1 GEOPHYSICAL FLUID DYNAMICS-I OC512/AS509 2015 LECTURES 5-6 Week 3: Coriolis Effects: Introduction to Dynamics of Fluids on a Rotating Planet
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Navier-Stokes Equation
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Vorticity Production Through Rotation, Shear, and Baroclinicity
A&A 528, A145 (2011) Astronomy DOI: 10.1051/0004-6361/201015661 & c ESO 2011 Astrophysics Vorticity production through rotation, shear, and baroclinicity F. Del Sordo1,2 and A. Brandenburg1,2 1 Nordita, AlbaNova University Center, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden e-mail: [email protected] 2 Department of Astronomy, AlbaNova University Center, Stockholm University, 10691 Stockholm, Sweden Received 31 August 2010 / Accepted 14 February 2011 ABSTRACT Context. In the absence of rotation and shear, and under the assumption of constant temperature or specific entropy, purely potential forcing by localized expansion waves is known to produce irrotational flows that have no vorticity. Aims. Here we study the production of vorticity under idealized conditions when there is rotation, shear, or baroclinicity, to address the problem of vorticity generation in the interstellar medium in a systematic fashion. Methods. We use three-dimensional periodic box numerical simulations to investigate the various effects in isolation. Results. We find that for slow rotation, vorticity production in an isothermal gas is small in the sense that the ratio of the root-mean- square values of vorticity and velocity is small compared with the wavenumber of the energy-carrying motions. For Coriolis numbers above a certain level, vorticity production saturates at a value where the aforementioned ratio becomes comparable with the wavenum- ber of the energy-carrying motions. Shear also raises the vorticity production, but no saturation is found. When the assumption of isothermality is dropped, there is significant vorticity production by the baroclinic term once the turbulence becomes supersonic. In galaxies, shear and rotation are estimated to be insufficient to produce significant amounts of vorticity, leaving therefore only the baroclinic term as the most favorable candidate. -
The Coriolis Effect in Meteorology
THE CORIOLIS EFFECT IN METEOROLOGY On this page I discuss the rotation-of-Earth-effect that is taken into account in Meteorology, where it is referred to as 'the Coriolis effect'. (For the rotation of Earth effect that applies in ballistics, see the following two Java simulations: Great circles and Ballistics). The rotating Earth A key consequence of the Earth's rotation is the deviation from perfectly spherical form: there is an equatorial bulge. The bulge is small; on Earth pictures taken from outer space you can't see it. It may seem unimportant but it's not: what matters for meteorology is an effect that arises from the Earth's rotation and Earth's oblateness together. A model: parabolic dish Thinking about motion over the Earth's surface is rather complicated, so I turn now to a model that is simpler but still presents the feature that gives rise to the Coriolis effect. Source: PAOC, MIT Courtesy of John Marshall The dish in the picture is used by students of Geophysical Fluid Dynamics for lab exercises. This dish was manufactured as follows: a flat platform with a rim was rotating at a very constant angular velocity (10 revolutions per minute), and a synthetic resin was poured onto the platform. The resin flowed out, covering the entire area. It had enough time to reach an equilibrium state before it started to set. The surface was sanded to a very smooth finish. Also, note the construction that is hanging over the dish. The vertical rod is not attached to the table but to the dish; when the dish rotates the rod rotates with it. -
Surface Circulation2016
OCN 201 Surface Circulation Excess heat in equatorial regions requires redistribution toward the poles 1 In the Northern hemisphere, Coriolis force deflects movement to the right In the Southern hemisphere, Coriolis force deflects movement to the left Combination of atmospheric cells and Coriolis force yield the wind belts Wind belts drive ocean circulation 2 Surface circulation is one of the main transporters of “excess” heat from the tropics to northern latitudes Gulf Stream http://earthobservatory.nasa.gov/Newsroom/NewImages/Images/gulf_stream_modis_lrg.gif 3 How fast ( in miles per hour) do you think western boundary currents like the Gulf Stream are? A 1 B 2 C 4 D 8 E More! 4 mph = C Path of ocean currents affects agriculture and habitability of regions ~62 ˚N Mean Jan Faeroe temp 40 ˚F Islands ~61˚N Mean Jan Anchorage temp 13˚F Alaska 4 Average surface water temperature (N hemisphere winter) Surface currents are driven by winds, not thermohaline processes 5 Surface currents are shallow, in the upper few hundred metres of the ocean Clockwise gyres in North Atlantic and North Pacific Anti-clockwise gyres in South Atlantic and South Pacific How long do you think it takes for a trip around the North Pacific gyre? A 6 months B 1 year C 10 years D 20 years E 50 years D= ~ 20 years 6 Maximum in surface water salinity shows the gyres excess evaporation over precipitation results in higher surface water salinity Gyres are underneath, and driven by, the bands of Trade Winds and Westerlies 7 Which wind belt is Hawaii in? A Westerlies B Trade -
