In their never-ending quest to show that all findings of standard climate science are wrong, Willis tackles ocean warming, in "Forcing the Ocean to Confess."
But not Willis. His work leads him to the following conclusion:
"...neither the average forcing, nor the trend in that forcing, are significantly different from zero. It’s somewhat of a surprise.What was Willis able to whip up in a little spreadsheet that disproves what all the professionals have determined? Some bad physics, that's what.
"The third is that in addition to the mean not being significantly different from zero, only a few of the individual years have a forcing that is distinguishable from zero.
"Those were a surprise because with all of the hollering about Trenberth’s missing heat and the Levitus ocean data, I’d expected to find that we could tell something from the Levitus’s numbers."
To see if ocean heat content (OHC) is increasing, most people would just calculate its trend, which is an approximate for its first derivative d(OHC)/dt, where t is time.
Instead, Willis first takes the annual difference in OHC, calls that the forcing (which it isn't really, but that's another story), and calculates the trend of that. He finds it slightly positive, though not significant at the 95% confidence level:
But that's equivalent to calculating the second derivative of OHC, i.e. the acceleration of the ocean's heat gain.
Willis gets himself all confused including autocorrelation in the statistics of the trend, which is a topic for another day. What's more relevant is that he doesn't understand what he's calculating, and why it says little about ocean heat content.
The trend of a function and the trend of its first derivative can have very different uncertainties, because the latter is a difference of differences, which often can fluctuate more than a first difference.
Here is an clean example that makes Willis's error explicit, without the malarky of all the statistics. Suppose the ocean heat content for year Y is
OHC(Y) = kY
where k is a positive constant. Obviously this ocean is gaining heat year after year, with a slope of k units per year.
Then what Willis is calling the "forcing" (again, it isn't really), and what he's plotting above using the real OHC data, is
F(Y) = {OHC(Y) - OHC(Y-1)}/A
where A is the area of the ocean. In this example F(Y) = k/A for all years, i.e the "forcing" is a constant. Hence it has a trend of zero (0).
But that doesn't mean the ocean isn't warming, viz. gaining heat. Clearly it is.
A constant forcing means warming. Only a zero forcing means no warming.
Willis sort of tries to cover his huge error by saying the "forcing" he's calculating isn't statistically different from zero. He finds the mean of the annual values to be 0.2 ± 0.3 W/m2.
But again, that number has a different interpretation that the one Willis tries to give it, because it's a mean and not a trend. The average forcing is positive, i.e. most years have a "positive" flow of heat into the ocean. Some years don't, and you could use the standard deviation above to find out what percentage of years don't.
But the average forcing is positive. The ocean is, on average, gaining heat. But remember, this is the second derivative of OHC(t), not the first. The first is clearly positive, and has strong statistics.
I find, using the pentadal temperature anomalies (the OHC results are here) a temperature trend of 1.25 ± millikelvin/yr, which translates into a OHC trend of
0.34 ± 0.02 W/m2
(1-sigma, no autocorrelation, total area of the ocean)
(1-sigma, no autocorrelation, total area of the ocean)
Now, autocorrelation will increase the uncertainty by a small factor (usually it's around 3), but hardly by a factor of 17! (Willis messes that up, too, but it's tangential to the main point.)
The ocean is gaining heat, i.e. warming. Willis's methodology is wrong, confused, and...well, too-often typical.
First derivative, second derivative, watts the difference?
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