Langmuir turbulence and surface heating in the ocean surface boundary layer

This study uses large-eddy simulation to investigate the structure of the ocean surface boundary layer (OSBL) in the presence of Langmuir turbulence and stabilizing surface heat fluxes. The OSBL consists of a weakly stratified layer, despite a surface heat flux, above a stratified thermocline. The weakly stratified (mixed) layer is maintained by a combination of a turbulent heat flux produced by the wave-driven Stokes drift and downgradient turbulent diffusion. The scaling of turbulence statistics, such as dissipation and vertical velocity variance, is only affected by the surface heat flux through changes in the mixed layer depth. Diagnostic models are proposed for the equilibrium boundary layer and mixed layer depths in the presence of surface heating. The models are a function of the initial mixed layer depth before heating is imposed and the Langmuir stability length. In the presence of radiative heating, the models are extended to account for the depth profile of the heating.

Multiyear measurements of the oceanic and atmospheric boundary layers at the Brazil-Malvinas confluence region

This study analyzes and discusses data taken from oceanic and atmospheric measurements performed simultaneously at the Brazil-Malvinas Confluence (BMC) region in the southwestern Atlantic Ocean. This area is one of the most dynamical frontal regions of the world ocean. Data were collected during four research cruises in the region once a year in consecutive years between 2004 and 2007. Very few studies have addressed the importance of studying the air-sea coupling at the BMC region. Lateral temperature gradients at the study region were as high as 0.3 degrees C km(-1) at the surface and subsurface. In the oceanic boundary layer, the vertical temperature gradient reached 0.08 degrees C m(-1) at 500 m depth. Our results show that the marine atmospheric boundary layer (MABL) at the BMC region is modulated by the strong sea surface temperature (SST) gradients present at the sea surface. The mean MABL structure is thicker over the warmside of the BMC where Brazil Current (BC) waters predominate. The opposite occurs over the coldside of the confluence where waters from the Malvinas (Falkland) Current (MC) are found. The warmside of the confluence presented systematically higher MABL top height compared to the coldside. This type of modulation at the synoptic scale is consistent to what happens in other frontal regions of the world ocean, where the MABL adjusts itself to modifications along the SST gradients. Over warm waters at the BMC region, the MABL static instability and turbulence were increased while winds at the lower portion of the MABL were strong. Over the coldside of the BC/MC front an opposite behavior is found: the MABL is thinner and more stable. Our results suggest that the sea-level pressure (SLP) was also modulated locally, together with static stability vertical mixing mechanism, by the surface condition during all cruises. SST gradients at the BMC region modulate the synoptic atmospheric pressure gradient. Postfrontal and prefrontal conditions produce opposite thermal advections in the MABL that lead to different pressure intensification patterns across the confluence.

Stability of numerical schemes on staggered grids

This paper considers the stability of explicit, implicit and Crank-Nicolson schemes for the one-dimensional heat equation on a staggered grid. Furthemore, we consider the cases when both explicit and implicit approximations of the boundary conditions arc employed. Why we choose to do this is clearly motivated and arises front solving fluid flow equations with free surfaces when the Reynolds number can be very small. in at least parts of the spatial domain. A comprehensive stability analysis is supplied: a novel result is the precise stability restriction on the Crank-Nicolson method when the boundary conditions are approximated explicitly, that is, at t =n delta t rather than t = (n + 1)delta t. The two-dimensional Navier-Stokes equations were then solved by a marker and cell approach for two simple problems that had analytic solutions. It was found that the stability results provided in this paper were qualitatively very similar. thereby providing insight as to why a Crank-Nicolson approximation of the momentum equations is only conditionally, stable. Copyright (C) 2008 John Wiley & Sons, Ltd.

Cascades of Hopf bifurcations from boundary delay

We consider a 1-dimensional reaction-diffusion equation with nonlinear boundary conditions of logistic type with delay. We deal with non-negative solutions and analyze the stability behavior of its unique positive equilibrium solution, which is given by the constant function u equivalent to 1. We show that if the delay is small, this equilibrium solution is asymptotically stable, similar as in the case without delay. We also show that, as the delay goes to infinity, this equilibrium becomes unstable and undergoes a cascade of Hopf bifurcations. The structure of this cascade will depend on the parameters appearing in the equation. This equation shows some dynamical behavior that differs from the case where the nonlinearity with delay is in the interior of the domain. (C) 2009 Elsevier Inc. All rights reserved.