Intro to Tidal Theory
Introduction to Tidal Theory Ruth Farre (BSc. Cert. Nat. Sci.) South African Navy Hydrographic Office, Private Bag X1, Tokai, 7966 1. INTRODUCTION Tides: The periodic vertical movement of water on the Earth’s Surface (Admiralty Manual of Navigation) Tides are very often neglected or taken for granted, “they are just the sea advancing and retreating once or twice a day.” The Ancient Greeks and Romans weren’t particularly concerned with the tides at all, since in the Mediterranean they are almost imperceptible. It was this ignorance of tides that led to the loss of Caesar’s war galleys on the English shores, he failed to pull them up high enough to avoid the returning tide. In the beginning tides were explained by all sorts of legends. One ascribed the tides to the breathing cycle of a giant whale. In the late 10 th century, the Arabs had already begun to relate the timing of the tides to the cycles of the moon. However a scientific explanation for the tidal phenomenon had to wait for Sir Isaac Newton and his universal theory of gravitation which was published in 1687. He described in his “ Principia Mathematica ” how the tides arose from the gravitational attraction of the moon and the sun on the earth. He also showed why there are two tides for each lunar transit, the reason why spring and neap tides occurred, why diurnal tides are largest when the moon was furthest from the plane of the equator and why the equinoxial tides are larger in general than those at the solstices. -
Coriolis Effect
Project ATMOSPHERE This guide is one of a series produced by Project ATMOSPHERE, an initiative of the American Meteorological Society. Project ATMOSPHERE has created and trained a network of resource agents who provide nationwide leadership in precollege atmospheric environment education. To support these agents in their teacher training, Project ATMOSPHERE develops and produces teacher’s guides and other educational materials. For further information, and additional background on the American Meteorological Society’s Education Program, please contact: American Meteorological Society Education Program 1200 New York Ave., NW, Ste. 500 Washington, DC 20005-3928 www.ametsoc.org/amsedu This material is based upon work initially supported by the National Science Foundation under Grant No. TPE-9340055. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the National Science Foundation. © 2012 American Meteorological Society (Permission is hereby granted for the reproduction of materials contained in this publication for non-commercial use in schools on the condition their source is acknowledged.) 2 Foreword This guide has been prepared to introduce fundamental understandings about the guide topic. This guide is organized as follows: Introduction This is a narrative summary of background information to introduce the topic. Basic Understandings Basic understandings are statements of principles, concepts, and information. The basic understandings represent material to be mastered by the learner, and can be especially helpful in devising learning activities in writing learning objectives and test items. They are numbered so they can be keyed with activities, objectives and test items. Activities These are related investigations. -
Theoretical Oceanography Lecture Notes Master AO-W27
Theoretical Oceanography Lecture Notes Master AO-W27 Martin Schmidt 1Leibniz Institute for Baltic Sea Research Warnem¨unde, Seestraße 15, D - 18119 Rostock, Germany, Tel. +49-381-5197-121, e-mail: [email protected] December 3, 2015 Contents 1 Literature recommendations 3 2 Basic equations 5 2.1 Flowkinematics................................. 5 2.1.1 Coordinatesystems........................... 5 2.1.2 Eulerian and Lagrangian representation . 6 2.2 Conservation laws and balance equations . 12 2.2.1 Massconservation............................ 12 2.2.2 Salinity . 13 2.2.3 The general conservation of intensive quantities . 15 2.2.4 Dynamic variables . 15 2.2.5 The momentum budget of a fluid element . 16 2.2.6 Frictionandmeanflow......................... 17 2.2.7 Coriolis and centrifugal force . 20 2.2.8 Themomentumequationsontherotatingearth . 24 2.2.9 Approximations for the Coriolis force . 