Prediction Rules for Regime Changes and Length in a New Regime for the Lorenz Model

Abstract Despite the widespread use of the Lorenz model as a conceptual model for predictability studies in meteorology, only Evans et al. seem to have studied the prediction of occurrence of regime changes and their duration. In this paper, simpler rules are presented for forecasting regime changes and their lengths, with near-perfect forecasting accuracy. It is found that when |x(t)| is greater than a critical value xc, the current regime will end after it completes the current orbit. Moreover, the length n of the new regime increases monotonically with the maximum value xm of |x(t)| in the previous regime. A best-fit cubic expression provides a very good estimate of n for the next regime, given xm for the previous regime. Similar forecasting rules are also obtained for regime changes in the forced Lorenz model. This model was introduced by Palmer and used as a conceptual model to explore the effects of sea surface temperature on seasonal mean rainfall. It was found that for the forced Lorenz model, the critical value xc changed linearly with the forcing parameter providing bias to one of the regimes. Similar regime prediction rules have been found in some other two-regime attractors. It seems these forecasting rules are a generic property of a large class of two-regime attractors. Although as a conceptual model, the Lorenz model cannot be taken very literally, these results suggest a relationship between magnitudes of maximum anomaly in one regime, for example, the active spell, and duration of the subsequent break spell.

Download Full-text

Nonlinear feedback in a six-dimensional Lorenz Model: impact of an additional heating term

Nonlinear Processes in Geophysics Discussions ◽

10.5194/npgd-2-475-2015 ◽

2015 ◽

Vol 2 (2) ◽

pp. 475-512

Author(s):

B.-W. Shen

Keyword(s):

Rayleigh Number ◽

Numerical Solutions ◽

Critical Rayleigh Number ◽

Critical Value ◽

Small Scale ◽

Solution Stability ◽

Nonlinear Feedback ◽

Lorenz Model ◽

Additional Heating ◽

The Impact

Abstract. In this study, a six-dimensional Lorenz model (6DLM) is derived, based on a recent study using a five-dimensional (5-D) Lorenz model (LM), in order to examine the impact of an additional mode and its accompanying heating term on solution stability. The new mode added to improve the representation of the steamfunction is referred to as a secondary streamfunction mode, while the two additional modes, that appear in both the 6DLM and 5DLM but not in the original LM, are referred to as secondary temperature modes. Two energy conservation relationships of the 6DLM are first derived in the dissipationless limit. The impact of three additional modes on solution stability is examined by comparing numerical solutions and ensemble Lyapunov exponents of the 6DLM and 5DLM as well as the original LM. For the onset of chaos, the critical value of the normalized Rayleigh number (rc) is determined to be 41.1. The critical value is larger than that in the 3DLM (rc ~ 24.74), but slightly smaller than the one in the 5DLM (rc ~ 42.9). A stability analysis and numerical experiments obtained using generalized LMs, with or without simplifications, suggest the following: (1) negative nonlinear feedback in association with the secondary temperature modes, as first identified using the 5DLM, plays a dominant role in providing feedback for improving the solution's stability of the 6DLM, (2) the additional heating term in association with the secondary streamfunction mode may destabilize the solution, and (3) overall feedback due to the secondary streamfunction mode is much smaller than the feedback due to the secondary temperature modes; therefore, the critical Rayleigh number of the 6DLM is comparable to that of the 5DLM. The 5DLM and 6DLM collectively suggest different roles for small-scale processes (i.e., stabilization vs. destabilization), consistent with the following statement by Lorenz (1972): 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. The implications of this and previous work, as well as future work, are also discussed.

Download Full-text

A New Conceptual Model for Adiabatic Fog

10.5194/acp-2020-1314 ◽

2021 ◽

Author(s):

Felipe Toledo ◽

Martial Haeffelin ◽

Eivind Wærsted ◽

Jean-Charles Dupont

Keyword(s):

