A previous question asked why the road sometimes appears wet on hot days.
The reason is that when there's a temperature gradient in the air, it causes a gradient in the index of refraction, causing the light to bend. This diagram (from Lagerbaer's answer to the previous question) is
I have two questions about this.
First, Wikipedia says that in order for the effect to appear, there must be a temperature gradient on the order of a few degrees per hundred meters above the asphalt.
To investigate this claim, I assumed the air near the surface of the asphalt is at constant pressure, so the density is inverse-proportional to the temperature (i.e. I assumed the density variation due to the weight of the air is small compared to the density variation induced by temperature gradient. Otherwise we would see mirages all the time.)
Then I assumed the difference of index of refraction of air from one is proportional to the density, i.e. $n = 1 + a\frac{\rho}{\rho_0}$ with $n$ the index of refraction of air, $a$ some dimensionless constant, and $\rho_0$ some reference density. Looking up the index of refraction of air online, I guessed $a = 0.0003$.
Then I assumed the temperature above the Earth is modeled by $T = T_0 + gy$ with $y$ the height and $g$ a temperature gradient in ${}^\circ\mathrm C/\mathrm m$.
This completes the model. Fermat's principle gives a variational problem to solve for the path of the light. However, the resulting differential equation was hard to work with, so to first order I approximated that the light starts at a horizontal distance $x = -L$ at height $y =h$, slopes down towards the ground with some slope $m$ until it gets to $x=0$, then slopes back up at the same slope $m$ until it gets to $x = L$ and $y = h$ again. Then I chose $m$ to minimize the travel time of this path.
To first order for small $h$, I found
$$m = \frac{agL}{2T_0}$$
The problem is that when I plug in $g = 5\ \mathrm{^\circ C}/100\ \mathrm m$, $a = 0.0003$, $T_0 = 300\ \mathrm K$, and $L = 100\ \mathrm m$ I get $m = 2.5\times10^{-6}$, which is too small to explain the mirage. It would only allow light to dip a quarter millimeter in the path from a car $200\ \mathrm m$ away to me. In the Wikipedia image of a mirage, the light clearly dips down at least 1000 times that much (check out the blue car).
So my first question is: what's wrong? Why are there mirages when this model predicts that there are not?
My second is: why do we see them only when it's hot? This model depends only weakly on absolute temperature, and hot absolute temperatures actually decrease the effect. Evidently, there are only high temperature gradients on hot days, by why should that be? A temperature gradient over the asphalt comes from sun heating the asphalt directly more than it heats the air. Shouldn't that happen on cold days as well as on hot days?