Local stochastic subgrid-scale modeling in efficient simulations of geophysical fluid dynamics
Final Report Abstract
The limited computational resources do not allow to represent all relevant scales in the climate system models simultaneously, raising the problem of parameterizing the unresolved scales. At the same time, as model resolution is continuously increasing, deterministic parameterization approaches might fail, since they often require the averaging over a sufficiently large ensemble of subgrid-scales (SGS). Errors in the representation of SGS are transferred to larger scales and might thus deteriorate the model skills. Stochastic parameterizations provide here an alternative. They can generate a particular SGS realization and are a valuable tool for probabilistic forecast and model error representation. However, the ad hoc nature of some stochastic parameterizations leaves room for mathematically rigorous approaches. One important application here is the construction of scale-aware parameterizations with a minimal number of tuning parameters. Such closures should be more reliable, if one is interested in climate change scenarios, because one might exit the regime where the parameters are optimally tuned. One systematic method, where the parameterization is derived from the governing equations, provides the stochastic mode reduction (SMR) procedure. This procedure requires a time-scale separation and a fitting of the unresolved modes only. In the project we address the question of parameterizing the SGS in simulations of geophysical flows by applying SMR. For this purpose we consider the one-dimensional dynamics of a shallow fluid layer. The problem is formulated in physical space by defining resolved variables as local spatial averages over finite-volume cells and unresolved variables as corresponding residuals. Based on the assumption of a time-scale separation between the slow spatial averages and the fast residuals, the SMR procedure is used to obtain a low-resolution model for the spatial averages alone with local stochastic SGS parameterization coupling each resolved variable only to a few neighboring cells. The closure improves the results of the low-resolution model and outperforms purely empirical stochastic parameterizations. Not only the representation of the energy spectrum, but also the time autocorrelation function and the scale-awareness are improved. By adjusting only a single coefficient (the strength of the noise), we observe that there is a potential for further enhancement of the performance. The present study was extended to two spatial dimensions, rotation was also included. By separating the dynamics into geostrophic and ageostrophic modes, it is found that the geostrophic component (even the SGS part) is characterized by large decorrelation times. This might limit the applicability of the SMR to ageostrophic modes only. However, the exact contribution of the different modes to the SGS dynamics requires a further investigation. The corresponding SMR parameterization is currently in development. It will provide a perspective for parameterization of 3D compressible turbulence. Beyond this, the project has contributed to several other activities: i) It was demonstrated that the fluctuation-dissipation theorem can be used to introduce a climate dependence in objectively tuned empirical parameters in closures. ii) A novel response theory approach for parameterization was applied to multi-scale systems of stochastic differential equations. iii) The potential of increasing computational efficiency by combining inexact hardware and stochastic parameterizations was exploited. iv) The theory for the largest scale atmospheric dynamics was supplemented by the evolution of barotropic flow.
Publications
- (2015). Rounding errors may be beneficial for simulations of atmospheric flow: Results from the forced 1D Burgers equation. Theoretical and Computational Fluid Dynamics, 29(4):311–328
Düben, P. D. and Dolaptchiev, S. I.
(See online at https://doi.org/10.1007/s00162-015-0355-8) - (2016). Parameterization of stochastic multiscale triads. Nonlinear Processes in Geophysics, 23(6):435–445
Wouters, J., Dolaptchiev, S. I., Lucarini, V., and Achatz, U.
(See online at https://doi.org/10.5194/npg-23-435-2016) - (2018). Climate Dependence in Empirical Parameters of Subgrid-Scale Parameterizations using the Fluctuation–Dissipation Theorem. Journal of the Atmospheric Sciences, 75(11):3843–3860
Pieroth, M., Dolaptchiev, S. I., Zacharuk, M., Heppelmann, T., Gritsun, A., and Achatz, U.
(See online at https://doi.org/10.1175/JAS-D-18-0022.1) - (2018). Stochastic subgrid-scale parametrization for one-dimensional shallow-water dynamics using stochastic mode reduction. Quarterly Journal of the Royal Meteorological Society, 144(715):1975–1990
Zacharuk, M., Dolaptchiev, S. I., Achatz, U., and Timofeyev, I.
(See online at https://doi.org/10.1002/qj.3396) - (2019). Planetary geostrophic Boussinesq dynamics: Barotropic flow, baroclinic instability and forced stationary waves. Quarterly Journal of the Royal Meteorological Society, 145(725):3751–3765
Dolaptchiev, S. I., Achatz, U., and Reitz, T.
(See online at https://doi.org/10.1002/qj.3655)
