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Theoretical Oceanography Lecture Notes Master AO-W27

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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 ................................ 61 4.2 Surfacegravity waves in thenon-rotatingocean . .... 65 4.2.1 Thedeepwatercase .......................... 65 4.2.2 Theinfluenceofthebottom . 69 4.2.3 TheStokesdrift............................. 71 4.2.4 Dispersion of surface gravity waves . 73 4.2.5 Theenergybudgetofwaves . 73 4.2.6 Windwavesspectra........................... 76 4.2.7 InterneSchwerewellen . .. .. 79 4.3 Theshallowwaterequations .. .. .. 84 4.4 Wavesestablishingflowfields . 90 5 Shallow water waves on a rotating planet 93 5.1 The shallow water equations for the f-plane . .. 93 5.1.1 Thesteadycase-geostrophicflow. 95 5.1.2 More conservation laws - potential vorticity . 96 5.1.3 The linearised shallow water equations . 97 5.1.4 An application - the Rossby adjustment problem . 100 5.2 Wavesinachannel ...............................109 5.2.1 A meridionally oriented channel, fplane approximation . 109 5.2.2 Crosschannelslopingbottom . 116 5.3 Waves on the β-plane..............................121 5.4 Equatorialwaves ................................126 A Wind driven currents - details 133 Bibliography 136 0 2 Chapter 1 Literature recommendations This script is not a textbook, but shall serve as a guide into the field of theoretical oceanography. The discipline in the foreground is physics. We demonstrate how the complex phenomena of oceanic flow can be derived and understood from basic laws of hy- drodynamics and thermodynamics. Hence, textbooks on “descriptive oceanography” are of limited help for this purpose, but may serve as an excellend supplement, since a detailed description of observations cannot be given in the limited frame of this lecture. Here the book Descriptive Physical Oceanography of L. Talley et al. [7] is strongly recommended. This book provides impressive insight in the beauty of our earth not to be gained with pure eyes. Also the book of M. Tomczak and J. Godfrey [8] gives a good overview. There is also a web based version, http://www.es.flinders.edu.au/ mattom/regoc/pdfversion.html.2 The lecture does not follow a specific textbook.∼ Many sections are inpired by the recently published book Ocean Dynamics of D. Olbers, J. Willebrand and C. Eden, [6]. The analytical theory based on formal solutions of linear partial differential equations (Green functions) follows W. Fennel and H. U. Lass, [2]. Stimulating ideas but also some of the figures are from previous lectures of Wolfgang Fennel. The books of - G. Vallis, Atmosphere and Ocean Fluid Dynamics, [9], - J. Apel, Principles of Ocean Physics, [1], - A. Gill, Atmosphere-Ocean Dynamics, [3], - P. Kundu, Fluid Mechanics, [4] - J. Lighthill, Waves in Fluids, [5] are also to be recommended. This is only a minor selection. For a deeper understanding of the principles of small scale hydrodynamics, see any of the numerous textbooks on theoretical physics. 3 4 CHAPTER 1. LITERATURE RECOMMENDATIONS Chapter 2 Basic equations The fluid water in the ocean or the gaseous air are continuous deformable bodies. Here we ask for macroscopic properties and mostly we do not consider the microscopic fun- damentals. So water and air have given thermodynamic properties like density, salinity, compressibility, specific heat or viscosity. This suggests that the field of thermodynamics should be a basic ingredient of theoretical oceanography. Water in the ocean is moving and we will base our consideration on the appropriate theory, the field of hydrodynamics. Both branches of theoretical physics will not be introduced here, but only those essen- tially needed to understand ocean or atmosphere physics will be repeated shortly on an introducing level. 2.1. Flow kinematics This section introduces the basic kinematic variables suitable to describe the motion of a fluid. We use mostly geophysical coordinates on a sphere or a local Cartesian coordinate system. Eu- lerian and Lagrangian representation are introduced. The Lagrangian view is more appropriate when considering the spreading of water masses within the ocean, the Eulerian view is mathe- matically simpler for most applications, especially for numerical ocean or atmosphere models. 