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T. N. Palmer

Abstract

Properties of the quasi-geostrophic Eliassen-Palm (EP) flux for planetary scale motions are discussed, in order to clarify how these properties generalize from their beta-plane counterparts when no restriction on the variation of the Coriolis parameter is imposed. These properties include the relationships between the divergence of the EP flux and the meridional flux of potential vorticity, and between the EP flux, group velocity and refractive index.

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T. N. Palmer

Abstract

The intense wavenumber-2 stratospheric warming of February 1979 is analyzed in a transformed Eulerian-mean formalism, and compared with diagnostics generated by the model warming of Dunkerton et al. (1981). Significant differences in the evolution of the zonal mean flow are found. The corresponding differences in wave, mean-flow interaction are examined by studying planetary wave activity in the troposphere and stratosphere, as measured by the Eliassen-Palm flux and its divergence. It is found that in the stratosphere, the direction of this flux changes several times during the warming. Zonal flow deceleration is most intense when the midlatitude stratospheric flux has positive poleward and upward components. Conversely, deceleration is smallest when the flux is directed equatorward. Some mechanisms that may account for this switching are discussed. However, unlike the model, the high-latitude zonal flow reversal does not arise from nonlinear critical layer interaction with the waves.

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C. A. Reynolds
and
T. N. Palmer

Abstract

The full set of kinetic energy singular values and singular vectors for the forward tangent propagator of a quasigeostrophic potential vorticity model is examined. In contrast to the fastest growing singular vectors, the fastest decaying vectors exhibit a downward and downscale transfer of energy and an eastward tilt with height. The near-neutral singular vectors resemble small-scale noise with no localized structure or coherence between levels.

Post-time forecast and analysis correction techniques are examined as a function of the number of singular vectors included in the representation of the inverse of the forward tangent propagator. It is found that for the case when the forecast error is known exactly, the best corrections are obtained when using the full inverse, which includes all of the singular vectors. It is also found that the erroneous projection of the analysis uncertainty onto the fastest decaying singular vectors has a significant detrimental effect on the estimation of analysis error. Therefore, for the more realistic case where the forecast error is known imperfectly, use of the full inverse will result in an inaccurate estimate of analysis errors, and the best corrections are obtained when using an inverse composed only of the growing singular vectors. Running the tangent equations with a negative time step is a very good approximation to using the full inverse of the forward tangent propagator.

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R. Buizza
and
T. N. Palmer

Abstract

The local phase-space instability Of the atmospheric global circulation is Characterized by its (nonmodal) singular vectors. The formalism of singular vector analysis is described. The relations between singular vectors, normal modes, adjoint modes, Lyapunov vectors, perturbations produced by the so-called breeding method, and wave pseudomomentum are outlined. Techniques to estimate the dominant part of the singular spectrum using large-dimensional primitive equation models are discussed. These include the use of forward and adjoint tangent propagators with a Lanczos iterative algorithm. Results are described, based first on statistics of routine calculations made between December 1992 and August 1993, and second on three specific case studies.

Results define three dominant geographical areas of instability in the Northern Hemisphere: the two regions of storm track cyclogenesis, and the North African subtropical jet Singular vectors can amplify as much as tenfold over 36 hours, and in winter there are typically at least 35 independent singular vectors, which quadruple in amplitude over this timescale. Qualitatively, the distribution of singular vectors can be associated with a simple diagnostic of baroclinic instability from the basic-state flow. However, this relationship is not quantitatively reliable, as, for example, the chosen diagnostic takes no account of the horizontal or time-varying structure of the basic-state flow.

Three basic types of singular vector are identified The most important and most frequent is located in mid latitudes. At initial time, the singular vector is localized in the horizontal, with most amplitude in the lower troposphere. Energy growth can be interpreted qualitatively in terms of wave pseudomomentum propagation into the jet, resulting in peak amplitudes in the upper troposphere at optimization time. During evolution the dominant horizontal wavenumber of the singular vector decreases. Singular vector growth is therefore fundamentally nonmodal. Singular vectors 1ocalized first in the tropical upper troposphere. and second with equivalent barotropic structure in the high-latitude troposhpere, are also identified.

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T. N. Palmer
and
C-P. F. Hsu

Abstract

In a, series of idealized numerical experiments, Butchart et al (1982) have recently established that the configuration of the mean zonal wind occurring immediately before the wavenumber-2 major stratospheric warming of February 1979 was crucial in subsequently focusing upward propagating planetary wave-activity into the high latitude stratosphere. In this sense, it was concluded that the stratospheric circumpolar flow should evolve to some preconditioned state before a wavenumber-2 major warming could occur.

