scholarly journals Multistability and rare spontaneous transitions in barotropic β-plane turbulence

2021 ◽  
Author(s):  
Eric Simonnet ◽  
Joran Rolland ◽  
Freddy Bouchet
Keyword(s):  
Large Deviation ◽  
Numerical Models ◽  
Rare Event ◽  
Transition Time ◽  
Arrhenius Law ◽  
Zonal Jets ◽  
Effective Dynamics ◽  
Beta Plane ◽  
Exponential Decrease ◽  
Quasigeostrophic Model

AbstractWe demonstrate that turbulent zonal jets, analogous to Jovian ones, which are quasi-stationary, are actually metastable. After extremely long times, they randomly switch to new configurations with a different number of jets. The genericity of this phenomenon suggests that most quasi-stationary turbulent planetary atmospheres might have many climates and attractors for fixed values of the external forcing parameters. A key message is that this situation will usually not be detected by simply running the numerical models, because of the extremely long mean transition time to change from one climate to another. In order to study such phenomena, we need to use specific tools: rare event algorithms and large deviation theory. With these tools, we make a full statistical mechanics study of a classical barotropic beta-plane quasigeostrophic model. It exhibits robust bimodality with abrupt transitions. We show that new jets spontaneously nucleate from westward jets. The numerically computed mean transition time is consistent with an Arrhenius law showing an exponential decrease of the probability as the Ekman dissipation decreases. This phenomenology is controlled by rare noise-driven paths called instantons. Moreover, we compute the saddles of the corresponding effective dynamics. For the dynamics of states with three alternating jets, we uncover an unexpectedly rich dynamics governed by the symmetric group of permutations, with two distinct families of instantons, which is a surprise for a system where everything seemed stationary in the hundreds of previous simulations of this model. We discuss the future generalization of our approach to more realistic models.

A conditional limit theorem for high-dimensional ℓᵖ-spheres

2018 ◽  
Vol 55 (4) ◽  
pp. 1060-1077 ◽  
Author(s):  
Steven S. Kim ◽  
Kavita Ramanan
Keyword(s):  
Limit Theorem ◽  
Large Deviation ◽  
Convex Geometry ◽  
Rare Event ◽  
High Dimensional ◽  
Random Projections ◽  
Dimensional Euclidean Space ◽  
High Dimensions ◽  
Geometric Setting ◽  
Conditional Limit

Abstract The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓp-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓp-spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓp-balls in a high-dimensional Euclidean space.

State-dependent importance sampling for regularly varying random walks

2008 ◽  
Vol 40 (04) ◽  
pp. 1104-1128 ◽  
Author(s):  
Jose H. Blanchet ◽  
Jingchen Liu
Keyword(s):  
Importance Sampling ◽  
Large Deviation ◽  
Rare Event ◽  
Parametric Family ◽  
Finite Variance ◽  
Or Efficiency ◽  
State Dependent ◽  
Regularly Varying ◽  
Heavy Tailed ◽  
Path Dependent

Consider a sequence (X k : k ≥ 0) of regularly varying independent and identically distributed random variables with mean 0 and finite variance. We develop efficient rare-event simulation methodology associated with large deviation probabilities for the random walk (S n : n ≥ 0). Our techniques are illustrated by examples, including large deviations for the empirical mean and path-dependent events. In particular, we describe two efficient state-dependent importance sampling algorithms for estimating the tail of S n in a large deviation regime as n ↗ ∞. The first algorithm takes advantage of large deviation approximations that are used to mimic the zero-variance change of measure. The second algorithm uses a parametric family of changes of measure based on mixtures. Lyapunov-type inequalities are used to appropriately select the mixture parameters in order to guarantee bounded relative error (or efficiency) of the estimator. The second example involves a path-dependent event related to a so-called knock-in financial option under heavy-tailed log returns. Again, the importance sampling algorithm is based on a parametric family of mixtures which is selected using Lyapunov bounds. In addition to the theoretical analysis of the algorithms, numerical experiments are provided in order to test their empirical performance.

scholarly journals Computation of extreme heat waves in climate models using a large deviation algorithm

2017 ◽  
Vol 115 (1) ◽  
pp. 24-29 ◽  
Author(s):  
Francesco Ragone ◽  
Jeroen Wouters ◽  
Freddy Bouchet
Keyword(s):  
Statistical Physics ◽  
Climate Models ◽  
Large Deviation ◽  
Heat Waves ◽  
Rare Event ◽  
Teleconnection Pattern ◽  
Extreme Heat ◽  
Sampling Efficiency ◽  
Wide Range ◽  
The Impact

Studying extreme events and how they evolve in a changing climate is one of the most important current scientific challenges. Starting from complex climate models, a key difficulty is to be able to run long enough simulations to observe those extremely rare events. In physics, chemistry, and biology, rare event algorithms have recently been developed to compute probabilities of events that cannot be observed in direct numerical simulations. Here we propose such an algorithm, specifically designed for extreme heat or cold waves, based on statistical physics. This approach gives an improvement of more than two orders of magnitude in the sampling efficiency. We describe the dynamics of events that would not be observed otherwise. We show that European extreme heat waves are related to a global teleconnection pattern involving North America and Asia. This tool opens up a wide range of possible studies to quantitatively assess the impact of climate change.

