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50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models

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A special issue of Atmosphere (ISSN 2073-4433). This special issue belongs to the section "Meteorology".

Deadline for manuscript submissions: closed (28 February 2023) | Viewed by 32998

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Special Issue Editors

Dr. Bo-Wen Shen

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Guest Editor

Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182, USA
Interests: global hurricane modeling and multiscale modeling; chaos and predictability; multiscale analysis; high performance computing and visualizations

Prof. Dr. Roger A. Pielke Sr.

E-Mail Website1 Website2
Guest Editor

ATOC/CIRES, University of Colorado, Boulder, CO 80303, USA
Interests: land–atmosphere interactions; ocean–atmosphere interactions; climate system dynamics; nonlinear mathematical study of atmospheric and climate processes; large eddy simulation (LES) modeling; mesoscale modeling

Prof. Dr. Xubin Zeng

E-Mail Website
Guest Editor

Department of Hydrology and Atmospheric Sciences, 1133 E. James E. Rogers Way, The University of Arizona, Tucson, AZ 85721, USA
Interests: land-atmosphere-ocean interface processes; weather and climate modeling; hydrometeorology; remote sensing; nonlinear dynamics; big data analytics

Special Issue Information

Dear Colleagues,

A well-accepted definition of the butterfly effect is the sensitive dependence on initial conditions (SDIC) that was rediscovered by Lorenz in 1963 (Lorenz 1963). Usage of the term “butterfly” appeared in 1972 when Lorenz applied a metaphor to discuss the possibility of whether a tiny perturbation may eventually create a tornado with a three-dimensional organized coherent structure. Since 1972, the metaphorical butterfly effect has received increasing attention, but its exact relation with the original butterfly effect (i.e., SDIC) has also become controversial. Lorenz (1963, 1972) laid a foundation for chaos theory, which is viewed as the 3rd scientific achievement of the 20th century, after relativity and quantum mechanics. The metaphorical butterfly effect turns 50 next year.

During the presentation announcing the 2021 Nobel Prize in Physics, the pioneering chaos study by Lorenz (1963) was cited as a foundation for the awarded studies. The awarded studies proposed physics-based mathematical models that are capable of revealing fundamental, crucial physical processes, including coexisting rapidly and slowly varying systems and the metastability of spin glass, in order to improve our understanding of predictability in complex systems such as weather and climate that possess time-varying multistability with coexisting attractors. At a smaller scale, as published by the Bulletin of the American Meteorological Society in January 2021, Shen et al. (2021) applied the generalized Lorenz model (Shen 2019) to propose a revised view that “weather possesses chaos and order; it includes emerging organized systems (such as tornadoes) and recurrent seasons”, in contrast to the conventional view of “weather is chaotic”.

In celebrating the 50th anniversary of the metaphorical butterfly effect in 2022, this Special Issue calls for research and review articles that report improved understanding for original and metaphorical butterfly effects, as well as recent advances in idealized Lorenz models and real-world models that address multistability, multiscale predictability, and sensitivity. Potential topics include (but are not limited to) the following:

The validity of existing analogies and metaphors for butterfly effects.
Insightful analyses of various Lorenz models (e.g., Lorenz 1963, 1969, 1984, 1996/2005) for revealing the role of monostability with single types of solutions and multistability with attractor coexistence in contributing to the multiscale predictability of weather and climate.
The development of conceptual, theoretical, and real-world models for revealing fundamental physical processes and their multiscale interactions that contribute to the predictability of weather and climate.
Innovative machine-learning methods that (1) classify chaotic and non-chaotic processes and identify weather and climate systems at various spatial and temporal scales (e.g., sub-seasonal to seasonal time scales), and (2) detect computational chaos and saturation dependence on various types of solutions.
The impact of tiny perturbations on emergent pattern formation with self-organization (e.g., stripes and rolls), the formation of high-impact weather (e.g., tornadoes and hurricanes), etc.

