Guest Post by Willis Eschenbach
I’ve heard many times that whereas weather prediction is an “initial-value” problem, climate prediction is a “boundary problem”. I’ve often wondered about this, questions like “what is the boundary?”. I woke up today thinking that I didn’t have an adequately clear understanding of the difference between the two types of problems.
For these kinds of questions I find it’s hard to beat Wolfram Reference, which is a reference to the various functions in the computer program Mathematica. Wolfram is a total genius in my opinion, and the Wolfram site reflects that. Here’s what Wolfram Reference says (emphasis mine):
Introduction to Initial and Boundary Value Problems
DSolve [a Mathematica function] can be used for finding the general solution to a differential equation or system of differential equations. The general solution gives information about the structure of the complete solution space for the problem. However, in practice, one is often interested only in particular solutions that satisfy some conditions related to the area of application. These conditions are usually of two types.
• The solution x(t) and/or its derivatives are required to have specific values at a single point, for example, x(0)=1 and x’(0)=2. Such problems are traditionally called initial value problems (IVPs) because the system is assumed to start evolving from the fixed initial point (in this case, 0).
• The solution x(t) is required to have specific values at a pair of points, for example, x(0)=1 and x(1)=5. These problems are known as boundary value problems (BVPs) because the points 0 and 1 are regarded as boundary points (or edges) of the domain of interest in the application.
The symbolic solution of both IVPs and BVPs requires knowledge of the general solution for the problem. The final step, in which the particular solution is obtained using the initial or boundary values, involves mostly algebraic operations, and is similar for IVPs and for BVPs.
IVPs and BVPs for linear differential equations are solved rather easily since the final algebraic step involves the solution of linear equations. However, if the underlying equations are nonlinear, the solution could have several branches, or the arbitrary constants from the general solution could occur in different arguments of transcendental functions. As a result, it is not always possible to complete the final algebraic step for nonlinear problems. Finally, if the underlying equations have piecewise (that is, discontinuous) coefficients, an IVP naturally breaks up into simpler IVPs over the regions in which the coefficients are continuous.
Now, as I read that, it says that for an initial value problem (IVP) we need to know the initial conditions at the starting time, and for a boundary value problem (BVP) we need to know the future conditions at a particular boundary. For example, suppose we are interested in the future thermal behavior of an iron rod with one end in a ice-water bath. The boundary condition is that the end of the iron rod in the ice-water bath is at 0°C.
So my question is two-fold. IF predicting weather is an IVP and predicting climate is a BVP, then
1) What is the “boundary” in question?, and
2) Once we determine what the boundary is, how do we know the future value of the boundary?
Some investigation finds that for US$48 I can read the following:
Existence and regularity theorems for a free boundary problem governing a simple climate model
Xiangsheng Xua
Abstract
From a class of mean annual, zonally averaged energy–balance climate models of the Budyko‐Sellers type, we arrive at a free boundary problem with the free boundary being the interface between ice‐covered and ice-free areas. Existence and regularity properties are proved for weak solutions of the problem. In particular, the regularity of the free boundary is investigated.
Fortunately, I don’t need to read it to see that the boundary in question is the ice-water interface. Now, that actually seems like it might work, because we know that at any time in the future, the boundary is always at 0°C. Since we know the future temperature values at that boundary, we can treat it as a boundary problem.
But then I continue reading, and I find Dr. Pielke’s excellent work , which says (emphasis mine):
One set of commonly used definitions of weather and climate distinguishes these terms in the context of prediction: weather is considered an initial value problem, while climate is assumed to be a boundary value problem. Another perspective holds that climate and weather prediction are both initial value problems (Palmer 1998). If climate prediction were a boundary value problem, then the simulations of future climate will “forget” the initial values assumed in a model. The assumption that climate prediction is a boundary value problem is used, for example, to justify predicting future climate based on anthropogenic doubling of greenhouse gases. This correspondence proposes that weather prediction is a subset of climate predictions and that both are, therefore, initial value problems in the context of nonlinear geophysical flow. The consequence of climate prediction being an initial value problem is summarized in this correspondence.
The boundaries in the context of climate prediction are the ocean surface and the land surface. If these boundaries are fixed in time, evolve independently of the atmosphere such that their time evolution could be prescribed, or have response times that are much longer than the time period of interest in the climate prediction, than one may conclude that climate prediction is a boundary problem.
So Dr. Pielke says that there is an entirely different boundary in play, the boundary between the atmosphere and the surface.
But then my question is, how would we know the future conditions of that boundary? If it’s a BVP, we have to know future conditions.
Dr. Pielke takes an interesting turn. IF I understand his method in another paper, Seasonal weather prediction as an initial value problem, he shows that the chosen boundary (the atmosphere/surface interface) doesn’t “evolve independently of the atmosphere such that their time evolution could be prescribed” and thus seasonal weather prediction is shown to be an IVP rather than a BVP.
However … he’s using an entirely different boundary than that used by Xiangsheng Xua above. Which one is right? One, both, or neither?
And the underlying problem, of course, is that IF climate is an initial value problem just like weather, given the chaotic nature of both we have little hope of modeling or predicting the future evolution of the climate.
My conclusion from all of this, which I think is shared by Dr. Pielke, is that climate prediction is an initial value problem. I say this in part because I see no difference in “climate” and “weather” in that both seem to be self-similar, non-linear, and chaotic.
This view is also shared by Mandelbrot, as was discussed about a decade ago over at Steve McIntyre’s excellent blog … have we really been at it that long? Mandelbrot analyzed a number of long-term records and found no change in the fractal nature of the records with timespan. In other words, there’s no break between the chaotic nature of the short, medium, and long-term looks at weather.
Now, it’s often argued that weather prediction has gotten much better over the decades … and this is true. But remember, weather prediction is an initial value problem. That means that the more accurately and specifically and finely we can measure the initial conditions, the better our prediction will be. Much of the improvement in our weather predictions is a result of satellites which give us our initial conditions in exquisite detail. And despite all our advances in predictive ability, lots of weekend barbecues still get rained on.
And at the end of the day, I’m left with my initial questions:
• If modeling the future evolution of the climate a boundary problem, what exactly is the boundary?, and
• Having specified the boundary, how can we know the future conditions of the boundary?
