Predictability of weather and climate

Abstract:

An arbitrarily dense discretization of the Bloch sphere of complex Hilbert states is constructed, where points correspond to bit strings of fixed finite length. Number-theoretic properties of trigonometric functions (not part of the quantum-theoretic canon) are used to show that this constructive discretized representation incorporates many of the defining characteristics of quantum systems: completementarity, uncertainty relationships and (with a simple Cartesian product of discretized spheres) entanglement. Unlike Meyer's earlier discretization of the Bloch Sphere, there are no orthonormal triples, hence the Kocken-Specker theorem is not nullified. A physical interpretation of points on the discretized Bloch sphere is given in terms of ensembles of trajectories on a dynamically invariant fractal set in state space, where states of physical reality correspond to points on the invariant set. This deterministic construction provides a new way to understand the violation of the Bell inequality without violating statistical independence or factorization, where these conditions are defined solely from states on the invariant set. In this finite representation, there is an upper limit to the number of qubits that can be entangled, a property with potential experimental consequences.

Abstract:

The use of single-precision arithmetic in ECMWF’s forecasting model gave a 40% reduction in wall-clock time over double-precision, with no decrease in forecast quality. However, using reduced-precision in 4D-Var data assimilation is relatively unexplored and there are potential issues with using single-precision in the tangent-linear and adjoint models. Here, we present the results of reducing numerical precision in an incremental 4D-Var data assimilation scheme, with an underlying two-layer quasigeostrophic model. The minimizer used is the conjugate gradient method. We show how reducing precision increases the asymmetry between the tangent-linear and adjoint models. For ill-conditioned problems, this leads to a loss of orthogonality among the residuals of the conjugate gradient algorithm, which slows the convergence of the minimization procedure. However, we also show that a standard technique, reorthogonalization, eliminates these issues and therefore could allow the use of single-precision arithmetic. This work is carried out within ECMWF’s data assimilation framework, the Object Oriented Prediction System.