Here we generate and test a number of series with different integration order I(n) and polynomial order O(n). The test is the Augmented Dickey Fuller test, one of the most well known of the unit root tests. Beenstock used three tests, because the tests for unit roots are known to have low power.
The six series shown above are as follows (black, red, green):
In the first group is a random series I(0), the integrated random series I(1), and the twice integrated random series I(2).
The second group is a linear trend plus noise O(1), a quadratic trend plus noise O(2), and a cubic trend plus noise O(3).
The way I diagnose order is like the code below returning the number of differences before the adf.test rejects (adf.test is in R package tseries).
for (n in 0:2)
if (d<0.05) return n
The result for the 6 series:  0 1 2 0 0 1
The first 5 were guessed right, but the cubic series is a false positive. Note that even when I use a highly curved function such as the sixth power or exponential I still get I(1).
This reaction to curvature is I guess the reason that Beenstock examine the rfCO2 for breaks. They found that rfCO2 was I(2), but also showed signs of being I(1) with a break. On the basis of another test they reject that possibility.
Next, some real data.