ATM S 509/OCEAN 512     SLN: 1548 (ATM), 6180 (OCN)

Tues/Thurs 10.30-11.50, and labs at 1.30 Wednesday, room 107 Ocean Sciences
Lectures in room 205 Ocean Teaching Building (OTB), labs in Ocean Sciences 107, the GFD lab .

Geophysical Fluid Dynamics - I - Winter 2015


Professor P.B.Rhines
Ocean Sciences Building 319
tel: 543-0593
office hours: after class, and by appointment.

Teaching Assistant:

Miguel Jimenez (
Ocean Sciences Building, 206-543-5214.
office hours: to be determined.

Homework & quizzes Bulletin Board Grades

Discussions Lecture notes Reading Assignments Homework Labs Observational Data Course description Prerequisites Outline (syllabus) Textbook Links

Mt. Rainier with wave clouds



This week we will study Rossby waves. In a real sense they are the end-game of GFD-1, where ideas of wave propagation meet potential vorticity (PV) head-on. Until now we have looked at PV at small scale (for example in the geostrophic adjustment of a small 'blob' of fluid released with the fluid initially at rest, or in flow over a single mountain). If instead, there are 'background' variations in PV over a large region (i.e., the curvature of the Earth or the slope of the solid Earth topography), these make possible an entirely new kind of long wave motion, at sub-inertial frequency (σ < f). The two kinds of Rossby wave, one due to a bottom slope as in the coastal ocean and the other due to the Earth's spherical shape are described in Gill sections 10.12, 12.1 - 12.3 which are the next reading assignment.


Problem set 4 is now due on Tues 3 Mar rather than Monday. For reading, review the internal wave discussion using wavenumber space diagrams to see the group velocity graphically, transferring these to physical space where you can sketch the wave pattern.

In discussion PS4 we suggested setting Coriolis frequency f to zero to simply the calculation. Then say the vertical energy flux by internal waves is E Cgz, the product of energy density E and vertical group, velocity, Cgz. Assume this to be independent of frequency. Then calculate the dependence of E on the frequency σ by expressing Cgz in terms of k and σ. For simplicity say k = constant.

The classic 'Kelvin' ship wave problem described in class is the solution to (σ -Uk) = (g|k|)1/2 = g1/2(k2 + l2))1/4 with the choice σ = 0; stationary short surface gravity waves in the frame of reference moving with the ship. You can sketch the full ship-wave pattern, with its two families of waves, by plotting this curve in wavenumber (k,l) space, and the group velocity vectors normal to the curve. The dependence of the wave pattern on Froude number U2/gL comes out naturally when you do. Two extreme Froude number 'experiments' are shown here, from a small plane hovering over Puget Sound when we were looking for internal gravity waves. Zoom in to see what kind of boats they are.

Lecture notes on internal waves from week 7 and Tuesday of this week are posted below, including a few figures from the GFD lab, theory and observations. A new section was added Thursday following our class discussion on 'thinking in wavenumber space'.

Problem set 4 is posted below, along with an m-file that solves the internal wave propagation from a small source region.

We have just begun internal waves with and without rotation. See the reading assignment which now involves both chapters 6 and 8 of Gill's text. As in class, Gill shows the way waves and potential vorticity conserving circulation appear in our 'master equation' (in his Table 8.1, p261). Think of this linearized equation as giving us the complete solution for simple flows and the early stage of evolution of more complicated flows (nonlinear, maybe turbulent). The observed oceans and atmosphere exhibit both the simple and the complex.

While our treatment is mostly for an incompressible, stratified fluid, be aware of the effects of compressibility and non-Boussinesq effects: that is, fluids whose density ranges widely (like the atmosphere) rather than narrowly (like the ocean). The latter sections in chapter 8 (8.9-8.16) are not fully required in this course but you will want to be aware of them: waves propagating through non-constant stratification and non-constant wind fields. Waves propagating through non-uniform fluids are one of the most beautiful topics in GFD theory.

