Saturday, May 31, 2008

Climate Science a Brain of two halves.

Our brain has two halves: one is responsible for the multiplication of polynomials and languages, and the other half is responsible for orientation of figures in space and all the things important in real life. Mathematics is geometry when you have to use both halves.

Vladimir Arnold

Describing the mathematical problems (some years before the celebrated Hilbert’s list) Poincare divided them into two parts: the binary problems (similar to the Fermat problem, where the answer is a choice between the two possibilities: “yes” or “no”), and the interesting problems, where the progress is continuous, studying first of all the possibility of the variations of the problem (like, say, the variation of the boundary conditions for a differential equation) and investigating then the influences of these variations on the properties of the solutions (which would be hidden, if the problem were formulated as a binary one).

Vladimir Arnold in Bifurcation theory reduces it to an easy understanding.

Poincare bifurcation theory was elaborated by the Russian mathematicians Pontryagin
and Andronov already in the 20’s and in the 30’s (due to the need to apply these bifurcations to radiophysics). Andronov published (with all the proofs) the theory of the birth of a periodic motion of a dynamical system under the generic loss of stability of an equilibrium position, in the case when two eigenvalues of the linearised system cross the imaginary axis, moving from the stable to the unstable complex half-plane.

Andronov’s theorem claims that (depending on the sign of some higher term of the Taylor series) exactly two generic cases may occur: Either the stability of the equilibrium position is inherited by the new-born limit cycle (whose radius grows like the square root of the difference between the new value of the parameter and the value at the stability loss), or else the radius of the attraction domain, diminishing like the square root of the difference between the growing parameter value and the future value, at which the stability will be destroyed, disappears at the stability loss moment.

The first case is called the mild stability loss, the new-born periodic motion-attractor describes a small oscillation near the old stationary regime. The second case is called the hard stability loss, the behaviour of the system after this stability loss being very far from the equilibrium, loosing its stability. The proofs of these results of Andronov on the phase portraits bifurcations were based on the Pontryagin’s extension of Poincare’s results in the holomorphic case to that of the smooth systems of differential equations.

As we see with quasiperiodic systems the return to a “previous climate” state is regularly seen in the NAO.,PDO and IPO. These are sates with long periodicity for a climate regime of warmer then normal or cooler then normal climate states.

The ability for a recurrent periodic state such as the PDO or an inverse temperature
“state” is in essence a binary transformation or bifurcation.. The transformation as a velocity inversion has the same effect as a time inversion,(v to -v ) (t to -t)
An interesting property is the probability P+=P- = 0.5 To put this in perspective, for records generated by statistically independent processes with finite standard deviation for periodic bifurcations , where there can be either hard or soft stability loss the exponent is ½.

A fundamental consequence of the aperiodicity of the atmospheric and climate dynamics is the well-known difficulty to make reliable predictions.Contrary to simple periodic or multiperiodic phenomena(such as eclipses,tides ets) for which a long term prediction is possible, predictions in meteorology and climate are limited in time.

The most plausible (and currently admitted) explanation is based on the realization that a small uncertainty in the initial conditions used in a prediction scheme (usually referred as \error") seems to be amplified in the course of the evolution. Such uncertainties are inherent in the process of experimental measurement, The uncertainty being in the “closeness of the boundary to the point of bifurcation, and the error in linear equations.” An important aspect discussed by Arnold and Shuliminov.

As the “natural variability” and inverse regimes are often accompanied by inverse temperature states always and everywhere(except in GCM PREDICTIONS) one wonders if we have a problem of cataclysmic proportions.

Friday, May 09, 2008

Climate Prediction or how the soothsayers wept

For Dante soothsayers and diviners are practitioners of "magic frauds". In Inferno they are condemned to walk backwards because their heads are twisted so ... with tears running down their backs for false predictions

As the BBC reports. This week, about 150 of the world's top climate modellers have converged on Reading for a four day meeting to plan a revolution in climate prediction.

And they have plenty of work to do. So far modellers have failed to narrow the total bands of uncertainties since the first report of the Intergovernmental Panel on Climate Change (IPCC) in 1990.

And Julia Slingo from Reading University admitted it would not get much better until they had supercomputers 1,000 times more powerful than at present.

