Articles | Volume 22, issue 6
https://doi.org/10.5194/npg-22-749-2015
© Author(s) 2015. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
https://doi.org/10.5194/npg-22-749-2015
© Author(s) 2015. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
Research article
| 
21 Dec 2015
Research article |
| 21 Dec 2015
Nonlinear feedback in a six-dimensional Lorenz model: impact of an additional heating term
B.-W. Shen
CORRESPONDING AUTHOR
Department of Mathematics and Statistics, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182-7720, USA
Related authors
B.-W. Shen
Nonlin. Processes Geophys. Discuss., https://doi.org/10.5194/npg-2016-40, https://doi.org/10.5194/npg-2016-40, 2016
Revised manuscript not accepted
Short summary
Short summary
Using a nondissipative Lorenz model, we show that the nonlinear feedback loop alone may act a restoring forcing and the heating term alone can produce a saddle point. We illustrate that the nonlinear feedback loop and heating term collectively lead to three critical points and three types of solutions, which may contain a small or big energy cycle. The existence of the homoclinic orbit and two types of oscillatory solutions suggests the appearance of diverged trajectories.
Bo-Wen Shen
Nonlin. Processes Geophys., 23, 189–203, https://doi.org/10.5194/npg-23-189-2016, https://doi.org/10.5194/npg-23-189-2016, 2016
Short summary
Short summary
We construct a seven-dimensional Lorenz model (7DLM) to discuss the impact of an extended nonlinear feedback loop on solutions' stability and illustrate the hierarchical scale dependence of chaotic solutions. The 7DLM requires a much larger critical value for the Rayleigh parameter (rc ∼ 116.9) for the onset of chaos. For chaotic solutions with r = 120, high correlation coefficients among the modes at different scales indicate hierarchical scale dependence.
B.-W. Shen
Nonlin. Processes Geophys. Discuss., https://doi.org/10.5194/npgd-1-519-2014, https://doi.org/10.5194/npgd-1-519-2014, 2014
Revised manuscript not accepted
B.-W. Shen
Nonlin. Processes Geophys. Discuss., https://doi.org/10.5194/npg-2016-40, https://doi.org/10.5194/npg-2016-40, 2016
Revised manuscript not accepted
Short summary
Short summary
Using a nondissipative Lorenz model, we show that the nonlinear feedback loop alone may act a restoring forcing and the heating term alone can produce a saddle point. We illustrate that the nonlinear feedback loop and heating term collectively lead to three critical points and three types of solutions, which may contain a small or big energy cycle. The existence of the homoclinic orbit and two types of oscillatory solutions suggests the appearance of diverged trajectories.
Bo-Wen Shen
Nonlin. Processes Geophys., 23, 189–203, https://doi.org/10.5194/npg-23-189-2016, https://doi.org/10.5194/npg-23-189-2016, 2016
Short summary
Short summary
We construct a seven-dimensional Lorenz model (7DLM) to discuss the impact of an extended nonlinear feedback loop on solutions' stability and illustrate the hierarchical scale dependence of chaotic solutions. The 7DLM requires a much larger critical value for the Rayleigh parameter (rc ∼ 116.9) for the onset of chaos. For chaotic solutions with r = 120, high correlation coefficients among the modes at different scales indicate hierarchical scale dependence.
B.-W. Shen
Nonlin. Processes Geophys. Discuss., https://doi.org/10.5194/npgd-1-519-2014, https://doi.org/10.5194/npgd-1-519-2014, 2014
Revised manuscript not accepted
Related subject area
Subject: Predictability, probabilistic forecasts, data assimilation, inverse problems | Topic: Climate, atmosphere, ocean, hydrology, cryosphere, biosphere
A range of outcomes: the combined effects of internal variability and anthropogenic forcing on regional climate trends over Europe
Extending ensemble Kalman filter algorithms to assimilate observations with an unknown time offset
Guidance on how to improve vertical covariance localization based on a 1000-member ensemble
Weather pattern dynamics over western Europe under climate change: predictability, information entropy and production
Using a hybrid optimal interpolation–ensemble Kalman filter for the Canadian Precipitation Analysis
Applying prior correlations for ensemble-based spatial localization
Control Simulation Experiments of Extreme Events with the Lorenz-96 Model
A stochastic covariance shrinkage approach to particle rejuvenation in the ensemble transform particle filter
Control simulation experiment with Lorenz's butterfly attractor
Ensemble Riemannian data assimilation: towards large-scale dynamical systems
Inferring the instability of a dynamical system from the skill of data assimilation exercises
Reduced non-Gaussianity by 30 s rapid update in convective-scale numerical weather prediction
Multivariate localization functions for strongly coupled data assimilation in the bivariate Lorenz 96 system
A study of capturing Atlantic meridional overturning circulation (AMOC) regime transition through observation-constrained model parameters
Calibrated ensemble forecasts of the height of new snow using quantile regression forests and ensemble model output statistics
Enhancing geophysical flow machine learning performance via scale separation
Improving the potential accuracy and usability of EURO-CORDEX estimates of future rainfall climate using frequentist model averaging
Ensemble Riemannian data assimilation over the Wasserstein space
An early warning sign of critical transition in the Antarctic ice sheet – a data-driven tool for a spatiotemporal tipping point
Training a convolutional neural network to conserve mass in data assimilation
Behavior of the iterative ensemble-based variational method in nonlinear problems
Fast hybrid tempered ensemble transform filter formulation for Bayesian elliptical problems via Sinkhorn approximation
A methodology to obtain model-error covariances due to the discretization scheme from the parametric Kalman filter perspective
A method for predicting the uncompleted climate transition process
Statistical postprocessing of ensemble forecasts for severe weather at Deutscher Wetterdienst
Data-driven predictions of a multiscale Lorenz 96 chaotic system using machine-learning methods: reservoir computing, artificial neural network, and long short-term memory network
From research to applications – examples of operational ensemble post-processing in France using machine learning
Correcting for model changes in statistical postprocessing – an approach based on response theory
Brief communication: Residence time of energy in the atmosphere
Simulating model uncertainty of subgrid-scale processes by sampling model errors at convective scales
Data-driven versus self-similar parameterizations for stochastic advection by Lie transport and location uncertainty
Seasonal statistical–dynamical prediction of the North Atlantic Oscillation by probabilistic post-processing and its evaluation
Application of a local attractor dimension to reduced space strongly coupled data assimilation for chaotic multiscale systems
Order of operation for multi-stage post-processing of ensemble wind forecast trajectories
Generalization properties of feed-forward neural networks trained on Lorenz systems
Revising the stochastic iterative ensemble smoother
Joint state-parameter estimation of a nonlinear stochastic energy balance model from sparse noisy data
Non-Gaussian statistics in global atmospheric dynamics: a study with a 10 240-member ensemble Kalman filter using an intermediate atmospheric general circulation model
Fluctuations of finite-time Lyapunov exponents in an intermediate-complexity atmospheric model: a multivariate and large-deviation perspective
Data assimilation using adaptive, non-conservative, moving mesh models
Data assimilation as a learning tool to infer ordinary differential equation representations of dynamical models
A Bayesian approach to multivariate adaptive localization in ensemble-based data assimilation with time-dependent extensions
Exploring the sensitivity of Northern Hemisphere atmospheric circulation to different surface temperature forcing using a statistical–dynamical atmospheric model
Review article: Comparison of local particle filters and new implementations
Data assimilation of radar reflectivity volumes in a LETKF scheme
Application of ensemble transform data assimilation methods for parameter estimation in reservoir modeling
Nonlinear effects in 4D-Var
A novel approach for solving CNOPs and its application in identifying sensitive regions of tropical cyclone adaptive observations
Chaotic dynamics and the role of covariance inflation for reduced rank Kalman filters with model error
Ensemble variational assimilation as a probabilistic estimator – Part 1: The linear and weak non-linear case
Clara Deser and Adam S. Phillips
Nonlin. Processes Geophys., 30, 63–84, https://doi.org/10.5194/npg-30-63-2023, https://doi.org/10.5194/npg-30-63-2023, 2023
Short summary
Short summary
Past and future climate change at regional scales is a result of both human influences and natural (internal) variability. Here, we provide an overview of recent advances in climate modeling and physical understanding that has led to new insights into their respective roles, illustrated with original results for the European climate. Our findings highlight the confounding role of internal variability in attribution, climate model evaluation, and accuracy of future projections.
Elia Gorokhovsky and Jeffrey L. Anderson
Nonlin. Processes Geophys., 30, 37–47, https://doi.org/10.5194/npg-30-37-2023, https://doi.org/10.5194/npg-30-37-2023, 2023
Short summary
Short summary
Older observations of the Earth system sometimes lack information about the time they were taken, posing problems for analyses of past climate. To begin to ameliorate this problem, we propose new methods of varying complexity, including methods to estimate the distribution of the offsets between true and reported observation times. The most successful method accounts for the nonlinearity in the system, but even the less expensive ones can improve data assimilation in the presence of time error.
Tobias Necker, David Hinger, Philipp Johannes Griewank, Takemasa Miyoshi, and Martin Weissmann
Nonlin. Processes Geophys., 30, 13–29, https://doi.org/10.5194/npg-30-13-2023, https://doi.org/10.5194/npg-30-13-2023, 2023
Short summary
Short summary
This study investigates vertical localization based on a convection-permitting 1000-member ensemble simulation. We derive an empirical optimal localization (EOL) that minimizes sampling error in 40-member sub-sample correlations assuming 1000-member correlations as truth. The results will provide guidance for localization in convective-scale ensemble data assimilation systems.
Stéphane Vannitsem
Nonlin. Processes Geophys., 30, 1–12, https://doi.org/10.5194/npg-30-1-2023, https://doi.org/10.5194/npg-30-1-2023, 2023
Short summary
Short summary
The impact of climate change on weather pattern dynamics over the North Atlantic is explored through the lens of information theory. These tools allow the predictability of the succession of weather patterns and the irreversible nature of the dynamics to be clarified. It is shown that the predictability is increasing in the observations, while the opposite trend is found in model projections. The irreversibility displays an overall increase in time in both the observations and the model runs.
Dikraa Khedhaouiria, Stéphane Bélair, Vincent Fortin, Guy Roy, and Franck Lespinas
Nonlin. Processes Geophys., 29, 329–344, https://doi.org/10.5194/npg-29-329-2022, https://doi.org/10.5194/npg-29-329-2022, 2022
Short summary
Short summary
This study introduces a well-known use of hybrid methods in data assimilation (DA) algorithms that has not yet been explored for precipitation analyses. Our approach combined an ensemble-based DA approach with an existing deterministically based DA. Both DA scheme families have desirable aspects that can be leveraged if combined. The DA hybrid method showed better precipitation analyses in regions with a low rate of assimilated surface observations, which is typically the case in winter.
Chu-Chun Chang and Eugenia Kalnay
Nonlin. Processes Geophys., 29, 317–327, https://doi.org/10.5194/npg-29-317-2022, https://doi.org/10.5194/npg-29-317-2022, 2022
Short summary
Short summary
This study introduces a new approach for enhancing the ensemble data assimilation (DA), a technique that combines observations and forecasts to improve numerical weather predictions. Our method uses the prescribed correlations to suppress spurious errors, improving the accuracy of DA. The experiments on the simplified atmosphere model show that our method has comparable performance to the traditional method but is superior in the early stage and is more computationally efficient.
Qiwen Sun, Takemasa Miyoshi, and Serge Richard
Nonlin. Processes Geophys. Discuss., https://doi.org/10.5194/npg-2022-12, https://doi.org/10.5194/npg-2022-12, 2022
Revised manuscript accepted for NPG
Short summary
Short summary
This paper is a follow up of a work by T. Miyoshi and Q. Sun which was published as NPG letters in 2022. The Control Simulation Experiment is applied to the Lorenz 96 model for avoiding extreme events. The results show that extreme events of this partially and imperfectly observed chaotic system can be avoided by applying pre-designed small perturbations. These investigations may be extended to more realistic numerical weather prediction models.
