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Roberto Buizza

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Roberto Buizza

Abstract

The (linear) time evolution of singular vectors computed with a primitive equation model following a 36-h evolving trajectory is analyzed at horizontal triangular spectral truncations T21, T42, and T63.

First, for each resolution, the impact of horizontal diffusion on the singular vectors characteristics (amplification factors, total energy spectra) is analyzed. Forecast error and singular vectors computed with different horizontal diffusion damping times are compared to assess whether, at each resolution, forecast error projection onto the first 10 most unstable singular vectors is maximized for specific values. Results suggest that better projections are obtained with horizontal diffusion damping times on the smallest scale (on divergence) of 3 h at T42 and T63 resolution, and of 12 h at T21.

Then amplification factors, geographical locations, total energy vertical distributions, and spectra of T21, T42, and T63 singular vectors computed, respectively, with 12-, 3-, and 3-h damping time on the smallest scale are analyzed. The ratio among the singular vector amplification factors at T21:T42:T63 resolution is shown to be approximately 1:1.5:2.5. The geographical location and the total energy vertical distribution of T21, T42, and T63 singular vectors are quite similar. By contrast, total energy spectra differ substantially. Forecast error projection onto singular vectors is shown to be slightly larger if higher-resolution singular vectors are used. It is argued that the impact of horizontal resolution on the forecast error projection is marginal because of the lack of physical processes in the forward and adjoint tangent model versions. Moreover, the fact that forecast error projections onto the leading 10 singular vectors are rather small could be seen as an indication that more singular vectors are needed to capture the growing components of forecast error.

Finally, singular vectors and forecast errors are compared to quantify the relevance of the singular vectors of day d to capture the growing features of the error of the forecast started on day d. Results indicate that forecast error projection onto the leading 10 singular vectors decreases if singular vectors of a wrong date are used.

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Roberto Buizza

Abstract

The influence of topography on fluid instability has been studied in literature both in the beta-channel approximation and on the sphere mainly using normal modes. A different approach recently proposed is based on the identification of unstable singular vectors (i.e., structures that have the fastest growth over finite-time intervals). Systems characterized by neutral or damped normal modes have been shown to have singular vectors growing (e.g., in terms of kinetic energy) over finite-time intervals. Singular vectors do not conserve their shape during time evolution as normal modes do. Various aspects related to the identification of singular vectors of a barotropic flow are analyzed in this paper, with the final goal of studying the impact of the orography on these structures.

First, the author focuses on very idealized situations to verify if neutral and damped flows can sustain structures growing over finite-time intervals. Then, the author studies singular vectors of basic states defined as the super-position of a superrotation and a Rossby-Haurwitz wave forced by orographies that project onto one spectral component only or forced by very simple orographies with longitudinally or latitudinally elongated shapes. This first part shows that orography can alter the unstable subspace generated by the most unstable singular vectors, either directly through the action of the orographic term in the linear equation or indirectly by modifying the evolution of the basic state.

In the second part, the author considers a realistic basic state, defined as a mean winter flow computed from 3 months of observed vorticity field, forced by a realistic orography. It is shown that the orographic forcing can indirectly modify the singular vector structures. In fact, “orographically induced” instabilities can be identified only when considering time-evolving basic states.

These results show that unstable structures related to physical processes can be captured by the adjoint technique.

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Roberto Buizza and Franco Molteni

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Linear instability analysis is applied to study the role of barotropic dynamics in the evolution of blocking events during winter 1990/91. Finite-time interval instabilities (i.e., nonnormal-mode structures defined as the singular vectors of the tangent propagator) growing over periods of 4 days have been computed using adjoint methods. Correlation between large values of the singular vector amplification rate and the occurrence of blocking onset in the real atmosphere is studied.

A correspondence is found between periods with the largest singular vector amplification rates and periods either leading to blocking formation or covering the mature phase of blocks. It is shown that at final time the singular vectors tend to have largest amplitude in the same regions of planetary wave ridging where blocks develop. On average, singular vectors developing on the Pacific have larger growth rates than those in the Euro–Atlantic region.

The analysis of some case studies indicates a qualitative similarity between observed tendencies and their projections onto the five leading singular vectors, although correlation coefficients between actual and projected fields are small. The cases with largest tendency correlation are associated with the formation of blocking dipoles from preexisting planetary-scale ridges of larger meridional scale. Overall, our results indicate that barotropic instability is mostly driven by planetary wave amplification rather than being the cause of it, and mainly contributes to a rather mature stage of blocking development.

Energetics of barotropic perturbations indicate that dipole structures similar to blocking patterns can efficiently gain energy from the planetary-scale flow provided that the longitudinal gradient of the basic-state zonal wind ub in the jet exit has a comparable magnitude to the meridional gradient of ub near the jet core. It is shown that an anomaly reinforcing the basic-state ridge on the eastern side of the Pacific and/or Atlantic Ocean (therefore increasing the magnitude of the longitudinal wind gradient) is necessary for a dipole structure to emerge as the fastest growing perturbation.