24 2.3 Thermodynamics ................................ 26 2.3.1 Extensive and intensive quantities . 26 2.3.2 Traditional approach to sea water density . 27 2.3.3 Mechanical, thermal and chemical equilibrium . 27 2.3.4 Firstlawofthermodynamics. 28 2.3.5 Secondlawofthermodynamics . 28 2.3.6 Thermodynamicspotentials . 28 2.4 Energyconsiderations ............................. 30 2.5 Temperatureequations ............................. 32 2.5.1 Thein-situtemperature . .. .. 33 2.5.2 Theconservativetemperature . 33 2.5.3 Thepotentialtemperature . 35 2.6 Hydrostaticfluids................................ 36 2.6.1 Equilibrium conditions . 36 2.6.2 Stability of the water column . 37 i 1 3 Wind driven flow 43 3.1 Earlytheory-theZ¨oppritzocean . 43 3.2 TheEkmantheory ............................... 51 3.2.1 The classical Ekman solution . 51 3.2.2 Theconceptofvolumeforces . 52 3.2.3 Ekmantheorywithvolumeforces . 54 4 Oceanic waves 61 4.1 Wavekinematics ............................... -
CONTENTS: I. Sea/Land Breeze II. Pressure Gradient Force III. Angular Momentum and Coriolis Force IV
Science A-30 Section Handout 4 March 4-5, 2008 CONTENTS: I. Sea/Land Breeze II. Pressure Gradient Force III. Angular Momentum and Coriolis Force IV. Geostrophic Flow Relevant formulas and conversions: Scale Height: H = kT/(mairg ) = R'T/g -z/H Barometric Law: P(z) = P(zo)e P = ρ(k/mair)T = ρR'T m ΔP Pressure gradient force: F = × pg ρ Δx 1 day = 86400 seconds Circle: 360 degrees = 2π radians Angular velocity: ω = 2π/t Angular momentum: L = m r2 ω Coriolis force: Fc = 2 m Ω v sin(λ) Coriolis deflection: Δy = [ Ω (Δx)2 / v ] sin(λ) I. Sea/Land Breeze Onshore sea breeze Offshore land breeze Warm land, cool water Cool land, warm water During the day, land heats up faster than the sea. This makes the scale height, H larger over the land than over sea. Pressure at some altitude over land is greater than pressure over sea at same altitude, which induces flow from high pressure (land) to low pressure (sea). Now there is less mass over land than over the sea. This makes the pressure at the surface higher over the sea. At the ground, air flows from high pressure (sea) to low pressure (land). Conservation of mass completes the circulation. During nighttime, land cools off faster than the sea. The flow reverses. At the ground, air flows from inland towards the sea. 1 II. Pressure Gradient Force Pressure gradient force (Fpg) is the force due to a pressure difference (ΔP) across a horizontal distance (Δx) (directed from high pressure to low pressure). -
True Polar Wander: Linking Deep and Shallow Geodynamics to Hydro- and Bio-Spheric Hypotheses T
True Polar Wander: linking Deep and Shallow Geodynamics to Hydro- and Bio-Spheric Hypotheses T. D. Raub, Yale University, New Haven, CT, USA J. L. Kirschvink, California Institute of Technology, Pasadena, CA, USA D. A. D. Evans, Yale University, New Haven, CT, USA © 2007 Elsevier SV. All rights reserved. 5.14.1 Planetary Moment of Inertia and the Spin-Axis 565 5.14.2 Apparent Polar Wander (APW) = Plate motion +TPW 566 5.14.2.1 Different Information in Different Reference Frames 566 5.14.2.2 Type 0' TPW: Mass Redistribution at Clock to Millenial Timescales, of Inconsistent Sense 567 5.14.2.3 Type I TPW: Slow/Prolonged TPW 567 5.14.2.4 Type II TPW: Fast/Multiple/Oscillatory TPW: A Distinct Flavor of Inertial Interchange 569 5.14.2.5 Hypothesized Rapid or Prolonged TPW: Late Paleozoic-Mesozoic 569 5.14.2.6 Hypothesized Rapid or Prolonged TPW: 'Cryogenian'-Ediacaran-Cambrian-Early Paleozoic 571 5.14.2.7 Hypothesized Rapid or Prolonged TPW: Archean to Mesoproterozoic 572 5.14.3 Geodynamic and Geologic Effects and Inferences 572 5.14.3.1 Precision of TPW Magnitude and Rate Estimation 572 5.14.3.2 Physical Oceanographic Effects: Sea Level and Circulation 574 5.14.3.3 Chemical Oceanographic Effects: Carbon Oxidation and Burial 576 5..14.4 Critical Testing of Cryogenian-Cambrian TPW 579 5.14A1 Ediacaran-Cambrian TPW: 'Spinner Diagrams' in the TPW Reference Frame 579 5.14.4.2 Proof of Concept: Independent Reconstruction of Gondwanaland Using Spinner Diagrams 581 5.14.5 Summary: Major Unresolved Issues and Future Work 585 References 586 5.14.1 Planetary Moment of Inertia location ofEarth's daily rotation axis and/or by fluc and the Spin-Axis tuations in the spin rate ('length of day' anomalies). -
Lecture 4: OCEANS (Outline)
