**5.1 Problem description**

The minimum energy state for a given helicity inside a (not very elongated) cylindrical flux conserver, see Fig. 5, is stable against small MHD perturbations. There exists, however, a simple modification of this configuration which is MHD unstable, in particular *kink* unstable. Now we derive the equations that will allow us to compute as well as to better understand these modified configurations.

For simplicity we consider force-free configurations. In general this condition may be expressed as **J** = *λ*(**r**)**B**, where *λ* may be an arbitrary function. However, we will restrict our study to the case in which *λ* is a flux function, that is to say it takes the same value on each flux surface and can only change from one surface to another. This condition is expressed as

$$
\nabla \times \mathbf{B} = \lambda(\psi)\mathbf{B}.\tag{45}
$$

Since we consider axisymmetric configurations, we can express the poloidal magnetic field component (**B***p*) in terms of *ψ* using Eq. (4), while for the toroidal component we have

18 Will-be-set-by-IN-TECH

The relaxation theory as formulated by Taylor (1974) is a variational principle that can not give details on the dynamical aspects of the process. All the considerations we have made regarding the important role of localized reconnection in helicity conservation are only heuristic arguments that try to explain the remarkable success of the theory at predicting the

There are a number of reasons that motivate the study of the dynamics of relaxation. For instance, in the context of spheromak research it is observed that during sustainment the system does not remain at the lowest energy state. Small deviations from the relaxed state as well as the ubiquitous presence of fluctuations are crucial issues that are out of the scope of relaxation theory (Knox et al., 1986);(Willet et al., 1999). In this work we study these aspects using numerical solutions of the non linear resistive MHD equations described in Sec. 3 as an initial and boundary value problem in three spatial dimensions. The nondimensional version described in Sec. 3.5 of these equations is used. The details of the numerical method are not

In this Section we present a study of the dynamics of the kink mode in spheromak configurations. We will focus on the dynamics of systems that are only marginally unstable. Even when this may sound as a rather specific topic we will see that this is a simple setup in which we can study magnetic reconnection and helicity transfer between flux tubes. Firstly, we describe the kink unstable configurations used as initial condition and explain how they can be computed. Secondly, we study the dynamics of the kink instability in several cases and discuss in which cases it leads to a complete relaxation process (as described in the preceding Section) and in which cases the relaxation process is only partial. Thirdly, we introduce the concept of safety factor and resonant surfaces and explain their relevance to the partial relaxation behavior observed in marginally unstable configurations. Then, we analyze in detail the reconnection process that is driven by the dominant kink mode. Finally, we discuss

The minimum energy state for a given helicity inside a (not very elongated) cylindrical flux conserver, see Fig. 5, is stable against small MHD perturbations. There exists, however, a simple modification of this configuration which is MHD unstable, in particular *kink* unstable. Now we derive the equations that will allow us to compute as well as to better understand

For simplicity we consider force-free configurations. In general this condition may be expressed as **J** = *λ*(**r**)**B**, where *λ* may be an arbitrary function. However, we will restrict our study to the case in which *λ* is a flux function, that is to say it takes the same value on each flux surface and can only change from one surface to another. This condition is expressed as

Since we consider axisymmetric configurations, we can express the poloidal magnetic field component (**B***p*) in terms of *ψ* using Eq. (4), while for the toroidal component we have

∇ × **B** = *λ*(*ψ*)**B**. (45)

presented here but can be found elsewhere (Garcia-Martinez & Farengo, 2009b).

**5. Dynamics of magnetic relaxation in spheromak configurations**

self-organized final state of the plasma.

simple models to describe this reconnection process.

