Short-Time Accuracy Test of Climate Models?

The climate of the Earth results from turbulent flow of air in the atmosphere and turbulent flow of water in the ocean, which is heated by incoming solar radiation and cooled by outgoing infrared radiation, both of which critically depend on cloud formation, which requires presence of particles in the air or aerosols acting as condensation kernels.

Burning of fossil fuels produces aerosols which can enhance cloud formation. Low clouds  can decrease incoming radiation and thus act as negative feedback to global warming from fossil fuels, while high clouds can decrease outgoing radiation and thus act as positive feed-back. The net effect appears to be unknown and uncertainties in modeling of cloud formation propagate to uncertainties in global climate modeling.

A main goal of climate modeling is to predict climate sensitivity, that is the increase of global temperature from doubling of CO2 in the atmosphere. In the  IPCC 4th Assessment Report climate sensitivity is predicted to likely be between 2 and 4.5 degrees Celcius, which thus could be the global warming in 2100 without CO2 emission control. The UK Met Office explains to us:

• There have been major advances in the development and use of models over the last 20 years and the current models give us a reliable guide to the direction of future climate change.
• Computer models cannot predict the future exactly...
• Current models enable us to attribute the causes of past climate change, and predict the main features of the future climate, with a high degree of confidence.
• As well as producing CO2, burning fossil fuels also produces small particles called aerosols which cool the climate by reflecting sunlight back into space. These have increased steadily in concentration over the 20th century, which has probably offset some of the warming we have seen.

IPCC and the Met Office thus inform us that even if climate model predictions are incorrect over year and decade, they can be correct over centennials. But is this really possible from a mathematical point of view? Are there dynamical systems which allow computational modeling with this surprising property? Long-time accurate while short-time inaccurate? We are familiar with models which are short-time pointwise accurate and long-time pointwise inaccurate, but the opposite?

There are trivial such systems, like modeling an oscillation between -1 and +1 by a constant 0 state, but is the climate such a trivial system? Probably not. 

There are dissipative systems which forget initial data over long-time, so if initial data are incorrect this affects accuracy only for short-time. Climate models are dissipative and thus partly forget initial data, but that is not enough to secure long-time accuracy. 

The previous blog Climate and Turbulence Modeling recalled the analysis of turbulent flow in Computational Turbulent Incompressible Flow, showing that long-time meanvalues of turbulent flow may be predictable even if pointvalues are predictable only over short time, because of cancellation effects.  It is likely that climate models can share this property, but it requires short-time accuracy.

Altogether, I see no real reason to expect that short-time inaccurate climate models can be long-time meanvalue accurate, as suggested by IPCC and the Met Office. 

If this observation is correct, one could require climate models to be short-time accurate, which can be tested in short-time,  as a necessary requirement for reliability of long-time meanvalue prediction of climate sensitivity. 

Concerning the predictive capabilities of current climate models, see the US Senate Report: Dissent over Global Warming Claims.

The lecture Considering the Human Influence on Climate by R. A. Pielke Sr, is also informative. 

Mathematics and Religion vs Self-Help

There are similarities between traditional education in religion and contemporary education in mathematics based on an idea that God = Mathematics. To see this, consider the following common statements which express the fundamental beliefs which religious/mathematics education seeks to imprint in the blank minds of the students:

• Religion/Math is a supreme creation of human thought.
• Religion/Math is a high perfect art as daring as the most secret dreams of imagination.
• Religion/Math is a fundament of society.
• Religion/Math helps us to understand the World.
• Religion/Math helps us to cope with the problems we face in our lives.
• God/Math is everywhere present but invisible.
• God/Math is allmighty but with very limited practical utility.
• Religion/Math can be properly understood only by a selected few; the others have to accept it without understanding.
• A life without Religion/Math is a miserable life.
• Religion/Math has an intrinsic beauty and coherence.
• Religion/Math Mathematics plays a pivotal role in the progress of society and its continued growth relies on the encouragement and teaching of the next generation religious/mathematical thinkers, and outreach to the public and schools. 
  

Traditional education in religion was very successful in implanting these beliefs, but nevertheless today confessional religion has vanished from the school system, at least in the Western World. 

Similarly, even if the school system still succeeds in making all students believe in the greatness and importance of mathematics, whether they succeed or fail miserably, mathematics education is today in a state of permanent crisis in the footpaths of religious education. 

