Solution of the Hartree-Fock equations for atoms and diatomic molecules with the finite element method

The finite element method (FEM) is now developed to solve two-dimensional Hartree-Fock (HF) equations for atoms and diatomic molecules. The method and its implementation is described and results are presented for the atoms Be, Ne and Ar as well as the diatomic molecules LiH, BH, N_2 and CO as examples. Total energies and eigenvalues calculated with the FEM on the HF-level are compared with results obtained with the numerical standard methods used for the solution of the one dimensional HF equations for atoms and for diatomic molecules with the traditional LCAO quantum chemical methods and the newly developed finite difference method on the HF-level. In general the accuracy increases from the LCAO - to the finite difference - to the finite element method.

Relativistic LCAO with Minimax Principle and New Balanced Basis Sets

Relativistic density functional theory is widely applied in molecular calculations with heavy atoms, where relativistic and correlation effects are on the same footing. Variational stability of the Dirac Hamiltonian is a very important field of research from the beginning of relativistic molecular calculations on, among efforts for accuracy, efficiency, and density functional formulation, etc. Approximations of one- or two-component methods and searching for suitable basis sets are two major means for good projection power against the negative continuum. The minimax two-component spinor linear combination of atomic orbitals (LCAO) is applied in the present work for both light and super-heavy one-electron systems, providing good approximations in the whole energy spectrum, being close to the benchmark minimax finite element method (FEM) values and without spurious and contaminated states, in contrast to the presence of these artifacts in the traditional four-component spinor LCAO. The variational stability assures that minimax LCAO is bounded from below. New balanced basis sets, kinetic and potential defect balanced (TVDB), following the minimax idea, are applied with the Dirac Hamiltonian. Its performance in the same super-heavy one-electron quasi-molecules shows also very good projection capability against variational collapse, as the minimax LCAO is taken as the best projection to compare with. The TVDB method has twice as many basis coefficients as four-component spinor LCAO, which becomes now linear and overcomes the disadvantage of great time-consumption in the minimax method. The calculation with both the TVDB method and the traditional LCAO method for the dimers with elements in group 11 of the periodic table investigates their difference. New bigger basis sets are constructed than in previous research, achieving high accuracy within the functionals involved. Their difference in total energy is much smaller than the basis incompleteness error, showing that the traditional four-spinor LCAO keeps enough projection power from the numerical atomic orbitals and is suitable in research on relativistic quantum chemistry. In scattering investigations for the same comparison purpose, the failure of the traditional LCAO method of providing a stable spectrum with increasing size of basis sets is contrasted to the TVDB method, which contains no spurious states already without pre-orthogonalization of basis sets. Keeping the same conditions including the accuracy of matrix elements shows that the variational instability prevails over the linear dependence of the basis sets. The success of the TVDB method manifests its capability not only in relativistic quantum chemistry but also for scattering and under the influence of strong external electronic and magnetic fields. The good accuracy in total energy with large basis sets and the good projection property encourage wider research on different molecules, with better functionals, and on small effects.

Estimators and Tests based on Likelihood-Depth with Application to Weibull Distribution, Gaussian and Gumbel Copula

