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This is a set of differential equations originally produced by Stommel (1961) has been simplified in this source (https://serc.carleton.edu/integrate/teaching_materials/earth_modeling/student_materials/unit8_article1.html)

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The values of these parameters were picked (somewhat) arbitrarily. So, I am instead trying to think of an intuitive way to understand this through temperature first (without taking into account salinity). It makes sense that, in an absence of water flow, the system would tend toward the equilibrium temperature difference (ΔT_{equilibrium} - ΔT). When there is ocean flow, I am curious about describing this in an intuitive sense. Does it make sense (with just focusing on the temperature first) that the rate of circulation/ocean flow is directly proportional to these temperature differences (if we ignore salinity first)? How could this be adapted from these ODEs? Is this okay as a conceptual model? Are there too many things behing left unaccounted for even though this is meant to be a simple model? Could it simply be a constant times deltaT for the first equation instead of QdelT if only temperature is the focus at first? This is not meant to be predictive.

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  • $\begingroup$ If the terms involving $Q$ are not taken into account there is no circulation. I'm having a hard time understanding what you want to know. $\endgroup$ Commented Mar 17 at 15:58
  • $\begingroup$ @JoschaFregin Yeah, without Q that models what would happen without ocean flow (similar to Newton's law of cooling). What I mean is to have an intuitive way of understanding what happens once you introduce flow (a way that logically flows from the structure of the equation). Assuming no salinity difference, this would imply that the rate of flow is delT/lambda. However, in the differential equation for delT, this means the last term will be delT²/lambda, but it seems like a constant times delT would make more sense. I know the model is a simplification, but I feel like it's missing something. $\endgroup$
    – AAAA
    Commented Mar 18 at 16:13
  • $\begingroup$ The way it is written, the model treats $Q$ as a diagnostic variable. Hence you do not substitute for $\Delta T/\lambda$ and actually can treat it as a constant. At least that is how I read it, might be wrong. $\endgroup$ Commented Mar 18 at 19:37
  • $\begingroup$ @JoschaFregin Unfortunately, Q is not a constant and depends on delT and delS. Because Q takes in delRho we have Q = |R*delS - delT|/lamba. If we set delS equal to zero, we have Q = |delT|/lamba. Then, when putting this into the differential equation for delT/dt, we get delT^2/lambda, which doesn't make as much sense as just a constant multiplied by delT. When I say "doesn't make as much sense", I mean in terms of being able to explain this idea to anyone first encountering the model. $\endgroup$
    – AAAA
    Commented Mar 18 at 20:58

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