1. VISCOUS HEATING in the Earth‘s Mantle Induced 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 Republic 2Department of Geology and Geophysics, University of Minnesota, Minneapolis, USA

2. IDEA How efficient can be the shear heating in the Earth’s mantle due to glacial forcing, i.e., internal energy source with exogenic origin? (“energy pumping into the Earth’s 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

3. 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

4. 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

5. 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)

6. DISSIPATIVE HEATING φ (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
∂ τS / ∂ t = 2 μ ∂ eS / ∂ t – μ / η τS ,
τS = τ – K div u I ,
eS = e – ⅓ div u I ,
where 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)
= ∂ eSel / ∂ 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) .

7. EARTH MODELS M
PREM
isoviscous mantle
elastic lithosphere M
PREM
LM viscosity hill
elastic lithosphere M
PREM
LM viscosity hill
low-viscosity zone
elastic lithosphere

8. SHAPE OF THE LOAD parabolic cross-sections radius 15
max. height 3500 m

9. LOADING HISTORIES L
glacial cycle kyr
linear unloading yr L
glacial cycle kyr
linear unloading kyr L
glacial cycle kyr
linear unloading kyr

10. DISSIPATIVE HEATING φ (r )
Earth model M1 (isoviscous)
Loading
History L1
(100 yr) L2
(1 kyr) L3
(10 kyr)

11. DISSIPATIVE HEATING φ (r )
Earth Model M1
Loading History L1

12. DISSIPATIVE HEATING φ (r )
Earth model M2 (LM viscosity hill)
Loading
History L1
(100 yr) L2
(1 kyr) L3
(10 kyr)

13. DISSIPATIVE HEATING φ (r )
Earth Model M2
Loading History L1

14. DISSIPATIVE HEATING φ (r )
Earth model M3 (LM viscosity hill & LVZ)
Loading
History L1
(100 yr) L2
(1 kyr) L3
(10 kyr)

15. DISSIPATIVE HEATING φ (r )
Earth Model M3
Loading History L1

16. TIME EVOLUTION OF MAX LOCAL HEATING maxr φ(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 ►

17. EQUIVALENT MANTLE HEAT FLOW qm(θ) M1 ► Earth Model M2 ► M3 ►
peak values time averages
[mW/m2] [mW/m2]
M1 ►
Earth Model M2 ►
Loading histories
L1 ... solid lines
L2 ... dashed lines
L3 ... dotted lines
M3 ►

18. CONCLUSIONS • explored (for the first time ever) the magnitude of viscous dissipation in the mantle induced by glacial forcing • peak values 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)