Finals: Or how I should take this cyanide pill

Finals in:
Advanced Vector Cal
Theory Gravity & Magnetics
Structural Geology.

Here's my take-home final for Computational Methods in Geophysics. It's not very fun... ;(

;___;:

Geop 470 Final Exam
Due: Monday, May1, 2006, 5 P.M.

You may use your book or notes, but may not work together on this.

1. Consider the fluid layer in T&S Figure 6-41, with the approximate velocities given by (6-355) and (6-356).
(a) Show that these equations describe an incompressible fluid; also, explain and show mathematically why the relationship (6-357) has to be true.
(b) Suppose that these are the velocities resulting from thermal convection. If you knew the velocity, u0, how would you write an expression for the thickness of the thermal boundary layer as a function of distance along the top wall?
Find the mean thickness of this boundary layer, by integrating your expression.
(c) Assume that you can describe the horizontally-averaged temperature profile as two thermal boundary layers over an isothermal core. With a uniform viscosity and no internal heating, the top and bottom boundary layers are the same thickness an must have the same heat flux. Sketch this temperature profile.
(d) Use your expression from (b) to find an expression for Nusselt number, in terms of the velocity, u0, box size and fluid properties. Use their calculated relationship (6-362) to write a relationship between Nu and Rayleigh number.

2. (a) Write very simple energy conservation equations for a planet that has differentiated into an iron core and silicate mantle. Write your equations in terms of spherically-averaged mantle and core temperatures, surface heat flux and the heat flux between core and mantle. Sketch the problem, and sketch a plot of temperature vs. radius. You will need to assume all of the following:
i. spherical symmetry
ii. the mantle is uniform in it’s properties (density, conductivity, viscosity, etc.)
iii. the mantle has a uniform source of radioactive heating, caused by a single long-half-life element (i.e., can be described by a single decay constant)
iv. the mantle is completely solid (no phase changes are occurring)
v. the mantle is convecting, and the boundary layers are relatively thin
vi. the core has uniform properties (but they are not necessarily equal to those of the mantle).
vii. there is no heat production in the core
viii. the core is completely liquid, convecting at such a high rate that it’s temperature can be approximated by a single value.
ix. the surface temperature is at a fixed value


You get to pick your own variables for everything. That means you need to define everything so I know what they are. Include a table, where you list the variable name, a verbal description and units.

Manipulate your equations until they have units of temperature/time (deg/s).

(b) Write an expression for conductive heat flux across the thermal boundary layer at the base of the core. Write an expression for surface heat flux in terms of the Nusselt number. Remember the Nusselt number is the ratio of the surface heat flux across the top cold thermal boundary layer to the heat flux that would arise from just conduction (no convection) from the core to surface (for this latter value, treat the mantle as a flat layer, rather than a spherical shell, for simplicity).

Substitute these expressions for heat flux into your conservation equation from (a).

(c) Assume that we can approximate the mantle as an iso-viscous fluid, that is primarily heated from below. In this sort of fluid, experiments tell us that thickness of the thermal boundary layer, , is related to the Raleigh Number by

where D is the thickness of the mantle, Racr is the critical Rayleigh Number and  a dimensionless exponent.
Given your definition of Nusselt number above, write an expression relating Nu to Ra.
Substitute for  and Nu in your equations.

(d) Non-dimensionalize your system of equations. HINT: I think it is a bit easier if you introduce a parameter that is the ratio of core radius to outer mantle radius. Choose your time and temperatures scales.

(e) You should now have two equations of this form:


where Tm and Tc are the average mantle and core temperatures, respectively, and the P’s are dimensionless parameters that you have calculated along the way.

Write out expressions for the P’s.

Now use the MATLAB script Convect1.m to solve these equations for core and mantle temperatures as a function of time. The parameter values to use for a silicate mantle and iron core are

mantle density = 3400 kg/m3
core density = 8800 kg/m3
mantle specific heat = 1200 J/kg-deg
core specific heat = 1200 J/kg-deg
thermal expansion coefficient=3x10-5 deg-1
mantle thermal conductivity = 4.2 J/m-deg-s
initial heat production rate = 3.4x10-11 J/kg-deg
time constant for radioactive decay = 1.3x10-17 s-1

For a first problem, let’s do the Earth: look up the radius and core radius in your book.
Set the initial mantle and core temperatures both equal to 3700 ˚C. Using these parameters, calculate Ra for internally-heated fluids (T&S equation 6-234), using a viscosity of 1021 Pa-s.
Assume that =1/3, and Racr=1000.
Calculate the P parameters.