Andrey A. Popov, Amit N. Subrahmanya, and Adrian Sandu
Nonlin. Processes Geophys., 29, 241–253, https://doi.org/10.5194/npg-29-241-2022, https://doi.org/10.5194/npg-29-241-2022, 2022
Short summary
Short summary
Numerical weather prediction requires the melding of both computational model and data obtained from sensors such as satellites. We focus on one algorithm to accomplish this. We aim to aid its use by additionally supplying it with data obtained from separate models that describe the average behavior of the computational model at any given time. We show that our approach outperforms the standard approaches to this problem.
Takemasa Miyoshi and Qiwen Sun
Nonlin. Processes Geophys., 29, 133–139, https://doi.org/10.5194/npg-29-133-2022, https://doi.org/10.5194/npg-29-133-2022, 2022
Short summary
Short summary
The weather is chaotic and hard to predict, but the chaos implies an effective control where a small control signal grows rapidly to make a big difference. This study proposes a control simulation experiment where we apply a small signal to control
naturein a computational simulation. Idealized experiments with a low-order chaotic system show successful results by small control signals of only 3 % of the observation error. This is the first step toward realistic weather simulations.
Sagar K. Tamang, Ardeshir Ebtehaj, Peter Jan van Leeuwen, Gilad Lerman, and Efi Foufoula-Georgiou
Nonlin. Processes Geophys., 29, 77–92, https://doi.org/10.5194/npg-29-77-2022, https://doi.org/10.5194/npg-29-77-2022, 2022
Short summary
Short summary
The outputs from Earth system models are optimally combined with satellite observations to produce accurate forecasts through a process called data assimilation. Many existing data assimilation methodologies have some assumptions regarding the shape of the probability distributions of model output and observations, which results in forecast inaccuracies. In this paper, we test the effectiveness of a newly proposed methodology that relaxes such assumptions about high-dimensional models.
Yumeng Chen, Alberto Carrassi, and Valerio Lucarini
Nonlin. Processes Geophys., 28, 633–649, https://doi.org/10.5194/npg-28-633-2021, https://doi.org/10.5194/npg-28-633-2021, 2021
Short summary
Short summary
Chaotic dynamical systems are sensitive to the initial conditions, which are crucial for climate forecast. These properties are often used to inform the design of data assimilation (DA), a method used to estimate the exact initial conditions. However, obtaining the instability properties is burdensome for complex problems, both numerically and analytically. Here, we suggest a different viewpoint. We show that the skill of DA can be used to infer the instability properties of a dynamical system.
Juan Ruiz, Guo-Yuan Lien, Keiichi Kondo, Shigenori Otsuka, and Takemasa Miyoshi
Nonlin. Processes Geophys., 28, 615–626, https://doi.org/10.5194/npg-28-615-2021, https://doi.org/10.5194/npg-28-615-2021, 2021
Short summary
Short summary
Effective use of observations with numerical weather prediction models, also known as data assimilation, is a key part of weather forecasting systems. For precise prediction at the scales of thunderstorms, fast nonlinear processes pose a grand challenge because most data assimilation systems are based on linear processes and normal distribution errors. We investigate how, every 30 s, weather radar observations can help reduce the effect of nonlinear processes and nonnormal distributions.
Zofia Stanley, Ian Grooms, and William Kleiber
Nonlin. Processes Geophys., 28, 565–583, https://doi.org/10.5194/npg-28-565-2021, https://doi.org/10.5194/npg-28-565-2021, 2021
Short summary
Short summary
In weather forecasting, observations are incorporated into a model of the atmosphere through a process called data assimilation. Sometimes observations in one location may impact the weather forecast in another faraway location in undesirable ways. The impact of distant observations on the forecast is mitigated through a process called localization. We propose a new method for localization when a model has multiple length scales, as in a model spanning both the ocean and the atmosphere.
Zhao Liu, Shaoqing Zhang, Yang Shen, Yuping Guan, and Xiong Deng
Nonlin. Processes Geophys., 28, 481–500, https://doi.org/10.5194/npg-28-481-2021, https://doi.org/10.5194/npg-28-481-2021, 2021
Short summary
Short summary
A general methodology is introduced to capture regime transitions of the Atlantic meridional overturning circulation (AMOC). The assimilation models with different parameters simulate different paths for the AMOC to switch between equilibrium states. Constraining model parameters with observations can significantly mitigate the model deviations, thus capturing AMOC regime transitions. This simple model study serves as a guideline for improving coupled general circulation models.
Guillaume Evin, Matthieu Lafaysse, Maxime Taillardat, and Michaël Zamo
Nonlin. Processes Geophys., 28, 467–480, https://doi.org/10.5194/npg-28-467-2021, https://doi.org/10.5194/npg-28-467-2021, 2021
Short summary
Short summary
Forecasting the height of new snow is essential for avalanche hazard surveys, road and ski resort management, tourism attractiveness, etc. Météo-France operates a probabilistic forecasting system using a numerical weather prediction system and a snowpack model. It provides better forecasts than direct diagnostics but exhibits significant biases. Post-processing methods can be applied to provide automatic forecasting products from this system.
Davide Faranda, Mathieu Vrac, Pascal Yiou, Flavio Maria Emanuele Pons, Adnane Hamid, Giulia Carella, Cedric Ngoungue Langue, Soulivanh Thao, and Valerie Gautard
Nonlin. Processes Geophys., 28, 423–443, https://doi.org/10.5194/npg-28-423-2021, https://doi.org/10.5194/npg-28-423-2021, 2021
Short summary
Short summary
Machine learning approaches are spreading rapidly in climate sciences. They are of great help in many practical situations where using the underlying equations is difficult because of the limitation in computational power. Here we use a systematic approach to investigate the limitations of the popular echo state network algorithms used to forecast the long-term behaviour of chaotic systems, such as the weather. Our results show that noise and intermittency greatly affect the performances.
