4 the Shallow Water System
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Well-Balanced Schemes for the Shallow Water Equations with Coriolis Forces
Well-balanced schemes for the shallow water equations with Coriolis forces Alina Chertock, Michael Dudzinski, Alexander Kurganov & Mária Lukáčová- Medvid’ová Numerische Mathematik ISSN 0029-599X Volume 138 Number 4 Numer. Math. (2018) 138:939-973 DOI 10.1007/s00211-017-0928-0 1 23 Your article is protected by copyright and all rights are held exclusively by Springer- Verlag GmbH Germany, part of Springer Nature. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Numer. Math. (2018) 138:939–973 Numerische https://doi.org/10.1007/s00211-017-0928-0 Mathematik Well-balanced schemes for the shallow water equations with Coriolis forces Alina Chertock1 · Michael Dudzinski2 · Alexander Kurganov3,4 · Mária Lukáˇcová-Medvid’ová5 Received: 28 April 2014 / Revised: 19 September 2017 / Published online: 2 December 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2017 Abstract In the present paper we study shallow water equations with bottom topog- raphy and Coriolis forces. The latter yield non-local potential operators that need to be taken into account in order to derive a well-balanced numerical scheme. -
Dispersion of Tsunamis: Does It Really Matter? and Physics and Physics Discussions Open Access Open Access S
EGU Journal Logos (RGB) Open Access Open Access Open Access Advances in Annales Nonlinear Processes Geosciences Geophysicae in Geophysics Open Access Open Access Nat. Hazards Earth Syst. Sci., 13, 1507–1526, 2013 Natural Hazards Natural Hazards www.nat-hazards-earth-syst-sci.net/13/1507/2013/ doi:10.5194/nhess-13-1507-2013 and Earth System and Earth System © Author(s) 2013. CC Attribution 3.0 License. Sciences Sciences Discussions Open Access Open Access Atmospheric Atmospheric Chemistry Chemistry Dispersion of tsunamis: does it really matter? and Physics and Physics Discussions Open Access Open Access S. Glimsdal1,2,3, G. K. Pedersen1,3, C. B. Harbitz1,2,3, and F. Løvholt1,2,3 Atmospheric Atmospheric 1International Centre for Geohazards (ICG), Sognsveien 72, Oslo, Norway Measurement Measurement 2Norwegian Geotechnical Institute, Sognsveien 72, Oslo, Norway 3University of Oslo, Blindern, Oslo, Norway Techniques Techniques Discussions Open Access Correspondence to: S. Glimsdal ([email protected]) Open Access Received: 30 November 2012 – Published in Nat. Hazards Earth Syst. Sci. Discuss.: – Biogeosciences Biogeosciences Revised: 5 April 2013 – Accepted: 24 April 2013 – Published: 18 June 2013 Discussions Open Access Abstract. This article focuses on the effect of dispersion in 1 Introduction Open Access the field of tsunami modeling. Frequency dispersion in the Climate linear long-wave limit is first briefly discussed from a the- Climate Most tsunami modelers rely on the shallow-water equations oretical point of view. A single parameter, denoted as “dis- of the Past for predictions of propagationof and the run-up. Past Some groups, on persion time”, for the integrated effect of frequency dis- Discussions the other hand, insist on applying dispersive wave models, persion is identified. -
Fluid Mechanics
FLUID MECHANICS PROF. DR. METİN GÜNER COMPILER ANKARA UNIVERSITY FACULTY OF AGRICULTURE DEPARTMENT OF AGRICULTURAL MACHINERY AND TECHNOLOGIES ENGINEERING 1 1. INTRODUCTION Mechanics is the oldest physical science that deals with both stationary and moving bodies under the influence of forces. Mechanics is divided into three groups: a) Mechanics of rigid bodies, b) Mechanics of deformable bodies, c) Fluid mechanics Fluid mechanics deals with the behavior of fluids at rest (fluid statics) or in motion (fluid dynamics), and the interaction of fluids with solids or other fluids at the boundaries (Fig.1.1.). Fluid mechanics is the branch of physics which involves the study of fluids (liquids, gases, and plasmas) and the forces on them. Fluid mechanics can be divided into two. a)Fluid Statics b)Fluid Dynamics Fluid statics or hydrostatics is the branch of fluid mechanics that