Efficient solutions to robust, semi-implicit discretizations of the immersed boundary method

The immersed boundary method is a versatile tool for the investigation of flow-structure interaction. In a large number of applications, the immersed boundaries or structures are very stiff and strong tangential forces on these interfaces induce a well-known, severe time-step restriction for explicit discretizations. This excessive stability constraint can be removed with fully implicit or suitable semi-implicit schemes but at a seemingly prohibitive computational cost. While economical alternatives have been proposed recently for some special cases, there is a practical need for a computationally efficient approach that can be applied more broadly. In this context, we revisit a robust semi-implicit discretization introduced by Peskin in the late 1970s which has received renewed attention recently. This discretization, in which the spreading and interpolation operators are lagged. leads to a linear system of equations for the inter-face configuration at the future time, when the interfacial force is linear. However, this linear system is large and dense and thus it is challenging to streamline its solution. Moreover, while the same linear system or one of similar structure could potentially be used in Newton-type iterations, nonlinear and highly stiff immersed structures pose additional challenges to iterative methods. In this work, we address these problems and propose cost-effective computational strategies for solving Peskin`s lagged-operators type of discretization. We do this by first constructing a sufficiently accurate approximation to the system`s matrix and we obtain a rigorous estimate for this approximation. This matrix is expeditiously computed by using a combination of pre-calculated values and interpolation. The availability of a matrix allows for more efficient matrix-vector products and facilitates the design of effective iterative schemes. We propose efficient iterative approaches to deal with both linear and nonlinear interfacial forces and simple or complex immersed structures with tethered or untethered points. One of these iterative approaches employs a splitting in which we first solve a linear problem for the interfacial force and then we use a nonlinear iteration to find the interface configuration corresponding to this force. We demonstrate that the proposed approach is several orders of magnitude more efficient than the standard explicit method. In addition to considering the standard elliptical drop test case, we show both the robustness and efficacy of the proposed methodology with a 2D model of a heart valve. (C) 2009 Elsevier Inc. All rights reserved.

ON THE SUPERCRITICALITY OF THE FIRST HOPF BIFURCATION IN A DELAY BOUNDARY PROBLEM

The goal of this paper is to analyze the character of the first Hopf bifurcation (subcritical versus supercritical) that appears in a one-dimensional reaction-diffusion equation with nonlinear boundary conditions of logistic type with delay. We showed in the previous work [Arrieta et al., 2010] that if the delay is small, the unique non-negative equilibrium solution is asymptotically stable. We also showed that, as the delay increases and crosses certain critical value, this equilibrium becomes unstable and undergoes a Hopf bifurcation. This bifurcation is the first one of a cascade occurring as the delay goes to infinity. The structure of this cascade will depend on the parameters appearing in the equation. In this paper, we show that the first bifurcation that occurs is supercritical, that is, when the parameter is bigger than the delay bifurcation value, stable periodic orbits branch off from the constant equilibrium.

A FREE BOUNDARY ISOPERIMETRIC PROBLEM IN HYPERBOLIC 3-SPACE BETWEEN PARALLEL HOROSPHERES

We investigate the isoperimetric problem of finding the regions of prescribed volume with minimal boundary area between two parallel horospheres in hyperbolic 3-space (the part of the boundary contained in the horospheres is not included). We reduce the problem to the study of rotationally invariant regions and obtain the possible isoperimetric solutions by studying the behavior of the profile curves of the rotational surfaces with constant mean curvature in hyperbolic 3-space. We also classify all the connected compact rotational surfaces M of constant mean curvature that are contained in the region between two horospheres, have boundary partial derivative M either empty or lying on the horospheres, and meet the horospheres perpendicularly along their boundary.

Lobatto deferred correction for stiff two-point boundary value problems

An iterated deferred correction algorithm based on Lobatto Runge-Kutta formulae is developed for the efficient numerical solution of nonlinear stiff two-point boundary value problems. An analysis of the stability properties of general deferred correction schemes which are based on implicit Runge-Kutta methods is given and results which are analogous to those obtained for initial value problems are derived. A revised definition of symmetry is presented and this ensures that each deferred correction produces an optimal increase in order. Finally, some numerical results are given to demonstrate the superior performance of Lobatto formulae compared with mono-implicit formulae on stiff two-point boundary value problems. (C) 1998 Elsevier B.V. Ltd. All rights reserved.