Scaling, spectra and zonal jets in beta-plane turbulence

2004 ◽  
Vol 16 (7) ◽  
pp. 2592-2603 ◽  
Author(s):  
Sergey Danilov ◽  
David Gurarie
Keyword(s):  
Zonal Jets ◽  
Beta Plane

scholarly journals Role of Eddies in the Maintenance of Multiple Jets Embedded in Eastward and Westward Baroclinic Shears

Fluids ◽  
2018 ◽  
Vol 3 (4) ◽  
pp. 91 ◽  
Author(s):  
Hemant Khatri ◽  
Pavel Berloff
Keyword(s):  
Large Scale ◽  
Bottom Layer ◽  
Relative Vorticity ◽  
Global Ocean ◽  
Dynamic Interactions ◽  
Zonal Jets ◽  
Eddy Dynamics ◽  
Quasigeostrophic Model ◽  
The Impact

Multiple zonal jets observed in many parts of the global ocean are often embedded in large-scale eastward and westward vertically sheared background flows. Properties of the jets and ambient eddies, as well as their dynamic interactions, are found to be different between eastward and westward shears. However, the impact of these differences on overall eddy dynamics remains poorly understood and is the main subject of this study. The roles of eddy relative vorticity and buoyancy fluxes in the maintenance of oceanic zonal jets are studied in a two-layer quasigeostrophic model. Both eastward and westward uniform, zonal vertically sheared cases are considered in the study. It is shown that, despite the differences in eddy structure and local characteristics, the fundamental dynamics are essentially the same in both cases: the relative-vorticity fluxes force the jets in the entire fluid column, and the eddy-buoyancy fluxes transfer momentum from the top to the bottom layer, where it is balanced by bottom friction. It is also observed that the jets gain more energy via Reynolds stress work in the layer having a positive gradient in the background potential vorticity, and this is qualitatively explained by a simple reasoning based on Rossby wave group velocity.

scholarly journals A Method of Successive Corrections of the Control Subspace in the Reduced-Order Variational Data Assimilation*

2009 ◽  
Vol 137 (9) ◽  
pp. 2966-2978 ◽  
Author(s):  
Max Yaremchuk ◽  
Dmitri Nechaev ◽  
Gleb Panteleev
Keyword(s):  
Data Assimilation ◽  
Numerical Models ◽  
Control Space ◽  
Strong Constraint ◽  
Reduced Order ◽  
Direct Model ◽  
Quasigeostrophic Model ◽  
Low Dimensional ◽  
The Cost ◽  
Better Than

Abstract A version of the reduced control space four-dimensional variational method (R4DVAR) of data assimilation into numerical models is proposed. In contrast to the conventional 4DVAR schemes, the method does not require development of the tangent linear and adjoint codes for implementation. The proposed R4DVAR technique is based on minimization of the cost function in a sequence of low-dimensional subspaces of the control space. Performance of the method is demonstrated in a series of twin-data assimilation experiments into a nonlinear quasigeostrophic model utilized as a strong constraint. When the adjoint code is stable, R4DVAR’s convergence rate is comparable to that of the standard 4DVAR algorithm. In the presence of strong instabilities in the direct model, R4DVAR works better than 4DVAR whose performance is deteriorated because of the breakdown of the tangent linear approximation. Comparison of the 4DVAR and R4DVAR also shows that R4DVAR becomes advantageous when observations are sparse and noisy.

scholarly journals Basin and Channel Contributions to a Model Antarctic Circumpolar Current

2009 ◽  
Vol 39 (4) ◽  
pp. 986-1002 ◽  
Author(s):  
Louis-Philippe Nadeau ◽  
David N. Straub
Keyword(s):  
Wind Stress ◽  
Antarctic Circumpolar Current ◽  
Robustness Analysis ◽  
Drake Passage ◽  
Western Boundary ◽  
Analytic Model ◽  
Saturation Regime ◽  
Zonal Jets ◽  
Circumpolar Current ◽  
Quasigeostrophic Model

Abstract The idea that basinlike dynamics may play a major role in determining the Antarctic Circumpolar Current (ACC) transport is revisited. A simple analytic model is developed to describe the relationship between the wind stress and transport. At very low-wind stress, a nonzero minimum is predicted. This is followed by two distinct dynamical regimes for stronger forcing: 1) a Stommel regime in which transport increases linearly with forcing strength; and 2) a saturation regime in which the transport levels off. The baroclinic structure of the Sverdrup flux into the Drake Passage latitude band is central to the analytic model, and the geometry of characteristics, or geostrophic contours, is key to predicting the transition between the two regimes. A robustness analysis is performed using an eddy-permitting quasigeostrophic model in idealized geometries. Many simulations were carried out in large domains across a range of forcing strengths. The simulations agree qualitatively with the analytic model, with two main discrepancies being related to zonal jet structures and to a western boundary inertial recirculation. Eddy fluxes associated with zonal jets modify the baroclinic structure and lower the saturation transport value. Inertial effects increase the transport, although this effect is mainly limited to smaller domains.

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