If the sea horizon appeared double, which of the two was the right one to use? Their separation might be several minutes of arc, corresponding to an error of several miles in a ship's position — a serious error in navigation. And, if the horizon was behaving strangely, would the usual correction for its dip be correct? These were practical problems for navigators.
Such problems often occur when the observer is in, or near, a duct. And the effects of ducts on the apparent horizon have to be understood in making mirage simulations. So I've had to figure out the details of these effects.
In this case, the duct will have two parts. The upper part is super-refracting (i.e., the curvature of a horizontal ray there exceeds the Earth's curvature); the lower part refracts normally, whether its lapse rate is positive or weakly negative. There may also be a normal (or even convective) surface layer below the bottom of the duct. Each of these parts affects the appearance of the horizon differently.
A ray that's horizontal just below the top of the inversion leaves the base of the inversion at an angle; so it continues to descend for some distance below the bottom of the inversion, before the curve of the Earth bends away from it, making the ray locally horizontal again. So the bottom of the duct is always below the bottom of the inversion. (See the ray diagrams and the discussion of the depth of the duct, on the duct page, if you haven't been there recently. The bottom of the duct is also shown as a dashed line in my simplified presentation of Wegener's model.)
As I explained on the duct page, several atmospheric regions are stacked up when there is a duct. At the top is the free atmosphere above the duct; then comes the top of the super-refracting thermal inversion that produces the duct. (The top of the inversion is also the top of the duct.) Some distance below this is the bottom of the super-refracting region; for convenience, I usually call this just “the base of the inversion”, although the lapse rate often remains mildly inverted below the bottom of the super-refracting region. Lower still, if the duct is elevated, we can have the bottom of the duct. Finally, if the duct itself is entirely above ground level, there is the surface of the Earth.
These pieces are always stacked up in this order. Some of the lower ones may be missing, if the inversion is low enough: if the duct is “surface-based”, the ground intervenes before a ray that is horizontal just below the top of the inversion becomes horizontal again below the inversion. Or, if the inversion itself extends down to the surface, the inversion is surface-based; then every ray below the top of the inversion meets the surface obliquely. Because the base of the duct is always lower than the base of the inversion, there cannot be an elevated duct if the inversion is surface-based. Hasse (1960; 1964) investigated the optics of surface-based inversions, and found that the top of the inversion produces a false horizon that he called a Kimmfläche or “horizon surface”.
The optical effects produced depend on geometric relationships among the duct, the observer, and the surface of the Earth. Let's begin with the relative positions of the duct and the surface, and then see what happens as the height of the eye changes in each arrangement. The diagrams in the subsection that follows show the relative heights above the ground of the atmospheric layers involved. (To see the correspondence between these atmospheric structures and the images they present to an observer, we will need to use transfer curves and the duct diagram, which are discussed in the section after that.)
Note: this is the only case treated by
The diagram at the left shows the temperature profile in the lowest 100 m of this model. As is usual in meteorology, height is plotted on the vertical axis.
For comparison, here is the cartoon showing the layers of the Type-1
(elevated-duct) model again, but with the layer boundaries
marked, instead of the layers themselves. The top of the inversion is at
T, corresponding to the right-hand corner in the temperature
profile; the bottom of the inversion is B, corresponding to the
left corner; and the bottom of the duct is D, which is near 45
meters — but no feature exists in the temperature profile
at that height.
The density can be calculated from the temperature by assuming hydrostatic equilibrium — i.e., the pressure at each level is just the weight per unit area of the overlying column of gas. (The well-known RGO program shows how this can be done for both isothermal and polytropic layers.)
The adjacent graph shows the resulting density plot for the duct model. As before, height is on the vertical axis, and the actual dependent variable is on the horizontal axis. The letters T, B, and D denote the Top of the inversion, the Base of the inversion, and the bottom of the Duct, respectively. As before, there is no feature in the profile that corresponds to the bottom of the duct.
The rapid decrease in density with height shows that this atmosphere is stably stratified. Notice the stronger stratification in the inversion, between B and T.
Now that we have the density profile, we can turn it into a refractivity
profile, as the refractivity (n − 1) is very nearly
proportional to the density. Adding 1 to the refractivity gives n,
the refractive index. And multiplying n by R, the distance
from the center of curvature, gives the product nR, which is the
ordinate of the dip diagram.
Here's the dip diagram for the same model: a plot of the refractive invariants of horizontal rays as a function of their heights. Notice that the height scale here is on the horizontal axis. Don't let this change of orientation confuse you.
As before, the top and base of the inversion are marked by T and B, and the bottom of the duct is at D. In the dip diagram, the point D on the model-atmosphere curve is at the same ordinate as the top of the duct, as shown by the dashed horizontal line in the figure. The duct covers the whole range of heights from D to T.
The dip scale, in minutes of arc, which is inserted just at the right of the model curve, is explained on the dip diagram page. It can be used to show that the dip of the duct edges D and T (where the dashed line meets the solid curve) — as seen from B, the base of the inversion — is 3 or 4 minutes of arc. That's the half-width of Wegener's “blank strip”, which is centered on the astronomical horizon.
Finally, because nR has the same value at the top (T) and bottom (D) of the duct, any ray passing obliquely across the duct has the same inclination to the local horizon at both those heights. (This follows from the refractive invariant.)
I intended to offer ray diagrams and simulated images here, but found that my ray-tracing programs have numerical difficulties when the ray curvature is close to the Earth's curvature. So this page offers just a hint of the complications that ducts can produce. I hope it's helpful, even though woefully incomplete.
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