Presenter: N. R. Sheeley
1.Introduction: As Yi-Ming described yesterday, we use a model in which new flux erupts in active regions and is transported on the photosphere by supergranular diffusion, differential rotation, and meridional flow. The field extends upward in a current-free configuration and open flux is simulated by requiring the non-radial components to vanish at a spherical source surface located at 2.5 R. Today, I am going to consider how the open flux changes in response to variations of the surface field. The general answer is that the open flux increases when opposite polarities are separated and the open flux decreases when the opposite polarities are mixed together.
2.Four Examples: It is easy to verify this answer for the source surface model because the open flux is precisely defined. Let’s consider four examples: a horizontal dipole, a tilted dipole, an equatorial configuration, and the eruption of a new doublet.
[a] HORIZONTAL DIPOLE: Differential rotation winds the horizontal dipole into stripes of alternating magnetic polarity. (FIG. 1) The source surface does not see such mixtures and its field is determined by the dwindling remnant of unwound flux at the equator. In the language of spherical harmonics, power is transferred from the dipole to the higher order harmonics (l = 3, 5, 7, …) which have little contribution at 2.5 R. The open flux mimics the dipole amplitude, hovering unchanged for a brief moment and then falling rapidly to half its initial value in the time it takes to wind the meridional boundary through 360 degrees (4.8 rotations). (FIG. 2) Thereafter, the amplitude decays inversely as the square root of the elapsed time. Because differential rotation does not change the total amount of flux on the Sun, open flux must have been converted to closed flux by reconnecting at the source surface and collapsing downward over the growing photospheric neutral line. Adding supergranular diffusion to the problem has little effect during the first windup time because the diffusion time scale is 15 yrs for a dipole. (FIG3) It’s main contribution is to decrease the total flux by removing the narrow-spaced stripes. (FIG. 4) Meridional flow also does not contribute during the first windup time because its time scale is about 1 yr. Thus, in general, we expect reconnection and collapse during the first windup time as open flux is converted to closed flux.
[b] TILTED DIPOLE: Now, let’s add a vertical component to make a tilted dipole. The photospheric neutral line starts as a sinusoid, but is quickly wound up into stripes as before. (FIG. 5) However, in this case, the dwindling remnant of unwound flux at the equator is accompanied by unwound flux at the poles, and the amplitude of the tilted dipole decays more slowly than the amplitude of its horizontal component. (FIG. 6) In fact, the net change of open flux is reduced by the factor tan(tilt_angle/2), which approaches zero for very strong vertical fields. Thus, we would expect the Sun to shed less open flux during the declining years of the sunspot cycle when the polar field is strengthening than in the years around sunspot maximum when the polar field is weak. As before, the addition of diffusion and flow do not change this behavior appreciably during the first windup time (when half of the change occurs).
[c] A LATITUDINALLY COMPRESSED DISTRIBUTION: If the non-axisymmetric flux were concentrated near the equator, then it would be unaffected by differential rotation, and the field would not change with time. (FIG. 7) The source-surface field would maintain a tilted dipole configuration with no change of open flux, and the meridional lobe of the coronal hole would not be sheared by differential rotation. The same thing would happen if the equatorial flux were compressed in longitude to simulate a low-latitude emerging flux region. (FIG. 8) In this case, one would obtain a large meridional coronal hole like Skylab CH#1 in 1973, which resisted the shearing effect of differential rotation for several months.
[d] FLUX ERUPTIONS: Because most newly erupting doublets are tilted relative to the parallels of latitude, they contribute to both the vertical and the horizontal components of the large-scale field. The vertical contribution is small, but systematically opposite the polar field at the start of the sunspot cycle, and therefore will weaken it. The horizontal contribution is larger, but may be out of phase with the background field or in phase, depending on the longitude and north-south hemisphere of the eruption. If the eruption is in phase with the horizontal dipole, it will strengthen it and cause the total dipole to tip further toward the equator. If it is out of phase, the eruption will weaken the horizontal dipole and tilt the total dipole back toward the pole. This is why some large active regions increase the warp of the coronal streamer belt, while other equally large regions do not. In the source-surface model, these changes occur in step with the growing active regions and do not depend on the rates of photospheric flux transport.
It is instructive to consider what would happen during a sunspot cycle if we were to neglect flux transport and simply add together the dipole contributions of all of the erupting active regions. There would be a systematic erosion and reversal of the polar fields. But the amplitude of the horizontal dipole would increase without limit in a random walk produced by the longitudinal fluctuations of the sources. The flux transport provides an effective dissipation that limits this growth and allows the evolving field to come into equilibrium with its sources. It would strengthen the horizontal dipole around sunspot maximum and weaken it again at minimum, as observed.
But this concept of a global dipole tipping over has two more problems. First, it leaves out quadrupole terms, which result from a variety of active region configurations (FIG. 10) and have a major contribution to the open flux in the years around sunspot maximum. (FIG. 9) Second, it ignores interactions between multipoles, such as occur when differential rotation increases the pole separation of a newly erupted doublet. In desperation, one might hope to salvage this idea of `something’ tipping over during the sunspot cycle by considering a more general concept like `open flux’ or `streamer belt topology’, and then averaging over transients and interactions between multipoles. But in doing so, he would be throwing away the opportunity for new discoveries that the flux-transport/source-surface model provides.
So where do we stand? We have a model that is useful for discovering new things about the Sun. We found that open flux decreases when the horizontal dipole is wound up by differential rotation, as often occurs in the years around sunspot maximum. And we learned that much less open flux is shed when the non-axisymmetric field is accompanied by a strong axisymmetric component, as occurs during the declining phase of the cycle.
The accompanying movies show coronal inflows at sector boundaries where opposite-polarity field lines are being pushed together by shearing motions of their footpoints.(http://lasco-www.nrl.navy.mil/public_movies.html ) Most inflows occur in long-lived streams where low-latitude coronal holes are being sheared by solar differential rotation and their oppositely directed coronal field lines are being pushed together (FIG. 11). This suggest that they indicate magnetic field reconnection associated with the decreasing open flux of shearing coronal holes.
Most of these inflows lack visible outward components, perhaps due to their extreme faintness. However, some inflows do have outward components. It is possible that these in/out pairs involve reconnection between open field lines and closed loops. If so, we might expect to see more of them during the declining phase of the sunspot cycle when the polar fields are stronger and coronal holes begin to rotate rigidly again.
References
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