"We've reached the end of the road of being able to improve models significantly so we can provide the sort of information that policymakers and business require," she told BBC News.

"In terms of computing power, it's proving totally inadequate. With climate models we know how to make them much better to provide much more information at the local level... we know how to do that, but we don't have the computing power to deliver it."

But can bigger and more powerful computers help with predictions, in a word no.

In an interesting chapter entitled Engineers Dreams from his book Infinite in all directions, Freeman Dyson explains the reasons for the failings of Von Neumann and his team for the prediction and control of Hurricanes.

Von Neumann’s dream

“As soon as we have good enough computers we will be able to divide the phenomena of meteorology cleanly into two categories, the stable and the unstable”, The unstable phenomena are those that are which are upset by small disturbances, and the stable phenomena are those that are resilient to small disturbances. All disturbances that are stable we will predict, all processes that are unstable we will control”

Freeman Dyson page 183.

What went wrong? Why was Von Neumann’s dream such a total failure. The dream was based on a fundamental misunderstanding of the nature of fluid motions. It is not true that we can divide cleanly fluid motions into those that are predictable and those that are controllable. Nature as usual is more imaginative then we are. There is a large class of classical dynamic systems, including non-linear electrical circuits as well as fluids, which easily fall into a mode of behavior that is described by the word “chaotic” A chaotic motion is generally neither predictable nor controllable. It is unpredictable because a small disturbance will produce exponentially growing perturbation of the motion .It is uncontrollable because small disturbances lead only to other chaotic motions, and not to any stable and predictive alternative.

A fact also addressed by Vladimir Arnold

"For example, we deduce the formulas for the Riemannian curvature of a group endowed with an invariant Riemannian metric. Applying these formulas to the case of the infinite-dimensional manifold whose geodesics are motions of the ideal fluid, we find that the curvature is negative in many directions. Negativeness of the curvature implies instability of motion along the geodesics (which is well-known in Riemannian geometry of infinite-dimensional manifolds). In the context of the (infinite-dimensional) case of the diffeomorphism group, we conclude that the ideal flow is unstable (in the sense that a small variation of the initial data implies large changes of the particle positions at a later time).Moreover, the curvature formulas allow one to estimate the increment of the exponential deviation of fluid particles with close initial positions and hence to predict the time period when the motion of fluid masses becomes essentially unpredictable.

For instance, in the simplest and utmost idealized model of the earth’s atmosphere (regarded as two-dimensional ideal fluid on a torus surface), the deviations grow by the factor of 10^5 in 2 months. This circumstance ensures that a dynamical weather forecast for such a period is practically impossible (however powerful the computers and however dense the grid of data used for this purpose)"

The modellers continue,...

“One trouble is that as some climate uncertainties are resolved, new uncertainties are uncovered.Some modellers are now warning that feedback mechanisms in the natural environment which either accelerate or mitigate warming may be even more difficult to predict than previously assumed.”

Yes this is also well known the mathematics (Fokker plamk equation) already tell us this. The Kolgomorov backwards equation (kpe) naturally tends towards global cooling in any bi-stable simulation (hot-cold)

Lorenz called it almost intransitive, this is why climate models have biased towards heating(over paramatization).

NICOLIS, G also showed this

Stochastic aspects of climatic transitions additive fluctuations

The Fokker-Planck equation, corresponding to a zero-dimensional climatic model showing bistable behavior, is analyzed. A climatic potential function is introduced, whose variational properties determine the most probable states of the stationary probability distribution. Both the static and the time dependent properties of the fluctuations are monitored by two basic quantities: (1) the climatic potential, U, and (2) the variance of the noise. A sensitivity analysis of U with respect to system parameters, particularly the temperature feedback coefficient, shows a distinction between a regime where present climate dominates and a regime where a deep freeze climate dominates. Conditions of coexistence of these two regimes in terms of the characteristic parameters were also determined. For small variance, the stationary probability distribution is very sharply peaked around the dominant state while the time scale of evolution becomes exceedingly slow. Moreover, an increase in the temperature feedback coefficient tends to favor the deep freeze climate.

Heck and they are telling us to shut the fires down.

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