As someone intrigued by chaos theory, and in view of the fact that world-wide climate system are both chaotic and poorly understood, this piece resonated loudly for me:
No new strange attractors: strong evidence against both positive feedback and catastrophe
Posted on January 9, 2012 by Anthony Watts
This is a comment by Dr. Robert Brown on the What we don’t know about Earth’s energy flow post. I thought it was so insightful on the topic of climate stability being “pushed” by CO2 forcing that I’ve elevated it to a separate post. – Anthony
Is it fair to say that the two systems would oscillate within the same parameters but the probability of them being synchronized is nil?
Sadly, no, not over long times. The systems could be as different as a ferromagnet magnetized up and an “identical” ferromagnet magnetized down. Or in the case of the Earth, as different as Glacial Earth and Interglacial Earth. The point is that both of these latter possibilities can be “stable” states for exactly the same insolation, etc, because feedbacks in the global system can themselves reconfigure to make them stable.
If you look at the link to chaos theory I provided, and look at the figure that shows two loopy braids of lines, that provides an heuristic picture of the kind of possibilities available to coupled nonlinear differential systems.
A plot of the Lorenz attractor for values r = 28, σ = 10, b = 8/3 Image via Wikipedia
At the heart of each loop is something called a “strange attractor”, which is typically a limit point. The x and y axes are coordinates in a generalized (phase) space that represent the state of the system at any given time, x(t),y(t). The lines themselves are the trajectory of the system over time moving under the influence of the underlying dynamics. The point of the figure is that instead of their being a single “orbit” the way the earth orbits a regular attractor like the sun, the system oscillates around one attractor for a time, then the other, then both. Instead of nice closed orbits the orbits themselves are almost never the same.
Two trajectories that are started close to one another will usually start out, for a while, orbiting the attractors the same general way. But over time — often a remarkably short time — the two trajectories will diverge. One will flip over to the other attractor and the other won’t. After a remarkably short time, the two trajectories are almost completely decorrelated in that the knowledge of where one lies (in the general accessible phase space) provides one with no help at all in guessing the location of the other.
It’s only in this final sense that you are correct. Either system has to be found in the space of physically consistent states, states that are accessible via the differential process from the starting points. There is no guarantee that the trajectories will “fill phase space”. So in this sense they are both going to be found within the phase space accessible from the starting points. If those two starting points are close enough, they will probably sample very similar phase spaces, but there is no guarantee that they will be identical — especially if there are (many) more than two attractors, and if some simple parameter. In stat mech, with different assumptions, there is a theorem to that regard, but in the general case of open system dynamics in a chaotic system, IFAIK no.
If you are interested in this sort of thing (which can be fun to play with, actually) you can look up things like the “predator-prey differential equations”, e.g.
http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equationIIRC this is one of the simplest systems exhibiting an attractor and limit cycle, and illustrates many of the features of more complicated dynamical systems. The attractor/fixed point in this case is the population of e.g. foxes and rabbits that remains in perfect equilibrium from year to year. Note well that this equation is deterministic, but of course a real population — even being modelled — always has random (or at least, “unpredictable”) variations — a certain amount of noise — and is actually discretized and not continuous as one cannot have half a cheetah eating \pi baboons.
A better continuous “kind” of differential equation for describing systems like this with noise is something called a Langevin equation in physics — a system with “fast” microscopic degrees of freedom that one accounts for on average with a stochastic term, and slower degrees of freedom one integrates out like the predator prey equation. In physics it is a special limiting case of something called a generalized Master equation, which is the full integrodifferential description of a many body open quantum system and is really, really difficult. The general approach, however, is not inapplicable here — and is a presumed part of most of the simplified climate models. When you “smooth” the temperature by e.g. doing a running average, you are giving up information (the short time variation) and trying to reduce the complexity of the system by focussing on the slower time scale dynamics.
If the system really is simple — has a single attractor and is in a very regular oscillation around it where the “noise” one is smoothing out really is irrelevant and just adds small variation to a single trajectory — this is probably OK. If the system is multistable and has many locally stable points, or worse if some of the degrees of freedom are things like the Sun whose time evolution is completely outside of “the system” and whose future you cannot predict and whose effect you do not precisely know, so that the attractors themselves can be moving around as the system evolves locally — it is probably not OK.
