by Tomas Milanovic
There are few scientific concepts that are more often misunderstood in blog debates than Determinism and Predictability. For many commenters, these two concepts are considered to be in fact equivalent, which leads to faulty or irrelevant arguments.
Having read Dan Hughes’ interesting post about climate models and many but not all of the comments, I thought that it might be interesting for interested Climate Etc denizens to obtain further insight into these problems.
The post is structured in the following way:
The first section examines some particular classes of natural laws and draw from them lessons about predictability.
In the second section we consider the Navier-Stokes equations and review what we know about them and their solutions.
In the third and last section we extend the insights from the second section to the climate system, to see what hypotheses about the predictability of this system can be addressed.
If you have an allergy for Hilbert spaces, uncountably infinite dimensions and operator’s spectra you should stop reading at this sentence.
I. Deterministic Laws of Nature
The following four equations and systems are well known dynamical laws.
A first observation is that all four equations describe dynamical variables as unknown functions:
- X is the position in (1);
- X, Y, and Z are variables related to the fluid flow in (2);
- T is the temperature in (3);
- and Ψ is the wave function in (4).
A second observation is that the equations have been grouped into two groups of two. In the first group the dynamical variables are functions only of time. In the second group the dynamical variables are functions of time and space. This difference has huge mathematical consequences that we will find throughout the whole post. The laws of nature of the first group, having one independent variable, are expressed as Ordinary Differential Equations (ODEs) while the laws of nature of the second group, having four independent variables, are expressed as Partial Differential Equations (PDEs).
The third observation is that all four equations and systems are deterministic. This means that if we know the value of the dynamical variables at time t for (1) and (2) the equations yield a unique value of the value at a later time (t + dt). For (3) and (4) if we know the value of the dynamical variable at time t and at a position x, then the equations yield the value of the dynamical variable at a later time (t + dt) and/or at another position (x + dx).
These observations are enough to start considering the predictability of the dynamical variable.
Indeed the equations allow to compute future states that are dt later but dt is an infinitesimal. Yet what we would like is to know the value at a macroscopically later time t+t1 and ideally until eternity when t1 is infinite. The mathematical process which allows exactly that is called integration.
When the integration is analytically possible (and we suspect that there is no reason why it should always be possible) we will obtain functions which give the values of the dynamical variables for eternity and for any positions. In this case, and only in this case, we will be able to uniquely predict the values of the dynamical variables in the whole space-time.
So do unique solutions of our four natural laws (1) exist and, (2) can we find them?
The answer on the first question is easy. It can be shown that a unique solution exists for all four equations under some conditions. For the first two ODEs, there is a general proof that under some conditions of continuity these kinds of equations has always a unique solution provided that we know the required number of initial values. As (1) is of second order, we need two initial values (X(0) and dX/dt(0)) while three initial values X(0), Y(0) and Z(0) are enough for (2) which is of first order.
The situation is much more complex for the PDEs of equations (3) and (4). We still need initial conditions but as this time the system is extended over some spatial domain D, we need an initial condition for every single point of D.
In other words, for example in the case of equation (3), we need the solution to satisfy the condition T(0,x)=a(x) for all x, where a(x) is a function representing the temperature at every point at time t=0. In addition the domain D is not always the whole space. For example, if the physical domain D is a rod, we are only interested in the temperature distribution inside the rod. This implies that in addition to initial conditions which are always necessary, for finite domains we need to also impose boundary conditions which specify the properties of the dynamical variable at the boundary of D. Finally, given the initial conditions in the whole D (always) and boundary conditions (only if D is finite) we can prove that equations (3) and (4) have a unique solution .
The second question, can we find the solutions, is very difficult and we are at the heart of our problem. Each of the equations is considered in turn in the following discussions.
Equation (1) In many cases of practical interest it is integrable. So for these cases this natural law is both deterministic and predictable with arbitrary accuracy. We will see a special case of this law below.
Equation (2) These equations have an obvious property of being non-linear and a subtle property of being sensitive to initial conditions for some values of the parameters (S, R, B). Of important note is that these two properties are not equivalent. A sensitivity to initial conditions always implies non-linearity, and in the case of ODEs three or more dependent (phase space) variables, but non-linearity does not always imply sensitivity to initial conditions, analytical solutions to non-linear cases of equation (1) exist. This has far reaching consequences because even if we know that a unique solution exists, we are unable to compute it.
