Mirages Produced by a Zigzag Temperature Profile:

Height-of-Eye Variations


After Wegener's discontinuity model, the next more realistic temperature profile is a piecewise-linear one. This is the kind usually used by meteorologists. I'll call it a zigzag profile for brevity.

The particular example used here is a 2° thermal inversion between heights of 50 and 60 m. That's the simple model used to make some ray-trace diagrams on the introductory mirage page. This model produces a duct that extends down to about 45.3 m height — almost exactly where the bottom of the duct is for Wegener's model with the same 2° temperature difference. (The temperatures for the two models differ only in the 10-m interval between 50 and 60 m, and the mean temperatures in this interval differ by only 1°; so the pressures and densities are practically equal in both models above 60 m, as well as being identical below 50 m.) As on the introductory mirage page, the inversion layer appears as a shaded band between 50 and 60 m height in the ray diagrams here. (Remember that the duct actually extends below the bottom of this band.)

So the duct has practically the same thickness in both models. The only difference is that the zigzag model has a continuous transition through the inversion, instead of the Wegener model's discontinuity. But this difference drastically alters the mirages.

The changes in appearance with height of the eye are more complex here than they were for Wegener's toy model; so, once again, it's useful to see what happens as we move the observer up through the duct at two distances from the target. And, as for Wegener's model, I'll choose 20 km and 40 km as the near and far target distances.

Instructions for using the [align] links are found here.

Target 20 km from observer

Once again, we start out just below the duct, and work our way up.

200m target seen at 20km in ZZ2 model from 45m ht.

Eye at 45 m   (zigzag model)

As you might expect from the rule that refraction rapidly becomes less sensitive to atmospheric structure with height above the observer, this doesn't look too different from the corresponding case in Wegener's model. The discussion there equally well explains what we see here.

Notice that the horizon ray here is bent down continuously, instead of being refracted abruptly at Wegener's density discontinuity. But it ends up at about the same place on the target.


However, once we enter the duct, there's an immediate difference: the view from 45.5 m height showed a narrow miraging strip for Wegener's model; but for the zigzag model, there's no strip — indeed, there's no mirage at all.

200m target seen at 20km in ZZ2 model from 46m ht.

Eye at 46 m   (zigzag model)   [align]

Even at 46 m, where the Wegener model showed a well-developed miraging strip, the zigzag model almost looks the same as at 45 m.

The ray diagram shows why. In Wegener's model, there's a sharp reflection of the horizon ray when it strikes the top of the duct. But here, the steady density change through the inversion simply bends the ray down a little. As the bending is moderate, the ray's intersection at the target is only a little lower than when the observer was a meter lower. The net result is just slightly steeper sides on the towered section around the astronomical horizon.

The lesson here is that a discontinuous atmospheric model can produce discontinuous changes in the mirage with height; while a continuous model produces continuous changes with height.

Instructions for using the [align] links are found here.

200m target seen at 20km in ZZ2 model from 47m ht.

Eye at 47 m   (zigzag model)   [align]

And even at 47 m, the sides of the stretched zone are simply vertical at 20 km target distance; there's still no mirage. (And certainly nothing like the spectacular mirage produced at this geometry in Wegener's toy model.)

However, you can see that the horizon ray is still above the middle of the inversion at the target here; so we can expect much more impressive results from this refraction at 40 km target distance than at the present range of 20 km. Though this page concentrates on height rather than distance effects, you can see this expectation confirmed by jumping ahead to 40 km target distance at this same eye height.

Incidentally, the vertical sides here show that the height where the horizon ray meets the target is an optical conjugate of the observer's eye, at this distance (20 km).

200m target seen at 20km in ZZ2 model from 48m ht.

Eye at 48 m   (zigzag model)

At last, at 48 m height, we see a mirage of the towering zone just above 50 m. (Notice that the horizon ray and the −4′ ray meet just at the target.) This 3-image mirage is relatively simple.

If we had plotted more rays (which would have produced a very cluttered ray diagram), you'd have seen rays between 0′ and −2′ intersecting before the target, producing an inverted image in that altitude zone.

200m target seen at 20km in ZZ2 model from 49m ht.

Eye at 49 m   (zigzag model)

However, just a meter higher, the mirage is becoming complex. Why? Because we are approaching the lower boundary of the duct-producing inversion, so Wegener's periodicity effect is becoming important: notice the sharp bend in the transfer curve (and the corresponding corner in the miraged zone) just at the astronomical horizon.

