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.
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.
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: Statistical Mechanics and its Applications.
Gonchenko, S. V., and Ovsyannikov, I. I. (2017). Homoclinic tangencies to resonant saddles and discrete Lorenz attractors. Discrete and Continuous Dynamical Systems Series, Vol 10 (2), p. 273-288, doi: 10.3934/dcdss.2017013.
Franzke, C. L. (2017). Extremes in dynamic-stochastic systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(1), 012101.
Franzke, C. L., and O'Kane, T. J. (Eds.). (2017). Nonlinear and Stochastic Climate Dynamics. Cambridge University Press.
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.