The sensitivity to initial conditions means that the smallest difference in initial conditions will lead to solutions which will separate exponentially with time. No computer can help here – as every computer works with a finite accuracy. Choosing two initial conditions which can’t be resolved by the computer we will obtain two wildly different solutions even if the computer sees only one. The computer’s “solution” will only be reasonably correct at the beginning but will become totally wrong for later times. Such a system is deterministic but unpredictable on long time scales and is called a temporally chaotic deterministic system.
The word temporal is very important because the dynamical variables depend only on time. If we apply the equation (1) to a system of three bodies and F is the gravitational force among them we obtain a non linear differential equation which is also sensitive to initial conditions and all conclusions drawn from equation (2) apply.
A system of three (or more) bodies interacting gravitationally is also deterministic but unpredictable on longer time scales. In this particular case of (1) we deal again with temporal deterministic chaos. Temporal chaos is relatively well understood, and there are many useful results especially concerning attractors (strange or not), but it is not the subject of this post to study temporal chaos.
Equation (3) It is linear so there are no particular problems (unless the initial conditions are extremely savage, discontinuous and non-differentiable, for examples). The solution can be computed and this natural law is both deterministic and predictable with arbitrary accuracy, both analytically and numerically.
Equation (4) This equation is formally identical to equation (3) – a linear PDE and the conclusions of equation (3) apply. The solution can be computed and this natural law is both deterministic and predictable with arbitrary accuracy. The reason why I mentioned this particular natural law governing the quantum mechanics is not because the equation is special. It is because the dynamical variable has a very specific role not seen in any other law of classical physics. The presence of “i” in the equation shows that Ψ is a complex number so it cannot represent any usual quantity in the physical domain as these are necessarily real.
Actually Ψ has a property that doesn’t exist in classical physics. It plays a role of a probability density which allows to calculation of probabilities of measure of all physically significant variables (energy, momentum ….). The only knowable information we can have about the system are the probabilities of the outcome of measure. The Hermitian operators playing the role of physical variables and Ψ impose the spectrum of authorised values for the physical variable and the result of a measure is any arbitrary number belonging to the set of allowed values. We have then here a very special case of a deterministically computable equation which allows only prediction of probabilities.
II. NAVIER STOKES EQUATIONS
The N-S equations govern the fluid mechanics. They are basically an expression of energy, momentum and mass conservation applied to a fluid.
We will use the variational form which is the most used and will focus only on the momentum equations which concentrates all the difficulties.
The advantage of this form is that by a suitable projection we have ensured a divergence-free velocity field and eliminated the pressure so that we are left only with the velocity vector as dynamical variable. The purpose is not to study N-S so that the reader doesn’t need the details of the operators . It is enough to see that is a linear operator comparable to a Laplacian and that the operator is obviously nonlinear because it contains products of the form .
What do we know about the existence, unicity and regularity of solutions of 3-D N-S equations? Well basically not much.
The existence, unicity and regularity of solutions in the general case has been an open problem for 2 centuries and remains so still today. There was a reason why I didn’t take the N-S equations as an example in section 1. It is because we have here the worst case of what we could get so that it would not be a simple example. The N-S equation is a nonlinear PDE with sensibility to initial conditions. It cannot be solved in the general case. It is an example of deterministic spatio-temporal chaos.
We will now examine more closely what that means and why spatio-temporal chaos is very different from the simple temporal chaos that we have seen in section I. The best start is with the phase space because it is there that the dynamical variables live.
In temporal chaos the phase space is simply R^N. For instance the phase space of the Lorenz system (2) has 3 degrees of freedom X, Y and Z so that the evolution of the system can be described by a curve in the ordinary 3-D space. More generally the temporally chaotic systems have their phase spaces in a finite and often low dimensional Euclidean space. The number of dimensions is equal to the number of degrees of freedom and each degree of freedom is a dynamical variable.
