VISCOUS%20HEATING%20in%20the%20Earth

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VISCOUS HEATINGin the Earths MantleInduced by
Glacial Loading
L. Hanyk1, C. Matyska1, D. A. Yuen2 and B. J.
Kadlec2 1Department of Geophysics, Faculty of
Mathematics and Physics, Charles University in
Prague, Czech Republic2Department of Geology and
Geophysics, University of Minnesota, Minneapolis,
USA
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IDEAHow efficient can be the shear heating in
the Earths mantle due to glacial forcing, i.e.,
internal energy source with exogenic
origin?(energy pumping into the Earths
mantle)APPROACH to evaluate viscous
heating in the mantle during a glacial cycle
by Maxwell viscoelastic modeling to compare
this heating with background radiogenic heating
to make a guess on the magnitude of surface heat
flow below the areas of glaciation
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PHYSICAL MODEL a prestressed selfgravitating
spherically symmetric Earth Maxwell
viscoelastic rheology arbitrarily stratified
density, elastic parameters and viscosity both
compressible and incompressible models cyclic
loading and unloading
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MATHEMATICAL MODEL momentum equation
Poisson equation Maxwell constitutive
relation boundary and interface conditions
formulation in the time domain (not in the
Laplace domain) spherical harmonic
decomposition a set of partial differential
equations in time and radial direction
discretization in the radial direction a set
of ordinary differential equations in time
initial value problem
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NUMERICAL IMPLEMENTATION method of lines
(discretization of PDEs in spatial directions)
high-order pseudospectral discretization
staggered Chebyshev grids multidomain
discretization almost block diagonal (ABD)
matrices (solvers in NAG) numerically stiff
initial value problem (Rosenbrock-Runge-Kutta
scheme in Numerical Recipes)
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DISSIPATIVE HEATING f (r )
In calculating viscous dissipation, we are not
interested in the volumetric deformations as
they are purely elastic in our models and no
heat is thus dissipated during volumetric
changes. Therefore we have focussed only on the
shear deformations. The Maxwellian constitutive
relation (Peltier, 1974) rearranged for the
shear deformations takes the form ? tS / ? t 2
µ ? eS / ? t µ / ? tS , tS t K div u I , eS
e ? div u I , where t, e and I are the
stress, deformation and identity tensors,
respectively, and u is the displacement vector.
This equation can be rewritten as the sum of
elastic and viscous contributions to the total
deformation, ? eS / ? t 1 / (2 µ) ? tS / ? t
tS / (2 ?) ? eSel / ? t ? eSvis / ? t
. The rate of mechanical energy dissipation f
(cf. Joseph, 1990, p. 50) is then f tS ?
eSvis / ? t (tS tS) / (2 ?) .
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EARTH MODELS
M1 . . . . . . . . PREM isoviscous mantle
elastic lithosphere
M2 . . . . . . . . PREM LM viscosity
hill elastic lithosphere
M3 . . . . . . . . PREM LM viscosity
hill low-viscosity zone elastic lithosphere
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SHAPE OF THE LOAD
parabolic cross-sections radius 15? max.
height 3500 m
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LOADING HISTORIES
L1 . . . . . . . . . . . . . glacial cycle 100
kyr linear unloading 100 yr
L2 . . . . . . . . . . . . . glacial cycle 100
kyr linear unloading 1 kyr
L3 . . . . . . . . . . . . . glacial cycle 100
kyr linear unloading 10 kyr
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DISSIPATIVE HEATING f (r )
Earth model M1 (isoviscous)
Loading History L1 (100 yr)
L2 (1 kyr)
L3 (10 kyr)
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DISSIPATIVE HEATING f (r )
Earth Model M1 Loading History L1
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DISSIPATIVE HEATING f (r )
Earth model M2 (LM viscosity hill)
Loading History L1 (100 yr)
L2 (1 kyr)
L3 (10 kyr)
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DISSIPATIVE HEATING f (r )
Earth Model M2 Loading History L1
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DISSIPATIVE HEATING f (r )
Earth model M3 (LM viscosity hill
LVZ)
Loading History L1 (100 yr)
L2 (1 kyr)
L3 (10 kyr)
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DISSIPATIVE HEATING f (r )
Earth Model M3 Loading History L1
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TIME EVOLUTION OF MAX LOCAL HEATING maxr f(t)
normalized by the chondritic radiogenic heating
of 3x10-9 W/m3
M1 ?
Earth Model M2 ?
Loading histories L1 ... solid lines L2
... dashed lines L3 ... dotted lines
M3 ?
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EQUIVALENT MANTLE HEAT FLOW qm(?)
peak values time averages
mW/m2 mW/m2
M1 ?
Earth Model M2 ?
Loading histories L1 ... solid lines L2
... dashed lines L3 ... dotted lines
M3 ?
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CONCLUSIONS explored (for the first time
ever) the magnitude of viscous dissipation
in the mantle induced by glacial forcing peak
values 10-100 higher than chondritic radiogenic
heating (below the center and/or edges of
the glacier of 15? radius) focusing of energy
into the low-viscosity zone, if present
magnitude of the equivalent mantle heat flow at
the surface up to mW/m2 after averaging over
the glacial cycle extreme sensitivity to the
choice of the time-forcing function
(equivalent mantle heat flow more than 10 times
higher)