A new paper is out by Foster and Rahmstorf (2011), and while I may later do a more in-depth analysis, I want to point out a rather interesting implication of this paper, if indeed one were to take it at face value — it supports Spencer and Braswell (2008, 2010, and 2011 to some degree). Allow me to explain. (Note: to avoid confusion, there is Grant Foster, a.k.a. Tamino, and Piers Fo*r*ster, whose papers I reference below attempt to measure sensitivity from radiation fluxes).

As you may recall, I performed some sensitivity tests related to the multiple regression a while back . Looking that post over again, there are a few errors on my part (I believe I used actual surface T for the S-B/Planck response), but there are a few interesting tidbits: 1) leaving the adjustments for TSI/solar out affects the conclusions, and 2) the estimated solar response is around 4 times greater than the volcanic response.

Let’s take a closer look at #2, which may have changed a bit from the post to the paper. From figure #3 of the FR11 paper, we see the coefficient for TSI at around 0.1 C for the land data, which, after adjusting for planetary albedo and shadow area / surface area, results in around a **0.57 C/(W/m^2)** instantaneous surface temperature response for the actual solar forcing. Note that in Tamino’s original post, he had estimated about 0.39 C/(W/m^2) for solar, but that was when the solar influence range was only 0.08 C rather than the 0.12 C mentioned in the new paper.

*[Aside]*

For Aerosol Optical Depth (volcanic), the coefficient is around 2 deg.c / tau. If we look up the approximate efficacy, we see that it is around -25 W/m^2/tau. Such an estimate would yield the instantaneous sensitivity of around 0.08 C/(W/m^2), which would put it at around 1/7 the efficacy of a solar forcing, both in W/m^2. Certainly, there are reasons to believe that the instantaneous surface temperature response to the larger forcings may be damped (thank you SteveF) by the ocean heat uptake, but it seems that a factor of 7x (or 4 times) remains far too big of a discrepancy to be considered a reasonable physical result. Furthermore, the longer-term response may be expected to manifest itself over the course of say 8-12 years, but for the FR11 paper anything beyond the instantaneous response is ignored.

*[End Aside]*

Anyhow, according to FR11, the time between the solar forcing anomaly to the surface temperature response is estimated to be around 1 month. Remember that for later.

**Relation to Spencer and Braswell**

For more on the background of attempting to measure climate sensitivity and where the Spencer and Braswell arguments fit in, please see this page . But as a quick summary, I’ll note that in Forster and Gregory (2006), the authors comment (my insert in **bold**):

The X terms

[radiative noise or unknown radiative forcings]are likely to contaminate the result for short datasets, but provided the X terms are uncorrelated to Ts, the regression should give the correct value for Y, if the dataset is long enough.

Spencer and Braswell argue that the unknown radiative forcing (fluctuations in cloud cover, which we know to exist AT LEAST on short timescales, per Dessler (2010)) would necessarily influence the Ts and hence lead to an underestimate of the radiative response. The counter-argument has been two-fold:

The decorrelation time of this radiative noise is shorter than the surface temperature response time. From Murphy et al. (2009), we read:

If temperature variations are changing outgoing radiation then temperature should be the independent variable whereas if radiation variations are affecting temperature then temperature should be the dependent variable. Although both are true to some extent, they can be partially separated by time response: outgoing radiation changes are mostly immediate whereas surface temperatures lag radiative forcing. Autocorrelation analyses of global temperatures suggest that the surface ocean portion of the Earth’s climate response has a time constant of about 8–12 years [Scafetta, 2008; Schwartz, 2008].

I believe this response misses the mark, as you might very well expect significant surface temperature responses to forcings on much shorter time-scales, even if the full forcing response is not realized for several more years. A better argument might be that the decorrelation time of this noise used by SB is too long, and that for cloud fluctuations it is actually on the scale of 2-3 months, whereas the temperature response is (for example) about 5 months later. However,** the Fo rster and Rahmstorf (2011) paper implies a lag in temperature response of only around 1 month **for these smaller fluctuations, which, even with only intraseasonal fluctuations (such as the Madden–Julian oscillation) in cloud cover, would suggest a strong correlation between these unknown radiative fluctuations (X) and T_surface!

