Atmospheric physics 7: Lapse rate

The mass density $\rho$ of the air parcel is inversely proportional to the volume $V$ : $\rho = \frac{M n}{V} \propto \frac{1}{V}$ Therefore, the adiabatic condition $PV^\kappa = \text{const}$ can be rewritten to $\begin{aligned} & & \frac{P}{\rho^\kappa} &= \text{const}\\ \Rightarrow & & \rho &= \text{const} \cdot P^{1/\kappa} & & | P \propto e^{-z/H}\\ & & &= \rho_0 \exp\left[ - \frac{z}{\kappa H} \right] \end{aligned}$ with the density $\rho_0 = \rho(z=0)$ at sea level. Here we have used the barometric height formula from the last section. Thus, the density also depends exponentially on the height, but drops faster than the pressure $P$ by the factor $\kappa = 1.4$ . Since we know pressure and density as a function of altitude, we can also use the general gas equation to determine the temperature curve: $\begin{aligned} & & P &= \text{const} \cdot \rho T \\ \Rightarrow & & T &= \text{const} \cdot \frac{P}{\rho} & & | P \propto e^{-z/H}, \rho \propto e^{-z/\kappa H} \\ & & &= T_0 \exp\left[ - \frac{\kappa - 1}{\kappa} \frac{z}{H} \right] \end{aligned}$ with the temperature $T_0 = T(z = 0)$ . At sea level, therefore, the temperature gradient gives the result $\begin{aligned} \Gamma &= -\left.\frac{dT}{dz}\right|_{z=0} = \frac{\kappa - 1}{\kappa} \frac{T_0}{H} \\ &= \frac{1.4 - 1}{1.4} \frac{288 \,\text{K}}{8417 \,\text{m}} \approx 9.8 \cdot 10^{-3} \,\frac{\text{K}}{\text{m}} \end{aligned}$ The quantity $\Gamma$ is also known as the dry adiabatic lapse rate.