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Guided flows in coronal magnetic flux tubes

There is evidence for coronal plasma flows to break down into fragments and to be laminar. We investigate this effect by modeling flows confined along magnetic channels. We consider a full MHD model of a solar atmosphere box with a dipole magnetic field. We compare the propagation of a cylindrical flow perfectly aligned to the field to that of another one with a slight misalignment. We assume a flow speed of 200 km/s, and an ambient magnetic field of 30 G. We find that while the aligned flow maintains its cylindrical symmetry while it travels along the magnetic tube, the misaligned one is rapidly squashed on one side, becoming laminar and eventually fragmented because of the interaction and backreaction of the magnetic field. This model could explain an observation of erupted fragments that fall back as thin and elongated strands and end up onto the solar surface in a hedge-like configuration, made by the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory. The initial alignment of plasma flow plays an important role in determining the possible laminar structure and fragmentation of flows while they travel along magnetic channels.


Introduction
Plasma flows are ubiquitous in the solar corona. They can be generated by many different processes characterized by the interaction between the strong and complex magnetic field of the corona and its plasma. The interaction can be relatively weak and it can lead to confined flows, e.g., spicules (Beckers 1968;de Pontieu et al. 2007), siphon flows (Rueedi et al. 1992;Brekke et al. 1997;Mariska & Dowdy 1992;Winebarger et al. 2001Winebarger et al. , 2002Teriaca et al. 2004), coronal rain (Parker 1953;Field 1965;Antolin & Rouppe van der Voort 2012) or it can be strong leading to violent ejection of plasma outside of the Sun as in Coronal Mass Ejections (CMEs, Chen 2011;Webb & Howard 2012).
In an impressive eruption event that occurred on 7 June 2011, the ejected material fell back along parabolic trajectories and impacted the solar surface (Carlyle et al. 2014;van Driel-Gesztelyi et al. 2014;Innes et al. 2012), producing brightenings .
In this case, the dynamics is strongly influenced by the intensity and the complexity of the magnetic field close to the impact region. In particular, impacts where the magnetic field is weak brighten because of the shock heating of their outer shells (Reale et al. , 2014, other plasma blobs are channelled by a strong magnetic field and the shocks brighten the whole channel ahead (Petralia et al. 2016). Petralia et al. (2017) show that the misalignment between the velocity of the blobs and the magnetic field lines determines a significant disruption of the blobs.
The misalignment between plasma flows and the ambient magnetic field can therefore be important. Magnetohydrodynamic modeling of the solar chromosphere up to the corona has shown that chromospheric flows can be not well aligned with the magnetic field (Martínez-Sykora et al. 2016). As we will show, there is evidence for this misalignment also from the observations.
In this work, we address specifically the possible role of the initial field alignment or misalignment of a persistent outflow ejected in a magnetized atmosphere. Our approach is to model flows which are pushed upward from the chromosphere along closed magnetic flux tubes in the corona by means of 3D-MHD simulations able to capture the relevant physics behind the process and to take into account other important effects, such as the natural expansion of the magnetic channel cross section with the height and its effect on the dynamics, or possible transversal motions of the flow (eg. Petralia et al. 2016). We first shows an observed sample case of laminar fragmented flow in Section 2. The model is described in Section 3. The simulations and the results are presented in Section 4, and we discuss them in Section 5.

An observed sample case
After a solar eruption on 4 November 2015, SDO/AIA observed dense fragments to fall back onto the solar surface. During the fall they elongated, spread and then impacted the Article number, page 2 of 11 solar surface in a row with an hedge-like configuration, as shown in Figure 1 and in the associated movie. This evolution might appear not obvious but our present model may explain it. The fragments are observed in absorption and this allows us to estimate their density and temperature according to the method in (Landi & Reale 2013). We obtain a density of ∼ 1.5 × 10 10 cm −3 and a temperature of ∼ 4 × 10 4 K. We also estimate the impact velocity perpendicular to the solar surface by assuming a parabolic motion. We measure the vertical distance (H) covered from the apex of the trajectory to the impact region and we obtained for the final velocity v = 2g ⊙ H ∼ 200 km/s, where g ⊙ is the solar gravity. The observed evolution and destiny of the erupted fragments suggest this as an example of an outflow that is initially not perfectly aligned to the magnetic field coming out of the solar surface, as illustrated in the following Sections.

