Instructor: |
Professor P.B.Rhines Ocean Sciences Building 319 tel: 543-0593 rhines@washington.edu office hours: after class, and by appointment. |
Teaching Assistant: |
Miguel Jimenez (jimenezm@uw.edu) 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 |
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} =
g^{1/2}(k^{2} +
l^{2}))^{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 U^{2}/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
WEEK 7:
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.
WEEK 6:
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
WEEK 5:
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.
WEEK 4:
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.
WEEK 3:
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.
WEEK 2:
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.
WEEK 1:
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 atmos.washington.edu 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:
There are in addition other fluid dynamics and GFD textbooks and each has its merits:
Vallis' text is available as a .pdf for your laptop or Kindle or IPad, for $80
here.
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Reading Assignments
It would be good to review Lectures 1,2,3 from the fall Fluids course, which are very similar to the 2010 version linked below. The Fluid Dynamics website with course material from 2011 is posted here and the website from the 2014 course just completed should still be on Canvas. But it's handy to have a single .pdf file of the lectures which is linked below.
Chap. 1 of Kundu & Cohen's text has similar material to the above sections of Gill and Vallis.
Chap. 2 of Kundu & Cohen is a review of vector calculus and basic
tensor notation useful in FD and GFD. Of particular interest are the divergence
theorem and Stokes' theorem, and vector/gradient identities like those in Prof.
Bretherton's Fall Q. Fluid Dynamics course: Useful
math identities (vectors and differentiation). These can be derived using
tensor notation.
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.
GFD-1 carries on fairly continuously from Fluid Dynamics, AtmosSci505/AMath505/Ocean511, Fall 2014. We will revisit much of that material so keep Prof. Bretherton's notes handy.
We will be setting up a GoPost bulletin board. We have not made much use of GoPost in the past, but it may be worth developing this year, for online discussions.