M3: Towards Consistent Subgrid Momentum Closures

Principal investigators: Dr. Almut Gassmann (Leibnitz Institute of Atmospheric Sciences), Dr. Stephan Juricke (Jacobs University), Prof. Marcel Oliver (Jacobs University), Dr. Peter Korn (Max Planck Institute for Meteorology)

Objectives

Computational models of atmosphere and ocean can only resolve a limited range of scales. Dynamical processes on scales smaller than the grid scale are parameterized or simply truncated.

When the grid scale lies within one of the classical turbulent inertial ranges, the use of viscous closures is well-understood, but in practice, simulations tend to be over-dissipative because of insufficient scale separation between numerical dissipation and the forcing scale. This problem is particularly severe when forcing occurs near the grid scale, for example through the process of baroclinic instability which converts available potential energy into kinetic energy in eddy-permitting or barely eddy-resolving simulations of the ocean. In general, momentum closures, as well as momentum advection, must aim at being minimally dissipative, which may require active reinjection of energy into the resolved kinetic energy range, a process termed in the context of oceanic momentum closures as kinetic energy backscatter.

This project will evaluate existing closure and advection schemes and develop new ones with a particular focus on (i) properly analysing their precise discrete behaviour, especially the energy budget near the resolution scale and the propagation of linear waves on unstructured meshes, (ii) developing closures that work on unstructured and variable grids (the B-grid in FESOM, the hexagonal and triangular C-grids in ICON-IAP and ICON-o), (iii) gaining physical understanding of numerically induced processes, and (iv) developing and assessing stochastic closures.

We will, first, continue the development of ocean kinetic energy backscatter which, as we demonstrated in the first project phase, is an effective means of improving energy consistency of eddy-permitting or barely eddy-resolving ocean simulations. Second, we will investigate the behaviour of models near the grid scale (i) top-down, by analysing the effective resolution of different choices of grids, discrete operators, and momentum closures empirically in the context of realistic model runs, and (ii) bottom-up, by building physical and mathematical understanding of the explicit as well as implicit choices of momentum closures. The combination of these methods and the application to the two different main models of the TRR181 – i.e., FESOM and ICON – will ensure a holistic approach to the development of efficient, energetically consistent and optimal discretizations of the momentum equations.

Phase 1

Highlights Phase I

Result I: Ocean kinetic energy backscatter

Ocean kinetic energy backscatter reinjects overdissipated kinetic energy via subgrid energy equation into resolved flow to reduce total (unphysical) energy dissipation via classical viscosity closures.

➢ 10% to 50% reduced SSH mean and variability biases (Fig. 1), as well as temperature and salinity mean biases in global ¼° simulations with the FESOM2 model

Figure 1: Bias reduction in sea surface height (SSH) due to backscatter: (top) temporal standard deviation of SSH anomalies from the AVISO observational estimates (1993-2009); (bottom) ratio of SSH standard deviation between AVISO and (left) the reference simulation and (right) the simulation with the new backscatter parametrization; Red corresponds to an underestimation of the variability by the simulation, blue to an overestimation. (Adapted from Juricke et al., 2020).

Result II: Stochastic atmosphere-ocean coupling for climate models

A stochastic coupling scheme is introduced communicate underestimated surface fluxvariability between the ocean and atmosphere. Fluxes are based on randomly drawn ocean surface fields for meshes with higher resolution in the ocean compared to the atmosphere.

➢ 10% to 50% reduced pricipitation mean and variability biases in the tropical Pacific

Seasonal ENSO phase locking of temperature anomalies. Black dots indicate the standard deviation of the observed Niño3.4 index (1870–2018) per month as provided by NOAA; the standard deviations of the simulated Niño3.4 indices are plotted as orange and blue bars (from Rackow et al., 2019).

Result III: Spurious waves and spectral artifacts on unstructured meshes

Differences in continuous, structured and unstructured models are clearly seen on, e.g., Floquet-Bloch dispersion diagrams for a 1D shallow water model (w, k, ℎ are frequency, wavenumber and discretization step, respectively):

➢ Spectral gaps need to be estimated since they imply absence of normal propagating waves at frequencies lying in the gaps. Such gaps create unwanted directional bias, spurious waves and other undesirable artefacts.

➢ Some structured, e.g., triangular, meshes also lead to spectral gaps.

