M2 aims at systematically deriving new numerical and stochastic methods for the energyconsistent representation of subgrid-scale processes of geophysical flows. We will systematically develop two new approaches for energy-consistent subgrid-scale parameterizations. One approach is stochastic, while the second is a multi-scale finite element approach, which can also include stochastic elements. While we have in mind two concrete complex models, ICON and FESOM, our methods will be general enough for use in other Earth System Models too.
Our research in M2 aims at developing new methods: We ask whether the newly developed methods are capable of providing some generic form of consistency and conservation. Our major aim is to provide methods for energy-consistent parameterizations to reduce the need for tuning.
M2 complements the approaches of M1 & M7 which use asymptotic, analytic and dynamical systems approaches within certain scale ranges. Our work packages focus, on the one hand, on questions of scale separation and scale interaction necessitated by the existence of a numerical subgrid in practical simulations of geophysical flows and climate. In particular, M2 aims at systematically developing accurate, conservative, and consistent subgrid parameterizations. By conservative, we mean that conservation properties across scales are discretely preserved. By consistent, we refer to mathematical consistency, while energetic consistency can also refer to the use of a subgrid-scale energy model which determines the amount of energy to backscatter. On the other hand, we also systematically investigate the effect of noise on balanced and unbalanced motions and also on the persistence of nonlinear waves and generation of gravity waves.
The composition of project personnel sees the following changes. C. Franzke will continue his successful work on stochastic backscatter as a project postdoc, extending the work toward looking at the mathematical structure in close collaboration with PL Rademacher, and also with former PL Oliver who will contribute his experience from first-phase work on variational methods to be applied in the context of conservation laws for stochastic subgrid models and with the AWI-FESOM group (M3/S2) regarding implementation and testing in a full primitive-equation ocean model. New PL Behrens from UHH-CEN joins M2, bringing in his expertise in numerical multi-scale methods. M. Oliver will continue to collaborate in particular through subproject M3 and serve as a co-advisor for the doctoral researcher.
We combine methods from formal asymptotics, mathematical analysis, dynamical systems, and stochastic analysis. Our primary focus lies on understanding phenomena and mechanisms – we believe that a better understanding of the nonlinear interactions between scales, waves and the flow regimes will be essential in evaluating and improving numerical weather and climate prediction models. Only in the nonlinear regime can we develop energy consistent schemes. Here we will approach this problem from 3 different directions:
- variational balanced model reduction,
- a dynamical systems approach of nonlinear waves and balance conditions and
- stochastic balanced model reduction.
While 1. and 3. will concentrate on foundational aspects of the problem, 2. will already check its validity using the ICON model in idealised set-ups.
Work package 1: Variational model reduction
The purpose of this work package is to look at balance and, more generally, multiscale phenomena, addressing the following questions.
- How do concepts of balance persist if the analysis is based directly upon the full set of equations of motion that form the dynamical core of global ocean or atmospheric models?
- How can we transition from one asymptotic regime to another while maintaining uniform validity of the balance relation?
The first question aims at the classical derivation of balance models via a succession of approximations, e.g. Boussinesq, hydrostatic, thin-layer, and small pressure gradient approximations lead to quasi-geostrophic models.
To answer the second question, common techniques in perturbation analysis involve matched asymptotic expansions. This is not very likely to be practical except for toy situations. Our work will therefore go into two extreme directions. First, when using balance relations as a pure diagnostic, active matching is not required, so the remaining difficulty is detecting the regime switch. At the other end of the scale, we propose to look into variational approaches where the matching is intrinsic to the geometric construction. An important advantage of variational asymptotics is the natural preservation of conservation laws for energy, momentum, and vorticity in the limit.
Work package 2: Equatorial quasi-geostrophic balances, dissipation and nonlinear waves
In this work package we propose to study nonlinear waves and bifurcations associated with the linear wave modes in equatorial quasi-geostrophic balance and adjustment, including energetically consistent dissipation parameterisations. Based on the recent progress in studying inviscid and (molecularly) viscous nonlinear waves, new approaches to quasi-geostrophic balance will be used to derived reduced models and study the wave phenomena therein. Dissipation modelling and parametrisation towards energy consistency will be included.
