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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
∂ τS / ∂ t = 2 μ ∂ eS / ∂ t – μ / η τS ,
where
τS = τ – K div u I , eS = e – ⅓ div u I ,
τ, 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) ∂ τS / ∂ t + τS / (2 η) = ∂ eS
el / ∂ t + ∂ eSvis / ∂ t .
The rate of mechanical energy dissipation φ (cf. Joseph, 1990, p. 50) is then
φ = τS : ∂ eSvis / ∂ t = (τS : τS) / (2 η) .
To obtain another view of the magnitude of dissipative heating, we introduce the quantity
qm(θ) = ∫CMBa φ(r, θ) r2 dr / a2 ,
where a is the Earth‘s 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.
Ladislav Hanyk1, Ctirad Matyska1 and David A. Yuen2 e-mail: [email protected], 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
Viscous Heating in the Mantle Induced by Glacial Viscous Heating in the Mantle Induced by Glacial ForcingForcing
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
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
SPATIAL DISTRIBUTION OF DISSIPATIVE HEATING φnormalized with respect to the chondritic radiogenic heating of 3x10-9 W/m3
TIME EVOLUTION OF NORMALIZED
MAXIMAL LOCAL HEATING max φ (t)
EQUIVALENT MANTLE HEAT FLOW qm(θ) [mW/m2]
peak values values averaged in time
MOVIES ON THE WEBhttp://www.msi.umn.edu/~lilli … links to larry_movieshttp://geo.mff.cuni.cz/~lh … links to references
REFERENCESHanyk 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/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, doi:10.1029/2002JE001943.
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.
Viscosity
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