"We find a relationship between sea level and temperature and find that temperature causally depends on the sea level, which can be understood as a consequence of the large heat capacity of the ocean."
They find that sea level is an exogenous variable and drives atmospheric temperature. I think this is a plausible explanation for this result given that the data is annual and as they say later in the abstract:
"We hypothesize that this is due to a long adjustment time scale of the ocean and show that the number of years of data needed to build statistical models that have the relationship expected from physics exceeds what is currently available by a factor of almost ten."
The slow adjustment of the ocean to changes in radiative forcing is a major challenge for time series modelling of the global climate system using historical data. I debated Michael Beenstock on this issue last year on this blog and in an e-mail exchange with him. The problem is that as Schmith et al. find here the atmospheric temperature depends on the temperature of the ocean. Directly modelling the effect of radiative forcing on the atmospheric temperature using a short time series can result in biased estimates. My approach was to use ocean heat content as an additional variable that acts as a second transmission path for the effects of radiative forcing changes on atmospheric temperature. Using sea level might work in a similar way though it depends on glacier melt and other factors potentially as well as thermal expansion of the ocean. An advantage of sea level is that a much longer time series is available.
Schmith et al.'s stranger result is:
"In a second step, we use the total radiative forcing as an explanatory variable, but unexpectedly find that the sea level does not depend on the forcing."
I would have expected them to find an effect of radiative forcing on sea level. But using a Monte Carlo analysis they argue that it would take 1000 years of data to get a significant result.
I'm not sure that this is the final word that can be said on this. They find only one long relationship between the three key variables.* This relationship is as follows:
zt = Tt - 1.4 St - 0.3 Ft
zt is the residual in year t, T is atmospheric temperature, S is sea level, and F is radiative forcing. My problem is that this doesn't make sense to me physically as an equilibrium relationship. It could make sense that there are long-run relationships between T and S, between S and F, and between T and F. But, as written, sea level can substitute for forcing in affecting atmospheric temperature and vice versa. I suspect that this result is because the volcanic events render the forcing series relatively stationary and, therefore, cointegration cannot be rejected for this relationship. The forcing series looks like this:
In my work, I instead assumed which variables had long-run relationships based on physical theory and I used a multicointegration type approach. So I'm not surprised that sea level doesn't react to this disequilibrium in their analysis. I would be interested in what results they get if they also test a forcing series without the volcanic effects or disaggregate the stationary (volcanics) and non-stationary (GHGs, aerosols, solar irradiation) forcing components.
Of course, this relationship might actually make sense and I could be totally wrong here. Let me know what you think.
I'm really happy, though, that time series analysis is increasingly being used now to investigate these questions. Robert Kaufmann and I pretty much pioneered this but I gave up as I found it hard to get much positive interest from climate modellers. I never published my final paper on the topic. Robert Kaufmann has continued to work on this with greater success.
* In the paper there is a typo where they say that the p-value for the test of one cointegrating vector against two is 0.01. It is in fact 0.9117 the authors tell me in an e-mail.
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