In spatio-temporal chaotic systems the phase space is an infinite dimensional Hilbert space. This can be intuitively easily understood. If we fix a point M of the spatial domain D, then the PDE in the point M is an ODE because only t may vary in a fixed point. The solution of the ODE in the fixed point M is then one degree of freedom of the system. But as there is an infinity of points in D, the PDE is equivalent to an infinity of coupled ODEs so that there is an infinity of degrees of freedom and the phase space is an infinite dimensional functional space.
Another way to characterize the phase space is to consider the example of solutions by separation of variables. When looking for solutions f(x,t) of a PDE, one looks for solutions of the form
f(x, t) = X(x) T(t). For linear PDEs like (3) and (4) this allows to obtain 2 ODEs – one for X with variable x and one for T with variable t. Solving these 2 ODEs for given initial and boundary conditions leads to a general solution which is an infinite sum of Xi(x) Tj(t).
These “elementary” functions Xi (x) and Tj (t) can be seen as vectors of an orthogonal basis of the set of solutions of that given PDE. Here again the basis contains an infinity of vectors so that the space of solutions is infinite dimensional. For example the Eigenfunctions Xi(x) of the hydrogen atom form an orthogonal infinite basis of the phase space and are called spherical harmonics.
Finally the phase space where the solutions of N-S live is an infinite dimensional Hilbert space of integrable functions f(x, t) which is very different from the phase space of a temporal chaotic system which is a simple finite dimensional Euclidean space.
This prompts a word of caution that analogies between temporal chaos (relatively well understood) and spatio-temporal chaos (badly understood) should not be used because they are likely wrong and misleading.
At this stage as we know that N-S equations cannot be solved and exhibit unpredictable behaviour, it is time to ask: “And what about Computational Fluid Dynamics (CFD) ?”
Well, there are many reasons why N-S is difficult but one of the most important is the dependence on the initial conditions and on the nature of the boundary of the physical domain. One can simplify both by constraining the former in a very small volume of the phase space and the latter in very simple geometries (a cube, 2 parallel planes etc). Aditionnaly one must eliminate the turbulence at very small spatial scales. This happens in industrial applications mostly by using Reynolds Averaged Navier Stokes (RANS) which is a transformation of the original N-S. The problem with RANS is that it is not closed (there are more unknowns than equations ). The only solution is to make up empirically new equations and that boils down to a sophisticated curve fitting. These equations can then be only used in special cases for which the curve fitting was validated.
After all this work CFD will give reasonable results but only for very small volumes (a few meters), strictly defined initial/boundary conditions, and simple geometries. It is obvious that these particular applications don’t give any insight about the solutions of N-S when the strong constraints are not respected.
To give a practical example – CFD can be used on a simple, smooth, small, horizontal wing moving at constant speed in calm air with constant temperature and give reasonable results on air pressures and velocities near to the wing. However if the speed exits the specifications (the plane stalls) the velocities and pressures become strongly chaotic and the CFD model is no more able to compute accurately drag and lift and the trajectory of the plane becomes unpredictable.
CFD is extremely taxing the computer resources. To realize how taxing it is, for example the CFD studies of combustion engines with a typical scale of 10 cm, need a space mesh size of 0,1 mm and a time step of approximately 1µs. This explains why CFD cannot be used for systems whose typical size is larger than a few meters. If we were to compare a CFD domain with a climate model then the spatial resolution of CFD is one billion times finer and the time step is 100 millions time shorter what gives an idea how coarse a climate model grid is.
Last important question about N-S that we will examine is “And what about asymptotic behaviour ?”
This question is inspired by the fact that N-S deals with a dissipative dynamical system. As it is known that volumes in phase spaces decrease when a system is dissipative, there could be hope that the solutions will settle on some manifold after a long time instead of exploring the whole infinite dimensional phase space forever.
Recent research answers that this hope could be, at least under some conditions, justified. Indeed it has been proven for 2-D N-S that there exists a global finite dimensional attractor and an upper bound for the dimension was found. Let us not be mislead by the expression “finite dimensional” – the number of dimensions depends on many factors and is counted in billions. Nothing such has been proven for 3-D N-S even if some encouraging results for weak topologies and with some constraints have been obtained.
