Out At The Boundaries

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.

242 thoughts on “Out At The Boundaries

  1. Climate prediction is a boundary value problem if you’ve already decided what the climate will look like at the end of the duration.
    Or am I being too cynical?

    • A climate model solution is both an initial value and a boundary value problem. One can not have a meaningful vector with out the affine connection. Differential Equations are basically linear transformations mod the kernel.

  2. Much thanks to you, Willis, for a very good discussion of the differences between an IVP and a BVP. More thanks to you for your explanation of Pielke’s claim that climate is an IVP.

  3. Living several miles inland from the Pacific Ocean, on a hill, in a valley I can speak with great authority that any boundary being contemplated twixt land and ocean is likely to be so large and/or vague as to render measure/reporting of initial conditions effectively meaningless.

  4. I’m surprised. I would think the boundary would really be the space/atmosphere. Land and water are part of the “inside” of the system. At the very least you have to include it along with land and water if you are just considering atmosphere and, if your taking that approach, possibly you should include ice.

    • Oh, and I’ve always considered initial value problems to be closed system problems where there is no extermal influence and boundary where you have multiple transferes of state at the edges of the system.

    • Bingo. Or if we are looking at f(t) then the boundary is the end of the Universe be it some final dissipation and then nothingness, or, an inflection point leading to a huge singular black hole.

  5. Weather predictions are largely fluid dynamics problems whereas climate prediction is largely a thermodynamics/heat transfer problem. Weather predictions require transient solutions (predictions marching through distinct time steps) and require boundary conditions in both time and space, including an initial value at all gridpoints as well as rules for interacting with boundaries, primarily the land/air interface, i.e. boundary conditions. Climate models, to my knowledge, are largely steady-state solutions for the entire closed system of the earths atmosphere at some future date- the boundary conditions in this case are the interaction between the upper atmosphere and outer space, as well as the atmosphere and the land/sea. In this sense boundary conditions are not actually a set of conditions in the future but a set of rules by which the model operates at the defined boundary.

    • “Climate models, to my knowledge, are largely steady-state solutions for the entire closed system of the earths atmosphere at some future date”
      Not GCMs. They are transient solutions of the Navier-Stokes equations. But as with a lot of CFD, that is a means to an end. The transient features on an hourly scale are not of interest – rather the progress of slower varying progress of variables with some conservation properties, that respond to forcings.

      • Right- but that is only because all CFD code is based on N-S. My point is that the transients that I used to look at when I did this stuff were all related to turbulence on scales that have no importance in GCMs. Im not sure what the time steps are for a GCM but I expect they are large, and in that sense you are simply finding one steady state solution and then the next one based on new initial conditions.

      • No, timesteps are about 20 minutes. The transient features are not of interest, but have to be modelled, even sound has to be resolved on the grid scale (which determines the 20 min step). It’s not pseudo-transient.

      • Why does it have to be such a small time scale (relatively speaking)? Surely there have been GCMs run at larger timescales- do they differ appreciably from the 20 min delta t runs? What phenomenon are being picked up at these time scales that affect the end result?

      • JPS,
        You have to resolve sound waves. They are the mechanism by which pressure gradients are created to stop fluid just collapsing. Actually, they impose hydrostatic pressure in the vertical, but that can’t be done in the horizontal. If you don’t get the grid-length waves right, the solution blows up.
        Sound wave transmission of pressure gradient is what makes the difference between subsonic and supersonic flow (with shocks etc).
        You can try various implicit mechanisms, but it’s hard to extend the time step much beyond that minimum.

      • Nick
        OK so the smallish time step is required for the stability of the solver given the obvious grid limitations and pressure gradients- Im still not sure why GCMs are solving a fluid problem at all. So back to the original point- for weather predictions it makes perfect sense that we need to track fluid movement over a defined space- climate in 50 or 100 years not so much. I mean to come to the conclusion that the Earth’s atmosphere will be 2C warmer in x years isnt that really just a thermodynamics/heat transfer problem?

      • JPS,
        GCM’s originated with weather forecasting programs. People saw that when they weren’t getting weather right aqny more, they were still showing a model with proper physics that responded to thermodynamic inputs.
        Here is an example showing just the SST component. It’s just Navier Stokes, with topography, sun etc. But it gets the recurrent patterns right – familiar currents – and this is important for climate heat transport.
        [youtube http://www.youtube.com/watch?v=aX9nHyMP4L0&w=420&h=315%5D

      • Sure, fluid dynamics are very important in the case you presented (Colorful Fluid Dynamics indeed!), but I feel like you are dodging my question- all of the GCMs I have seen predict a higher global spatially averaged temperature. Assuming the solution grid is reasonably fine, the mixing of the closed system that is the atmosphere should have no effect on the final result (spatial average), unless I am missing something.

      • The atmosphere isn’t a closed system. One of the big issues is the poleward transport of heat, which changes the total amount that is radiated at a given average temperature. Sea currents are an aspect of this.
        Grid size is related to timestep. You can’t diminish one without the other. Very roughly, the ratio is the speed of sound.

      • Nick writes “It’s just Navier Stokes, with topography, sun etc. But it gets the recurrent patterns right – familiar currents – and this is important for climate heat transport.”
        This is just straight out wishful thinking Nick. The “familiar” NS solutions have nothing to do with how climate changes and is why calculating climate change has nothing to do with calculating believable weather patterns. They’re two quite different questions.

  6. Climate prediction is palm reading. This is the loosest scam ever. As if anybody could control the climate or really cared about their hypothetical great grandchildren. What a bizarre time to exist.

    • How dare you insult the palm readers !!! They occasionally get something right, name me something that CAGW has. The odds of getting a 100% failure rate means they are using the wrong 1) equations 2) boundaries or 3) continuous twisted counterintuitive logic. Purely guessing would yield some correct results. Worse they contradict themselves all the time… depending on the argument.

  7. In climate predictions the boundary condition is the change of forcing over time, the most important is considered CO2. f= (5.35 ln ([CO2](t)/[CO2](0)), To test climate software therefore storylines (SRES, RCP) have been introduced. We are led to believe that the previously worst case scenario SRES A1FI is now a business as usual scenario called RCP8.5. These scenario’s are virtually identical, so whilst climate sensitivity is coming down, the previously worst case is now sold as business as usual.
    Look, look, be afraid: we are on the RCP8.5 track! Hiding that we are also still on the SRES A1B track (the real business as usual)

  8. Climate prediction is not a boundary value problem (BVP), irrespective of whether the boundary is time (between now and the end of the century, say), or atmosphere/surface, or ice/water, or anything else. This is because the problems are not well-posed.
    Well-posed BVPs have unique solutions that vary continuously with their inputs. No climate projection has ever been shown to be (or claimed to be, as far as I know) a unique solution to a well-posed BVP. This is why ensemble modelling is used: compute a load of things that could be BVP solutions, then take some sort of average.
    Well-posed BVPs do get solved all the time in areas like designing wings for aircraft, geophysics and electrostatics. Real science (where there’s a problem if the wings fall off, you drill in the wrong place, or your industrial painting doesn’t work), rather than climate pseudoscience (where observations are never matched to projections).
    In summary, whoever said that whereas weather prediction is an “initial-value” problem, climate prediction is a “boundary problem” wasn’t a mathematician.

    • “In summary, whoever said that whereas weather prediction is an “initial-value” problem, climate prediction is a “boundary problem” wasn’t a mathematician.”
      They certainly weren’t an engineer or anyone responsible for applying science to the real world. The problem is that the present climate “system” is a slowly moving target – it changes as the continents move around, it changed as the Earth way back in its formation picked up more mass, more water, etc., it changes as the “weather” slowly erodes mountain ranges, as rivers gouge out gorges, as ice dams break after ice ages, etc. All this begs the obvious questions: WHEN would the initial conditions be applied? Do they really think there was a point in time long, long ago when the “climate system” first fired up and started running from a standstill?
      It appears to me from the quotes in the original post that the climate scientists want to have a demarcation of “climate” as being the system that is produced by a “solution” to a set of equations given a certain set of boundary conditions, and mere “weather” as being the solution to the “climate” system given a certain set of initial conditions applied to the “solved” climate system, so that they never have to worry about specifying “initial conditions” – because they CAN’T specify those pesky things. The demarcation is one that is presupposed rather than observed, and done entirely for convenience rather than some actual reasoning. What they want is to justify conclusions as to how the climate (theorized to be completely represented by a finite set of known equations) will react to a given input without having to solve for weather, and to excuse any failure to predict actual climate states or outcomes.

  9. a good day of wondering and quite graphic for our imaginations,
    a skill I fear is lost by many who don’t make the efforts to read

  10. I have always been under the impression that when people described the climate problem as a “boundary value” problem it was with respect to the atmosphere/space interface and the principles of conservation of energy and spontaneous increase of entropy. Given that the shortwave flux incoming is known, the long wave flux outgoing must equal it in order to have long term temperature stability and satisfy the principle of conservation of energy; and the difference in entropy between the incoming and outgoing fluxes is determined by the difference in frequency of the fluxes and the transition from parallel incoming flux to radially distributed outgoing flux; so the total work done within the system and the total amount of entropy created within the system must be finite and determinable, assuming no long-term storage of energy is taking place and that there is no other significant source of energy than the solar flux. The resulting climate(s) over the face of the globe are constrained by the energy and entropy gradients available within the system together with the material constitution of the system. In short, what is possible within the system and what is inevitable within the system is determined by what happens at the boundary of the system, given that all the energy available to the system passes through the system without accumulation.

    • The entire AGW rests on incoming and out going energy. Which is based on the retention of heat by co2. See the arguments by ‘the science of doom’ on the greenhouse effect. And if you go by the numbers and this were the year 2000, then with the formula it is easy to predict so much warming from so much co2 added to the atmosphere. And I would agree with that, if it happened, it didn’t. The problem for CAGW is that co2 levels have skyrocketed and temps have fallen below their lowest estimates of rise. It is at this point impossible for them to rule out natural variation, at least if politics weren’t involved. With politics and God all things are possible.

      • No, you cannot possibly predict with a formula how much warming will result from increased CO2. The climate is too complex to solve analytically. One can only simulate with models. But the models have consistently over predicted as you say.

        • My point is that if the formulas were correct and this were the year 2000, then everything that the IPCC predicted in that year should have happened in view of the fact that co2 levels have consistently risen. AGW Theory is wrong. I maintain that the only way you can be 100% wrong is by using the wrong formula. It’s not a matter of fine tuning, AGW predictions aren’t even in the ballpark. They’re playing in a sandlot across town.

      • I disagree with this: “Given that the shortwave flux incoming is known, the long wave flux outgoing must equal it in order to have long term temperature stability and satisfy the principle of conservation of energy:
        You must account for the biological transformation of energy involved. The huge layers of carbonates, coal, and shale represent energy that has been transformed by biological processes. We release some of this heat through combustion of fuels. Thus the outgoing long wave energy MUST be lower than the incoming.

    • “…Given that the shortwave flux incoming is known, the long wave flux outgoing must equal it in order to have long term temperature stability and satisfy the principle of conservation of energy…”
      But because of significant and continual variations in both solar irradiation and albedo, the system never reaches true equilibrium, therefore climate stability is a chimera and energy conservation is irrelevant. From the fact that system temperatures remain in a relatively tight zone, we can conclude that feedbacks are net negative over the possible forcing regimes.

      • “From the fact that system temperatures remain in a relatively tight zone, we can conclude that feedbacks are net negative over the possible forcing regimes.”
        This is where I landed shortly after beginning my “engineer’s examination” of AGW theory and it’s where I’m stuck 20 years later. Were it not so, our earth would either have long since burned to a crisp or frozen onto a permanent lump of ice. The fact of “negative system feedback” to all climate temperature forcing functions is written with bold letters in the reconstituted temperature history of the planet.
        At the foundation of AGW theory lies the demonstrably false contention that increased greenhouse warming driven by man made CO2 is met with net “positive” climate system feedback (primarily via increased atmospheric water vapor). That contention simply cannot be true. If it were true, life as we know it would not exist today.
        Why this observation of reality did not constitute a stake in the heart of AGW theory shortly following the theory’s introduction mystifies me to this day.

  11. Thanks, Willis.
    I agree that Wolfram is a total genius. Dr. Pielke is another one.
    I think weather forecasting will be improved by satellite observations. Climate predictions, not so much, because of the chaotic nature of weather, with becomes climate in aggregate form.
    I have seen incredible advances in weather forecasting as practiced by Joe Bastardi. Another genius.
    The NWS does an acceptable job with 3-days forecasts, the 7-days forecasts, not so much.
    The hurricane seasons forecasts by Dr. Philip Klotzbach and Dr. William Gray of Colorado State University, are quite good. NOAA-NWS, not even in the same league.

    • If climate is weather averaged over 30 years, and if climate scientists first started paying attention to global warming in the mid-1980’s, then where did all this alleged fountain of climate knowledge come from? If I hypothesize that the amount of berries eaten by a bear affects the length of its hibernation, and I start watching a bear eat a bunch of berries in September, watch it go to sleep in November, jump up and yell “Eureka” in April right after it wakes up, and rush to publish my Master’s thesis, what does that say about the quality of my conclusions?
      If climate is weather averaged over 30 years, human beings can’t know squat about the climate system.

