M2: Mathematical, Numerical and Datadriven Approaches to Ocean Parameterisations

Principal investigators: Prof. Jörn Behrens (University of Hamburg), Dr. Peter Korn (Max Planck Institute for Meteorology), Prof. Jens Rademacher (University of Hamburg)

Objectives

The core research question of subproject M2 is how do resolved and unresolved scales in discrete modeling approaches interact and how can these interactions be represented in a mathematically consistent, quantitatively correct and numerically stable way?

To this end we will integrate mathematical analysis, numerics, and data-driven approaches such as machine learning to advance ocean parameterisations. In particular we will

• analyse and mathematize parameter patchiness in ocean models;
• develop multi-scale finite element methods for the upscaling and regularisation of parameterisations;
• apply physics-informed machine learning to subgrid representation of relevant processes;
• assess properties of machine learning approaches with a special focus on multi-scale physics.

In recent works focussing on the emerging topic of ‘mathematics of parameterisations’, it was demonstrated that parameterisations potentially challenge the regularity of the solution of the PDE. In numerical experiments, this manifests as ‘patchiness’ of mixing coefficients and we refer to this patchiness as ‘parameterisation noise’. This is a robust phenomenon across different atmosphere and ocean models and it increases with higher resolution. Examples of such parameterisations in ocean models comprise the mesoscale eddy parameterisation of Gent-McWilliams or vertical mixing schemes.

During the previous phase of the TRR, novel methods for representing scale interactions were developed. These multi-scale finite element based methods (MsFEM) serve as a mathematically consistent tool to lift unresolved features or processes into the resolved scale. The methods will be further developed to provide a physically consistent regularization mechanism that prevents parameter patchiness numerically and allows for a mathematically consistent formulation that admits global well-posedness.

In a complementary approach, employing machine learning approaches, we will generate representations of (small-scale) variations in order to provide novel tools for diagnostics and improvements of vertical mixing parameterisations. For neural networks, we will, on the one hand, systematically use underlying continuum differential equations and desired conservation properties in the sense of physics informed machine learning with application to the IDEMIX parameterisation. On the other hand, we will study variations of the network architecture to better represent the multi-scale nature.

Invited Guests

November 2016

Dr. Terence O'Kane (CSIRO Australia)

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

July 2016

Prof. Edgar Knobloch (UC Berkeley)

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.

Reports

Research Stay in Douai by Mouhanned Gabsi (Oct 23)

After completing a 1-month research stay at the Centre for Materials & Processes (CERI MP)- IMT Nord Europe in Douai, I feel it is an important moment to reflect on how the stay has informed my work, including key research findings, implications and emerging questions.
My intention in working in Douai as a visiting researcher was to experience another academic reality, build a professional network and identify possibilities for future research collaborations.

From October 2nd to October 31st, I had the great opportunity to visit Prof. Modesar shakoor at the IMT Nord Europe’s Centre for Education, Research and Innovation in Materials and Processes. The IMT (Institut Mines Télécom) Nord Europe is a French graduate school of engineering. It is located in the Hauts-de-France region, shared between 2 campuses: the science campus of the University of Lille (Villeneuve-d'Ascq, European Metropolis of Lille); and the city of Douai. The school trains high-level engineers and scientists (Master and PhD level) in various technological fields including Digital Sciences, Energy and Environment Eco-Materials, Industry and Civil Engineering.

The working environment in the research center was very vivid and enabled a productive exchange of knowledge. I received a lot of useful input regarding my PhD topic and was supported in any possible way. This leads me to clarify some doubts related to my current work. I have been able to further develop and improve a big part of my PhD.

I shared the office with Sarabilou, a Phd Student who is working in the same field of interest. This provided me an opportunity to have some interesting discussions with him on various topics.

During my research stay, Prof. Modesar recommended that I learn and experience some deep learning tools and for this, he proposed some online tutorials related to specific class of artificial recurrent neural network (RNN) architecture called the Long Short-Term Memory (LSTM) neural networks implemented in Python with the TensorFlow library. I used the LSTM to solve the 1D wave equation and 2D heat equation. Our regular meeting and discussions lead to a possibility for future research collaborations in this direction.

Although my stay abroad was shortened, the experience was still great and the journey was worthwhile. I got to know interesting personalities. The people I met in the Student Residence were extremely welcoming, supportive and easy to talk with.

Lastly, I believe the networking I have done throughout this experience will become valuable in the future and am very grateful to have made so many valuable contacts.

I would like to thank the TRR181 for enabling and financially supporting this research stay abroad.

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.
Mouhanned Gabsi, PhD M2

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.

Publications

Holst, P., Rademacher, J.D.M. & Yang, J. (2024). Rotating Convection and Flows with Horizontal Kinetic Energy Backscatter. In: Henry, D. (eds) Nonlinear Dispersive Waves. Advances in Mathematical Fluid Mechanics. Birkhäuser, Cham. doi: https://doi.org/10.1007/978-3-031-63512-0_7.
Brecht, R. & Bihlo, A. (2024). M-ENIAC: A Physics-Informed Machine Learning Recreation of the First Successful Numerical Weather Forecasts. Geophysical Research Letters, 51, e2023GL107718, doi: https://doi.org/10.1029/2023GL107718.
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.
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. & 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, H. (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.
Nian, D., Franzke, C.L.E., 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.
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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 .
Badin, G., Behrens, J., Franzke, C., Oliver, M. & Rademacher, J. (2019). Introduction, Geophys. Astro. Fluid, 113:5-6, 425-427, DOI: 10.1080/03091929.2019.1655259.
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., 1-43, doi.org/10.1007/s00332-019-09563-2.
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
Hu, G. and Franzke, C. L. (2017). Data Assimilation in a Multi-Scale Model.Math. Clim. Weather Forecasting, 3(1), 118-139, https://doi.org/10.1515/mcwf-2017-0006.
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.
Bódai, T. and Franzke, C. (2017). Predictability of fat-tailed extremes. Phys. Rev. E, 96(3), 032120, doi.org/10.1103/PhysRevE.96.032120.
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.
Franzke, C. L. (2017). Extremes in dynamic-stochastic systems.Chaos, 27(1), 012101, doi.org/10.1063/1.497354.
Franzke, C. L. and O'Kane, T. J. (Eds.) (2017). Nonlinear and Stochastic Climate Dynamics. Cambridge University Press, doi: 10.1017/9781316339251.
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.