The symptom of the latter kind of multistable system where it is probably not OK is a series of punctuated equilibria, visible in the smoothed data. The 30 year satellite data and SST data fairly clearly shows this kind of behavior.
One final very important point — systems that oscillate almost always have negative feedback. In fact, that is the fundamental thing that defines an oscillatory system — it has attractors in it. Attractors are themselves stable (equilibrium) points such that if the system is perturbed from them it is pulled back towards equilibrium, not pushed away from it. In the general case of attractors in high dimensional spaces, this leads to the (Poincare) cycles around the attractors visible in the predator-prey equations or the Chaos figure with two strange attractors, except that they can get very, very complicated (and difficult to visualize) in 3+ dimensional spaces (where I’m not talking about physical spaces, note well, but parametric “phase” spaces, state spaces). Within some neighborhood of an attractor there is generally a fair bit of local stability — trajectories in that neighborhood will oscillate tightly around the one attractor and will be relatively unlikely to switch over to other attractors. Hence glacial and interglacial periods tend to last a fairly long time (compared to all of the many shorter timescales available to the system.
Moving a single underlying external parameter — e.g. anthropogenic CO_2 concentration, Solar state, geomagnetic state — can be thought of as moving the fixed points of the multistable system. If we linearize, we can often guess at least the direction of the first order direction of the movement. For example, more CO_2, given the greenhouse effect, should increase heat trapping, hence increase average global temperature. The stable fixed point should thus move a bit up in the warming direction.
Nearly all of the argument “revolves” (in more ways than one:-) around two simple problems, and note that I’m presenting them in a very different way than usual:
a) Is this linear response assumption valid? This is not a trivial question. Increased CO_2 in a multistable system doesn’t just move the local attractor, it moves all the attractors, and not necessarily in simple linear ways in a really complicated system with many negative feedbacks (there by hypothesis all over the place because the system is dominated by attractors). In many systems, there are conservation principles at work (not necessarily known ones) that act as constraints so that moving one attractor up moves another one down or increases the “barrier height” between two attractors and hence deforms all of the limit cycles.
b) Is the response the order of the mean difference between attractors being predominantly sampled within the system already? If it is greater, then it is likely not just to move the current attractor but to kick the system over to a new attractor. And it may not be the attractor you expect, one on the warmer side of the previous one. More warming, as warmists state in more heuristic terms, can make the system oscillate more wildly and hence be both warmer at the warmest part of the oscillation and colder at the coldest part of the oscillation. If the new excursion of the oscillation is great enough, it can kick the system into oscillation around a new attractor altogether on either side of things.
Note that this latter statement is still oversimplified as it makes it sound like there are only two directions, warmer and cooler. But that is not true. There is warmer with morewater vapor in the atmosphere, warmer with less water vapor in the atmosphere, warmer with the sun active, warmer with the sun not active, warmer with sea ice increasing, warmer with sea ice decreasing, warmer with more clouds, warmer with less clouds, and the clouds in question can be day side or night side clouds, arctic or antarctic clouds, in the summer, fall, winter or spring, really month by month if not day by day, with feedbacks everywhere — tweaking any single aspect of this cycle affects all of the rest, and I haven’t even begun to list all of the important dimensions or note that there are really important time scales with nearly periodic oscillation of many of these drivers, or noted that the underlying dynamics takes place on a spinning globe that generates airflow vortices as standard operating procedure that have lifetimes ranging from days to decades.
I have argued in posts above that the punctuated quasi-equilibrium evident in the climate record makes it very likely that the answer to b) is yes. The anthropogenic CO_2 shifts the system by order of or more than the distance between attractors, simply because the system jumped around between attractors even during time periods when there was no anthropogenic CO_2. Furthermore, the excursion of the system as it wandered among the attractors was as great as it is today, and not qualitatively different.
This strongly suggests that while the the linear response assumption made in a) may be valid (per attractor) — or may not, but it will be a huge problem to prove it — the effect is less than the natural excursion, not greater than the natural excursion, and the negative feedback factors that make the multistable attractors (locally) attractive also act as negative feedback on the CO_2 induced shift!
