Comparison of uncertainty quantification methods for cloud simulation

Quantification of evolving uncertainties is required for both probabilistic forecasting and data assimilation in weather prediction. In current practice, the ensemble of model simulations is often used as a primary tool to describe the required uncertainties. In this work, we explore an alternative approach, the so-called stochastic Galerkin (SG) method, which integrates uncertainties forward in time using a spectral approximation in stochastic space. In an idealized two-dimensional model that couples nonhydrostatic weakly compressible Navier–Stokes equations to cloud variables, we first investigate the propagation of initial uncertainty. Propagation of initial perturbations is followed through time for all model variables during two types of forecast: the ensemble forecast and the SG forecast. Series of experiments indicate that differences in performance of the two methods depend on the system state and truncations used. For example, in more stable conditions, the SG method outperforms the ensemble of simulations for every truncation and every variable. However, in unstable conditions, the ensemble of simulations would need more than 100 members (depending on the model variable) and the SG method more than a truncation at five to produce comparable but not identical results. As estimates of the uncertainty are crucial for data assimilation, secondly we instigate the use of these two methods with the stochastic ensemble Kalman filter. The use of the SG method avoids evolution of a large ensemble, which is usually the most expensive component of the data assimilation system, and provides results comparable with the ensemble Kalman filter in the cases investigated

Reduced-order hybrid multiscale method combining the molecular dynamics and the discontinuous-galerkin method

We present a new reduced-order hybrid multiscale method to simulate com- plex fluids. continuum and molecular descriptions. We follow the framework of the heterogeneous multi-scale method (HMM) that makes use of the scale separation into macro- and micro-levels. On the macro-level, the governing equations of the incompressible flow are the continuity and momentum equations. The equations are solved using a high-order accurate discontinuous Galerkin Finite Element Method (dG) and implemented in the BoSSS code. The missing information on the macro-level is represented by the unknown stress tensor evaluated by means of the molecular dynam- ics (MD) simulations on the micro-level. We shear the microscopic system by applying Lees-Edwards boundary conditions and either an isokinetic or Lowe-Andersen thermostat. The data obtained from the MD simulations underlie large stochastic errors that can be controlled by means of the least-square approximation. In order to reduce a large number of computationally expensive MD runs, we apply the reduced order approach. Nume al experiments confirm the robustness of our newly developed hybrid MD-dG method

Stochastic Galerkin method for cloud simulation. Part II: a fully random Navier-Stokes-cloud model

This paper is a continuation of the work presented in [Chertock et al., Math. Cli. Weather Forecast. 5, 1 (2019), 65--106]. We study uncertainty propagation in warm cloud dynamics of weakly compressible fluids. The mathematical model is governed by a multiscale system of PDEs in which the macroscopic fluid dynamics is described by a weakly compressible Navier-Stokes system and the microscopic cloud dynamics is modeled by a convection-diffusion-reaction system. In order to quantify uncertainties present in the system, we derive and implement a generalized polynomial chaos stochastic Galerkin method. Unlike the first part of this work, where we restricted our consideration to the partially stochastic case in which the uncertainties were only present in the cloud physics equations, we now study a fully random Navier-Stokes-cloud model in which we include randomness in the macroscopic fluid dynamics as well. We conduct a series of numerical experiments illustrating the accuracy and efficiency of the developed approach

An application of 3-D kinematical conservation laws: propagation of a 3-D wavefront

Three-dimensional (3-D) kinematical conservation laws (KCL) are equations of evolution of a propagating surface Omega(t) in three space dimensions. We start with a brief review of the 3-D KCL system and mention some of its properties relevant to this paper. The 3-D KCL, a system of six conservation laws, is an underdetermined system to which we add an energy transport equation for a small amplitude 3-D nonlinear wavefront propagating in a polytropic gas in a uniform state and at rest. We call the enlarged system of 3-D KCL with the energy transport equation equations of weakly nonlinear ray theory (WNLRT). We highlight some interesting properties of the eigenstructure of the equations of WNLRT, but the main aim of this paper is to test the numerical efficacy of this system of seven conservation laws. We take several initial shapes for a nonlinear wavefront with a suitable amplitude distribution on it and let it evolve according to the 3-D WNLRT. The 3-D WNLRT is a weakly hyperbolic 7 × 7 system that is highly nonlinear. Here we use the staggered Lax–Friedrichs and Nessyahu–Tadmor central schemes and have obtained some very interesting shapes of the wavefronts. We find the 3-D KCL to be suitable for solving many complex problems for which there presently seems to be no other method capable of giving such physically realistic features