Convect1 will read 12 numbers from the file cparams.txt. The first six are the P parameters, in order. The seventh is your time scale, the eighth is your temperature scale. The ninth and tenth are the planet radius and core radius. The eleventh is the initial temperature of the core and mantle, and the twelfth is Ra.
Change the dummy values in cparams.txt with your calculated values, and run the script. You will get a plot of core and mantle temperature vs. time, for 4.5 By.

From this output:
- print your plot
- from your plot (or by typing Tm(N) after the script finishes) get mantle temperature at the present day, based on your model. Using the previously calculated thermal boundary layer thickness, what is the average surface heat flow for your model?
- sketch mantle temp. vs. radius at 4 By ago, 2 By ago and today. Explain what the difference in your plots imply for heat transfer through the mantle.

(f) Another dimensionless number used in convection studies is the Urey number (Ur) which is the ratio of the total heat flow out of the surface of a planet, to the total of heat being produced within the planet. This is a measure of “secular cooling” or loss of the planet’s original heat. Write an expression for Urey number, and calculate it’s present day value according to the run in part (e). How does Ur change over time?

(g) Rerun your model with a Ra 10 times higher and 10 times lower than your original value. Show your plots, and describe how this changes the results.

(h) Mars: redo your calculation, with planet radius=3400km, and core radius 1400 km. Keep everything else the same, but recalculate anything that depends on the radii (like Ra). Print your cparams files and plot. Describe the differences from Earth.

(j) The Moon: planet radius=1740 km, core radius=400 km. Print your cparams files and plot. What has happened here and why? We discussed this problem with the Moon before. What characteristic of the mantle is likely to be different for the Moon? Change that value and rerun.

(k) For each of the three planets, recalculate the Ra for the present day (some of the numbers you put in the original Ra should have changed). As the heat source decays, the Ra will decrease, which should decrease the rate of cooling.

What if you account for temperature-dependent viscosity?
Suppose that the mantle viscosity is given by


where T0 is the initial mantle temperature, and Tm is the current mantle temperature. The script Convect2.m automatically adjust the Ra for increasing viscosity and decreasing heat production. You just give it the initial value.

Rerun your Earth calculation using the script Convect2.m. It reads all the same parameters, but it modifies your input Ra during the run, as the heat production decreases, and the viscosity increases. It will also plot the variation of Ra with time in a separate figure: you might need to use log axes to see this better.

Run a few trial cases with planets of different radii, but a core/planet radius ratio of 0.5; with all else equal, try to find the size planet that hit Ra=1 right about the present day. Just show your final run.

Jam it back in, in the dark.

My past Physics profs gave us freakin' HANDOUTS of formulas. About 99% of them were irrelevant. If you didn't have the formula you needed memorized, you'd spend 5-10 minutes reading just for that one. I remember I did that once and found out the hard way that they forgot to put that formula on.

The Physics Department at Texas A&M is a joke. Plain and simple ;(

There's nowhere I can't reach.

Take home final is going horribly for our whole class. 14 people... You can go to the prof for help. The paper says we can ask him quesitons, yet we can't work together... But collaboration is the only way to do this shit. Hardest assignment yet and it is monstrous.

Problem 1. c) and d) took from 9am => 6pm straight just to do. They are that fucked up.

I've still got to start on problem 2, which is 3.5 of the 4 page assignment ;__;

The whole class has one thing on their mind: we are fucked. Assignment is due Monday. Prof won't be around to answer questions on the weekend or Monday. Tomorrow is our last push. The rest of it is on our own.

This one prick is almost done. Because he camps out the prof so god damn much. When I've gone for questions, I hafta wait 30-40 minutes for him to finish with a round of questions he has lined up for the prof. I finish within 20 and on my way out the door, he is headed RIGHT on back. The whole class is quite pissed at him. He's inhaling shitloads of the time the prof is available.

This thing is sticky, and I don't like it. I don't appreciate it.