No abstract

New global periodic orbit collision/separatrix reconnection scenarios in the standard nontwist map in different regions of parameter space are described in detail, including exact methods for determining reconnection thresholds that are implemented numerically. The results are compared to a break-up diagram of shearless invariant curves. The existence of meanders (invariant tori that are not graphs) is demonstrated numerically for both odd and even period reconnection for certain regions in parameter space, and some of the implications on transport are discussed.In recent years, area-preserving maps that violate the twist condition locally in phase space have been the object of interest in several studies in physics and mathematics. These nontwist maps show up in a variety of physical models, e.g., in magnetic field line models for reversed magnetic shear tokamaks. An important problem is the determination and understanding of the transition to global chaos (global transport) in these models. Nontwist maps exhibit several different mechanisms: the break-up of invariant tori and separatrix reconnections. The latter may or may not lead to global transport depending on the region of parameter space. In this paper we conduct a detailed study of newly discovered reconnection scenarios in the standard nontwist map, investigating their location in parameter space and their impact on global transport.

SUMMARYThe bulk of this paper contains a concise mathematical overview of the subject of data assimilation, highlighting three primary ideas: (i) the standard optimization approaches of 3DVAR, 4DVAR and weak constraint 4DVAR are described and their interrelations explained; (ii) statistical analogues of these approaches are then introduced, leading to filtering (generalizing 3DVAR) and a form of smoothing (generalizing 4DVAR and weak constraint 4DVAR) and the optimization methods are shown to be maximum a posteriori estimators for the probability distributions implied by these statistical approaches; and (iii) by taking a general dynamical systems perspective on the subject it is shown that the incorporation of Lagrangian data can be handled by a straightforward extension of the preceding concepts.We argue that the smoothing approach to data assimilation, based on statistical analogues of 4DVAR and weak constraint 4DVAR, provides the optimal solution to the assimilation of space-time distributed data into a model. The optimal solution obtained is a probability distribution on the relevant class of functions (initial conditions or time-dependent solutions). The approach is a useful one in the first instance because it clarifies the notion of what is the optimal solution, thereby providing a benchmark against which existing approaches can be evaluated. In the longer term it also provides the potential for new methods to create ensembles of solutions to the model, incorporating the available data in an optimal fashion.Two examples are given illustrating this approach to data assimilation, both in the context of Lagrangian data, one based on statistical 4DVAR and the other on weak constraint statistical 4DVAR. The former is compared with the ensemble Kalman filter, which is thereby shown to be inaccurate in a variety of scenarios.

International audienceThe characteristics of the model dynamics are critical in the performance of (ensemble) Kalmanfilters. In particular, as emphasized in the seminal work of Anna Trevisan and coauthors, theerror covariance matrix is asymptotically supported by the unstable-neutral subspace only, i.e., itis spanned by the backward Lyapunov vectors with nonnegative exponents. This behavior is at thecore of algorithms known as assimilation in the unstable subspace, although a formal proof was stillmissing. This paper provides the analytical proof of the convergence of the Kalman filter covariancematrix onto the unstable-neutral subspace when the dynamics and the observation operator are linearand when the dynamical model is error free, for any, possibly rank-deficient, initial error covariancematrix. The rate of convergence is provided as well. The derivation is based on an expression thatexplicitly relates the error covariances at an arbitrary time to the initial ones. It is also shown thatif the unstable and neutral directions of the model are sufficiently observed and if the column spaceof the initial covariance matrix has a nonzero projection onto all of the forward Lyapunov vectorsassociated with the unstable and neutral directions of the dynamics, the covariance matrix of theKalman filter collapses onto an asymptotic sequence which is independent of the initial covariances.Numerical results are also shown to illustrate and support the theoretical findings

Extending the work of del-Castillo-Negrete, Greene, and Morrison, Physica D 91, 1 (1996) and 100, 311 (1997) on the standard nontwist map, the breakup of an invariant torus with winding number equal to the inverse golden mean squared is studied. Improved numerical techniques provide the greater accuracy that is needed for this case. The new results are interpreted within the renormalization group framework by constructing a renormalization operator on the space of commuting map pairs, and by studying the fixed points of the so constructed operator.In recent years, area-preserving maps that violate the twist condition locally in phase space have been the object of interest in several studies in physics and mathematics. These nontwist maps show up in a variety of physical models. An important problem from the physics point of view is the understanding of the breakup of invariant tori, which show remarkable resilience in the region where the twist condition is violated, called shearless tori. In terms of the physical system modelled, these tori represent transport barriers, and their breakup corresponds to the transition to global chaos. Mathematically, nontwist maps present a challenge since the standard proofs of celebrated theorems in the theory of area-preserving maps rely heavily on the twist condition. In this paper, we study the breakup of the shearless torus with winding number 1/γ 2 , where γ is the golden mean. This torus serves as a test case for improved techniques we developed. At the point of breakup the shearless torus exhibits universal scaling behavior which leads to a renormalization group interpretation.