<p>Modifiing my previous answer thanks to you comment, I realized I not clear at all. Here it's better.</p>
<p>First, I want to state that the notations in your material seem weird to me as the normalizing and adimentionnality of the values are not properly presented. For the eq. 1 and 2, <span class="math-container">$T_{eq}$</span> and <span class="math-container">$S_{eq}$</span> are said to be dimensionless yet they are added to <span class="math-container">$\Delta T$</span> and <span class="math-container">$\Delta S$</span> that are not introduced as dimensionless.</p>
<p>In any case, it seems akin to the <em>density anomaly</em> (see <a href="https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1040&amp;context=codee" rel="nofollow noreferrer">THIS LINK</a>, page 6). The same pdf file the link leads to bears a thorough explanation of the model (page 9 and following).</p>
<p>Without going trough the nondimensionalization process, by taking the density equation of salty waters, <span class="math-container">$\rho=\rho_0(1-\alpha T+\beta S)$</span> (increases with increasing S and decreases with increasing T with rate respectively <span class="math-container">$\alpha$</span> and <span class="math-container">$\beta$</span>), you have that <span class="math-container">\begin{align}\Delta \rho&amp;=\rho_{A}-\rho_{B}\\&amp;=\rho_0(1-\alpha T_{A}+\beta S_{A})-\rho_0(1-\alpha T_{B}+\beta S_{B})\\&amp;=\rho_0(-\alpha \Delta T+\beta \Delta S).\end{align}</span> Using <span class="math-container">$R=\frac{\beta}{\alpha}$</span> and <span class="math-container">$\Delta \rho_*=\frac{\Delta \rho}{\alpha\rho_0}$</span> leads to <span class="math-container">$$\Delta\rho_*=-\Delta T+R\Delta S.$$</span> I'm not sure about why the absolute value is introduced, maybe because the textbook authors weren't interested in wich part of the ocean was the most dense?</p>
<p>You can take a look at the original paper by Stommel at he following <a href="https://onlinelibrary.wiley.com/doi/epdf/10.1111/j.2153-3490.1961.tb00079.x" rel="nofollow noreferrer">LINK</a>.</p>

CC BY-SA 4.0
2024-02-03T18:13:06.827Z
A higher temperature difference across two regions of the ocean also means a higher density difference.

The detlaT and deltaS are between 0 and 1 (always positive):

T_equatorial-T_polar = Positive value;
S_equatorial - S_polar = Positive value.
AAAA
CC BY-SA 4.0
2024-02-04T23:15:07.403Z
Hi @AAAA , I modified my answer to be more thorough . I'm not absolutely sure of being right as it's been a while since I've studied this model. However, your base material is perplexity inducing as it jumps over large parts of the explanation and seems to neglect some precisions - I recommand you to look at other material for a finer understanding.
Varazak
CC BY-SA 4.0
2024-02-06T22:00:43.517Z
Great! Thanks. 
Yes, I have been using my mathematics texts but was hoping to see if there was a more intuitive way of understanding this model.  Thank you for the explanation! I will keep working on it to figure out how else it can be presented.
AAAA
CC BY-SA 4.0
2024-02-03T11:46:00.440Z
2024-02-04T23:16:16.017Z
2024-02-04T23:16:16.017Z
Varazak
<p>I have a question about <a href="https://serc.carleton.edu/integrate/teaching_materials/earth_modeling/student_materials/unit8_article1.html" rel="nofollow noreferrer">a module on Stommel's Two-Box model</a>.</p>
<p>I am uncertain about the following density equation:
<span class="math-container">$ \Delta  \rho =  | R \Delta S - \Delta T| $</span></p>
<p>A larger difference between temperatures/salinity levels across the two regions of the North Atlantic should both increase the density difference. Why is this a subtraction? Is this because of the opposite relationship between salinity/temperature at each pole (i.e., high salinity increasing the density at the equatorial region with high temp decreasing density at the equatorial region, and vice versa at the poles)?</p>

