On a Babcock-Leighton Solar Dynamo Model with a Deep-seated Generating Layer for the Toroidal Magnetic Field. IV.

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. 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 ηr equals 5 × 109 cm2 s-1, and the horizontal diffusivity ηθ is adjusted to achieve marginal stability. The initial conditions are: a negligible poloidal field, and a maximum value of |B| in the GL equal to 1.5 × Bcr, where Bcr is a prescribed field. For every time step the maximum value of |B| in the GL is computed. If this value exceeds Bcr, 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 (Δt) and B 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 Δt, and since the problem is linear, the solutions scale with Bcr. Therefore, the equations need to be solved for one value of Bcr only. Since only one eruption is allowed per time step, the dependence of the solutions on Δt needs to be studied. Let Ft be an arbitrary numerical factor (= 3 for example) and compare the solutions of the equations for (Bcr, Δt) and (Bcr, Δt/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 (Δt 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 Δt fixed while multiplying the erupted flux by an appropriate factor. In the numerical calculations Δt was set equal to 105 s. The factor Ft can then be interpreted as the number of eruptions per 105 s. The integration of the equations shows that there is a transition in the nature of the solutions for Ft ≈ 2.5. For Ft 2.5, the eruptions occur for θ greater than ≈ π/4, where θ is the polar angle. Furthermore, for Ft 2.5. The factor Ft is an arbitrary parameter in the model and an appeal to observations is necessary. We set Bcr = 103 G. In the model, the magnetic flux of erupting magnetic tubes, is then about 3 × 1021 G, of the order of the solar values. For this value of Bcr and for the value of Ft (≈ 2.5) at which the transition takes place, the total erupted flux in 10 years is about 0.85 × 1025 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.