References

Shen, B.-W., R. A. Pielke Sr., X. Zeng, J.-J. Baik, S. Faghih-Naini#, J. Cui#, and R. Atlas, Is Weather Chaotic? Coexistence of Chaos and Order within a Generalized Lorenz Model. Bulletin of the American Meteorological Society, 102(1), E148-E158. Retrieved Jan 29, 2021a, Available online: https://journals.ametsoc.org/view/journals/bams/102/1/BAMS-D-19-0165.1.xml
Shen, B.-W., Aggregated Negative Feedback in a Generalized Lorenz Model. International Journal of Bifurcation and Chaos: World Scientific, Singapore, Vol. 29, No. 3, 2019. https://doi.org/10.1142/S0218127419500378
Shen, B.-W., R. A. Pielke Sr., X. Zeng, J.-J. Baik, S. Faghih-Naini, J. Cui, R. Atlas, T.A. Reyes: Is Weather Chaotic? Coexisting Chaotic and Non-Chaotic Attractors within Lorenz Models. In: Christos H. Skiadas, Yiannis Dimotikalis (eds) The 13th Chaos International Conference CHAOS 2020. Springer Proceedings in Complexity. Springer, Cham. 2021b. https://doi.org/10.1007/978-3-030-70795-8_57

Dr. Bo-Wen Shen
Prof. Dr. Roger Pielke Sr.
Prof. Dr. Xubin Zeng
Guest Editors

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Keywords

butterfly effect
sensitive dependence on initial conditions
attractor coexistence
chaos and order
computational chaos
Lorenz model
predictability
monostability
multistability
machine learning

Published Papers (11 papers)

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Editorial

Jump to: Research, Review, Other

22 pages, 1726 KiB

Open AccessEditorial

The 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models

by Bo-Wen Shen, Roger A. Pielke, Sr. and Xubin Zeng

Atmosphere 2023, 14(8), 1279; https://doi.org/10.3390/atmos14081279 - 12 Aug 2023

Cited by 4 | Viewed by 2261

Abstract

Lorenz rediscovered the butterfly effect, which is defined as the sensitive dependence on initial conditions (SDIC), in 1963. In 1972, he used the term “butterfly” as a metaphor to illustrate how a small perturbation can lead to a tornado with a complex structure. The metaphorical butterfly effect, which celebrated its 50th anniversary in 2022, is not precisely the same as the original butterfly effect with SDIC. To commemorate the 50th anniversary, a Special Issue was launched and invited the submission of research and review articles that can help to enhance our understanding of both the original and metaphorical butterfly effects. The Special Issue also sought recent developments in idealized Lorenz models and real-world models that address multistability, multiscale predictability, and sensitivity. The call for papers was opened 15 months prior to the completion of the Special Issue and features nine selected papers. This editorial provides a brief review of Lorenz models, introduces the published papers, and summarizes each one of them. Full article

(This article belongs to the Special Issue 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models)

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Research

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13 pages, 955 KiB

Open AccessArticle

Can the Flap of a Butterfly’s Wings Shift a Tornado into Texas—Without Chaos?

by Yoshitaka Saiki and James A. Yorke

Atmosphere 2023, 14(5), 821; https://doi.org/10.3390/atmos14050821 - 02 May 2023

Cited by 3 | Viewed by 2969

Abstract

In our title, “chaos” means there is a positive Lyapunov exponent that causes the tornado to move. We are asserting that a positive Lyapunov exponent is not always needed to have a butterfly effect. Lorenz’s butterfly effect initially appeared in meteorology and has captured the imaginations of people for applications to all kinds of fields. We feel it is important to understand simpler non-meteorological models to understand the additional aspects of the butterfly effect. This paper presents simple linear map models that lack “chaos” but exhibit a butterfly effect: our simplest model does not have any positive Lyapunov exponents but still exhibits a butterfly effect, that is, temporary exponential growth from a tiny perturbation such as one infected mosquito setting off an epidemic outbreak. We focus on a 24-dimensional version of the map where a significant butterfly effect is observed even though the only Lyapunov exponent is 0. We introduce a linear “infected mosquito” model that shows how off-diagonal matrix entries can cause a finite-time growth rate. We argue that the degree of instability in our systems can be better measured by its finite-time growth rate. Our findings suggest that even in linear systems, off-diagonal matrix entries can significantly impact the system’s behavior and be more important than the Lyapunov exponents in higher-dimensional systems. A focus on finite-time growth rates can yield valuable insights into the system’s dynamics. Full article