**5.1 Problem description**

these modified configurations.

$$J\_z = (\nabla \times \mathbf{B})\_z = \frac{1}{r} \frac{\partial}{\partial r} (rB\_\theta) = \lambda B\_z. \tag{46}$$

Using this, along with Eq. (3) we obtain

$$B\_{\theta} = \frac{1}{2\pi r} \int\_{0}^{r} \lambda B\_{z} \, 2\pi \vec{r} d\vec{r} = \frac{1}{2\pi r} \int\_{0}^{\psi} \lambda (\tilde{\psi}) d\tilde{\psi}.\tag{47}$$

Thus, expressing the magnetic field in terms of *ψ* we can rewrite the toroidal component of Eq. (45) as

$$
\frac{\partial^2 \psi}{\partial r^2} - \frac{1}{r} \frac{\partial \psi}{\partial r} + \frac{\partial^2 \psi}{\partial z^2} + \lambda(\psi) \int\_0^\psi \lambda(\tilde{\psi}) d\tilde{\psi} = 0 \tag{48}
$$

which is the force-free version of the Grad-Shafranov equation. We are interested in solving Eq. (48) in the rectangle Ω : (*r*, *z*)=[0, *a*] × [0, *h*], i.e. a cylinder of radius *a* and height *h*.

The most simple option for *λ*(*ψ*) would be *λ* = 0, which corresponds to the vacuum solution (currentless magnetic field). The solution vanishes in this case if homogeneous boundary conditions (*ψ*|*∂*<sup>Ω</sup> = 0) are applied.

A more interesting case is obtained by setting *λ* = *λ<sup>n</sup>* (constant) which gives

$$-\Delta^\* \psi = \lambda\_n^2 \psi \tag{49}$$

where we have introduced the Grad-Shafranov operator defined as <sup>Δ</sup><sup>∗</sup> <sup>=</sup> *<sup>∂</sup>*2/*∂r*<sup>2</sup> <sup>−</sup> (1/*r*)*∂*/*∂r* + *∂*2/*∂z*2. If we impose homogeneous boundary conditions we obtain an eigenvalue problem which has non trivial solutions only for a discrete set of real and positive values of *λn*. The lowest value (*λ*1) is given by Eq. (44) and its associated eigenfunction is the minimum energy state described in detail in Sec. 4.3. Thus, if the appropriate boundary conditions are imposed, we can also regard the spheromak as the lowest eigenfunction of the Grad-Shafranov operator.

In this study we will consider initial equilibria having

$$
\lambda(\psi) = \bar{\lambda} \left[ 1 + \alpha \left( 2 \frac{\psi}{\psi\_{ma}} - 1 \right) \right] \tag{50}
$$

which is a linear *λ*(*ψ*) profile with slope *α* and mean value *λ*¯ . When this linear profile is injected in Eq. (48) a generalized non-linear eigenvalue problem is obtained. Some mathematical considerations as well as a basic numerical scheme to solve this problem were given by Kitson & Browning (1990). Note that even if one is able to solve the non-linear Grad-Shafranov equation, the profile given by Eq. (50) includes *ψma* which is not know *a priori*. The procedure adopted here is to set *ψma* = 1, fix the desired value of *α* and iterate over *λ*¯ until *ψ* is equal to one at the magnetic axis. With this procedure we obtain the values of *λ*¯ listed in Table 1. Note that each *α* value uniquely defines a configuration.

In Fig. 6 (a) we show two linear *λ*(*ψ*) profiles and (b) *ψ* contours and the *λ* colormap for the *α* = −0.4 case. The reason why we have chosen negative values for *α* is the following. It is evident that for negative values of the slope the configuration will have larger *λ* values in the outer flux surfaces (at lower *ψ* values) and *vice versa*. Since *λ* is proportional to the current

The instability that arises has dominant toroidal number *n* = 1 (where *n* stands for the number of the coefficient of the Fourier decomposition in the toroidal direction). This current driven *n* = 1 mode is the kink mode. It is well known that the kink mode triggers the relaxation process in spheromaks during sustainment. It has been shown that when the initial unstable configuration has an *α* value close to the stability threshold, the relaxation process is not complete (Garcia-Martinez & Farengo, 2009a);(Garcia-Martinez & Farengo, 2009b). This means that the final state of the evolution is not a minimum energy state. In particular, the *λ* profile is not uniform. This partial relaxation behavior can be observed in Fig. 7. In the

Dynamics of Magnetic Relaxation in Spheromaks 105

(a) (b)

Fig. 7. (a) Toroidal and poloidal magnetic field profiles at *t* = 0 and *t* = 200 (final time). The dashed line shows the fully relaxed profiles. (b) *λ*(*ψ*) profiles at three times for the same *α*

*α* = −0.6 case it is clear that the final state does not have neither the same radial magnetic field profiles than the minimum energy state (shown in dashed lines) nor a uniform *λ* profile. On the other hand, the most unstable case, *α* = −0.8, exhibits a fully relaxed final state.

