corresponds to a right-hand twisted flux tube. The base of the computational domain is set atz=−25. The major radius of the torus isR0= 15(2550km) and the minor radius isa= 2.5(425km). The initial set-up
of the experiment is summarised in Fig.3.9b. A summary of the parameter choice is given in Table4.1. The total flux through a cross-section of the flux tube is6.6×1011Wb (6.6×1019Mx), typical of a large ephemeral region or small active region.
Table 4.1: Parameter choice.
Magnetic field parameters Global parameters
B0= 9,α= 0.4 5123grid points
R0= 15,a= 2.5 η= 0.005everywhere
4.2
General analysis
Before we analyse the rotational motion of the sunspots at the photosphere in this experiment, let us consider a general overview of the evolution of the flux tube as it rises through the interior and emerges into the atmosphere.
4.2.1
Rise through solar interior
The density deficit introduced (see Eq.3.13) disrupts the equilibrium and allows the flux tube to start to rise buoyantly to the photosphere. This deficit is implemented by setting the temperature of the tube equal to the temperature of the surroundings and maintaining the negative pressure excess (or pressure deficit) found by balancing the pressure gradient with the Lorentz force. Further details of this can be found in Chapter3. The flux tube continues to rise through the solar interior due to the buoyancy instability until it reaches the convectively stable photosphere. The isothermal stratification in this layer results in anN2>0.
See Section1.4for a definition of the buoyancy frequencyN2and further details on both the buoyancy and magnetic buoyancy instability.
The height-time profiles for the axis and leading edge of the system are shown in Fig.4.1. The rise of the flux tube to photospheric heights is governed by the buoyancy instability as shown in the height-time plot up untilt= 25. Following the method ofHood et al.(2009), the axis can be identified by plotting the zero contour ofBxandBz in the mid plane, and identifying the intersection of the two contours. In this
particular caseBzis zero along the line atx= 0andy = 0. Thus, we track the axis by any zero ofBx
along the line prescribed byx= 0andy= 0. This has also been checked against tracing the field from the centre of both footpoints and is found to agree for most of the experiment. We believe the formation of a new flux rope and, in turn, new axis is responsible for the divergence of the two methods. New flux ropes form due to shearing flows and reconnection (seeMactaggart,2010for an explanation of the mechanism). We warn the reader that tracing from the left and right footpoints produce almost identical results and as such the blue and pink colourings are difficult to identify. The leading edge of the expanding volume is
4.2 General analysis 85
calculated as the height where the field strength first increases above10−7. The leading edge of the system is determined by the pressure balance boundary, where the total pressure within the tube equals the gas pressure. Initially, the flux tube rises relatively slowly until the leading edge reaches the photosphere. The tube then expands more quickly due to the density drop off at the photosphere. Later, the magnetic bubble expands very quickly due to the initiation of the magnetic buoyancy instability. The divergence of the two methods at later times is likely to be due to the kinking of the axis from they= 0plane.
Figure 4.1: The height-time profiles of the axis of the flux tube traced in thex = 0,y = 0plane using
Bx = 0(x symbol), the leading edge of the flux system (dashed), the intersection of the magnetic field
with they = 0plane as traced from the centre of the left footpoint (blue) and from the centre of the right footpoint (pink). The three horizontal lines atz= 0,z= 10andz= 20signify the solar surface, the start of the transition region and the start of the corona, respectively.
4.2.2
Arrival at the photosphere
The rise of the axis of the flux tube appears to slow when the tube reaches photospheric heights due to the change in stratification. At the photosphere, the plasma is stably stratified with a constant temperature and the flux tube is no longer able to rise by means of the buoyancy instability. The temperature gradient is no longer sufficiently decreasing and is therefore strongly sub-adiabatic. Therefore, the magnetic field must find another way to rise and expand into the corona, and it does, specifically, through the magnetic buoyancy instability. In order to initiate this instability, a criterion must be satisfied as derived in Chapter1. Typically, the onset of this instability occurs when the plasmaβ, defined as the gas pressure divided by the magnetic pressure, drops to one (Murray et al.,2006). Therefore, this can only occur if the initial field strength is large enough. This suggests that the properties of the emerging flux are highly dependent on the strength of the original interior field, a concept which we will investigate in Chapter5. At this stage, the plasmaβdropping below one means the magnetic pressure exceeds the gas pressure and the field expands into the atmosphere. As a reminder, the criterion for the magnetic buoyancy instability is shown below in terms of the plasmaβ,
−1 β d d˜zlogB0> γk˜|| 2 β 1 + ˜ kz 2 ˜ k⊥2 ! +γ 2δ, (4.1)
4.2 General analysis 86
where we have divided the criterion given in Eq.1.45by β and all variables used here are outlined in Section1.4. Before the flux tube reaches the photosphere the criterionN2<0determines whether the flux
tube rises by means of the buoyancy instability. However, atz = 0,N2 >0and the buoyancy instability
can no longer be triggered. Hence, the magnetic buoyancy instability is the only instability that could allow the flux tube to rise. Note from Eq.1.43that even ifN2>0, as at the photosphere, the criterion can still
be satisfied if the magnetic field gradient is significantly steep.