Stephen Jewson, Giuliana Barbato, Paola Mercogliano, Jaroslav Mysiak, and Maximiliano Sassi
Nonlin. Processes Geophys., 28, 329–346, https://doi.org/10.5194/npg-28-329-2021, https://doi.org/10.5194/npg-28-329-2021, 2021
Short summary
Short summary
Climate model simulations are uncertain. In some cases this makes it difficult to know how to use them. Significance testing is often used to deal with this issue but has various shortcomings. We describe two alternative ways to manage uncertainty in climate model simulations that avoid these shortcomings. We test them on simulations of future rainfall over Europe and show they produce more accurate projections than either using unadjusted climate model output or statistical testing.
Sagar K. Tamang, Ardeshir Ebtehaj, Peter J. van Leeuwen, Dongmian Zou, and Gilad Lerman
Nonlin. Processes Geophys., 28, 295–309, https://doi.org/10.5194/npg-28-295-2021, https://doi.org/10.5194/npg-28-295-2021, 2021
Short summary
Short summary
Data assimilation aims to improve hydrologic and weather forecasts by combining available information from Earth system models and observations. The classical approaches to data assimilation usually proceed with some preconceived assumptions about the shape of their probability distributions. As a result, when such assumptions are invalid, the forecast accuracy suffers. In the proposed methodology, we relax such assumptions and demonstrate improved performance.
Abd AlRahman AlMomani and Erik Bollt
Nonlin. Processes Geophys., 28, 153–166, https://doi.org/10.5194/npg-28-153-2021, https://doi.org/10.5194/npg-28-153-2021, 2021
Short summary
Short summary
This paper introduces a tool for data-driven discovery of early warning signs of critical transitions in ice shelves from remote sensing data. Our directed spectral clustering method considers an asymmetric affinity matrix along with the associated directed graph Laplacian. We applied our approach to reprocessing the ice velocity data and remote sensing satellite images of the Larsen C ice shelf.
Yvonne Ruckstuhl, Tijana Janjić, and Stephan Rasp
Nonlin. Processes Geophys., 28, 111–119, https://doi.org/10.5194/npg-28-111-2021, https://doi.org/10.5194/npg-28-111-2021, 2021
Short summary
Short summary
The assimilation of observations using standard algorithms can lead to a violation of physical laws (e.g. mass conservation), which is shown to have a detrimental impact on the system's forecast. We use a neural network (NN) to correct this mass violation, using training data generated from expensive algorithms that can constrain such physical properties. We found that, in an idealized set-up, the NN can match the performance of these expensive algorithms at negligible computational costs.
Shin'ya Nakano
Nonlin. Processes Geophys., 28, 93–109, https://doi.org/10.5194/npg-28-93-2021, https://doi.org/10.5194/npg-28-93-2021, 2021
Short summary
Short summary
The ensemble-based variational method is a method for solving nonlinear data assimilation problems by using an ensemble of multiple simulation results. Although this method is derived based on a linear approximation, highly uncertain problems, in which system nonlinearity is significant, can also be solved by applying this method iteratively. This paper reformulated this iterative algorithm to analyze its behavior in high-dimensional nonlinear problems and discuss the convergence.
Sangeetika Ruchi, Svetlana Dubinkina, and Jana de Wiljes
Nonlin. Processes Geophys., 28, 23–41, https://doi.org/10.5194/npg-28-23-2021, https://doi.org/10.5194/npg-28-23-2021, 2021
Short summary
Short summary
To infer information of an unknown quantity that helps to understand an associated system better and to predict future outcomes, observations and a physical model that connects the data points to the unknown parameter are typically used as information sources. Yet this problem is often very challenging due to the fact that the unknown is generally high dimensional, the data are sparse and the model can be non-linear. We propose a novel approach to address these challenges.
Olivier Pannekoucke, Richard Ménard, Mohammad El Aabaribaoune, and Matthieu Plu
Nonlin. Processes Geophys., 28, 1–22, https://doi.org/10.5194/npg-28-1-2021, https://doi.org/10.5194/npg-28-1-2021, 2021
Short summary
Short summary
Numerical weather prediction involves numerically solving the mathematical equations, which describe the geophysical flow, by transforming them so that they can be computed. Through this transformation, it appears that the equations actually solved by the machine are then a modified version of the original equations, introducing an error that contributes to the model error. This work helps to characterize the covariance of the model error that is due to this modification of the equations.
Pengcheng Yan, Guolin Feng, Wei Hou, and Ping Yang
Nonlin. Processes Geophys., 27, 489–500, https://doi.org/10.5194/npg-27-489-2020, https://doi.org/10.5194/npg-27-489-2020, 2020
Short summary
Short summary
A system transiting from one stable state to another has to experience a period. Can we predict the end moment (state) if the process has not been completed? This paper presents a method to solve this problem. This method is based on the quantitative relationship among the parameters, which is used to describe the transition process of the abrupt change. By using the historical data, we extract some parameters for predicting the uncompleted climate transition process.
Reinhold Hess
Nonlin. Processes Geophys., 27, 473–487, https://doi.org/10.5194/npg-27-473-2020, https://doi.org/10.5194/npg-27-473-2020, 2020
Short summary
Short summary
Forecasts of ensemble systems are statistically aligned to synoptic observations at DWD in order to provide support for warning decision management. Motivation and design consequences for extreme and rare meteorological events are presented. Especially for probabilities of severe wind gusts global logistic parameterisations are developed that generate robust statistical forecasts for extreme events, while local characteristics are preserved.
Ashesh Chattopadhyay, Pedram Hassanzadeh, and Devika Subramanian
Nonlin. Processes Geophys., 27, 373–389, https://doi.org/10.5194/npg-27-373-2020, https://doi.org/10.5194/npg-27-373-2020, 2020
Short summary
Short summary
The performance of three machine-learning methods for data-driven modeling of a multiscale chaotic Lorenz 96 system is examined. One of the methods is found to be able to predict the future evolution of the chaotic system well from just knowing the past observations of the large-scale component of the multiscale state vector. Potential applications to data-driven and data-assisted surrogate modeling of complex dynamical systems such as weather and climate are discussed.