(This article belongs to the Special Issue 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models)

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15 pages, 3823 KiB

Open AccessArticle

An Expanded Sensitivity Study of Simulated Storm Life Span to Ventilation Parameterization in a Cloud Resolving Model

by Yen-Liang Chou and Pao-Kuan Wang

Atmosphere 2023, 14(4), 720; https://doi.org/10.3390/atmos14040720 - 15 Apr 2023

Cited by 1 | Viewed by 1053

Abstract

We performed a sensitivity study on the life span of a numerically simulated storm using the parameterization of the ventilation coefficient. This is an expanded sequel to our previous study, where the ventilation effect of precipitation particles (snow, rain, and hail) was either halved or doubled as a whole. In this study, we tested the sensitivity of the ventilation coefficient for different precipitation particles and compared that with the previous results. In the present study, we changed the ventilation coefficient in two scenarios: (1) only the rain category was changed; (2) only the snow and hail categories were changed. The results show that these different scenarios lead to different evolution paths for the storm. In general, reducing the ventilation effect of rain leads to quick dissipation, whereas enhancing the ventilation of either rain or snow/hail leads to the development of multicellular storms. An analysis of the physical mechanisms leading to such results is provided. This study shows yet another example of how a change in a cloud’s microphysical parameterization can lead to a profound change in its larger-scale dynamical process. Full article

(This article belongs to the Special Issue 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models)

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35 pages, 8058 KiB

Open AccessArticle

Atmospheric Instability and Its Associated Oscillations in the Tropics

by Xiping Zeng

Atmosphere 2023, 14(3), 433; https://doi.org/10.3390/atmos14030433 - 21 Feb 2023

Cited by 3 | Viewed by 1555

Abstract

The interaction between tropical clouds and radiation is studied in the context of the weak temperature gradient approximation, using very low order systems (e.g., a two-column two-layer model) as a zeroth-order approximation. Its criteria for the instability are derived in the systems. Owing to the connection between the instability (unstable fixed point) and the oscillation (limit cycle) in physics (phase) space, the systems suggest that the instability of tropical clouds and radiation leads to the atmospheric oscillations with distinct timescales observed. That is, the instability of the boundary layer quasi-equilibrium leads to the quasi-two-day oscillation, the instability of the radiative convective equilibrium leads to the Madden–Julian oscillation (MJO), and the instability of the radiative convective flux equilibrium leads to the El Niño–southern oscillation. In addition, a linear model as a first-order approximation is introduced to reveal the zonal asymmetry of the atmospheric response to a standing convective/radiative heating oscillation. Its asymmetric resonance conditions explain why a standing ~45-day oscillation in the systems brings about a planetary-scale eastward travelling vertical circulation like the MJO. The systems, despite of their simplicity, replicate the oscillations with the distinct timescales observed, providing a novel cloud parameterization for weather and climate models. Their instability criteria further suggests that the models can successfully predict the oscillations if they properly represent cirrus clouds and convective downdrafts in the tropics. Full article

(This article belongs to the Special Issue 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models)

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16 pages, 4817 KiB

Open AccessEditor’s ChoiceArticle

The Dual Nature of Chaos and Order in the Atmosphere

by Bo-Wen Shen, Roger Pielke, Sr., Xubin Zeng, Jialin Cui, Sara Faghih-Naini, Wei Paxson, Amit Kesarkar, Xiping Zeng and Robert Atlas

