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.
“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.
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.
This discussion reminds me of the Avett brothers song so a bit of light relief should be welcome.
Enjoy
http://www.vagalume.com.br/avett-brothers/ten-thousand-words.html
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).
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.
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.
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
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.
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.
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.
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.
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.
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?
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.
I am so glad the science is settled. Think of the confusion if it were not. All, a wonderful post and discussion.
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.
Are IVP’s, BVP’s are not based on physical boundaries.
Here is a link that has what I think is a good descriptions:
http://tutorial.math.lamar.edu/Classes/DE/BoundaryValueProblem.aspx
For EE’s it’s the difference between a state dependent simulation, and a non-state dependent timing verification.
nickreality65:
“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.”
The measurements of solar radiation in the satellite era 1979 onwards show very little variation. What is not certain is the absolute value. These are two different factors.
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).
scienceofdoom May 26, 2015 at 12:44 am
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:
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.
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.
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.
@ur momisugly Monkton of Brenchley
It would be good if you could attend http://www.royalsoced.org.uk/events/event.php?id=394 which is local to you (a few hundred yards walk).
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.
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.
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?….
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.
mr willis you forgot science is settled.
We know we don’t know much.
How lillte we know , how certain people can be …