Linear and Nonlinear Signatures in the Planetary Wave Dynamics of an AGCM: Probability Density Functions

2007 ◽  
Vol 64 (1) ◽  
pp. 117-136 ◽  
Author(s):  
Judith Berner ◽  
Grant Branstator
Keyword(s):  
Phase Space ◽  
Probability Density ◽  
General Circulation ◽  
Planetary Wave ◽  
Circulation Model ◽  
Gaussian Mixture ◽  
Empirical Orthogonal Functions ◽  
Planetary Waves ◽  
Linear And Nonlinear ◽  
Non Gaussian

Abstract To identify and quantify indications of linear and nonlinear planetary wave behavior and their impact on the distribution of atmospheric states, characteristics of a very long integration of an atmospheric general circulation model (GCM) in a four-dimensional phase space are examined. The phase space is defined by the leading four empirical orthogonal functions of 500-hPa geopotential heights. First it is established that nonlinear tendencies similar to those reported in an earlier study of the phase space behavior in this GCM have the potential to lead to non-Gaussian features in the probability density function (PDF) of planetary waves. Then using objective measures it is demonstrated that the model’s distribution of states has distinctive non-Gaussian features. These features are characterized in various subspaces of dimension as high as four. A key feature is the presence of three radial ridges of enhanced probability emanating from the mode, which is shifted away from the climatological mean. There is no evidence of multiple maxima in the full PDF, but the radial ridges lead to three distinct modes in the distribution of circulation patterns. It is demonstrated that these key aspects of non-Gaussianity are captured by a two-Gaussian mixture model fitted in four dimensions. The two circulation states at the centroids of the component Gaussians are very similar to those associated with two nonlinear features identified by Branstator and Berner in their analysis of the trajectories of the GCM. These two dynamical features are locally linear, so it is concluded that the behavior of planetary waves can be conceptualized as being approximately piecewise-linear, leading to a two-Gaussian mixture with three preferred patterns.

Linear and Nonlinear Signatures in the Planetary Wave Dynamics of an AGCM: Phase Space Tendencies

2005 ◽  
Vol 62 (6) ◽  
pp. 1792-1811 ◽  
Author(s):  
Grant Branstator ◽  
Judith Berner
Keyword(s):  
Phase Space ◽  
General Circulation ◽  
Planetary Wave ◽  
Gaussian White Noise ◽  
Circulation Model ◽  
Lag Time ◽  
Empirical Orthogonal Functions ◽  
Linear Component ◽  
Least Squares Fit ◽  
Linear And Nonlinear

Abstract To identify and quantify indications of linear and nonlinear planetary wave behavior, characteristics of a very long integration of an atmospheric general circulation model in a four-dimensional phase space are examined. The phase space is defined by the leading four empirical orthogonal functions of 500-hPa geopotential heights, and the primary investigated characteristic is the state dependence of mean phase space tendencies. Defining the linear component of planetary wave tendencies as that part which can be captured by a least squares fit linear operator driven by additive Gaussian white noise, the study finds that there are distinct linear and nonlinear signatures. These signatures are especially easy to see in plots of mean tendencies projected onto phase space planes. For some planes the mean tendencies are highly linear, while for others there are strong departures from linearity. The results of the analysis are found to depend strongly on the lag time used to estimate tendencies with the linear component monotonically increasing with lag time. This is shown to result from the ergodicity of the system. Using the theory of Markov models it is possible to remove the lag-dependent component of the tendencies from the results. When this is done the projected mean dynamics in some planes is found to be almost exclusively nonlinear, while in others it is nearly linear. In the four-dimensional space the linear component of the dynamics is largely a reflection of a westward propagating Northern Hemisphere pattern concentrated over the Pacific and North America. The nonlinear signature can be approximated by two linear functions, each operating in a different region of phase space. One region is centered around a Pacific blocking pattern while the other is centered on a state with enhanced zonal symmetry. It is concluded that reduced models of the planetary waves should strive to include these state-dependent dynamics.

