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Mathematics of Climate and Weather Forecasting

Ed. by Khouider, Boualem

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2353-6438
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Numerical Development and Evaluation of an Energy Conserving Conceptual Stochastic Climate Model

Federica Gugole
  • Corresponding author
  • Meteorological Institute and Center for Earth System Research and Sustainability, University of Hamburg, Hamburg, Germany
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/ Christian L. E. Franzke
  • Meteorological Institute and Center for Earth System Research and Sustainability, University of Hamburg, Hamburg, Germany
  • Other articles by this author:
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Published Online: 2019-05-24 | DOI: https://doi.org/10.1515/mcwf-2019-0004

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.

Keywords: stochastic parameterization; energy conservation; projection operator; spatial noise structure; Empirical Orthogonal Functions

MSC 2010: 65C20; 68U20

References

  • [1] A. Arakawa. “Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part I.” In: Journal of Computational Physics 1.1 (1966), pp. 119–143.Google Scholar

  • [2] G. Badin and F. Crisciani. Variational formulation of fluid and geophysical fluid dynamics - Mechanics, symmetries and conservation laws. Springer Berlin, 2018.Google Scholar

  • [3] E. A. Barnes and D. L. Hartmann. “The Global Distribution of Atmospheric Eddy Length Scales.” In: Journal of Climate 25.9 (2012), pp. 3409–3416.CrossrefGoogle Scholar

  • [4] M. Belkin and P. Niyogi. “Laplacian Eigenmaps for Dimensionality Reduction and Data Representation.” In: Neural Computation 15.6 (2003), pp. 1373–1396.CrossrefGoogle Scholar

  • [5] J. Berner, G. Shutts, M. Leutbecher, and T. Palmer. “A spectral stochastic kinetic energy backscatter scheme and its impact on flow-dependent predictability in the ECMWF ensemble prediction system.” In: J. Atmos. Sci. 66 (2009), pp. 603–626.CrossrefGoogle Scholar

  • [6] J. Berner, U. Achatz, L. Batte, L. Bengtsson, A. Camara, H. Christensen, M. Colangeli, D. Coleman, D. Crommelin, S. Dolaptchiev, C. Franzke, P. Friederichs, P. Imkeller, H. Jarvinen, S. Juricke, V. Kitsios, F. Lott, V. Lucarini, S. Mahajan, T. Palmer, C. Penland, M. Sakradzija, J. von Storch, A. Weisheimer, M. Weniger, P. Williams, and J. Yano. “Stochastic parameterization: Toward a new view of weather and climate models.” In: Bull. Amer. Meteorol. Soc. 98.3 (2017), pp. 565–588.Google Scholar

  • [7] R. Buizza, M. Miller, and T. Palmer. “Stochastic representation of model uncertainties in the ECMWF ensemble prediction system.” In: Quart. J. R. Meteorol. Soc. 125 (1999), pp. 2887–2908.Google Scholar

  • [8] G. F. Carnevale. “Statistical features of the evolution of two-dimensional turbulence.” In: Journal of Fluid Mechanics 122 (1982), 143–153.Google Scholar

  • [9] G. F. Carnevale and J. S. Frederiksen. “Nonlinear stability and statistical mechanics of flow over topography.” In: Journal of Fluid Mechanics 175 (1987), 157–181.Google Scholar

  • [10] R. R. Coifman and S. Lafon. “Diffusion maps.” In: Applied and Computational Harmonic Analysis 21 (2006), pp. 5–30.Google Scholar

  • [11] C. J. Cotter, D. Crisan, D. D. Holm,W. Pan, and I. Shevchenko. “Modelling uncertainty using circulation-preserving stochastic transport noise in a 2-layer quasi-geostrophic model.” In: arXiv:1802.05711 (2018).Google Scholar

  • [12] C. J. Cotter, D. Crisan, D. D. Holm, W. Pan, and I. Shevchenko. “Numerically modelling stochastic Lie transport in fluid dynamics.” In: arXiv:1801.09729 (2018).Google Scholar

