Intro to Tidal Theory
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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. -
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 -
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 ............................... -
Oceanography 200, Spring 2008 Armbrust/Strickland Study Guide Exam 1
Oceanography 200, Spring 2008 Armbrust/Strickland Study Guide Exam 1 Geography of Earth & Oceans Latitude & longitude Properties of Water Structure of water molecule: polarity, hydrogen bonds, effects on water as a solvent and on freezing Latent heat of fusion & vaporization & physical explanation Effects of latent heat on heat transport between ocean & atmosphere & within atmosphere Definition of density, density of ice vs. water Physical meaning of temperature & heat Heat capacity of water & why water requires a lot of heat gain or loss to change temperature Effects of heating & cooling on density of water, and physical explanation Difference in albedo of ice/snow vs. water, and effects on Earth temperature Properties of Seawater Definitions of salinity, conservative & non-conservative seawater constituents, density (sigma-t) Average ocean salinity & how it is measured 3 technology systems for monitoring ocean salinity and other properties 6 most abundant constituents of seawater & Principle of Constant Proportions Effects of freezing on seawater Effects of temperature & salinity on density, T-S diagram Processes that increase & decrease salinity & temperature and where they occur Generalized depth profiles of temperature, salinity, density & ocean stratification & stability Generalized depth profiles of O2 & CO2 and processes determining these profiles Forms of dissolved inorganic carbon in seawater and their buffering effect on pH of seawater Water Properties and Climate Change 2 main causes of global sea level rise, 2 main causes of local sea level rise Physical properties of water that affect sea level rise Icebergs & sea ice: Differences in sources, and effects of freezing & melting on sea level How heat content of upper ocean & Arctic sea ice extent have changed since about 1970s Invasion of fossil fuel CO2 in the oceans and effects on pH The Sea Floor & Plate Tectonics General differences in rock type & density of oceanic vs. -
Oceanography Lecture 12
Because, OF ALL THE ICE!!! Oceanography Lecture 12 How do you know there’s an Ice Age? i. The Ocean/Atmosphere coupling ii.Surface Ocean Circulation Global Circulation Patterns: Atmosphere-Ocean “coupling” 3) Atmosphere-Ocean “coupling” Atmosphere – Transfer of moisture to the Low latitudes: Oceans atmosphere (heat released in higher latitudes High latitudes: Atmosphere as water condenses!) Atmosphere-Ocean “coupling” In summary Latitudinal Differences in Energy Atmosphere – Transfer of moisture to the atmosphere: Hurricanes! www.weather.com Amount of solar radiation received annually at the Earth’s surface Latitudinal Differences in Salinity Latitudinal Differences in Density Structure of the Oceans Heavy Light T has a much greater impact than S on Density! Atmospheric – Wind patterns Atmospheric – Wind patterns January January Westerlies Easterlies Easterlies Westerlies High/Low Pressure systems: Heat capacity! High/Low Pressure systems: Wind generation Wind drag Zonal Wind Flow Wind is moving air Air molecules drag water molecules across sea surface (remember waves generation?): frictional drag Westerlies If winds are prolonged, the frictional drag generates a current Easterlies Only a small fraction of the wind energy is transferred to Easterlies the water surface Westerlies Any wind blowing in a regular pattern? High/Low Pressure systems: Wind generation by flow from High to Low pressure systems (+ Coriolis effect) 1) Ekman Spiral 1) Ekman Spiral Once the surface film of water molecules is set in motion, they exert a Spiraling current in which speed and direction change with frictional drag on the water molecules immediately beneath them, depth: getting these to move as well. Net transport (average of all transport) is 90° to right Motion is transferred downward into the water column (North Hemisphere) or left (Southern Hemisphere) of the ! Speed diminishes with depth (friction) generating wind. -
Tidal and Wind-Driven Currents from Oscr
FEATURE TIDAL AND WIND-DRIVEN CURRENTS FROM OSCR By David Prandle TWO IMPORTANTASPECTS of tidal currents are (1) be seen by comparing calculations of M, tidal vor- their temporal coherence and (2) their constancy ticity distributions <OV/OX- c?U/OY> from OSCR (over centuries). The first rigorous evaluation of an measurements with corresponding model calcula- Ocean Surface Current Radar (OSCR) system ex- tions (Prandle 1987). ploited these characteristics using sequential de- Tidal Residuals ployments of the one available unit with subse- The propagation of tidal energy from the quent combination of radial components to ocean into shelf seas produces an attendant net construct tidal ellipses (Prandle and Ryder 1985). residual current Uo of 0.5(OUD)cos 0 (0, oscil- Specifications for Tidal Mapping lating current amplitude; c, elevation amplitude: The Rayleigh criterion for separation of closely D, water depth: O, phase difference between lJ spaced constituents in tidal analysis suggests ob- and {). In U.K. waters Uo is typically 0-3 cm s ' servational periods exceeding the related beat fre- compared with 0 of 40-100 cm s ', thus conven- • . enhanced reso- quency, this dictates 15 d of observations to sepa- tional current meters often fail to resolve U~,. lution of the instru- rate the two largest constituents M~ and St. For Moreover, numerical models that accurately sim- tidal elevations this criterion is often relaxed: ulate M z may not resolve U,, with the same accu- mentation reveals however, while elevations show a noise:tidal sig- racy. Year-long deployments of OSCR, in the finer scale dynamical nal ratio of 0(0.1-0.2), the same ratio for currents Dover Straits (Prandle et al., 1993) and the North is 0(0.5). -
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). -
Physical Oceanography - UNAM, Mexico Lecture 3: the Wind-Driven Oceanic Circulation
Physical Oceanography - UNAM, Mexico Lecture 3: The Wind-Driven Oceanic Circulation Robin Waldman October 17th 2018 A first taste... Many large-scale circulation features are wind-forced ! Outline The Ekman currents and Sverdrup balance The western intensification of gyres The Southern Ocean circulation The Tropical circulation Outline The Ekman currents and Sverdrup balance The western intensification of gyres The Southern Ocean circulation The Tropical circulation Ekman currents Introduction : I First quantitative theory relating the winds and ocean circulation. I Can be deduced by applying a dimensional analysis to the horizontal momentum equations within the surface layer. The resulting balance is geostrophic plus Ekman : I geostrophic : Coriolis and pressure force I Ekman : Coriolis and vertical turbulent momentum fluxes modelled as diffusivities. Ekman currents Ekman’s hypotheses : I The ocean is infinitely large and wide, so that interactions with topography can be neglected ; ¶uh I It has reached a steady state, so that the Eulerian derivative ¶t = 0 ; I It is homogeneous horizontally, so that (uh:r)uh = 0, ¶uh rh:(khurh)uh = 0 and by continuity w = 0 hence w ¶z = 0 ; I Its density is constant, which has the same consequence as the Boussinesq hypotheses for the horizontal momentum equations ; I The vertical eddy diffusivity kzu is constant. ¶ 2u f k × u = k E E zu ¶z2 that is : k ¶ 2v u = zu E E f ¶z2 k ¶ 2u v = − zu E E f ¶z2 Ekman currents Ekman balance : k ¶ 2v u = zu E E f ¶z2 k ¶ 2u v = − zu E E f ¶z2 Ekman currents Ekman balance : ¶ 2u f k × u = k E E zu ¶z2 that is : Ekman currents Ekman balance : ¶ 2u f k × u = k E E zu ¶z2 that is : k ¶ 2v u = zu E E f ¶z2 k ¶ 2u v = − zu E E f ¶z2 ¶uh τ = r0kzu ¶z 0 with τ the surface wind stress. -
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. -
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.