DOCSLIB.ORG

Shallow Water Waves on the Rotating Sphere

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Shallow Water Waves on the Rotating Sphere PHYSICAL RBVIEW E VOLUME SI, NUMBER S MAYl99S Shallowwater waves on the rotating sphere D. MUller andJ. J. O'Brien Centerfor Ocean-AtmosphericPrediction Studies,Florida State Univenity, TallahDSSee,Florida 32306 (Received8 September1994) Analytical solutions of the linear wave equation of shallow waters on the rotating, spherical surface have been found for standing as weD as for inertial waves. The spheroidal wave operator is shown to playa central role in this wave equation. Prolate spheroidal angular functions capture the latitude- longitude anisotropy, the east-westasymmetry, and the Yoshida inhomogeneity of wave propagation on the rotating spherical surface. The identification of the fundamental role of the spheroidal wave opera- tor permits an overview over the system's wave number space. Three regimes can be distinguished. High-frequency gravity waves do not experience the Yoshida waveguide and exhibit Cariolis-induced east-westasymmetry. A second, low-frequency regime is exclusively populated by Rossby waves. The familiar .8-plane regime provides the appropriate approximation for equatorial, baroclinic, low- frequency waves. Gravity waves on the .8-plane exhibit an amplified Cariolis-induced east-westasym- metry. Validity and limitations of approximate dispersion relations are directly tested against numeri- caDy calculated solutions of the full eigenvalueproblem. PACSnumber(s): 47.32.-y. 47.3S.+i,92.~.Dj. 92.10.Hm L INTRODUcnON [4] in the framework of the .B-pJaneapproximation. In this Cartesian approximation of the spherical geometry The first theoretical investigation of the large-scale near the equator Matsuno identified Rossby, mixed responseof the ocean-atmospheresystem to external tidal Rossby-gravity, as well as gravity waves, including the forces was conducted by Laplace [I] in terms of the KelVin wave. He moreover confirmed Yoshida's hy- linearized Euler equations on the rotating sphere. Since pothesis [5] of an equatorial waveguide. A subsequent then. these equations are known as "the tidal equations." numerical study of the spherical problem by Longuet- although their significance widely exceedsthe tidal prob- Higgins [6] provided a quantitative representation of the lem: the free. linearized Euler equations on the rotating eigenvalues and eigenfunctions of the tidal equation. sphere govern the basic small-amplitude dynamics of the These calculations demonstrated that Matsuno's theory shallow ftuid envelope of a rotating planet. As such they was qualitatively complete. However, in spite of its pose the fundamental linear problem of planetary circula- eleganceand fundamental role for the terrestrial climate tion theory with applications to the atmosphereas well as problem, the ,B-pJaneapproximation is not entirely free to the ocean. These applications include short term. from ambiguities. In contrast to Laplace's tidal equation large-scalephenomena like the tides in the narrow sense. it does not provide a physically meaningful nonrotating and long term. large-scalephenomena like the terrestrial limit and postulates, furthermore, an apparently paradox- climate problem and the circulation of the atmospheres ical east-west asymmetry of gravity of waves. For given of other planets. The generic dynamics of the mode number and frequency a westward traveling gravity overwhelming majority of contemporary numerical circu- wave has a different phase speed than its eastward coun- lation models are represented by Laplace's tidal equa- terpart. Such an asymmetry suggests a Doppler-type tions. effect due to a relative motion of medium and observer. In contrast to the nonrotating case. Laplace observed However, the "tidal" equations represent a situation that the special form of the rotation vector rendered the where both medium and observer are corotating and at three-dimensional (3D) spherical problem nonseparable. rest relative to each other. In the framework of Doppler With "the tidal equation" he established a 2D approxi- theory, these gravity waves should thus be symmetric. mation, which is separable in spherical coordinates. In Recently, exact, analytical solutions of the spherical ti- this framework the discussed what is now called the dal equation have been presented [7] for the special cases Lamb wave under the inftuence of external tidal forces. of standing waves and inertial waves. These results show Laplace was not able to evaluate the eigensolutionsof his that the eigenfunctions of the tidal equation are closely equation. A century later. Margules [2] identified asymp- related to spheroidal wave functions and underscore the totically two classesof solutions: gravity waves and the baroclinic, low-frequency nature of the ,B-pJaneapproxi- now called Rossby waves. An extensive investigation of mation. At high frequencies, the .B-pJaneapproximation these asymptotics by Hough [3] led to the notion of cannot appropriately accommodate the effectsof rotation "Hough functions" for the eigensolutionsof Laplace's ti- on the dispersion of gravity waves. In this regime, an al- dal equations. ternative approximation of the dispersion relation ap- The most comprehensivediscussion of the dynamics of plies. Presently, a complete analytical theory of the "tidal" problem to date has been given by Matsuno Laplace's tidal equations is still not available. 