  • [13] J. Demaeyer and S. Vannitsem. “Comparison of stochastic parameterizations in the framework of a coupled ocean– atmosphere model.” In: Nonlinear Processes in Geophysics 25.3 (2018), pp. 605–631.CrossrefGoogle Scholar

  • [14] H. A. Dijkstra, A. Tantet, J. Viebahn, E. Mulder, M. Hebbink, D. Castellana, H. van den Pol, J. Frank, S. Baars, F. Wubs, M. Chekroun, and C. Kuehn. “A numerical framework to understand transitions in high-dimensional stochastic dynamical systems.” In: Dynamics and Statistics of the Climate System 1.1 (2016).Google Scholar

  • [15] S.Dwivedi, C. L. E. Franzke, and F. Lunkeit. “Energetically Consistent Stochastic and Deterministic Kinetic Energy Backscatter Schemes for Atmospheric Models.” In: Q. J. Roy. Meteorol. Soc. (2019), submitted.Google Scholar

  • [16] J. E. Frank and G. A. Gottwald. “Stochastic homogenization for an energy conserving multi-scale toy model of the atmosphere.” In: Physica D 254 (2013), pp. 46–56.Google Scholar

  • [17] C. L. E. Franzke. “Extremes in dynamic-stochastic systems.” In: Chaos: An Interdisciplinary Journal of Nonlinear Science 27.1 (2017), p. 012101.Google Scholar

  • [18] C. L. E. Franzke and A. J. Majda. “Low-Order Stochastic Mode Reduction for a Prototype Atmospheric GCM.” In: J. Atmos. Sci. 63 (2006), pp. 457–479.CrossrefGoogle Scholar

  • [19] C. L. E. Franzke, A. J.Majda, and E. Vanden-Eijnden. “Low-order stochastic mode reduction for a realistic barotropic model climate.” In: J. Atmos. Sci. 62 (2005), pp. 1722–1745.CrossrefGoogle Scholar

  • [20] C. L. E. Franzke, T. J. O’Kane, J. Berner, P. D.Williams, and V. Lucarini. “Stochastic Climate Theory and Modelling.” In: WIREs Climate Change 6 (2015), pp. 63–78.Google Scholar

  • [21] J. S. Frederiksen, T. J. O’Kane, and M. J. Zidikheri. “Stochastic subgrid parameterizations for atmospheric and oceanic flows.” In: Physica Scripta 85.6 (2012), p. 068202.Google Scholar

  • [22] J. S. Frederiksen, V. Kitsios, T. J. O’Kane, and M. J. Zidikheri. “Stochastic Subgrid Modelling for Geophysical and Three-Dimensional Turbulence.” In: Nonlinear and Stochastic Climate Dynamics. Ed. by C. L. E. Franzke and T. J.Editors O’Kane. Cambridge University Press, 2017, 241–275.Google Scholar

  • [23] C. W. Gardiner. Stochastic Methods: A Handbook for the Natural and Social Sciences. Vol. 4. Springer Berlin, 2009.Google Scholar

  • [24] D. Giannakis and A. J. Majda. “Comparing low-frequency and intermittent variability in comprehensive climate models through nonlinear Laplacian spectral analysis.” In: Geophysical Research Letters 39.10 ().Google Scholar

  • [25] D. Giannakis and A. J. Majda. “Nonlinear Laplacian spectral analysis for time series with intermittency and low-frequency variability.” In: Proceedings of the National Academy of Sciences 109.7 (2012), pp. 2222–2227.Google Scholar

  • [26] D. Givon, R. Kupferman, and A. Stuart. “Extracting macroscopic dynamics: model problems and algorithms.” In: Nonlinearity 17.6 (2004), R55.Google Scholar

  • [27] G. A. Gottwald, D. Crommelin, and C. L. E. Franzke. “Stochastic Climate Theory.” In: Nonlinear and Stochastic Climate Dynamics. Ed. by C. L. E. Franzke and T. O’Kane. Cambridge: Cambridge University Press, 2017.Google Scholar

