Tuesday, July 17, 2012

Ray, Bart and Eli

Back there on the Andy Lacis provides a good answer thread Ray Pierrehumbert said

raypierre said...
A slightly more mathematical way of putting it is that in the absence of trends in forcing, the temperature is not brownian motion (random walk) but something more like an AR(1) process, with a tendency for the state to relax back to an equilibrium with a certain time constant. What's more, that time constant is proportional to the climate sensitivity, so if somebody is arguing for large natural fluctuations, they are simultaneously arguing for high climate sensitivity. It's all a consequence of the key role of top-of-atmosphere energy balance.

Of course, the linear AR(1) process is not a good quantitative fit to the real climate system, since one has multiple time scales of ocean heat uptake, plus various nonlinearities. But it does serve to connect the natural variability issue with the climate sensitivity issue
which motivated Bart to ask
You wrote that, after an unforced change in climate state, there is

"a tendency for the state to relax back to an equilibrium with a certain time constant. What's more, that time constant is proportional to the climate sensitivity"

Intuitively it makes sense that if a random change can push the system far away from its equilibrium, it must mean the system is very sensitive to any changes in state (whether forced or unforced), but I don't quite grasp the explanation you gave that the timescale of equilibration is proportional to climate sensitivity. Could you elaborate?

 Now Eli might venture out on the thin ice (have the bunnies noticed the Arctic Ice Disappearance Act?) and note that at least for the fast feedbacks the system is tightly coupled, so that with the exception of solar each forcing couples back and becomes a feedback only, as Jay Zimmerman points out, being limited by the Stefan Boltzman law, therefore, QED the larger any single feedback, the faster the system returns to equilibrium and the larger the overshoot, because it ain't gonna be a soft landing.

PS:  On local and global scales the driving negative feedbacks are IEHO convection and radiation.

Also some fine poetry there abouts

Anonymous Anonymous said...
Random Walks (part II)
-- by Horatio Algeranon

ARy traipses
On airy mesas
Will put you on your butte

But random spills
Down random hills
Will ruin your new suit
12/7/12 12:13 PM
Anonymous raypierre said...
With AR(1) I had just begun
With AR(2) it was still brand new
I use AR(3) for rings of a tree
And AR(4) for much, much more
Then with AR(5) I was really alive
But that AR(6), it's just clever as clever
I think I'll use AR(6) forever and ever
12/7/12 5:54 PM


Tom Curtis said...

Ray Pierre said that the time constant constant is proportional to the climate sensitivity. In contrast, you have said that the larger the feedback the more rapidly the system returns to equilibrium, but that means a smaller time constant. So your explanation contradicts Ray Pierre's claim.

Instead, I would suggest that feedbacks are feedbacks on changes in temperature rather than on the forcings themselves. Ergo, if a random fluctuation in temperature warms the Earth, that will generate a feedback which will further warm the Earth, and so on. With an initial fluctuation of 0.1 C and a climate sensitivity equivalent to 3 degrees C per doubling of CO2, we can expect further increases in temperature up to 0.3 C before the feedbacks self damp. The temperature will tend to return to equilibrium because the initial temperature increase was unforced and hence represents a negative energy balance.

Given this, we can consider two cases. One in which there is an initial perturbation without any following perturbations. In this case, because temperature increases due to the feedback do not result in a net energy imbalance, the large feedback results in a small energy imbalance for a large perturbation in temperature, and hence a longer time to relax back to equilibrium.

In the second, more realistic case there are ongoing random fluctuations. In this case, however, the larger the feedback, the less likely is a new random fluctuation to be larger than the feedback response to an earlier fluctuation, and hence the less likely that a new perturbation will reverse the direction of temperature response resulting form the feedback.

I think these two features combined will result in the connection between climate sensitivity and a longer relaxation time.

Anonymous said...

While I think that the system will usually return to equilibrium, I do think that given enough time there might be an excursion sufficient to actually drive the system into a new stable state. Eg, if we froze orbital parameters in today's state, and assumed preindustrial GHG levels, I wouldn't be surprised if given enough time (millions of years) a random cooling fluctuation was sufficient to trigger a new glacial maximum. This requires being fairly close to such a tipping point... the further the tipping point, the longer it would take for a random excursion to reach it.


Arthur said...

I've always thought this (or something close to this) was a natural consequence of the Fluctuation-dissipation theorm. Linear response of a physical system is very generally correlated with the scale of un-forced variations.

Lloyd Flack said...

Tamino has argued that the temperatures are an ARMA(1,1) process rather than AR(1). I would expect such a process to be more difficult to move away from equilibrium but return to it more slowly than AR(1).

raypierre said...