In dieser Arbeit werden mithilfe der Likelihood-Tiefen, eingeführt von Mizera und Müller (2004), (ausreißer-)robuste Schätzfunktionen und Tests für den unbekannten Parameter einer stetigen Dichtefunktion entwickelt. Die entwickelten Verfahren werden dann auf drei verschiedene Verteilungen angewandt. Für eindimensionale Parameter wird die Likelihood-Tiefe eines Parameters im Datensatz als das Minimum aus dem Anteil der Daten, für die die Ableitung der Loglikelihood-Funktion nach dem Parameter nicht negativ ist, und dem Anteil der Daten, für die diese Ableitung nicht positiv ist, berechnet. Damit hat der Parameter die größte Tiefe, für den beide Anzahlen gleich groß sind. Dieser wird zunächst als Schätzer gewählt, da die Likelihood-Tiefe ein Maß dafür sein soll, wie gut ein Parameter zum Datensatz passt. Asymptotisch hat der Parameter die größte Tiefe, für den die Wahrscheinlichkeit, dass für eine Beobachtung die Ableitung der Loglikelihood-Funktion nach dem Parameter nicht negativ ist, gleich einhalb ist. Wenn dies für den zu Grunde liegenden Parameter nicht der Fall ist, ist der Schätzer basierend auf der Likelihood-Tiefe verfälscht. In dieser Arbeit wird gezeigt, wie diese Verfälschung korrigiert werden kann sodass die korrigierten Schätzer konsistente Schätzungen bilden. Zur Entwicklung von Tests für den Parameter, wird die von Müller (2005) entwickelte Simplex Likelihood-Tiefe, die eine U-Statistik ist, benutzt. Es zeigt sich, dass für dieselben Verteilungen, für die die Likelihood-Tiefe verfälschte Schätzer liefert, die Simplex Likelihood-Tiefe eine unverfälschte U-Statistik ist. Damit ist insbesondere die asymptotische Verteilung bekannt und es lassen sich Tests für verschiedene Hypothesen formulieren. Die Verschiebung in der Tiefe führt aber für einige Hypothesen zu einer schlechten Güte des zugehörigen Tests. Es werden daher korrigierte Tests eingeführt und Voraussetzungen angegeben, unter denen diese dann konsistent sind. Die Arbeit besteht aus zwei Teilen. Im ersten Teil der Arbeit wird die allgemeine Theorie über die Schätzfunktionen und Tests dargestellt und zudem deren jeweiligen Konsistenz gezeigt. Im zweiten Teil wird die Theorie auf drei verschiedene Verteilungen angewandt: Die Weibull-Verteilung, die Gauß- und die Gumbel-Copula. Damit wird gezeigt, wie die Verfahren des ersten Teils genutzt werden können, um (robuste) konsistente Schätzfunktionen und Tests für den unbekannten Parameter der Verteilung herzuleiten. Insgesamt zeigt sich, dass für die drei Verteilungen mithilfe der Likelihood-Tiefen robuste Schätzfunktionen und Tests gefunden werden können. In unverfälschten Daten sind vorhandene Standardmethoden zum Teil überlegen, jedoch zeigt sich der Vorteil der neuen Methoden in kontaminierten Daten und Daten mit Ausreißern.

