Winter 2014
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TuTh 10:30-11:50: Lectures in Gould 436
We 1:30-2:20: Lab demonstrations in GFD Lab), OSB 107, by Peter Rhines (rhines@uw.edu)

Instructor: Prof. Chris Bretherton breth@uw.edu ATG 704, x5-7414 Office hours: MF 1:30-2:20, or by appointment. Course description Prerequisites Syllabus Textbook Grading Schedule Homework and Exams Lecture notes Animations Lab descriptions Matlab scripts Course Description Dynamics of rotating stratified fluid flow in the atmosphere/ocean and laboratory analogues. Equations of state, compressibility, Boussinesq approximation. Geostrophic balance, Rossby number. Poincare, Kelvin, Rossby waves, geostrophic adjustment. Ekman layers, spin-up. Continuously stratified dynamics: inertia gravity waves, potential vorticity, quasigeostrophy. Prerequisites A course in basic fluid mechanics, such as AMATH 505/ATMS 505/OCEAN 511 Learning objectives Fluid flow in the atmosphere, ocean, sun, and many other geophysical and engineering systems is stratified and/or rotating, and may be very slow and broad in horizontal scale. These features can lead to fascinating and unexpected behaviors that require special approaches to understand, but also can lead to powerful simplifying approximations to the mathematical governing equations. We will develop an understanding of rotating and stratified fluid flow using strategies learned in introductory fluid dynamics, (1) scale analysis to simplify the governing equations for particular situations, (2) studying linear wave motions, (3) learning how to reason with vorticity, and (4) observing fluids in the lab, videos and computer animations. At the end of the course, you should understand when ambient rotation and fluid stratification are important, what Coriolis acceleration, hydrostatic and geostrophic balance, and potential vorticity are, understand the concept of effective gravity and in what sense the shallow water equations are a reasonable analogue to continuously stratified flows, and be able to describe and find natural examples of important wave types such as inertia-gravity, Rossby and Kelvin waves. Textbook Scanned course notes will be provided, and no textbook is required, but the following excellent textbook is recommended: Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics, Cambridge Univ. Pres. (More comprehensive than needed for this class but a good reference for later) Other useful texts: Gill, A.E., 1982: Atmosphere-Ocean Dynamics. Academic Press (old, but particularly good for gravity waves) Cushman-Roisin, B., 1994: Introduction to Geophysical Fluid Dynamics, Prentice-Hall, 320 pp. (a good basic treatment) Pedlosky, J., 1979: Geophysical Fluid Dynamics (2nd Ed.). Springer-Verlag (good discussions of vorticity, PV, and quasigeostrophic scaling). Syllabus Governing equations; rotation, Boussinesq and hydrostatic approximations, pressure coordinates, primitive equations on a sphere Rotating linear SWE on an f-plane. Rossby adjustment problem. Potential vorticity. Poincare waves, Kelvin waves. Flow over a ridge. Rossby waves on a beta plane. Quasigeostrophy. Ekman layers, Ekman pumping, and Sverdrup transport. Linear internal inertia-gravity waves in a continuously stratified fluid. Mountain waves. Grading Homework (50%), posted to class web page and assigned on a quasi-weekly basis. Collaboration with your fellow students is encouraged, but everyone needs to write out the homework in their own words, and with their own Matlab scripts where necessary. Late homework (after 5pm on due date) accepted only by arrangement with instructor. Midterm (20%), in-class, one page of hand-written notes, Tu 11 Feb. Final (30%), take-home, open book and note, handed out Th 13 Mar, due We 19 Mar by 5 pm, in Bretherton's mailbox in ATG410, or under his office door in ATG 704. Special days Tu/Th 7/9 Jan: No class: Instructor at meeting in LA Mo 27 Jan: Makeup class, ATG 310c, 12:30-1:20 Mo 3 Feb: Makeup class, ATG 310c, 12:30-1:20 Tu 11 Feb: In-class midterm (Instructor in Boston). Th 13 Mar: Last day of class; take-home final posted. Homework and Exams Item Due Date Download Solutions Homework 1 due Th 23 Jan HW1 Solutions Homework 2 due Th 30 Jan HW2 solutions Homework 3 due Th 6 Feb HW3 solutions Homework 4 due Fr 28 Feb HW4 solutions Homework 5 due Tu 11 Mar HW5 solutions Take-home final due 5 pm We 19 Mar Final solutions Lecture notes Lecture 1 (1/14): What is GFD? Lecture 2 (1/16): Rotating reference frame and Coriolis force; inertial oscillations Lecture 3 (1/16,21): Boussinesq approximation Lecture 4 (1/21,23): Hydrostatic approximation and pressure coordinates Lecture 5: (1/23,27) Hydrostatic