This week I’ll interview another physicist who successfully made the transition from gravity to climate science: Tim Palmer.
JB: I hear you are starting to build a climate science research group at Oxford. What led you to this point? What are your goals?
TP: I started my research career at Oxford University, doing a PhD in general relativity theory under the cosmologist Dennis Sciama (himself a student of Paul Dirac). Then I switched gear and have spent most of my career working on the dynamics and predictability of weather and climate, mostly working in national and international meteorological and climatological institutes. Now I’m back in Oxford as a Royal Society Research Professor in climate physics. Oxford has a lot of climate-related activities going on, both in basic science and in impact and policy issues. I want to develop activities in climate physics. Oxford has wonderful Physics and Mathematics Departments and I am keen to try to exploit human resources from these areas where possible.
The general area which interests me is in the area of uncertainty in climate prediction; finding ways to estimate uncertainty reliably and, of course, to reduce uncertainty. Over the years I have helped develop new techniques to predict uncertainty in weather forecasts. Because climate is a nonlinear system, the growth of initial uncertainty is flow dependent. Some days when the system is in a relatively stable part of state space, accurate weather predictions can be made a week or more ahead of time. In other more unstable situations, predictability is limited to a couple of days. Ensemble weather forecast techniques help estimate such flow dependent predictability, and this has enormous practical relevance.
How to estimate uncertainty in climate predictions is much more tricky than for weather prediction. There is, of course, the human element: how much we reduce greenhouse gas emissions will impact on future climate. But leaving this aside, there is the difficult issue of how to estimate the accuracy of the underlying computer models we use to predict climate.
To say a bit more about this, the problem is to do with how well climate models simulate the natural processes which amplify the anthropogenic increases in greenhouse gases (notably carbon dioxide). A key aspect of this amplification process is associated with the role of water in climate. For example, water vapour is itself a powerful greenhouse gas. If we were to assume that the relative humidity of the atmosphere (the percentage of the amount of water vapour at which the air would be saturated) was constant as the atmosphere warms under anthropogenic climate change, then humidity would amplify the climate change by a factor of two or more. On top of this, clouds — i.e. water in its liquid rather than gaseous form — have the potential to further amplify climate change (or indeed decrease it depending on the type or structure of the clouds). Finally, water in its solid phase can also be a significant amplifier of climate change. For example, sea ice reflects sunlight back to space. However as sea ice melts, e.g. in the Arctic, the underlying water absorbs more of the sunlight than before, again amplifying the underlying climate change signal.
We can approach these problems in two ways. Firstly we can use simplified mathematical models in which plausible assumptions (like the constant relative humidity one) are made to make the mathematics tractable. Secondly, we can try to simulate climate ab initio using the basic laws of physics (here, mostly, but not exclusively, the laws of classical physics). If we are to have confidence in climate predictions, this ab initio approach has to be pursued. However, unlike, say temperature in the atmosphere, water vapour and cloud liquid water have more of a fractal distribution, with both large and small scales. We cannot simulate accurately the small scales in a global climate model with fixed (say 100km) grid, and this, perhaps more than anything, is the source of uncertainty in climate predictions.
This is not just a theoretical problem (although there is some interesting mathematics involved, e.g. of multifractal distribution theory and so on). In the coming years, governments will be looking to spend billions on new infrastructure for society to adapt to climate change: more reservoirs, better flood defences, bigger storm sewers etc etc. It is obviously important that this money is spent wisely. Hence we need to have some quantitative and reliable estimate of certainty that in regions where more reservoirs are to be built, the climate really will get drier and so on.
There is another reason for developing quantitative methods for estimating uncertainty: climate geoengineering. If we spray aerosols in the stratosphere, or whiten clouds by spraying sea salt into them, we need to be sure we are not doing something terrible to our climate, like shutting off the monsoons, or decreasing rainfall over Amazonia (which might then make the rainforest a source of carbon for the atmosphere rather than a sink). Reliable estimates of uncertainty of regional impacts of geoengineering are going to be essential in the future.
