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Physical Oceanography Geosci 233: Physical Oceanography Douglas R. MacAyeal Department of Geophysical Sciences University of Chicago Chicago, Illinois March 28, 2001 Figure 1: The Atrevida discovering the Aurora Islands. c 1994 Douglas R. MacAyeal. All rights reserved. 1 Contents 1 Maps 8 1.1 Three Classes of Maps . ..................... 8 1.2 Spherical Coordinates . ..................... 10 1.3 Map Projections ......................... 12 1.4 Polar Stereographic Projection . ................ 12 1.4.1 Lambert Equal-Area Projection . ............ 13 1.4.2 Mercator’s Projection . ................ 15 1.4.3 Hammer-Aitoff and Mollweide Projections . .... 19 2 Shape of the Ocean Basins 25 2.1 Sea-Floor Spreading and Continental Drift . ........ 26 2.1.1 Kelvin’s Solution for the Cooling of the Earth . .... 29 2.1.2 Laplace Transform .................... 30 2.1.3 Solution of the Subsidiary Equation . ........ 31 2 2.1.4 Inverse Laplace Transform ................ 31 2.1.5 Integrating the Bromwich Integral ............ 33 2.1.6 Geothermal Gradient . ................ 37 2.1.7 Kelvin’s Edinburgh Calculation . ............ 38 2.2 Theory of Sea-Floor Subsidence . ................ 38 2.3 A Plate Model of Oceanic Crust . ................ 42 2.4 Numerical Methods . ..................... 47 2.4.1 Finite-Difference Solution as a Matrix Algebra Problem 50 2.5 Hypsometry . ......................... 51 2.6 Assignment: Analysis of Ocean Bathymetry . ........ 54 2.6.1 Assignment: Hypsometry of the Earth . ........ 55 2.6.2 Assignment: Finite-difference model of cooling plate . 56 3 Hydrography I 59 3.1 Temperature . ......................... 60 3.2 Sea Surface Temperature ..................... 62 3.2.1 Freezing Point of Seawater ................ 62 3.3 Salinity . ............................. 63 3.4 Density . ............................. 65 3.5 Assignment . ......................... 66 3 4 Hydrography II 69 4.1 Stratification . ......................... 69 4.1.1 Available Potential Energy ................ 70 4.1.2 Buoyancy ......................... 71 4.2 A Bouncing Beachball . ..................... 73 4.2.1 Internal Gravity Waves . ................ 77 4.3 Temperature and Salinity Diagrams . ............ 82 4.3.1 Mixing . ......................... 82 4.4 Bulk Ocean Properties . ..................... 83 4.5 Exercises . ............................. 84 5 Bogus Forces 90 5.1 The Rotating Plane . ..................... 92 5.1.1 Assignment - Inertial oscillations ............ 94 5.2 Foucault Pendulum . ..................... 95 5.2.1 Assignment - Foucault pendulum ............ 96 5.3 Description of Motion at the Earth’s Surface . ........ 97 5.3.1 Assignment – Rotating earth coordinates. .... 98 6 Geostrophy 103 6.1 Equations Governing Horizontal Motion ............103 4 6.2 The Geostrophic Balance .....................105 6.2.1 “Thermal-wind” relationship . ............105 6.3 Dynamic Topography . .....................106 6.4 Assignment . .........................107 7 Ekman Pump 110 7.1 Ekman Layer: Conceptual View . ................110 7.2 Ekman Pumping .........................111 7.3 Ekman Pumping in Geographic Coordinates . ........113 7.4 Upwelling .............................114 7.5 The Ekman Spiral . .....................114 7.6 Ekman Transport .........................117 7.7 Assignment . .........................118 8 Wind-Driven Ocean Circulation 122 8.1 Two Possible Balances . .....................122 8.2 Basal Ekman Layer Only .....................123 8.2.1 Exercise: An Wind-Driven Circulation Predicted by Incomplete Physics ....................125 8.3 Basal Ekman Layer Plus Divergence of Geostrophic Flow (Sver- drup Balance) . .........................125 8.4 Exercise: A Predicted Wind-Driven Circulation That Satisfies Our Preconceptions . .....................131 5 9 Shallow Water Gravity Waves 133 9.1 Overview . .............................133 9.2 Hydrostatic Balance . .....................134 9.3 Shallow Water Equations .....................135 9.4 Shallow-Water Gravity Waves . ................136 9.5 Particle Motions .........................137 9.6 Geostrophic Adjustment .....................142 9.6.1 An analytic example . ................143 9.7 Applications . .........................144 10 Tides 146 10.1 overview . .............................146 10.2 A Conceptual View . .....................146 10.3 Tide Generating Potential ....................148 10.4 Tidal Response . .........................152 10.5 Assignment . .........................153 6 Chapter 1 Maps Maps provide a means to visually comprehend the ocean’s temperature, salin- ity and circulation. Our first step in the study of oceanography will be to consider the mathematical and graphical tools which allow us to make maps. It is easy to appreciate the main question of cartography: How is an essentially spherical body such as the earth represented on a two-dimensional map? It is impossible, for example, to peel a thin paper covering from a globe without tearing, stretching and distorting the paper. Thus, in making a map of the earth, the cartographer must accept various forms of distortion that are inherent in mapmaking. In some circumstances the cartographer will choose a map projection which minimizes or eliminates some forms of distortion (such as the magnification of area). In making such a choice, the cartographer will often be forced to compromise between various other forms of distortion (such as preservation of shape and direction). 