This means concretely that if we speculate on the existence of such a global attractor (for some sufficiently regular initial conditions and domains) then every solution could be expressed as a combination of a few billions of well defined functions Fi(x,t). These functions can then be considered as the basic dynamical modes which fully describe the asymptotic behaviour of the N-S solutions.
For both questions asked above it is important to stress again that we are studying here the momentum conservation N-S equation only. The kinetic viscosity being considered constant, this implies that we deal with a newtonian, incompressible and isothermal fluid – for example water.
Indeed for isothermal fluids the internal energy is approximately constant so that the momentum conservation is approximately equivalent to energy conservation and no additional equation is needed. When the fluid is not isothermal, it is necessary to add a specific energy conservation equation which adds a new variable, the temperature.
III. WEATHER AND CLIMATE
If the N-S equations are difficult, weather analysis is much more difficult and climate analysis is infinitely more difficult, the word infinitely being meant literally.
Weather is studied basically with the N-S equations discussed above including the energy conservation equation. Additional complexity is that weather analysis must deal with polyphasic flows (liquid, vapour, solid). Following the discussion of N-S, weather is deterministic, sensitively dependent on initial conditions, spatio-temporal chaotic and therefore unpredictable beyond a very short time scale (days). The quality of the prediction is extremely sensitive on initial conditions, too – an anticyclonic system gives regular, slowly varying velocity fields which are rather stable while depression fronts and strongly varying velocity fields can not be predicted sometimes even within 24 hours.
The weather models, like CFD and for the same reasons, are useful in conditions where the velocity field is regular. These conditions can be described by the popular saying “The best prediction for tomorrow’s weather is that it will be the same like today.” They are however relatively useless for extreme weather, irregular (stormy) systems where the numerical simulation cannot find the right solution.
A didactical example is given by the storm of the century over western Europe in December 1999. The ECWMF model saw a stormy weather but failed to predict a storm of the century. Fifty different forecasts were realized by perturbing the measured initial conditions. The resulting forecasts varied anywhere between “nothing happens” and “there will be a severe storm”. Approximately 10 forecasts among the 50 looked like the reality 40 hours later. From the discussion about N-S we know that the system is governed by deterministic equations therefore we know that the “probability” of a storm of the century in the real world was 100 %.
This observation is very important for the notion of “probability” that we will examine now. Indeed one might want to consider the 50 forecasts as a representative sample of what the weather could be, from where it is only one step to suppose that each individual forecast has a 2 % probability and thus deduce that a storm of the century has about 10×2 = 20 % probability to happen.
Nothing could be more wrong.
First we know that the storm had to happen because the equations are deterministic. So the probability of the event was 100 %.
Second, the sample of 50 initial conditions is arbitrary both in its size (50) and in the choice of each specific perturbation. Let us recall that we have an infinite dimensional phase space so that if we randomly choose 50 perturbations of the initial conditions, the probability that we find the same forecasts that were obtained with the initial sample would be 0 ! Therefore the sample doesn’t represent in any way the whole space of possible final states.
Third, there is no reason that the final states (forecasts) corresponding to a given perturbation of the initial conditions have all the same probability (2% in our example). On the contrary, the Lyapounov coefficient that measures the divergence of orbits in the phase space depends on the initial conditions. From there follows that a perturbation with a “large” Lyapounov coefficient will occupy a “large” proportion of the final states. However as the Lyapounov coefficients are unknown, it is impossible to associate a “probability” to a given final state and certainly not by naively dividing 100 by the number of perturbations selected arbitrarily.
The only fundamental difference between CFD and weather forecasts on one side and climate on the other side is the size of the grid used for the numerical scheme. Climate models can therefore be seen as a weather model with a much larger spatial grid – hundred(s) of kilometer(s).
NUMERICAL CLIMATE MODELS
In this last part we will examine what is a numerical scheme and what consequences we can derive for the predictability of climate models.
