Study of Three-Dimensional Magnetic Reconnection Phenomena in the Experiment PROTO-SPHERA

Macroarea di Scienze MM FF NN Corso di Laurea in Fisica

Master Thesis

Study of three-dimensional Magnetic Reconnection Phenomena in the experiment PROTO-SPHERA

Student: Supervisor: Samanta Macera Prof. Marco Tavani

Co-Supervisor: Dott. Franco Alladio

Academic Year 2019/2020

Contents

Introduction 4

1 Magnetic Reconnection 6 1.1 General theory of reconnection ...... 6 1.2 Steady state reconnection ...... 9 1.3 Fast Magnetic Reconnection ...... 14 1.4 Collisionless reconnection ...... 17 1.5 Three-dimensional reconnection ...... 24 1.6 Magnetic Reconnection in Astrophysics ...... 27

2 Laboratory Plasma 34 2.1 Experimental setup to study reconnection ...... 34 2.1.1 Magnetic Conﬁnement ...... 35 2.1.2 Tokamak ...... 39 2.1.3 Reversed Field Pinch (RFP) and Spheromak . . . . . 41 2.2 Magnetic Reconnection and kink instability ...... 43 2.3 Experimental evidence of reconnection ...... 46

3 PROTO-SPHERA 51 3.1 From the screw pinch to the spherical torus ...... 54 3.1.1 Phase 1.0 ...... 55 3.1.2 Phase 1.25 (January 2018 - September 2019) . . . . . 57 3.1.3 Phase 1.5 (September 2019 - December 2021) . . . . . 58 3.1.4 The role of magnetic reconnection ...... 59 3.1.5 Towards phase 2.0 and beyond ...... 59

4 Data Analysis 63 4.1 Hall sensor ...... 63 4.2 Langmuir probe ...... 64 4.3 Fast cameras ...... 66 4.4 Fast camera image analysis ...... 68 4.5 Cross Correlations analysis ...... 74 4.6 Plasma phenomenology ...... 81

2 Conclusions 85

Bibliography 89

Ringraziamenti Introduction

In the last eighty years, a lot of efforts have been made in Plasma Physics in order to understand magnetic reconnection, which is a physical process consisting of a topological rearrangement of magnetic field lines in a plasma. Magnetic fields pervade the most of the objects throughout the Universe. Depending on the object, the magnetic field will have different features and complexity. For example, the ones emanating from the planets in our s So- lar System are essentially dipolar and are relatively weak compared to the strengths that can be found on the Sun’s magnetic field. Reconnection occurs in regions of strong magnetic shear, that are regions in which the direction of the magnetic field changes significantly over a short distance. Under certain circumstances, magnetic field lines can “break” and re-assemble in order to minimize the magnetic energy stored in the magnetic field and in order to achieve a new equilibrium configuration with lower energy. The big amount of energy released during the process is converted into kinetic energy through acceleration or heating of charged particles in the plasma: the electrons through Ohmic heating and the ions through acceleration of Alfvénicjets. Reconnection can be both collisional or collisionless, depending on the characteristics of the magnetized plasma. Collisional reconnection is usually too slow to explain the data observed from both astrophysical and laboratory plasmas. Therefore, fast magnetic reconnection requires collisionless models. This process is of great importance in many astrophysical processes such as the heating of stellar coronae, the acceleration of stellar winds and astrophysical jets, γ-ray burst and magnetospheric substorms, being the mechanism behind the auroras. Earth’s tear-shaped magnetic field contin- uously oscillates and responds to the changing intensity of the solar wind. The solar wind particles funnel around to the long tail of the magnetosphere, where they become trapped. When magnetic reconnection occurs, the particles are accelerated toward Earth’s poles. Reconnection could be involved also in the formation of stars. Stellar activity is magnetically driven, and raconnections can be expected also in strongly magnetized neutron stars (classed as anomalous X-ray pulsars and as soft gamma repeaters). More- over the observed hard X-ray variability of some accretion disks could be due to flares powered by reconnection. It is also of fundamental important

4 Contents in laboratory plasma, since it has been observed in numerous laboratory experiments and it seems to be the key process for self-organization of fusion plasmas in devices as tokamaks, spheromaks and reversed field pinches. Although it concerns several phenomena in nature, this process is challeng- ing to observe. In fact reconnection might not always have the signatures which we expect. Moreover, a change of the magnetic structure might also occur with a lower energy output or on a slower time scale. Over the years, several models have been proposed: two-dimensional, three- dimensional, collisional and collisionless models and so on. In this thesis work, the data obtained from the PROTO-SPHERA (PROTOtype of a Spherical Plasma for HElicity Relaxation Assessment) fusion experiment, located at the CR Enea in Frascati were analyzed. This experiment has a magnetic setup that exploits magnetic reconnection mechanisms to confine the plasma in a spherical torus. The purpose of this thesis is to characterize the episodes of magnetic reconnection within the experiment. In chapter 1 the theory of magnetic reconnections is presented. Collisional reconnection models, such as Sweet Parker and Petschek model, and collisionless models are explored. Some applications in Astrophysics are exposed in section 1.6. Chapter 2 describes laboratory experiments aimed at studying nuclear fusion, in which reconnection processes can be studied. Different types of magnetic configurations used to confine plasmas and the role of magnetic reconnections in these laboratory plasmas are presented. In Sec- tion 2.2 kink instability and its connection with Astrophysics are exposed. In chapter 3 the PROTO-SPHERA experiment and its differences from the configurations described in Chapter 2 are described. Chapter 4 presents the diagnostics used and the data analysis. Finally, section 4.6 presents the PROTO-SPHERA plasma phenomenological description that derives from this analysis.

5 Chapter 1

Magnetic Reconnection

1.1 General theory of reconnection

Magnetic reconnection is a topological rearrangement of magnetic field that converts magnetic energy to plasma energy. This phenomenon affects plasma dynamics, energetics and transport, and it couples global and local scales. There is a unique relationship between the topology of a magnetic field and the corresponding equilibrium: any change in topology has a significant im- pact on the entire equilibrium [1]. Therefore, all equilibria are characterized by their own topology. If the current topology of the plasma is characterized by an equilibrium with energy E, and a change in the aforementioned topology would lead to an equilibrium with energy E0 < E , the plasma will evolve towards this latter state of equilibrium by modifying the topology of the magnetic field through magnetic reconnection processes. A topological change of the magnetic structure can release huge amounts of the energy stored in the magnetic field. During this process there is a rapid conversion of magnetic energy by viscous processes into heat, radiation and particle acceleration. Therefore, in its most general formulation, magnetic reconnection requires only a change in the magnetic connectivity of plasma elements. In the ideal case of highly conducting plasma the electric field is given by Ohm’s law E + v × B = 0 (1.1) where v is the bulk plasma velocity field. In this condition Alfvén theorem holds: without dissipation terms on the right hand side of the previous equation, the topology of field lines is left invariant, which means that the flux is invariant for all the times, and therefore magnetic field lines are frozen to the plasma. Hence magnetic flux and magnetic field lines are conserved [2]. By taking the curl of ideal Ohm’s law (equation 1.1), the induction equation for the magnetic field can be obtained: ∂B = ∇ × (v × B). (1.2) ∂t

6 1.1. General theory of reconnection

In order to allow field lines to change their connection, there must be a dissipative term which breaks the frozen-in condition, even if only in a small diffusion region. In fact, a change in a region small compared to the size of the system still has global effects. A non ideal term R must be added to the right side of Ohm’s law 1.1:

E + v × B = R (1.3) where R can contain resistivity, Hall current, electron inertia and pressure and ambipolar diﬀusion. If R can be represented as

R = ∇φ + u × B, (1.4) equation 1.3 becomes E + (u + v) × B = ∇φ (1.5) and equation 1.2 becomes dB − ∇ × w × B = 0 (1.6) dt with w = u + v. So even if the connection of plasma elements might change (the field lines move with velocity u, different from the velocity of plasma element v), the topology of field lines remains the same since the field is frozen-in respect to the velocity w, so no reconnection can occour, provided that u and w are globally regular. As already mentioned, R can be the resitive term in Ohm’s law ηj, or can contain the electron inertia or the electron pressure tensor. Depending on the terms it contains, different reconnection models can be obtained. R can be decomposed in the two direction parallel and orthogonal to the magnetic field: Rk = ∇kφ (1.7)

R⊥ = ∇⊥φ + u × B. (1.8) We can now solve 1.7 by integrating the potential φ along magnetic field lines and then use this solution to determine u from 1.8. It can be shown that the solution for u is B × (R − ∇φ) u = . (1.9) B2 At magnetic null points the direction of B is undefined. Here the splitting into 1.7 and 1.8 breaks down. An analysis of flow u near the null point shows that certain R require diverging derivatives of u at the null (Hornig and Schindler, 1996). A bifurcation of null points is a change of topology and requires an infinite velocity u at the time of the bifurcation. Therefore

7 1.1. General theory of reconnection magnetic null points are locations where the topology of the magnetic field can change. Another important quantity connected to the magnetic field which could play a role in reconnection and which will be used later on is the helicity density. Let A be the magnetic vector potential; the helicity is defined as Z H(B) = A · Bd3x (1.10) V with B · n = 0 on the boundaries of the volume V . The helicity density h = A · B becomes well defined, i.e. a gauge invariant quantity when integrated over a finite volume V . It is a measure of the linkage of magnetic flux in the volume and therefore a topological measure of the magnetic field in V . If the plasma is perfectly conducting helicity is conserved.

8 1.2. Steady state reconnection

1.2 Steady state reconnection

The magnetohydrodynamic model describes the plasma as a one- or two- fluid object under the effect of pressure and electromagnetic forces. If the collisionality is high enough to minimize the relative drift velocity distribu- tions of both ions and electrons, and if the difference in inertia between the two species is negligible, a single fluid model can be used. This model can be obtained assuming mass and momentum conservation. Therefore, the following set of equations is needed to describe magnetic reconnection: • Mass conservation ∂ρ + ∇ · (ρv) = 0 (1.11) ∂t • Momentum conservation ∂v ρ + v · ∇v = −∇p + J × B (1.12) ∂t • Ohm’s law E + v × B = ηJ (1.13) • Induction equation1 ∂B η = ∇ × (v × B) + ∇2B (1.14) ∂t µ0 where ρ is the plasma density, v is the plasma velocity, µ0 is the vacuum magnetic permeability and η the plasma resistivity. The first theoretical model for reconnection is the Sweet-Parker model, which describes a collisional steady state reconnection in which inflow of plasma with density ρ and conductivity σ is connected with outflow plasma from a diffusion region. As shown in figure 1.2, two opposite directed magnetic fields are carried toward a neutral line at speed vin. At the center of the neutral line there is a X-point in which magnetic field is null. Here a resistive layer of width 2δ is formed. This layer is called current sheet or diffusion region, and it is the place in which reconnection takes place: magnetic field lines with opposite direction near the X-point break and form new connection, and after that they are expelled with the plasma from the diffusion region with velocity vout. From this simple model, the reconnection rate and the basic energetics can be extimated. Let’s assume that vout is almost equal to the Alfvén speed B vA = √ . (1.15) µ0ρ

1If the ideal Ohm’s law holds, the induction equation is given by 1.2, but if resistivity η is taken into account, then by taking the curl of the equation 1.13 the modiﬁed induction equation 1.14 can be obtained.

9 1.2. Steady state reconnection

This is true if one assumes that magnetic energy is converted into plasma kinetic energy through resistive heating and magnetic tension force associated with the field lines near the X-point. As depicted in figure 1.1 and figure 1.2 the magnetic field lines are very folded near the X-point, which means that near that point there is a high magnetic tension; when the lines break, a large amount of magnetic tension energy is released leading to a sort of magnetic sliding effect on plasma particles. Mass is conserved, and for an incompressible flow

vinL = vAδ. (1.16)

One can also assume that the electric field given by Ohm’s law (equation 1.3) with R = ηJ is perpendicular to the plane of the flow. Moreover E is constant in a steady state and it is primarly inductive, E = vinB, in the most of the configuration, but not near the X-point, where it is especially resistive. From equation 1.13 it follows that η vinB ∼ J. (1.17) µ0 In this model we have a highly conducting plasma, but there is a narrow current region with high electrical current density: if two field lines in opposite direction approach each other they will realize a regime in between where 1 J = ∇ × B (1.18) µ0 will generally increase.

Figure 1.1: Formation of X-type null point. a) initial topology of magnetic field lines; b) two oppositely directed magnetic field are pushed towards each other by the plasma inflow; c) the geometric lines cross, with a X- shape, dividing four topologically distinct domains; d) field lines reconnect and form a new topology.

10 1.2. Steady state reconnection

Figure 1.2: Current sheet in the Sweet-Parker model. The red coloured region is the current sheet; there is a null point at the center of the conﬁguration, and the layer has a width of 2δ and a length of 2L. From [3].

