In Kundu's book , 4ed, P21, they define the potential density $\rho_\theta$ like this:
However, later in P22, they define the potential density gradient as
It seems to me that the potential density shall also be $\rho_\theta=\rho-\rho_a$. But I cannot derive this from equation (1.34). Could you please give me any hints on how to build the link between them?
Equation (1.34) is derived from the ideal gas laws, and refers only to an ideal gas. Equation (1.38) is more general, and is valid for any fluid--in particular for seawater.
Considering the relation you want to derive, $\rho_\theta=\rho-\rho_a$, let's first define the terms: $\rho$ is the density (mass per unit volume). $\rho_\theta$ is the potential density. It is defined relative to a reference pressure, and at every point is equal to the density the fluid there would have it it were compressed or expanded the required amount to reach that reference pressure. $\rho_a$ is defined relative to a reference pressure and density: it's what the density would be if the fluid were well mixed--Kundu calls it the "neutrally stable reference state". From these definitions, you can see that the relation is not correct, but is off by a constant. We can write the actual relation thus: $\rho_\theta(z)=\rho(z)-\rho_a(z)+\rho_0$. Here, $\rho_0$ is the density at the reference point, and I have added $(z)$'s to emphasize that, unlike $\rho_0$, the other terms are functions of depth. This relation can be regarded as a definition of potential density; as such is does not really make sense to try to derive it.
