Dear Ferreters
To fully expand this query is to compute e.g. thermal expansion coefficient (alpha)
alpha = -(1/rho) * d(rho)/d(theta)
d(rho)/d(theta) means a partial derivative of potential density wrt potential temperature.
Not knowing a better (and easy) way to do that, I would calculate "numerical derivatives".
To calculate how in-situ density changes if the local in-situ temperature changes under constant pressure (depth) and constant salinity,
∂ρ/∂T under constant p and S ≈ [rho_un(S, Tplus, p) - rho_un(S, Tminus, p) ]/ (2 * delT),
where Tplus = T + delT and Tminus = T - delT. If delT is too small, the result will be noisy. If delT is too large, the result will be inaccurate. So, you would test several values. You would plot graphs (curves) showing the calculated derivative versus various delT values at several sample points. As you decrease delT, the graph would become flat, which is where good delT values are.
If you want a derivative w.r.t. potential temperature, you would need one more step: Under constant p and S,
∂ρ/∂θ = (∂ρ/∂T) (∂T/∂θ)