The reference
The temperatures and depth of origin of the Lutynia mantle xenoliths (960–1,000°C, 50 km, for details see The lithospheric mantle in the region of the Niedźwiedź massif) may serve as a reference point for temperature distribution modelling. The xenoliths represent the thermal state of the sampled mantle at ca. 4.5 Ma. Since the Lutynia volcano was isolated, and no intense volcanic activity occurred in the neighbouring area of eastern part of West Sudetes and Fore-Sudetic Block, we assume that the sampled mantle was not rejuvenated thermally at the time of volcanic activity and is therefore representative for the lithospheric mantle in the area. We use the 50 ± 5 km depth for the calculated temperature range, to account for the uncertainties in pressure estimates.
The equation
The numerical methods are used for the calculation of geotherms. The thermal steady state is assumed and consequently our model is based on the following equation:
$$ \frac{\text{d}}{{{\text{d}}z}}\left( {k(T,p)\frac{{{\text{d}}T}}{{{\text{d}}z}}} \right) + \rho (p)H(z) = 0, $$
(3)
where k [W °C−1 m−1] is the coefficient of thermal conductivity, T [°C] is the temperature, p [Pa] is the pressure, z [m] is the depth, ρ [kg m−3] is the density and H [W kg−1] is the rate of the radiogenic heat production per unit mass (note that A = ρ H is the rate of radiogenic heat production per unit volume, A [W m−3]). The thermal diffusivity κ is defined as κ = k/(ρ c
p
).
The geotherms and position of LAB are often discussed using isotherms (e.g. the 1,300°C isotherm is chosen by Majorowicz 2004, the 1,200°C by Tesauro et al. 2009) and/or potential adiabatic temperatures of 1,300°C.
A simple parametric model
We use here a simple parametric model to present the basic thermal properties of the crust-lithosphere system. According to Whittington et al. (2009), “many thermal models of the Earth’s lithosphere assume constant values of κ (~1 mm2 s−1) and/or k (~3–5 W m−1 K−1) owing to large experimental uncertainties […]. Recent advances permit accurate (±2%) measurement […]”. Moreover, they indicate that κ strongly decreases from 1.5 to 2.5 mm2 s−1 at ambient conditions to 0.5 mm2 s−1 at mid-crustal temperatures. Their results as well as the results of Abdulagatov et al. (2006) suggest that the average k of the crust is ~2 W m−1 K−1.
In our simplified model, the crust is assumed to be a homogenous layer of thickness D, thermal conductivity k
c and, heat generation rate per unit volume A. The total heat generation rate for the crust per unit surface area is AD [W m−2]. The surface heat flow density and the surface temperature are Q
s [W m−2] and T
S [°C], respectively. For lithosphere below the MOHO, we assume k = const and A = 0. So, the model has the following parameters: D, k
c, A, k. Temperature distribution in the mantle is given by (e.g. Czechowski 1993):
$$ T(z) = \frac{1}{k}\left( {Q_{\text{s}} - hD} \right)(z - D) + G\quad {\text{for}}\quad D \le z. $$
(4)
where
$$ G = - \frac{{AD^{2} }}{{2k_{\text{c}} }} + \frac{{Q_{\text{s}} D}}{{k_{\text{c}} }} + T_{\text{s}} . $$
(5)
Let us discuss the properties of the model. The following starting parameters were chosen for calculations: D = 31,000 m, k = 3 W m−1 K−1, A = 1.3 × 10−6 W m−3, k
c = 1.9 W m−1 K−1, Q
s = 69.5 mW m−2, and T
s = 4°C. Four panels of Fig. 9 give geotherms for different values of some chosen values of parameters. The unspecified parameters are not changed comparing to above set.
Figure 9a gives geotherms for the following values of the mantle conductivity: k = 2.5, 3, 3.5, 4 W m−1 K−1. One can see that most of them fulfil the Lutynia xenoliths p–T range. The depth of the 1,300°C isotherm varies from 74 to 98 km.
The mean heat generation in the crust A is changed in the Fig. 9b. It is the least known parameter of the models depending mainly on the effective granite participation (or other rocks with high content of radioactive isotopes) and its composition. Note that lower values of A in the crust result in higher mantle temperature and smaller depth of the 1,300°C isotherm. Note also that for the model A of the crust (Fig. 2), the value A = 1.3 × 10−6 W m−3. The geotherm for this value is also the best fit of the Lutynia xenoliths p–T range.
Figure 9c presents results of special manipulation of the mean heat generation in the crust h and thermal conductivity of the mantle k. For four values of A: 1.1 × 10−6, 1.2 × 10−6, 1.3 × 10−6, 1.4 × 10−6 W m−3), the four values of k are chosen: 5.06, 3.91, 3.06, 2.42 W m−1 K−1, respectively. Such pairs of A and k give geotherms crossing the centre of the Lutynia xenoliths p–T range (i.e. the point: T = 950°C and z = 50,000 m). For this panel, k
c = 2 W m−1 K−1. The depth of the 1,300°C isotherm varies from 83 to 100 km.
The geotherms for different values of the crust conductivity k
c are given in Fig. 9d. In this case, the geotherms differ one from another for the whole range of z (in the crust as well as in the mantle). The k
c = 2 W m−1 K−1 is the best fit to the Lutynia xenoliths p–T range. The 1,300°C isotherm is located in the range of depth from 75 to 90 km.
This discussion indicates that even a simple model could give realistic behaviour of the temperature distribution in the lithosphere.