Maxime Taillardat and Olivier Mestre
Nonlin. Processes Geophys., 27, 329–347, https://doi.org/10.5194/npg-27-329-2020, https://doi.org/10.5194/npg-27-329-2020, 2020
Short summary
Short summary
Statistical post-processing of ensemble forecasts is now a well-known procedure in order to correct biased and misdispersed ensemble weather predictions. But practical application in European national weather services is in its infancy. Different applications of ensemble post-processing using machine learning at an industrial scale are presented. Forecast quality and value are improved compared to the raw ensemble, but several facilities have to be made to adjust to operational constraints.
Jonathan Demaeyer and Stéphane Vannitsem
Nonlin. Processes Geophys., 27, 307–327, https://doi.org/10.5194/npg-27-307-2020, https://doi.org/10.5194/npg-27-307-2020, 2020
Short summary
Short summary
Postprocessing schemes used to correct weather forecasts are no longer efficient when the model generating the forecasts changes. An approach based on response theory to take the change into account without having to recompute the parameters based on past forecasts is presented. It is tested on an analytical model and a simple model of atmospheric variability. We show that this approach is effective and discuss its potential application for an operational environment.
Carlos Osácar, Manuel Membrado, and Amalio Fernández-Pacheco
Nonlin. Processes Geophys., 27, 235–237, https://doi.org/10.5194/npg-27-235-2020, https://doi.org/10.5194/npg-27-235-2020, 2020
Short summary
Short summary
We deduce that after a global thermal perturbation, the Earth's
atmosphere would need about a couple of months to come back to equilibrium.
Michiel Van Ginderachter, Daan Degrauwe, Stéphane Vannitsem, and Piet Termonia
Nonlin. Processes Geophys., 27, 187–207, https://doi.org/10.5194/npg-27-187-2020, https://doi.org/10.5194/npg-27-187-2020, 2020
Short summary
Short summary
A generic methodology is developed to estimate the model error and simulate the model uncertainty related to a specific physical process. The method estimates the model error by comparing two different representations of the physical process in otherwise identical models. The found model error can then be used to perturb the model and simulate the model uncertainty. When applying this methodology to deep convection an improvement in the probabilistic skill of the ensemble forecast is found.
Valentin Resseguier, Wei Pan, and Baylor Fox-Kemper
Nonlin. Processes Geophys., 27, 209–234, https://doi.org/10.5194/npg-27-209-2020, https://doi.org/10.5194/npg-27-209-2020, 2020
Short summary
Short summary
Geophysical flows span a broader range of temporal and spatial scales than can be resolved numerically. One way to alleviate the ensuing numerical errors is to combine simulations with measurements, taking account of the accuracies of these two sources of information. Here we quantify the distribution of numerical simulation errors without relying on high-resolution numerical simulations. Specifically, small-scale random vortices are added to simulations while conserving energy or circulation.
André Düsterhus
Nonlin. Processes Geophys., 27, 121–131, https://doi.org/10.5194/npg-27-121-2020, https://doi.org/10.5194/npg-27-121-2020, 2020
Short summary
Short summary
Seasonal prediction of the of the North Atlantic Oscillation (NAO) has been improved in recent years by improving dynamical models and ensemble predictions. One step therein was the so-called sub-sampling, which combines statistical and dynamical predictions. This study generalises this approach and makes it much more accessible. Furthermore, it presents a new verification approach for such predictions.
Courtney Quinn, Terence J. O'Kane, and Vassili Kitsios
Nonlin. Processes Geophys., 27, 51–74, https://doi.org/10.5194/npg-27-51-2020, https://doi.org/10.5194/npg-27-51-2020, 2020
Short summary
Short summary
This study presents a novel method for reduced-rank data assimilation of multiscale highly nonlinear systems. Time-varying dynamical properties are used to determine the rank and projection of the system onto a reduced subspace. The variable reduced-rank method is shown to succeed over other fixed-rank methods. This work provides implications for performing strongly coupled data assimilation with a limited number of ensemble members on high-dimensional coupled climate models.
Nina Schuhen
Nonlin. Processes Geophys., 27, 35–49, https://doi.org/10.5194/npg-27-35-2020, https://doi.org/10.5194/npg-27-35-2020, 2020
Short summary
Short summary
We present a new way to adaptively improve weather forecasts by incorporating last-minute observation information. The recently measured error, together with a statistical model, gives us an indication of the expected future error of wind speed forecasts, which are then adjusted accordingly. This new technique can be especially beneficial for customers in the wind energy industry, where it is important to have reliable short-term forecasts, as well as providers of extreme weather warnings.
Sebastian Scher and Gabriele Messori
Nonlin. Processes Geophys., 26, 381–399, https://doi.org/10.5194/npg-26-381-2019, https://doi.org/10.5194/npg-26-381-2019, 2019
Short summary
Short summary
Neural networks are a technique that is widely used to predict the time evolution of physical systems. For this the past evolution of the system is shown to the neural network – it is
trained– and then can be used to predict the evolution in the future. We show some limitations in this approach for certain systems that are important to consider when using neural networks for climate- and weather-related applications.
Patrick Nima Raanes, Andreas Størksen Stordal, and Geir Evensen
Nonlin. Processes Geophys., 26, 325–338, https://doi.org/10.5194/npg-26-325-2019, https://doi.org/10.5194/npg-26-325-2019, 2019
Short summary
Short summary
A popular variational ensemble smoother for data assimilation and history matching is simplified. An exact relationship between ensemble linearizations (linear regression) and adjoints (analytic derivatives) is established.
Fei Lu, Nils Weitzel, and Adam H. Monahan
Nonlin. Processes Geophys., 26, 227–250, https://doi.org/10.5194/npg-26-227-2019, https://doi.org/10.5194/npg-26-227-2019, 2019
Short summary
Short summary
ll-posedness of the inverse problem and sparse noisy data are two major challenges in the modeling of high-dimensional spatiotemporal processes. We present a Bayesian inference method with a strongly regularized posterior to overcome these challenges, enabling joint state-parameter estimation and quantifying uncertainty in the estimation. We demonstrate the method on a physically motivated nonlinear stochastic partial differential equation arising from paleoclimate construction.