Aiming at mathematical rigour, we will build upon a combination of asymptotic analysis and bifurcation based numerical approaches. Mathematical well-posedness will be addressed, and nonlinear solutions will be numerically tracked by numerical path-following and bifurcation analysis. Starting with simple single layer problems in one-dimensional extended domains we will work towards higher space dimension, stratification and spherical geometry.
Work package 3: Stochastic model reduction
In this work package we will use asymptotic multi-scale models to systematically derive effective dynamic-stochastic models for the resolved scales. These model hierarchies will provide a framework for the systematic treatment of the interaction between different flow regimes and representation of unresolved scales in numerical models and will constitute a systematic framework for stochastic parameterisations. In this setting, fast synoptic- and meso-scale interactions will be systematically represented by a stochastic Ansatz.
While the full unforced and inviscid equations conserve energy and momentum, this is not necessarily the case for the truncated systems. There will be energy, enstrophy and momentum transfers between the resolved and unresolved scales. Most current deterministic parameterisation schemes do not reinject energy into the resolved scales, instead they are effectively an energy drain. Similarly, current stochastic parameterisations are operated mainly ad hoc without consideration of energy and momentum consistency. This likely contributes to the significant observed model bias.
While it is possible to derive energy conserving reduced stochastic models for some systems, here we will focus on conservation in an ensemble sense so that energy and momentum fluctuate around a mean value in individual model simulations. In the real atmosphere and ocean, energy would flow between the scales we can resolve and the ones we cannot. So in reduced order or truncated systems—climate models are clearly in this category—the energy should fluctuate around a mean value.
Main results from the first phase of the CRC
- Development of systematic energy-consistent stochastic and deterministic backscatter schemes for the atmosphere
- Development of variational balance model with full Coriolis term
- Identification of several non-linear wave phenomena
Systematic Stochastic Kinetic Backscatter Schemes (copy 1)
- Development of energy consistent kinetic backscatter schemes for
- Primitive equation model (Portable University Model of the Atmosphere (PUMA):
Published in Dwivedi, S., C. Franzke and F. Lunkeit, 2019: Energetically Consistent Stochastic and Deterministic Kinetic Energy Backscatter Schemes for Atmospheric Models. Q. J. Roy. Meteorol. Soc., 145, 3376-3386.
- 2-layer QG model:
Published in Gugole, F. and C. Franzke, 2019: Numerical Development and Evaluation of an Energy Conserving Conceptual Stochastic Climate Model. Math. Clim. Weather Forecast., 5, 45-64.
- Primitive equation model (Portable University Model of the Atmosphere (PUMA):
- Noise covariance modelling using Koopman Operator and Dynamic Mode Decomposition
Explicit internal waves and flows in nonlinear fluid models (copy 1)
- New explicit wave-type solutions in various fluid models in the whole space
- Existence for adapted forcing via linear variation of constants in the nonlinear equations
- Specific case: rotating shallow water equations with simple backscatter (cf. Figure)
- energy can accumulate in exponentially growing solutions due to backscatter
- linear combinations as explicit solutions imply instability of some stationary waves
Variational Balance model with full Coriolis force (copy 1)
- Variational balance model with full Coriolis force is derived.
- Validity of variational principle for anisotropic scaling to equatorial long-wave dynamics is in progress.
Dr. Terence O‘Kane from CSIRO Atmospheres and Ocean, Hobart, Australia, is an expert on stochastic subgrid modeling, coupled data assimilation, predictability, turbulence, geophysical fluid dynamics and advanced time series analysis. He has pionered subgrid modeling using statistical physics methods. His research is very relevant to project M2 but also projects M3 and M4.
As part of his visit he gave a TRR seminar entitled „Statistical Dynamical Subgrid Scale Parameterisation“. First he introduced the overall methodology and then he showed how this approach can be employed to atmospheric as well as ocean models with very good results. His seminar spark a lot of interest and subsequently Terry had many meetings with other TRR scientists but also non-TRR scientists from the Universität Hamburg and the Max-Planck-Institute.
