# Can a small change in the Earth temperature give rise to large-scale climate changes?

Earth's atmosphere is a chaotic system. In such systems arbitrarily small changes the conditions can give rise to very large effects.

There are many rumors about the physical and large scale environmental impacts that global warming will cause in the future. (for example, here)

After a remark made by one of the users here, I thought to ask this question:

Can a small change in Earth's temperature (that can't be felt) give rise to large-scale climate changes?

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A small change in the average temperature of the Earth's entire oceans and atmosphere would be a very large change in the total heat content of the system. Or are you asking whether a small local temperature change would be likely to spread and have an effect on global climate. –  Mark Rovetta Jan 25 '14 at 19:58
Why did you choose to refer this as 'rumors'? Did you mean to express doubt about the proposition - or was that merely a choice of words? –  Mark Rovetta Jan 25 '14 at 20:07
@user215721: I did not make any such claims - all I objected to was the idea that causes must somehow be sufficiently large in an arbitrary human temperature scale to have significant consequences, and that's bullshit (see Luboš last comment - evolution just doesn't work that way) –  Christoph Jan 25 '14 at 20:19
@ColinMcFaul the question asks about the fact that complex systems that are often capable of showing chaotic behavior can be kicked out of an unstable steady state by in principle infinitesimally small perturbations. The climate system can be investigated by these methods too. So for people familiar with complex systems, there is nothing unclear about this question, and any close flags / votes are not justified. Voting to leave open. –  Dilaton Jan 26 '14 at 21:25
@ColinMcFaul: I think the fact that there are several answers on this post would suggest that the question is not unclear. –  Kyle Kanos Jan 26 '14 at 22:06

Let's look at a toy model, obeying the differential equation $$\ddot x(t) + \alpha\dot x(t) + x^3(t) = \beta\sin\omega t + Y(t)$$ where the dampening coefficient $\alpha$ on its own would make the system evolve towards an equilibrium state, the $x^3$ term corresponds to the non-linear system response and the right-hand side includes a periodic external driving term as well as an as of now undetermined term modelling human influence.

This is a variant of the Duffing equation, which is known to exhibit chaotic behaviour.

A common parametrization for the Ueda attractor would correspond to $$\alpha = 0.05\qquad \beta = 7.5\qquad \omega = 1\qquad Y = 0$$ which looks like this for initial conditions $x=0,\dot x=0$:

However, a more interesting parametrization for our purpose is $$\beta = 0.5\qquad \omega = 0.1$$ which looks like this:

Now we'll get some human action going, in particular $$Y(t) = 0.3\cdot(t-t_i) \;\text{for}\;t\in[t_i,t_i+\Delta t]$$

Let's choose $t_i=75$ so we're right in the middle of a 'warm' period.

For $\Delta t = 1$, nothing much happens - we started in a downwards branch and basically just decrease the amplitude a bit:

For $\Delta t=2$, we're starting to see some warming and corresponding backlash:

This is where it gets interesting, as for $\Delta t=3$ we end up with this:

That's the point I was trying to make.

And yes, I obviously needed to fiddle with the parameters to end up with this particularly illustrative result. And no, this is not due to the fact the the human forcing goes from 'small' to 'large' after a sufficient amount of time has passed:

If we go on for a bit longer to $\Delta t=4$, this is what we'll see:

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Luboš' answer here is a good one to the question as posed, but as the question is in response to something I said, I think I should explain a bit at what I was actually getting at.

Microclimate (also known as weather) is often taken as a prime example for chaotic systems and that in turn as the reason behind our inability to accurately make forecasts beyond a certain point in time.

However, looking at temperature records on the scale of 100,000s of years (let's call this macroclimate), we do not see chaos, but a sequence of cold and warm intervals with a period of about 100k years despite a lot of seemingly random noise (spectral analysis of ice core data confirms significant spikes at periods 111kyr, 41kyr, 23kyr).

At the scale of 10.000 years, we're still near the peak of a warm period following the end of the last glacial one. Our best estimate on global temperature shows at an (again, highly scientific) eyeball analysis a periodic overlay over a slight downward trend.