Fig. 8 shows the evolution of the magnetic field lines during the kink instability. A magnetic island is formed due to the helical distortion of the magnetic axis. This island then moves toward the central position while the flux surfaces originally placed around the magnetic axis are gradually pushed outward. A localized magnetic reconnection layer can be observed in the region where the inner flux surfaces come into contact with the outer flux surfaces. This is indicated in the small box drawn in Fig. 8 (a). After a reconnection process, a system with

The Poincaré maps showing the evolution of the *α* = −0.6 case can be observed in Fig. 9. The overall behavior is analogous to the previously studied case. A magnetic island is formed at an outer position (relative to the magnetic axis) which then moves and occupies

values (Garcia-Martinez & Farengo, 2009b).

axisymmetric nested flux surfaces is recovered (see Fig. 8 (e)).


Table 1. *λ*¯ values for some prescribed *α* values.

Fig. 6. (a) Two *λ*(*ψ*) profiles. (b) *ψ* contours and *λ* colormap for the case with *α* = −0.4. A hollow current profile is obtained for negative *α* values.

density it is said that the configuration has a *hollow* current profile. This is actually the case for spheromak configurations during sustainment.

Real spheromaks have some amount of open magnetic surfaces (i. e. there is some magnetic flux crossing the walls) along which current is driven. This injects magnetic helicity. Then the system relies on magnetic relaxation to drive the current in the inner flux surfaces. In order to sustain this current drive process in (quasi) steady state, some current (or *λ*) gradient is required. In fact, experiments show that sustained spheromaks are better approximated by a force-free state with *α* = −0.3 rather than by the lowest energy state (having *α* = 0) (Knox et al., 1986); (Willet et al., 1999).

#### **5.2 Complete relaxation vs partial relaxation**

Up to this point we know that the minimum energy state is MHD stable and that we can modify the configuration by giving the *λ*(*ψ*) profile a non zero slope. Now we consider the stability of configurations having negative *α* values. A linear MHD stability analysis has determined that there exists a threshold value for the slope at which the system becomes unstable (Knox et al., 1986). Configurations with *λ*(*ψ*) profiles that are steeper than the threshold (lower *α* values) are unstable while configurations with less steep profiles are stable. The value of this threshold (which lies between −0.3 and −0.4 for the geometry used here) was also verified using non-linear simulations of spheromak configurations (Garcia-Martinez & Farengo, 2009b).

20 Will-be-set-by-IN-TECH

*α* 0 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 *λ*¯ 4.95 5.08 5.18 5.32 5.51 5.78 6.23

Fig. 6. (a) Two *λ*(*ψ*) profiles. (b) *ψ* contours and *λ* colormap for the case with *α* = −0.4. A

density it is said that the configuration has a *hollow* current profile. This is actually the case for

Real spheromaks have some amount of open magnetic surfaces (i. e. there is some magnetic flux crossing the walls) along which current is driven. This injects magnetic helicity. Then the system relies on magnetic relaxation to drive the current in the inner flux surfaces. In order to sustain this current drive process in (quasi) steady state, some current (or *λ*) gradient is required. In fact, experiments show that sustained spheromaks are better approximated by a force-free state with *α* = −0.3 rather than by the lowest energy state (having *α* = 0) (Knox et

Up to this point we know that the minimum energy state is MHD stable and that we can modify the configuration by giving the *λ*(*ψ*) profile a non zero slope. Now we consider the stability of configurations having negative *α* values. A linear MHD stability analysis has determined that there exists a threshold value for the slope at which the system becomes unstable (Knox et al., 1986). Configurations with *λ*(*ψ*) profiles that are steeper than the threshold (lower *α* values) are unstable while configurations with less steep profiles are stable. The value of this threshold (which lies between −0.3 and −0.4 for the geometry used here) was also verified using non-linear simulations of spheromak configurations (Garcia-Martinez

Table 1. *λ*¯ values for some prescribed *α* values.

hollow current profile is obtained for negative *α* values.

spheromak configurations during sustainment.