In the present paper, the mechanisms responsible for the transition of the circumpolar flow from its normal midwinter state to this preconditioned state are investigated through a combination of observational numerical and theoretical studies. For the 1978–79 winter, this transition occurred during the substantial wavenumber-1 minor warming of January 1979, and the characteristic structure associated with the preconditioned mean zonal flow was established four days after the peak of this warming, during a period of intense high latitude acceleration. This latter phenomenon is referred to as a stratospheric sudden cooling. Observations of Eliassen-Palm flux cross-sections indicate that while wave, zonal mean-flow interaction theory could account for the qualitative evolution of the circumpolar flow during the warming, substantial nonlinear wave interactions were active during the cooling period, and these interactions significantly influenced the evolution of the circumpolar flow.

In a series of numerical experiments using a truncated semi-spectral model, we show that this sudden cooling phenomenon can be realistically reproduced in an idealized integration in which wave-wave interactions are present. By contrast, we were unable to simulate this phenomenon with these interactions removed.

Two different mechanisms are put forward to account for these nonlinearities. One mechanism is that of wave-breaking and associated potential vorticity mixing, as suggested by McIntyre (1982). The second mechanism is based on the notion of wave-activity, forced in the troposphere, propagating relative to isopleths of potential vorticity of some zonally asymmetric basic state.

Results of the observational and numerical study suggest that the first mechanism was dominant, and that potential vorticity mixing in the outer regions of the polar vortex was central to the process of preconditioning. Nevertheless, we believe that the second mechanism plays an important role in the dynamics of the stratosphere.

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T. N. Palmer
,
R. Gelaro
,
J. Barkmeijer
, and
R. Buizza

Abstract

Singular vectors of the linearized equations of motion have been used to study the instability properties of the atmosphere–ocean system and its related predictability. A third use of these singular vectors is proposed here: as part of a strategy to target adaptive observations to “sensitive” parts of the atmosphere. Such observations could be made using unmanned aircraft, though calculations in this paper are motivated by the upstream component of the Fronts and Atlantic Storm-Track Experiment. Oceanic applications are also discussed. In defining this strategy, it is shown that there is, in principle, no freedom in the choice of inner product or metric for the singular vector calculation. However, the correct metric is dependent on the purpose for making the targeted observations (to study precursor developments or to improve forecast initial conditions). It is argued that for predictability studies, where both the dynamical instability properties of the system and the specification of the operational observing network and associated data assimilation system are important, the appropriate metric will differ from that appropriate to a pure geophysical fluid dynamics (GFD) problem. Based on two different sets of calculations, it is argued that for predictability studies (but not for GFD studies), a first-order approximation to the appropriate metric can be based on perturbation energy. The role of observations in data assimilation procedures (constraining large scales more than small scales) is fundamental in understanding reasons for the requirement for different metrics for the two classes of problems. An index-based tensor approach is used to make explicit the role of the metric.

The strategy for using singular vectors to target adaptive observations is discussed in the context of other possible approaches, specifically, based on breeding vectors, potential vorticity diagnosis, and sensitivity vectors. The basic premises underlying the use of breeding and singular vectors are discussed. A comparison of the growth rates of breeding and singular vectors is made using a T21 quasigeostrophic model.

Singular vectors and subjective potential vorticity (PV) diagnosis are compared for a particular case study. The areas of sensitivity indicated by the two methods only partially agree. Reasons for disagreement hinge around the fact that subjective PV diagnosis emphasizes Lagrangian advection, whereas singular vector analysis emphasizes wave propagation. For the latter, areas of sensitivity may be associated with regions of weak PV gradient, for example, mid to lower troposphere. Amplification of singular vectors propagating from regions of weak PV gradient to regions of strong PV gradient is discussed in terms of pseudomomentum conservation. Evidence is shown that analysis error may be as large in the lower midtroposphere as in the upper troposphere.

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R. Gelaro
,
R. Buizza
,
T. N. Palmer
, and
E. Klinker

Abstract

The sensitivity of forecast errors to initial conditions is used to examine the optimality of perturbations constructed from the singular vectors of the tangent propagator of the European Centre for Medium-Range Weather Forecasts model. Sensitivity and pseudo-inverse perturbations based on the 48-h forecast error are computed as explicit linear combinations of singular vectors optimizing total energy over the Northern Hemisphere. It is assumed that these perturbations are close to the optimal perturbation that can be constructed from a linear combination of these singular vectors. Optimality is measured primarily in terms of the medium-range forecast improvement obtained by adding the perturbations a posteriori to the initial conditions. Several issues are addressed in the context of these experiments, including the ability of singular vectors to describe forecast error growth beyond the optimization interval, the number of singular vectors required, and the implications of nonmodal error growth. Supporting evidence for the use of singular vectors based on a total energy metric for studying atmospheric predictability is also presented.

