Wind Mixing and Restratification in a Lake near the Temperature of Maximum Density

Abstract

The cooling of a freshwater take provides an opportunity for studying wind mixing and restratification under the peculiar conditions associated with a density maximum. The concepts are explored using a mixing-layer model that incorporates both nonlinearity and pressure dependence in the equation of state; the results provide a basis for interpreting temperature structure in Babine Lake and for making some more general observations on mixing and restratification. Following destruction of summer stratification by wind mixing and convection in autumn, the lake is essentially isothermal as it cools through 4°C. Near this temperature, the coefficient of expansion becomes so small that pressure effects, which play little part in the dynamics at higher temperatures, can dominate the stability. In effect, the depth at which the local temperature equals the temperature of maximum density at constant pressure marks the transition between forced and free convection. Above this transition depth, the wind must work against buoyancy forces during cooling, below it the water is gravitationally unstable. The existence of a transition depth allows conditional instabilities to occur in which a downward movement of initially stable water can lead to gravitational instability. As the lake cools the stability becomes less sensitive to pressure and a given heat flux produces a progressively greater buoyancy flux leading to the process of restratification. The essential physics are contained in the specification of a Monin-Obukhov mixing length, for a cooling lake near the temperature of maximum density this length can assume complex values, both real and imaginary components having distinct physical interpretations. The theory predicts that if the boat flux and the work done by the wind remain relatively constant the mixing layer properties will change exponentially with time, a prediction that is supported by the observations. Since the temperature profile that evolves beneath a retreating mixed layer retains information on the conditions that led to its formation, the theory also provides a consistent basis for interpreting winter profiles in terms of the relative strength of wind mixing and buoyancy flux during the cooling period. It is shown that, in general, lakes subjected to relatively more intense wind mixing during winter restratification will have lower interior temperatures. The observations lead to a calculation of the fraction of energy that is used to redistribute buoyancy. Estimates based on 33 days of data yield a value of 0.26r0u*3, with standard deviation 0.04, where u* is the friction velocity of the water. Measurable temperature gradients exist within the mixing layer and these gradients increase as the mixing layer retreats,