primitive equations on a rotating sphere; geostrophic and thermal wind balance. Lecture 6: (1/27,28) Shallow water equations (SWE); potential vorticity equation. Lecture 7: (1/28,30) Linearized SWE; Poincare waves Lecture 8: (2/3) The dam break problem and Rossby adjustment Lecture 9: (2/4) Solution of 1D LSWE IVP using Fourier analysis and discrete Fourier transform; dam break example. Lecture 10: (2/6) Two-layer LSWE; barotropic and baroclinic modes Lecture 11: (2/13) Boundary-trapped and equatorial Kelvin waves Lecture 12: (2/18) LSWE energy equation; KE/APE partitioning Lecture 13: (2/20,25) LSWE Rossby waves Lecture 14: (2/27) Bottom Ekman layers; spindown Lecture 15: (3/4) Top Ekman layers; Sverdrup transport Lecture 16: (3/4,6) Standing Poincare waves over topography Lecture 17: (3/6) Inertia-gravity waves in a continuously stratified fluid Lecture 18: (3/11) Upward-propagating inertia-gravity waves over a mountain ridge Lecture 19: (3/13) Potential vorticity in continuously stratified fluids Rhines GFD lab images (page down to section "Postings for GFD-1 lab demos") Animations and Videos Coriolis force. Linear shallow water equations Slab-symmetric (no y-variation) LSWE examples on an f-plane. Colors= v velocity (red = away, blue = toward you). Time is in units of 1/f. Monochromatic right-moving Poincare wave with wavenumber kR = 1. Note that fluid moves in wave direction (u>0) under crests, opposite wave direction in troughs, and is turned toward you by Coriolis forces after a crest passes, away from you after a trough passes. Also note how horizontal convergence and divergence support the free surface height changes. Black points are material points halfway up the fluid columns. Short Poincare wavepacket catching up to long-wavelength Poincare packet with slower group velocity . Notice the effect of dispersion; waves grow in the rear of the packet, propagate forward through it, and dissipate in front because their phase velocity exceeds their group velocity. Dambreak problem LSWE in a channel. Perspective plots with free-surface height perturbation color-shaded and arrows on bottom indicating horizontal fluid velocity vector (visualize as being like seaweed with one end anchored to the bottom and the other pulled by the current). Poincare wave in a channel Counter-propagating Kelvin waves in a channel (3.5MB) Geostrophic adjustment of a Gaussian hump in mid-channel. Geostrophic adjustment of a Gaussian hump near channel edge, which produces a Kelvin wave. TOPEX view of oceanic equatorial Kelvin waves(10 MB) during 1997-2001 using satellite altimetry that sensitively measures the average sea-surface height to within a few cm. The waves are seen as rapid eastward-propagating pulses of changed sea-surface height along the equator; a prominent Kelvin wave is right at the beginning. If you look carefully when a Kelvin wave hits the S American coast, you can often see coastally-trapped Kelvin waves flit poleward in each hemisphere. Slower changes are associated with the evolution of El Nino and midlatitude ocean circulations. Quasigeostrophic (Rossby wave) adjustment in 1D LSWE on an beta-plane. Colors= v velocity (red = away, blue = toward you), which is geostrophically balanced with the free surface height gradients. Time is in units of 1/(beta*R), which is typically much longer than 1/f. Long and short wavelength Rossby wave packets with carrier wavenumbers k = 0.5/R and 2/R respectively. The long wave packet has westward phase and group velocity; the short wave packet has slower westward phase velocity and eastward group velocity. Dispersion of a geostrophically balanced hump. With the beta effect, a hump will disperse into long Rossby waves, which have westward group velocity so are seen west of the original hump position, and shorter Rossby waves, which have eastward group velocity and hence are seen east of the original hump position. Matlab scripts Matlab scripts relevant to class material: Dambreak problem (uses waveplot.m and makemovie.m). Makes animation showing only the left half of the periodic domain used to construct solution with FFTs. Note Poincare waves propagating away from the dambreak, with short waves having fastest group velocity, and geostrophically balanced steady state left behind preserving the initial PV distribution. Matlab scripts used in homework solutions: rossby waveplot(0) movie(RPmovie,5,2) generated with makemovie('waveplot',(0:1:30)); rossbyhump waveplot(0) movie(RHmovie,5,2) generated with makemovie('waveplot',(0:0.4:8)); hw1p2b_soln.m: Damped inertial oscillation hw1p3_soln.m: Mixed layer densification hw2p2b_soln.m: Response to linearly ramped wind stress.