Now, Luboš' claim that triggered my comment was that the human-induced global warming (according to the Berkeley Earth analysis of land-surface temperature data, $1.5K$ over the past 250 years, of which about $0.9K$ happened during the last 50 years) is too small to affect anything.

But as far as I can see, that's not really the case. As I described in the second to last paragraph, we've been in a sort of meta-stable state, and on that scale, the temperature increase is significant:

It disrupts the 'natural' cycle that emerged from the various positive and negative feedback mechanism (ice albedo, water vapour, cloud cover, ocean dynamics, ...) and external forcing factors (solar activity, planetary orbit).

Going by heuristics alone and not detailed modelling (last I looked at global warming with more than a cursory glance was in 2007 or so), I do not find it too far fetched that the recent introduction of a new forcing mechanism ($CO_2$ emission) of the observed magnitude could move a rather complex and messy dynamical system with peridoc forcing, noise, chaotic subsystems away from it's current meta-stable state.

In fact, such things are even possible for actual chaotic systems. Take the Lorenz system: If you're lucky (as humans possibly have been for the last 10kyr - which is, perhaps not incidentally, also the timescale on which human civilization really got going), you can stay orbiting around a given fixed point for a significant amount of time - but a seemingly slight perturbation will prematurely force you into an effectively random (as far as predictability goes) domain.

I can imagine (temporarily) runaway solutions in either direction due to passing some critical point after an inconsequential temperature change, but as I'm a non-expert, that's somewhat idle speculation, and was just one of the three claims I made in that particular comment: The more moderate claim was about the accumulated effect of inconsequential changes on the ecosystem after a sufficiently long timespan under the assumption non-chaotic behaviour.

Feel free to correct any gross mistakes in my heuristic arguments.

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Dear @Christoph. Thanks but: in your 3rd paragraph, you talk about periodic functions that have random minima or maxima. But such functions can't exist. Periodic functions have completely predictable maxima and minima. Periodic and randomly fluctuating (with some dominant frequencies) functions are important to be distinguished, especially in a question that asks about chaos and therefore (un)predictability! –  Luboš Motl Jan 25 '14 at 18:05
Your "runaway solution" that is claimed to be a consequence of the initial small perturbation of the initial state is impossible in a chaotic system; that's what the term "chaotic system" means. - Otherwise this question was about general concepts in physics, thermodynamics, and chaos theory. Being a "climate scientist" isn't the appropriate expertise to answer such questions. ... I've studied the climate for 15 years now but I don't claim that any of the things I learned in that context are helpful to settle questions involving chaos theory. –  Luboš Motl Jan 25 '14 at 18:07
@LubošMotl: re (1) temperature records extracted from ice cores show spectral peaks at $111kyr$, $41kyr$, $23kyr$; the former is the most pronounced and is the one you can eyeball without doing spectral analysis; and of course you can have random fluctuations on top of a periodic signal - we call that noise –  Christoph Jan 25 '14 at 19:07
re (2) the runaway solution is not due to chaos, but positive feedback and initially triggered by human $CO_2$ emission; we're dealing with a complex dynamical system with peridoc forcing, noise, chaotic subsystems, recent $CO_2$ forcing due to human activity; that system appears to be in a meta-stable phase; even in a fully chaotic system we can have intervals of apparent stability (eg strange attractors), and external forcing can end such periods prematurely –  Christoph Jan 25 '14 at 19:16
@LubošMotl: I incorporated my comments into the answer and clarified the argument –  Christoph Jan 25 '14 at 21:13

First we have to consider what is meant by the word temperature. If we look towards classical thermodynamics, temperature is an intensive property of a system. Which means it is independent of the amount of material or system size. It is defined as the partial derivative of the internal energy of a system with respect to the entropy of the system.

$$T=\dfrac{\partial U}{\partial S}$$

Which means simply that temperature is the slope, or gradient, of how internal energy changes when entropy changes. Internal energy is the total energy of a thermodynamic system, which includes both its kinetic and potential terms. Entropy is a measure of the disorder of the system. We can also understand entropy as a measure of how close a system is to a uniform distribution of energy across its accessible microstates (e.g. whether energy is distributed equally across all potential configurations of the underlying particles of the system).