Organische Mikro- und Nanodrähte auf Basis von Perylendiimiden

Diese Arbeit thematisiert die optimierte Darstellung von organischen Mikro- und Nanodrähten, Untersuchungen bezüglich deren molekularen Aufbaus und die anwendungsorientierte Charakterisierung der Eigenschaften. Mikro- und Nanodrähte haben in den letzten Jahren im Zuge der Miniaturisierung von Technologien an weitreichendem Interesse gewonnen. Solche eindimensionalen Strukturen, deren Durchmesser im Bereich weniger zehn Nanometer bis zu einigen wenigen Mikrometern liegt, sind Gegenstand intensiver Forschung. Neben anorganischen Ausgangssubstanzen zur Erzeugung von Mikro- und Nanodrähten haben organische Funktionsmaterialien aufgrund ihrer einfachen und kostengünstigen Verarbeitbarkeit sowie ihrer interessanten elektrischen und optischen Eigenschaften an Bedeutung gewonnen. Eine wichtige Materialklasse ist in diesem Zusammenhang die Verbindungsklasse der n-halbleitenden Perylentetracarbonsäurediimide (kurz Perylendiimide). Dem erfolgreichen Einsatz von eindimensionalen Strukturen als miniaturisierte Bausteine geht die optimierte und kontrollierte Herstellung voraus. Im Rahmen der Doktorarbeit wurde die neue Methode der Drahterzeugung „Trocknen unter Lösungsmittelatmosphäre“ entwickelt, welche auf Selbstassemblierung der Substanzmoleküle aus Lösung basiert und unter dem Einfluss von Lösungsmitteldampf direkt auf einem vorgegebenen Substrat stattfindet. Im Gegensatz zu literaturbekannten Methoden ist kein Transfer der Drähte aus einem Reaktionsgefäß nötig und damit verbundene Beschädigungen der Strukturen werden vermieden. Während herkömmliche Methoden in einer unkontrolliert großen Menge von ineinander verwundenen Drähten resultieren, erlaubt die substratbasierte Technik die Bildung voneinander separierter Einzelfasern und somit beispielsweise den Einsatz in Einzelstrukturbauteilen. Die erhaltenen Fasern sind morphologisch sehr gleichmäßig und weisen bei Längen von bis zu 5 mm bemerkenswert hohe Aspektverhältnisse von über 10000 auf. Darüber hinaus kann durch das direkte Drahtwachstum auf dem Substrat über den Einsatz von vorstrukturierten Oberflächen und Wachstumsmasken gerichtetes, lokal beschränktes Drahtwachstum erzielt werden und damit aktive Kontrolle auf Richtung und Wachstumsbereich der makroskopisch nicht handhabbaren Objekte ausgeübt werden. Um das Drahtwachstum auch hinsichtlich der Materialauswahl, d. h. der eingesetzten Ausgangsmaterialien zur Drahterzeugung und somit der resultierenden Eigenschaften der gebildeten Strukturen aktiv kontrollieren zu können, wird der Einfluss unterschiedlicher Parameter auf die Morphologie der Selbstassemblierungsprodukte am Beispiel unterschiedlicher Derivate betrachtet. So stellt sich zum einen die Art der eingesetzten Lösungsmittel in flüssiger und gasförmiger Phase beim Trocknen unter Lösungsmittelatmosphäre als wichtiger Faktor heraus. Beide Lösungsmittel dienen als Interaktionspartner für die Moleküle des funktionellen Drahtmaterials im Selbstassemblierungsprozess. Spezifische Wechselwirkungen zwischen Perylendiimid-Molekülen untereinander und mit Lösungsmittel-Molekülen bestimmen dabei die äußere Form der erhaltenen Strukturen. Ein weiterer wichtiger Faktor ist die Molekülstruktur des verwendeten funktionellen Perylendiimids. Es wird der Einfluss einer Bay-Substitution bzw. einer unsymmetrischen Imid-Substitution auf die Morphologie der erhaltenen Strukturen herausgestellt. Für das detaillierte Verständnis des Zusammenhanges zwischen Molekülstruktur und nötigen Wachstumsbedingungen für die Bildung von eindimensionalen Strukturen zum einen, aber auch die resultierenden Eigenschaften der erhaltenen Aggregationsprodukte zum anderen, sind Informationen über den molekularen Aufbau von großer Bedeutung. Im Rahmen der Doktorarbeit konnte ein molekular hoch geordneter, kristalliner Aufbau der Drähte nachgewiesen werden. Durch Kombination unterschiedlicher Messmethoden ist es gelungen, die molekulare Anordnung in Strukturen aus einem Spirobifluoren-substituierten Derivat in Form einer verkippten Molekülstapelung entlang der Drahtlängsrichtung zu bestimmen. Um mögliche Anwendungsbereiche der erzeugten Drähte aufzuzeigen, wurden diese hinsichtlich ihrer elektrischen und optischen Eigenschaften analysiert. Neben dem potentiellen Einsatz im Bereich von Filteranwendungen und Sensoren, sind vor allem die halbleitenden und optisch wellenleitenden Eigenschaften hervorzuheben. Es konnten organische Transistoren auf der Basis von Einzeldrähten mit im Vergleich zu Dünnschichtbauteilen erhöhten Ladungsträgerbeweglichkeiten präpariert werden. Darüber hinaus wurden die erzeugten eindimensionalen Strukturen als aktive optische Wellenleiter charakterisiert. Die im Rahmen der Dissertation erarbeiteten Kenntnisse bezüglich der Bildung von eindimensionalen Strukturen durch Selbstassemblierung, des Drahtaufbaus und erster anwendungsorientierter Charakterisierung stellen eine Basis zur Weiterentwicklung solcher miniaturisierter Bausteine für unterschiedlichste Anwendungen dar. Die neu entwickelte Methode des Trocknens unter Lösungsmittelatmosphäre ist nicht auf den Einsatz von Perylendiimiden beschränkt, sondern kann auf andere Substanzklassen ausgeweitet werden. Dies eröffnet breite Möglichkeiten der Materialauswahl und somit der Einsatzmöglichkeiten der erhaltenen Strukturen.