    • Yes, very much like at the dog and horse tracks, but with fewer horses’ front ends.

    • But why 30 years?
      Why not, 50 or 80 or 100 or 300 years etc?
      What is the scientific justification for defining climate as weather averaged over 30 years?
      That strikes at the heart of the cAGW debate. It seems to me that it stands on very shakey ground on that point alone, ie., climate is not weather averaged over a 30 year period.

      • IPCC AR5 Annex III: Glossary
        Climate
        Climate in a narrow sense is usually defined as the average weather, or more rigorously, as the statistical description in terms of the mean and variability of relevant quantities over a period of time ranging
        from months to thousands or millions of years. The classical period for averaging these variables is 30 years, as defined by the World Meteorological Organization. The relevant quantities are most often surface variables
        such as temperature, precipitation and wind. Climate in a wider sense is the state, including a statistical description, of the climate system.

      • Richard
        1) 30 years is the “productive” time between a PhD climate scientist getting a job and when he starts to retire.
        2) It was about the length of the “global warming” from ~ 1972 to ~1992- aka the rising portion of the 60 year PDO cycle.
        3) It is how far most people remember major weather.
        4) It is about how far predictions can be made that the predictor won’t be around to have them tested.

      • The definition of climate goes back to Classical Greece and Rome where “climate” was the overall environment an individual experienced living in a given location over their life time. Climate was considered as a summation of weather, plant community, and other biological and geological phenomena that characterized a region where one lived. You can see that a figure like “30 years” is a reasonable term for attempting to quantify a perceived change in climate as a “genuine” change. It is neither scientifically nor empirically justified. The climate “changes” if over your life time, things that you took for granted when you were young were dissipated, depleted or changed in some manner as you aged. The short of it is that “climate” is a reification of a social generalization about where we live. That doesn’t mean that there are not long term weather changes that lead to changes in climate, but the relation between “climate” and weather is inverted in climate science.

    • If that is so, then would not ‘climate’ also be an IV problem? I’m not sure how simply averaging the results of an IV problem over 30 years (still not clear on exactly what is being ‘averaged’) would transform it into a boundary problem. Additionally, since the IPCC also defines ‘climate’ as a coupled, non-linear, chaotic system, which are not ‘predictable’ in any practical sense, how does this description of the system enable you (generic reference) to declare ‘climate’ to be a boundary system?

      • ‘warrenlb’ will be along shortly to answer your cogent question with a few choice papers from the ‘peer reviewed’ literature he always uses for his argumentum ad verecundiam.
        /sarc

    • “IPCC & WMO define climate as weather averaged over 30 years.”
      Which is utterly unsuitable for a function which appears to follow a cycle of approximately sixty years, of course.
      More generally, it never fails to amaze me that “climate scientists” are obsessed with attempting to impose linear trends on phenomena that are clearly cyclic, the technique is to cherry-pick a suitable portion of the wave form, linearly regress it to Armageddon, declare that we’re all doomed and can only be saved by giving the Government more money.
      Nice work if you can get it.

  12. Solar energy makes it through the atmosphere (other than ozone and clouds and sulfur dioxide from volcanoes and some other more minor intercepters 15% may be making it through) and then it mostly gets absorbed by the surface liquid and solid surfaces (less how much gets reflected by those liquid and solid surfaces which is another 20% of the 75% that makes it through).
    After that, it is supposed to get simpler. But now we have a number of boundary problems to deal with. How does the gaseous atmosphere interact with those liquid and solid surfaces. We are talking about a tirillion, trillion, trillion, trillion molecules. How much energy in the terms of 10^27 joules do those solid and liquid surfaces already hold. What happens if a small 10^23 joules radiation forcing changes when the surfaces are already holding 10^27 joules. Atmospheric windows, water vapor etc.
    I mean, that is all a very big boundary problem which sounds impossible to solve in my opinion.

  13. Willis – This is a very important topic; thanks for posting on it. For another perspective, see this paper by Shaun Lovejoy –
    Lovejoy, S., 2013: What is Climate?: EOS, 94, No. 1, 1 January 2013, p1-2. http://onlinelibrary.wiley.com/doi/10.1002/2013EO010001/pdf
    His more recent work seeks to ferret out the human part where he assumes the climate models accurately replicate the natural climate system as well as that the human role is dominated by CO2. Both of those assumptions are, in my view, wrong.
    However, his basic analysis on the chaotic character of the climate system is correct. It is an initial value problem (or to be more precise an initial-boundary value problem as some components of the climate system are nearly invariant on multi-decadal time scales (e.g. terrain).
    This paper of ours might be of use on this subject:
    Rial, J., R.A. Pielke Sr., M. Beniston, M. Claussen, J. Canadell, P. Cox, H. Held, N. de Noblet-Ducoudre, R. Prinn, J. Reynolds, and J.D. Salas, 2004: Nonlinearities, feedbacks and critical thresholds within the Earth’s climate system. Climatic Change, 65, 11-38. http://pielkeclimatesci.wordpress.com/files/2009/10/r-260.pdf
    See also my posts
    https://pielkeclimatesci.wordpress.com/2011/01/23/recommended-reading-on-whether-climate-is-an-initial-value-problem/
    https://pielkeclimatesci.wordpress.com/2006/12/22/further-comments-demonstrating-that-climate-prediction-is-an-initial-value-problem/
    https://pielkeclimatesci.wordpress.com/2011/04/29/climate-science-myths-and-misconceptions-post-4-on-climate-prediction-as-an-boundary-value-problem/
    https://pielkeclimatesci.wordpress.com/2006/10/27/the-consequences-of-nonlinearities-in-the-earths-climate-system/
    [there are more posts, but this will provide examples of why climate is not just a boundary value problem as the IPCC and others assume.
    Roger Sr.

    • My thanks to you as always, Dr. P, and I hope I have not misrepresented your findings. In particular, thanks for the list of your posts on the subject I can see I have my reading cut out for me for a bit of time …
      w.

    • Initial value or boundary problem ?
      Dr. Pielke
      In the N. Atlantic area it appears to be both, as far as I understand it.
      boundary value – During summer months the Arctic atmospheric pressure component moves to the north of Iceland following the ice retreat.
      initial value – The N. Atlantic SST amplitude oscillation (throughout the year) appear to ‘follow’ the Arctic atmospheric pressure (summer) amplitude oscillations but with variable delay, which has been progressively increasing, presumably linked to some past initial state of the Arctic ice boundary (perhaps the deep ocean floor current ‘northern icelandic jet’ takes longer and longer to reach the critical subpolar gyre area?) .
      In this Illustration
      – graph A shows the ‘real time’ relationship between Arctic atmospheric pressure and the N. Atlantic SST
      – graph B shows relationship as it would exist if the ‘boundary condition’ was time invariant.
      note: the above does not mean that two variables are subject of a direct cause-consequence relationship.

      • What is the basis of your ‘tectonic records’ plot in the 3rd graph in your illustration?

      • I intend to write an article sometime, sources will be quoted and data tabulated, unless Chinese ménage to bug my PC before I do it.

    • At risk of sounding a pedant, the meaning of a boundary is well established.
      If a partial differential equation is solved in a filed, the boundary value ddefines how the equations are solved at the boundary of the field. These are Direchelet (the value of the solved variable is specified at the boundary), Neumann (The derivative normal to the boundary is specified) or Cauchy (the value and normal derivative are specified).
      If the boundary conditions are specified, this does not mean that the values at the boundaries cannot change, except in the Dirichelet case. The boundary conditions specify how the model interacts with what is ouside it and solution of the model involves solution of what happens at its boundaries.
      If one was going to model something to do with Oceans: the shore would be one boundary, the surface another boundary and the sea bed another boundary. If, say heat flux was modelled, the flux across the shore and sea bed would be considered to be zero and hence the flux along the spatial derivative would be zero (Neumann) or the temperature of the sea bed could be considered to be a constant with the heat flow determined by the assumed value and the water temperature (Dirichelet). The heat flux along the surface of the Ocean would be normal and specified (Neumann). There could be many effects that are created on land and affect the oceans but these only affect the boundaries of the ocean model.
      In otherwords, the boundary value is an assumption about what happens at the edge of the field you are considering.
      The initial conditions are the “start up” values of the system both within the field of the model and at its boundaries/ One could assume the oceans to be frozen and let them heat up due to energy flux across the surface boundary. Alternatively, one could set a mean temperature gradient throughout the model and let it run to a steady state. This would be a faster method. Again, one could make the initial conditions the measured temperature and bondary fluxes in the Oceans (assuming one could measure them) and then suddenly impose a change in the surface flux to see what happens.

  14. I cannot answer your questions, but can add something related to your excellent post. The weather is mathematically chaotic, and so even a very small difference in assumed initial conditions can result in a very large difference in the weather forecast. The way they (weather forecasters) have greatly improved weather forecasts in recent years is to use ensembles of initial conditions and run many weather forecasts and then throw out the outliers. The problem is, even with small errors in current atmospheric conditions like barometer readings, humidity, etc., or even small inaccuracies in the extrapolation of that data from weather station points across areas, large deviations from reality may result in forecast models because of the chaotic property of the weather.

  15. The boundary value referred to is the changing IR cooling of the climate system due to increasing co2.

    • Willis, this comment ( Thanks Doc) would relate to my belated reply to your prior response in a recent post.
      My question related to a saturation point with respect to CO2 levels and IR/LWR relationships.

    • Roy Spencer May 25, 2015 at 5:55 pm

      The boundary value referred to is the changing IR cooling of the climate system due to increasing co2.

      Thanks for that, Dr. Roy. We now have (I think) four candidates for the boundary:
      • The ice-water boundary
      • The boundary between the surface and the atmosphere
      • The boundary between the atmosphere and outer space
      • The boundary condition of a doubling of CO2 leading to a change in absorbed upwelling radiation
      The problem that I see with all of these is well described in the Wolfram Reference:

      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.

      It’s also not clear to me that 2XCO2 = 3.7 W/m2 increased absorption is a sufficiently strong boundary condition to allow it to be the secret key to unraveling the climate future. If that is truly a boundary condition, then so is G = 9.8 m/sec2, the force of gravity. Knowing the boundary condition of the force of gravity doesn’t magically make future climates predictable …
      Regards,
      w.

      • Anthony should shut you down. “Knowing the boundary condition of the force of gravity”, seriously?

      • Dinostratus May 25, 2015 at 10:04 pm Edit

        Anthony should shut you down. “Knowing the boundary condition of the force of gravity”, seriously?

        Thanks, Dinostratus. Yes, quite seriously. As I understand it, Dr. Roy proposed that the relevant boundary condition is
        2 x CO2 = 3.7 W/m2
        If such a simple mathematical statement can be a “boundary condition”, I wondered, then what other simple mathematical statements could serve as a boundary condition?
        I saw that the obvious one was that G = 9.8 m/sec2. And in fact, as I understand it, this is as valid a boundary condition as is the change in forcing from a doubling of CO2.
        My question to Dr. Roy was not whether the change in forcing from a doubling was indeed a boundary condition. It was whether it is a sufficient boundary condition to unlock the mystery of the future of climate evolution.
        w.

      • I think that you should have said the “acceleration” of gravity, which is 9.8 m/s2 at the Earth’s surface.

      • Is there another boundary condition between incoming solar and the bio-mass. Does creation of more bio-mass store the missing heat? Is heat lost through other natural chemical reactions taking place because of increased concentration of Co2

      • I’m not saying that changing one of the boundary conditions of the climate system makes it predictable. Only trying to clear up the confusion regarding why global warming is considered a boundary value problem: if the rules by which the atmosphere operates changes (a 1-2% forced reduction in its IR cooling rate), then the system will change. It might not change by MUCH compared to all the other chaotic variations going on. But that’s the boundary value that’s being referred to.

      • Ice-water, land atmosphere etc should more accurately be called interfacial locations with fluxes both ways. In a mathematical sense, calling them boundaries is misleading.

      • Gravity is a very pertinent boundary issue and influences many aspects of reality on planet Earth that are not immediately obvious, every thing from the location and shape of the mean surface of the oceans to the “angle of repose” in unconsolidated material, and IIRC the atmospheric lapse rates.

      • “If such a simple mathematical statement can be a “boundary condition”
        It’s not a boundary condition. It’s an equation. Beyond that a better but less precise description would be “constitutive relation” but since the units don’t match it’s not really that. Maybe “short hand for for something in Hottel’s book” would be a better monicker.
        A boundary condition is something typically applies to a harmonic or even biharmonic equation which is why I brought up “j x omega” in another reply.
        Seriously though, it’s not the internet’s job to explain to you what someone might have meant by “weather prediction is an “initial-value” problem, climate prediction is a “boundary problem”. There are books. You can read them. Some of them will have Euler’s formula. If you’re a double smarty pants you’ll figure out how to use the Lapace transform tables to solve BVP’s using Euler’s formula. If you’re really, really good, you’ll probably figure out that climate predictions are quasi-equilibrium problems and then maybe you’ll think about how weather problems require explicit solver mechanisms while climate problems are amenable to implicit solvers. Then who knows, maybe you’ll figure out that weather problems are parabolic in nature while climate problems are more elliptical, etc. etc. etc.
        But again, it’s not the internet’s job to teach you and I suppose it’s the arrogance of assuming that anyone should give a darn about your misunderstandings that gets on my nerves. In other words, do your own homework.