The second major argument against the Spencer and Braswell result, as advanced by Murphy and Forster (2010) and Dessler (2011), is that the effective heat capacity of the ocean on these timescales is too high for the unknown radiative forcing to have any significant effect on surface temperatures. They attribute almost all of the surface temperature fluctuations during this recent decade to internal, non-radiative forcings (e.g., heat exchange between the ocean layers). I have explained before why their estimates of heat capacity are inappropriate for these monthly timescales. Nonetheless, using a similar method to Dessler (2011), I’ll point out that the standard deviation in surface temperature anomaly from 2000-2010 is around 0.1 C. Dessler (2011) calculates the standard deviation of the cloud forcing/noise to be around 0.5 W/m^2. So, can this 0.5 W/m^2 cloud fluctuation force any significant amount of the 0.1 C surface temperature changes? According to Dessler (2011), the answer is a strong NO (~5%). But according to Foster and Rahmstorf (2011), with its **0.57 C/(W/m^2)** instantaneous response to solar forcing, such cloud forcing fluctuations (if the response scales) could result in 0.28 C changes! Now one may argue that the responses to slower solar cycles don’t experience the same damping, but even if the cloud forcing efficacy is only 1/5 that amount, this would imply that the measurements of climate sensitivity from radiative fluxes has been greatly overestimated.

Overall, I don’t think a proper analysis will support either the high Dessler (2011) heat capacity over these short period, or the high instantaneous effect of changes in TSI from FR11 that contradicts it. Indeed, I suspect the latter is likely an artifact of fitting to an underlying linear trend, as the effect of the solar minimum is overestimated in order to counter the flattening in the early 21st century. I think this point highlights the larger problems with such a methodology. Nonetheless, if one were to take the FR11 results at face value, Spencer and Braswell could very well point out that this peer-reviewed paper suggests a short lag time for the large surface temperature response (1 month) to a small forcing, lending credence to their argument that unknown radiative forcing “noise” will correlate with surface temperature. Heck, even using a T_s response midway between the FR11 values for TSI and AOT per W/m^2 would strongly support the SB case.

When I saw FR11’s 1 month lag in TSI, I was surprised. Given a 2+ month lag in mid-latitude SSTs for the annual cycle, and a ~4 month lag for the ~4 year ENSO cycle, I was expecting a lag of >4 months on the ~11 year TSI cycle. (BTW… both the seasonal and ENSO lags imply a much shorter “time constant” than 8 yrs in a simple one box model).

I hadn’t considered that different forcings result in different dampings… interesting.

Comment by AJ — December 13, 2011 @ 12:30 pm

Hi AJ,

Regarding the 1-box model and the short single time constant, I believe it is why many (see, for example, the Isaac Held post #3) have pointed out the issues with using such a simple model…it doesn’t take into account the ocean heat uptake that will affect temperatures later. This is why Tamino’s analysis is curious…either the instantaneous volcanic response (~ 0.08 C/W/m^2) represents the bulk of total response, in which case the climate sensitivity is exceedingly low, or the bulk of the response must be considered over the remainder of the period, which is completely missing from this method of analysis. IMO, there is not one set “lag” time, which is why both the instantaneous and later responses are underestimated for AOD.

However, I do believe the short time constant in the one-box models DOES imply that the largest response occurs soon after (even if there is more to come), which is why the Murphy et. al (2009) quote posted seems rather weak, along with the subsequent justification for pure OLS. If even 50% of the response comes within a few months, their 8-12 years comment does not seem to make much sense (unless the “immediate” response time is larger than the forcing decorrelation time, but that’s a different argument).

Comment by troyca — December 13, 2011 @ 7:58 pm

Hi Troy… I agree that a 1-box model does not represent the climate system’s equilibration of a radiative imbalance. Given a fast and slow reponse, however, I would naively expect that the longer the forcing cycle, the longer the apparent time constant should be.

I can only think of a few cyclical forcings: annual, ENSO, 11yr Solar, and glacial. The first two forcings I would consider internal and the last two external. This is where I need your help. Could you check my math?

The R script below is my attempt to calculate that apparent time constant for the annual, ENSO, and glacial cycles. “In defense of Milankovitch” by Gerard Roe, finds a zero lag between Milankovitch cycles and temperature. Given the lack of time resolution, however, I’ll estimate that the actual lag is between 2 and 250 years. Does a short lag, relative to the period, approximate the apparent time constant?

