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I have a question about a module on Stommel's Two-Box model.

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)?

AAAA
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  • 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 Feb 03 '24 at 18:25

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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, 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.

Varazak
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  • 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 Feb 03 '24 at 18:13
  • 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 Feb 04 '24 at 23:15
  • 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 Feb 06 '24 at 22:00