The equilibrium geotherm is a temperature profile that balances the heat flow from the Earth's interior and the cooling that happens at the surface. It is difficult to evaluate because of variations in the composition and thermal properties of Earth's crust.
The equilibrium geotherm for the model Archaean crust can be determined by utilizing Fourier's Law of heat conduction and taking the rate of heat production into consideration.
The equilibrium geotherm equation is given by: q = k (dT/dz) + H, where q is the heat flow, k is the thermal conductivity, dT/dz is the temperature gradient, and H is the heat-generating internal heat source.
We can calculate the geotherm with the given data by rearranging the above equation. The temperature gradient is determined as dT/dz = (q - H)/k, where H is the heat-generating internal heat source. By integrating the temperature gradient, the temperature at any depth can be determined.
10. Depth of temperature influence on the Earth's surface: According to the question, the thermal conductivity is 2.5 W/m°C, and the specific heat is 10³ J/kg°C.
We know that temperature, depth, thermal conductivity, and heat flow are all interconnected and follow a relationship which is given by: q = k (dT/dz), where q is the heat flow, k is the thermal conductivity, and dT/dz is the temperature gradient.
From this equation, we can get the value of dT/dz = q/k = (20 × 10-³)/2.5 = 8°C/km. The temperature at the surface is assumed to be 0°C. We can determine the temperature at a depth of 2 km by utilizing the given equation: dT/dz = (T2 - T1)/(z2 - z1).
Hence, T2 = (dT/dz) × (z2 - z1) + T1 = (8 × 2) + 0 = 16°C. Similarly, the temperature at a depth of 5 km would be T2 = (dT/dz) × (z2 - z1) + T1 = (8 × 5) + 0 = 40°C.
So, the temperature difference between the surface and the depth of 2 km is 16°C, and the temperature difference between the surface and the depth of 5 km is 40°C.
Therefore, the depth of temperature influence is about 5 km.
11. Calculation of the equilibrium geotherm for a two-layered crust: We are given the following data: Heat flow at the base of the crust = 20 × 10-³ W/m², Thermal conductivity = 2.5 W/m°C, Internal heat generation of the upper layer = 2.5 μW/m, Internal heat generation of the lower layer = 0. The thickness of the upper layer = 10 km.
The thickness of the lower layer = 25 km. To calculate the equilibrium geotherm for a two-layered crust, we will utilize the same formula as we did in problem 9, which is given by q = k (dT/dz) + H. The temperature gradient will be different for the two layers as the upper layer has an internal heat generation of 2.5 μW/m and the lower layer has no internal heat generation.
The temperature gradient for the upper layer is dT/dz = (q - H)/k = (20 × 10-³ - 2.5 × 10-⁶)/(2.5) = 7.99°C/km, while the temperature gradient for the lower layer is dT/dz = (q - H)/k = (20 × 10-³)/(2.5) = 8°C/km.
Now, we will integrate the temperature gradient to get the temperature at any depth. For the upper layer, the temperature at the base of the crust would be T = (dT/dz) × (z - 10) + T1.
Substituting the values, we get T = (7.99 × 15) + 0 = 120°C. For the lower layer, the temperature at the base of the crust would be T = (dT/dz) × (z - 35) + T2. Substituting the values, we get T = (8 × 35) + 120 = 400°C.
So, the equilibrium geotherm for a two-layered crust is shown below.
12. Calculation of the equilibrium geotherm for a two-layered crust with different internal heat generation: We are given the following data: Heat flow at the base of the crust = 20 × 10-³ W/m², Thermal conductivity = 2.5 W/m°C, Internal heat generation of the upper layer = 0, Internal heat generation of the lower layer = 1 pW/m³.The thickness of the upper layer = 10 km, The thickness of the lower layer = 25 km..
Now, the temperature gradient for the upper layer is dT/dz = (q - H)/k = (20 × 10-³)/(2.5) = 8°C/km, while the temperature gradient for the lower layer is dT/dz = (q - H)/k = (20 × 10-³ - 1 × 10-⁹)/(2.5) = 7.99°C/km.
Now, we will integrate the temperature gradient to get the temperature at any depth. For the upper layer, the temperature at the base of the crust would be T = (dT/dz) × (z - 10) + T1.
Substituting the values, we get T = (8 × 15) + 0 = 120°C. For the lower layer, the temperature at the base of the crust would be T = (dT/dz) × (z - 35) + T2. Substituting the values, we get T = (7.99 × 25) + (120 + (1 × 10-¹² × 25 × 25)) = 284°C. Therefore, we see that the distribution of heat-generating elements has an effect on geotherms.
In this example, the temperature of the lower layer is lower than in the previous example, where the lower layer had no internal heat generation.
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