My goals? To bring quantitative methods from physics and maths into climate decision making. One area that particularly interests me is the application of nonlinear stochastic-dynamic techniques to represent unresolved scales of motion in the ab initio models. If you are interested to learn more about this, please see this book:
• Tim Palmer and Paul Williams, editors, Stochastic Physics and Climate Modelling, Cambridge U. Press, Cambridge, 2010.
JB: Thanks! I’ve been reading that book. I’ll talk about it next time on This Week’s Finds.
Suppose you were advising a college student who wanted to do something that would really make a difference when it comes to the world’s environmental problems. What would you tell them?
TP: Well although this sounds a bit of a cliché, it’s important first and foremost to enjoy and be excited by what you are doing. If you have a burning ambition to work on some area of science without apparent application or use, but feel guilty because it’s not helping to save the planet, then stop feeling guilty and get on with fulfilling your dreams. If you work in some difficult area of science and achieve something significant, then this will give you a feeling of confidence that is impossible to be taught. Feeling confident in one’s abilities will make any subsequent move into new areas of activity, perhaps related to the environment, that much easier. If you demonstrate that confidence at interview, moving fields, even late in life, won’t be so difficult.
In my own case, I did a PhD in general relativity theory, and having achieved this goal (after a bleak period in the middle where nothing much seemed to be working out), I did sort of think to myself: if I can add to the pool of knowledge in this, traditionally difficult area of theoretical physics, I can pretty much tackle anything in science. I realize that sounds rather arrogant, and of course life is never as easy as that in practice.
JB: What if you were advising a mathematician or physicist who was already well underway in their career? I know lots of such people who would like to do something "good for the planet", but feel that they’re already specialized in other areas, and find it hard to switch gears. In fact I might as well admit it — I’m such a person myself!
TP: Talk to the experts in the field. Face to face. As many as possible. Ask them how your expertise can be put to use. Get them to advise you on key meetings you should try to attend.
JB: Okay. You’re an expert in the field, so I’ll start with you. How can my expertise be put to use? What are some meetings that I should try to attend?
TP: The American Geophysical Union and the European Geophysical Union have big multi-session conferences each year which include mathematicians with an interest in climate. On top of this, mathematical science institutes are increasingly holding meetings to engage mathematicians and climate scientists. For example, the Isaac Newton Institute at Cambridge University is holding a six-month programme on climate and mathematics. I will be there for part of this programme. There have been similar programmes in the US and in Germany very recently.
Of course, as well as going to meetings, or perhaps before going to them, there is the small matter of some reading material. Can I strongly recommend the Working Group One report of the latest IPCC climate change assessments? WG1 is tasked with summarizing the physical science underlying climate change. Start with the WG1 Summary for Policymakers from the Fourth Assessment Report:
• Intergovernmental Panel on Climate Change, Climate Change 2007: The Physical Science Basis, Summary for Policymakers.
and, if you are still interested, tackle the main WG1 report:
• Intergovernmental Panel on Climate Change, Climate Change 2007: The Physical Science Basis, Cambridge U. Press, Cambridge, 2007.
There is a feeling that since the various so-called "Climategate" scandals, in which IPCC were implicated, climate scientists need to be more open about uncertainties in climate predictions and climate prediction models. But in truth, these uncertainties have always been openly discussed in the WG1 reports. These reports are absolutely not the alarmist documents many seem to think, and, I would say, give an extremely balanced picture of the science. The latest report dates from 2007.
JB: I’ve been slowly learning what’s in this report, thanks in part to Nathan Urban, whom I interviewed in previous issues of This Week’s Finds. I’ll have to keep at it.
You told me that there’s a big difference between the "butterfly effect" in chaotic systems with a few degrees of freedom, such as the Lorenz attractor shown above, and the "real butterfly effect" in systems with infinitely many degrees of freedom, like the Navier-Stokes equations, the basic equations describing fluid flow. What’s the main difference?
