Alfred Wegener noticed that the length of the period (between corresponding parts of the same ray path) depends in a remarkable way on the ray's altitude at the observer. He then used this relationship to give a qualitative explanation of the distortions in superior mirages.

Wegener's relationship can also be used to infer other properties of superior mirages, so it's worth detailed examination.

Wegener,
in his
**1918 paper**,
stated that

… the two points of reflection of the limiting rays at E, A and D must … always be equally far apart, while the reflection point of the horizontal ray is shifted back more at each new reflection. The path of the rays is therefore not exactly symmetric at the reflection point A, and at C only the two limiting rays intersect, while the horizontal ray here runs somewhat above this intersection.Notice that the two limiting rays — the two solid arcs that intersect at the observer's position, at B — have the same shape; they're just one another's reflections about the vertical line. (This follows from the symmetry properties of refracted rays.)

As I've noted elsewhere, Wegener's figure is a little confusing; so I'll try to explain his idea with simpler diagrams here.

It's well known that the ray curvature is less than the Earth's curvature. So, to be sure, one would obtain quantitatively false values if one regarded the rays as straight; but qualitatively, the phenomena must occur just the same way. If we imagine a cross-section through the Earth in the plane of the ray, we can without difficulty “stretch” the Earth's surface (i.e., increase the radius of curvature) until the previously curved ray has become straight.This is a very useful transformation; in fact, all the phenomena depend locally just on the

So here's Wegener's cross-section through the Earth, with center at C.
In my diagram at the left, the outer heavy circle
(BAA′B′A″B″…) is Wegener's density
discontinuity, and the dashed circle inside it is the bottom of the duct
— or, as Wegener described it, the envelope of the limiting rays.
The diagram at the left shows two ducted rays, AA′A″… and
BB′B″…, with different
*perigee*
heights.
BB′B″… is one of Wegener's limiting rays, which is
just barely trapped in the duct; any ray below BB′ is refracted at
Wegener's density discontinuity, instead of being reflected back into the duct.

The lowest point on a ray is its perigee, where it's closest to C. At the perigee point, the ray is locally horizontal (i.e., perpendicular to the local vertical.) The perigee points on the two ray segments AA′ and BB′ are their midpoints, which fall on the dashed vertical line that bisects them. The period of the upper ray is just the interval AA′, or A′A″; and the period of the lower ray is from B to B′ = B′B″.

It should be obvious that the segment AA′ with the higher perigee
is shorter than BB′, so that the period of the higher ray is shorter
than the period of the lower ray. As BB′ here is a *limiting* ray,
its perigee is lower than that of any other ducted ray, and
its period is longer than that of any other ducted ray.
Notice that it's horizontal at the bottom of the duct (the heavy dashed
circle).

If we imagine an observer placed at the midpoint of AA′ (i.e. its intersection with the dashed vertical), this ray is at the observer's astronomical horizon. But to draw the observer's limiting rays, at the edges of the blank strip, we have to rotate the limiting ray BB′… about the center C until this ray passes through the observer's position.

Here's what you get when you do that. In the figure at the right,
the observer is at the intersection of all the straight lines, at the
midpoint of the horizon ray AA′. The two limiting rays that
define
Wegener's blank strip
are B_{1}B_{1}′
and B_{2}B_{2}′; the B_{1} ray is at the
top of the blank strip, and the B_{2} ray is at its lower edge,
if the observer looks to our right.
(Already, with just these two limiting rays, you can see why Wegener
described the bottom of the duct as the envelope of the limiting rays.)

It's obvious in these last two diagrams that the period — i.e.,
the interval between reflections at the discontinuity —
of a ducted ray depends on the ray's
perigee height: the lower the perigee, the longer the period. As the
limiting ray BB′… is the *lowest* possible ducted ray, it
has the *longest* inter-reflection period.

So an observer's horizon ray always has a shorter period than the limiting rays. This shorter period makes successive reflections of the horizon ray steadily fall behind, relative to the limiting rays — and this mis-match between the horizon and limiting rays produces an increasing distortion of the miraged image, as Wegener noted.

Furthermore, the perigee height of an observer's horizon ray is just
the observer's own height (see AA′…). But, while observers
at different heights have different horizon rays, **all** observers
have the **same** limiting rays, at the edges of the blank strip;
it's just the phasing (rotation about C) of the limiting rays, not their
shape, that changes with observer height. So, the higher the observer
is above the bottom of the duct, the shorter the period of the horizon
ray, and the bigger the difference between the periods of the horizon
ray and the limiting rays. That makes the distortions larger for
an observer nearer the top of the duct.

To put this another way: the closer the observer is to the top of the duct, the bigger are the distortions produced by the periodicity effect.

This is exactly what we see in the simulations. The complexity of the miraged image grows with the number of reflections of the horizon ray. So for an observer near the bottom of the duct, the mirage is a simple inverted image for nearby objects, and gradually becomes distorted and multiple for distant ones. For an observer near the top of the duct, the period discrepancy is large; so the complex images appear at shorter distances.

Of course, in the real atmosphere, we have to replace Wegener's
“reflections” with a continuous
bending back down of the rays within the inversion. So the period-discrepancy
effect is more complicated for an observer *inside* the inversion.
However, for observers between the base of the inversion and the bottom of
the duct, the effects predicted by Wegener, and discussed here, are still
observed.

A particularly important consequence of the periodicity effect is that,
as the rays at the center of the miraging zone at the astronomical horizon
get increasingly out of step with those at the edges, additional
transitions between inverted and erect images begin to appear in the
mirage, when the target is sufficiently far away. Thus, *multiple*
alternations of erect and inverted zones — a hallmark of the Fata
Morgana — are produced, even if there is only a single inversion.
The observer simply needs to be quite close beneath it.

The persistence of these properties of Wegener's simple model in realistic situations makes Wegener's periodicity effect a useful tool in understanding superior mirages, particularly those that are spectacularly complex.

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

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