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]:
 | (9) |
 | (10) |
 | (11) |
 | (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
 | (13) |
 | (14) |
 | (15) |
where
I is the moment of inertia of the continent about the axis
z
passing
through its center of gravity. Its value is calculated in a moving frame of
reference with the axes
x
and
y
directed along the principal axes of inertia
of the continent:
 | (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/m
10-23.
[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):
 | (17) |
 | (18) |
 | (19) |
 | (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:
 | (21) |
where
Qc is the density of heat sources inside the continent.

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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