Keiichi Kondo and Takemasa Miyoshi
Nonlin. Processes Geophys., 26, 211–225, https://doi.org/10.5194/npg-26-211-2019, https://doi.org/10.5194/npg-26-211-2019, 2019
Short summary
Short summary
This study investigates non-Gaussian statistics of the data from a 10240-member ensemble Kalman filter. The large ensemble size can resolve the detailed structures of the probability density functions (PDFs) and indicates that the non-Gaussian PDF is caused by multimodality and outliers. While the outliers appear randomly, large multimodality corresponds well with large analysis error, mainly in the tropical regions and storm track regions where highly nonlinear processes appear frequently.
Frank Kwasniok
Nonlin. Processes Geophys., 26, 195–209, https://doi.org/10.5194/npg-26-195-2019, https://doi.org/10.5194/npg-26-195-2019, 2019
Short summary
Short summary
The stability properties as characterized by finite-time Lyapunov exponents are investigated in an intermediate-complexity atmospheric model. Firstly, the dominant patterns of collective excitation are identified by an empirical orthogonal function analysis of the fluctuation field of all of the finite-time Lyapunov exponents. Secondly, a large-deviation principle is established for all of the Lyapunov exponents and the large-deviation rate functions are estimated.
Ali Aydoğdu, Alberto Carrassi, Colin T. Guider, Chris K. R. T Jones, and Pierre Rampal
Nonlin. Processes Geophys., 26, 175–193, https://doi.org/10.5194/npg-26-175-2019, https://doi.org/10.5194/npg-26-175-2019, 2019
Short summary
Short summary
Computational models involving adaptive meshes can both evolve dynamically and be remeshed. Remeshing means that the state vector dimension changes in time and across ensemble members, making the ensemble Kalman filter (EnKF) unsuitable for assimilation of observational data. We develop a modification in which analysis is performed on a fixed uniform grid onto which the ensemble is mapped, with resolution relating to the remeshing criteria. The approach is successfully tested on two 1-D models.
Marc Bocquet, Julien Brajard, Alberto Carrassi, and Laurent Bertino
Nonlin. Processes Geophys., 26, 143–162, https://doi.org/10.5194/npg-26-143-2019, https://doi.org/10.5194/npg-26-143-2019, 2019
Short summary
Short summary
This paper describes an innovative way to use data assimilation to infer the dynamics of a physical system from its observation only. The method can operate with noisy and partial observation of the physical system. It acts as a deep learning technique specialised to dynamical models without the need for machine learning tools. The method is successfully tested on chaotic dynamical systems: the Lorenz-63, Lorenz-96, and Kuramoto–Sivashinski models and a two-scale Lorenz model.
Andrey A. Popov and Adrian Sandu
Nonlin. Processes Geophys., 26, 109–122, https://doi.org/10.5194/npg-26-109-2019, https://doi.org/10.5194/npg-26-109-2019, 2019
Short summary
Short summary
This work has to do with a small part of existing algorithms that are used in applications such as predicting the weather. We provide empirical evidence that our new technique works well on small but representative models. This might lead to creation of a better weather forecast and potentially save lives as in the case of hurricane prediction.
Sonja Totz, Stefan Petri, Jascha Lehmann, Erik Peukert, and Dim Coumou
Nonlin. Processes Geophys., 26, 1–12, https://doi.org/10.5194/npg-26-1-2019, https://doi.org/10.5194/npg-26-1-2019, 2019
Alban Farchi and Marc Bocquet
Nonlin. Processes Geophys., 25, 765–807, https://doi.org/10.5194/npg-25-765-2018, https://doi.org/10.5194/npg-25-765-2018, 2018
Short summary
Short summary
Data assimilation looks for an optimal way to learn from observations of a dynamical system to improve the quality of its predictions. The goal is to filter out the noise (both observation and model noise) to retrieve the true signal. Among all possible methods, particle filters are promising; the method is fast and elegant, and it allows for a Bayesian analysis. In this review paper, we discuss implementation techniques for (local) particle filters in high-dimensional systems.
Thomas Gastaldo, Virginia Poli, Chiara Marsigli, Pier Paolo Alberoni, and Tiziana Paccagnella
Nonlin. Processes Geophys., 25, 747–764, https://doi.org/10.5194/npg-25-747-2018, https://doi.org/10.5194/npg-25-747-2018, 2018
Short summary
Short summary
Accuracy of numerical weather prediction forecasts is strongly related to the quality of initial conditions employed. To improve them, it seems advantageous to use radar reflectivity observations because of their high spatial and temporal resolution. This is tested in a high-resolution model whose domain covers Italy. Results show that the employment of reflectivity observations improves precipitation forecast accuracy, but the positive impact is lost after a few hours of forecast.
Sangeetika Ruchi and Svetlana Dubinkina
Nonlin. Processes Geophys., 25, 731–746, https://doi.org/10.5194/npg-25-731-2018, https://doi.org/10.5194/npg-25-731-2018, 2018
Short summary
Short summary
Accurate estimation of subsurface geological parameters is essential for the oil industry. This is done by combining observations with an estimation from a model. Ensemble Kalman filter is a widely used method for inverse modeling, while ensemble transform particle filtering is a recently developed method that has been applied to estimate only a small number of parameters and in fluids. We show that for a high-dimensional inverse problem it is superior to an ensemble Kalman filter.
Massimo Bonavita, Peter Lean, and Elias Holm
Nonlin. Processes Geophys., 25, 713–729, https://doi.org/10.5194/npg-25-713-2018, https://doi.org/10.5194/npg-25-713-2018, 2018
Short summary
Short summary
This paper deals with the effects of nonlinearity in a state-of-the-art atmospheric global data assimilation system. It is shown that these effects have become increasingly important over the years due to increased model resolution and use of nonlinear observations. The ability to deal with nonlinearities has thus become a crucial asset of data assimilation algorithms. At ECMWF this is done in a perturbative fashion. Advantages and limitations of this technique are discussed.