1063-6SlX/9S/S I(S)/4418(14)/506.00 n 4418 @ 1995 The American Physical Society n SHALLOW WATER WAVES ON nIB R.OT A TIN G SPHERE 4419 The "tidal" equations occupy their fundamental posi- tal dynamical nature of the large-scalecirculation. Verti- tion in planetary circulation theory essentially due to cal structure has to comply with covariance requirements their consistency with respect to geometry, dynamics, and for multilayer versions this is the case. Shallow wa- and inertial forces, i.e., their covariance with respect to ter theory with the covariant inclusion of vertical varia- the 2D non-Euclidean geometry of the spherical surface. bility as (prognostic or diagnostic) internal variability of a It is noteworthy t~t Laplace establishedhis form of the spatially 2D fluid representsthe dynamically appropriate equations some 60 years before Coriolis's work on framework for planetary circulation theory. Newton dynamics in rotating systems and-although The lack of a complete analytical theory of the linear spherical trigonometry was known at the time-some 75 shallow water equations on the rotating spheremore than years before Riemann's systematic discussion of non- two centuries after their formulation provides a measure Euclidean geometries. Laplace considered the diagnos- of their complexity. This is partly due to the necessarily tics of a barotropic gas. Structurally the same diagnos- curvilinear nature of the problem. Partial differential tics hold for shallow water theory in the vertically in- operators in curvilinear coordinates carry coordinate- tegrated ("transport") form. Linear, as well as nonlinear dependent coefficients, which complicate the derivation shallow water theories in transport form thus preserve of higher-order wave equations from the first order equa- the dynamically essential covariance of Laplace's tidal tions of motion. However, these problems are drastically equations. In view of their role for the large-scalecircu- simplified by utilizing tensor analysis in Riemann space, ~tion, the "tidal" equations are thus primarily the linear invoking in particular the concept of covariant shallow water equations on the rotating sphere. Howev- differentiation. In contrast to the partial derivative, the er, it has always been questioned whether shallow water covariant derivative "automatically" accounts for the theory can provide a sufficiently rich framework to cap- coefficient functions so that the usual differentiation rules ture the relevant features of the large-scalecirculation. for products and other combinations of tensors apply to Shallow water theory in its generic form has an unreal- covariant differentiation. The manipulation of istically trivial vertical structure. But, interestingly differential expressions then almost proceeds as in the enough, the incorporation of vertical structure, namely, caseof Cartesian coordinates. baroclinicity, into circulation models has been essentially The additional feature to be taken into account is the guided by the spirit of boundary layer theory with little non-Euclidean nature of the intrinsic geometry of the 2D regard for covariance requirements. The prototype of the spherical surface (it may be noted that the "3D spheri- resulting models is the set of the so called "primitive cal" geometry of the "primitive equations" with vertical equations," constructed from three basic ingredients: (1) coordinate z = r -a, where r is the radial coordinate of hydrostatics in the vertical, (2) generally a Boussinesq- 3D Euclidean, spherical geometry and a the Earth's ra- type formulation of stratification, and (3) Laplace's dius, is also non-Euclidean). For a non-Euclidean Coriolis parameter for the representation of inertial geometry the Riemann tensor no longer vanishes. This forces. As a mixture of 3D and 2D features, the "primi- fourth order tensor is a functional of the metric, measur- tive equations" are strictly noncovariant. The derivation ing the anticommutator of the covariant second order of physically meaningful results from these equations re- derivatives of a vector. As a consequenceof a nontrivial quires a sophisticated set of scaling conditions [8] and in Riemannian, the covariant second order derivatives
Load more

Recommended publications
  • 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.