From equation 1.18 we can estimate the current density B J ≈ . (1.19) µ0δ From equation 1.16 reconnection rate can be extimated:

vin δ − 1 = = S 2 (1.20) vA L where S is the Lundquist number

µ Lv S = 0 A (1.21) η 2 and represents the ratio of global ohmic diffusion time τdiff ≡ L /η to global Alfvén time τA ≡ L/vA. This model also shows that there is a close relation between the reconnection velocity vrec ≡ vin and the opening angle of the field [3]. If xb is the direction of the inflow, zb the direction of the outflow and θ the angle between these two directions, it follows that

Bx vin vrec − 1 tanθ = = = ∼ S 2 . (1.22) Bz vout vA Energetics can be extimated from equation 1.20: the Poynting ﬂux into the layer, given by v LB2 in (1.23) µ0 is of the same order of the kinetic energy ﬂux out ρv3 δ K = A (1.24) out 2

11 1.2. Steady state reconnection and Ohmic dissipation rate

2 DOhm = J δLη (1.25) inside the layer. This means that energy is equally distributed into acceleration of Alfvénic jets and heating of electrons. An interesting feature is that near the X-point the particles can be accelerated also by the electric field parallel to the reconnection plane. If the extension of this region is D, the maximum energy gain for the particles is eED (in the case of an ion the maximum gain is ZeED, where Z is the ion’s atomic number). However most of the particles are drifted away from the diffusive region before reaching this energy. Moreover, spectra produced by this mechanism do not fit very well astrophysical spectra. In fact, Sweet-Parker model, although it contains the essential elements to describe reconnection, is inapplicable. In this theory a steady state is assumed, but dynamics exhibits an impulsiveness. Considering the solar flares or magnetosphere substorms, the magnetic configuration evolves slowly for a long period of time, to then undergo a sudden dynamical change over a short period of time. Finally, if S is very large, like in most astrophysical cases, reconnection is very slow, since the entire fluid carried to the reconnection site must flow away through a thin layer. This is in contrast with experimental evidence of reconnection, since very short reconnection rate are found from observations and experiments. A first quantitative test of Sweet-Parker model was performed at MRX (Magnetic Reconnection Experiment) at Princeton Plasma Physics Labora- tory, in a high-collisionality regime. The typical rectangular shape of the reconnection layer was confirmed, but the reconnection rate measured was very different from the one predicted by the Sweet-Parker model. However, it is necessary to deviate a little from the basic model and include for example effects due to plasma compressibility. That is, replace the reconnection velocity of the model δ v = v rec L z with the more accurate reconnection velocity which takes into account the fact that the density n increases within the reconnection layer: δ L ∂n v = v + . (1.26) rec L z n ∂t This effect accelerates reconnection during the density build-up phase. Other important effects are due to the higher downstream pressure and to

12 1.2. Steady state reconnection an effective resistivity, which seems to be larger than the Spitzer’s resistivity. In fact, from experiments it can be seen that outside the current sheet the term v × B balances with the electric field E, but inside the current sheet it must be balanced by other terms. In magnetohydrodynamics the balancing term is the resistivity, so an effective resistivity of the form E η∗ = 0 (1.27) J0 can be defined. If plasma is collisional, this resistivity agrees with the Spitzer’s resistivity and ∗ − 3 η ∝ T 2 . Including these effects in the Sweet-Parker model, it can be seen that this model applies with and without a guide field in a stable 2D reconnection neutral sheet with axisymmetric geometry.

13 1.3. Fast Magnetic Reconnection

1.3 Fast Magnetic Reconnection

In 1964 Petschek proposed a new model which could account for a faster reconnection. By fast magnetic reconnection, we mean magnetic field lines changing their connections on a time scale determined by Alfvénic, not resistive, physics. Since − 1 vin = vrec ∝ L 2 , (1.28) a faster reconnection can be obtained with a shorter resistive layer and if the flow, instead of passing entirely through the layer, can pass by standing shock waves. Outside the Petschek diffusive region the acceleration up to about vA is accomplished almost instantaneously by the slow shocks. In this way, replacing L with a shorter length L∗ the reconnection rate q L increases by a factor L∗ . A family of solution with progressively smaller length can be found, down to a smaller limit at L( 8√πS )2, which corrisponds π S to the maximum reconnection rate π v = v . rec A 8lnS

This rate is of the order of a few percent of vA and it is reasonable for the most of astrophysical situation. The most of energy conversion comes from the shocks, which accelerate and heat the plasma and form two hot outflow jets. In this model most of the energy is converted to ion kinetic energy of the outflow and to heat, with a small percentage of the energy going into resistive heating of electrons. However, if one tries to verify Petschek model with numerical MHD simulation, the result is that the model in unstable unless the resistivity η increases near the X-point: the larger the resistivity is, the faster field lines are reconnected in the layer, and if η has a maximum at the X-point and rapidly decreases away from it a larger angle θ can be formed, leading to a faster reconnection. Numerical simulations also show that this type of reconnection does not develop by itself and needs to be switched on by external forces acting on the system. Petschek model is sketched in figure 1.3. Despite Sweet-Parker and Petschel models seem to have different problems, both of them take into account the most important features of reconnection, that is small dissipation region, strong ouflows and hot electrons, but it is not explained under which circumstances steady state reconnection can occour. As alredy mentioned, magnetic reconnection can take place when some resistivity appears in the plasma, changing the topology of the magnetic field and accessing to lower energy states. Therefore resistive instabilities can develop.

14 1.3. Fast Magnetic Reconnection

Figure 1.3: Petschek fast reconnection model

A resistive instability is characterized by a growth time greater than the ideal one and less than the resistive diffusion time. Tearing modes are the most important example of resistive instabilities. Let’s consider the reconnection of a scalar quantity, for example the detachment of a drop in a system characterized by a scalar quantity P [2]. Suppose P discerns two different phases: the black (P = 1) and white (P = 0) phases depicted in figure 1.29, and that there is a smooth transition between these two phases, which means that P can be treated as a continuous quantity. The system can be characterized by the evolution of the velocity field, that is by a Navier-Stokes equation, and by the transport of the scalar quantity P by the flow: ∂P (x, t) + v · ∇P (x, t) = 0. (1.29) ∂t When the drops detaches, the velocity field will show a stagnation flow, but the time needed to reach this situation goes to infinity since for any smooth velocity field it takes an infinitely long time to transport a fluid element to the stagnation point. Therefore the previous equation for the evolution of P shows that the drop cannot detach in a fine time in the ideal case. In order to obtain the detachment a term must be added to the right side of equation 1.29. This term must have two important features: it must be negligible compared to v · ∇P (x, t) and it must be of the right sign in order to have the deatachment of the drop, which means reconnection. The value or the profile is not important, since it will be sufficient in any case to trigger reconnection. In the magnetic case, there is a very similar situation, since in the presence of resistivity, the field lines can tear and reconnect, and a non ideal term must be added to the ideal Ohm’s law, as already explained at the end of

15 1.3. Fast Magnetic Reconnection

Figure 1.4: The detachment of a drop as an example of reconnection of a scalar quantity. From [2]

Figure 1.5: Sketch of magnetic field geometry for the tearing mode. There is a large, nearly constant guide field perpendicular to the page. The sheared component has reconnected, forming a chain of magnetic islands. From [3] the previous section. As shown in figure 1.5 there is a strong component of the field, called guide field perpendicular to the plane of the figure. The By component of the field reverses at x = 0. As a consequence the magnetic tension force due to the bending of the field lines goes to zero. There should be a gradient in the current density in order to have instability, otherwise magnetic tension tends to stabilize the mode. That’s why these kind of instabilities are called current-driven instabilities. The gradient length scale must be smaller than the perturbation length scale. It is found that the growth time of the most unstable mode is of order 3 τAS 5 and the width of the resistive layer relative to the global scale is of − 2 order S 5 . It must be noted that in the Sweet Parker model the growth time scales as 1 − 1 τAS 2 while the width of the resistive layer scales as S 2 . The driving energy of the tearing mode comes from the unstable current gradient within the layer. As shown in figure 1.5, when reconnection oc- cours magnetic island are formed. As the tearing progresses, these islands widen exponentially, until when their width is larger than the resistive layer width. At this point non linear forces J × B start to be significant and the exponential growth is replaced by an algebraic growth at a rate proportional

16 1.4. Collisionless reconnection to η; the current profile, which was initially unstable, flattens. Current instabilities can lead also to kink mode, a plasma instability in which a plasma columns transverse displacement from its center of mass. 1 The growth time of the kink instability is of order τAS 3 , which is faster than the tearing mode, but still slow in most astrophysical systems. Like the tearing mode, the resistive kink makes a transition from exponential to algebraic growth. In the last years much progress has been made at compu- tational level. Therefore it was possible to simulate numerically some of the models of reconnection, like Petschek and Sweet-Parker models and other fast reconnection models to test their validity. Simulation required specific conditions, like local anomalously large resistivity or ad-hoc boundary conditions in order to work well. These numerical studies showed that what appear to be the most suitable theories for magnetic reconnection are actually wrong. In particular, Biskamp tried to reproduce numerically both Petschek and Sweet-Parker models, and with a constant and uniform resistivity he could not find Petschek’s solution. An important instability raises is due to the evolution of the reconnection layer itself. The latter is subject to both large scale instabilities and microinstabilities, and can undergo to tearing modes. Magnetic island are thus formed, and different islands can interact because of the attractive force between neighboring current filaments, as shown in figure 1.6. Finally, reconnection layer can also show Kelvin-Helmotz instability, which make it turbulent. The models mentionated above need to be modified in order to account for more specific conditions that have been neglected in first approximation.

1.4 Collisionless reconnection

In the previous sections, it is assumed that the spatial extent of the reconnection site is much larger than the mean free path of the plasma particles. MHD form of Ohm’s law has been used. MHD is valid only on scales larger than the ion gyroradius outside the ion inertial region that is centered on the current sheet. Thus, even though one may speak about the diffusion of the field into the current sheet, the process of reconnection itself lies outside of MHD. MHD merely covers the post-reconnection effects of collisionless reconnection. In steady state, the Lorentz force on the electrons is balanced by frictional drag due to collision. The velocity that appears in equation 1.3 is the electron velocity ve. Since the current density is given by

J = (vi − ve)ene (1.30)

17 1.4. Collisionless reconnection

Figure 1.6: Simulation in the MHD case with S = 106 at two diﬀerent times: just before a tearing instability is triggered in the secondary current sheets (left panels) and during the secondary reconnection events. The de- fragmentation of the current sheet into magnetic islands can be observed [4].

vi beeing the ion velocity, in absence of a guide field Ohm’s law reads J × B E + v × B − = ηJ. (1.31) ene The third term on the left represents the Hall effects. Near the reconnecting current sheet, electron and ion motions decouple almost completely on the inertial scale of the ions. The electron flow (in the diffusive region and out of it) forms an in-plane current. This is the basic difference with the Sweet-Parker model, in which the current is completely perpendicular to the reconnection plane. The out of plane component of the Hall term can be written as follows:

J × B B · ∇B = z . (1.32) ene z µ0ene Last equation shows that the in-plane current generates an out-of-plane field, Bz. The form of this field can be derived by noting that Jx,Jy, which represent the electron inflow and outflow from the diffusive region, are proportional respectively to x, −y. Hence Bz ∝ xy, so the out of plane field has a quadrupole nature. Hall term dominates in equation 1.31 on scales δ < δi, δi beeing the ion skin

18 1.4. Collisionless reconnection

Figure 1.7: Sketch of magnetic field geometry in Hall reconnection, a collisionless model of reconnection. The ions decouple from the electrons at a distance δi ≡ c/ωpi from the neutral line. The electrons continue flowing inward and the field is reconnected within the much thinner electron diffusion layer. Hall currents JH are centered on the separatrices forming four half-loops. They generate the quadrupolar

Hall magnetic field BH . The convection electric field points out of the plane. Ions are accelerated in this field out of the plane thereby contributing to the sheet current. This contribution effectively causes a broadening of the current layer. See [3].

depth, so Hall reconnection is expected when δSP < δi:

1 1 δ L 2 m 4 SP = e (1.33) δi λmfp mi where λmfp is the electron mean free path and me, mi are respectively the electron mass and the ion mass. Since if the layer width δ is almost equal to λmfp we are in collisionless regime, Hall reconnection is known as collisionless reconnection. A sketch of collisionless reconnection is shown in ﬁgure 1.7. The fundamental scale at which electrons and ions decuple is δi. This can be seen by considering the equation of motion for a particle of species α:

δvα qα + ωcα × vα = E (1.34) δt mα with the cyclotron frequency

qαB ωcα = . (1.35) mα

δ If δt is replaced by a characteristic frequency ω, for ω << ωcα the second term on the left in equation 1.34 dominates: particle motion is due to the E × B drift and the particles are frozen to the ﬁeldlines, so MHD conditions hold.

19 1.4. Collisionless reconnection

Instead, when ω > ωcα inertia dominates, particles are not frozen and they slip off the fieldlines. In order to extimate the length scale at which this happens, by looking at the dispersion relation for MHD waves one can find that for ions ω = kvA, so −1 the length scale for the decoupling of ions and electrons is exactly k = δi. This is a microscopical scale in most of the astrophysical systems of interest. On the other hand, the electron skin depth is smaller than the ion skin depth q by a factor me ; so while ions decouple, the electrons are still tied to the mi fieldlines for longer distances after the ions drop out. When magnetic field lines approach the X-point, the area per unit flux expands and diverges at the X-point and along the separatrices. Plasma must remain quasi-neutral, so there must be a flow of electrons along the fieldlines which forms an in-plane current that together with the ion flow support the By field. Since area per unit flux diverges, also J must diverge at X in order to have continuity. This divergence is “cancelled” by the electron pressure and inertia. This model of reconnection can be seen as a two-fluid model, since electrons and ions behave differently. Two fluid reconnection is much faster than Sweet-Parker and Petschek reconnection if η is small enough. This basic difference is due to the physical difference between two kind of waves that can propagate in a plasma: the Alfvén waves and Whistlers. While Alfvén waves are non-dispersive, which means that they propagate with velocity vA, Whistlers’s dispersion relation is

2 ω = k δivA (1.36) so a faster propagation is possible at small scales with a velocity v ∝ k. The introduction of a magnetic field in the out-of-plane direction (the guide field), along the current direction, changes the structure of the dissipation region even at rather low values of the guide field. With this additional magnetic field the out-of-plane inductive electric field has a component parallel to the magnetic field and the resulting parallel acceleration of electrons produces strong out-of-plane currents, in contrast to the cross-field currents. The in-plane components of the parallel electron flows along newly reconnected field lines drive a pronounced density asymmetry across the reconnection layer that couples reconnection to a kinetic Alfvén wave. Thus, the kinetic Alfvéen wave drives the electron outflow from the X-line. The guide field also suppresses the unmagnetized bounce motion of electrons that defines the width of the electron current layer. The result is that the electron current layer narrows substantially and has a width that scales with the electron Larmor radius. A consequence is that the nongyrotropic behavior of electrons survives even when the guide field is very large. Anyway not all the astrophysical systems can be treated in collisionless regime. Collisionless reconnection can be applied to systems like stellar

20 1.4. Collisionless reconnection coronae and accretion disks around compact object, while it is not applicable in situation like interstellar and intracluster gas. As already mentioned, reconnection can lead to plasma instabilities, which can increase the reconnection rate. Reconnection rate can be increased by anomalous resistivity η∗, which arises due to an interaction of a multitude of current sheets resulting in an abrupt increase of resistivity of the medium: this is due to the transfer of electrons momentum to small scale ﬂuctuation (that can be electomagnetic or electrostatic) which leads to an eﬀective friction force. The reconnection rate is thus enhanced by a factor rη∗ . η

Moreover, if resistivity is a function of position, Petschek reconnection can occour. Plasma instabilities observed in reconnecting plasmas include lower hybrid waves (whose frequency is almost equal to the lower hybrid frequency) observed in the Earth’s magnetotail, which are high frequency electrostatic and electromagnetic waves and whistlers waves. From numerical simulation, it can be seen that microinstabilities in the reconnection layer can grow, leading to modes that can interact with each other and with the basic layer itself. This also can lead to an enhancment of the reconnection rate. Since current-driven instabilities have a threesold electron drift speed vDc under which the wave energy is absorbed by the thermal ions and the instability cannot grow, and since we can express the width of the reconnection layer as a function of the velocity

vA δc ≈ δi , (1.37) vDc anomalous resistivity becomes important in astrophysical plasma when δ ≈ δi. However, a lot of important aspects cannot be properly described by fluid models. In particular, kinetic effect must be introduced, and this marks a net transition from fluid models to particles models. The kinetic terms which must be introduced include:

• charged particle acceleration

• charged particle heating

• nongyrotropic pressure

• microinstabilities due to inhomogenities in velocity space.

