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Rupert Klein
and
Olivier Pauluis

Abstract

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.

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Nikki Vercauteren
and
Rupert Klein

Abstract

Atmospheric boundary layers with stable stratification include a variety of small-scale nonturbulent motions such as waves, microfronts, and other complex structures. When the thermal stratification becomes strong, the presence of such motions could affect the turbulent mixing to a large extent, and common similarity theory that is used to describe weakly stable conditions may become unreliable. The authors apply a statistical clustering methodology based on a bounded variation, finite-element method (FEM-BV) to characterize the interaction between small-scale nonturbulent motions and turbulence. The clustering methodology achieves a multiscale representation of nonstationary turbulence data by approximating them through an optimal sequence of locally stationary multivariate autoregressive factor model (VARX) processes and some slow hidden process switching between them. The clustering method is used to separate periods with different influence of the nonturbulent motions on the vertical velocity fluctuations. The methodology can be used in a later stage to derive a stochastic parameterization for the interactions between nonturbulent and turbulent motions.

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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.

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Stamen I. Dolaptchiev
and
Rupert Klein

Abstract

A reduced asymptotic model valid for the planetary and synoptic scales in the atmosphere is presented. The model is derived by applying a systematic multiple-scales asymptotic method to the full compressible-flow equations in spherical geometry. The synoptic-scale dynamics in the model is governed by modified quasigeostrophic equations, which take into account planetary-scale variations of the background stratification and of the Coriolis parameter. The planetary-scale background is described by the planetary geostrophic equations and a new closure condition in the form of a two-scale evolution equation for the barotropic component of the background flow. This closure equation provides a model revealing an interaction mechanism from the synoptic scale to the planetary scale.

To obtain a quantitative assessment of the validity of the asymptotics, the balances on the planetary and synoptic scales are studied by utilizing a primitive equations model. For that purpose, spatial and temporal variations of different terms in the vorticity equation are analyzed. It is found that, for planetary-scale modes, the horizontal fluxes of relative and planetary vorticity are nearly divergence free. It is shown that the results are consistent with the asymptotic model.

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Andrew J. Majda
and
Rupert Klein

Abstract

Systematic multiscale perturbation theory is utilized to develop self-consistent simplified model equations for the interaction across multiple spatial and/or temporal scales in the Tropics. One of these models involves simplified equations for intraseasonal planetary equatorial synoptic-scale dynamics (IPESD). This model includes the self-consistent quasi-linear interaction of synoptic-scale generalized steady Matsuno–Webster–Gill models with planetary-scale dynamics of equatorial long waves. These new models have the potential for providing self-consistent prognostic and diagnostic models for the intraseasonal tropical oscillation. Other applications of the systematic approach reveal three different balanced weak temperature gradient (WTG) approximations for the Tropics with different regimes of validity in space and time: a synoptic equatorial-scale WTG (SEWTG); a mesoscale equatorial WTG (MEWTG), which reduces to the classical models treated by others; and a new seasonal planetary equatorial WTG (SPEWTG). Both the SPEWTG and MEWTG model equations have solutions with general vertical structure, yet have the linearized dispersion relation of barotropic Rossby waves; thus, these models can play an important role in theories for midlatitude connections with the Tropics. The models are derived both from the equatorial shallow water equations in a simplified context and also as distinguished limits from the compressible primitive equations in general.

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Daniel Bäumer
,
Sabine Hittmeir
, and
Rupert Klein

Abstract

Quasigeostrophic (QG) theory is of fundamental importance in the study of large-scale atmospheric flows. In recent years, there has been growing interest in extending the classical QG plus Ekman friction layer model (QG–Ekman) to systematically include additional physical processes known to significantly contribute to real-life weather phenomena. This paper lays the foundation for combining two of these developments, namely, Smith and Stechmann’s family of precipitating quasigeostrophic (PQG) models on the one hand, and the extension of QG–Ekman for dry air by a strongly diabatic layer (DL) of intermediate height (QG–DL–Ekman) on the other hand. To this end, Smith and Stechmann’s PQG equations for soundproof motions are first corroborated within a general asymptotic modeling framework starting from a full compressible flow model. The derivations show that the PQG model family is naturally embedded in the asymptotic hierarchy of scale-dependent atmospheric flow models introduced by one of the present authors. Particular emphasis is then placed on an asymptotic scaling regime for PQG that accounts for a generic Kessler-type bulk microphysics closure and is compatible with QG–DL–Ekman theory. The detailed derivation of a moist QG–DL–Ekman model is deferred to a future publication.

