Josh Willis, in response to a request for information for a short article that I am writing, wrote to me on May 6 2008 the e-mail below [reproduced with his permission]. This e-mail contains important new insight into recent upper ocean heat variations, as well as the uncertainty in the data. The analyses performed by Josh Willis and colleagues should be the gold standard used to monitor global climate system heat changes (e.g., as discussed on the Climate Science weblog of May 26 2008).
Josh Willis’s e-mail [edited to focus on his new data analysis]
“…After thinking this over some, I really think that the best plot for you is the first one attached to this message. This is just a plot of ocean heat content with errorbars, with the red lines illustrating the average, 4-year rate of ocean warming (plus or minus one standard error in the slope). Note that in all plots and in the discussion below, I have used one standard error, NOT 95% confidence limits.
Although I’ve attached a couple of other plots to illustrate some of my points, I think the first plot is the best one to use for several reasons.
First, this pretty much just differs by a scale factor from the steric curve we published before, and I think this will make the explanation clearer. Also, the size of the error bars is of critical importance if we want to use these data to constrain the radiative imbalance, and this gives a clear illustration of how well we can measure the average temperature of the upper ocean in a single month. As you’ll see below, that one-month error is sort of the building block for how accurately we can use ocean temperature data for this problem and what time scales we can expect it to be useful over.
Finally, this plot does not use any other assumptions or contain any other data about the radiative imbalance, the area of the Earth, the time derivative or any other factors to confuse its meaning.
So, to equate this plot to a radiative imbalance, we need to do two things. First, we assume that all of the radiative imbalance at the top of the atmosphere goes toward warming the ocean (this is not exactly true of course, but we think it is correct to first order at these time scales). So in normalizing this, we divide by the surface area of the Earth instead of the surface area of the Ocean. For the purposes of this calculation, I’ve approximated this as 5.1 x 10^14 m^2. The second thing we have to do is take the time derivative of the heat content curve to arrive at the rate of warming, or radiative imbalance. In other words, we have to turn Joules into Watts.
This second part is were the estimate gets really noisy. If I do the simplest thing and take a first difference of my monthly time series of ocean heat content, the noise is very large, as you might expect. This is shown in the second plot, and you can see that errors in a single one-month difference often exceed 20 W/m^2. For all of the following time derivative calculations, I computed the errors as follows:
error_rate = sqrt( error(month 1)^2 + error(month 2)^2 ) / time_difference
We can do a bit better, if instead of taking a one-month difference, we take one-year differences. This has the added benefit of removing the seasonal cycle. In other words, we subtract:
warming rate = hc(July, 2004) – hc(July, 2003) / (1 year)
The error bar is now smaller because the time difference has gotten larger.
In the example above, the estimate of the one-year warming rate is centered on January, 2004. If we do this for every pair of points separated by one year in the time series, we get the last figure attached.
However, what you would really like to know is the 4-year warming rate over this period. We have to be careful with the second two figures attached.
We cannot simply average over these and get a “mean warming rate” because we have taken a time derivative and the points are not independent (and neither are their errors). So, to get the 4-year rate illustrated by the red lines in the first plot, I took the 7 months in the time series that have 4-year pairs. That is, July, 2003 through Jan., 2004 and July, 2007 through Jan, 2008. Each of these pairs gives a 4-year rate and error bar as follows (in W/m^2):
-0.1132 +/- .5820
-0.4417 +/- .5807
-0.0102 +/- .5693
-0.0154 +/- .5673
-0.0231 +/- .5636
-0.0799 +/- .5410
0.1519 +/- .5468
Each one of these estimates is completely independent, so if we take their mean, we get an average 4-year warming rate of:
-0.075941 +/- 0.2139 W/m^2
Again, this is a one standard error estimate. Also, this is the number used for the slope of the red line in the first figure. I think it is very important to note that this is NOT a least squares fit of a straight line to the heat content curve. In my experience, whenever you fit a line to a time series, the meaning of the line and its errorbars is often muddled (even for scientists). For this reason, I try to avoid fitting lines to time series whenever possible.
Another thing that is important to note is that this excludes any warming in the deep ocean. Over four years, it is reasonable to expect at least some warming (or cooling) below 900 m as well as in regions that Argo does not sample such as under the sea ice. Not to mentions the small changes in heat content due to melting ice, warming land, and a warming atmosphere (with more or less water vapor). For these reasons, it would probably be wise to add at least a few more tenths of a W/m^2 to the errorbar on this 4-year warming rate.
Anyway, I hope this is useful. Sorry for the long email, but I wanted to be clear about the errors and I think there are a few subtlties there.
Josh Willis, Ph.D.
Jet Propulsion Laboratory”
My Follow Up E-mail On May 9 2008
“Hi Josh, I have started to go through your analysis and have several comments and suggestions. I agree Figure 1 is the clearest for the …. article. It clearly shows, for example, the intrannual variation in heat with an interesting variation in the time of the peaks and troughs between the years. Within a year, there is both a period with positive radiative imbalance and a period with negative radiative imbalance (e.g., see this also for the lower troposphere in the April 18, 2008 weblog).
There is also another way to assess the heat content change over the period and that is to bin the data in yearly blocks and test statistically whether any of the blocks are different from each other. If the last block is not statistically different from the first block, then there is no statistically significant change. This would add more data points to test (within the blocks) and also reduces the question to whether or not there has been a statistically significant addition or loss of heat over the time period.
With respect to how to allocate the heat changes to the global scale, I suggest that there is no need to scale up by area. The assumption that since the oceans are about a 70% sample of the Earth, the computed value of the heat changes could be assumed to be the same for the other 30% (but, of course, are not sampled). This is an approximation, but in lieu of better ways to estimate, this seem reasonable.
On the other heat reservoirs, the global sea ice is actually near its long-term average; see
In any case, this data can be used to estimate the heat changes in the sea ice by assuming a reasonable thickness (it will be a small contribution in any case).
The troposphere has actually been flat and has now cooled; e.g. see Figure 7 in
I agree with you on the deeper ocean. The only other explanation for continuing sea level rise is a rise in the ocean bottom on these time scales (which is a topic outside of my expertise).
It also would be useful to compute what would be the maximum and minimum global annually averaged radiative imbalance using the 95% confidence value. This would permit a direct comparison to the way the IPCC present their data (as global annually averaged TOA radiative forcing), in order to assess if the sum of the radiative forcings and feedbacks are less than or greater than the IPCC estimates of the radiative forcing. If they are negative under any reasonable estimate, this means the radiative feedbacks are negative which would conflict with the IPCC assumption of an amplification of warming by the water vapor feedback. If the feedbacks are positive, this supports the IPCC’s view….
Best Regards, Roger”
The analysis being completed by Josh Willis and colleagues is central to the issue of assessing global warming and cooling. Climate Science recommends that upper ocean heat changes in Joules become the primary assessment tool for global climate system heat changes, as the data, with the introduction of the Argo network, is now robust to this evaluation. A website with the latest upper ocean heat content analyses should be funded and made widely available to the clmiate community, policymakers and, of course, the public.