Egads … a post without a single graphic … curious.
w.
My Usual Request: If you think something is incorrect, please have the courtesy to quote the exact words that you disagree with so that everyone can understand your objections.
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Given that climate is weather’s strange attractor, consider using the Brouwer Fixed Thinking Theorem.
Compared to statistics, calculus, and ordinary & partial differential equations, we are in our infancy with regards to chaos theory and fractals. Are there yet-to-be-discovered tools in chaos theory that would provide more predictive power than what we currently use with the mathematics that has been around since the 16- and 17-hundreds?
Willis,
You have looked at the terminology of ordinary (1D) differential equations. I do not think that is what is meant. Dr Spencer is on the right track.
The right analogy might be flow along a river or channel (PDE). Seen as an initial value problem, you calculate the velocity cross-section going in, solve for momentum etc. But that doesn’t get you very far.
As a BVP, you forget about initial conditions, and look at what happens along the way. There is loss of altitude, boundary friction, maybe cascades, and of course the constantly varying depth. These are what you use to determine the river flow in the longer term. You can’t usefully forecast the flow from the initial conditions, but that doesn’t mean you know nothing.
If you do a CFD model, you’ll need turbulence (weather). But it is random, and you are only interested in what it does to the mean flow.
I gave him the same example. He doesn’t seem to understand that macro and micro always have completely different approaches.
Mr Stokes presents an analogy of river flow prediction that is not complete He says that:
“The right analogy might be flow along a river or channel (PDE). Seen as an initial value problem, you calculate the velocity cross-section going in, solve for momentum etc. But that doesn’t get you very far.
As a BVP, you forget about initial conditions, and look at what happens along the way. There is loss of altitude, boundary friction, maybe cascades, and of course the constantly varying depth. These are what you use to determine the river flow in the longer term. You can’t usefully forecast the flow from the initial conditions, but that doesn’t mean you know nothing.”
With the information he suggests we record we can determine “the river flow in the long term”. The limited processes he quotes would not allow you to determine the long-term flow. You need to determine the relationship between measured flow and the stage (stream height) and also for simulation the catchment area, it antecedent groundwater and soil moisture storage (and account for these stores and losses over time) and most important of all the long-term rainfall over the catchment area which after all determines the ultimate flow in the stream. The groundwater store, soil moisture store and rain will vary continuously over time. In addition you need to conduct a calibration of the actual and modeled flow over time if you are to have any chance of being able to predict future flows.
It is true that the influence of ground and soil moisture store and flow initial conditions will tend to dissipate over time as the simulation proceeds during calibration provided the flow record during calibration is sufficiently long.
Nick writes “As a BVP, you forget about initial conditions, and look at what happens along the way. There is loss of altitude, boundary friction, maybe cascades, and of course the constantly varying depth.”
But GCMs are spun up with initial values and allowed to run until stable. Stable means they just wander around a bit. Then we add a forcing and they wander around but tend in one direction according to assumed energy retention.
The fact is that if no forcing was added, GCMs would never simulate most of the features of our climate. No ice ages punctuated by short interglacials for example.
GCMs fundamentally dont model climate and you can argue all you like they aren’t powerful enough and run long enough to simulate climate but that makes no difference. They are what they are and its not at all obvious to me that they fall in the “useful” category of models.
I would add a third question: Does it matter if climate is an initial value problem or a boundary value problem?
Although it is likely possible to set up a simple climate model such that it is a boundary value problem, I think that the GCM climate models are numerically set up as initial value problems. So that seems to suggest that the modellers think that it does not really matter if the problem is initial value or boundary value.
It seems to me that the key issue is whether, on long enough times scales, the chaotic aspects of weather average out in the way that random noise averages out. That sort of averaging seems to be a central assumption of climate modelling for the simple reason that if it does not occur, then climate modelling is hopeless.
It is certainly possible for such averaging to occur, but has anyone demonstrated that it does occur for the climate system? If the problem can be cast as a boundary value problem, does that guarantee that such averaging occurs? I doubt that anyone can prove that. And I think it very likely that chaotic systems can be designed for which such averaging is invalid.
One possibility is that this is not meant to be a rigorous mathematical statement. So one might argue that since there is a boundary (TOA) for which a particular condition (energy balance) must apply in the long run, then the set of possible solutions must be bounded in the sense that only a certain range of solutions is allowed. Then sufficient sampling could give an estimate of most likely values. But that is only useful if the range of solutions is fairly tightly constrained and if a reasonable sampling occurs on a meaningful time scale. Given the ice ages, that is far from obvious.
Another possibility is that the “climate is a boundary value problem” meme is just mathematical bullying. “Do you have an answer to this profound statement? No? Then shut up and let those of us initiated into these mysteries get on with the job of telling you what should be done.”
“It seems to me that the key issue is whether, on long enough times scales, the chaotic aspects of weather average out in the way that random noise averages out.”
That would imply that there is a chaos-free frequency band. In other words, that we can represent weather as the sum of a chaotic high-frequency-only system plus a chaos-free low-frequency-only system – which would likely be linear and periodic.
That’s of course something that looks rather artificial and it has never been proposed by any warmunist.
Also, as a moving average is a weak (6 dB / octave) low pass filter it does not cut off steeply at the boundary frequency. Warmunists would have to propose using a higher order filter. Not that they EVER mention anything about power spectra, periodicity, or filtering, as they seem to have no knowledge of time series analysis.
Mike M. May 25, 2015 at 7:44 pm
You seem to have overlooked the point I made in the head post, which was that Mandelbrot has demonstrated that such averageing DOES NOT OCCUR for the climate system. No matter how long the time series he studied, Mandelbrot showed that there was no time when “the chaotic aspects of weather average out”.
w.
Mandelbrot was writing in 1969, at the absolute beginning of computer solution of Navier-Stokes solutions. And he wasn’t even writing about weather measurements. He was writing about paleo proxy records, such as they were then.
Willis, no one has overlooked that statement, but it’s not compelling. One, there is chaos everywhere we look. By your logic, we shouldn’t have any laws of science. Two, chaos is not random. We call it chaos when the details are too complex to handle. Three, Mandelbrot wrote that in 1969, and a lot has happened since. Four, the statement no time when “the chaotic aspects of weather average out” has been falsified by history and climate reconstructions.