By including all these quantities, oﬀ diagonal terms in the electron pressure tensor can be studied.

21 1.4. Collisionless reconnection

Ohm’s law becomes ∇P m ∂ E + v × B = − e − e + v · ∇ v . (1.38) e en e ∂t e e

In fluid models the electron pressure Pe degenerates in a scalar pressure Pe. Since ∇Pe = ∇Pe is curl-free, it has no effects on the magnetic field evolution in the electrons reference frame: if the pressure is isotropic, only the electronic inertia remains to allow the reconnection. A kinetic approach is needed in order to obtain detailed information about reconnection events, but a full analytic theory is extremely difficult to carry out. Fully kinetic models require PIC (Particle In Cell) simulation. As mentionated above, a reconnection angle can be defined. When the guide field is approximately zero, which means that the angle is near 180◦, the reconnection speed is maximized. As the angle is reduced with the increasing guide field, reconnection speed decreases substantially. This happens for different reasons. First of all because with a guide field there is a smaller resistivity for a neutral current parallel to the guide field. This is explained by the fact that the current flow along the field lines cause less microturbu- lence and less Hall effects. Moreover, the guide field suppresses plasma flow and causes less compressibility of the plasma. This is due to the fact that the guide field confines plasma locally, increasing downstream pressure and reducing compressibility. Nevertheless in tokamak sawtooth crash (see section 2.2), fast reconnections are observed despite the presence of very strong guide fields. This is maybe due to the 3D global magnetohydrodynamic instabilities that can drive fast reconnections in localized regions. It is worth mentioning something more about nongyrotropic pressure, which is related to the winding motion of particles near the center line. In fact particle’s gyromotion reverses direction when they reach the neutral line, so particles drift in the out of plane direction: there is therefore a breakdown of gyrotropic symmetry in the inflow direction, and this can explain one part of the force due to nongyrotropic pressure. Moreover nongyrotropic pressure can be also associated to the Speiser orbits, which describe the ejection of charged particle’s from the current sheet, due to the transverse field in the reconnection plane. The presence of a guide field change substantially the kinematic, since it magnetizes the electrons within the electron diffusion region and reduces the electron nongyrotropic effects. From PIC simulations, a decrease of the electron layer thickness can be observed. If the guide field is strong enough, electrons start to be magnetized in their diffusion region. As the strength of the field increases, the thickness of the diffusion region decreases and scales as the electron gyroradius in the guide field. The electron nongyrotropic pressure balances the reconnecting electric field within this scale.

22 1.4. Collisionless reconnection

Also a tilt of the separatrices can be observed due to the guide field. This can be attributed to the Hall effect. An important source for the nongyrotropic pressure is the electron heat flux: this flux is due to the mixing of the less energetic incoming electrons and the accelerated outgoing electrons in the diffusion layer. This remains valid in three dimensions, despite the observation of waves propagating along the third direction.

23 1.5. Three-dimensional reconnection

1.5 Three-dimensional reconnection

Over the past fifty years, the main focus of researchers has been on the two-dimensional aspects of magnetic reconnection. Three-dimensional (3D) reconnection has different characteristics to 2D reconnection and it is a much richer and more varied process. Indeed, even a slight departure from an exactly 2D configuration leads to considerably different behaviour. In 3D, reconnection occurs in a range of locations that can, but do not have to be, associated with null points (points at which all components of the magnetic field are zero), unlike in 2D, in which reconnection can only occur at X-type null points. Reconnection in 3D occurs in a finite volume, known as a diffusion region, within which the plasma and the field lines become unfrozen. In this diffusion volume, the field lines continually and contin- uously diffuse through plasma and, as long as some portion of a field line is passing through the diffusion region volume, then it will reconnect with other field lines (figure 1.8). Therefore it is not possible, in general, to find pairs of field lines that, after reconnection, match to form two new pairs of field lines, as occurs in 2D reconnection, but it is possible to find two surfaces (or volumes) of field lines that reconnect to form two new surfaces (or volumes). A necessary and sufficient condition for 3D reconnection is that there exists a region where the ideal magnetohydrodynamic assumption breaks down, i.e., a diffusion region through which Z Ekdl 6= 0 (1.39) fl where fl is the field line path and Ek is the component of the electric field parallel to the field line. From the dot product of Ohm’s law in MHD with the magnetic field, j E = k (1.40) k σ Hence, the presence of electric currents are essential for 3D reconnection, just as they are for 2D reconnection, but in 3D it is the parallel component of current that plays the crucial role. In 3D, strong accumulations of current and current layers, can arise in a wide variety of locations and are not just associated with magnetic nulls, as they are 2D. The locations for current layer formation in 3D may be divided into those that are associated with topological features and those associated with geometrical features. Quasi separatrix layers (QSLs) are an example of a geometric feature about which reconnection can occur: these are regions, usually long and narrow, identified on a plane in a magnetic domain which is threaded by field lines whose footpoints significantly diverge at one end. Naturally, if two field lines, which start off running along a similar path, end up in very different places, as do QSL field lines, then these lines will be associated with electric currents.

24 1.5. Three-dimensional reconnection

If the divergence of the field is dramatic then the associated currents may be significant and reconnection (termed either QSL or slip-running reconnection) may result. Currents also accumulate when magnetic flux tubes are twisted or braided. Neither twisting nor braiding has to be excessive for strong currents to form. As is shown in the braiding experiments conducted by Wilmot-Smith et al. 2010; Pontin et al. 2011. Their experiment consists of magnetic field that runs in the same direction, i.e., out from the bottom and into the top of the numerical box. The field is braided and the initial force-free field involving this braid is associated with a large-scale current. However, as this force-free system is allowed to resistively relax, it first col- lapses to form intense currents at which reconnection occurs leading to a cascade process in which the original large-scale homogeneous current frag- ments to smaller and smaller scales. This process is associated with rapid reconnection that occurs at multiple small-scale intense current accumulations throughout the domain. It is not simple for the magnetic field in this experiment to untangle itself and, since the plasma is not clever enough to work out the fastest way to untangle itself with the minimum amount of reconnection, magnetic flux is found to reconnect multiple times. The consequence of this type of behaviour is widespread reconnection throughout the flux tube, that lasts a long time. 3D magnetic null points are one type of topological feature which are prone to collapse to form a current layer, just like 2D nulls are. From a positive (negative) 3D null point (figure 1.9) there are a set of field lines that extend out of (or into) the null forming what is known as a fan surface and a pair of field lines that extend into (out of) the null forming a curve known as the spine. How the magnetic null is perturbed determines the nature of the collapse and the resulting current layer formed. For instance, a rotational disturbance in planes perpendicular to the spine result in accumulations of current around the spine and/or fan. Two types of reconnection are found to be associated with these sorts of disturbances, namely, torsional-spine reconnection which occurs in response to a rotational disturbance of the fan plane, and torsional-fan reconnection which occurs in response to a rotational disturbance of the spine (Priest Pontin 2009). However, the most common type of null-point reconnection, is spine-fan reconnection, which occurs as a result of any shearing motion in which the angle between the spine and fan are altered leading to a collapse of the spine and fan creating a current layer lying along both structures. 3D null points are not the only topological feature at which reconnection can occur. Fan surfaces from 3D nulls extend far out from the nulls themselves separating the magnetic field from topologically distinct field regions. Thus they are more generally known as separatrix surfaces and are bounded by either the edge of the domain in- vestigated or by spine field lines from other nulls. If two separatrix surfaces intersect, special field lines called separators arise 1.9. This manifestation of a separator is stable and resides at the intersection of four topologically

25 1.5. Three-dimensional reconnection distinct flux domains. This means they are in many ways the 3D equivalent of a 2D null point, although, of course, the magnetic field is only zero at the ends of the separators not along its length. It also means that reconnection at separators has global consequences and can lead to global restructuring of the magnetic field [5].

Figure 1.8: Illustrations highlighting the characteristics of (a) 2D and (b,c) 3D reconnection where the thick tubes represent flux tubes with arrows indicating the direction of the field, the block arrows represent the direction of the outflowing plasma and the purple shaded spheres are the diffusion volume in 3D with the arrows indicating the direction of plasma flow on its surface. (Images from [5]).

26 1.6. Magnetic Reconnection in Astrophysics

Figure 1.9: Magnetic ﬁeld structure of (a) a positive 3D potential null point and (b) a separator formed by the intersection of two separatrix surfaces. Image taken from Pontin (2011).

1.6 Magnetic Reconnection in Astrophysics

Magnetic reconnections are very important in Astrophysics as they seem to play a role in many particle acceleration mechanisms. Traditionally, magnetic reconnection was associated mostly with solar flares. In reality, the process must be ubiquitous as astrophysical fluids are magnetized and motions of fluid elements necessarily entail crossing of magnetic frozen-in field lines and magnetic reconnection [6]. For what concernes the Sun, its surface is threaded by a dynamic patchwork of features with fluxes through which magnetic fields are directed into, or out from, the Sun. This environment, in which magnetic field lines are subjected to dynamic and complex processes, is a breeding ground for reconnections. Solar flares are a sudden explosion of energy in the solar corona caused by the rearrangement of the magnetic field lines, usually accompained by ejection of plasmas and particles into outer space. The process is characterized basi- cally by opposite directed magnetic field lines that converge and form a sort of X-shaped point. At the center of this configuration the magnetic field is null. Cusp or X-shaped structures are seen at the tops of flare loops, as showed in figure 1.10. The distribution of flare energy emission rates follows a power- law over several decades. This could be due to the fact that individual reconnection events can trigger others. Flares show many of the key signatures of magnetic reconnection (fast particles, topological changes in the magnetic field, evidence for release

27 1.6. Magnetic Reconnection in Astrophysics

Figure 1.10: Solar flares: formatin of X-type neutral point in the Solar corona. The blue arrows show the direction of the inflow, while the red arrows show the direction of the outflow after the reconnection of field lines. From [7]. of magnetic energy), even if these events are characterized by large size scales and short timescales, so they are difficult to explain with the basic models of reconnection, which assume a two-dimensional and steady state configuration. In fact, Sweet-Parker model for solar flares provides reconnection times of the order of months, while from observations the duration of the phenomenon, estimated by the duration of the increase in x-ray flux, seems to be of about 15 minutes. The nearest natural environment in which reconnection can be studied in situ is the interaction between the terrestrial magnetosphere and the magnetic field of the solar wind. Magnetic reconnection occurs in two primary locations in Earth’s magnetosphere in response to driving from solar wind: dayside magnetopause, at which solar wind plasma reconnects with magnetospheric plasma, and magnetotail, in response to magnetic energy building up in lobes due to solar wind driving (see Figure 1.11). Magnetic neutral points are observed in both magnetopause and magnetotail, and in these points reconnection takes place. The thickness of these regions is of the order of the ion skin depth δi (or the ion gyroradius). In the magnetosphere, measurements of the reconnection region have been made, and a quadrupolar out-of-plane magnetic

28 1.6. Magnetic Reconnection in Astrophysics

field has been found in the electron diffusion region. This is a clear signature of collisionless reconnection. At the magnetopause reconnection occurs between two topologically distinct regions, the shocked solar wind and the magnetosphere, which also have quite different plasma properties. Recon- nection generates a magnetic field component normal to the magnetopause and thereby leads to an interconnection between the two regions. Thus both the heliosphere and terrestrial magnetosphere are rich environments for studying collisionless reconnection in natural plasmas. It was pro-

Figure 1.11: Cross section of magnetosphere. The picture shows two different location at which reconnection can occour in the magnetosphere and in the magnetotail. posed earlier that the relativistic ejections observed in microquasars could be produced by violent magnetic reconnection episodes at the inner disk coronal region. In microquasars and AGNs, violent reconnection episodes between the magnetic field lines of the inner disk region and those that are anchored in the black hole are able to heat the coronal/disk gas and accelerate the plasma to relativistic velocities through a diffusive first-order Fermi-like process within the reconnection site that will produce intermittent relativistic ejections or plasmons. The resulting power-law electron distribution is compatible with the syn- chrotron radio spectrum observed during the outbursts of these sources [8]. Magnetic reconnections play a key role in the evolution of astrophysical jets kink instabilities. Jets are well-collimated outflows produced by compact, accreting objects (protostars, neutron stars, black holes etc.). They are usually bipolar and oriented perpendicularly to an accretion disk. They are a fundamental aspect of star formation. The sizes and velocities of these jets vary over many orders of magnitude. Two particular classes of jets have

