A one-step blended soundproof-compressible model with balanced data assimilation: theory and idealised tests

A challenge arising from the local Bayesian assimilation of data in an atmospheric flow simulation is the imbalances it may introduce. Acoustic fast-mode imbalances of the order of the slower dynamics can be negated by employing a blended numerical model with seamless access to the compressible and the soundproof pseudoincompressible dynamics. Here, the blended modelling strategy by Benacchio et al., MWR, vol. 142 (2014) is upgraded in an advanced numerical framework and extended with a Bayesian local ensemble data assimilation method. Upon assimilation of data, the model configuration is switched to the pseudo-incompressible regime for one time-step. After that, the model configuration is switched back to the compressible model for the duration of the assimilation window. The switching between model regimes is repeated for each subsequent assimilation window. An improved blending strategy for the numerical model ensures that a single time-step in the pseudoincompressible regime is sufficient to suppress imbalances coming from the initialisation and data assimilation. This improvement is based on three innovations: (i) the association of pressure fields computed at different stages of the numerical integration with actual time levels; (ii) a conversion of pressure-related variables between the model regimes derived from low Mach number asymptotics; and (iii) a judicious selection of the pressure variables used in converting numerical model states when a switch of models occurs. Idealised two-dimensional travelling vortex and buoyancy-driven bubble convection experiments show that acoustic imbalances arising from data assimilation can be eliminated by using this blended model, thereby achieving balanced analysis fields. Significance statement Weather forecasting models use a combination of physics-based algorithms and meteorological measurements. A problem with combining outputs from the model with measurements of the atmosphere is that insignificant signals may generate noise and compromise the physical soundness of weatherrelevant processes. By selecting atmospheric processes through the toggling of parameters in a mixed model, we propose to suppress the undesirable signals in an efficient way and retain the physical features of solutions produced by the model. The approach is validated here for acoustic imbalances using a compressible/pseudo-incompressible model pair. This development has the potential to improve the techniques used to bring observations into models and with them the quality of atmospheric model output.

[1]  A. Arakawa,et al.  Unification of the Anelastic and Quasi-Hydrostatic Systems of Equations , 2009 .

[2]  D. Durran.  Improving the Anelastic Approximation , 1989 .

[3]  Colin J. Cotter,et al.  Probabilistic Forecasting and Bayesian Data Assimilation , 2015 .

[4]  William C. Skamarock,et al.  Efficiency and Accuracy of the Klemp-Wilhelmson Time-Splitting Technique , 1994 .

[5]  P. Houtekamer,et al.  Ensemble size, balance, and model-error representation in an ensemble Kalman filter , 2002 .

[6]  Abdessamad Qaddouri,et al.  Implementation of a semi-Lagrangian fully-implicit time integration of the unified soundproof system of equations for numerical weather prediction , 2021, Monthly Weather Review.

[7]  P. Smolarkiewicz,et al.  On Forward-in-Time Differencing for Fluids: Extension to a Curvilinear Framework , 1993 .

[8]  W. Skamarock,et al.  Damping Acoustic Modes in Compressible Horizontally Explicit Vertically Implicit (HEVI) and Split-Explicit Time Integration Schemes , 2018, Monthly Weather Review.

[9]  J. Wishart.  Statistical tables , 2018, Global Education Monitoring Report.

[10]  Lili Lei,et al.  A Four-Dimensional Incremental Analysis Update for the Ensemble Kalman Filter , 2016 .

[11]  R. Klein.  Asymptotics, structure, and integration of sound-proof atmospheric flow equations , 2009 .

[12]  Milija Zupanski,et al.  Theoretical and Practical Issues of Ensemble Data Assimilation in Weather and Climate , 2009 .

[13]  Ichiro Fukumori,et al.  A Partitioned Kalman Filter and Smoother , 2002 .

[14]  T. Shepherd,et al.  Four-Dimensional Data Assimilation and Balanced Dynamics , 2006 .

[15]  J. Holton.  Geophysical fluid dynamics. , 1983, Science.

[16]  Pierre Bénard,et al.  Semi‐implicit integration of the unified equations in a mass‐based coordinate: model formulation and numerical testing , 2019, Quarterly Journal of the Royal Meteorological Society.

[17]  J. Whitaker,et al.  Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter , 2001 .

[18]  Rupert Klein,et al.  A Semi-Implicit Compressible Model for Atmospheric Flows with Seamless Access to Soundproof and Hydrostatic Dynamics , 2019, Monthly Weather Review.

[19]  Joanna Szmelter,et al.  FVM 1.0: A nonhydrostatic finite-volume dynamical core formulation for IFS , 2018 .

[20]  P. Smolarkiewicz,et al.  Conservative integrals of adiabatic Durran's equations , 2008 .

[21]  S. Cohn,et al.  Assessing the Effects of Data Selection with the DAO Physical-Space Statistical Analysis System* , 1998 .

[22]  Michael L. Minion,et al.  A fourth-order auxiliary variable projection method for zero-Mach number gas dynamics , 2008, J. Comput. Phys..

[23]  Roger Daley,et al.  The normal modes of the spherical non‐hydrostatic equations with applications to the filtering of acoustic modes , 1988 .

[24]  Fuqing Zhang,et al.  Review of the Ensemble Kalman Filter for Atmospheric Data Assimilation , 2016 .