Problem set 4 is imminent and I will email you when it is ready. -PBR Fri 20 Feb

above: surface temperature anomaly, 20 Feb 2015; internal wave lab

Problem set 3 (due Weds.) has a few comments added in blue, regarding testing thermal wind balance against the observations. Note that the circulation model for the NCEP reanalysis has a bit more resolution in the vertical than the 17 pressure levels of our data. That is why your testing of thermal wind balance cannot be as accurate as the model output, which uses a T62 lateral grid with 28 sigma levels (terrain-following coordinate surfaces) in the vertical. T62 resolution is roughly 2.5 degrees, which misses much detailed dynamics particularly at high latitude and near extreme mountain terrain. The T799 model runs I showed in class (with gravity waves and downslope winds over Greenland) correspond to about 25km lateral grid resolution. More detail is in Jung & Rhines, JAS 2007

This week we are doing examples of development geostrophic flows with heating and topography, using the 'master' equation for pressure p' and potential vorticity equation. The other branch of the equation is internal waves. The reading for these waves is posted below.

Tuesday we will try an in-class problem involving the response of a uniformly stratified (initial N=constant) , rotating fluid to a compact (small) heat source. Think about the PV distribution caused by heating the stratified fluid suddenly, before any flow develops: this is the key to predicting the geostrophic flow.

Lecture notes for week 7 on internal waves will be posted today.

If all goes well, in the GFD lab Wednesday we will do a quick geostrophic adjustment problem in a 1 1/2 layer model, and then internal gravity waves, which is a memorable experiment.


Problem set 3 is rather open-ended, asking you to explore NCEP data and the thermal wind balance related to zonal winds and developing cyclonic eddies. Hovmoeller plots (contour plots of, for example, 250HPa meridional wind with axes longitude and time, at one latitude) are particularly interesting. For spatial maps, the m_map set of m-files provides good sense of the geography, as in the figure below (250 HPa height contours and 1000 HPa temperature from winter 1993; note relation with 20Feb2015 surface temperature anomaly, shown above).

Solutions to the quiz problems are posted under 'homework'.

Wei points out a sign error in the notes for Week 5: the thermal wind equations are correct up to p.4 but there it should read
δu/δp = -1/ρ^2 δρ/δy
δv/δp = +1/ρ^2 δρ/δx . You can figure out these signs quickly by imagining a jet stream flowing eastward along the polar front.

Miguel's office hours are Mondays 1.00-3.00 pm and Tuesdays 2.00-5.00 pm, and he or I can talk immediately after class or lab periods.

Lecture notes and problem set 3 for week 6 are posted below

For the problems working with NCEP data, where you will be calculating derivatives like du/dz, use a finite difference approximation Δu/Δz. Since u is stored on pressure levels rather than z-levels, you can use the dynamic height z(p) to give Δz or can use du/dz = du/dp x dp/dz which is -(du/dp)(gρ) =-(du/dp)(gp/RT).

We meet Weds. 1.30 in the teaching lab, OTB 206 just across from the classroom. Bring a laptop or a friend with a laptop. Power is available.

We have posted NCEP reanalysis data for the global atmosphere below under 'observational data'. There are 4 times daily full global data for 1Nov2014-1Feb2015, daily mean data for the same interval, monthly mean data from 1996-2015 and the same for just 13 months May2003-May2004. A short m-file with sample plotting routines is also posted (ncep_plots_gfd1_2015.m). This describes a mapping toolbox available for free download to give us good map projections (M_Map).

More later, including MIMOC ocean data.

Lecture notes for Week 5 are now posted below, with small updates in blue. They augment what was done in class and in Gill's text. We will be working on stratified rotating dynamics, with the full potential vorticity equation. The 'master equation' includes both internal gravity waves and geostrophic flows with vorticity; together these give us the full 3-dimensional adjustment to a balanced, geostrophic flow. Internal gravity waves (with or without rotation) will then be our next focus. The reading encompasses all of Gill's Ch. 7 and then will jump back to Ch. 6.4-6.8


Solutions for problem set 1 and 2 are posted below.

The lecture notes from week 4 have been revised with some added comments (in blue), posted below under Lectures; this was done on 2 Feb.

This week's Weds. lab will be mostly a single experiment with a stratified, rotating flow to see thermal wind balance. We will try to use some of that period for quiz review.


In response to questions I've posted some notes on problems 2.7,2.8 of problem set 2 below (under Homework).

Our mid-term quiz will tentatively be handed out on Thurs 5 Feb as a take-home, to be returned Friday by noon. We will discuss possible conflicts in class this week.

Week 4 lecture notes for 1-layer geostrophic adjustment are posted below. This parallels Gill's treatment in Ch. 7 so far.

