Heat flow measurements under some lakes in the Superior Province of the Canadian Shield: Discussion
In their recent paper Allis and Garland (1979) have attempted to make use of dimictic lakes for heat flow measurements. While recognizing that to allow for the effects of temperature changes at lake bottom it is necessary to apply large corrections to the temperatures and temperature gradients in the sediments, the form of the correction they have applied may not be sufficiently accurate to give reliable results. They show, in their Fig. 2, a temperature-time function for the bottoms of lakes which is essentially a square wave, with sudden temperature changes at 6 month intervals, on which is superimposed either a longer period change or an aperiodic change, and use a 30 term Fourier series to approximate the function. There are two problems with this approach which may lead to serious error in the estimate of the correction required. Firstly, since only 18 months of records are shown it is assumed that only 18 months of data are incorporated in the Fourier series, which implies that the longest term component has a period of less than 18 months; secondly, such a procedure implies a continuous temperature-time function for each term, whatever its period, which conflicts with their statement that the 2 K temperature rise in May 1976 was due to a period of unusual warmth and, therefore, should presumably be treated as a unique event. In what follows I shall, for the sake of simplicity, assume sediment surface (i.e., lake bottom) temperature disturbances of unit amplitude. Figure 1, which is included as a form of baseline against which to compare other sets of curves, illustrates the disturbances to the temperature gradient due to an annual sinusoidal variation of V = Vo cos (2n tlP), where P (= 1 year) is the period, at the surface of sediments of diffusivity = 0.17 mm2/s; the numbers on the curves represent the time, in months, at which the gradients are measured using the time at which V = V, as the reference zero time. Sinusoidal changes of different period and amplitude would have a similar symmetrical appearance but would, of course, be of different amplitude at different depths. For the basic square wave function shown by Allis and Garland, the disturbance to gradients in the lake sediment should be computed on the basis of repetitive step functions (Beck 1977) and it is easy to show that even with equal amplitude steps the resultant computation based upon data for only 1 year will be very different from that based upon data for 2 to several years. This is illustrated in Fig. 2 using a sediment diffusivity of 0.17 mm2/s. Figure 2a shows curves for corrections to the temperature gradient due to unit change of temperature at t = 0 followed by a similar change of opposite sign at t = 6 months; the number on a curve represents the number of months (1 month = 30 days) that have passed, after the initial disturbance, before a measurement is made. This approach can be used for