# How is horizontal atmospheric refraction explained?

I can understand vertical atmospheric refraction due to a gradual change in the refractive index of air. The distances measured in this model are often great. Sun moon planets etc.. now the question I have is about horizontal atmospheric refraction.( eye level over less than 100kms for example). As I understand it, the refractive index of air Is practically zero, and visual information travelling parallel to the plane will go through sections of air masses, which may have varrying refractive indexes. say +or- 10 or 20% changes in barometric pressures, depending on local atmospheric conditions along the line of sight, (a lot less than vertical gradient anyway) the "light" will be traversing the interfaces of this changing medium practically head on, or parallel to the normal and may not be subjected to much angular refraction. If some refraction might occur, Sub and super refraction however minute they may be, would largely negate each other over distance. I can't "see" how refraction can "bend" light in such a system, 8"/ mile or 12 cms/ kilometer following the curve of earth or oceans. My question is: how is this possible?

First: refrative index of air $n\approx 1 \neq 0$, let's say $n=1$. Now we take $12cm/km$ "bending" of light which results in $\beta = \arctan\left(0.12/1000\right) = 0.006875^\circ$. That is small... Now we say a small differene in air density results in change of the refractive index of $\Delta n = 0.000392$ and we have $n = 1.000392$, thus the incident angle for getting 12cm/km is: $\alpha = \arcsin\left(n \sin \left(\arctan \left(0.12/1000\right)\right)\right) \approx 0.00688^\circ$. I don't see any problem why this shouldn't be realistic...