Suppose that you had access to a table showing the average temperature in Salt
Lake City for every day for the last one hundred years. You could easily
average over those hundred years to produce an average daily temperature
for each day of the year. If you then plotted these averages by using the
horizontal axis for days of the year and the vertical axis for temperatures,
you would come out with an oscillation curve. The high point would occur
some time in the summer, and the low point would occur some time in the winter.
Now suppose that you had access to a table showing the average length of each
day of the year in Salt Lake City. If you graphed this, you would see a
similar curve. The high point would occur on June 21, and the low point would
occur on December 21.
There is, of course, a relationship between these two curves. To make an
analogy with what we've been studying in this lesson, the average day lengths
are like the bumps in a road, and the average daily high temperatures are like
the bouncing of a car. We can model the climate of Salt Lake City as a linear
system that is responding to the oscillation in the amount of sunlight. Notice
that the amount of sunlight and the average daily temperature oscillate
at the same frequency: one year. This is what we'd expect from a linear
system.
You now know that to predict how a car will vibrate as it rolls over the bumps
in a road, you only need to know the car's transfer function. Similarly, to
predict how the average daily high temperature will change over the course of a
typical year, you need to know a transfer function that characterizes the
climate of Salt Lake City.
Salt Lake City and San Francisco are at roughly the same latitude, which means
that they see roughly the same annual variation in day length. But the two
cities have very different climates. What this means is that Salt Lake City
and San Francisco have very different transfer functions!
A place that is far from the oceans, like Salt Lake City, is said to have a
continental climate. A typical transfer function for a continental climate
might be
continental[s_]:= 1 / (4.4*s*10^6 + 1)
A place that is near an ocean, like San Francisco, is said to have a coastal
climate. A typical transfer function for a coastal climate might be
coastal[s_]:= 1 / (3.7*s*10\^7 + 1)
Let's see how we can use these transfer functions to understand the difference
in the average high temperatures in Salt Lake City and San Francisco.
The transfer functions given above (just like all the transfer functions in
this lesson) assume that oscillations are given in units Hz. What this means
is that if we were to calculate the gain or phase delay at a frequency of 1 Hz,
we would be looking at what the world would be like if the earth revolved
around the sun once per second. Obviously, we are more interested in what
happens at frequency of 1 revolution/year rather than 1 revolution/second.
Write a short report answering the following questions. When is
the hottest day of the year in the coastal climate? What about in the
continental climate? How much hotter is the hottest day of the year in the
continental climate than in the coastal climate?
Explain how you drew your conclusions by including pieces of your Mathematica session
into your report in appropriate places. Your report should incorporate your
answers to questions 1 and 2 below, and should refer to the plots from
questions 3--6 below. Be sure to label the plots sensibly.
(1) If a frequency of 1 Hz corresponds to one revolution/second, what
frequency corresponds to one revolution/year? The answer will be a tiny
fraction of 1. (Hint: there are 60 seconds in a minute, 60 minutes in an hour,
24 hours in a day, and 365.257 days in a year.)
(2) At a frequency of one revolution/year, as you calculated above, how
many radians corresponds to a phase delay of a single day?
(3) Graph the gain of the continental climate transfer function over
frequencies ranging from 0.25 revolutions/year to 4 revolutions/year. (You'll
need to convert these frequencies back to revolutions/sec in order to determine
the proper ranges to plot.)
(4) Graph the gain of the coastal climate transfer function over frequencies
ranging from 0.25 revolutions/year to 4 revolutions/year.
(5) Graph the phase delay of the continental climate transfer function over
the same frequency range as in question 3.
(6) Graph the phase delay of the coastal climate transfer function over the
same frequency range as in question 4.