In general, less than 30 singular vectors capture a large fraction of the variance of the Northern Hemisphere sensitivity pattern obtained from a T63 adjoint model integration, especially in cases of low forecast skill. The sensitivity patterns for these cases tend to be highly localized with structures determined by the dominant singular vectors. Forecast experiments with these perturbations show significant improvements in skill in the medium range, indicating that singular vectors optimized for a short-range forecast continue to provide a useful description of error growth well beyond this time. The results suggest that ensemble perturbations based on 10–30 singular vectors should provide a reasonable description of the medium-range forecast uncertainty, although the inclusion of additional singular vectors is likely to be beneficial.

Nonmodality is a key consideration in the construction of optimal perturbations. There is virtually no projection between the contemporaneous unstable subspaces at the end of one forecast trajectory portion and the beginning of a second, consecutive portion. Sensitivity and ensemble perturbations constructed using the evolved singular vectors from a previous (day−2) forecast are suboptimal for the current (day+0) forecast initial conditions. It is argued that these results have implications for a range of issues in atmospheric predictability including ensemble weather prediction, data assimilation, and the development of adaptive observing techniques.

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H. M. Christensen
,
I. M. Moroz
, and
T. N. Palmer

Abstract

It is now acknowledged that representing model uncertainty in atmospheric simulators is essential for the production of reliable probabilistic forecasts, and a number of different techniques have been proposed for this purpose. This paper presents new perturbed parameter schemes for use in the European Centre for Medium-Range Weather Forecasts (ECMWF) convection scheme. Two types of scheme are developed and implemented. Both schemes represent the joint uncertainty in four of the parameters in the convection parameterization scheme, which was estimated using the Ensemble Prediction and Parameter Estimation System (EPPES). The first scheme developed is a fixed perturbed parameter scheme, where the values of uncertain parameters are varied between ensemble members, but held constant over the duration of the forecast. The second is a stochastically varying perturbed parameter scheme. The performance of these schemes was compared to the ECMWF operational stochastic scheme, stochastically perturbed parameterization tendencies (SPPT), and to a model that does not represent uncertainty in convection. The skill of probabilistic forecasts made using the different models was evaluated. While the perturbed parameter schemes improve on the stochastic parameterization in some regards, the SPPT scheme outperforms the perturbed parameter approaches when considering forecast variables that are particularly sensitive to convection. Overall, SPPT schemes are the most skillful representations of model uncertainty owing to convection parameterization.

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Peter A. G. Watson
,
H. M. Christensen
, and
T. N. Palmer

Abstract

Important questions concerning parameterization of tropical convection are how should subgrid-scale variability be represented and which large-scale variables should be used in the parameterizations? Here the statistics of observational data in Darwin, Australia, are compared with those of short-term forecasts of convection made by the European Centre for Medium-Range Weather Forecasts Integrated Forecast System. The forecasts use multiplicative-noise stochastic physics (MNSP) that has led to many improvements in weather forecast skill. However, doubts have recently been raised about whether MNSP is consistent with observations of tropical convection. It is shown that the model can reproduce the variability of convection intensity for a given large-scale state, both with and without MNSP. Therefore MNSP is not inconsistent with observations, and much of the modeled variability arises from nonlinearity of the deterministic part of the convection scheme. It is also shown that the model can reproduce the lack of correlation between convection intensity and large-scale CAPE and an entraining CAPE, even though the convection parameterization assumes that deep convection is more intense when the vertical temperature profile is more unstable, with entrainment taken into account. Relationships between convection and large-scale convective inhibition and vertical velocity are also correctly captured.

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J. Berner
,
G. J. Shutts
,
M. Leutbecher
, and
T. N. Palmer

Abstract

Understanding model error in state-of-the-art numerical weather prediction models and representing its impact on flow-dependent predictability remains a complex and mostly unsolved problem. Here, a spectral stochastic kinetic energy backscatter scheme is used to simulate upscale-propagating errors caused by unresolved subgrid-scale processes. For this purpose, stochastic streamfunction perturbations are generated by autoregressive processes in spectral space and injected into regions where numerical integration schemes and parameterizations in the model lead to excessive systematic kinetic energy loss. It is demonstrated how output from coarse-grained high-resolution models can be used to inform the parameters of such a scheme. The performance of the spectral backscatter scheme is evaluated in the ensemble prediction system of the European Centre for Medium-Range Weather Forecasts. Its implementation in conjunction with reduced initial perturbations results in a better spread–error relationship, more realistic kinetic-energy spectra, a better representation of forecast-error growth, improved flow-dependent predictability, improved rainfall forecasts, and better probabilistic skill. The improvement is most pronounced in the tropics and for large-anomaly events.

It is found that whereas a simplified scheme assuming a constant dissipation rate already has some positive impact, the best results are obtained for flow-dependent formulations of the unresolved processes.

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