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

TL;DR

Imbalance from local data assimilation can excite fast acoustic modes that degrade slow atmospheric dynamics. The authors develop a one-step blended soundproof–compressible framework that switches to the pseudo-incompressible regime after assimilation and back, using low Mach number–inspired pressure-variable conversions and careful time-level alignment to suppress these imbalances. When coupled with untuned ensemble data assimilation (LETKF), the method yields balanced analyses and improved RMSE in idealised 2D tests (travelling vortex and rising bubble), demonstrating robust suppression of acoustic noise. The approach has potential to enhance forecast-model data assimilation and can be extended to hydrostatic and moist three-dimensional dynamics in operational cores, offering a dynamics-driven path to balanced, multi-regime numerical weather prediction.

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 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 pseudo-incompressible 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.

Figures (15)

• Figure 1: Summary of the time-levels of $\pi$ for the pseudo-incompressible (psinc) and the compressible (comp) solutions in the numerical scheme. $\Delta t$ indicates time-step size. The dashed lines relate the $\pi$'s at the same time-level between the two models. The two valid choices of $\pi$ in the pseudo-incompressible to compressible blending, at time $t^n$ and $t^{n+1}$, are depicted with arrows.
• Figure 2: Schematic of data assimilation with blending for data assimilated at time $t^n$. Blending time interfaces are in red, and the time-step spent in the pseudo-incompressible regime is shaded. See main text for full description. Numbers in brackets refer to equations, and (§) denotes the section.
• Figure 3: Blended data assimilation workflow with the sections (§) of this paper describing the algorithmic components. The initial condition (dashed outline) is used only once to start the simulation. For each assimilation window, externally obtained observations / data are assimilated into the forecast and the algorithm loops through the components following the direction of the two-headed arrows.
• Figure 4: Travelling vortex initial balanced states: Exner pressure perturbation $\pi$; dimensionless contours in the range $[-5,0] \times 10^{-4}$ with interval of $10^{-4}$ (top left), horizontal momentum $\rho u$; contours in the range $[50, 115]$ kg m$^{-1}$ s$^{-1}$ with a $5$ kg m$^{-1}$ s$^{-1}$ interval (top right), vorticity; contours in the range $[-2.2,2.8] \times 10^{-2}$ s$^{-1}$ with a 1.0 $\times 10^{-2}$ s$^{-1}$ interval (bottom left), and potential temperature $\Theta$; contours in the range $[1.1,1.9] \times 300$ K with a 0.1 $\times ~300$ K interval (bottom right). Negative contours dashed.
• Figure 5: Travelling vortex: effect of blending for imbalanced initial states on the time series of temporal increments of the full pressure $\delta p$ at location $(x,z) = (0 \text{ km},0 \text{ km})$. Top: comparison between a blended run using $\pi_{\text{half}}$ (orange), a run without blending (blue), and the reference solution from the pseudo-incompressible model (black). Bottom: comparison of blended runs using $\pi_{\text{half}}$ (orange) and $\pi_{\text{full}}$ (purple), and the compressible solution with balanced initial states (green). The blended runs are with one time-step spent in the pseudo-incompressible regime.