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Yanzhao Cao, Max Gunzburger, Fei Hua, Xiaoming Wang
Commun. Math. Sci. 8 (1), 1-25, (March 2010)
KEYWORDS: Stokes and Darcy system, fluid and porous media flow, Beavers-Joseph interface boundary condition, well-posedness, time discretization, finite element approximation, 35Q35, 65M60, 76D07, 76S05, 86A05
We investigate the well-posedness of a coupled Stokes-Darcy model with Beavers-Joseph interface boundary conditions. In the steady-state case, the well-posedness is established under the assumption of a small coefficient in the Beavers-Joseph interface boundary condition. In the time-dependent case, the well-posedness is established via an appropriate time discretization of the problem and a novel scaling of the system under an isotropic media assumption. Such coupled systems are crucial to the study of subsurface flow problems, in particular, flows in karst aquifers.
Two types of filtering failure are the well known filter divergence where errors may exceed the size of the corresponding true chaotic attractor and the much more severe catastrophic filter divergence where solutions diverge to machine infinity in finite time. In this paper, we demonstrate that these failures occur in filtering the L-96 model, a nonlinear chaotic dissipative dynamical system with the absorbing ball property and quasi-Gaussian unimodal statistics. In particular, catastrophic filter divergence occurs in suitable parameter regimes for an ensemble Kalman filter when the noisy turbulent true solution signal is partially observed at sparse regular spatial locations.
With the above documentation, the main theme of this paper is to show that we can suppress the catastrophic filter divergence with a judicious model error strategy, that is, through a suitable linear stochastic model. This result confirms that the Gaussian assumption in the Kalman filter formulation, which is violated by most ensemble Kalman filters through the nonlinearity in the model, is a necessary condition to avoid catastrophic filter divergence. In a suitable range of chaotic regimes, adding model errors is not the best strategy when the true model is known. However, we find that there are several parameter regimes where the filtering performance in the presence of model errors with the stochastic model supersedes the performance in the perfect model simulation of the best ensemble Kalman filter considered here. Secondly, we also show that the advantage of the reduced Fourier domain filtering strategy, A. Majda and M. Grote, Proceedings of the National Academy of Sciences, 104, 1124-1129, 2007, E. Castronovo, J. Harlim and A. Majda, J. Comput. Phys., 227(7), 3678-3714, 2008, J. Harlim and A. Majda, J. Comput. Phys., 227(10), 5304-5341, 2008 is not simply through its numerical efficiency, but significant filtering accuracy is also gained through ignoring the correlation between the appropriate Fourier coefficients when the sparse observations are available in regular space locations.
A three-mode nonlinear slow-fast system with fast forcing is studied here as a model for filtering turbulent signals from partial observations. The model describes the interaction of two externally driven fast modes with a slow mode through catalytic nonlinear coupling. The special structure of the nonlinear interaction allows for the analytical solution for the first and second order statistics even with fast forcing. These formulas are used for testing the exact Nonlinear Extended Kalman Filter for the slow-fast system with fast forcing. Various practical questions such as the influence of the strong fast forcing on the slowly varying wave envelope, the role of observations, the frequency and variance of observations, and the model error due to linearization are addressed here.
Squall lines are coherent turbulent traveling waves on scales of order 100 km in the atmosphere that emerge in a few hours from the interaction of strong vertical shear and moist deep convection on scales of order 10 km. They are canonical coherent structures in the tropics and middle latitudes reflecting upscale conversion of energy from moist buoyant sources to horizontal kinetic energy on larger scales. Here squall lines are introduced through high resolution numerical simulations which reveal a new self-similarity with respect to the shear amplitude. A new multi-scale model on mesoscales which allows for large vertical shears, appropriate for squall lines, is developed here through systematic multi-scale asymptotics. Mathematical and numerical formulations of the new multi-scale equations are utilized to illustrate both new mathematical and physical phenomena captured by these new models. In particular, non-hydrostatic Taylor-Goldstein equations govern the upscale transports of momentum and temperature from the order 10 km microscales to the order 100 km mesoscales; surprisingly, upright single mode convective heating without tilts can lead to significant upscale convective momentum transport from the microscales to the mesoscales due to the strong shear. The multi-scale models developed here should be especially useful for dynamic parameterizations of upscale transports as well as for new theory in three-dimensions with a transverse shear component, where contemporary theoretical understanding is meager.
