Cabelling calculations

January 15, 2014
By

(This article was first published on Dan Kelley Blog/R, and kindly contributed to R-bloggers)

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.

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# 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).

cabelling TS diagram

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

  1. Alter the R code to calculate Sb in terms of Tb.
  2. Consider (calculate or measure) the convection associated with sidewall heat conduction.

To leave a comment for the author, please follow the link and comment on their blog: Dan Kelley Blog/R.

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