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CHAPTER 5 Dynamics

The Thermal Wind

In this section we consider how and why the winds change as we rise up through the atmosphere. Since we know from observations that the winds aloft in the middle latitudes are westerly (+u direction) we will begin our consideration with geostrophic winds in the +u direction.

(EQ 259)

In this equation we see that if pressure declines (-dP) as we go north (+dy), given that density is positive and f is positive in the Northern Hemisphere, that u must be positive as well, i.e. a wind to the east. It is this wind that we will consider in our exploration of wind changes with altitude. We wish to know how the wind (u) changes with changing altitude (dP), i.e.,

(EQ 260)

We will differentiate Eq. 259 with respect to pressure.

(EQ 261)

We learn from Eq. 261 that the winds change with altitude in proportion to how the north-south slope of the pressure surface changes as we go up. Next we will replace the dP in the numerator with dP from the hydrostatic equation (dP = -rgdz).

(EQ 262)

First, we simplify Eq. 262, i.e. assuming constant density in the y direction, and because, for a + u direction wind dz must get smaller (-) as one proceeds north (+dy). As a result it is necessary at this point to assign a - to dz/dy (see Figure 97).

(EQ 263)

In Eq. 263 we see that the winds change with elevation in proportion how the slope of the pressure surface changes with altitude (Figure 97). One of the causes of a slope to a pressure surface is differential heating Figure 96. In warm places the height to a given pressure level is greater than in cool places (h < h').

FIGURE 96 : Thermal Fields and Thickness

FIGURE 97 : Slope of a Tilted Pressure Surface

Next we change the order of differentiation on the right hand side of the equation.

(EQ 264)

Next we substitute the hydrostatic equation in the numerator of the right hand side of Eq. 264.

(EQ 265)

The equation can now be simplified by canceling g and substitute the equation of state r = RT/P for r.

(EQ 266)

Here R is the gas constant for dry air and for a layer of the air we are concerned with. We will use the average pressure of the layer making P a constant as well. Our equation now reduces to

(EQ 267)

This equation is known as the thermal wind equation. It states that the winds change with altitude in proportion to the change in temperature in the north-south direction. More specifically, it states that for a positive f (Northern Hemisphere) when it is cold in the north and warm in the south the +u winds increase as you gain altitude and pressure falls (dP negative). By substituting dz the hydrostatic equation on the left and R from the equation of state on the right, Eq. 267can be converted to a form in which the vertical reference in z in meters.

(EQ 268)

This form of the Thermal Wind Equation can also be written as

(EQ 269)

The thermal wind equation helps us understand the seasonal variations in the strength of the westerlies and the jet stream aloft. Summer and winter the tropical latitudes have very similar temperatures. In the high latitudes, however, temperatures are warm and in the winter very cold. Thus dT/dy is small and negative in summer and very large and negative in winter. This means that the westerly winds and the jet stream increase in speed from summer to winter. We also know that the location where the north-south temperature contrast is greatest is far to the north in the summer and far to the south in winter. We then expect that the westerlies and jet stream aloft will undergo seasonal variation with the latitude. The jet stream is in a northerly position (at the latitude of the US/Canada border) in summer and in a southerly position (at the latitude Cape Hatteras) in winter.

Since 1949 the U.S. Weather Service has recorded the wind speed an direction with altitude at a number of Class A weather stations. Such a sounding of wind speed and direction is shown in Figure 98. The data for this figure come from Green Bay, Wisconsin on 19 April 1963 at 6 P.M. The illustration indicates that wind speeds increase rapidly with elevation above the surface and reach 52 m s-1 at 10 km altitude. This increase in winds with altitude indicates that there exists a horizontal temperature gradient.

The numbers at various points on the profile of wind speeds indicate the direction of the wind at each level (90o is from the East, 180o is form the South and 270o is from the West). At the surface the winds are out of the SE (143o) and the winds veer (clockwise change in direction with height) to the WSW at 10 km where the westerly winds are strongest. Given the maximum winds at 10 km from the WSW, we can now conclude that the lowest temperatures are found in the direction NNW and the warmest temperatures in the SSE. This direction of temperature gradient (dT/dy) is 90o to the direction of the WSW winds.

FIGURE 98 : Wind Speed and Direction Change with Altitude

It is useful to illustrate the changes in wind speed and wind direction with height as a schematic of vectors and the addition of vectors. Consider a circumstance where the winds at Charlottesville are from the SW and the clouds aloft are moving from the NW. We can diagram this condition (Figure 99). Three vectors are shown. The winds at the surface (thinnest arrow) are from the SW. The winds aloft (medium thickness arrow) are from the NW. The vector sum of these two vectors is the thickest arrow and represents the thermal wind. In this case the thermal wind is the wind that would have to be added to the surface winds to get the winds aloft. Also shown in Figure 99 are isobars (parallel to surface winds with low pressure to the upper left and high pressure on the lower right) and isotherms (parallel to the thermal wind vector with cold air on the left and warm on the right). If you know the surface thermal field and the surface pressure field, you can then estimate the winds aloft by construction of a diagram like that in Figure 99.

FIGURE 99 : Thermal Wind by Vector Addition

We can now use this convention of vector addition and consider a high and a low pressure at the surface with a temperature gradient where it is cold in the north and warm in the south (Figure 101).

In Figure 101 wind vectors representing the surface wind flow (light shade), the thermal wind vector (horizontal shade) and the flow aloft (black). The sinusoidal isolines represent flow that is everywhere parallel to the winds aloft, i.e. streamlines of the winds aloft. It is clear from this analysis that the flow aloft takes on a sinusoidal character.

Climate Dynamics - 05 FEB 96
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