Introduction to Superior-Mirage Simulations


Superior mirages are complicated. Though the basic idea — that the mirage is something like a “reflection” produced by a steep thermal inversion  whose top is above the observer — is simple, mirages in the real world are affected by the thickness and temperature gradient of the inversion layer, the smooth thermal transitions at the top and bottom of the inversion, repeated “reflections”, and deviations from horizontal uniformity in the atmospheric structure. On top of all these variables, all  mirages depend strongly on both the height of the eye and the observer's distance from the miraged object(s).
CAUTION: Don't confuse image inversion (i.e., something appearing upside down — the common property of all mirages) with the thermal inversion  (an atmospheric structure) that produces mock mirages and superior mirages.

I put “reflection” in quotes here, because these are really refraction  phenomena, not true reflections. Although Wegener's simplistic model of superior mirages does reproduce some of their characteristics by using total internal reflections to represent the effects of thermal inversions, it fails to capture other features of these mirages. And its use of “reflection” is potentially confusing. [Perhaps a better term is J. D. Everett's (1873) phrase “quasi  reflection”.]

As Humphreys says of these mirages on p. 451 of his textbook:

The image nearest the object always is inverted and therefore appears as though reflected from an overhead plane mirror — hence the name “superior mirage” —and, indeed, many seem to assume that this image really is due to a certain kind of reflection, that is, total reflection, such as occurs at the undersurface of water. It is obvious, however, that this assumption is entirely erroneous, since the atmosphere can never be sufficiently stratified in nature to produce the discontinuity in density (adjacent layers are always interdiffusible) this explanation of the origin of the proximate inverted image presupposes.

Furthermore, the phrase “superior mirage” is an observational term that encompasses a wide variety of optical situations. It literally means any mirage in which an inverted image appears above  an erect one; but this includes both 2-image and 3-image mirages, as well as Fata Morganas. Even some terrestrial mock mirages may be called “superior mirages”. (This use of a single name for several different optical phenomena proved to be a hindrance in understanding green flashes, and the same is likely to be true for mirages.)

As I've dealt with terrestrial mock mirage simulations elsewhere, I'll confine my attention here to mirages produced by ducts , and seen by an observer below the top of the duct. This includes examples of Wegener's Nachspiegelung , as well as simple quasi-reflections at the top of the duct (including the multiple “reflections” that produce multiple-image mirages). There's really a continuous transition between the Nachspiegelung and ducted mock mirages, as the observer moves vertically across the duct.

In such complex situations, it's often useful to start with a “toy” model that contains only a few important effects, and then gradually relax its unrealistic assumptions. That's the approach I take here. This page provides an overview of the illustrative models, from simple to complex.

Wegener's toy model

Wegener (1918) published an extremely simple model for mirages that, while quite unphysical, captures a surprising range of real-world phenomena. (Indeed, it was this model that led him to discover the Nachspiegelung — the first purely refractive mirage — even though his model for superior mirages is based on total internal reflection.)

The basic feature of the model is to substitute a density discontinuity for a strong thermal inversion. Like a real-world inversion, Wegener's discontinuity produces a duct; so it can simulate some (but not all) of the optical phenomena ducts really produce. However, it replaces refraction with reflection, and so misses several important effects.

Quantitative simulations for Wegener's model are shown here.

I have followed Wegener's practice of describing his model in terms of “reflections”, because this is how his model  works, although this is not the way mirages  work in the real world. Please bear in mind that this aspect of his model is unrealistic, and is a defect of the model.

Despite this weakness, Wegener's model illustrates several optical phenomena of ducts in a very simple way; so it's definitely worth examination. But because of this weakness, it misses some other, real-world effects. To understand those, we need to proceed to more realistic models.

Zigzag profile

As a first step in making the model more realistic, let's consider a steep inversion with a constant lapse rate. (This is the kind of piecewise-linear temperature profile usually used by meteorologists.)