    [Show full text]
  • THE HOMOGENEITY SCALE of the UNIVERSE 1 Introduction On
    THE HOMOGENEITY SCALE OF THE UNIVERSE P. NTELIS Astroparticle and Cosmology Group, University Paris-Diderot 7 , 10 rue A. Domon & Duquet, Paris , France In this study, we probe the cosmic homogeneity with the BOSS CMASS galaxy sample in the redshift region of 0.43 <z<0.7. We use the normalised counts-in-spheres estimator N (<r) and the fractal correlation dimension D2(r) to assess the homogeneity scale of the universe. We verify that the universe becomes homogenous on scales greater than RH 64.3±1.6 h−1Mpc, consolidating the Cosmological Principle with a consistency test of ΛCDM model at the percentage level. Finally, we explore the evolution of the homogeneity scale in redshift. 1 Introduction on Cosmic Homogeneity The standard model of Cosmology, known as the ΛCDM model is based on solutions of the equations of General Relativity for isotropic and homogeneous universes where the matter is mainly composed of Cold Dark Matter (CDM) and the Λ corresponds to a cosmological con- stant. This model shows excellent agreement with current data, be it from Type Ia supernovae, temperature and polarisation anisotropies in the Cosmic Microwave Background or Large Scale Structure. The main assumption of these models is the Cosmological Principle which states that the universe is homogeneous and isotropic on large scales [1]. Isotropy is well tested through various probes in different redshifts such as the Cosmic Microwave Background temperature anisotropies at z ≈ 1100, corresponding to density fluctuations of the order 10−5 [2]. In later cosmic epochs, the distribution of source in surveys the hypothesis of isotropy is strongly supported by in X- ray [3] and radio [4], while the large spectroscopic galaxy surveys such show no evidence for anisotropies in volumes of a few Gpc3 [5].As a result, it is strongly motivated to probe the homogeneity property of our universe.
    [Show full text]
  • 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
    [Show full text]
  • Fluid Simulation Using Shallow Water Equation
    Fluid Simulation using Shallow Water Equation Dhruv Kore Itika Gupta Undergraduate Student Graduate Student University of Illinois at Chicago University of Illinois at Chicago [email protected] [email protected] 847-345-9745 312-478-0764 ABSTRACT rivers and channel. As the name suggest, the main In this paper, we present a technique, which shows how characteristic of shallow Water flows is that the vertical waves once generated from a small drop continue to ripple. dimension is much smaller as compared to the horizontal Waves interact with each other and on collision change the dimension. Naïve Stroke Equation defines the motion of form and direction. Once the waves strike the boundary, fluids and from these equations we derive SWE. they return with the same speed and in sometime, depending on the delay, you can see continuous ripples in In Next section we talk about the related works under the the surface. We use shallow water equation to achieve the heading Literature Review. Then we will explain the desired output. framework and other concepts necessary to understand the SWE and wave equation. In the section followed by it, we Author Keywords show the results achieved using our implementation. Then Naïve Stroke Equation; Shallow Water Equation; Fluid finally we talk about conclusion and future work. Simulation; WebGL; Quadratic function. LITERATURE REVIEW INTRODUCTION In [2] author in detail explains the Shallow water equation In early years of fluid simulation, procedural surface with its derivation. Since then a lot of authors has used generation was used to represent waves as presented by SWE to present the formation of waves in water.
