Stability Equation

mix the lake to an isothermal condition (actually uniform density). Zero stability is usually encountered
during isothermal conditions. Negative stability indicates that the rate of energy loss from near the
surface is sufficiently great that mixing may proceed spontaneously due to the buoyancy of underlying
waters. Negative stability therefore is seldom encountered on a large scale and it is usually a short-term
phenomenon.
1.2.6.2 Stability Equation
1   zm
mz  0
S'
z & Az z & z dz
A0
WHERE:
S = stability,
A0 = Area of the lake surface (depth = 0),
Az = area of lake at depth
zm = maximum depth
z0 = lake surface
z- = depth at which water temperature corresponds to average density
ρz = density at depth
-
ρ = volume-weighted average density
Heat budgets may be derived using two different approaches. The simplest approach is to employ
morphometry with known temperature distributions (which are used to calculate densities in the stability
calculation, see the relationship?) This is an empirical approach because it uses knowledge in hand to
further describe characteristics which already exist. Using known specific heats for water and the
temperature and the depth-volume relationships, it is simple to calculate the amount of thermal energy in
calories or other units contained in the various depths of the lake. These are then summed and the total
energy (caloric) content can be calculated relative to some reference value. This reference value is
usually chosen as the minimum temperature experienced by the lake during the year. Calculating the
changes in heat content, it is easy to see the effect of various processes on the entire lake as the seasons
progress.
Another approach to the heat budget is more complex but this is the approach used in many
sophisticated models. This analytical approach uses the rates of energy exchange combined with lake
morphometry to predict the outcome of any modification to energy exchange processes. It can also
predict the same characteristics that can be calculated empirically using the method previously
described. Obviously, this is the more powerful approach and these models are very successful in
making useful, accurate predictions about thermal characteristics.
1.2-17