Basically every numerical scheme consists to choose a time step and a spatial grid and replace the partial derivatives of a variable U by finite fractions . The equation is then rewritten with indexes for the discrete time variable and the discrete space variable. For instance the time discretisation of incompressible N-S yields:
It appears clearly that if we know the initial value U(0) and boundary conditions of U then we can easily compute U(t+1) from (6) and by substitution in (5) we solve for P(t+1). To fully discretise the equations we have still to express the spatial operator . There are many choices for example for a 2-D square grid where the index I is in the x direction and the index J is in the y direction we can choose .
For those interested in more details, a good synthesis and discussion of discretisation techniques for incompressible Navier-Stokes can be found in this paper Numerical Methods for Incompressible Viscous Flow.
By substituting the discretisation of the spatial variables in (5) and (6) we obtain a linear system:
where :
- M is a matrix with elements depending on U(t, I, J) and constants
- (U, P) is a vector containing U(t+1, I, J) and P(t+1, I, J) at all points of the grid
- (f, 0) is the vector of the external force at all points of the grid.
The whole N-S problem has now been reduced to the question of invertibility of the matrix M. If M is invertible then the system has a unique solution U(t+1, I, J) and P(t+1, I, J). The iteration of this method yields then U(t+2, I, J) and P(t+2, I, J) knowing U(t+1, I, J) and P(t+1, I, J) so that a solution of (8) and (9) in [0,T] can be computed for all T.
In fact all problems start really here.
The 2 convergence problems
“Do the solutions of (8) and (9) converge when the grid step and the time step decrease to 0 ?”
This question cannot be tested for climate models because a minimum scale is a few hundreds of km and it is impossible to decrease the space and time steps to 0. So the answer is “We don’t know“.
“Do the solutions of (8) and (9) converge to the solutions of N-S ?”
Here the answer is a clear “No” because a single value in a grid 100km x100km (with or without subgrid parametrization) cannot represent in any way the continuous solution which is anyway unknown.
A weaker form of this question is “Do the solutions of (8) and (9) have some similarity to the solutions of N-S at spatial scales greater than the grid scale ?”
Again this cannot be tested because the N-S solutions are unknown.
However even low dimensional approximations of large scale atmospheric circulation, for instance using the constructal theory, and assuming only 3 partitions of a rotating planet, get the large scale circulation in Hadley, Ferrel and polar cells correctly. This is also what Large Eddy Simulation (LES) methods, or subgrid parametrisation methods, which is a synonym, try. The basic idea here is to simply average (filter) everything that is below the grid scale L. The N-S solution can then be written as
S(x,t) = Sa(x,t) + Sna(x,t)
where:
- Sna is the solution of the non-averaged part (on scales > L)
- Sa is the solution of the averaged part (on scales < L)
Sna could then be numerically computed by equations similar to (8) and (9) IF one knows the initial and boundary conditions for the scale L. But as these depend on the solution Sa, we are back to the problem that we cannot solve N-S for scales smaller than L.
The problem is here identical to the RANS problem– the nonlinear advection term effectively couples the averaged (small) scales to the non-averaged (large) scales so that it is impossible to consider Sna and Sa as independent. This leads to the necessity to model the interactions between all small scales up to the grid scale where the interaction between Sna and Sa happens.
Unfortunately the subgrid scales contain phenomena (boundary layer dissipation, clouds, storm convection, phase changes, biological reactions, . . . ) that impact significantly the large scale dynamics and that are not well understood. For that reason there is a very large number of possible subgrid parametrizations which lead each to different large scale dynamics.
Finally the answer on the initial question can only be “Maybe sometimes.”
The stability problem
There is a unique solution for (8) and (9) if the matrix M is invertible. However as its elements depend on initial conditions, there is no guarantee that M is invertible. Actually there is an infinite number of initial conditions for which M is not invertible and this leads to numerical instabilities.
The behaviour of M also depends on the size and topology of the grid so that purely numerical artifacts may appear when the grid is badly chosen. Some of these artifacts can be easily identified but those that are not identified (unknown unknowns) cannot be corrected.
The chaos problem
This problem is severe and unsolved.
We know that the N-S equations exhibit spatio-temporal chaotic behaviour. The consequence is that 2 initially very close orbits in the (infinite dimensional functional) phase space will diverge exponentially, and that property leads to unpredictability beyond some time T dependent on the system considered. Yet we have also seen that the solution of (8) and (9) is unique and predictable for all times when it exists. From these it follows that the solutions of (8) and (9) can never represent a solution of N-S because they lack the defining feature which is the chaotic behaviour.
