VISCOUS HEATING in the Earth‘s Mantle Induced by Glacial Loading L. Hanyk 1 , C. Matyska 1, D. A. Yuen 2 and B. J. Kadlec 2 1 Department of Geophysics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic 2 Department of Geology and Geophysics, University of Minnesota, Minneapolis, USA
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VISCOUS HEATINGin 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
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
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
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
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)
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)= ∂ eS
el / ∂ t + ∂ eSvis / ∂ t .
The rate of mechanical energy dissipation φ (cf. Joseph, 1990, p. 50) is then
• explored (for the first time ever) the magnitude of viscous dissipation in the mantle induced by glacial forcing• peak values 10-100 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)