Cont'd from here.
The non-linearity in the model is clear from this scatter
plot. For small perturbations in the inputs from the mean values (depicted as
the square dots below), the solutions can end up being anywhere in the space
depicted by the scatter plots.
Stated in Lorenz’s words, a flap of the butterfly’s wings
for the input conditions can create a drastic change in outputs, possibly
resulting in a tornado. In Meteorological terms, this is one of the main
reasons why longer term weather forecasts remain a difficulty. Even a
statistical analysis such as the one performed here is worthless in predicting
the longer term effects of weather, since conditions such as temperatures can
lie anywhere in the contour of the scatter plots and thus mean nothing in terms
of accurate predictions. This is the crux of Lorenz’s work; that accurate
longer term predictions cannot be made when it comes to weather. Further, he
leaves the question of the butterfly creating a tornado in a different
hemisphere of the world unanswered.
Predictability of phenomena is more reliable for linear
systems. While short term predictions can be made for non-linear systems, exact
longer term predictions are both unreasonable and impossible. All that one can
predict precisely about the location of the electronic butterfly in response to a tap on the jar is that it can be found some place inside the
bottle after a few seconds. In a similar (and a rather cynical) vein, all that
the Meteorologist can predict with certainty for longer term forecasts is that
it will be summer in July in the Northern hemisphere. It only takes a mental
extrapolation to extend this analogy to even longer term predictions with
climate change, and the reasons for the discussions that ensue!
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