**5.2 Complete relaxation vs partial relaxation**

al., 1986); (Willet et al., 1999).

& Farengo, 2009b).

The instability that arises has dominant toroidal number *n* = 1 (where *n* stands for the number of the coefficient of the Fourier decomposition in the toroidal direction). This current driven *n* = 1 mode is the kink mode. It is well known that the kink mode triggers the relaxation process in spheromaks during sustainment. It has been shown that when the initial unstable configuration has an *α* value close to the stability threshold, the relaxation process is not complete (Garcia-Martinez & Farengo, 2009a);(Garcia-Martinez & Farengo, 2009b). This means that the final state of the evolution is not a minimum energy state. In particular, the *λ* profile is not uniform. This partial relaxation behavior can be observed in Fig. 7. In the

Fig. 7. (a) Toroidal and poloidal magnetic field profiles at *t* = 0 and *t* = 200 (final time). The dashed line shows the fully relaxed profiles. (b) *λ*(*ψ*) profiles at three times for the same *α* values (Garcia-Martinez & Farengo, 2009b).

*α* = −0.6 case it is clear that the final state does not have neither the same radial magnetic field profiles than the minimum energy state (shown in dashed lines) nor a uniform *λ* profile. On the other hand, the most unstable case, *α* = −0.8, exhibits a fully relaxed final state.

Fig. 8 shows the evolution of the magnetic field lines during the kink instability. A magnetic island is formed due to the helical distortion of the magnetic axis. This island then moves toward the central position while the flux surfaces originally placed around the magnetic axis are gradually pushed outward. A localized magnetic reconnection layer can be observed in the region where the inner flux surfaces come into contact with the outer flux surfaces. This is indicated in the small box drawn in Fig. 8 (a). After a reconnection process, a system with axisymmetric nested flux surfaces is recovered (see Fig. 8 (e)).

The Poincaré maps showing the evolution of the *α* = −0.6 case can be observed in Fig. 9. The overall behavior is analogous to the previously studied case. A magnetic island is formed at an outer position (relative to the magnetic axis) which then moves and occupies

In contrast with the marginally unstable case analyzed previously (*α* = −0.4) where the activity was milder, here the larger level of fluctuations causes the field lines to wander through the whole domain. This facilitates the helicity transfer and enable a more effective

Dynamics of Magnetic Relaxation in Spheromaks 107

It is important to keep in mind that these stochastic regions can be produced even by relatively low wave number magnetic fluctuations. In fact, few toroidal Fourier modes with a rather gentle dependence along the poloidal plane are enough to produce the disorder observed in

These observations are in agreement with the discussion presented in Sec. 4.2. As remarked there, a significant amount of small scale MHD activity (fluctuations) leading to the formation of numerous small current sheets is required to obtain the full relaxation behavior. In the marginal unstable case (*α* = −0.4) the dominant kink mode produce a regular evolution in which a single localized current sheet is observed. This is not enough to produce a complete

Fig. 11. *λ* profiles at *t* = 0 (*λ*0) and at *t* = 100 (after reconnection, *λ<sup>f</sup>* ). In this plot the abscissa measures the distance to the magnetic axis. A partial relaxation behavior is evident, since *λ<sup>f</sup>*

As *α* is lowered (*λ* profile is steepened), the kink mode becomes stronger and activates higher order modes. Only when a significant level of activity is induced the Taylor's relaxation theory becomes applicable to obtain a good approximation of the final state of the system. Interestingly, the full relaxation behavior is recovered even for a modest separation of scales

Now we focus on the partial relaxation behavior of the marginally kink unstable configurations where relaxation theory is not applicable. A very useful concept developed in the context of the study of MHD modes (in particular the kink mode) is the safety factor *q*. The safety factor is the number of times a field line on a flux surface goes around toroidally

(Garcia-Martinez & Farengo, 2009a);(Garcia-Martinez & Farengo, 2009b).

relaxation behavior with uniform *λ* in the final state, as it can be observed in Fig. 11.

flattening of the *λ*(*ψ*) profile (as shown in Fig. 7).

is still far away from the eigenvalue *λ*1.

**5.3 Kink onset and resonant surfaces**

Fig. 10 (d).