Furthermore, we discussed energy consistent subgrid modeling and stochastic modeling approaches which are of particular importance to project M2. We started a joint project in which we will examine how stochastic subgrid scale parameterizations will affect coupled data assimilation and predictability.
We also finalized a book we are together editing on „Nonlinear and Stochastic Climate Dynamics“ to be published by Cambridge University Press later this year.
written by Dr. Christian Franzke
Prof. Edgar Knobloch from UC Berkeley visited Jens Rademacher in Bremen to discuss the M2 project related questions. An expert in nonlinear fluid phenomena, multiscale analysis and modelling, Prof. Knobloch’s work is very relevant for the M2 area in general and project M2-2 in particular. His work on the influence of viscosity, scaling regimes and models in geophysical flow directly overlaps with the projects’ fundamental questions. Inviscid models fail to accont for the sometimes profound effect that viscous layers have on the bulk flow. Moreover, the energy flow through scales ultimately requires viscous dissipation and suitable driving. During the visit also the question of the role of nonlinear waves in the enery flow were discussed and it seems that many questions remain open at this point. This is especially true in the context of geostrophic balance.
Combining the multi-scale finite element with stochastic subgrid informations
I defined my PhD reaserch project within the goal to combine Multi scale numerics with stochastic subgrid informations.
My name is Mouhanned Gabsi and I work as a PhD student at the University of Hamburg under the supervision of Prof. Dr. Jörn Behrens (University of Hamburg). I am part of the TRR subproject M2: Systematic Multi-Scale Modelling and Analysis for Geophysical Flows. M2 aims at systematically deriving new numerical and stochastic methods for the energyconsistent representation of subgrid-scale processes of geophysical flows. Beginning with a bit about myself, I got a bachelor degree in Mathematics and Applications at the University of Monastir (Tunisia), after that I persued a Master degree in Applied Analysis and Mathematical Physics at the University of Toulon (France) that I acquired with an internship of 6 months at the University of Paris Saclay under the supervision of Danielle Hilhorst and Ludovic Goudenège. The goal was to present numerical studies of iterative coupling for solving flow and geomechanics in a porous Medium. I started my work as part of TRR in April 2021. At the beginning, I spent more time in literature and reading papers to dicover the new environment that I am working on. Within this, I started to understand new scientific terms, phenomena and mechanisms related to Oceans, Atmosphere and Climate models and I found RTG course that I took in Mathematics, Oceanography, Meteorology and TRR meeting very helpful to me to acquire new knowledge and skills. After that, I defined my PhD reaserch project within the goal to combine Multi scale numerics with stochastic subgrid informations. Multi-scale numerical methods will address the research questions by providing a framework for coupling small-scale processes to the large-scale. Subgrid-scale parametrization is the mathematical procedure describing the statistical effect of sub-grid- scale processes on the mean flow that is expressed in terms of the resolved-scale parameters. In global atmospheric models, the range of processes which have to be parametrized is large and the characteristics of the different parametrized processes vary, e.g., atmospheric convection, gravity wave drag, vertical diffusion. The resolvedand the subgrid-scale processes in the Earth's atmosphere are the response to mechanical andthermal forcing, associated with the distribution of solar incoming radiation, topography, continents and oceans. There are several methods to improve the process of transferring information from the subgrid-scale to the coarse grid in a mathematically consistent way such as numerical multi-scale methods which are based on homogenization or the multi-scale finite element approach. This method is well established in porous media. The second method is stochastic, and in particular stochastic parametrization exploit the time scale difference between the slow resolved scale and the fast-unresolved scale to model the latter with random noise terms. This has many advantages such gain in computational timecompared to higher resolved simulations, reduction of model errors and systematic representation of uncertainties. A first task is to combine these two methods and to see if this combination inherently address conservation properties, or it pose an unnecessary overhead.
Subgrid-scale processes of geophysical flows using machine learning
I am working on applying machine learning tasks such as image super resolution to geophysical data.
I am a postdoc at Universität Bremen and I work on new sub-grid methods as part of the project M2. My research focuses on applying machine learning algorithms to geophysical fluid dynamics. Moreover, I organize the TRR Machine Learning Seminar, which takes place Tuesdays at 13:00 (everyone is welcome to join). Here, experts and newcomers meet to discuss project ideas or research results related to machine learning.
