The Blackboard:

Over at Climate Audit, Steve Mosher has been showing readers who Model E hindcasts HadCrut data. He suggested I show how “Lumpy” hindcasts. And, as we’ve been discussing trying to tease out time constants using assumptions made by Schwartz in his 2007, I thought I’d show you the method I’d really prefer to estimate the time constant and climate sensitivity: that’s Lumpy.

Here’s how she well she hindcasts compared to Model E:

If you need metrics that describe the fit, the only ones I have are these: Standard error for Lumpy, from 1880-1999: σ_lumpy =0.16C. The standard error for Model E during the same time period: σ_modelE =0.17C.

Of course, this isn’t quite fair, as Lumpy is literally fit to GISS data which tracks the HadCrut data well. However, she only has two parameters: Time constant and climate sensitivity. I won’t reveal those until after we’ve discussed the “splendidness” or “crumminess” of Lumpy. That way, you can decide if you love her or hate her without knowing too much about what she says about the future climate.

What is Lumpy?

“Lumpy” is the simplest possible lumped parameter model to describe the earth’s climate. It assumes the earth’s climate is isothermal (i.e. the same temperature, T, exists everywhere) and applies a linearized form of conservation of energy. This results in the same ordinary differential equation suggested by Schwartz (2007). Here I’ve re-organized to express constant, τ and the climate sensitivity S:

(1)dT/dt = -T/τ+ (S/τ) F

To estimate the time constant τ, and sensitivity, S, I “drove” the model using forcings used in the GISS Model E and fit the monthly Land/Ocean temperature data from 1880-1999. (I stopped in 1999 because I have no monthly forcings for later times.)

Why are Lumpy’s temperature predictions so smooth?

Lumpy is driven by smoothed forcing data. Also, because she’s such a simple model, she has no internal dynamics. So, her projections are smoothed. To obtain “noisy” predictions, I would either need “noisy” forcing or a more complex model. (Ideally, perfection requires both.)

What sort of noisy forcings might I apply to Lumpy to make things “noisy”? Well, for example, the GISS forcings show the Total Solar Irradiance as an more or less periodic smoothly varying function with a period of about 11 years. Measured irradiances look like this:

Figure 2: Image from ‘pmod wrc: Solar Constant’. In this figure we see forcing from the sun that few would describe as smoothly varying over a period of 11 years. Certainly, averaged over periods like a month, one sees more or less smooth variation, but over shorter periods, the Total Solar Irradiance (TSI) is sizable.

Based on known theoretical behavior of equation (1), the noise in the TSI would be sufficient to create visible noise in Lumpy’s temperature predictions. (It might not make them any better, but it would make them noisy. )

It’s worth noting that model E, which does contain physics to create some weather, does shown some “noisy” variability. However, it is also smoother than Hadcrut data. This could be because of measurement noise, is also due to averaging over 5 model runs, but some excess smoothness might be due to the smoothed forcing functions. These could include the smoothed solar irradiance, smoothed behavior or aerosols after eruptions, smoothing of the effect of minor volcanos, and pretty much anything smoothed in the input files.

So, which is better?

Well….

Obviously, Lumpy is not a sophisticated model. She doesn’t explain much. But, she does show how we can, in principle, use physics when curve fitting. Also, there is some glimmer of hope we can get a more empirically based estimate of the time constant and climate sensitivity with a tool like this. (Or not. But if we can’t, then we can’t. )

Unfortunately, I also didn’t do the fitting in a sophisticated way, so I don’t have a clue what the uncertainties on my parameters are. But, learning issues I need to make Lumpy better (and/or figure out the error bars on the parameters) is one of the real reasons I’m fiddling with the even simpler (Schwartz + Meaurement noise) model and trying to learn how to best estimate the uncertainty bounds (as well as figure out if there are ways to look at that data to make the uncertainties smaller.)

While some may prefer other courses, I actually prefer to learn how to use the various statistical tools on the simplest possible method where I don’t expect to get a precise final answer. Then, later, I can apply the methods to the version of the model that I think more likely to give better answers. (Though, of course, Lumpy may be totally hopeless too.)

Update: I should never write these before coffee. I couldn’t sleep, wrote as I drank coffee and hit “publish” as my husband was hurrying me out the door to meet the inlaws for breakfast.
We went an hour earlier than usual because the inlaws golf outing was canceled due to rain. All through breakfast I thought “Dang! Comments will be full of people reminding me Model E is an ensemble average.” Indeed they were. I nserted about 10 words to reflect this.)

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