A time‐dependent model of the lake‐averaged, vertical temperature distribution of lakes1,2

A model for predicting the time-dependent vertical thermal structure of lakes is presented. Radiative heating with depth, mixing induced by the surface wave field, and turbulent energy exchanges arc included as lake-averaged processes. The model was designed to be applicable to a wide range of lake sizes. Comparison of predictions with horizontally averaged observations from Lake Ontario during IFYGL and unaveraged (local) observations from Llyn Cwellyn, Wales, show good agreement. The depth of the summer thermocline in lakes is correlated with the size of the lake because of the more vigorous wind induced surface motion resulting from increased fetch. We present here a one-dimensional lake thermocline model in which this fetch dependence is parameterized in terms of the fetch-limited wave field. The model has a basic structure similar to those of Kraus and Turner (1967) and Denman ( 1973). Although our theoretical framework and general assumptions are similar to theirs, we have changed considerably the representation of important physical processes such as mixing due to surface waves and turbulent kinetic energy dissipation. Input variables required for the model are basin size (shoreline configuration, mean depth), solar radiation attenuation coefficient for the lake, wind speed and direction, air temperature and atmospheric water vapor pressure (or any convenient humidity variable) at some specified height above the lake surface, cloud cover, and an initial temperature profile. As has been noted by others (e.g. Kitaigorodskiy and Miropolskiy 1970; Linden 1975; Sundaram 1973) the KrausTurner model involves several sweeping assumptions not generally confirmed either in nature or in experimental studies. The ‘Study supported in part by the University of Michigan Sea Grant Program, maintained by NOAA, U.S. Dept. of Commerce. ’ Contribution 229 from the Department of Atmospheric and Oceanic Science, University of Michigan. key assumptions are neglect of mean flow, the assumption of a homogeneous surface layer, and the assumption that the energy available to increase the potential energy of the water column is proportional to the surface energy input, rwsfc, where r is surface stress and wsfc is a surface velocity scale, The turbulent exchange processes appear to depend also on lake size (Blanton 1973). Previous models of lake thermoclines have not included the effects of lake size on turbulence in the epilimnion; we describe methods to introduce this dependence. Also we present an alternative to the third assumption of the Kraus-Turner model, based on consideration of the dissipation of turbulent energy gained from the shear flow, wave breaking, and convection; implications and consequences of the first two assumptions are discussed later. To demonstrate the generality of the model for different sized basins, we have compared its predictions with observations from two lakes, one with a surface area of 1 km2, the other of 1.8 x lo* km2. For the smaller lake ( Llyn Cwellyn : Darbyshire and Colclough 1972), the available data only allow a l-day simulation over a period when strong winds significantly altered thermal structure. For the larger lake (L,. Ontario: IFYGL data), a much longer integration was possible. The basic parameters of the model are identical for both simulations. Symbols used in the text are given in Table 1. We thank R, L. Pickett and F. Rodante for providing edited versions of the IFYGL data. LIMNOLOGY AND OC.EANOGRAPEIY 687 JULY 1977, V. 22(4) 688 Tucker and Green Table 1. List of symbols. Table 1. Continued. __._ -.-_--_-----I--.-. --. .-. . . --.._ _ -__--_.-.. ..-.--...--. ____ __ ___.._. -_--_.----.__ -----_-.---