    • Roy – I agree. They interpret the addition of CO2 from human activities as a boundary forcing which dominates the subsequent multi-decadal changes in regional and global climate. This changing climate moves toward a new “equllibrium”. Efforts are, therefore, needed to “stabilize” the climate system.
      However, this simplistic view is incorrect. The climate is never stable even in the absence of human intervention, as we documented in our paper
      Rial, J., R.A. Pielke Sr., M. Beniston, M. Claussen, J. Canadell, P. Cox, H. Held, N. de Noblet-Ducoudre, R. Prinn, J. Reynolds, and J.D. Salas, 2004: Nonlinearities, feedbacks and critical thresholds within the Earth’s climate system. Climatic Change, 65, 11-38. http://pielkeclimatesci.wordpress.com/files/2009/10/r-260.pdf
      The abstract starts with
      “The Earth’s climate system is highly nonlinear: inputs and outputs are not proportional,
      change is often episodic and abrupt, rather than slow and gradual…”
      The CMIP5 and other climate projection models fail to properly replicate this natural climate’s chaotic behavior. When human’s perturb the system (from added, CO2, aerosols, land use change, ect), we, therefore, should have concerns about the robustness and realism of the model forecasts.
      Roger Sr.

  16. It might be an issue of the scale on which the prediction is made.
    If you are trying to predict climate for the next 100 years, its unlikely the land / sea boundary will change very much, probably even the ice cover / ice free boundary won’t change very much.
    But if you are trying to built a climate model which can handle 10s or 100s of millions of years, there will be substantial changes over this period to the land / sea distribution.
    Trying to build a special case climate model which can handle the next 100 years, is probably a different class of problem to trying to build a general climate model which can accurately replicate climate change over millions of years – though without understanding of the general case, its difficult to see how you could create a reliable special case model, without a lot of empirical tuning (say 100 years worth 😉 ).

    • AFAIK, no one is trying to model the climate for 10s of millions of years. Nor is anyone trying to predict the climate in 100 years. The only ongoing effort is to confuse people sufficiently long for Big Socialism to take control of global power (with the assistance of their MSM lickeys.)

  17. The symbolic solution of both IVPs and BVPs requires knowledge of the general solution for the problem.
    Do we have the requisite knowledge of the general solution to know?

  18. “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. ”
    That’s all meaningless waffling. As long as they define climate as a 30 year average of weather, it’s the same thing, only run through a lowpass filter. An average is just a lowpass filter. So it’s the same problem. You wanna know what the climate in 2100 is; run a weather model for 85 years and run a 30 year average over it.
    They are just waffling to sidestep the tiny issue that weather prediction beyond 10 days is impossible. Too bad for them.
    And, an average is a weak low pass filter, and, a chaotic nonlinear system will shift energy from high to low and from low to high frequencies as it is NONLINEAR. So you will have chaos in the low frequencies as well, rendering the entire modeling enterprise predictive-skill-less.

    • Well Dirk, I would hardly call an “average” to be a “weak” low pass filter.
      I can’t Imagine a stronger low pass filter, if there is one.
      The “average” algorithm simply removes ALL frequencies from the signal, having any frequency value greater that zero frequency.
      The average value of any signal is simply the DC component of that signal.
      A low pass filter through which no signal having a frequency greater than zero, can propagate with an amplitude greater than zero, is hardly a “weak” filter.

    • Dirk. A moving average scheme introduces false periodicities. This is called the Slutsky-Yule Effect. So when I see periodic behavior in a moving average fitted to data I always wonder how much of the periodicity is not real, but just periodicity introduced by the moving average method.

    • DirkH
      Your main point is that climate and weather are both IVPs and you say

      They are just waffling to sidestep the tiny issue that weather prediction beyond 10 days is impossible. Too bad for them.

      You say

      As long as they define climate as a 30 year average of weather, it’s the same thing, only run through a lowpass filter. An average is just a lowpass filter. So it’s the same problem. You wanna know what the climate in 2100 is; run a weather model for 85 years and run a 30 year average over it.

      The issue you raise is both more clear and stronger than you state.
      This is because “they” do NOT “define climate as a 30 year average of weather”.

      For example, the IPCC glossary gives this definition of climate.

      Climate
      Climate in a narrow sense is usually defined as the average weather, or more rigorously, as the statistical description in terms of the mean and variability of relevant quantities over a period of time ranging from months to thousands or millions of years. The classical period for averaging these variables is 30 years, as defined by the World Meteorological Organization. The relevant quantities are most often surface variables
      such as temperature, precipitation, and wind. Climate in a wider sense is the state, including a statistical description, of the climate system. In various chapters in this report different averaging periods, such as a period of 20 years, are also used.

      A “period of time ranging from months to thousands or millions of years” provides an infinite number of possible averages for description of a climate.
      A different definition of climate needs to be specified, agreed and adopted if the true boundary conditions of a climate are to be known. Until that postulated new definition exists, climate will remain a loosely specified extension of weather and, therefore, any description of the evolution of climate will be an initial value problem (IVP) and NOT a boundary level problem (BVP).
      Richard

      • This is what I was getting at, in my comments on the Dr Ball’s recent article on climate change d******
        Given tht climate consists of an array of parameters (temperature being just one of many factors) that constantly change, it follows that change in itself is not climate change, nor even evidence of climate change (although it could be evidence depending upon the extent or rate of change).
        Presently, in the late 20th/early 21st century, we are not seeing climate change. All we are seeing/witnessing is climate, and nothing more than that.
        Obviously, the Holocene epoch has its own climate. I am not fully convinced that the MWP and/or the LIA are different climate regimes. Sure one was warm and the other rather cool, but even within these periods there was significant variability. For example, not every year in the LIA was a cold year etc. But minor variations in temperature of a few tenths of a degree is not climate change. If it were, in the case of the UK, one would be forced to conclude that Scotland has a different climate to the boarders/north of England, that has a different climate to Wales, that has a different climate to central England, that has a different climate to south west England, that has a different climate to south east England etc. Of course there are variations in the UK climate because of topography and geographical location etc, but it would be absurd to claim that the UK has 6 or 7 distict climate zones. There are minor regional variations to some parameters, but that is all.
        We cannot begin to discuss climate change until we first fully understand and are able to define precisely what is and what we mean by climate. That is the starting point to any debate on climate change.
        This is one reason why the warmists should not be permitted to move the goal posts. The so called physics and settled science of CO2 is that it is a so called greenhouse gas and that an increase in concentration of this gas in the Earth’s atmosphere MUST ALWAYS lead to warming (all other factors remaining constant).
        That is a scientific proposition and one that can be observed and tested, but one that will always be difficult to test since we first have to know the extent of “all other factors”, we need to be able to measure “all other factors” to see what if any changes have occurred and we need to know the effect of chnages in the “other factors” to see whether this counteracts or amplifies the effect of rising concentrations of CO2.
        The so called physics and settled science of CO2 is not that it inevitably leads to climate change, and that is why the debate ought not to centre on climate change or climate disruption, or weather weirdening.

      • “Climate in a wider sense is the state, including a statistical description, of the climate system. ”
        That seems like a circular definition, at least to me.

      • dccowboy
        Yes, you are emphasising the point that I and richard verney have been making.
        I repeat my earlier conclusion.
        A different definition of climate needs to be specified, agreed and adopted if the true boundary conditions of a climate are to be known. Until that postulated new definition exists, climate will remain a loosely specified extension of weather and, therefore, any description of the evolution of climate will be an initial value problem (IVP) and NOT a boundary level problem (BVP).
        Richard

      • Here’s a definition of Climate from 1901, a time of innocence and relative simplicity…
        “Outlines of Physiography” by the geographer Andrew John Herbertson [OPAH]:
        “By climate we mean the average weather as ascertained by many years’ observations. Climate also takes into account the extreme weather experienced during that period. Climate is what on an average we may expect, weather is what we actually get.”
        http://quoteinvestigator.com/2012/06/24/climate-vs-weather/

      • Yirgach
        Yes, but how many years are to be averaged and what kind of average?
        If those parameters are not defined then there are an infinite number of climates that can be calculated as existing at any time and place.
        Richard

  19. Beware, beware, their computer models,
    their society chairs.
    Weave a challenge around them thrice,
    And advertise by whom they’re led,
    For they on research funds have fed,
    And drunk the milk of UN advice
    with apologies to Samuel Coleridge

  20. How would a dog define climate?
    How can a creature, Homo Sapiens Sapiens who has existed less than 0.004 of 1 percent of Earth History define climate?
    Answer: Not with any respect to or by any definition of Hell.

  21. How can climate be a boundary value problem if climate is the average of weather, which is an initial value problem?
    Time can’t be the boundary. I don’t see any boundary to the climate of my mid-west USA location or any other location, given enough time. After my neighborhood has put the glacial/interglacial cycles in the rearview mirror, the climate in my neck of the woods could become desert, Mediterranean, tropical; who knows? Now that’s just my regional climate. What will the global climate be, if there is such a beastie; snowball earth, a lava lamp?

    • How can climate be a boundary value problem if climate is the average of weather, which is an initial value problem?

      That is exactly the point. Because it is a long-term average, the initial values are not important. The boundary conditions are the overall constants that keep the system constrained. So short-term predictions depend on initial values, long-term predictions depend on constraining values

  22. Weather predictions require the exact weather conditions as a starting point. I.e. they take the most recent measurements, put them on a grid (with as small a mesh as practically possible) and let a supercomputer calculate how the system will progress step by step.
    Climate predictions don’t care whether or not it rains today or not. The goal is to predict the long-term average and then it does not matter whether your starting point happens to be a cold autumn day or an extreme heat wave or whatever. Much more important are the factors that keep things stable in the long term. These are the boundaries that are referred to. It may be a bit of a confusing term, since you shouldn’t so much think of them as boundaries in space or time, but rather as the constants or constraints that the system is subjected to (constraints would probably be a better term imho). For instance the average power the earth receives from the sun, the rotational speed of the earth, et cetera. So basically all the factors that the determine what the average conditions will be irrespective of the extremes that may happen from time to time.
    I find this a pretty good explanation (also gives a good explanation of forcings and feedback):
    http://www.easterbrook.ca/steve/2010/01/initial-value-vs-boundary-value-problems/

    • Aran,
      Interesting link. The crux of the matter seems to me to be the author’s claim that “But if the boundary conditions are right, eventually the simulation will settle down into a stable climate.” I don’t think we know that this is true for the climate, although it does seem to be true for climate models.
      If you look at the power spectrum of temperature, at short time scales (weeks or less) it looks like red noise (random or chaotic) but at longer times scales, up to at least a decade or three, it looks close to white noise (variation around a stable value). That is true for real Earth and model Earth. But if you go out to centuries or longer (and possibly as short as a few decades), the spectrum for real Earth reverts to red noise while that of model Earth remains close to white noise.
      So although the models exhibit the well bounded solutions that make the problem solvable, it is not clear that they are solving the correct problem.

      • Yes, it is this “settling down to a stable climate” that is a leap of faith. There are some aspects of the world where the aggregate behavior of lower level phenomenon averages out to simpler behavior at a higher level of observation. e.g. we can come up with equations dealing with the flow of fluids even if the motion of the individual molecules involved may be chaotic, the same with the flow of heat. However it isn’t always the case in every aspect of the world that the behavior of lower level phenomenon can sort of average out over the long run. Instead in some cases , the end results of a system can vary quite a bit depending on small changes at a lower level. There can be tipping points where a system will head in one direction or another. Climate modelers make a leap of faith that merely because they build models that in a way that things tend to average out, that the real world will match that. It isn’t something they have proven, its merely something they wish to believe because it justifies their approach to modeling. It is an assumption that they don’t question but don’t wish to acknowledge is merely a leap of faith. Unfortunately there are other implicit leaps of faith they make, such as that somehow uncertain and incomplete models of lower level processes like clouds can still magically lead to the right answer for the wrong reasons. They merely try to obfuscate these leaps of faith so those that aren’t paying close attention miss the existence of assumptions that are subject to being questioned. They seem to assume that “if an assumption allows us to create a model that will vaugely match the data, the assumption must be ok to make”, which isn’t the case. For instance their assumption they can create models without an understanding of various long term climate processes because they assume without evidence that those long term processes never have an effect on shorter time scales.