TP: Everyone knows, or at least think they know, what the butterfly effect is: the exponential growth of small initial uncertainties in chaotic systems, like the Lorenz system, after whom the butterfly effect was named by James Gleick in his excellent popular book:
• James Gleick, Chaos: Making a New Science, Penguin, London, 1998.
But in truth, this is not the butterfly effect as Lorenz had meant it (I knew Ed Lorenz quite well). If you think about it, the possible effect of a flap of a butterfly’s wings on the weather some days later, involves not only an increase in the amplitude of the uncertainty, but also the scale. If we think of a turbulent system like the atmosphere, comprising a continuum of scales, its evolution is described by partial differential equations, not a low order set of ordinary differential equations. Each scale can be thought of as having its own characteristic dominant Lyapunov exponent, and these scales interact nonlinearly.
If we want to estimate the time for a flap of a butterfly’s wings to influence a large scale weather system, we can imagine summing up all the Lyapunov timescales associated with all the scales from the small scales to the large scales. If this sum diverges, then very good, we can say it will take a very long time for a small scale error or uncertainty to influence a large-scale system. But alas, simple scaling arguments suggest that there may be situations (in 3 dimensional turbulence) where this sum converges. Normally, we thinking of convergence as a good thing, but in this case it means that the small scale uncertainty, no matter how small scale it is, can affect the accuracy of the large scale prediction… in finite time. This is quite different to the conventional butterfly effect in low order chaos, where arbitrarily long predictions can be made by reducing initial uncertainty to sufficiently small levels.
JB: What are the practical implications of this difference?
TP: Climate models are finite truncations of the underlying partial differential equations of climate. A crucial question is: how do solutions converge as the truncation gets better and better? More practically, how many floating point operations per second (flops) does my computer need to have, in order that I can simulate the large-scale components of climate accurately. Teraflops, petaflops, exaflops? Is there an irreducible uncertainty in our ability to simulate climate no matter how many flops we have? Because of the "real" butterfly effect, we simply don’t know. This has real practical implications.
JB: Nobody has proved existence and uniqueness for solutions of the Navier-Stokes equations. Indeed Clay Mathematics Institute is offering a million-dollar prize for settling this question. But meteorologists use these equations to predict the weather with some success. To mathematicians that might seem a bit strange. What do you think is going on here?
TP: Actually, for certain simplifications to the Navier-Stokes equations, such as making them hydrostatic (which damps acoustic waves) then existence and uniqueness can be proven. And for weather forecasting we can get away with the hydrostatic approximation for most applications. But in general existence and uniqueness haven’t been proven. The "real" butterfly effect is linked to this. Well obviously the Intergovernmental Panel on Climate Change can’t wait for the mathematicians to solve this problem, but as I tried to suggest above, I don’t think the problem is just an arcane mathematical conundrum, but rather may help us understand better what is possible to predict about climate change and what not.
JB: Of course, meteorologists are really using a cleverly discretized version of the Navier-Stokes equations to predict the weather. Something vaguely similar happens in quantum field theory: we can use "lattice QCD" to compute the mass of the proton to reasonable accuracy, but nobody knows for sure if QCD makes sense in the continuum. Indeed, there’s another million-dollar Clay Prize waiting for the person who can figure that out. Could it be that sometimes a discrete approximation to a continuum theory does a pretty good job even if the continuum theory fundamentally doesn’t make sense?
TP: There you are! Spend a few years working on the continuum limit of lattice QCD and you may end up advising government on the likelihood of unexpected consequences on regional climate arising from some geoengineering proposal! The idea that two so apparently different fields could have elements in common is something bureaucrats find it hard to get their heads round. We at the sharp end in science need to find ways of making it easier for scientists to move fields (even on a temporary basis) should they want to.
