Guest Post by Willis Eschenbach
I’ve heard many times that whereas weather prediction is an “initial-value” problem, climate prediction is a “boundary problem”. I’ve often wondered about this, questions like “what is the boundary?”. I woke up today thinking that I didn’t have an adequately clear understanding of the difference between the two types of problems.
For these kinds of questions I find it’s hard to beat Wolfram Reference, which is a reference to the various functions in the computer program Mathematica. Wolfram is a total genius in my opinion, and the Wolfram site reflects that. Here’s what Wolfram Reference says (emphasis mine):
Introduction to Initial and Boundary Value Problems
DSolve [a Mathematica function] can be used for finding the general solution to a differential equation or system of differential equations. The general solution gives information about the structure of the complete solution space for the problem. However, in practice, one is often interested only in particular solutions that satisfy some conditions related to the area of application. These conditions are usually of two types.
• The solution x(t) and/or its derivatives are required to have specific values at a single point, for example, x(0)=1 and x’(0)=2. Such problems are traditionally called initial value problems (IVPs) because the system is assumed to start evolving from the fixed initial point (in this case, 0).
• The solution x(t) is required to have specific values at a pair of points, for example, x(0)=1 and x(1)=5. These problems are known as boundary value problems (BVPs) because the points 0 and 1 are regarded as boundary points (or edges) of the domain of interest in the application.
The symbolic solution of both IVPs and BVPs requires knowledge of the general solution for the problem. The final step, in which the particular solution is obtained using the initial or boundary values, involves mostly algebraic operations, and is similar for IVPs and for BVPs.
IVPs and BVPs for linear differential equations are solved rather easily since the final algebraic step involves the solution of linear equations. However, if the underlying equations are nonlinear, the solution could have several branches, or the arbitrary constants from the general solution could occur in different arguments of transcendental functions. As a result, it is not always possible to complete the final algebraic step for nonlinear problems. Finally, if the underlying equations have piecewise (that is, discontinuous) coefficients, an IVP naturally breaks up into simpler IVPs over the regions in which the coefficients are continuous.
Now, as I read that, it says that for an initial value problem (IVP) we need to know the initial conditions at the starting time, and for a boundary value problem (BVP) we need to know the future conditions at a particular boundary. For example, suppose we are interested in the future thermal behavior of an iron rod with one end in a ice-water bath. The boundary condition is that the end of the iron rod in the ice-water bath is at 0°C.
So my question is two-fold. IF predicting weather is an IVP and predicting climate is a BVP, then
1) What is the “boundary” in question?, and
2) Once we determine what the boundary is, how do we know the future value of the boundary?
Some investigation finds that for US$48 I can read the following:
Existence and regularity theorems for a free boundary problem governing a simple climate model
Xiangsheng Xua
Abstract
From a class of mean annual, zonally averaged energy–balance climate models of the Budyko‐Sellers type, we arrive at a free boundary problem with the free boundary being the interface between ice‐covered and ice-free areas. Existence and regularity properties are proved for weak solutions of the problem. In particular, the regularity of the free boundary is investigated.
Fortunately, I don’t need to read it to see that the boundary in question is the ice-water interface. Now, that actually seems like it might work, because we know that at any time in the future, the boundary is always at 0°C. Since we know the future temperature values at that boundary, we can treat it as a boundary problem.
But then I continue reading, and I find Dr. Pielke’s excellent work , which says (emphasis mine):
One set of commonly used definitions of weather and climate distinguishes these terms in the context of prediction: weather is considered an initial value problem, while climate is assumed to be a boundary value problem. Another perspective holds that climate and weather prediction are both initial value problems (Palmer 1998). If climate prediction were a boundary value problem, then the simulations of future climate will “forget” the initial values assumed in a model. The assumption that climate prediction is a boundary value problem is used, for example, to justify predicting future climate based on anthropogenic doubling of greenhouse gases. This correspondence proposes that weather prediction is a subset of climate predictions and that both are, therefore, initial value problems in the context of nonlinear geophysical flow. The consequence of climate prediction being an initial value problem is summarized in this correspondence.