VikingExplorer
You say
You could not be more wrong.
Taking each of your statements in turn.
“no one has overlooked that statement,”
Really? You know that? You asked everybody?
“it’s not compelling.”
Rubbish! Of course it is: prove me wrong.
“One, there is chaos everywhere we look.”
So what? At issue is whether there is sufficient knowledge of the climate system to understand the effects of its strange attractors: there is NOT such knowledge.
“Two, chaos is not random.”
So what? Chaos is unpredictable.
“We call it chaos when the details are too complex to handle.”
No, you and other warmunists may “call it chaos when the details are too complex to handle” but mathematicians don’t. You are confusing the general understanding of chaos with chaos theory.
Chaos Theory is the mathematical subject which deals with nonlinear things that are effectively impossible to predict or control; e.g. turbulence, weather, the stock market, brain states, etc.. Chaos Theory often uses fractal mathematics to describe such non-linear things.
“Three, Mandelbrot wrote that in 1969,”
So what?
“a lot has happened since.”
True, e.g. my children were born and grew up. So what?
“Four, the statement no time when “the chaotic aspects of weather average out” has been falsified by history and climate reconstructions.”
No. That is absolutely untrue as your link shows: anything can be “averaged” in many ways, but there is no evidence that such averaging negates “chaotic aspects”. And there are no “climate reconstructions” sufficient to have displayed predictive ability; e.g. a change of attractor from glacial to interglacial states cannot be predicted.
Richard
“VikingExplorer
May 26, 2015 at 12:09 am
One, there is chaos everywhere we look . . . Two, chaos is not random.”
Wow, a Nietzsche disciple. In any case, I’m not seeing anything in your linked climate reconstruction that would disprove the point, which is that there is no time interval above which which you can discern a stable, predictable pattern called “climate” around which a separately driven chaotic process is superimposed (like EM noise around a signal through a wire). Climate and weather seem to be controlled by the exact same physical processes, the only difference being some arbitrarily defined time scale, hence there is no way that you can determine whether, or more appropriately to what degree, a trend in a variable is due to some new input or simply the chaotic natural response of the system to prior inputs.
Willis,
“You seem to have overlooked the point I made in the head post, which was that Mandelbrot has demonstrated that such averageing DOES NOT OCCUR for the climate system.”
Actually, I did overlook that but I think it does not matter. My point was that we don’t know either way. One study does not change that.
Nick Stokes May 25, 2015 at 10:58 pm
Nick, if you think saying it was early days means something, then we should forget Newton.
If you think Mandelbrot “wasn’t even writing about weather measurements”, since when is precipitation not a weather record?
And finally, if you think Mandelbrot is wrong, please quote what he said that was wrong, and show us where he went of the rails.
Because you simply claiming he was wrong? Well, that’s just the typical Stokes bluster, worth absolutely nothing. You’d claim he was wrong even if he was right, if it fit your fancy or if you didn’t like the person who made the claim.
You’ve called wolf too many times, Nick, your credibility is shot … if you think Mandelbrot made a mistake, I fear you’ll have to show us.
w.
Sometimes I wonder if Nick Stokes is related to Doug Cotton
“Sometimes I wonder if Nick Stokes is related to Doug Cotton”.
You mean like the Arnold Schwarzenegger and Danny DeVito characters in “Twins”? -:)
Huh? I’m more anti AGW than anyone. You are foolish to not realize that it is YOU who are arguing for AGW in this case. If the climate is truly chaotic, then it is very fragile, and we need to be very careful, because even small changes (like trace gases) can cause a dramatic change (“tipping point”).
Even if so, circuit theory and quantum electrodynamics describe the same underlying physical processes.
However, if weather is defined as atmospheric, and climate science is defined as concerning the thermodynamics of the ocean, and all external factors that affect the ocean, then they are quite different physical processes. Climate would NOT be an average of weather, and climate theory & models would not be extensions of weather theory / models.
Actually, there isn’t much consensus about definition of chaos.
“a kind of order without periodicity.” However, the climate reconstruction shows periodic wave-like behavior. This implies predictability. When one is on top of a wave, a trough is coming.
Except that all of these are predictable to some extent.
“Most physical systems are chaotic. It takes something special for a system to not be chaotic” -Leo C. Stein, Ph.D. from MIT
If chaos is grounds for declaring defeat and unpredictability, all science is impossible.
A good example of chaos is gravity. Two bodies are easy, but as soon as we add a 3rd body, it becomes chaotic, which is why our solar system is chaotic.. We can’t really predict the position of Venus 10,000 years from now. However, like the climate, we don’t need to. In both cases, we just need to predict the next state.
Here is an excellent answer to the question: Are continuous chaotic systems necessarily uncomputable?
“chaoticity can appear as soon as you can’t predict some behaviour”. In this sense, “chaos” is partly as much about us as it is about the system itself. It is a reflection of our ignorance.
“Computations can be made to define the next state of the system, but you can’t answer some question like “Will this planet crash into this other one ?”
For climate, we just need to predict the next state.
Mike M,
More like the Kardashians.☺
Willis,
“And finally, if you think Mandelbrot is wrong, please quote what he said that was wrong, and show us where he went of the rails.”
First you should quote what he said that you think was right. You haven’t. I can’t find anything in Mandelbrot and Wallis 1969 that resembles:
“Mandelbrot analyzed a number of long-term records and found no change in the fractal nature of the records with timespan. In other words, there’s no break between the chaotic nature of the short, medium, and long-term looks at weather.”
He wasn’t writing about chaos; it is not mentioned. He was writing about long term interdependence and Hurst’s law.
VikingExplorer
May 26, 2015 at 9:22 am
A good example of chaos is gravity. Two bodies are easy, but as soon as we add a 3rd body, it becomes chaotic, which is why our solar system is chaotic.. We can’t really predict the position of Venus 10,000 years from now.
What a ridiculous statement. If you need to know where Venus will be 10,000 years time you need only refer to the JPL Development Ephemeris DE431 which has been numerically integrated up to the year 17191 Mar 15.
http://ssd.jpl.nasa.gov/?planet_eph_export
http://ipnpr.jpl.nasa.gov/progress_report/42-196/196C.pdf
At least the celestial mechanicians do know that they are dealing with an IVP and also what all the relevant physical laws are, along with the physical parameters of the 343 asteroids included in the dynamical model.