The Model
We use the model presented in Petralia et al. (2016) and the numerical code PLUTO (Mignone et al. 2012) to describe the dynamics of a persistent flow injected upwards and confined inside a closed coronal magnetic flux tube. This recalls closely the dynamics of siphon flows. We consider two slightly different flow directions, one perfectly aligned to the Article number, page 3 of 11 field and the other slightly inclined with respect to the field lines. In the following, we will call the former 'Aligned Flow' and the latter 'Misaligned Flow'.
The ambient atmosphere is made by a stratified million-degree corona linked to a much denser chromosphere through a steep transition region. The corona is a hydrostatic atmosphere (Rosner et al. 1978;Serio et al. 1981) that extends upwards for 10 10 cm. The chromosphere below is hydrostatic and isothermal at 10 4 K with a density at the base of ∼ 10 16 cm −3 . The atmosphere is made plane-parallel along the vertical direction (Z).
The closed magnetic loop has been obtained by setting a dipole magnetic field centred on and parallel to the solar surface (XY ). The field intensity is ∼ 30G at the top of the chromosphere and rapidly decreasing with the height (Fig. 2). The field confines the flow but can be perturbed by it; indeed the ratio between the ram pressure carried by the flow and the magnetic pressure is ρv 2 B 2 /8π ∼ 0.7 at the top of the transition region. This ratio may be expected in an active region loop.
We consider the same ambient atmosphere as the 'Dense Model' described in Petralia et al. (2016), that -combined to the field intensity -gives an Alfvén speed of ∼ 2000 km/s at the top of the chromosphere close to the footpoint of the loop.
Figures 2 and 3 show our initial conditions. The flow is injected upwards from a localized area of the chromosphere and is continuously fed from the lower domain boundary, i.e., the injection is set as a boundary condition inside the flow area at the base of the chromosphere.
The flow area is a circle with a radius R F = 2 × 10 8 cm. The flow is present since the beginning inside the domain above the lower boundary, as shown in Fig. 2, with the shape of a cylinder with a vertical length of L F = 10 9 cm. Its initial density and temperature are uniformly 3×10 10 cm −3 and 4×10 4 K, respectively. Since the thickness of the chromosphere is ≈ 7 × 10 8 cm, the tip of the jet protrudes in the corona already since the beginning.
The misaligned flow is injected at a distance X = 1.3 × 10 9 cm from the left boundary where the magnetic field lines rapidly curve and leads to a misalignment of the flow.
The flow propagates in the quasi semi-circular flux tube that has a distance between the footpoints ∼ 4 × 10 9 cm and a height of 2.5 × 10 9 cm above the chromosphere.
As shown in Fig.3, the velocity of the flow is uniform along the Z direction v Z = 200 km/s inside an inner circular area with radius R = 10 8 cm, and then linearly decreasing to zero in an outer shell to the cylinder boundary R = 2 × 10 8 cm. This initial velocity is such that the flow has enough kinetic energy to reach and surpass the apex of the closed flux tube.
The aligned flow is injected instead at a smaller distance X = 0.9 × 10 9 cm from the left boundary, where the magnetic field lines are nearly open and thus vertical because closer to the magnetic pole and aligned with the vertical direction of the flow. This flow will travel to a much larger height along the flux tube (Fig. 3).
Article number, page 4 of 11 The computational box is three-dimensional and Cartesian (X, Y , Z) and extends over 6×10 9 cm in the X direction, 1×10 9 cm in the Y direction and 6×10 9 cm in the Z direction, that is perpendicular to the solar surface. The mesh of the 3D domain is adaptively refined to strong gradients of density. The roughest level has 120 × 20 × 120 number of cells and it is refined up to 3 levels. Each level refines locally the mesh by a factor of two, giving a cell size of ∼ 60 km at the highest level of refinement. The geometry of the system is symmetric with respect to the Y = 0 plane, allowing us to simulate only half of the domain. Therefore, at the Y = 0 plane boundary we impose reflecting conditions. At all the other boundaries the magnetic field is fixed. For the other physical variables, we impose outflow conditions at all boundaries except for the plane at the base of the chromosphere, i.e. at Z = 0. There, we have fixed values and zero velocity outside the flow area, and inflow condition inside the flow area, i.e. a fixed velocity with the same radial dependence as in the inner domain. Thus, the flow velocity at the boundary is also constant in time.  cm −2 ) is much larger than inside the flow (p f low ∼ 10 −1 dyn cm −2 , p mag ∼ 4 × 10 2 dyn cm −2 ). Therefore, at its origin, both the flow and the magnetic field inside it are rapidly squeezed by the outside pressure and the flow is de facto quenched after ∼ 40 s of evolution. This might not be entirely realistic, although we may expect that such rapid flows do not have enough time to reach pressure equilibrium while crossing the chromosphere.