While this neglects some important real-world effects — namely, the smoothness of real temperature profiles, and the resulting phenomena produced by circulating rays — it does introduce many that are missing from Wegener's oversimplification. So the zigzag model really isn't too unrealistic.

To make a duct, the ray curvature in the thermal inversion must be greater than the Earth's curvature. Then upward-sloping rays in the inversion can be bent back down again, and trapped below the top of the inversion. (This bending — Everett's quasi-reflection — is the actual effect that Wegener approximated with an actual reflection in his toy model.)

Note that only the limiting rays reach the top of the thermal inversion. Rays that are horizontal within the inversion are already at their highest point, so an observer within the inversion sees a very different mirage than Wegener's toy model produces. In Wegener's model, the duct effectively lies completely below the inversion, so it cannot show the phenomena observed by an observer inside the inversion.

Quantitative simulations for a zigzag profile are shown on two pages. Probably the more informative is the one that shows the effects of changing the height of the observer. On another page, you can see the effects of changing the distance between the observer and the target, for different heights.

Fortunately, it turns out that the zigzag profile captures most  (though certainly not all) of the effects that occur in real-world mirages. So, if you have a limited amount of time to spend, and want to understand about 85 per cent of superior-mirage phenomena, these zigzag-model pages are the ones to read.

Smoothed duct

To make still more realistic simulations, it's necessary to round off the corners of the zigzag profile a little. This gives the smooth temperature profile that was used to make ducted sunset simulations. It's also the model that was used to make the simulations of ducted mock mirages.

Quantitative simulations for the smoothed duct are arranged in two groups: one to show the effect of varying eye-height at fixed target distances, and one to show the effect of varying target distance at fixed eye heights.

Because these simulations differ mainly in rather subtle ways from the zigzag-profile simulations, I have limited the discussions on the smoothed-duct pages to the differences  between them and the zigzag ones. Each page has links that allow you to compare the results for the smoothed profile and the zigzag one at the same geometry.

Also, because the main mirage phenomena are already displayed and discussed in detail on the pages for the zigzag model, you should read the zigzag pages first, before proceeding to the subtler effects introduced by smoothing the edges of the inversion.

The differences between the smoothed and zigzag simulations are so small that a side-by-side comparison often fails to show them. To make the changes more apparent, an old astronomers' trick is useful: switch back and forth quickly between one and the other. (In the days of photographic plates, we used a machine called a “blink microscope” or “blink comparator” to switch between two matched photographic plates.)

The main requirement for successful blinking is perfect geometric alignment of the two images. It isn't easy to blink two Web pages without a little help; even if the images are correctly positioned horizontally, it's hard to match them up vertically.

So I've added an alignment tool to the smooth-profile simulation pages here. Whenever there's a smooth simulation with an exact zigzag counterpart, you'll see a funny-looking link after each simulation's heading on the smooth page that looks like this:   [align] . If you click on that “align” link, your browser should adjust the current page vertically so that it will be in nearly perfect register with the corresponding zigzag page, when you click on the “zigzag” link in the nearby text.

Then, you can blink between the two correctly-registered simulations by using your browser's “Forward” and “Back” buttons — or, better yet, by using the corresponding keyboard shortcuts. (On my Debian boxes, those are the <Alt><Right-arrow> and <Alt><Left-arrow> key combinations, respectively, for the browser derived from Mozilla's Firefox. Your system may be different.)

There may be a short delay when you follow the link to the zigzag member of the pair of simulations to be blinked; but, once both pages are in the browser's cache, the blinking should occur very quickly. In a few cases, the two images aren't quite in perfect register, but seem to be off vertically by about one pixel on the screen, This isn't bad enough to spoil the blinking effect, though.


Because there are many pages here dealing with superior-mirage simulations, and it's often difficult to see how they all fit together, you may find the Mirage section of the Table of Contents helpful.


Copyright © 2008 – 2009, 2012, 2020, 2022 Andrew T. Young


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