The latter is the fluctuation-dissipation theorem, as I already noted in one thread or another (two tired of writing to go see if it was this one). In an open system in a locally stable phase, the oscillations (fluctuations) couple to the dissipation so that more fluctuation makes more dissipation — negative feedback. If this is not true, the locally stable phase is not stable.
This is a strong argument against catastrophe! The point is that given that CO_2 is making only small, slow, local shifts of the attractors compared to the large shifts of the system between the attractors, if there was a point where the system was likely to fall over to a much warmer stable point — the “catastrophe” threatened by the warmists — it almost certainly would have already done it, as the phase oscillations over the last ten thousand years have on numerous occasions made it as warm as it is right now.
The fact that this has not happened is actually enormously strong evidence against both positive feedback and catastrophe. Yes, anthropogenic CO_2 may have shifted all the attractor temperatures a bit higher, it may have made small rearrangements of the attractors, but there is no evidence that suggests that it is probably going to suddenly create at new attractor far outside of the normal range of variation already visible in the climate record. Is it impossible? Of course not. But it is not probable.
I’ll close with an analogy. When physicists were getting ready to test the first nuclear bomb, there was some concern expressed by the less gifted physicists present that in doing so they might “ignite the Earth’s atmosphere” or somehow turn the Earth into a Sun (note that this was before there was any understanding of fusion — the sun’s energy cycle was still not understood). I’ve read (far more recently) some concern that collisions at the LHC could have the same effect — create a mini-black hole or the like that swallows the Earth.
Both of these are silly fears (although offered up, note well, by real scientists, because they could see that these outcomes were possible, at least in principle) and here’s why.
The temperature and pressure created by the nuclear bomb is not unique! Although it is rare, asteroids fall to the earth, and when they do they create pressures and temperatures much higher than those produced by nuclear bombs. A very modest sized asteroid can release more energy in a few milliseconds than tens of thousands of times the total explosive energy of all of the man-made explosives, including nuclear bombs, on Earth! In a nutshell, if it could happen (with any reasonable probability), it already would have happened.
Ditto the fears associated with the LHC, or other “super” colliders. Sure, it generates collisions on the order of electron-teravolts, but this sort of energy in nuclear collisions is not unique! The Earth is constantly being bombarded by high energy particles given off by extremely energetic events like supernovae that happened long ago and far away. The energies of these cosmic rays are vastly greater than anything we will ever be able to produce in the laboratory until the laboratory in question contains a supernova. The most energetic cosmic ray ever observed (so far) was a (presumably) proton with the kinetic energy of a fastball-pitched baseball, a baseball travelling at some 150 kilometers per hour. Since we’ve seen one of these in a few decades of looking, we have to assume that they happen all the time — literally every second a cosmic ray of this sort of energy is hitting the Earth (BIG target) somewhere. If such a collision could create a black hole that destroyed planets with any significant probability, we would have been toast long, long ago.
Hence it is silly to fear the LHC or nuclear ignition. If either were probable, we wouldn’t be here to build an LHC or nuclear bomb.
It is not quite that silly to fear CAGW. The truth is that we haven’t been around long enough to know enough about the climate system to be able to tell what sorts of feedbacks and factors structure the multistable climate attractors, so one can create a number of doomsday scenarios — warming to a critical point that releases massive amounts of methane that heats things suddenly so that the ocean degasses all of its CO_2 and the ice caps melt and the oceans boil and suddenly there we are, Venus Earth with a mean temperature outside of 200 C. If we can imagine it and write it down, it must be possible, right? Science fiction novels galore explore just that sort of thing. Or movies proposing the opposite — the appearance of attractors that somehow instantly freeze the entire planet and bring about an ice age. Hey! It could happen!
But is it probable?