Tom Curtis -- you have to be careful there about the sign convention for the feedback. The restorative feedback for climate is that the warmer it gets, the greater the rate of radiation to space. Linearize and call the proportionality between temperature deviation and radiation to space A. With fixed greenhouse substance composition, that's the Planck feedback (about 4*5.67e-8*(255K)^3 W/m*2/K). The larger A is, the more the restorative forcing, and the faster the return to equilibrium. The climate sensitivity is proportional to 1/A. Destabilizing feedbacks, like water vapor feedback, make A smaller, and climate sensitivity larger. So, the higher the climate sensitivity, the greater the tendency to variability as well. This has certain affinities with the fluctuation-dissipation theorem.

So the upshot is that anybody arguing that climate is like Brownian Motion is also arguing for perilously high climate sensitivity.

raypierre said...

Actually, I had overlooked Bart's question. The equation to consider is:

mu*dT/dt = -A*T + dF + e(t)

where e(t) is white noise, dF is radiative forcing, mu is the heat storage per Kelvin (thermal inertia), and A is the sensitivity coefficient. The relaxation time is mu/A. The temperature change due to dF is dT is dF/A. Klar, eller hur? Thermal inertia is discussed in Chapter 7 of Principles etc., though in the next edition I think I'll make the connection to climate sensitivity more explicit.

I think all this was first written down by Hansen (see the 1980's paper in The Warming Papers), but it's a simple idea and probably was known earlier.

Dallas said...

mu*dT/dt = -A*T + dF + e(t)

But e(t) may be useful noise. With fixed greenhouse substance composition is a fairly large assumption.

mu1*dT1/dt+mu2*dT2/dt+mun*dTn/dt=-A*T + dF or is it unreasonable to assume that different ocean layers would have different thermal inertia?

raypierre said...

The assumption of "fixed greenhouse gas composition" just applies to what you hold fixed when computing A. You can still have a specified increase in GHG in the forcing term dF. When doing feedbacks, as in water vapor feedback, there is a part of dF that is linked to temperature: dF = dF0 + r*T, where T is the temperature perturbation. Then, you can lump r in with A, and get a modified a. If r is negative, then this reduces A and makes the climate more sensitive.

The one-layer thermal inertia model (i.e. single T) suffices to explain the connection between climate sensitivity and fluctuations. The multiple time scales of the deep ocean are of course important in the real temporal behavior. I've found you can capture most of the essence of the problem with a two-layer ocean, though.

Anybody still out there?

David B. Benson said...

Yes, a two layer ocean makes for a convenient 3 box model which well matches the instrumental record.

Dallas said...

"I've found you can capture most of the essence of the problem with a two-layer ocean, though."

Yes, two layers is better, two asymmetrical layers even better, due to land/ocean ratio, solar insolation differences and the range of sea ice variability possible. You should have two different time constants which would likely produce the internal pseudo-cyclic oscillations.

It would be impossible to isolate all the noise sources, but it looks like at least two time constants with asymmetrical charge discharge rates may be possible.

Dallas said...


Here is a little non-standard view of the noise. The plots are 60 month linear regressions of each month of the time series. Using the tropical oceans as "standard" you can see the synchronization of all of the series forming at the end of the plot. Some of that "noise" may be useful.

Anonymous said...

Hi Ray Pierre,

"mu is the heat storage per Kelvin (thermal inertia)"

Thermal Inertia is I think a poor metaphor for Thermal Capacitance (Heat Capacity), insofar as it encourages an extension to some notion of Thermal Momentum.

I know it is common usage but to any audience that might be more familiar with mechanics than thermodynamics it might lead to some poor analogies. Some sense that once started a heat flux should keep going with the possibility of overshoot and resonant behaviour by itself, e.g. without bulk flows.

Thermal diffusion and conduction are commonly modelled as flows with zero momentum/inertia, whereby at all interior points the flux is dependent only on the current local temperature gradient. (I beleive that in the limit the inertia of the quanta involved break this symmetry but the effect is very small but does save us from instantaneous action at a distance implied.)

I don't imagine that we will ever cease to talk about thermal inertia, or thermal mass for that matter, but where it is used, particularly in textbooks, it might be beneficial to indicate that is a metaphor and not analogous to mechanical inertia.


Dallas said...

Alex, "I don't imagine that we will ever cease to talk about thermal inertia, or thermal mass for that matter, but where it is used, particularly in textbooks, it might be beneficial to indicate that is a metaphor and not analogous to mechanical inertia."

It is a useful metaphor in a complex system though. As a convection process decreases to a minimum temperature differential, viscosity and conduction become more significant and can produce the overshoot. Bridging the gap between a viscous/conductive dominated process to a radiant dominated process, thermal inertia and thermal momentum appear to be good descriptions of the major issues.