Homework Questions for Chapter 13 - Weather Forecasting

Consult Syllabus for Due Date

Questions 1 and 2 will be turned in for a grade.

Follow the Problem solving steps discussed in class

1. In class we discussed that cloud cover at night will increase the minimum temperature observed. Another important factor to consider is the height of the clouds when forecasting the minimum temperature. The following relationship shows how much radiation is emitted between the ground and cloud layer above.

Enet is the net radiation transferred between the ground and clouds.  Remember that both the ground and cloud layer are emitting radiation.  Enet represents the amount of radiation emitted by the cloud layer minus the amount emitted by the ground.  Tcloud is the cloud temperature, and Tground is the temperature of the ground. s is the Stefan-Boltzman Constant.

(a)   If Enet is positive, will the ground warm or cool with time?  Explain.  If Enet is negative, will the ground warm or cool with time?  Explain.

(b)  From the above equation, will you observe a higher minimum temperature with high clouds or low clouds present during the nighttime hours? Explain.

(c)  Using data from the U.S Standard Atmosphere, if the net radiation lost by the ground is 50 W m-2, what is the ground temperature if:

1.  low clouds at 2.0 km AGL are overhead

2.  high clouds at 7 km AGL are overhead.

HINT: (s = 5.67 x 10-8 W m-2 ēK-4)

Also make sure that your temperature values are in ēK, not ēC

2. The equations that govern the state of the atmosphere are fundamentally chaotic. Another words, the forecast that the equations produce are very sensitive to the model initial conditions. We can replicate this situation with the simple example of 2-D convection in a tank of water, where the bottom is heated. Due to the heating at the bottom of the tank, the warm water near the bottom will rise and the cooler water at the top of the tank will sink. The equations that govern this motion can be expressed as:

where C gives the circulation of the fluid. Circulation is a measure of rotation in the fluid which will be larger for more vigorous convection. L represents the left-right temperature distribution and M indicates the amount of vertical mixing. All three of these variables are dimensionless. s, b, and r are dimensionless constants while Dt is the time increment.

a. Let's focus on a forecast of the circulation, C. Using the above equations, create a time series plot of circulation of the fluid versus time from time = 0 to time = 10. Use the following initial conditions (at time = 0): C(0) = 13.0, L(0) = 8.1, and M(0) = 45. Values for the constants are: s = 10.0 b = 8/3, and r = 28. Use a time step (Dt) = 0.01.

b. Now, on the same plot, create another time series of circulation versus time for M(0) = 44.

c. Create a final time series of circulation on the same plot with M(0) = 46.

d. Out to approximately what time can you have some confidence in your forecast. Explain.

You do not need to turn in all the numbers generated to make the plots.  Please turn in some discussion of how you solved the equations.

3. Climatology shows that the skies over Vancouver, Canada are cloudy 327 days a year. If you were to forecast cloudy skies for every day of the year, how accurate will your forecasts be? Do your forecasts show skill? Explain.

4. Suppose that is is presently warm and raining. A cold front will pass your area in three hours. Behind the front it is cold and snowing. Make a persistence forecast for your area 6 hours from now. Would you expect this forecast to be correct? Explain. Now, make a forecast for your area using the steady-state, or trend method.

5. Explain how the phrase "sensitive dependence on initial conditions" relates to the final outcome of a computer-based weather forecast.

6. Describe with the aid of a diagram the movement of low and mid-level clouds for: