### Project Summary

Atmospheric flows cover a scale range from ~ 0.1 mm (cloud droplets) to ~ 10 000 km (planet), and a comparable range of time scales. Computational models can only resolve part of this spectrum and their numerical discretizations modify the dynamics of scale interactions through associated truncation errors. In addition, any systematic impact of the unresolved scales on the resolved ones needs to be modeled by parametrization schemes, which introduce further model errors. At the same time, our insight into the current state of the atmosphere is limited by the spatio-temporal distribution of meteorological observations. In order to cope with the resulting uncertainties arising from discretization and modelling errors as well as from underdetermined initial and boundary conditions, data assimilation (DA) enables controlled adjustments of model-based forward simulations using incoming observational data in a judicious fashion.

DA combines such simulations with statistical views of observations and model output to minimize the model-to-data distance in a suitable norm. DA algorithms need to make explicit use of the underlying multi-scale nature of atmospheric .ows in order to be applicable in the presence of limited data sets and poor statistical resolutions. Currently this is achieved through adhoc techniques such as localization and ensemble in.ation, for which a satisfactory mathematical framework is as yet missing. At the same time, DA procedures also modify a model’s scale-dependent dynamics and may, e.g., affect the model’s balancing behavior with respect to fast internal gravity and acoustic wave modes – a problem with extensive history in numerical weather prediction. How can we quantify and correct for misrepresentations of scale interactions arising from underresolution, numerical truncation, and DA procedures?

Speaking in meteorological terms, this project aims at advanced data assimilation methods by synergetically connecting techniques of scale analysis, computational .uid dynamics, and advanced data .ltering. Methodologically speaking, we address the predictive modelling of the large scales of a system whose root model is known but inaccessible to direct computation due to a cascade of unresolvable spatio-temporal scales. We aim to exploit observational data, asymptotic characterizations of both the root model and the DA procedures, and asymptotically balanced numerical methods to control the resulting uncertainties.

### Project publications

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

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
.
pp. 1-49.
ISSN Online: 1520-0493; Print: 0027-0644
(Submitted)

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