Ulrich Achatz

A Two-Layer Model with Empirical Linear Corrections and Reduced Order for Studies of Internal Climate Variability

Abstract This work discusses the formulation and testing of a simplified model of atmospheric dynamics. The model, which has only 200- and 700-mb streamfunctions as its prognostic fields, is designed to have a climate that approximates that of a comprehensive perpetual January general circulation model. Its governing equations are based on a Lorenz-type filtered two-layer model, but its linear terms are replaced by an empirically determined operator; the simplified model is semiempirical. Its basis consists of three-dimensional empirical orthogonal functions that are calculated using a total energy metric. The linear operator is intended to serve as a parameterization of fields, patterns, and dynamics not explicitly represented in the model. The operator is found through an optimization procedure that ensures that the semiempirical model optimally predicts streamfunction tendencies observed to occur in an extended control integration of the general circulation model. It turns out that a model determined i...

Regime of Validity of Soundproof Atmospheric Flow Models

Ogura and Phillips derived the original anelastic model through systematic formal asymptotics using the flow Mach number as the expansion parameter. To arrive at a reduced model that would simultaneously represent internal gravity waves and the effects of advection on the same time scale, they had to adopt a distinguished limit requiring that the dimensionless stability of the background state be on the order of the Mach number squared. For typical flow Mach numbers of M ; 1/30, this amounts to total variations of potential temperature across the troposphere of less than one Kelvin (i.e., to unrealistically weak stratification). Various generalizations of the original anelastic model have been proposed to remedy this issue. Later, Durran proposed the pseudoincompressible model following the same goals, but via a somewhat different route of argumentation. The present paper provides a scale analysis showing that the regime of validity of two of these extended models covers stratification strengths on the order of (hsc/u)du/dz , M 2/3 , which corresponds to realistic variations of potential temperature u across the pressure scale height hsc of Duj h sc 0 ,30K. Specifically, it is shown that (i) for (hsc/u)du/dz , M m with 0 , m , 2, the atmosphere features three asymptotically distinct time scales, namely, those of advection, internal gravity waves, and sound waves; (ii) within this range of stratifications, the structures and frequencies of the linearized internal wave modes of the compressible, anelastic, and pseudoincompressible models agree up to the order of M m ; and (iii) if m , 2 /3, the accumulated phase differences of internal waves remain asymptotically small even over the long advective time scale. The argument is completed by observing that the three models agree with respect to the advective nonlinearities and that all other nonlinear terms are of higher order in M.

Stochastic parameterization: Towards a new view of weather and climate models

AbstractThe last decade has seen the success of stochastic parameterizations in short-term, medium-range, and seasonal forecasts: operational weather centers now routinely use stochastic parameterization schemes to represent model inadequacy better and to improve the quantification of forecast uncertainty. Developed initially for numerical weather prediction, the inclusion of stochastic parameterizations not only provides better estimates of uncertainty, but it is also extremely promising for reducing long-standing climate biases and is relevant for determining the climate response to external forcing. This article highlights recent developments from different research groups that show that the stochastic representation of unresolved processes in the atmosphere, oceans, land surface, and cryosphere of comprehensive weather and climate models 1) gives rise to more reliable probabilistic forecasts of weather and climate and 2) reduces systematic model bias. We make a case that the use of mathematically stri...

The Primary Nonlinear Dynamics of Modal and Nonmodal Perturbations of Monochromatic Inertia–Gravity Waves

Abstract The breaking of an inertia–gravity wave (IGW), initiated by its leading normal modes (NMs) or singular vectors (SVs), and the resulting small-scale eddies are investigated by means of direct numerical simulations of a Boussinesq fluid characterizing the upper mesosphere. The focus is on the primary nonlinear dynamics, neglecting the effect of secondary instabilities. It is found that the structures with the strongest impact on the IGW and also the largest turbulence amplitudes are the NM (for a statically unstable IGW) or short-term SV (statically and dynamically stable IGW) propagating horizontally transversely with respect to the IGW, possibly in agreement with observations of airglow ripples in conjunction with statically unstable IGWs. In both cases these leading structures reduce the IGW amplitude well below the static and dynamic instability thresholds. The resulting turbulent dissipation rates are within the range of available estimates from rocket soundings, even for IGWs at amplitudes lo...

Primitive-Equation-Based Low-Order Models with Seasonal Cycle. Part I: Model Construction

Abstract In a continuation of previous investigations on deterministic reduced atmosphere models with compact state space representation, two main modifications are introduced. First, primitive equation dynamics is used to describe the nonlinear interactions between resolved scales. Second, the seasonal cycle in its main aspects is incorporated. Stability considerations lead to a gridpoint formulation of the basic equations in the dynamical core. A total energy metric consistent with the equations can be derived, provided surface pressure is treated as constant in time. Using this metric, a reduction in the number of degrees of freedom is achieved by a projection onto three-dimensional empirical orthogonal functions (EOFs), each of them encompassing simultaneously all prognostic variables (winds and temperature). The impact of unresolved scales and not explicitly described physical processes is incorporated via an empirical linear parameterization. The basis patterns having been determined from 3 sigma le...

