## 1. Introduction

The full compressible (FC) flow equations rank as the most comprehensive model for representing atmospheric fluid flows. They support sound waves, yet acoustic effects are often considered to be of little relevance for weather and climate—although this is not entirely obvious for very-large-scale modes. Atmospheric motions at small-scale and mesoscale are therefore frequently modeled by approximate “soundproof” equations that suppress elastic effects and are justified by low-Mach-number scalings involving small density or pressure perturbations around a background state (Ogura and Phillips 1962; Lipps and Hemler 1982; Durran 1989). In contrast, global numerical weather prediction codes have largely relied upon the hydrostatic (HY) primitive equations, which remove vertically propagating sound waves only and are justified by scaling arguments involving small vertical-to-horizontal-scale aspect ratios (e.g., White et al. 2005).

Recent efforts focused on bridging these small- and large-scale models in unified soundproof analytical formulations that have shown competitive behavior with respect to established approaches (Durran 2008; Arakawa and Konor 2009; Konor 2014; Dubos and Voitus 2014). While the general operational viability of reduced analytical models has been questioned on the grounds of inferior performance in normal-mode analyses (Davies et al. 2003; Dukowicz 2013), other studies found that numerical errors incurred with different discretizations applied to a single set of equations may outweigh analytical model-to-model errors (Smolarkiewicz and Dörnbrack 2008). In an effort to facilitate like-to-like comparison of soundproof and compressible formulations and controlled treatment of acoustics triggered by unbalanced initial data, Benacchio et al. (2014) devised a continuously blended multimodel discretization where thermodynamically consistent pseudoincompressible (PI) and fully compressible dynamics are accessed by simple switching within a single numerical framework [see also Klein et al. (2014); Benacchio (2014); Benacchio et al. (2015)].

We note that Gatti-Bono and Colella (2006) and Smolarkiewicz et al. (2014) propose related approaches to designing unified computational frameworks for different sets of governing equations. While these authors implement different model equations based on the same fundamental numerical operators and thus achieve like-to-like comparability as outlined above, they do not pursue a continuous blending of models at the analytical or coding level as proposed here.

In this short note, we broaden the scope of the blending approach by Benacchio (2014) and Benacchio et al. (2015) to include larger-scale dynamics. We devise a two-parameter family of models that accesses pseudoincompressible, hydrostatic primitive, unified anelastic and quasi-hydrostatic, as well as fully compressible dynamics, depending on the binary choice of switches. For simplicity and brevity, we have excluded the Coriolis term in the sequel, as its inclusion would call for a third parameter controlling geostrophic balance.

We first revise in section 2 the blending between the FC and PI models suggested by Benacchio et al. (2014). The blending switch now features as a parameter in the equation of state instead of appearing explicitly in the pressure equation. Energy conservation of the resulting blended model family is discussed. Section 3 describes a blended full compressible–hydrostatic model and its energy conservation. The suppression of vertical acoustics is achieved again through a switch in the equation of state, whereas control of hydrostasy requires a second switching component in the vertical momentum balance. Suppressing vertical acoustics by the same mechanism but allowing for nonhydrostatic motions leads to a model that is conceptually very close to the unified anelastic and quasi-hydrostatic system of equations (AK) of Arakawa and Konor (2009). Section 4 presents this model variant and demonstrates that it is—in fact—equivalent to the AK model. Section 5 presents a new two-parameter blended model family that combines the FC, PI, HY, and AK models in one and the same formulation. Conclusions are drawn in section 6.

## 2. FC–PI: Governing equations blended via the state equation

### a. FC–PI blended equations

**u**and

*w*, respectively;

*g*is the gravitational acceleration,

**k**is the vertical unit vector. For simplicity, we restrict our analysis to adiabatic flows. In (1), pressure and density are nondimensionalized by standard values

*P*is used here to formulate the internal energy equation in conservation form in (1b). We note in passing that (1a) and (1b) imply that the potential temperature

*P*around

*α*appears in front of the (Exner) pressure time derivative.

