5 research outputs found
Strategies for Reduced-Order Models for Predicting the Statistical Responses and Uncertainty Quantification in Complex Turbulent Dynamical Systems
Turbulent dynamical systems characterized by both a high-dimensional phase
space and a large number of instabilities are ubiquitous among many complex
systems in science and engineering. The existence of a strange attractor in the
turbulent systems containing a large number of positive Lyapunov exponents
results in a rapid growth of small uncertainties, requiring naturally a
probabilistic characterization for the evolution of the turbulent system.
Uncertainty quantification in turbulent dynamical systems is a grand challenge
where the goal is to obtain statistical estimates such as the change in mean
and variance for key physical quantities in their nonlinear responses to
changes in external forcing parameters or uncertain initial data. One central
issue in contemporary research is the development of a systematic methodology
that can recover the crucial features of the natural system in statistical
equilibrium (model fidelity) and improve the imperfect model prediction skill
in response to various external perturbations (model sensitivity). A general
mathematical framework to construct statistically accurate reduced-order models
that have skill in capturing the statistical variability in the principal
directions with largest energy of a general class of damped and forced complex
turbulent dynamical systems is discussed here. The methods are developed under
a universal class of turbulent dynamical systems with quadratic nonlinearity
that is representative in many applications in applied mathematics and
engineering. The validity of general framework of reduced-order models is
demonstrated on instructive stochastic triad models. Recent applications to
two-layer baroclinic turbulence in the atmosphere and ocean with combinations
of turbulent jets and vortices are also surveyed.Comment: 43 pages, 18 figure
A Flux-Balanced Fluid Model for Collisional Plasma Edge Turbulence: Numerical Simulations with Different Aspect Ratios
We investigate the drift wave -- zonal flow dynamics in a shearless slab
geometry with the new flux-balanced Hasegawa-Wakatani model. As in previous
Hasegawa-Wakatani models, we observe a sharp transition from a turbulence
dominated regime to a zonal jet dominated regime as we decrease the plasma
resistivity. However, unlike previous models, zonal structures are always
present in the flux-balanced model, even for high resistivity, and strongly
reduce the level of particle and vorticity flux. The more robust zonal jets
also have a higher variability than in previous models, which is further
enhanced when the computational domain is chosen to be elongated in the radial
direction. In these cases, we observe complex multi-scale dynamics, with
multiple jets interacting with one another, and intermittent bursts. We present
a detailed statistical analysis which highlights how the changes in the aspect
ratio of the computational domain affect the third-order statistical moments,
and thus modify the turbulent dynamics.Comment: 20 pages, 16 figure
Zonal Jet Creation from Secondary Instability of Drift Waves for Plasma Edge Turbulence
A new strategy is presented to explain the creation and persistence of zonal
flows widely observed in plasma edge turbulence. The core physics in the edge
regime of the magnetic-fusion tokamaks can be described qualitatively by the
one-state modified Hasegawa-Mima (MHM) model, which creates enhanced zonal
flows and more physically relevant features in comparison with the familiar
Charney-Hasegawa-Mima (CHM) model for both plasma and geophysical flows. The
generation mechanism of zonal jets is displayed from the secondary instability
analysis via nonlinear interactions with a background base state. Strong
exponential growth in the zonal modes is induced due to a non-zonal drift wave
base state in the MHM model, while stabilizing damping effect is shown with a
zonal flow base state. Together with the selective decay effect from the
dissipation, the secondary instability offers a complete characterization of
the convergence process to the purely zonal structure. Direct numerical
simulations with and without dissipation are carried out to confirm the
instability theory. It shows clearly the emergence of a dominant zonal flow
from pure non-zonal drift waves with small perturbation in the initial
configuration. In comparison, the CHM model does not create instability in the
zonal modes and usually converges to homogeneous turbulence.Comment: 16 pages, 9 figure
A Flux-Balanced Fluid Model for Collisional Plasma Edge Turbulence: Model Derivation and Basic Physical Features
We propose a new reduced fluid model for the study of the drift wave -- zonal
flow dynamics in magnetically confined plasmas. Our model can be viewed as an
extension of the classic Hasegawa-Wakatani (HW) model, and is based on an
improved treatment of the electron dynamics parallel to the field lines, to
guarantee a balanced electron flux on the magnetic surfaces. Our flux-balanced
HW (bHW) model contains the same drift-wave instability as previous HW models,
but unlike these models, it converges exactly to the modified Hasegawa-Mima
model in the collisionless limit. We rely on direct numerical simulations to
illustrate some of the key features of the bHW model, such as the enhanced
variability in the turbulent fluctuations, and the existence of stronger and
more turbulent zonal jets than the jets observed in other HW models, especially
for high plasma resistivity. Our simulations also highlight the crucial role of
the feedback of the third-order statistical moments in achieving a statistical
equilibrium with strong zonal structures. Finally, we investigate the changes
in the observed dynamics when more general dissipation effects are included,
and in particular when we include the reduced model for ion Landau damping
originally proposed by Wakatani and Hasegawa.Comment: 23 pages, 9 figure
Linear Response Based Parameter Estimation in the Presence of Model Error
Recently, we proposed a method to estimate parameters of stochastic dynamics
based on the linear response statistics. The method rests upon a nonlinear
least-squares problem that takes into account the response properties that stem
from the Fluctuation-Dissipation Theory. In this article, we address an
important issue that arises in the presence of model error. In particular, when
the equilibrium density function is high dimensional and non-Gaussian, and in
some cases, is unknown, the linear response statistics are inaccessible. We
show that this issue can be resolved by fitting the imperfect model to
appropriate marginal linear response statistics that can be approximated using
the available data and parametric or nonparametric models. The effectiveness of
the parameter estimation approach is demonstrated in the context of molecular
dynamical models (Langevin dynamics) with a non-uniform temperature profile,
where the modeling error is due to coarse-graining, and a PDE (non-Langevin
dynamics) that exhibits spatiotemporal chaos, where the model error arises from
a severe spectral truncation. In these examples, we show how the imperfect
models, the Langevin equation with parameters estimated using the proposed
scheme, can predict the nonlinear response statistics of the underlying
dynamics under admissible external disturbances.Comment: 27 pages, 20 figure