Fig. 8. Poincaré maps at several times showing the evolution of kink instability for the *α* = −0.4 case. The black contour shows the initial position of the *q* = 1 surface.

the magnetic axis position. However, in this case a large region of stochastic magnetic field lines emerges between the two magnetic o-points and we are no longer able to identify a well defined localized reconnection layer.

The situation is even more drastic in the case with *α* = −0.7 shown in Fig. 10. Most of the initially regular surfaces are quickly destroyed and large regions of stochastic field lines are observed. Though, a small coherent structure can still be devised even at times of strong activity (the saturation of the instability takes place at *t* = 100). After the instability saturation the toroidal modes decay and new regular nested flux surfaces are formed (*t* = 200).

Fig. 9. Poincaré maps showing the evolution of the kink instability in the *α* = −0.6 case.

Fig. 10. Poincaré maps showing the evolution of the kink instability in the *α* = −0.7 case.

22 Will-be-set-by-IN-TECH

Fig. 8. Poincaré maps at several times showing the evolution of kink instability for the *α* = −0.4 case. The black contour shows the initial position of the *q* = 1 surface.

defined localized reconnection layer.

the magnetic axis position. However, in this case a large region of stochastic magnetic field lines emerges between the two magnetic o-points and we are no longer able to identify a well

The situation is even more drastic in the case with *α* = −0.7 shown in Fig. 10. Most of the initially regular surfaces are quickly destroyed and large regions of stochastic field lines are observed. Though, a small coherent structure can still be devised even at times of strong activity (the saturation of the instability takes place at *t* = 100). After the instability saturation

the toroidal modes decay and new regular nested flux surfaces are formed (*t* = 200).

Fig. 9. Poincaré maps showing the evolution of the kink instability in the *α* = −0.6 case.

Fig. 10. Poincaré maps showing the evolution of the kink instability in the *α* = −0.7 case.

In contrast with the marginally unstable case analyzed previously (*α* = −0.4) where the activity was milder, here the larger level of fluctuations causes the field lines to wander through the whole domain. This facilitates the helicity transfer and enable a more effective flattening of the *λ*(*ψ*) profile (as shown in Fig. 7).

It is important to keep in mind that these stochastic regions can be produced even by relatively low wave number magnetic fluctuations. In fact, few toroidal Fourier modes with a rather gentle dependence along the poloidal plane are enough to produce the disorder observed in Fig. 10 (d).

These observations are in agreement with the discussion presented in Sec. 4.2. As remarked there, a significant amount of small scale MHD activity (fluctuations) leading to the formation of numerous small current sheets is required to obtain the full relaxation behavior. In the marginal unstable case (*α* = −0.4) the dominant kink mode produce a regular evolution in which a single localized current sheet is observed. This is not enough to produce a complete relaxation behavior with uniform *λ* in the final state, as it can be observed in Fig. 11.

Fig. 11. *λ* profiles at *t* = 0 (*λ*0) and at *t* = 100 (after reconnection, *λ<sup>f</sup>* ). In this plot the abscissa measures the distance to the magnetic axis. A partial relaxation behavior is evident, since *λ<sup>f</sup>* is still far away from the eigenvalue *λ*1.

As *α* is lowered (*λ* profile is steepened), the kink mode becomes stronger and activates higher order modes. Only when a significant level of activity is induced the Taylor's relaxation theory becomes applicable to obtain a good approximation of the final state of the system. Interestingly, the full relaxation behavior is recovered even for a modest separation of scales (Garcia-Martinez & Farengo, 2009a);(Garcia-Martinez & Farengo, 2009b).

