ALBERT

Description: As present weather forecast codes and increasingly many atmospheric climate models resolve at least part of the mesoscale flow, and hence also internal gravity waves (GWs), it is natural to ask whether even in such configurations subgrid-scale GWs might impact the resolved flow and how their effect could be taken into account. This motivates a theoretical and numerical investigation of the interactions between unresolved submesoscale and resolved mesoscale GWs, using Boussinesq dynamics for simplicity. By scaling arguments, first a subset of submesoscale GWs that can indeed influence the dynamics of mesoscale GWs is identified. Therein, hydrostatic GWs with wavelengths corresponding to the largest unresolved scales of present-day limited-area weather forecast models are an interesting example. A large-amplitude WKB theory, allowing for a mesoscale unbalanced flow, is then formulated, based on multiscale asymptotic analysis utilizing a proper scale-separation parameter. Purely vertical propagation of submesoscale GWs is found to be most important, implying inter alia that the resolved flow is only affected by the vertical flux convergence of submesoscale horizontal momentum at leading order. In turn, submesoscale GWs are refracted by mesoscale vertical wind shear while conserving their wave-action density. An efficient numerical implementation of the theory uses a phase-space ray tracer, thus handling the frequent appearance of caustics. The WKB approach and its numerical implementation are validated successfully against submesoscale-resolving simulations of the resonant radiation of mesoscale inertia GWs by a horizontally as well as vertically confined submesoscale GW packet.

Description: 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.

Description: 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 “p–ρ 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.

Description: In this study the authors apply a recently developed clustering method for the systematic identification of metastable atmospheric regimes in high-dimensional datasets generated by atmospheric models. The novelty of this approach is that it decomposes the phase space in, possibly, overlapping clusters and simultaneously estimates the most likely switching sequence among the clusters. The parameters of the clustering and switching are estimated by a finite element approach. The switching among the clusters can be described by a Markov transition matrix. Possible metastable regime behavior is assessed by inspecting the eigenspectrum of the associated transition probability matrix. The recently introduced metastable data-analysis method is applied to high-dimensional datasets produced by a barotropic model and a comprehensive atmospheric general circulation model (GCM). Significant and dynamically relevant metastable regimes are successfully identified in both models. The metastable regimes in the barotropic model correspond to blocked and zonal states. Similar regime states were already previously identified in highly reduced phase spaces of just one and two dimensions in the same model. Next, the clustering method is applied to a comprehensive atmospheric GCM in which seven significant flow regimes are identified. The spatial structures of the regimes correspond to, among others, both phases of the Northern Annular Mode and Pacific blocking. The regimes are maintained predominantly by transient eddy fluxes of low-pass-filtered anomalies. It is demonstrated how the dynamical description of the slow process switching between the regimes can be acquired from the analysis results, and an investigation of the resulting simplified dynamical model with respect to predictability is performed. A predictability study shows that a simple Markov model is able to predict the regimes up to six days ahead, comparable to the ability of high-resolution state-of-the-art numerical weather prediction models to accurately predict the onset and decay of blockings. The implications of the results for derivation of reduced models for extended-range predictability are discussed.

Description: 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.

Description: 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.

Description: This paper presents an extension of the recently developed method for simultaneous dimension reduction and metastability analysis of high-dimensional time series. The modified approach is based on a combination of ensembles of hidden Markov models (HMMs) with state-specific principal component analysis (PCA) in extended space (guaranteeing that the overall dynamics will be Markovian). The main advantage of the modified method is its ability to deal with the gaps in the high-dimensional observation data. The proposed method allows for (i) the separation of the data according to the metastable states, (ii) a hierarchical decomposition of these sets into metastable substates, and (iii) calculation of the state-specific extended empirical orthogonal functions simultaneously with identification of the underlying Markovian dynamics switching between those metastable substates. The authors discuss the introduced model assumptions, explain how the quality of the resulting reduced representation can be assessed, and show what kind of additional insight into the underlying dynamics such a reduced Markovian representation can give (e.g., in the form of transition probabilities, statistical weights, mean first exit times, and mean first passage times). The performance of the new method analyzing 500-hPa geopotential height fields [daily mean values from the 40-yr ECMWF Re-Analysis (ERA-40) dataset for a period of 44 winters] is demonstrated and the results are compared with information gained from a numerically expensive but assumption-free method (Wavelets–PCA), and the identified metastable states are interpreted w.r.t. the blocking events in the atmosphere.

Description: 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.