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where k is a drag coefficient that depends on the roughness of the underlying surface. Note that the minus sign ensures that F acts as a drag on the flow. Then the momentum equations, Eq. 7-25, written out in component form along and across the isobars, become:

Note that v is entirely ageostrophic, being the component directed across the isobars. But u has both geostrophic and ageostro-phic components.

Solving Eq. 7-29 gives:

Note that the wind speed is less than its geostrophic value, and if u> 0, then v> 0 and vice versa; v is directed down the pressure gradient, from high to low pressure, just as in the laboratory experiment and in Fig. 7.24.

conditions, and

In typical meteorological ~ 1 km, k is (1

k/f 5 ~ 0.1. So the wind speed is only slightly less than geostrophic, but the wind blows across the isobars at an angle of 6° to 12°. The cross-isobaric flow is strong over land (where k is large), where the friction layer is shallow (5 small), and at low latitudes (f small). Over the ocean, where k is small, the atmospheric flow is typically much closer to its geostrophic value than over land.

Ekman developed a theory of the boundary layer in which he set F = nd2u/dz2 in Eq. 7-25, where n was a constant eddy viscosity. He obtained what are now known as Ekman spirals, in which the current spirals from its most geostrophic to its most ageostrophic value (as will be seen in Section 10.1 and Fig. 10.5). But such details depend on the precise nature of F, which in general is not known. Qualitatively, the most striking and important feature of the Ekman layer solution is that the wind in the boundary layer has a component directed toward lower pressure; this feature is independent of the details of the turbulent boundary layer.

### Vertical motion induced by Ekman layers

Unlike geostrophic flow, ageostrophic flow is not horizontally nondivergent; on the contrary, its divergence drives vertical motion, because in pressure coordinates, Eq. 6-12 can be written (if f is constant, u

FIGURE 7.25. Surface pressure field and surface wind at 12 GMT on June 21, 2003, at the same time as the upper level flow shown in Fig. 7.4. The contour interval is 4mbar. One full quiver represents a wind of 10ms-1; one half quiver a wind of 5ms-1. The thick black line marks the position of the meridional section shown in Fig. 7.21 at 80° W.

FIGURE 7.25. Surface pressure field and surface wind at 12 GMT on June 21, 2003, at the same time as the upper level flow shown in Fig. 7.4. The contour interval is 4mbar. One full quiver represents a wind of 10ms-1; one half quiver a wind of 5ms-1. The thick black line marks the position of the meridional section shown in Fig. 7.21 at 80° W.

so that geostrophic flow is horizontally nondivergent):

This has implications for the behavior of weather systems. Fig. 7.26 shows schematics of a cyclone (low-pressure system) and an anticyclone (high-pressure system). In the free atmosphere, where the flow is

FIGURE 7.26. Schematic diagram showing the direction of the frictionally induced ageostrophic flow in the Ekman layer induced by low pressure and high pressure systems. There is flow into the low, inducing rising motion (the dotted arrow), and flow out of a high, inducing sinking motion.

Atmospheric Surface Pressure (mb)

120"E 150'E 180' 150'W 120"W flO'W 60'W 30'WI 0' 30'E 60'E 90'E 120'E 150'E

Longitude

FIGURE 7.27. The annual-mean surface pressure field in mbar, with major centers of high and low pressure marked. The contour interval is 5 mbar.

120"E 150'E 180' 150'W 120"W flO'W 60'W 30'WI 0' 30'E 60'E 90'E 120'E 150'E

Longitude

FIGURE 7.27. The annual-mean surface pressure field in mbar, with major centers of high and low pressure marked. The contour interval is 5 mbar.

geostrophic, the wind just blows around the system, cyclonically around the low and anticyclonically around the high. Near the surface in the Ekman layer, however, the wind deviates toward low pressure, inward in the low, outward from the high. Because the horizontal flow is convergent into the low, mass continuity demands a compensating vertical outflow. This Ekman pumping produces ascent, and, in consequence, cooling, clouds, and possibly rain in low pressure systems. In the high, the divergence of the Ekman layer flow demands subsidence (through Ekman suction). Therefore high pressure systems tend to be associated with low precipitation and clear skies.

### 7.4.3. Planetary-scale ageostrophic flow

Frictional processes also play a central role in the atmospheric boundary layer on planetary scales. Fig. 7.27 shows the annual average surface pressure field in the atmosphere, ps. We note the belt of high pressure in the subtropics (latitudes ± 30°) of both hemispheres, more or less continuous in the southern hemisphere, confined mainly to the ocean basins in the northern hemisphere. Pressure is relatively low at the surface in the tropics and at high latitudes (± 60°), particularly in the southern hemisphere. These features are readily seen in the zonal-average ps shown in the top panel of Fig. 7.28.

To a first approximation, the surface wind is in geostrophic balance with the pressure field. Accordingly (see the top and middle panels of Fig. 7.28 since dps/dy < 0 in the latitudinal belt between 30° and 60° N, then from Eq. 7-4, us> 0 and we observe westerly winds there; between 0° and 30° N, ps increases, dps/dy> 0 and we find easterlies, us < 0—the trade winds. A similar pattern is seen in the southern hemisphere (remember f < 0 here); note the particularly strong surface westerlies

FIGURE 7.28. Anually and zonally averaged (top) sea level pressure in mbar, (middle) zonal wind in ms-1, and (bottom) meridional wind in ms-1. The horizontal arrows mark the sense of the meridional flow at the surface.

Latitude

FIGURE 7.28. Anually and zonally averaged (top) sea level pressure in mbar, (middle) zonal wind in ms-1, and (bottom) meridional wind in ms-1. The horizontal arrows mark the sense of the meridional flow at the surface.

around 50° S associated with the very low pressure observed around Antarctica in Figs. 7.27 and 7.28 (top panel).

Because of the presence of friction in the atmospheric boundary layer, the surface wind also shows a significant ageostrophic component directed from high pressure to low pressure. This is evident in the bottom panel of Fig. 7.28 which shows the zonal average of the meridional component of the surface wind vs. This panel shows the surface branch of the meridional flow in Fig. 5.21 (top panel). Thus in the zonal average we see vs which is entirely ageostrophic, feeding rising motion along the inter-tropical convergence zone at the equator, and being supplied by sinking of fluid into the subtropical highs of each hemisphere, around ± 30°, consistent with Fig. 5.21.

We have now completed our discussion of balanced dynamics. Before going on to apply these ideas to the general circulation of the atmosphere and, in subsequent chapters, of the ocean, we summarize our key equations in Table 7.1.