DISCUSION Y COMENTARIOS

To build on our analysis of the horizontal photospheric magnetic field, a calculation of the angle of rotation at the photosphere is necessary. As stated in Section1.6, the rotation angle is an observable quantity and is therefore of interest to both modellers and observers. In order to calculate this, we have again traced three fieldlines from the base of the simulation domain to the photosphere in an attempt to track the general behaviour of fieldlines threading the sunspots. The axis of the flux tube has been traced from the lower negative footpoint as well as two fieldlines either side of the axis in they−direction, i.e. we have traced fieldlines from(0,−14.5,−25),(0,−15,−25), and(0,−15.5,−25)coloured in blue, black, and red respectively. Note, the axis fieldline threads through the centre of the sunspot. A schematic of the traced fieldlines is shown in Fig.4.8aand Fig.4.8bfor timest = 40and timest = 80respectively. As evident from Fig.4.8, the outer fieldlines coloured in red and blue appear to move around the central black fieldline over the course of the experiment. Both the red and blue fieldlines (henceforth referred to as the outer fieldlines) appear to have rotated through an angle of at leastπradians over40normalised time units. This visual estimation is not sufficient and hence we calculate the rotation angle more rigorously.

In order to track selected fieldlines undergoing this rotation, we have traced thexandycoordinates of the locations of the red, black, and blue fieldlines as they pass through the photospheric plane with time. The fieldlines are traced using a fourth-order Runge-Kutta scheme (a numerical technique used to solve ordinary differential equations). The(x, y)trajectories of these fieldlines are shown in Fig.4.9a. Initially,

4.3 Rotation analysis 91

(a) (b)

Figure 4.8: Visualisation of the axis of the flux tube (black fieldline) as well as two fieldlines (red and blue) spaced either side of the axis for comparison at (a)t= 40and (b)t= 80. A movie of this figure is included in the electronic version.

the three fieldlines drift outwards in a line as the sunspots separate. Subsequently, the fieldlines start to move across the photosphere more slowly then start to move around one another. This figure is not very helpful in quantifying the rotation as it is hard to envisage how the outer fieldlines move with respect to the central fieldline as they all translate across the photosphere as the sunspots separate. To rectify this issue, we consider the relative positions of the outer fieldlines with respect to the central axis by redefining

¯

x=x−xaxisandy¯=y−yaxis, as presented in Fig.4.9b. This plot is much more helpful in visualising the rotation and indicates that the outer fieldlines have in fact rotated around the central fieldline axis by almost

360◦. This is a significant rotation similar in magnitude to those seen in observations (see Section1.6). With thexandycoordinates of the photospheric intersections of select fieldlines stored, we can easily calculate the angle of rotation as

tanφ= y0−yaxis

x0−xaxis

, (4.2)

wherexaxis andyaxis are thexandy coordinates of the axis of the tube (black fieldline) andx0 andy0

are the coordinates of the outer fieldline we are investigating, e.g. the red or blue fieldline. In Fig.4.10a schematic has been included to help us visualise the rotation angleφ. This shows that as the blue fieldline moves clockwise the angleφwill decrease passing through zero when it is in line with the axis. The rotation angle for the red fieldline is calculated in exactly the same way. Through the use of Eq.4.2, the angleφ

is calculated for the outer red and blue fieldlines as displayed in Fig.4.11a. As the red and blue fieldlines are initially equally spaced on either side of the axis, the rotation angles areπout of phase at the beginning as expected. Both fieldlines undergo a rotation of between7π/4 and2πover90normalised time units. Specifically, the red fieldline undergoes a rotation of340◦and the blue fieldline undergoes a rotation of

353◦before the experiment is terminated. These rotations can certainly be seen as significant given that the motion is not prescribed and is a direct result of the twist contained within the tube. The specific driver of this rotation is investigated in Section4.3.3.

4.3 Rotation analysis 92

(a) (b)

Figure 4.9: (a) The trajectories of the fieldlines as they pass through the photospheric plane coloured with increasing intensity as time progresses and (b) the relative trajectories with the location of the black axis subtracted. The colour scale on the right shows the times during the evolution.

Figure 4.10: Representation of angleφwith the black, red, and blue dots representing the photospheric intersections of the respective fieldlines.