Above the astronomical horizon, the +4′ ray is now rather sharply bent down by its long path in the super-refracting inversion, so it falls much lower on the target than you'd expect from the rays above it. That compresses the target zone between +4′ and +8′, just above the corner in the image (and transfer curve) near +3′ altitude. Successively lower rays have more and more of their length in the inversion, and so continue to fall relatively far apart on the target, producing a strongly compressed (stooped) image zone.

But just a little lower, around +3′, the rays are completely in the inversion. Here the stooped zone ends abruptly, and is replaced by a short segment with almost normal slope.

At and below the horizon ray, which meets the target near the base of the inversion, the process is reversed: successively lower rays escape from the bottom of the inversion before reaching the target, and experience progressively less refraction. Because of the rapidly decreasing refraction as we look lower, the rays actually intersect the target higher ; that is, the image is inverted. (As we are inside the duct, even though below the base of the inversion, this would be an example of Wegener's Nachspiegelung.) This is essentially the mock-mirage mechanism at work.

This continues down to an altitude of about −3′, where the decrease in refraction within the inversion is no longer rapid enough to overcome the increasing depression of the ray at the observer. So there is a smooth change of slope in the transfer curve around this altitude, and the normal erect image is finally seen below the distorted zone.

200m target seen at 20km in ZZ2 model from 50m ht.

Eye at 50 m   (zigzag model)

At 50 m, the eye is at the base of the inversion. Now a 3-image mirage has appeared, with some additional fine structure around the astronomical horizon — another indication of Wegener's periodicity effect at work. The strong towering in the horizon zone is typical of this effect just below a refracting boundary. Its complex structure is a consequence of the physically impossible discontinuous change in slope of the temperature profile at 50 m.

The corners in the transfer curve (and the image) that were noted above are now sharper. The miraged zone continues to produce 3 (or perhaps more) images.

Notice that the 0′ and −4′ rays, which intersected about 19 km from the 49-m observer, now intersect even closer to the 50-m observer here.

200m target seen at 20km in ZZ2 model from 54m ht.

Eye at 54 m   (zigzag model)

As the observer moves up into the inversion, the stooped and near-normal zones above the astronomical horizon change only slowly. The stooping around +3′ altitude is steadily increasing. However, the complex region of towering near the astronomical horizon has been squeezed out by the Nachspiegelung, which is increasing in altitude, and now adjoins the normal zone above it.

Notice that the 0′ and −4′ rays no longer intersect before the target.

200m target seen at 20km in ZZ2 model from 57m ht.

Eye at 57 m   (zigzag model)

As we approach the top of the inversion, the distorted zone shrinks in width. The stooped zone has become almost flat in the image. The normal zone just above the astronomical horizon is becoming narrower. Here, the Nachspiegelung has become simply a towering (vertical) zone in the image. At this short target distance (20 km), we no longer have enough thermal inversion above eye level to produce a mirage.

200m target seen at 20km in ZZ2 model from 59m ht.

Eye at 59 m   (zigzag model)

Just below the top of the inversion, these trends continue. The “normal” zone has almost disappeared; above it, the stooped zone is quite flat. Below it, the towering is diminishing.


And above…

The continuation of these effects with increasing eye height can be seen on the ducted mock-mirage page. There's no simulation there for a target distance of 20 km, but you can interpolate between the ones at 15 km and 30 km to see what happens when the eye is 65 m high.

Target 40 km from observer

200m target seen at 40km in ZZ2 model from 44m ht.

Eye at 44 m   (zigzag model)   [align]

For a target 40 km away, it's useful to start out a little lower than at 20 km, for reasons that will shortly become apparent. Just as at 20 km target distance, this image looks a lot like the one for Wegener's model, when the observer was just below the duct (although a meter higher than here).

Once again, there is no mirage, and the image distortions are what we have come to expect when looking up through an inversion. The bottom 50 m of the target is practically undistorted, as it's seen through a piece of the Standard Atmosphere. The top is somewhat stooped — notice its obtuse apex angle — because its towering decreases with altitude, due to the decrease in incidence angle at the inversion as we look farther from the astronomical horizon. And the towering and looming of the middle part are generally similar to what was seen from just below the duct at this distance in Wegener's toy model.

None of that is surprising. What is surprising is what we see at 45 m height, one meter above this viewpoint.

200m target seen at 40km in ZZ2 model from 45m ht.

Eye at 45 m   (zigzag model)

Here we are at 45 meters, just below the bottom of the duct (at 45.3 m). At this height, we saw no mirage when the target was 20 km away in the present (zigzag) model atmosphere. Nor was there a mirage at this height and target distance in Wegener's toy model.

