# Viscosity of the Earth's Mantle

LAWRENCE M. CATHLES
Pages: 403
https://www.jstor.org/stable/j.ctt13x0t47

1. Front Matter
(pp. [i]-[vi])
2. ACKNOWLEDGMENTS
(pp. [vii]-[viii])
(pp. [ix]-[xxii])
4. ONE Introduction
(pp. 1-8)

At the close of the last ice age, climatic changes caused vast loads to shift on the earth’s surface. Starting roughly 12,000 years ago, the three kilometers of ice that had covered Canada, the last European glaciers in Fennoscandia and Siberia, as well as other minor glaciers, melted quickly and dumped their meltwaters into the oceans, increasing the average depth of the world’s oceans by about 110 meters. The earth’s response to this load redistribution was one of fluid flow. By studying the way in which that flow occurred, we can learn much about the long term (~ 1000 yrs)...

5. TWO Physical and Mathematical Foundations
(pp. 9-34)

In this section we derive the equations of motion for simple elastic and simple viscous self-gravitating spheres. After a brief introduction to Lagrangian and Eulerian coordinates, we start by writing down the usual Lagrangian equation for conservation of momentum. We then identify a certain material element and its surface in Eulerian co ordinates and convert the Lagrangian equation to a Eulerian one. This identification is valid only for a very short time interval bt about the time t0at which we desire a valid equation. That is, the Eulerian equation holds only in the interval

${t_0} - \frac{{\delta t}}{2} < t < {t_0} + \frac{{\delta t}}{2}$

whereas the original Langrangian...

6. THREE Formulation of the Theory For Application to the Earth
(pp. 35-108)

In the first part of this section we solve several simple cases involving a flat earth withg= constant and${\partial _z}{\rho _0} = 0$.¹ The resuhs of these calculations will be of use to us in developing the full self-gravitating viscoelastic solutions which we shall do in Section III. B.

Let g = constant =$- \nabla {\phi _0} = - {g_0}\hat z$. Equations (II-22) and (II-23) then become:

$\nabla \tau - {\rho _0}{g_0}\nabla {u_z} + {g_0}{\rho _0}\nabla u\hat z$Elastic (III-1)

$\nabla \tau + ({g_0}{\rho _0}\nabla u + {g_0}{u_z}{\partial _z}{\rho _0})\hat z = 0$. Viscous (III-2)

If the material is assumed to be incompressible,$\nabla u = 0$Only the hydrostatic pre-stress term remains in (III-1), and its meaning is easily determined.

$\nabla \tau - {\rho _0}{g_0}\nabla {u_z} = 0$. Elastic Incompressible (III-3)

Suppose, for example,${\partial _x}{u_z} = neg.const. = - C$. Then since elastic...

7. FOUR Application of Theory: The Earth’s Viscosity Structure
(pp. 109-276)

We have calculated decay spectra that describe how various earth models would adjust to the sudden application of one dyne harmonic load of different order number. For convenience these earth models are summarized in Table IV-1. If these models are combined with a model of the melting of the late Wisconsin ice and the attendant filling of the ocean basins, i.e., a model of the post-Wisconsin load redistribution, isostatic adjustment for the various earth viscosity models can be pre dicted. Comparison of these predictions to the adjustment observed geologically can indicate which mantle viscosity model is most appropriate.

The general...

8. APPENDIX I Sketch Derivation of the Navier-Stokes Equation after Eringen
(pp. 277-284)
9. APPENDIX II Flat Space Runge-Kutta Equations
(pp. 285-290)
10. APPENDIX III Reduction of Equation of Motion in Spherical Coordinates to Scaler Form after Backus (1967)
(pp. 291-298)
11. APPENDIX IV Boundary Conditions at the Fluid Core
(pp. 299-306)
12. APPENDIX V Spherical Harmonics
(pp. 307-318)
13. APPENDIX VI The Isostatic Adjustment of a Layered Viscous Half-Space
(pp. 319-337)
14. APPENDIX VII Glacial Uplift in Canada
(pp. 338-358)
15. BIBLIOGRAPHY
(pp. 359-372)
16. INDEX
(pp. 373-387)
17. Back Matter
(pp. 388-390)