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Rupert Klein
and
Tommaso Benacchio

Abstract

The compressible flow equations for a moist, multicomponent fluid constitute the most comprehensive description of atmospheric dynamics used in meteorological practice. Yet, compressibility effects are often considered weak and acoustic waves outright unimportant in the atmosphere, except possibly for Lamb waves on very large scales. This has led to the development of “soundproof” models, which suppress sound waves entirely and provide good approximations for small-scale to mesoscale motions. Most global flow models are based instead on the hydrostatic primitive equations that only suppress vertically propagating acoustic modes and are applicable to relatively large-scale motions. Generalized models have been proposed that combine the advantages of the hydrostatic primitive and the soundproof equation sets. In this note, the authors reveal close relationships between the compressible, pseudoincompressible (soundproof), hydrostatic primitive, and the Arakawa and Konor unified model equations by introducing a continuous two-parameter (i.e., “doubly blended”) family of models that defaults to either of these limiting cases for particular parameter constellations.

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Tommaso Benacchio
and
Rupert Klein

Abstract

When written in conservation form for mass, momentum, and density-weighted potential temperature, and with Exner pressure in the momentum equation, the pseudoincompressible model and the hydrostatic model only differ from the full compressible equations by some additive terms. This structural proximity is transferred here to a numerical discretization providing seamless access to all three analytical models. The semi-implicit second-order scheme discretizes the rotating compressible equations by evolving full variables, and, optionally, with two auxiliary fields that facilitate the construction of an implicit pressure equation. Time steps are constrained by the advection speed only as a result. Borrowing ideas on forward-in-time differencing, the algorithm reframes the authors’ previously proposed schemes into a sequence of implicit midpoint step, advection step, and implicit trapezoidal step. Compared with existing approaches, results on benchmarks of nonhydrostatic- and hydrostatic-scale dynamics are competitive. The tests include a new planetary-scale gravity wave test that highlights the scheme’s ability to run with large time steps and to access multiple models. The advancement represents a sizeable step toward generalizing the authors’ acoustics-balanced initialization strategy to also cover the hydrostatic case in the framework of an all-scale blended multimodel solver.

Free access
Tommaso Benacchio
,
Warren P. O’Neill
, and
Rupert Klein

Abstract

A blended model for atmospheric flow simulations is introduced that enables seamless transition from fully compressible to pseudo-incompressible dynamics. The model equations are written in nonperturbation form and integrated using a well-balanced second-order finite-volume discretization. The semi-implicit scheme combines an explicit predictor for advection with elliptic corrections for the pressure field. Compressibility is implemented in the elliptic equations through a diagonal term. The compressible/pseudo-incompressible transition is realized by suitably weighting the term and provides a mechanism for removing unwanted acoustic imbalances in compressible runs.

As the gradient of the pressure is used instead of the Exner pressure in the momentum equation, the influence of perturbation pressure on buoyancy must be included to ensure thermodynamic consistency. With this effect included, the thermodynamically consistent model is equivalent to Durran’s original pseudo-incompressible model, which uses the Exner pressure.

Numerical experiments demonstrate quadratic convergence and competitive solution quality for several benchmarks. With the inclusion of an additional buoyancy term required for thermodynamic consistency, the “pρ formulation” of the pseudo-incompressible model closely reproduces the compressible results.

The proposed unified approach offers a framework for models that are largely free of the biases that can arise when different discretizations are used. With data assimilation applications in mind, the seamless compressible/pseudo-incompressible transition mechanism is also shown to enable the flattening of acoustic imbalances in initial data for which balanced pressure distributions are unknown.

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Ray Chew
,
Tommaso Benacchio
,
Gottfried Hastermann
, and
Rupert Klein

Abstract

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 pseudo-incompressible dynamics. Here, the blended modeling strategy by Benacchio et al. 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 pseudo-incompressible regime is sufficient to suppress imbalances coming from the initialization 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. Idealized two-dimensional traveling 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 weather-relevant 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.

Open access