Life Cycle ◽

Conceptual Model ◽

Real Time ◽

Liquid Water ◽

Critical Value ◽

Liquid Water Path ◽

Horizontal Visibility ◽

Fog Water ◽

Water Path ◽

The Way

Abstract. We propose a new paradigm to describe the temporal evolution of continental fog layers. This paradigm defines fog as a layer saturated from the surface to a known upper boundary, and whose liquid water path (LWP) exceeds a critical value, the critical liquid water path (CLWP). When the LWP is less than the CLWP the fog water cannot extend all the way to the surface, leading to a surface horizontal visibility greater than 1 km. On the opposite, when the LWP is larger than the CLWP, the fog water extends all the way to the surface, inducing a horizontal visibility less than 1 km. The excess water with respect to the critical value is then defined as the reservoir liquid water path (RLWP). The new fog paradigm is formulated as a conceptual model that relates the liquid water path of adiabatic fog with its thickness and surface liquid water content, and allows the critical and reservoir liquid water paths to be computed. Both variables can be tracked in real time using vertical profiling measurements, enabling a real time diagnostic of fog status. The conceptual model is tested using data from seven years of measurements performed at the SIRTA observatory, combining cloud radar, microwave radiometer, ceilometer, scatterometer and weather station measurements. In this time period we found 80 fog events with reliable measurements, with 56 of these lasting more than three hours. The paper presents the conceptual model and its capability to derive the LWP from the fog CTH and surface horizontal visibility with an RMS uncertainty of 10.5 g m−2. The impact of fog liquid water path and fog top height variations on fog life cycle (formation to dissipation) is presented based on four case studies, and statistics derived from 56 fog events. Our results show in particular that the reservoir liquid water path is consistently positive during the mature phase of the fog and that it starts to decrease quasi monotonously about one hour before dissipation, reaching a near-zero value at the time of dissipation. The reservoir liquid water path and its time derivative could hence be used as an indicator for life cycle stage and support short range forecasting of fog dissipation.

Download Full-text

A nudging-based data assimilation method: the Back and Forth Nudging (BFN) algorithm

Nonlinear Processes in Geophysics ◽

10.5194/npg-15-305-2008 ◽

2008 ◽

Vol 15 (2) ◽

pp. 305-319 ◽

Cited By ~ 56

Author(s):

D. Auroux ◽

J. Blum

Keyword(s):

Data Assimilation ◽

Variational Method ◽

Burgers Equation ◽

Reconstruction Error ◽

Sea Surface ◽

Reference Trajectory ◽

Discrete Observations ◽

Lorenz Model ◽

Time Period ◽

The Difference

Abstract. This paper deals with a new data assimilation algorithm, called Back and Forth Nudging. The standard nudging technique consists in adding to the equations of the model a relaxation term that is supposed to force the observations to the model. The BFN algorithm consists in repeatedly performing forward and backward integrations of the model with relaxation (or nudging) terms, using opposite signs in the direct and inverse integrations, so as to make the backward evolution numerically stable. This algorithm has first been tested on the standard Lorenz model with discrete observations (perfect or noisy) and compared with the variational assimilation method. The same type of study has then been performed on the viscous Burgers equation, comparing again with the variational method and focusing on the time evolution of the reconstruction error, i.e. the difference between the reference trajectory and the identified one over a time period composed of an assimilation period followed by a prediction period. The possible use of the BFN algorithm as an initialization for the variational method has also been investigated. Finally the algorithm has been tested on a layered quasi-geostrophic model with sea-surface height observations. The behaviours of the two algorithms have been compared in the presence of perfect or noisy observations, and also for imperfect models. This has allowed us to reach a conclusion concerning the relative performances of the two algorithms.

Download Full-text

On the Optimal Use of Isotherm Models for the Characterization of Biosorption of Lead Onto Algae

10.1101/017426 ◽

2015 ◽

Author(s):

F. Brouers ◽

Tariq Al-Musawi

Keyword(s):

Algal Biomass ◽

Removal Rate ◽

Isotherm Models ◽

Good Representation ◽

Experimental Isotherm ◽

Nonlinear Fitting ◽

Maximum Value ◽

Complete Set ◽

Best Fit

In this study the experimental isotherm data of biosorption of Pb(II) onto algae was modeled using several models. These models are: Langmuir, Hill-Sips, Brouers-Sotolongo, Brouers-Gaspard, and Redlich-Peterson models. The coefficients of each model were determined by non-linear fitting using Mathematica9 program. The maximum Pb(II) removal rate increased with the increase of temperature and reached the maximum value (98%) at the temperature of 40°C. Even if the R2 error quantity is not the unique and always the best measure for nonlinear fitting, the Brouers-Sotolongo model gives in all cases the best fit and is definitely the most suitable one to satisfactorily describe bioisorption of Pb(II) on the algal biomass. In addition, this study shows that a complete set of data is necessary to have a good representation of the isotherm.