As we have seen in the discussion about LES, the subgrid parametrization may, but must not, introduce random behaviour (especially for stochastic models) which could give an illusion of chaotic behaviour. However this is only an illusion because randomness and chaotic behaviour can be readily distinguished and each leads to different solutions.
A word for the sempiternal but artificial distinction between initial and boundary conditions. It is simple – no PDE and no numerical scheme like (8) and (9) can be solved if initial and boundary conditions are not prescribed. It is irrelevant whether the system is chaotic or not – the heat equation, N-S, weather, climate they all depend on initial and boundary conditions. In addition for chaotic systems it is also irrelevant whether the instantaneous values, their discretization or their averages are used because as we have seen, if the defining continuous equations are chaotic then their discretization, and averages, are also chaotic and they are generally different from the solution of the continuous equations.
Sometimes the fundamental unpredictability of chaotic systems is challenged by examples like seasonal averages – for example “We can predict that a summer temperature average in [insert place] will be greater than a winter temperature average.” This pseudo argument is based on a deep misunderstanding of spatio-temporal chaos. Of course chaos must not be equated with exactly zero predictability. There is sometimes a number of phenomena and parameters’ ranges which can be predicted at least approximately even in a fully chaotic system. They are almost all based on the existence and invariance of a finite dimensional attractor. Indeed in this case we can predict that the system will always stay inside the attractor. If we know its number of dimensions and its topology we can analyse the properties of its subsets and predict what will happen when the system will visit a particular subset.
For the sake of clarity imagine an ergodic chaotic system whose attractor looks like: O=O. The system wanders quasi-periodically from left to right and back again – for example it is periodically driven. While it is impossible to predict where the system will exactly go and how long it will stay there, we know that it will spend broadly half of its time in the left part and half of its time in the right part of the attractor.
As we know the topology of the attractor we can compute Ts, the average of a degree of freedom T in the volume of the phase space which represents the right hand “O” and compare it to Tw, the phase space average of T in the left hand “O”. If we find for example that Ts>Tw then the ergodicity hypothesis will allow to conclude that because the system spends approximately half of its time W left and half of its time S right, then the time average of T during the period S will be greater than the time average during time period W.
Of course this conclusion stays true only as long as the system stays ergodic and the attractor invariant. Eventually this conclusion may only be valid for some ranges of some parameter (let us call it latitude for example). We must stress that ergodicity is the necessary condition to draw this conclusion for all variables and all parts of the attractor. The ergodicity of the climate system has not been proven and this proof is a challenge that still waits to be met.
Over large time scales the attractor will change and even this simple conclusion will no more be valid. Also, trivially, the ability to compute phase space averages for particular attractor topologies changes nothing on the fact that the system is still chaotic and will react on perturbations in an unpredictable way over larger time scales.
The probability problem
An argument often heard is that climate models do not predict the future states but the probability of future states.
This statement would be true for equation (4) (Schrödinger) because the unknown function Ψ represents explicitly the probability of future states. However there exists no such equation for fluid dynamics and by extension for climate. So as it is impossible to compute the probabilities of future climate states explicitly, the claim cannot be understood literally.
What the climate models do is to compute a SINGLE future state instead. The argument makes also no sense empirically because it is not possible to observe a large number N of climate realizations and to define statistics on the set of these realizations.
The only way to give sense to this argument is to consider that multiple runs of a climate model are equivalent to potential multiple realizations of climate and that a probability density may be defined over the finite set of these runs even if the results can never be observed in reality.
Can this hypothesis be true? We have already shortly discussed this hypothesis for the “storm of the century” in 1999 and the answer was no. The same answer holds for the climate.
A necessary but not sufficient condition is that all computer runs correctly solve N-S with an arbitrarily low accuracy for all times and all initial conditions. This condition is not fulfilled as we have discussed above and that is enough to reject the hypothesis.