A Matlab m-file is posted under Lecture notes, which animates the 1-dimensional (η(x,t),u(x,t),v(x,t)) adjustment to an unbalanced initial height field (u=0=v at t=0). You can rewrite the initial η-field to mimick problems 2.7 and 2.8. A similar code of mine also plots the paths of fluid particles (seen from above): see figures at end of lectures 7,8 which show how near-inertial oscillations occur in the gravity/inertial waves emerging from the adjusting flow.


Reading for week 4 and Problem set 2 are posted below. I retooled it because we have not fully described geostrophic adjustment in class; I put a due date of Friday 30 Jan to give an extra day (if you can email or deliver it then; if logistically difficult see me).

Again we've asked you for this Thursday 22 Jan to write on one page (2 sides) notes on the readings for Coriolis effects without density stratification, mostly. This comes from reading and lectures and labs. Reading, posted below is mostly the same in Ch. 7 as assigned last week, but with two sections 7.9, 7.10 added (which are mostly review material). It would be good to include questions, impressions and possibly an example applying these rather abstract ideas to atmosphere and ocean flows, more than repeating equations from the text. Think about raising a question in class Thursday from your reading etc.

The 2d and 3d sets of lecture notes are now posted below.

Lab 3 this week will be on geostrophic flow and its development when the fluid is forced into motion in some way. Given that Coriolis effects push the fluid at right angles to its horizontal velocity, the pressure force that allows the fluid to break out of this constraint needs thought.


Problem set #1 is posted below (under 'homework'). In problem 1 an error has been corrected (in red). Due Tues 20 Jan.
We will continue spending some class time working out problems. This Thursday try to bring questions based on this week's reading (see below for that). This does not need to be handed in.

Slides from week 1 are now posted under 'lectures' below, and a typo was corrected in Week 1 lecture notes (in red font).
The grading policy is now shown under 'Grading' below.
This week's reading is posted below: continuing with basic equations and beginning Coriolis effects of the rotating Earth. In wlectures we will say some more about heat engines and convection, and the Earth's energy balance and then work on rotating Earth GFD.
A problem set is forthcoming, and we will try to do some of it during classtime.
Our 2d lab on Weds. 14 Jan (1.30-2.20) will be about Coriolis effects, angular momentum and geostrophic balance.


We scheduled the first lab for Wednesday 7 Jan at 1.30-2.20 in the GFD lab. Images and notes from this lab are posted below (linked under 'Labs' at the top of this page).

Lecture notes for Week 1 are posted below also (under 'Lectures'). These parallel the sections in Gill on thermodynamics and equations of state. Much of the detail has not been given in class, but this is meant to add extra ideas to your developing ideas of the thermal aspects of atmospheres and oceans.

With the two lecture meetings per week, 1.5 periods long, we will try to dedicate some time to in-class discussion. This week's 'assignment' is below under 'discussions'.

Observations of large-scale circulations (jet stream, storm track, cyclonic development (at the 1000km scale), ocean eddies and boundary currents) and smaller scale waves (internal ocean tides, which are internal gravity waves influence by Earth's rotation) were examples of GFD in action.

The weather loops are a good place to spend time: we looked at cold air outbreaks in the central US (this week's weather), which involve strongly developing waves in the jet stream, a low-pressure trough carrying frigid air to Texas, east of the Rocky Mountains. Rossby waves in the ocean take several different forms, one being the westward marching mesoscale eddies (~100km diameter) which are highly nonlinear waves: these we saw in satellite altimetry videos of the surface ocean currents of the Atlantic. The newly established global observing system for the oceans provides 'ground truth' for newly capable numerical models of the ocean circulation.

The mathematics used in GFD is important, yet in some cases the equations can be very simple, one example being the wave equation for Rossby waves/mesoscale eddies in the upper few hundred m of the oceans, where the same equation also describes the wind-driven gyre circulation of the ocean (the Sverdrup transport).

Thermodynamics is important yet often neglected in GFD textbooks. Of course there are thermodynamics books..a classic series being by Francis Sears (in library or Amazon). Dennis Hartmann's Global Physical Climatology is a very good introduction to atmospheric circulation related to thermodynamics, moist and dry, and radiation. We don't have time for more than the brief introduction and one lab unfortunately, but I hope the important ideas of heat engines, 1st- and 2d law of thermodynamics introduced there will help in understanding the buoyancy effects of GFD.

