Water balance for Crater Lake, Oregon by Manuel Nathenson

 Water Balance Using Yearly Data

The yearly water balance is comprised of 1) water supply from precipitation qp and inflow from the crater walls qi and 2) water loss from leakage qo and evaporation qe. The change in volume of the lake is the change in lake level A for one year times the area of the lake A and may be expressed as:

Dividing by the lake area, each term may be expressed as a volume per unit area d, and the water balance may be written as:

The inflow from the crater walls mostly flows in the spring and early summer when snow accumulated in the fall and winter melts. Over the period of a year, it seems reasonable to assume that the inflow should be proportional to the precipitation p measured at Park Headquarters. The water supply from precipitation should also be proportional to the precipitation at Park Headquarters, and the total water supply can expressed as a function of precipitation as:

Although Phillips (1968) found a suggestive relationship between leakage and lake level, Redmond (1990) could not verify the relation. As a first approximation, the leakage is assumed proportional to the long-term average precipitation po that keeps the lake level unchanged. Evaporation is likely to vary from year to year depending on temperature, wind, and humidity, but evaporation will also be assumed to be proportional to the long term average precipitation po. If evaporation is a small fraction of total water loss, the yearly variation will have a small effect on the water balance. Over a period of many years when the average change in water level A is zero, the water supply will be B po and so will the water loss. Thus equation (2) may be rewritten as:

Equation (4) may be rearranged to:

Redmond (1990) used least-squares correlation based on equation (4) with precipitation as the independent variable and B po as a single parameter and obtained the results given in Table 1 along with those of Phillips (1968) and Simpson (1970). For the three studies, the long-term average water supply ranges from 208 to 247 cm/y and the coefficient B from 1.2 to 1.457. The range constitutes about 20 % for both quantities.

Although not an unreasonable range for a water supply calculation, the range results in nearly a factor of two range in evaporation derived by differencing leakage from long-term water supply (Table 1).