MODIFICACIÓN DEL ARTÍCULO 184 DE LA LEY GENERAL

The majority of the atmospheric velocities that arise are due to the error associated with the first order error method used to calculate the gas pressure and density in the photosphere. The re- maining velocities occur because of sharp gradient changes in the magnetic field profile and the temperature profile. The velocities are comparable in magnitude to the velocity of the buoyant flux tube and, therefore, should be removed to ensure the results from the emerging flux experiments

are dependent upon the rising tube and not the non-hydrostatic atmosphere.

We have shown that by modifying the profile for the increasing field strength in the photosphere from a tanh function to a sine profile, we can ensure there is a smooth transition in the field strength between the top of the photosphere and the transition region. This modification, solution A, eradicated the peak in velocity atz= 10. Using the sine profile for the magnetic field strength in the photosphere also enables an analytical expression to be derived for the gas pressure and density here. The use of the analytical expression over the finite difference method dramatically decreases the errors in calculating hydrostatic equilibrium and, therefore, the unwanted vertical velocities in the photosphere are greatly reduced. The reduction in vertical velocity due to the implementation of solution B is evident through the comparison of figures4.7and4.16. These confirm that the solution is valid for domains with either a magnetised or unmagnetised atmo- sphere.

Solution B has also had a positive effect on the size of time step between consecutive iterations ofDiffin3d. As discussed in section 3.3.2, the size of the time step is varied to ensure that information does not propagate out of the numerical stencil during a single iteration. By removing the sharp peaks and troughs in the vertical velocity, the size of the time step can be increased. For example, in the experiment with no tube and an atmospheric field the original numerical model uses a step size of0.00156time units. With the modifications of solution B, the time step size increases to0.00169time units. We note that the new numerical model for solution B may contain more calculations and will, therefore, require slightly more computational time. However, since this step is only carried out once per experiment the gain in fewer iterations from a larger time step size will easily out weigh any additional computational time required for the setup calculations. Finally, modifying the temperature profile in the transition region from the power law to a sine function enables zero gradients to be imposed at the interfaces with the bounding regions. This modification removed the vertical velocity discontinuity at the interface between the photosphere and the transition region but was less successful at the interface between the transition region and the corona. This may be because the gradient in the temperature profile is greater at the top of the transition region and, thus, the turn over to a zero gradient must occur over a very small height, as shown in figure4.17. If there are not enough gridpoints to accurately model the turn over, then errors can be introduced into the model. In addition, solution C actually leads to a decrease in the size of the time step to0.00038time units. This is smaller than that of the original numerical model and the model produced by solution B and, thus, a greater number of iterations would be required to reach the same point in time as the other models. In light of both of these facts, it does not seem advantageous to employ the new temperature profile and associated pressure and density profiles.

(a) Flux tube and an unmag- netised atmosphere. (b) An unmagnetised atmo- sphere. (c) A magnetised atmo- sphere.

Figure 4.16: Vertical velocity against height for three different experiment scenarios at different times, using the numerical model of solution B.

Figure 4.17: Temperature profile in the transition region using solution C,(4.40).

model and should, ideally, be used in any future flux emergence simulations using this model domain. However, it should be noted that the original numerical model will be used in the simu- lations in chapters5,6and7. By using the original model, clear comparisons will be able to be made between the results from previous simulations that have used the same model.

Effects of Twist & Strength

Note: The material in this chapter has been published in Astronomy & Astrophysics. The reference is:

Murray, M. J., Hood, A. W., Moreno-Insertis, F., Galsgaard, K., and Archontis,V. (2006). 3D simulations identifying the effects of varying the twist and field strength of an emerging flux tube.Astronomy & Astrophysics, 460:909-923.

Previous emerging flux simulations have considered only a small region of the parameter space for the initial flux tube, as discussed in chapter2. In this chapter, we will concentrate on two of the parameters associated with the constant twist flux tube, namely twist and axial field strength. By independently varying the values of these parameters, we aim to determine the individual effects of each parameter and their roles in the emergence process.