> # functions and calculations have reference of 1 year

> #

> # function to calculate rate given cycle period and lag

>

> ryr

> # function to calculate time constant given cycle period and lag

>

> tcyr

> # function to calculate lag given cycle period and time constant

>

> lgyr

> # calculate apparent time constant given mid-latitude temp lag of 65 days (guesstimate) to the annual forcing

>

> tcyr(1,65/365)

[1] 0.3279063

>

> # calculate apparent time constant given global temp lag of 4 months (guess) to 4 yr ENSO cycle (guess)

>

> tcyr(4,4/12)

[1] 0.3675526

>

> # calculate months lag needed to get an apparent time constant of .4 years on the 11 year solar cycle

>

> lgyr(11,.4)*12

[1] 4.718998

>

> # calculate apparent time constant given 6 month lag on 100,000 yr glacial cycle

>

> tcyr(100000,6/12)

[1] 0.5

>

> # calculate apparent time constant given 250 yr lag on 100,000 yr glacial cycle

>

> tcyr(100000,250)

[1] 250.0206

>

Comment by AJ — December 17, 2011 @ 7:43 pm

Crap… WordPress ate my R code!

Comment by AJ — December 17, 2011 @ 7:49 pm

Let’s try this again. Apparently WordPress interprets angle brackets as HTML operators, so I’ll replace them.

} # functions and calculations have reference of 1 year

} #

} # function to calculate rate given cycle period and lag

}

} ryr = function(peryr,lagyr){-2*pi*(1/peryr)/tan(2*pi*lagyr/peryr)}

}

} # function to calculate time constant given cycle period and lag

}

} tcyr = function(peryr,lagyr){-1/ryr(peryr,lagyr)}

}

} # function to calculate lag given cycle period and time constant

}

} lgyr = function(peryr,tc){atan(2*pi*tc/peryr)*peryr/(2*pi)}

}

} # calculate apparent time constant given mid-latitude temp lag of 65 days (guesstimate) to the annual forcing

}

} tcyr(1,65/365)

[1] 0.3279063

}

} # calculate apparent time constant given global temp lag of 4 months (guess) to 4 yr ENSO cycle (guess)

}

} tcyr(4,4/12)

[1] 0.3675526

}

} # calculate months lag needed to get an apparent time constant of .4 years on the 11 year solar cycle

}

} lgyr(11,.4)*12

[1] 4.718998

}

} # calculate apparent time constant given 6 month lag on 100,000 yr glacial cycle

}

} tcyr(100000,6/12)

[1] 0.5

}

} # calculate apparent time constant given 250 yr lag on 100,000 yr glacial cycle

}

} tcyr(100000,250)

[1] 250.0206

}

Comment by AJ — December 18, 2011 @ 6:04 am

AJ,

Sorry for the late response. To be honest, I’m not sure there is necessarily a time constant to be found. The lag we see until the “maximum response” likely emerges as a result of how/where the particular forcing affects the system, the strength of the forcing, and the heat capacity of the area affected…cyclical ones might exhibit a shorter lag time if there exists some sort of inertia in the system, but I wouldn’t bet that there is necessarily a time constant relating the period alone of a cycle to the lag time (except that perhaps longer periods may generally exhibit stronger forcings?). The place I would start is with a simple energy balance model: Cp *dT/dt = forcing – lambda * T, described in Spencer and Braswell (2011) for example, and you can experiment to see the expected lag times with different forcing signatures. Of course, the effective heat capacity for the surface temperature “layer”, Cp, will vary based on the frequency of the forcing, so I suppose it is still not a trivial task.

Comment by troyca — December 20, 2011 @ 9:03 am

Troy, I agree that there isn’t a time constant to be found either. I just expected that given a longer cycle, that the lag and resulting time constant would be longer. That’s why I found the FR11 paper (or at least their references) puzzling. I would have expected a 11yr cycle to have a lag of +4 months, while they show 0-1 months. Anyway, thanks for the response.

Comment by AJ — December 22, 2011 @ 2:54 pm

Interesting analysis, albeit one that won’t please Tamino very much! I think you mixed up your Forsters and Fosters in your last bolded statement.

Comment by matt — December 13, 2011 @ 5:22 pm

Thanks matt, fixed!

Comment by troyca — December 13, 2011 @ 7:37 pm

As usual with climate skepticism, we have to go beyond personal motivations and analyze the arguments in their own right.

Comment by Aiden Hardman z — December 31, 2011 @ 6:26 pm