Yet there's a plain 3-image mirage here. Look how the horizon (0′) and −3′ rays cross, about 35 km from the observer.

How can this be? We're clearly below the duct: even the horizon ray passes through the inversion, striking the target about 84 m above the ground (well above the top of the duct). There's no ray-trapping, no reflection; yet there is a mirage.

The answer is that the horizon ray has a longer path in the inversion layer than the −3′ ray does. (The horizon ray enters the inversion closer to the observer; but the two rays leave the top of the inversion almost together.) So it gets bent a little more than the lower ray.

These two rays leave the top of the inversion between 25 and 30 km from the observer. The horizon ray is still a little above the −3′ ray at 30 km, but it's sloping down a little more; so they cross a few kilometers before reaching the target. The ray diagram shows all these details, if you look closely.

This is the terrestrial counterpart to the mirage that's associated with the sub-duct green flash. Take a look at the simulated sunset for this same height, and you'll see this same mirage effect.

The mechanism for this mirage is the familiar one of the mock mirage: a longer path in the inversion, and so more bending, for a ray that's higher at the observer. In fact, it turns out that this is just the continuation of Wegener's Nachspiegelung below the duct.

200m target seen at 40km in ZZ2 model from 46m ht.

Eye at 46 m   (zigzag model)   [align]

At 46 meters, the observer is definitely inside the duct: the ray diagram vividly shows the trapping of the horizon ray, which is bent down so strongly that it meets the target near the base of the inversion, about where the −9′ ray ends. It has crossed the −3′ ray near 24 km from the observer, and the −6′ ray at 32 km.

A narrow zone of rays around the astronomical horizon is likewise trapped, producing an inverted, towered image delimited by two horizontal lines parallel to the horizon. These lines mark Wegener's blank strip or “reflecting strip”, in which only ducted rays are visible to the observer in the duct, and which hides the Sun and other celestial objects. The inverted image below the lower edge of the strip is the Nachspiegelung.

Only two images are seen. The Nachspiegelung matches an erect image zone immediately below it; and the inverted zone in the strip matches a much lower section of the target's image. This zone is a textbook superior mirage.

Already at this height, we see that the superior mirage is appreciably distorted — the straight sides of the target appear curved — because Wegener's periodicity effect is at work.

That invites comparison with Wegener's toy model at the same eye height and target distance. The differences are striking: the toy model produced an erect mirage, while the zigzag model here gives an ordinary inverted image. We also notice how much more prominent the Nachspiegelung is in the zigzag model. Other differences include greater distortions here in the stooped zone just above Wegener's strip.

These differences basically arise from the milder action of the periodicity effect in the zigzag model. All the ducted rays have similar curvatures here inside the duct, so their refractions depend mostly on their path lengths in the inversion: the mock-mirage mechanism once again. But the path lengths differ only because the observer is below the base of the inversion; higher rays enter the inversion closer to the observer, so they bend down more. When the target is far enough away from the observer — as it it here — the extra bending can make the higher ray end up slightly lower on the target. (The differences are small, so only a narrow slice of the target appears in the superior mirage; this accounts for the towering in the strip.)

But because all the ducted rays have similar curvatures in the inversion, the ones that enter at greater altitudes also reach greater heights before being turned back down. (This differs from Wegener's toy model, where all rays reach the discontinuity at the top of the duct.) This greatly diminishes the variation in periods of different rays. So the periodicity effect acts mainly to bend the transfer curve; it isn't strong enough to re-invert the image and make the erect mirage we saw in the toy model.

200m target seen at 40km in ZZ2 model from 47m ht.

Eye at 47 m   (zigzag model)   [align]

At 47 meters, the miraged strip is a little wider.

Comparison with the toy model shows a very different appearance: it shows hardly any Nachspiegelung ; an erect instead of an inverted mirage strip, etc. These differences illustrate a basic rule of refraction phenomena: it is the thermal structure near and below eye level that has the greatest influence on the image.

Here, the distortion of the miraged strip is strong enough to produce almost infinite magnification in its upper half (i.e., above the astronomical horizon). That's due to the periodicity effect, as we approach the lower boundary of the thermal inversion. So we can expect more complexity to develop rapidly with height.

200m target seen at 40km in ZZ2 model from 48m ht.

Eye at 48 m   (zigzag model)

Sure enough, just a meter higher is enough to introduce another image inversion, producing a small erect-image zone just below the astronomical horizon.

Multiple ray crossings are now beginning to clutter the ray diagram. Notice the confluence of rays at the target near the base of the inversion (50 m height). The towering is increasing.

