Lake Water Body in the Ice Season - Freezing of Lakes and the Evolution of their Ice Cover - page 209

and took the air temperature piecewise as daily means. Knowing the actual/predicted air
temperatures in days 1,

… , n, the prediction for the surface temperature at the time t =n ʔ t,

or on the nth day, is

X n

T ð t Þ ¼T ð 0 Þ e k t þð 1 e k t Þ T ð i D t)e kð n i ÞD t

i¼1

This is known as the method of the weighted air temperature sum. It is easy to see that
the solution is a weighted average of initial temperature and predicted daily mean air
temperatures with exponentially decaying weights. In other words, the air temperature is
low pass filtered for the mixed layer temperature, and the filter weights depend on the

ʻ − 1 r is added to the solution.

depth of the water layer. For r ≠ 0, the term [1 − exp(

- ʻ t)]

Example 7.4 . Freezing date delay (Simojoki 1940). Simojoki (1940) examined the

freezing date of lakes in Finland (Fig. 7.4 ). Taking a linear atmospheric cooling rate,
T a = t and r = constant, the slab model can be directly integrated as shown by

Eq. ( 7.11 ). Leaving the transient terms out, the solution is T(t)=T a (t)+

- ʱ

ʻ - 1

ʱ + ʔ T. The

freezing date t F is then obtained from T(t F )=T f , where T f =0 ° C is the freezing point

temperature. We have

t F k 1 þ a 1 D T

Fig. 7.4 Freezing date delay after 0 ° C downcrossing of air temperature as the function of lake
depth. Simojoki (1940) shows data from Finnish lakes, and the lines T = 2H and T = 4H refer to
delays of 2 and 4 days/m, respectively