When a numerical simulation or data for a numerical simulation does not resolve the full dynamical scales, we need to simulate these missing dynamics. Unlike landscape or face pictures, geophysical data follows self-similarity such that learning the unresolved dynamics from data is a reasonable task. Especially for geophysical flow simulations an enormous amount of data has been stored in the last decades. Moreover, machine learning performs well when there is enough data available. Thus, I am working on applying machine learning tasks such as image super resolution to geophysical data.
Last year, I completed my PhD at Memorial University of Newfoundland, Canada. For my thesis, I used the shallow water equations to develop structure preserving discretization methods and a stochastic sub-grid model for efficient ensemble forecasting.
Prugger, A., Rademacher, J.D.M. & Yang, J. (2023). Rotating Shallow Water Equations with Bottom Drag: Bifurcations and Growth Due to Kinetic Energy Backscatter. SIAM Journal on Applied Dynamical Systems 22(3), doi: https://doi.org/10.1137/22M152222X.
Brecht, R., Bakels, L., Bihlo, A. & Stohl, A. (2023). Improving trajectory calculations by FLEXPART 10.4+ using single-image super-resolution. Geosci. Model Dev. 16(8), 2181–2192, doi: https://doi.org/10.5194/gmd-16-2181-2023.
Darbenas, Z., van der Hout, R. & Oliver, M. (2023). Conditional uniqueness of solutions to the Keller–Rubinow model for Liesegang rings in the fast reaction limit. J. Differential Equations 347, 212–245, doi: https://doi.org/10.1016/j.jde.2022.11.038.
Brecht, R. & Bihlo, A. (2023). Computing the Ensemble Spread From Deterministic Weather Predictions Using Conditional Generative Adversarial Networks. Geophys. Res. Lett. 50(2), e2022GL101452, doi: https://doi.org/10.1029/2022GL101452.
Franzke, C.L.E., Gugole, F. & Juricke, S. (2022). Systematic multi-scale decomposition of ocean variability using machine learning. Chaos: An Interdisciplinary Journal of Nonlinear Science 32(7), 073122, doi: https://doi.org/10.1063/5.0090064.
Darbenas, Z., van der Hout, R. & Oliver, M. (2022). Long-time asymptotics of solutions to the Keller–Rubinow model for Liesegang rings in the fast reaction limit. Ann. Inst. H. Poincaré Anal. Non Linéaire 39(6), 1413–1458, doi: https://doi.org/10.4171/AIHPC/34.
Prugger, A., Rademacher, J. D. M., & Yang, J. (2022). Geophysical fluid models with simple energy backscatter: explicit flows and unbounded exponential growth, Geophysical & Astrophysical Fluid Dynamics 116(5-6), doi: https://doi.org/10.1080/03091929.2021.2011269.
Franzke, C.L.E. (2021). Towards the development of economic damage functions for weather and climate extremes. Ecological Economics 189, 107172, doi: https://doi.org/10.1016/j.ecolecon.2021.107172.
Prugger, A. and Rademacher, J. D. M. (2021). Explicit superposed and forced plane wave generalized Beltrami flows. IMA J. Appl. Math., doi: https://doi.org/10.1093/imamat/hxab015.
Özden, G. & Oliver, M. (2021). Variational balance models for the three-dimensional Euler–Boussinesq equations with full Coriolis force. Phys. Fluids 33, 076606, doi: https://doi.org/10.1063/5.0053092.
Ma, Q., Lembo, V. & Franzke, C. L. (2021). The Lorenz energy cycle: trends and the impact of modes of climate variability. Tellus A, doi: https://doi.org/10.1080/16000870.2021.1900033.
Ma, Q. & Franzke, C.L.E. (2021). The role of transient eddies and diabatic heating in the maintenance of European heat waves: a nonlinear quasi-stationary wave perspective. Clim. Dyn. 56: 2983–3002, doi: https://doi.org/10.1007/s00382-021-05628-9.
Darbenas, Z. & Oliver, M. (2021). Breakdown of Liesegang precipitation bands in a simplified fast reaction limit of the Keller–Rubinow model. Nonlinear Differ. Equ. Appl. 28(4), doi: https://doi.org/10.1007/s00030-020-00663-7.
