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, EL and PL are the evaporation and precipitation rates over the lake surface (in units of depth per unit time), AL is the lake-surface area, and Gi and Go 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 ALe is determined by (EL - PL) and R, which reflect the surface water balance over the lake and its surrounding catchment, respectively. Hence,
At equilibrium, the ratio of the lake-surface area ALe to the total catchment area AC (including the water surface) is given by
where C is a dimensionless index dependent only on R, EL, and PL (Mason et al., 1994):
On time scales longer than a year, R may be approximated by the expression
where AB is the area draining into the lake (excluding the water surface), and PB and EB are, respectively, the mean precipitation and evapotranspiration rates over the basin (in units of depth per unit time). Since by definition AC = AL + AB, 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) te, defined as the time taken to reach a fraction (1 - 1/e), or 63 percent, of the total change in area. The response time te is given by
where L is lake level (Mason et al., 1985, 1994). Values of te calculated by Mason et al. (1994) for modern closed lakes vary from 1.5 to 350 years. te 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 te 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 te. 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 te. For low-frequency (LF) variations in climate of period p 2pte, the lake is approximately in equilibrium and exhibits negligible phase shift (curve e). For high-frequency (HF) variations of period p 2pte, the lake lags the variations in climate with a phase shift of -p/ 2 (curve f); the maximum and minimum rates of increase in area (or level) correspond to the maxima and minima in C, respectively (Mason et al., 1994). For practical purposes, the limits of the LF and HF bands can be taken as p > 20te and p < 2te. In theory, it should be possible to invert a time series of water-surface area or water level derived from a lake with known characteristics to obtain a record of variations in C (I. M. Mason, pers. commun.).
In summary, a lake acts as a simple, low-pass signal filter with a characteristic time constant te. 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