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7 Amplification of a Toroidal Magnetic Field by Conversion of Potential Energy

Thin flux tube models of emerging flux loops through the solar convective envelope (Section 5.1) have inferred a strong super-equipartition field strength of order 5 10 G for the toroidal magnetic field at the base of the solar convection zone. Generation of such a strong field is dynamically difficult since the magnetic energy density of a 105G field is about 10 – 100 times the kinetic energy density of the differential rotation (Parker, 1994Jump To The Next Citation PointRempel and Schüssler, 2001Jump To The Next Citation Point). An alternative mechanism for amplifying the toroidal magnetic field has been proposed which converts the potential energy associated with the stratification of the convection zone into magnetic energy. It is shown that upflow of high entropy plasma towards the inflated and “exploding” top of a rising Ω-loop developed from an initial toroidal field of equipartition field strength (∼ 104G) can significantly intensify the submerged part of the field by extracting plasma out of it (Parker, 1994Jump To The Next Citation PointMoreno-Insertis et al., 1995Jump To The Next Citation PointRempel and Schüssler, 2001Jump To The Next Citation Point). This process is a barometric effect and is caused by the entropy gradient in the solar convection zone maintained by the energy transport.

In the thin flux tube simulations of rising Ω-loops in the solar convection zone, it is found that flux loops with a low initial field strength of ∼ 104G do not reach the upper half of the solar convection zone before the apexes of the loops loose pressure confinement and effectively “explode” (Moreno-Insertis et al., 1995Jump To The Next Citation Point). This loss of pressure confinement at the top of the emerging loop occurs as plasma inside the tube establishes hydrostatic equilibrium along the tube which happens if the emerging loop rises sufficiently slowly (Moreno-Insertis et al., 1995Jump To The Next Citation Point). The loop rises adiabatically carrying the high entropy plasma from the base of the solar convection zone while the entropy outside the flux tube decreases with height in the superadiabatically stratified convection zone. Hydrostatic equilibrium therefore dictates that the plasma pressure inside the flux tube decreases with height slower than the outside and becomes equal to the external pressure at a certain height where the magnetic field can no longer be confined. This “explosion” height for the emerging loop is found to be a function of the initial tube field strength at the base of the solar convection zone (see Figure 23View Image).

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Figure 23: Depth from the surface where the apex of an emerging flux loop with varying initial field strength rising from the bottom of the convection zone looses pressure confinement or “explodes” as a result of tube plasma establishing hydrostatic equilibrium along the tube. The explosion height is computed by considering an isentropic thin flux loop with hydrostatic equilibrium along the field lines (see Moreno-Insertis et al., 1995) in a model solar convection zone of Christensen-Dalsgaard (Christensen-Dalsgaard et al., 1993).
For loops with an initial field strength ∼ 104G, the explosion height is at about the middle of the solar convection zone. When the loop apex approaches the explosion height, it expands drastically and the buoyancy of the high entropy material in the tube is expected to drive an outflow which extracts plasma out of the lower part of the flux tube at the base of the solar convection zone. This process has been demonstrated by Rempel and Schüssler (2001Jump To The Next Citation Point), who performed MHD simulations of exploding magnetic flux sheets in two-dimensional Cartesian geometry.

The simulations of Rempel and Schüssler (2001Jump To The Next Citation Point) start with a magnetic sheet with a higher value of entropy placed at the bottom of an adiabatically stratified layer (constant entropy layer). This setup avoids the complication of involving convective flows in the simulations while keeping the essential effect of the entropy decrease in the solar convection zone by assuming a constant entropy difference between the flux sheet and the isentropic layer. The central portion of the flux sheet is perturbed upward which subsequently forms a rising loop as a result of its magnetic buoyancy (see Figure 24View Image).

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Figure 24: From Rempel and Schüssler (2001). Evolution of magnetic field strength (gray scale: darker gray denotes stronger field) and velocity field (arrows) during the flux loop explosion. The horizontal part of the field is amplified by a factor of 3.
The apex of the rising loop explodes into a cloud of weak magnetic field as it crosses the predicted explosion height (middle panel) and high entropy plasma, driven by buoyancy, continues to flow out of the “stumps”, draining mass from the lower horizontal part of the flux sheet (bottom panel). The field strength of the horizontal part of the flux sheet is visibly intensified. It is found that the final field strength the horizontal part of the field can reach is roughly the value for which the explosion height is close to the top of the stratification. For the solar convection zone, this value corresponds to 5 ∼ 10 G (see Figure 23View Image). This implies that the large field strength of order 5 10 G may be achieved by the process of flux “explosion”, which draws energy from the potential energy associated with the stratification of the solar convection zone. A coherent picture maybe as follows. Differential rotation in the tachocline shear layer generates and amplifies the toroidal magnetic field to an equipartition value of about 4 10 G, at which point magnetic buoyancy becomes important dynamically, and magnetic buoyancy instability, radiative heating or other convective perturbations drive the formation of buoyant flux loops rising into the solar convection zone. These rising loops explode in the middle of the convection zone and fail to rise all the way to the surface. These “failed” eruptions “pump” out the plasma from the toroidal field at the base of the solar convection zone (Parker, 1994) and amplify the toroidal field until it reaches strongly super-equipartition field strength of order 5 10 G, whose eruptions then lead to the emergence of solar active regions at the surface.


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