4

You may be forgetting that pressure also decreases with height (exponentially). Also, because $P=\rho R T$, $\frac{dP}{dT}=\rho R$ (that is, $c_p$ does not appear). But I digress in answering your question. Let's break down why potential temperature increases with height. Let's start with the equation: $$\theta=T\left(\frac{P_0}{P}\right)^{\frac{R_d}{c_p}}\...


4

If we'd be living in a dry atmosphere your reasoning is indeed correct. Air would rise adiabatically and air would loose about 9.8 °C/km (dry adiabatic lapse rate). This means constant potential temperature. However, Earths atmosphere isn't dry. As soon as a rising, moist air parcel reaches saturation, it will rise with a moist adiabatic lapse rate (6-7 °C/...


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