29 1.6. Magnetic Reconnection in Astrophysics come in for substantial study: protostellar jets, which arise from stars in the process of forming, and which have sizes of order l ≈ 0.01−1pc and velocities of a few hundred km/s, and extragalactic jets, found in radio galaxies and quasars, which have sizes of order l ≈ 0.01−1Mpc and relativistic velocities. Young Stellar Object (YSO) jets are typically nonrelativistic whereas black hole and AGN jets are relativistic. Jets are believed to be launched and accelerated by magnetohydrodynamic mechanism, but it is possible that far from the star MHD ceases to be important in which case the jets become predominantly hydrodynamic. Since astrophysical jets seem always to be associated with accreting objects, it is possible to conclude that some feature of the accretion process must be able to give rise to a jet. Moreover, the fact that jets are observed to start very close to the accreting objects, in a regime where we expect the effects of rotation to be important and accretion to be occurring through an accretion disk, suggests that the disk accretion in particular has features that lead to jet formation. Suppose we have a system which is accreting mass at a rate M˙ acc and that this system powers a jet that carries away mass at a rate M˙ jet. If the infall velocity of the gas is vin and the jet velocity is vout, energetically the system can power the jet purely with accretion provided that vin M˙ jet ≤ M˙ acc, (1.41) vout therefore even if vin is less than vout the jet can be powered with accretion since it removes only a small fraction of the accretion mass. But how does the accretion flow energy power the outflow? The leading theory is that jets are powered by magneto-centrifugal processes. Consider a magnetic field line anchored in the accretion disk at a distance r0 from the central object, and oriented at an angle α with respect to the plane of the disk. Suppose that the disk is rotating with a Keplerian angular velocity, so at the base of the field line, we have GM 1/2 Ω0 = 3 . (1.42) r0 Let’s introduce the Alfvenic Mach number vr MA = . (1.43) vA Close to the star and disk, the behaviour of the outflow is similar to the Parker wind, and so we know that the radial velocity is initially small. In order to satisfy ∇ · B = 0, the radial component of the magnetic field 2 must satisfy Brr = constant. From the continuity equation, we know that 2 M˙ = 4πr ρvr = constant, and Alfven velocity therefore scales as 1 v1/2 v ∝ · rv1/2 ∝ r (1.44) A r2 r r

30 1.6. Magnetic Reconnection in Astrophysics and the Alfvenic Mach number of the out ﬂow scales as

1/2 MA ∝ rvr . (1.45)

We therefore see that MA 7→ 0 as we approach the star, i.e. the flow is sub-Alfvenic and the field has plenty of time to change its configuration in response to the flow. From this analysis it is clear that MA increases with increasing distance from the star, and that there will be a point rA at which MA = 1 and the outflow velocity exceeds the Alfven velocity: this is the point at which the field can no longer respond to changes in the flow. Outside of this radius (the Alfven radius) the field is dragged along by the flow. However, inside rA the field resists winding and remains approximately radial. Consequently, if a magnetic field line is anchored in the disk at a point r < rA, it will rotate with the same angular velocity as its anchor point. For what concernes the gas associated with the field, if ideal MHD applies, then the gas is tied to the field, and cannot move away from the field line, but it can move freely along the field line without feeling any magnetic force. The net force acting on a fluid element located on the field line is therefore 2 the sum of two forces: the centrifugal force Fc = Ω0r directed perpendicularly away from the rotation axis, and the gravitational force Fg oriented towards the central mass M. It can been demonstrated that net force is directed outward along the field line if α < 60◦. Therefore, provided that the field is not too steeply angled with respect to the accretion disk, the rotational motion caused by the magnetic field is able to drive matter outwards from the disk, powering an outflow. Once the gas passes the Alfven radius, the magnetic field starts to wind up, becoming increasingly dominated by its azimuthal component: gas in the outflow that has a non-zero velocity in the z direction therefore starts to feel a Lorentz force acting towards the axis of rotation. This “pinch” effect therefore channels the upwards-flowing gas towards the polar axis of the system, collimating it into a jet. We therefore see that the same mechanism can explain jet production, jet collimation and the fact that a wide variety of different systems produce jets. One of the most fundamental questions is where and how jets dissipate their magnetic energy. This has important im- plications on particle acceleration and emission mechanisms in the jets, the fraction of magnetic energy carried by the jets at large distances, and on the stability properties of the jets. In toroidal-field dominated jets, the fastest growing instability is known as kink instability. This instability, which will be explored more in Chapter 2, is a current-driven instability (CDI) that generates helical deformations in the jet and can lead to an efficient dissipation of the magnetic energy of the jets and may even disrupt the jet

3https://www.esa.int/ESA_Multimedia/Images/2003/05/Extragalactic_jets_ from_a_black_hole_accretion_disk

31 1.6. Magnetic Reconnection in Astrophysics

Figure 1.12: Artist’s impression of the formation region of M 87’s jet. An accretion disc (red-yellow) surrounds the black hole, and its magnetic field lines twist tightly to channel the subatomic particles pouring out into a narrow jet. The jet opens widely near the black hole, then is shaped into a narrower beam within a light-year of the black hole. Image taken from the website3. altogether. The resulting reconnection takes place in a high Reynolds number medium, where turbulence is further enhanced and in turn accelerates the reconnection process. This boot-strap reconnection gives rise to bursts of reconnection events, through which the free energy of magnetic field is transformed into a gamma-ray burst [9]. Moreover, the magnetic reconnection events cause the heating of the coronal gas, which can be conducted back to the disk to enhance its thermal soft X-ray emission as observed during outbursts in microquasars. Kink instability has a linear and a non-linear phase. Durig the latter one the dissipation of electromagnetic (EM) energy takes place through reconnection and stochastic/turbulent dissipation. Although the magnetic configuration changes during the dissipation process, the total helicity is roughly conserved. The magnetic field configuration gradually relaxes into a minimal energy state, known as a Taylor state.

32 1.6. Magnetic Reconnection in Astrophysics

Figure 1.13: Kink instability of the magnetic ﬁeld in the GRB jet [9].

Jets have been studied via direct observation, analytical models and numerical models. Most recently laboratory configurations which simulate important features of jets have been studied. One of these laboratory meth- ods is based on spheromaks, a toroidal magnetic confinement configuration involving self-organization. The physics of spheromak formation has much in common with magnetohydrodynamically driven astrophysical jets. Both spheromaks and proposed models of astrophysical jets involve an initial poloidal magnetic field which is then inflated outwards by a toroidal field associated with a poloidal current. This parallelism will be explored further in the following sections.

33 Chapter 2

Laboratory Plasma

2.1 Experimental setup to study reconnection

Evidence of magnetic reconnection can be found in space plasma as well as in laboratory plasma. The first laboratory experiments were short-pulsed pinch-type experiments in collision dominated regime. More recent experiments were performed in less collisional regime with a strong guide field, and they allow to study in more detail local structure of non-MHD features of the reconnection region. Anyway, it is not clear how the diffusive neutral sheet is formed in a MHD plasma with δi << L and S >> 1. A class of interesting experiments for magnetic reconnection is fusion research experiments, in which the magnetic field generated by internal plasma can help in confining the plasma itself thanks to stable compressing forces (pinch forces). Fusion experiments require toroidal plasma in order to obtain magnetic confinement, density of the order of 2 · 1023cm−3 and high temperature. These kinds of plasma systems can be described by MHD. Different plasma configuration are used in fusion research experiments, as Tokamak, Reversed-Field-Pinch (RFP) and Spheromak. All of these configuration generate self-pinching poloidal fields, while toroidal field are supplied differently depending on the configuration. However, for all of them the basic equilibrium configuration is reached thanks to magnetic self organization and relaxation phenomena: if we define β as the ratio of the plasma pressure to the magnetic pressure, a low β plasma tends to reach the state with minimum energy thanks to magnetic reconnection. During magnetic relaxation processes helicity must be conserved.

34 2.1. Experimental setup to study reconnection

2.1.1 Magnetic Confinement As already mentioned, fusion experiments are a suitable place to study reconnection. The goal of fusion experiments is of course to obtain controlled fusion of nuclei, but the evolution of the plasma can undergo to reconnection phenomena, or better reconnection phenomena are the key processes for maintaining plasma stability and confinement, as we will see later on. The least difficult fusion reaction that one can try to obtain is the Deuterium- Tritium reaction, which requires a plasma temperature of the order of 10-20 KeV. In 1957 John D. Lawson introduced a criterion for controlled fusion experiments in order to characterize the set of parameters that allow a fusion reactor to produce more energy than it absorbs: the heating due to charged particles should compensate the energetic loss (at least losses due to thermal radiation). Given a certain temperature, at least greater that 4.3 KeV, plasma density −3 measured in m and the energetic and confinement time τE measured in seconds s, these parameters must satisfy the triple product 21 −3 nτET ≈ 4 · 10 KeV · s/m ≈ 10Atm · s. (2.1) In practice, this means that a reactor should have T ≈ 20KeV , n ≈ 2 · 20 −3 10 m and τE ≈ 1s to obtain a burning plasma. These values correspond to a plasma pressure of the order of 10 Atm. Since a fusion collision can take place every 20 seconds, the mean free path is of the order of 20000 km: a magnetic field is required to obtain such mean free paths. Since a 10 km long linear machine is not the best choice, a toroidal shape is required. In order to have stability, a magnetic confinement pressure of 10 times the plasma pressure is deemed necessary. Therefore the β parameter, given by plasma kinetic pressure β = , (2.2) magnetic pressure of the confinement magnetic field should be at least 1/10 to ensure stability. However there are problems with this geometry of magnetic field: since the fieldlines are closed, charged particles are subject to drift motion. Moreover, in a field with circular lines of force ions and electrons drift in opposite direction. The drift due to the curvature is k = (ˆb · ∇)ˆb (2.3) where ˆb is the direction of the magnetic field ˆb = B/|B|. The charged particles velocity is given by ˆb v = v ˆb + × (v2k) (2.4) g k Ω k

35 2.1. Experimental setup to study reconnection where the second term on the right is the drift velocity due to the centrifugal force and Ω = qB/m is the gyrofrequency. Since the centrifugal force is equal in modulus for particles with diﬀerent charge, ions and electrons are subject to the same drift but in opposite directions, as shown in ﬁgure 2.1.

Figure 2.1: Drift motion for ions and electrons

Toroidal configuration requires a central vertical current, which produces a non-uniform magnetic field. The gradient of the magnetic field produces a further drift motion: ˆb v2 ∇B v = v ˆb + × v2k + ⊥ . (2.5) g k Ω k 2 B The separation of charges creates a vertical electric field inside the torus, so equation 2.5 becomes:

ˆb v2 ∇B E × B v = v ˆb + × v2k + ⊥ + . (2.6) g k Ω k 2 B 2 The component E × B destroys the confinement, since ejects plasma out from the torus. In order to obtain a stronger confinement, 2D magnetic surfaces are necessary and another magnetic field component must be introduced in the poloidal direction. Field lines must have a rotational transform on the magnetic surfaces, so that the magnetic field wraps around the torus. In this way drift motions can be contained. Unlike the case of hydrostatic equilibrium, which requires

∇p = ρg magnetohydrostatic equilibrium requires

∇p = J × B (2.7) where B contains the field generated by plasma and the and the external confining magnetic field. Equation 2.7 describes the balance between magnetic and pressure gradient forces: the force exerted from the current which

36 2.1. Experimental setup to study reconnection

flows perpendicular to the magnetic field on the fluid element is balanced by the plasma kinetic pressure. It follows that B · ∇p = 0. (2.8) Likewise J · ∇p = 0 (2.9) so both B and J lines lie on surfaces with constant pressure. As mentionated above, lines of force lie on magnetic surfaces and close by wrapping m times toroidally and n times poloidally. The quantity n l = (2.10) m can be introduced. If l is a rational number, field lines close in on themselves. If l is an irrational number, field lines are surface filling and fill the magnetic surfaces. If φ is the toroidal angle and θ the poloidal one, lines of force satisty

Figure 2.2: Magnetic ﬁeld lines are shown: in green the case of rational l, in violet the surface ﬁlling case.

θ + lφ = constant. (2.11) In the case of closed lines (green part of figure 2.2) an electric current can be directed along the lines, but instabilities can occour, in particular when perturbation of the magnetic field with δB resonant with l = n/m are present. Magnetic island are formed and rotate along the plasma torus. Between these islands, X-type neutral points are formed, as well as O-type neutral points. The island’s width δnm is proportional to the perturbation intensity and, defining ψp, ψt as the poloidal and toroidal flux, to the variation of ψt with time: s dt δnm = 4(ψep,nm)/ . (2.12) dψt

37 2.1. Experimental setup to study reconnection

Magnetic islands cause ergodicity, so when islands form the magnetic field starts to fill the entire volume, and confinement is lost.

Figure 2.3: Section of a tokamak with three magnetic islands.

38 2.1. Experimental setup to study reconnection

2.1.2 Tokamak A tokamak is composed from doughnut-shaped vacuum chamber. Inside the chamber, a gas is transformed into plasma under the influence of extreme heat and pressure. Plasma is confined thanks to the action of three different magnetic fields: a toroidal field (created by toroidal coils doubly connected with the plasma), a vertical field (generated by external coils that surround the whole plasma), and a poloidal field produced by the plasma itself (see figure 2.4) A charged particles in a magnetic field will move according to

Figure 2.4: Magnetic conﬁguration inside a tokamak. Source: EFDA-JET (now EUROfusion)

Larmor equation following a helicoidal trajectory with radius mv ρ = ort . (2.13) ZeB It is thus clear that, since a particle cannot have a radius greater than ρ, an opportune magnetic field can efficiently confine efficiently a plasma, the motion of a charged particle being linked to the dynamic of magnetic field lines. If these lines are closed around the torus, particles follow ring orbits around it. However, the toroidal magnetic field has inhomogeneity in intensity and curvature, and this causes a drift motion for the particle as explained in the previous section, moving them to the wall of the tokamak. If the toroidal magnetic field lines are curved enough (helical-shaped), the drift velocity points outwards and inwards alternatively, so that the global

39 2.1. Experimental setup to study reconnection effect is almost zero, with an excursion form each magnetic surface which is less than a Larmor gyroradius in the poloidal field. How much the field lines are curved is expressed by the safety factor 2π q = , (2.14) l where l is the rotational transform of toroidal magnetic field lines introduced in Section 2.1.1. The operating mechanism of a tokamak can be summarized as follows: first of all air and impurities are evacuated from the vacuum chamber; after that, the magnetic systems are charged up and the gas is introduced; electrical current runs into the central solenoid and, through its inductively produced toroidal electric field, the gas breaks down, becoming ionized and forming the plasma. Plasma particles start to collide, the temperature increases and if it is high enough, particles can potentially start to fuse, releasing huge amounts of energy. An axisymmetric tokamak plasma consists of nested toroidal flux surfaces on each of which electron temperature Te is easily balanced by the rapid transport along the magnetic field and becomes constant. Peaked temperature profiles lead to peaked current profiles because of the lower Spitzer resistivity at the center of the plasma. As a result, plasma is unstable to helical MHD kink modes which develop near to resonant flux surfaces. Plasma is thus helically deformed and as a consequence reconnection can take place near the resonant q = 1, 2, 3, m/n surfaces, rearranging the topology of magnetic field lines. These reconnection events lead to a uniform current-density configuration and to a flat Te profile within the most critically resonant q = 1 surface. This cyclic evolution of relaxation and reconnection is repeated in time and is called Sawtooth crash and will be discussed in the following.

