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

Appendix A3: Boundary Conditions

[86]  The equations describing mantle convection (1-3), motion of a continent (9-11), and heat transfer in the continent (21) are interrelated though boundary conditions.

[87]  The impermeability and free-slip conditions are set for mantle flows at the lower and lateral boundaries of the model region (the normal component of the fluid velocity and tangential components of viscous forces vanish at these boundaries):

eq022.gif(22)

where nk ti are, respectively, unit vectors normal and tangential to a given surface.

[88]  The impermeability and no-slip conditions are set at the boundaries of moving continents (velocities of the fluid mantle are equal to those of the continents):

eq023.gif(23)

at continental surfaces contacting with the mantle.

[89]  The temperature at the lower boundary of the region is fixed, T = 1. The zero heat flux condition is accepted at lateral boundaries:

eq024.gif(24)

where nk is a unit vector normal to the boundary. At the upper free surface, the temperature vanishes ( T = 0) only in the oceanic part, outside the continents. The continuity condition is accepted for the temperature and heat flux at the mantle-continent contact:

eq025.gif(25)

The temperature is set to zero at the upper surface of a continent:

eq026.gif(26)

[90]  Thus, the mathematical problem can be summarized as follows. There three unknown functions of coordinates and time describing mantle convection: the mantle flow velocity vector Vi(x, y, z, t), the temperature distribution T(x, y, z, t), and the pressure distribution p(x, y, z, t); we also have four unknown functions of time describing the motion of a continent as a whole: two components of the instantaneous velocity of the center of gravity u0(t) and v0(t), the instantaneous angular velocity of the rotation of a continent around its center of gravity w(t), and the temperature distribution in the continent Tc(x, y, z, t). These unknowns should be found by solving a system of interrelated equations: the differential equations of convection (1-3), three integral equations (17-19) resulting from the Euler equations, and the equation for heat transfer in the continent (21). Knowing at a given time moment the position of the continent and its velocities u0(t), v0(t) and w(t), its position at the next time moment can be found from (20). Boundary conditions (22-26) are used to determine constants of integration of the differential equations.

[91]  The above problem with a freely floating continent differs from the known problem with a fixed continent by that the conditions for flow velocities and temperature at the boundary with a continent are set for the instantaneous position of a floating continent whose velocity and position are not known a priori but are determined by solving a system of interrelated differential equations.

[92]  In the case of a few continents, the equations of motion (17-19) and the equation for temperature (21) are written out for each continent. In addition, the mutual impermeability condition is set for colliding continents. To this end, the forces of viscous friction of the continents are supplemented at the time of a local contact of the continents by the force pushing them away from each other; this additional force is applied at the contact area of the continents and is directed opposite to their relative velocity.


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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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