Here is where the argument above provides us with a great deal of comfort. There is little in the climate record to suggest the existence of another major stable state, another major attractor, well above the current warm phase attractor. Quite the opposite — the record over the last few tens of millions of years suggest that we are in the middle of a prolonged cooling phase of the planet, of the sort that has happened repeatedly over geological time, such that we are in the warm phase major attractor, and that there is literally nothing out there above it to go to. If there were, we would have gone there, instead, as local variations and oscillation around the many> minor warm phase attractors has repeatedly sampled conditions that would have been likely to cause a transition to occur if one was at all likely. At the very least, there would be a trace of it in the thermal record of the last million years or thereabouts, and there isn’t. We’re in one of the longest, warmest interglacials of the last five, although not at the warmest point of the current interglacial (the Holocene). If there were a still warmer attractor out there, the warmest point of the Holocene would have been likely to find it.
Since it manifestly did not, that suggests that the overall feedbacks are safely negative and all of the “catastrophe” hypotheses but one are relatively unlikely.
The one that should be worrisome? Catastrophic Global Cooling. We know that there is a cold phase major attractor some 5-10C cooler than current temperatures. Human civilization arose in the Holocene, and we have not yet advanced to where it can survive a cold phase transition back to glacial conditions, not without the death of 5 billion people and probable near-collapse of civilization. We know that this transition not only can occur, but will occur. We do not know when, why, or how to estimate its general probability. We do know that the LIA — a mere 400-500 years ago — was the coolest period in the entire Holocene post the Younger Dryas excursion; in general the Holocene appears to be cooling from its warmest period, and the twentieth century was a Grand Solar Maximum, the most active sun in 11,000 years, a maximum that is now clearly past.
IMO we are far more likely to be hanging out over an instability in which a complete transition to cold phase becomes uncomfortably likely than we are to be near a transition to a superwarm phase that there is no evidence of in the climate record. The probability is higher for two reasons. One is that unlike the superwarm phase, we know that the cold phase actually exists, and is a lot more stable than the warm phase. The “size” of the quasistable Poincare cycle oscillations around the cold phase major attractor is much larger than that around the warm phase attractors, and brief periods of warming often get squashed before turning into actual interglacials — that’s how stable they are.
The other is that we spend 90% of the time in glacial phase, only 10% in interglacial, and the Holocene is already one of the longer interglacials! There is dynamics on long timescales that we do not understand at work here. We have only the foggiest idea of what causes the (essentially chaotic) transition from warm phase to cold phase or vice versa — very crude ideas involving combinations of Milankovich cycles, the tipping of the ecliptic, the precession of the poles, orbital resonances, and stuff like that, but there is clearly a strong feedback within the climate cycle that enables cold phase “tipping”, probably related to albedo.
It could be something as simple as a quiet sun; the LIA-Maunder minimum suggests that we should actively fear a quiet sun, because something in the nonlinear differential system seems to favor colder attractors (still in the warm phase major attractor) during Maunder-type minima. One has to imagine that conversion to glaciation phase is more likely at the bottom of e.g. the LIA than at any other time, and the Holocene is probably living on borrowed time at this point, where a prolonged LIA-like interval could tip it over.
To be honest, even a LIA would be a disaster far greater than most of the warmist catastrophic imaginings. The population of the world is enormous compared to what it was in the last ice age, and a huge fraction of it lives and grows food on temperate zone land. Early frost and late spring could both reduce the available land and halve the number of crops grown on the land that survives, even before full blown glaciation. Cold (warm) phases are often associated with temperature/tropic droughts, as well, at least in parts of the world. IMO, the “rapid” onset of a LIA could kill a billion people as crops in Siberia and China and Canada and the northern US fail, and could easily destabilize the world’s tenuous political situation to where global war again becomes likely to add to our woes.
We may ultimately discover that AGW was our salvation — the CO_2 released by our jump to civilization may ameliorate or postpone the next LIA, it may block cold-phase excursion that could begin the next REAL ice age for decades or even a century. In the meantime, perhaps we can get our act together and figure out how to live together in a civilized world, not a few civilized countries where people are well off and all the rest where they are poor and more or less enslaved by a handful of tyrants or religious oligarchs.
Note well, this latter bit is itself “speculative fiction” — I don’t fully understand climate cycles either (it’s a hard problem). But at least there I can provide evidence for a lurking catastrophe in the actual climate record, so it is a lot less “fiction” than CAGW.
http://wattsupwiththat.com/2012/01/09/strange-new-attractors-strong-evidence-against-both-positive-feedback-and-catastrophe/