Result IV: Local diagnostics of entropy production

Diagnosed rates of entropy production calculated from resolved wind fluctuations take positive or negative signs, with only a slight bias to positive values (part of M4). The dynamics is therefore, on average, thermodynamically consistent.

➢ Restrictions to the dynamic Smagorinsky model arise that also take into account the need for model stability.

Outlook

Next phase outlook:

➢ Spurious interfacial waves on unstructured meshes

➢ Further development and adjustment of kinetic energy backscatter schemes in global ocean models

➢ Investigation of effective resolution of general circulation models on various grids

The union of different grids (B) and the local inclusions of refined mesh areas (C) imply the appearance of a new type of waves, namely guided waves propagating at the interface, and local waves. These types of waves, sometimes called Rayleigh-Stoneley waves, considered to be "spurious" in the current context and should be muffled, since there are no such waves in the original uniform grid (A) reflecting the homogeneity of real models.

Reports

First Steps in Stochastic Ocean Modelling

To find a good stochastic kinetic energy backscatter parameterization is my local aim so far.
Ekaterina Bagaeva, PhD M3

My name is Ekaterina and I’m a PhD Student of project M3 “Towards Consistent Subgrid Momentum Closures”. I work at Jacobs University in Bremen and at AWI in Bremerhaven as well. Due to Corona pandemic the personal contacts are quite limited, but I was lucky enough to meet already my colleagues in AWI and also regularly meet the colleagues in Jacobs University.

Before joining to TRR I’ve got a master degree in Mathematical Modelling following an Erasmus Program in Europe. My master thesis was related to stochastic modelling of the evolution of an epidemic. The stochastic area of my thesis brings me to the current work in TRR.

My global aim in the projects is to improve the representation of ocean variability that represents by ocean eddies. Some ocean eddies are already resolved for certain degrees of resolution, but still there are variations where explicit simulation is not possible. Some of these variabilities could be resolved by bringing back to the model kinetic energy, so called kinetic energy backscatter, obtained through the stochastic parametrization. And to find a good stochastic kinetic energy backscatter parameterization is my local aim so far.

To date, we are on the second stage of the project, so the first months of my work were devoted to acquaintance with the FESOM model, its configurations and code itself. The other significant part was related to understanding of papers which were published by members of M3 during the first stage of the project. Currently I start to implement the idea of identifying the process based on EOF (Empirical Orthogonal Functions) analysis of kinetics energy difference between realization of the model on the fine and coarse grids.

I see the evolution of my work in the application of smarter and more sophisticated approached of stochastic parametrization, preserving the model intuitively understandable and available for computation.

Stochastic superparametrization (SSP) for ocean models

We are working on adapting SSP for applying it to ocean primitive equations (PE).
Anton Kutsenko, Postdoc in M3

Please download Anton's report here, since we cannot display his equations with our CMS.