Religious education is today being replaced by a flood of self-help psychology and traditional mathematics is being replaced by self-help laptop technology:

• how to handle words, pictures, sound and videos
• how handle social life using facebook and blog 
• how to get information and get to understand the world, using search engines
• how to play computer games and music
• how handle gps, mobile and bank account. 

If not mathematics education is going to vanish from the school system, it has to be reformed into self-help laptop technology, and come down form the higher spheres of religion. This is further motivated on my blogs about mathematics education.

Climate and Turbulence Modeling

Global climate models are based on turbulence models, since the slightly viscous flow of air in the atmosphere and water in the oceans is turbulent. Turbulence modeling, in the form of  analytical mathematical models, is a main unsolved problem of fluid mechanics.

In our book Computational Turbulent Incompressible Flow, with prel. version for download, Johan Hoffman and I present a new approach to turbulence modeling based on ab initio numerical computation with turbulence automatically modeled by the stabilization of the numerics, thus without any explicit analytical turbulence model.  

We show that mean-value quantities of turbulent flow such as drag and mean temperature can be accurately computed without analytical turbulence model, thus circumventing the main unsolved problem of fluid mechanics. We plan to test this approach on climate modeling with hopefully a connection between turbulence and the main unsolved problem of climate modeling: cloud formation. 

We will report as soon as we have something to report on...hopefully before the Copenhagen meeting in December...since the outcome of this meeting critically depends on computational modeling of turbulence and cloud formation...and dark clouds over the meeting are already forming... 

Coin Tossing: Cold or Warm?

Newscientist reports:

• Forecasts of climate change are about to go seriously out of kilter. One of the world's top climate modellers said Thursday we could be about to enter "one or even two decades during which temperatures cool."People will say this is global warming disappearing," he told more than 1500 of the world's top climate scientists gathering in Geneva at the UN's World Climate Conference.
• "I am not one of the sceptics," insisted Mojib Latif of the Leibniz Institute of Marine Sciences at Kiel University, Germany. "However, we have to ask the nasty questions ourselves or other people will do it.
• "Few climate scientists go as far as Latif, an author for the Intergovernmental Panel on Climate Change. But more and more agree that the short-term prognosis for climate change is much less certain than once thought.
•  "In many ways we know more about what will happen in the 2050s than next year," said Vicky Pope from the UK Met Office.

The message is that global climate models cannot predict year or decade meanvalues, but can predict centennial meanvalues. How can this be? What is the mathematics behind such a belief? 

The first idea that come to mind is the law of large numbers of statistics offering prediction of the meanvalue 0.5 of many cointosses between 0 and 1, but no prediction of the meanvalue of a few tosses. But is climate modeling the same as coin tossing between cold and warm? 

Newscientist concludes:

• The world may badly want reliable forecasts of future climate. But such predictions are proving as elusive as the perfect weather forecast.
  

The future of mankind thus seems to lie in the hands of mathematicians running the climate models...but coin tossing statistics does not seem to be enough...what can be done or said? 

Well, let us recall that the 0.5 probability of heads in coin tossing is computed mathematically using the fact that a rotating coin has head up half of the time, that is using a short-time-accurate mathematical model, see the discussion in Chapter 13 Turbulence and Chaos in Computational Turbulent Incompressible Flow. Without a short-time-accurate model, nothing can be be predicted about long-time...Compare with the UK Met Office assurement:

• There have been major advances in the development and use of models over the last 20 years and the current models give us a reliable guide to the direction of future climate change.
• Computer models cannot predict the future exactly...
• Current models enable us to attribute the causes of past climate change, and predict the main features of the future climate, with a high degree of confidence.

What are "the advances in the development and use of models"? What is meant by "direction of future climate change"? Colder or warmer? Does "direction" indicate that the size of the change cannot be predicted? What is the meaning of "computer models cannot predict the future exactly"? That computer models can predict the future almost exactly? Who is the inventor of this form of newspeak? Note the clever construction of the following key statement by Met Office:

• As well as producing CO2, burning fossil fuels also produces small particles called aerosols which cool the climate by reflecting sunlight back into space. These have increased steadily in concentration over the 20th century, which has probably offset some of the warming we have seen.
  

Note the clever use of "probably" and "some of the warming". Very clever doublespeak: Clearly suggesting something, without saying anything! This is not the language of science. 

Can really these semantic tricks help save the World?

