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Visualization of Viscous Heating in the Earths
Mantle Induced by Glacial Loading
Ladislav Hanyk1, Ctirad Matyska1, David A. Yuen2
and Ben J. Kadlec2 e-mail ladislav.hanyk_at_mff.cun
i.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
2003 NG11A-0166
SUMMARY We have studied the a possible mechanism
of transferring gravitational potential energy
into viscous heating in the mantle via glacial
loading during the ice ages. Shear heating
associated with the transient flow occurring over
a short timescale on the order of tens of
thousand of years can cause a non-negligible
amount of heat production in the mantle. We have
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 ice sheets of the size of the
Laurentide ice mass and cyclic loading with a
fast unloading phase two orders of magnitude less
than that associated with mountain building and
vertical tectonics. Much to our surprise, we
have found that this kind of internal heating can
represent a non-negligible internal energy source
with, however, an exogenic origin. The volumetric
heating by this fast rate of deformation can be
locally higher than the chondritic radiogenic
heating during peak events with short timescales.
In the presence of an abrupt change in the
ice-loading, its time average of the integral
over the depth corresponds to equivalent mantle
heat flow of the order of magnitude of milliwatts
per m2 below the periphery of ancient glaciers or
below their central areas. However, peak
heat-flow values in time are almost by about two
orders higher. On the other hand, 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 employed the powerful 3-D
visualization system Amira (http//www.amiravis.co
m). With our data format we can animate very
easily the temporal evolution of the data fields
on a moving curvilinear mesh, which spreads over
outer and inner mantle boundaries and mantle
cross-sections. Amira movies can reveal the
complex nature of dissipative heating of the PREM
model with a lower-mantle viscosity hill at the
end of the recent Pleistocene ice age. This
viscoelastic model can be employed in other
dynamical situations with fast dynamical
timescales, such as the aftermath of a meteoritic
impact or other global cryospheric events.
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(?)
(r, ?) r 2 dr / a 2 , where a is the Earths
radius. qm can be formally considered as an
equivalent mantle heat flow due to dissipation.
TIME EVOLUTION OF NORMALIZED MAXIMAL LOCAL
HEATING max f (t) r
EQUIVALENT MANTLE HEAT FLOW qm(?)
peak values time averages mW/m2
mW/m2
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. In this way,
realistic forcing in time can substantially
increase time-averaged heating due to the
presence of several abrupt changes in the loading
function within glacial cycles (e.g., Siddall et
al., 2003). 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.
Loading histories solid lines L1
dashed lines L2 dotted lines L3
Earth Model ? M1 ?
REFERENCES Amira v. 3.0. http//www.amiravis.com.
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/servi
ce/journals/10069/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. Siddall, M.,
Rohling, E. J. et al., Sea-level fluctuations
during the last glacial cycle, Nature, 423,
853-858. Ward, S. N. and E. Asphaug, 2003.
Asteroid impact tsunami of 2880 March 16,
Geophys. J. Int., 153, F6--F10.
? M2 ?
? M3 ?
AMIRA MOVIES http//www.msi.umn.edu/lilli
http//geo.mff.cuni.cz/lh
Earth Model M1 . ? ? ?
PREM ? isoviscous mantle ?
elastic lithosphere
Loading Histories L1, L2, L3 ? ?
parabolic cross-section ? radius 15?, max.
height 3500 m ? loading cycle period of 100
kyr ? loading history L1 ... linear unloading
100 yr ? loading history L2 ... linear unloading
1 kyr ? loading history L3 ... linear unloading
10 kyr
Earth Model M2 . ? ? ?
PREM ? lower-mantle viscosity hill
? elastic lithosphere
Earth Model M3 . ? ? ?
lower-mantle viscosity hill ? a
low-viscosity zone ? elastic
lithosphere
DISSIPATIVE HEATING f (r ) ? normalized by the
chondritic radiogenic heating of 3x10-9 W/m3 ?
displayed at three close time instants (before,
at and after max f)
Loading History ? L1 ?
? L2 ?
? L3 ?