Amit Apte

Meanders and reconnection–collision sequences in the standard nontwist map

New global periodic orbit collision and separatrix reconnection scenarios exhibited by the standard nontwist map are described in detail, including exact methods for determining reconnection thresholds, methods that are implemented numerically. Results are compared to a parameter space breakup diagram for 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. Implications for transport are discussed.

A Bayesian approach to Lagrangian data assimilation

Lagrangian data arise from instruments that are carried by the flow in a fluid field. Assimilation of such data into ocean models presents a challenge due to the potential complexity of Lagrangian trajectories in relatively simple flow fields.We adopt a Bayesian perspective on this problem and thereby take account of the fully non-linear features of the underlying model. In the perfect model scenario, the posterior distribution for the initial state of the system contains all the information that can be extracted from a given realization of observations and the model dynamics. We work in the smoothing context in which the posterior on the initial conditions is determined by future observations. This posterior distribution gives the optimal ensemble to be used in data assimilation. The issue then is sampling this distribution. We develop, implement, and test sampling methods, based on Markov-chain Monte Carlo (MCMC), which are particularly well suited to the low-dimensional, but highly non-linear, nature of Lagrangian data. We compare these methods to the well-established ensemble Kalman filter (EnKF) approach. It is seen that the MCMC based methods correctly sample the desired posterior distribution whereas the EnKF may fail due to infrequent observations or non-linear structures in the underlying flow.

Renormalization and destruction of 1/γ2 tori in the standard nontwist map

Extending the work of del-Castillo-Negrete, Greene, and Morrison [Physica D 91, 1 (1996); 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.Extending the work of del-Castillo-Negrete, Greene, and Morrison, Physica D {\bf 91}, 1 (1996) and {\bf 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.

A unified approach to the Darwin approximation

There are two basic approaches to the Darwin approximation. The first involves solving the Maxwell equations in Coulomb gauge and then approximating the vector potential to remove retardation effects. The second approach approximates the Coulomb gauge equations themselves, then solves these exactly for the vector potential. There is no a priori reason that these should result in the same approximation. Here, the equivalence of these two approaches is investigated and a unified framework is provided in which to view the Darwin approximation. Darwin’s original treatment is variational in nature, but subsequent applications of his ideas in the context of Vlasovs theory are not. We present here action principles for the Darwin approximation in the Vlasov context, and this serves as a consistency check on the use of the approximation in this setting.

Breakup of shearless meanders and "outer" tori in the standard nontwist map.

The breakup of shearless invariant tori with winding number omega=(11+gamma)(12+gamma) (in continued fraction representation) of the standard nontwist map is studied numerically using Greenes residue criterion. Tori of this winding number can assume the shape of meanders [folded-over invariant tori which are not graphs over the x axis in (x,y) phase space], whose breakup is the first point of focus here. Secondly, multiple shearless orbits of this winding number can exist, leading to a new type of breakup scenario. Results are discussed within the framework of the renormalization group for area-preserving maps. Regularity of the critical tori is also investigated.

A Hybrid Particle–Ensemble Kalman Filter for Lagrangian Data Assimilation

Lagrangian measurements from passive ocean instruments provide a useful source of data for estimating and forecasting the ocean’s state (velocity field, salinity field, etc.). However, trajectories from these instrumentsare often highly nonlinear, leading to difficulties with widely used data assimilationalgorithms such as the ensemble Kalmanfilter (EnKF). Additionally, the velocityfield is often modeled as a high-dimensional variable, which precludes the use of more accurate methods such as the particle filter (PF). Here, a hybrid particle‐ensemble Kalman filter is developed that applies the EnKF update to the potentially highdimensional velocity variables, and the PF update to the relatively low-dimensional, highly nonlinear drifter positionvariable. This algorithmis tested with twin experiments on the linearshallow water equations. In experiments with infrequent observations, the hybrid filter consistently outperformed the EnKF, both by better capturing the Bayesian posterior and by better tracking the truth.

Regularity of critical invariant circles of the standard nontwist map

We study critical invariant circles of several noble rotation numbers at the edge of break-up for an area-preserving map of the cylinder, which violates the twist condition. These circles admit essentially unique parametrizations by rotational coordinates. We present a high accuracy computation of about 10 7 Fourier coefficients. This allows us to compute the regularity of the conjugating maps and to show that, to the extent of numerical precision, it only depends on the tail of the continued fraction expansion.

On reconnection phenomena in the standard nontwist map

Separatrix reconnection in the standard nontwist map is described, including exact methods for determining the reconnection threshold in parameter space. These methods are implemented numerically for the case of oddperiod orbit reconnection, where meanders (invariant tori that are not graphs) appear. Nested meander structure is numerically demonstrated, and the idea of meander transport is discussed.

Degenerate Kalman Filter Error Covariances and Their Convergence onto the Unstable Subspace

The characteristics of the model dynamics are critical in the performance of (ensemble) Kalman filters. In particular, as emphasized in the seminal work of Anna Trevisan and co-authors, the error covariance matrix is asymptotically supported by the unstable-neutral subspace only, i.e., it is spanned by the backward Lyapunov vectors with non-negative exponents. This behavior is at the core of algorithms known as Assimilation in the Unstable Subspace, although a formal proof was still missing. This paper provides the analytical proof of the convergence of the Kalman filter covariance matrix onto the unstable-neutral subspace when the dynamics and the observation operator are linear and when the dynamical model is error-free, for any, possibly rank-deficient, initial error covariance matrix. The rate of convergence is provided as well. The derivation is based on an expression that explicitly relates the error covariances at an arbitrary time to the initial ones. It is also shown that if the unstable and neutral directions of the model are sufficiently observed and if the column space of the initial covariance matrix has a non-zero projection onto all of the forward Lyapunov vectors associated with the unstable and neutral directions of the dynamics, the covariance matrix of the Kalman 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.

Rank Deficiency of Kalman Error Covariance Matrices in Linear Time-Varying System With Deterministic Evolution

We prove that for linear, discrete, time-varying, deterministic system (perfect model) with noisy outputs, the Riccati transformation in the Kalman filter asymptotically bounds the rank of the forecast and the analysis error covariance matrices to be less than or equal to the number of non-negative Lyapunov exponents of the system. Further, the support of these error covariance matrices is shown to be confined to the space spanned by the unstable-neutral backward Lyapunov vectors, providing the theoretical justification for the methodology of the algorithms that perform assimilation only in the unstable-neutral subspace. The equivalent property of the autonomous system is investigated as a special case.