## A02 - Multiscale data and asymptotic model assimilation for atmospheric flows

**Head(s):** Prof. Dr.-Ing. Sebastian Reich (U Potsdam), Prof. Dr.-Ing. Rupert Klein (FU Berlin)

**Project member(s):** Ray Chew, Gottfried Hastermann, Dr. Nikolas Nüsken, Maria Reinhardt

**Participating institution(s):** FU Berlin, U Potsdam

### Project Summary

Computational flow models can only resolve part of the vast range of spatio-temporal scales found in the atmosphere. Consequently, their numerical discretisations modify scale interactions through associated truncation errors, and parameterisations of the net effects of unresolved scales introduce further model errors. At the same time, insight into the current state of the atmosphere is limited by the sparsity of meteorological observations. To cope with the resulting uncertainties, data assimilation (DA) enables controlled adjustments of model-based forward simulations using incoming observational data by minimizing the model-to-data distances in suitable norms. DA algorithms require explicit use of the multi-scale nature of atmospheric flows to be applicable in the presence of limited data and poor statistical resolution.

This project aims at DA methods connecting scale analysis, computational fluid dynamics, and advanced data filtering. Methodologically speaking, we address the predictive modelling of weather systems whose root model is known but computationally inaccessible due to a cascade of partially unresolvable scales.

More specifically, we will exploit observational data and asymptotic characterisations of both the root model and the DA procedures, to (i) derive efficient and robust data assimilation techniques, to (ii) extend the DA approach from the first funding period for providing physically consistent analysis fields to meteorological applications, and to (iii) provide a mathematical and computational framework for multi-level DA applicable to model hierarchies involving moist atmospheric processes.

### Project publications

Hittmeir, Sabine and Klein, Rupert and Li, Jinkai and Titi, Edriss
(2020)
*Global well-posedness for the primitive equations coupled to nonlinear moisture dynamics with phase changes.*
Nonlinearity, 33
(7).
pp. 3206-3236.

Nüsken, N. and Richter, L.
(2020)
*Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using
neural networks: perspectives from the theory of controlled diffusions and measures on path space.*
SFB 1114 Preprint in arXiv
.
pp. 1-40.
(Submitted)

Duncan, A. and Nüsken, N. and Szpruch, L.
(2019)
*On the geometry of Stein variational gradient descent.*
SFB 1114 Preprint in arXiv:1912.00894
.
(Unpublished)

Garbuno Inigo, A. and Nüsken, N. and Reich, S.
(2019)
*Affine invariant interacting Langevin dynamics for Bayesian inference.*
SFB 1114 Preprint in arXiv:1912.02859
.
pp. 1-29.
(Unpublished)

Leung, T.Y. and Leutbecher, M. and Reich, S. and Shepherd, Th.G.
(2019)
*Atmospheric Predictability: Revisiting the Inherent Finite-Time Barrier.*
Journal of the Atmospheric Sciences, 76
(12).
pp. 3883-3892.
ISSN Online: 1520-0469 Print: 0022-4928

Benacchio, T. and Klein, R.
(2019)
*A semi-implicit compressible model for atmospheric flows with seamless access to soundproof and hydrostatic dynamics.*
Monthly Weather Review, 147
(11).
pp. 4221-4240.
ISSN Online: 1520-0493; Print: 0027-0644

Hittmeir, Sabine and Klein, Rupert and Li, Jinkai and Titi, Edriss
(2019)
*"Global well-posedness for the primitive equations coupled to nonlinear moisture dynamics with phase changes" by Hittmeir, Sabine; Klein, Rupert; Li, Jinkai; Titi, Edriss.*
Analysis of PDEs
.
pp. 1-28.
ISSN 1907.11199
(Submitted)

Nüsken, N. and Reich, S. and Rozdeba, P.J.
(2019)
*State and parameter estimation from observed signal increments.*
Entropy, 21
(5).
-505.
ISSN 1099-4300

Kühnlein, C. and Deconinck, W. and Klein, R. and Malardel, S. and Piotrowski, Z. and Smolarkiewicz, P.K. and Szmelter, J. and Wedi, N.
(2019)
*FVM 1.0: a nonhydrostatic finite-volume dynamical core for the IFS.*
Geosci. Model Dev., 12
(2).
pp. 651-676.
ISSN 1991-959X, ESSN: 1991-9603

Vater, S. and Klein, R.
(2018)
*A Semi-Implicit Multiscale Scheme for Shallow Water Flows at
Low Froude Number.*
Communications in Applied Mathematics & Computational Science, 13
(2).
pp. 303-336.
ISSN 1559-3940

Müller, A. and Névir, P. and Klein, R.
(2018)
*Scale Dependent Analytical Investigation of the Dynamic State Index Concerning the Quasi-Geostrophic Theory.*
Mathematics of Climate and Weather Forecasting, 4
(1).
pp. 1-22.
ISSN 2353-6438 (online)

Taghvaei, A. and de Wiljes, J. and Mehta, P.G. and Reich, S.
(2017)
*Kalman Filter and Its Modern Extensions for the Continuous-Time Nonlinear Filtering Problem.*
J. Dyn. Sys., Meas., Control, 140
(3).
030904.

Acevedo, W. and de Wiljes, J. and Reich, S.
(2017)
*Second-order accurate ensemble transform particle filters.*
SIAM J. Sci. Comput., 39
(5).
A1834-A1850.
ISSN 1095-7197 (online)

Hittmeir, S. and Klein, R. and Li, J. and Titi, E.
(2017)
*Global well-posedness for passively transported nonlinear moisture dynamics with phase changes.*
Nonlinearity, 30
(10).
pp. 3676-3718.
ISSN 0951-7715

Reinhardt, M. and Hastermann, G. and Klein, R. and Reich, S.
(2017)
*Balanced data assimilation for highly-oscillatory mechanical systems.*
Journal of Nonlinear Science
.
pp. 1-23.
(Submitted)

O'Kane, T.J. and Monselesan, D.P. and Risbey, J.S. and Horenko, I. and Franzke, Ch.L.E.
(2017)
*On memory, dimension, and atmospheric teleconnection patterns.*
Math. Clim. Weather Forecast, 3
(1).
pp. 1-27.

Horenko, I. and Gerber, S. and O'Kane, T.J. and Risbey, J.S. and Monselesan, D.P.
(2017)
*On inference and validation of causality relations in climate teleconnections.*
In:
Nonlinear and Stochastic Climate Dynamics.
Cambridge University Press, pp. 184-208.
ISBN 9781107118140

Chustagulprom, N. and Reich, S. and Reinhardt, M.
(2016)
*A Hybrid Ensemble Transform Particle Filter for Nonlinear and Spatially Extended Dynamical Systems.*
SIAM/ASA Journal on Uncertainty Quantification, 4
(1).
pp. 592-608.
ISSN 2166-2525

Klein, R. and Benacchio, T.
(2016)
*A doubly blended model for multiscale atmospheric dynamics.*
Journal of the Atmospheric Sciences, 73
.
pp. 1179-1186.
ISSN Online: 1520-0469 Print: 0022-4928

Feireisl, E. and Klein, R. and Novotný, A. and Zatorska, E.
(2016)
*On singular limits arising in the scale analysis of stratified fluid flows.*
Mathematical Models and Methods in Applied Sciences, World Scientific, 26
(3).
pp. 419-443.
ISSN Print: 0218-2025 Online: 1793-6314

Horenko, I. and Gerber, S.
(2015)
*Improving clustering by imposing network information.*
Science Advances, 1
(7).
ISSN 2375-2548