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FIGURE 10.2. Annual mean wind stress on the ocean. The green shading and contours represent the magnitude of the stress. Stresses reach values of 0.1 to 0.2 Nm-2 under the middle-latitude westerlies, and are particularly strong in the southern hemisphere. The arrow is a vector of length 0.1 Nm-2. Note that the stress vectors circulate around the high and low pressure centers shown in Fig. 7.27, as one would expect if the surface wind, on which the stress depends, has a strong geostrophic component.

FIGURE 10.2. Annual mean wind stress on the ocean. The green shading and contours represent the magnitude of the stress. Stresses reach values of 0.1 to 0.2 Nm-2 under the middle-latitude westerlies, and are particularly strong in the southern hemisphere. The arrow is a vector of length 0.1 Nm-2. Note that the stress vectors circulate around the high and low pressure centers shown in Fig. 7.27, as one would expect if the surface wind, on which the stress depends, has a strong geostrophic component.

10.1.1. Balance of forces and transport in the Ekman layer

If the Rossby number is small, we can neglect the D/Dt terms in the horizontal momentum Eq. 9-7, reducing them to a three way balance between the Coriolis force, the horizontal pressure gradient, and the applied wind stress forcing. This is just Eq. 7-25 in which F is interpreted as an applied body force due to the action of the wind on the ocean. First we need to express F in terms of the wind stress, Twind.

Consider Fig. 10.3 showing a stress that varies with depth, acting on a body of ocean. The stress component of interest here, tx (z), is the x-component of force acting at depth z, per unit horizontal area on the layer beneath. Note that the units of t are Nm-2. The slab of thickness Sz at level z is subjected to a force per unit horizontal area tx (z + Sz) at its upper surface, but also subjects the layers beneath it to a force tx (z) per unit horizontal area. Therefore the net force per unit horizontal area felt by the layer is tx(z + Sz) - tx(z). Since the slab has thickness Sz, it has volume Sz per unit horizontal area; and if the slab has uniform density pref, it has

mass pref Sz per unit horizontal area. Therefore the force per unit mass, Fx, felt by the slab is

Fx force per unit area mass per unit area

Pref dz

Pref Sz for small slab thickness. We can obtain a similar relationship for Fy and hence write:

Pref dz

for the horizontal stress vector t = (tx,tv). Hence our momentum equation for the steady circulation becomes Eq. 7-25 with

F defined above which, for convenience, we write out in component form here:

Pref dx Pref dz '

Pref fy Pref dz

Equation 10-3 describes the balance of forces in the directly wind-driven circulation, but it does not yet tell us what that circulation is. The stress at the surface is known—it is the wind stress, Twind, plotted in Fig. 10.2—but we do not know the vertical distribution of stress beneath the surface. The wind stress will be communicated downward by turbulent, wind-stirred motions confined to the near-surface layers of the ocean. The direct influence of wind-forcing decays with depth (rather rapidly, in a few tens of meters or so, depending on wind strength) so that by the time a depth z = -5 has been reached, the stress has vanished, t = 0, as sketched in Fig. 10.4. As discussed in Chapter 7, this is the Ekman layer.

Conveniently, we can bypass the need to know the detailed vertical distribution of t by focusing on the transport properties of the layer by integrating vertically across it. As in Section 7.4, we split the flow into geostrophic and ageostrophic parts. With F given by Eq. 10-2, the ageostrophic component of Eq. 7-25 is

Pref dz

Multiplying Eq. 10-4 by pf and integrating across the layer from the surface where t = Twind, to a depth z = -5, where t = 0 (see Fig. 10.4) we obtain:

Pref uagdz

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