studies fluids at rest. It embraces the study of the conditions under which fluids are at rest in stable equilibrium Hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids. Hydrostatics offers physical explanations for many phenomena of everyday life, such as why atmospheric pressure changes with altitude, why wood and oil float on water, and why the surface of water is always flat and horizontal whatever the shape of its container. Fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow— the natural science of fluids (liquids and gases) in motion. It has several subdisciplines itself, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). -
Fluid Viscosity and the Attenuation of Surface Waves: a Derivation Based on Conservation of Energy
INSTITUTE OF PHYSICS PUBLISHING EUROPEAN JOURNAL OF PHYSICS Eur. J. Phys. 25 (2004) 115–122 PII: S0143-0807(04)64480-1 Fluid viscosity and the attenuation of surface waves: a derivation based on conservation of energy FBehroozi Department of Physics, University of Northern Iowa, Cedar Falls, IA 50614, USA Received 9 June 2003 Published 25 November 2003 Online at stacks.iop.org/EJP/25/115 (DOI: 10.1088/0143-0807/25/1/014) Abstract More than a century ago, Stokes (1819–1903)pointed out that the attenuation of surface waves could be exploited to measure viscosity. This paper provides the link between fluid viscosity and the attenuation of surface waves by invoking the conservation of energy. First we calculate the power loss per unit area due to viscous dissipation. Next we calculate the power loss per unit area as manifested in the decay of the wave amplitude. By equating these two quantities, we derive the relationship between the fluid viscosity and the decay coefficient of the surface waves in a transparent way. 1. Introduction Surface tension and gravity govern the propagation of surface waves on fluids while viscosity determines the wave attenuation. In this paper we focus on the relation between viscosity and attenuation of surface waves. More than a century ago, Stokes (1819–1903) pointed out that the attenuation of surface waves could be exploited to measure viscosity [1]. Since then, the determination of viscosity from the damping of surface waves has receivedmuch attention [2–10], particularly because the method presents the possibility of measuring viscosity noninvasively. In his attempt to obtain the functionalrelationship between viscosity and wave attenuation, Stokes observed that the harmonic solutions obtained by solving the Laplace equation for the velocity potential in the absence of viscosity also satisfy the linearized Navier–Stokes equation [11]. -
Part II-1 Water Wave Mechanics
Chapter 1 EM 1110-2-1100 WATER WAVE MECHANICS (Part II) 1 August 2008 (Change 2) Table of Contents Page II-1-1. Introduction ............................................................II-1-1 II-1-2. Regular Waves .........................................................II-1-3 a. Introduction ...........................................................II-1-3 b. Definition of wave parameters .............................................II-1-4 c. Linear wave theory ......................................................II-1-5 (1) Introduction .......................................................II-1-5 (2) Wave celerity, length, and period.......................................II-1-6 (3) The sinusoidal wave profile...........................................II-1-9 (4) Some useful functions ...............................................II-1-9 (5) Local fluid velocities and accelerations .................................II-1-12 (6) Water particle displacements .........................................II-1-13 (7) Subsurface pressure ................................................II-1-21 (8) Group velocity ....................................................II-1-22 (9) Wave energy and power.............................................II-1-26 (10)Summary of linear wave theory.......................................II-1-29 d. Nonlinear wave theories .................................................II-1-30 (1) Introduction ......................................................II-1-30 (2) Stokes finite-amplitude wave theory ...................................II-1-32 -