Figure 2.5: Sawtooth crash and reconnection [1]

40 2.1. Experimental setup to study reconnection

2.1.3 Reversed Field Pinch (RFP) and Spheromak While in a tokamak a strong field is supplied externally, in RFP configuration the toroidal field is the result of a combination of internal currents and a small external toroidal field. In Spheromak there are no external field coils, and the toroidal field is created only by internal plasma current. In RFP and Spheromak, the plasma is confined by both poloidal and toroidal field, and in the case of RFP the strength of these fields is almost compara- ble with the external one. In particular RFP plasma is confined by sheared magnetic field. Magnetic field pitch changes from the center of the plasma to the outer region, that is from toroidal field to poloidal field (see figure 2.6). The spheromak is the minimum energy state for an oblate shaped spheroidal

Figure 2.6: Reversed Field Pinch configuration scheme: the figure shows the sheared magnetic field and the transition from toroidal field (at the center) to poloidal field (at the boundary) volume. This equilibrium can have closed nested flux surfaces that should have good confinement with no material or coils linking the boundary. Thus, from an engineering point of view, it leads to a smaller much more cost effective reactor compared to configurations that have a toroidal vacuum vessel linked by large toroidal field coils. Magnetic reconnection phenomena occour during self organization processes and can be impulsive or continuous. From figure 2.6 it is clear that there are different magnetic flux surfaces. Sudden rearrangements of field lines on an inner flux surface can produce other rearrangements in the outer flux surfaces, leading to global magnetic relaxation events in which magnetic reconnection plays a major role, taking place at multiple radii in the plasma torus. These radii correspond to the magnetic flux surfaces at which the safety factor q is equal to 1: for this value, MHD instabilities start to develop, leading to reconnection. During these events, magnetic flux and magnetic energy are converted from poloidal to toroidal. In RFP relaxation magnetic energy is converted to ion kinetic energy very quickly. The spheromak contains large internal electric currents, and the magne-

41 2.1. Experimental setup to study reconnection tohydrodynamic forces within the spheromak are almost balanced. As a result, there is a long-lived confinement times (of the order of millisecond) without external field. Spheromak’s relaxation phenomena are very similar to the case of RFP. The difference is that in Spheromak the different surfaces are more separated than those of RFP plasma, and this implies that reconnection events can be more easily isolated.

Figure 2.7: Diﬀerence between tokamak and spheromak. As can be seen, one of the foundamental diﬀerence is in the safety factor, which is almost 4 for the tokamak and 0.03 for a spheromak plasma.

42 2.2. Magnetic Reconnection and kink instability

2.2 Magnetic Reconnection and kink instability

A cylindrical equilibrium configuration containing both axial and azimuthal fields is subject to a class of current driven instabilities known as kink modes. Kink instability is a current-driven plasma instability characterized by transverse displacements of a plasma column’s cross-section from its center of mass without any change in the characteristics of the plasma This instability results in a displacement of the magnetic axis into a helix with a pitch similar to the twist of the fieldlines themselves. Once the axis and the circular boundary of the flux tube has developed a helical pitch, the flux tube itself is said to be kinked. Though the linear growth of the kink mode has a rather weak dependence on the pitch profile, its evolution in the nonlinear regime changes with the pitch profile. The non-linear evolution of the kink instability can distort the axis of the tube severely.

Figure 2.8: One of the earliest photos of the kink instability in action - the 3 by 25 cm pyrex tube at Aldermaston. From [10].

If we look at solar corona, its most active regions do not show strong signs of magnetic twist, and appear consistent with the emergence of simple loops, but there are small subset of loops that are significantly twisted and kinked. These regions are linked to the occurrence of large solar flares. Magnetic reconnection underlies critical dynamics of magnetically confined plasmas in both nature and the laboratory. Moreover, reconnection involves localized diffusion of the magnetic field across plasma. It is generally proposed that the field diffusion underlying fast reconnection results from some combination of non-magnetohydrodynamic processes that become important

43 2.2. Magnetic Reconnection and kink instability on the ‘microscopic’ scale of the ion Larmor radius or the ion skin depth. A recent laboratory experiment by Auna L. Moser and Paul M. Bellan [11] demonstrated a transition from slow to fast magnetic reconnection when a current channel narrowed to a microscopic scale (but still did not address how a macroscopic magnetohydrodynamic system accesses the microscale). In their experiment, Moser and Bellan reproduced a magnetized plasma jet using the coplanar disk and annulus electrodes and the background poloidal magnetic field coil. The jet front travels at almost 10km · s−1, increasing the jet length until the current-driven kink instability sets in and deforms the plasma jet into a helical structure the amplitude of which grows in time. In the case of an exponentially growing kink, the jet segment develops a periodic fine structure. The fine-structure growth rate, location and spatial periodicity are consistent with the magnetized plasma Rayleigh–Taylor instability (see figure 2.9). This instability develops in a gravitational field at an interface where a heavy fluid with density ρ2 lies above a light fluid with density ρ1. The acceleration of the exponentially growing kink segment creates an effective gravitational field in the plasma frame. The fastest-growing mode of the magnetized plasma Rayleigh–Taylor instability has k · B = 0 (where k is the instability wavevector and B is the magnetic field vector). The results obtained from this experiment support the conclusion that the fine-scale instability is a Rayleigh–Taylor instability, the portion of the jet beyond the break-up region retains its magnetic structure and separates from the remaining jet base demonstrating a clear magnetic reconnection. Therefore, reconnection is observed in the nominal parameter regime of these experiments only when preceded by the Rayleigh–Taylor instability; this could imply that the Rayleigh–Taylor instability is necessary for the observed reconnection to occur. This could be one possible mechanism by which a macroscopic MHD system can couple to the microscale processes necessary for magnetic reconnection. This could work also in nature plasma. Returning to the example of the solar corona, current-carrying magnetic flux tubes confining plasma with density higher than the ambient value are common, as is kinking of such flux tubes, and Rayleigh–Taylor instabilities have been observed (see [12]).

44 2.2. Magnetic Reconnection and kink instability Figure 2.9: Timethe series kinked images jet of quicklystructure plasma narrows on jet to the evolution. a trailingthe For thin side filament exponential of filament, diameteruntil kink the the which amplitude radially filament then breaks outward-accelerating growth, brightens filament. up. shown while For here, As developing more the a a details fine-structure segment sharp, see amplitude of [11]. distinctive, grows, periodic it fine erodes

45 2.3. Experimental evidence of reconnection

2.3 Experimental evidence of reconnection

In the study of magnetic reconnection rates and the energy release during these events, the plasma dynamics is of fundamental importance near and within the diffusion region. Reconnection can take place for different reasons, but mostly because there is a need for magnetic field to release its excessive energy stored on global scales. During plasma evolution, if external forces are applied to the plasma, the magnetic configuartion gradually changes, until when the system is no more in equilibrium. At this point the unstable plasma reorganizes itself rapidly to a new MHD equilibrium state through the formation of current sheet. During this process plasma parameters change, trying to adapt to the new configuration. Magnetic reconnection takes place and a new topology configuration with lower energy is reached, and the excess magnetic energy is converted to plasma kinetic energy. The particle energization layer is thin, but the topological rearrangement is global. The signatures of these events have been observed in both space and astrophysical plasmas. For example, solar flares were defined by localized sudden brightening of the chromosphere in Hα, and Zeeman measurements of the line-of-sight photospheric magnetic field showed the presence of positive and negative polarities. This suggested that the overlying coronal fieldlines formed closed loops. Moreover, radio emission from the high corona is a signature of the outward travelling disturbances. During the years space observatories like SMM (Solar Maximum Mission), Yohkoh, SOHO (Solar and Heliospheric Observatory) and others have studied solar flares at other wavelengths: extreme ultraviolet (EUV), X-rays and γ-rays. These observations provided imaging and spectroscopy at high spatial and temporal resolution. The most of the emission comes from the chromospheric footpoints of the coronal loops. The emission results sustained for hundreds to thousands of seconds, but varies on timescales as short as several microseconds. This could be explained by an intermittent acceleration mechanisms or to propagation effects. Reconnection phenomena can be observed also in laboratory plasma by mon- itoring quantitatively the global and local plasma parameters. A clear manifestation of global magnetic reconnection in high-temperature fusion plasmas, as mentioned above, is given by Sawtooth relaxation oscillations, a cyclic evolution of relaxation and reconnection repeated in time. The sawtooth instability is generally associated with the growth of the large- scale MHD perturbations around the q = 1 magnetic surfaces. In general, modes can grow unstable if the wave vector k satisfies the resonance condition k · B0 = 0 (2.15)

46 2.3. Experimental evidence of reconnection

where B0 is the equilibrium magnetic field. The resonance condition can be also written as a function of the mode numbers m n k · B = Bθ − Bφ. (2.16) r R0 Therefore the safety factor q, which has been introduced in section 2.1.2, can also be defined approximately, for large aspect ratio toroidal surfaces (R >> r) as: rB q = φ (2.17) RBθ where r, R0 are respectively the minor and the major radii and Bφ,Bθ the toroidal and poloidal magnetic fields. Non-thermal electrons can be produced due to the strong electric fields induced during the reconnection of the magnetic field around the resonant surfaces. The temperature profile is initially almost axisymmetric, but then is deformed by a helical instability before a very rapid temperature collapse re- estabilishes an axisymmetric profile with a lower value at the magnetic axis. More precisely a sawtooth oscillation is characterized by a periodic collapse or crash of the central plasma pressure. There are several theoretical models for the sawtooth crash, but Kadomt- sev’s model, developed more than 20 years ago, is one of the most accepted. According to this model, an m = n = 1 resistive internal mode is unstable for q < 1 (m and n are the poloidal and toroidal mode numbers), and grows until a full magnetic reconnection process flattens the pressure and current profiles, causing the central q to increase to unity. A lot of efforts have been made in the years to measure the central q values. There is agreement among all experiments that the relative change of the central q during a sawtooth crash is small (the change is less than 0.1), so it cannot explain the onset of the crash. Alternative crash models include resistive two fluids MHD [14], collisionless kinetic effects [15][16], accelerated complete reconnection due to nonlinear collisionless effects [17], stocastization leadind to enhanced perpendicular transport [18] and triggering of secondary instabilities [19]. The conventional diagnostics for this phenomena are soft-X-ray diodes mea- suring bremsstrahlung emission along different chords across the plasma, while the electron cyclotron emission (ECE) diagnostics have accurately measured the evolution of the temperature profiles. As mentionated in section 2.1.2, an axisymmetric tokamak plasma consists of nested toroidal flux surfaces on each of which the electron temperature Te. A peaked electron temperature profile will produce highly peaked current profile because of higher Spitzer conductivity. This will result in a helical MHD kink mode near a resonant flux surface. Kink instability is sometimes referred to as the m = 1 mode. It can be

47 2.3. Experimental evidence of reconnection studied very well in fusion plasmas with Z-pinch configurations. If a kink mode begins to develop in a column the magnetic forces on the inside of the kink become larger than those on the outside, which leads to growth of the perturbation. As it develops at fixed areas in the plasma, kinks belong to the class of “absolute plasma instabilities”. Moreover, fast electron heat transfer has been documented in several experiments (see for example [1]). Just before the crash, a shrinking circular hot peak shows up and a crescent-shaped flat island grows inside the q=1 region with a kink structure of m/n = 1/1 [1]. During the crash phase, a fast heat transfer from inside to outside the q = 1 surface was observed and was attributed to magnetic reconnection. The electron temperature profile inside the q = 1 radius becomes flat after the crash (which is consistent with the Kadomtsev prediction). The measured central q values indicate that q values increase by 5–10%, typically from 0.7 to 0.75, during the sawtooth crash phase but do not relax to unity even while the pressure gradient disappears inside the q = 1 region. A heuristic model was proposed for the sawtooth crash. The plasma is viewed as two concentric toroidal plasmas separated by the q=1 flux surface. A kink mode develops due to a strong peaking of toroidal current and displaces the pressure contours on an ideal MHD time scale with a helical m=1, n=1 poloidal and toroidal mode numbers structure. If ballooning modes (which can become unstable with high mode numbers) trigger reconnection, it could occur preferentially in the outer part of the displaced surface. This nonaxisymmetric deformation of toroidal plasma destroys the nested flux surfaces of different electron temperatures inside the q=1 flux surface making the electron temperature profile uniform inside q=1. Simultaneously, a rapid reflux of thermal energy occurs through the reconnection region along newly connected field lines which connect the inside and the outside of the q=1 surface. Summarizing, what is clear so far is: • Magnetic reconnection is often driven by an ideal kink-type MHD instability excited after a gradual change of the equilibrium -at least in tokamak- and the reconnection time is much faster than the Sweet- Parker time. With the recent understanding of two-fluid physics in the collisionless plasmas, this is not surprising since the Sweet-Parker model is only applicable to collisional plasmas, while tokamak plasmas are collisionless or halfway between the two regimes. • Heat diffusion transport can occur much faster than the magnetic reconnection, namely, on the time scale of parallel electron heat conduction, and influence the evolution of global reconnection phenomena or magnetic self-organization. Kadomtsev-type full reconnection is trun- cated because the high pressure gradient that drives a kink mode is reduced due to fast heat conduction through reconnection region.

