Inferior Mirages Research Articles

Light refraction in the Earth’s atmosphere III. Inferior mirages: images locus

Our main goal is to find the locus of images formed as a result of an inferior mirage. To achieve our goal we show, at first, that, provided the beam entering a detection system has a small aperture, the image of a point object formed by that system in a vertical plane passing through the object is a point, regardless of whether the image formed by an optical system previously traversed by the beam has generated a non-point image of the object. Secondly, we show that the different images formed by the detection system, as its position relative to the object varies, are located on the caustic curve corresponding to the previously traversed optical system. Next, we have found the analytical expressions of the caustic curves corresponding to two particular cases, one of them being the inferior mirages. These expressions have been found by means of the Legendre antitransform of the asymptotic lines to the paths of light rays reaching the detector. For the case of inferior mirages, we have studied in detail the locus of the images in each vertical plane passing through the object as a function of the position of the object relative to the ground, its position relative to the detector, and the atmospheric conditions. Finally, we get somewhat into the matter of the images position as they would be seen by a casual observer of an inferior mirage.

Did Monge really explain inferior mirages?

Gaspard Monge has received undue credit for his observations of mirages in 1798 in Egypt. He did not discover inferior mirages; they were known to sailors before 1687, and the term mirage had entered the French literature in 1753. Monge’s “total reflection” explanation for them was seriously flawed; it had already been considered and rejected by Gruber and by Büsch, and was promptly criticized by others. Yet he continues to be given credit for the first scientific explanation of mirages; and his false total-reflection mechanism survives today in some textbooks. Apparently this is due to its superficial simplicity, which appeals to novices.

Light refraction in the earth's atmosphere I. Inferior mirages: analytic solution of ray paths

We aim to reach an analytic expression that describes the path of light rays through the Earth’s atmosphere in the particular situation in which an inferior mirage is occurring. To achieve our goal, we assume an exponential refractive index profile close to the ground, as suggested by empirical and theoretical studies on the state of air when an inferior mirage is taking place. We consider a parallel-plane atmosphere and assume that the laws of geometric optics apply. Since Fermat’s principle holds, we solve the Euler’s equation and, from the solution we obtain an analytic expression that describes the ray paths in a plane perpendicular to the ground. Given that we focus on the particular case of inferior mirages, we were able to find a very simple and easy-to-use expression to describe the ray paths, allowing us to extract additional valuable information with minimal computational effort. We take advantage of it to impose a limit to the thickness of the air layer next to the ground where appropriate conditions exist to bend the rays upwards, and produce an inferior mirage.

A Simple Way to Demonstrate and Explain Different Facets of Mirages

As intriguing optical phenomena that occur in nature, mirages are widely known and often mentioned in physics classes when introducing the basics of optics. In the atmosphere, a mirage occurs under the condition of vertical variations in air density, which are usually caused by a temperature gradient. The light rays are continuously refracted and follow a curved path when passing through the air, because the refractive index of air changes with its density. Hence, the image observed in an optical manner is displaced from the location of the object. An upward-shifted image is known as a superior mirage, whereas a downward-shifted one is known as an inferior mirage. There are several simple experiments that can create both superior and inferior mirages, using a non-uniform liquid solution to simulate the refractive index gradient in the air. These experiments not only demonstrate the curved paths of light rays, but also produce artificial multiple-image mirages. This paper introduces a simplified method of creating inferior mirages in a liquid solution. Using this method, we further demonstrate several different facets of mirages and explain their origins through computer simulations.

Sunset mirages involving ducts

Sunset mirages are often observed in coastal California. Cold ocean currents and warm air from inland create great conditions for temperature inversions, prerequisite required for producing mirages. Most sunset mirages seen in California are so called mock mirages. Inferior mirages are also common during winter time. On a rare occasions California sky watchers are treated by some special mirages that involve ducting, including Novaya Zemlya effect sunsets, blank strip sunsets and others.