The aligned flow
However, the initial supply is large enough to sustain the flow along the magnetic channel.
This applies also to the case of misaligned flow described in Section 4.2.
Since the initial speed of the flow (v f low = 200km/s) exceeds the (isothermal) sound speed (c s = γ 2K b T µmp ∼ 130 km/s, with T ∼ 8×10 5 K), a shock front is generated. This is a slow-mode shock and moves ahead of the flow (see the pale blue cocoon in Fig. 5) along the magnetic field lines. Its initial compression ratio of the density is ρ post−shock /ρ pre−shock ≈ 2.8, and the temperature grows by ∼ 15% in the post-shock. The overall propagation is similar to Petralia et al. (2016).
The flow moves upwards without deviating from the initial direction and as a uniform cylinder. Since its thermal pressure is much greater than that of the surrounding corona, the flow expands at the tip and pushes apart the magnetic field where it propagates.
However, since the magnetic field is tightly anchored at the footpoint of the flux tube, its expansion rapidly stops (with Alfvénic time scales) and the tension constrains back the flow. As a result, a thin shell of overdense plasma forms all around the head of the flow, as shown in Fig. 5 and Fig. 6, which is progressively smoothed out. Overall, the flow keeps its uniformity and cylindrical symmetry while it propagates upwards.
Article number, page 6 of 11   (not visible) is generated and moves ahead of the flow along the tube and the injection is quenched from below. Both the compression ratio of density ( 1.7) and the temperature increase ( 10%) are smaller than in the aligned case. This is mainly due to the misalignment, as only the component of the velocity along the field lines contributes. The shock propagates in the hot ambient corona and is potentially visible in the EUV band. Although this is the case closer to the observation described in Section 2, we are actually unable to detect the shock in the data, probably because it is not dense enough to stand out in the thick bright solar limb. Fig. 7. Snapshots from our simulation of a misaligned flow: Rendering of the density (10 9 cm −3 , logarithmic scale) at times t=80, 160, 320, 480s in which a bundle of magnetic field lines (Gauss) is shown (see the associated movie).
Since the magnetic flux tube is closed on the chromosphere, the shock hits the other footpoint, at t=260 s (when the flow has just crossed the apex of the magnetic channel).
Its average speed is therefore v sh ∼ 220km/s. Figure 7 and the associated movie has a small component perpendicular to the magnetic field, already since the beginning. Although the plasma is anyway forced to move along the field lines, its misaligned momentum distorts the field above the chromosphere. The flux tube still expands, but the expansion is no longer symmetric. In turn, the deformed field deviates the flow in the direction of the tube curvature, already at t = 20 s, as we can see in Fig. 5. Since the dense plasma is forced to flow in a direction different from the initial one, its structure becomes asymmetric. It converges and thus is squashed on the side of the curvature, where the magnetic field offers more resistance to deformation. It expands more freely on the other side.

The bulk plasma motion visible in
The interplay between the magnetic back-effects (magnetic pressure and tension) and the upward push of the flow is very strong until the flow reaches the apex of the magnetic channel at t ∼ 200 s. After this time, the magnetic field starts to relax to the initial configuration while the flow falls on the other side of the magnetic channel. The relaxation of the field lines acts as a press that flattens the flux tube and deforms the flow inside it even more. The flow expands laterally at t ∼ 320 s (Fig. 8). The relaxation of the field is untidy and frays the flow. Due to this effect, as the flow moves along the tube, it splits becoming highly sub-structured at the end of the evolution at t ∼ 480s (Fig. 7).
Overall, we observe an initially cylindrical and uniform flow becoming laminar and filamentary, basically because of the initial misalignment of its direction to the field lines.

Discussion and Conclusions
In this work, we analyse the dynamics of a continuous flow perfectly or not perfectly aligned to the field lines of a magnetic channel anchored to the solar surface by means of a 3D MHD model analogous to that presented in Petralia et al. (2016).
Summarizing, we find that the aligned flow travels mostly unperturbed along the magnetic channel, except for some expansion at the tip and for the formation of a dense outer shell. If instead the field lines are strongly curved and the flow is forced to change its direction to follow the magnetic channel, the evolution is significantly different. Because of the stiffness of the field, the flow is squashed to the side of the curved lines and converts into a dense laminar flow, strikingly different from the symmetric dense shell found when the alignment is perfect. As the flow travels through regions of weaker field, it causes some deformation of the field and its untidy back-reaction makes the flow split. The propagation has been monitored until the flow hits the chromosphere at the other side of the closed channel.
We expect that the result can somewhat change for different conditions in the model, but we are unable to quantify without a relevant exploration of the parameter space. For instance, we might expect that a weaker field and/or a more tenuous ambient atmosphere should make the flow expansion easier and perhaps smooth down the dense shell of the aligned flow. A faster outflow in the curved loop should determine a stronger deformation of the field and we might expect an even larger backreaction and flow splitting.
The transformation of the initially cylindrical, symmetric and uniform flow into a strongly laminar and substructured one might explain the puzzling evidence of hedgelike downfalls observed with SDO/AIA (ex. Figure 1 and the associated movie). We might be simply in the presence of a relatively long-fed flow that is not launched in the direction of the local magnetic field, probably not particularly strong, and therefore considerably deviated by the field during its journey. Our estimate of the flow conditions supports this hypothesis and also gives us an idea of the intensity of the ambient magnetic field, which might be similar to that in our model, e.g. ∼ 30G. This scenario may not be uncommon (Martínez-Sykora et al. 2016).
The results we found are general and could be applied to many flows moving in a magnetized medium: if the flow motion is not aligned to the magnetic field, the flow tends to be flattened and split by the deviation of the direction of motion. If the alignment is nearly perfect the flow simply travels along the field lines, without significant deformations, as expected for plasma strictly confined in magnetic coronal loops. This effect might determine Article number, page 10 of 11 also the morphology of coronal flows out into the interplanetary space, as those detectable and to be studied by Solar Orbiter instruments.