# Numerical Fit Modeling for Temperature Mitigation in
## 2 Numerical Fit Model for Arid Regions
### Choice of Method  
A new numerical fit modeling process that is more generalizable will be developed here. The heating rate is taken as a function of the solar insolation at the top of the atmosphere. The radiation reaching the surface is reduced by clouds and calcite. Clouds nonlinearly reduce radiant transfer from the Beer-Lambert Law.  
The cooling rate is a function of the ground temperature. Although the air at a height of 2 m is modeled, the ground temperature is assumed to be correlated to the modeled temperature. Air masses have characteristic heating and cooling rates due to several effects, such as typical cloud cover within the air mass. Also, the cooling rate is a function of humidity levels. Most of the diurnal change in humidity is in the lowest atmospheric layers, and the upper layers will not usually change as much.  
Air normally cools as altitude increases. The adiabatic lapse rate is the rate of temperature decrease with height when the temperature change with height is solely due to pressure changes. Data from radio balloon measurements was found for Las Vegas, NV for 24 May 2023 [33]. Temperature versus height is shown in Figure 1 for 0 UTC. The air temperature decreases from the ground to the top of the troposphere, which is shown here at 12,000 m.  
An additional factor is that air warms nonlinearly with increasing solar energy arriving at the surface. Figure 2 is a hypothetical temperature-versus-height (T-H) plot for the early mornings through mid-day. The leftmost T-H curve is the hypothetical temperature in  
Figure 1: Air temperature versus height over Las Vegas, NV on 24 May 2023 0 UTC. The temperatures drop consistently for 12 km.  
the early morning. In that curve, the temperatures rise to the inversion, which is shown at 600 m.  
In Figure 2, each curve to the right corresponds to increments in the surface temperature. As the air near the ground is heated during the day, the air above it will have to warm to the point where the line of adiabatic lapse intersects with the existing T-H curve. Figure 2 shows that incremental heating of air at the surface requires an increasing amount of thermal energy to heat the air column. The effect of this is that temperature rises rapidly in the mornings.  
Non-linear heating of the air is a function of the amount of cooling the night before, the height of the PBL, and the density and humidity of the air at each level. It is complex to model it with a simple numerical equation. However, a simple model can account for the most predominant effects.  
A ramp function captures the nonlinearity in the heating rate. Notice that in Figure 2, in early mornings, the heat required for each temperature increment is outlined by a triangle of increasing size. The heat required for each temperature increase goes up with the area of this triangle. Therefore, the heat required is roughly a ramp function, and the temperature change for each level of heating would ramp down:  
\[F=2-t_{1}/13\geq 1 \tag{2}\]  
where \(F\) is the heating effectiveness factor, and \(t_{1}\) is the time (hours) since sunrise. This is determined entirely empirically for the best fit under these conditions.  
It will be shown that the numerical fit matches the actual temperatures when a square function is used, such that the heating rate is scaled to be double in the mornings. In the model, this is performed by including a linear ramp in the heating rate that doubles it in the mornings, but decreases to zero later. Figure 2 shows that at higher temperatures, the areas are roughly rectangular. Therefore, in afternoons, the temperature changes are roughly linear with the heating rate.  
Figure 2: Hypothetical T-H plot as it changes during the day. Increases in ground temperatures require heating of the air column above. The blue line at the left is the initial condition. Each incremental colored line to the right is an increase of 2.5 C in ground temperature.