Climate modelers have a very long way to go.
The n-body problem is another classic example of chaos. We do not know the initial conditions well enough. We do not know precisely the attributes of the bodies in our solar system. We measure mass by placing an object into a known gravity field. We can’t be precisely sure of the mass of the sun or any body in our solar system. An slight error in the decimal places will be a problem for long term prediction.
It’s really the same as the climate model situation. There is no analytical solution to the n-body problem, so in general, n-body problems must be solved or simulated using numerical methods. As the simulation runs further into the future, its accuracy degrades.
You say we know what “all the relevant physical laws are”. However, recent empirical results under very tightly controlled conditions to determine the value of G to something better than 4 decimal places (least accurate constant) found large errors (~2%).
Also similar to climate models is the fact that it’s going to take a really long time to test whether our simulations are correct. I find it amazing and psychologically interesting that you are so sure of an n-body simulation 10,000 years into the future, but not any climate model simulations. You can’t have it both ways.
VikingExplorer
May 26, 2015 at 2:02 pm
If you’d bothered to read the pdf you would have discovered that the authors deal with the uncertainties in the integration:
DE431 is suitable for the analysis of earlier historical observations of the Sun, Moon, and planets. The DE431 time span from the year –13,200 to the year 17,191 extends far beyond historical times and caveats are offered. For the planets, uncertainties in the initial conditions of the orbits will cause errors in the along-track directions that increase at least linearly with time away from the present. Resonances including, but not limited to, those between Jupiter and Saturn, and between Uranus, Neptune, and Pluto, may complicate the propagation of errors. Typically, the along-track component will degrade faster than the other two components. For the Moon, the uncertainty given for the tidal acceleration causes a 28 m/century2 along-track uncertainty. But there are other concerns, e.g., the theory for the orientation of Earth includes polynomial expressions that are adequate for thousands of years, but are not designed for much longer times.
I don’t know where to start with the rest of your statement. Seems the solar system dynamicists do know rather a lot about the gravity fields of non-spherical bodies:
The modeled accelerations of bodies due to interactions of point masses with the gravitational field of nonspherical bodies include: (a) the interaction of the zonal harmonics of the Earth (through fourth degree) and the point mass Moon, Sun, Mercury, Venus, Mars, and Jupiter; (b) the interaction between the zonal, sectoral, and tesseral harmonics of the Moon (through sixth degree) and the point mass Earth, Sun, Mercury, Venus, Mars, and Jupiter; (c) the second-degree zonal harmonic of the Sun (J2) interacting with all other bodies.
The zonal harmonics of the earth are well enough known to enable the detection of frame-dragging and the geodetic effect predicted by Einstein by Gravity Probe B.
http://einstein.stanford.edu/content/press-media/results_news_2011/C_Will-Physics.4.43-Viewpoint.pdf
As for the value of G, a recent paper (2015) by Anderson et al in EPL show systematic variations in the value:
Figure 1 appears to provide convincing evidence that there exists a 5.9 year periodicity in the macroscopic determinations of G in the laboratory with variations at the level of ΔG/G ∼ 2.4 × 10−4 about a mean value of 6.673899 × 10−11m3 kg−1 s−2, close to the value recommended by CODATA in 2010 but with a much smaller standard error of 10.3 ppm instead of the CODATA recommended error of 120 ppm.
These are much smaller than you suggest, 240 parts per million rather than the 2% figure you quoted. The authors are also quite clear that these errors are attributed to the measurement process, not the value of G itself.
Climate simulations are as toys compared with these solar system dynamical models. Climate modelers need to put as much effort into collecting accurate data and determining relevant parameters to include in their models as the solar system modelers to have any hope of predicting climate.
The 2% was the error found in a certain highly controlled experiment. Never said that was about the accuracy of G. Of course the errors are attributed to the measurement process. The accuracy of G is now known up to 5 decimal places.
The bottom line is that the n-body gravity problem is chaotic. G and mass are very difficult to measure, and G is the least accurate of all constants. This makes any long term prediction problematic.
However, the point is that they didn’t throw up their hands and declare it impossible.
BVP or IVP, any medium- or long-term prediction (‘projection,’ whatever) of climate necessarily assumes the following: “As long as nothing currently unknown, unexpected or not properly factored in happens along the way.” In other words, it assumes no more nor less than a glorified form of ceteris parabus.
Good luck with that.
Well, it seems that Mike M. and DirkH above make a similar point in more technical fashion (but I swear I didn’t see them until after posting my own!). Not surprisingly, I agree strongly with both.
Brad Crawford
1) What is the “boundary” in question?
The sun.
2) Once we determine what the boundary is, how do we know the future value of the boundary?
We can’t.
One could construct a case for arguing that the Solar System is the boundary.
After all, things going on in the Solar System can and do influence climate here on planet Earth, as any dinosaur well knows (not that I am claiming that the impact of 65 million years ago was the exclussive cause of their extinction).
Personally, I am more concerned about an asteroid impact than rising levels of CO2, and of course, I know that you are concerned about the quiet sun and it may well be that over the next 20 or so years we get a chance to investigate the consequences of a quiet sun .
It is all piffle, artifact of the construction of a model. The initial condition is just the initial boundary, and eventual boundaries are just initial conditions for the next phase.
“As a result, it is not always possible to complete the final algebraic step for nonlinear problems.”
This is the root of the problem. The whedefugawe effect. Make your own model and you will see…
I met Dr. Fugawe once at a seminar. He co-presented a paper with Dr. Coliwasa, the inventor of the eponymous toxic waste study device and a recipient of the Nobel-Crackerjack Prize.
There is an interesting discussion of this topic at http://scienceofdoom.com/2014/11/29/natural-variability-and-chaos-four-the-thirty-year-myth/
Given that the algorithm “the weather in 4 days time will be the same as today’s” is almost as accurate as any other forecast, weather forecasts are clearly an initial value problem.
http://weather.slimyhorror.com/
My father used to joke: the best weather forecast: the same as today.
It’s funny because there is a truth to it. Rather than indicate inherent unpredictability, it shows that there are strong boundaries and tendencies. We may not know exactly, but we know it won’t be far off the average for that time of year.