#### **5.3 Kink onset and resonant surfaces**

Now we focus on the partial relaxation behavior of the marginally kink unstable configurations where relaxation theory is not applicable. A very useful concept developed in the context of the study of MHD modes (in particular the kink mode) is the safety factor *q*. The safety factor is the number of times a field line on a flux surface goes around toroidally

magnetic reconnection of flux surfaces having different *λ* values, as confirmed in Fig. 11. The

Dynamics of Magnetic Relaxation in Spheromaks 109

Here we describe the magnetic reconnection process that redistributes currents in the case with *α* = −0.4. Consider the Poincaré map inside the box shown in Fig. 8 (a). This is

Fig. 13. (a) Poincaré map inside the box shown in Fig.8 (a). Two points, one red and one blue, are manually selected. (b) The same Poincaré plot including two additional magnetic lines followed from the selected points. (c) *n* = 1 component of the poloidal velocity (black

shown in Fig. 13 (a). A reconnection layer is clearly identified, in the middle of which we have drawn a point (in red). We follow the magnetic field line that passes through this point for a long distance (ten thousand times the cylinder radius). The results for this single line are shown in Fig. 13 (b) and (c) (the red points) and in Fig. 14. The flow induced by the instability, shown with vectors in Fig. 13 (c), produces the helical distortion of the central flux surfaces. Eventually, one (or more) of these surfaces gets in contact with an outer surface. This is clearly observed in Fig. 14 where a single field line spans both surfaces. Note that the inner surface has a lower *λ* value than the outer one. At the helical reconnection layer *λ* adopts an

As a result of this reconnection a new magnetic structure is formed. This structure has a crescent shape cross section as shown by the blue dots in Fig. 13 (b) and (c). This is basically the closed surface that encloses the volume between the two reconnecting toroidal flux surfaces. Fig. 15 shows another visualization of this new magnetic entity. It has been constructed by following the magnetic field line that passes through the blue point indicated in Fig. 13 (a). It is interesting to note that this surface has a lower *λ* value in its inner face

magnetic reconnection process is further studied in the next Section.

**5.4 Magnetic reconnection process**

vectors) and the two field lines also shown in (b).

intermediate value.

for a single poloidal turn. Based on the equation for a field line

$$\frac{r d\theta}{ds} = \frac{B\_{\theta}}{B\_{p}}\tag{51}$$

where *ds* is the distance in the poloidal direction while moving a toroidal angle *dθ*, the safety factor can be defined as

$$q = \frac{1}{2\pi} \oint \frac{1}{r} \frac{B\_\theta}{B\_p} ds \tag{52}$$

where the integral is taken over a single poloidal circuit. Note that *q* adopts the same value for every field line lying on the same flux surface and thus it is a flux function *q* = *q*(*ψ*). In

Fig. 12. (a) Safety factor profiles for several configurations. Note that configurations having a *q* = 1 surface are unstable. (b) Poincaré map showing ten field lines during the instability onset in the *α* = −0.4 case. The dashed line shows the *q* = 1 surface, where the formation of a magnetic island is observed.

Fig. 12 (a) the *q* profiles for several configurations are shown. We already mentioned that the kink instability threshold lies between *α* = −0.3 and *α* = −0.4. In Fig. 12 (a) we can see that the kink instability is associated to the appearance of a rational surface with *q* = 1. Rational surfaces are those where *q* = *m*/*n* being *m* and *n* integer numbers and thus *q* has a rational value. The field lines lying in such surfaces can not span a closed toroidal surface and are particularly prone to develop different MHD modes. That is why these surfaces are also called resonant surfaces. In Fig. 12 (b) we clearly see that it is at the *q* = 1 surface where the first modification to the flux surfaces occurs. This crescent shaped structure (which shows the onset of the island observed in Fig. 8) has a *n* = 1 toroidal dependence.

Note that, in the *α* = −0.4 case, all the relevant MHD activity triggered by the kink takes place inside the *q* = 1 surface of the initial condition (Fig. 8). Thus, we can not expect this evolution to cause a complete relaxation process. However, some partial relaxation occurs due to the magnetic reconnection of flux surfaces having different *λ* values, as confirmed in Fig. 11. The magnetic reconnection process is further studied in the next Section.

