viscous heating in the earths mantle induced by glacial loading l. hanyk 1, c. matyska 1, d. a. yuen...
TRANSCRIPT
![Page 1: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/1.jpg)
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
![Page 2: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/2.jpg)
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
![Page 3: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/3.jpg)
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
![Page 4: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/4.jpg)
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
![Page 5: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/5.jpg)
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)
![Page 6: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/6.jpg)
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) .
![Page 7: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/7.jpg)
EARTH MODELS
M1 . . . . . . . .PREMisoviscous mantle elastic lithosphere
M2 . . . . . . . .PREM LM viscosity hillelastic lithosphere
M3 . . . . . . . .PREM LM viscosity hilllow-viscosity zoneelastic lithosphere
![Page 8: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/8.jpg)
SHAPE OF THE LOAD
parabolic cross-sections
radius 15
max. height 3500 m
![Page 9: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/9.jpg)
LOADING HISTORIES
L1 . . . . . . . . . . . . .glacial cycle 100 kyrlinear unloading 100 yrL2 . . . . . . . . . . . . .glacial cycle 100 kyrlinear unloading 1 kyrL3 . . . . . . . . . . . . .glacial cycle 100 kyrlinear unloading 10 kyr
![Page 10: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/10.jpg)
Video Clip
L2(1 kyr)
LoadingHistory
L1(100 yr)
L3(10 kyr)
Earth model M1 (isoviscous)
DISSIPATIVE HEATING φ (r )
![Page 11: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/11.jpg)
DISSIPATIVE HEATING φ (r )
Earth Model M1
Loading HistoryL1
![Page 12: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/12.jpg)
Video Clip
Earth model M2 (LM viscosity hill)
DISSIPATIVE HEATING φ (r )
L2(1 kyr)
LoadingHistory
L1(100 yr)
L3(10 kyr)
![Page 13: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/13.jpg)
DISSIPATIVE HEATING φ (r )
Earth Model M2
Loading HistoryL1
![Page 14: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/14.jpg)
Video Clip
Earth model M3 (LM viscosity hill & LVZ)
DISSIPATIVE HEATING φ (r )
L2(1 kyr)
LoadingHistory
L1(100 yr)
L3(10 kyr)
![Page 15: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/15.jpg)
DISSIPATIVE HEATING φ (r )
Earth Model M3
Loading HistoryL1
![Page 16: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/16.jpg)
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 ►
![Page 17: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/17.jpg)
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 ►
![Page 18: VISCOUS HEATING in the Earths 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](https://reader035.vdocument.in/reader035/viewer/2022081602/55174b05550346a3338b4834/html5/thumbnails/18.jpg)
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)