Atmospheric optics in the near infrared

Digital near-infrared photography opens up new observation possibilities and applications for atmospheric optics. We discuss necessary conditions and requirements for observing a variety of atmospheric optical phenomena in the infrared spectral range and report for the first time near-infrared photographs of 22° ring halos and inferior mirages. Our emphasis is on optical phenomena observable in the troposphere, excluding the large body of work addressing near-infrared airglow and aurora.

Inferior mirages: an improved model.

A quantitative model of the inferior mirage is presented, based on a realistic temperature profile in the convective boundary layer, using Monin-Obukhov similarity theory. The top of the inverted image is determined by the logarithmic part of the profile; the bottom is the apparent horizon, which depends on optical obstruction by roughness elements. These effects of surface roughness are included in the model, which is illustrated with a simulation. The vertical magnification varies throughout the mirage, becoming infinite at Minnaert's ill-named "vanishing line"-which makes green flashes apparent to the naked eye.

Visible and invisible mirages: comparing inferior mirages in the visible and thermal infrared.

Visible (VIS)-light and thermal infrared (IR) inferior mirages in the 8-14μm waveband have been observed simultaneously for the takeoff and landing of various airplanes at distances of several kilometers. Similarities as well as differences between the VIS and IR mirages are discussed.

Noninverted images in inferior mirages

Inferior mirages over sun-exposed roads often appear in isolated strips at their near sides and the reflected scenery exhibits multiple images. This effect is explained as due to slight undulations of the road's surface. At the same time, some of these images, although they are reflections, are not inverted. Photographic material illustrates this phenomenon and a ray tracing study is presented that confirms these conclusions.

Simulation of the Inferior Mirage

A mirage can occur when a continuous variation in the refractive index of the air causes light rays to follow a curved path. As a result, the image we see is displaced from the location of the object.1 If the image appears higher in the air than the object, it is called a “superior” mirage, while if it appears lower it is called an “inferior” mirage.2 The most common example of an inferior mirage is when, on a hot day, a stretch of dry road off in the distance appears to be wet (see Fig. 1). Many lab activities have been described that simulate the formation of superior mirages. In these demonstrations light beams curve downward as they pass through a nonuni-form fluid.3–6 Much less common are laboratory demonstrations of upward-curving light rays of the kind responsible for inferior mirages. This paper describes a simple version of such a demonstration.

Halo and mirage demonstrations in atmospheric optics.

Some laboratory demonstrations on atmospheric optics are presented. The focus is on dispersion effects in mirages, lateral mirages, and inferior mirages produced with small hot plates. We also show a demonstration of the upper-tangent-arc halo, produced with a hexagonal prism, rotating about two axes.

Simulation of inferior mirages observed at the Halligen Sea

Two unusual forms of inferior mirage are observed and photographed at the Halligen Sea. With heuristic analytic functions for the temperature profiles, numerical integration of the refraction differ-ential equation on a flat earth is performed. The simulation shows that a double inferior mirage can appear if a light wind carries hot air from above dry sandbanks in the mud flats. Horizontal stripes can appear in the mirage image if a water channel crosses the line of sight between the observer and the object.

Simulation of inferior mirages observed at the Halligen Sea

Two unusual forms of inferior mirage are observed and photographed at the Halligen Sea. With heuristic analytic functions for the temperature profiles, numerical integration of the refraction differential equation on a flat earth is performed. The simulation shows that a double inferior mirage can appear if a light wind carries hot air from above dry sandbanks in the mud flats. Horizontal stripes can appear in the mirage image if a water channel crosses the line of sight between the observer and the object.

A Three-Parameter Inferior Mirage Model for Optical Sensing of Surface Layer Temperature Profiles

Inferior mirages provide a sensitive and fairly accurate probe for determining vertical temperature distributions in the atmospheric surface layer. Optical measurements on the image can be used to calculate the parameters in a temperature profile model, in this case a function with three adjustable parameters. The function contains an exponential term (two parameters) and an additive linear term (one parameter). The optical observations, for which a known target is required, consist of the elevation angles of the apparent peak, caustic, and horizon. Analytic expressions that must be simultaneously satisfied are derived for all three conditions. The parameter values are extracted numerically by minimizing a positive definite function of the three conditions. The model is tested on a set of images for which nearly simultaneous photographs, theodolite readings, and temperature profiles were available. For each image the three calculated elevations matched the measured values very closely. The complete images also match well in most of the cases. The results, a distinct improvement over previous two-parameter models, also provide a more accurate reconstruction than is obtained from the thermodynamic model for unstable stratification.

