy dTx dx dy

North

North

Fig. 2.26 Sketch of the Ekman pumping in a subtropical basin.

Fig. 2.26 Sketch of the Ekman pumping in a subtropical basin.

Therefore, the Ekman pumping rate consists of two parts: the first part is due to the windstress curl, and the second part is due to the beta effect. Assuming that the scale of wind stress is t ~ 0.1 N/m2, and the length scale is 1,000 km, the pumping velocity is estimated as we ~ -10-6 m/s. Over the world's oceans, westerly winds prevail at mid latitudes and easterlies at low latitudes. The equatorward Ekman transport driven by westerlies at mid latitudes and the poleward Ekman transport driven by easterlies at low latitudes converge in the subtropical basin. As a result, Ekman pumping prevails over the subtropical basin (Fig. 2.26). Similarly, strong westerlies at mid latitudes and weak westerlies or easterlies at high latitudes give rise to Ekman upwelling for the subpolar basin. Thus, wind-stress curl in the world's oceans gives rise to basin-scale Ekman pumping/upwelling. As will be discussed in relation to the wind-driven circulation theory, Ekman pumping is the primary driving force for the upper ocean circulation.

There are also other mechanisms which can induce upwelling/downwelling. As shown in the last term of Eqn. (2.197), the change in the Coriolis parameter, the so-called "beta effect," can give rise to Ekman upwelling/downwelling. In addition, strong along-shore winds can drive strong upwelling/downwelling in coastal areas. For example, near the coast of California, strong equatorward trade winds drive strong offshore Ekman transport in the upper ocean. Since there is no mass flux across the coastline, water must be drawn from depth near the coast in order to sustain the strong offshore Ekman flux. The upwelling/downwelling near coastal areas is a critically important component of the water mass cycle in the world's oceans.

2.9.2 Ekman spiral with inhomogeneous diffusivity

Although the Ekman theory has been widely regarded as the backbone of modern dynamical oceanography, the Ekman spiral predicted by classical theory was not verified through oceanic observations for a long time. According to the theory, for a steady wind stress, the velocity vector on the sea surface from classical theory is 45° to the right of wind stress (in the Northern Hemisphere), and the velocity vector rotates in the form of a spiral in a vertical direction. Observations, however, indicate that the angle between surface wind stress and surface drift velocity vector is in the range between 5° and 20° (Cushman-Roisin, 1994).

Recent observations also indicate that the surface velocity lies at more than the predicted 45° to the right of the wind. More importantly, the observed current amplitude decreases at a faster rate than it turns to the right, i.e., the observed velocity profiles in the Ekman layer seem "flat" (Chereskin and Price, 2001).

Despite great efforts to find solutions that fit the observations, most models within the framework of the classical laminar theory (with a diffusivity which is spatially isotropic and independent of time) fail to produce the observed "flat" spiral. It seems that other essential dynamical processes, such as buoyancy flux through the air-sea interface, stratification, the diurnal cycle, or even the Stokes drift, may have to be included in order to explain the observed structure of the Ekman layer (Price and Sundermeyer, 1999).

Owing to complicated dynamical processes in the upper ocean, parameterization of turbulent dissipation in the upper ocean remains a great challenge. Observations indicate that in a thin surface layer immediately below the sea surface, waves and turbulent activities are rather strong, so dissipation is nearly constant or increases slightly with depth; however, the dissipation rate declines with depth below this shallow layer. Direct observations in the California Current indicate that turbulent diffusivity declines exponentially below 20 m depth (Chereskin, 1995). Terray et al. (1996) carried out field observations and found that the dissipation rate is higher and roughly constant in a near-surface layer, but below this layer the dissipation rate decays as |z| 2. This is further refined as a -2.3 power law, i.e., dissipation rate decays as |z|-23, by Terray et al. (1999).

A crude model that can incorporate this complexity is a two-layer model with a power law of vertical diffusivity in each layer. In addition, the same approach can be used to find solutions for the case where the vertical diffusivity changes with depth exponentially. In such cases the solutions are in form of Bessel functions. A simple linear profile A = a|z| was used in previous studies, e.g., Madsen (1977). Such a profile is questionable because it is inconceivable that turbulent diffusivity is zero at the sea surface (N.E. Huang, 1979). Therefore, it seems more reasonable to choose a linear profile in the surface layer, n1 = 1, starting with a finite diffusivity on the sea surface. For the second layer, it is found that an inverse power profile with n2 = -0.7 has a best fit for the diffusivity diagnosed by Chereskin (1995). The application of this two-layer model is referred to in the study by Wang and Huang (2004a).

Another way to narrow the gap between observations and the classical Ekman spiral is to relax the isotropic assumption of vertical diffusivity. Although vertical diffusivity in the previous example is assumed to be isotropic, it is possible that, due to surface waves and other processes, the turbulence diffusivity may be non-isotropic. Assuming that the turbulent motions in the Ekman layer are non-isotropic, the structure of the Ekman spiral can easily be derived as follows. For simplicity, we will assume that vertical diffusivity is non-isotropic and remains constant in the vertical direction, i.e., Ax = const., Ay = const., but Ax = Ay. The corresponding momentum equations for the ageostrophic velocity components are d ( due\

2.10 Sverdrup relation, island rule, and the f-spiral 137

Integrating Eqn. (2.198a) and using boundary conditions, we obtain

Introducing a coordinate transform:

where A = AxAy is the geometric mean of the vertical diffusivity. Parallel to the previous approach, we can introduce the complex velocity

Eqations (2.198a, 2.198b) are reduced to d 2M

dz'2

The solution which satisfies the condition of no flow as z ^ -x is

Applying the stress boundary condition at the sea surface leads to

P0 V 2fA

The structure of the Ekman spiral under the assumption of non-isotropic vertical diffusivity is shown in Figure 2.24. It is clear that if the diffusivity is spatially non-isotropic, the Ekman spiral should have a structure quite different from the classical one.

2.10 Sverdrup relation, island rule, and the P-spiral

Many aspects of the fundamental structure of the wind-driven circulation in the ocean have been successively described by some classical theories, as discussed in many introductory textbooks about dynamical oceanography. In this section, we give a brief presentation of the dynamical laws. As discussed shortly, these laws are essentially various forms of the potential vorticity balance in the upper ocean.

2.10.1 Sverdrup relation

For large-scale steady wind-driven circulation in the ocean interior, the nonlinear advection terms are negligible, so the time-dependent term in Eqns. (2.133) and (2.134) can be omitted. Therefore, in a j-plane it can be described in terms of depth-integrated equations

where (U, V) are the depth-integrated volume flux, (tx, ty) are the wind-stress components, (Fx, Fy) are the forces due to bottom or lateral friction, which are assumed to be negligible except within the narrow western boundary, and (Px, Py) are the pressure gradient terms, which can be calculated from the density structure. As discussed in Section 2.6, the hydrostatic relation can be reduced to a relation between the perturbations of pressure and the density.

Assuming that both the bottom and lateral friction are negligible, cross-differentiating and subtraction of Eqns. (2.205) and (2.206) leads to

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