  • [28] I. Grooms and L. Zanna. “A note on ‘Toward a stochastic parameterization of ocean mesoscale eddies’.” In: Ocean Modelling 113 (2017), pp. 30 –33.Google Scholar

  • [29] M. J. Grote, A. J. Majda, and C. Grotta Ragazzo. “Dynamic Mean Flow and Small-Scale Interaction through Topographic Stress.” In: Journal of Nonlinear Science 9.1 (Feb. 1999), pp. 89–130.CrossrefGoogle Scholar

  • [30] J. Harlim, A.Mahdi, and A. J.Majda. “An ensemble Kalman filter for statistical estimation of physics constrained nonlinear regression models.” In: Journal of Computational Physics 257 (2014), pp. 782 –812.Google Scholar

  • [31] K. Hasselmann. “Stochastic climate models part I. Theory.” In: Tellus 28.6 (1976), pp. 473–485.Google Scholar

  • [32] D. D. Holm. “Variational principles for stochastic fluid dynamics.” In: Proc. R. Soc. A 471 (2015), p. 20140963.Google Scholar

  • [33] M. F. Jansen and I. M. Held. “Parameterizing subgrid-scale eddy effects using energetically consistent backscatter.” In: Ocean Modelling 80 (2014), pp. 36–48.Google Scholar

  • [34] E. Kalnay. Atmospheric modeling, data assimilation and predictability. Cambridge university press, 2003.Google Scholar

  • [35] D. Kondrashov, S. Kravtsov, and M. Ghil. “Empirical mode reduction in a model of extratropical low-frequency variability.” In: J. Atmos. Sci. 63.7 (2006), pp. 1859–1877.CrossrefGoogle Scholar

  • [36] S. Kravtsov, D. Kondrashov, and M. Ghil. “Multilevel regression modeling of nonlinear processes: Derivation and applications to climatic variability.” In: J. Climate 18 (2005), pp. 4404–4424.CrossrefGoogle Scholar

  • [37] E. Mémin. “Fluid flow dynamics under location uncertainty.” In: Geophysical and Astrophysical Fluid Dynamics 108 (2014), pp. 119–46.Google Scholar

  • [38] A. J.Majda, C. L. E. Franzke, and D. Crommelin. “Normal forms for reduced stochastic climate models.” In: Proc. Natl. Acad. Sci. USA 106 (2009), pp. 3649–3653.Google Scholar

  • [39] A. J. Majda, C. L. E. Franzke, and B. Khouider. “An Applied Mathematics Perspective on Stochastic Modelling for Climate.” In: Phil. Trans. R. Soc. A 366 (2008), pp. 2429–2455.Google Scholar

  • [40] A. J. Majda and J. Harlim. “Physics constrained nonlinear regression models for time series.” In: Nonlinearity 26.1 (2013), p. 201.CrossrefGoogle Scholar

  • [41] A. J. Majda and D. Qi. “Strategies for Reduced-Order Models for Predicting the Statistical Responses and Uncertainty Quantification in Complex Turbulent Dynamical Systems.” In: SIAM Review 60.3 (2018), pp. 491–549.CrossrefGoogle Scholar

  • [42] A. J. Majda, I. Timofeyev, and E. Vanden-Eijnden. “A mathematical framework for stochastic climate models.” In: Communications on Pure and Applied Mathematics 54.8 (2001), pp. 891–974.Google Scholar

  • [43] A. J.Majda, I. Timofeyev, and E. Vanden-Eijnden. “Models for stochastic climate prediction.” In: Proceedings of the National Academy of Sciences 96.26 (1999), pp. 14687–14691.Google Scholar

  • [44] A. J. Majda, I. Timofeyev, and E. Vanden-Eijnden. “Systematic strategies for stochastic mode reduction in climate.” In: J. Atmos. Sci. 60.14 (2003), pp. 1705–1722.CrossrefGoogle Scholar

  • [45] A. J. Majda and X. Wang. Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows. Cambridge University Press, 2006.Google Scholar

