## N Wt

d) If the slippery boundary condition applies, fB,nn = 0, at n = 0; thus, C4 = -03/^/3. The final solution is fB = fi \ 1 - e 2

e) Structure of the western boundary current: the layer thickness of the boundary current can be obtained from the semi-geostrophic relation h2 = 2f fi + hW, hW = hi - 2f fi/g' (4.67)

The corresponding velocity field can be calculated accordingly as v = — = ^^.

h dMh inertial western boundary current The existence of an inertial western boundary current was postulated first by Stommel (1954). His basic idea is that within the western boundary the inertial term associated with horizontal advection balances the planetary vorticity term. In this way the ambiguity of the friction parameters used in models with interfacial friction or lateral friction can be avoided. The accurate formulation of this problem was first presented by Charney (1955) and Morgan (1956).

### General solution

The basic equations for this case include the nonlinear advection terms, but the interfacial and lateral friction terms are omitted h(uux + vuy) - fhv = -g'hhx + rx/p0 (4.68)

Scaling analysis leads to a simpler set of equations. In particular, the x-momentum equation is reduced to geostrophy in the cross-stream direction fhv = g'hhx (4.71)

Combining Eqns. (4.71), (4.69), and (4.70) leads to

B = 2 v + g h = F (f), energy conservation (4.72) f + vx

Q =-= G(f), potential vorticity conservation (4.73)

where F(f) and G(f) are functions completely determined from the interior solution at the outer edge of the western boundary current.

where Y is the meridional coordinate at the outer edge of the western boundary layer. From Eqn. (4.42):

Assuming this function is invertible, we can write

Thus, both F and G are completely determined from the interior solution.

The one-to-one inversion of Eqn. (4.76) breaks down at the latitude where f j (Y) reaches its maximum; this is the northern limit of the purely inertial boundary layer. North of this limit, other mechanisms are needed in order to maintain a steady boundary current.

The semi-geostrophic relation leads to the relation between the streamfunction and the layer depth, the same as in the case with interfacial friction, Eqn. (4.40):

The meridional velocity can be calculated using the Bernoulli law v = V2 [F (f) - g'h] (4.78)

In this way both the layer thickness and meridional velocity are determined in the streamfunction coordinates. Solution in the physical coordinates can be obtained through coordinate transformation f f d f x = (4.79)

Jo hv where hv is the function of f, defined in Eqns. (4.40') and (4.78). The transformation from the physical coordinates x to the streamfunction coordinates is the well-known von Mises (1927) transformation used in fluid mechanics. This coordinate transformation was first used by Charney (1955) to solve the inertial western boundary current.

The structure of the solution obtained from a model with the same parameters (for the interior solution) as shown in the previous model with a Stommel boundary layer (Fig. 4.6) is shown in Figures 4.9 and 4.10, including the inertial western boundary current in the southern half of the western boundary.

a Western boundary b Basin interior a Western boundary b Basin interior

The inertial western boundary current has features quite different from frictional boundary layers. First, a purely inertial western boundary current is allowed for the southern half of the western boundary only, but in the northern half of the basin, a purely inertial western boundary current is not a valid solution.

Second, the width of the boundary currents may depend on parameters used in the models. In general, for the parameters suitable for simulating the ocean, the inertial western boundary current is wider than the frictional boundary layer. Simulating the western boundary current in terms of a pure inertial boundary current or a purely interfacial frictional boundary layer is idealization only. In reality, the inertial terms, the interfacial frictional terms, and the lateral frictional terms should all contribute to the dynamic balance of the western boundary current. Since the inertial boundary layer is wider than the frictional boundary layer, the western boundary layer can be separated into different regimes dominated by different dynamical processes. The outer part of the boundary layer is dominated by the inertial terms, while the frictional effect is mostly limited to the relatively narrow sub-layer near the wall (Pedlosky, 1987a).

### Special case when is constant

The inertial western boundary current has simple analytical form when potential vorticity is a constant, G(fi) = f /hj = const. In such a case the vorticity equation is reduced to fh vx + f = « (4.80)

Using the semi-geostrophic condition, this leads to hxx - h = f (4.81)

The general solution of this equation is h = a • e-x/* + b • ex/* + hj where

is the radius of deformation. Assuming h« ~ 400 m and g' ~ 0.015 m/s2, we have Sj ~ * ~ 250 km; thus, the scale width of the inertial western boundary is wider than the frictional boundary layer discussed in the previous subsection. Under the following conditions h(0) = hw, h(«) = hj (4.83)

where hw is calculated from Eqn. (4.40').

