The evolution of the moon: A finite element approach |
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| Authors: | Soman Chacko J Cl De Bremaecker |
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| Institution: | (1) Department of Geology, Rice University, Houston, Texas, U.S.A.;(2) Present address: Exxon Production Research Company, Houston, Texas, U.S.A.;(3) Department of Geology, Rice University, Houston, Texas, U.S.A. |
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| Abstract: | Thermal convection has considerable influence on the thermal evolution of terrestrial planets. Previous numerical models of planetary convection have solved the system of partial differential equations by finite difference methods, or have approximated it by parametrized methods. We have evaluated the applicability of a finite element solution of these equations. Our model analyses the thermal history of a self-gravitating spherical planetary body; it includes the effects of viscous dissipation, internal melting, adiabatic gradient, core formation, variable viscosity, decay of radioactive nucleides, and a depth dependent initial temperature profile. Reflecting current interest, physical parameters corresponding to the Moon were selected for the model.Although no initial basalt ocean is assumed for the Moon, partial melting is observed very early in its history; this is presumably related to the formation of the basalt maria. The convection pattern appears to be dominated by an L-2 mode. The present-day lithospheric thickness in the model is 600 km, with core-mantle temperatures close to 1600 K. Surface heat flux is 25.3 mW m–2, higher than the steady state-value by about 16%.The finite element method is clearly applicable to the problem of planetary evolution, but much faster solution algorithms will be necessary if a sufficient number of models are to be examined by this method.Notation 
coefficient of thermal expansion
- 
ij
Kronecker delta
- 
absolute or dynamic viscosity
-  
perturbation in temperature
- 
thermal diffusivity
- 
kinematic viscosity
- 
density
- 
stress tensor
- B.P.
before present
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c
specific heat at constant pressure or volume (Boussinesq approximation)
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d
depth of convection
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E
*
activation energy for creep
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g
gravity
-
Ga
billions of years
-
H(t)
heat generation per unit mass per unit time at timet
-
k
Boltzmann's constant
- K
mean thermal conductivity
-
Ma
millions of years
-
p
pressure
-
q
heat flux
-
q
ss
steady-state heat flux
- Ra
Rayleigh number
-
S
volumetric heat sources, includes radioactivity and viscous dissipation
-
t
time
-
T
temperature
-
u
verocity vector
-
V
*
activation volume for creep |
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