This reminds me of a story. When I was finishing my PhD, my supervisor, Dennis Sciama announced one day that the process of Hawking radiation, from black holes, could be understood using the Principle of Maximum Entropy Production in non-equilibrium thermodynamics. I had never heard of this Principle before, no doubt a gap in my physics education. However, a couple of weeks later, I was talking to a colleague of a colleague who was a climatologist, and he was telling me about a recent paper that purported to show that many of the properties of our climate system could be deduced from the Principle of Maximum Entropy Production. That there might be such a link between black hole theory and climate physics, was one reason that I thought changing fields might not be so difficult after all.
JB: To what extent is the problem of predicting climate insulated from the problems of predicting weather? I bet this is a hard question, but it seems important. What do people know about this?
TP: John Von Neumann was an important figure in meteorology (as well, for example, as in quantum theory). He oversaw a project at Princeton just after the Second World War, to develop a numerical weather prediction model based on a discretised version of the Navier-Stokes equations. It was one of the early applications of digital computers. Some years later, the first long-term climate models were developed based on these weather prediction models. But then the two areas of work diverged. People doing climate modelling needed to represent lots of physical processes: the oceans, the cryosphere, the biosphere etc, whereas weather prediction tended to focus on getting better and better discretised representations of the Navier-Stokes equations.
One rationale for this separation was that weather forecasting is an initial value problem whereas climate is a "forced" problem (e.g. how does climate change with a specified increase in carbon dioxide?). Hence, for example, climate people didn’t need to agonise over getting ultra accurate estimates of the initial conditions for their climate forecasts.
But the two communities are converging again. We realise there are lots of synergies between short term weather prediction and climate prediction. Let me give you one very simple example. Whether anthropogenic climate change is going to be catastrophic to society, or is something we will be able to adapt to without too many major problems, we need to understand, as mentioned above, how clouds interact with increasing levels of carbon dioxide. Clouds cannot be represented explicitly in climate models because they occur on scales that can’t be resolved due to computational constraints. So they have to be represented by simplified "parametrisations". We can test these parametrisations in weather forecast models. To put it crudely (to be honest too crudely) if the cloud parametrisations (and corresponding representations of water vapour) are systematically wrong, then the forecasts of tomorrow’s daily maximum temperature will also be systematically wrong.
To give another example, I myself for a number of years have been developing stochastic methods to represent truncation uncertainty in weather prediction models. I am now trying to apply these methods in climate prediction. The ability to test the skill of these stochastic schemes in weather prediction mode is crucial to having confidence in them in climate prediction mode. There are lots of other examples of where a synergy between the two areas is important.
JB: When we met recently, you mentioned that there are currently no high-end supercomputers dedicated to climate issues. That seems a bit odd. What sort of resources are there? And how computationally intensive are the simulations people are doing now?
TP: By "high end" I mean very high end: that is, machines in the petaflop range of performance. If one takes the view that climate change is one of the gravest threats to society, then throwing all the resources that science and technology allows, to try to quantify exactly how grave this threat really is, seems quite sensible to me. On top of that, if we are to spend billions (dollars, pounds, euros etc.) on new technology to adapt to climate change, we had better make sure we are spending the money wisely — no point building new reservoirs if climate change will make your region wetter. So the predictions that it will get drier in such a such a place better be right. Finally, if we are to ever take these geoengineering proposals seriously we’d better be sure we understand the regional consequences. We don’t want to end up shutting off the monsoons! Reliable climate predictions really are essential.
I would say that there is no more computationally complex problem in science than climate prediction. There are two key modes of instability in the atmosphere, the convective instabilites (thunderstorms) with scales of kilometers and what are called baroclinic instabilities (midlatitude weather systems) with scales of thousands of kilometers. Simulating these two instabilities, and their mutual global interactions, is beyond the capability of current global climate models because of computational constraints. On top of this, climate models try to represent not only the physics of climate (including the oceans and the cryosphere), but the chemistry and biology too. That introduces considerable computational complexity in addition to the complexity caused by the multi-scale nature of climate.
