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

φ = τS : ∂ eSvis / ∂ t = (τS : τS) / (2 η) .

EARTH MODELS

M1 . . . . . . . .PREMisoviscous mantle elastic lithosphere

M2 . . . . . . . .PREM LM viscosity hillelastic lithosphere

M3 . . . . . . . .PREM LM viscosity hilllow-viscosity zoneelastic lithosphere

SHAPE OF THE LOAD

parabolic cross-sections

radius 15

max. height 3500 m

LOADING HISTORIES

L1 . . . . . . . . . . . . .glacial cycle 100 kyrlinear unloading 100 yrL2 . . . . . . . . . . . . .glacial cycle 100 kyrlinear unloading 1 kyrL3 . . . . . . . . . . . . .glacial cycle 100 kyrlinear unloading 10 kyr

Video Clip

L2(1 kyr)

LoadingHistory

L1(100 yr)

L3(10 kyr)

Earth model M1 (isoviscous)

DISSIPATIVE HEATING φ (r )

DISSIPATIVE HEATING φ (r )

Earth Model M1

Loading HistoryL1

Video Clip

Earth model M2 (LM viscosity hill)

DISSIPATIVE HEATING φ (r )

L2(1 kyr)

LoadingHistory

L1(100 yr)

L3(10 kyr)

DISSIPATIVE HEATING φ (r )

Earth Model M2

Loading HistoryL1

Video Clip

Earth model M3 (LM viscosity hill & LVZ)

DISSIPATIVE HEATING φ (r )

L2(1 kyr)

LoadingHistory

L1(100 yr)

L3(10 kyr)

DISSIPATIVE HEATING φ (r )

Earth Model M3

Loading HistoryL1

TIME EVOLUTION OF MAX LOCAL HEATING maxr

φ(t)

Loading histories

L1 ... solid linesL2 ... dashed linesL3 ... dotted lines

normalized by the chondritic radiogenic heating of 3x10-9 W/m3

Earth Model M2 ►

M3 ►

M1 ►

EQUIVALENT MANTLE HEAT FLOW qm(θ)

Loading histories

L1 ... solid linesL2 ... dashed linesL3 ... dotted lines

peak values time averages [mW/m2] [mW/m2]

Earth Model M2 ►

M3 ►

M1 ►

CONCLUSIONS

• 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)