MATERIALS AND METHODS

The experiment was carried out at the Agricultural Faculty Experimental Field in Ondokuz Mayýs University, Samsun (41°21.86' N; 36°11.41'E; 187 mH), Turkey in June 2005 in order to compare the theoric values obtained from the solution of heat conductivity equation with the values obtained from experiment. Some physical and chemical properties of the soil profile were determined as following: particle size distribution by hydrometer method (Day, 1965), soil reaction, pH, 1:1 (w:v) soil water suspension by pH meter, electrical conductivity (EC₂₅°C) in the same soil suspension by EC meter, Organic Matter (OM) contents by modified Walkley-Black method (Kacar, 1994), CaCO₃ by Scheibler calsimeter (Kacar, 1994) and total N by Kjeldahl method (Kacar, 1994). Exchangeable K, Ca, Na and Mg were determined with 1 N NH₄OAc (Sağlam, 1997). Micro elements (Mn, Zn and Cu) were estimated in the extraction solution of 0.005M DTPA+0.01 M CaCl₂ + 0.1 M TEA by using an Atomic Absorption Spectrometer (Lindsay and Norvell, 1978). Available phosphorus was measured by the spectrophotometer with molybdophosphoric blue colour method (Olsen et al., 1954).

Measurements of soil temperature between 1st and 17th of June, 2005 were also done at 0, 10, 20, 30, 40, 50 and 60 cm depths by using mercury-in-glass thermometer (Sterling and Jackson, 1986). Temperature readings were taken six times in a day at 9 am, 11 am, 1 pm, 3 pm, 5 pm and 7 pm for 16 days. Heat diffusivity (b²) at that depth (x) can be estimated using 16 days frequency (ω = 2π/1382400sn = 4.54x10^-6 sec^-1) by the following equations (Hillel, 1982; Nerpin and Chudnovskii, 1984; Gülser and Ekberli, 2004):

where, T₀ is the amplitude of the surface temperature fluctuation, which is the range from maximum or minimum to average temperature and T_0(x) is the amplitude of the surface temperature fluctuation at any given depth.

The solution of soil’s heat conductivity equation has been determined by taking into consideration that initial unconditional limit problem of soil’s heat conductivity equation for 0≤x≤l ve 0<τ, generally, is as following:

(here, μ₁ (τ), μ₂ (τ) - are the functions expressing warmth values in τ time (sec.) on soil surface or any x = l layer (cm) of soil.

RESULTS AND DISCUSSION

Experimental results: The results showed that the textural class of soil is clay, low in organic matter (OM), neutral in pH, non saline according to EC value (Soil Survey Staff, 1993). In experimental soils, the amount of CaCO₃ diminishes along soil layer and demonstrates accumulation in soil depth of 40-50 cm. Total N and exchangeable K, Ca, Na, Mg vary in each layer downwards throughout the soil. Also, the amount of available phosphorus and micro elements (Mn, Zn and Cu) demonstrate decrease in each layer downwards throughout the soil (Table 1).

As a result of temperature measurements conducted during the research period (01-17 June, 2005), average temperature, amplitude and diffusivity values in soil surface and its lower layers enabling the control of initial unconditional solution of equation of heat conductivity in finite soil layer have been obtained (Table 2).

Table 1:	Some physical and chemical properties of the soil used in the study

Table 2:	Measured average values of soil temperature (T, °C), amplitude (To, °C) and diffusivity (b2, cm2 sec-1) at the different soil depths (01-17 June, 2005)

Due to the fact that soil surface gets warmer and cools downs comparatively faster, average temperature varies in larger interval (24.3-41.9°C). In lower soil layers temperature variance occurs in narrow interval. In 60 cm soil depth, average temperature varies between 19.4-19.7°C. The values of average temperature amplitude on soil surface vary between 7.1-12.5°C and this interval gets narrower in lower soil layers. The average diffusivity values vary between 0.13x10^-3-2.51x10^-3 (cm² sec^-1). This interval is getting more stretched in lower soil layers, therefore, results in diminishing temperature level. This increase occurs more in >30 cm soil depth. Along with other factors (climate, plant cover, grass cover, etc.), the variance of physical and chemical properties of soil in >30 cm soil depth also have significant effect on diffusivity.