LectureLecture 44 :: OCEANSOCEANS (Outline)(Outline) Basic Structures and Dynamics Ekman transport Geostrophic currents Surface Ocean Circulation Subtropicl gyre Boundary current Deep Ocean Circulation Thermohaline conveyor belt ESS200A Prof. Jin -Yi Yu BasicBasic OceanOcean StructuresStructures Warm up by sunlight! Upper Ocean (~100 m) Shallow, warm upper layer where light is abundant and where most marine life can be found. Deep Ocean Cold, dark, deep ocean where plenty supplies of nutrients and carbon exist. ESS200A No sunlight! Prof. Jin -Yi Yu BasicBasic OceanOcean CurrentCurrent SystemsSystems Upper Ocean surface circulation Deep Ocean deep ocean circulation ESS200A (from “Is The Temperature Rising?”) Prof. Jin -Yi Yu TheThe StateState ofof OceansOceans Temperature warm on the upper ocean, cold in the deeper ocean. Salinity variations determined by evaporation, precipitation, sea-ice formation and melt, and river runoff. Density small in the upper ocean, large in the deeper ocean. ESS200A Prof. Jin -Yi Yu PotentialPotential TemperatureTemperature Potential temperature is very close to temperature in the ocean. The average temperature of the world ocean is about 3.6°C. ESS200A (from Global Physical Climatology ) Prof. Jin -Yi Yu SalinitySalinity E < P Sea-ice formation and melting E > P Salinity is the mass of dissolved salts in a kilogram of seawater. Unit: ‰ (part per thousand; per mil). The average salinity of the world ocean is 34.7‰. Four major factors that affect salinity: evaporation, precipitation, inflow of river water, and sea-ice formation and melting. (from Global Physical Climatology ) ESS200A Prof. Jin -Yi Yu Low density due to absorption of solar energy near the surface. DensityDensity Seawater is almost incompressible, so the density of seawater is always very close to 1000 kg/m 3. -
Appendix a Glossary
BASICS OF RADIO ASTRONOMY Appendix A Glossary Absolute magnitude Apparent magnitude a star would have at a distance of 10 parsecs (32.6 LY). Absorption lines Dark lines superimposed on a continuous electromagnetic spectrum due to absorption of certain frequencies by the medium through which the radiation has passed. AM Amplitude modulation. Imposing a signal on transmitted energy by varying the intensity of the wave. Amplitude The maximum variation in strength of an electromagnetic wave in one wavelength. Ångstrom Unit of length equal to 10-10 m. Sometimes used to measure wavelengths of visible light. Now largely superseded by the nanometer (10-9 m). Aphelion The point in a body’s orbit (Earth’s, for example) around the sun where the orbiting body is farthest from the sun. Apogee The point in a body’s orbit (the moon’s, for example) around Earth where the orbiting body is farthest from Earth. Apparent magnitude Measure of the observed brightness received from a source. Astrometry Technique of precisely measuring any wobble in a star’s position that might be caused by planets orbiting the star and thus exerting a slight gravitational tug on it. Astronomical horizon The hypothetical interface between Earth and sky, as would be seen by the observer if the surrounding terrain were perfectly flat. Astronomical unit Mean Earth-to-sun distance, approximately 150,000,000 km. Atmospheric window Property of Earth’s atmosphere that allows only certain wave- lengths of electromagnetic energy to pass through, absorbing all other wavelengths. AU Abbreviation for astronomical unit. Azimuth In the horizon coordinate system, the horizontal angle from some arbitrary reference direction (north, usually) clockwise to the object circle. -
An Essay on the Coriolis Force
James F. Price Woods Hole Oceanographic Institution Woods Hole, Massachusetts, 02543 http:// www.whoi.edu/science/PO/people/jprice [email protected] Version 3.3 January 10, 2006 Summary: An Earth-attached and thus rotating reference frame is almost always used for the analysis of geophysical flows. The equation of motion transformed into a steadily rotating reference frame includes two terms that involve the rotation vector; a centrifugal term and a Coriolis term. In the special case of an Earth-attached reference frame, the centrifugal term is exactly canceled by gravitational mass attraction and drops out of the equation of motion. When we solve for the acceleration seen from an Earth-attached frame, the Coriolis term is interpreted as a force. The rotating frame perspective gives up the properties of global momentum conservation and invariance to Galilean transformation. Nevertheless, it leads to a greatly simplified analysis of geophysical flows since only the comparatively small relative velocity, i.e., winds and currents, need be considered. The Coriolis force has a simple mathematical form, 2˝ V 0M , where ˝ is Earth’s rotation vector, V 0 is the velocity observed from the rotating frame and M is the particle mass. The Coriolis force is perpendicular to the velocity and can do no work. It tends to cause a deflection of velocity, and gives rise to two important modes of motion: (1) If the Coriolis force is the only force acting on a moving particle, then the velocity vector of the particle will be continually deflected and rotate clockwise in the northern hemisphere and anticlockwise in the southern hemisphere.