New linear response formulas for unperturbed chaotic (stochastic) complex dynamical systems with time periodic coefficients are developed here. Such time periodic systems arise naturally in climate change studies due to the seasonal cycle. These response formulas are developed through the mathematical interplay between statistical solutions for the time-periodic dynamical systems and the related skew-product system. This interplay is utilized to develop new systematic quasi-Gaussian and adjoint algorithms for calculating the climate response in such time-periodic systems. These new formulas are found in section 4. New linear response formulas are also developed here for general time-dependent statistical ensembles arising in ensemble prediction including the effects of deterministic model errors, initial ensembles, and model noise perturbations simultaneously. An information theoretic perspective is developed in calculating those model perturbations which yield the largest information deficit for the unperturbed system both for climate response and finite ensemble predictions.
We characterize the high intensity limits of minimal free energy states for interacting corpora — that is, for objects with finitely many degrees of freedom, such as articulated rods. These limits are measures supported on zero-level-sets of the interaction potential. We describe a selection mechanism for the limits that is mediated by evanescent entropic contributions.
A stochastic model for representing the missing variability in global climate models due to unresolved features of organized tropical convection is presented here. We use a Markov chain lattice model to represent small scale convective elements which interact with each other and with the large scale environmental variables through convective available potential energy (CAPE) and middle troposphere dryness. Each lattice site is either occupied by a cloud of a certain type (congestus, deep or stratiform) or it is a clear sky site. The lattice sites are assumed to be independent from each other so that a coarse-grained stochastic birth-death system, which can be evolved with a very low computational overhead, is obtained for the cloud area fractions alone. The stochastic multicloud model is then coupled to a simple tropical climate model consisting of a system of ODEs, mimicking the dynamics over a single GCM grid box. Physical intuition and observations are employed here to constrain the design of the models. Numerical simulations showcasing some of the dynamical features of the coupled model are presented below.
A methodology is developed to assign, from an observed sample, a joint-probability distribution to a set of continuous variables. The algorithm proposed performs this assignment by mapping the original variables onto a jointly-Gaussian set. The map is built iteratively, ascending the log-likelihood of the observations, through a series of steps that move the marginal distributions along a random set of orthogonal directions towards normality.
We study a semi-implicit time-difference scheme for magnetohydrodynamics of a viscous and resistive incompressible fluid in a bounded smooth domain with a perfectly conducting boundary. In the scheme, the velocity and magnetic fields are updated by solving simple Helmholtz equations. Pressure is treated explicitly in time, by solving Poisson equations corresponding to a recently de- veloped formula for the Navier-Stokes pressure involving the commutator of Laplacian and Leray projection operators. We prove stability of the time-difference scheme, and deduce a local-time well- posedness theorem for MHD dynamics extended to ignore the divergence-free constraint on velocity and magnetic fields. These fields are divergence-free for all later time if they are initially so.
We present a time-dependent semiclassical transport model for coherent pure-state scattering with quantum barriers. The model is based on a complex-valued Liouville equation, with interface conditions at quantum barriers computed from the steady-state Schrödinger equation. By retaining the phase information at the barrier, this coherent model adequately describes quantum scattering and interference at quantum barriers, with a computational cost comparable to that of classical mechanics. We construct both Eulerian and Lagrangian numerical methods for this model, and validate it using several numerical examples, including multiple quantum barriers.
The Navier-Stokes-Voigt (NSV) model of viscoelastic incompressible fluid has been recently proposed as a regularization of the 3D Navier-Stokes equations for the purpose of direct numerical simulations. In this work we investigate its statistical properties by employing phenomeno- logical heuristic arguments, in combination with Sabra shell model simulations of the analogue of the NSV model. For large values of the regularizing parameter, compared to the Kolmogorov length scale, simulations exhibit multiscaling inertial range, and the dissipation range displaying low inter- mittency. These facts provide evidence that the NSV regularization may reduce the stiffness of direct numerical simulations of turbulent flows, with a small impact on the energy containing scales.
An idealized framework to study the impacts of phase transitions on atmospheric dynamics is described. Condensation of water vapor releases a significant amount of latent heat, which directly affects the atmospheric temperature and density. Here, phase transitions are treated by assuming that air parcels are in local thermodynamic equilibrium, which implies that condensed water can only be present when the air parcel is saturated. This reduces the number of variables necessary to describe the thermodynamic state of moist air to three. It also introduces a discontinuity in the partial derivatives of the equation of state. A simplified version of the equation of state is obtained by a separate linearization for saturated and unsaturated parcels. When this equation of state is implemented in a Boussinesq system, the buoyancy can be expressed as a piecewise linear function of two prognostic thermodynamic variables, D and M, and height z. Numerical experiments on the nonlinear evolution of the convection and the impact of latent heat release on the buoyant flux are presented.