# How to derive the total potential energy of the atmosphere?

In one of Lorenz’s paper (I mean the meteorologists Edward Lorenz), he stated that the total potential energy of the whole atmosphere $$P + I$$ (means the sum of potential energy and internal energy) is

$$$$P + I = c_p p_{00}^{-x} \int p^x \Theta dM$$$$

where $$p$$ is pressure, $$p_{00}$$ is a standard pressure, and $$x$$ is the ratio $$(c_p - c_v)/c_p$$, $$c_v$$ and $$c_p$$ are the specific heats of air,$$\Theta$$ is the temperature and M is the mass.

Based on my knowledge, the internal energy should be a simple form that $$$$I = \int c_v T dM$$$$ and the potential energy is actually the gravitational potential energy, which is $$$$P = \int g h dM$$$$

where $$g$$ is the gravitational acceleration and $$h$$ is the height. So my question is how did Lorenz derived his form of the total potential energy of the atmosphere? The original paper is Energy and numerical weather predictionLorenz 1960, and this equation was the first equation in the paper.

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 we rearrange (1) we can get

$$P+I=\int{c_v T+RT dM}\tag{4}$$. You observed correctly that $$I=\int{c_v T}dM$$. This leaves $$P=\int{RT dM}\tag{5}$$.

Using the ideal gas law,$$P=\int{P\alpha dM}$$, where $$\alpha$$ is specific volume. This makes (4) look like $$P+I=\int{c_v T+P\alpha dM}\tag{4}$$ which is just a restatement of the first law of thermodynamics: $$\delta q = c_v dT + p d\alpha$$

• what is x here ?
– user1066
Commented Feb 10, 2021 at 1:16
• @gansub The original problem has $x=(c_p-c_v)/c_p$ which is actually $x=R/c_p$ in disguise. Commented Feb 10, 2021 at 1:18
• Can you tell me what does standard pressure p00 means? Is it a constant?
–  Hou
Commented Feb 11, 2021 at 7:16
• Typically, $p_{00}$ is 1000 hPa. But if you are thinking of applying to a different planet, then I would recommend a different value. Commented Feb 11, 2021 at 7:40
• Thanks, but I still have one more question, where is the gravitational energy? In your equations it seems that it doesn’t represent the gravitational energy, it’s just the work of the pressure. What does total potential energy means? Does it includes the gravitational energy, or it is just the heat?
–  Hou
Commented Feb 14, 2021 at 11:03