#### **5.4 Magnetic reconnection process**

24 Will-be-set-by-IN-TECH

where *ds* is the distance in the poloidal direction while moving a toroidal angle *dθ*, the safety

where the integral is taken over a single poloidal circuit. Note that *q* adopts the same value for every field line lying on the same flux surface and thus it is a flux function *q* = *q*(*ψ*). In

Fig. 12. (a) Safety factor profiles for several configurations. Note that configurations having a *q* = 1 surface are unstable. (b) Poincaré map showing ten field lines during the instability onset in the *α* = −0.4 case. The dashed line shows the *q* = 1 surface, where the formation of a

Fig. 12 (a) the *q* profiles for several configurations are shown. We already mentioned that the kink instability threshold lies between *α* = −0.3 and *α* = −0.4. In Fig. 12 (a) we can see that the kink instability is associated to the appearance of a rational surface with *q* = 1. Rational surfaces are those where *q* = *m*/*n* being *m* and *n* integer numbers and thus *q* has a rational value. The field lines lying in such surfaces can not span a closed toroidal surface and are particularly prone to develop different MHD modes. That is why these surfaces are also called resonant surfaces. In Fig. 12 (b) we clearly see that it is at the *q* = 1 surface where the first modification to the flux surfaces occurs. This crescent shaped structure (which shows the

Note that, in the *α* = −0.4 case, all the relevant MHD activity triggered by the kink takes place inside the *q* = 1 surface of the initial condition (Fig. 8). Thus, we can not expect this evolution to cause a complete relaxation process. However, some partial relaxation occurs due to the

onset of the island observed in Fig. 8) has a *n* = 1 toroidal dependence.

 1 *r Bθ Bp* (51)

*ds* (52)

*rdθ ds* <sup>=</sup> *<sup>B</sup><sup>θ</sup> Bp*

*<sup>q</sup>* <sup>=</sup> <sup>1</sup> 2*π*

for a single poloidal turn. Based on the equation for a field line

factor can be defined as

magnetic island is observed.

Here we describe the magnetic reconnection process that redistributes currents in the case with *α* = −0.4. Consider the Poincaré map inside the box shown in Fig. 8 (a). This is

Fig. 13. (a) Poincaré map inside the box shown in Fig.8 (a). Two points, one red and one blue, are manually selected. (b) The same Poincaré plot including two additional magnetic lines followed from the selected points. (c) *n* = 1 component of the poloidal velocity (black vectors) and the two field lines also shown in (b).

shown in Fig. 13 (a). A reconnection layer is clearly identified, in the middle of which we have drawn a point (in red). We follow the magnetic field line that passes through this point for a long distance (ten thousand times the cylinder radius). The results for this single line are shown in Fig. 13 (b) and (c) (the red points) and in Fig. 14. The flow induced by the instability, shown with vectors in Fig. 13 (c), produces the helical distortion of the central flux surfaces. Eventually, one (or more) of these surfaces gets in contact with an outer surface. This is clearly observed in Fig. 14 where a single field line spans both surfaces. Note that the inner surface has a lower *λ* value than the outer one. At the helical reconnection layer *λ* adopts an intermediate value.

As a result of this reconnection a new magnetic structure is formed. This structure has a crescent shape cross section as shown by the blue dots in Fig. 13 (b) and (c). This is basically the closed surface that encloses the volume between the two reconnecting toroidal flux surfaces. Fig. 15 shows another visualization of this new magnetic entity. It has been constructed by following the magnetic field line that passes through the blue point indicated in Fig. 13 (a). It is interesting to note that this surface has a lower *λ* value in its inner face

Fig. 16. Inner reconnecting magnetic flux surface. A zoom near the zone of higher *λ* value

has a mainly helical structure, however, a zoom around the region with the highest *λ* values (shown in red) reveals the presence of higher toroidal components. It is not clear, at this point, if higher harmonics play an important role or this process could be recovered considering a

Dynamics of Magnetic Relaxation in Spheromaks 111

The magnetic reconnection process described so far leads to a flux rearrangement in the region where *q* > 1. This process involves a rather regular evolution of the magnetic surfaces with only one helical current sheet. Without a significant level of MHD activity the magnetic relaxation theory becomes inapplicable. Now we seek for a simple but adequate model to

In the context of tokamak research, the evolution of the resistive kink has been intensively studied. In particular, it is believed that this mode is responsible for a phenomenon called *sawtooth oscillations* that limits in practice the maximum temperature reachable at the core. One of the first models to describe the final state of the non linear resistive kink mode was proposed by Kadomtsev (1975) (see also the explanation of Wesson (2004)). In this Section we describe the Kadomtsev's model and discuss its applicability to the results of our simulations. Then, a modification to the model that significantly improves the agreement with our results