(a) (b) (c)
Figure 4.2: The left (red) and right (blue) hand terms of Eq.4.1plotted against height for (a)t = 11, (b)
t= 17and, (c)t= 20. Also overplotted is the plasmaβas shown in green. The grey term in Eq.4.1is not plotted as it is not comparable with other terms. A movie of this figure is attached in the electronic version.
Figure 4.3: The change in the density excess at the axis of the flux tube, as a fraction of the unsigned initial density excess at the axis, plotted against height.
In Fig.4.2, the different terms in Eq.4.1are shown against height for selected times. The terms of the criterion are coloured as they are in the equation. The red line displays the magnetic field gradient divided byβ and the blue line shows the superadiabatic excessδmultiplied byγ/2. We note the term involving wave-numbers is shown in grey as this term is negligible in comparison to the rest of the equation, and in turn has been excluded from the plot. Note, the red term is positive as the magnetic field strength decreases with height. As the magnetic field reaches the photosphere, the magnetic field gradient increases, and at the same time the plasma beta (β =p/(|B|2/2)shown in green) decreases. This combined effect causes
the red term to increase and allows the criterion to be satisfied. There is often a delay in the initiation of this instability as the magnetic field builds up atz = 0and spreads horizontally allowing the plasmaβ to lower. Here we can see that the criterion for the magnetic buoyancy instability is satisfied when the plasma
4.2 General analysis 87
βdrops to unity as predicted byMurray et al.(2006). Only at this point, when the instability is satisfied, does the field rise above the photosphere and emerge into the corona. As the plasmaβ has now dropped below one, the magnetic pressure exceeds the background pressure, allowing the field to expand rapidly into the corona. This is made easier by the exponential decrease in pressure with height in the atmosphere.
Due to the expansion of the magnetic field into the corona, plasma drains from the top of the emerging bubbles and flows down fieldlines to the photospheric plane. To explore this, a plot of the weighted density excess, which is calculated asρ(x= 0, y= 0, zaxis)−ρ(x=−50, y=−50, zaxis)divided by its initial
magnitude, is shown in Fig.4.3. Moving up through the solar interior, the surrounding plasma density decreases, and hence the excess decreases in magnitude until the tube is over-dense with its surroundings when it reaches the photosphere. This is an improvement on the cylindrical case, where in the experiments performed byMurray et al.(2006) the tube became over-dense much lower in the solar interior. This is a consequence of the geometry of the toroidal loops allowing for efficient draining of plasma.
One important distinction between the cylindrical and toroidal model is that the cylindrical tube was initiated by a density deficit that made the tube maximally buoyant at the centre and reduced towards the edges. However due to the exponential profile of the density deficit in the cylindrical case, the edges of the tube are made weakly buoyant which allows the sunspots to drift continually until they reach the edge of the box. This is not the case in the toroidal simulations, whereby the sunspots drift to a fixed distance, namely the major diameter of the torus. The separation of the sunspots is shown in Fig.4.4. To estimate the separation of the sunspots, we have plotted the separation in they direction between the maximum and minimum ofBz, and find that it levels off after t = 60. This result is corroborated by
observational studies, includingKosovichev and Stenflo(2008) andWallace Hartshorn(2012). Through the analysis of a sample of active regions from a study of715active regions,Kosovichev and Stenflo(2008) found a general trend where polarity separation increases to a maximum and then starts to very gradually decrease.Wallace Hartshorn(2012) analysed the polarity separation of57active regions in her PhD thesis and found a similar trend. The polarity separation reached a maximum and started to level off with a slight decrease. This highlights one of the advantages of the toroidal model over the corresponding cylindrical model.
Figure 4.4: They−separation in time of the maximum and minimum ofBzat the base of the photosphere