Atmospheric flows feature length scales from 10⁻⁵to 10⁵m and timescales from microseconds to weeks or more. For scales above several kilometers and minutes, there is a natural scale separation induced by the atmosphere's thermal stratification, together with the influences of gravity and Earth's rotation, and the fact that atmospheric-flow Mach numbers are typically small. A central aim of theoretical meteorology is to understand the associated scale-specific flow phenomena, such as internal gravity waves, baroclinic instabilities, Rossby waves, cloud formation and moist convection, (anti-)cyclonic weather patterns, hurricanes, and a variety of interacting waves in the tropics. Single-scale asymptotics yields reduced sets of equations that capture the essence of these scale-specific processes. For studies of interactions across scales, techniques of multiple-scales asymptotics have received increasing recognition in recent years. This article recounts the most prominent scales and associated scale-dependent models and summarizes recent multiple-scales developments.

We present a nonhydrostatic finite-volume global atmospheric model formulation for numerical weather prediction with the Integrated Forecasting System (IFS) at ECMWF and compare it to the established operational spectral-transform formulation. The novel Finite-Volume Module of the IFS (henceforth IFS-FVM) integrates the fully compressible equations using semi-implicit time stepping and non-oscillatory forward-in-time (NFT) Eulerian advection, whereas the spectral-transform IFS solves the hydrostatic primitive equations (optionally the fully compressible equations) using a semi-implicit semi-Lagrangian scheme. The IFS-FVM complements the spectral-transform counterpart by means of the finite-volume discretization with a local low-volume communication footprint, fully conservative and monotone advective transport, all-scale deep-atmosphere fully compressible equations in a generalized height-based vertical coordinate, and flexible horizontal meshes. Nevertheless, both the finite-volume and spectral-transform formulations can share the same quasi-uniform horizontal grid with co-located arrangement of variables, geospherical longitude–latitude coordinates, and physics parameterizations, thereby facilitating their comparison, coexistence, and combination in the IFS. We highlight the advanced semi-implicit NFT finite-volume integration of the fully compressible equations of IFS-FVM considering comprehensive moist-precipitating dynamics with coupling to the IFS cloud parameterization by means of a generic interface. These developments – including a new horizontal–vertical split NFT MPDATA advective transport scheme, variable time stepping, effective preconditioning of the elliptic Helmholtz solver in the semi-implicit scheme, and a computationally efficient implementation of the median-dual finite-volume approach – provide a basis for the efficacy of IFS-FVM and its application in global numerical weather prediction. Here, numerical experiments focus on relevant dry and moist-precipitating baroclinic instability at various resolutions. We show that the presented semi-implicit NFT finite-volume integration scheme on co-located meshes of IFS-FVM can provide highly competitive solution quality and computational performance to the proven semi-implicit semi-Lagrangian integration scheme of the spectral-transform IFS.

We study the low Mach low Freude numbers limit in the compressible Navier–Stokes equations and the transport equation for evolution of an entropy variable — the potential temperature $\mathrm{\Theta }$ . We consider the case of well-prepared initial data on “flat” torus and Reynolds number tending to infinity, and the case of ill-prepared data on an infinite slab. In both cases, we show that the weak solutions to the primitive system converge to the solution to the anelastic Navier–Stokes system and the transport equation for the second-order variation of $\mathrm{\Theta }$ .

In soundproof model equations for geophysical fluid dynamics, the momentum and mechanical energy budgets decouple from the thermodynamics for adiabatic flows. In applying such models to nonadiabatic flows of fluids with general equations of state, thermodynamic consistency of the soundproof approximations needs to be ensured. Specifically, a physically meaningful total energy conservation law should arise as an integral of adiabatic dynamics, while for diabatic flows the effective energy source terms should be related through thermodynamic relationships to the rates of change of entropy and other pertinent internal degrees of freedom. Complementing earlier work by one of the authors on the Lipps and Hemler-type anelastic approximation, this paper discusses the thermodynamic consistency of an extension of Durran’s pseudoincompressible model to moist atmospheric motions allowing for a general equation of state.

We study a moisture model for warm clouds that has been used by Klein and Majda (2006 Theor. Comput. Fluid Dyn . 20 525–551) as a basis for multiscale asymptotic expansions for deep convective phenomena. These moisture balance equations correspond to a bulk microphysics closure in the spirit of Kessler (1969 Meteorol. Monogr . 10 1–84) and Grabowski and Smolarkiewicz (1996 Mon. Weather Rev . 124 487–97), in which water is present in the gaseous state as water vapor and in the liquid phase as cloud water and rain water. It thereby contains closures for the phase changes condensation and evaporation, as well as the processes of autoconversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. Phase changes are associated with enormous amounts of latent heat and therefore provide a strong coupling to the thermodynamic equation. In this work we assume the velocity field to be given and prove rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture model with viscosity, diffusion and heat conduction. To guarantee local well-posedness we first need to establish local existence results for linear parabolic equations, subject to the Robin boundary conditions on the cylindric type of domains under consideration. We then derive a priori estimates, for proving the maximum principle, using the Stampacchia method, as well as the iterative method by Alikakos (1979 J. Differ. Equ . 33 201–25) to obtain uniform boundedness. The evaporation term is of power law type, with an exponent in general less or equal to one and therefore making the proof of uniqueness more challenging. However, these difficulties can be circumvented by introducing new unknowns, which satisfy the required cancellation and monotonicity properties in the source terms.

We consider the theoretical model of Bergmann and Lebowitz for open systems out of equilibrium and translate its principles in the adaptive resolution simulation molecular dynamics technique. We simulate Lennard-Jones fluids with open boundaries in a thermal gradient and find excellent agreement of the stationary responses with the results obtained from the simulation of a larger locally forced closed system. The encouraging results pave the way for a computational treatment of open systems far from equilibrium framed in a well-established theoretical model that avoids possible numerical artifacts and physical misinterpretations.