24 research outputs found
Global Regularity for an Inviscid Three-dimensional Slow Limiting Ocean Dynamics Model
We establish, for smooth enough initial data, the global well-posedness
(existence, uniqueness and continuous dependence on initial data) of solutions,
for an inviscid three-dimensional {\it slow limiting ocean dynamics} model.
This model was derived as a strong rotation limit of the rotating and
stratified Boussinesg equations with periodic boundary conditions. To establish
our results we utilize the tools developed for investigating the
two-dimensional incompressible Euler equations and linear transport equations.
Using a weaker formulation of the model we also show the global existence and
uniqueness of solutions, for less regular initial data
Assimilation of nearly turbulent Rayleigh-B\'enard flow through vorticity or local circulation measurements: a computational study
We introduce a continuous (downscaling) data assimilation algorithm for the
2D B\'enard convection problem using vorticity or local circulation
measurements only. In this algorithm, a nudging term is added to the vorticity
equation to constrain the model. Our numerical results indicate that the
approximate solution of the algorithm is converging to the unknown reference
solution (vorticity and temperature) corresponding to the measurements of the
2D B\'enard convection problem when only spatial coarse-grain measurements of
vorticity are assimilated. Moreover, this convergence is realized using data
which is much more coarse than the resolution needed to satisfy rigorous
analytical estimates
Long-time Behavior of a Two-layer Model of Baroclinic Quasi-geostrophic Turbulence
We study a viscous two-layer quasi-geostrophic beta-plane model that is
forced by imposition of a spatially uniform vertical shear in the eastward
(zonal) component of the layer flows, or equivalently a spatially uniform
north-south temperature gradient. We prove that the model is linearly unstable,
but that non-linear solutions are bounded in time by a bound which is
independent of the initial data and is determined only by the physical
parameters of the model. We further prove, using arguments first presented in
the study of the Kuramoto-Sivashinsky equation, the existence of an absorbing
ball in appropriate function spaces, and in fact the existence of a compact
finite-dimensional attractor, and provide upper bounds for the fractal and
Hausdorff dimensions of the attractor. Finally, we show the existence of an
inertial manifold for the dynamical system generated by the model's solution
operator. Our results provide rigorous justification for observations made by
Panetta based on long-time numerical integrations of the model equations
Continuous Data Assimilation for the 2D B\'enard Convection through Velocity Measurements Alone
An algorithm for continuous data assimilation for the two- dimensional
B\'enard convection problem is introduced and analyzed. It is inspired by the
data assimilation algorithm developed for the Navier-Stokes equations, which
allows for the implementation of variety of observables: low Fourier modes,
nodal values, finite volume averages, and finite elements. The novelty here is
that the observed data is obtained for the velocity field alone; i.e. no
temperature measurements are needed for this algorithm. 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, the unique exact unknown
solution of the B\'enard convection problem associated with the observed
(finite dimensional projection of) velocity