  • [46] M.Matsumoto and T. Nishimura. “Mersenne Twister:A 623-Dimensionally Equidistribuited Uniform Pseudo-Random Number Generator.” In: ACM Transactions on Modelling and Computer Simulation 8.1 (1998), pp. 3–30.Google Scholar

  • [47] T. J. O’Kane and J. S. Frederiksen. “Statistical dynamical subgrid-scale parameterizations for geophysical flows.” In: Physica Scripta 2008.T132 (2008), p. 014033.Google Scholar

  • [48] G. A. Pavliotis and A. Stuart. Multiscale methods: averaging and homogenization. Springer Science & Business Media, 2008.Google Scholar

  • [49] P. Porta Mana and L. Zanna. “Toward a stochastic parameterization of ocean mesoscale eddies.” In: Ocean Modelling 79 (2014), pp. 1 –20.Google Scholar

  • [50] D. Qi and A. J. Majda. “Low-Dimensional Reduced-Order Models for Statistical Response and Uncertainty Quantification: Two-Layer Baroclinic Turbulence.” In: Journal of the Atmospheric Sciences 73.12 (2016), pp. 4609–4639.Google Scholar

  • [51] V. Resseguier, E. Mémin, and B. Chapron. “Geophysical flows under location uncertainty, Part I Random transport and general models.” In: Geophysical & Astrophysical Fluid Dynamics 111.3 (2017), pp. 149–176.Google Scholar

  • [52] V. Resseguier, E. Mémin, and B. Chapron. “Geophysical flows under location uncertainty, Part II Quasi-geostrophy and efficient ensemble spreading.” In: Geophysical & Astrophysical Fluid Dynamics 111.3 (2017), pp. 177–208.Google Scholar

  • [53] P. D. Sardeshmukh and P. Sura. “Reconciling non-Gaussian climate statistics with linear dynamics.” In: J. Climate 22.5 (2009), pp. 1193–1207.Google Scholar

  • [54] M. J. Senosiain and A. Tocino. “Two-step strong order 1.5 schemes for stochastic differential equations.” In: Numerical Algorithms 75.4 (Aug. 2017), pp. 973–1003.Google Scholar

  • [55] T. G. Shepherd. “Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics.” In: Advances in Geophysics 32 (1990), pp. 287–338.CrossrefGoogle Scholar

  • [56] G. Shutts. “A kinetic energy backscatter algorithm for use in ensemble prediction systems.” In: Quart. J. Roy. Meteorol. Soc. 131 (2005), pp. 3079–3102.Google Scholar

  • [57] D. J. Stensrud. Parameterization schemes. Cambridge University Press, 2007.Google Scholar

  • [58] H. von Storch and F. W. Zwiers. Statistical analysis in climate research. Cambridge University Press, 2003.Google Scholar

  • [59] G. K. Vallis. Atmospheric and oceanic fluid dynamics: fundamentals and large-scale circulation. Cambridge University Press, 2006.Google Scholar

  • [60] S. Vannitsem. “Predictability of large-scale atmospheric motions: Lyapunov exponents and error dynamics.” In: Chaos: An Interdisciplinary Journal of Nonlinear Science 27.3 (2017), p. 032101.CrossrefGoogle Scholar

  • [61] J. Wouters and V. Lucarini. “Disentangling multi-level systems: averaging, correlations and memory.” In: Journal of Statistical Mechanics: Theory and Experiment 2012.03 (2012), P03003.Google Scholar

  • [62] L. Zanna, P. Porta Mana, J. Anstey, T. David, and T. Bolton. “Scale-aware deterministic and stochastic parametrizations of eddy-mean flow interaction.” In: Ocean Modelling 111 (2017), pp. 66 –80.Google Scholar

About the article

Received: 2018-11-08

Accepted: 2019-04-18

Published Online: 2019-05-24

Published in Print: 2019-02-01


Citation Information: Mathematics of Climate and Weather Forecasting, Volume 5, Issue 1, Pages 45–64, ISSN (Online) 2353-6438, DOI: https://doi.org/10.1515/mcwf-2019-0004.

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© 2019 Federica Gugole et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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