Problem solving for each chapter on the CD-ROM are consists of 2 steps:
(a) problem formulation: state the numerical exercise in terms of one or more equations
(b) problem solution: solving the equation(s).
(a) The first step in problem solving is to decide which equation(s) to use. It is useful to keep in mind a few key equations, such as the hydrostatic balance (Note 1.G). This does not mean that one needs to memorise all the equations given in the Notes. Basically, one should be able to express some fundamental principles in the form of a simple equation. A fundamental principle, for instance, is the surface energy balance (Chapter 5). To be more specific, consider this hypothetical problem:
Imagine a lake that evaporates at a rate of 17mm/d. As discussed in Chapter 4, an evaporating lake loses heat through the expense of latent heat. Estimate the net radiation received by the lake, if it is assumed that the water temperature remains unchanged and in equilibrium with the environment.
In this case, the amount of heat lost through evaporation must equal the amount of heat received by solar radiation. This statement is step (a) in your solution. The equation that underlies this physical principle is fairly trivial once we have set the frame of surface energy balance calculations Chapters 4 and 5, but in the mean time, you managed to understand the basic principle on which these calculations are based, i.e. the conservation of energy.
(b) Once the problem is formulated, then one can proceed to solve the equations. Two golden rules for solving the numerical equations are:
1) if necessary, convert the given non-SI units to the SI; and
2) perform the calculations simultaneously on the numerical values and on the units of these values.
Only this way can you ensure that your result is physically plausible and dimensionally correct. Consider this problem, as an example:
An approximate equation to estimate the evaporation (E , m/s) of water from a free water surface was given by Dalton two centuries ago (Note 4.E) as:
E = K u (esat- e)
where u is the wind speed (m s-1) and (esat- e) the saturation vapour pressure deficit (Pa). The Dalton equation says that more evaporation occurs when it is windier and drier. K is a proportionality constant, and field experiments have indicated that K is about 0.2 10-10 Pa-1. Now presume that a wind of 10 m s-1 with a saturation vapour pressure deficit of 10 mb blows over a lake. Estimate how fast the water level of the lake would drop (in mm/d) due only to evaporation.
Answer:
E = K u (esat- e) (1)
= 0.2 x 10-10 x Pa-1 x 10 m s-1 x 10 x 100 Pa (2)
= 0.2 x 10-10 x 10 m s-1 x 10 x 100 Pa-1 Pa
= 0.2 x 10-6 m s-1 (3)
= 0.2 x 10-6 m s-1 x 103 mm m-1 x 3.6 x 103 s h-1 x 2.4 x 10 h d-1 (4)
= 1.7 mm d-1
(note: h = hour)
In general, a safe way to solve problems is by means of the following steps:
(i) By means of the information given in the problem, write the expression in the form that the unknown variable is on the left and the known variables on the right. In the above example, this step is straightforward; imagine that in the above example the evaporation rate (E) was given, and that the problem asked for a wind speed which would produce so much evaporation. In that case, (1) would have to be converted to get an expression for u. Often, this step is the most challenging, and also the most heavily assessed. Some students may feel quite comfortable solving problems. Others may be less comfortable, and only practice will solve that.
(ii) Plug in the numbers as well as the unit dimensions, as done in (2) above. Simplify both numbers and dimensions, as shown in (3). This step is usually easy. Make sure the resulting unit dimensions are the ones you expect.
Note: If you need to convert from one unit to another, be it for an input variable or the output variable (as above), it is wise to write out the desired dimensions as a function of the original dimensions, as is shown in (4). Then the expression can be simplified again, as in step (b). Note 1.J lists all unit conversions encountered in this CD-ROM.