RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 7, ES3001, doi:10.2205/2005ES000179, 2005

Appendix A2: Equations of Motion of a Freely Floating Continent

[81]  The Euler equations describing the horizontal motion of a solid continent of an arbitrary shape and its rotation around an instantaneous vertical axis reduce to a system of three equations [Trubitsyn, 2000]:

eq009.gif(9)

eq010.gif(10)

eq011.gif(11)

eq012.gif(12)

where Iik is the tensor of moments of inertia of a solid

Iik = rc [(x1 - x10)2 dlj - (xi - xi0) (xk - xk0)] d V,

xc(t) and yc(t) are the coordinates of the center of gravity of the continent, and j is the rotation angle of the continent.

[82]  In the particular case of horizontal motion of solid and infinitely thin continents, the viscous force is applied only to the base of the continent, for which we have nk = (0, 0, -1). Then the Euler equations are simplified and take the form

eq013.gif(13)

eq014.gif(14)

eq015.gif(15)

where I is the moment of inertia of the continent about the axis zprime passing through its center of gravity. Its value is calculated in a moving frame of reference with the axes xprime and yprime directed along the principal axes of inertia of the continent:

eq016.gif(16)

[83]  Dimension relations indicate that, as well as in the case of equation (1) describing the momentum transfer in a viscous fluid, the inertial terms on the left-hand side of Equations (9-11) are on the order of kr/mapprox10-23.

[84]  Neglecting the inertial terms, the Euler equations yield six relations to find six unknowns: the coordinates of the center of gravity of the continent xc(t) and yc(t), its rotation angle j and velocities u0(t), v0(t), and w3(t):

eq017.gif(17)

eq018.gif(18)

eq019.gif(19)

eq020.gif(20)

[85]  The equation for the temperature ( Tc ) distribution inside a solid continent in the initial fixed reference frame reduces to the equation of thermal conduction with advective heat transfer:

eq021.gif(21)

where Qc is the density of heat sources inside the continent.


RJES

Citation: Trubitsyn, V. P., and A. P. Trubitsyn (2005), Evolution of mantle plumes and uplift of continents during the Pangea breakup, Russ. J. Earth Sci., 7, ES3001, doi:10.2205/2005ES000179.

Copyright 2005 by the Russian Journal of Earth Sciences

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