1. Visualization of Viscous Heating in the Earth’s Mantle Induced by Glacial Loading Visualization of Viscous Heating in the Earth’s Mantle Induced by Glacial Loading Ladislav Hanyk 1, Ctirad Matyska 1, David A. Yuen 2 and Ben J. Kadlec 2 e-mail: ladislav.hanyk@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 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 m 2 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.com). 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. Earth Model M1 ……………………. ► ▼ PREM isoviscous mantle elastic lithosphere ◄ L3 ► ◄ L2 ► Loading History ◄ L1 ► 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 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 ∂ τ S / ∂ t = 2 μ ∂ e S / ∂ t – μ / η τ S, where τ S = τ – K div u I, e S = 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, ∂ e S / ∂ t = 1 / (2 μ) ∂ τ S / ∂ t + τ S / (2 η) = ∂ e S el / ∂ t + ∂ e S vis / ∂ t. The rate of mechanical energy dissipation φ (cf. Joseph, 1990, p. 50) is then φ = τ S : ∂ e S vis / ∂ t = (τ S : τ S ) / (2 η). To obtain another view of the magnitude of dissipative heating, we introduce the quantity q m ( θ ) = (r, θ ) r 2 dr / a 2, where a is the Earth‘s radius. q m can be formally considered as an „equivalent mantle heat flow“ due to dissipation. Earth Model M3 ……………………. ► ▼ lower-mantle viscosity hill a low-viscosity zone elastic lithosphere TIME EVOLUTION OF NORMALIZED MAXIMAL LOCAL HEATING max φ (t) r EQUIVALENT MANTLE HEAT FLOW q m ( θ ) peak values time averages [mW/m 2 ] [mW/m 2 ] Earth Model ◄ M1 ► ◄ M2 ► ◄ M3 ► Loading histories:solid lines …L1 dashed lines …L2 dotted lines …L3 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. 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/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. 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. DISSIPATIVE HEATING φ (r ) normalized by the chondritic radiogenic heating of 3x10 -9 W/m 3 displayed at three close time instants (before, at and after max φ ) AMIRA ® MOVIES http://www.msi.umn.edu/~lilli http://geo.mff.cuni.cz/~lh