1.1 Three Classes of Maps Our goal is to understand the geometry of the three basic classes of map projections (cylindrical, conical and azimuthal). As shown in Fig. (1.2), 7 Figure 1.1: The voyages of Captain James Cook. 8 these classes are differentiated by the type of surface onto which the map of the earth is eventually to be drawn. These so-called developable surfaces are the cylynder, cone and plane. The cylinder and cone can be cut and unrolled into a flat map. To make a map using one of the three classes of projections, we must transfer points from the surface of the earth to the surface of the map. This transformation, the map projection, can be done in one of two ways: mathe- matically, or physically (by plotting the location of shadows of points on the globe generated a small light source). While the physical map projections are easiest to comprehend, the most useful projections are often designed mathematically to produce maps with optimal distortion characteristics. In the examples below, we shall consider three projections. The polar stereographic projection is an example of an azimuthal projection involving a physical transformation of points from the globe to the plane. The Lambert equal area projection is very similar to the polar stereographic projection, but is constructed mathematically to preserve the area of closed regions on the globe (but not their shape). The Mercator projection is an example of a cylindrical projection which involves a rather intricate mathematical construction designed to preserve compass headings for mariners (i.e., is a loxodromic projection). 1.2 Spherical Coordinates The natural coordinates for the representation of position on a spherical, or near-spherical, surface are the spherical coordinates (φ, λ, r). As shown in Fig. (1.3), specification of φ and λ alone are sufficient to uniquely describe the location of the point on a sphere of constant radius r = R. We refer π π to φ as the latitude, and it falls in the range [ 2 , 2 ]. Points in the Southern hemisphere are described by φ<0. The Equator is designated as the circle of points satisfying φ = 0. We refer to λ as the longitude. It falls in the range [−π, π]. Points in the Western hemisphere (e.g., here in Chicago), satisfy λ<0. The Greenwich meridian is designated as the half-circle of points satisfying λ =0. 9 Figure 1.2: The three classes of map projections. Figure 1.3: Sphericalcoordinates. 10 The transformations between the (x, y, z)-coordinate system and the (φ, λ, r)- coordinate system involves simple trigonometry. r = x2 + y2 + z2 (1.1) z φ = tan−1 √ (1.2) x2 + y2 y λ = tan−1 (1.3) x and, x = r cos(φ) cos(λ) (1.4) y = r cos(φ) sin(λ) (1.5) z = r sin(φ) (1.6) In most of what follows, we will not make reference to the cartesian coor- dinates (x, y, z), or to the radial coordinate r. Our concern will principally involve the spherical coordinates (φ, λ), which we shall refer to as the geo- graphic coordinates. 1.3 Map Projections Even a modest scan of the reference literature on map projections (e.g., Snyder and Boxland, 1989), reveals a vast variety of map projections. One can easily become overwhelmed by this variety. Our goal is to learn about just a few projections that will help us in our study of oceanography. 1.4 Polar Stereographic Projection As shown in Fig. (1.4), the polar-stereographic map is an azimuthal projec- tion that is constructed geometrically, i.e., using an imaginary point source of light to transfer points on the globe to points on the map. The simple 11 geometrical origin of the polar-stereographic projection has made it popular for representing the polar regions. The mapping from (φ, λ)to(r, θ) is readily obtained by inspection of Fig. (1.4). First, we note that θ = λ (1.7) Next, we define the angle α which separates the rotational axis and the line segment passing through the South geographic pole and the (φ, λ)-point in question. The triangle which has the South pole, the center of the earth, and the (φ, λ)-point as vertices has two equal sides; thus, two of its angles are α. π The other angle is 2 + φ. The value of α is
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