28 research outputs found

### Continuous Data Assimilation for a 2D B\'enard Convection System through Horizontal Velocity Measurements Alone

In this paper we propose a continuous data assimilation (downscaling)
algorithm for a two-dimensional B\'enard convection problem. Specifically we
consider the two-dimensional Boussinesq system of a layer of incompressible
fluid between two solid horizontal walls, with no-normal flow and stress free
boundary condition on the walls, and fluid is heated from the bottom and cooled
from the top. In this algorithm, we incorporate the observables as a feedback
(nudging) term in the evolution equation of the horizontal velocity. We show
that under an appropriate choice of the nudging parameter and the size of the
spatial coarse mesh observables, and under the assumption that the observed
data is error free, the solution of the proposed algorithm converges at an
exponential rate, asymptotically in time, to the unique exact unknown reference
solution of the original system, associated with the observed data on the
horizontal component of the velocity. Moreover, we note that in the case where
the observational measurements are not error free, one can estimate the error
between the solution of the algorithm and the exact reference solution of the
system in terms of the error in the measurements.Comment: arXiv admin note: text overlap with arXiv:1506.0867

### Abridged continuous data assimilation for the 2D Navier-Stokes equations utilizing measurements of only one component of the velocity field

We introduce a continuous data assimilation (downscaling) algorithm for the
two-dimensional Navier-Stokes equations employing coarse mesh measurements of
only one component of the velocity field. This algorithm can be implemented
with a variety of finitely many observables: low Fourier modes, nodal values,
finite volume averages, or finite elements. We provide conditions on the
spatial resolution of the observed data, under the assumption that the observed
data is free of noise, which are sufficient to show that the solution of the
algorithm approaches, at an exponential rate asymptotically in time, to the
unique exact unknown reference solution, of the 2D Navier-Stokes equations,
associated with the observed (finite dimensional projection of) velocity

### Data Assimilation algorithm for 3D B\'enard convection in porous media employing only temperature measurements

In this paper we propose a continuous data assimilation (downscaling)
algorithm for the B\'enard convection in porous media using only coarse mesh
measurements of the temperature. In this algorithm, we incorporate the
observables as a feedback (nudging) term in the evolution equation of the
temperature. We show that under an appropriate choice of the nudging parameter
and the size of the mesh, and under the assumption that the observed data is
error free, the solution of the proposed algorithm converges at an exponential
rate, asymptotically in time, to the unique exact unknown reference solution of
the original system, associated with the observed (finite dimensional
projection of) temperature data. Moreover, we note that in the case where the
observational measurements are not error free, one can estimate the error
between the solution of the algorithm and the exact reference solution of the
system in terms of the error in the measurements.Comment: arXiv admin note: text overlap with arXiv:1504.0597

### Launching Drifter Observations in the Presence of Uncertainty

Determining the optimal locations for placing extra observational
measurements has practical significance. However, the exact underlying flow
field is never known in practice. Significant uncertainty appears when the flow
field is inferred from a limited number of existing observations via data
assimilation or statistical forecast. In this paper, a new computationally
efficient strategy for deploying Lagrangian drifters that highlights the
central role of uncertainty is developed. A nonlinear trajectory diagnostic
approach that underlines the importance of uncertainty is built to construct a
phase portrait map. It consists of both the geometric structure of the
underlying flow field and the uncertainty in the estimated state from
Lagrangian data assimilation. The drifters are deployed at the maxima of this
map and are required to be separated enough. Such a strategy allows the
drifters to travel the longest distances to collect both the local and global
information of the flow field. It also facilitates the reduction of a
significant amount of uncertainty. To characterize the uncertainty, the
estimated state is given by a probability density function (PDF). An
information metric is then introduced to assess the information gain in such a
PDF, which is fundamentally different from the traditional path-wise
measurements. The information metric also avoids using the unknown truth to
quantify the uncertainty reduction, making the method practical. Mathematical
analysis exploiting simple illustrative examples is used to validate the
strategy. Numerical simulations based on multiscale turbulent flows are then
adopted to demonstrate the advantages of this strategy over some other methods.Comment: 45 pages, 11 figure