We may not be able to predict the path of a leaf, but we’re pretty sure the path of the river will stay the same.
Climatology is, or should be, about understanding the factors that affect the big picture, external input, output, etc.
VikingExplorer May 26, 2015 at 12:21 am
Sometimes, you guys are hilarious. You’re pretty sure the path of the river will stay the same?
So, you’re pretty sure you can you predict the future path of the river shown above?
You (and Nick Stokes) are welcome to try to sell that nonsense to e.g. the folks living along the Ganges River in Bangladesh … “don’t worry, folks, we can’t predict the leaf, but we’re pretty sure that the river will never change its path” …
The problem, Viking, is that the climate is no less chaotic than the weather. In the case of the climate, we can’t predict the path of either the weather or the climate … but then, in many cases we can’t predict the path of either the leaf or the river, so that should be no surprise.
w.
Actually, it was pretty late when I wrote that. I meant to say: the path of the river will stay the same [within certain boundaries].
The path of the river is a function of the topography and characteristics of the land, NOT the small scale fluid dynamics affecting the leaf.
We know that the steepness of the grade affects the behavior. The flatter, the more the river will start to snake, as the water picks the path of least resistance. As time goes on, it will wear down the rivers edge on the outside of turns, and may eventually break through, taking a shortcut. This is what your image shows.
My whole point is that the science of a river’s path is dramatically different than the science of the leaf’s path. This is a good example where the unpredictability of a leaf is NOT evidence of the unpredictability of the river.
>> The problem, Viking, is that the climate is no less chaotic than the weather.
Actually, as has been hinted at in several comments, the problem is that the terms weather and climate are subjectively defined.
Let’s put it more concretely:
The land & oceans are less chaotic than the atmosphere. The lithosphere & hydrosphere are different physical processes than the atmosphere, and are not the average of atmospheric variability. In fact, the atmosphere is thermodynamically dependent on the land & ocean.
The predictability (subjective definition) of the atmosphere implies nothing about the nature of the lithosphere, hydrosphere and other certain external factors.
VikingExplorer writes “The path of the river is a function of the topography and characteristics of the land, NOT the small scale fluid dynamics affecting the leaf.”
They are the same. Its just a question of time.
I see climate BVP as the boundary of the range of output values for all initial conditions, where the pause is an excellent example of an IVP, the output is dependent on initial conditions. In this case BVD’S are used in worst-case and sensitivity analysis of electronic equipment.
And for this example most will find it obvious as to why this is even a topic of discussion.
When I was thinking of boundaries, I was thinking of the drought in California. Southern California is a desert and is definitely in drought. But the North West boundary of CA is as far as I can tell not in a drought. And since they get most of the rain in California there, you would think that they would have most of the dams to catch the plentiful water which they could pump to the dry regions. But if you look at the damn dam map of dams in California, it looks like there are not many dams in the part of CA that gets the most rain – the northwest corner:
WUWT?:
http://www.kqed.org/news/science/climatewatch/waterandpower/map.jsp
What was the boundary of the “Dust Bowl”??
And if you look at the dams in that area, they are very small. Maybe Jerry Brown might want to build a new large dam in that area to help supply California with the water it needs since doubling it’s population since 1980. Is there someone with common sense that can run for governor and win??
Last I heard, they were still dumping millions of gallons of fresh water every day into the Sacramento Delta. I “smelt” a rat. Do you?
I smelt irony, which you often get from political ores.
There are areas within the “boundary” of California that average over 100″ of rain per year. (no dams in that area) It is near the upper northwest boundary of CA:
http://www.eldoradocountyweather.com/calprecip-full-size.html
Isn’t data from 1900 to 1960 close to a climate trend?
“If you think something is incorrect, please have the courtesy to quote the exact words that you disagree with so that everyone can understand your objections.”
Just the part between “I’ve” and “curious”.
But seriously, if someone could tell me the difference between j x omega and s then I’d be happy to explain the difference between BVP’s and IVP’s.
” Mandelbrot analyzed a number of long-term records and found no change in the fractal nature of the records with timespan. In other words, there’s no break between the chaotic nature of the short, medium, and long-term looks at weather.”
A few weeks ago, you linked a video here; the title was something like ‘5 things to believe before breakfast’ (?) – Near the end of that presentation, the were a few examples of time-lapses that were supposed to illustrate that some things that seem to be chaotic, e.g. changing occupation of a car park, traffic on a street do not look chaotic at all in the long run, and the point to take away from that analogy was that we simply haven’t observed climate for long enough to make a call as to whether changes in climate that we can detect in our short lives are even relevant.
That seems to go contrary to what Mandlbrot has found, because he he could only observe the rather short climate record, isn’t it? Our records are not “long term” by any stretch of the meaning…
Matt May 25, 2015 at 10:03 pm says
Sorry, but I have no idea what video you’re talking about. In any case, yes, I have to assume that there are types of records which are chaotic at short timescales and not chaotic at long timescales. The question is whether climate is of that type. Mandelbrot says no.
Thanks, Matt. If you followed the link to Climate Audit, Steve noted that:
So he was looking at records hundreds and thousands of years long, and he found the exact opposite of your claim. He found out that no, the chaotic nature of the weather does NOT change as you look at longer and longer time intervals. It’s turtles all the way down.
w.
“Big whorls have little whorls
That feed on their velocity.
And little whorls have lesser whorls
And so on to viscosity”
If I remember correctly, Steve was not convinced and was arguing against such a conclusion.
I am highly dubious of attempts to detect chaos mathematically (especially back in 1969).
>> the chaotic nature of the weather does NOT change as you look at longer and longer time intervals
The chaotic nature starts to disappear at rather short time scales:
The average temperature for today is X
April showers bring may flowers
Summer is hot, winter is cold.
As the reconstruction shows, the average temperature is almost a constant going back 10,000 years.
Something is causing the ice ages. It is NOT a random event produced by an unknowable cause. It is not caused by the chaos of fluid dynamics.
VikingExplorer
May 26, 2015 at 12:46 am
“The chaotic nature starts to disappear at rather short time scales:
The average temperature for today is X
April showers bring may flowers
Summer is hot, winter is cold.”
