Analyzing the term $\frac{1}{c_s^2}\frac{dP}{dz}$ for atmospheric phenomena?

Question

Some background

I am reading through a textbook on atmospheric fluid dynamics. The author is discussing something called the Boussinesq relations, and eventually the following expression arises:

$$ \frac{1}{\theta}\frac{d\theta}{d z}=\frac{1}{\rho}\left(\frac{1}{c_s^2}\frac{dP}{dz}-\frac{d\rho}{dz}\right) $$

where $c_s$ is the speed of sound and $\theta$ is the potential temperature. The potential temperature is the temperature a parcel of air would have if it were compressed (or expanded) adiabatically from its initial pressure to a pressure of 1000 mbar. By using the definition of potential temperature and the ideal gas law, you can arrive at the relation above through differentiating with respect to $z$.

Question

The author states the following:

Because the rates of motion of the atmosphere and gravity waves are generally much less than the speed of sound, the term $\frac{1}{c_s^2}\frac{dP}{dz}$ can be neglected.

My question is how I can reason about the term $\frac{1}{c_s^2}\frac{dP}{dz}$ to convince myself that it should be small for most atmospheric phenomena? Or, in other words, how can I show that $\frac{dP}{dz}$ is small compared to $c_s^2$? In what sense is the derivative of pressure comparable to a speed squared?

Answer 1

You might find more satisfying the justification given by Holden and Hakim in An Introduction to Dynamic Meteorology: "For buoyancy wave motions [$\left|\frac{\rho}{\theta}\frac{d\theta}{dz}\right|\gg \left|\frac{1}{c_s^2}\frac{dP}{dz}\right|]$; that is, density fluctuations due to pressure changes are small compared with those due to temperature changes." (I've used Nappo's notation for the relation in brackets.)

On the other hand, Satoh's Atmospheric Circulation Dynamics and General Circulation Models simply asks us to consider the simplification arising "[i]n the case that the speed of sound $c_s$ approaches infinity".