200m target seen at 40km in ZZ2 model from 49m ht.

Eye at 49 m   (zigzag model)

Only one meter below the base of the inversion, the eye sees a still more complicated image, again due to the periodicity effect close to the astronomical horizon. (Notice the general symmetry of the strip about the horizon at its mid-line.) However, this symmetry is partly due to the artificial sharpness of the base of the inversion.

At this height, the miraged strip is wide enough that the rays 3′ from the observer's astronomical horizon are trapped in the duct. This makes the ray diagram rather congested.

200m target seen at 40km in ZZ2 model from 50m ht.

Eye at 50 m   (zigzag model)   [align]

At the inversion base itself, the miraging strip is strongly towered, and filled with symmetrical details that are the rather artificial result of the strong periodicity effect. We will compare this later with a more realistic model.

The top of Wegener's miraging strip is still well defined, but its lower edge is becoming much less prominent. The strip has its maximum angular width here, at the base of the inversion.

Of course, the appearance of the miraged strip in the toy model for this geometry is completely different.

200m target seen at 40km in ZZ2 model from 54m ht.

Eye at 54 m   (zigzag model)   [align]

The observer is now near the middle of the duct. As the eye is no longer near a refracting boundary, the periodicity effect is relatively weak, and the symmetry we saw at 50 m has disappeared. However, there is still enough periodicity effect to produce an erect-image zone just above the astronomical horizon; so 4 or 5 images of a target strip near the base of the inversion are still visible.

Because less of the inversion is available above eye level to trap inclined rays, the width of the miraging strip has appreciably decreased: the rays 3′ from the astronomical horizon are no longer ducted. The lower edge of the miraged strip has almost disappeared here.

The contrast with Wegener's toy model at the same geometry is quite pronounced, because the lapse rates at eye level are so different in the two models.

200m target seen at 40km in ZZ2 model from 57m ht.

Eye at 57 m   (zigzag model)

As we approach the upper edge of the thermal inversion, the lower edge of the miraging strip disappears. The inverted image in the strip blends smoothly into the inverted image of the Nachspiegelung. The transfer curve shows that the boundary between these two inverted images survives only as a gentle change of slope at −2′ altitude.

The ray diagram shows that the 0′, −3′, and −6′ rays all arrive at nearly the same height at the target. Thus, a small part of the target is enlarged to fill a wide band of the observer's field of view. This is why the image is so greatly stretched (towered).

Only two images of the miraged part of the target are apparent. Actually, this is a 3-image mirage; but the upper erect image is so flattened that it is not resolvable from the upper edge of the strip.

200m target seen at 40km in ZZ2 model from 59m ht.

Eye at 59 m   (zigzag model)   [align]

Just one meter below the top of the inversion, the mirage has become very simple. The inverted image is dominated by the Nachspiegelung. Again, this is a 3-image mirage that seems to show only two images.

Wegener's toy model at the same geometry looks completely different, because it's dominated by the periodicity effect, while the zigzag model's mirage is primarily a large Nachspiegelung.

The transfer curve shows how the top of the miraged strip approaches the astronomical horizon as the observer approaches the top of the inversion.

And still higher…

Of course, once we climb above the top of the inversion, we have a ducted mock mirage; the simulation for an observer at 65 m height is shown here. The flat top of the inverted image there has moved below the astronomical horizon.

With increasing height above the inversion, the inverted image gradually shrinks in size, and becomes more and more compressed (stooped).

These two cases show the transformation of the Nachspiegelung into a ducted mock mirage above the top of the inversion.

Concluding remarks

The zigzag model captures the effects of an inverted lapse rate at eye level, when the observer lies inside the inversion layer. Below the inversion, the zigzag model also differs strongly from Wegener's toy model in the lower part of the duct. However, both models produce similar results a little way below the duct — yet another illustration of how insensitive refraction is to details of atmospheric structure even a little way above eye level.

However, even the zigzag model is not sufficiently realistic to capture all the behavior of superior mirages. Its temperature profile is continuous, but not smooth. The physical constraints of heat transfer compel actual temperature profiles to be smooth (i.e., have a continuous first derivative).

So the corners of the zigzag profile must be rounded off. When this is done, there is a level at both the top and the bottom of the inversion where the curvature of a horizontal ray exactly matches that of the Earth; such a ray follows the curve of the planet at constant height. These orbiting rays, and their neighbors, produce some additional phenomena that actually appear in ducted mirages.


Copyright © 2008 – 2009, 2012, 2014 Andrew T. Young


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