Basic Numerical Methods in Meteorology and Oceanography

The purpose of this book is to provide an introduction to numerical modelling of the ocean and the atmosphere. It originates from courses given at Stockholm University and is intended to serve as a textbook for students in meteorology and oceanography with a background in mathematics and physics. Focus is on numerical schemes for the most commonly used equations in oceanography and meteorology as well as on the stability, precision and other properties of these schemes. Simple equations capturing the properties of the primitive equations employed in models of the ocean and atmosphere will be used. These model equations are solved numerically on a grid by discretisation, the derivatives of the differential equations being replaced by finite-difference approximations. The focus will be on the basic numerical methods used for oceanographic and atmospheric modelling. These models are based on the Navier-Stokes equations (including the Coriolis effect) and a tracer equation for heat in both the atmosphere and ocean and tracer equations for humidity and salt in the atmosphere and ocean, respectively. A coupled atmospheric and oceanic general circulation model represents the core part of an Earth System climate model. The book starts by presenting the most common types of partial differential equations and finite difference schemes used in meteorology and oceanography. Subsequently the limitations of these numerical schemes as regards stability, accuracy, presence of computational modes and accuracy the computationally determined phase speed are discussed. The shallow-water equations are discretised for different spatial grids and friction and diffusion terms are introduced. Hereafter implicit and semi-implicit schemes are discussed as well as the semi-Lagrangian technique. Coordinates for atmospheric as well as oceanic models are presented as well as a highly simplified 3D model. A brief description is given of how some atmospheric general circulation models use spectral methods as ""horizontal coordinates"". Finally, some ""pen-and-paper"" theoretical exercises and a number of GFD computer exercises are given

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

Simulating rolling noise on ballasted and slab tracks: vibration, radiation, and pass-by signals

Shifting to rail-bound freight and passenger traffic is key in Europe\u27s strategy towards transport decarbonisation. However, increasing railway traffic can increase environmental noise pollution. Rolling noise is often the dominant noise source. It originates from the interaction of the rough running surfaces of wheel and rail. Predicting rolling noise and performing acoustic optimisation of existing and new tracks requires validated, flexible, and physics-based prediction tools. This is especially relevant for the different designs of ballastless tracks, which are increasingly used for high-speed lines. Therefore, this thesis aims to develop and implement a modelling approach for rolling noise in the time and frequency domain to increase understanding of sound radiation, investigate noise mitigation measures, and allow research of the perception of transients in rolling noise.To achieve this, models for vibration in wheels and several types of ballasted and slab tracks have been implemented using the Waveguide Finite Element method. This method allows an efficient prediction of the track vibration up to high frequencies. Next, models for the sound radiation from wheel and track were implemented using adaptions of the Boundary Element method (BEM), such as the Fourier series BEM and the wavenumber domain BEM.The computational efficiency was addressed in multiple ways. Finally, an approach to simulate the sound produced at a stationary track-side receiver has been developed and implemented based on moving Green\u27s functions. The simulations were largely implemented in in-house Python code. The ballasted and slab track dynamic models have been tuned and compared with measurements on full-scale tracks.The developed models have been used to analyse the vibrations in track and wheel and the acoustic radiation from these vibrations. This allowed the investigation of noise mitigation measures. Further, the necessary complexity of the dynamic track model for predicting rolling noise in time domain was investigated. Two parameter studies were carried out with a focus on track design with lower noise emission. Slab tracks with booted sleepers showed a potential for noise reduction without increasing loads on the track structure. A continuous rail support lowered the radiated sound power at high frequencies. The contributions of different wheel modes to the radiated sound were investigated considering the directivity of each mode, and dominant modes were identified. The established models produce time signals usable for auralisation, which, among others, has the potential to research human perception of transients in rolling noise

MATHEMATICS OF HUNG-PING TSAO

Tsao, Hung-ping (2020). Mathematics of Hung-ping Tsao. In: "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", Wang, Lawrence K. and Tsao, Hung-ping (editors). Volume 2, Number 11, November 2020; 336 pages. Lenox Institute Press, Newtonville, NY, 12128-0405, USA. No. STEAM-VOL2-NUM11-NOV2020; ISBN 978-0-9890870-3-2.............ABSTRACT: I would like to share some of my ideas in Number Theory, Actuarial Mathematics, Sudoku Solving and Optimization Teaching with college students and colleagues. ............KEYWORDS: Natural sequence, AP-sequence, Power-sum, Product-sum, Sorting, Combination, Permutation, Cycle, Subset, Binomial coefficient, Stirling number, Pascal triangle, Bernoulli coefficient, Eulerian number, Bell number, Ordered Bell polynomial, Eulerian Bell polynomial, Recursive formula, q-Gaussian coefficient, Life insurance, Life annuity, Interest, Mortality, Contingency, Premium, Reserve, Sudoku, Puzzle, Row, Column, Box, Unique solution, Flipflops chain, Residue

A Magic Cube Approach for Crashworthiness and Blast Protection Designs of Structural and Material Systems.

