Lie symmetry analysis and exact solutions of the quasi-geostrophic two-layer problem

The quasi-geostrophic two-layer model is of superior interest in dynamic meteorology since it is one of the easiest ways to study baroclinic processes in geophysical fluid dynamics. The complete set of point symmetries of the two-layer equations is determined. An optimal set of one- and two-dimensional inequivalent subalgebras of the maximal Lie invariance algebra is constructed. On the basis of these subalgebras we exhaustively carry out group-invariant reduction and compute various classes of exact solutions. Where possible, reference to the physical meaning of the exact solutions is given. In particular, the well-known baroclinic Rossby wave solutions in the two-layer model are rediscovered.Comment: Extended version, 24 pages, 1 figur

Copula-Based Multivariate Hydrologic Frequency Analysis

Multivariate frequency distributions are being increasingly recognized for their role in hydrological design and risk management. The conventional multivariate distributions are severely limited in that all constituent marginals have to be from the same distribution family. The copula method is a newly emerging approach for deriving multivariate distributions which overcomes this limitation. Use of copula method in hydrological applications has begun only recently and ascertaining the applicability of different copulas for combinations of various hydrological variables is currently an area of active research. Since there exists a variety of copulas capable of characterizing a broad range of dependence, the selection of appropriate copulas for different hydrological applications becomes a non-trivial task. This study evaluates the relative performance of various copulas and methods of parameter estimation as well as of recently developed statistical inference procedures. Potential copulas for multivariate extreme flow and rainfall processes are then identified. Multivariate hydrological frequency analysis typically utilizes only the concurrent parts of observed data, leaving a lot of non-concurrent information unutilized. Uncertainty in distribution parameter estimates can be reduced by simultaneously including such non-concurrent data in the analysis. A new copula-based “Composite Likelihood Approach” that allows all available multivariate data of varying lengths to be combined and analyzed in an integrated manner has been developed. This approach yields additional information, enhancing the precision of parameter estimates that are otherwise obtained from either purely univariate or purely multivariate considerations. The approach can be advantageously employed in limited hydrological data situations in order to provide significant virtual augmentation of available data lengths by virtue of increased precision of parameter estimates. The effectiveness of a copula selection framework that helps in an a priori short listing of potentially viable copulas on the basis of dependence characteristics has been examined using several case studies pertaining to various extreme flow and rainfall variables. The benefits of the composite likelihood approach in terms of significant improvement in the precision of parameter estimates of commonly used distributions in hydrology, such as normal, Gumbel, gamma, and log-Pearson Type III, have been quantified

Blind Multiridge Detection and Reconstruction Using Ultrasonic Signals

Time-frequency signal analysis has been widely applied in the modern radar, acoustic, sonar and ultrasonic signal processing techniques. Recently, the nondestructive testing (NDT) techniques via the ultrasonic instrumentation have shown the striking capability of the quality control for the material fabrication industry. In this thesis, we first provide a general mathematical model for the ultrasonic signals collected by pulse-echo sensors and then design a totally blind, novel, signal processing NDT technique relying on neither a priori signal information nor any manual effort. The signature signal can be blindly extracted by using the automatic optimal frame size selection for further modeling and characterization of the ultrasonic signal using Gabor analysis. This modeled signature signal is used for multiridge detection and for reconstruction of the signal. The detected ridge information can be used to estimate the transmission and attenuation coefficients, shear modulus, and Young’s modulus associated with any arbitrary material sample for fabrication quality control. Thus, our algorithm can be applied for ultrasonic signal characterization and ridge detection in non-destructive testing for new material fabrication. Experimental results show that the ridge detection performance by our proposed method is superior to that of the existing techniques

An assembly gap control method based on posture alignment of wing panels in aircraft assembly

The gaps between two mating surfaces should be strictly controlled in precision manufacturing. Oversizing of gaps will decrease the dimensional accuracy and may reduce the fatigue life of a mechanical product. In order to reduce the gaps and keep them within tolerance, the relative posture (orientation and position) of two components should be optimized in the assembly process. This paper presents an optimal posture evaluation model to control the assembly gaps in aircraft wing assembly.Based on the step alignment strategy, i.e. preliminary alignment and refined alignment, the concept of a small posture transformation (SPT) is introduced. In the preliminary alignment, an initial posture is estimated by a set of auxiliary locating points (ALPs), with which the components can be quickly aligned near each other. In the refined alignment, the assembly gaps are calculated and the formulation of the gaps with component posture is derived by the SPT. A comprehensive weighted minimization model with gap tolerance constraints is established for redistributing the gaps in multi-regions. Powell-Hestenes-Rockafellar (PHR) optimization, Singular Value Decomposition (SVD) and KD-tree searching are introduced for the solution of the optimal posture for localization. Using the SPT, the trigonometric posture transformation is linearized, which benefits the iterative solution process. Through the constrained model, overall gaps are minimized and excess gaps are controlled within tolerance. Practical implications – This method has been tested with simulated model data and real product data, the results of which have shown efficient coordination of mating components.This paper proposed an optimal posture evaluation method for minimizing the gaps between mating surfaces through component adjustments. This will promote the assembly automation and variation control in aircraft wing assembly