First-principles electronic theory of non-collinear magnetic order in transition-metal nanowires

The structural, electronic and magnetic properties of one-dimensional 3d transition-metal (TM) monoatomic chains having linear, zigzag and ladder geometries are investigated in the frame-work of first-principles density-functional theory. The stability of long-range magnetic order along the nanowires is determined by computing the corresponding frozen-magnon dispersion relations as a function of the 'spin-wave' vector q. First, we show that the ground-state magnetic orders of V, Mn and Fe linear chains at the equilibrium interatomic distances are non-collinear (NC) spin-density waves (SDWs) with characteristic equilibrium wave vectors q that depend on the composition and interatomic distance. The electronic and magnetic properties of these novel spin-spiral structures are discussed from a local perspective by analyzing the spin-polarized electronic densities of states, the local magnetic moments and the spin-density distributions for representative values q. Second, we investigate the stability of NC spin arrangements in Fe zigzag chains and ladders. We find that the non-collinear SDWs are remarkably stable in the biatomic chains (square ladder), whereas ferromagnetic order (q =0) dominates in zigzag chains (triangular ladders). The different magnetic structures are interpreted in terms of the corresponding effective exchange interactions J(ij) between the local magnetic moments μ(i) and μ(j) at atoms i and j. The effective couplings are derived by fitting a classical Heisenberg model to the ab initio magnon dispersion relations. In addition they are analyzed in the framework of general magnetic phase diagrams having arbitrary first, second, and third nearest-neighbor (NN) interactions J(ij). The effect of external electric fields (EFs) on the stability of NC magnetic order has been quantified for representative monoatomic free-standing and deposited chains. We find that an external EF, which is applied perpendicular to the chains, favors non-collinear order in V chains, whereas it stabilizes the ferromagnetic (FM) order in Fe chains. Moreover, our calculations reveal a change in the magnetic order of V chains deposited on the Cu(110) surface in the presence of external EFs. In this case the NC spiral order, which was unstable in the absence of EF, becomes the most favorable one when perpendicular fields of the order of 0.1 V/Å are applied. As a final application of the theory we study the magnetic interactions within monoatomic TM chains deposited on graphene sheets. One observes that even weak chain substrate hybridizations can modify the magnetic order. Mn and Fe chains show incommensurable NC spin configurations. Remarkably, V chains show a transition from a spiral magnetic order in the freestanding geometry to FM order when they are deposited on a graphene sheet. Some TM-terminated zigzag graphene-nanoribbons, for example V and Fe terminated nanoribbons, also show NC spin configurations. Finally, the magnetic anisotropy energies (MAEs) of TM chains on graphene are investigated. It is shown that Co and Fe chains exhibit significant MAEs and orbital magnetic moments with in-plane easy magnetization axis. The remarkable changes in the magnetic properties of chains on graphene are correlated to charge transfers from the TMs to NN carbon atoms. Goals and limitations of this study and the resulting perspectives of future investigations are discussed.

Explorations of the Practical Issues of Learning Prediction-Control Tasks Using Temporal Difference Learning Methods

There has been recent interest in using temporal difference learning methods to attack problems of prediction and control. While these algorithms have been brought to bear on many problems, they remain poorly understood. It is the purpose of this thesis to further explore these algorithms, presenting a framework for viewing them and raising a number of practical issues and exploring those issues in the context of several case studies. This includes applying the TD(lambda) algorithm to: 1) learning to play tic-tac-toe from the outcome of self-play and of play against a perfectly-playing opponent and 2) learning simple one-dimensional segmentation tasks.

Tempus Post Mortem?* Adaptação portuguesa da transcendental-future time perspective scale (TFTPS)

Integrado no conceito mais geral da perspetiva temporal, o futuro transcendental tem sido conceptualizado como uma dimensão que abrange as crenças sobre o futuro a partir da morte imaginada do corpo físico até ao infinito. A transcendental-future time perspective scale (TFTPS) é uma escala unidimensional composta por 10 itens que avalia as cognições relacionadas com este espaço temporal. O objetivo deste estudo é apresentar a adaptação à língua e cultura portuguesa desta escala, assim como a sua estrutura fatorial e características psicométricas numa amostra de 346 participantes com idades compreendidas entre os 17 e os 54 anos (M = 19.87, DP = 4.27). Os resultados encontrados através da análise fatorial exploratória validaram a unidimensionalidade da escala (65.94 % de variância total explicada, α = 0.87).

