The problem with nonlinearity (AKA why I cannot predict the weather)

Being from an engineering background, and having mainly engineers for friends, I normally get asked why I cannot predict tomorrow’s weather, and jibes as to how weather prediction is a pseudo-science etc etc., Thus, I decided to just rant about how life is so difficult for me.

Engineers of all kinds like to work with computationally friendly methods of analysis. One way to ensure this is to use mathematical maps that are linear in nature, and preferably orthogonal. What I mean by this is that it should be representable by a matrix, and all columns should have a zero inner product with every other column but itself. The classic example is the Discrete Fourier Transform. One of the most important properties (atleast to me!) of a linear system is that of superposition, i.e, if x and y are any two ‘signals’ or vectors, and F is a linear transform, then F(x+y) = F(x) + F(y). This property tremendously simplifies the analysis of the behavior of the transform. It is easy to identify ‘problematic’ vectors and do away with them.

For example, if im building a music system and I have a linear amplifier which I know goes nuts if I input music containing the 2 Khz frequency, I can remove that frequency in advance so that there are no problems in performance. Thus, a signal localised in a certain frequency band will not ‘leak’ to other bands. The case is not so in nonlinear systems. There is a transfer of energy from one part of the spectrum to another (eg: the Kolmogorov spectrum in turbulence), and thus there is no guarantee that your amplifier will be well behaved for all time.

This also implies that the superposition principle no longer applies. Since energy in one frequency invariably finds its way to other places, there is interaction between different frequencies and thus the resulting behavior of the system is not just the addition of the behavior of the system with the individual frequencies as inputs, i.e, F(x+y) \neq F(x) + F(y). Thus, the resulting behavior is not easy to predict in advance, and pretty much impossible if the number of interacting components is huge, like in an ecosystem or the climate. This is called emergent behavior, since it cannot be predicted by looking at the individual components themselves.

If losing superposition was a problem, the problem of chaos is as bad, if not worse. Chaos is a fancy way of saying that nonlinear systems are extremely sensitive to their inputs and their mathematical formulation. For example, if you had perfect knowledge about every quantity but not a perfect model of the phenomenon being observed, you will make errors in prediction, which are huge. Similarly, if your models were perfect, but you were not able to measure accurately enough, the same fate. In real life, both are true. We don’t understand natural phenomena well enough (Of course, dam builders will disagree), nor do we have measurements that are accurate enough. Thus, even the fact that we can say whether tomorrow will be cloudy or not with reasonable confidence is a testament to how well weathermen have learnt to live with nonlinearity.

And if all this was not enough, there is the problem of phenomena occuring at multiple scales. A typical cyclone has a horizontal extent of around 1000 km, while the convection that drives it is of the order of 1 km. There are planetary waves that have a wavelength of 10000 km, and they are dissipated by turbulence acting at the micrometer level. Any model that tries to incorporate the largest and the smallest scales will probably tell us about tomorrow’s weather sometime in the next century!!

And coming to the worst problem of all, rain.While one can say with reasonable confidence about whether it will rain or not, since that is constrained by the first law of thermodynamics and behavior of water vapor, it probably is next to impossible to predict when or how much. Quite amazingly, there still does not seem to have been found a sufficient condition for rainfall to occur: the necessary conditions are known, and still we don’t know when it will rain.

Interestingly, average behavior is more predictable, since averaging ‘smooths” out the nonlinearity in the system, and thus we are able to reasonably estimate climate, which is a long time-average of weather. The constraints of thermodynamics, which seem to be the only thing that will never be violated, are stronger as we go into longer and longer time scales.

Handling nonlinear systems is hard, but we are getting there! (In a century or so.)

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