## APPENDix B FLoW iN An Ekman Layer

Most large-scale flow in the ocean is in geostrophic balance, meaning that, away from the direct influence of the wind and so beneath the Ekman layer, the pressure gradient force is balanced by the coriolis force. Mathematically we have, as in equation 3.11, ftvg fPUg I' (3.26a, b)

where ug and vg are the fluid speeds in the zonal and meridional directions, respectively. The first of this pair of equations is the momentum balance in the zonal direction (with x the distance toward the east): the Coriolis force -fv balances the pressure gradient force. Similarly, the second equation is the momentum balance in the meridional (y) direction. These equations may be regarded as defining the geostrophic velocities ug and vg (note the subscript g on the variables).

Now suppose that we add a stress, x, to this balance. The stress is provided by the wind, and it diminishes with depth. The force is the vertical derivative of the stress so that equations 3.26a and b become dp 2X dp dTy „„ , , fPv dx + dx> fpU dy + T (3.27a, b)

where u and v are the zonal and meridional components of the total velocity (not just the geostrophic flow) in the upper ocean, and xx and xy are the corresponding components of the stress. We rewrite equations 3.27a and b as dT1 dTy fp (v vg) dZ' fp {U Ug) it' (3.2^ b)

having used equations 3.26a and b as the definitions of geostrophic velocity. If we integrate equations 3.28a and b over the depth of the Ekman layer (i.e., from the surface to the depth at which the stress vanishes), we obtain fv xw, fU xy, (3.29a, b)

where xxw and xyw are the zonal and meridional components of the wind stress at the surface, and V = jp(v — vg) dz and U = jp(u — ug)dz are the meridional and zonal components of the wind-induced, nongeostrophic mass transports, integrated over the Ekman layer. Thus, we see that a zonal wind stress (i.e., a nonzero TWW) induces a meridional flow in the ocean, and a meridional wind stress induces a zonal flow. That is, the average induced velocity in the Ekman layer—the Ekman transport—is perpendicular to the imposed wind stress at the surface. In the Northern Hemisphere where f is positive, if the stress is eastward (i.e., T^ is positive), then the Ekman transport is southward (V is negative).

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