Atmospheric downwelling longwave radiation at the surface during cloudless and overcast conditions. Measurements and modeling

Atmospheric downwelling longwave radiation is an important component of the terrestrial energy budget; since it is strongly related with the greenhouse effect, it remarkably affects the climate. In this study, I evaluate the estimation of the downwelling longwave irradiance at the terrestrial surface for cloudless and overcast conditions using a one-dimensional radiative transfer model (RTM), specifically the Santa Barbara DISORT Atmospheric Radiative Transfer (SBDART). The calculations performed by using this model were compared with pyrgeometer measurements at three different European places: Girona (NE of the Iberian Peninsula), Payerne (in the East of Switzerland), and Heselbach (in the Black Forest, Germany). Several studies of sensitivity based on the radiative transfer model have shown that special attention on the input of temperature and water content profiles must be held for cloudless sky conditions; for overcast conditions, similar sensitivity studies have shown that, besides the atmospheric profiles, the cloud base height is very relevant, at least for optically thick clouds. Also, the estimation of DLR in places where radiosoundings are not available is explored, either by using the atmospheric profiles spatially interpolated from the gridded analysis data provided by European Centre of Medium-Range Weather Forecast (ECMWF), or by applying a real radiosounding of a nearby site. Calculations have been compared with measurements at all sites. During cloudless sky conditions, when radiosoundings were available, calculations show differences with measurements of -2.7 ± 3.4 Wm-2 (Payerne). While no in situ radiosoundings are available, differences between modeling and measurements were about 0.3 ± 9.4 Wm-2 (Girona). During overcast sky conditions, when in situ radiosoundings and cloud properties (derived from an algorithm that uses spectral infrared and microwave ground based measurements) were available (Black Forest), calculations show differences with measurements of -0.28 ± 2.52 Wm2. When using atmospheric profiles from the ECMWF and fixed values of liquid water path and droplet effective radius (Girona) calculations show differences with measurements of 4.0 ± 2.5 Wm2. For all analyzed sky conditions, it has been confirmed that estimations from radiative transfer modeling are remarkably better than those obtained by simple parameterizations of atmospheric emissivity.

A Teoria Crítica e a Educação

O artigo parte do conceito de emancipação para fazer uma “leitura” de alguns teóricos críticos cujas teses rejeitam a razão iluminista que evoluiu de um ideal transformador da sociedade no sentido do “progresso do espírito humano” para uma “razão instrumental” que pretende ter o domínio da natureza e o controlo das relações sociais. E fá-lo enfatizando o poder da razão, mas de uma “razão” subvertida. Conforme Marcuse, a instrumentalização da tecnologia, ao assumir um carácter racional, deixa de ser percebida como dominadora e exploradora, para ser entendida como promotora de progresso, legitimando o sistema de produção e troca capitalista. Para este teórico, o resultado é um “one dimensional man”, sem pensamento crítico, nem capacidade de argumentação. Habermas encontra na competência argumentativa, nos processos de interacção sem coacções, o potencial emancipatório do indivíduo e da sociedade. Estamos perante um conceito de autodeterminação que, na contemporaneidade, terá que corresponder à concepção de um espaço público “a mil vozes” em que os cidadãos terão um papel activo assente na comunicação. A existência de um espaço público de democracia redistribuitiva implica, para Boaventura Sousa Santos, a emergência de um novo contrato social em que o Estado deve assumir-se como elemento articulador, Estado experimental, democrático na observância do direito às experiências alternativas institucionais democráticas e da garantia de padrões de inclusão, fomentador da participação activa e contínua dos cidadãos. Entende-se, assim, a contribuição destes teóricos para uma visão em que a escola se perspectiva como espaço público gerido por relações dialógicas e em que a educação tem como fim primordial a competência para a auto determinação, para a participação democrática e para a solidariedade.

A ética protestante do trabalho e o commitment organizacional : um estudo comparativo entre protestantes e não protestantes

O commitment é um conceito que tem sido desenvolvido e estudado nas últimas décadas. Só, ou com outras variáveis, o commitment tem sido objecto de estudo em várias culturas. Partindo do princípio que o commitment é afectado pelos valores de cada indivíduo, este estudo procura verificar a sua relação com um sistema de valores de trabalho específico: a Ética Protestante do Trabalho. O estudo tem como contexto a classe trabalhadora, composta por protestantes e não protestantes. Os resultados permitiram concluir que a Ética Protestante do Trabalho não é um construto unidimensional. Contudo, algumas das componentes criadas apresentaram uma relação com as componentes do commitment.

