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First, let's acknowledge this fact: $$c_p=R+c_v \tag{1}$$, where $R$ is the specific gas constant. This means that $x=R/c_p$. Rearranging the equation, we can see that $$P+I=\int{c_p\left(\frac{P}{P_{00}}\right)^{R/c_p}\Theta dM}\tag{2}$$ Notice that $\left(\frac{P}{P_{00}}\right)^{R/c_p}$ is the Exner function. By extension,$$P+I=\int{c_p T dM}\tag{3}$$ If ...


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