On this website lecture material will be posted, which gives an independent treatment of GFD in parallel with Gill's (and Vallis') textbook sections.

First class: 10.30 Tuesday Jan 6, 2015 in Room 205 Ocean Teaching Building.

We will ask you to describe:

  • Your fluid dynamics background (courses etc)
  • Your math background
  • Your Matlab experience level (likely greater than ours!).
  • What would you like to get out of this GFD course?


    Atmosphere-Ocean Dynamics by Adrian Gill (Academic Press 1982) is our primary text. We have a 'suggested' text, Atmospheric and Oceanic Fluid Dynamics by Geoffrey Vallis (Cambridge University Press, 2006), which is not required. Vallis' text is newer and includes many modern topics, particularly involving vorticity dynamics of synoptic-scale flows. We will give a list of useful sections in Vallis that parallel our lectures and Gill's text.

    There are in addition other fluid dynamics and GFD textbooks and each has its merits:

    • An Introduction to Dynamical Meteorology by James Holton and Greg Hakim (5th Ed., Academic Press 2013).
    • An Introduction to Geophysical Fluid Dynamics - 2d Edition: Physical and Numerical Aspects by Benoit Cushman-Roisin & Jean-Marie Beckers (Internat'l Geophysics 2011).
    • Geophysical Fluid Dynamics by Joseph Pedlosky (Springer Verlag),
    • sections in Fluid Dynamics by Pijush Kundu and Ira Cohen (4th Edition, Academic Press),
    • Lectures on Geophysical Fluid Dynamics by Rick Salmon (Oxford University Press),
    • Introduction to Circulating Atmospheres by Ian James (Cambridge University Press 1994),
    • Global Physical Climatology by Dennis Hartmann (Academic Press);
    • Waves in Fluids by James Lighthill (Cambridge University Press, 1978),
    • An Informal Introduction to Theoretical Fluid Mechanics by James Lighthill (Clarendon/Oxford University Press 1986),
    • Fluid Mechanics, 2d Edition by L.D. Landau and L.M. Lifshitz (Butterworth-Heinemann div or Reed Publishing, Ltd. 1959-2000),
    • Atmosphere, Ocean and Climate Dynamics, an Introductory Text by John Marshall and Allan Plumb (Elsevier Academic Press, 2008);
    • Fundamentals of Atmospheric Physics by Murray Salby (Academic Press, 1996).
    • Thermodynamics, while not a major activity in this course, is important. An excellent text is Thermodynamics, Kinetic Theory and Statistical Thermodynamics, by Francis Sears and Gerhard Salinger.
    • On the same subject, an introduction by a Nobel laureate physicist is Thermodynamics by Enrico Fermi, (Dover!)

    Vallis' text is available as a .pdf for your laptop or Kindle or IPad, for $80 here. 

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    Reading Assignments

    Lecture notes
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    • Thurs 8 Jan
      Write down (on no more than one piece paper) notes on your impressions, questions, examples relating to the following. Add notes to it during class and hand in as you leave. These could be bullet points or descriptive paragraphs. Better to concentrate on one or two than do all three.

      In reading Gill's sections in Ch.3 and 4 on thermodynamics and the basic equations of motion, prepare questions for discussion in 3 areas:
      o Energy equations, particularly focusing on how the internal thermodynamic energy (1st law of thermo) and the external mechanical energy (KE and PE, kinetic and potential energy) interact. How do they interact, and what are examples in atmos and ocean?
      o The geopotential field: what is it and how does it combine true gravity and effects of being on a spinning planet?
      o Potential temperature: how is it derived and how does it appear in equations of state for an ideal gas?

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    Observational Data

    GFD Labs: Winter 2015

    These links above are under construction. Meanwhile to see some images from similar labs in earlier incarnations of GFD1, see past GFD1 webpages linked here
    for example: (internal gravity waves)

    In the UW teaching program We do several varieties of hour-long lab demonstrations, as well as term projects: for graduate courses in GFD-1, GFD-2, Waves, Stirring and Mixing, and others; for undergraduate courses in Oceanography, Earth & Space Sciences, Atmospheric Sciences and global environment. Here is a general outline of GFD-1 labs developed at UW over three decades; details vary year-to-year. See other class websites for other demo offerings.