An advanced thermal model
More advanced model is presented here. The heat production and the thermal conductivity for the considered crustal rock are measured and given in Table 1, Figs. 10 and 11. The layered structures of the crust and their thermal properties are adopted according to Fig. 2. The effect of temperature and pressure on the crustal rock conductivity is included in the model. According to Abdulagatov et al. (2006), a sharp increase in the conductivity is found for rocks at low pressures (i.e. between 0.1 and 100 MPa) along various isotherms (between 0 and 150°C). At higher pressures (p > 100 MPa), weak linear dependence of the conductivity with the pressure was observed. Abdulagatov et al. (2006) used the following function to describe their experimental results:
$$ k(T,p) = k_{0} (T,p = \infty )\frac{(1 - 0.661\,\Upphi (p))}{(1 + 41.3\Upphi (p))} $$
(6)
where k
0(T, p = ∞) is the thermal conductivity of the rock at high pressure (i.e. for p → ∞), when all of the internal cracks are assumed to be closed. The function Φ(p) is:
$$ \Upphi \left( p \right) = \Upphi_{0} \exp \left( { - p/p_{0} } \right), $$
(7)
where Φ0 and p
0 are some constants determined by the experiments (see Table 1).
To describe the temperature dependence of k(T), we follow the paper of Clauser and Huenges (1995) and Zoth and Hänel (1988). They express the conductivity of the typical rocks by the function:
$$ k(T) = \left( {E + \frac{B}{350 + T}} \right) $$
(8)
where the coefficients E and B are given in Table 2. Note, however, that the absolute value of the thermal conductivity of the rocks considered here is measured independently (at ambient condition, see Table 1). Therefore, we use the above formula to determine the temperature dependence only (not the values). Eventually, combining formulas (6) and (8), we get:
$$ k(T,p) = k_{0} \left( {\frac{{1 + 41.3\,\Upphi_{0} }}{{1 - 0.661\,\Upphi_{0} }}} \right)\left( {\frac{1 - 0.661\,\Upphi (p)}{1 + 41.3\,\Upphi (p)}} \right)\left( {\frac{{E + \frac{B}{350 + T}}}{{E + \frac{B}{350 + 20}}}} \right), $$
(9)
where k
0 is the coefficient of thermal conductivity of a given rock at ambient conditions (given in Table 1) and T is expressed in [°C]. We believe that the above formula describes well T–p dependence of k in the crust (see also Seipold 1998).
Table 2 Constants: E, B, k
0
,Φ
0, p
0 used in formulas (7) and (9)
Let us now discuss the thermal properties of the rocks below the MOHO. Most of the results (e.g. Clauser and Huenges 1995; Seipold 1998; Tommasi et al. 2001; Abdulagatov et al. 2006; Katsur 2007; Whittington et al. 2009) indicate that thermal conductivity and thermal diffusivity stabilize for the p–T range of upper lithospheric mantle (i.e. for: 1–4 GPa, and 700–1,500°C). The same behaviour of k results from Eq. 9. Moreover, Katsur (2007) states: “thermal diffusivity in the upper mantle has almost constant values of 7–8 × 10−7 m2 s−1”. Therefore, we assume in the model that k is constant below the MOHO. Eventually, the calculations are performed for k = 2.5, 3, 3.5 W m−1 K−1. Note that Hofmeister (1999) results for forsterite give 4.5 W m−1 K−1 for 100°C, but 2.2 W m−1 K−1 only for 1,300°C. Other data suggest that k in the upper mantle is in the range 2–3 W m−1 K−1.
The radiogenic heat production chosen for the mantle (2.3 × 10−12 W kg−1) corresponds to depleted peridotite. The compressibility of the rocks is described by:
$$ \rho (p) = \rho (0)\left( {1 + \frac{p}{K}} \right), $$
(10)
where ρ(0) is the density for zero pressure given in Table 1. The bulk modulus K is calculated from seismic velocity of longitudinal waves V
p
from the formula:
$$ K = \frac{5}{9}\frac{{V_{p}^{2} }}{\rho (0)}, $$
(11)
It is assumed here that both Lame’s modules are equal, i.e., V
s
= (1/3)1/2
V
p
. We used this approximate formula because density is not a critical factor for our model.
The following boundary conditions are used:
-
1.
Heat flow at the surface, i.e., Q
s(z = 0) = 69.5 or 62.5 mW m−2.
-
2.
Temperature at the surface, i.e., T(0) = −4°C for z = 0 m.
The Eq. 3 with k(T, p) given by (9) is nonlinear. To solve it, we used the software developed by Czechowski (1993).
Results of the model
The calculations were performed for two models of the crustal structure: model A and model B (Fig. 2). The discrepancy of the results is negligible, so model A was used for the rest of the calculations. The main results are geotherms, i.e., functions T(z). The results presented in Fig. 12 indicate that:
-
1.
Correction of the heat flow in the Niedźwiedź borehole based on the paleoclimatic data (see the previous chapter) results in considerable increase in temperature in the lower crust and in the upper mantle (compare model A for Q
s = 69.5 mW m−2 and model A for Q
s = 62.5 mW m−2 and the appropriate lines in Fig. 12).
-
2.
The critical test is a comparison with the Lutynia xenoliths data. The Lutynia xenoliths p–T range is given by the rectangle on the Fig. 12. The geotherms calculated for model A and B fit well this constraint if k = 3 W m−1 K−1 is used as the mantle conductivity. Lower and higher values of k in the range from 2.5 to 3.5 W m−1 K−1 are also possible.
-
3.
The models A and B correspond to essentially the same mantle heat flow, i.e., 27.6 mW m−2.
Depth of the 1,300°C isotherm for paleoclimatically corrected heat flow is less than 100 km (Fig. 12) For uncorrected heat flow, the isotherm would be considerably deeper (130–140 km). Note that this isotherm is used often as the lithosphere–asthenosphere boundary. The same conclusion holds for LAB defined by the potential adiabatic temperature.