Fourth-Order Nonlinear Evolution Equations for Surface Gravity Waves in the Presence of a Thin Thermocline
J. Austral. Math. Soc. Ser. B 39(1997), 214-229 FOURTH-ORDER NONLINEAR EVOLUTION EQUATIONS FOR SURFACE GRAVITY WAVES IN THE PRESENCE OF A THIN THERMOCLINE SUDEBI BHATTACHARYYA1 and K. P. DAS1 (Received 14 August 1995; revised 21 December 1995) Abstract Two coupled nonlinear evolution equations correct to fourth order in wave steepness are derived for a three-dimensional wave packet in the presence of a thin thermocline. These two coupled equations are reduced to a single equation on the assumption that the space variation of the amplitudes takes place along a line making an arbitrary fixed angle with the direction of propagation of the wave. This single equation is used to study the stability of a uniform wave train. Expressions for maximum growth rate of instability and wave number at marginal stability are obtained. Some of the results are shown graphically. It is found that a thin thermocline has a stabilizing influence and the maximum growth rate of instability decreases with the increase of thermocline depth. 1. Introduction There exist a number of papers on nonlinear interaction between surface gravity waves and internal waves. Most of these are concerned with the mechanism of generation of internal waves through nonlinear interaction of surface gravity waves. Coherent three wave interactions of two surface waves and one internal wave have been investigated by Ball [1], Thorpe [22], Watson, West and Cohen [23] and others. Using the theoretical model of Hasselman [12] for incoherent three-wave interaction, Olber and Hertrich [18] have reported a mechanism of generation of internal waves by coupling with surface waves using a three-layer model of the ocean. -
Lecture 18 Ocean General Circulation Modeling
Lecture 18 Ocean General Circulation Modeling 9.1 The equations of motion: Navier-Stokes The governing equations for a real fluid are the Navier-Stokes equations (con servation of linear momentum and mass mass) along with conservation of salt, conservation of heat (the first law of thermodynamics) and an equation of state. However, these equations support fast acoustic modes and involve nonlinearities in many terms that makes solving them both difficult and ex pensive and particularly ill suited for long time scale calculations. Instead we make a series of approximations to simplify the Navier-Stokes equations to yield the “primitive equations” which are the basis of most general circu lations models. In a rotating frame of reference and in the absence of sources and sinks of mass or salt the Navier-Stokes equations are @ �~v + �~v~v + 2�~ �~v + g�kˆ + p = ~ρ (9.1) t r · ^ r r · @ � + �~v = 0 (9.2) t r · @ �S + �S~v = 0 (9.3) t r · 1 @t �ζ + �ζ~v = ω (9.4) r · cpS r · F � = �(ζ; S; p) (9.5) Where � is the fluid density, ~v is the velocity, p is the pressure, S is the salinity and ζ is the potential temperature which add up to seven dependent variables. 115 12.950 Atmospheric and Oceanic Modeling, Spring '04 116 The constants are �~ the rotation vector of the sphere, g the gravitational acceleration and cp the specific heat capacity at constant pressure. ~ρ is the stress tensor and ω are non-advective heat fluxes (such as heat exchange across the sea-surface).F 9.2 Acoustic modes Notice that there is no prognostic equation for pressure, p, but there are two equations for density, �; one prognostic and one diagnostic. -
Assessment of DUACS Sentinel-3A Altimetry Data in the Coastal Band of the European Seas: Comparison with Tide Gauge Measurements