Önskog, T., Franzke, C.L.E. & Hannachi, A. (2020). Nonlinear time series models for the North Atlantic Oscillation. Adv. Stat. Clim. Meteorol. Oceanogr. 6(2), 141–157, doi: https://doi.org/10.5194/ascmo-6-141-2020.
Franzke, C.L.E. & Torelló i Sentelles (2020). Risk of extreme high fatalities due to weather and climate hazards and its connection to large-scale climate variability. Climatic Change 162,507–525, doi: https://doi.org/10.1007/s10584-020-02825-z.
Franzke, C.L.E., Nian, D., Yuan, N. et al. (2020). Identifying the sources of seasonal predictability based on climate memory analysis and variance decomposition. Clim. Dyn. 55, 3239–3252. doi: https://doi.org/10.1007/s00382-020-05444-7.
Huang, Y., Franzke, C.L.E., Yuan, N. & Fu, Z. (2020). Systematic identification of causal relations in high-dimensional chaotic systems: application to stratosphere-troposphere coupling. Clim. Dyn. 55, 2469–2481. doi: https://doi.org/10.1007/s00382-020-05394-0.
Gugole, F. & Franzke, C.L.E. (2020). Spatial Covariance Modeling for Stochastic Subgrid-Scale Parameterizations Using Dynamic Mode Decomposition. J. Adv. Model Earth Sy. 12(8), e2020MS002115, doi: https://doi.org/10.1029/2020MS002115.
Huang, Y., Fu, Z., & Franzke, C.L.E. (2020). Detecting causality from time series in a machine learning framework. Chaos: An Interdisciplinary Journal of Nonlinear Science 30(6), doi: https://doi.org/10.1063/5.0007670.
Akramov, I. & Oliver, M. (2020). On the existence of solutions to a bi-planar Monge–Ampère equation. Acta Math. Sci. 40, 379–388, doi: https://doi.org/10.1007/s10473-020-0206-6.
Yang, L., Franzke, C. L., & Fu, Z. (2020). Evaluation of the Ability of Regional Climate Models and a Statistical Model to Represent the Spatial Characteristics of Extreme Precipitation. Int. J. Clim., doi: https://doi.org/10.1002/joc.6602.
Yang, L., Franzke, C. L., & Fu, Z. (2020). Power‐law behaviour of hourly precipitation intensity and dry spell duration over the United States. I. J. Clim., 40(4), 2429-2444, https://doi.org/10.1002/joc.6343.
Franzke, C. L., Barbosa, S., Blender, R., Fredriksen, H. B., Laepple, T., Lambert, F., ... & Vannitsem, S. (2020). The Structure of Climate Variability Across Scales. Rev. Geophys., e2019RG000657, https://doi.org/10.1029/2019RG000657 .
Darbenas, Z. & Oliver, M. (2019). Uniqueness of solutions for weakly degenerate cordial Volterra integral equations. J. Integral Equ. Appl. 31, 307–327, doi: https://doi.org/10.1216/JIE-2019-31-3-307.
Yang, L. Franzke, C.L., & Fu, Z. (2019). Power-law behaviour of hourly precipitation intensity and dry spell duration over the United States. International Journal of Climatology doi: https://doi.org/10.1002/joc.6343.
De Luca, P., Harpham, C., Wilby, R. L., Hillier, J. K., Franzke, C. L., & Leckebusch, G. C. (2019). Past and projected weather pattern persistence with associated multi-hazards in the British Isles. Atmosphere, 10(10), 577, https://doi.org/10.3390/atmos10100577 .
Chirilus-Bruckner, M., van Heijster, P., Ikeda, H., & Rademacher, J. D. (2019). Unfolding symmetric Bogdanov-Takens bifurcations for front dynamics in a reaction-diffusion system. J. Nonlin. Sc.
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Dwivedi, S., Franzke, C. L., & Lunkeit, F. (2019). Energetically Consistent Scale Adaptive Stochastic and Deterministic Energy Backscatter Schemes for an Atmospheric Model. Q. J. Roy. Meteorolo. Soc. https://doi.org/10.1002/qj.3625.
