689 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
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
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
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
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
Moisture - Gravity Wave Interactions in a Multiscale Environment
Starting from the conservation laws for mass, momentum and energy together with
a three species, bulk microphysic model, a model for the interaction of internal gravity waves and
deep convective hot towers is derived by using multiscale asymptotic techniques.
From the resulting leading order equations, a closed model is obtained by applying weighted
averages to the smallscale hot towers without requiring further closure approximations. The resulting
model is an extension of the linear, anelastic equations, into which moisture enters as the area fraction
of saturated regions on the microscale with two way coupling between the large and small scale.
Moisture reduces the effective stability in the model and defines a potential temperature sourceterm
related to the net effect of latent heat release or consumption by microscale up- and downdrafts.
The dispersion relation and group velocity of the system is analyzed and moisture is found to have
several effects: It reduces energy transport by waves, increases the vertical wavenumber but decreases
the slope at which wave packets travel and it introduces a lower horizontal cutoff wavenumber, below
which modes turn into evanescent. Further, moisture can cause critical layers.
Numerical examples for steady-state and time-dependent mountain waves are shown and the effects
of moisture on these waves are investigated
A Variational Principle Based Study of KPP Minimal Front Speeds in Random Shears
Variational principle for Kolmogorov-Petrovsky-Piskunov (KPP) minimal front
speeds provides an efficient tool for statistical speed analysis, as well as a
fast and accurate method for speed computation. A variational principle based
analysis is carried out on the ensemble of KPP speeds through spatially
stationary random shear flows inside infinite channel domains. In the regime of
small root mean square (rms) shear amplitude, the enhancement of the ensemble
averaged KPP front speeds is proved to obey the quadratic law under certain
shear moment conditions. Similarly, in the large rms amplitude regime, the
enhancement follows the linear law. In particular, both laws hold for the
Ornstein-Uhlenbeck process in case of two dimensional channels. An asymptotic
ensemble averaged speed formula is derived in the small rms regime and is
explicit in case of the Ornstein-Uhlenbeck process of the shear. Variational
principle based computation agrees with these analytical findings, and allows
further study on the speed enhancement distributions as well as the dependence
of enhancement on the shear covariance. Direct simulations in the small rms
regime suggest quadratic speed enhancement law for non-KPP nonlinearities.Comment: 28 pages, 14 figures update: fixed typos, refined estimates in
section
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