Thermomechanical Properties of the Lithosphere (for understanding deformation) prior to any investigation on the mechanical stability of continental lithosphere it is important to have some ideas of the distribution of density, strength, and temperature with depth , and how those profiles are affected and affect deformation
Temperature distribution in the continental lithosphere The steady state geotherm depends on (i) the mantle heat flow entering the crust at the Moho, (ii) the radiogenic heat production in the lithosphere (the crust is the radiogenic layer as there is the mantle is extremely depleted in radiogenic elements), (iii) the distribution of the radiogenic elements, and (iv) the rate at which the heat is conducted away from the lithosphere
The heat flow (Q) is the rate at which heat is conducted through a solid, it is proportional to the temperature gradient: (Eq. -1)
Let's L t' assume that th t Q(z) Q( ) is i the th heat h t flow fl entering t i one end d off a smallll cylinder li d or rock of section a. The heat flow coming out at the other ends of the cylinder is Q(z+dz) and can be defined with a Taylor series (Eq. -2)
The internal heat production due to radioactive decay (U, Th, K) in the cylindrical body is given by A a δz and thus the total heat gained in the cylindrical mass as (A is the radioactive heat generation in ¾Wm-3) (Eq. -3)
Equation of heat conduction in one-dimension Th heat The h t content t t off a mass off rockk can also l be b expressed d as (Eq. -4)
Equating eq. 3 and 4 we get (Eq. - 5)
Combining with eq-1 we get
(Eq. - 6)
Equation – 6 is the one-D heat conduction equation. If we consider that our representative volume of rock moves up or down (burial or exhumation), we can add a heat advection term
(Eq. - 7)
Calculation of simple geotherms (without advection) Equilibrium Geotherms
(Eq. - 8)
To solve this, we impose two boundary conditions : 1 – let temp T = T0 at z=0 z 0 (surface) 2 – let the surface heat flow Q = -Q0 at z=0
Integrating once (Eq. - 9)
As per the second boundary condition C1 = Q0/K so the equation reduces to (Eq. - 10)
After the second integration it becomes
(Eq. - 11)
With the boundary condition T=T0 at z=0, the equation can be rewritten as (Eq. - 12)
Consider the depth of the Moho as zc and Qm as the heat flow from mantle to the base of the crust. Now a different set of boundary condition can be put as T=T0 at z=0, and
In that case
And the solution becomes (E - 13) (Eq.
This equation allows to determine the temperature profile for the domain 0<z<zc as a function of (i) the heat flow entering the crust at the Moho,(ii) the crustal thickness, and (iii) the thermal properties of the crust. This equation is valid for an homogeneous h di distribution t ib ti off radiogenic di i element l t ii.e. A does d nott depend d d on z. Because the second terms of the equations (12) and (13) equal we can write:
(Eq. - 14)
Equation (14) indicates that the contribution to the surface heat flow of a radiogenic heat production A is A.zc. Similarly, the contribution of the mantle heat flow Qm to the temperature at a depth z is (Qm/K)z If the production of radiogenic heat is zero in the mantle then the temperature profile beneath the Moho is linear and then:
((Eq. q - 15))
Which in turn can be written as
(Eq. - 16)
Temperature at Moho is (Eq. - 17)
With zc=35 000 m, zl=100 000 m, Tl=1250 ºC, K=3 Wm-1°C-1 and A=0.75 x 10-6 Wm-3 we get Tm ~ 563 C
Steady State Geotherms in Multi Layers models A=A1 for 0<=z<z1, A=A2 for z1<=z<z2, T=0 on z=0. With a b basall h heatt flflow Q Q=-Q2 Q2 on z=z2. 2 In the first layer, 0<=z<z1, the equilibrium heat conduction equation is
In the second layer, z1<=z<z2, the equilibrium heat conduction equation is
The solution of these two differential equations equations, subject to the boundary conditions and matching both temperature T and temperature gradient on the boundary z=z1, is
((Eq. q - 18))
for 0<=z<z1
(Eq. - 19)
for z1<=z<z2
Derive the equations for different thermal conductivities of the t two layers. l Besides, B id radiogenic di i h heatt generation ti can also l b be modeled.
Thermal consequences of lithospheric deformations Potential Geotherm of Deformed Lithospheres Let's assume that we change g the thickness of the crust and the thickness of the whole lithosphere by a factor fc and fl respectively. After deformation and complete thermal relaxation the temperature profile is given by the potential steady state geotherm. Then, the temperature profile when the thermal equilibrium is reached is given by (Eq. - 20)
With zpl the thickness of the thermally equilibrated continental lithosphere. That thickness is calculated assuming that the heat flow entering the Moho recovers its original value: (Eq - 21) (Eq.
Therefore (Eq. - 22)
Incorporating (22) in (21) and rearranging we get the Moho temperature after deformation and thermal relaxation
(Eq. - 23)
Transient Geotherms Take the 1-D heat conduction equation and impose boundary condition as discussed. Solve the equation numerically by using a C N scheme. C-N h