Predictability of weather and climate

Abstract:

Posit numbers, a recently proposed alternative to floating-point numbers, claim to have smaller arithmetic rounding errors in many applications. By studying weather and climate models of low and medium complexity (the Lorenz system and a shallow water model) we present benefits of posits compared to floats at 16 bit. As a standardised posit processor does not exist yet, we emulate posit arithmetic on a conventional CPU. Using a shallow water model, forecasts based on 16-bit posits with 1 or 2 exponent bits are clearly more accurate than half precision floats. We therefore propose 16 bit with 2 exponent bits as a standard posit format, as its wide dynamic range of 32 orders of magnitude provides a great potential for many weather and climate models. Although the focus is on geophysical fluid simulations, the results are also meaningful and promising for reduced precision posit arithmetic in the wider field of computational fluid dynamics.

Abstract:

Abstract Feynman asserted that interference is the only mystery in quantum mechanics (QM). It is concluded that the reason for this mystery, and thereby the related mysteries of complementarity, non-commutativity of observables, the uncertainty principle and violation of Bell's equality, is that the Hilbert space continuum of QM is unphysical. We describe a theory of quantum physics—rational QM (RaQM)—in which Hilbert space is (gravitationally) discretized. The key to solving the mysteries of QM in RaQM is a ubiquitous number-theoretic property of the cosine function, concealed in QM when angles range over the continuum. This property implies that the laws of physics are profoundly holistic, but not non-local. In the former, correlations between remote parts of the universe exist, but an intervention at one place cannot instantaneously influence affairs at the other. In the latter, such interventions can spread non-causally. Examples of holistic theories include Mach's principle, Penrose spin networks or the fractal geometry of chaotic attractors. Indeed, RaQM solves the deepest mystery of all; why nature makes use of complex numbers. It is to permit realistic incommensurate discretizations of incompatible observables. Such a solution is wholly concealed in the continuum theory of QM.