US Senators Freeze Despite Global Warming

The Guardian reports that The US Freezes on Climate Change:

• The stalled US climate change debate has killed the hope of reaching a final agreement at the Copenhagen summit
• Without concrete action in the Senate, there will not be an actual deal ready to sign in Copenhagen. With no Senate action, there's no guarantee that the US will commit to binding targets. And with no US targets, there will be no firm agreement from China, India or other emerging powers.
• It won't be a failure if there's no deal in Copenhagen, but it will be hard to gauge success with no new expectations.

What is the true reason that US Senators hesitate? Is it because most US Senators are not convinced about the reliability of present computational climate models predicting catastrophical global warming from CO2 emission? Is this because the Senators knowledge of mathematics and computer simulation is not deep enough? If so, would some education help?

Or is it because the Senators realize that they can only convince voters about the necessity of costly emission controls, if they themselves are convinced about the predictive power of present climate models, and they do not feel convinced? 

It seems that the only possibility is that the Senators sit down and carefully study mathematics and computational simulation, and after this experience check if they are convinced or not. It should be possible to bring them to the research front in a couple of weeks, with good math teachers. I guess this effort cannot be spared? 

I guess we here touch the essence of democracy? You cannot dictate, only convince others by first convincing yourself by first thinking yourself. Not even President Obama can get around this predicament. In his campaing Obama talked like a convinced: "the science is beyond dispute, the facts are clear", but today he has to face a disputable reality of unclear facts. How is Obama going to handle this situation? By joining the Senators math class?

Financial Times reports:

• The cost of reducing China’s total greenhouse gas emissions is likely to reach $438bn a year within 20 years, and developed economies will have to bear much of that cost, according to a group of Beijing’s leading climate economists.The figure, equivalent to about 7.5 per cent of China’s estimated gross domestic product in 2030, is likely to be deployed to support Beijing’s argument at December’s climate change summit in Copenhagen that industrialised nations must share the cost of cutting emissions in developing countries.
  

Another math lesson for both US President Obama and EU Chairman Reinfelt? 

Scoping the 5th Assessment Report of IPCC

                                                       The core of a climate model.

The upcoming 5th Assessment Report of IPCC AR5 was scoped in IPCCs 30th Session in Antalya, April 21-23, 2009, as a Proposal for an IPCC Expert Meeting on Assessing and Combining Multi Model Climate Projections:

• Climate model results provide the basis for IPCC projections of future climate change. Previous  assessment reports included model evaluations but avoided weighting or ranking models.  
• Defining a set of minimum criteria for a model to be 'credible' or  agreeing on a metric of performance is therefore difficult and the criteria are likely to depend on the  variable and timescale of interest. Combined with an estimated data volume exceeding 1000  Terabytes, the AR5 faces immense obstacles in trying to make sense of the deluge of model runs and  data that it will produce. 
• Recent studies have started to address these issues by proposing ways to weight or rank models, based  on process evaluation, agreement with present day observations, past climate or observed trends.  While there is agreement that 'the end of model democracy' may be near, there is no consensus on how  such a model selection or weighting process could be agreed upon. 

Reading these reservations, we get a warning that we should not expect any significant improvement of climate model reliability from AR4 to AR5. World leaders preparing far-reaching reductions in CO2 emissions thus have to look to the stars for guidance, or simply rely on the predictions of IPCC based on climate models of unknown but most likely low reliability.

 

Bara En Liten Miljon Till, till Matematik

I gårdagens SvD föreslår fem matematikprofessorer en blygsam satsning på matematiska vetenskaper genom adekvat stöd till Institut Mittag-Leffler på en miljon eller två. Motiveringen beskriver den rådande krisen inom matematik som akademiskt ämne:

• Den accellererande utvecklingen mot ett alltmer kunskapsbaserat samhälle tar stöd i matematiken.
• Idag är en satsning på matematik accepterad som ett fundamentalt villkor för ett lands utveckling. 
• För att bevara vår position som kunskapsnation ... måste Institut Mittag-Leffler få ett rimligt och långsiktigt stöd.     
  

Huvudargumentet är att matematik är fundamentet till vårt moderna Sverige, ett fundament som emellertid är osynligt och därför tyvärr lätt glöms bort i den nuvarande forskningspolitiken styrd av strategiska satsningar på lättbegripliga breda ämnesområden.

Således: Eftersom matematik är detta allestädes närvarande, men tyvärr osynliga och något obegripliga fundament, måste Sverige satsa inte en utan två miljoner på Institut Mittag-Leffler.