By and large individual countries don’t have the financial resources (or at least they claim they don’t!) to fund such high end machines dedicated to climate. And the current economic crisis is not helping! On top of which, for reasons discussed above in relation to the "real" butterfly effect, I can’t go to government and say: "Give me a 100 petaflop machine and I will absolutely definitely be able to reduce uncertainty in forecasts climate change by a factor of 10". In my view, the way forward may be to think about internationally funded supercomputing. So, just as we have internationally funded infrastructure in particle physics, astronomy, so too in climate prediction. Why not?
Actually, very recently the NSF in the US gave a consortium of climate scientists from the US, Europe and Japan, a few months of dedicated time on a top-end Cray XT4 computer called Athena. Athena wasn’t quite in the petaflop range, but not too far off, and using this dedicated time, we produced some fantastic results, otherwise unachievable, showing what the international community could achieve, given the computational resources. Results from the Athena project are currently being written up — they demonstrate what can be done where there is a will from the funding agencies.
JB: In a Guardian article on human-caused climate change you were quoted as saying "There might be a 50% risk of widespread problems or possibly only 1%. Frankly, I would have said a risk of 1% was sufficient for us to take the problem seriously enough to start thinking about reducing emissions."
It’s hard to argue with that, but starting to think about reducing emissions is vastly less costly than actually reducing them. What would you say to someone who replied, "If the risk is possibly just 1%, it’s premature to take action — we need more research first"?
TP: The implication of your question is that a 1% risk is just too small to worry about or do anything about. But suppose the next time you checked in to fly to Europe, and they said at the desk that there was a 1% chance that volcanic ash would cause the aircraft engines to fail mid flight, leading the plane to crash, killing all on board. Would you fly? I doubt it!
My real point is that in assessing whether emissions cuts are too expensive, given the uncertainty in climate predictions, we need to assess how much we value things like the Amazon rainforest, or of (preventing the destruction of) countries like Bangladesh or the African Sahel. If we estimate the damage caused by dangerous climate change — let’s say associated with a 4 °C or greater global warming — to be at least 100 times the cost of taking mitigating action, then it is worth taking this action even if the probability of dangerous climate change was just 1%. But of course, according to the latest predictions, the probability of realizing such dangerous climate changes is much nearer 50%. So in reality, it is worth cutting emissions if the value you place on current climate is comparable or greater than the cost of cutting emissions.
Summarising, there are two key points here. Firstly, rational decisions can be made in the light of uncertain scientific input. Secondly, whilst we do certainly need more research, that should not itself be used as a reason for inaction.
Thanks, John, for allowing me the opportunity to express some views about climate physics on your web site.
JB: Thank you!
The most important questions of life are, for the most part, really only problems of probability. – Pierre Simon, Marquis de Laplace
Posted by John Baez 
is a probability distribution on a finite set, its 
is defined to be
. This looks pretty weird at first, and we need 
to avoid dividing by zero, but you can show that the Rényi entropy approaches the 
to be the Shannon entropy… and then the Rényi entropy generalizes the Shannon entropy by allowing an adjustable parameter 
. And that Hamiltonian is unique. Starting with that Hamiltonian, we can then compute the free energy 
at any temperature 
, and up to a certain factor this free energy turns out to be the Rényi entropy 
, where 
. More precisely:
, Rényi entropy is the same as free energy. It seems like a good thing to know — but I haven't seen anyone say it anywhere! Have you?
. We can always write
. Let’s think of these numbers 
is the 
when 
, the Gibbs state reduces to our original probability distribution at 
. 
is the expected energy in the Gibbs state at temperature 
is the usual Shannon entropy of this Gibbs state. I also show that all this stuff works quantum-mechanically as well as classically. But so far, it seems the main benefit is that Rényi entropy has become a lot less mysterious. It’s not a mutant version of Shannon entropy: it’s just a familiar friend in disguise.
whose elements are called vertices, together with a set 
of unordered pairs of distinct vertices, which we call edges. So, these are undirected finite graphs without self-loops or multiple edges connecting vertices. 