But there are many additional arguments to reject the hypothesis too. The most important is related to the finite sample of initial conditions. Indeed if we define a finite number N of arbitrary perturbations Pi applied to a fixed initial condition we obtain a set {ICi} of N initial conditions. But as the system is chaotic, we know that for any neighbourhood of a given initial condition ICi, the orbits will diverge exponentially for any 2 points chosen in this neighbourhood. Thus we may choose a perturbation Ri as close to Pi as we wish and obtain a final state FS(Ri) as far from FS(Pi) as we wish, eventually bounded by the size of the attractor if it exists.
Finally from the unicity of solutions follows that the intersection of the sets of final states {FS(Ri)} and {FS(Pi)}is empty so that if we define a probability in both sets, for example P(FSi) = 1/N, then we have P(FS(Pi)) = 1/N over {FS(Pi)} and P(FS(Pi)) = 0 over{FS(Ri)} what is a contradiction. It follows that no consistent probability can be defined over a finite set of initial conditions (or scenarios which is the same thing) and therefore climate models cannot predict any probabilities.
There is an even more bizarre argument related to the issue of probabilities and averages. Real Climate says: “Multi-model Ensemble – a set of simulations from multiple models. Surprisingly, an average over these simulations gives a better match to climatological observations than any single model.”
It is indeed surprising because it is most certainly wrong, or misleading, or both.
Now as there are dozens of papers dealing with Multi-model Ensembles, it seems that there are people who take the above Real Climate statement seriously. Yet as everybody could have seen, it fails already at the basic logic level.
The Real Climate statement is equivalent to the implication: “If model A says 1 = 0 AND model B says 5 = 0 then the model (A+B)/2 saying 3= 0 gives a better answer“.
This implication is false and therefore useless. The observation is simply explained by the trivial statistical fact that the variance of the average of Xi is smaller than the variance of the Xi. This fact, however, doesn’t allow any useful conclusion about the validity of solutions obtained by averaging different models which are known to give wrong results.
ARE NUMERICAL CLIMATE MODELS USEFUL ?
I do not think so.
Not because climate models are wrong and they are indeed wrong. But because they drain financial and human ressources to the least efficient and most cost intensive research direction. When one invents concepts like “ensemble averages” which have no rational fundaments and when there are more papers studying why model A doesn’t behave like model B than papers studying the climate itself, you suspect that something must have gone wrong.
Also if after 30 + years of huge investments we are still unable to robustly describe and predict the defining features of the system (pressures, velocities, precipitations, temperatures, cloudiness) at the only scales of interest which are regional, then it is reasonable to suppose that this research direction is not adapted to explain and predict regional features. However if this post only criticized the shortcomings of numerical climate models what is quite easy, it would miss the mark .
There are other research directions actually unfortunately understudied. I believe that the weather and therefore the climate have a global finite dimensional attractor. As the boundary conditions of the system are given by the shape and location of the continents and of the ocean floor on one side and the orbital parameters as well as the energy output of the Sun on the other side, this attractor can be considered as invariant over the time scales of interest – e.g hundreds or thousands of years.
Considering that almost all energy of the system is in the oceans and in the water cycle (ice, water, water vapour), the characteristic spatio-temporal functions defining the attractor would mostly describe oceanic dynamics. These characteristic functions would appear like spatio-temporal quasi periodic oceanic oscillations and currents. Even if the attractor had millions of dimensions, by analogy with a Taylor expansion, only a small number of them could be enough to explain the system’s behaviour at the scales of interest. For instance the observations suggest that ENSO is the leading order oscillation with other large scale features like the Gulf Stream and the Circumpolar Antarctic stream following. Of course such dominating features like ENSO are certainly not a single oscillation but rather a composite of a number of smaller and shorter oscillations, but these can be looked for.
Techniques allowing to reconstruct the attractor properties from lower diemnsional projections exist for temporal chaos. They could be extended to spatio-temporal chaos. I am convinced that the direction of research aiming to understand oceanic oscillations and their interactions as they are observed could lead to a real breakthrough in our understanding of climate. My personnal hope is that realizing the lack of results, sooner or later resources will be diverted from numerical models and super computers towards theoretical work on spatio-temporal chaotic attractors and their applications which would identify the dominating oceanic modes.
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