Travelling waves in two-dimensional plane Poiseuille flow

The asymptotic structure of laminar modulated travelling waves in two-dimensional high-Reynolds-number plane Poiseuille flow is investigated on the upper-energy branch. A finite set of independent slowly varying parameters are identified which parameterize the solution of the Navier–Stokes equations in this subset of the phase space. Our parameterization of the weakly stable modes describes an attracting manifold of maximum-entropy configurations. The complementary modes, which have been neglected in this parameterization, are strongly damped. In order to seek a closure, a countably infinite number of modulation equations are derived on the long viscous time scale: a single equation for averaged kinetic energy and momentum; and the remaining equations for averaged powers of vorticity. Only a finite number of these vorticity modulation equations are required to determine the finite number of unknowns. The new results show that the evolution of the slowly varying amplitude parameters is determined by the vorticity field and that the phase velocity responds to these changes in the amplitude in accordance with the kinetic energy and momentum. The new results also show that the most crucial physical mechanism in the production of vorticity is the interaction between vorticity and kinetic energy, this interaction being responsible for the existence of the attractor

Parameterization of travelling waves in plane Poiseuille flow

© The authors 2012. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This is a pre-copyedited, author-produced PDF of an article accepted for publication in [IMA Journal of Applied Mathematics ] following peer review. The version of record [ IMA Journal of Applied Mathematics (2014) 79(1): 22-32.] is available online at: http://imamat.oxfordjournals.org/content/79/1/22The first finite-dimensional parameterization of a subset of the phase space of the Navier-Stokes equations is presented. Travelling waves in two-dimensional plane Poiseuille flow are numerically shown to approximate maximum-entropy configurations. In a coordinate system moving with the phase velocity, the enclosed body of the flow exhibits a hyperbolic sinusoidal relationship between the vorticity and stream function. The phase velocity and two-amplitude parameters describe the stable manifold on the slow viscous time scale. This original parameterization provides a valuable visualization of this subset of the phase space of the Navier-Stokes equations. These new results provide physical insight into an important intermediate stage in the instability process of plane Poiseuille flow

Oscillatory behavior of two nonlinear microbial models of soil carbon decomposition

A number of nonlinear models have recently been proposed for simulating soil carbon decomposition. Their predictions of soil carbon responses to fresh litter input and warming differ significantly from conventional linear models. Using both stability analysis and numerical simulations, we showed that two of those nonlinear models (a two-pool model and a three-pool model) exhibit damped oscillatory responses to small perturbations. Stability analysis showed the frequency of oscillation is proportional to √(ε⁻¹-1) Ks/Vs in the two-pool model, and to √(ε⁻¹-1) Kl/Vl in the three-pool model, where ε is microbial growth efficiency, Ks and Kl are the half saturation constants of soil and litter carbon, respectively, and /Vs and /Vl are the maximal rates of carbon decomposition per unit of microbial biomass for soil and litter carbon, respectively. For both models, the oscillation has a period of between 5 and 15 years depending on other parameter values, and has smaller amplitude at soil temperatures between 0 and 15°C. In addition, the equilibrium pool sizes of litter or soil carbon are insensitive to carbon inputs in the nonlinear model, but are proportional to carbon input in the conventional linear model. Under warming, the microbial biomass and litter carbon pools simulated by the nonlinear models can increase or decrease, depending whether ε varies with temperature. In contrast, the conventional linear models always simulate a decrease in both microbial and litter carbon pools with warming. Based on the evidence available, we concluded that the oscillatory behavior and insensitivity of soil carbon to carbon input are notable features in these nonlinear models that are somewhat unrealistic. We recommend that a better model for capturing the soil carbon dynamics over decadal to centennial timescales would combine the sensitivity of the conventional models to carbon influx with the flexible response to warming of the nonlinear model.15 page(s

Asymptotic analysis of the attractors in two-dimensional Kolmogorov flow

The high-Reynolds-number structure of the laminar, chaotic and turbulent attractors is investigated in a two-dimensional Kolmogorov flow. The laminar attractors include the families of multi-phased travelling waves and quasi-periodic standing waves both of which form the backbone of the transition to a turbulent flow. At leading order, each laminar attractor under study is obtained by solving the Euler equations on a manifold subject to the appropriate periodicity and symmetry conditions. The manifold is determined by a finite number of vorticity equations, these being required to suppress the secular terms at the next order. Our results show that, for the multi-phased travelling waves, the first phase velocity can be determined by an integral conservation law for kinetic energy and the subsequent phase velocities can be evaluated by a nonlinear eigenvalue problem. The results also reveal that whereas viscosity determines the smallest scales and controls the amplitude of the flow, the inertial terms govern the shape and form of the flow. The com-parison of our analytical predictions for evaluating the stable single-phased travelling wave with the direct numerical simulation of the Navier–Stokes equations has been undertaken, the agreement being excellent. For sufficiently high Reynolds number, after the bifurcation to chaotic flow, all of the multi-phased travelling waves and quasi-periodic standing waves become unstable non-wandering sets. Based on the above new findings for these unstable non-wandering sets and other travelling and standing waves of this kind in phase space, necessary conditions for the invariant manifolds of the chaotic and turbulent attractors are obtained, these necessary conditions being conjectured to be also sufficient

