Currents and field-aligned flows

Although the suppression of vortex formation by twisted magnetic field could result in the instability being difficult to observe in the solar corona, the formation of currents still persists and may have interesting consequences for the energetics of the system. Indeed, as we shall show in this section, the maximum magnitude of the current density that forms during the instability can be significantly larger in the twisted field cases. Ultimately, this may allow the magnetic Kelvin-Helmoltz instability to enhance the rate of wave energy dissipation even if only small vortices are able to form.

As we demonstrated in Chapter 2, the formation of the KHI is associated with the generation of significant currents as the vortices stress the magnetic field. Meanwhile, the inclusion of a background, azimuthal field ensures that there are currents present in the initial conditions. Since Bφ is a non-trivial function of R, we see that these

currents are loop-aligned. We aim to quantify how the development of the azimuthal wave and subsequent growth of the KHI modifies and enhances the magnitude of these currents.

In Figure 3.13, we display the magnitude of the current density integrated over the cross-section of the loop at different heights and as a function of time for case 2c. The analagous plot for the straight field simulation is shown in the lower panel of Figure 2.14. In both cases, we see the formation of larger currents as the simulation progresses; initially, as resonant absorption injects energy into the azimuthal Alfv´en waves and subsequently (t > 600 s), as the KHI develops. In the straight field case, the magnitude of the currents remains very small at the loop apex, however, in the twisted field simulations, the presence of currents in the background field ensures that the area integral of|j|is always non-zero at any value ofy along the length of the loop. Despite this, in Figure 3.13 we can observe that the currents that form during res- onant absorption and the growth of the KHI, are much larger than the initial currents. In case 2c, we also see significant current formation at the loop apex as the instability

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Figure 3.13: Magnitude of current density integrated over loop cross-section as a func- tion of time and position along the length of the loop. The figure was produced using the results of case 2c.

develops. This is a significant departure from the straight field regime. Although we still expect, in a non-ideal simulation, that magnetic wave energy dissipation will be largest at the loop foot points, in the twisted field simulations we might expect some contribution from currents along the entire length of the flux tube, including at the loop apex.

In Figure 3.14, we display the magnitude of the current density that forms on the ρ = 2ρe surface for three simulations. The surface plots correspond to a time at

which the KHI is well-developed. This can be seen by the deformation of the initially cylindrical surface. In all three cases we see that the currents that form are highly localised and this highlights the difficulty in resolving the turbulent-like aftermath of the KHI on a finite numerical grid.

In the straight field case (first panel), we see a band around the loop apex in which no significant currents have formed. This is not the case, however, in the twisted field simulations. In all three simulations, the large currents tend to form in field- aligned strands, which suggests any Ohmic heating (withη6= 0) at a given time would

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(a) Case 1. (b) Case 2b. (c) Case 2c.

Figure 3.14: Magnitude of current density on the surface defined by ρ = 2ρe during

the development of the KHI.

cause temperature increases on along individual field lines rather than across the entire surface. Despite this, these figures mask the highly dynamic nature of the small scales that form, and over the course of a wave period, the majority of the surface experiences the formation of energetic current sheets.

In Chapter 2, we argued that, for the straight field case, the currents that do form at the loop apex following the onset of the KHI are dominated by the horizontal com- ponentsjx andjz. However, this is not true for the twisted magnetic field simulations.

In Figure 3.15, we show the maximum magnitude of the current density (and each component) at the loop apex for each case as the simulations progress. Each curve displayed in the top left-hand panel has been normalised to the pre-KHI formation current density (at t = 560 s) in the respective simulation. In this manner, we aim to represent the relative growth of the current density during the development of the instability.

