Correcting Numerical Simulations for Known Expectations

Abstract

Predictions of the future climate require long-time simulation of a chaotic dynamical system. This poses a challenge for numerical simulations, as these do not necessarily capture the correct long-term behaviour of chaotic systems. This problem is exacerbated by the wide range of length scales present in atmospheric and oceanic dynamics. The modeling choices for small scale processes have a large impact on long term statistics of the scales of interest. This thesis studies the dynamics of two different fluid models as a proxy for atmospheric dynamics: a point vortex flow on the sphere and two-dimensional turbulent flow on a torus. We apply gentle dynamical perturbations commonly used in molecular dynamics to these fluid dynamics problems as a means for improving the statistical veracity of low fidelity simulations. For the point vortex system we study a system consisting of both strong and weak vortices. The strong vortex dynamics are mildly influenced by the weak vortices on a short time scale, but the presence of the weak vortices introduces a variability in the strong vortex energy. We mimic this behaviour in a model with only the strong vortices by gentle perturbations to the equations of motion. The perturbations have a stochastic forcing such that the modified dynamics ergodically sample an invariant measure consistent with observations from the strong vortex system in contact with the weak vortices. We choose the invariant measure as the minimal entropy density consistent with observations. The required Lagrange multipliers can be computed either a priori using a sample set in some prior distribution, or they can be computed on-the-fly using the simulation history as an ensemble. The latter method allows the observations to be updated during runtime, providing the opportunity for data-assimilation. We construct a Poisson integration method for the aforementioned point vortex dynamics by splitting the Hamiltonian into its constituent vortex pair terms. The method provides exact solutions to a Poisson system with the same bracket as the original dynamical system, but with a modified Hamiltonian function. Different orderings of the pairwise interactions are considered and are also used for the construction of higher order methods. The energy and momentum conservation of the splitting schemes is demonstrated for several test cases. For particular orderings of the pairwise interactions, the schemes allow scalable parallelization. This results in a linear -- as opposed to quadratic -- scaling of computation time with system size. We also explore the direct modification of the pseudo-spectral truncation of two-dimensional, incompressible fluid dynamics on a torus to maintain a prescribed kinetic energy spectrum. The method provides a means of simulating fluid states with defined spectral properties, for the purpose of matching simulation statistics to given information, arising from observations, theoretical predictions or high fidelity simulations. In the scheme outlined here, Nos\'{e}-Hoover thermostats -- commonly used in molecular dynamics -- are introduced as feedback controls applied to energy bands of the Fourier-discretized Navier-Stokes equations. As we demonstrate in numerical experiments, the dynamical properties -- quantified using autocorrelation functions -- are only modestly perturbed by our device, while ensemble dispersion is significantly enhanced in comparison with simulations of a corresponding truncation using hyperviscosity.