Saturday, June 02, 2007

The Diophantine approximations theory of Climate science

The Diophantine approximations theory appears in these problems because of the crucial influences of the resonances between the frequencies of the unperturbed problems on the perturbations evolution. One of the first observed manifestations of these resonances is the approximated commensurability of the years of Saturn and of Jupiter, whose periods ratio is approximately 5:2 (Jupiter’s angular motion is about 299” a day, that of Saturn - about 120”).
The Poincare averaging in the case of such resonance leads to the large “secular perturbation”,whose period is of order 10^3 years, but which is still periodic (like the pendulum oscillation), near the unperturbed motion. It leads to the evolution of the orbit in one direction during several centuries, which would destroy the Solar system, being continued forever.

V Arnold

That there is a causal connection between the observed variations in the forces of the Sun,the terrestrial magnetic field, and the meteorological elements has been the conclusion of every research into this subject for the past 50 years. The elucidation of exactly what the connection is and the scientific proof of it is to be classed among the most difficult problems presented in terrestrial physics. The evidence adduced in favor of this conclusion is on the whole of a cumulative kind, since the direct sequence of cause and effect is so far masked in the complex interaction of the many delicate forces in operation as to render its immediate measurement quite impossible in the present state of science.

F.H. Bigelow
US Dept. Agriculture Weather Bureau
Bulletin No.21, 1898



The complexities of meteorology and its “generalized pupil “Climate science” are often misrepresented as being able to understand and measure (model) the changes of differentials in thermodynamic equilibrium from an initial state to a predictive state.

The misrepresentation of scientists to predict the changes that will initiate catastrophic climatic events is far from reality, as the predictive capabilities are beyond the existing systems to predict the simplest differentials ie changes in convective thermodynamics over a short period.

In thermodynamics, the Gibbs free energy (IUPAC recommended name: Gibbs energy or Gibbs function) is a thermodynamic potential which measures the "useful" or process-initiating work obtainable from an isothermal, isobaric thermodynamic system. Technically, the Gibbs free energy is the maximum amount of non-expansion work which can be extracted from a closed system, and this maximum can be attained only in a completely reversible process. When a system changes from a well-defined initial state to a well-defined final state, the Gibbs free energy ΔG equals the work exchanged by the system with its surroundings, less the work of the pressure forces, during a reversible transformation of the system from the same initial state to the same final state.

Gibbs energy is also the chemical potential that is minimised when a system reaches equilibrium at constant pressure. As such, it is a convenient criterion of spontaineity for isobaric processes.

The Gibbs free energy, originally called available energy, was developed in the 1870s by the American mathematical physicist Willard Gibbs. In 1873, Gibbs defined what he called the “available energy” of a body as such:
“The greatest amount of mechanical work which can be obtained from a given quantity of a certain substance in a given initial state, without increasing its total volume or allowing heat to pass to or from external bodies, except such as at the close of the processes are left in their initial condition.

The initial state of the body, according to Gibbs, is supposed to be such that "the body can be made to pass from it to states of dissipated energy by reversible processes."

To understand the complexities of the simple change from an initial state to a subsequent state one can view the variables (parameters) that need to be assigned to the differential equations.

First level

Second level

Third level.

Simple is it not.

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