Limitation and extension of the reduced-gravity models

### Limitation of the reduced-gravity models

The major assumption used in the layered reduced-gravity models is the existence of a stagnant lower layer. The major advantages of such an assumption are as follows. First, these models filter out the external gravity modes, with phase speed */gH, where H is the depth of the ocean. Instead, these models retain only internal gravity waves, whose wave speed is ^fg7h, where h is the thickness of the upper layer. Since the upper layer is typically a few hundreds of meters, h ^ H .In addition, g'/g ~ 0.001, so that large time steps can be used in reduced-gravity models. Second, the reduced-gravity models can avoid the complication due to flow over bottom topography.

The reduced-gravity models are based on the assumption that flow underneath the main thermocline is very slow and negligible. These models can be good tools for the study of circulation in the upper ocean; however, they cannot represent the complicated three-dimensional circulation associated with flows in the abyssal ocean very accurately. For example, in the study of thermohaline circulation or coastal circulation, flows in the bottom layer are an important component of the whole circulation system; thus, models ignoring bottom circulation are not suitable.

For problems associated with time scales shorter than the time scale for the first baroclinic Rossby waves to move across the basin at the latitude of interest, the errors implied by the reduced-gravity formulation may not be totally negligible. Nevertheless, reduced-gravity models have been used to simulate the seasonal cycle, such as the free surface elevation and other properties, for the subtropical basin. Furthermore, reduced-gravity models have been extensively used to study equatorial circulation on seasonal to interannual time scales. For a more accurate description of the time evolution of the sea surface, a two-layer model can be used. In such a model, the lower layer is in motion, so that the contribution due to fast-moving Rossby waves can be included in the calculation as well (Qiu, 2002b).

### Layer outcropping and the Parsons model

A major difference between the reduced-gravity models discussed above and the linearized layer model used in many previous studies is that we allow the layer thickness to vary greatly. When the forcing is strong, the interface can outcrop. A layered model with outcropping requires careful treatment both analytically and numerically. Parsons (1969) discussed such a model and made the connection between the Gulf Stream and the outcropping of the main thermocline in a subtropical basin. The application of models with outcropping isopycnals to the world's oceans has been discussed in many papers, e.g., Veronis (1973), Huang and Flierl (1987). We discuss the Parsons model in Section 4.1.4.

### 4.1.3 The physics of wind-driven circulation

The discussion of wind-driven circulation in the previous section is based on detailed dynamical analysis which can be complicated for some cases. For a better understanding of the circulation, it is more revealing to build up a physical picture of the circulation without involving the mathematical details. Thus, in this section we will try to interpret the basic structure of the circulation in terms of the fundamental physics.

The interior solution

### Meridional flow driven by Ekman pumping

The wind-driven circulation in the ocean interior can be illustrated as follows (Fig. 4.11). In order to maintain a relatively steady rotation of the Earth, the globally integrated frictional torque exerted by the atmosphere on the solid Earth and oceans should be zero; thus, both the westerlies and easterlies are necessary components of the atmospheric circulation, i.e., the prevailing westerlies at mid latitudes coexist with easterlies at low latitudes and polar regimes.

This wind stress pattern drives poleward Ekman flows at both low and high latitudes, but drives an equatorward Ekman flow at mid latitudes. As a result, the meridional convergence of Ekman flux in the upper ocean gives rise to the Ekman pumping and upwelling below the base of the Ekman layer. In the subtropical basin, a downward Ekman pumping induces a compression of the water column. In the basin interior, the relative vorticity is negligible, so potential vorticity for a water column is f /h. Ekman pumping at the base of the Ekman layer compresses the water column height h. In order to conserve potential vorticity f /h, the individual water column moves toward the equator, where the Coriolis parameter f is smaller. Thus, Ekman pumping in the subtropical basin drives an equatorward flow in the

Wind stress

Equaton

North East

Ekman layer

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