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1
Viscous Heating in the Mantle Induced by Glacial
Forcing
Ladislav Hanyk1, Ctirad Matyska1 and David A.
Yuen2 e-mail ladislav.hanyk_at_mff.cuni.cz,
www http//geo.mff.cuni.cz/lh 1 Department of
Geophysics, Faculty of Mathematics and Physics,
Charles University, Prague, Czech Republic 2
University of Minnesota Supercomputing Institute
and Department of Geology and Geophysics,
Minneapolis
SPATIAL DISTRIBUTION OF DISSIPATIVE HEATING
f normalized with respect to the chondritic
radiogenic heating of 3x10-9 W/m3
SUMMARY We have studied the possibility of
energy transfer from glacial forcing to the
Earth's interior via viscous dissipation of the
transient flow. We applied our initial-value
approach to the modelling of viscoelastic
relaxation of spherical compressible
self-gravitating Earth models with a linear
viscoelastic Maxwellian rheology. We have
focussed on the magnitude of deformations, stress
tensor components and corresponding dissipative
heating for glaciers of Laurentide extent and
cyclic loading with a fast unloading phase of
various lengths. We have surprisingly found that
this kind of heating can represent a
non-negligible internal energy source with
exogenic origin.The volumetric heating by fast
deformation can be locally higher than the
chondritic radiogenic heating during peak events.
In the presence of an abrupt change in the
ice-loading, the time average of the integral of
the heating over the depth corresponds to
equivalent mantle heat flow of milliwatts per m2
below the periphery of ancient glaciers or below
their central areas. However, peak heat-flow
values in time are by about two orders of
magnitude higher. Note that nonlinear rheological
models can potentially increase the magnitude of
localized viscous heating. To illustrate the
spatial distribution of the viscous heating for
various Earth and glacier models, we have invoked
the 3-D visualization system Amira
(http//www.amiravis.com). Our file format allows
us to animate effectively time evolution of data
fields on a moving curvilinear mesh, which
spreads over outer and inner mantle boundaries
and vertical mantle cross-sections.
EQUATIONS 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 , where tS
t K div u I , eS e ? div u I , 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 ?) . To obtain
another view of the magnitude of dissipative
heating, we introduce the quantity qm(?)
?CMBa f(r, ?) r2 dr / a2 , where a is the
Earths radius. qm can be formally considered as
an equivalent mantle heat flow due to
dissipation. CONCLUSIONS We have demonstrated
that the magnitude of viscous dissipation in the
mantle can be comparable to chondritic heating
below the edges of the glacier of Laurentide
extent and/or below the center of the glacier.
During the time interval of maximal heating after
deglaciation, the magnitude increases
approximately thirty times. The magnitude and the
spatial distribution of shear heating is
extremely sensitive to the choice of the
time-forcing function because its jumps result in
heating maxima. The presence of the low-viscosity
zone enables focusing of energy into this layer.
Glacial forcing need not be the only source of
external energy pumping in the planetary system,
e.g., tidal dissipation is known to play
a substantial role in the dynamics of Io (Moore,
2003). An asteroid impact (Ward and Asphaug,
2003) could also generate substantial dissipative
heating inside the Earth.
EARTH MODELS ? PREM ? viscosity model M1
isoviscous mantle, elastic lithosphere ?
viscosity model M2 lower-mantle viscosity
hill, elastic lithosphere ? viscosity model M3
lower-mantle viscosity hill, a
low-viscosity zone, elastic lithosphere
Model M1
L1 ?
L2 ?
L3 ?
Model M2
LOADING ? paraboloid ? radius 15?, max. height
3500 m ? loading cycle of 100 kyr length ?
loading history L1 ... linear unloading 100 yr ?
loading history L2 ... linear unloading 1 kyr ?
loading history L3 ... linear unloading 10 kyr
L1 ?
L2 ?
L3 ?
L1
Model M3
L1 ?
L2
L2 ?
L3
L3 ?
TIME EVOLUTION OF NORMALIZED MAXIMAL LOCAL
HEATING max f (t)
EQUIVALENT MANTLE HEAT FLOW qm(?) mW/m2
peak values values averaged in time
solid, dashed and dotted lines for the L1, L2 and
L3 loading history, respectively
Viscosity model ? M1 ?
REFERENCES Hanyk L., Yuen D. A. and Matyska C.,
1996. Initial-value and modal approaches for
transient viscoelastic responses with complex
viscosity profiles, Geophys. J. Int., 127,
348-362. Hanyk L., Matyska C. and Yuen D. A.,
1998. Initial-value approach for viscoelastic
responses of the Earth's mantle, in Dynamics of
the Ice Age Earth A Modern Perspective, ed. by
P. Wu, Trans Tech Publ., Switzerland, pp.
135-154 Hanyk L., Matyska C. and Yuen D. A.,
1999. Secular gravitational instability of
a compressible viscoelastic sphere, Geophys.
Res. Lett., 26, 557-560. Hanyk L., Matyska C. and
Yuen D.A., 2000. The problem of viscoelastic
relaxation of the Earth solved by a matrix
eigenvalue approach based on discretization in
grid space, Electronic Geosciences, 5,
http//link.springer.de/link/service/journals/1006
9/free/discussion/evmol/evmol.htm. Hanyk L.,
Matyska C. and Yuen D.A., 2002. Determination of
viscoelastic spectra by matrix eigenvalue
analysis, in Ice Sheets, Sea Level and the
Dynamic Earth, ed. by J. X. Mitrovica and B. L.
A. Vermeersen, Geodynamics Research Series
Volume, American Geophysical Union, pp.
257-273. Joseph D. D., 1990. Fluid Dynamics of
Viscoelastic Liquids, Springer, New York
etc. Moore,W. B., 2003. Tidal heating and
convection in Io, J. Geophys. Res., 108,
doi10.1029/2002JE001943. Peltier, W. R., 1974.
The impulse response of a Maxwell earth, Rev.
Geophys. Space Phys., 12, 649-669. Ward, S. N.
and E. Asphaug, 2003. Asteroid impact tsunami of
2880 March 16, Geophys. J. Int., 153, F6--F10.
? M2 ?
? M3 ?
MOVIES ON THE WEB http//www.msi.umn.edu/lilli
links to larry_movies http//geo.mff.cuni.cz/lh
links to references