Essay 1: Dimensional Analysis of Models and Data Sets: Similarity Solutions

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1 Table of Contents Essay 1: Dimensional Analysis of Models and Data Sets: Similarity Solutions and Scaling Analysis 1 About dimensional analysis Thegoalandtheplan Aboutthisessay Models of a simple pendulum Aphysicalmodel Amathematicalmodel Modelsgenerally An informal dimensional analysis Invariancetoachangeofunits Naturalunits Extraandomittedvariables A basis set of nondimensional variables Themathematicalproblem Thenullspace A basis set for the simple, inviscid pendulum The viscous pendulum A physical model of the viscous pendulum Dragonamovingsphere Zeroordersolution The other nondimensional variables: remarks on the Reynolds number Anumericalsimulation An approximate model of the decay rate A similarity solution for diffusion in one dimension Honingthephysicalmodel Asimilaritysolution Scaling analysis Anonlinearprojectileproblem Small parameter! smallterm? Scalingthedependentvariable Approximate and iterated solutions Summary and closing remarks 39

2 Essay 2: The Coriolis force 1 Large-scale flows of the atmosphere and ocean Classical mechanics observed from a rotating Earth The goal and the plan of this essay About this essay Part I: Rotating reference frames and the Coriolis force Kinematics of a linearly accelerating reference frame Kinematics of a rotating reference frame Transforming the position, velocity and acceleration vectors Stationary Inertial; Rotating Earth-Attached Remarks on the transformed equation of motion Inertial and noninertial descriptions of elementary motions Switching sides To get a feel for the Coriolis force Zero relative velocity With relative velocity An elementary projectile problem From the inertial frame From the rotating frame Appendix to Section 3: Circular motion and polar coordinates A reference frame attached to the rotating Earth Cancelation of the centrifugal force Earth s (slightly chubby) figure Vertical and level in an accelerating reference frame The equation of motion for an Earth-attached frame Coriolis force on motions in a thin, spherical shell Why do we insist on the rotating frame equations? Inertial oscillations from an inertial frame Inertial oscillations from the rotating frame A dense parcel on a slope Inertial and geostrophic motion

3 1 LARGE-SCALE FLOWS OF THE ATMOSPHERE AND OCEAN Energy budget Part II: Geostrophic adjustment and potential vorticity The shallow water model Solving and diagnosing the shallow water system Energy balance Potential vorticity balance Linearized shallow water equations Models of the Coriolis parameter Case 1, f = 0, non rotating Case 2, f = constant, an f-plane, Case 3, f = f o + βy, a β-plane, Beta-plane phenomena Rossby waves; low frequency waves on a beta plane Modes of potential vorticity conservation Some of the varieties of Rossby waves Summary of the essay Supplementary material Matlab and Fortran source code Additional animations

4 Essay 3: Lagrangian and Eulerian Representations of Fluid Flow: Kinematics and the Equations of Motion 1 The challenge of fluid mechanics is mainly the kinematics of fluid flow Physical properties of materials; what distinguishes fluids from solids? The response to pressure in linear deformation liquids are not very different from solids The response to shear stress solids deform and fluids flow A first look at the kinematics of fluid flow Two ways to observe fluid flow and the Fundamental Principle of Kinematics The goal and the plan of this essay; Lagrangian to Eulerian and back again The Lagrangian (or material) coordinate system ThejoyofLagrangianmeasurement Transforming a Lagrangian velocity into an Eulerian velocity The Lagrangian equations of motion in one dimension Mass conservation; mass is neither lost or created by fluid flow Momentum conservation; F = Ma in a one dimensional fluid flow The one-dimensional Lagrangian equations reduce to an exact wave equation The agony of the three-dimensional Lagrangian equations The Eulerian (or field) coordinate system Transforming an Eulerian velocity field to Lagrangian trajectories Transforming time derivatives from Lagrangian to Eulerian systems; the material derivative Transforming integrals and their time derivatives; the Reynolds Transport Theorem TheEulerianequationsofmotion Mass conservation represented in field coordinates The flux form of the Eulerian equations; the effect of fluid flow on properties at a fixed position Momentum conservation represented in field coordinates Fluid mechanics requires a stress tensor (which is not as difficult as it first seems) Energy conservation; the First Law of Thermodynamics applied to a fluid A few remarks on the Eulerian equations Depictions of fluid flows represented in field coordinates Trajectories (or pathlines) are important Lagrangian properties Streaklines are a snapshot of parcels having a common origin Streamlines are parallel to an instantaneous flow field Eulerian to Lagrangian transformation by approximate methods. 60 6

5 5.1 Tracking parcels around a steady vortex given limited Eulerian data The zeroth order approximation, or PVD A first order approximation, and the velocity gradient tensor Trackingparcelsingravitywaves The zeroth order approximation, closed loops The first order approximation yields the wave momentum and Stokes drift Aspects of advection, the Eulerian representation of fluid flow The modes of a two-dimensional thermal advection equation The method of characteristics implements parcel tracking as a solution method A systematic look at deformation due to advection; the Cauchy-Stokes Theorem Therotationratetensor Thedeformationratetensor The Cauchy-Stokes Theorem collects it all together Lagrangian observation and diagnosis of an oceanic flow Concluding remarks; where next? 86 9 Appendix: A Review of Composite Functions Definition Rules for differentiation and change of variables in integrals ;

6 MIT OpenCourseWare Resource: Online Publication.Fluid Dynamics James Price The following may not correspond to a particular course on MIT OpenCourseWare, but has been provided by the author as an individual learning resource. For information about citing these materials or our Terms of Use, visit:

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