    [Show full text]
  • Shallow Water Waves and Solitary Waves Article Outline Glossary
    Shallow Water Waves and Solitary Waves Willy Hereman Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado, USA Article Outline Glossary I. Definition of the Subject II. Introduction{Historical Perspective III. Completely Integrable Shallow Water Wave Equations IV. Shallow Water Wave Equations of Geophysical Fluid Dynamics V. Computation of Solitary Wave Solutions VI. Water Wave Experiments and Observations VII. Future Directions VIII. Bibliography Glossary Deep water A surface wave is said to be in deep water if its wavelength is much shorter than the local water depth. Internal wave A internal wave travels within the interior of a fluid. The maximum velocity and maximum amplitude occur within the fluid or at an internal boundary (interface). Internal waves depend on the density-stratification of the fluid. Shallow water A surface wave is said to be in shallow water if its wavelength is much larger than the local water depth. Shallow water waves Shallow water waves correspond to the flow at the free surface of a body of shallow water under the force of gravity, or to the flow below a horizontal pressure surface in a fluid. Shallow water wave equations Shallow water wave equations are a set of partial differential equations that describe shallow water waves. 1 Solitary wave A solitary wave is a localized gravity wave that maintains its coherence and, hence, its visi- bility through properties of nonlinear hydrodynamics. Solitary waves have finite amplitude and propagate with constant speed and constant shape. Soliton Solitons are solitary waves that have an elastic scattering property: they retain their shape and speed after colliding with each other.
    [Show full text]
  • Shallow-Water Equations and Related Topics
    CHAPTER 1 Shallow-Water Equations and Related Topics Didier Bresch UMR 5127 CNRS, LAMA, Universite´ de Savoie, 73376 Le Bourget-du-Lac, France Contents 1. Preface .................................................... 3 2. Introduction ................................................. 4 3. A friction shallow-water system ...................................... 5 3.1. Conservation of potential vorticity .................................. 5 3.2. The inviscid shallow-water equations ................................. 7 3.3. LERAY solutions ........................................... 11 3.3.1. A new mathematical entropy: The BD entropy ........................ 12 3.3.2. Weak solutions with drag terms ................................ 16 3.3.3. Forgetting drag terms – Stability ............................... 19 3.3.4. Bounded domains ....................................... 21 3.4. Strong solutions ............................................ 23 3.5. Other viscous terms in the literature ................................. 23 3.6. Low Froude number limits ...................................... 25 3.6.1. The quasi-geostrophic model ................................. 25 3.6.2. The lake equations ...................................... 34 3.7. An interesting open problem: Open sea boundary conditions .................... 41 3.8. Multi-level and multi-layers models ................................. 43 3.9. Friction shallow-water equations derivation ............................. 44 3.9.1. Formal derivation ....................................... 44 3.10.
    [Show full text]
  • Waves in Water from Ripples to Tsunamis and to Rogue Waves
    Waves in Water from ripples to tsunamis and to rogue waves Vera Mikyoung Hur (Mathematics) (Image from Flickr) The motion of a fluid can be very complicated as we know whenever we see waves break on a beach, (Image from the Internet) The motion of a fluid can be very complicated as we know whenever we fly in an airplane, (Image from the Internet) The motion of a fluid can be very complicated as we know whenever we look at a lake on a windy day. (Image from the Internet) Euler in the 1750s proposed a mathematical model of an incompressible fluid. @u + (u · r)u + rP = F, @t r · u = 0. Here, u(x; t) is the velocity of the fluid at the point x and time t, P(x; t) is the pressure, and F(x; t) is an outside force. The Navier-Stokes equations (adding ν∆u) allow the fluid to be viscous. It is concise and captures the essence of fluid behavior. The theory of fluids has provided source and inspiration to many branches of mathematics, e.g. Cauchy's complex function theory. Difficulties of understanding fluids are profound. e.g. the global-in-time solution of the Navier-Stokes equations in 3 dimensions is a Clay Millennium Problem! Waves, jets, drops come to mind when thinking of fluids. They involve one or more fluids separated by an unknown surface. In the mathematical community, they go by free boundary problems. (Image from the Internet) Free boundaries are mathematically challenging in their own right. They occur in many other situations, such as melting of ice stretching a membrane over an obstacle.