Despite this normalisation, which is more significant in the simulations with larger magnetic twist (they have greater background current densities), we see that the weakly twisted simulations (case 2a and 2b) exhibit much larger currents than the straight field case. By considering the remaining three panels, we see that this is associated with differences in the loop-aligned current density, jy. Indeed, in the lower left-hand panel

of Figure 3.15, it is clear that the twisted field simulations generate much larger values of |jy| than the straight magnetic field case. Meanwhile, with the exception of |jz| in

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Figure 3.15: Growth of currents during the development of the KHI. The top left- hand panel displays the maximum magnitude of the current density in the horizontal cross-section at the loop apex. For each simulation, it is normalised by the maximum current density in this plane at t = 560 s. This corresponds to a time just before the first Kelvin-Helmholtz vortices can be observed. The remaining three panels display the maximum magnitude of each component in the same cross-section. The units are dimensionless but are the same for each of the components.

case 3 (which we discuss below), the horizontal current densities, jx and jz, remain

similar in magnitude in all simulations.

From Amp`ere’s law (1.7), we see that the y-component of the current density is associated with radial gradients in Bφ and azimuthal gradients in BR. In the case 1

simulation (straight magnetic field), at the loop apex, there is no background horizontal field component. Furthermore, the standing kink and Alfv´en wave modes do not induce any horizontal field perturbations in they= 0 plane. Thus, with the exception of small variations induced by the KHI, |jy| is necessarily small. In the twisted field cases, on

the other hand, there is a non-zero component of jy, even prior to the onset of the

instability (t < 600 s; Figure 3.15, lower left-hand panel). As the magnitude of the radial gradient is a function of twist factor (τ or ψ) , we see that, for example, Case 2c has more significant currents than Case 2b.

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Figure 3.16: Horizontal components of the magnetic field (vectors) and the deformed density profile (coloured contour plot). The plane displayed is the upper half (z ≥ 0) of the horizontal cross section at the loop apex once the KHI is well-developed. The figure is produced using the results of case 2b.

Subsequently, we see that during the development of the KHI, this difference is enhanced. We attempt to explain this phenomenon in Figure 3.16. Here, we display the horizontal components of the magnetic field along with the deformed density profile for Case 2b. We can clearly see the azimuthal field which is well-structured within the loop’s core but has become stressed within the vortex-forming region. We consider a plasma element that is moving radially outwards on account of the instability and recall that the magnetic Reynolds number is much larger than unity. Thus, the motion of this plasma element will advect magnetic field away from the centre of the loop. Hence, field with a large azimuthal component will be transported into a region with lower twist. Therefore, we can see that this process reduces the length scales on which the azimuthal component varies and so increases the current density. Furthermore, vortical motions produced by the KHI introduce radial components of the field and generate additional loop-aligned currents.

Although the presence of azimuthal magnetic field can suppress the formation of vortices, it can also produce locally enhanced currents. Indeed, the lower left-hand panel of Figure 3.15 demonstrates that the KHI can be more energetic over (possibly small) regions of the loop. Consequently, we may expect more powerful Ohmic heating events in the most energetic vortices in a loop with weakly twisted field. We also note that despite the very small magnitude of Bφ implemented in case 2a, there are

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still significant loop-aligned currents generated. This suggests that even though the magnetic field is locally, almost straight, it may produce different heating signatures than the truly straight field case.

As mentioned above, in the lower right-hand panel of Figure 3.15, we see that the alternative magnetic twist profile explored in case 3 generates larger values of|jz|than

the other simulations. Further, the maximum forjx (top right-hand panel) occurs later

and is larger than the values observed in each of the other simulations. We see more significant values of the horizontal current density in this case because the Kelvin- Helmholtz vortices form in regions of stronger twist than in the case 2 simulations. As a result, the transverse gradients generated during the instability are much larger than those in the other experiments.

Another consequence of the inclusion of azimuthal magnetic field is the excitation of loop-aligned velocities, vy, at the loop apex. In the straight field case, at y = 0,

there is no vertical gradient in the gas or magnetic pressures and there is no magnetic tension force. Thus, in the absence of gravity, there are no forces to drive loop-aligned flows at the loop apex (we note that ponderomotive forces will drive vertical flows close to the loop foot points). However, if an azimuthal component of the magnetic field is included, the radial current density (associated the KHI) induces ay-component of the Lorentz force. This effect occurs for even extremely weakly twisted field (case 2a). Both positive and negative radial currents are generated during the KHI and thus both upwards and downward flows are generated.