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Edoardo Mazza
,
Uwe Ulbrich
, and
Rupert Klein

Abstract

The processes leading to the tropical transition of the October 1996 medicane in the western Mediterranean are investigated on the basis of a 50-member ensemble of regional climate model (RCM) simulations. By comparing the composites of transitioning and nontransitioning cyclones it is shown that standard extratropical dynamics are responsible for the cyclogenesis and that the transition results from a warm seclusion process. As the initial thermal asymmetries and vertical tilt of the cyclones are reduced, a warm core forms in the lower troposphere. It is demonstrated that in the transitioning cyclones, the upper-tropospheric warm core is also a result of the seclusion process. Conversely, the warm core remains confined below 600 hPa in the nontransitioning systems. In the baroclinic stage, the transitioning cyclones are characterized by larger vertical wind shear and intensification rates. The resulting stronger low-level circulation in turn is responsible for significantly larger latent and sensible heat fluxes throughout the seclusion process.

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Daniel Ruprecht
,
Rupert Klein
, and
Andrew J. Majda

Abstract

Starting from the conservation laws for mass, momentum, and energy together with a three-species bulk microphysics model, a model for the interaction of internal gravity waves and deep convective hot towers is derived using multiscale asymptotic techniques. From the leading-order equations, a closed model for the large-scale flow is obtained analytically by applying horizontal averages conditioned on the small-scale hot towers. No closure approximations are required besides adopting the asymptotic limit regime on which the analysis is based. The resulting model is an extension of the anelastic equations linearized about a constant background flow. Moist processes enter through the area fraction of saturated regions and through two additional dynamic equations describing the coupled evolution of the conditionally averaged small-scale vertical velocity and buoyancy. A two-way coupling between the large-scale dynamics and these small-scale quantities is obtained: moisture reduces the effective stability for the large-scale flow, and microscale up- and downdrafts define a large-scale averaged potential temperature source term. In turn, large-scale vertical velocities induce small-scale potential temperature fluctuations due to the discrepancy in effective stability between saturated and nonsaturated regions.

The dispersion relation and group velocity of the system are analyzed and moisture is found to have several effects: (i) it reduces vertical energy transport by waves, (ii) it increases vertical wavenumbers but decreases the slope at which wave packets travel, (iii) it introduces a new lower horizontal cutoff wavenumber in addition to the well-known high wavenumber cutoff, and (iv) moisture can cause critical layers. Numerical examples reveal the effects of moisture on steady-state and time-dependent mountain waves in the present hot-tower regime.

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Antony Z. Owinoh
,
Bjorn Stevens
, and
Rupert Klein

Abstract

This paper presents the derivation of a model to explore the coupling between the dynamic and thermodynamic processes of a cloud-topped boundary layer on mesoscales using a formal multiscale asymptotic approach. The derived equations show how the anomalies in the heat, moisture, and mass budgets in the boundary layer affect boundary layer motions, and how these motions can organize and amplify (or damp) such anomalies.

The thermodynamics equations are similar to those that have been suggested in mixed layer studies; that is, the evolution of the thermodynamics variables depends on the surface heat and moisture fluxes, cloud-top radiative cooling rate, temperature, and moisture jumps across the capping inversion. However, these equations are coupled to the dynamics equation through the entrainment rate at the top of the cloud deck. The entrainment rate is parameterized from results obtained in laboratory experiments and clearly shows the dependence on the velocity perturbation, which in turn strongly depends on the horizontal gradient of the thermodynamics variables. The derived entrainment rate is applicable when the thermal jump at cloud top is sufficiently weak and the velocity jump is on the order of the velocity perturbation.

Aside from some initial analyses of the main balances in steady-state solutions, the mathematical properties and physical characteristics of the system of equations will be explored in future papers.

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