    [Show full text]
  • Modified Shallow Water Equations for Significantly Varying Seabeds Denys Dutykh, Didier Clamond
    Modified Shallow Water Equations for significantly varying seabeds Denys Dutykh, Didier Clamond To cite this version: Denys Dutykh, Didier Clamond. Modified Shallow Water Equations for significantly varying seabeds: Modified Shallow Water Equations. Applied Mathematical Modelling, Elsevier, 2016, 40 (23-24), pp.9767-9787. 10.1016/j.apm.2016.06.033. hal-00675209v6 HAL Id: hal-00675209 https://hal.archives-ouvertes.fr/hal-00675209v6 Submitted on 26 Apr 2016 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 - NonCommercial| 4.0 International License Denys Dutykh CNRS, Université Savoie Mont Blanc, France Didier Clamond Université de Nice – Sophia Antipolis, France Modified Shallow Water Equations for significantly varying seabeds arXiv.org / hal Last modified: April 26, 2016 Modified Shallow Water Equations for significantly varying seabeds Denys Dutykh∗ and Didier Clamond Abstract. In the present study, we propose a modified version of the Nonlinear Shal- low Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the case when the bottom undergoes some significant variations in space and time. The model is derived from a variational principle by choosing an appropriate shallow water ansatz and imposing some constraints.
    [Show full text]
  • ATM 500: Atmospheric Dynamics End-Of-Semester Review, December 4 2018 Here, in Bullet Points, Are the Key Concepts from the Second Half of the Course
    ATM 500: Atmospheric Dynamics End-of-semester review, December 4 2018 Here, in bullet points, are the key concepts from the second half of the course. They are not meant to be self-contained notes! Refer back to your course notes for notation, derivations, and deeper discussion. Static stability The vertical variation of density in the environment determines whether a parcel, disturbed upward from an initial hydrostatic rest state, will accelerate away from its initial position, or oscillate about that position. For an incompressible or Boussinesq fluid, the buoyancy frequency is g dρ~ N 2 = − ρ~dz For a compressible ideal gas, the buoyancy frequency is g dθ~ N 2 = θ~dz (the expressions are different because density is not conserved in a small upward displacement of a compressible fluid, but potential temperature is conserved) Fluid is statically stable if N 2 > 0; otherwise it is statically unstable. Typical value for troposphere: N = 0:01 s−1, with corresponding oscillation period of 10 minutes. Dry adiabatic lapse rate g −1 Γd = = 9:8 K km cp The rate at which temperature decreases during adiabatic ascent (for which Dθ Dt = 0) \Dry" here means that there is no latent heating from condensation of water vapor. Static stability criteria for a dry atmosphere Environment is stable to dry con- vection if dθ~ dT~ > 0 or − < Γ dz dz d and otherwise unstable. 1 Buoyancy equation for Boussinesq fluid with background stratification Db0 + N 2w = 0 Dt where we have separated the full buoyancy into a mean vertical part ^b(z) and 0 2 d^b 2 everything else (b ), and N = dz .
    [Show full text]
  • Multilayer Shallow Water Equations with Complete Coriolis Force. Part I: Derivation on a Non-Traditional Beta-Plane
    Accepted for publication in the Journal of Fluid Mechanics °c 2010 Cambridge University Press 1 doi:10.1017/S0022112009993922 Multilayer shallow water equations with complete Coriolis force. Part I: Derivation on a non-traditional beta-plane By ANDREW L. STEWART and PAUL J. DELLAR OCIAM, Mathematical Institute, 24–29 St Giles’, Oxford, OX1 3LB, UK [email protected], [email protected] (Received 22 May 2009; revised 9 December 2009; accepted 9 December 2009) We derive equations to describe the flow of multiple superposed layers of inviscid, incompressible fluids with con- stant densities over prescribed topography in a rotating frame. Motivated by geophysical applications, these equations incorporate the complete Coriolis force. We do not make the widely used “traditional approximation” that omits the contribution to the Coriolis force from the locally horizontal part of the rotation vector. Our derivation is performed by averaging the governing Euler equations over each layer, and from two different forms of Hamilton’s variational principle that differ in their treatment of the coupling between layers. The coupling may be included implicitly through the map from Lagrangian particle labels to particle coordinates, or explicitly by adding terms representing the work done on each layer by the pressure exerted by the layers above. The latter approach requires additional terms in the Lagrangian, but extends more easily to many layers. We show that our equations obey the expected conservation laws for energy, momentum, and potential vorticity. The conserved momentum and potential vorticity are modified by non-traditional effects. The vertical component of the rotation vector that appears in the potential vorticity for each layer under the traditional approximation is replaced by the component perpendicular to the layer’s midsurface.