Gugole, F., and C.L. Franzke (2019). Numerical Development and Evaluation of an Energy Conserving Conceptual Stochastic Climate Model. Math. Climate Weather Forecasting, 5(1), 45-64, doi: https://doi.org/10.5194/os-15-601-2019.
Franzke, C. L., D. Jelic, S. Lee, and S. B. Feldstein (2019). Systematic Decomposition of the MJO and its Northern Hemispheric Extra‐Tropical Response into Rossby and Inertio‐Gravity Components. Q. J. Roy. Meteor. Soc., doi: https://doi.org/10.1002/qj.3484.
Franzke, C. L., Oliver, M., Rademacher, J. D., & Badin, G. (2019). Multi-scale methods for geophysical flows. In Energy Transfers in Atmosphere and Ocean (pp. 1-51). Springer, Cham., doi: https://doi.org/10.1007/978-3-030-05704-6_1.
Mohamad, H., and Oliver, M. (2019). A direct construction of a slow manifold for a semilinear wave equation of Klein–Gordon type. J. Differ. Eq., doi: https://doi.org/10.1016/j.jde.2019.01.001.
Mohamad, H. and Oliver, M. (2018). H s-class construction of an almost invariant slow subspace for the Klein-Gordon equation in the non-relativistic limit, J. Math. Phys., 59, 051509, https://doi.org/10.1063/1.5027040.
Gonchenko, M., Gonchenko, S., Ovsyannikov, I. & Vieiro, A.(2018). On local and global aspects of the 1:4 resonance in the conservative cubic Hénon maps. Chaos, 28, 043123, 2018. https://doi.org/10.1063/1.5022764
Önskog, T., Franzke, C. L. & Hannachi, A. (2018). Predictability and Non-Gaussian Characteristics of the North Atlantic Oscillation. J. Climate, 31(2), 537-554, doi: https://doi.org/10.1175/JCLI-D-17-0101.1.
Gao, M. and Franzke, C. L. (2017). Quantile Regression–Based Spatiotemporal Analysis of Extreme Temperature Change in China. J. Climate, 30(24), 9897-9914, doi: https://doi.org/10.1175/JCLI-D-17-0356.1.
Graves, T., Gramacy, R., Watkins, N. & Franzke, C. (2017). A brief history of long memory: Hurst, Mandelbrot and the road to ARFIMA, 1951–1980. Entropy, 19(9), 437, doi: https://doi.org/10.3390/e19090437.
Dritschel, D. G., Gottwald, G. A. & Oliver, M. (2017). Comparison of variational balance models for the rotating shallow water equations. J. Fluid Mech., 822, 689-716, doi: https://doi.org/10.1017/jfm.2017.292.
Franzke, C. L. (2017). Impacts of a changing climate on economic damages and insurance.Econ. Dis. Clim. Cha., 1(1), 95-110, doi: https://doi.org/10.1007/s41885-017-0004-3.
Gonchenko, S. V. and Ovsyannikov, I. I. (2017). Homoclinic tangencies to resonant saddles and discrete Lorenz attractors.Discret Contin. Dyn. S., Vol 10 (2), p. 273-288, doi: 10.3934/dcdss.2017013.
Gonchenko, M., Gonchenko, S. & Ovsyannikov, I. (2017). Bifurcations of Cubic Homoclinic Tangencies in Two-dimensional Symplectic Maps. Math. Model. Nat. Phenom., Vol. 12, No. 1, 2017, pp. 41-61. doi: 10.1051/mmnp/20171210.
Graves, T., Franzke, C. L., Watkins, N. W., Gramacy, R. B. & Tindale, E. (2017). Systematic inference of the long-range dependence and heavy-tail distribution parameters of ARFIMA models.Physica A, 473, 60-71, doi: https://doi.org/10.1016/j.physa.2017.01.028.
Ovsyannikov, I. I. & Turaev, D. V. (2016). Analytic proof of the existence of the Lorenz attractor in the extended Lorenz model. Nonlinearity, 30(1), 115, doi: https://doi.org/10.1088/1361-6544/30/1/115.