Weather_Models

Pressure + Temperature model

From the barometric formula: $$\frac{dP}{P} = -\frac{g M}{RT(h)}dh$$

We get that: $$P(h) = A exp\left(-\frac{g M}{R}\int_0^h\frac{1}{T(h')}dh'\right)$$

If we have a simple lapse_rate model $T(h) = T_o + LR(h - h_0)$, we find that

$$\int_0^h\frac{1}{T_o + LR(h' - h_o)}dh' = -\frac{1}{LR}\log\left(1 - \frac{h LR}{h_o LR-T0}\right)$$