Saraz (29 Aprilie 2012 06:42)
nothing about the flow itself (infinite diensmional phase space and generally no ergodicity) and its predictability.I find here the much too frequent overhyping of the Lorenz' system which people with insufficient non linear dynamics experience consider consciously or inconsciously as a proxy for ALL chaotic systems.And this is clearly wrong as the true discoverer of chaos, not Lorenz but Poincarre9, knew already 100 years ago.I can also answer the wondering of the authors This lack of spread suggested that the perturbations to the initial conditions were sub-optimal and did not necessarily capture the most rapidly growing modes . Considerable research into identifying those modes using singular vectortechniques was undertaken , and significant progress achieved in increasing the spread of the ensemble. Nevertheless, the ensemble remained under-dispersive so something else must be missing.In this case the phase space is the infinite diensmional Hilbert space of functions f(x,y,z,t). The dynamics of the system are represented by a vector in the phase space.Now what does it mean to generate ensembles by random perturbations?Well it clearly means that the phase space is supposed isotropic and that the attractor (a Hilbert sub space) is isotropic too e.g any random perturbation will do.However this is apparently wrong any attractor if it exists has privileged directions what means that the Hilbert vector in some initial position, will have some privileged directions too. The evolution/perturbation that will happen will not be random some will have a very high probability while some other will have a probability 0.If the region of the attractor is very rugged , the dispersion may be very high what a smooth isotropic probability distribution could never simulate.Actually my conclusion from the authors' comment is that the models are obviously unable to correctly reproduce the anisotropy of the attractor.A word about the figure 3 in the paper which nicely illustrates the above comment.Top left we have the initial spatio-temporal pressure field with a quite dramatic depression.Below we have 50 computer runs obtained with random(isotropic) perturbations giving the result a few days later.18 of them show also a dramatic depression on different places. 32 show that it dissipates.A naive approach would be to predict a hurricane with a 36% (18/50) probability.Of course this prediction would base on totally unjustifiable assumptions that :a) 50 runs are enoughb) the probability of the result of every run is iid, e.g 2 % (1/50)c) the distribution of the 50 runs is independent of the arbitrary choice of perturbations which generate them.In reality the highly energetical initial state in which the system is, is likely also highly anisotropic.For instance that could mean that the direction in the phase space to the final state shown in run 13 has a probability of 0 while the direction to the final state shown in run 11 has a probability of 80%. The distribution of these probabilities could be of course dependent on the initial state (e.g the system is not ergodic) and the attractor highly anisotropic.Consequence is that the probability of a hurricane would be no more 36% but near 80%.This short example shows that this kind of hypothesis doesn't play at margins it may change the order of magnitude of the result!I also add a technical comment related to figure 4 the only correct metric for a Hilbert space is the Hilbert scalar product. That's why all dispersions must be expressed with this metrics and the notion of ensemble average is irrelevant.Therefore it appears paramount that before one rushes in some (often stupid) climatic computer runs and averagings over too long time scales, it is necessary to clear the very fundamental theoretical basis what is the attractor? How anisotropic is it? Is there any hope for strong ergodicity? Or any weaker kind? What about pseudo periodical spatial structures at all time scales?For the conclusion I would join Theo Goodwin who said it in a much shorter and relevant way :Models not specified by theories are tinker toys. The research money has to go into theories, sets of physical hypotheses, and not models.