The boundaries in the context of climate prediction are the ocean surface and the land surface. If these boundaries are fixed in time, evolve independently of the atmosphere such that their time evolution could be prescribed, or have response times that are much longer than the time period of interest in the climate prediction, than one may conclude that climate prediction is a boundary problem.
So Dr. Pielke says that there is an entirely different boundary in play, the boundary between the atmosphere and the surface.
But then my question is, how would we know the future conditions of that boundary? If it’s a BVP, we have to know future conditions.
Dr. Pielke takes an interesting turn. IF I understand his method in another paper, Seasonal weather prediction as an initial value problem, he shows that the chosen boundary (the atmosphere/surface interface) doesn’t “evolve independently of the atmosphere such that their time evolution could be prescribed” and thus seasonal weather prediction is shown to be an IVP rather than a BVP.
However … he’s using an entirely different boundary than that used by Xiangsheng Xua above. Which one is right? One, both, or neither?
And the underlying problem, of course, is that IF climate is an initial value problem just like weather, given the chaotic nature of both we have little hope of modeling or predicting the future evolution of the climate.
My conclusion from all of this, which I think is shared by Dr. Pielke, is that climate prediction is an initial value problem. I say this in part because I see no difference in “climate” and “weather” in that both seem to be self-similar, non-linear, and chaotic.
This view is also shared by Mandelbrot, as was discussed about a decade ago over at Steve McIntyre’s excellent blog … have we really been at it that long? Mandelbrot analyzed a number of long-term records and found no change in the fractal nature of the records with timespan. In other words, there’s no break between the chaotic nature of the short, medium, and long-term looks at weather.
Now, it’s often argued that weather prediction has gotten much better over the decades … and this is true. But remember, weather prediction is an initial value problem. That means that the more accurately and specifically and finely we can measure the initial conditions, the better our prediction will be. Much of the improvement in our weather predictions is a result of satellites which give us our initial conditions in exquisite detail. And despite all our advances in predictive ability, lots of weekend barbecues still get rained on.
And at the end of the day, I’m left with my initial questions:
• If modeling the future evolution of the climate a boundary problem, what exactly is the boundary?, and
• Having specified the boundary, how can we know the future conditions of the boundary?
Egads … a post without a single graphic … curious.
w.
My Usual Request: If you think something is incorrect, please have the courtesy to quote the exact words that you disagree with so that everyone can understand your objections.
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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?
I totally agree. So why have models, just guess. Which is what happens anyway.
Climate models must be both initial and boundary. To do otherwise is like a vector without an affine point.
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.
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.
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.
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.
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.
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.
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)
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.
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
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.
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.
IPCC & WMO define climate as weather averaged over 30 years. So it’s simply data and statistics, just like at the dog & horse tracks.
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.
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.
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.
Thanks, Dr. Pielke. Your papers in this matter are my most valued references.
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.
Willis – please also see this paper
Pielke, R.A. and X. Zeng, 1994: Long-term variability of climate. J. Atmos. Sci., 51, 155-159. https://pielkeclimatesci.files.wordpress.com/2009/09/r-120.pdf
Roger Sr.
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.
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.
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.
That sounds right Roy, as usual.
Roy Spencer May 25, 2015 at 5:55 pm
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:
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
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.
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.)
Lackeys. I meant lackeys.
@jorge
Sure you did, sure…
If you tell the truth so bluntly, it makes it more difficult to pretend like we’re talking about science. 🙂
Wilis, brilliant exposition.
From a Mathematica user, never failed yet even if some is unsolvable.
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?
“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.
Well I should have said moving average.
“I can’t Imagine a stronger low pass filter, if there is one.”
First derivative? Or maybe the second.
There are some differences (see graph) when comparing moving averages (here 11 year) and a low pas filter (here Butterworth 6 Db down at 22 year)
http://www.vukcevic.talktalk.net/SSN-MA-LPF.gif
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
You say
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.
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.
richard verney
YES!
Richard
“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
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
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.
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?
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
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
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
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
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
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
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
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
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
And in my next post to you I wrote
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
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)
Your post I am replying concludes saying.
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:
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’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:
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:
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:
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:
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:
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:
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:
Thank you Aran for a very clear explanation. Useful link too.
Aran, well said. I agree with you.
MikeB and Viking Explorer
Days have passed since I asked you
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.
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