      • I would say the author’s claim (that the simulation will settle down to a stable solution) is true. The big question is whether this solution will be a good approximation of what happens in reality. As with all models, the climate models are far from complete, and some of the boundary conditions are not well known. This is even more of a problem in many other fields. For instance in economics the track records for predictive models are way worse. I suppose one of the reasons some climate modelers are confident that the climate system can be modeled with at least some accuracy, is that at least the underlying physical processes are pretty well known, which is why we can make fairly accurate short term predictions. This is not the case in economics. The main problem for climate modelling is getting the boundary conditions right, which is quite a challenge since we’ve only been monitoring the entire system to some degree for a very short time (compared to the time over which we try to predict).

    • Aran
      You say

      Climate predictions don’t care whether or not it rains today or not. The goal is to predict the long-term average and then it does not matter whether your starting point happens to be a cold autumn day or an extreme heat wave or whatever.

      Sorry, but No.
      If you have no clear definition of “the long-term average” then you only have evolution of weather to assess future climate. Please see my above post that is here.
      Richard

      • Sorry, but yes. If your prediction is in the form of a long term average, the initial values become pretty much irrelevant. Whether this long term is 30 years, 100 years or 10,000 years makes no real difference in that respect.
        This is simply a mathematical statement that goes beyond the specific case of climate predictions. The longer the period you aggregate over, the less important initial values become wrt boundary conditions. It also holds vice versa, for short term predictions, the boundary values are pretty much irrelevant compared to the initial values.

      • Aran
        You say to me

        If your prediction is in the form of a long term average, the initial values become pretty much irrelevant. Whether this long term is 30 years, 100 years or 10,000 years makes no real difference in that respect.
        This is simply a mathematical statement that goes beyond the specific case of climate predictions.

        No and no!
        Each of your different time periods (30 years, 100 years or 10,000 years) would provide a different result. This is because climate as you describe it IS evolution of weather. You are describing a long-term weather forecast similar to “a wet week”, “a hot year”, “a cold decade”. etc.. Weather is a IVP.
        And without an agreed meaning of “average” your statement is NOT ” a mathematical statement”: it is handwaving.
        Richard

      • MikeB and Viking Explorer
        In addition to your cheerleading, perhaps you could say what you found cogent about Aran’s assertions?
        Richard

      • Richard,
        I agree they would give different results, but that was not the point I made. I did not claim they would give the same results. The point was that the initial values will be irrelevant to the results. For all those cases it will be the boundary values that determine the outcome, not the initial values, which makes it a BVP not an IVP.
        There is no need for a definition of average (which by the way, is well-defined as the sum of observations divided by the number of observations), since the statement holds true not only for the average, but for any aggregate function. Some of the outcomes of a climate model will not be averages, but frequencies or variances etc. For all those predictions the statement will still hold: The longer the period over which you predict, the more the importance of boundary values will grow and the more the importance of initial values will diminish.
        This is not a hand-waving statement it is a generic statement. It holds for any prediction (not just climate) and any aggregation (not just averaging). Therefore, however you define your climate average is irrelevant, as long as it is a long-term aggregation it will always be a BVP.

      • Aran
        It seems we are ‘talking past’ each other, and both of us need to reconsider if we are to progress. We are discussing to gain understanding and evaluation: if we succeed then every body wins, but if we try to defeat each other then nobody can win.
        Clearly, we need to understand what each other is saying. I will spell out what I see as our disagreements and champion my understanding.
        I agree that you had said

        I agree they would give different results, but that was not the point I made. I did not claim they would give the same results. The point was that the initial values will be irrelevant to the results.

        My point was about WHY they would give different results, and it is BECAUSE those different results demonstrate the initial values DETERMINE there is not one unique value.
        And you continue

        For all those cases it will be the boundary values that determine the outcome, not the initial values, which makes it a BVP not an IVP.

        And that is the crux of our dispute.
        I say the boundary values of climate differ depending on the length of time assessed because the system evolves with time. That is why “average weather” differs depending on the length of time assessed. And at any point in time the boundary conditions have evolved from a previous time.
        All future time starts from now and, therefore, the boundary conditions at any point in time have evolved from the system that exists now. Determination of that evolution of the system from today is weather forecasting. And weather forecasting is an IVP.

        You reply goes on

        There is no need for a definition of average (which by the way, is well-defined as the sum of observations divided by the number of observations), since the statement holds true not only for the average, but for any aggregate function. Some of the outcomes of a climate model will not be averages, but frequencies or variances etc. For all those predictions the statement will still hold: The longer the period over which you predict, the more the importance of boundary values will grow and the more the importance of initial values will diminish.

        Sorry, but NO!
        You have stated the definition of the arithmetic mean. The median and the mode are also averages. There are an infinite number of possible averages.
        If your model is considering “aggregate functions” then the definition of whatever you mean by “average” is crucially important.
        And the values of the boundary conditions alter as the system evolves but their “importance” does not change.
        Please note (n.b. this is not point scoring) that I understand you to admit that climate is an IVP when you say “the importance of initial values will diminish”.
        Climate is an IVP or it is not. I say it is and you – perhaps by use of unfortunate wording – admit it is.
        You conclude saying

        This is not a hand-waving statement it is a generic statement. It holds for any prediction (not just climate) and any aggregation (not just averaging). Therefore, however you define your climate average is irrelevant, as long as it is a long-term aggregation it will always be a BVP

        That is pure handwaving assertion into which you have inserted the falsehood that “however you define your climate average is irrelevant”: different definitions of “average” provide different results.
        I hope I have clearly stated our disagreements.
        Richard

      • Richard,
        Reading your post I think I may have found we have been arguing over a definition rather than anything else. So I will not comment on all parts of your post but rather try to get something clear first. Do you consider the time over which the prediction is made to be an initial value? I get that impression from your first comment. If so we will never agree.
        The definition of initial values in the context of IVP and BVP is: the values of your variables at t=0, i.e. at the starting point of the time period you are modeling over.
        (If you disagree, at the very least this is definition as I intended in my original post and I’m sure from the text that it is the definition used in the link in my original)
        Let me try to make a few other things clear, which I may not have expressed clearly before, causing some confusion:
        1. The solution to every problem depends on both the initial values in the boundary conditions
        2. The terms IVP and BVP are used for cases where the outcome is determined predominantly by initial values or boundary conditions respectively.
        3. For an IVP, changing the boundary conditions will affect the outcome, but only to a very small degree, i.e. the umpteenth decimal or something like that. Changing the initial values, however, will affect the outcome very strongly.
        4. Same for BVP but vice versa, so strong effect from boundary values, small effect from initial values.
        5. If I want to predict the weather, say for tomorrow, the initial values (i.e. the weather today) are extremely important. The situation today is the major contributor to the outcome, making it an IVP.
        6. If I want to predict climate, say over the next 30 years, the initial values (i.e. the weather today) are hardly important, since I will aggregate over 30 years. What will be important are the processes that regulate the system, i.e. the boundary conditions, making climate prediction a BVP.
        Take any climate model, change the initial values and the outcome will be almost the same.

      • Aran
        You ask me

        Do you consider the time over which the prediction is made to be an initial value?

        NO! I do not. And I fail to understand how you claim to understand I do think that.
        When predicting climate the initial value is the state of the climate system at the time from which climate is predicted.
        In my first post addressed to you I said

        Each of your different time periods (30 years, 100 years or 10,000 years) would provide a different result. This is because climate as you describe it IS evolution of weather. You are describing a long-term weather forecast similar to “a wet week”, “a hot year”, “a cold decade”. etc.. Weather is a IVP.

        And in my next post to you I wrote

        I say the boundary values of climate differ depending on the length of time assessed because the system evolves with time. That is why “average weather” differs depending on the length of time assessed. And at any point in time the boundary conditions have evolved from a previous time.
        All future time starts from now and, therefore, the boundary conditions at any point in time have evolved from the system that exists now. Determination of that evolution of the system from today is weather forecasting. And weather forecasting is an IVP.

        I do not know how I could have been more clear.
        But it seems we are still talking past each other because your post I am answering says

        2. The terms IVP and BVP are used for cases where the outcome is determined predominantly by initial values or boundary conditions respectively.

        NO. An IVP problem and a BVP problem are different in nature.
        If a problem is IVP then it is limited in the time it can forecast. The degree of initial value uncertainty affects the length of time that can be reliably forecast (about a week for weather). A problem that is BVP is not affected by the initial values and, therefore, is not limited in the time it can forecast (assuming the system is adequately understood).
        This is what I meant when I wrote (and you have not addressed)

        Climate is an IVP or it is not. I say it is and you – perhaps by use of unfortunate wording – admit it is.

        Your post I am replying concludes saying.

        6. If I want to predict climate, say over the next 30 years, the initial values (i.e. the weather today) are hardly important, since I will aggregate over 30 years. What will be important are the processes that regulate the system, i.e. the boundary conditions, making climate prediction a BVP.
        Take any climate model, change the initial values and the outcome will be almost the same.

        Sorry, but the result DOES depend on “the initial values (i.e. the weather today)”.
        Please consider that the 30 year prediction could be a forecast or a hindcast. So, from within a climate time series, take a starting time and predict the boundary conditions of
        (a) the previous 30 years
        and
        (b) the following 30 years.
        The two predictions give the boundary conditions that can be combined to give the boundary conditions for the total of 60 years from within the time series.
        Now start at the beginning of the 60 year period and predict the boundary conditions of the following 60 years.
        The two results of boundary conditions predicted from the same data are not the same. This is because predicting climate boundary conditions is an Initial Value Problem (IVP) and the two analyses used different initial values.
        QED

        And the inabilities of existing climate models that don’t work are not relevant to this.
        Richard

      • @moderator: I have accidentally submitted this same post as a reply to an earlier post. Feel free to delete that one, but leave this one, because it is in the right place.
        [Done. -w.]
        Richard,
        I will try to answer everything you say as well as I can.
        I asked you whether you considered the time over which the climate is modeled to be an inital value. To which you replied:

        NO! I do not. And I fail to understand how you claim to understand I do think that.

        I thought that, because you use the fact that models over different times give different results to argue that climate is an IVP. If climate were an IVP, as you claim, the proof would be to show that the results are predominantly determined by the initial values. The argument that different periods give different result gives no credibility to your IVP claim, unless you consider this time period an initial value. Which you have now clearly stated you do not. That means the argument makes no sense. The fact that different time periods give different results says nothing about the problem being an IVP or BVP.
        Next you say:

        I say the boundary values of climate differ depending on the length of time assessed because the system evolves with time. That is why “average weather” differs depending on the length of time assessed. And at any point in time the boundary conditions have evolved from a previous time.
        All future time starts from now and, therefore, the boundary conditions at any point in time have evolved from the system that exists now. Determination of that evolution of the system from today is weather forecasting. And weather forecasting is an IVP.

        I’ve really tried to understand what you are saying here, but this makes absolutely no sense. Boundary conditions are fixed by definition. They don’t evolve. They are needed to reduce the solution space, so allowing them to evolve would be nonsensical.
        You also say:

        You have stated the definition of the arithmetic mean. The median and the mode are also averages. There are an infinite number of possible averages.
        If your model is considering “aggregate functions” then the definition of whatever you mean by “average” is crucially important.

        I have said before, it really doesn’t matter which one you take. The problem will always be a BVP, whatever definition of average you choose.
        You also say:

        And the values of the boundary conditions alter as the system evolves but their “importance” does not change.
        Please note (n.b. this is not point scoring) that I understand you to admit that climate is an IVP when you say “the importance of initial values will diminish”.
        Climate is an IVP or it is not. I say it is and you – perhaps by use of unfortunate wording – admit it is.

        Let me try to phrase it as clearly as possible. What I said is that weather (short-term modeling) is an IVP and that as you increase the time over which you model, the importance of initial values diminishes and the importance of the boundary conditions increases. For long term predictions such as climate the importance of initial values ahs diminished to practically zero, whereas the boundary conditions have become crucial. Making climate modeling a (textbook example of a) BVP.
        You admit yourself that weather can not be reliably predicted for more than a week ar so ahead, showing the initial values no longer adequately constrain the possible solutions at timescales longer than that. If they can not do so over more than a week, they definitely can’t over years or decades.
        You further say:

        That is pure handwaving assertion into which you have inserted the falsehood that “however you define your climate average is irrelevant”: different definitions of “average” provide different results.

        You repeat yourself. I had already answered that the definition of average is irrelevant to climate modeling being a BVP or IVP. It is obvious that different definitions of average will give different answers, but the problem will always be a BVP, no matter what definition you choose.
        You say:

        An IVP problem and a BVP problem are different in nature.

        What do you mean ‘different in nature’? In the particular case of weather and climate, the differential equations will be more or less the same since they are basically just the laws of physics the system is subjected to. The only difference between IVP and BVP is then the way in which the possible solutions are constrained.
        You also say:

        If a problem is IVP then it is limited in the time it can forecast. The degree of initial value uncertainty affects the length of time that can be reliably forecast (about a week for weather). A problem that is BVP is not affected by the initial values and, therefore, is not limited in the time it can forecast (assuming the system is adequately understood).

