What is "small layer thickness" defined as in terms of bulk Richardson number approaching gradient Richardson number?

Question

In many sources (https://glossary.ametsoc.org/wiki/Bulk_richardson_number), bulk Richardson number is defined as an approximation to gradient Richardson number. The prior only approaches the ladder when layer thickness, $\Delta z$ becomes "small". What is small defined as? For example, comparison of simulation domain to $\Delta z$, overall length $\Delta z$ approaches 0, or etc?

Answer 1

Strictly speaking, $\lim_{\Delta z \to 0} Ri=Ri_b$. This is because the gradient Richardson number: $$Ri=\frac{\frac{g}{T_v}\frac{\partial \theta_v}{\partial z}}{\left(\frac{\partial U}{\partial z}\right)^2++\left(\frac{\partial V}{\partial z}\right)^2}$$ can be approximated as $$Ri \approx\frac{\frac{g}{T_v}\frac{\Delta \theta_e}{\Delta z}}{\left(\frac{\Delta U}{\Delta z}\right)^2++\left(\frac{\Delta V}{\Delta z}\right)^2} \tag{1}$$

which can be rewritten as the Bulk Richardson Number: $$Ri_b=\frac{\frac{g\Delta z \Delta \theta_v}{T_v}}{(\Delta U)^2+(\Delta V)^2} \tag{2}$$. The derivation of (2) from (1) is left as an exercise for the reader.