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Fly with the Wind

S uppose you want to visit New York City from San Francisco. According to Google Maps, this trip would take 47 hours to drive, each way! More likely, you would decide to save time and fly. The eastbound flight times from San Francisco to New York City are approximately 5.5 hours, and the return flights are roughly 6.5 hours. Why is the eastbound flight so much shorter than the westbound? It is, after all, the same distance. The explanation is probably not a surprise—it’s the wind. But have you ever asked why the winds blow from west to east across the United States? And what do wind patterns have to do with multivariable calculus? Upper-Level Charts Before we investigate the calculus behind wind patterns, we need to understand one of the standard tools used in aviation meteorology: upper-level charts. Figure 1 shows an upper-level chart of elevation for a fixed pressure of 250 mb (millibars are a measure of pressure) from April 15, 2012. You probably have seen maps like this while looking up weather information, but you have also seen diagrams resembling this in multivariable calculus. In multivariable calculus, we analyze functions by looking at level curves. For example, in figure 2, we see a topographical map from Palmer Lake, Colorado. The brown elevation contours indicate points that are at the same elevation; they are level curves for the elevation function. If you were to hike along one of those contours, the trail would be horizontal and easy to walk. If you walk perpendicular to a contour toward higher altitude, like from the Lower Reservoir toward the top of Sundance Mountain, you would be hiking straight uphill. In multivariable calculus, we learn that the direction of steepest ascent is given by the gradient of the elevation function. The gradient vector is orthogonal to the level curve, and its magnitude is the maximum rate of increase. The contours are very close together between the Lower Reservoir and the top of Sundance Mountain, meaning that the elevation changes quickly there and the slope of the land is very steep; this corresponds to a large gradient vector. Near the highway, indicated by the red-and-white line, the height contours are far apart, indicating that the surface is relatively level; the gradient is small. Twice each day, at hundreds of sites around the world, technicians launch weather balloons carrying a light, compact instrument package called a radiosonde. As the balloon rises, the equipment measures pressure, temperature, humidity, and location. From the weather balloon data we obtain maps showing height and temperature contours at standard pressure levels throughout the atmosphere. For example, figure 1 shows the elevations where the pressure is 250 mb. This is the reverse of what we might expect; the map shows the altitude at a constant pressure, instead of the pressure at a constant altitude. The maps show altitude differences using height contours just like the topographical map in figure 2 does. That is, figure 1 shows the level curves of the function h, where h(x,y) is the altitude of the 250 mb pressure surface over the point (x,y). The spacing between the contours is 120 m, or just less than 400 ft. The lowest altitudes are in Canada, Fly with the Wind