        Finally something I can largely agree to. (just as a sidenote: I can imagine there are IVP’s where the initial values give rise to only one unique solution, in which case they are not limited in the time over which can be forecasted)
        Now if you admit that a BVP is not affected by the initial values: do you think that if we were to model the climate for the next 30 years from today, the model would give a very different answer depending on whether we told it it rained today or not? I don’t think that rain falling today will have much influence in the evolution of the weather over 30 years, therefore I claim climate modeling is a BVP. The initial values only influence the solution over a couple of days, but not on the long term.
        Finally your grande finale:

        Sorry, but the result DOES depend on “the initial values (i.e. the weather today)”.
        Please consider that the 30 year prediction could be a forecast or a hindcast. So, from within a climate time series, take a starting time and predict the boundary conditions of
        (a) the previous 30 years
        and
        (b) the following 30 years.
        The two predictions give the boundary conditions that can be combined to give the boundary conditions for the total of 60 years from within the time series.
        Now start at the beginning of the 60 year period and predict the boundary conditions of the following 60 years.
        The two results of boundary conditions predicted from the same data are not the same. This is because predicting climate boundary conditions is an Initial Value Problem (IVP) and the two analyses used different initial values.
        QED

        Again, I have tried, but I really can’t make any sense from this. Firstly you claim the result depends on the initial values and then to prove this you change everything about the problem EXCEPT the initial values. On top of that you are again talking about things like ‘predicting boundary values’, which really makes no sense whatsoever. Boundary conditions are fixed and an INput to the model, not an OUTput. One simply does not predict boundary conditions, they are defined a priori. It becomes really hard to follow what you are saying if you are misusing terminology like this. Meybe read the wikipedia entry for boundary value problem, where the concept of boundary conditions is quite clearly explained imho.
        Oh and I agree completely with this:

        And the inabilities of existing climate models that don’t work are not relevant to this.

      • MikeB and Viking Explorer
        Days have passed since I asked you

        In addition to your cheerleading, perhaps you could say what you found cogent about Aran’s assertions?

        I am still waiting/
        Richard

      • Richard, you are actually a troll, because you’re not interested in rational discussion. I’m way more skeptical than you, but you never add anything useful or constructive to the discussion. Further discussion with you would be irrational.

  23. Saying climate/weather is a “XYZ problem” is like saying the national debt is an “addition/subtraction problem”. The labels “IVP” and “BVP” merely characterize the constraints on various parameters for solving the ‘primitive equations’ in the mathematical systems used to model these systems. But that is not a complete characterization of weather/climate analysis and forecasting.
    These labels don’t convey any information about the probabilistic aspects of the problems. They also don’t convey any notion of the actual physics of weather and climate. Moreover, the full details of weather at every point of time and space would be too complex for the kinds of models we currently use to try understand, explain and predict these phenomena.
    I think the problem of weather/climate prediction can be more completely characterized as a Bayesian inference process, where the observations and forecasts are probability distributions. Yes, numerical methods are applied to solve differential equations, but overall it is more like an application of Bayes Theorem and recursive Bayesian estimation.
    http://en.wikipedia.org/wiki/Data_assimilation

    • 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?

  24. 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.

  25. 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

      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?

      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

        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.

        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

        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.

        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”.
        You mean like the Arnold Schwarzenegger and Danny DeVito characters in “Twins”? -:)

      • No, you and other warmunists may

        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”).

        Climate and weather seem to be controlled by the exact same physical processes

        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.

        call it chaos when the details are too complex to handle” but mathematicians don’t

        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.

        are effectively impossible to predict or control; e.g. turbulence, weather, the stock market, brain states

        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.

      • 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.

      • What a ridiculous statement. … 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

        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.

      • 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.

        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.

  26. 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

  27. 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 .

  28. 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.

  29. “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.

    • 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

        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.

        Sometimes, you guys are hilarious. You’re pretty sure the path of the river will stay the same?
        https://wattsupwiththat.files.wordpress.com/2009/12/oxbow-lakes1.png
        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.

  30. 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.

  31. 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”??

  32. “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.

  33. ” 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

      ” 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.

      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.

      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…

      Thanks, Matt. If you followed the link to Climate Audit, Steve noted that:

      Mandelbrot calculates Hurst indices and 3rd and 4 th moments for 12 varve series, 27 tree ring series from western U.S. (no bristlecones), 9 precipitation series, 1 earthquake frequency series, 11 river series and 3 Paleozoic sediment series.

      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.

      • 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

        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

      • Something is causing the ice ages. …. You don’t know that, VikingExplorer

        I do know that Something is causing the ice ages, because I believe in causality.

      • 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.

        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

        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.

        No, Viking is not correct in the slightest, and this can easily be shown. As Dan Hughes commented below,

        The following three statements are indisputable:
        1. Averages of chaotic response are chaotic. You can demonstrate this for yourself by use of the algebraic logistics map or the original Lorenz system from 1963, or your favorite system that exhibits chaotic response.

        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.

  34. 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.

  35. 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.

  36. 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!

  37. 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).

  38. 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.

  39. 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.

  40. 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.

    • 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..”

  41. 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.

  42. 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.

  43. “A boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions.” (Wiki)
    So sez wiki…
    Now: pause and consider. If the world is deterministic, and therefore climate is deterministic, its behaviour is ipso facto represented by a suite of differential equations that describe its behaviour.
    That says nothing about the linearity of non linearity of those equations, or whether or not there will be IV issues arising..
    So let’s look at boundaries. We have none really. All we can say that IF the earth doesn’t go into thermal runaway, the energy leaving it must equal the energy being generated within it plus the energy arriving at it.
    I guess that’s some sort of a boundary, but not much.
    Any other boundary is entirely artificial, because there is nothing – certainly not in climate science the myth, that says that the ocean – air or land – air interface must and will be constrained at any particular value: In fact the whole essence of AGW the myth, is the fear that it will be NOT so constrained.
    Perhaps we could say that in the past it has been so constrained – despite massive fluctuations in atmospheric CO2 and dramatic rearrangement of the height and position of the continental land masses, the earth has stayed within about a +-5% variation in absolute average temperature.
    But that doesn’t give an alarming (or a useful picture), either.
    Look the issue of boundaries in solving equations is simply that it gives you an extra bit of information. It says ‘solutions beyond here can be discarded’ leaving you with a set that may present possible system states in the future.
    So e.g world futures that include impossibilities, like more renewable energy than is actually falling on the earth (or a reasonable amount of its orbit – one accepts Dyson spheres etc) as sunlight – represent boundaries beyond which you simply cannot go.
    A LOT of my analysis of e.g. renewable energy options (are there any?) involves this. Put simply: if proposition X when extrapolated to a real world system results in complete nonsense, one can reject proposition X.
    What seems to me to be the case is a lot of mealy mouthed BBB* by the climate community. What is the case is that climate is in fact a long term integral of weather. They are on fact no different and suffer from identical problems of analysis. The BBB statement is and has been for years ‘climate isn’t weather’
    Well of course it is. It’s a lot of weather averaged out.
    Its a bit like saying ‘wealth isn’t income or expenditure’. No, but it is the long term integral of their difference..
    What it boils down to I suspect, is that climate ‘science’ is desperate to keep itself meaningful. The terrifying prospect is that if it is chaotic – I am more and more convinced that it is – no amount of modelling is going to come up with a politically or socially useful prediction of future climate whatsoever, which begs the question of why we are spending so much money on it in the first place.
    THAT is the question the alarmists and their coat-tails do not want on the agenda. Weather, with its IV problems, is patently chaotic in nature and its absolute fallibility in terms of prediction, must be separated from climate, in order that climate is not the Curryian ‘wicked problem’ that could probably be proved to be incapable of any practical prediction whatsoever.
    Hence the BBB. By giving a spurious reason why its different, the status quo is preserved – at least for now.
    *Bullshit Baffles Brains Taurus excreta cerebrum vincit see many on line definitions.

