911 research outputs found
Hybrid deterministic stochastic systems with microscopic look-ahead dynamics
We study the impact of stochastic mechanisms on a coupled hybrid system consisting of a general advection-diffusion-reaction partial differential equation and a spatially distributed stochastic lattice noise model. The stochastic dynamics include both spin-flip and spin-exchange type interparticle interactions. Furthermore, we consider a new, asymmetric, single exclusion pro- cess, studied elsewhere in the context of traffic flow modeling, with an one-sided interaction potential which imposes advective trends on the stochastic dynamics. This look-ahead stochastic mechanism is responsible for rich nonlinear behavior in solutions. Our approach relies heavily on first deriving approximate differential mesoscopic equations. These approximations become exact either in the long range, Kac interaction partial differential equation case, or, given sufficient time separation con- ditions, between the partial differential equation and the stochastic model giving rise to a stochastic averaging partial differential equation. Although these approximations can in some cases be crude, they can still give a first indication, via linearized stability analysis, of the interesting regimes for the stochastic model. Motivated by this linearized stability analysis we choose particular regimes where interacting nonlinear stochastic waves are responsible for phenomena such as random switching, convective instability, and metastability, all driven by stochasticity. Numerical kinetic Monte Carlo simulations of the coarse grained hybrid system are implemented to assist in producing solutions and understanding their behavior
Nonlinear stability and ergodicity of ensemble based Kalman filters
The ensemble Kalman filter (EnKF) and ensemble square root filter (ESRF) are
data assimilation methods used to combine high dimensional, nonlinear dynamical
models with observed data. Despite their widespread usage in climate science
and oil reservoir simulation, very little is known about the long-time behavior
of these methods and why they are effective when applied with modest ensemble
sizes in large dimensional turbulent dynamical systems. By following the basic
principles of energy dissipation and controllability of filters, this paper
establishes a simple, systematic and rigorous framework for the nonlinear
analysis of EnKF and ESRF with arbitrary ensemble size, focusing on the
dynamical properties of boundedness and geometric ergodicity. The time uniform
boundedness guarantees that the filter estimate will not diverge to machine
infinity in finite time, which is a potential threat for EnKF and ESQF known as
the catastrophic filter divergence. Geometric ergodicity ensures in addition
that the filter has a unique invariant measure and that initialization errors
will dissipate exponentially in time. We establish these results by introducing
a natural notion of observable energy dissipation. The time uniform bound is
achieved through a simple Lyapunov function argument, this result applies to
systems with complete observations and strong kinetic energy dissipation, but
also to concrete examples with incomplete observations. With the Lyapunov
function argument established, the geometric ergodicity is obtained by
verifying the controllability of the filter processes; in particular, such
analysis for ESQF relies on a careful multivariate perturbation analysis of the
covariance eigen-structure.Comment: 38 page
Improved linear response for stochastically driven systems
The recently developed short-time linear response algorithm, which predicts
the average response of a nonlinear chaotic system with forcing and dissipation
to small external perturbation, generally yields high precision of the response
prediction, although suffers from numerical instability for long response times
due to positive Lyapunov exponents. However, in the case of stochastically
driven dynamics, one typically resorts to the classical fluctuation-dissipation
formula, which has the drawback of explicitly requiring the probability density
of the statistical state together with its derivative for computation, which
might not be available with sufficient precision in the case of complex
dynamics (usually a Gaussian approximation is used). Here we adapt the
short-time linear response formula for stochastically driven dynamics, and
observe that, for short and moderate response times before numerical
instability develops, it is generally superior to the classical formula with
Gaussian approximation for both the additive and multiplicative stochastic
forcing. Additionally, a suitable blending with classical formula for longer
response times eliminates numerical instability and provides an improved
response prediction even for long response times
Systematic Multiscale Models for Deep Convection on Mesoscales
This paper builds on recent developments of a unified asymptotic approach to meteorological modelling (Klein (2000), Klein (2003)), which was used successfully in the development of “Systematicmultiscale models for the tropics” in (Majda & Klein (2003), Majda & Biello (2004), Biello & Majda(2005)). Here we account for typical bulk microphysics parameterizations of moist processes within
this framework. The key steps are careful nondimensionalization of the bulk microphysics equations
and the choice of appropriate distinguished limits for the various nondimensional small parameters that appear.
