On a Babcock-Leighton solar dynamo model with a deep-seated generating layer for the toroidal magnetic field .4.

The study is continued of a dynamo model of the Babcock-Leighton type (i.e., the surface eruptions of toroidal magnetic field are the source for the poloidal field) with a thin, deep seated layer (GL), for the generation of the toroidal field, B-phi. The partial differential equations satisfied by B and by the vector potential for the poloidal field are integrated in time with the help of a second order time- and space-centered finite different scheme. Axial symmetry is assumed; the gradient of the angular velocity in the GL is such that within this layer a transition to uniform rotation takes place; the meridional motion, transporting the poloidal field to the GL, is poleward and about 3 m s(-1) at the surface; the radial diffusivity eta(r) equals 5 x 10(9) cm(2) s(-1), and the horizontal diffusivity eta(theta) is adjusted to achieve marginal stability. The initial conditions are: a negligible poloidal field, and a maximum value of \B-phi\ in the GL equal to 1.5 x B-cr, where B-cr is a prescribed field. For every time step the maximum value of \B-phi\ in the GL is computed. If this value exceeds B-cr, then there is eruption of a flux tube (at the latitude corresponding to this maximum) that rises radially to the surface. Only one eruption is allowed per time step (Delta t) and B-phi in the GL is unchanged as a consequence of the eruption. The ensemble of eruptions is the source for the poloidal field, i.e., no use is made of a mean field equation relating the poloidal with the toroidal field. For a given value of Delta t, and since the problem is linear, the solutions scale with B-cr. Therefore, the equations need to be solved for one value of B-cr only. Since only one eruption is allowed per time step, the dependence of the solutions on Delta t needs to be studied. Let F-t be an arbitrary numerical factor (=3 for example) and compare the solutions of the equations for (B-cr, Delta t) and (B-cr, Delta t/3). It is clear that there will be 3 times as many eruptions in the second case (with the shorter time step) than in the first case. However, if the erupted flux in case one is multiplied by 3, then the solutions for this case become nearly identical to those of case two (Delta t is shorter than any typical time of the system, and the difference due to the unequal time steps is negligible). Therefore, varying the time step is equivalent to keeping At fixed while multiplying the erupted flux by an appropriate factor. In the numerical calculations At was set equal to 10(5) s. The factor F-t can then be interpreted as the number of eruptions per 10(5) s. The integration of the equations shows that there is a transition in the nature of the solutions for F-t approximate to 2.5. For F-t < 2.5, the eruptions occur only at high latitudes, whereas for F-t > 2.5, the eruptions occur for theta greater than approximate to pi/4, where theta is the polar angle. Furthermore, for F-t < 2.5, the toroidal field, \B-phi\, in the GL can become considerably larger than B-cr, while this ceases to be the case for F-t > 2.5. The factor F-t is an arbitrary parameter in the model and an appeal to observations is necessary. We set B-cr = 10(3) G. In the model, the magnetic flux of erupting magnetic tubes, is then about 3 x 10(21) G, of the order of the solar values. For this value of B-cr and for the value of F-t(approximate to 2.5) at which the transition takes place, the total erupted flux in 10 years is about 0.85 x 10(25) Mx in remarkable agreement with the total erupted flux during a solar cycle. Concerning the dynamo models studied here, a major drawback encountered in previous papers has been the eruptions at high latitudes, which entail unrealistically large values for the radial magnetic field at the poles. The results of this paper provide a major step forward in the resolution of this difficulty.