Symmetry methods in the atmospheric sciences

Zahlreiche Symmetriemethoden werden auf Differentialgleichungen der Atmosphärendynamik angewandt. Die Lie-Punktsymmetrien der barotropen Vorticitygleichung, der barotropen potentiellen Vorticitygleichung und des baroklinen Zweischichtmodels werden berechnet. Ein- und zweidimensionale inäquivalente Subalgebren der jeweiligen maximalen Lie-Invarianzalgebren werden klassifiziert und dazu verwendet, exakte Lösungen der jeweiligen Gleichungen zu bestimmen. Die physikalische Bedeutung dieser Lösungen wird untersucht und diskutiert. Mittels der Symmetrien der barotropen potentiellen Vorticitygleichung auf der beta-Ebene und der barotropen Vorticitygleichung auf der rotierenden Kugel können Punkttransformationen gefunden werden, die beide Gleichungen in die jeweiligen Gleichungen im Inertialsystem transformieren. Zwei erweiterte Techniken zur Berechnung der gesamten Punktsymmetriegruppe von Differentialgleichungen werden vorgestellt, die im Rahmen der direkten Methode angewandt werden können. Die erste Technik basiert auf der Invarianz von Megaidealen der maximalen Lie-Invarianzalgebra unter von Punktsymmetrien erzeugten Automorphismen. Die zweite Technik verwendet Kenntnisse über admissible transformations von Klassen von Differentialgleichungen, die die untersuchte Gleichung enthalten. Weiters wird gezeigt wie Symmetrien dazu verwendet werden können, Schließungen im Zuge des Parameterisierungsproblems zu definieren. Für diesen Zweck werden Verfahren der direkten und inversen Gruppenklassifikation benützt. Als Beispiel werden verschiedene Parameterisierungen für den Eddy-Vorticityfluß in der Reynolds-gemittelten Vorticitygleichung konstruiert, die unterschiedliche Symmetrieeigenschaften besitzen. In einem weiteren Schritt werden die Symmetrien der barotropen Vorticitygleichung und der Saltzman'schen Konvektionsgleichungen dazu verwendet um spektrale, niedrigdimensionale Approximationen dieser Gleichungen zu erzeugen. Dazu werden Lie-Punkt- und diskrete Symmetrien als Kriterium zur Auswahl der Fouriermoden verwendet. Es wird bewiesen dass das Lorenz--1960 Modell systematisch unter Zuhilfenahme der Punktsymmetrien der Vorticitygleichung ableitbar ist. Auf ähnliche Weise wird demonstriert dass die Wahl der Moden des Lorenz--1963 Modells der thermischen Konvektion nicht mittels Symmetrien begründbar ist. Zudem wird gezeigt dass sowohl die Hamiltonsche als auch die Nambu Form des Lorenz--1963 Modells nicht mit der entsprechenden Hamiltonschen bzw. Nambu-Darstellung der Saltzman'schen Konvektionsgleichungen zusammenhängen. Aus diesem Grund wird ein sechskomponentiges Modell der Konvektionsgleichungen abgeleitet. Die Modenwahl dieses neuen Modells basiert vollständig auf Punktsymmetrien der Saltzman'schen Gleichungen. Durch geeignetes Skalieren dieser Moden ist es möglich eine Hamiltonsche bwz. Nambu-Darstellung dieses sechskomponentigen Modells zu finden, die der Hamilton- bzw. Nambuformulierung der kontinuierlichen Konvektionsgleichungen vollständig analog ist.Wide ranges of symmetry methods are applied to several differential equations arising in the atmospheric sciences. Lie point symmetries of the barotropic vorticity equation, the barotropic potential vorticity equation and the two-layer baroclinic model are computed. One- and two-dimensional inequivalent subalgebras of the respective maximal Lie invariance algebras are classified. Based on this classification, we determine various group-invariant solutions of the investigated differential equations. The physical relevance of these particular solutions is evaluated. Symmetries are used to find point transformations that map the barotropic potential vorticity equation on the beta-plane and the barotropic vorticity equation on the rotating sphere to the respective equations in the inertial frame. Two refined techniques for the computation of the complete point symmetry group of differential equations are proposed within the framework of the direct method. The first technique is based on the invariance of megaideals of the maximal Lie invariance algebra under automorphisms generated by point symmetries. The second technique involves knowledge on the admissible transformations of classes of differential equations containing the given equation. It is shown how symmetries can be employed to determine closure schemes in the course of the parameterization problem. The methods we apply rest on techniques of direct and inverse group classifications. These methods are exemplified by parameterizing the eddy vorticity flux in the Reynolds averaged vorticity equation. This leads to several invariant parameterization schemes possessing different degrees of symmetry. The symmetries of the barotropic vorticity equation and the Saltzman convection equations are used to derive spectral finite-mode approximations. This is done using both Lie and discrete point symmetries as a criterion for the selection of Fourier modes. It is proved that the Lorenz--1960 model can be systematically re-derived with the aid of point symmetries of the vorticity equation. In a similar manner, it is demonstrated that the selection of modes for the Lorenz--1963 convection model is not compatible with the symmetries of the Saltzman equations. It is shown that the Hamiltonian and Nambu structures of the Lorenz--1963 model are not related to the Hamiltonian and Nambu forms of the Saltzman convection equations. A new six-component truncation of the convection equation is proposed. The selection of modes for this model is based on point symmetries of the convection equations. These modes are suitably scaled to allow the six-component model to be of Hamiltonian and Nambu forms analog to those of the original Saltzman equations