Shear and Static Instability of Inertia–Gravity Wave Packets: Short-Term Modal and Nonmodal Growth

Abstract The problem of nonmodal instabilities of inertia–gravity waves (IGW) in the middle atmosphere is addressed, within the framework of a Boussinesq model with realistic molecular viscosity and thermal diffusion, by singular-vector analysis of horizontally homogeneous vertical profiles of wind and buoyancy obtained from IGW packets at their statically least stable or most unstable horizontal location. Nonmodal growth is always found to be significantly stronger than that of normal modes, most notably at wave amplitudes below the static instability limit where normal-mode instability is very weak, whereas the energy gain between the optimal perturbation and singular vector after one Brunt–Vaisala period can be as large as two orders of magnitude. Among a multitude of rapidly growing singular vectors for this optimization time, small-scale (wavelengths of a few 100 m) perturbations propagating in the horizontal parallel to the IGW are most prominent. These parallel optimal perturbations are amplified b...

Optimal Growth in Inertia–Gravity Wave Packets: Energetics, Long-Term Development, and Three-Dimensional Structure

Abstract Using a hierarchy of three models of increasing realism and complexity, and expanding on a previous study, optimal perturbations of inertia–gravity wave (IGW) packets are studied with respect to several aspects. It is shown that normal modes are comparatively less able to extract energy from the IGW over finite time due to their time-invariant structure, while singular vectors (SVs) can adjust their dynamical fields flexibly so as to optimize the statically enhanced roll and Orr mechanisms by which they grow. On longer time scales, where the time dependence of the IGW packet precludes a normal-mode analysis, optimal growth is found to further amplify suitable perturbations. The propagation characteristics of these exhibit critical layer interactions for horizontal propagation directions transverse with respect to the IGW, preventing significant vertical propagation, while parallel and obliquely propagating perturbations of sufficiently long horizontal scales are found to radiate gravity waves int...

Fluctuation–Dissipation Supplemented by Nonlinearity: A Climate-Dependent Subgrid-Scale Parameterization in Low-Order Climate Models

AbstractClimate-system models use a multitude of parameterization schemes for small-scale processes. These should respond to externally forced climate variability in an appropriate manner so as to reflect the response of the parameterized process to a changing climate. The most attractive route to achieve such a behavior would certainly be provided by theoretical understanding sufficiently deep to enable the a priori design of climate-sensitive parameterization schemes. An alternative path might, however, be helpful when the parameter tuning involved in the development of a scheme is objective enough so that these parameters can be described as functions of the statistics of the climate system. Provided that the dynamics of the process in question is sufficiently stochastic, and that the external forcing is not too strong, the fluctuation–dissipation theorem (FDT) might be a tool to predict from the statistics of a system (e.g., the atmosphere) how an objectively tuned parameterization should respond to e...

Modal and Nonmodal Perturbations of Monochromatic High-Frequency Gravity Waves: Primary Nonlinear Dynamics

Abstract The primary nonlinear dynamics of high-frequency gravity waves (HGWs) perturbed by their most prominent normal modes (NMs) or singular vectors (SVs) in a rotating Boussinesq fluid have been studied by direct numerical simulations (DNSs), with wave scales and values of viscosity and diffusivity characteristic for the upper mesosphere. The DNS is 2.5D in that it has only two spatial dimensions, defined by the direction of propagation of the HGW and the direction of propagation of the perturbation in the plane orthogonal to the HGW phase direction, but describes a fully 3D velocity field. Many results of the more comprehensive fully 3D simulations in the literature are reproduced. So it is found that statically unstable HGWs are subject to wave breaking ending in a wave amplitude with respect to the overturning threshold near 0.3. It is shown that this is a result of a perturbation of the HGW by its leading transverse NM. For statically stable HGWs, a parallel NM has the strongest effect, quite in l...

Parameterization of stochastic multiscale triads

We discuss applications of a recently developed method for model reduction based on linear response theory of weakly coupled dynamical systems. We apply the weak coupling method to simple stochastic differential equations with slow and fast degrees of freedom. The weak coupling model reduction method results in general in a non-Markovian system; we therefore discuss the Markovianization of the system to allow for straightforward numerical integration. We compare the applied method to the equations obtained through homogenization in the limit of large timescale separation between slow and fast degrees of freedom. We numerically compare the ensemble spread from a fixed initial condition, correlation functions and exit times from a domain. The weak coupling method gives more accurate results in all test cases, albeit with a higher numerical cost.