### Lagrangian Descriptors with Uncertainty

Lagrangian descriptors provide a global dynamical picture of the geometric
structures for arbitrarily time-dependent flows with broad applications. This
paper develops a mathematical framework for computing Lagrangian descriptors
when uncertainty appears. The uncertainty originates from estimating the
underlying flow field as a natural consequence of data assimilation or
statistical forecast. It also appears in the resulting Lagrangian trajectories.
The uncertainty in the flow field directly affects the path integration of the
crucial nonlinear positive scalar function in computing the Lagrangian
descriptor, making it fundamentally different from many other diagnostic
methods. Despite being highly nonlinear and non-Gaussian, closed analytic
formulae are developed to efficiently compute the expectation of such a scalar
function due to the uncertain velocity field by exploiting suitable
approximations. A rapid and accurate sampling algorithm is then built to assist
the forecast of the probability density function (PDF) of the Lagrangian
trajectories. Such a PDF provides the weight to combine the Lagrangian
descriptors along different paths. Simple but illustrative examples are
designed to show the distinguished behavior of using Lagrangian descriptors in
revealing the flow field when uncertainty appears. Uncertainty can either
completely erode the coherent structure or barely affect the underlying
geometry of the flow field. The method is also applied for eddy identification,
indicating that uncertainty has distinct impacts on detecting eddies at
different time scales. Finally, when uncertainty is incorporated into the
Lagrangian descriptor for inferring the source target, the likelihood criterion
provides a very different conclusion from the deterministic methods.Comment: 51 pages, 17 figure

### Spectral scaling of the Leray-$\alpha$ model for two-dimensional turbulence

We present data from high-resolution numerical simulations of the
Navier-Stokes-$\alpha$ and the Leray-$\alpha$ models for two-dimensional
turbulence. It was shown previously (Lunasin et al., J. Turbulence, 8, (2007),
751-778), that for wavenumbers $k$ such that $k\alpha\gg 1$, the energy
spectrum of the smoothed velocity field for the two-dimensional
Navier-Stokes-$\alpha$ (NS-$\alpha$) model scales as $k^{-7}$. This result is
in agreement with the scaling deduced by dimensional analysis of the flux of
the conserved enstrophy using its characteristic time scale. We therefore
hypothesize that the spectral scaling of any $\alpha$-model in the sub-$\alpha$
spatial scales must depend only on the characteristic time scale and dynamics
of the dominant cascading quantity in that regime of scales. The data presented
here, from simulations of the two-dimensional Leray-$\alpha$ model, confirm our
hypothesis. We show that for $k\alpha\gg 1$, the energy spectrum for the
two-dimensional Leray-$\alpha$ scales as $k^{-5}$, as expected by the
characteristic time scale for the flux of the conserved enstrophy of the
Leray-$\alpha$ model. These results lead to our conclusion that the dominant
directly cascading quantity of the model equations must determine the scaling
of the energy spectrum.Comment: 11 pages, 4 figure

### Analysis of a General Family of Regularized Navier-Stokes and MHD Models

We consider a general family of regularized Navier-Stokes and
Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian
manifolds with or without boundary, with n greater than or equal to 2. This
family captures most of the specific regularized models that have been proposed
and analyzed in the literature, including the Navier-Stokes equations, the
Navier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alpha
model, the Simplified Bardina model, the Navier-Stokes-Voight model, the
Navier-Stokes-alpha-like models, and certain MHD models, in addition to
representing a larger 3-parameter family of models not previously analyzed. We
give a unified analysis of the entire three-parameter family using only
abstract mapping properties of the principle dissipation and smoothing
operators, and then use specific parameterizations to obtain the sharpest
results. We first establish existence and regularity results, and under
appropriate assumptions show uniqueness and stability. We then establish
results for singular perturbations, including the inviscid and alpha limits.
Next we show existence of a global attractor for the general model, and give
estimates for its dimension. We finish by establishing some results on
determining operators for subfamilies of dissipative and non-dissipative
models. In addition to establishing a number of results for all models in this
general family, the framework recovers most of the previous results on
existence, regularity, uniqueness, stability, attractor existence and
dimension, and determining operators for well-known members of this family.Comment: 37 pages; references added, minor typos corrected, minor changes to
revise for publicatio