<p>Strictly speaking, <span class="math-container">$\lim_{\Delta z \to 0} Ri=Ri_b$</span>. This is because the <a href="https://glossary.ametsoc.org/wiki/Gradient_richardson_number" rel="nofollow noreferrer">gradient Richardson number</a>: <span class="math-container">$$Ri=\frac{\frac{g}{T_v}\frac{\partial \theta_v}{\partial z}}{\left(\frac{\partial U}{\partial z}\right)^2++\left(\frac{\partial V}{\partial z}\right)^2}$$</span> can be approximated as <span class="math-container">$$Ri \approx\frac{\frac{g}{T_v}\frac{\Delta \theta_e}{\Delta z}}{\left(\frac{\Delta U}{\Delta z}\right)^2++\left(\frac{\Delta V}{\Delta z}\right)^2} \tag{1}$$</span></p>
<p>which can be rewritten as the <a href="https://glossary.ametsoc.org/wiki/Bulk_richardson_number" rel="nofollow noreferrer">Bulk Richardson Number</a>: <span class="math-container">$$Ri_b=\frac{\frac{g\Delta z \Delta \theta_v}{T_v}}{(\Delta U)^2+(\Delta V)^2} \tag{2}$$</span>. The derivation of (2) from (1) is left as an exercise for the reader.</p>

CC BY-SA 4.0
2023-02-08T19:52:17.260Z
Thank you for your clarification! This was very helpful.
DustinMa
CC BY-SA 4.0
2021-04-18T20:53:05.970Z
2021-04-18T20:53:05.970Z
BarocliniCplusplus
<p>In many sources (<a href="https://glossary.ametsoc.org/wiki/Bulk_richardson_number" rel="nofollow noreferrer">https://glossary.ametsoc.org/wiki/Bulk_richardson_number</a>), bulk Richardson number is defined as an approximation to gradient Richardson number. The prior only approaches the ladder when layer thickness, <span class="math-container">$\Delta z$</span> becomes &quot;small&quot;. What is small defined as? For example, comparison of simulation domain to <span class="math-container">$\Delta z$</span>, overall length <span class="math-container">$\Delta z$</span> approaches 0, or etc?</p>

<meteorology><atmosphere><fluid-dynamics>
What is "small layer thickness" defined as in terms of bulk Richardson number approaching gradient Richardson number?
DustinMa
# What is "small layer thickness" defined as in terms of bulk Richardson number approaching gradient Richardson number?

(Asked by: username_0 on 2021-04-09)

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?

**Tags:** meteorology, atmosphere, fluid-dynamics


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## 1 answers

### Answer by username_1 on 2021-04-18. Score: 2 | Accepted answer

Strictly speaking, $\lim\_{\Delta z \to 0} Ri=Ri\_b$. This is because the [gradient Richardson number](https://glossary.ametsoc.org/wiki/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](https://glossary.ametsoc.org/wiki/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.

> **Comments:**
> **username_0**: Thank you for your clarification! This was very helpful.  

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