Download Full-text

Boundary-layer separation of rotating flows past surface-mounted obstacles

Journal of Fluid Mechanics ◽

10.1017/s0022112092003446 ◽

1992 ◽

Vol 237 ◽

pp. 343-371 ◽

Cited By ~ 5

Author(s):

K. J. Richards ◽

D. A. Smeed ◽

E. J. Hopfinger ◽

G. Chabert D'Hières

Keyword(s):

Flow Separation ◽

Three Dimensional ◽

Separated Flow ◽

Boundary Layer Separation ◽

Critical Value ◽

Complex Stress ◽

Rotating Fluid ◽

Separation Line ◽

Maximum Value ◽

The Hill

This paper describes laboratory experiments on the flow over a three-dimensional hill in a rotating fluid. The experiments were carried out in towing tanks, placed on rotating tables. Rotation is found to have a strong influence on the separation behind the hill. The topology of the separation is found to be the same for all the flows examined. The Rossby number R in the experiments is of order 1, the maximum value being 6. The separated flow is dominated by a single trailing vortex. In the majority of cases the surface stress field has a single separation line and there are no singular points. In a few experiments at the highest Rossby numbers the observations suggest more complex stress fields but the results are inconclusive.A criterion for flow separation is sought. For values of D/L > 1, where D is the depth of the flow and L the lengthscale of the hill, separation is found to be primarily dependent on R. At sufficiently small values of R separation is suppressed and the flow remains fully attached.Linear theory is found to give a good estimate for the critical value of R for flow separation. For hills with a moderate slope (slope ≤ 1) this critical value is around 1, decreasing with increasing slope. It is postulated that the existence of a single dominant trailing vortex is due to the uplifting and subsequent turning of transverse vorticity generated by surface pressure forces upstream of the separation line.

Download Full-text

A critical humidity threshold for monsoon transitions

Climate of the Past Discussions ◽

10.5194/cpd-7-1737-2011 ◽

2011 ◽

Vol 7 (3) ◽

pp. 1737-1765 ◽

Cited By ~ 6

Author(s):

J. Schewe ◽

A. Levermann ◽

H. Cheng

Keyword(s):

Conceptual Model ◽

Monsoon Rainfall ◽

Reanalysis Data ◽

Sea Surface ◽

Specific Humidity ◽

Critical Threshold ◽

Monsoon Region ◽

Proxy Records ◽

Moisture Advection ◽

Monsoon Strength

Abstract. Monsoon systems around the world are governed by the so-called moisture-advection feedback. Here we show that, in a minimal conceptual model, this feedback implies a critical threshold with respect to the atmospheric specific humidity qo over the ocean adjacent to the monsoon region. If qo falls short of this critical value qoc, monsoon rainfall over land cannot be sustained. Such a case could occur if evaporation from the ocean was reduced, e.g. due to low sea surface temperatures. Within the restrictions of the conceptual model, we estimate qoc from present-day reanalysis data for four major monsoon systems, and demonstrate how this concept can help understand abrupt variations in monsoon strength on orbital timescales as found in proxy records.

Download Full-text

Impact of global cooling on Early Cretaceous high pCO2 world during the Weissert Event

Nature Communications ◽

10.1038/s41467-021-25706-0 ◽

2021 ◽

Vol 12 (1) ◽

Author(s):

Liyenne Cavalheiro ◽

Thomas Wagner ◽

Sebastian Steinig ◽

Cinzia Bottini ◽

Wolf Dummann ◽

...

Keyword(s):

Early Cretaceous ◽

Weddell Sea ◽

Sea Surface ◽

High Pco2 ◽

Polar Ice ◽

Surface Cooling ◽

Climate Simulations ◽

Global Cooling ◽

Best Fit ◽

Global Mean

AbstractThe Weissert Event ~133 million years ago marked a profound global cooling that punctuated the Early Cretaceous greenhouse. We present modelling, high-resolution bulk organic carbon isotopes and chronostratigraphically calibrated sea surface temperature (SSTs) based on an organic paleothermometer (the TEX86 proxy), which capture the Weissert Event in the semi-enclosed Weddell Sea basin, offshore Antarctica (paleolatitude ~54 °S; paleowater depth ~500 meters). We document a ~3–4 °C drop in SST coinciding with the Weissert cold end, and converge the Weddell Sea data, climate simulations and available worldwide multi-proxy based temperature data towards one unifying solution providing a best-fit between all lines of evidence. The outcome confirms a 3.0 °C ( ±1.7 °C) global mean surface cooling across the Weissert Event, which translates into a ~40% drop in atmospheric pCO2 over a period of ~700 thousand years. Consistent with geologic evidence, this pCO2 drop favoured the potential build-up of local polar ice.