We have chosen to use a flux tube that has been used in multiple previous flux emergence simula- tions. In section5.1, we will define the magnetic field of the flux tube and identify the parameter space to be tested for the two variables under consideration.

In chapter4, we presented the hydrostatic simulation domain, which models the layers of the Sun from the solar interior up to the low corona. We place the flux tube in the domain’s solar interior and choose it to be in radial force balance with the background plasma. In section5.2, we will consider how the gas pressure, temperature and density profiles of the tube are chosen such that the tube is buoyant in comparison to the surrounding medium. Here, we will also detail the effects of changing the twist and field strength on the initial structure of the flux tube.

The first results we consider, in section5.3, are those for the case with parameter values for the twist and field strength that most closely reflect the experiments of existing literature. These illustrate the general evolution of the tube during its rise through the solar interior and emergence

into the atmosphere. Following this, we consider the effects on the rise and emergence processes when the values of these parameters are varied. In section5.4, we will present the results of solely varying the field strength and, in section5.5, we will present the results of varying only the twist associated with the tube.

Finally, in section5.6, we will summarise the modifications in the emergence of flux caused by varying the flux tube’s field strength and twist and, using our findings, we will draw conclusions as to the robustness of the results from previous emergence simulations.

5.1

Defining the Magnetic Flux Tube

Most simulations of buoyant, twisted, flux tubes use a constant twist magnetic field for the flux tube. The concept of constant twist flux tubes was introduced in section1.3.3. A large collection of papers actually use the same equations to define the constant twist magnetic field of the tube. In the cylindrical coordinate system,(r, θ, y), the popular magnetic field is given byB= (Br, Bθ, By),

where Br = 0, (5.1) Bθ = αrBy, (5.2) By = B0e−r 2_/R2 . (5.3)

These equations ensure the field strength of the tube falls to zero at large radial distances from the axis. Hence, the tube can be considered as a distinct magnetic body in relation to the sur- rounding unmagentised environment. On the axis, the magnetic field has solely aBy component,

with direction given by the sign ofB0 and strength by|B0|. At a distanceR from the axis, the

component of the field in the axial direction is37%its original value. The value ofRis held fixed at2.5over the series of experiments andB0andαare varied. The magnetic fieldlines of the tube

twist around its axis and, using the definition in section1.3.3, the twist is given byα.

The experiments are split into two groups, group 1 for those with fixedα and varying B0 and

group 2 for fixedB0with varyingα. A summary of the values ofB0 andαunder consideration

is given in table 5.1. α is constant with radius in each of the experiments so that the tubes are uniformly twisted. Additionally, the group 1 cases, withαfixed at 0.4, are marginally unstable to the kink instability sinceα > 1/R(Fan et al. 1998b), however, we do not see the instability develop during the tube’s rise through the solar interior in any of our experiments.

An immediate effect of varying these two parameters is seen in figure5.1. Reducingαincreases the rate at which the field strength of the tube fall off with radius. IncreasingB0 increases the

Group 1 Group 2

B0 ={1.0,2.0,3.0,5.0,9.0} B0 = 2.9

α= 0.4 α={0.1,0.2,0.3} Table 5.1: Summary of the parameter space under investigation.

(a)αcases. (b) B0cases.

Figure 5.1: Radial distribution of the initial magnetic field strength,|B|.

magnetic strength of the tube at all radii. As we will see in section5.2, variations in the values of these parameters cause further modifications in the buoyancy of the tube.

We have identified seventeen papers that use exactly this field definition and a further four papers that modify the transverse field distribution(5.2) and so have a non-constant twist profile with radius. Each paper considers different combinations of values within the parameter space forB0,

αandR. To demonstrate how our study fits with and differs from those already carried out, we briefly outline the contents of these specific papers, many of which have been discussed in full in chapter2.

