thermomechanical prop of lithosp [Compatibility Mode]

Thermomechanical Properties of the Lithosphere (for understanding deformation)understanding deformation)

prior to any investigation on the mechanical stability of continental lithosphere it is important to have some ideas of the distribution oflithosphere 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 onThe 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)

L t' th t Q( ) i th h t fl t i d f ll li dLet's assume that Q(z) is the heat flow entering one end of a small cylinder 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 µWm-3)

(Eq. -3)

Equation of heat conduction in one-dimensionTh h t t t f f k l b dThe heat content of a mass of rock can also be expressed 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), werepresentative 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 (surface)1 let temp T T0 at 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 (Eq. - 11)integration it becomes

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. - 13)

This equation allows to determine the temperature profile for the domain 0<z<zcas 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

h di t ib ti f di i l t i A d t d dan homogeneous distribution of radiogenic element i.e. A does not depend 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 aEquation (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 temperatureIf the production of radiogenic heat is zero in the mantle then the temperature profile beneath the Moho is linear and then:

(Eq. - 15)( q )

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 b l h t fl Q Q2 2With a basal heat flow Q=-Q2 on z=z2.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 subject to the boundaryThe solution of these two differential equations, subject to the boundary conditions and matching both temperature T and temperature gradient on the boundary z=z1, is

(Eq. - 18)( q )

for 0<=z<z1for 0<=z<z1

(Eq. - 19)

for z1<=z<z2

Derive the equations for different thermal conductivities of the t l B id di i h t ti l btwo layers. Besides, radiogenic heat generation can also be modeled.

Thermal consequences of lithospheric deformations

Potential Geotherm of Deformed Lithospheres

Let's assume that we change the thickness of the crust and the thickness of the gwhole 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 byequilibrium 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 recoversthickness is calculated assuming that the heat flow entering the Moho recovers its original value:

(Eq - 21)(Eq. 21)

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 hC-N scheme.