The Betz limit

There is an interesting discussion to be had about the possible efficiency of wind turbines which presents opportunities for the instructor to look at several different kinds of computations.  The discussion ends up with an optimization problem which could be approached by the classical methods of calculus – if the students have that available to them – or could be approximated simply by drawing a graph of one critical function. Continue reading “The Betz limit”

Sample question and response

A reader asked “Why did the chicken cross the road?”

It is well known that this is a hard question to analyze rigorously.  However, by starting with the 1-dimensional chicken diffusion equation

\[ \frac{\partial^2\chi}{\partial x^2} + \frac{1}{4\sqrt{\pi}}\frac{\partial \chi}{\partial t} \approx 0, \]

where \(\chi\) is the local density of chickens, we may arrive at the answer, “Because it wanted to get to the other side (to a first order approximation).”

Page 336 Sea Ice Data

A couple of exercises for chapter 5 refer to a dataset of Arctic sea ice extent, and we also plan to add an online case study on regression to the mean where this will be one of the examples.  The data comes from the National Snow and Ice Data Center Arctic Sea Ice News and Analysis.   The dataset is more completely summarized in the graphic below.

Sea Ice Extent, from Arctic Sea Ice News
All NSIDC data is available for download here.

Page 315 Extreme value calculator

This post is a supplement to the discussion of extreme value statistics at the end of Section 5.3 of the book.  You can find an online extreme value distribution calculator, provided by South Dakota State University, at

This calculator fits a Gumbel distribution (a form of generalized extreme value distribution) to a data set.  It uses the language of river floods because that is what the authors are interested in, but the underlying mathematics applies to many different situations.

To use the calculator one provides a data series consisting of extreme values.  For instance, one might provide the data series

12.1; 11; 2; 1.8; 16.4; 6.7; 8; 3;  4; 9;

which represents the biggest value of some variable (the depth of the deepest flood, the windspeed of the strongest storm, or whatever) in each of 10 successive years.  The output of the calculator is a table giving a probability distribution.  It has five columns: the key ones are labeled T (return period), P (probability) and Q (“flood discharge” for this calculator, but it refers to whatever variable we are modeling).  Here is part of the output for the data series above:


Return period T, year Probability P, percent Value Q
25 4 21
50 2 25
100 1 28

This tells us (based on the data provided) that, for instance, the value \(Q=28\) will be exceeded only once in a hundred years; the value \(Q=25\) will be exceeded only once in fifty years; and so on.