Potential-based Formulations of the Navier-Stokes Equations and their Application

Based on a Clebsch-like velocity representation and a combination of classical variational principles for the special cases of ideal and Stokes flow a novel discontinuous Lagrangian is constructed; it bypasses the known problems associated with non-physical solutions and recovers the classical Navier-Stokes equations together with the balance of inner energy in the limit when an emerging characteristic frequency parameter tends to infinity. Additionally, a generalized Clebsch transformation for viscous flow is established for the first time. Next, an exact first integral of the unsteady, three-dimensional, incompressible Navier-Stokes equations is derived; following which gauge freedoms are explored leading to favourable reductions in the complexity of the equation set and number of unknowns, enabling a self-adjoint variational principle for steady viscous flow to be constructed. Concurrently, appropriate commonly occurring physical and auxiliary boundary conditions are prescribed, including establishment of a first integral for the dynamic boundary condition at a free surface. Starting from this new formulation, three classical flow problems are considered, the results obtained being in total agreement with solutions in the open literature. A new least-squares finite element method based on the first integral of the steady two-dimensional, incompressible, Navier-Stokes equations is developed, with optimal convergence rates established theoretically. The method is analysed comprehensively, thoroughly validated and shown to be competitive when compared to a corresponding, standard, primitive-variable, finite element formulation. Implementation details are provided, and the well-known problem of mass conservation addressed and resolved via selective weighting. The attractive positive definiteness of the resulting linear systems enables employment of a customized scalable algebraic multigrid method for efficient error reduction. The solution of several engineering related problems from the fields of lubrication and film flow demonstrate the flexibility and efficiency of the proposed method, including the case of unsteady flow, while revealing new physical insights of interest in their own right

Fluorescent particle tracers for surface hydrology

Surface water processes control downstream runoff phenomena, waste and pollutant diffusion, erosion mechanics, and sediment transport. However, current observational methodologies do not allow for the identification and kinematic characterization of the physical processes contributing to catchment dynamics. Traditional methodologies are not capable to cope with extreme in-situ conditions, including practical logistic challenges as well as inherent flow complexity. In addition, available observational techniques are non-exhaustive for describing multiscale hydrological processes. This research addresses the need for novel observations of the hydrological community by developing pioneer flow characterization approaches that rely on the mutual integration of traditional tracing techniques and state-of-the-art image-based sensing procedures. These novel methodologies enable the in-situ direct observation of surface water processes through remote and unsupervised procedures, thus paving the way to the development of distributed networks of sensing platforms for catchment-scale environmental sensing. More specifically, the proposed flow characterization methodology is a low-cost measurement system that can be applied to a variety of real-world settings spanning from few centimeters rills in natural catchments to riverine ecosystems. The technique is based on the use of in-house synthesized environmentally-friendly fluorescent particle tracers through digital cameras for direct flow measurement and travel time estimations. Automated image analysis-based procedures are developed for real-time flow characterization based on image manipulation, template-based correlation, particle image velocimetry, and dimensionality reduction methodologies. The feasibility of the approach is assessed through laboratory-designed experiments, where the accuracy of the methodology is investigated with respect to well-established flow visualization techniques. Further, the transition of the proposed flow characterization approach to natural settings is studied through paradigmatic observations of natural stream flows in small scale channel and riverine settings and overland flows in hillslope environments. The integration of the proposed flow sensing system in a stand-alone, remote, and mobile platform is explored through the design, development, and testing of a miniature aerial vehicle for environmental monitoring through video acquisition and processing