Sure, by that logic, chaos disappears in the climate system after about, say a second, since in that interval, whether it’s raining in the 1st hundredth of a second is highly correlated with whether its raining in the second hundredth of a second. It’s not whether a system has some degree of memory, it’s whether changes in the system are predictable or unpredictable. Stipulating some degree of resistance to change, over any arbitrarily defined interval of a second, a day, a year, etc, doesn’t address the issue.
“As the reconstruction shows, the average temperature is ALMOST a constant going back 10,000 years.” (emphasis added)
Now indicate whether those fluctuations about the constant are chaotic or predictable? If predictable let’s have the equation.
“Something is causing the ice ages. It is NOT a random event produced by an unknowable cause. It is not caused by the chaos of fluid dynamics.”
The question is whether the timing, duration, and depth of the ice age occurs chaotically. And why can’t it be caused by the chaos of fluid dynamics, when over such time scales the Earth’s crust moves fluidly due to continental drift, which according to some theories is what causes ice ages. And if continental drift is a chaotic fluid system, well there you go.
VikingExplorer wrote: “The chaotic nature starts to disappear at rather short time scales”.
That is correct, the chaos starts to average out on a time scale of a few weeks and largely disappears on time scales of a few months to a few decades. That is why we have the concept of climate and why it is possible for people to convince themselves that climate can be modeled. The problem is that when you go to even longer times scales it gets chaotic again, which appears to be what Mandelbrot was looking at. The modellers seem to ignore that.
I suspect that what is going on is that climate really is a boundary value problem with reasonably well constrained solutions on human time scales *IF* one boundary is taken as the surface of the ocean. But that boundary exhibits chaotic behavior on multidecadal and longer time scales, thus forcing the climate to be chaotic on those time scales.
“Something is causing the ice ages. … It is not caused by the chaos of fluid dynamics.”
You don’t know that, VikingExplorer. We don’t know what causes ice ages. There are many theories, none of which seem to be generally accepted. Some have chaotic aspects, usually with astronomical influences acting as a pacemaker but not the primary driver.
I meant atmospheric fluid dynamics. My point was that climate is not necessarily the “weather” (which is associated with atmospheric fluid dynamics) at longer timescales.
If the theory you mention is correct, it would confirm this idea. Now, if it turns out that the completely different physical processes (whether continental drift or solar dynamics) that control climate turn out also to be unpredictable, it will be bad luck. It will NOT be because weather (atmospheric fluid dynamics) was unpredictable at all time scales.
The bad logic would be:
premise: Physical Process A is chaotic and hard to predict
conclusion: An unknown physical process B is chaotic and hard to predict
non sequitur
I do know that Something is causing the ice ages, because I believe in causality.
This brings up a great point. I’ve been waiting for this.
The issue is what does “predictable” mean? It’s extremely subjective. Some crazy climate models are off by .1%, and anti AGW people universally declare them WRONG. Let’s look at the Boolean logic:
float ModelPrediction, Observation;
bool wrong = ModelPrediction != Observation;
Should we wonder why it’s always wrong?
With a different definition ofpredictable, I assert without fear of contradiction:
April showers bring may flowers
Summer is hot, winter is cold.
In fact, the person who asserts process X is unpredictable with the above Boolean definition of predictable is saying more about their own agenda and desires than they are about science.
They are constructing an argument which is actually:
premise: nothing is predictable (bool wrong = ModelPrediction != Observation;)
conclusion: climate is unpredictable
Mike M. May 26, 2015 at 8:52 am
No, Viking is not correct in the slightest, and this can easily be shown. As Dan Hughes commented below,
I invite both of you to actually do the indicated average calculations before you go further off the rails.
w.
Willis,
It is you that has gone off the rails in a chaotic manner.
“Averages of chaotic response are chaotic” is falsified by the sciences of Thermodynamics and Circuit Theory.
For some psychological reason, you want it to be true. But it isn’t.
Viking Explorer wrote “I am highly dubious of attempts to detect chaos mathematically (especially back in 1969).”
Why? I think there is a strong argument that back in 1969 people had to think through problems carefully in ways that’s simply not done now. These days we simply do enough thought to form it into a computable problems and let the computers spit out the answer.
I’m not sure that these days we’re getting better answers for everything …especially the questions that are essentially intractable.
“…..Keep changing their minds, the best way to confuse.
Turning around the terms they like to use;
Climate’s not weather, well, more often than not,
But weather can be climate! Are you still with the plot?…..”
Read more: http://wp.me/p3KQlH-cq
Thanks Willis, an informative discussion in the comments also.
It’s a boundary problem. The boundaries are the solid land surface, including sea floor, and the top of the atmosphere. Inside the boundaries we have the oceans, atmosphere, and a can of worms.
Fernando, as you may be aware, the First Law of Worms is that once you open any can of worms, you’ll need a bigger can to contain them again.
And sadly a corollary, the Recursive Law of Worms, projects that given the First Law of Worms, eventually the universe will be completely filled with nothing but huge cans of worms.
This has led to interesting scientific speculation on the true nature of the so-called “Dark Matter” …
w.
Very high cosmic radiation. It is better not to sunbathe in Canada.
http://sol.spacenvironment.net/raps_ops/current_files/rtimg/dose.15km.png
Yikes, is this a one time occurrence or is this normal?
I’m concerned for my fellow Norwegians…
Good example of alarmism. It turns out that orange is about 1.8 mRem, which is equivalent to watching a TV or computer screen.
Still got CRT’s have you?
Current usage isn’t necessary to put the data into perspective…
This will work in a long time. Add more UV light.
ren, what on earth does have to do with the topic at hand? It seems like a random interjection.
It has very much, because we enter into the solar minimum.
http://services.swpc.noaa.gov/images/goes-xray-flux.gif
Willis,
The “boundary value problem” here is the fact that for energy balance (stable temperatures over period t1) incoming absorbed solar radiation must balance the outgoing longwave radiation.
One of the problems is “what is t1?” as discussed in Natural Variability and Chaos – Four – The Thirty Year Myth.
Boundary value problems, as typically described in thermodynamics or heat transfer, usually mean “calculate the steady state solution given conditions A, B, C, etc”. Dynamic problems (initial value problems) usually mean “calculate the response as a function of time”.