  44. The following three statements are indisputable:
    1. Averages of chaotic response are chaotic. You can demonstrate this for yourself by use of the algebraic logistics map or the original Lorenz system from 1963, or your favorite system that exhibits chaotic response.
    Calculate a very long series response for your favorite chaotic demonstration. Calculate an average of the response. Using any one of the available methods, calculate the Lyapunov exponents for the series of averages. There will be a positive exponent.
    The Google, standard or Scholar, will lead you to canned, off-the-shelf software for calculating Lyapunov exponents for long series of numbers, including con-current calculations for systems of ODEs.
    2. Temporal chaotic response for autonomous ODEs cannot display trends; it’s impossible. The region of phase space on the attractor that will be occupied at any time cannot be predicted. And this refers to the relatively simple case of low-dimensional temporal chaos. The real-world case of infinite-dimensional temporal-spatial chaos is a little more difficult.
    3. It is a fact that the model equations and numerical solution methods used in all GCMs are formulated as an Initial Boundary Value Problem ( IBVP ). The model equations are integrated forward in time. Yet we frequently read that the physical domain is actually a boundary Value Problem ( BVP ).
    Yet another failed analogy that has been invoked to reduce climate process modeling to exceedingly simplistic, and incorrect, terms. The analogy uses the, Weather is an initial value problem, climate is a boundary value problem ( BVP ), approach. The boundary value problem argument is invoked so as to ensure us that climate process modeling is very straight-forward relative to numerical weather prediction ( NWP ). One important aspect of the argument is that the chaotic nature seen in NWP does not negatively impact climate calculations. These arguments are attempts to reduce climate process modeling to the steady-state/stationary case.
    Here is the concluding paragraph:
    ” We cannot predict what the weather will do on any given day far into the future. But if we understand the boundary conditions and how they are altered, we can predict fairly accurately how the range of possible weather patterns will be affected. Climate change is a change in the boundary conditions on our weather systems .” [ bold by edh ]
    The analogy in this case is especially flawed because a very limited fluid flow condition using a “system”, the balloon, that is more correctly described as a problem with a constraint, and not a BVP. The presented argument is meant to ensure us that the boundary conditions for the climate system process modeling solely and completely determine the solution of the IBVP-formulation of the climate process models.
    The boundary conditions at the top and bottom of the Earth’s climate systems include specification of the in-coming SW radiative energy from the Sun, plus energy transport considerations at the interfaces between the atmosphere and the material on the surface of the earth. Note that the out-going radiative energy at the top cannot be specified. Note, too, that the energy fluxes at the interfaces within Earth’s climate systems cannot be specified. These both are calculated by the process models in the GCMs. The out-going radiative energy is an out-come from the process-modeling formulations. In this sense, the boundary condition at the top does not, because it cannot, specify the steady-state/stationary conditions at the top. Relative to energy, Earth’s climate systems are an open system.
    It is very important to note that the out-going energy is determined by the process models that are used to calculate the states internal to the Earth’s climate system; especially including the energy interactions between the sub-systems of the complete system. The balloon, in contrast, is constrained by the string to which it is attached, and cannot affect through feedback the hydrodynamics of its environment. This constraint, very roughly, maybe is supposed to allow representations of the effects of processes internal to the Earth’s climate systems. In this sense the constraint does not model the important effects of the processes internal to the system that cause changes in the radiative energy balance at the top of the atmosphere.
    The BVP assumption requires that balance between in-coming and out-going radiative energy at the top not be affected by the energy exchanges and internal processes that occur within the climate system. I think that the assumption requires that energy exchanges at all the interfaces within the complete system also be in balance. At least to the extent that these processes are not significant relative to the balance at the top. It is difficult for me to envision that the required degree of balance at all the energy-important interfaces within Earth’s climate systems can attain a state of balance.
    Finally note that, in contrast to the characterization in the final paragraph quoted above, the effects of CO2 in the atmosphere are not boundary conditions at the top of the atmosphere and thus are not directly altering the boundary conditions. The CO2 instead causes effects internal to the systems that are included in the climate process modeling, and these effects alter the radiative-energy transport state at the top. The balloon cannot be an analog for the energy content of Earth’s climate systems.
    Finally, really this time, because processes internal to the systems affect states at the boundary of the systems, it is impossible for modeling of Earth’s climate systems to be a BVP in the classic sense of the term for which the specified conditions at the boundary alone determine completely the states within the systems. The Earth’s climate systems are open relative to energy exchanges with their surroundings.
    It is a fact that the model equations and numerical solution methods used in GCMs are formulated as an Initial Boundary Value Problem ( IBVP ). Yet we frequently read that the physical domain is actually a boundary Value Problem ( BVP ). This argument is usually presented as a defense against the known limitations encountered in Numerical Weather Prediction ( NWP ). The severe degradation in the fidelity of the NWP results relative to the physical domain is attributed to the chaotic response exhibited by NWP models and methods. The Climate Science argument is that the NWP problem is an Initial Value Problem ( IVP ), that the chaotic response is expected and the short time frame of weather forecasts are completely dominated by the chaotic response. In order to invalidate the argument that, If the weather can’t be accurately forecast for even a few days how can the climate be forecast for a period of 100 years. ( I don’t know where or when this argument was introduced. )
    It is at this point that the BVP concept of climate modeling is invoked. The fundamental hypothesis of the CO2-climate issue is that an equality between the out-going and in-coming radiative energy at the Top of the Atmosphere ( ToA ), when averaged over some, unspecified, time period will attain at some, unspecified, future time. My interpretation of the BVP argument is that this equality imposes a constraint on the climate problem. In essence, the argument says that the initial values and early-time chaotic response are immaterial to calculation of the response of the climate. It is also my understanding that nothing beyond the hypothesized radiative-energy equality at the ToA is given to support the argument.
    There are a few problems with the BVP argument in both the physical and mathematical domains. In the physical domain I think the argument means that the physical phenomena and processes occurring within Earth’s climate systems do not affect the out-going radiative energy. ( This statement ignores that the state of the atmosphere in fact affects the amount of energy that penetrates into the atmosphere and that is rejected from the atmosphere. ) In other words, the climate-change problem is solely and purely a radiative-energy transport problem in a practically non-participating medium and is unaffected by conditions at interfaces within the climate systems. If this is the case, I think the problem would have been solved several decades ago.
    The non-isotropic, inhomogeneous time-variations of radiative-energy transport interactions within Earth’s atmosphere must somehow average out over the time-averaging period. Chaotic response does not decrease, so long as the conditions for chaotic response obtain, no matter the length of time of interest. The average of chaotic response is also chaotic. Long term averages of chaotic response are chaotic.
    The initial state of Earth’s climate systems do in fact affect the response to changes internal to the systems. The initial surface albedo, for example. The initial temperature level is important relative to changes in the phases of water; liquid-to-vapor, liquid-to-solid, and versa vise. The different calculated responses whenever the boundary conditions are changed in the modeling approach; fixed Sea Surface Temperature vs. coupled atmosphere-ocean modeling, etc. Cloud covered vs. cloud free, water vapor present or not, and etc.
    In the mathematical domain, the energy leaving a solution domain cannot be specified. A useful rule of thumb is that if you can’t built a physical realization of the boundary condition in the laboratory, it’s not a valid boundary condition. If you don’t want to go with a rule of thumb, you can work out the compatibility conditions and associated eigenvalues and eigenvectors for the model equation system. Generally, any physical quantity that can be affected by the processes occurring within the solution domain cannot be specified at the exit from the domain; temperature, density, internal energy, enthalpy, among others. You will find that the eigenvectors for these quantities always point out of the solution domain at the exit surfaces and into the domain at the entrance surfaces.
    Mathematically the purely radiative-energy transport problem for a grey ( isotropic, homogeneous ) interacting media can be set up as a boundary-value problem by setting the out-going energy equal to the in-coming energy at the ToA. An analytical solution to the Schwarzchild equation, under sufficient simplifications including a 1D solution domain, does exactly this. The approach introduces discontinuities into the solution at both boundaries.
    Given the measured data at the ToA it is highly unlikely that the equality is set in GCMs. Doing so when data indicate that out-going exceeds in-coming would ‘trap’ excess energy internal to Earth’s climate systems, and when the data indicate in-coming exceeds out-going allow ‘to much’ energy to be rejected.
    The radiative energy balance at the ToA constraint cannot be applied during the integration of the model equations in GCMs. To do so will force the response of the materials internal to the domain to adjust to the incorrect use of the constraint.
    The constraint at the ToA is used during the tuning process and spin-up to an initial state. Some of the many parameterizations in the models are changed in a way so that the model equations more nearly attain the ToA balance.
    Climate is not the Average of Weather
    Climate
    The climate at a location is fundamentally determined by the radiative energy of the Sun, the relationships between the geometry of the earth, the geometry of the revolution of the earth around the Sun ( the yearly cycle ), the relationship between Earth’s axis of rotation and the plane of Earth’s orbit ( the seasons ), and the rotation of the earth about its axis ( the daily cycle ).
    The climate at a location is determined to first order by these factors and the latitude and altitude of the location. The climate also can be influenced by significant, more-or-less thermally stable, bodies of liquid or solid water, primarily near the oceans but including also other large bodies of liquid or solid water.
    Meso-scale ( larger than local, smaller than global ) topology of Earth’s surface can also affect local climate. Mountain ranges that significantly affect specific regions relative to precipitation are an example; rain shadows, monsoons.
    Weather at a location is the time-varying thermodynamic and hydrodynamic states of the atmosphere. Weather can be viewed as perturbations, deviations from some kind of norm, in the local climate. In this sense, one could argue that climate is some kind of temporal average of the weather at a location. Note, however, the descriptions in the previous three paragraphs of the basic factors that determine the climate at a location. These factors are independent of the temporal variations of the states of the atmosphere.
    The climate at a location is not determined by the weather. Local climate is determined by factors outside the domain of the states of the atmosphere.
    Climate and weather are both local and neither is global. There is no need to focus on any aspects of “global climate”: such averages are useless for decision support. Primary focus should be on the advantages and dis-advantages, if any, of the status and changes in local weather. It cannot be over-emphasized the extent to which focus on global changes in the “global climate” is mis-guided. Local decision support demands solely local information. It also cannot be over-emphasized that the complete lack of focus on local states is a major failing.
    The hour-by-hour, day-to-day, and month-by-month variations in local weather are determined by the effects of the net of the radiative energy into the physical phenomena and processes occurring within and between the thermodynamic and hydrodynamic sub-systems. All phenomena and processes, ( thermodynamic, hydrodynamic, chemical, biological, all ), occurring within the Earth’s systems of interest are driven by this net energy. Basically, weather is the distribution, and internal redistribution, of the energy supplies of sensible and latent thermal energy within Earth’s thermodynamic and hydrodynamic systems.
    The solar-system and Earth’s geometric relationships, and local altitude/latitude, are the primary reasons that we can know that the temperature, and the weather in general, at a location will be different, for example, at January and July. The degree of differences between the seasons primarily is determined by the local latitude and altitude. The degree of differences over the seasonal / yearly cycle is a strong function of location. Variations over the seasonal cycle are more or less distinct; the variations are either small or large depending on the location.
    Weather and Chaos
    Weather is thought to be chaotic. And the numerical solutions of the mathematical models for both weather (NWP) and climate (GCM) are classified as ill-posed, in the sense of Hadamard, initial-value problems; lack of continuous dependence on the initial data. The average of a chaotic response is itself chaotic. Thus, if climate is the average of weather, then climate is also chaotic. Again, the descriptions of climate given above preclude the chaotic nature of weather being a part of chaotic climate. It is the Earth-Sun geometry and the axis of Earth’s rotation that determines that January and July are easily differentiated. That differentiation is independent of, and is not a function of, the chaotic nature of weather.
    Chaotic is not random. Chaotic is the antithesis of random. Random fills phase space, whereas temporal chaotic trajectories are limited to the attractor and so by definition, cannot fill phase space. Weather is not random and is not noise; neither pink or white. Especially not white noise which has equal power at all frequencies. Averaging the trajectories from multiple runs of a single GCM, or one or more runs from several GCMs, does not in any way ensure that the so-called noise will be ‘averaged away’. The averaging is instead an averaging of different trajectories. Additionally, there is no way to ensure that the different trajectories are associated with ‘an attractor’ for the real-world case of spatio-temporal chaotic response, which is the case of finite-difference approximations to partial differential equations.
    GCM Validation
    Validation, fidelity of simulations relative to the physical domain, of GCMs thus firstly requires that the calculations be shown to be correctly simulating the distribution and internal redistribution of the internal variations that are responsible for the local weather. Validation relative to effects of increasing concentrations of CO2 must then require that the GCMs are correctly simulating how the distribution and internal redistribution of the internal variations have been altered by the increasing concentration of CO2 in the atmosphere. This is a very difficult problem.
    A first major difficulty will be in devising and development of procedures and processes that can be used to determine that changes in the phenomena and processes that are responsible for changes in local weather are in fact due primarily or solely to changes in CO2 concentration. A third order delta that will be exceedingly difficult to (1) observe and (2) model and calculate.
    The case of extreme weather events requires the same series of accounting if climate change is invoked as the fundamental cause. Extreme weather events are generally very localized. Thus if the invoked driving source of the event is a significant distance away from the observed occurrence, the effects of changes in the composition of the atmosphere are required to be shown to obtain over the distance of the course from the source to the location of occurrence. If the event is a mighty downpour of rain and the source of the rain is said to be the Oceans far away from the location of the downpour, it must be shown that the changes in the composition of the atmosphere are directly related to the fact that the water vapor survived its path from the Oceans to the downpour location, and that the previous composition of the atmosphere would have prevented the water vapor from surviving its journey. Sounds very difficult to me.
    Global Metrics are Useless
    Global metrics for assessing GCMs fidelity to the real world are of no use whatsoever relative to assessing the correctness of simulations of local weather. The changes in local weather are required for decision support. Additionally, none of the fundamental laws describing weather and climate can be usefully expressed in terms of global quantities. The state of the atmosphere and the state of liquid and solid phases of water, and changes in these states, are determined by local conditions. No physical phenomena and processes, governed by the fundamental natural laws, have been demonstrated to scale with global averages of anything.
    The temperature of the atmosphere, which has been chosen to represent changes in climate due to increasing concentrations of CO2 in the atmosphere, on the other hand, is determined by the path of the thermodynamic processes that the atmosphere experiences at the locations of interest. As the initial states are different, and the changes in weather are different, so will the temperature be different. Again, no dependency on the global-average state.
    Climate is local, weather is local. Weather is the variations in local climate. For decision support, the variations in local weather are what must be correctly simulated by GCMs. The models are required to be able to correctly simulate the changes in the weather variations due to changes in the concentration of CO2.
    Based on my comment here.
    Corrections for incorrectos will be appreciated.

    • Dan, that was a really long comment. I’m only going to address the first thing:
      “Averages of chaotic response are chaotic”
      This is falsified by the fact that everything is chaotic. The science of Thermodynamics (supreme of physical laws) and Circuit Theory are examples where the underlying physics are extremely chaotic, but at a macro scale, they are very predictable (when one uses a reasonable definition of predictable.

    • Bravo!!! This is a terrific exposition of the problems with modeling climate.
      One other point about nonlinear systems. If they are complicated enough there can be many attractors and you typically have no knowledge of which one you will be modeling when you start a calculation. A single set of equations may describe many different climates and it’s unknowable in advance which one you are studying.
      Your observation that global metrics are meaningless is especially important. I’ve long thought that the real purpose of the global averages is to hide the severe deficiencies in the models. Suppose you wrapped the globe in cellophane and wrote a temperature map on it. Now rotate the cellophane so that its poles are on the earth’s equator and its equator passes through the earth’s poles, i.e., polar bears on the equator and sharks at the poles. The global averages are completely unchanged by the crazy orientation of the cellophane map. The models aren’t this bad, but it’s well known that they do a bad job with local climate. It’s much easier to get global averages right, which apparently can be done by knob twiddling, than to get the whole map right. Or as anyone who has studied physical science or engineering learned, getting the right answer on an exam question for the wrong reason didn’t get you any credit.

  45. It might be useful to consider the difference between initial value and boundary value problems in terms of more familiar electrical systems. It’s a simple matter to calculate the dissipation of such a system from boundary parameters without any knowledge of what’s inside the box, i.e. the internal distribution of potential and flux – but, only if the system is in a steady state. This holds true even for an internally chaotic system such as a high-pressure, gas-discharge lamp. Hypothetically, given a circuit diagram, one might calculate internal distributions solving differential equations and hope to arrive at the same result.
    It might be supposed that a similar situation exists for the case of thermal potentials and energy fluxes. The current approach seems focused on Navier-Stokes differential equations which, even if solved, presume a flux-invariant viscosity tensor with no consideration of thermal dissipation due to energy fluxes in thermal gradients. Thermodynamics provides an alternative boundary value solution, the Carnot equation. The favored definition of temperature is based on the assumption of a steady state independent of the system’s history (Caratheodory).

  46. Seems to me that any problem that requires a prediction that extrapolates to a future time period is an initial value problem. Any problem that evaluates the relationships within a system and uses the limits of the system as constants is a boundary problem. If you use a model that’s formulated for a closed system with fixed boundary conditions (structural model, causal model, etc., etc. …), and try to go outside its bounds you are going to be wrong.
    Because many climate modelers solve a boundary value problem, and then extrapolate beyond the specified boundary they have problems
    Pielke is correct in stating that a legitimate climate prediction is an initial value problem.

  47. Both weather and climate are non-linear and chaotic, the primary difference being time scale.
    Often the argument against evolution is that micro evolution is acceptable and macro evolution is not. The argument is frivolous since evolution theory works the same regardless of the time scale.
    In terms of predicting atmospheric conditions it seems as arbitrary differentiating climate and weather as differentiating micro and macro evolution.
    It does not help that the term climate is vaguely differentiated from weather; adaptable to any situation, it could refer to anything between 20 years to a thousand to many thousands of years.