48 2.3. Experimental evidence of reconnection

One more thing which must be analyzed is whether magnetic helicity (which has been introduced at the end of section 1.6) is still conserved in a highly conducting plasma undergoing reconnection. According to the Taylor relaxation theory [20] the plasma has a tendency to relax toward the minimum magnetic energy state while conserving total magnetic flux and helicity. Experimentally, magnetic helicity was observed in several experiments to change little compared to magnetic energy which decreases substantially during RFP relaxation [21]. A simple estimate of helicity change due to magnetic reconnection can show that the total magnetic helicity is a well conserved quantity. Magnetic reconnection is perhaps the only process that can release magnetic free energy while conserving flux and helicity. Thus, magnetic reconnection is strongly implied, although not explicitly specified, in the process of Taylor relaxation. The underlying instabilities for reconnection and relaxation are driven by excess internal current within the plasma. In spheromaks, a kink instability can be destabilized. The instability is no longer localized to the central region but occupies the entire plasma and causes global reorganization. Inside the reconnection layer various dissipative and non-linear effects (such as finite resistivity, viscosity, and inertia, as well as magnetic turbulence) can lead to effective decoupling of the plasma flows from the magnetic fields. Diffusion of the magnetic fluxes leads to interconnection of the field lines inside the reconnection layer with subsequent release of the magnetic energy and its transformation into radiation, plasmaheating, and particles acceleration. Particle acceleration during the reconnection can bealso provided due to the shock waves [7], plasma turbulence (see [3]), and whistler waves [8].

49 2.3. Experimental evidence of reconnection

Figure 2.10: (A) The eruptive flare observed in EUV and X-ray wavelengths by the Atmospheric Imaging Assembly (AIA) 171 A˙ (red), X-Ray Telescope (XRT; aboard the Hinode satellite) Be-thin (yellow contours, showing the eruption), and AIA 131 A˙ (green, showing the newly reconnected flare loops) passbands, which are respectively sensitive to plasma temperatures of 0.8, ¿2, and 10 MK. (B) Closer view of the flaring region [box in (A), rotated clockwise to an upright orientation]. A radio source (blue; at 1.2 GHz) is observed at the top of hot flaring loops ( 10 MK), which is nearly cospatial with a nonthermal HXR source (white contours; at 15 to 25 keV) seen by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI). [13]

50 Chapter 3

PROTO-SPHERA

The PROTO-SPHERA experiment, located at CR-ENEA Frascati, is an in- novative magnetic confinement plasma experiment for controlled thermonu- clear fusion research. As described in the previous section, magnetic fusion configurations mostly used in tokamaks and other toroidal pinches are doubly connected (fig. 3.1). If one could obtain a simply connected configuration with the same confinement properties as the tokamak, the design of fusion reactor would be much easier, due to the absence of the critical central post and the ease of access to a cylindrical reactor chamber. Moreover, by the physical point of view, if there is an edge magnetic separatrix with singular magnetic X-points on the symmetry axis at the two ends of the simply connected configuration, a confinement system with two ‘ends’ could be obtained, and this eases the refuelling/exhausting of the plasma and the emerging field lines could help in controlling the electric potential within the plasma. This magnetic configuration characterizes the PROTO-SPHERA system, which is showed in figure 3.2: this is a simply connected magnetic configuration composed of a spherical torus (ST) with closed flux surfaces and a toroidal plasma current Ist and a screw pinch (SP) fed by electrodes with open flux surfaces and plasma electrode current Ie. The screw pinch replaces the central conductor of a tokamak. This combined plasma configuration is called “bumpy Z-pinch” or “flux-core-spheromak” (FCS). This new configuration is based on magnetic helicity injection. Any initial plasma configuration in absence of external connected volume power sources will self-organize in a relaxed state: ∇ × B = µB (3.1) where µ is the relaxation parameter defined as µ j · B µ = 0 (3.2) B2 and it is constant all over the plasma after a sufficient time. In a more realistic case there will be a volume Va containing magnetized

51 Figure 3.1: Tokamak-doubly connected configuration: this configuration requires a toroidal-shaped vacuum chamber, which is not very advantageous from an engineering and physical point of view. plasma with open field lines passing through the boundary. In this situation it is possible to remodel the helicity content of the magnetized plasma. Therefore it is more useful to define the relative helicity Z δK = (A + Av)(B − Bv)dV (3.3) Va with Bv = ∇ × Av (3.4) the vacuum potential field. Inside the volume Va, ∇ × Bv = 0 with the boundary condition

Bv · na = B · na where na is the outward pointing normal unit vector of the boundary surface of the volume Va. The plasma formation and sustainment can occur if the magnetic helicity can be injected through the boundary by driving current along the lines of force more quickly than it is dissipated inside the domain by resistive processes. The electric current forced to move along the B-lines generates an orthogonal magnetic ﬂux and causes B-lines to kink-up with a helical pattern, and this is the origin of the helicity injection mechanism. Not only helicity, but also plasma current and magnetic energy are injected, and magnetic reconnections convert part of the magnetic energy into kinetic

52 Figure 3.2: PROTO-SPHERA -simply connected conﬁguration: this con- ﬁguration requires a cylindrical vacuum chamber, which had enormous ad- vantages with respect to a toroidal-one. Moreover, the central solenoid and magnet have been eliminated, which is also an advantage in terms of costs of the experiment. energy of the plasma particles. If the helicity source is separated from the helicity sink, then

∇µ 6= 0 (3.5) and therefore resistive magnetic MHD instabilities cause a helicity flow from region of larger µ to region with lower µ. Helicity injection, using coaxial electrodes, has been used to form and sustain spheromaks and spherical tori in differet experiments. Helicity conservation has been confirmed experimentally.

53 3.1. From the screw pinch to the spherical torus

3.1 From the screw pinch to the spherical torus

A predictive MHD code has been developed in order to simulate the ST formation in the PROTO-SPHERA experiment (see [22]). The formation sequence can be described as follows. Firstly a pure pinch in equilibrium is formed, with a pinch current of the order of 8-10 kA. This configuration is a stable configuration as long as the pinch current is low, because with a low current the helix safety factor is less than 2. If the pinch current rises up to 60-70 kA, the magnetic field lines start to bend and knot, and a kink instability is formed in the central plasma column. This instability in its non-linear phase gives rise to the spherical torus. The experiment has been divided in different phases in order to refine and improve each step of the spherical torus formation process.

Figure 3.3: The ST formation process as reproduced from the MHD equilibrium code: the central plasma column (pinch) is at first in equilibrium with a low current. As the current rises, the plasma column develops a kink instability, characterized from windings and knots of magnetic field lines and magnetic flux tubes. When the instability goes through its non-linear phase, reconnections confine plasma in magnetic islands, thus forming a spherical torus whith closed magnetic surfaces.

54 3.1. From the screw pinch to the spherical torus

3.1.1 Phase 1.0 In the first phase the goal is to produce a plasma column which can effectively replace both the central post of the tokamak configuration (which closes the electric current of the toroidal external magnets that generate the toroidal magnetic field) and the central solenoidal tansformer which with a variable current induces the toroidal current in the plasma. The main components of the experiment are listed below: • Vacuum Chamber: The first vacuum chamber used for PROTO- SPHERA is the one from the START-experiment (Small Tight Aspect Ratio Tokamak), a spherical tokamak built in Culham (Oxfordshire, UK) in 1991. This chamber has been completely emptied, leaving unchanged only the external part, consisting of 4 cm of aluminium, with a diameter of 2 meters and height of 2 meters. Therefore, the vacuum chamber is composed of only one cylinder, whose upper and lower part are occupied by electrodes which close hermetically the chamber. The produced vacuum is almost 0.6 · 10−7bar. • Power supply group: cathode power supply consisting of a six trans- former secondary with primary voltage regulation. Each of the secon- daries feeds a sector that contains 3 modules in molybdenum with 1.7 kA and 25 V; 2 kA and 350 V coil power supply; 10 kA and 350 V discharge power supply. • Electrodes. In order to obtain the PROTO-SPHERA configuration two electrodes inside the vacuum chamber are needed. The annular anode is divided in six sectors, each with five perforated plates made of 90% tungsten and 10% of copper. Through an annular tube placed on the outside of the anode, the gas is introduced directly into the perforated plates. The particular shape of the plates allows to increase the interaction surface between metal and plasma. The central arc is caused by the potential difference, created by the discharge power supply, between the two electrodes. With 10 kA of plasma current reached, a toroidal magnetic field is induced on the plasma column. As in the Tokamak, the combination of the two field components is obtained again: azimuthal (toroidal) and longitudinal (poloidal), which is a consequence of the lines of force wound in a helix (figure 3.5). The annular cathode is located in the lower part of the machine and is formed by 4 copper disks: the two external disks serve as water- cooled heat sinks; while the two central discs house 18 Molybdenum modules with each 3 tungsten filaments capable of emitting 150 A each. The cathode is divided into six sectors and powered with alternating voltage, thus creating a rotating magnetic field that could have helped in avoiding anode anchoring.

55 3.1. From the screw pinch to the spherical torus

Figure 3.4: On the top: The anode-cathode system is showed. Bottom: on the left a section of the anode, on the right a section of the cathode.

• Coils. In this phase there are 4 copper magnetic coils, armored in steel, inside the vacuum chamber: PF2; PF3-1; PF4-1 and PF4-2 in the upper part and 4 other coils in symmetrical configuration in the lower part. PF4-1 and PF4-2 ensure that a homogeneous shaping of the plasma is obtained near the electrodes, spreading the plasma current, and therefore also the power, over a larger area. PF2 and PF3-1 with which the vertical magnetic field, also called poloidal, is formed. The poloidal field shapes the plasma current centrepost on open lines of force, which start from the anode and end on the cathode.

The basic operation of the PROTO-SPHERA experiment can be described as follows. First of all the cathode tungsten ﬁlaments are heated up to 2600 Celsius degree in 20 s approximately, while the current in the PF-shaping coils of Group-B reaches its constant value IB = 1875A in less than 1 s. Gas is introduced in through the hollow anode in order to ﬁll the SP discharge region at a pressure 10−2 − 10−1 mbar. In less than 2 ms a total voltage Ve ≈ 100 − 350V is applied between the electrodes. The force-free screw pinch is then formed by the hot cathode breakdown.

56 3.1. From the screw pinch to the spherical torus

Figure 3.5: Illustration of the origin of the helical magnetic ﬁeld lines in a tokamak. The toroidal Bφ and poloidal Bθ contributions to the total Btot magnetic ﬁeld. From [23]

3.1.2 Phase 1.25 (January 2018 - September 2019) In this phase some modifications were made to test the magnetic confinement of the first plasma tori obtained in the experiment. Four PF-Ext coils have been added externally to the vacuum chamber, one on the top, one on the bottom and two near to the equatorial plane. The equatorial PF-Ext coils generate a vertical magnetic field directed from top to bottom, which moves the plasma interaction zone from near to the electrodes to a wrong position. Therefore the top and bottom coils (in which the current flows in opposite direction respect to the equatorial ones) are needed to bring the two electrode plasmas back to the right position. With the introduction of the new coils, it was necessary to add the new power supply SuperCapacitor, capable of supplying a current of 2 kA at 90 V for a few seconds. Finally, two divertor plates were introduced in order to protect and isolate both the upper and the lower group of the PF2 and PF3.1 coils, which would be exposed to the passage of plasma. The discharge procedure is slightly modified respect to the phase 1.0: the external and internal coils are turned on together, but due to the alluminium vacuum chamber, whose thickness is 4 cm, the external ones need almost 600 ms to stabilize the internal field. Once the discharge is triggered, the torus is formed almost almost immediately and is sustained until the power supply stops delivering the current.

57 3.1. From the screw pinch to the spherical torus

3.1.3 Phase 1.5 (September 2019 - December 2021) The alluminium vacuum chamber is replaced with a chamber in PMMA (Plexiglas) 1.7 meters high, with 2 meters of diameters and a thickness of 9 cm. This new vacuum chamber has 12 window (port) in which diagnostics can be entered. To avoid the direct contact between the vacuum chamber with the plasma and protect the chamber from the UV radiation emitted from the plasma, the chamber is internally coated with a thin layer of 2 mm of polycarbonate. Six more coils have been added (PFint-A, figure 3.6), three in the upper part and three in the lower one. The four external coils have been replaced with four new coils embedded with a bigger cable which has a lower resistance. With the help of these six coils the purpose is to confine the plasma in a spherical torus, bringing the confinement volume at almost 70% of the total volume. The toroidal current estimated is almost 5 kA, and the potential needed by electrodes is almost 350 V. The discharge procedure is composed of two step: first the phase 1.25 is repeated, feeding the internal coils PFint-B for the formation of the central column. In the following 20 seconds there is a slow current rise in the external coils PFext with the SuperCapacitor power supply, and the torus is formed. Second, in order to obtain a spherical torus with a bigger volume, the external coils and the internal ones PFext-B PFint-B are powered in series from the power supply group, while the SuperCapacitor powers the new six internal coils PFint-A.

Figure 3.6: Poloidal ﬁeld coils of PROTO-SPHERA, divided in groups ‘A’ (red), ‘B’ (green) and ’Ext’ (ochre) [22].

