The central idea of The Fluid Catastrophe (henceforth TFC) is that, in all but a few special cases, deterministic models do not provide a realistic picture of reality; stochastic models are more powerful. Determinism works well for the motions of the planets but not for fluid dynamics
A closely related idea is the idea of the steady-state whereby the equations describing quantities are assumed to be independent of time. The more sophisticated idea is that of stationarity whereby the statistics of a system are assumed independent of time; values of measured quantities may vary over time and it is only the statistics and model parameters which are assumed to be static. This is one step removed from steady-state; it is a higher level of abstraction.
We assume steady-state in order to come to grips with reality, in order to make reality conceivable and manageable. It is only ever a working hypothesis. We can never be sure that a system is steady-state but we can be sure when it isn’t. There is a natural tendency to assume that those processes which are long compared with a human lifetime are steady-state: wilderness, undisturbed ecosystems, ocean circulation. Events which disturb such systems are then sufficiently rare to be regarded as accidents or aberrations, outside the cosy delusion we have constructed for ourselves. It then follows that someone or something must be to blame: the gods, God, capitalism, humanity. Non-indigenous Australians seem to believe this of bushfires and talk of the “devastation” wrought by bushfires in wilderness areas. We could equally well talk of “renewal” or “regeneration” following such fires. Even qualified ecologists go along with this hand-wringing, while well aware that numerous native species could only have evolved in a regime of regular burning. This is the human lifetime fallacy.
Most people have an intuitive grasp of the Law of Large Numbers: that the average of a sample gets closer and closer to the true value as the sample size gets bigger.
But this is only true of the average! It is not true of the sum!
The average converges only because the sum is divided by the number in the sample, N. The sum of a number of zero mean, random fluctuations becomes larger as N becomes larger. Its standard deviation is the the square root of N even though the mean tends to zero. The belief that the variance of the sum of random quantities tends to zero is the Law of Large Numbers fallacy.
For a physical system in which a large number of small rapid fluctuations are added, the outcome will not be a steady state, it will be a variation which is both larger and slower than that of the input fluctuations. This sort of slow variation is called red noise because there is more variance at lower frequencies compared to white noise. An example is the temperature of a body of water heated and cooled by radiation and evaporation in the course of a year (TFC pp 64-65).
Another example is ocean circulation. At short time scales we see the surface currents driven by wind and tides. At longer time scales we see the big warm core eddies which spin off promontories like the Cape of Good Hope and Cape Horn. At even longer time scales we see the great ocean gyres and ocean currents such as the Gulf Stream, the Kurashio and the Aghulas. It would be comforting to assume, because these large gyres are the sum of component smaller currents and eddies, that they are steady-state and not subject to variation over time. This would be wrong. It is an example of the human lifetime fallacy and the law of large numbers fallacy. Large scale ocean currents are not the average of smaller variations, they are more like their sum and most likely have a red-noise frequency spectrum with significant variability at low frequencies.
Ocean currents are not steady-state. Concerns about variations in the Gulf Stream are unwarranted.