remote sensing Article Assessment of DUACS Sentinel-3A Altimetry Data in the Coastal Band of the European Seas: Comparison with Tide Gauge Measurements Antonio Sánchez-Román 1,* , Ananda Pascual 1, Marie-Isabelle Pujol 2, Guillaume Taburet 2, Marta Marcos 1,3 and Yannice Faugère 2 1 Instituto Mediterráneo de Estudios Avanzados, C/Miquel Marquès, 21, 07190 Esporles, Spain; [email protected] (A.P.); [email protected] (M.M.) 2 Collecte Localisation Satellites, Parc Technologique du Canal, 8-10 rue Hermès, 31520 Ramonville-Saint-Agne, France; [email protected] (M.-I.P.); [email protected] (G.T.); [email protected] (Y.F.) 3 Departament de Física, Universitat de les Illes Balears, Cra. de Valldemossa, km 7.5, 07122 Palma, Spain * Correspondence: [email protected]; Tel.: +34-971-61-0906 Received: 26 October 2020; Accepted: 1 December 2020; Published: 4 December 2020 Abstract: The quality of the Data Unification and Altimeter Combination System (DUACS) Sentinel-3A altimeter data in the coastal area of the European seas is investigated through a comparison with in situ tide gauge measurements. The comparison was also conducted using altimetry data from Jason-3 for inter-comparison purposes. We found that Sentinel-3A improved the root mean square differences (RMSD) by 13% with respect to the Jason-3 mission. In addition, the variance in the differences between the two datasets was reduced by 25%. To explain the improved capture of Sea Level Anomaly by Sentinel-3A in the coastal band, the impact of the measurement noise on the synthetic aperture radar altimeter, the distance to the coast, and Long Wave Error correction applied on altimetry data were checked. -
Internal Gravity Waves: from Instabilities to Turbulence Chantal Staquet, Joël Sommeria
Internal gravity waves: from instabilities to turbulence Chantal Staquet, Joël Sommeria To cite this version: Chantal Staquet, Joël Sommeria. Internal gravity waves: from instabilities to turbulence. Annual Review of Fluid Mechanics, Annual Reviews, 2002, 34, pp.559-593. 10.1146/an- nurev.fluid.34.090601.130953. hal-00264617 HAL Id: hal-00264617 https://hal.archives-ouvertes.fr/hal-00264617 Submitted on 4 Feb 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License INTERNAL GRAVITY WAVES: From Instabilities to Turbulence C. Staquet and J. Sommeria Laboratoire des Ecoulements Geophysiques´ et Industriels, BP 53, 38041 Grenoble Cedex 9, France; e-mail: [email protected], [email protected] Key Words geophysical fluid dynamics, stratified fluids, wave interactions, wave breaking Abstract We review the mechanisms of steepening and breaking for internal gravity waves in a continuous density stratification. After discussing the instability of a plane wave of arbitrary amplitude in an infinite medium at rest, we consider the steep- ening effects of wave reflection on a sloping boundary and propagation in a shear flow. The final process of breaking into small-scale turbulence is then presented. -
Waves and Weather
Waves and Weather 1. Where do waves come from? 2. What storms produce good surfing waves? 3. Where do these storms frequently form? 4. Where are the good areas for receiving swells? Where do waves come from? ==> Wind! Any two fluids (with different density) moving at different speeds can produce waves. In our case, air is one fluid and the water is the other. • Start with perfectly glassy conditions (no waves) and no wind. • As wind starts, will first get very small capillary waves (ripples). • Once ripples form, now wind can push against the surface and waves can grow faster. Within Wave Source Region: - all wavelengths and heights mixed together - looks like washing machine ("Victory at Sea") But this is what we want our surfing waves to look like: How do we get from this To this ???? DISPERSION !! In deep water, wave speed (celerity) c= gT/2π Long period waves travel faster. Short period waves travel slower Waves begin to separate as they move away from generation area ===> This is Dispersion How Big Will the Waves Get? Height and Period of waves depends primarily on: - Wind speed - Duration (how long the wind blows over the waves) - Fetch (distance that wind blows over the waves) "SMB" Tables How Big Will the Waves Get? Assume Duration = 24 hours Fetch Length = 500 miles Significant Significant Wind Speed Wave Height Wave Period 10 mph 2 ft 3.5 sec 20 mph 6 ft 5.5 sec 30 mph 12 ft 7.5 sec 40 mph 19 ft 10.0 sec 50 mph 27 ft 11.5 sec 60 mph 35 ft 13.0 sec Wave height will decay as waves move away from source region!!! Map of Mean Wind -
