# The link between two definitions of "potential density"

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?

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.