  48. we arrive at a free boundary problem with the free boundary being the interface between ice‐covered and ice-free areas.
    ===============
    that doesn’t make sense. in a BVP, the boundary is a temperature, not something physical.
    Ice covered areas are not bounded by 0C, They can still get colder. Ice free areas are not bounded by any temperature.
    The bounds on earth’s temperature are the background radiation of space and the surface of the sun. Our climate lies somewhere in between those temperatures.
    In reality however, climate is much more stable.
    http://www.geocraft.com/WVFossils/PageMill_Images/image277.gif

  49. One of the problems I see in this thread, is that assumptions of “smooth behavior” are repeatedly made. Willis’ isn’t the guilty party, it’s those of you who make comments about the “fluid motion” of the earth’s crust or the rainfall rates, etc. It’s necessary to ask with the existence of 9+ magnitude earthquakes does anyone really believe in “fluid motion” of the earth’s crust? If you are measuring rainfall rates in a thunderstorm, does the rainfall taper off smoothly, or suddenly stop?
    My experience is that sometimes things behave smoothly, but often enough jerks and surges occur. Averaging destroys information, in particular it destroys information regarding fractal boundaries.
    As an aside, Mandelbrot was right about much of nature, and also right about the price distribution and behavior of markets. Read his book, “The Misbehavior of Markets,” 2005, for a prescient warning about long tailed distributions and Black Swans in the stock market, but I digress.

    • My comment about the Earth’s crust moving fluidly may have been worded poorly, but it’s essentially correct – continental drift is caused by the fluid flow of molten rock convecting upwards, cooling and becoming denser, then sinking, or subducting to melt again. See the videos at the following link, which clearly show that continent’s move due to fluid motion.
      http://maggiesscienceconnection.weebly.com/mantle-convection-plate-tectonics-earthquakes–volcanoes.html
      I’m sure that on a molecular level, the fluid flow of toothpaste doesn’t look too smooth, and by same token, on epoch time scales, those 9.0 earthquakes don’t look too abrupt.
      Nor was there any assumption of smooth behavior anywhere in my post. To the contrary, the point I was making was that it was erroneous to infer predictability from a limited period of stability. Predictability requires prediction of changes in behavior, not predicting erratic behavior around a temporarily and perhaps coincidentally stable average. I don’t care whether change in a system is smooth or abrupt. If you can’t predict those changes, you can’t claim to understand what drives the system.

  50. See floor is not static (tectonic movements, submarine eruptions, hot vents all have some effect on the currents) thus it could be a boundary too.
    Went to NOAA’s website
    http://www.noaanews.noaa.gov/stories2012/20120917_pacificvolcanomission.html
    and was greeted by:
    “We need your help!
    By giving us your feedback, you can help improve your http://www.NOAA.gov experience. This short, anonymous survey only takes just a few minutes to complete 11 questions. Thank you for your input!”
    Mr. NOAA if you are really interested to improve our experience, as a keeper of the global geomagnetic data, you need to alter the algorithm for the field calculations since 1880. It is too complex to elaborate here.
    As a reference I would bring the CET annual temperature calculations by the UK MetOffice. I emailed them on 30th of July 2014 suggesting alternative method, and hey presto, from the 1st January 2015 the new method was implemented.
    Get in touch Mr. NOAA.

  51. Surely climate is a Boundary Value Problem. Because weather is short term and a microscopic part of the system (less grainy than climate), I can see that knowing initial conditions is important. If you have just broken a high temperature record, you can forecast that warm weather should continue for some days and then it will cool. Initial conditions of climate makes no sense and I don’t think it has any relevance. The inability to forecast very far is because “initial conditions” are a fleeting part in time of the much larger bourndary condition situation in the background that confounds the weather forecast beyond a week or so.
    I vote for Xua. We are dealing with a heat engine and what’s better to consider than the cold and the hot ends of the system. It’s even what we are seeking to predict. The geological record of the ice ages (yeah, I know that idle academia changed the terminology to all one ice age and thereby destroys the most important meaning of the term, and incidentally makes it okay to call global warming both hot and cold) is a swing back and forth of the two extremes. These are both boundaries of the problem. What happens to initial conditions when a large bolide strikes the earth? The earth, inexorably, eventually, gets back under control of the boundary conditions. It is perfectly sensible to assume for the purpose of weather forecasting that initial conditions are approximately correct over the short run even if the big picture is a boundary condition problem.
    Cold and hot. Now, if I’m in the middle of an ice age (my definition has ice as the defining deal) I could confidently make a weather forecast if I’m in Manitoba that for the next 100 years, its going to be bitterly cold and low humidity. If I’m at the equator, I’m going to forecast the next 100 years as warm but not too hot. If I’m in an interglacial, such a forecast is of no use. I have to guess whether it will be a few degrees warmer or colder than today, whether it will rain, snow, blow, etc. It’s part of the same system as climate but its wrong to choose initial conditions for climate.

  52. 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

    That is correct Willis, not to mention as far as the climate is concerned the models do not know or do not account for solar variability and the associated secondary effects , geo magnetic variability ,and assign wrong values to items they think may influence the climate.
    The climate is stable but not stable because the difference between ice age conditions versus inter-glacial conditions is razor thin. 4c to 5c at most.

  53. Very interesting essay, once again showing how out of touch the climate community is with the mathematics needed to handle the problems involved. No one can show which, if any, boundaries apply to climate modelling(top of atmosphere, the ice line, the ocean or land surface, the volume of the ocean involved, etc). Get past that and we hit a boundary, the limits on calculating how fluids behave. The scale for calculation needs to be sub millimeter(the limit to vortex size in air and water), no hundreds of kilometers. Once you might figure that out there are numerical computation limits- the minimum finite difference computers can calculate, which sets another impossibly small size for measurements.
    There is a million dollar bet out there for the person who can solve the Navier-Stokes equation, or even just show that a solution is possible. No potential claimants are in sight.

  54. Boundaries are attractors, which may be modified with stochastic perturbations, hysteresis, and even some random attractors thrown in. Though from my frame of reference that appears to be an apt set of metaphors for the thinking and behaviour of proponents of natural variability being internal, chaotic, or random, rather than a climate that is largely governed by atmospheric responses and oceanic negative feedbacks to solar variability at down to daily scales.
    But if that is currently beyond your boundaries, you could still ask, that if the globe warms by 1°C, what difference does that make to major teleconnections globally? do the AO&NAO become increasingly positive? are negative extremes in AO&NAO reduced? are El Nino conditions reduced?

  55. I suppose the idea that climate is a BVP comes from thinking that initial conditions decay away leaving the solution dependent only on boundary conditions such as is true of elliptical and parabolic PDEs with Dirichlet or Von Neumann boundaries. But as the “boundaries” of the climate system have time dependent characteristics, then each instant possibly becomes t=0 again. And statistically if the time dependence contains 1/f noise, one can hardly appeal to the “mean value” trope. Willis has found, yet again, something interesting to ponder…
    Wolfram is indeed a genius, but that tome of his promoting a new science was just plain wacky.

  56. I am so glad the science is settled. Think of the confusion if it were not. All, a wonderful post and discussion.

  57. I’ve seen solar constants cited from 1,358 to 1,366 W/m^2, a range of 8 W/m^2. Compare that to the radiative forcing of GHGs at 3.0 W/m^2. Simply lost in the uncertainty and noise.
    Some consensus.

    • Dan Hughes gets the prize for mentioning the paper that established very long-term weather prediction (a.k.a. climate prediction) as an initial-value problem. The late Edward N. Lorenz, in the magisterial and landmark paper “Deterministic non-periodic flow” (1963), established that even the smallest perturbation in the value of an initial condition defining the climate at a starting moment t(0) could radically alter the future evolution of the climate in a manner that was deterministic but non-periodic and hence not determinable without sufficiently well resolved initial data as well as sufficient knowledge of the object’s evolutionary processes.
      The climate, in mathematical terms, behaves as a chaotic object. For this reason, we cannot know with sufficient reliability the future value of any of its defining variables that are sufficiently far from t(0). This consideration firmly renders climate prediction (which by definition concerns time-series, commencing with perhaps-known initial values and unknown subsequent or out-boundary values) an initial-value and not, repeat not, a boundary-value problem, for time-series boundary-value problems in the climate cannot even be described, let alone solved, unless we knew or could predict with reasonable precision the values of all relevant variables at t(n), where n is a value indicating a period greater than about ten days (that, in predicting the climate, is the very long term).
      Papers talking of climate as a physical-boundary value problem are, therefore, by definition considering only the present state of the climate, whose key relevant boundaries are the troposphere above, the surface where we live and move and have our being, and the ocean floor. Here, it is in theory possible to establish relevant values: however, we have far too little knowledge of the ocean floor. Of particular interest are the mid-ocean divergence boundaries, which are almost entirely unmonitored, though variations in magmatic upwelling through these boundaries may well have a more profound effect on climate variability than our puny perturbation of the atmospheric composition.

      • Sir Monckton,
        If that is true, then the climate is inherently unstable. It could reach a “tipping point” at any time. If that were true, then maybe we need illegalize anything that could disturb the fragile state of the climate.

      • It is good to see you posting on this site since your comments add much to the debate.
        I have looked at your linked page. You state: “We can see the solar cycles as the 11-year cycle of increase and decrease in TSI. One item of note is that the change in annual mean TSI from minimum to maximum of these cycles is less than 0.08%, or less than 1.1 W/m2.”
        However, when I eyeball the various plots, I see a variation of about double the figure claimed by you. For example, NIMBUS7/ERB peaks at just over 1374, (1980) and dips to just below 1372 W/m2 (1987). The same can be seen with SOHO/VIRGO which peaks at a little over 1367 (2002/3) and dips down to a little below 1365 W/m2 (2009).
        When there are errors/uncertainties of only 0.5W/m2 on a number of different factors it soon adds up when you are looking for less tha the 3.7W/m2 said to be driven by a doubling of CO2 (bearing in mind that we are presently about 50% over the claimed pre-industrial level of CO2).

  58. scienceofdoom May 26, 2015 at 12:44 am

    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.

    Thanks, SOD. While I’m always reluctant to disagree with you because you are so dang often right, the necessity for a TOA energy balance says nothing about the temperature. We could have 340 incoming sunlight watts/m2 balanced by 40 W/m2 of reflected sunlight and 300 W/m2 of upwelling thermal radiation, a very hot planet.
    Or we could have 340 watts/m2 incoming sunlight balanced by 300 W/m2 reflected sunlight and 40 W/m2 of upwelling thermal radiation, a very cold planet.
    Look, the freakin’ moon has exactly the same TOA boundary conditions as the earth—340 watts/m2 in, 340 watts/m2 …
    And there’s a further problem. Due to the poorly-named “greenhouse effect”, the surface is thermally isolated, and energy accumulates there until the surface is emitting about 50 W/m2 MORE radiation than is hitting the top of the atmosphere. Same TOA boundary conditions … and yet another temperature.
    Since the boundary condition can tell us absolutely nothing about the surface temperature, which is the main variable of interest in the whole deal, I fear that your proposed boundary condition doesn’t help the analysis much. Yes, it is a necessary condition for understanding the system, but it is not a sufficient condition to tell us about future temperatures.
    And the same is true regarding Dr. Roy’s statement that the boundary condition is that a doubling of CO2 (“2XCO2”) = 3.7 W/m2 increased absorption. Venus certainly has some version of that as a boundary condition, it’s universal … and immediately after the last deglaciation we saw a thousand years of falling temperatures and increasing CO2 levels. Like the TOA radiation balance, the 2XCO2 = 3.7 W/m2 boundary condition provides us with necessary but not sufficient understanding to model a hideously messy chaotic system composed of a half-dozen constantly interacting subsystems. A change of 3.7 W/m2 from a doubling of CO2 would be counterbalanced if tropical clouds formed about an hour earlier on average, and there could be no change in the global average temperature. Alternatively, 2XCO2 would be exactly counterbalanced by an average albedo change from ~30% to ~31%, again with no change in the global temperature. In other words, the surface temperature is NOT just a function of the CO2 level.
    So the idea that to solve the climate conundrum it is sufficient to specify the change in forcing that would result from a doubling in CO2 is … well … let me call it “highly optimistic”.
    What I’m starting to think is that the distinction between an IVP and a BVP is artificial, and that most practical real-world problems of any complexity are some mix of the two. And in a LINEAR SYSTEM, if we know enough of some combination of the initial values and the boundaries of some system, we can model that system to some specified level of exactness.
    However, we have to bear in mind the wise words of the Wolfram Reference, who said:

    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.

    Since we know for a fact that the underlying equations are nonlinear and that the solutions certainly give every appearance of having multiple branches, the idea that simply saying ‘it’s a boundary value problem’ and naming some boundary makes the massively complex and chaotic climate system solvable is not just “a bridge too far”. It’s a bridge to nowhere.
    w.

    • Re multiple branches, see http://yorke.umd.edu/Yorke_papers_most_cited_and_post2000/1987_06_GOY_Science_Chaos_Strange_Attractors_Fractal_BasB.pdf and its discussion of fractal basin boundaries. Not only are there multiple branches but you can’t tell which one you are going to get. This has been understood for 30 years.
      Regarding IVP versus BVP the best example I can think of is flow through a circular pipe by a viscous fluid, ie, all real fluids. The boundary value is that the fluid velocity has to be zero at the surface of the pipe. The initial conditions are the pressure on the pipe and the velocity of the fluid at time zero. The solution, starting wth the same initial conditions, will depend on whether the pipe is smooth with a circular cross section everywhere or whether there are bumps on the wall of the pipe.