In Figure 3.17, we display the maximum value of |vy| at the loop apex for all the

simulations within our parameter space. We note that on this scale case 2a (green line) does not seem too different from the straight field simulation (black line). However, the vertical velocities obtained in the weakly twisted case are in fact many orders of magnitude larger than the very small loop-aligned flows observed in the straight case. The velocities observed in the twisted field simulations for t ≤ 600 s, are associated with the Alfv´en mode wave vector not being parallel to the horizontal plane. Thus, any excited Alfv´en wave will have a velocity component in the vertical direction. Sub- sequently, we see the radial current density that forms during the instability generates much larger flows. The stabilising effects associated with the magnetic twist ensure that for a short period the vertical flows are greater in case 2b than case 2c despite the stronger twist component present in the latter simulation.

In theory the excitation of these loop-aligned velocities could act as a proxy for detecting magnetic twist within a transversely oscillating coronal loop. Indeed, there is a significant difference between the vertical flows observed in each case. However,

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Figure 3.17: Maximum value of |vy| at the loop apex as a function of time for all

simulations.

the velocities are highly localised and will likely exist over scales significantly beyond current observing capabilities. Furthermore, the differences are only significant at the loop apex for a fundamental standing mode. The ponderomotive force excites loop- aligned flows along much of the rest of the loop in all cases. Therefore difficulties in identifying the loop apex, flux tube asymmetries and the existence of higher harmonics will likely further impede the applicability of this proxy.

3.4.3

Vorticity

As we discussed in Chapter 2, the development of the KHI also produces small scales in the velocity field which can be monitored by considering the vorticity, ω. In non-ideal regimes, gradients in the velocity are susceptible to the effects of viscosity which, in turn, lead to the dissipation of wave energy. In the straight field case, we highlighted that the excitation of azimuthal Alfv´en waves enhances the loop-aligned component of vorticity, ωy. Additionally, the subsequent development of Kelvin-Helmholtz vortices

further increases the magnitude of velocity gradients.

In Figure 3.18, we display the magnitude of the vorticity,|ω|in the horizontal cross- section at the loop apex. We show two different stages during the simulation for the

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Figure 3.18: Magnitude of the vorticity at two different times, t = 420 s (left) and t = 990 s (right) in the horizontal cross-section through the loop apex. The first row corresponds to case 1, the straight field simulation, and the second row corresponds to case 2c. Here we have normalised the vorticity by the same value for both simulations.

straight field case (first row) and the twisted field case 2c (final row). In the left-hand column, significant resonant absorption has occurred, however, no vortices have formed in either simulation. In the right-hand column, on the other hand, we can see that the instability has developed. Prior to the formation of the KHI, since resonant absorption is only weakly modified by the azimuthal field, the profiles of|ω|are very similar. Once the KHI forms, however, differences become apparent as the characteristic vortices are suppressed in the twisted field case. Despite this, the maximum magnitude of the vorticity is very similar in both of these simulations, suggesting that the vortices that form in case 2c are at least as energetic as those observed in the straight field simulation. In Figure 3.19, we consider the three components of vorticity in more detail. In the top row, we display the three components of ω integrated over the loop cross-section. In particular, at each time we calculate,

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Figure 3.19: First row - the three components of the vorticity integrated over the horizontal plane at the loop apex. Second row - maximum size of each component of the vorticity in this plane. For both of the rows, the black line corresponds to the straight field simulation (case 1) and the red line corresponds to a twisted field simulation (case 2c). Z A ωxdA= Z A ∂vz ∂y − ∂vy ∂z dA, (3.9) Z A ωydA= Z A ∂vx ∂z − ∂vz ∂x dA, (3.10) Z A ωzdA= Z A ∂vy ∂x − ∂vx ∂y dA, (3.11)

for the panels from left to right. Here, A is a circle of radius 1.5 Mm that tracks the cross-section of the flux tube as it moves with the global kink mode. For all three components of ω, the straight field simulation (black lines) displays very little net vor- ticity when integrated across the loop cross-section. In other words, any contributions from regions with positive vorticity are cancelled by regions with negative vorticity. On the other hand, the inclusion of magnetic twist permits the non-linear generation of additional harmonics that are associated with the oscillating profiles in the panels