R

Equation 7-24 expresses the thermal wind relationship in pressure coordinates. By analogy with Eq. 7-8, just as height contours on a pressure surface act as streamlines for the geostrophic flow, then we see from Eq. 7-24 that temperature contours on a pressure surface act as streamlines for the thermal wind shear. We note in passing that one can obtain a relationship similar to Eq. 7-24 in height coordinates (see Problem 9 at end of chapter), but it is less elegant because of the p factors in Eq. 7-4. The thermal wind can also be written down in terms of potential temperature (see Problem 10, also at the end of this chapter).

The connection between meridional temperature gradients and vertical wind p p p

9In practice, friction caused by the cone of fluid rubbing over the bottom brings currents there toward zero. The thermal wind shear remains, however, with cyclonic flow increasing all the way up to the surface.

10In a stratified fluid the buoyancy frequency (Section 4.4) is given by N2 = -g/p dp/dz or N2 ~ g"/H where g = g^p/p and Ap is a typical variation in density over the vertical scale H.

shear expressed in Eq. 7-24 is readily seen in the zonal-average climatology (see Figs. 5.7 and 5.20). Since temperature decreases poleward, dT/dy < 0 in the northern hemisphere, but dT/dy> 0 in the southern hemisphere; hence f -1öT/öy< 0 in both.

Then Eq. 7-24 tells us that du/dp < 0: so, with increasing height (decreasing pressure), winds must become increasingly eastward (westerly) in both hemispheres (as sketched in Fig. 7.19), which is just what we observe in Fig. 5.20.

FIGURE 7.19. A schematic of westerly winds observed in both hemispheres in thermal wind balance with the equator-to-pole temperature gradient. (See Eq. 7-24 and the observations shown in Figs. 5.7 and 5.20.)

FIGURE 7.20. The temperature, T, on the 500-mbar surface at 12 GMT on June 21, 2003, the same time as Fig. 7.4. The contour interval is 2°C. The thick black line marks the position of the meridional section shown in Fig. 7.21 at 80° W extending from 20° N to 70° N. A region of pronounced temperature contrast separates warm air (pink) from cold air (blue). The coldest temperatures over the pole get as low as -32°C.

FIGURE 7.20. The temperature, T, on the 500-mbar surface at 12 GMT on June 21, 2003, the same time as Fig. 7.4. The contour interval is 2°C. The thick black line marks the position of the meridional section shown in Fig. 7.21 at 80° W extending from 20° N to 70° N. A region of pronounced temperature contrast separates warm air (pink) from cold air (blue). The coldest temperatures over the pole get as low as -32°C.

The atmosphere is also close to thermal wind balance on the large scale at any instant. For example, Fig. 7.20 shows T on the 500-mbar surface on 12 GMT on June 21, 2003, the same time as the plot of the 500-mbar height field shown in Fig. 7.4. Remember that by Eq. 7-24, the T contours are streamlines of the geostrophic shear, dug/dp. Note the strong meridional gradients in middle latitudes associated with the strong meandering jet stream. These gradients are also evident in Fig. 7.21, a vertical cross section of temperature T and zonal wind u through the atmosphere at 80° W, extending from 20° N to 70° N at the same time as in Fig. 7.20. The vertical coordinate is pressure. Note that, in accord with Eq. 7-24, the wind increases with height where T surfaces slope upward toward the pole and decreases with height where T surfaces slope downwards. The vertical wind shear is very strong in regions where the T surfaces steeply slope; the vertical wind shear is very weak where the T surfaces are almost horizontal. Note also the anomalously cold air associated with the intense low at 80° W, 40° N marked in Fig. 7.4.

In summary, then, Eq. 7-24 accounts quantitatively as well as qualitatively for the observed connection between horizontal temperature gradients and vertical wind shear in the atmosphere. As we shall see in Chapter 9, an analogous expression of thermal wind applies in the ocean too.

Instantaneous Section of Wind and Temperature along 80 W

FIGURE 7.21. A cross section of zonal wind, u (color-scale, green indicating away from us and brown toward us, and thin contours every 5 ms-1), and temperature, T (thick contours every 5°C ), through the atmosphere at 80° W, extending from 20° N to 70° N, on June 21, 2003, at 12 GMT, as marked on Figs. 7.20 and 7.4. Note that du/dp < 0 in regions where dT/dy < 0 and vice versa.

FIGURE 7.21. A cross section of zonal wind, u (color-scale, green indicating away from us and brown toward us, and thin contours every 5 ms-1), and temperature, T (thick contours every 5°C ), through the atmosphere at 80° W, extending from 20° N to 70° N, on June 21, 2003, at 12 GMT, as marked on Figs. 7.20 and 7.4. Note that du/dp < 0 in regions where dT/dy < 0 and vice versa.

7.4. SUBGEOSTROPHIC FLOW: THE EKMAN LAYER

Before returning to our discussion of the general circulation of the atmosphere in Chapter 8, we must develop one further dynamical idea. Although the large-scale flow in the free atmosphere and ocean is close to geostrophic and thermal wind balance, in boundary layers where fluid rubs over solid boundaries or when the wind directly drives the ocean, we observe marked departures from geostrophy due to the presence of the frictional terms in Eq. 6-29.

The momentum balance, Eq. 7-2, pertains if the flow is sufficiently slow (Ro << 1) and frictional forces F sufficiently small, which is when both F and Du/Dt in Eq. 6-43 can be neglected. Frictional effects are indeed small in the interior of the atmosphere and ocean, but they become important in boundary layers. In the bottom kilometer or so of the atmosphere, the roughness of the surface generates turbulence, which communicates the drag of the lower boundary to the free atmosphere above. In the top one hundred meters or so of the ocean the wind generates turbulence, which carries the momentum of the wind down into the interior. The layer in which F becomes important is called the Ekman layer, after the famous Swedish oceanographer who studied the wind-drift in the ocean, as will be discussed in detail in Chapter 10.

If the Rossby number is again assumed to be small, but F is now not negligible, then the horizontal component of the momentum balance, Eq. 6-43, becomes:

To visualize these balances, consider Fig. 7.22. Let's start with u: the Coriolis force per unit mass, -fz x u, must be to the right of the flow, as shown. If the frictional force per unit mass, F, acts as a "drag" it will be directed opposite to the prevailing flow. The sum of these two forces is

FIGURE 7.22. The balance of forces in Eq. 7-25: the dotted line is the vector sum F-fz x u and is balanced by -1/pVp.

depicted by the dashed arrow. This must be balanced by the pressure gradient force per unit mass, as shown. Thus the pressure gradient is no longer normal to the wind vector, or (to say the same thing) the wind is no longer directed along the isobars. Although there is still a tendency for the flow to have low pressure on its left, there is now a (frictionally-induced) component down the pressure gradient (toward low pressure).

Thus we see that in the presence of F, the flow speed is subgeostrophic (less than geostrophic), and so the Coriolis force (whose magnitude is proportional to the speed) is not quite sufficient to balance the pressure gradient force. Thus the pressure gradient force "wins," resulting in an ageostrophic component directed from high to low pressure. The flow "falls down'' the pressure gradient slightly.

It is often useful to explicitly separate the horizontal flow uh in the geostrophic and ageostrophic components thus:

where uag is the ageostrophic current, the departure of the actual horizontal flow from its geostrophic value ug given by Eq. 7-3. Using Eq. 7-25, Eq. 7-26, and the geostrophic relation Eq. 7-2, we see that:

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