Quasi-geostrophic Atmospheric Model Research Articles

Linear Inverse Modeling of Large-Scale Atmospheric Flow Using Optimal Mode Decomposition

Abstract Linear inverse modeling or principal oscillation pattern (POP) analysis is a widely applied tool in climate science for extracting from data dominant spatial patterns together with their dynamics as approximated by a linear Markov model. The system is projected onto a principal linear subspace and the system matrix is estimated from data. The eigenmodes of the system matrix are the POPs, with the eigenvalues providing their decay time scales and oscillation frequencies. Usually, the subspace is spanned by the leading principal components (PCs) and empirical orthogonal functions (EOFs). Outside of climate science, this procedure is now more commonly referred to as dynamic mode decomposition (DMD). Here, we use optimal mode decomposition (OMD) to address the full linear inverse modeling problem of simultaneous optimization of the principal subspace and the linear operator. The method is illustrated on two pedagogical examples and then applied to a three-level quasigeostrophic atmospheric model with realistic mean state and variability. The OMD models significantly outperform the EOF/DMD models in predicting the time evolution of the large-scale flow modes. The advantage of the OMD models stems from finding more persistent modes as well as from better capturing the nonnormality of the linear operator and the associated nonmodal growth. The dynamics of the large-scale flow modes turn out to be markedly non-Markovian and the OMD modes are superior to the EOF/DMD modes also in a modeling setting with a higher-order vector autoregressive process. The OMD modes could also be used as basis functions for a nonlinear dynamical model although they are not optimized for that purpose. Potential applications of the OMD method in weather and climate science include ENSO or MJO prediction, reduced-rank data assimilation, and generation of initial perturbations for ensemble prediction.

Fluctuations of finite-time Lyapunov exponents in an intermediate-complexity atmospheric model: a multivariate and large-deviation perspective

Abstract. The stability properties as characterized by the fluctuations of finite-time Lyapunov exponents around their mean values are investigated in a three-level quasi-geostrophic atmospheric model with realistic mean state and variability. Firstly, the covariance structure of the fluctuation field is examined. In order to identify dominant patterns of collective excitation, an empirical orthogonal function (EOF) analysis of the fluctuation field of all of the finite-time Lyapunov exponents is performed. The three leading modes are patterns where the most unstable Lyapunov exponents fluctuate in phase. These modes are virtually independent of the integration time of the finite-time Lyapunov exponents. Secondly, large-deviation rate functions are estimated from time series of finite-time Lyapunov exponents based on the probability density functions and using the Legendre transform method. Serial correlation in the time series is properly accounted for. A large-deviation principle can be established for all of the Lyapunov exponents. Convergence is rather slow for the most unstable exponent, becomes faster when going further down in the Lyapunov spectrum, is very fast for the near-neutral and weakly dissipative modes, and becomes slow again for the strongly dissipative modes at the end of the Lyapunov spectrum. The curvature of the rate functions at the minimum is linked to the corresponding elements of the diffusion matrix. Also, the joint large-deviation rate function for the first and the second Lyapunov exponent is estimated.

Convergence of Extreme Value Statistics in a Two-Layer Quasi-Geostrophic Atmospheric Model

We search for the signature of universal properties of extreme events, theoretically predicted for Axiom A flows, in a chaotic and high-dimensional dynamical system. We study the convergence of GEV (Generalized Extreme Value) and GP (Generalized Pareto) shape parameter estimates to the theoretical value, which is expressed in terms of the partial information dimensions of the attractor. We consider a two-layer quasi-geostrophic atmospheric model of the mid-latitudes, adopt two levels of forcing, and analyse the extremes of different types of physical observables (local energy, zonally averaged energy, and globally averaged energy). We find good agreement in the shape parameter estimates with the theory only in the case of more intense forcing, corresponding to a strong chaotic behaviour, for some observables (the local energy at every latitude). Due to the limited (though very large) data size and to the presence of serial correlations, it is difficult to obtain robust statistics of extremes in the case of the other observables. In the case of weak forcing, which leads to weaker chaotic conditions with regime behaviour, we find, unsurprisingly, worse agreement with the theory developed for Axiom A flows.