We are then in the position to study scale interactions in the atmosphere involving moist physics. We demonstrate this by developing two systematic multiscale models that are motivated by our interest in mesoscale organized convection. The emphasis here is on multiple length, but common time scales. The first of these models describes the short time evolution of slender, deep convective “hot towers” with horizontal scale 1 km interacting with the linearized momentum balance on length and time scales of (10km / 3 min). We expect this model to describe how convective inhibition may be overcome near the surface, how the onset of deep convection triggers convective scale gravity waves, and that it will also yield new insight into how such local convective events may conspire to create larger scale strong storms. The second model addresses the next larger range of length and time scales (10 km, 100 km,
and 20min) and exhibits mathematical features that are strongly reminiscent of mesoscale organized convection. In both cases, the asymptotic analysis reveals how the stiffness of condensation/evaporation processes induces highly nonlinear dynamics.
Besides providing new theoretical insights, the derived models may also serve as a theoretical devices for analyzing and interpreting the results of complex moist process model simulations, and they may stimulate the development of new, theoretically grounded subgrid scale parameterizations
Viscous evolution of point vortex equilibria: The collinear state
When point vortex equilibria of the 2D Euler equations are used as initial
conditions for the corre- sponding Navier-Stokes equations (viscous), typically
an interesting dynamical process unfolds at short and intermediate time scales,
before the long time single peaked, self-similar Oseen vortex state dom-
inates. In this paper, we describe the viscous evolution of a collinear three
vortex structure that cor- responds to an inviscid point vortex fixed
equilibrium. Using a multi-Gaussian 'core-growth' type of model, we show that
the system immediately begins to rotate unsteadily, a mechanism we attribute to
a 'viscously induced' instability. We then examine in detail the qualitative
and quantitative evolution of the system as it evolves toward the long-time
asymptotic Lamb-Oseen state, showing the sequence of topological bifurcations
that occur both in a fixed reference frame, and in an appropriately chosen
rotating reference frame. The evolution of passive particles in this viscously
evolving flow is shown and interpreted in relation to these evolving streamline
patterns.Comment: 17 pages, 15 figure
Local and nonlocal pressure Hessian effects in real and synthetic fluid turbulence
The Lagrangian dynamics of the velocity gradient tensor A in isotropic and
homogeneous turbulence depend on the joint action of the self-streching term
and the pressure Hessian. Existing closures for pressure effects in terms of A
are unable to reproduce one important statistical role played by the
anisotropic part of the pressure Hessian, namely the redistribution of the
probabilities towards enstrophy production dominated regions. As a step towards
elucidating the required properties of closures, we study several synthetic
velocity fields and how well they reproduce anisotropic pressure effects. It is
found that synthetic (i) Gaussian, (ii) Multifractal and (iii) Minimal Turnover
Lagrangian Map (MTLM) incompressible velocity fields reproduce many features of
real pressure fields that are obtained from numerical simulations of the Navier
Stokes equations, including the redistribution towards enstrophy-production
regions. The synthetic fields include both spatially local, and nonlocal,
anisotropic pressure effects. However, we show that the local effects appear to
be the most important ones: by assuming that the pressure Hessian is local in
space, an expression in terms of the Hessian of the second invariant Q of the
velocity gradient tensor can be obtained. This term is found to be well
correlated with the true pressure Hessian both in terms of eigenvalue
magnitudes and eigenvector alignments.Comment: 10 pages, 4 figures, minor changes, final version, published in Phys.
Fluid
ANOMALOUS SCALING OF THE PASSIVE SCALAR
We establish anomalous inertial range scaling of structure functions for a
model of advection of a passive scalar by a random velocity field. The velocity
statistics is taken gaussian with decorrelation in time and velocity
differences scaling as in space, with . The
scalar is driven by a gaussian forcing acting on spatial scale and
decorrelated in time. The structure functions for the scalar are well defined
as the diffusivity is taken to zero and acquire anomalous scaling behavior for
large pumping scales . The anomalous exponent is calculated explicitly for
the 4^{\m\rm th} structure function and for small and it differs
from previous predictions. For all but the second structure functions the
anomalous exponents are nonvanishing.Comment: 8 pages, late
The Exotic Statistics of Leapfrogging Smoke Rings
The leapfrogging motion of smoke rings is a three dimensional version of the
motion that in two dimensions leads to exotic exchange statistics. The
statistical phase factor can be computed using the hydrodynamical Euler
equation, which is a universal law for describing the properties of a large
class of fluids. This suggests that three dimensional exotic exchange
statistics is a common property of closed vortex loops in a variety of quantum
liquids and gases, from helium superfluids to Bose-Einstein condensed alkali
gases, metallic hydrogen in its liquid phases and maybe even nuclear matter in
extreme conditions.Comment: 9 pages 1 figur
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