Download Full-text

Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model

10.5194/npg-2016-10 ◽

2016 ◽

Author(s):

Bo-Wen Shen

Keyword(s):

Stability Analysis ◽

Prandtl Number ◽

Analytical Solutions ◽

Feedback Loop ◽

Critical Value ◽

Scale Dependence ◽

Nonlinear Feedback ◽

Lorenz Model ◽

Onset Of Chaos ◽

Hierarchical Scale

Abstract. In this study, 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. Compared to the 5DLM, the 7DLM includes two additional high wavenumber modes that are selected based on an analysis of the nonlinear temperature advection term, a Jacobian term (J(ψ, θ)), where, ψ and θ represent the streamfunction and temperature perturbations, respectively. Fourier modes that represent temperature in the 7DLM can be categorized into three major scales as the primary (the largest scale), secondary, and tertiary (the smallest scale) modes. Further extension of the nonlinear feedback loop within the 7DLM can provide negative nonlinear feedback to stabilize solutions, thus leading to a much larger critical value for the Rayleigh parameter (rc ∼ 116.9) for the onset of chaos, as compared to an rc of 42.9 for the 5DLM as well as an rc of 24.74 for the 3DLM. The rc is determined by an analysis of ensemble Lyapunov exponents (eLEs) with Prandtl number (σ) of 10. To examine the dependence of rc on the value of the Prandtl number, a linear stability analysis is performed near the nontrivial critical point using a wide range of Rayleigh parameter (40 ≤ r ≤ 195) and Prandtl number (5 ≤ σ ≤ 25). Then an eLE analysis is conducted using selected values of the Prandtl number. The linear stability analysis is done by solving for the analytical solutions of the critical points, by linearizing the 7DLM with respect to the analytical solutions, and by calculating the eigenvalues of the linearized system. Within the range of (5 ≤ σ ≤ 25), the 7DLM requires a larger rc for the onset of chaos than the 5DLM. In addition to the negative nonlinear feedback illustrated and emulated by the quasi-equilibrium state solutions for high wavenumber modes, the 7DLM reveals the hierarchical scale dependence of chaotic solutions. For chaotic solutions with r = 120, the Pearson correlation coefficients (PCCs) between the primary and secondary modes (i.e., Z and Z1) and between the secondary and tertiary modes (i.e., Z1 and Z2) are 0.988 and 0.998, respectively. Here, Z, Z1, and Z2 represent the time-varying amplitudes of the primary, secondary, and tertiary modes, respectively. High PCCs indicate a strong linear relationship among the modes at various scales and a hierarchy of scale dependence. Future work will be undertaken to examine how higher dimensional LMs may produce a larger critical value for the Rayleigh parameter for the onset of chaos and reveal stronger hierarchical scale dependence.

Download Full-text

Testing for the Equality of Variance for the Four-parameter Logistic (4PL) Model for the Detection of Vibrio cholerae DNA with Polystyrene-coacrylic Acid Composite Nanospheres

Bulletin of Environmental Science and Sustainable Management (e-ISSN 2716-5353) ◽

10.54987/bessm.v5i2.647 ◽

2021 ◽

Vol 5 (2) ◽

pp. 5-7

Author(s):

Mohd Yunus Shukor

Keyword(s):

Vibrio Cholerae ◽

Statistical Tests ◽

Critical Value ◽

P Value ◽

Effective Technique ◽

Bartlett’S Test ◽

Homogeneity Of Variance ◽

The Difference ◽

Residual Variances ◽

Best Fit

When it comes to finding the best fit of nonlinear curves to acceptable models, linear regression with least squares is the most effective technique. Because residuals (the difference between observed and predicted data) must follow a normal distribution and the data must be free of outliers and uniform variance, statistical tests are used to identify the most appropriate model for a given situation (homoscedasticity). If all of these characteristics are satisfied, the system is said to be robust. In parametric nonlinear regression, one of the numerous assumptions is that the within-group variances of the groups are all the same, which is one of several assumptions (exhibit homoscedasticity). If the variances vary from one another (show heteroscedasticity), then the model is not statistically competent to describe the data as a whole. Data on the detection of Vibrio cholerae DNA with polystyrene-coacrylic acid composite nanospheres as modelled using the nonlinear four-parameter logistic (4PL) regression was preliminary check for homogeneity of variance using the Bartlett’s and Levene’s tests. It was found that the critical value of 2 was 28.869, according to Bartlett's test findings. Excel's CHIDIST function yielded a probability of 0.389 (not significant), suggesting that the variances of the residuals did not change significantly. The p-value for Levenes's test was 0.917, indicating that there were no distinct changes between the residual variances meaning that the use of the 4-PL model in fitting the data was adequate statistically.

Download Full-text