2.1.1. Coordinate systems For two persons who want to meet somewhere on the earth it should be sufficient to exchange the coordinates of the meeting point and to find a way to navigate to this place. As long as they meet at the solid or fluid surface of the earth, the two geographical coordinates longitude, λ, and latitude, ϕ, are sufficient to arrange a meeting. If diving into the ocean or flying into the air is allowed, a vertical coordinate z is needed too. Not to forget that both should be on the spot at the same time t. Assuming that longitude λ is positive eastward, latitude ϕ is positive northward and height z orients positive upward, the three spatial coordinates form a right handed system. In oceanography it is common to count z negative below the sea surface. 5 6 CHAPTER 2. BASIC EQUATIONS So simple this concept seems to be so complex it is. The earth is not a sphere but is deformed from inhomgeneous mass distribution, earth rotation and tides. The ocean sur- face is mostly elevated from its equilibrium position by winds and non-uniform heating. At longer time scales the average sea level is changing, continents are drifting and the geophysical coordinate system is well defined only by a complex procedure. Such a gen- eral coordinate system is not suitable for an introduction into basic theoretical concepts. Hence, for global considerations we approximate the earth by a sphere which is sufficient for this lecture. In most cases even a local Cartesian co-odinate system x = (x, y, z) is used. It should be mentioned that other systems than geographical coordinates may be useful for numerical models of the ocean or the atmosphere. Especially the depth z as vertical coordinate may be replaced by another coordinate, for example the relative depth in relation to the total depth or by the water density. Advantages and disadvantages of a special choice is beyond the scope of the lecture. 2.1.2. Eulerian and Lagrangian representation Euler 1 and Lagrange 2 developed a complementary approach to describe the motion of fluids. Lagranges approach considers fluid elements, marked by a vector of properties, a. It is located at position x, fluid motion is the change of its position with time, x = x(a, t). (2.1) A possible way to mark a fluid element is its position at time t0, x(a, t0)= a. (2.2) The Lagrangian velocity a is ∂ uL(a, t)= x(a, t). (2.3) ∂t In turn the position of the fluid elements at time, t, can be found if the velocity at all times before t is known, t ′ L ′ x(a, t)= x(a, t0)+ dt u (a, t ). (2.4) Zt0 Note, uL is solely a function of time! Hence, the varying set of positions of fluid elements in principle describes the fluid motion. 1Leonhard Euler (* 15. April 1707 in Basel; 7. September 1783 in Sankt Petersburg. Swiss mathemati- cian and physicist. Basel, Berlin, Sankt Petersburg.† 2Joseph-Louis de Lagrange (* 25. January 1736, Turin (Giuseppe Lodovico Lagrangia); 10. April 1813, Paris. Italian Mathematician. Turin, Berlin, Paris. † 2.1. FLOW KINEMATICS 7 This form is of little use for practical applications. Marking a fluid element is diffi- cult (aspects of quantum statistics that elementary particles are indistinguishable need consideration). Much simpler is the idea to consider the velocity of the fluid as function of position, x, and time t, u = uE(x, t). (2.5) This velocity field is called Eulerian velocity. With modern ADCPs or LASER-Doppler anemometers snapshots of the velocity field can be gained. Both representations of motion are equivalent. At time t a fluid element marked with a with x = x(a, t) is found at x. The local motion is the velocity of the fluid elements, hence, uE(x, t)= uL(a, t). (2.6) The task to find the trajectory of a fluid element, X(t), requires the solution of the partial differential equation ∂ X = uE(X, t), ∂t X(t0) = a. (2.7) The equivalence of Eulerian and Lagrangian view applies for all quantities, say A, AE(X(a, t), t)= AL(a, t). (2.8) The time tendency of A within the moving fluid element reads ∂AL(a, t) ∂AE(X, t) ∂AE (X, t) ∂X = + ∂t ∂t ∂X · ∂t X t ∂AE(X, t) = + uE(x, t) ∇ AE(X, t) .
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