Mathematics of Climate and Weather Forecasting

Open Access Published by De Gruyter Open Access

Volume 5 Issue 1

February 2019

Issue of Mathematics of Climate and Weather Forecasting

Contents
Journal Overview

Open Access February 19, 2019

Fractional Order on the Impact of Climate Change With Dominant Earth’s Fluctuations

S.C. Eze, M.O. Oyesanya

Page range: 1-11

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Abstract

In this investigation, fractional order model on the impact of climate change with dominant Earth’s fluctuations is given. The solution of the modelwas obtained using modified LaplaceAdomian decomposition method. The result is compared with the result obtained from integer solution.We observed that what is seen in the fractional part takes longer to be seen in the integer part.We also observed that regardless of any choice we make to mitigate climate change, the impact will still persist due to the effect of Earth’s fluctuations.

Open Access April 5, 2019

Spectral stability of nonlinear gravity waves in the atmosphere

Mark Schlutow, Erik Wahlén, Philipp Birken

Page range: 12-33

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Abstract

We apply spectral stability theory to investigate nonlinear gravity waves in the atmosphere. These waves are determined by modulation equations that result from Wentzel-Kramers-Brillouin theory. First, we establish that plane waves, which represent exact solutions to the inviscid Boussinesq equations, are spectrally stable with respect to their nonlinear modulation equations under the same conditions as what is known as modulational stability from weakly nonlinear theory. In contrast to Boussinesq, the pseudo-incompressible regime does fully account for the altitudinal varying background density. Second,we show for the first time that upward-traveling non-plane wave fronts solving the inviscid nonlinear modulation equations, that compare to pseudo-incompressible theory, are unconditionally unstable. Both inviscid regimes turn out to be ill-posed as the spectra allow for arbitrarily large instability growth rates. Third, a regularization is found by including dissipative effects. The corresponding nonlinear traveling wave solutions have localized amplitude. As a consequence of the nonlinearity, envelope and linear group velocity, as given by the derivative of the frequency with respect to wavenumber, do not coincide anymore. These waves blow up unconditionally by embedded eigenvalue instabilities but the instability growth rate is bounded from above and can be computed analytically. Additionally, all three types of nonlinear modulation equations are solved numerically to further investigate and illustrate the nature of the analytic stability results.

Open Access April 9, 2019

Optimal Algorithms for Computing Average Temperatures

S. Foucart, M. Hielsberg, G. L. Mullendore, G. Petrova, P. Wojtaszczyk

Page range: 34-44

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Abstract

A numerical algorithm is presented for computing average global temperature (or other quantities of interest such as average precipitation) from measurements taken at speci_ed locations and times. The algorithm is proven to be in a certain sense optimal. The analysis of the optimal algorithm provides a sharp a priori bound on the error between the computed value and the true average global temperature. This a priori bound involves a computable compatibility constant which assesses the quality of the measurements for the chosen model. The optimal algorithm is constructed by solving a convex minimization problem and involves a set of functions selected a priori in relation to the model. It is shown that the solution promotes sparsity and hence utilizes a smaller number of well-chosen data sites than those provided. The algorithm is then applied to canonical data sets and mathematically generic models for the computation of average temperature and average precipitation over given regions and given time intervals. A comparison is provided between the proposed algorithms and existing methods.

Open Access May 24, 2019

Numerical Development and Evaluation of an Energy Conserving Conceptual Stochastic Climate Model

Federica Gugole, Christian L. E. Franzke

Page range: 45-64

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Abstract

In this study we aim to present the successful development of an energy conserving conceptual stochastic climate model based on the inviscid 2-layer Quasi-Geostrophic (QG) equations. The stochastic terms have been systematically derived and introduced in such away that the total energy is conserved. In this proof of concept studywe give particular emphasis to the numerical aspects of energy conservation in a highdimensional complex stochastic system andwe analyzewhat kind of assumptions regarding the noise should be considered in order to obtain physical meaningful results. Our results show that the stochastic model conserves energy to an accuracy of about 0.5% of the total energy; this level of accuracy is not affected by the introduction of the noise, but is mainly due to the level of accuracy of the deterministic discretization of the QG model. Furthermore, our results demonstrate that spatially correlated noise is necessary for the conservation of energy and the preservation of important statistical properties, while using spatially uncorrelated noise violates energy conservation and gives unphysical results. A dynamically consistent spatial covariance structure is determined through Empirical Orthogonal Functions (EOFs). We find that only a small number of EOFs is needed to get good results with respect to energy conservation, autocorrelation functions, PDFs and eddy length scale when comparing a deterministic control simulation on a 512 × 512 grid to a stochastic simulation on a 128 × 128 grid. Our stochastic approach has the potential to seamlessly be implemented in comprehensive weather and climate prediction models.

Open Access November 20, 2019

Stochastic Galerkin method for cloud simulation

A. Chertock, A. Kurganov, M. Lukáčová-Medvid’ová, P. Spichtinger, B. Wiebe

Page range: 65-106

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Abstract

We develop a stochastic Galerkin method for a coupled Navier-Stokes-cloud system that models dynamics of warm clouds. Our goal is to explicitly describe the evolution of uncertainties that arise due to unknown input data, such as model parameters and initial or boundary conditions. The developed stochastic Galerkin method combines the space-time approximation obtained by a suitable finite volume method with a spectral-type approximation based on the generalized polynomial chaos expansion in the stochastic space. The resulting numerical scheme yields a second-order accurate approximation in both space and time and exponential convergence in the stochastic space. Our numerical results demonstrate the reliability and robustness of the stochastic Galerkin method. We also use the proposed method to study the behavior of clouds in certain perturbed scenarios, for examples, the ones leading to changes in macroscopic cloud pattern as a shift from hexagonal to rectangular structures.

About this journal

THE JOURNAL IS CLOSED FOR SUBMISSIONS

MCWF was an open access journal that published articles that use mathematical methods in climate change science and weather forecasting. The scope included numerical methods, stochastic processes, PDE analysis, time series analysis, data filtering and assimilation, applied to any topic of atmosphere and ocean sciences.

The journal was published in Open Access model and is archived in Portico.

All authors retain copyright, unless – due to their local circumstances – their work is not copyrighted. The copyrights are governed by the Creative-Commons license.

The journal was edited by: Prof. Boualem Khouider, University of Victoria, Canada