De fem matematikprofessorerna söker alltså skjuta en mygga med kanon: Om nu matematik är så väsentligt för vårt samhälle, vilket är riktigt om man tolkar matematik som dator/beräknings-matematik = basen för informationssamhället,  så räcker det inte med ytterligare en miljon till ett matematiksintititut i Djursholm, utan då måste matematikutbildningen på alla nivåer reformeras så att den motsvarar dagens behov.

Om detta behov finns att läsa på mina Knols on Mathematics/Science Education.

A Modest Proposal for Research Money

The Royal Society has published a report on the feasibility and possible dangers of technologies for cooling down the Earth, known as geoengineering. The ideas include artificial trees that draw CO2 from the air, mimicking volcanoes by spraying sulphate particles a few miles above the Earth to deflect the Sun’s rays, cloud seeding or launching trillions of small mirrors into space to act as a sunshield. For more ideas see news article in DN. 

The panel of 12 scientists who produced the report called for a £100 million annual global research fund to study geoengineering technologies and said that Britain should contribute £10 million a year, ten times the amount being spent now on such research. Professor John Shepherd, who chaired the panel, said:

• It is an unpalatable truth that unless we can succeed in greatly reducing carbon dioxide emissions we are heading for a very uncomfortable and challenging climate future, and geoengineering will be the only option left to limit further temperature increases.
  

However, Professor Shepherd admitted that he had no firm opinion on how likely it was that the world would need some form of geoengineering: 

• My opinion ranges from maybe to possibly to probably, depending on what I had for breakfast.
  

In the report Professor Shepherds breakfast ambivalence is expressed as:

• It is likely that global warming will exceed 2°C this century  unless global greenhouse gas emissions are cut by at least  50% of 1990 levels by 2050, and by more thereafter. There  is no credible emissions scenario under which global mean  temperature would peak and then start to decline by 2100.  
  

What to say about this? Is the Royal Society joking or not? Well, let us take a look at its Climate Change Controversies, A Simple Guide, in particular how the Society counters what it presents as Misleading Argument 5: "Global warming computer models which predict the

future climate are unreliable":

• Modern climate models have become increasingly accurate in reproducing how the real climate 'works'. They are based on our understanding of basic scientific principles, observations of the climate and our understanding of how it functions.
• Using this understanding of the climate system, scientists are then able to project what is likely to happen in the future, based on various assumptions about human activities.
  
• It is important to note that computer models cannot exactly predict the future, since there are so many unknowns concerning what might happen.  
• While climate models are now able to reproduce past and present changes in the global climate rather well, they are not, as yet, sufficiently well-developed to project accurately all the detail of the impacts we might see at regional or local levels. 
  
• They do, however, give us a reliable guide to the direction of future climate change.  The reliability also continues to be improved through the use of new techniques and technologies.  
  

These statements seem to support rather than refute Misleading Argument 5 stating that climate models are unreliable. How is it possible that the Royal Society without blushing can present nonsense like: Modern climate models have become increasingly accurate...the reliability also continues to be improved...they do, however, give us a reliable guide to the direction of future climate change... 

Notice in particular the Misleading Statement: the climate models are not, as yet, sufficiently well-developed to project accurately all the detail of the impacts we might see at regional or local levels... intended to give the impression that global climate models are accurate, without in fact saying anything. Clever or just stupid? Which is the audience addressed by the Royal Society?

The Royal Society, in good company with the Royal Swedish Academy of Sciences, seems to uncritically repeat whatever the political organ IPCC says, but why?

Illusions of Theories of Everything

The ultimate dream of theoretical physicists is a Grand Unified Theory GUT or a Theory Of Everything TOE  as a mathematical equation in the form of a system of partial differential equations, the solutions of which would represent all there is in the World. We find this dream partially realized in specific areas of mechanics and physics identified by a specific system of partial differential equations, such as 

• fluid mechanics: Navier-Stokes equations 
• quantum mechanics: Schrödinger's equation
• celestial mechanics: Newton's equations of motion/gravitation.

One can argue that in a certain sense all of celestial mechanics including the motion of the planets, comets, asteroids et cet in our Solar system, is represented as solutions to Newton's equations of motion/gravitation, that all of quantum mechanics is represented as solutions of Schrödinger's equation, and that all of fluid mechanics is represented as solutions of the Navier-Stokes equations. 

We can thus view Newton's equations, Navier-Stokes equations and Schrödinger's equation as different forms of TOE with Everything representing the totality of a certain part of the World, like a continent of the Earth. 

Newton seemed to be able to describe all of celestial mechanics by his equations of motion and gravitation in his TOE,  which made Newton immensly famous attributed with godlike power. 