That’s what I assumed the boundary condition was – with one caveat. Over long enough periods the heat rereleased by the nuclear reaction in the Earth’s core has to escape as well. Therefore, incoming absorbed solar radiation must balance the outgoing longwave radiation and the Earth’s internal heat escape.
But all this tells us is that, over a long enough time-period, negative feedbacks keep the climate stable. This tells us nothing about how long that required time-period is. And for practical purposes we need that time-period to be shorter than the lifetime of our infrastructure (e.g. about 50 years).
Can anyone explain why we think that the boundary conditions dominate over the initial conditions, over the period of half a century?
Thanks, SOD. That now gives (by my count) no less than five proposals for the nature of the “boundary” in the climate boundary problem. Your answer is very different from those of Dr. Spencer and Dr. Pielke, which in turn are different from each other.
I’m sure that you can see why, at the top of the head post, I said that I “woke up today thinking that I didn’t have an adequately clear understanding of the difference between the two types of problems” … and why this discussion is giving me more heat than light. You, Dr. Roy, and Dr. P are all among my scientific heroes, and yet you each give a totally different answer to the question.
What’s a po’ boy to do?
w.
Willis Eschenbach wrote:
“• The boundary between the atmosphere and outer space
• The boundary condition of a doubling of CO2 leading to a change in absorbed upwelling radiation”
and
“You, Dr. Roy, and Dr. P …each give a totally different answer to the question.”
I think SOD, Spencer, and Pielke are all giving essentially the same answer: The boundary radiation flux through the top of atmosphere and changing CO2 changes the boundary.
Pielke has clearly criticized the usefulness of that assumption and I think that both SOD and Spencer are also doubtful. That boundary should constrain the possible solutions, but does it constrain the solutions to a useful degree on a useful time scale? The computer models say “yes” but “observations seem to say “no”.
Going back to my experience with sensitivity analysis (which I didn’t do, but helped the engineer doing it), I think to formalize a BVP, you have to have an equation that defines the output based on the inputs, and then you can replace the terms with partial differentials, isolate the term, then you can solve it.
What’s missing for climate is the equation. You can use 2XCO2 = 3.7 W/m2, but we already know that isn’t enough to determine the sate of the climate, otherwise we would not have a pause to discuss.
Interesting. The “boundary” solution advocated by models of course has nothing to do with the freezing/melting point of water, but rather the theoretical fixed thermal enhancement properties of CO2. This is climatology’s version of a cosmological constant from which all other variables can be worked backwards from. Thus they argue that the non linear nature of the rest of the system is irrelevant because this fits inside the fixed calculation of greenhouse gasses hear trapping qualities which can be derived from x=y
This is a problem for all you “Luke Warmers” because deviation from x=y can be attributed to a fluctuation of the non linear “initial” state of weather/ natural variability, whilst the x=y “boundary” of GHG is adding a fixed input on top. 18 years no warming? No problem they say, we got the natural variability wrong. Sooner or later though that will swing the other way and x=y will pay you back with interest!
Happily for me, I am a proud “denier”. I refute the assertion that CO2 or any other gas has a magic heat trapping quality x=y (insert greenhouse equations instead) is a false premise to begin with. The boundary conditions are infact as follows:
1) Solar energy received
2) Albedo
3) Atmospheric Pressure.
Now you may add your initial state calculations for weather!
Excuse typos. Touch screen phone. I’m sure you can work out what I actually mean where typo has occurred.
Hello Willis,
¿Why does the “boundary” have to be in the future? Climate model simulations used by IPCC do not start “today”. The simulated climate starts around 1900. The boundary can be set with the known conditions in 2 different times IN THE PAST, say, for instance, 1900 and 2000 (which would explain why they “more or less” replicate correctly the 20th century climate but fail miserably with the climate in the 21st).
Willis,
try to solve the energy balance model on your PC. Then you know the difference between IVP and BVP. Unfortunately, the energy balance model doesn’t describe the surface temperature profile of the earth very well. So you know nothing.
The link l see between weather and climate is the amount of variation there is within the weather over the longer term. As long as there is a high degree of variation within the weather patterns over the longer term, then this will block any extreme change in the climate from taking place. Because any extremes that do happen are too short lived to have any real impact over the longer term. Extreme climate change is much more likely to happen when the weather patterns become far less variable over the longer term. As this would allow weather patterns to turn up often enough to cause large changes in the climate. lts this lack of variation in the weather patterns is what l believe leads to ice ages.
The biggest problem with the CAGW hypothesis is the assumption that 3.7watts/M^2 of CO2 forcing per CO2 doubling will cause a runaway positive feedback loop involving ever increasing levels of atmospheric water vapor GHG forcing.
To keep this positive runaway feedback loop from going to infinity, the CAGW hypothesis assumes that airborne particulates from fossil fuel combustion (always man the center of the universe) will miraculously offset the sum of the feedbacks from exceeding 1, which is the point where global temps go all Buzz Lightyear “to infinity and beyond”…
Nature abhors runway positive feedback loops as our very existence proves… Nature seeks equilibrium– not its own destruction.
Earth doesn’t become Venus because Earth’s mass is about 23% greater than Venus, which generates enough additional gravity to keep ocean water from being blown out to space, which is apparently what happened to Venus. Venus is also closer to Sun so it gets more solar radiation, stronger solar winds, and its atmospheric CO2 concentration is roughly 965,000ppm compared to Earth’s 400ppm…
Earth’s climate is basically sinusoidal with various large-scale fluxes involving Milankovitch Cycles (Obliquity, Orbital Eccentricity, Axial precession, etc.) and solar cycles (Grand Solar Minima, 1,000-yr super cycles, etc.) that usually (not always) bring about significant climatic changes.
I’m sure I’m missing something important, but I don’t see the problem in thinking of climate as a BVP system that stays within 2 standard deviations of mean seasonal values, until some major variable shift (obliquity/orbital eccentricity/axial precession/solar cycle,etc., and their infinite permutations) bring about significant energy imbalances that cause significant climate changes.
CO2’s paltry logarithmic forcing effect of 3.7 watts/M^2 per doubling just doesn’t seem to provide enough energy to create any significant change other than perhaps 0.5C~1C per CO2 doubling, which seems to be what’s happening..
Willis’ excellent posts on ocean temps/ocean evaporation/cloud formation/albedo flux have been very helpful in understanding the natural bounds within which the climate works..