Atmosphere 2022, 13(11), 1892; https://doi.org/10.3390/atmos13111892 - 12 Nov 2022

Cited by 10 | Viewed by 3978

Abstract

In the past, the Lorenz 1963 and 1969 models have been applied for revealing the chaotic nature of weather and climate and for estimating the atmospheric predictability limit. Recently, an in-depth analysis of classical Lorenz 1963 models and newly developed, generalized Lorenz models suggested a revised view that “the entirety of weather possesses a dual nature of chaos and order with distinct predictability”, in contrast to the conventional view of “weather is chaotic”. The distinct predictability associated with attractor coexistence suggests limited predictability for chaotic solutions and unlimited predictability (or up to their lifetime) for non-chaotic solutions. Such a view is also supported by a recent analysis of the Lorenz 1969 model that is capable of producing both unstable and stable solutions. While the alternative appearance of two kinds of attractor coexistence was previously illustrated, in this study, multistability (for attractor coexistence) and monostability (for single type solutions) are further discussed using kayaking and skiing as an analogy. Using a slowly varying, periodic heating parameter, we additionally emphasize the predictable nature of recurrence for slowly varying solutions and a less predictable (or unpredictable) nature for the onset for emerging solutions (defined as the exact timing for the transition from a chaotic solution to a non-chaotic limit cycle type solution). As a result, we refined the revised view outlined above to: “The atmosphere possesses chaos and order; it includes, as examples, emerging organized systems (such as tornadoes) and time varying forcing from recurrent seasons”. In addition to diurnal and annual cycles, examples of non-chaotic weather systems, as previously documented, are provided to support the revised view. Full article

(This article belongs to the Special Issue 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models)

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27 pages, 4662 KiB

Open AccessArticle

Role of the Observability Gramian in Parameter Estimation: Application to Nonchaotic and Chaotic Systems via the Forward Sensitivity Method

by John M. Lewis and Sivaramakrishnan Lakshmivarahan

Atmosphere 2022, 13(10), 1647; https://doi.org/10.3390/atmos13101647 - 10 Oct 2022

Cited by 1 | Viewed by 1421

Abstract

Data assimilation in chaotic regimes is challenging, and among the challenging aspects is placement of observations to induce convexity of the cost function in the space of control. This problem is examined by using Saltzman’s spectral model of convection that admits both chaotic and nonchaotic regimes and is controlled by two parameters—Rayleigh and Prandtl numbers. The problem is simplified by stripping the seven-variable constraint to a three-variable constraint. Since emphasis is placed on observation positioning to avoid cost-function flatness, forecast sensitivity to controls is needed. Four-dimensional variational assimilation (4D-Var) is silent on this issue of observation placement while Forecast Sensitivity Method (FSM) delivers sensitivities used in placement. With knowledge of the temporal forecast sensitivity matrix

V

, derivatives of the forecast variables to controls, the cost function can be expressed as a function of the observability Gramian

V^{T} V

using first-order Taylor series expansion. The goal is to locate observations at places that force the Gramian positive definite. Further, locations are chosen such that the condition number of

V^{T} V

is small and this guarantees convexity in the vicinity of the cost function minimum. Four numerical experiments are executed, and results are compared with the structure of the cost function independently determined though arduous computation over a wide range of the two nondimensional numbers. The results are especially good based on reduction in cost function value and comparison with cost function structure. Full article

(This article belongs to the Special Issue 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models)

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30 pages, 11752 KiB

Open AccessArticle

Time-Lagged Ensemble Quantitative Precipitation Forecasts for Three Landfalling Typhoons in the Philippines Using the CReSS Model, Part I: Description and Verification against Rain-Gauge Observations

by Chung-Chieh Wang, Chien-Hung Tsai, Ben Jong-Dao Jou and Shirley J. David

Atmosphere 2022, 13(8), 1193; https://doi.org/10.3390/atmos13081193 - 28 Jul 2022