58 3.1. From the screw pinch to the spherical torus

3.1.4 The role of magnetic reconnection With the introduction of the external vertical magnetic field a discontinuty in the field is generated. As can be seen in figure 3.7, a region in which magnetic fields are subjected to abrupt spatial changes of polarity is now present, and this leads to reconnection. As showed in figure 3.7, the SP and ST have a common embedded magnetic

Figure 3.7: Magnetic configuration at the X-points. separatrix with regular X-points (which means a poloidal magnetic field null point): resistive instabilities drive magnetic reconnections, injecting magnetic helicity, poloidal flux and plasma current from the electrode-driven SP into the ST and converting into plasma kinetic energy a fraction of the injected magnetic energy. The SP is magnetically given a disc-shape near each electrode, with a singular magnetic X-point (X-point with non null magnetic field) on the symmetry axis. Magnetic reconnection is therefore the crucial mechanism which traps plasma particles inside a magnetic island (Figure 3.8). Since the system is 3-dimensional, and thanks to its rotational invari- ance, this magnetic island is actually a torus in which plasma is confined. The two X-points at which the poloidal field is null fall into the intersection areas of the two magnetic surfaces of the SP and ST.

3.1.5 Towards phase 2.0 and beyond The ﬁnal phase of the experiment involves the construction of all internal armored coils and power supplies capable of generating 70 kA in the central column. With the aim of reaching spherical tori, from 95% of the total volume, with about 300 kA of toroidal current.

59 3.1. From the screw pinch to the spherical torus

Figure 3.8: Plasma conﬁnement: magnetic reconnection conﬁnes plasma inside magnetic islands, that are tori in 3D

PROTO-SPHERA could also be used as a space thruster. Up to ∝ 15% of total charged products from fusion are emitted at an energy that is about 1 million times higher of the products of chemical reactions, obtaining a specific thrust (per unit of mass expelled) much higher. Using both of the X-type points one could generate electricity with a thermionic converter in the point with non-predominant ejection and in the other point (with predominant ejection) the rocket propulsion effect could be obtained. An experiment like PROTO-SPHERA can be extremely useful for the study of magnetic reconnection phenomena and acceleration of particles in astrophysical jets. In fact, in the last experimental campaign, pinch-only discharges were also studied, in which the central plasma column experiences kink instability. The results were surprising, as what has been obtained is very close to what is observed in some astrophysical sources (fig. 3.10). Furthermore, the magnetic configuration of this experiment is similar to the magnetic configurations found in Astrophysics if one thinks, for example, of jets, accretion discs or magnetized stars. Therefore obtaining a model of

60 3.1. From the screw pinch to the spherical torus

Figure 3.9: In front of the electrodes there are open magnetic field lines. The open magnetic field lines are wound in a circular direction. Magnetic reconnections convert open B lines into closed B and J lines wrapped around the spherical torus. this experiment (even if very simplified) could be very useful for the study of astrophysical objects, mostly sources of strong radiation (AGN, GRB) that cannot be studied in laboratories.

61 3.1. From the screw pinch to the spherical torus

Figure 3.10: Side-by-side images of the jet from the Crab Nebula show its directional change between Nov. 5, 2008 (left) and May 11, 2011. Credit: NASA / CXC / SAO

Figure 3.11: This is a Helium shot (shot 1885) with only the central column produced by PROTO-SPHERA. As the current rises, the central column is subjected to a kink instability, which is clearly seen in the image. In the non-linear phase of this instability, the reconnections conﬁne the plasma to a torus.

62 Chapter 4

Data Analysis

Thanks to its configuration, based precisely on the mechanism of magnetic reconnections, PROTO-SPHERA is a fertile ground for the study of reconnection phenomena and their consequences on the entire plasma. By exploiting the data of the probes and the images of the fast cameras, an attempt was made to characterize these phenomena. However, a clarifica- tion must be made: we already know that with the acquisition frequencies of the current diagnostics it will be hard to observe the reconnection phenomena taken individually, but it is possible to study their overall effect on the plasma. By combining the data from the cameras and probes together, important informations about the geometry and dynamics of the plasma have been obtained. Several shots with Helium, Argon and Hydrogen gases have been produced during the last experimental campaign, but for the purposes of this thesis only few shots in Helium have been analyzed. Furthermore, it has been observed that for some shots the current of the plasma column, which should close on the cathode, actually discharges on the divertor or in any case on other components. As a result, the plasma is displaced downward from the center of the vacuum chamber. However, despite the current reaching the cathode is about half of that which comes from the anode, the experiment works well and the configuration remains stable throughout the discharge. This shows the solidity of this magnetic configuration, and is a good omen for the next phase of the experiment, in which the pinch current will rise to values of about 10 times the current one. The diagnostics used are presented in the following.

4.1 Hall sensor

Hall effect sensor measures the vertical component of the magnetic field in the equatorial part of the plasma using the Hall effect. An example of a

63 4.2. Langmuir probe signal obtained from these probes is shown in the figure 4.1. As can be seen from the figure, the intensification of the magnetic field (which becomes more negative when the plasma is turned on) confirms the presence of a toroidal current of about 5 kA. The signal is averaged at 500 Hz, that is to the millisecond. This could be a problem, as reconnection events cannot be identified with an average of this magnitude. Moreover, already with averages at the fraction of a millisecond, the information contained in the signal is completely lost in the noise.

Figure 4.1: Signal of the vertical component of the magnetic field for the Helium shot 2141. The lowering of the signal indicates a current input, which shows the presence of toroidal current and therefore confirms plasma confinement.

4.2 Langmuir probe

Langmuir probe measures temperature and plasma density near the outer edge of the equatorial plane of the plasma. The probe consists of a Tungsten wire placed in a plasma which collects current and voltage. The wire is connected to a power supply which applies a voltage between the probe and the chamber. The measure is based upon the I-V characteristic of the Debye sheath1. Figure 4.2 shows the plot of the applied voltage from the power supply and the current collected from the probe. The negative region on the left is called ion-saturation region, and it allows the ion density to be determined. The point at which the curve crosses the zero-current is called

1The Debye sheath is a layer in a plasma which has a greater density of positive ions (negative electrons) and hence an excess of positive (negative) charge, that balances the opposite negative (positive) charge on the surface of a material with which it is in contact.

64 4.2. Langmuir probe

Figure 4.2: An idealized I-V curve. The left curve is expanded 10X to show the ion current. the ﬂoating potential Vf , that is when the electron current equals the ion current, and the net current is zero. Vs is the plasma potential, which is just before the electron saturation region. This last enables the electron density to be determined. Ion current density is deﬁned as I J = i = en u (4.1) i A s B 1/2 kTe uB = (4.2) mi where uB is the Bohm speed, ns the ion density at the sheath edge and A is the surface of the probe. Therefore, the ion current is given by 1/2 kTe Ii = ensA (4.3) mi and using the Boltzmann relation for ns, which is eφe ns = n0exp (4.4) kBTe the current can be written as 1/2 1 kTe Ii = en0A (4.5) 2 mi where n0 is plasma density. When the probe is inserted in the plasma, the latter does behave like a resistance and forms a circuit. Therefore, the voltage applied to the probe must satisfy V = Vp + Vs. The electron current density can be calculated from Z Je = e f(v)dv (4.6)

65 4.3. Fast cameras where f(v) is the Maxwellian distribution function. Therefore, the electron current is given by en0Ave eVp Ie = exp . (4.7) 4 kBTe The temperature is obtained by taking the logarithm of both sides of the previous equation, and is the slope of the curve I-V between the floating potential and the plasma potential. If we now replace Vp in the expression for the electron current with V − Vs, and if we chose V = Vs, then en Av I = 0 e (4.8) e 4 and the plasma density can be determined. An example of density and temperature signals is given in figure 4.3. The spikes correspond to sudden variations of some quantity within the plasma. The problem of the Langmuir probe is that it must be polarized with a potential that corresponds to 3 times the temperature which has to be measured in order to give an accurate result. In the case of PROTO-SPHERA this was not possible for technical reasons. As a result, the numerical values of spikes observed at temperatures higher than 20 eV are not reliable. However, the position of the spikes, not related to any kind of disturbance or malfunction of the acquisition system, is reliable and indicates that in this time interval the plasma was subjected to a rapid fluctuation of some kind. Possible interpretations will be explained later on.

Figure 4.3: Temperature and Density signal for shot 2141. The temperature is given in eV, the density in particles per cubic meter.

4.3 Fast cameras

Six fast cameras with a resolution of 640x480 pixels and 600 fps, are displaced on the equatorial plane at 60 degrees apart. An Ultra-Fast camera, with a resolution of 1024x1024 pixels and 3600 fps, is present. The cameras conﬁguration is depicted in ﬁgure 4.4. The images from the cameras are

66 4.3. Fast cameras

Figure 4.4: Section of the vessel seen from above. The six cameras are arranged on the equatorial plane of the vacuum chamber, 60 degrees apart. The Ultra Fast Camera is positioned at about 290 degrees.

RGB (Red, Green and Blue) images. In the following data analysis from Helium shot number 2141 and 2143 of the last experimental campaign are presented.

67 4.4. Fast camera image analysis

4.4 Fast camera image analysis

Images from Fast cameras and Ultra-Fast camera have been used to calculate the diameter of the plasma. To do this, a horizontal line passing under the ports and under the Langmuir probe was selected, and the brightness along this line was studied. The brightness gradient was then calculated, and the left and right edges were identiﬁed with the maximum and minimum points of the gradient, as can be seen from ﬁgure 4.5. The same methodology

Figure 4.5: On the top, the brightness curve along the equatorial line. On the bottom, the gradient of the curve: since the curve has abrupt variations at the edges of the plasma, the choice of the absolute maximum and minimum points of the gradient is a good criterion for obtaining the edges of the plasma. was applied to ﬁve of the six cams around the vacuum chamber, since for one of them the presence of the Langmuir probe disturbed the brightness signal. As can be seen from ﬁgure 4.7, the value obtained for the diameter

68 4.4. Fast camera image analysis oscillates from 400 to 500 pixels, which corresponds to an average diameter of 450 pixels, that is about 50 centimeters.

Figure 4.6: Zone selected for the brightness and diameter calculation.

Figure 4.7: Diameter calculation from the six fast cams. On the ordinate axes the number of pixels is represented.

69 4.4. Fast camera image analysis

Another information obtained from the images is the brightness of the central area of the plasma. First of all, the image was ﬁrst ﬁltered in orange/gold, selecting the RGB values corresponding to the exact color of this Helium plasma, that is 100% Red, 86% Green, 25% Blue. After, the central area of the plasma was selected and the brightness was calculated as the sum of the color intensity of each pixel. By doing this for all the frames related to a single shot, it was possible to see how the brightness varies during the shot itself. Figure 4.8

Figure 4.8: Brightness evolution of Shot 2141 from the Ultra-Fast Camera. shows the brightness evolution as seen from the Ultra-Fast camera, while ﬁgure 4.9 shows the brightness seen from the six Fast Cams.

70 4.4. Fast camera image analysis

Figure 4.9: On the top, brightness evolution from Shot 2141 from the six cams. On the bottom, the overlap between the brightness: the trend is almost identical, the diﬀerence in module depends on the fact that the cameras had diﬀerent exposures.

71 4.4. Fast camera image analysis

Important informations have been obtained by studying the brightness equi-surfaces of the filtered images. The equi-surfaces have been obtained by studying the level curves of the color intensity of the previously filtered image. In many frames it can be noted that in the equatorial plane two more intense zones are formed with closed equisurfaces nested one inside the other: this could be a confirmation of the fact that this magnetic configuration is able to confine the plasma in a torus. This is supported by the fact that the same behavior can be seen in the images of the six cameras arranged around the vacuum chamber (figure 4.10): the plasma is confined in 3D along the entire azimuth, resulting in a torus.

Figure 4.10: On the top, the ﬁltered image obtained from the Ultra-Fast Camera. On the bottom, the ﬁltered images obtained from the six cameras around the vacuum chamber.

If we now look at the areas where the magnetic conﬁguration foresees the presence of the X-points, we notice one thing: the brightness gradient

72 4.4. Fast camera image analysis presents a minimum in this area, or rather a valley of minima. This is what one would expect to find by looking at the magnetic equisurfaces in the vicinity of the X-points: since the magnetic field is null at X-points, one expects to find a valley of minima for the magnetic field, surrounded by zones in which magnetic field increases. Based on this, an algorithm was developed that studies the equisurfaces of the plasma and the variation of the gradient along these surfaces, in a specific area around the estimated position of the X point. Once the valley of the minima is found, the estimated X point will be given by the average of these points. However, it must be taken into account that in the case of the brightness equisurfaces the behavior is different, because in addition to the valley there will be another descent towards the outer edge of the plasma, where the brightness reaches the minimum. Therefore it is necessary to ensure that the analyzed area always falls inside the plasma and does not fall with the outer edge.

Figure 4.11: The figure shows how the algorithm works: once the portion of the filtered image to be analyzed has been selected, the gradient of the color intensity along x and along y will be calculated. Once this is done, the valley of minimum points will be found in the specified area, the midpoint of which will be calculated.

73 4.5. Cross Correlations analysis

4.5 Cross Correlations analysis

To understand whether or not there were physically important relationships between the various signals, cross correlations between different quantities were calculated. Cross-correlation is a measure of similarity of two series as a function of the time displacement of one relative to the other. For continuous functions f and g, the cross-correlation is defined as: Z ∞ (f ~ g)(τ) = f(t)g(t + τ)dt (4.9) −∞ where f(t) denotes the complex conjugate of f(t), and τ is the displacement, known as lag (a feature in f at t occurs in g at t + τ). Similarly, for discrete functions, the cross-correlation is defined as: ∞ X (f ~ g)[n] = f(m)g(m + n). (4.10) m=−∞ It is very useful to normalize the correlation coefficient: in this way the coefficient will be between -1 (maximum anti-correlation) and 1 (maximum correlation). The normalized cross correlation can be obtained using the time series with zero mean and dividing by the variance of the two signals and by the number n of the samples (which must be the same for both the time series). After calculating the cross-correlation between the two signals, the maximum (or minimum if the signals are negatively correlated) of the cross- correlation function indicates the point in time where the signals are best aligned. The first signals for which the correlation was calculated are the magnetic field and density from shot 2141: the correlation is shown in figure 4.12.

Figure 4.12: Cross correlation between the signals of density and magnetic ﬁeld from the Helium shot 2141.