Stationary Atmospheric Responses to an Idealized Sea Surface Temperature Anomaly in the Southern Ocean

Abstract The stationary atmospheric response to an idealized sea surface temperature anomaly (SSTA) is studied with a quasigeostrophic atmospheric model of the Southern Hemisphere. Sensitivity of the stationary response to the midlatitude SSTA location is determined and responses are decomposed on vertical modes. The SSTA almost directly forces baroclinic responses, inducing warm-air anomalies 40°–50° downstream, eastward, to the SSTA. These baroclinic responses arise from an equilibrium between the SSTA-induced anomalous vortex stretching and (i) advection by the quasi-stationary flow and (ii) dissipation by high-frequency eddies. The barotropic response consists of a midlatitude ridge (trough) and a South Pole trough (ridge) for SSTAs localized from the Drake Passage to the western Indian Ocean (from south of Australia to the center of the Pacific Ocean). This response can be further decomposed into (i) a zonally asymmetric component, a quasi-stationary wave train forced by a barotropic ridge downstream of the SSTA; and (ii) a zonal-mean component similar to a meridional shift of westerlies and hence a southern annular mode (SAM)-like pattern. The former component is phase locked with the SSTA position, while the latter has a phase that depends on the relative SSTA position with regard to the background quasi-stationary wave pattern. The study shows that the barotropic downstream ridge response is responsible for modifying the low-frequency eddy–mean flow interactions through relative vorticity fluxes and inducing the bipolar projection of the zonal-mean response.

On Some Aspects of the Definition of Initial Conditions for Ensemble Prediction

Abstract Four methods for initialization of ensemble forecasts are systematically compared, namely the methods of singular vectors (SV) and bred modes (BM), as well as the ensemble Kalman filter (EnKF) and the ensemble transform Kalman filter (ETKF). The comparison is done on synthetic data with two models of the flow, namely, a low-order model introduced by Lorenz and a three-level quasigeostrophic atmospheric model. For the latter, both cases of a perfect and an imperfect model are considered. The performance of the various initialization methods is assessed in terms of the statistical reliability and resolution of the ensuing predictions. The relative performance of the four methods, which is statistically significant to a range of about 6 days, is in the order EnKF > ETKF > BM > SV. The difference between the former two methods and the latter two is on the whole more significant than the differences between EnKF and ETKF, or between BM and SV separately. The general conclusion is that, if the quality of ensemble predictions is assessed by the degree to which the predicted ensembles statistically sample the uncertainty on the future state of the flow, the best initial ensembles are those that best statistically sample the uncertainty on the present state of the flow.

Adaptive observations and assimilation in the unstable subspace by breeding on the data-assimilation system

ABSTRACTResults of targeting and assimilation experiments in a quasi-geostrophic atmospheric model are presented and discussed. The basic idea is to assimilate observations in the unstable subspace (AUS) of the data-assimilation system. The unstable subspace is estimated by breeding on the data-assimilation system (BDAS). The analysis update has the same local structure as the observationally forced bred modes.Use of adaptive observations, taken at locations where bred vectors have maximum amplitude, enhances the efficiency of the procedure and allows the use of a very limited number of observations and modes. The performance of the targeting and assimilation design is tested in an idealized context, under perfect model conditions. It is shown that the process of driving the control solution toward the true trajectory accomplished by the assimilation reduces the number and growth of unstable modes. By observing and assimilating the unstable structures it is then possible to stabilize the assimilation system so that few observations are sufficient to keep the analysis error within very low bounds, even in the presence of observational noise.In an idealized limited area model configuration the number and frequency of observations necessary to control the system is shown to be related to the properties of its unstable subspace.

The Role of Scales in the Dynamics of Parameterization Uncertainties

Abstract The dynamics of model error due to parameterization uncertainties are investigated in the context of two spatially distributed systems: the one-dimensional convection system known as the extended Kuramoto–Sivashinsky equation and a quasigeostrophic atmospheric model. In addition to the different phases of error growth already reported for low-order systems, unexpected behaviors associated with the spectral characteristics of the model perturbation sources have been brought out. Notably, the predictability of the system is less affected by model uncertainties acting at small scales than at larger ones. An interpretation in terms of the spectral properties of the Lyapunov vectors is advanced.

Using Neutral Singular Vectors to Study Low-Frequency Atmospheric Variability

Abstract The authors explore the use of the “neutral vectors” of a linearized version of a global quasigeostrophic atmospheric model with realistic mean flow in the study of the nonlinear model's low-frequency variability. Neutral vectors are the (right) singular vectors of the linearized model's tendency matrix that have the smallest eigenvalues; they are also the patterns that exhibit the largest response to forcing perturbations in the linear model. A striking similarity is found between neutral vectors and the dominant patterns of variability (EOFs) observed in both the full nonlinear model and in the real world. The authors discuss the physical and mathematical connection between neutral vectors and EOFs. Investigation of the “optimal forcing patterns”—the left singular vectors—proves to be less fruitful. The neutral modes have equivalent barotropic vertical structure, but their optimal forcing patterns are baroclinic and seem to be associated with low-level heating. But the horizontal patterns of th...