The Relationship between Statistically Linear and Nonlinear Feedbacks and Zonal-Mean Flow Variability in an Idealized Climate Model

2009 ◽  
Vol 66 (2) ◽  
pp. 353-372 ◽  
Author(s):  
Sergey Kravtsov ◽  
John E. Ten Hoeve ◽  
Steven B. Feldstein ◽  
Sukyoung Lee ◽  
Seok-Woo Son
Keyword(s):  
Phase Space ◽  
General Circulation ◽  
Climate Model ◽  
Circulation Model ◽  
Zonal Flow ◽  
Empirical Orthogonal Functions ◽  
Nonlinear Flow ◽  
Mean Flow ◽  
Orthogonal Functions ◽  
Flow Variability

Abstract Simulations using an idealized, atmospheric general circulation model (GCM) subjected to various thermal forcings are analyzed via a combination of probability density function (PDF) estimation and spectral analysis techniques. Seven different GCM runs are examined, each model run being characterized by different values in the strength of the tropical heating and high-latitude cooling. For each model run, it is shown that a linear stochastic model constructed in the phase space of the ten leading empirical orthogonal functions (EOFs) of the zonal-mean zonal flow provides an excellent statistical approximation to the simulated zonal flow variability, which includes zonal index fluctuations, and quasi-oscillatory, poleward, zonal-mean flow anomaly propagation. Statistically significant deviations from the above linear stochastic null hypothesis arise in the form of a few anomalously persistent, or statistically nonlinear, flow patterns, which occupy particular regions of the model’s phase space. Some of these nonlinear regimes occur during certain phases of the poleward propagation; however, such an association is, in general, weak. This indicates that the regimes and oscillations in the model may be governed by distinct dynamical mechanisms.

scholarly journals Tropospheric Planetary Wave Dynamics and Mixture Modeling: Two Preferred Regimes and a Regime Shift

2007 ◽  
Vol 64 (10) ◽  
pp. 3521-3541 ◽  
Author(s):  
A. Hannachi
Keyword(s):  
General Circulation ◽  
Regime Shift ◽  
Planetary Wave ◽  
Clustering Algorithm ◽  
Spatial Clustering ◽  
Gaussian Mixture Models ◽  
Circulation Model ◽  
Gaussian Mixture ◽  
Stochastic Dynamical Systems ◽  
Wave Dynamics

Abstract Investigation of preferred structures of planetary wave dynamics is addressed using multivariate Gaussian mixture models. The number of components in the mixture is obtained using order statistics of the mixing proportions, hence avoiding previous difficulties related to sample sizes and independence issues. The method is first applied to a few low-order stochastic dynamical systems and data from a general circulation model. The method is next applied to winter daily 500-hPa heights from 1949 to 2003 over the Northern Hemisphere. A spatial clustering algorithm is first applied to the leading two principal components (PCs) and shows significant clustering. The clustering is particularly robust for the first half of the record and less for the second half. The mixture model is then used to identify the clusters. Two highly significant extratropical planetary-scale preferred structures are obtained within the first two to four EOF state space. The first pattern shows a Pacific–North American (PNA) pattern and a negative North Atlantic Oscillation (NAO), and the second pattern is nearly opposite to the first one. It is also observed that some subspaces show multivariate Gaussianity, compatible with linearity, whereas others show multivariate non-Gaussianity. The same analysis is also applied to two subperiods, before and after 1978, and shows a similar regime behavior, with a slight stronger support for the first subperiod. In addition a significant regime shift is also observed between the two periods as well as a change in the shape of the distribution. The patterns associated with the regime shifts reflect essentially a PNA pattern and an NAO pattern consistent with the observed global warming effect on climate and the observed shift in sea surface temperature around the mid-1970s.

scholarly journals The Origin of Nonlinear Signatures of Planetary Wave Dynamics: Mean Phase Space Tendencies and Contributions from Non-Gaussianity

2007 ◽  
Vol 64 (11) ◽  
pp. 3987-4003 ◽  
Author(s):  
Christian Franzke ◽  
Andrew J. Majda ◽  
Grant Branstator
Keyword(s):  
Phase Space ◽  
General Circulation ◽  
General Framework ◽  
Planetary Wave ◽  
Low Frequency ◽  
Planetary Waves ◽  
Small Deviations ◽  
Wave Dynamics ◽  
Gaussian Case ◽  
The Impact

