3.1 The mechanical equilibria for an isolated toroidal flux tube or an extended magnetic layer

The Hale’s polarity rule of solar active regions indicates a subsurface magnetic field that is highly organized, of predominantly toroidal direction, and with sufficiently strong field strength (super-equipartition compared to the kinetic energy density of convection) such that it is not subjected to strong deformation by convective motions. It is argued that the weakly subadiabatically stratified overshoot layer at the base of the solar convection zone is the most likely site for the storage of such a strong coherent toroidal magnetic field against buoyant loss for time scales comparable to the solar cycle period (see Parker, 1979Jump To The Next Citation Pointvan Ballegooijen, 1982Jump To The Next Citation Point).

It is not clear if the toroidal magnetic field is in the state of isolated flux tubes or stored in the form of a more diffuse magnetic layer. Moreno-Insertis et al. (1992) have considered the mechanical equilibrium of isolated toroidal magnetic flux tubes (flux rings) in a subadiabatic layer using the thin flux tube approximation (Section 2.1). The forces experienced by an isolated toroidal flux ring at the base of the convection zone is illustrated in Figure 5View Image(a).

View Image

Figure 5: Schematic illustrations based on Schüssler and Rempel (2002) of the various forces involved with the mechanical equilibria of an isolated toroidal flux ring (a) and a magnetic layer (b) at the base of the solar convection zone. In the case of an isolated toroidal ring (see the black dot in (a) indicating the location of the tube cross-section), the buoyancy force has a component parallel to the rotation axis, which cannot be balanced by any other forces. Thus mechanical equilibrium requires that the buoyancy force vanishes and the magnetic curvature force is balanced by the Coriolis force resulting from a prograde toroidal flow in the flux ring. For a magnetic layer (as indicated by the shaded region in (b)), on the other hand, a latitudinal pressure gradient can be built up, so that an equilibrium may also exist where a non-vanishing buoyancy force, the magnetic curvature force and the pressure gradient are in balance with vanishing Coriolis force (vanishing longitudinal flow).

The condition of total pressure balance (1View Equation) and the presence of a magnetic pressure inside the flux tube require a lower gas pressure inside the flux tube compared to the outside. Thus either a lower density or a lower temperature (or a combination of the two) inside the flux tube is needed to achieve the lower gas pressure required for pressure balance. If the flux tube is in thermal equilibrium with the surrounding, then the density inside needs to be lower and the flux tube is buoyant. The buoyancy force associated with a magnetic flux tube in thermal equilibrium with its surrounding is often called the magnetic buoyancy (Parker, 1975). It can be seen in Figure 5View Image(a) that a radially directed buoyancy force has a component that is parallel to the rotation axis, which cannot be balanced by any other forces associated with the toroidal flux ring. Thus for the toroidal flux ring to be in mechanical equilibrium, the tube needs to be in a neutrally buoyant state with vanishing buoyancy force, and with the magnetic curvature force pointing towards the rotation axis being balanced by a Coriolis force produced by a faster rotational speed of the flux ring (see Figure 5View Image(a)). Such a neutrally buoyant flux ring (with equal density between inside and outside) then requires a lower internal temperature than the surrounding plasma to satisfy the total pressure balance. If one starts with a toroidal flux ring that is initially in thermal equilibrium with the surrounding and rotates at the same ambient angular velocity, then the flux ring will move radially outward due to its buoyancy and latitudinally poleward due to the unbalanced poleward component of the tension force. As a result of its motion, the flux ring will lose buoyancy due to the subadiabatic stratification and attain a larger internal rotation rate with respect to the ambient field-free plasma due to the conservation of angular momentum, evolving towards a mechanical equilibrium configuration. The flux ring will undergo superposed buoyancy and inertial oscillations around this mechanical equilibrium state. It is found that the oscillations can be contained within the stably stratified overshoot layer and also within a latitudinal range of Δ ðœƒ ≲ 20∘ to be consistent with the active region belt, if the field strength of the toroidal flux ring B ≲ 105 G and the subadiabaticity of the overshoot layer is sufficiently strong with −5 δ ≡ ∇ − ∇ad ≲ − 10, where ∇ ≡ d lnT ∕dln P is the logarithmic temperature gradient and ∇ad is ∇ for an adiabatically stratified atmosphere. Flux rings with significantly larger field strength cannot be kept within the low latitude zones of the overshoot region.

Rempel et al. (2000Jump To The Next Citation Point) considered the mechanical equilibrium of a layer of an axisymmetric toroidal magnetic field of 105 G in a subadiabatically stratified region near the bottom of the solar convection zone in full spherical geometry. In this case, as illustrated in Figure 5View Image(b), a latitudinal pressure gradient can be built up, allowing for force balance between a non-vanishing buoyancy force, the magnetic curvature force, and the pressure gradient without requiring a prograde toroidal flow. Thus a wider range of equilibria can exist. Rempel et al. (2000Jump To The Next Citation Point) found that under the condition of a strong subadiabatic stratification such as the radiative interior with δ ∼ − 0.1, the magnetic layer tends to establish a mechanical equilibrium where a latitudinal pressure gradient is built up to balance the poleward component of the magnetic tension, and where the net radial component of the buoyancy and magnetic tension forces is efficiently balanced by the strong subadiabaticity. The magnetic layer reaches this equilibrium solution in a time scale short compared to the time required for a prograde toroidal flow to set up for the Coriolis force to be significant. For this type of equilibrium where a latitudinal pressure gradient is playing a dominant role in balancing the poleward component of the magnetic curvature force, there is significant relative density perturbation (≫ 1∕β) in the magnetic layer compared to the background stratification. On the other hand, under the condition of a very weak subadiabatic stratification such as that in the overshoot layer near the bottom of the convection zone with −5 δ ∼ − 10, the magnetic layer tends to evolve towards a mechanical equilibrium which resembles that of an isolated toroidal flux ring, where the relative density perturbation is small (≪ 1 ∕β), and the magnetic curvature force is balanced by the Coriolis force induced by a prograde toroidal flow in the magnetic layer. Thus regardless of whether the field is in the state of an extended magnetic layer or isolated flux tubes, a 105 G toroidal magnetic field stored in the weakly subadiabatically stratified overshoot region is preferably in a mechanical equilibrium with small relative density perturbation and with a prograde toroidal flow whose Coriolis force balances the magnetic tension. The prograde toroidal flow necessary for the equilibrium of the 105 G toroidal field is about 200 ms–1, which is approximately 10% of the mean rotation rate of the Sun. Thus one may expect significant changes in the differential rotation in the overshoot region during the solar cycle as the toroidal field is being amplified (Rempel et al., 2000). Detecting these toroidal flows and their temporal variation in the overshoot layer via helioseismic techniques is a means by which we can probe and measure the toroidal magnetic field generated by the solar cycle dynamo.


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