Por media vida deslumbrados (Dossier: Narradores ecuatorianos de la década de 1950)

En este trabajo se plantea que acaso con mayor intensidad que en Huasipungo (1934) y El Chulla Romero y Flores (1958), es en tres obras poco estudiadas de Jorge Icaza donde puede apreciarse con claridad –tal que deslumbra– su propuesta estética: en la pieza de teatro Flagelo (1936), la novela Media vida deslumbrados (1942) y el cuento «El nuevo San Jorge» (1952). Sin apartarse de la denuncia social, Icaza cierra con Flagelo su propuesta dramática, constituyen una suerte de manifiesto literario, por las repercusiones estéticas que alcanzan a la narrativa del autor. En Media vida deslumbrados se puede apreciar un equilibrio efectivo entre las propuestas del autor y su afán de denuncia social. «El nuevo San Jorge», según el autor de este ensayo, sería la obra que alberga lo más radical, en cuanto propuesta estética, de toda la obra icaciana, destaca los elementos neo-barrocos de la misma, con insistencia en los juegos de máscaras, las intermitencias entre las luces y las sombras, y en el múltiple y a la vez unívoco rostro de quien detenta el poder –algunos de estos elementos comunes a las dos obras comentadas, y a El Chulla Romero y Flores.

Analysis of convectively-generated gravity waves in mesoscale model simulations and wind profiler observations

The characteristics of convectively-generated gravity waves during an episode of deep convection near the coast of Wales are examined in both high resolution mesoscale simulations [with the (UK) Met Oce Unified Model] and in observations from a Mesosphere-Stratosphere-Troposphere (MST) wind profiling Doppler radar. Deep convection reached the tropopause and generated vertically propagating, high frequency waves in the lower stratosphere that produced vertical velocity perturbations O(1 m/s). Wavelet analysis is applied in order to determine the characteristic periods and wavelengths of the waves. In both the simulations and observations, the wavelet spectra contain several distinct preferred scales indicated by multiple spectral peaks. The peaks are most pronounced in the horizontal spectra at several wavelengths less than 50 km. Although these peaks are most clear and of largest amplitude in the highest resolution simulations (with 1 km horizontal grid length), they are also evident in coarser simulations (with 4 km horizontal grid length). Peaks also exist in the vertical and temporal spectra (between approximately 2.5 and 4.5 km, and 10 to 30 minutes, respectively) with good agreement between simulation and observation. Two-dimensional (wavenumber-frequency) spectra demonstrate that each of the selected horizontal scales contains peaks at each of preferred temporal scales revealed by the one- dimensional spectra alone.

On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

An adaptive finite element water column model using the Mellor-Yamada level 2.5 turbulence closure scheme

A one-dimensional water column model using the Mellor and Yamada level 2.5 parameterization of vertical turbulent fluxes is presented. The model equations are discretized with a mixed finite element scheme. Details of the finite element discrete equations are given and adaptive mesh refinement strategies are presented. The refinement criterion is an "a posteriori" error estimator based on stratification, shear and distance to surface. The model performances are assessed by studying the stress driven penetration of a turbulent layer into a stratified fluid. This example illustrates the ability of the presented model to follow some internal structures of the flow and paves the way for truly generalized vertical coordinates. (c) 2005 Elsevier Ltd. All rights reserved.

A comparison of two methods for developing the linearization of a shallow-water model

We develop the linearization of a semi-implicit semi-Lagrangian model of the one-dimensional shallow-water equations using two different methods. The usual tangent linear model, formed by linearizing the discrete nonlinear model, is compared with a model formed by first linearizing the continuous nonlinear equations and then discretizing. Both models are shown to perform equally well for finite perturbations. However, the asymptotic behaviour of the two models differs as the perturbation size is reduced. This leads to difficulties in showing that the models are correctly coded using the standard tests. To overcome this difficulty we propose a new method for testing linear models, which we demonstrate both theoretically and numerically. © Crown copyright, 2003. Royal Meteorological Society