Thanks, Willis! Your posts are always thought provoking, informative and interesting.
SAMURAI:
“The biggest problem with the CAGW hypothesis is the assumption that 3.7watts/M^2 of CO2 forcing per CO2 doubling will cause a runaway positive feedback loop involving ever increasing levels of atmospheric water vapor GHG forcing.
To keep this positive runaway feedback loop from going to infinity, the CAGW hypothesis assumes that airborne particulates from fossil fuel combustion (always man the center of the universe) will miraculously offset the sum of the feedbacks from exceeding 1, which is the point where global temps go all Buzz Lightyear “to infinity and beyond”…”
That is not the CAGW hypothesis.
The climate science general consensus on the result of CO2x2 forcing of 3.7 W/m^2 is that positive feedbacks from water vapor (primarily) and also the ice albedo effect (secondarily) will amplify the effect of 2xCO2. In the meantime, higher temperatures will cause more radiation, leading to a negative feedback. The overall result, as identified by said climate science general consensus, is a positive feedback, i.e., amplification, not a “runaway positive feedback loop”.
Can you cite even one paper published in any climate science journal in the last 30 years that claims a “runaway positive feedback loop” from doubled CO2?
ScienceOfDoom– CO2 forcing/doubling logarithmic equation is: 5.35watts/M^2*ln(560ppm/280ppm)=3.7watts/M^2* 0,31 (Stefan-Boltzmann constant)=1.1C gross net potential global warming per CO2 doubling.
CAGW then multiplies this 1.1C by a factor of 3 to 4 (runaway H2O feedback loop) to come up with their “best guess” estimate of 3C~4C after fiddling with negative airborne particulate negative feedback a to orient the runaway H20 feedback from going to infinity.
Here is an excellent lecture by Dr. Lindzen going over this and many other aspects climatology:
http://wattsupwiththat.com/2014/06/13/dr-richard-lindzens-talk-at-eike/
Enjoy!
I’ve read most of Lindzen’s papers. To save me the 50 minutes of watching the video – at what point does he say something to the effect of:
“CAGW then multiplies this 1.1C by a factor of 3 to 4 (runaway H2O feedback loop) to come up with their “best guess” estimate of 3C~4C after fiddling with negative airborne particulate negative feedback a to orient the runaway H20 feedback from going to infinity.”
And if he does, what papers does he cite in support of this claim?
Because – regardless of whether CAGW is correct or not – your description of CAGW is not correct. Here is an excellent – and free – review paper for everyone to read:
Water Vapor Feedback and Global Warming, I.M. Held & B.J. Soden, Annual Review of Energy and the Environment (2000)
Many erroneous ideas that spread around on blogs can be dispelled with just one good paper.
ScienceOfDoom–
Please watch Dr. Lindzen’s lecture from around 17 minutes onward.
Thanks for your links, which I’ll read.
Cheers.
Thanks for the link, Science of Doom. The Held and Sodden paper is very informative and easy to read. But now I’ve got to the maths bit and slowed down.
According to IPCC clouds have a negative feedback of -20 W/m^2, ten times more negative than the positive RF of CO2 at 2 W/m^2.
nickreality65:
“According to IPCC clouds have a negative feedback of -20 W/m^2, ten times more negative than the positive RF of CO2 at 2 W/m^2.”
That is the net effect of clouds. Not a negative feedback on increasing temperature. As you can see in Clouds and Water Vapor – Part One:
“Clouds reflect solar radiation by 48 W/m2 but reduce the outgoing longwave radiation (OLR) by 30 W/m2, therefore the average net effect of clouds – over this period at least – is to cool the climate by 18 W/m2. Note that these values are the global annual average..”
As the things stand now, neither weather prediction nor climate prediction fit the paradigm assumed in the article. Full set of hydrodinamic equations known as Navier-Stokes system is so complicated that it is even not known if it has a solution in general case – no theorem of existence or uniquness exists for this system. And also there are more than one boundary for coupled ocean-atmosphere circulation: ocean bed is one boundary and land surface is another, and temperature should be set at ocean surface as well. We simply cannot pose these boundary conditions, so such problem is mathematically intractable. In reality models for weather prediction and climate prediction are not formulated from these first principles, but from greatly simplified description without real knowledge how this simplification affects the results.
What is more important issue is whether the non-linear dynamics of ocean-ocean interaction is regular or chaotic, both cases are possible. If it is chaotic, no adequate mathematical modelling and no prediction is possible in principle for sufficiently long period of time. This is certainly the case with weather prediction (that is how the first strange attractor, the Lorentz attractor was discovered by a meteorologist Edward Lorentz during a routine weather prognosis calculation). For climate prediction this question of regular or chaotic dynamics is still open.
Weather prediction — relates to wind and pressure patterns; cold wave, heat wave, precipitation, etc — at short and medium range predictions — long range prediction is a hypothetical statistical prediction, the can predict within the coefficient of variation range and rarely predict extremes [like excess rainfall or deficit rainfall ranges]
Climate prediction — relates to homogeneous zones; with the area of study the coefficient of variation reduces and thus the extremes changes. For example, All-India Southwest Monsoon rainfall presents 60-year cycle and within the 30-year above the average period the excess rainfall will be expected in 5 to 7 years and deficit in 2 to 3 years and in the 30-year below the average period the deficit rainfall will be expected in 5 to 7 years and deficit in 2 to 3 years; and in each thus the rainfall is within normal. Also, same way with El Nino, out of 126 years, 18 are El Nino years; of which 7 are deficit years and 10 are normal years and one year is excess rainfall year. In the case of 24 La Nina years, 10 are excess rainfall years, 14 are normal years. In the case of undivided Andhra Pradesh annual rainfall [with two monsoons and cyclonic activity] present 132 years cycle. In the 66-year below the average period 24 years are deficit years and 12 are surplus years with 30 normal years; and in the case of 66-year above the average rainfall period 24 years are surplus, 12 years are deficit and 30-years are normal. So, these clearly specify the need to homogenizaton of the regions for predicting climate. However, these will be modified with the time with changes in land use and land cover.
Dr. S. Jeevananda Reddy
Those cyclical rainfall observations do seem to fit with Mandelbrot’s more general observations.