Simple solution for obtaining a temperature profile from the inferior mirage

The first-known, explicit, analytic optical inversion for a refractive-index profile with curvature is given. It enables a quasi-parabolic profile of height vs temperature to be calculated from observations of the inferior mirage of natural objects. Given sufficient fetch, an inferior mirage will occur anytime the heat flux is away from a horizontal surface such as a large body of warm water. All that is needed to obtain the data for a temperature profile over such a surface is a theodolite, a tape measure, and a topographic map. A total of four measurements and a pocket calculator are sufficient to determine the temperature profile on the spot. The resulting profile represents a weighted horizontal mean over the surface. Not all inferior mirages are amenable to the technique, but only those where the temperature gradient at the eye is no less than half the mean gradient, a situation that seems to require some minimum wind. The predictions of the theory are verified with measurements from thermocouples.

Mirages on Mars

The possibility of observing mirages on Mars from the Viking lander cameras is examined. A simple model for the production of both inferior and superior mirages is developed. Assuming the atmospheric index of refraction to be a linear function of density (i.e., temperature), ray curvatures are calculated through layers of large, expected thermal gradient. Assuming the Martian morning inversions of Gierasch and Goody (1968), calculations of ray curvature show the superior mirage to be an unlikely occurrence on Mars since the downward curvature of the ray through the inversion layer is less than the downward curvature of the planet. In order to examine the nature of inferior mirages we select a reasonable expression for temperature profile in the surface layer fitted to the midafternoon, midlatitude summer results of Gierasch and Goody. Integration of the expression for ray curvature yields a relation for the minimum distance between the lander cameras and an inferior mirage as a function of the surface superadiabatic lapse rate. Such calculations indicate that the Viking lander cameras will record inferior mirages at horizontal distances of a kilometer or so from the lander. Given the appearance of an inferior mirage at a measured minimum distance from the observer it should be a simple matter to calculate the corresponding mean temperature lapse rate at the surface.

A Comparison of Two Optical Methods for Measuring Line Averages of Thermal Exchanges Above Warm Water Surfaces

Abstract Line averages of the vertical transports of sensible and latent heats in the surface boundary layer are determined above a warm water surface by the use of two optical, line-of-sight, remote-sensing techniques. In one method, the amount of atmospherically-induced blurring of images is observed visually with the aid of a small portable astronomical telescope. The other method utilizes unaided observations of the formation of inferior mirages, in particular the heights of distant objects that appear to be partially hidden behind the mirage. In the present application above industrial cooling ponds, lines of sight are within 1 m of the water surface and 0.4 to 1.5 km in length. The evaluation of the heat fluxes also requires estimates of the surface friction velocity and the ratio of the sensible to latent heat fluxes, each of which can be obtained with sufficient accuracy over a warm water surface from relatively simple measurements of temperatures and wind speeds. The direct visual measurements ar...

Recurrent mirages at Puerto Peñasco, Sonora

Mirage behavior at Puerto Peñasco, on the Sonoran shore of the Gulf of California, is not only systematic, but is fully compatible with conditions of atmospheric refraction common in that arid area adjacent to a tropical sea. Images of mountains in Baja California, normally hidden below the visual horizon, are regularly loomed into visibility as the lapse rate in the lower layers of the atmosphere increases; and sink again below the horizon with the onset of nocturnal inversions. False sea horizons, due to refraction, are a standard midday condition; as are inferior mirages over adjacent land areas. Air stratification is normally accompanied by “paper doll” duplications of distant images; Fata Morgana is a common occurrence when cold air drainages from high mountains reach the warm waters of the Gulf of California. Photographic differentiation between mirage lakes and actual bodies of water is sometimes possible by use of deep red filters.