Cited by 6 | Viewed by 1862

Abstract

In this study, the 2.5 km Cloud-Resolving Storm Simulator was applied to forecast the rainfall by three landfalling typhoons in the Philippines at high resolution: Mangkhut (2018), Koppu (2015), and Melor (2015), using a time-lagged strategy for ensemble. The three typhoons penetrated northern Luzon, central Luzon, and the middle of the Philippine Archipelago, respectively, and the present study verified the track and quantitative precipitation forecasts (QPFs) using categorical statistics against observations at 56 rain-gauge sites at seven thresholds up to 500 mm. The predictability of rainfall is the highest for Koppu, followed by Melor, and the lowest for Mangkhut, which had the highest peak rainfall amount. Targeted at the most-rainy 24 h of each case, the threat score (TS) within the short range (≤72 h) could reach 1.0 for Koppu at 350 mm in many runs (peak observation = 502 mm), and 1.0 for Mangkhut and 0.25 for Melor (peak observation = 407 mm) both at 200 mm in the best member, when the track errors were small enough. For rainfall from entire events (48 or 72 h), TS hitting 1.0 could also be achieved regularly at 500 mm for Koppu (peak observation = 695 mm), and 0.33 at 350 mm for Melor (407 mm) and 0.46 at 200 mm for Mangkhut (786 mm) in the best case. At lead times beyond the short range, one third of these earlier runs also produced good QPFs for both Koppu and Melor, but such runs were fewer for Mangkhut and the quality of QPFs was also not as high due to larger northward track biases. Overall, the QPF results are very encouraging, and comparable to the skill level for typhoon rainfall in Taiwan (with similar peak rainfall amounts). Thus, at high resolution, there is a fair chance to make decent QPFs even at lead times of 3–7 days before typhoon landfall in the Philippines, with useful information on rainfall scenarios for early preparation. Full article

(This article belongs to the Special Issue 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models)

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12 pages, 9542 KiB

Open AccessArticle

An ISOMAP Analysis of Sea Surface Temperature for the Classification and Detection of El Niño & La Niña Events

by John Chien-Han Tseng

Atmosphere 2022, 13(6), 919; https://doi.org/10.3390/atmos13060919 - 06 Jun 2022

Cited by 6 | Viewed by 3344

Abstract

Isometric feature mapping (ISOMAP) is a nonlinear dimensionality reduction method used for extracting features from spatiotemporal data. The traditional principal component analysis (PCA), a linear dimensionality reduction method, measures the distance between two data points based on the Euclidean distance (line segment), which cannot reflect the actual distance between the data points in a nonlinear space. By contrast, the ISOMAP measures the distance between two data points based on the geodesic distance, which more closely reflects the actual distance by the view of tracing along the local linearity in the original nonlinear structure. Thus, ISOMAP-reconstructed data points can reflect the features of real structures and can be classified more accurately than traditional PCA-reconstructed data points. Moreover, these ISOMAP-reconstructed data points can be used for cluster analysis by emphasizing the differences among the points more than those by the traditional PCA. In this study, sea surface temperature (SST) data points reconstructed using the traditional PCA and ISOMAP were compared. The classification based on these reconstructed SST points was tested using the Niño 3.4 index, which labels El Niño, La Niña, or normal events. The mean differences from the ISOMAP data points were larger than those from the traditional PCA data points. The ISOMAP not only helped differentiate the points in two different events but also provided better difference measurement of the points belonging to the same class (e.g., 82/83 and 97/98 El Niño events). On examining the evolution of the leading three temporal eigen components of the SST PCA, or especially the SST ISOMAP, we found that the trajectories were similar to the Lorenz 63 model on a phase space figure. This implies that NWP perturbations can be traced using the ISOMAP to measure growing unstable behaviors. Spatial eigenmodes (empirical orthogonal function) between the traditional PCA and ISOMAP were also determined and compared herein. Full article

(This article belongs to the Special Issue 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models)