    • Willis,
      We could have 340 incoming sunlight watts/m2 balanced by 40 W/m2 of reflected sunlight and 300 W/m2 of upwelling thermal radiation, a very hot planet.
      Or we could have 340 watts/m2 incoming sunlight balanced by 300 W/m2 reflected sunlight and 40 W/m2 of upwelling thermal radiation, a very cold planet.

      You are correct. I should have written more.
      Given only the boundary value of energy in = energy out, of course there are a large number of completely difference possible climates. Given the constraints of our actual climate – the concentration of GHGs, the size and location of the oceans, etc etc, the general meaning of “this is a boundary value problem” is – given the boundary values and the material properties of the system – find the “long term” or “steady state solution”.
      Or solutions. There is no a priori knowledge that only one stable solution exists.
      Since we know for a fact that the underlying equations are nonlinear and that the solutions certainly give every appearance of having multiple branches, the idea that simply saying ‘it’s a boundary value problem’ and naming some boundary makes the massively complex and chaotic climate system solvable is not just “a bridge too far”..
      I agree, writing “it’s a boundary value problem” doesn’t mean it is solvable.
      However, I think understanding the nature of simple chaotic systems gives us one useful insight which has generated the (annoying trite) phrase that sparked your article – we can see that while weather forecasting beyond a certain point may be impossible, this does not necessarily lead to the conclusion that climate predictions are impossible (where climate is the statistics of weather over a “long enough period”). It doesn’t means that reliable climate predictions are possible either.

      • Given only the boundary value of energy in = energy out, of course there are a large number of completely difference possible climates.
        That is not a boundary value, in the classical mathematical sense, that can be imposed on the mathematical model equations. It is also not a boundary value that can be imposed in the physical domain. The energy out cannot be specified under any circumstances.
        It is used as a hypothesized state of the climate systems at a single location, the ToA, without mentioning the ramifications of the hypothesis relative to all the interfaces within the climate systems, and without mentioning the internal state of the materials that make up the sub-systems.
        Given that that is the case, what is left to justify invoking the (annoying trite) phrase.
        It is much more than an em>(annoying trite) phrase. It is used to minimize the ramifications of chaotic weather. Much as you some-whatly attempted in your second paragraph. However, proponents of application of the, plainly wrong, boundary-value problem phrase very likely will never utter these two sentences;
        “I agree, writing “it’s a boundary value problem” doesn’t mean it is solvable.”
        “It doesn’t means that reliable climate predictions are possible either.”
        See, for one example among hundreds, the post by Steve Easterbrook linked in my comment above. In which, as I read it, he explicitly states the converse of your conclusion; reliable climate predictions are possible solely because of this trite phrase.
        It is invoked in Climate Science so as to avoid discussions of the conundrum that they themselves set when it was first revealed that, Weather is chaotic but climate is not, and at the same time they state, Climate is the average of weather.

  59. VikingExplorer misunderstands the nature of chaotic objects whose relevant behavior (in the present instance, temperature change) is confined within asymptotic bounds. In the past 810,000 years, global temperatures have varied by little more than 3 K either side of the long-run mean – about the same variance as that which a home thermostat permits. It is reasonable to infer that in modern conditions even the combustion of all affordably-recoverable fossil fuels will not push the temperature beyond the upper bound indicated by cryostratigraphy – about 1-1.5 K above today’s global mean surface temperature. The reason for the near-perfect thermostasis of the Earth is that the atmosphere is sandwiched between two substantial heat-sinks – outer space above and the ocean below.
    An analogy: the height to which a tennis ball will bounce if dropped from a fixed altitude will vary in a mathematically-chaotic fashion owing to the many influences on it: the springiness of the ball, the compressibility of its surface, the variability of air density, the variable composition and hardness of the ground, etc. But in practice the ball cannot bounce higher than its starting point, however chaotic the influences upon it, for the laws of thermodynamics impose an asymptotic upper bound on its bound, as it were.

    • Sir Monckton,
      Although I admire and applaud the great work you’ve done to derail the AGW hysteria, I’ve got a science degree, and you are a journalist. I think it’s bad when my fellow anti-AGWers advance ideas which are untrue. I especially don’t like it when they advocate ideas which are untrue AND help the AGW cause, like your comment.
      If your comment at (May 26, 2015 at 11:26 am) were in fact true, then you could not possibly know that “even the combustion of all affordably-recoverable fossil fuels will not push the temperature beyond the upper bound”. You can’t have it both ways.
      Your comment can be summarized as the climate is unpredictable, yet I’ve just manipulated you into admitting that the climate is predictable: “is confined within asymptotic bounds”.
      “But in practice the ball cannot bounce higher than its starting point, however chaotic the influences upon it, for the laws of thermodynamics impose an asymptotic upper bound on its bound, as it were”.
      Exactly. Climatology is (or should be) the study of those asymptotic boundaries.
      You indicate that thermodynamics is providing predictability (which it is), yet Thermodynamics is the average result of chaotic molecular phenomena.
      So, we can dispense with silly notions such as “Averages of chaotic response are chaotic”. On average, V = IR. There is no chaos in Ohm law, yet it is the average result of chaotic electrodynamics.

      • You indicate that thermodynamics is providing predictability (which it is), yet Thermodynamics is the average result of chaotic molecular phenomena. [bold by edh]
        Thermodynamics is the average result of random molecular phenomena.
        Random is the antithesis of chaotic. Random fills all of phase space. Chaotic cannot leave the attractor and so cannot fill phase space.

      • Thank you Dan and you are correct, random and not chaotic, and that does get right into the topic of phase space characteristics. These terms so often get tossed about and even their use are not technically correct. Good call.

      • I’ve got a science degree
        ==============
        You didn’t get your money’s worth. You are confusing constant, random and chaotic to calculate nonsense.

      • Random behavior is truly unpredictable. Chaotic behavior is deterministic, so it is theoretically predictable. “You cannot have chaos without determinism. Chaos is not the lack of order. Chaos is order that is very sensitive to initial conditions; it’s not random at all”.
        Some other interesting links:
        Quantum Mechanics is not random.
        Molecular Chaos.
        “Interestingly, when noise, which is random, is added to an otherwise deterministic dynamical system, it can actually SUPPRESS the dynamical instability which underlies chaos. Thus randomness actually can quench chaos.” -Patrick Diamond, PhD ,MIT; Distinguished Professor, UCSD; APS Fellow, Two Int’l Prizes.

  60. scienceof
    doom said:
    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.

    Boundary and Initial, as in BVP and IVP and IBVP, usually refer to information that is specified at the boundaries. The Earth’s climate systems are open relative to energy. ‘the outgoing longwave radiation’ cannot be specified. The outgoing radiative energy is set by the states of the material internal to the physical and mathematical domains.
    Consider a 1-D transient heat conduction problem. (The heat conduction equation is almost always stated as a parabolic equation, but that’s not an issue.) Boundary conditions of the second kind cannot be specified at both ends of the physical and mathematical domains. That is not a well-posed mathematics problem.
    GCMs do not attempt to specify the outgoing longwave radiation. That would not be a well-posed mathematics problem.
    There are excellent reasons that GCMs are all formulated as an IBVP.

  61. The key is to acknowledge that science is a frame-based philosophy that through the application of the scientific method necessarily and intentionally is designed to constrain its application and thereby utility in both time and space. This need arises from the fact that a quasi-stable state (e.g. linear, uniform, closely bounded) can only be reasonably assumed over a finite region of time and space in a system characterized by chaotic (i.e. uncharacterized and unwieldy) and nonlinear processes. While inference (i.e. created knowledge) can direct the application of the scientific method, it is not itself a valid form of scientific logic; and correlation is not equal to causation and is a source of weak, albeit often sufficient (in quasi-stable frames), evidence.
    The key to analyzing nonlinear systems with any measure of accuracy is to establish a frame over which quasi-stability or linearity can be reasonably assumed. This is the foundation of weather forecasts that cast its predictive skills into the scientific domain with a statistically determined certainty.
    That said, there is a need to distinguish between the logical domains: science, philosophy, faith, and fantasy. A “scientific” theory only begins in the philosophical domain when there is a probable path that will lead to the scientific domain where the scientific method (i.e. observation, replication, and deduction) can be applied. Other theories, despite substantial circumstantial evidence, and evidence of punctuated continuity, do not rightly belong in the scientific domain (and perhaps not even the philosophical domain), because there does not exist a probable path where the scientific method could ever be applied.
    The Earth system is comprised of chaotic processes with indefinite but extended periods of quasi-stability. Since it is both incompletely, and by all measures insufficiently, characterized, the scientific method cannot be applied outside of a strictly limited frame in both time and space. While this assures a certain accuracy in short-term forecasts, and does not preclude application of risk management practices in the long-term, it does limit predictions of the system over indefinite and long ranges of time and space.

  62. Since climate is basically “the long term weather of a certain location” and since that is an initial boundary problem I just always figured it was the same thing with climate. You can average out some day in and day out variance with climate, but you can not really predict long-term climate without predicting weather, because weather is how warmth and cold circulate. And if that changes, heat content in the atmosphere changes which yes effects your model, so ergo it is an initial state problem that people have improperly called a boundary condition problem.
    the ironic thing I find with this bad classification that they use is that in the end they tell us “that we are disrupting the climate” – Holdren and so now they are telling us that the climate models which are bounded are supposedly going out of bounds…which yes is very ironic to me.
    The truth here?
    There is no boundaries to the climate and there never will be. All we have is an extremely chaotic initial state problem that is extremely difficult to predict. And we might never do it. But to insist that without evidence that climate models have any kind of prediction skill is the ultimate in hubris to me. It takes a special kind of arrogance to assume that you know something without actually proving it.

  63. How can we know whither climate is going without knowing where it is now? And how can we know where it is now without knowing where it has been?
    ISTM that the fundamental problem with climate models is that they have a “start date”. But how can we know how the climate is going to proceed from the start date if we don’t know how it got to the start date? That is, doesn’t the state of the climate in, say, 1979, depend on the state of the climate in 1978? (Or, if you’d prefer, 1948?) If we don’t know how the climate of 1978 (or, if you’d prefer, 1948) got to be what it was, how can we know how the climate of 1979 got to be what it was? So, the models don’t work because, at least in part, they pick an arbitrary initial value which actually isn’t the initial value at all: they sort of pretend that climate started in, say, 1979, though of course it did not.
    And ISTM that a lot of the discussion here takes it for granted that my questions can be asked about weather, but are inappropriate for climate. Why would that be? That is the same question, in slightly different form, as that posed in the article, isn’t it? Or am I misunderstanding? Or what?….

  64. Most of the laws of physics are usually expressed as differential equations. In the general case these are partial differential equations. In some special cases they are ordinary differential equations. The general solution to an ODE (ordinary differential equation) involves one arbitrary constant for each derivative, thus two for the common simple circuit or moving particle problem. Most PDEs (partial differential equations) of physics are of the second order and in three dimensions. Their general solution involves 3 times 2 arbitrary functions. Almost all such problems have no known solution. So, the examples you can find are very simple. The most common example seems to be the vibrating string with the ends constrained to not move. In that case the boundary is the two ends of the string. Solutions are known for the problem of a round vibrating membrane with the edge constrained, as in a drum. The circle is the boundary. That is just one boundary condition, and a very simple one.
    An initial value problem is just a simple boundary problem such as finding the solution for a particle with a certain position and velocity at time t=0.

  65. mr willis you forgot science is settled.
    We know we don’t know much.
    How lillte we know , how certain people can be …

  66. Excellent article Willis. I’ve only worked through 60 or so comments so far. They are also remarkably well thought out BTWr. Clearly, this article and the associated comments needs a lot of time. Three things stand out.
    1. I personally do not really understand BVPs. Not your fault. You’ve defined them. I just can’t get my mind about them. Something for me to work on.
    2. This discussion (and many others) seems to be hampered by the lack of meaningful metrics for climate/climate change. I understand what global temperature is supposed to be. But I can’t relate it in any useful way to what goes on outside my windows. And a frankly do not think that anyone else can either.
    3. I think maybe many commenters have the “boundaries” in BVP confused with the limits that climate operates within. The limits are things like it’s difficult with the current sun to freeze ALL the water vapor out of the atmosphere (yes, I know about the Snowball Earth hypothesis) and it is probably impossible to boil the oceans and achieve a Venus like state. I don’t think BVP boundaries and pragmatic limits on weather/climate are the same thing.
    I shall spend a few months thinking on all this.
    Again: thanks

  67. Thermodynamic boundaries are propagated between thermodynamic systems with divergent lapse rates (entropy distributions), The boundary layer is the zone of broken entropy symmetry where distribution of entropy is non-linear. Symmetry can also be broken by entropic effects in the zone where state change occurs (latent heat flux) such as the convective boundary layer. Turbulence in the boundary layer reflects the chaotic solution of the non-linear entropy state. Initial value conditions can usually be described as an absolute value. Boundary value conditions will be best described through statistical probability.

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