Mathematical solution: Heat conductivity problem in finite soil depth, in case of harmonic variance of temperature on soil surface and in case of zero temperature level in any l soil depth, will be as following:

(1)

(2)

where, T (x, τ) is the soil temperature when x = 0, x is the vertical coordinate, τ is the time computed from a given initial moment (°C), b² is the diffusivity (cm² sec^-1), T_or is the average temperature of the soil surface, T₀ is the amplitude of the surface temperature fluctuation, which is the range from maximum or minimum to average temperature and ω = 2π/P - is radian variance (1/sec) of temperature, where P- is period (sec).

Let’s solve Eq. 1 in:

(3)

form by expressing limit conditions (2) as:

(4)

If

expressions that are obtained from solution (3) are taken into account,

(5)

will derive. Here,

The additional conditions derived from the expressions (3) and (4) will be as following:

(6)

From λ² + γ² = 0 characteristic equation of equation (5), λ₁ = γi ve λ₂ = -γi, therefore the solution will be in the following form: X (x) = C₁e^γix + C₂e^-γix (C₁, C₂ - are constants). The general solution can be written as following, by using Eyler’s formula:

(7)

are obtained from the solution of (7) by taking conditions (6) into account. Then, the solution (7) of equations (5)-(6) will be expressed in the following form:

(8)

Here, X₁ (x), X₂ (x) functions are real and imaginary parts of X (x) solution, consecutively.

The solution (4) is written as following by using the expression (8):

(9)

The solution of initial unconditional Eq. 1 will be obtained as following by spliting real and imaginary parts of solution (9):

(10)

The functions of X₁ (x) = ReX (x) and X₂ (x) = ImX (x) are calculated from the expression (8) by using equations of sin z = (e^iz - e^-iz)/2i, cos z = (e^iz + e^-iz)/2, e^z = e^x+iy = e^x (cos y + i sin y), for ∀z ε C.

(11)

(12)

are obtained.

The general solution of equation (1) will be as following, if we place the expressions (11) and (12) in (10) to their appropriate spots:

(13)

The solution (13) can be written without any units in the following form:

(14)

Here,

Theoretical testing of the solution: The limit conditions (2) are justified, when the solutions of (13) or (14) are x = 0 and x = l. Based on expression (10):

(15)

From expression (11) [ch (2kl) - cos (2kl)] X₁ (x) = ch [(2l - x) k] cos (kx) - ch (kx) cos [2l - x)k] is derived and used in expression (15) as following:

(16)

And from expression (12)

[ch (2kl) - cos (2kl)] X₂ (x) = sh [(2l - x) k] sin (kx) - sh (kx) sin [2l - x) k] is derived and used in expression (15) as following:

(17)

If the expressions (16) and (17) are placed in expression

(18)

Table 3:	Comparison of values (Tm, °C) obtained from experiment and calculations

will be derived.

As seen from the expression (15) and (18), the solution (13) or (14) justifies Eq. 1.

Comparison of values obtained from calculation and experiment: Based on solution (13) of heat conductivity equation, the average absolute (Δ = |T - T_m|) and relative (δ = (Δ/|T_m|)x100, %) errors between calculated temperature values by soil layers and temperature values obtained from experiment vary between 0.5-1.9 and %2.41-5.66, consecutively (Table 3). And limit values for agricultural researches are in (< %10) interval (Dospexov, 1973).

The temperature variance in soil effects biological development of plant and soil properties in significant way. Therefore, the estimation of temperature along soil layer and regulation of temperature regime is one of the main problems in Agriculture. According to heat conductivity equation’s solution (13) or (14), the estimation of temperature depending on time in any soil layer is possible, when average temperature by soil layer, amplitude and diffusivity values are known.