A mechanism of decadal variability of the sea-ice volume in the Northern Hemisphere

A long-term simulation performed with a coarse-resolution, global, atmosphere-ocean-sea-ice model displays strong decadal variability of the sea-ice volume in the Northern Hemisphere with a significant peak at about 15-18 years. This model results from the coupling of ECBILT, a spectral T21, 3-level quasi-geostrophic atmospheric model, and CLIO, a sea-ice-ocean general circulation model. First, the mechanism underlying the variability of ice volume in the model was studied by performing correlation analyses between the simulated variables. In a second step, a series of additional sensitivity experiments was performed in order to illustrate the role of specific physical processes. This as allowed us to identify a feedback loop in the ice-ocean system, which proceeds as follows: an increase in At tic sea-ice volume induces an increase in the salinity there. This salinity anomaly is transported to the Greenland Sea where it promotes convective activity. This warms up the surface oceanic layer and the atmosphere in winter and induces a decrease of the ice volume, completing half a cycle. The changes in ice volume are driven by a geopotential height pattern characterised by centres of action of opposite signs over Greenland and the Barents-Kara-Central Arctic area. Thermodynamic feedback between the ice and the atmosphere appear also to be very important for the persistence of the oscillation. The dynamical response of the atmosphere to sea-ice and temperature anomalies at surface plays a smaller role.

Climate model attractors: chaos, quasi-regularity and sensitivity to small perturbations of external forcing

Abstract. In this paper we discuss some theoretical results obtained for climate models (theorems for the existence of global attractors and inertial manifolds, estimates of attractor dimension and Lyapunov exponents, symmetry property of Lyapunov spectrum). We define the conditions for "quasi-regular behaviour" of a climate system. Under these conditions, the system behaviour is subject to the Kraichnan fluctuation-dissipation relation. This fact allows us to solve the problem of determining a system's sensitivity to small perturbations to an external forcing. The applicability of the above approach to the analysis of the climate system sensitivity is verified numerically with the example of the two-layer quasi-geostrophic atmospheric model.

Seasonal Predictability in a Model Atmosphere

The predictability of atmospheric mean-seasonal conditions in the absence of externally varying forcing is examined. A perfect-model approach is adopted, in which a global T21 three-level quasigeostrophic atmospheric model is integrated over 21 000 days to obtain a reference atmospheric orbit. The model is driven by a time-independent forcing, so that the only source of time variability is the internal dynamics. The forcing is set to perpetual winter conditions in the Northern Hemisphere (NH) and perpetual summer in the Southern Hemisphere. A significant temporal variability in the NH 90-day mean states is observed. The component of that variability associated with the higher-frequency motions, or climate noise, is estimated using a method developed by Madden. In the polar region, and to a lesser extent in the midlatitudes, the temporal variance of the winter means is significantly greater than the climate noise, suggesting some potential predictability in those regions. Forecast experiments are performed to see whether the presence of variance in the 90-day mean states that is in excess of the climate noise leads to some skill in the prediction of these states. Ensemble forecast experiments with nine members starting from slightly different initial conditions are performed for 200 different 90-day means along the reference atmospheric orbit. The serial correlation between the ensemble means and the reference orbit shows that there is skill in the 90-day mean predictions. The skill is concentrated in those regions of the NH that have the largest variance in excess of the climate noise. An EOF analysis shows that nearly all the predictive skill in the seasonal means is associated with one mode of variability with a strong axisymmetric component.

The Quality of Skill Forecasts for a Low-Order Spectral Model

Abstract A skill forecast gives the probability distribution for the error in the forecast. The purpose of this paper is to develop a skill-forecasting method. The method is applied to a spectral two-layer quasigeostrophic atmospheric model with a triangular truncation at wavenumber 5. The analysis is restricted to internal error growth. It is investigated how observational errors lead to errors in the analysis. It appears that climatological distributions can be used for the errors in the analysis. In the forecast run the evolution of these distributions is computed. For that purpose the tangent-linear equations for the errors are used. Because of this linearization, the results are valid for short-range skill forecasts only. The Lanczos algorithm is used to find the structures that dominate the forecast error. This algorithm is intended to be applicable in a realistic model.

Numerical integration of a quasi-geostrophic atmospheric model with an asymmetric zonal current

A two-level quasi-geostrophic model on a β-plane, with an initial state characterized by an asymmetric zonal current was integrated numerically for a period of 14 days. Various field variables were computed and compared with the corresponding distributions obtained in two experiments with the same model and with initial states characterized by a constant zonal flow and by a flow varying sinusoidally with latitude.