The general water balance of a lake can be written as

where V is the lake volume, *R* and *D* are the surface runoff and discharge rates into and out of the lake, *E*_{L} and *P*_{L} are the evaporation and precipitation rates over the lake surface (in units of depth per unit time), *A*_{L} is the lake-surface area, and *G*_{i} and *G*_{o} are the groundwater inflows and outflows from the lake.

For closed, sealed lakes, defined as those that lack a surface outflow and for which the subsurface fluxes are negligible, equation (1) reduces to

At equilibrium, *dV/dt =* 0, and the equilibrium lake-surface area *A*_{Le} is determined by *(E*_{L} - *P*_{L}*)* and *R,* which reflec_{t} the surface water balance over the lake and its surrounding catchment, respectively. Hence,

At equilibrium, the ratio of the lake-surface area *A*_{Le} to the total catchment area *A*_{C} (including the water surface) is given by

where *C* is a dimensionless index dependent only on *R, E*_{L}*,* and *P*_{L} (Mason et al., 1994):

On time scales longer than a year, *R* may be approximated by the expression

where *A*_{B} is the area draining into the lake (excluding the water surface), and *P*_{B} and *E*_{B} are, respectively, the mean precipitation and evapotranspiration rates over the basin (in units of depth per unit time). Since by definition *A*_{C} = *A*_{L} + *A*_{B}, *C* can now be rewritten as follows:

*C* is an inverse measure of aridity, ranging from near 0 in extremely arid conditions (small *P* and large *E)* to a theoretical maximum of 1 in very humid climates.

So far, only the equilibrium case has been considered. The response of the area of a closed, sealed lake to a perturbation in its inputs or outputs is determined by its characteristic response time (*e*-folding constant) t_{e}**,** defined as the time taken to reach a fraction (1 - 1/*e),* or 63 percent, of the total change in area. The response time t_{e} is given by

where *L* is lake level (Mason et al., 1985, 1994). Values of t_{e} calculated by Mason et al. (1994) for modern closed lakes vary from 1.5 to 350 years. t_{e} is greatest for extensive, steep-sided lakes in relatively moist climates, such as Lake Malawi in the historical past (see below).

For an open lake (a lake possessing an outflow) with negligible groundwater fluxes,

(Hutchinson, 1975). Calculated values of t_{e} for open lakes are in general much smaller than those for closed lakes, varying from 10 ^{-2} to about 5 years (Mason et al., 1994).

The theoretical response of a closed, sealed lake to small perturbations in the aridity index *C* with time is summarized in Figure 1 (after Mason et al., 1994). Three simple types of change in *C* are illustrated: a step increase or decrease, a "spike" (short-lived fluctuation), and a sinusoidal oscillation of period *p*. A step increase (shown in curve a) or decrease (curve b) in *C* results in an asymptotic approach of lake area (or level) to its new equilibrium value over a time span dependent on t_{e}. A spike produces a rapid increase (curve c) or decrease (curve d) in lake area (or level), followed by a slower, asymptotic recovery. In contrast, a sinusoidal oscillation in climate produces an oscillating response with a phase shift dependent on the relative magnitudes of *p* and t_{e}* .* For low-frequency (LF) variations in climate of period

In summary, a lake acts as a simple, low-pass signal filter with a characteristic time constant t_{e}. A wide range of lakes can act as climatic indicators on the decade-to-century time scale, although not necessarily over the whole range. Small, shallow, closed lakes or large, open lakes