The weather

Almost all the intricate variations of atmospheric temperature, pressure, velocity and humidity that define the weather we see are in the end determined by fairly simple underlying rules for fluid behavior. (Details of phase changes in water are also important, as are features of topography, ocean currents, solar radiation levels and presumably land use.) Our everyday personal attempts to predict the weather tend to be based just on looking at local conditions and then recalling what happened when we saw these conditions before. But ever since the mid-1800s synoptic weather maps of large areas have been available that summarize conditions in terms of features like fronts and cyclones. And predictions made by looking at simple trends in these features tend at least in some situations to work fairly well. Starting in the 1940s more systematic efforts to predict weather were made by using computers to run approximations to fluid equations. The approximations have improved as larger computers have become available. But even though millions of discrete samples are now used, each one typically still represents something much larger then for example a single cloud. Yet ever since the 1970s, the approach has had at least some success in making predictions up to several days in advance. But although there has been gradual improvement it is usually assumed that—like in the Lorenz equations—the phenomenon of chaos must make forecasts that are based on even slightly different initial measurements eventually diverge exponentially (see page 972). Almost certainly this does indeed happen in at least some critical situations. But it seems that over most of a typical weather map there is no such sensitivity—so that in the end the difficulties of weather prediction are probably much more a result of computational irreducibility and of the sophisticated kinds of computations that the Principle of Computational Equivalence implies should often occur even in simple fluids.