Stability of Filters for the Navier-Stokes Equation

Data assimilation methodologies are designed to incorporate noisy observations of a physical system into an underlying model in order to infer the properties of the state of the system. Filters refer to a class of data assimilation algorithms designed to update the estimation of the state in a on-line fashion, as data is acquired sequentially. For linear problems subject to Gaussian noise filtering can be performed exactly using the Kalman filter. For nonlinear systems it can be approximated in a systematic way by particle filters. However in high dimensions these particle filtering methods can break down. Hence, for the large nonlinear systems arising in applications such as weather forecasting, various ad hoc filters are used, mostly based on making Gaussian approximations. The purpose of this work is to study the properties of these ad hoc filters, working in the context of the 2D incompressible Navier-Stokes equation. By working in this infinite dimensional setting we provide an analysis which is useful for understanding high dimensional filtering, and is robust to mesh-refinement. We describe theoretical results showing that, in the small observational noise limit, the filters can be tuned to accurately track the signal itself (filter stability), provided the system is observed in a sufficiently large low dimensional space; roughly speaking this space should be large enough to contain the unstable modes of the linearized dynamics. Numerical results are given which illustrate the theory. In a simplified scenario we also derive, and study numerically, a stochastic PDE which determines filter stability in the limit of frequent observations, subject to large observational noise. The positive results herein concerning filter stability complement recent numerical studies which demonstrate that the ad hoc filters perform poorly in reproducing statistical variation about the true signal

On the dynamics of correlations in Scaling limits of interacting particle systems

In this work, we analyze the properties of two-particle correlations in the weak-coupling and plasma limit of interacting particle systems motivated by Bogolyubov's formal derivation of kinetic equations. We prove that the leading order evolution in the weak-coupling scaling limit is stable on the macroscopic timescale, and yields the nonlinear Landau equation as kinetic equation. This result shows the transition from the non-Markovian dynamics of the interacting particle system to the Markovian, parabolic evolution in the kinetic limit. Since the system is non-dissipative before taking the limit, we introduce a time-averaged notion of stability to derive an a priori estimate on the solution. Moreover, we prove the global stability of the truncated correlation function in the plasma limit, for a time independent background distribution and soft potentials. In the case of systems with Coulombian interaction, we prove the onset of the Debye screening length and show that the limit evolution is driven by the interaction of particles with impact parameter much smaller than the Debye length

Damage spreading transition in glasses: a probe for the ruggedness of the configurational landscape

We consider damage spreading transitions in the framework of mode-coupling theory. This theory describes relaxation processes in glasses in the mean-field approximation which are known to be characterized by the presence of an exponentially large number of meta-stable states. For systems evolving under identical but arbitrarily correlated noises we demonstrate that there exists a critical temperature $T_0$ which separates two different dynamical regimes depending on whether damage spreads or not in the asymptotic long-time limit. This transition exists for generic noise correlations such that the zero damage solution is stable at high-temperatures being minimal for maximal noise correlations. Although this dynamical transition depends on the type of noise correlations we show that the asymptotic damage has the good properties of an dynamical order parameter such as: 1) Independence on the initial damage; 2) Independence on the class of initial condition and 3) Stability of the transition in the presence of asymmetric interactions which violate detailed balance. For maximally correlated noises we suggest that damage spreading occurs due to the presence of a divergent number of saddle points (as well as meta-stable states) in the thermodynamic limit consequence of the ruggedness of the free energy landscape which characterizes the glassy state. These results are then compared to extensive numerical simulations of a mean-field glass model (the Bernasconi model) with Monte Carlo heat-bath dynamics. The freedom of choosing arbitrary noise correlations for Langevin dynamics makes damage spreading a interesting tool to probe the ruggedness of the configurational landscape.Comment: 25 pages, 13 postscript figures. Paper extended to include cross-correlation

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 2. pseudoenergy-based theory

This paper represents the second part of a study of semi-geostrophic (SG) geophysical fluid dynamics. SG dynamics shares certain attractive properties with the better known and more widely used quasi-geostrophic (QG) model, but is also a good prototype for balanced models that are more accurate than QG dynamics. The development of such balanced models is an area of great current interest. The goal of the present work is to extend a central body of QG theory, concerning the evolution of disturbances to prescribed basic states, to SG dynamics. Part 1 was based on the pseudomomentum; Part 2 is based on the pseudoenergy. A pseudoenergy invariant is a conserved quantity, of second order in disturbance amplitude relative to a prescribed steady basic state, which is related to the time symmetry of the system. We derive such an invariant for the semi-geostrophic equations, and use it to obtain: (i) a linear stability theorem analogous to Arnol'd's ‘first theorem’; and (ii) a small-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit. The results are analogous to their quasi-geostrophic forms, and reduce to those forms in the limit of small Rossby number. The results are derived for both the f-plane Boussinesq form of semi-geostrophic dynamics, and its extension to β-plane compressible flow by Magnusdottir & Schubert. Novel features particular to semi-geostrophic dynamics include apparently unnoticed lateral boundary stability criteria. Unlike the boundary stability criteria found in the first part of this study, however, these boundary criteria do not necessarily preclude the construction of provably stable basic states. The interior semi-geostrophic dynamics has an underlying Hamiltonian structure, which guarantees that symmetries in the system correspond naturally to the system's invariants. This is an important motivation for the theoretical approach used in this study. The connection between symmetries and conservation laws is made explicit using Noether's theorem applied to the Eulerian form of the Hamiltonian description of the interior dynamics