Crashworthiness design is one of the most challenging tasks in automotive product development, and blast protection design is crucial for military operations. The goal is to design an optimal crashworthy or blast-protective structure in terms of topology, shape, and size, for both structural and material layouts. Due to the difficulties in the crash analyses and the complexity of the design problems, previous studies were limited to component-level examinations, or considered only a simple design aspect. In this research, an advanced approach entitled the Magic Cube (MQ) approach is proposed, which for the first time, provides a systematic way to examine general crashworthiness and blast protection designs in terms of both structural and material aspects. The MQ developed in this research consists of three major dimensions: decomposition, design methodology, and general consideration. The decomposition dimension includes the major decomposition approaches developed for the crashworthiness design problems, and it can be applied to the blast protection design. It has three layers: time (process) decomposition, space decomposition, and scale decomposition. The design methodology dimension is related to the methodologies employed in the design process; three layers in this dimension are: target cascading, failure mode management, and the optimization technique. The general consideration dimension has three layers, which are multidisciplinary objectives, loadings, and uncertainties. All these layers are coupled with each other to form a 27-element magic cube. A complicated crashworthiness or blast protection design problem can be solved by employing the appropriate approaches in the MQ, which can be represented by the corresponding elements of the MQ. Examples are given to demonstrate the feasibility and effectiveness of the proposed approach and its successful application in real vehicle crashworthiness, blast protection, and other related design problems. The MQ approach developed in this research can be readily applied to other similar design problems, such as those related to active safety and vehicle rollover.Ph.D.Mechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/58392/1/cqi_1.pd

Hydrodynamics, Stability and Accretion: from planets, to stars, to supermassive black holes

In this golden era of multi-messenger astronomy, with highly sensitive telescopes detecting spectacular transients almost nightly, a whole new window is now wide open to study the universe on all timescales. Typically, these events are generated from the total or partial destruction of an astrophysical object and emit electromagnetic waves of all different wavelengths, neutrinos and gravitational waves--carrying important physics that was previously inaccessible. Therefore, it is of utmost importance to take advantage of this tremendous progress in observation with analytic and simulation tools to explain the physics of exotic astrophysical events. This thesis aims to do so by studying the hydrodynamics of some of the most exotic high-energy astrophysical phenomena, ranging from tidal disruption events, shock physics, accretion, and gravitational wave emission from core collapse supernovae, and dynamic stability of the giant planetary atmosphere. We have developed novel analytical tools, primarily using classical hydrodynamics and general relativity, and have utilized computational techniques (numerical and simulations). Our study on giant planets shows how the presence of a solid core can save the planet from being unstable when due to ionization it is expected to be, and thus solves a puzzle in the ``core-accretion” theory of giant planet formation. We have presented a general relativistic modification for the accretion solution on a neutron star through stalled shock, which is useful in understanding weak or failed supernovae and can potentially impact the much-discussed standing accretion shock instability--which we restudied and in the process uncovered a new variant impacting the explosion mechanism and gravitational-wave signature. We have also studied the oscillation modes of a nascent proto-neutron star to show how they can contribute to the gravitational wave signature emitted. Our study of deep-tidal disruption events (events in which stars approach black holes closely) refutes the widely speculated possibility of nuclear detonation arising in such events

Hydrodynamic analysis of mooring lines based on optical tracking experiments

Due to the complexity of body-shape, the investigation of hydrodynamic forces on mooring lines, especially those comprised of chain segments, has not been conducted to a sufficient degree to properly characterize the hydrodynamic damping effect of mooring lines on the global motions of a moored offshore platform. In the present study, an experimental investigation of the hydrodynamic characteristics of various mooring elements is implemented through free and forced oscillation tests. Since no direct measurement capability for distributed hydrodynamic forces acting on mooring line segments such as chain and wire rope is available yet, an indirect measurement technique is introduced. The technique is based on the fact that hydrodynamic forces acting on a body oscillating in still water and on a stationary body in an oscillatory flow are equivalent except for the additional inertia force, the so-called Froude-Krylov force, present in the latter condition. The time-dependent displacement of a slender body moving in calm water is acquired through optical tracking with a high speed camera. The distributed hydrodynamic measurements are then used to obtain the force by solving the equation of motion with the boundary condition provided from tension measurements. Morison’s equation is employed along with Fourier analysis to separate the inertia and drag components out of the total fluid force. Given the experimentally-derived information on hydrodynamic behavior, the resistance provided by a mooring line to a floating structure is briefly studied in terms of damping and restoring force in a coupled dynamic system

Wave Interactions with Coastal Structures

Due to the ongoing rise in sea level and increases in extreme wave climates, which consequently change the wave climate, coastal structures such as sea dikes and seawalls are exposed to severe and frequent sea storms. Even though much research related to wave–structure interactions has been carried out, it remains one of the most important and challenging topics in the field of coastal engineering. The recent publications in the Special Issue “Wave Interactions with Coastal Structures” in the Journal of Marine Science and Engineering include a wide range of research, including theoretical/mathematical, experimental, and numerical work related to the interaction between sea waves and coastal structures. These publications address conventional coastal hard structures in deep water zones as well as those located in shallow water zones, such as wave overtopping over shallow foreshores with apartment buildings on dikes. The research findings presented help to improve our knowledge of hydrodynamic processes, and the new approaches and developments presented here will be good benchmarks for future work