The magnetic field lines on the *q* = 1 surface form a helix around the magnetic axis. The Kadomtsev's model describes the reconnection process in terms of the flux perpendicular to this helix, called helical flux *ψh*. This flux can be computed from the helical magnetic field

> *r*ma *r*

*Bh* = *Bz*(1 − *q*) (53)

*Bz*(*x*, *z*ma)(1 − *q*)*xdx* (54)

reveals the presence of higher toroidal harmonics (*n* > 1).

two dimensional problem with helical symmetry.

will be introduced.

as

**5.5 Reconnection model for the resistive kink mode**

describe the final state of the non-linear evolution of the resistive kink.

*ψh*(*r*) = 2*π*

Fig. 14. A single magnetic field line showing two reconnecting flux surfaces. Its color is proportional to the local *λ* value (the color scale is indicated on right). The outer surface has a higher *λ* than the inner surface. The helical reconnection layer adopts an intermediate value.

Fig. 15. Magnetic structure formed by the reconnection of the flux surfaces shown in Fig. 14. The color scale indicates local *λ* value.

(corresponding to the *λ* value of the original inner flux surface) and a higher *λ* value in its outer face. This clearly shows that the reconnection is a localized process. It is also evident that the mean *λ* value of this structure will lie between the *λ* values of the original surfaces.

With these considerations in mind we can reinterpret Fig. 8. The motion of the island toward the magnetic axis involves the reconnection of inner and outer surfaces having low and high *λ* values, respectively. The new surfaces formed adopt intermediate *λ* values. The result of this redistribution is shown in Fig. 11. Note that all this activity takes place in the region where *ψ* ≥ 0.8 (the region inside the original location of the *q* = 1 surface). In Fig. 6 (a) we see that within this region *λ* 4 and thus we can not expect a full relaxation process.

A final comment is made regarding the symmetry of this process. The kink mode has a *n* = 1 toroidal dependence and thus the reconnection layer shown in Fig. 13 has a dominant helical shape. However, we want to mention that there are also higher harmonics (*n* > 1) present in the reconnection process. This can be observed in Fig. 16 where the inner flux surface of Fig. 14 is shown. The high *λ* region (mainly yellow) shows the reconnection layer. It 26 Will-be-set-by-IN-TECH

Fig. 14. A single magnetic field line showing two reconnecting flux surfaces. Its color is proportional to the local *λ* value (the color scale is indicated on right). The outer surface has a higher *λ* than the inner surface. The helical reconnection layer adopts an intermediate value.

Fig. 15. Magnetic structure formed by the reconnection of the flux surfaces shown in Fig. 14.

(corresponding to the *λ* value of the original inner flux surface) and a higher *λ* value in its outer face. This clearly shows that the reconnection is a localized process. It is also evident that the mean *λ* value of this structure will lie between the *λ* values of the original surfaces. With these considerations in mind we can reinterpret Fig. 8. The motion of the island toward the magnetic axis involves the reconnection of inner and outer surfaces having low and high *λ* values, respectively. The new surfaces formed adopt intermediate *λ* values. The result of this redistribution is shown in Fig. 11. Note that all this activity takes place in the region where *ψ* ≥ 0.8 (the region inside the original location of the *q* = 1 surface). In Fig. 6 (a) we see that

A final comment is made regarding the symmetry of this process. The kink mode has a *n* = 1 toroidal dependence and thus the reconnection layer shown in Fig. 13 has a dominant helical shape. However, we want to mention that there are also higher harmonics (*n* > 1) present in the reconnection process. This can be observed in Fig. 16 where the inner flux surface of Fig. 14 is shown. The high *λ* region (mainly yellow) shows the reconnection layer. It

within this region *λ* 4 and thus we can not expect a full relaxation process.

The color scale indicates local *λ* value.

Fig. 16. Inner reconnecting magnetic flux surface. A zoom near the zone of higher *λ* value reveals the presence of higher toroidal harmonics (*n* > 1).

has a mainly helical structure, however, a zoom around the region with the highest *λ* values (shown in red) reveals the presence of higher toroidal components. It is not clear, at this point, if higher harmonics play an important role or this process could be recovered considering a two dimensional problem with helical symmetry.