Linlin Zhang, Bin Mu, Shijin Yuan, and Feifan Zhou
Nonlin. Processes Geophys., 25, 693–712, https://doi.org/10.5194/npg-25-693-2018, https://doi.org/10.5194/npg-25-693-2018, 2018
Short summary
Short summary
We propose a novel approach to solve conditional nonlinear optimal perturbation for identifying sensitive areas for tropical cyclone adaptive observations. This method is free of adjoint models and overcomes two obstacles, not having adjoint models and having dimensions higher than the problem space. All experimental results prove that it is a meaningful and effective method for solving CNOP and provides a new way for such research. This work aims to solve CNOP and identify sensitive areas.
Colin Grudzien, Alberto Carrassi, and Marc Bocquet
Nonlin. Processes Geophys., 25, 633–648, https://doi.org/10.5194/npg-25-633-2018, https://doi.org/10.5194/npg-25-633-2018, 2018
Short summary
Short summary
Using the framework Lyapunov vectors, we analyze the asymptotic properties of ensemble based Kalman filters and how these are influenced by dynamical chaos, especially in the context of random model errors and small ensemble sizes. Particularly, we show a novel derivation of the evolution of forecast uncertainty for ensemble-based Kalman filters with weakly-nonlinear error growth, and discuss its impact for filter design in geophysical models.
Mohamed Jardak and Olivier Talagrand
Nonlin. Processes Geophys., 25, 565–587, https://doi.org/10.5194/npg-25-565-2018, https://doi.org/10.5194/npg-25-565-2018, 2018
Short summary
Short summary
Ensemble variational assimilation (EnsVAR) has been implemented on two small-dimension non-linear chaotic toy models, as well as on a linearized version of those models. In the linear case, EnsVAR is exactly Bayesian and produced
highly reliable ensembles. In the non-linear case, EnsVAR, implemented on temporal windows on the order of magnitude of the predictability time of the systems, shows as good performance as in the exactly linear case. EnsVar is as good an estimator as EnKF and PF.
Cited articles
Anthes, R.: Turning the tables on chaos: is the atmosphere more predictable than we assume?, UCAR Magazine, available at: https://www2.ucar.edu/atmosnews/opinion/turning-tables-chaos-atmosphere-more-predictable-we-assume-0, last access: 14 December 2015, 2011.
Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J. M.: Lyapunov Characteristic Exponents fro Smooth Dynamical Systems and for Hamiltonian Systems; A method for computing all of them. Part 1: Theory, Meccanica, 15, 9–20, 1980.
Blender, R. and Lucarini, V.: Nambu representation of an extended Lorenz model with viscous heating, Physica D, 243, 86–91, 2013.
Chen, Z.-M. and Price, W. G.: On the relation between Raleigh-Benard convection and Lorenz system, Chaos Soliton. Fract., 28, 571–578, 2006.
Christiansen, F. and Rugh, H.: Computing Lyapunov spectra with continuous Gram–Schmid orthonormalization, Nonlinearity, 10, 1063–1072, 1997.
Curry, J. H.: Generalized Lorenz systems, Commun. Math. Phys., 60, 193–204, 1978.
Curry, J. H., Herring, J. R., Loncaric, J., and Orszag, S. A.: Order and disorder in two- and three-dimensional Benard convection, J. Fluid. Mech., 147, 1–38, 1984.
Ding, R. O. and Li, J. P.: Nonlinear finite-time Lyapunov exponent and predictability, Phys. Lett., 354, 396–400, 2007.
Eckhardt, B. and Yao, D.: Local Lyapunov exponents in chaotic systems, Physica D, 65, 100–108, 1993.
Franceschini, V. and Tebaldi, C.: Truncations to 12, 14 and 18 Modes of the Navier-Stokes Equations on a Two-Dimensional Torus, Meccanica, 20, 207–230, 1985.
Franceschini, V., Giberti, C., and Nicolini, M.: Common Period Behavior in Larger and Larger Truncations of the Navier Stokes Equations, J. Stat. Phys., 50, 879–896, 1988.
Froyland, J. and Alfsen, K. H.: Lyapunov-exponent spectra for the Lorenz model, Phys. Rev. A, 29, 2928–2931, 1984.
Gleick, J.: Chaos: Making a New Science, Penguin, New York, 360 pp., 1987.
Grassberger, P. and Procaccia, I.: Characterization of strange attractors, Phys. Rev. Lett., 5, 346–349, 1983.
Hermiz, K. B., Guzdar, P. N., and Finn, J. M.: Improved low-order model for shear flow driven by Rayleigh–Benard convection, Phys. Rev. E, 51, 325–331, 1995.
Howard, L. N. and Krishnamurti, R. K.: Large-scale flow in turbulent convection: a mathematical model, J. Fluid Mech., 170, 385–410, 1986.
IPCC: Climate Change 2007: The Physical Science Basis, in: Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, edited by: Solomon, S., Qin, D., Manning, M., Chen, Z., Marquis, M., Averyt, K. B., Tignor, M., and Miller, H. L., Cambridge University Press, Cambridge, UK and New York, NY, USA, 996 pp., available at: http://www.ipcc.ch/pdf/assessment-report/ar4/wg1/ar4-wg1-faqs.pdf, last access: 14 December 2015, 2007.
Kaplan, J. L. and Yorke, J. A.: Chaotic behavior of multidimensional difference equations, in: Functional Differential Equations and the Approximations of Fixed Points, edited by: Peitgen, H. O. and Walther, H. O., Springer-Verlag, New York, Lect. Notes Math., 730, 228–237, 1979.
Kazantsev, E.: Local Lyapunov exponents of the quasi-geostrophic ocean dynamics, Appl. Math. Comput., 104, 217–257, 1999.
