|Figure 2. Baseline temperature of
5 degrees and linear dependence
between temperature (x) and
development rate M(x)
According to the theory of Degree Days (or Effective Heat Sum) applied in Finland the growing season for boreal or arctic terrestrial plants advances only in the temperatures above the baseline value of 5 degrees. The daily accumulated Degree Day units are then calculated by subtracting the baseline temperature from the daily average temperature. In other words, the development rate, M(x) has been normalized so that it gets value 1 when temperature is 6 degrees. When temperature is 5 degrees the M(x) gets value zero. The equation for the development rate (Eq.1) becomes thus
(1) M(x) = x – 5
Applying the Finnish method for Degree Day calculation between bud burst and full leaf, the following measures result:
Average duration 28.3 DD units
Range 5.1 – 20.0 DD units
Standard deviation 12.1 DD units
Coefficient of Variation 42.8 per cent
The most important measure in this case is the Coefficient of Variation
(CV) as it is a dimensionless number, and it can be used to compare the amount of variance between populations with different means. As the CV in the dataset for calendar days is 64.3 per cent and in the dataset of Degree Days only 42.8 per cent, the Degree Days as a concept is nearer to the invariant time value than the concept of calendar days.
The baseline temperature of five degrees, commonly used in Finnish DD value calculation is a convenient measure, as the disappearance of melting snow and rise of daily average temperature into five degrees coincide with each other. The baseline value, however, presupposes that during the days of no snow but temperature under five degrees, say between zero and five, there is no development in Downy birch. This presupposition has no physiological background, as the plant metabolism is in slow function in chilly temperatures.
Therefore, is there more ambient baseline temperature b for the development of Downy birch, lower than 5 degrees? This can be studied with the help of Coefficient of Variation. Let us maintain the normalization condition for the development rate, Thus
M(6) = 1, and
M(b) = 0
A more general equation for the development rate thus becomes (Eq. 2)
(2) M(x) = (x – b) / (6 – b)
Next, let us vary the baseline temperature, calculate a new measure DDb for Degree Days and study how the Coefficient of Variation changes along with the temperature. It minimizes in a new baseline temperature of 1.98 degrees (Figure 3).
The corresponding measures for the minimum solution result
Baseline temperature, b 1.98 degrees
Average duration 11.1 DDb units
Range 5.3 – 20.9 DDb units
Standard deviation 3.99 DDb units
Coefficient of Variation 36.0 per cent
|Figure 3. Minimum Coefficient of
Variation for Optimum baseline
temperature of Downy birch.
It is notable that the new value of DDb units (11.1) is considerably lower as compared to the original DD5 units (28.3). This is logical as the calendar time is included as boundary value in Equation 2. The DDb value approaches the calendar time duration 6.1 days as b approaches minus infinity.
In calculation of the Degree Day units for Downy birch the baseline temperature of 2 degrees (rounded from 1.98) gives more appropriate development rate measures than the widely applied baseline of 5 degrees. This corresponds to the experience on ecological phenomena near to the arctic timber line. It is, however, a biological and meteorological fact, that many different plant species follow different baseline temperatures
. Most commonly used baseline temperature is 10 degrees, as it suits best for instance for maize. For potato 8 degrees is used, whereas 5.5 degrees seems to be the best for temperate cereals wheat, barleay, oats and rye. Pohjonen (1975) determined optimum baseline temperature of 1.699 degrees for Italian ryegrass (Lolium multiflorum) grown for fodder in Finnish Lapland. The optimum baseline temperature for boreal or for arctic timber trees has not been studied at similar intensity.
Another notable issue is that the development rate function M = M(x) cannot be linear, increasing function of temperature. It is bound to bend, level and finally drop to zero as temperature increases high enough. Such high temperatures, however, seldom occur in arctic timberline, that this would happen. Linear function of development rate seems to be good approximation for practical agricultural and silvicultural prediction purposes.
Pohjonen, V. 1975. A dynamic model for determining the optimum cutting schedule of Italian ryegrass. J. Scient. Agric. Soc. Finland. 47:71-137.
Sarvas, R. 1972. Investigations on the annual cycle of development of forest trees. Comm. Inst. For. Fenn. 76.3.
text by Veli Pohjonen 8.1.2009, update 12.1.2009.