74 4.5. Cross Correlations analysis

Near the zero lag the correlation exhibits an oscillatory behavior. This could be due to the fact that the plasma moves away and closer to the Hall and Langmuir probes. The two signals show an important autocor- relation with zero lag. A possible explanation could be the following: the increase in density is caused by the arrival on the probe of energetic particles produced by magnetic reconnections. Consequently, the arrival of these particles causes an increase in current in the torus, and therefore an increase in the modulus of the Hall magnetic ﬁeld, which becomes more negative. From the cross correlation between the diameter and brightness signals another important feature emerges. Figure 4.13 shows the correlation for two diﬀerent cameras, the one situated at 117o and the one immediately following, placed at 177o. At zero lag, one shows a positive correlation and the

Figure 4.13: Cross correlation between the brightness and the diameter calculated from Camera 117 (top) and Camera 177 (bottom). The peaks that are observed with too large lag have not been taken into account. In fact, since the plasma discharge lasts about half a second, it makes no sense to consider correlation coeﬃcients that correspond to a delay of 20 milliseconds.

75 4.5. Cross Correlations analysis other a negative correlation, and the same goes for the other positive and nagative peaks: this is the sign of a bulk rotation of the plasma. In fact, if the plasma rotates in a rigid way moving away from the center of symmetry of the conﬁguration, when it is closer to a camera, this will see it brighter and larger, while the other cameras will see a lower brightness and a smaller diameter. From the same two cameras, 117o and 177o, the correlation between density and brightness is illustrated in ﬁgure 4.14. The same trend is observed for the shots 2143 and 2092 and for the rest of the cameras.

Figure 4.14: Cross correlation between density and brightness for camera 117 (top) and camera 177 (bottom) relative to the Helium shot 2141.

76 4.5. Cross Correlations analysis

Figure 4.15: Brightness and diameter (measured from the Ultra-Fast Cam- era) correlation for the Helium shots 2143 (top) and 2141 (bottom). Both the plots show two major positive peaks near the zero lag and a negative peak at zero lag.

77 4.5. Cross Correlations analysis

Moreover, looking at the correlation between the diameter and the brightness from the Ultra-Fast Camera (ﬁgure 4.15), the presence of two positive peaks close to the zero lag can be noted. This information, coupled with the rotation of the plasma, can be interpreted as follows: when the plasma is closer to one of the cameras, this will see an increase in brightness; the fact that there are two positive peaks with delay and advance in the correlation between brightness and diameter, tells us that the plasma is not perfectly spherical. Therefore in a single rotation the plasma passes twice closer to a camera. The shape of an ellipsoid could explain what is observed. In fact, when one of the two edges corresponding to the major axis of the ellipse passes in front of the camera, this will record a maximum of brightness and a minimum in diameter. Since in a single rotation both the points corresponding to the major axis will pass in front of each camera, this could explain the presence of the two positive peaks. Figure 4.16 shows the cross correlation of the horizontal distance between the upper X-points and the diameter, and the vertical distance of the X-point on the right and the diameter. In both cases there is a negative correlation, which means that when the plasma in larger on the equatorial plane the X-point are they are closer together, both horizontally and vertically. For what concernes the magnetic ﬁeld and temperature signals, a very

Figure 4.16: Cross correlation between the horizontal distance of the upper X-points and the diameter (top) and the vertical distance of the pair of X-points on the right and the diameter (bottom). interesting feature can be noted. If temperature peaks greater than 25 eV

78 4.5. Cross Correlations analysis are projected onto the Hall probe signal, we note that these peaks correspond to abrupt variations in the magnetic field (figure 4.17). Recalling the sawtooth oscillation phenomenon explained in Section 2, if the magnetic field fluctuations are actually due to reconnections, we would expect a sawtooth shape. Instead, what is observed is a rather triangular shape of the peaks. What one would expect is a current input, and therefore a sudden decrease of the magnetic field (a decrease corresponds to an increase in the toroidal current), and then a slow dissipation of the current in the torus, which would correspond to a slow and gradual rise of the magnetic field. Instead, the signal shows the abrupt decrease and raising of the field. This means that the current is not only thrown into the torus, but also out of the torus. Moreover, peaks in the temperature always correspond to an

Figure 4.17: Projection of the temperature peaks greater than 25 eV (red dots) onto the Hall-magnetic ﬁeld signal. increase in brightness in the areas where the X-points fall. Often, peaks are also accompanied by deformations always in the areas of the X-points, which persist for a few frames (two-three frames). By studying these deformations on the six cameras, it was possible to deduce the plasma direction of rotation. The deformation seems to be contained in 60 azimuth degrees, since only four of the six cameras can see it: two consecutive cameras and the two diametrically opposite ones (ﬁgure 4.19).

79 4.5. Cross Correlations analysis

Figure 4.18: From the top: density, temperature and B-Hall signals from shot 2139. The horizontal red line shows the average temperature of the plasma. The current temperature peak corresponds to a higher brightness seen from camera 057 (top) and 237 (bottom), which are diametrically opposite.

Figure 4.19: On the left a mosaic of the six Fast Cameras is showed. The deformation in this case concerns an upper X-point on the left side, and it is seen by cams 117, 177 and the two opposite 297, 357, while the Cams 057 and 237 seem to see an axisymmetric plasma.

80 4.6. Plasma phenomenology

4.6 Plasma phenomenology

By combining all the information obtained from the data analysis, it is possible to derive a toy model of the plasma produced by PROTO-SPHERA. For what concernes the geometry and dynamics, we can say that plasma is not perfectly axisymmetrical, and appears to have a more oblate shape than a sphere. The plasma motions are mainly two: one of rotation around its own axis of symmetry (vertical axis, passing through the center of the plasma column) and one of rigid rotation, consisting of a rotation of the plasma center around the axis of symmetry. This picture could explain well the curves observed in the correlation plots of the previous section. The plasma appears to breathe, expanding and tightening systematically. This could be explained by the reconnection phenomena: first the plasma appears larger, less dense and poorly confined, and the X points are more distant from each other; then the reconnections confine the plasma to the torus, which becomes more visible again, and the plasma expands on the equatorial plane and tightens at the X points (figure 4.20). The deformations that are observed near the X points (high or low) seem

Figure 4.20: Toy model of PROTO-SPHERA: the evolution of the plasma is depicted; the plasma expands and tightens in response to reconnection events. to persist for a few milliseconds, and always involve only four of the six cameras. These deformations are accompanied by an increase in brightness in the same area and temperature peaks. Having ascertained that these peaks do not correspond to a malfunction of the acquisition system, we can inter-

81 4.6. Plasma phenomenology pret them in the following way: they correspond to the arrival on the probe of highly energetic and non-thermal particles deriving from magnetic reconnections. Therefore, if one could observe the velocity distribution of the plasma particles, one would expect a Maxwellian distribution corresponding to a temperature of about 20 eV with deformation at high energies. How- ever, for now we still do not have enough information to deduce the position and magnitude of the deformation in the Maxwellian distribution. These peaks also correspond to abrupt changes in the magnetic field. When the field decreases, that is, it becomes “more negative”, there has been a current input into the torus (current coming from the magnetic reconnections) equal to about 20% of the total toroidal current, that is an increase of ≈ 1kA. When the field rises, i.e. becomes “less negative”, the current has been thrown away from the torus. Therefore it seems that reconnection episodes throw current both in and out of the torus: as happens in phase changes or chemical reactions, in which there is always an exchange between the two sides of the reaction (reactants and products), there is a kind of dynamic equilibrium between the two processes of current input and output. This can be seen in figure 4.21: the parts of the figure colored in green correspond to current inputs, while the ones colored in red correspond to current outputs. The two colors are almost balanced, whit a little predom- inance of the input events. Another thing that can be noted is that the durations of the two phenomena are similar: there are both slow and fast current increase, and the same is true for current exit events. From the oscillations of quantities, such as magnetic field and density, an estimate of the duration of what we are interpreting as magnetic reconnection can be made: from figures it is clear that the period of the oscillations is about a few milliseconds. Let’s try to see if such a rate is consistent with the reconnection patterns described in the first chapter. From the physical quantities measured the PROTO-SPHERA plasma is more compatible with a collisional model. The estimated reconnection rate from the Sweet-Parker model is given by √ L τSP = S (4.11) τA where S is the Lundquist number, L is a characteristic length of the system and τA is the Alfven time. The Lundquist number is given as the ratio between two different time scales: the Alfven time and the resistive time. Alfven time is given by L τA = (4.12) vA

82 4.6. Plasma phenomenology

Figure 4.21: Top: density signal. Bottom: Temperature and B-Hall signals. The red zones represent events in which current has been ejected from the torus; the green ones events in which current entered in the torus. Courtesy of Franco Alladio.

83 4.6. Plasma phenomenology

Therefore, with B ≈ 80 Gauss, which is B ≈ 0.008 Tesla, n ≈ 1019/m3 and −27 mi ≈ 6, 6464 × 10 kg, the Alfven velocity is approximately cm m v ≈ 106 ≈ 104 A s s which corresponds to the Alfven time

−4 τA ≈ 10 s = 0, 1ms

The resistive time calculated for PROTO-SPHERA with a temperature of 100 eV and a higher diameter is about 78 ms. Since the resistive time scales 3/2 as Te , with a temperature of 30 eV and a length of 20 cm the resistive time is reduced by a factor of 20,

τR ≈ 3ms.

Therefore, the Lundquist number for PROTO-SPHERA is

S = 30 and the Sweet-Parker reconnection time is

−4 τSP ≈ 5 · 10 s = 0, 5ms.

Therefore, although the Sweet Parker time usually overestimates the reconnection times, in this case it is much smaller than the observed oscillation periods, which are of the order of 10 ms.

84 Conclusions

Magnetic reconnections appear to be the key to many particle acceleration mechanisms in Astrophysics. Despite the progress made in the last fifty years, these phenomena are still not fully understood. Laboratory plasmas can be used to thoroughly exploit the process of magnetic reconnections: in this context, reconnections are the mechanisms underlying the magnetic confinement and relaxation phenomena inside plasma confining machines such as tokamak and spheromak. In particular, in the case of the PROTO-SPHERA experiment, magnetic confinement and sus- taiment of the torus are direct consequences of magnetic reconnections. In this thesis work, a description and a preliminary analysis of the plasma produced by PROTO-SPHERA has been provided. From this analysis it results that the plasma, not completely axial symmetrical, exhibits dynamic behaviors guided in part by the magnetic reconnections: from the correlation graphs and from the study of the images of the fast cameras it appears that reconnection cause the plasma to expand and tighten in a systematic way. From the study of magnetic field, density and temperature signals it can be deduced that this sort of breathing is accompanied by phenomena of incoming and outgoing current from the spherical torus, which remains for the entire duration of the plasma discharge. Furthermore, putting together the temperature signals and images it can be seen that luminosity and temperature peaks correspond to localized deformations around the X-shaped points present in a stable manner in the magnetic configuration of the experiment itself. These deformations have an azimuthal extension of about 60 degrees, seem to persist for a maximum of 3 frames and rotate rigidly together with the plasma counterclockwise. However, the observed times do not seem to be compatible with the collisional and collisionless reconnection models considered in the following thesis. This may depend on the fact that the diagnostics in use have a sam- pling frequency that does not allow direct observation of the reconnection episodes, but only the effects of these episodes on the plasma. The long periods observed could, however, be linked to the intrinsic nature of the reconnections, which show a decidedly three-dimensional phenomenology. This analysis will be able to be deepened in the future with the use of further diagnostics such as X-ray detectors which will be able to confirm

85 4.6. Plasma phenomenology the presence of non-thermal particles coming from the reconnection mechanisms. Moreover, having available measurements of the temperature and density proﬁle in the most central part of the torus, a more accurate and quantitative model could be proposed.

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89 Ringraziamenti

Scrivere i ringraziamenti èper me sempre la parte piùdifficile della tesi, soprattutto perchéèl’unica che leggeranno i miei parenti.

Vorrei ringraziare prima di tutto il Professore Marco Tavani, che mi ha ac- compagnato in questo percorso di tesi trasmettendomi l’amore per la ricerca. I suoi insegnamenti e la sua curiosit`asono stati una guida per me, e lo saranno per molti anni ancora.

Un ringraziamento particolare va a Franco Alladio, Paolo Micozzi, Paolo Buratti, Luca Boncagni, Matteo Iafrati e a tutte le persone con cui ho avuto modo di lavorare all’interno del Centro di Ricerche Enea di Frascati. In questi mesi mi avete dato tutto il supporto di cui avevo bisogno, e grazie a voi ho imparato tantissimo.

A proposito dei miei parenti, ringrazio infinitamente i miei genitori. Grazie, dal profondo del mio cuore, perchémi avete permesso di realizzare un sogno, mi avete dato la forza e la grinta tutte le volte che ne avevo bisogno. Senza il vostro supporto niente di tutto ciòsarebbe stato possibile, questo giorno èsoprattutto vostro.

Un grazie speciale a Nadir, che si ètrovato a vivere (anzi, a rivivere) la parte piùdifficile e stressante di questo percorso. Mi hai aiutato in tutti i modi in cui potevi aiutarmi, e per questo non ti ringrazieròmai abbastanza. Senza di te questi mesi sarebbero stati almeno tre volte piùpesanti.

Un grande grazie alla mia buﬀa e variopinta famiglia, perch´eovunque io sia, mi fa sempre venire voglia di tornare a casa.

Ai miei amici/colleghi/compagni di avventure e di sventure, vecchi e meno vecchi (anche quelli che conosco da meno di 8 anni) vorrei dire che se c’èuna cosa che ho capito in questi anni di Magistrale, dopo dieci mesi passati in Germania e dopo piùdi un anno di pandemia, èche il legame che ho stretto con ognuno di loro èsolido e forte, e sono sicura che continueràad essere tale nel tempo e nello spazio, ovunque le nostre strade ci porteranno. Infine, questa volta un piccolo grazie lo voglio dire anche a me, per aver trovato sempre il coraggio di mettermi alla prova, misurandomi spesso con cose piùgrandi di me. Ho capito molto in questi anni, ho capito chi sono e dove voglio arrivare. Ma soprattutto ho capito quali sono i miei limiti, ora non mi resta che cercare di superarli.

Roma, 28 Maggio 2021