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Isla Gilmour, Leonard A. Smith, and Roberto Buizza

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Day-to-day variations in the growth of uncertainty in the current state of the atmosphere have led to operational ensemble weather predictions in which an ensemble of different initial conditions, each perturbed from the best estimate of the current state and yet still consistent with the observations, is forecast. Contrasting competing methods for the selection of ensemble members is a subject of active research; the assumption that the ensemble members represent sufficiently small perturbations so as to evolve within the “linear regime” is implicit to several of these methods. This regime, in which the model dynamics are well represented by a linear approximation, is commonly held to extend to 2 or 3 days for operational forecasts. It is shown that this is rarely the case. A new measure, the relative nonlinearity, which quantifies the duration of the linear regime by monitoring the evolution of “twin” pairs of ensemble members, is introduced. Both European and American ensemble prediction systems are examined; in the cases considered for each system (87 and 25, respectively), the duration of the linear regime is often less than a day and never extends to 2 days. The internal consistency of operational ensemble formation schemes is discussed in light of these results. By decreasing the optimization time, a modified singular vector–based formation scheme is shown to improve consistency while maintaining traditional skill and spread scores in the seven cases considered. The relevance of the linear regime to issues regarding data assimilation, adaptive observations, and model sensitivity is also noted.

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Nedjeljka Žagar, Roberto Buizza, and Joseph Tribbia

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A new methodology for the analysis of ensemble prediction systems (ENSs) is presented and applied to 1 month (December 2014) of ECMWF operational ensemble forecasts. The method relies on the decomposition of the global three-dimensional wind and geopotential fields onto the normal-mode functions. The ensemble properties are quantified in terms of the 50-member ensemble spread associated with the balanced and inertio-gravity (IG) modes for forecast ranges every 12 h up to 7 days. Ensemble reliability is defined for the balanced and IG modes comparing the ensemble spread with the control analysis in each scale.

Modal analysis shows that initial uncertainties in the ECMWF ENS are largest in the tropical large-scale modes and their spatial distribution is similar to the distribution of the short-range forecast errors. Initially the ensemble spread grows most in the smallest scales and in the synoptic range of the IG modes but the overall growth is dominated by the increase of spread in balanced modes in synoptic and planetary scales in the midlatitudes. During the forecasts, the distribution of spread in the balanced and IG modes grows toward the climatological spread distribution characteristic of the analyses. In the 2-day forecast range, the global IG spread reaches 60% of its asymptotic value while the same percentage of the global balanced spread is reached after 5 days of forecasts. An underdispersiveness of the system is suggested to be associated with the lack of tropical variability, primarily the Kelvin waves.

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Y. Qiang Sun, Fuqing Zhang, Linus Magnusson, Roberto Buizza, Jan-Huey Chen, and Kerry Emanuel

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In their comment, Žagar and Szunyogh raised concerns about a recent study by Zhang et al. that examined the predictability limit of midlatitude weather using two up-to-date global models. Zhang et al. showed that deterministic weather forecast may, at best, be extended by 5 days, assuming we could achieve minimal initial-condition uncertainty (e.g., 10% of current operational value) with a nearly perfect model. Žagar and Szunyogh questioned the methodology and the experiments of Zhang et al. Specifically, Žagar and Szunyogh raised issues regarding the effects of model error on the growth of the forecast uncertainty. They also suggested that estimates of the predictability limit could be obtained using a simple parametric model. This reply clarifies the misunderstandings in Žagar and Szunyogh and demonstrates that experiments conducted by Zhang et al. are reasonable. In our view, the model error concern in Žagar and Szunyogh does not apply to the intrinsic predictability limit, which is the key focus of Zhang et al. and the simple parametric model described in Žagar and Szunyogh does not serve the purpose of Zhang et al.

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Fuqing Zhang, Y. Qiang Sun, Linus Magnusson, Roberto Buizza, Shian-Jiann Lin, Jan-Huey Chen, and Kerry Emanuel

Abstract

Understanding the predictability limit of day-to-day weather phenomena such as midlatitude winter storms and summer monsoonal rainstorms is crucial to numerical weather prediction (NWP). This predictability limit is studied using unprecedented high-resolution global models with ensemble experiments of the European Centre for Medium-Range Weather Forecasts (ECMWF; 9-km operational model) and identical-twin experiments of the U.S. Next-Generation Global Prediction System (NGGPS; 3 km). Results suggest that the predictability limit for midlatitude weather may indeed exist and is intrinsic to the underlying dynamical system and instabilities even if the forecast model and the initial conditions are nearly perfect. Currently, a skillful forecast lead time of midlatitude instantaneous weather is around 10 days, which serves as the practical predictability limit. Reducing the current-day initial-condition uncertainty by an order of magnitude extends the deterministic forecast lead times of day-to-day weather by up to 5 days, with much less scope for improving prediction of small-scale phenomena like thunderstorms. Achieving this additional predictability limit can have enormous socioeconomic benefits but requires coordinated efforts by the entire community to design better numerical weather models, to improve observations, and to make better use of observations with advanced data assimilation and computing techniques.

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