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Ice thickness
Developed by Andy Rundquist  Published October 3, 2016
This set of exercises guides the student in exploring how to model the thickness of ice on a lake based on actual temperature data for a given winter. Students will have to learn how to access the data, reformat the data, and use it in a calculation.
Subject Area  Thermal & Statistical Physics 

Level  Beyond the First Year 
Available Implementation  Mathematica 
Learning Objectives 
Students will be able to
* Obtain data from a public source (**Exercise 1**)
* Derive the differential equation for heat flow through an ice sheet (**Exercise 2**)
* Analytically derive the thickness of ice for a constant temperature difference (**Exercise 3**)
* Learn how to use time series data in a numeric approach to integrating a differential equation (**Exercises 4 and 5**)

Time to Complete  60 min 
These exercises are not tied to a specific programming language. Example implementations are provided under the Code tab, but the Exercises can be implemented in whatever platform you wish to use (e.g., Excel, Python, MATLAB, etc.).
1. Go to [climate data](http://www.ncdc.noaa.gov/cdoweb/) and access temperature data for a recent winter for a city in a wintry climate. Download the data as a CSV file. Read the csv file into your computing environment and plot the daily temperature as a function of time, indicating when it is below freezing. Then cut out any temperatures at the beginning that are above freezing (up until the point where there is a long portion of time below freezing).
2. Derive the differential equation for ice thickness, taking into account heat flow through the ice from the water beneath and assuming the water is at the freezing point. As the ice gets thicker the heat flow slows.
Answer:
$$\frac{dx(t)}{dt}=\frac{\Delta T(t) k}{\rho L x(t)}$$
where x is the thickness of the ice, \(\Delta T(t)\) is the timedependent temperature difference across the ice, k is the thermal conductivity of the ice, \(\rho\) is the mass density of the water, and L is the heat of fusion for ice.
3. For a constant temperature difference of, say, 10 degrees Celsius, determine the thickness of ice after one month. Note that you can solve the differential equation in (2) analytically for this simplification, but you should also test your numeric differential equation solver to make sure you get the same answer.
answer: roughly 0.5 meters.
4. Develop an interpolating function for the data in (1) to be used in your differential equation solver. Apply that function and plot the thickness of ice as a function of time.
5. Building in the appropriate conditional statements to make sure x > 0 at all times, change your code for (4) to go beyond the contiguous freezing temperatures to determine when the ice begins and when it ends.
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Credits and Licensing
Andy Rundquist, "Ice thickness," Published in the PICUP Collection, October 2016.
The instructor materials are ©2016 Andy Rundquist.
The exercises are released under a Creative Commons AttributionNonCommercialShareAlike 4.0 license