Basic fluid mechanics questions about an oceanography paper

[Velatox]

Homework Statement

There is no precise problem statement since it's an homework to be done independently, but I will try my best to explain it concisely (I also apologize for my mistakes in English) :
I want to reproduce the simulation of the model presented in the pdf file attached to this post (direct link to it : https://journals.ametsoc.org/doi/pdf/10.1175/1520-0485%281976%29006%3C0029%3ATSEIAM%3E2.0.CO%3B2 )
It consists of the system of equations (6) to (10) shown on page 2, that i will simulate over a length of 10 days using a Runge-Kutta type integration scheme. I've clarified a lot of things so far, but I have 2 minor problems left that still bother me , and both are related to the values of certain parameters that i need in order to run the program, but i can't seem to find in the paper. (probably because it's so obvious to real oceanographs that they didn't bother to be more precise haha)

1) First, there are the two coefficients $α = (ρ _0)^-1\frac{\partial ρ }{\partial T}$ and $λ = (ρ _0)^-1\frac{\partial ρ }{\partial S}$ which are first introduced page 2 (just below the equation (5)). Given these expressions, I assumed both parameters to be time-dependant, but the paper never discuss that matter more in depth. Furthermore, on page 4, (just below the first graph), λ is (seemingly) considered a constant of value $0.7 * 10^{-3} ppt^{-1}.$
So i was wondering : which values for α and λ should I actually use as input for my simulation?

2) Finally, my second issue is related to the parameter G - D, whose value is never explicitely given in the article, but discussed at the end of page 2 (right column, below equation (6)). However, i simply do not understand what the author is saying there (probably because of language barrier plus the fact that some concepts tackled here are unknown to me) . Therefore, I'm wondering as well about the value I should use for that parameter.

Relevant Equations

$α = (ρ _0)^-1\frac{\partial ρ }{\partial T}$
$λ = (ρ _0)^-1\frac{\partial ρ }{\partial S}$
Where $ρ _0$ is the mean density of seawater, T the temperature and S the salinity

$m = -ρ _0αg(G - D)/(τU)$
Where g is the gravitationnal acceleration, τ the surface wind stress and U the wind speed at 10 meters.

Problem Statement: There is no precise problem statement since it's an homework to be done independently, but I will try my best to explain it concisely (I also apologize for my mistakes in English) :
I want to reproduce the simulation of the model presented in the pdf file attached to this post (direct link to it : https://journals.ametsoc.org/doi/pdf/10.1175/1520-0485(1976)006<0029:TSEIAM>2.0.CO;2 )
It consists of the system of equations (6) to (10) shown on page 2, that i will simulate over a length of 10 days using a Runge-Kutta type integration scheme. I've clarified a lot of things so far, but I have 2 minor problems left that still bother me , and both are related to the values of certain parameters that i need in order to run the program, but i can't seem to find in the paper. (probably because it's so obvious to real oceanographs that they didn't bother to be more precise haha)

1) First, there are the two coefficients α = (ρ _0)^-1\frac{\partial ρ }{\partial T} and λ = (ρ _0)^-1\frac{\partial ρ }{\partial S} which are first introduced page 2 (just below the equation (5)). Given these expressions, I assumed both parameters to be time-dependent, but the paper never discuss that matter more in depth. Furthermore, on page 4, (just below the first graph), λ is (seemingly) considered a constant of value $0.7 * 10^{-3} ppt^{-1}.$
So i was wondering : which values for α and λ should I actually use as input for my simulation?

2) Finally, my second issue is related to the parameter G - D, whose value is never explicitely given in the article, but discussed at the end of page 2 (right column, below equation (6)). However, i simply do not understand what the author is saying there (probably because of language barrier plus the fact that some concepts tackled here are unknown to me) . Therefore, I'm wondering as well about the value I should use for that parameter.
Relevant Equations: α = (ρ _0)^-1\frac{\partial ρ }{\partial T}
λ = (ρ _0)^-1\frac{\partial ρ }{\partial S}
Where $ρ _0$ is the mean density of seawater, T the temperature and S the salinity

m = -ρ _0αg(G - D)/(τU)
Where g is the gravitationnal acceleration, τ the surface wind stress and U the wind speed at 10 meters.

So I've done a bit of research before posting, it seems like alpha and lambda come from the "equation of state for seawater", and their respective names are "thermal expansion coefficient" and "haline contraction coefficient". They are nonlinear functions of density, salinity, and temperature, so i was wondering if the author of the paper considered them constant as an approximation? I found different mean values for both of these parameters depending on the different websites I checked, but all the values I saw for lambda are somewhat coherent with the one mentionned in the article. (always around $0.7 * 10^{-3} ppt^{-1}$). Regarding alpha, I found an average value of ~ $0.2 * 10^{-3} K^{-1}$.
If I use these values for my computation, will it heavily affect the outcome or will it be very similar to the result presented in the study?
As for the estimation of G - D, i found another paper, (on which is based the one I'm currently studying), and that matter is discussed in terms somewhat similar in both studies, here it is (page 7, top of the right column) : https://journals.ametsoc.org/doi/pdf/10.1175/1520-0485(1973)003<0185:ULMAOS>2.0.CO;2
It did not give me any insight on my problem though, I'm still stuck wondering what input I should use to estimate this particular parameter.

 

[fresh_42]

I have found $G-D$ (with values) in what looks like the original papers:

https://journals.ametsoc.org/doi/pdf/10.1175/1520-0485(1973)003<0185:ULMAOS>2.0.CO;2https://journals.ametsoc.org/doi/pdf/10.1175/1520-0485(1973)003<0173:ATDMOT>2.0.CO;2
and here is the table work for the dependencies density - temperature - salinity:

https://ig-meeresaquaristik.de/index.php/Attachment/47-Table-Salinity-pdf/
I'm not sure whether this helps since I have no idea about oceanography.

 

[pasmith]

I think the idea is to approximate \rho(T,S) about some reference state (T_0,S_0) by <br /> \frac{\rho(T,S)}{\rho_0} = \frac{\rho(T_0,S_0)}{\rho_0} + \frac{1}{\rho_0}\frac{\partial \rho}{\partial T}(T - T_0) + \frac1{\rho_0}\frac{\partial \rho}{\partial S}(S - S_0) with the partial derivatives evaluated at (T,S) = (T_0,S_0). It follows that \alpha and \lambda are constants. (Denman (1973) states on p 173 in the discussion of equation (3) that the Boussinesq assumption is in use, whereby this linear approximation is used only in the bouyancy force with \rho elsewhere treated as constant.)

Towards the bottom right of page 30, it is stated that a non-dimensional grouping proportional to (G - D) is to be given a fixed value:<br /> m = - \frac{\rho_0 \alpha g(G - D)}{\tau U} = 0.0012.<br />