Fig.4.11b. Although this illustrates that different regions of the sunspot are rotating at slightly different rates, both fieldlines are found to rotate most quickly from approximatelyt= 40tot= 70where we find a peak in the rotation rate. This peak in rotation rate occurs at aboutt= 44for the red fieldline and at about

t= 62for the blue fieldline. The rate of rotation diminishes as the experiment proceeds until it reaches zero indicating that the fieldlines have essentially stopped rotating. The reason behind the final rotation angle value is an interesting concept to explore. Is the interior field completely untwisting or is the twist per unit length tending to a constant along the field? This is investigated in later sections.

In order to investigate if the sunspot is rotating as a whole, i.e. that the rotation angle does not depend on the radius of the fieldline, we check the assumption of solid body rotation. To proceed, we note that the velocity in theφdirection, at radiusr, is given by

vφ=r

dφ

4.3 Rotation analysis 93

(a) (b)

Figure 4.11: Evolution of (a) the angle of rotationφ, and (b) the rate of change of angle, dφ

dt, for both the

red and blue fieldlines as depicted in Fig.4.8.

(a) (b)

Figure 4.12: Comparison of terms dφ

dt (solid line) and ωz

2 (dashed line) from Eq. 4.3for (a) the blue

fieldline traced from(0,−14.5,−25)and (b) the red fieldline traced from(0,−15.5,−25). and thez−component of the vorticity is given by

ωz = (∇ ×v)z= 1 r ∂ ∂r(rvφ)− 1 r ∂vr ∂φ ≈ 1 r ∂ ∂r(rvφ) = 1 r ∂ ∂r r2dφ dt .

It is worth noting that we have ignored the term∂vr/∂φas the sunspots are essentially axisymmetric. If

we assume that the rotation is solid body, and hence thatφdoes not depend onr, we can relate the vertical vorticity and the rate of change of the angleφby

ωz= 1 r2r dφ dt ⇒ dφ dt = ωz 2 . (4.3)

To check if the assumption of solid-body rotation is valid, we investigate Eq.4.3by plotting dφ/dtand

ωz/2for both the red and blue fieldlines. In both panels of Fig.4.12, the two terms are approximately in

phase with each other suggesting that Eq.4.3is approximately valid and the rotation angle may not have a large dependence on the radius from the axis of the tube. This analysis is very simple and brief and hence

4.3 Rotation analysis 94

further investigation is necessary in order to state whether the rotation is solid body given that different fieldlines appear to rotate at slightly different rates.

Figure 4.13: Schematic of starting locations of the traced fieldlines at four different radii, namelyr= 0.24

as shown in black,r= 0.48as shown in blue,r= 0.71shown in green andr= 0.95shown in red. Now that we have calculated the angle of rotation and rate of rotation for two specific fieldlines, we can generalise this to considering a larger selection of fieldlines traced from a footpoint within a given radius. This can then be averaged to gain a more accurate representation of the rotation rate and in turn we can use this to investigate further how the rotation rate varies with radius. To achieve this, we have traced100

fieldlines from within a circle on the base with centre (xc= 0,yc=−15) and radius one. We have, in fact,

traced fieldlines from four different radii on the base within the left footpoint. A schematic of the starting locations of the traced fieldlines is shown in Fig.4.13. The fieldlines are coloured by their starting radius as described in Fig.4.13. The time evolution of the rotation angle for the100traced fieldlines is shown in Fig.4.14aand displays that all fieldlines show the same general trend, though they all appear at different locations on the surface and hence have varying initial angles. The average of this set of fieldlines is shown in Fig.4.14bwhere we have subtracted off the initial angle and the final average angle through which the fieldlines rotate is394◦.

For completeness, we have also shown a plot of how the rotation angle and rate differ with radius in Fig.4.15. As evident from this plot, the radius within the sunspot has little effect on the rotation angle observed. This supports our earlier argument that the rotation is indeed a solid body motion. However, there is perhaps a slight trend with the fieldlines traced from the outer edge of the footpoint rotating slightly slower during the peak rotation phase. This is in support of observations fromYan and Qu(2007) where the authors found the greatest rotation rate in the umbra.