Abstract Mean phase space tendencies are investigated to systematically identify the origin of nonlinear signatures and the dynamical significance of small deviations from Gaussianity of planetary low-frequency waves. A general framework for the systematic investigation of mean phase space tendencies in complex geophysical systems is derived. In the special case of purely Gaussian statistics, this theory predicts that the interactions among the planetary waves themselves are the source of the nonlinear signatures in phase space, whereas the unresolved waves contribute only an amplitude-independent forcing, and cannot contribute to any nonlinear signature. The predictions of the general framework are studied for a simple stochastic climate model. This toy model has statistics that are very close to being Gaussian and a strong nonlinear signature in the form of a double swirl in the mean phase space tendencies of its low-frequency variables, much like recently identified signatures of nonlinear planetary wave dynamics in prototype and comprehensive atmospheric general circulation models (GCMs). As predicted by the general framework for the Gaussian case, the double swirl results from nonlinear interactions of the low-frequency variables. Mean phase space tendencies in a reduced space of a prototype atmospheric GCM are also investigated. Analysis of the dynamics producing nonlinear signatures in these mean tendencies shows a complex interplay between waves resolved in the subspace and unresolved waves. The interactions among the resolved planetary waves themselves do not produce the nonlinear signature. It is the interaction with the unresolved waves that is responsible for the nonlinear dynamics. Comparing this result with the predictions of the general framework for the Gaussian case shows that the impact of the unresolved waves is due to their small deviations from Gaussianity. This suggests that the observed deviations from Gaussianity, even though small, are dynamically relevant.

scholarly journals A Multivariate Empirical Orthogonal Function Method to Construct Nitrate Maps in the Southern Ocean

2018 ◽  
Vol 35 (7) ◽  
pp. 1505-1519 ◽  
Author(s):  
Yu-Chiao Liang ◽  
Matthew R. Mazloff ◽  
Isabella Rosso ◽  
Shih-Wei Fang ◽  
Jin-Yi Yu
Keyword(s):  
Southern Ocean ◽  
General Circulation ◽  
Model Simulation ◽  
Potential Temperature ◽  
Circulation Model ◽  
Empirical Orthogonal Functions ◽  
Cluster Method ◽  
Sampling Strategies ◽  
Data Sampling ◽  
Mean Square Errors

AbstractThe ability to construct nitrate maps in the Southern Ocean (SO) from sparse observations is important for marine biogeochemistry research, as it offers a geographical estimate of biological productivity. The goal of this study is to infer the skill of constructed SO nitrate maps using varying data sampling strategies. The mapping method uses multivariate empirical orthogonal functions (MEOFs) constructed from nitrate, salinity, and potential temperature (N-S-T) fields from a biogeochemical general circulation model simulation Synthetic N-S-T datasets are created by sampling modeled N-S-T fields in specific regions, determined either by random selection or by selecting regions over a certain threshold of nitrate temporal variances. The first 500 MEOF modes, determined by their capability to reconstruct the original N-S-T fields, are projected onto these synthetic N-S-T data to construct time-varying nitrate maps. Normalized root-mean-square errors (NRMSEs) are calculated between the constructed nitrate maps and the original modeled fields for different sampling strategies. The sampling strategy according to nitrate variances is shown to yield maps with lower NRMSEs than mapping adopting random sampling. A k-means cluster method that considers the N-S-T combined variances to identify key regions to insert data is most effective in reducing the mapping errors. These findings are further quantified by a series of mapping error analyses that also address the significance of data sampling density. The results provide a sampling framework to prioritize the deployment of biogeochemical Argo floats for constructing nitrate maps.

scholarly journals Effects of the planetary waves on the MLT airglow

2017 ◽  
Vol 35 (5) ◽  
pp. 1023-1032 ◽  
Author(s):  
Fabio Egito ◽  
Hisao Takahashi ◽  
Yasunobu Miyoshi
Keyword(s):  
General Circulation ◽  
Atomic Oxygen ◽  
Lower Thermosphere ◽  
Lower Boundary ◽  
Circulation Model ◽  
Vertical Transport ◽  
Transport Processes ◽  
Planetary Waves ◽  
Emission Rates ◽  
Middle Latitudes

