Each of the simulations described within this chapter is carried out in the dimensionless domain
(−60,60)×(−70,70)×(−22,70) in the (x, y, z) coordinate system. This corresponds to a horizontal domain of size20.4 Mm×23.8 Mmon the Sun, which extends to a depth of3.7 Mm
below the solar surface and an atmospheric height of11.9 Mm. The grid contains148×160×218
points, with uniform spacing in the horizontal plane and stretched spacing in the vertical direction. The numerical resolution is highest over the region from just below the initial location of the flux tube to just above the transition region. The gridspacing represents a physical distance of
4.77×104mat its smallest and2.49×105mat its largest. The boundaries of the box are periodic in the horizontal directions and closed on the top and bottom. A damping region is included at both the top and bottom of the box to limit the reflection of waves.
(a) Contours of pressure excess,pexc. (b)Contours of density excess,ρexc.
Figure 5.3: Variation in the excess quantities through the(x, z)plane aty= 0in the caseB0= 3.0
andα = 0.4. The white circle has a radius of7units and indicates an approximate boundary for the tube.
We set up the domain of our simulation using the numerical solar interior and atmosphere de- scribed in section4.2, where the solar surface is located atz = 0. The magnetic flux tube is then placed horizontally within the solar interior, with its axis initially located atz = −10. This cor- responds to a depth of1.7Mmbelow the solar surface. At the commencement of the simulation, we require the tube to be in radial force balance with the external plasma. This prevents a sudden radial expansion or compression of the tube when the simulation begins.
The internal gas pressure of the tube, pi, is defined as pe +pexc where pe is the external gas
pressure andpexcis the pressure excess. The pressure excess can be found by solving
dpexc
dr = (J×B)r, (5.4)
whereBis prescribed by(5.1),(5.2),(5.3). Integrating this with respect toryields
pexc=
1 2µ α
2 R2/2−r2
−1By2+C, (5.5)
whereCis a constant of integration. At large radial distances, the field strength diminishes to zero and, thus, we consider the tube boundary to have been reached. Hence, we also havepexc= 0at
these largerand, thus,C = 0. Therefore,
pexc=
1 2µ α
2 R2/2−r2
−1By2. (5.6)
Figure5.3(a)illustrates the pressure excess, which increases radially from the axis of the tube.
whereρeis the external density andρexcis the density excess. If we were to choose the whole of
the tube to be in thermal equilibrium with its surroundings then the density excess would be given by ρexc= ˜ µpexc ˜ RT (z) = ˜ µ 2µRT˜ (z) α 2 R2/2−r2 −1 By2. (5.7)
The whole tube would then be buoyant when a density deficit exists,ρexc<0.
We now consider what values the parameters are required to take in order to make the tube buoyant for all radii. If we want to haveρexc<0at all radii, then we must satisfy α2 R2/2−r2
−1
<
0at all radii. We note that the criterion for buoyancy is independent ofB0. This criterion can be
simplified to R2 2 − 1 α2 < r 2. (5.8)
Given thatRandαare constants, the tube will be buoyant at all radii providing the axis is buoyant. Thus, the criterion reduces to
α2< 2
R2. (5.9)
and, forR = 2.5, the tube will be buoyant at all radii providing |α| < 0.57. The values of α
chosen for the tubes within groups 1 and 2 satisfy this criterion and, therefore, will be buoyant at all radii. The cross-sectional density excess of a tube is indicated in figure5.3(b). The most buoyant region in the tube’s cross-section sits slightly above the axis of the tube as a result of the specific ambient temperature profile chosen for the tube.
This profile of density excess given by(5.7)is independent of position in the axial direction and, therefore, the whole length of the tube will exhibit the density excess illustrated in figure5.3(b). However, we wish to encourage the formation of an Ω-loop shape along the tube’s length to promote emergence in just a section of the tube. Thus, we choose only the cross-section of the tube aty= 0to have this complete buoyancy at all radii. The density deficit is reduced away from
y= 0following the Gaussian profile
ρm=ρexce−y
2/λ2
, (5.10)
whereλ = 20, such that the density in the tube is now given byρi = ρe+ρm. The smaller
density deficit for|y|>0implies that there is a corresponding increase in the temperature contrast between the tube and surroundings.
From(5.7)we can see that both the twist and the magnetic field strength play a role in determining how buoyant the flux tube will be. For group 2, the variation inαleaves the strength of the field at
the tube’s axis unchanged and has little effect on the intensity of the magnetic field at outer radii, as shown in figure 5.1(a). However, increasing the amount of twist will increase the inwardly acting tension force more than the outwardly acting magnetic pressure force and, therefore, will alter the buoyancy of the tube. Thus, the tubes in group 2 are buoyant to varying degrees despite the small variation in their overall magnetic field intensity.
From figure5.1(b)it can be seen that for the cases in group 1, there is a large variation in B0,
and, consequently, there will be a large variation in the initial buoyancy of the tube, both at the axis and at all radii until the edge of the tube is reached. Since, from(5.7), the density deficit is proportional toB02, the tube withB0 = 9.0is 9 times more buoyant than that withB0 = 3.0,
which in turn is 9 times more buoyant than the tube withB0 = 1.0.