Fan et al.(1998b) consider the rise of a highly twisted and, therefore, kink-unstable flux tube within the convection zone whileDorch et al.(2001) compares the rise of tubes in convective and non-convective flows. Moreno-Insertis and Emonet(1996),Emonet and Moreno-Insertis(1998),

Abbett et al.(2000),Cheung et al.(2006) andDorch(2003) vary the degree of twist of a tube rising though the convection zone. The first four papers aim to understand the fragmentation of the flux tube during its rise whilst the fifth considers the various instabilities arising from the different amounts of twist. The effects of variations in the magnetic field strength and twist of a tube rising though a convective flow are identified byFan et al.(2003), whileCheung et al.(2007) considers these parameters’ effects as the flux tube approaches and passes through the solar surface.Dorch et al.(1999) andDorch(2007) carry out comparisons between the rise of horizontal and undular twisted flux ropes in both 2D and 3D during their rise through an unmagnetised and magnetised

convection zone, respectively.

The remaining papers consider the full emergence of the flux tube from the solar interior into the overlying atmosphere. Abbett and Fisher(2003), using two different computational codes to simulate the full emergence process, find that decreasing the twist of the flux tube increases the degree to which the initial emergence is force-free.Fan(2001) considers a tube which is stable to the kink-instability and reports on the dynamic evolution of the tube during the emergence process into an unmagnetised atmosphere. A whole series of further studies have sprung from the results of this single experiment. A variety of values for the field strength, twist and radius are used but each subsequent study considered just one value for each parameter.

Archontis et al.(2004) have further investigated the results ofFan(2001), reporting them in more detail and advancing the experiment by adding a magnetised atmosphere above the emerging flux tube. A number of papers (Galsgaard et al. 2005,2007;Archontis et al. 2005,2006) have varied the structure of the magnetised atmosphere and these use the same parameter values asArchontis et al.(2004). With the aim of generating a more self-consistent model,Archontis et al. (2007) set two buoyant tubes in the subphotospheric region such that the first tube emerges and creates a magnetised atmosphere into which the second emerges.

Manchester et al.(2004) use the same parameter values as the original experiment byFan(2001) but they reduce the region over which the tube is buoyant and find a CME type event occurs once the tube has emerged into the atmosphere. Finally,Leake and Arber(2006) include a partially ionised region in the atmosphere of the domain to simulate the choromsphere. During the tube’s emergence through this new region, ion-neutral collisions cause all cross-field currents to be dis- sipated and, as a result, a force-free field emerges into the corona.

The extensive use of the field structure given by (5.1), (5.2), (5.3), in emergence experiments warrants a comprehensive study of the parameter space. Figure5.2illustrates the limited explo- ration of the parameter space to date. We will vary the twist and strength of a magnetic flux tube and, for the first time in an emergence study, evaluate the effects of the variations. Our aim is to understand the role of twist and field strength in shaping the emergence process.

It should be noted that a parameter space investigation has been carried out byMagara(2001), in which the radius, twist and strength of an emerging flux tube were varied. Our study is different on several fronts.

• We use the field structure defined by(5.1), (5.2), (5.3), and this is not used by Magara

(2001).

• The flux tube we define is fully buoyant and, therefore, prescribed with a buoyancy pertur- bation (described in section5.2) to encourage the formation of anΩ-shaped loop. Following

Figure 5.2: TheB0-α parameter space for the magnetic flux tube defined by(5.1), (5.2),(5.3).

The red triangles indicate the cases under investigation in this chapter and the black stars demon- strate theB0-αpairings used in previous flux emergence simulations.

this initialised perturbation we allow the system to evolve by itself. On the other hand,Ma- gara(2001) choose to drive the central portion of the tube upwards by imposing a vertical fluid velocity.

• The values of the parameters were altered simultaneously byMagara(2001) and, therefore, it is difficult to quantify the independent impact of each on the emergence process. We aim to provide a clear explanation of how varying the twist and field strength of the flux tube, independently of each other, will alter the dynamics of the rise of the flux tube in the upper layers of the convection zone and its subsequent expansion into the atmosphere.