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lab5_extra.rst

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Extra lab 5 problem -- due Friday Nov. 4

  • Solve the steady state temperature problem of Mini Project 1 (pdf)

    A_{max}, & d < h \\

    A_{depth} + \left[A_{max}-A_{depth}-A_{dip}(d-h)\right] \exp \left[-0.5(d-h)\right], & d > h. \\

    \frac{\partial T}{\partial t} =& 0 =  \frac {\partial }{\partial d}
   \left(A_h \frac {\partial T}{\partial d} \right) -
   \frac{ 1} {c_p} \frac {\partial I}{\partial d}

    using the standard ode integrator for Python (scipy.integrate.odeint) or matlab (ode45)

  • Use an exact form for \frac{\partial^2T}{\partial d^2}. You'll need to chain rule the first term in equation (3), which requires \frac{\partial A_h}{\partial d}. I will provide matlab and python versions of this function, along with A_h(d) and \frac{\partial I(d)}{\partial d}, and values for all the coefficients.

  • Initial conditions: T(d=0) = -1\ ^\circ C, T^\prime(d=0) = 0.00105563 ^\circ C/m

  • As a side effect, have your derivative function save the temperature and height for each time it is called

  • Turn in a script which makes a plot of temperature as a function of depth, along with the T,depth values returned by the ode integrator overlayed as dots or crosses (identified in your legend) Ask the integrator for 10 meter spacing between 0-50 meters, and 50 meter spacing below 50 meters.

  • Compare the solution you get using the odesolver with the fixed-grid solution you obtained for the miniproject. Are the two techniques solving the same problem?

Matlab and python files

Click on these to save

  • mini1_data.txt contains three columns with (z,Temp, Ah) from the solution with these values for the various parameters:

    co.Amax = 1e-2
co.Adepth = 1e-4
co.Adip = 1.5e-3
co.h = 10
co.beta=0.5
co.alpha=0.1
co.cp=4.e6
co.I0=100.
co.albedo=0.1

  • plot_derivs.m and plot_derivs.py are matlab and python programs that plot the analytic formulas for A_h and dA_h/dz and compare dA_h/dz with a finite difference version to show that I've actually done the derivative correctly

  • test_solve.m and test_solve.py are matlab and python programs that show how to use the built-in matlab and python ode integrators to solve a second-order ode and write the intermediate results to a file, with test plots showing the guesses the integrators took.

  • Your script should use the Ah and dAh/dz routines from plot_derivs and the structure of test_solve to setup and solve the miniproject equation. I found a good fit to the answer by starting with an initial temperature of -1 degC and an initial dT/dz=0.00105563 deg C/m