Abstract. The planetary-wave-induced airglow variability in the mesosphere and lower thermosphere (MLT) is investigated using simulations with the general circulation model (GCM) of Kyushu University. The model capabilities enable us to simulate the MLT OI557.7 nm, O2b(0–1), and OH(6–2) emissions. The simulations were performed for the lower-boundary meteorological conditions of 2005. The spectral analysis reveals that at middle latitudes, oscillations of the emission rates with the period of 2–20 days appear throughout the year. The 2-day oscillations are prominent in the summer and the 5-, 10-, and 16-day oscillations dominate from the autumn to spring equinoxes. The maximal amplitude of the variations induced by the planetary waves was 34 % in OI557.7 nm, 17 % in O2b(0–1), and 8 % in OH(6–2). The results were compared to those observed in the middle latitudes. The GCM simulations also enabled us to investigate vertical transport processes and their effects on the emission layers. The vertical transport of atomic oxygen exhibits similar periodic variations to those observed in the emission layers induced by the planetary waves. The results also show that the vertical advection of atomic oxygen due to the wave motion is an important factor in the signatures of the planetary waves in the emission rates.

scholarly journals Linking Nonlinearity and Non-Gaussianity of Planetary Wave Behavior by the Fokker–Planck Equation

2005 ◽  
Vol 62 (7) ◽  
pp. 2098-2117 ◽  
Author(s):  
Judith Berner
Keyword(s):  
Stochastic Model ◽  
Probability Density ◽  
Planetary Wave ◽  
Multiplicative Noise ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
State Dependent ◽  
Nonlinear Stochastic Model ◽  
Non Gaussian ◽  
Fokker Planck

Abstract To link prominent nonlinearities in the dynamics of 500-hPa geopotential heights to non-Gaussian features in their probability density, a nonlinear stochastic model of atmospheric planetary wave behavior is developed. An analysis of geopotential heights generated by extended integrations of a GCM suggests that a stochastic model and its associated Fokker–Planck equation call for a nonlinear drift and multiplicative noise. All calculations are carried out in the reduced phase space spanned by the leading EOFs. It is demonstrated that this nonlinear stochastic model of planetary wave behavior captures the non-Gaussian features in the probability density function of atmospheric states to a remarkable degree. Moreover, it not only predicts global temporal characteristics, but also the nonlinear, state-dependent divergence of state trajectories. In the context of this empirical modeling, it is discussed on which time scale a stochastic model is expected to approximate the behavior of a continuous deterministic process. The reduced model is then used to determine the importance of the nonlinearities in the drift and the role of the multiplicative noise. While the nonlinearities in the drift are crucial for a good representation of planetary wave behavior, multiplicative (i.e., state dependent) noise is not absolutely essential. It is found that a major contributor to the stochastic component is the Branstator–Kushnir oscillation, which acts as a fluctuating force for physical processes with even longer time scales, like those that project on the Arctic Oscillation pattern. In this model, the oscillation is represented by strongly correlated noise.

scholarly journals Adaptive Weight Update Algorithm for Target Tracking of UUV Based on Improved Gaussian Mixture Cubature Kalman Filter

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Hongjian Wang ◽  
Ying Wang ◽  
Cun Li ◽  
Juan Li ◽  
Qing Li ◽  
...  
Keyword(s):  
Kalman Filter ◽  
Probability Density Function ◽  
Target Tracking ◽  
Probability Density ◽  
Density Function ◽  
Gaussian Mixture ◽  
System State ◽  
Cubature Kalman Filter ◽  
Non Gaussian ◽  
Time Update

The Gaussian mixture filter can solve the non-Gaussian problem of target tracking in complex environment by the multimode approximation method, but the weights of the Gaussian component of the conventional Gaussian mixture filter are only updated with the arrival of the measurement value in the measurement update stage. When the nonlinear degree of the system is high or the measurement value is missing, the weight of the Gauss component remains unchanged, and the probability density function of the system state cannot be accurately approximated. To solve this problem, this paper proposes an algorithm to update adaptive weights for the Gaussian components of a Gaussian mixture cubature Kalman filter (CKF) in the time update stage. The proposed method approximates the non-Gaussian noise by splitting the system state, process noise, and observation noise into several Gaussian components and updates the weight of the Gaussian components in the time update stage. The method contributes to obtaining a better approximation of the posterior probability density function, which is constrained by the substantial uncertainty associated with the measurements or ambiguity in the model. The estimation accuracy of the proposed algorithm was analyzed using a Taylor expansion. A series of extensive trials was performed to assess the estimation precision corresponding to various algorithms. The results based on the data pertaining to the lake trial of an unmanned underwater vehicle (UUV) demonstrated the superiority of the proposed algorithm in terms of its better accuracy and stability compared to those of conventional tracking algorithms, along with the associated reasonable computational time that could satisfy real-time tracking requirements.

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