Kennamer, K. S.: Studies of the onset of chaos in the Lorenz and generalized Lorenz systems, MS thesis, The University of Alabama, Huntsville, USA, 1995.
Li, J. and Ding, R.: Temporal-Spatial Distribution of atmospheric predictability limit by local dynamical analogs, Mon. Weather Rev., 139, 3265–3283, 2011.
Lorenz, E.: Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130–141, 1963.
Lorenz, E.: Predictability: does the flap of a butterfly's wings in Brazil set off a tornado in Texas?, in: American Association for the Advancement of Science, 139th Meeting, 29 December 1972, Boston, Mass., AAAS Section on Environmental Sciences, New Approaches to Global Weather, GARP, available at: http://eaps4.mit.edu/research/Lorenz/Butterfly_1972.pdf, last access: 14 December 2015, 1972.
Lucarini, V. and Fraedrich, K.: Symmetry breaking, mixing, instability, and low-frequency variability in a minimal Lorenz-like system, Phys. Rev. E, 80, 026313, https://doi.org/10.1103/PhysRevE.80.026313, 2009.
Musielak, Z. E., Musielak, D. E., and Kennamer, K. S.: The onset of chaos in nonlinear dynamical systems determined with a new fractal technique, Fractals, 13, 19–31, 2005.
Nese, J. M.: Quantifying local predictability in phase space, Physica, 35, 237–250, 1989.
Nese, J. M., Dutton, J. A., and Wells, R.: Calculated attractor dimensions for low-order spectral models, J. Atmos. Sci., 44, 1950–1972, 1987.
Pelino, V., Maimone, F., and Pasini, A.: Energy cycle for the Lorenz attractor, Chaos Soliton. Fract., 64, 67–77, 2014.
Pielke, R.: The Real Butterfly Effect, available at: http://pielkeclimatesci.wordpress.com/2008/04/29/the-real-butterfly-effect/, last access: 14 December 2015, 2008.
Roy, D. and Musielak, Z. E.: Generalized Lorenz models and their routes to chaos, I. Energy-conserving vertical mode truncations, Chaos Soliton. Fract., 32, 1038–1052, 2007a.
Roy, D. and Musielak, Z. E.: Generalized Lorenz models and their routes to chaos, II. Energy-conserving horizontal mode truncations, Chaos Soliton. Fract., 31, 747–756, 2007b.
Roy, D. and Musielak, Z. E.: Generalized Lorenz models and their routes to chaos, III. Energy-conserving horizontal and vertical mode truncations, Chaos Soliton. Fract., 33, 1064–1070, 2007c.
Ruelle, D.: Chaotic Evolution and Strange Attractors, in: Lezioni Lincee, Cambridge University Press, Cambridge, https://doi.org/10.1017/CBO9780511608773, 1989.
Saltzman, B.: Finite amplitude free convection as an initial value problem, J. Atmos. Sci., 19, 329–341, 1962.
Shen, B.-W.: Nonlinear feedback in a five-dimensional Lorenz model, J. Atmos. Sci., 71, 1701–1723, https://doi.org/10.1175/JAS-D-13-0223.1, 2014a.
Shen, B.-W.: On the nonlinear feedback loop and energy cycle of the non-dissipative Lorenz model, Nonlin. Processes Geophys. Discuss., 1, 519–541, https://doi.org/10.5194/npgd-1-519-2014, 2014b.
Shen, B.-W.: Parameterization of Negative Nonlinear Feedback using a Five-dimensional Lorenz Model, Fractal Geometry and Nonlinear Analysis in Medicine and Biology, 1, 33–41, https://doi.org/10.15761/FGNAMB.1000109, 2015.
Shen, B.-W., Atlas, R., Reale, O., Lin, S.-J., Chern, J.-D., Chang, J., Henze, C., and Li, J.-L.: Hurricane forecasts with a global mesoscale-resolving model: Preliminary results with Hurricane Katrina (2005), Geophys. Res. Lett., 33, L13813, https://doi.org/10.1029/2006GL026143, 2006.
Shen, B.-W., Tao, W.-K., Lin, Y.-L., and Laing, A.: Genesis of twin tropical cyclones as revealed by a global mesoscale model: the role of mixed Rossby gravity waves, J. Geophys. Res., 117, D13114, https://doi.org/10.1029/2012JD017450, 2012.
Shen, B. W., DeMaria, M., Li, J.-L. F., and Cheung, S.: Genesis of hurricane Sandy (2012) simulated with a global mesoscale model, Geophys. Res. Lett., 40, 4944–4950, https://doi.org/10.1002/grl.50934, 2013.
Sprott, J. C.: Numerical Calculation of Largest Lyapunov Exponent, available at: http://sprott.physics.wisc.edu/chaos/lyapexp.htm, last access: 14 December 2015, 1997.
Sprott, J. C.: Chaos and Time-Series Analysis, Oxford University Press, 528 pp., 2003.
Thiffeault, J.-L. and Horton, W.: Energy-conserving truncations for convection with shear flow, Phys. Fluids, 8, 1715–1719, 1996.
Treve, Y. M. and Manley, O. P.: Energy conserving Galerkin approximations for 2-D hydrodynamic and MHD Benard Convection, Physica, 4, 319–342, 1982.
Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. A.: Determining Lyapunov exponents from a time series, Physica, 16, 285–317, 1985.
Zeng, X., Eykholt, R., and Pielke, R. A.: Estimating the Lyapunov-exponent spectrum from short time series of low precision, Phys. Rev. Lett., 66, 3229–3232, 1991.
Zeng, X., Pielke, R. A., and Eykholt, R.: Estimate of the fractal dimension and predictability of the atmosphere, J. Atmos. Sci., 49, 649–659, 1992.
Short summary
While the negative nonlinear feedback associated with two new modes in the 5DLM can stabilize solutions, additional resolved heating processes by a third mode in the 6DLM can destabilize solutions. The findings support the view of Lorenz (1972) on the role of small-scale processes: if the flap of a butterfly’s wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado.
While the negative nonlinear feedback associated with two new modes in the 5DLM can stabilize...
