d2 y dz

This equation shows that terrestrial mirages are a consequence of vertical refractive index gradients: if this gradient is zero, ray paths are straight lines.

For a constant refractive index gradient, which to good approximation occurs for a constant temperature gradient, the solution to Eq. (8.48) is a parabola. One such parabola, for a constant lapse rate more than 100 times the average, is shown in Fig. 8.19. Note the greatly different horizontal and vertical scales. If we had plotted the parabola to uniform scale its curvature would not have been noticeable. The image is displaced downward from what it would be in the absence of the atmosphere, strictly in the absence of a vertical refractive index gradient. That is, an observer at I sees light that originated from O coming from a direction below (in angle) the straight line between O and I: hence the designation inferior mirage. This is the familiar mirage seen over highways warmer than the air above them. The downward angular displacement is

2 dz

This was obtained by solving Eq. (8.48), then determining the two constants of integration by requiring the ray to go through the points (h, 0) and (h, s), where h is the height and s the horizontal distance between object (O) and image (I). The displacement is then tan (5 =( -j-) ^5. (8.50)

Even for temperature gradients 1000 times the average lapse rate, angular displacements of mirages are less than a degree at distances of a few kilometers.

If temperature increases with height, as it might, for example, in air over a colder sea, the resulting mirage is called a superior mirage. The refractive index gradient in Eq. (8.49) changes sign, as does S. Inferior and superior are not designations of lower and higher castes but rather of displacements downward and upward.

For a constant temperature gradient, one and only one parabolic ray trajectory connects an object point to an image point. Multiple images therefore are not possible. But temperature gradients close to the ground are rarely linear. The upward transport of energy from a hot surface occurs by molecular conduction through a stagnant boundary layer of air. Somewhat above the surface, however, energy is transported by air in motion. As a consequence the temperature gradient steepens near the ground if the energy flux is constant. This variable gradient can lead to two consequences: magnification and multiple images.

Figure 8.20: When the sun is at an angle 5 (a fraction of a degree) below the horizon it still can be seen because of atmospheric refraction. The relative value of the scale height H to Earth's radius R is greatly exaggerated in this figure.

According to Eq. (8.49) all image points at a given horizontal distance are displaced downward by the same amount proportional to the constant refractive index gradient. This suggests that the closer an object point is to a surface, where the temperature is greatest, the greater the downward displacement of the corresponding image point. Thus nonlinear vertical temperature profiles may magnify images. Magnification in the optical sense is an increase in angular size, which is all that human observers directly perceive. Transforming angular sizes into linear sizes (lengths) is a complicated perceptual process. There is far more to seeing than the formation of images on retinas.

Multiple images are seen frequently on highways. What often appears to be water on a highway but evaporates before the water is reached is the inverted image of either the horizon sky or horizon objects brighter than the highway.

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