CC BY-SA 4.0
2024-02-03T18:25:09.297Z
Have you looked at the paragraph "It is important to understand an important negative feedback in this system involving Q and the pole-equator temperature difference ΔT. When the temperature difference, ΔT, is great, then Q will be strong, but this transports more heat to the polar region, thus lessening ΔT, which in turn weakens Q. On the other hand, if Q is weak, then there is little heat transport to the polar region, allowing this region to grow colder, thus increasing ΔT, which can then in turn leads to a stronger Q and thus more heat to the polar region." on the page?
JeopardyTempest
<oceanography>
Questions about Stommel's Two-Box Model (North Atlantic Current)
AAAA
# Questions about Stommel's Two-Box Model (North Atlantic Current)

(Asked by: username_0 on 2024-02-03)

I have a question about [a module on Stommel's Two-Box model](https://serc.carleton.edu/integrate/teaching_materials/earth_modeling/student_materials/unit8_article1.html).

I am uncertain about the following density equation:
$ \Delta \rho = | R \Delta S - \Delta T| $

A larger difference between temperatures/salinity levels across the two regions of the North Atlantic should both increase the density difference. Why is this a subtraction? Is this because of the opposite relationship between salinity/temperature at each pole (i.e., high salinity increasing the density at the equatorial region with high temp decreasing density at the equatorial region, and vice versa at the poles)?

**Tags:** oceanography

> **Comments:**
> **username_1**: Have you looked at the paragraph "It is important to understand an important negative feedback in this system involving Q and the pole-equator temperature difference ΔT. When the temperature difference, ΔT, is great, then Q will be strong, but this transports more heat to the polar region, thus lessening ΔT, which in turn weakens Q. On the other hand, if Q is weak, then there is little heat transport to the polar region, allowing this region to grow colder, thus increasing ΔT, which can then in turn leads to a stronger Q and thus more heat to the polar region." on the page?  

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## 1 answers

### Answer by username_2 on 2024-02-03. Score: 1 | Accepted answer

Modifiing my previous answer thanks to you comment, I realized I not clear at all. Here it's better.

First, I want to state that the notations in your material seem weird to me as the normalizing and adimentionnality of the values are not properly presented. For the eq. 1 and 2, $T\_{eq}$ and $S\_{eq}$ are said to be dimensionless yet they are added to $\Delta T$ and $\Delta S$ that are not introduced as dimensionless.

In any case, it seems akin to the *density anomaly* (see [THIS LINK](https://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1040&context=codee), page 6). The same pdf file the link leads to bears a thorough explanation of the model (page 9 and following).

Without going trough the nondimensionalization process, by taking the density equation of salty waters, $\rho=\rho\_0(1-\alpha T+\beta S)$ (increases with increasing S and decreases with increasing T with rate respectively $\alpha$ and $\beta$), you have that \begin{align}\Delta \rho&=\rho\_{A}-\rho\_{B}\\&=\rho\_0(1-\alpha T\_{A}+\beta S\_{A})-\rho\_0(1-\alpha T\_{B}+\beta S\_{B})\\&=\rho\_0(-\alpha \Delta T+\beta \Delta S).\end{align} Using $R=\frac{\beta}{\alpha}$ and $\Delta \rho\_\*=\frac{\Delta \rho}{\alpha\rho\_0}$ leads to $$\Delta\rho\_\*=-\Delta T+R\Delta S.$$ I'm not sure about why the absolute value is introduced, maybe because the textbook authors weren't interested in wich part of the ocean was the most dense?

You can take a look at the original paper by Stommel at he following [LINK](https://onlinelibrary.wiley.com/doi/epdf/10.1111/j.2153-3490.1961.tb00079.x).

> **Comments:**
> **username_0**: A higher temperature difference across two regions of the ocean also means a higher density difference.  The detlaT and deltaS are between 0 and 1 (always positive):  T_equatorial-T_polar = Positive value; S_equatorial - S_polar = Positive value.  
> **username_2**: Hi @username_0 , I modified my answer to be more thorough . I'm not absolutely sure of being right as it's been a while since I've studied this model. However, your base material is perplexity inducing as it jumps over large parts of the explanation and seems to neglect some precisions - I recommand you to look at other material for a finer understanding.  
> **username_0**: Great! Thanks.  Yes, I have been using my mathematics texts but was hoping to see if there was a more intuitive way of understanding this model.  Thank you for the explanation! I will keep working on it to figure out how else it can be presented.  

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