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232 Geosci Problem Set 2 June 14, 2002 1 Adiabats, moist and dry Consider an atmosphere composed of a mixture of a noncondensing substance like air (referred to by the subscript "a") and a condensing substance like water (referred to by the subscript "c"). The heat balance defining the adiabat is: dT pc dT pc Q Ra dpa Rc dpc = CpaT ( )+ Cpc T ( )+L d - - Ma T Cpa pa pa T Cpc pc pa 0= (1) where = ma is the ratio of the molecular weights of the two species and mc Ma is the total mass of the noncondensing species under consideration. Assuming that the condensing species is saturated implies that pc is given by the saturation vapor pressure psat (T ), according to the Clausius-Clapeyron relation. (a) Complete the derivation of the equation for d(ln(pa ))/dT by collecting terms in dT and dpa /pa , making use of Clausius-Clapeyron, and using d pc pc = (d ln pc - d ln pa ) pa pa (2) (b) The current atmosphere of Earth consists of about 20% O2 by moles, and 80% N2 . Compute the dry adiabat for the pre-biotic atmosphere without any oxygen, and compare with the dry adiabat for the present atmosphere. Note: You can do this problem analytically. There should be no need to resort to a numerical solution. You will need to look up Cp for oxygen and nitrogen. 1 (c) Compute the moist adiabat for an atmosphere consisting of a mixture of air and water, like the present Earth's atmosphere. Keep the partial pressure of pa of air fixed at 1000mb at the ground level, and let the surface temperature range from very cold values to very warm values, so that you can show the evolution of the adiabat from the dry adiabat to the ClausiusClapeyron adiabat for a pure steam atmosphere. For the purposes of this problem, you may assume that water vapor has a constant latent heat of condensation of 2.5x106 J/kg. Even with this simplification, you will need integrate to the equation for the moist adiabat numerically (see instructions below). The best procedure is to integrate the equation for d(ln(pa ))/dT , and then use the results to make a plot of T (ln ptot ), where ptot = pa + psat . The saturation vapor pressure of water is 1008.1266 Pascals (about 10mb) at 280K, as you can determine for yourself by using the function satvpg(T) contained in the module phys (d) Do the same for a hypothetical early Martian atmosphere consisting of .5 bars of noncondensing N2 and a varying amount of condensing CO2 . The latent heat of condensation of CO2 into dry ice is 5.9x105 J/kg, and the saturation vapor pressure is 104 P a at 154.316K. The specific heat of CO2 gas is Cp = 820J/kgK. 1.1 How to do numerical Integration using Runge-Kutta To do problems (c) and (d) you will need to be able to numerically integrate a differential equation of the form dY/dx = f(x, Y ). In the course modules directory, there is a module called rk.py which makes this easy to do in Python. The following bit of code gives an example of how you would use the module to integrate dY/dx = -Y subject to the boundary condition Y = 1 at x = 0, using a step size x = .1, for values out to x = 10. import rk #First define the function giving the derivative. #You can use any name #you want for the function and its arguments. def f(x,Y): return -Y # #Now use the class rk.integrator_rk1d to 2 #create an object to integrate #the function. This is an "object oriented" #approach to numerical integration. # fi = rk.integrator_rk1d(f,0.,1.,.1) # #Now use your integrator to step through the solution # for i in range(100): result = fi.next() print result[0],result[1] # result[0] is x, result[1] is Y. The result should be approximately e-x . Experiment with smaller and larger values of x to see how it affects the accuracy. 3

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