Meteorology 301 - Lab 9

Introduction

In this lab, we will again exploit the capability of a computer to repeat computations endlessly (or until we tell it to stop). We will also construct a simple model of the earth's climate and see how sensitive surface temperature might be to changes in solar radiation and how other atmospheric behavior can modify the sensitivity through feedback

GOAL:
1. Compute the dependence of surface temperature on solar radiation for a simple climate model. 2. Compute and understand the effects of the water-vapor feedback.

Instructions

1. As with previous Matlab labs, follow the instructions for accessing meteorology computers to log into your Vincent account. Lab partners should open a window on the host machine and connect to Vincent in the window.

2. Before starting Matlab, copy three codes I wrote:
(1) Solar constant vs. surface temperature, fixed emissivity
(2) Solar constant vs. surface temperature, variable emissivity
(3) Emissivity vs. precipitable water

3. Some background:
As discussed in class, averaged over the year, the earth is in an equilibrium between incoming solar radiation and outgoing infrared radiation. The incoming radiation is

         Rin = (Q/4)*(1-alpha)
where Q is the solar constant, Q = 1360 W/m/m for the current climate, and alpha is the earth's albedo, alpha ~ 0.3. The outgoing radiation can be expressed in terms of surface temperature as
         Rout = epsilon*sigma*(Ts)^4
where epsilon is an effective emissivity, sigma is the Stefan-Boltzman constant, and Ts is surface temperature. (Note that I have used Matlab notation to symbolize T raised to the fourth power.) For the current climate, epsilon ~ 0.6. If there were no atmosphere, we would have epsilonn ~ 1. With an atmosphere present, the surface is less effective at radiating energy to space, so then epsilon < 1. (This is another way of stating the greenhouse effect.)

The balance of energy then implies

         Rout = Rin

We could solve this equation to get surface temperature as a function of Q. If epsilon is a fixed constant, this is rather simple. However, epsilon will depend on the greenhouse gases present in the atmosphere, especially water vapor, which we have seen varies qutie strongly with temperature.

A simpler way to proceed is to assume we know Ts and solve for Q. We are then asking:

What Q is needed for an energy balance to occur at the specified Ts?

We will compute answers to this question in this lab.

4. The two files, Q_Ts_nov.m and Q_Ts_v.m solve for Q when given Ts under different assumptions. The "nov" file assumes that the emissivity does not vary at all. The "v" file assumes that it does and that it depends on water vapor in the atmospheric column, PW. The related file Eps.m computes epsilon(PW).

Use emacs or your editor of choice to look at each "Q_Ts" file to see what type of input and output variables its function uses. Now start Matlab, from a Vincent prompt. (If you forget how, go to Lab 5 , step 4.)

5. Set up an array of surface temperature values that will be fed into both "Q_Ts" functions. This is done by running a "for" loop interactively:
>>for i = 1 : 71
Ts(i) = 200 + (i-1)*2 ;
end
 Note that after entering the first line, "for ...", you hit the carriage return and enter the second and third lines without getting a prompt. When you start a "for" loop interactively, Matlab waits for you to enter all the commands of the loop. When you enter the last line with the "end", then it executes the loop.

Also assign alpha = 0.3 .

6. Now compute Q(Ts) for the constant emissivity. I've chosen emissivity so that Ts = 288 K gives Q ~ 1360 W/m/m, both of which are current values. Enter:
>>for i = 1 : 71
Qnov(i) = Q_Ts_nov(you decide what goes here!) ;
end

Plot Ts vs. Q, with Q on the x-axis, and answer these questions:

  1. How does Ts vary with Q? Increasing, decreasing? At the same rate of change for all Q?
7. Now enter another loop like the one in step 6., only for the function "Q_Ts_v". Make sure you call it correctly, with appropriate input and output variables!

  1. Plot PW vs. Ts, with Ts on the x-axis.
    You may be placing this plot in your porfolio in a future lab, so it would be best to annotate it carefully by putting in axis labels, etc. Enter "help" in Matlab to review how this is done.
  2. Answer this question: How is PW changing with Ts?
  3. Print this plot to disk, for use in a future lab. We need to follow some steps that will save the file in "gif" format. Matlab allows us to save the file in "tiff" format. We then use another utility to convert it to "gif", which can be done following these steps:

    1. >>print -dtiff PW_Ts.tiff
    2. Exit Matlab or, better if you can, open another terminal window on your meteorology Linux machine. .
    3. At the terminal window's prompt, enter the two letters "xv".
      This starts an application known as XVue. A window will pop up with Xvue in it.
    4. Place the mouse pointer in the XVue window and click your right buttton.
      Another window should open with a menu of options.
    5. Click on "Load", select your tiff file from the list of files that appears, and click on "OK" to open your tiff file.
      A window with your plot should appear.
    6. Click on "Save". A new window appears. In the upper right corner is a "Format" menu. Place the mouse pointer on the menu and hold down the mouse button. A menu will apear; choose "GIF"
    7. Execute the save by clicking on "OK", then exit XVue by clicking on the "Quit" button in the lower right corner of your "xv controls" window.
      If you enter "ls" at the terminal window's prompt, your plot file should appear with the same root name but a .gif suffix, e.g., PW_Ts.gif .

  4. Plot the emissivity vs. Ts, and answer these questions:
  5. Why does epsilon get smaller as Ts increases? (Answer in terms of how PW varies with Ts)
  6. For what Ts is epsilon changing most rapidly?
  7. What values does epsilon approach for large and for small Ts?
  8. How do these limiting values compare with the epsilon used in "Q_Ts_nov"?
  9. Print this plot to disk, again using the -dtiff option, for reference in a future lab.

8. Plot two curves on one figure: Ts vs. Q as computed in both "Q_Ts" functions. Put Q on the x-axis. Answer these questions. You will find it helpful to think back to the questions in step 7. :

  1. Where do the two functions give nearly the same Ts for a given Q?
  2. Why does this make sense? (Think about epsilon values being used for each curve.)
  3. For small Q, for which function is d(Ts)/dQ largest? (That is, for which case is Ts increasing most rapidly with Q.) How is this behavior related to the epsilon values used for each curve?
  4. On the curve given by "Q_Ts_v", where is T increasing most rapidly with Q?
  5. Print this plot to disk also for future reference, using the -dtiff option and XVue to convert to gif. We will be placing this plot in your porfolio in a future lab, so it would be best to annotate it carefully by putting in axis labels, etc. Again, enter "help" in Matlab to review how this is done.

9. The location of rapidly increasing Ts vs. Q is an example of climatic feedback:
Imagine that an equilibrium exists between Q and Ts, with Ts = 260 K.

  1. If we increase Q slightly, then Rin increases, providing more energy to the planet. Then
  2. The increased energy will cause Ts to increase, which will bring Rout back into balance with Rin.
  3. Increasing Ts causes PW to increase, which
  4. causes epsilon to decrease, which
  5. requires further increase in Ts to bring Rout into balance with Rin.
This is an example of a positive feedback, because it amplifies the change that would have occurred without the feedback. In this case, the feedback is due to PW and epsilon changing with Ts in addition to Rout.

Answer this question:

10. Hand in your answers to the questions in parts 6, 7, 8 and 9 to complete this lab.

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