Similarly one can argue that all of fluid mechanics can be described by the Navier-Stokes equations, and all of quantum mechanics by Schrödinger's equation as different forms of TOE. 

This can give a physicist in control of a TOE the illusion of superhuman power, but there is a hook: Even if the equations can be written down in a couple of lines, like the Navier-Stokes equations, they can be impossible to solve by analytical mathematics representing solutions in terms of elementary functions such as polynomials and trigonometric functions: Analytical solutions of Newton's equations are known only for the two-body problem of one small body like the Earth orbiting a big body like the Sun. Already three bodies is beyond analytical representation, not to speak of the turbulent solutions of the Navier-Stokes equations. 

If we stop here, a TOE in the form of Navier-Stokes equations instead of a theory of everything, would seem to be rather a theory of nothing. This would be like a jeweler with diamonds still to be captured from the rock. Is a  jeweler without jewels, still a jeweler?

However, one can compute digital solutions of the Navier-Stokes equations using computers, for specific choices of data, and in this way gain insight case by case using a Computational Theory of Everything. Some diamonds thus can be brought to the surface and put into rings to be admired. But we cannot get full insight in one shot. We cannot capture all diamonds in one day in a true TOE.

One can argue that specific knowledge of e.g. fluid mechanics comes form specific computational solutions to the Navier-Stokes equations for specific data, and fluid mechanics is the totality of such specific knowledge. I give examples in my knols on fluid mechanics. See also

Soul as Simulation of Body, Is God Mathematician, Simulation Technology.

So even if a GUT or TOE unifying quantum mechanics and gravitation in the form of one set of equations, e.g. in the form of string theory, the main task of computing and studying specific solutions would remain.

We can make a parallel with Darwin's theory of evolution based on an equation expressing genetic variability + selection by survival of the fittest. Anybody can formulate this equation and the non-trivial part of evolution theory is the study of specific solutions. Still 150 years after Darwin,  Richard Dawkins struggles hard to compute specific solutions of Darwin's equation...

One can even argue that Darwins TOE of evolution as variability + selection, is trivial in the sense that it can be written down in one line, while the determination of specific solutions such as amoebas and human beings is highly non-trivial.

We often hear physicists claim that the atomic electron structure of the periodic table of elements is a consequence of quantum mechanics, but at a closer examination we find the electron structure for atoms with more than one electron, that is all elements except Hydrogen, is unknown as a solution to the Schrödinger equation, see my knols on quantum mechanics.

Likewise, one can argue that the secret of turbulence is hidden as solutions to the Navier-Stokes equations, a secret closed to analytical solution but open to exploration by computation, as well as the n-body problem of celestial mechanics.

The basic differential equations of celestial, fluid and quantum mechanics express basic physical laws of balance or conservation, such as Newton's 2nd law, conservation of mass, momentum and energy. These physical laws are what a blind Nature obeys in its evolution from one moment of time to the next in some form of analog computation, which can be mimicked by digital computation. To evolve according to physical laws does not require any intelligence, just work. In rare cases human intelligence allows shortcuts to analytical solutions, but in general only brute force computation is effective. This is what makes the world go round, whether it is understood by someone or not.

Will Mathematicians Save the World, Again?

The free world was saved from the threat of both nazism, fascism and communism, because of free world mathematicians were able to compute both how to make nuclear bombs and run Star Wars. 

Today we are told that mathematical climate models predict catastrophical global warming by CO2 emission from burning of fossil fuels, which represent 75% of the total energy production in the World. Based on these mathematical predictions US President Obama stated the G8 meeting in Aquila in July:

• The G8 nations agreed that by 2050, we'll reduce our emissions by 80 percent and that we'll work with all nations to cut global emissions in half. 
  

Realization of these goals will require a major reorganization of the industrial world and threatens to keep the developing world from development. The necessity of the drastic actions required come from predictions of mathematical climate models, and the question that the leaders of the world must pose concerns the reliability of these predictions. 

This is a question for mathematicians. Are mathematicians ready to once again save the World from catastrophy, by once again focussing on the most urgent question facing mankind?

Let us see what the International Union of Mathematicians IMU has to say about global warming? Nothing it seems. Strange! Are mathematicians not willing to save the World this time? The last International Congress of Mathematicians ICM organized by IMU in Madrid in 2006, had no section on mathematical climate modeling, and the upcoming IMC in Hyderabad, India, in 2010 seems no better. Why?