# Cabelling calculations

**Dan Kelley Blog/R**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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*Abstract* R code is provided in aide of laboratory demonstration of
cabelling.

# Introduction

Setting up a cabelling experiment requires creating two watermasses of equal density, and if only S and T can be measured, that means calculating densities. Using a TS diagram and graphical interpolation is one approach to that task, but another is to use R to do the calculation.

# Methods

The code given below will do the calculation for specified `Sa`

, `Ta`

and `Sb`

, where the second letter indicates the watermass. The code uses `uniroot()`

to find the temperature `Tb`

that yields equal densities for watermasses *a* and *b*.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | # Alter next three lines as desired; a and b are watermasses. Sa <- 30 Ta <- 10 Sb <- 40 library(oce) # Should not need to edit below this line rho0 <- swRho(Sa, Ta, 0) Tb <- uniroot(function(T) rho0-swRho(Sb,T,0), lower=0, upper=100)$root Sc <- (Sa + Sb) /2 Tc <- (Ta + Tb) /2 ## density change, and equiv temp change drho <- swRho(Sc, Tc, 0) - rho0 dT <- drho / rho0 / swAlpha(Sc, Tc, 0) if (!interactive()) png("cabelling.png", width=7, height=7, unit="in", res=200, pointsize=12) plotTS(as.ctd(c(Sa, Sb, Sc), c(Ta, Tb, Tc), 0), pch=20, cex=2) drawIsopycnals(levels=rho0, col="red", cex=0) segments(Sa, Ta, Sb, Tb, col="blue") text(Sb, Tb, "b", pos=4) text(Sa, Ta, "a", pos=4) text(Sc, Tc, "c", pos=4) legend("topleft", legend=sprintf("Sa=%.1f, Ta=%.1f, Sb=%.1f -> Tb=%.1f, drho=%.2f, dT=%.2f", Sa, Ta, Sb, Tb, drho, dT), bg="white") if (!interactive()) dev.off() |

If run non-interactively, the code will produce a PNG file like that given below.

# Results

The legend summarizes the results, indicating also the density change and the temperature change that would be equivalent to that density change (at the midpoint, *c*).

# Conclusions

If the design goal is that the density mismatch between watermasses *a* and *b* should be, say, 10 percent of the density difference of the mixture watermass (*c*), then in the illustrated case the temperature would have to be controlled to within a quarter of a degree Celcius – a task that is challenging enough to argue against this as an informal classroom demonstration.

# Exercises

- Alter the R code to calculate
`Sb`

in terms of`Tb`

. - Consider (calculate or measure) the convection associated with sidewall heat conduction.

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**Dan Kelley Blog/R**.

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