The Energy Lost by Differential Rotation in the Generation of the Solar Toroidal Magnetic Field |
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| Authors: | Durney Bernard R |
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| Institution: | (1) Physics Department, University of Arizona, Tucson, AZ, 85721, U.S.A. |
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| Abstract: | The integrals, Ii(t) =  GL ui
j × B
i
dv over the volume GL are calculated in a dynamo model of the Babcock–Leighton type studied earlier. Here, GL is the generating layer for the solar toroidal magnetic field, located at the base of the solar convection zone (SCZ); i=r,  ,  , stands for the radial, latitudinal, and azimuthal coordinates respectively; j = (4 )-1
 × B, where B is the magnetic field; ur,u are the components of the meridional motion, and u is the differential rotation. During a ten-year cycle the energy cycle I(t)dt needs to be supplied to the azimuthal flow in the GL to compensate for the energy losses due to the Lorentz force. The calculations proceed as follows: for every time step, the maximum value of |B| in the GL is computed. If this value exceeds Bcr (a prescribed field) then there is eruption of a flux tube that rises radially, and reaches the surface at a latitude corresponding to the maximum of |B| (the time of rise is neglected). This flux tube generates a bipolar magnetic region, which is replaced by its equivalent axisymmetric configuration, a magnetic ring doublet. The erupted flux can be multiplied by a factor Ft, i.e., by the number of eruptions per time step. The model is marginally stable and the ensemble of eruptions acts as the source for the poloidal field. The arbitrary parameters Bcr and Ft are determined by matching the flux of a typical solar active region, and of the total erupted flux in a cycle, respectively. If E(B) is the energy, in the GL, of the toroidal magnetic field B = B sin cos , B (constant), then the numerical calculations show that the energy that needs to be supplied to the differential rotation during a ten-year cycle is of the order of E(Bcr), which is considerably smaller than the kinetic energy of differential rotation in the GL. Assuming that these results can be extrapolated to larger values of Bcr, magnetic fields  104 G, could be generated in the upper section of the tachocline that lies below the SCZ (designated by UT). The energy required to generate these 104 G fields during a cycle is of the order of the kinetic energy in the UT. |
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