Download 12.003 Atmosphere, Ocean, and Climate Dynamics Problem set 1

12.003 Atmosphere, Ocean, and Climate Dynamics
Problem set 1
Due in class on Monday September 26, 2016
Need help?: Recitation Wed noon-1pm; Office hours TBD; Email as needed.
Collaboration is allowed, but write up the solution on your own. Show all work. Give
units for all numerical results. Put axis labels and units on any graphs.
1. In this problem, you will investigate the effect of moisture on the density of air.
(a) Estimate the zonal-mean and annual-mean temperature at 900hPa at the
Earth’s equator, and then calculate the density of dry air for this pressure
and temperature. The observed temperature distribution is shown at
http://paoc.mit.edu/labweb/atmos-obs/temperature.htm
Note that the zonal mean is an average over longitude. The gas constant for
dry air is 287.04 J/kg/K.
(b) Calculate the density for the same pressure and temperature but for moist
air in which the water vapor is at saturation (e = es ) and there is no liquid or
ice. You may use the simplified expression for the saturation vapor pressure
es from class. Base your answer on the ideal gas law but do not use “virtual
temperature” formulae that you may have learned about previously. The gas
constant for water vapor is 461.39 J/kg/K. Is the moist air more or less dense
than the dry air considered in part a?
(c) Now suppose the air also contains liquid water (cloud) with a mass mixing
ratio of 1 g of liquid water per kg of dry air. What is the density of the air
in this case? You may neglect the volume of the liquid water.
2. In this problem, you will consider radiative transfer for a human and then a planetary system.
(a) Compute the total power radiated by a person with a normal body temperature of 37C assuming for simplicity that the internal and skin temperatures
are the same. For the purposes of estimating the surface area needed in this
problem, you may assume that the person is shaped approximately like a rectangular prism, with height 2m, width 0.4m and depth 0.25m. Make a list of
5 other major terms in the energy budget of the human body (treat longwave
and shortwave radiation separately), and for each one indicate whether it is
an energy loss or gain.
(b) Consider a planet orbiting at a mere 0.07 A.U. (Astronomical Units) from a
red-dwarf star. The temperature of the photosphere of the star is 4000K and it
has a radius roughly one-third that of Earth’s sun. Compute the equilibrium
temperature of the planet, assuming it is an isothermal spherical body with
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an albedo of 0.3, and assuming that the atmosphere has no greenhouse effect.
What is the gap between the wavelength of maximum emission of the star
and that of the planet?
3. Regional or global nuclear war would lead to large fires and subsequent lofting of
smoke (absorbing aerosols) high in the atmosphere. Consider the simple radiative
model discussed in class in which there is a one-layer isothermal atmosphere with
an emissivity of one in the longwave and transparent in the shortwave. Modify
this one-layer model so that no solar radiation reaches the surface - it is either
absorbed or reflected by the atmosphere. Assume for simplicity that the fraction
of reflected solar radiation (the planetary albedo) remains unchanged.
(a) Solve for the surface temperature. How does it compare with the surface
temperature in the standard one-layer model?
(b) Is there a greenhouse effect in the nuclear winter case? Explain.
(c) Can you think of another effect of nuclear winter on the physical part of the
climate system (other than the change in temperature)? Give your reasoning.
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