Iterated deferred correction for linear two-point boundary value problems

An analysis of iterated deferred correction based on various classes of implicit Runge-Kutta formulae is given. Out of different possibilities considered, it is shown that those based purely on Lobatto formulae have the best stability. The enhanced stability of Lobatto schemes is very important for the efficient integration of excessively stiff boundary value problems and this is demonstrated by means of some numerical results.

Using transient and steady state considerations to investigate the mechanism of loss of stability of a dynamical system

This work presents the complete set of features for solutions of a particular non-ideal mechanical system near the fundamental and near to a secondary resonance region. The system comprises a pendulum with a horizontally moving suspension point. Its motion is the result of a non-ideal rotating power source (limited power supply), acting oil the Suspension point through a crank mechanism. Main emphasis is given to the loss of stability, which occurs by a sequence of events, including intermittence and crisis, when the system reaches a chaotic attractor. The system also undergoes a boundary-crisis, which presents a different aspect in the bifurcation diagram due to the non-ideal supposition. (c) 2004 Published by Elsevier B.V.

Stability analysis of Crank-Nicolson and Euler schemes for time-dependent diffusion equations

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Exponential stability for a plate equation with p-Laplacian and memory terms

This paper is concerned with the energy decay for a class of plate equations with memory and lower order perturbation of p-Laplacian type, utt+?2u-?pu+?0tg(t-s)?u(s)ds-?ut+f(u)=0inOXR+, with simply supported boundary condition, where O is a bounded domain of RN, g?>?0 is a memory kernel that decays exponentially and f(u) is a nonlinear perturbation. This kind of problem without the memory term models elastoplastic flows.

SUFFICIENT CONDITIONS ON DIFFUSIVITY FOR THE EXISTENCE AND NONEXISTENCE OF STABLE EQUILIBRIA WITH NONLINEAR FLUX ON THE BOUNDARY

A reaction-diffusion equation with variable diffusivity and non-linear flux boundary condition is considered. The goal is to give sufficient conditions on the diffusivity function for nonexistence and also for existence of nonconstant stable stationary solutions. Applications are given for the main result of nonexistence.

Mesoscale dynamics and boundary-layer structure in topographically forced low-level jets

Two types of mesoscale wind-speed jet and their effects on boundary-layer structure were studied. The first is a coastal jet off the northern California coast, and the second is a katabatic jet over Vatnajökull, Iceland. Coastal regions are highly populated, and studies of coastal meteorology are of general interest for environmental protection, fishing industry, and for air and sea transportation. Not so many people live in direct contact with glaciers but properties of katabatic flows are important for understanding glacier response to climatic changes. Hence, the two jets can potentially influence a vast number of people. Flow response to terrain forcing, transient behavior in time and space, and adherence to simplified theoretical models were examined. The turbulence structure in these stably stratified boundary layers was also investigated. Numerical modeling is the main tool in this thesis; observations are used primarily to ensure a realistic model behavior. Simple shallow-water theory provides a useful framework for analyzing high-velocity flows along mountainous coastlines, but for an unexpected reason. Waves are trapped in the inversion by the curvature of the wind-speed profile, rather than by an infinite stability in the inversion separating two neutral layers, as assumed in the theory. In the absence of blocking terrain, observations of steady-state supercritical flows are not likely, due to the diurnal variation of flow criticality. In many simplified models, non-local processes are neglected. In the flows studied here, we showed that this is not always a valid approximation. Discrepancies between simulated katabatic flow and that predicted by an analytical model are hypothesized to be due to non-local effects, such as surface inhomogeneity and slope geometry, neglected in the theory. On a different scale, a reason for variations in the shape of local similarity scaling functions between studies is suggested to be differences in non-local contributions to the velocity variance budgets.

Stability, viscous dissipation and local thermal non-equilibrium in fluid saturated porous media

In this thesis, the field of study related to the stability analysis of fluid saturated porous media is investigated. In particular the contribution of the viscous heating to the onset of convective instability in the flow through ducts is analysed. In order to evaluate the contribution of the viscous dissipation, different geometries, different models describing the balance equations and different boundary conditions are used. Moreover, the local thermal non-equilibrium model is used to study the evolution of the temperature differences between the fluid and the solid matrix in a thermal boundary layer problem. On studying the onset of instability, different techniques for eigenvalue problems has been used. Analytical solutions, asymptotic analyses and numerical solutions by means of original and commercial codes are carried out.