Publications

Kutsenko, A.A. (2023). Approximation of the Number of Descendants in Branching Processes. J. Stat. Phys. 190(68), doi: https://doi.org/10.1007/s10955-023-03079-6.
Chrysagi, E., Basdurak, N.B., Umlauf, L., Gräwe, U. & Burchard, H. (2022). Thermocline Salinity Minima Due To Wind-Driven Differential Advection. JGR Oceans 127(11), doi: https://doi.org/10.1029/2022JC018904.
Strommen, K., Juricke, S. & Cooper, F. (2022). Improved teleconnection between Arctic sea ice and the North Atlantic Oscillation through stochastic process representation. Weather Clim. Dynam. 3(3), 951–975, doi: https://doi.org/10.5194/wcd-3-951-2022.
Franzke, C. L., 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.
Gassmann, A. (2021). Inherent Dissipation of Upwind-Biased Potential Temperature Advection and its Feedback on Model Dynamics. J. Adv. Model Earth Sy., doi: https://doi. org/10.1029/2020MS002384.
Juricke, S., Danilov, S., Koldunov, N., Oliver, M.,Sein, D.V.,Sidorenko, D. & Wang, Q. (2020). A Kinematic Kinetic Energy Backscatter Parametrization: From Implementation to Global Ocean Simulations. J. Adv. Model Earth Sy., 12, e2020MS002175. doi: https://doi.org/10.1029/2020MS002175.
Kutsenko, A. (2020). An entire function connected with the approximation of the golden ratio. Am. Math. Monthly, preprint arXiv:1906.01059 (accepted).
Kutsenko, A. A. (2019). Programming Infinite Machines. Erkenntnis, doi: https://doi.org/10.1007/s10670-019-00190-7.
Juricke, S., Danilov, S., Koldunov, N., Oliver, M. & Sidorenko, D. (2020). Ocean kinetic energy backscatter parametrization on unstructured grids: Impact on global eddy‐permitting simulations, J. Adv. Model. Earth Sys., 12, e2019MS001855. https://doi.org/10.1029/2019MS001855.
Danilov, S., & Kutsenko, A. (2019). On the geometric origin of spurious waves in finite-volume discretizations of shallow water equations on triangular meshes. J. Comput. Phys.,https://doi.org/10.1016/j.jcp.2019.108891.
Kutsenko, A. A. (2019). Matrix representations of multidimensional integral and ergodic operators. Appl. Math. Comput., Vol. 369, https://doi.org/10.1016/j.amc.2019.124818.
Rackow, T., & Juricke, S (2019). Flow‐dependent stochastic coupling for climate models with high ocean‐to‐atmosphere resolution ratio. Q. J. Roy. Meteor. Soc., 1-17, https://doi.org/10.1002/qj.3674.
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.
Strommen, K., Christensen, H. M., MacLeod, D., Juricke, S., & Palmer, T. (2019). Progress Towards a Probabilistic Earth System Model: Examining The Impact of Stochasticity in EC-Earth v3. 2. Geosci. Model Dev., 12(7), doi: https://doi.org/10.5194/gmd-12-3099-2019.
Juricke, S., S. Danilov, A. Kutsenko and M. Oliver (2019). Ocean kinetic energy backscatter parametrizations on unstructured grids: Impact on mesoscale turbulence in a channel.Ocean Model., doi: https://doi.org/10.1016/j.ocemod.2019.03.009.
Kutsenko, A. A. (2019). A note on sharp spectral estimates for periodic Jacobi matrices. J. Approx. Theory, Vol. 242, p. 58-63, doi: https://doi.org/10.1016/j.jat.2019.03.003.
Danilov, S., Juricke, S., Kutsenko, A., & Oliver, M. (2019). Toward consistent subgrid momentum closures in ocean models. In Energy Transfers in Atmosphere and Ocean (pp. 145-192). Springer, Cham., doi: https://doi.org/10.1007/978-3-030-05704-6_5.
Juricke, S., MacLeod, D., Weisheimer, A., Zanna, L., & Palmer, T. (2018). Seasonal to annual ocean forecasting skill and the role of model and observational uncertainty. Q. J. Roy. Meteor. Soc., doi: https://doi.org/10.1002/qj.3394.
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.
Wang, Q., Wekerle, C., Danilov, S., Koldunov, N., Sidorenko, D., Sein, D., Rabe, B. & Jung, T. (2018). Arctic Sea Ice Decline Significantly Contributed to the Unprecedented Liquid Freshwater Accumulation in the Beaufort Gyre of the Arctic Ocean, Geophys. Res. Lett., 45, 4956-4964, doi: https://doi.org/10.1029/2018GL077901.
Kutsenko, A. A., Shuvalov, A. L. & Poncelet, O. (2018). Dispersion spectrum of acoustoelectric waves in 1D piezoelectric crystal coupled with 2D infinite network of capacitors , J. Appl. Phys., 123, 044902, https://doi.org/10.1063/1.5005165 .
Kutsenko, A. (2017). Mixed multidimensional integral operators with piecewise constant kernels and their representations. Lin. Multilin. Algebra, 67, 1-10, https://doi.org/10.1080/03081087.2017.1415294.
Oliver, M. (2017). Lagrangian averaging with geodesic mean. P. R. Soc. London, https://doi.org/10.1098/rspa.2017.0558.
Kutsenko, A. A. (2017). Application of matrix-valued integral continued fractions to spectral problems on periodic graphs with defect. J. Math. Phys. 58, 063516, doi:10.1063/1.4989987
Kutsenko, A. A., Nagy, A. J., Su, X., Shuvalov, A. L. & Norris, A. N. (2017). Wave Propagation and Homogenization in 2d and 3d Lattices: A Semi-Analytical Approach. Q. J. Mech. Appl. Math. (2017) 70 (2): 131-151. doi: 10.1093/qjmam/hbx002.