Stefan LLEWELLYN SMITH
EBUII 574
x23475
http://mae.ucsd.edu/~sgls
This is the homepage for MAE224A during the Fall Quarter 2010. Last
updated: December 4, 2010.
E-mail
Please make sure the e-mail address UCSD has on
file for you is correct.
Times
Lectures: MWF 10:00-10:50 am in CENTR 220. No formal
office hours; I sit in EBU II 574. You can e-mail me to make an
appointment or drop by (I may be busy - see my calendar).
Texts
I have placed a number of books on reserve: see here.
You
may
also
find
this
book
preprint useful.
The grade in this course is based on homeworks, a midterm, and a final
exam. An approximate division is 30% , 30% and 40%, but this is by no
means definite. Your final grade is hence the culmination of a
quarter-long effort. I do not like giving C grades and below for
graduate courses. Please try and keep me happy.
Homework
Homework policy: you may discuss problems among yourselves, but
everything you write and hand in should be your own work. Homework
should be typeset, preferably in LaTeX, and sent to me by e-mail on the
due date. For a primer on LaTeX, see here
and here.
The
goal
is
for
you
to
gain
experience
in
writing
technical
language
with
equations
and to present the arguments clearly and concisely. All
equations and all text are both wrong; look at good textbooks to see
how it is done.
I Due Oct 4. 1. Derive the Reynolds Transport Theorem. 2. Discuss
the physical meanings of the Rossby, Reynolds and Ekman numbers.
Calculate them for (i) mantle convection, (ii) the Great Red Spot of
Jupiter, and (iii) the outflow from the Point Loma Wastewater Treatment
Plant. 3. Show from the vertical equation of motion and the divergence
theorem that the force acting on a solid object immersed in a
stationary fluid is equal to the weight of the displaced fluid
(Archimedes' principle). 4. There is an experiment in the San Francisco
Exploratorium that allows you to turn a handle to rotate a turntable
with a tank on it and see the resulting shape of the surface of the
water in the tank. Assume the tank is two-dimensional and extends from
-L to L, with water of depth h when the tank is at rest. Find the
steady-state shape of the water surface when the turntable is rotating
with frequency Ω. Show there
is a critical value ΩC
such that when Ω>ΩC, the
free surface touches the bottom of the tank. Solutions (pdf and tex).
II Due Oct 18. 1. Derive the plume equations for two-dimensional
line plumes in the same way as was done in class. 2. Solve the plume
equations numerically (e.g. using Matlab) for a pure plume with
buoyancy flux B0 released
into
an
ambient
with
constant
stratification
N
and find the maximum plume rise
in terms of these two parameters. Now consider the case where N(z) ~ za and find the value of a separating regimes in which the
plume rises forever regime in which it reaches a maximum height. (See
Caulfield & Woods 1998.) 3. If a plume rises in a rotating
stratified ambient, eventually it will spread out enough for rotation
to become significant. Discuss at what height this might happen. Will
the plume still be rising or will it be spreading out as a gravity
plume? What happens then? (Scaling arguments are enough here.)
Solutions (pdf, tex and matlab).
III Due Nov 8. 1. Surface tension leads to a difference in
pressure between the atmospheric pressure and the pressure at the
surface of the fluid of T multiplied by the
curvature of the free surface. For linearized waves, show that the
curvature,σ, is
approximately
the
horizontal
Laplacian
of the surface displacement η and
that
the
dispersion relation becomesω2=(g + σTK2/ρ)
tanh KD. Discuss the group
velocity as a function of K.
2. Compute particle orbits for particles initially at (x0,z0).
Show
that
they
are
ellipses
and discuss the shallow- and deep-water
limits. 3. Fluid at rest occupies a semi-infinite channel for x>0
and -D<z<0. The left-hand wall is given velocity u = U(z).
Discuss the subsequent motion. (This is more difficult than the other
questions. I encourage you to discuss this amongst yourselves and think
about normal modes and quantum mechanics.) Solutions (pdf
and tex).
IV Due Nov 24. 1. Summarize the paper of Cahn (1945). 2. Examine
the behavior of internal waves in a wedge bounded by the planes y = -ax
and y = 0. (See Wunsch 1969). 3. Calculate the normal modes for a
stratified fluid of depth D with uniform buoyancy frequency N and a
free surface. If DN2/g
<< 1, explain why the modes fall into two classes: barotropic
modes that do not feel the stratification and baroclinic modes for
which the surface is effectively rigid. 4. Compute the dispersion
relation for internal gravity waves in the presence of rotation.
Discuss the properties of the relation as the frequency of the wave
approaches the Coriolis parameter. Solutions (pdf
and tex).
Monday November 1. Open-book 50 minutes exam. Solution.
Practice questions:
1a. Define the Ekman number. Give example values of it for a variety of
real-world situations.
1b. Explain how and when the centrifugal force can be absorbed into the
pressure gradient term.
1c. Explain the entrainment assumption.
2. Derive the compressible vorticity equation and express it in
non-dimensional form. Discuss the non-dimensional numbers that you
obtain.
3. Solve the equations for a thermal in an unstratified ambient for a
pure thermal with no initial momentum and buoyancy.
Take-home exam to be typset and sent to me by e-mail by the end of the
scheduled time, i.e. 11 am on Friday December 10.
Grading policy
I remind you of UCSD's policy
on academic dishonesty. I may rescale the three components (homework,
midterm, presentation and project) separately to arrive at the final
grade.