Retirement Behavior and the Global Financial Crisis

Recent economic conditions have vastly changed the retirement landscape. Declines in assets as well as high unemployment changed the retirement plans of many Americans. Shocks to employment and wealth have likely influenced retirement behavior. This chapter provides a survey of the current literature on the influence of employment and wealth shocks on retirement and then makes use of administrative records on benefit applications to provide a preliminary analysis of changes in early retirement (age 62) claiming resulting from the recent economic downturn and implications. Since early claiming can have long lasting implications for retirement well being, we address how Americans learn about their retirement options

Serial Monodomy in the Gypsy Ant, Aphaenogaster araneoides: Does Nest Odor Reduction Influence Colony Relocation?

Serial monodomy is the nesting behavior in which a colony of animals maintains multiple nests for its exclusive use, occupying one nest at a time. Among serially monodomous ants, the availability of unoccupied nests reduces the probability and costs of army ant attacks. It has been proposed that nest odors mediate serial monodomy in the gypsy ant, Aphaenogaster araneoides Emery (Hymenoptera: Formicidae), and that colonies avoid returning to previously occupied nests that harbor colony odors. To evaluate this hypothesis, the odors inside the nests of A. araneoides colonies were experimentally reduced through ventilation; the nest movement behaviors of treatment and control colonies were compared. Odor reduction was found to have increased the frequency of movements into and out of the treated nest, without a change in the total occupation duration in the treated nest. Nests with a more open architecture that permitted natural flow of air were reoccupied more quickly than nests with smaller nest entrances. In summary, the openness of the architecture of A. araneoides nests and the ventilation of air through nests alters the use of these nests. These findings further support the working hypothesis that nest-bound odors mediate the pattern of serial monodomy in A. araneoides

The Leeway of Shipping Containers at Different Immersion Levels

The leeway of 20-foot containers in typical distress conditions is established through field experiments in a Norwegian fjord and in open-ocean conditions off the coast of France with wind speed ranging from calm to 14 m/s. The experimental setup is described in detail and certain recommendations given for experiments on objects of this size. The results are compared with the leeway of a scaled-down container before the full set of measured leeway characteristics are compared with a semi-analytical model of immersed containers. Our results are broadly consistent with the semi-analytical model, but the model is found to be sensitive to choice of drag coefficient and makes no estimate of the cross-wind leeway of containers. We extend the results from the semi-analytical immersion model by extrapolating the observed leeway divergence and estimates of the experimental uncertainty to various realistic immersion levels. The sensitivity of these leeway estimates at different immersion levels are tested using a stochastic trajectory model. Search areas are found to be sensitive to the exact immersion levels, the choice of drag coefficient and somewhat less sensitive to the inclusion of leeway divergence. We further compare the search areas thus found with a range of trajectories estimated using the semi-analytical model with only perturbations to the immersion level. We find that the search areas calculated without estimates of crosswind leeway and its uncertainty will grossly underestimate the rate of expansion of the search areas. We recommend that stochastic trajectory models of container drift should account for these uncertainties by generating search areas for different immersion levels and with the uncertainties in crosswind and downwind leeway reported from our field experiments.Comment: 25 pages, 11 figures and 5 tables; Ocean Dynamics, Special Issue on Advances in Search and Rescue at Sea (2012

Estimating the furrow infiltration characteristic from a single advance point

Management and control of surface irrigation, in particular furrow irrigation, is limited by spatio-temporal soil infiltration variability as well as the high cost and time associated with collecting intensive field data for estimation of the infiltration characteristics. Recent work has proposed scaling the commonly used infiltration function by using a model infiltration curve and a single advance point for every other furrow in an irrigation event. Scaling factors were calculated for a series of furrows at two sites and at four points down the length of the field (0.25 L, 0.5 L, 0.75 L and L). Differences in the value of the scaling factor with distance were found to be a function of the shape of the advance curves. It is concluded that use of points early in the advance results in a substantial loss of accuracy and should be avoided. The scaling factor was also strongly correlated with the furrow-wetted perimeter suggesting that the scaling is an appropriate way of both predicting and accommodating the effect of the hydraulic variability

Using an imperfect photonic network to implement random unitaries

We numerically investigate the implementation of Haar-random unitarity transformations and Fourier transformations in photonic devices consisting of beam splitters and phase shifters, which are used for integrated photonics implementations of boson sampling. The distribution of reflectivities required to implement an arbitrary unitary transformation is skewed towards low values, and this skew becomes stronger the larger the number of modes. A realistic implementation using Mach-Zehnder interferometers is incapable of achieving the low values required and thus has limited fidelity. We show that numerical optimisation and adding extra beam splitters to the network can help to restore fidelity