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

Chertock, Alina; Kurganov, Alexander; Lukáčová-Medviďová, Mária; Spichtinger, Peter; Wiebe, B.

doi:10.1016/j.jcp.2023.111987

Cited by 1 publication

(6 citation statements)

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“…In order potentially to use the SG method (Chertock et al, 2023) for the evolution of uncertainty through time in atmospheric applications, we performed a first comparison study between the ensemble approach and the SG method on an idealized two-dimensional model that couples nonhydrostatic weakly compressible Navier-Stokes equations to cloud variables. The two methods are compared for probabilistic forecasting and in data assimilation settings.…”

Section: Discussionmentioning

confidence: 99%

“…In this section, we briefly introduce two uncertainty quantification approaches that we will be using in our study, the MC (ensemble) approach and the SG method. For a more detailed description of the SG method applied to model (M2), we refer the reader to (Chertock et al ., 2019; Wiebe, 2021; Chertock et al ., 2023).…”

Section: Uncertainty Quantification Methodsmentioning

confidence: 99%

“…To decribe the SG method in more detail, we follow Chertock et al . (2023). Projecting (M2) onto the finite‐dimensional space spanned by the Hermite basis polynomials, we obtain, for

k=0,\dots, M

$$ {\displaystyle \begin{array}{llll}& \left\langle {\left({\rho}^{\prime}\right)}_t+\nabla \cdotp \left(\rho \boldsymbol{u}\right),{\Phi}_k\right\rangle =0,& & \\ {}& \left\langle {\left(\rho \boldsymbol{u}\right)}_t+\nabla \cdotp \left(\rho \boldsymbol{u}\otimes \boldsymbol{u}+{p}^{\prime}\mathrm...

…”

Section: Uncertainty Quantification Methodsmentioning

confidence: 99%

“…(7) To decribe the SG method in more detail, we follow Chertock et al (2023). Projecting (M2) onto the finite-dimensional space spanned by the Hermite basis polynomials, we obtain, for k = 0, … , M,…”

Section: 2mentioning

confidence: 99%

“…Since the structure of the system of expansion coefficients is similar to that of the deterministic cloud model (Equation 1), one can adapt the deterministic numerical methods used for the space and time approximation in a suitable way. For more details on the resulting system and its approximation, we refer the reader to Chertock et al (2019), Wiebe (2021), Chertock et al (2023). From the coefficients obtained, one can compute the stochastic solution for each event 𝜔 by evaluating the respective expansions in Equations ( 6) and ( 7) and by using the Hermite polynomials from Equation (4).…”

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confidence: 99%

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Comparison of uncertainty quantification methods for cloud simulation

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et al. 2023

Quart J Royal Meteoro Soc

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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 the stochastic space.In an idealized two‐dimensional model that couples non‐hydrostatic 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 forecasts: 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 avoids evolution of the large ensemble that is usually the most expensive component of the data assimilation system and provides comparable results to the ensemble Kalman filter in the cases investigated.This article is protected by copyright. All rights reserved.

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Section: Discussionmentioning

confidence: 99%

Section: Uncertainty Quantification Methodsmentioning

confidence: 99%

“…To decribe the SG method in more detail, we follow Chertock et al . (2023). Projecting (M2) onto the finite‐dimensional space spanned by the Hermite basis polynomials, we obtain, for