The Internal Gravity Wave Spectrum: a New Frontier in Global Ocean Modeling
The internal gravity wave spectrum: A new frontier in global ocean modeling Brian K. Arbic Department of Earth and Environmental Sciences University of Michigan Supported by funding from: Office of Naval Research (ONR) National Aeronautics and Space Administration (NASA) National Science Foundation (NSF) Brian K. Arbic Internal wave spectrum in global ocean models Collaborators • Naval Research Laboratory Stennis Space Center: Joe Metzger, Jim Richman, Jay Shriver, Alan Wallcraft, Luis Zamudio • University of Southern Mississippi: Maarten Buijsman • University of Michigan: Joseph Ansong, Steve Bassette, Conrad Luecke, Anna Savage • McGill University: David Trossman • Bangor University: Patrick Timko • Norwegian Meteorological Institute: Malte M¨uller • University of Brest and The University of Texas at Austin: Rob Scott • NASA Goddard: Richard Ray • Florida State University: Eric Chassignet • Others including many members of the NSF-funded Climate Process Team led by Jennifer MacKinnon of Scripps Brian K. Arbic Internal wave spectrum in global ocean models Motivation • Breaking internal gravity waves drive most of the mixing in the subsurface ocean. • The internal gravity wave spectrum is just starting to be resolved in global ocean models. • Somewhat analogous to resolution of mesoscale eddies in basin- and global-scale models in 1990s and early 2000s. • Builds on global internal tide modeling, which began with 2004 Arbic et al. and Simmons et al. papers utilizing Hallberg Isopycnal Model (HIM) run with tidal foricng only and employing a horizontally uniform stratification. Brian K. Arbic Internal wave spectrum in global ocean models Motivation continued... • Here we utilize simulations of the HYbrid Coordinate Ocean Model (HYCOM) with both atmospheric and tidal forcing. • Near-inertial waves and tides are put into a model with a realistically varying background stratification. -
Potential Vorticity Dynamics and Tropopause Fold
POTENTIAL VORTICITY DYNAMICS AND TROPOPAUSE FOLD M.D. ANDREI1,2, M. PIETRISI1,2, S. STEFAN1 1 University of Bucharest, Faculty of Physics, P.O.BOX MG 11, Magurele, Bucharest, Romania Emails: [email protected]; [email protected] 2 National Meteorological Administration, Bucuresti-Ploiesti Ave., No. 97, 013686, Bucharest, Romania Received November 1, 2017 Abstract. Extra tropical cyclones evolution depends a lot on the dynamically and thermodynamically interactions between the lower and the upper troposphere. The aim of this paper is to demonstrate the importance of dynamic tropopause fold in the development of a cyclone which affected Romania in 12th and 13th of November, 2016. The coupling between a positive potential vorticity anomaly and the jet stream has caused the fall of dynamic tropopause, which has a great influence in the cyclone deepening. This synoptic situation has resulted in a big amount of precipitation and strong wind gusts. For the study, ALARO limited area spectral model analyzes (e.g., 300 hPa Potential Vorticity, 1,5 PVU field height, 300 hPa winds, mean sea level pressure), data from the Romanian National Meteorological Administration meteorological stations, cross-sections and satellite images (water vapor and RGB) were used. Accordingly, the results of this study will be used in operational forecast, for similar situations which would appear in the future, and can improve it. Key words: potential vorticity, tropopause fold, cyclone. 1. INTRODUCTION Dynamic tropopause is the 1.5 or 2 PVU (potential vorticity units) surface that separates the dry, stable, stratospheric air from the humid, unstable, tropospheric air [1]. The dynamic tropopause fold corresponds to a positive potential vorticity anomaly, with high amplitude in the upper troposphere [2, 3], which induces a cyclonic circulation that propagates to the surface and reduce, under it, static stability.