M3: Toward consistent subgrid momentum closures

Principal investigator: Prof. Marcel Oliver (Jacobs University Bremen), Dr. Sergey Danilov (Alfred Wegener Institute for Polar and Marine Research Bremerhaven)

Objectives

This project addresses energy backscatter from sub-grid scale motion, both theoretically and in the context of state-of-the-art global ocean circulation models at “eddy permitting” or barely eddy resolving resolutions. We systematically explore remedies by quantifying the energy budget near the grid scale in situations close to geostrophic turbulence, evaluating existing closure schemes, investigating new approaches to minimally dissipative sub-grid closures, and transferring the best approaches to full primitive equation ocean and earth system models.

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.

Next phase outlook:

➢ Spurious interfacial waves on unstructured meshes

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.

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.

Invited Guests

No available.

Reports

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. (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:10.1007/s10670-019-00190-7.
Juricke, S., Danilov, S., Koldunov, N., Oliver, M. & Sidorenko, D. (2019). Ocean kinetic energy backscatter parametrization on unstructured grids: Impact on global eddy‐permitting simulations, J. Adv. Model. Earth Sys., 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.