One-gas Models with Height-dependent Mean Molecular Weight: Effects on Gravity Wave Propagation

Many models of the thermosphere employ the one-gas approximation where the governing equations apply only to the total gas and the physical properties of the gas that depend on composition (mean molecular weight and specific heats) are height-dependent. It is further assumed that the physical properties of the gas are locally constant; thus motion-induced perturbations are nil. However, motion in a diffusively separated atmosphere perturbs local values of mean molecular weight and specific heats. These motion-induced changes are opposed by mutual diffusion of the constituent gases, which attempts to restore diffusive equilibrium. Assuming that composition is locally constant is equivalent to assuming that diffusion instantaneously damps the changes that winds attempt to produce. This is the limit of fast diffusion. In the limit of slow diffusion, gas properties are constant (conserved) following the motion but are perturbed locally by advection. An analysis of the static stability shows that composition effects significantly change the static stability, with greater changes for the slow-diffusion limit than for the fast-diffusion limit. We have used a one-gas full-wave model to examine the effects of wave-perturbed composition on gravity waves propagating through the lower thermosphere. We have augmented the conventional system (fixed gas properties) with predictive equations for composition-dependent gas properties. These equations include vertical advection and mutual diffusion. The latter is included in parameterized form as second-order scale-dependent diffusion. We have found that the fast diffusion implied by locally fixed properties has a significant effect on the dynamics. Predicted temperatures are larger for locally fixed composition than for conserved composition. The simulations with parameterized mutual diffusion gave results that are much closer to the results for conserved gas properties than for fixed properties. We found that the divergence between the fast and slow limits was greatest for fast waves and for colder thermospheres. This is because the propagation characteristics of fast waves are sensitive to changes in the static stability and because compositional gradients are stronger for colder thermospheres. We conclude that future models that use the one-gas approximation for fast waves in the lower thermosphere should include, at minimum, the simplification of conserved rather than fixed properties, especially for colder thermospheres

Glass and polycrystal states in a lattice spin model

We numerically study a nondisordered lattice spin system with a first order liquid-crystal transition, as a model for supercooled liquids and glasses. Below the melting temperature the system can be kept in the metastable liquid phase, and it displays a dynamic phenomenology analogous to fragile supercooled liquids, with stretched exponential relaxation, power law increase of the relaxation time and high fragility index. At an effective spinodal temperature Tsp the relaxation time exceeds the crystal nucleation time, and the supercooled liquid loses stability. Below Tsp liquid properties cannot be extrapolated, in line with Kauzmann's scenario of a `lower metastability limit' of supercooled liquids as a solution of Kauzmann's paradox. The off-equilibrium dynamics below Tsp corresponds to fast nucleation of small, but stable, crystal droplets, followed by extremely slow growth, due to the presence of pinning energy barriers. In the early time region, which is longer the lower the temperature, this crystal-growth phase is indistinguishable from an off-equilibrium glass, both from a structural and a dynamical point of view: crystal growth has not advanced enough to be structurally detectable, and a violation of the fluctuation-dissipation theorem (FDT) typical of structural glasses is observed. On the other hand, for longer times crystallization reaches a threshold beyond which crystal domains are easily identified, and FDT violation becomes compatible with ordinary domain growth.Comment: 25 page

A Stabilization of a Continuous Limit of the Ensemble Kalman Filter

The ensemble Kalman filter belongs to the class of iterative particle filtering methods and can be used for solving control--to--observable inverse problems. In recent years several continuous limits in the number of iteration and particles have been performed in order to study properties of the method. In particular, a one--dimensional linear stability analysis reveals a possible instability of the solution provided by the continuous--time limit of the ensemble Kalman filter for inverse problems. In this work we address this issue by introducing a stabilization of the dynamics which leads to a method with globally asymptotically stable solutions. We illustrate the performance of the stabilized version of the ensemble Kalman filter by using test inverse problems from the literature and comparing it with the classical formulation of the method