### b. Energy conservation for the FC–PI blended equations

*P*equation in (3b), and using the blended state equation in (4c) and the definition of the pseudodensity

## 3. FC–HY: Blending the full compressible and hydrostatic equations

### a. FC–HY blended equations

*α*of section 2. For

*π*and

*ρ*, namely

### b. The hydrostatic limit for β = 0

*w*is determined from (20b), which becomes a velocity divergence constraint once

### c. Energy conservation for the blended compressible–hydrostatic system

*p*is replaced with

**u**and multiplication of (20d) by

*w*,

## 4. FC–AK: Blending the full compressible and the Arakawa and Konor (2009) unified model

### a. FC–AK blended equations

*β*induces two different transitions at the same time. On the one hand, since for

*α*, suppresses compressibility altogether by replacing

*P*with

*P*equation (see section 2). On the other hand, the FC–HY blending parameter

*β*controls the influence of the perturbation pressure

*β*here as the notation for the blending parameter to indicate that here the parameter only constrains compressibility but does not imply hydrostasy. In fact, the parameter

*α*in (4).

### b. The limiting case β_{c} = 0 and comparison with Arakawa and Konor (2009)

*ρ*, using (31a), and employing (31e), yields the momentum equation in advective form:

Therefore, except for nondimensionalization and momentum and energy source terms, system (31) coincides with Arakawa and Konor’s (2009) unified anelastic and quasi-hydrostatic model. From the perspective of the present derivation, the label “unified pseudoincompressible and quasi-hydrostatic model” would seem more appropriate because the suppression of compressibility effects in constructing the model is the same as that underlying the derivation of the pseudoincompressible model by Durran (1989) (see section 2).

The reader is referred to Arakawa and Konor (2009) for an analysis of the model’s energy budget. They show that there is no *local* conservation law for total energy but that *global* conservation can be achieved through a suitable adjustment of the domain-averaged perturbation pressure.

### c. The hydrostatic pressure field

**u**, casting it into an implicit equation for an update

## 5. Two-parameter blended formulation

*α*,

*β*, and

*P*in

## 6. Conclusions

In this paper we have introduced a new two-parameter family of dynamical models for atmospheric flows. The family encompasses the pseudoincompressible, hydrostatic primitive, unified Arakawa and Konor (2009), and full compressible models and allows access to reduced soundproof dynamics by straightforward switching. As a byproduct of the derivation, we have found that, in the context of our blended formulation, the suppression of compressibility in the model by Arakawa and Konor (2009) mirrors the same process in the pseudoincompressible model. A normal-mode analysis, not reported here for conciseness, corroborates the equivalence of system (31) with the model by Arakawa and Konor (2009).

The two-way blended formulation (39)–(41) naturally lends itself to a continuously tunable numerical discretization along the lines of Benacchio et al. (2014), enabling an effective treatment of fast acoustic and gravity waves within an analytical framework that covers small-, meso-, synoptic-, and planetary-scale motions. In this approach, unbalanced modes are filtered (e.g., from assimilated data) by running the related balanced model for a few time steps and then tuning back smoothly to the full compressible model over a few more steps. An open issue in this context concerns the conservation properties of the blended model family for intermediate values

The development presented in this paper becomes all the more attractive in light not only of the current efforts to show viability of soundproof models at synoptic and planetary scales (Smolarkiewicz et al. 2014; Kurowski et al. 2015) but also of the imminent necessity to compare performance of hydrostatic and nonhydrostatic codes at global operational resolutions finer than 10 km afforded by next-generation exascale supercomputers (Wedi et al. 2012; Smolarkiewicz et al. 2015).

## Acknowledgments

R.K.’s research has been partially funded by Deutsche Forschungsgemeinschaft (DFG) through Grants CRC 1114 and SPP 1276 “MetStröm.” Constructive critical remarks by Nigel Wood, Met Office, and Christian Kühnlein, ECMWF, are kindly acknowledged. TB’s contribution is British Crown Copyright 2016.

## APPENDIX

### Pressure–Density Formulation of the FC–PI System

^{tc}system of Benacchio et al. (2014). Then, maintaining (3a) and (3b), we replace (3c) with (Benacchio et al. 2014)

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