    [Show full text]
  • A Particle-In-Cell Method for the Solution of Two-Layer Shallow-Water Equations
    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2000; 32: 515–543 A particle-in-cell method for the solution of two-layer shallow-water equations Benoit Cushman-Roisina,*, Oleg E. Esenkovb and Benedict J. Mathiasc a Thayer School of Engineering, Dartmouth College, Hano6er, NH 03755-8000, U.S.A. b Rosenstiel School of Marine and Atmospheric Science, Uni6ersity of Miami, Miami, FL 33149-1098, U.S.A. c i2 Technologies, 303 Twin Dolphin Dri6e, Redwood City, CA 94065, U.S.A. SUMMARY A particle-in-cell (PIC) numerical method developed for the study of shallow-water dynamics, when the moving fluid layer is laterally confined by the intersection of its top and bottom surfaces, is described. The effect of ambient rotation is included for application to geophysical fluids, particularly open-ocean buoyant vortices in which the underlying density interface outcrops to the surface around the rim of the vortex. Extensions to include the dynamical effect of a second moving layer (baroclinicity) and the presence of a lateral rigid boundary (sidewall) are also described. Although the method was developed for oceanographic investigations, applications to other fluid mechanics problems would be straightforward. Copyright © 2000 John Wiley & Sons, Ltd. KEY WORDS: open-ocean; particle-in-cell; shallow-water equation 1. INTRODUCTION There are a number of situations, mostly geophysical in nature, where fluid flows occur in layers that are much thinner than they are wide. In such cases, hydrostatic equilibrium is invoked in the vertical, and the equations of motion reduce to the so-called St. Venant or shallow-water equations [1].
    [Show full text]
  • Dark Energy and the Homogeneous Universe 暗能量及宇宙整體之膨脹
    Dark Energy and the Homogeneous Universe 暗能量及宇宙整體之膨脹 Lam Hui 許林 Columbia University This is a series of 3 talks. Today: Dark energy and the homogeneous universe July 11: Dark matter and the large scale structure of the universe July 18: Inflation and the early universe Outline Basic concepts: a homogeneous, isotropic and expanding universe. Measurements: how do we measure the universe’s expansion history? Dynamics: what determines the expansion rate? Dark energy: a surprising recent finding. There is by now much evidence for homogeneity and isotropy. There is also firm evidence for an expanding universe. scale ∼ 1023 cm Blandton & Hogg scale ∼ 1024 cm Hubble deep field scale ∼ 1025 cm CfA survey Sloan Digital Sky Survey scale ∼ 1026 cm Ned Wright There is by now much evidence for homogeneity and isotropy. There is also firm evidence for an expanding universe. There is by now much evidence for homogeneity and isotropy. There is also firm evidence for an expanding universe. But how should we think about it? Frequently Asked Questions: 1. What is the universe expanding into? 2. Where is the center of expansion? 3. Is our galaxy itself expanding? Frequently Asked Questions: 1. What is the universe expanding into? Nothing. 2. Where is the center of expansion? Everywhere and nowhere. 3. Is our galaxy itself expanding? No. C C A B A B dAC = 2 dAB ∆d ∆d AC = 2 AB Therefore: ∆t ∆t Hubble law: Velocity ∝ Distance Supernova 1998ba Supernova Cosmology Project (Perlmutter, et al., 1998) (as seen from Hubble Space Telescope) 3 Weeks Supernova Before Discovery (as seen from telescopes on Earth) Difference Supernova 超新星as standard candle FAINTER (Farther) (Further back in time) Perlmutter, et al.
    [Show full text]