[25]  Andrew C. Lorenc,et al.  The potential of the ensemble Kalman filter for NWP—a comparison with 4D‐Var , 2003 .

[26]  Colin J. Cotter,et al.  Ensemble filter techniques for intermittent data assimilation , 2013 .

[27]  M. Diamantakis,et al.  An inherently mass‐conserving semi‐implicit semi‐Lagrangian discretization of the deep‐atmosphere global non‐hydrostatic equations , 2014 .

[28]  Rupert Klein,et al.  A Blended Soundproof-to-Compressible Numerical Model for Small- to Mesoscale Atmospheric Dynamics , 2014 .

[29]  Peter Lynch,et al.  Initialization of the HIRLAM Model Using a Digital Filter , 1992 .

[30]  Peter Jan van Leeuwen,et al.  Nonlinear Data Assimilation , 2015 .

[31]  P. Houtekamer,et al.  Data Assimilation Using an Ensemble Kalman Filter Technique , 1998 .

[32]  Brian R. Hunt,et al.  Local Ensemble Transform Kalman Filter : An Efficient Scheme for Assimilating Atmospheric Data , 2006 .

[33]  E. Kalnay,et al.  Balance and Ensemble Kalman Filter Localization Techniques , 2011 .

[34]  J. Kepert.  Covariance localisation and balance in an Ensemble Kalman Filter , 2009 .

[35]  A. Staniforth,et al.  A new dynamical core for the Met Office's global and regional modelling of the atmosphere , 2005 .

[36]  S. Cohn,et al.  Ooce Note Series on Global Modeling and Data Assimilation Construction of Correlation Functions in Two and Three Dimensions and Convolution Covariance Functions , 2022 .

[37]  Rupert Klein,et al.  A moist pseudo-incompressible model , 2014 .

[38]  P. J. van Leeuwen,et al.  Data Assimilation in the Solar Wind: Challenges and First Results , 2017, Space weather : the international journal of research & applications.

[39]  Jonathan Flowerdew.  Towards a theory of optimal localisation , 2015 .

[40]  Lawrence L. Takacs,et al.  Data Assimilation Using Incremental Analysis Updates , 1996 .

[41]  Cathy Hohenegger,et al.  Predictability and Error Growth Dynamics in Cloud-Resolving Models , 2007 .

[42]  Rupert Klein,et al.  Scale-Dependent Models for Atmospheric Flows , 2010 .

[43]  Jeffrey L. Anderson,et al.  An adaptive covariance inflation error correction algorithm for ensemble filters , 2007 .

[44]  Christian Kühnlein,et al.  A consistent framework for discrete integrations of soundproof and compressible PDEs of atmospheric dynamics , 2014, J. Comput. Phys..

[45]  Ross N. Bannister,et al.  How is the Balance of a Forecast Ensemble Affected by Adaptive and Nonadaptive Localization Schemes , 2015 .

[46]  A. M. Clayton,et al.  On the Relationship between Incremental Analysis Updating and Incremental Digital Filtering , 2004 .

[47]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[48]  Piotr K. Smolarkiewicz,et al.  On Forward-in-Time Differencing for Fluids , 1991 .

[49]  Marc Bocquet,et al.  Ensemble Kalman filtering without the intrinsic need for inflation , 2011 .

[50]  Ann Almgren,et al.  A Low Mach Number Model for Moist Atmospheric Flows , 2014, 1409.2861.

[51]  R. Klein,et al.  Using the sound-proof limit for balanced data initialization , 2014 .

[52]  Rupert Klein,et al.  Thermodynamic Consistency of a Pseudoincompressible Approximation for General Equations of State , 2012 .

[53]  Mats Hamrud,et al.  A Fast Spherical Harmonics Transform for Global NWP and Climate Models , 2013 .

[54]  Adrian Sandu,et al.  A Bayesian Approach to Multivariate Adaptive Localization in Ensemble-Based Data Assimilation with Time-Dependent Extensions , 2018, Nonlinear Processes in Geophysics.

[55]  Sebastian Reich,et al.  Balanced data assimilation for highly oscillatory mechanical systems , 2017, 1708.03570.

[56]  Huan He,et al.  Impacts of Assimilation Frequency on Ensemble Kalman Filter Data Assimilation and Imbalances , 2020, Journal of Advances in Modeling Earth Systems.

[57]  J. Dudhia.  A Nonhydrostatic Version of the Penn State–NCAR Mesoscale Model: Validation Tests and Simulation of an Atlantic Cyclone and Cold Front , 1993 .

[58]  Colin J. Cotter,et al.  A mixed finite‐element, finite‐volume, semi‐implicit discretization for atmospheric dynamics: Cartesian geometry , 2019, Quarterly Journal of the Royal Meteorological Society.

[59]  Omar M. Knio,et al.  Regime of Validity of Soundproof Atmospheric Flow Models , 2010 .

[60]  Rupert Klein,et al.  A Doubly Blended Model for Multiscale Atmospheric Dynamics , 2016 .

[61]  T. Sonar,et al.  Asymptotic adaptive methods for multi-scale problems in fluid mechanics , 2001 .

[62]  L. M. Berliner,et al.  A Bayesian tutorial for data assimilation , 2007 .

[63]  Istvan Szunyogh,et al.  Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter , 2005, physics/0511236.

[64]  W. Skamarock,et al.  The stability of time-split numerical methods for the hydrostatic and the nonhydrostatic elastic equations , 1992 .