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21 pages, 5196 KiB

Open AccessArticle

One Saddle Point and Two Types of Sensitivities within the Lorenz 1963 and 1969 Models

by Bo-Wen Shen, Roger A. Pielke, Sr. and Xubin Zeng

Atmosphere 2022, 13(5), 753; https://doi.org/10.3390/atmos13050753 - 07 May 2022

Cited by 9 | Viewed by 4258

Abstract

The fact that both the Lorenz 1963 and 1969 models suggest finite predictability is well known. However, less well known is the fact that the mechanisms (i.e., sensitivities) within both models, which lead to finite predictability, are different. Additionally, the mathematical and physical relationship between these two models has not been fully documented. New analyses, along with a literature review, are performed here to provide insights regarding similarities and differences for these two models. The models represent different physical systems, one for convection and the other for barotropic vorticity. From the perspective of mathematical complexities, the Lorenz 1963 (L63) model is limited-scale and nonlinear; and the Lorenz 1969 (L69) model is closure-based, physically multiscale, mathematically linear, and numerically ill-conditioned. The former possesses a sensitive dependence of solutions on initial conditions, known as the butterfly effect, and the latter contains numerical sensitivities due to an ill-conditioned matrix with a large condition number (i.e., a large variance of growth rates). Here, we illustrate that the existence of a saddle point at the origin is a common feature that produces instability in both systems. Within the chaotic regime of the L63 nonlinear model, unstable growth is constrained by nonlinearity, as well as dissipation, yielding time varying growth rates along an orbit, and, thus, a dependence of (finite) predictability on initial conditions. Within the L69 linear model, multiple unstable modes at various growth rates appear, and the growth of a specific unstable mode (i.e., the most unstable mode during a finite time interval) is constrained by imposing a saturation assumption, thereby yielding a time varying system growth rate. Both models were interchangeably applied for qualitatively revealing the nature of finite predictability in weather and climate. However, only single type solutions were examined (i.e., chaotic and linearly unstable solutions for the L63 and L69 models, respectively), and the L69 system is ill-conditioned and easily captures numerical instability. Thus, an estimate of the predictability limit using either of the above models, with or without additional assumptions (e.g., saturation), should be interpreted with caution and should not be generalized as an upper limit for atmospheric predictability. Full article

(This article belongs to the Special Issue 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models)

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Review

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124 pages, 123747 KiB

Open AccessReview

Challenges and Progress in Computational Geophysical Fluid Dynamics in Recent Decades

by Wen-Yih Sun

Atmosphere 2023, 14(9), 1324; https://doi.org/10.3390/atmos14091324 - 22 Aug 2023

Cited by 2 | Viewed by 1426

Abstract

Here we present the numerical methods, applications, and comparisons with observations and previous studies. It includes numerical analyses of shallow water equations, Sun’s scheme, and nonlinear model simulations of a dam break, solitary Rossby wave, and hydraulic jump without smoothing. We reproduce the longitude and transverse cloud bands in the Equator; two-day mesoscale waves in Brazil; Ekman spirals in the atmosphere and oceans, and a resonance instability at 30° from the linearized equations. The Purdue Regional Climate Model (PRCM) reproduces the explosive severe winter storms in the Western USA; lee-vortices in Taiwan; deformation of the cold front by mountains in Taiwan; flooding and drought in the USA; flooding in Asia; and the Southeast Asia monsoons. The model can correct the small-scale errors if the synoptic systems are correct. Usually, large-scale systems are more important than small-scale disturbances, and the predictability of NWP is better than the simplified dynamics models. We discuss the difference between Boussinesq fluid and the compressible fluid. The Bernoulli function in compressible atmosphere conserving the total energy, is better than the convective available potential energy (CAPE) or the Froude number, because storms can develop without CAPE, and downslope wind can form against a positive buoyancy. We also present a new terrain following coordinate, a turbulence-diffusion model in the convective boundary layer (CBL), and a new backward-integration model including turbulence mixing in the atmosphere. Full article

(This article belongs to the Special Issue 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models)

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