If a system is at a constant temperature, this means that any change in entropy must cause a change in the internal energy of the system, and vice versa; a change in internal energy must accompany a change in entropy.

The first problem with the question is that one has to abuse thermodynamics slightly to claim that there is a global temperature for Earth by first assuming there is a well defined boundary for the Earth's climate system, and secondly assuming one can reliably average across all the cells of the system. Assuming this can be done one can possibly imagine such a thing as a global temperature.

The second problem is one of identifying what is meant by "chaotic" in the system. In general, it is defined as how a small change in initial conditions for a dynamically evolving system can lead to vastly different trajectories for the underlying particles.

Well this sort of problem is kind of why statistics was brought into discussions of thermodynamics in the first place. If one considers all possible initial conditions and all possible final conditions of the particles in a system, then one defines all potential microstates of the system as it evolves in time. A high entropy system is one where all the potential microstates have equal chance of being the actual microstate of the system. A low entropy system is one where some microstates are more likely then others. We simply find it too difficult to try to track all of the trajectories of all the particles of the system to determine what its actual state is.

Chaos is best described by the use of Lyapunov exponents, $\lambda$ which control the spreading of two trajectories in phase space.

$$| \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 | \,$$

There is a spectrum (e.g. more than one) of Lyapunov exponents dependent on the number of dimensions of the phase space, which itself is dependent on the number of particles in the system. The largest Lyapunov exponent (Maximal Lyapunov Exponent, or MLE) in the spectrum determines its predictability. If the MLE is positive (indicating a divergence in the separation of two trajectories over time), then the system is generally considered chaotic.

The Lyapunov spectrum gives an estimate of the rate of entropy production of the system, and basically also tells how fast we lose predictability of the system. The inverse of the MLE is the Lyapunov time which is the measure of time it takes to lose predictability. We generally speak of e-folding time, but there also exist 2-folding time and 10-folding time which are associated with losing a bit of information (knowledge) or a digit of precision respectively.

So a small change in temperature is effectively saying that there was a change in the slope associated with the rate internal energy changes with entropy. It has been argued in numerous outlets that extra energy is being released into the environment which is increasing the temperature of the Earth (or alternatively, energy is being trapped). This means that the rate of entropy change with respect to the amount of energy in the atmosphere has decreased since the internal energy and entropy are inversely related (assuming a positive slope). This in turn implies that the Lyapunov time has also increased implying the system is less chaotic and more predictable (e.g. less information about the system is lost within a given period of time).

I'd be happy to revise if there is a flaw identified.

Addendum: As far as the question of whether small changes temperature makes a difference, I would argue that it will depend on how close the system is to chaotic flow conditions. If the system is already "chaotic" then a small change in temperature would only have an major consequences if it was near some critical point where the system may potentially be completely predictable; the converse is also true. The example one has of this is in connection with fluid dynamics and the importance of the Reynolds Number as measure of which is the ratio of the contribution of inertial and viscous forces. In many cases it is possible to determine when a system exhibits turbulent or chaotic flow by using the Reynold's Number. If a system is near such a critical point, a small change in the energy of the system can lead to major changes. The argument that is frequently offered in climate discussion is that we are near some critical point, or tipping point, and some small change will cause us to suddenly shift to some new chaotic regime. The situation with the North Atlantic conveyor is similar but is tied to changes in salinity. The problem in these discussions with the critical point is that they are often very hard to derive.

So the answer to the question is yes, small changes in temperature can lead to major changes if one assumes one is close to some critical point whose value is dependent on temperature that would lead to a major phase change for the system. However, such a phase change does not necessarily imply the system is more or less chaotic even if the change in phase is viewed psychologically as being "chaotic".

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If we accept that the system (the Earth's atmosphere in this case) is chaotic and adopt the usual definitions of a chaotic system, e.g. one by Edward Lorenz

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.