I am trying to wrap my head around Stommel's Two-Box Model of ocean circulation (specifically of the North Atlantic).
The general idea of this model is as follows: Diagram of Stommel's Two-Box Model
$T_1$ is the temperature of box 1 and $T_2$ is the temperature of box 2. Also, $S_1$ is the salinity of box 1, and $S_2$ is the salinity of box 2. In a "normal" state, we have $T_2 > T_1$ since box 2 represents water near the equatorial regions. The $T_1^*, T_2^*, S_1^*$ and $S_2^*$ represent the temperature and salinity of the surrounding basins (see posted image above).
We also set $\rho_1$ as the density of box 1, and $\rho_2$ as the density of box 2. We can calculate the capillary flow $q$ as follows: $q = k \frac{(\rho_1 -\rho_2)}{\rho_0}$
$\rho_0$ is the density when $T=T_0$ and $S=S_0$ such that $T_0=\frac{1}{2}(T_1+T_2)$ and $S_0=\frac{1}{2}(S_1+S_2)$.
Then, we can compute the density of each box as: $\rho = \rho_0 (1 - \alpha (T-T_0) + \beta(S - S_0))$
We can compute the capillary flow as: $q = k(\alpha(T_2-T_1) + \beta(S_2-S_1))$
Here, we know that, $\alpha = 1.5 * 10^{-4}$ deg $^{-1}$, $\beta = 8*10^{-4}$ psu $^{-1}$, and $k = 1.5*10^{-6}$ s $^{-1}$.
So far, this makes sense. What I am having a hard time wrapping my head around is the second piece of this leading to a system of differential equations (see attached image showing how the system of differential equations were obtained). How do I find realistic values of c, H, and d? Where would I start? I tried looking at some published papers, which tend to still not give realistic values for these variables. This is not for an assignment. I'm just trying to wrap my head around this model.
It appears all of those constant are values chosen by the user and are not calculated anywhere in the model. Even if you consult Stommel's paper (see here: https://onlinelibrary.wiley.com/doi/epdf/10.1111/j.2153-3490.1961.tb00079.x ), he appears to be making an arbitrary choice so he can better understand the dynamics of the model. This is a pretty common practice with conceptual models such as this in the geosciences(as well as mathematical modeling in general) as the point is to better understand the behavior of the system. Section 6.4 in Chapter 6 of the book have a brief discussion about the realism and history of this model. If interested in learning about of thermohaline circulation and it models, I would recommend checking out a textbook on Physical Oceanography(Stewart's Intro to Physical Oceanography is excellent and free: https://github.com/introocean/introocean-en/releases/tag/v20220927) or looking at a book on Geophysical Fluid Dynamics(Vallis' Atmospheric and Oceanic Fluid Dynamics is a staple and Marhsall and Plumb's Atmosphere, Ocean and Climate Dynamics is a good introductory text). Hope this helps!