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

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

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 will be introduced.

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

$$B\_{\rm li} = B\_{\rm z}(1 - q) \tag{53}$$

as

$$\Psi\_{l}(r) = 2\pi \int\_{r}^{r\_{\text{ma}}} B\_{\text{z}}(\mathbf{x}, z\_{\text{ma}}) (1 - q)\mathbf{x} d\mathbf{x} \tag{54}$$

Several aspects of this model are in close agreement with the evolution of the marginally unstable case shown in Fig. 8. First of all, the fact that the reconnection process is restricted to the core, i.e. the region *q* 1, and does not affect the whole configuration (as assumed by relaxation theory). Secondly, in Fig. 8 we effectively see that the small island formed at *r*<sup>1</sup> (*q* = 1) moves until it occupies the position of the magnetic axis. Thirdly, in our simulation we

Dynamics of Magnetic Relaxation in Spheromaks 113

this means that *Bh* after reconnection does not change its sign. This means in turn that *q* does not cross 1 (in fact *q* is equal to 1 at *r* = 0). The absence of *q* = 1 surfaces prevents the appearance of magnetic islands just after the reconnection and thus it is said that the Kadomtsev's model predicts complete reconnection. Accordingly, we do not observe any island (other that the magnetic axis) after the reconnection (see Fig. 8) and the resulting *q* profile does not cross 1, as observed in Fig. 18 (a). Despite this agreement in the overall

Fig. 18. (a) Initial (solid) and reconnected, i.e. at *t* = 100, (dashed) *q* and toroidal field profiles.

behavior we will show that the results of the *α* = −0.4 case are better described by introducing

A better approximation can be obtained by looking at Fig. 8 more carefully and noting that the reconnection process takes place *inside* the *q* = 1 surface. Since little or no effect is observed outside *r*<sup>1</sup> we propose a modified procedure for the construction of the reconnected helical

*<sup>h</sup>* . Again, the flux surface at *<sup>r</sup>*<sup>1</sup> is reconnected with the magnetic axis so *<sup>ψ</sup>*<sup>m</sup>

surface at *r*<sup>0</sup> (c) with the surface at *r*<sup>f</sup> (d). The shaded regions have the same area.

a modification to the Kadomtsev's model. In Fig. 18 (b) the initial helical flux *ψ*<sup>0</sup>

*<sup>h</sup>* predicted by Kadomtsev are compared with the actual final helical flux *<sup>ψ</sup>*<sup>f</sup>

obtained at *t* = 100 for the *α* = −0.4 case. Note that the agreement is not good.

*<sup>h</sup>*), final (*ψ*<sup>f</sup>

(b) Several helical fluxes profiles: initial (*ψ*<sup>0</sup>

predicted by the modified model (*ψ*<sup>m</sup>

due to resistive decay (*ψ<sup>η</sup>*

*ψ*K

flux *ψ*<sup>m</sup>

*<sup>h</sup>* is monotonic

r 1

r 1

*h* ),

*<sup>h</sup>* and the final

*<sup>h</sup>* (the red curve)

*<sup>h</sup>* (0) = *<sup>ψ</sup>*<sup>0</sup>

*<sup>h</sup>*(*r*1).

r f r 0

(c) Before reconnection

r η

(d) After reconnection

*<sup>h</sup>* ) and predicted by the modified model with correction

*<sup>h</sup>* ). The modified model proposed involves the reconnection of the

*<sup>h</sup>*), predicted by Kadomtsev (*ψ*<sup>K</sup>

also observe what is called a complete reconnection process. Note that since *ψ*<sup>K</sup>

where (*r*ma, *z*ma) is the position of the magnetic axis and this definition is to be used with *r* ≤ *r*ma. In what follows we will use the minor radius *r*˜ = *r*ma − *r* as the abscissa. In order to

Fig. 17. (a) Initial *q* and poloidal field (*Bz*) profiles. (b) Initial (*ψ*<sup>0</sup> *<sup>h</sup>*) and final (*ψ*<sup>K</sup> *<sup>h</sup>* ) helical flux predicted by Kadomtsev's model. *ψ*<sup>K</sup> *<sup>h</sup>* is obtained by assuming that the area enclosed by the two reconnecting surfaces before reconnection (c) is equal to the area inside the final reconnected surface (d).

not overload the notation we will drop the tilde. Fig. 17 (a) shows *q* and *Bz* as a function of the minor radius. Note that *Bh* changes its sign at *r*1, where *q* = 1, producing a minimum in *ψ*0 *<sup>h</sup>* as shown in Fig. 17 (b). The Kadomtsev's model model provides a simple way to compute the helical flux after reconnection *ψ*<sup>K</sup> *<sup>h</sup>* (see Fig. 17 (b)) from which one can readily obtain the reconnected poloidal field profile.

The reconnection begins at the minimum value of *ψ*<sup>0</sup> *<sup>h</sup>*, i.e. at *<sup>ψ</sup>*<sup>0</sup> *<sup>h</sup>*(*r*1). It is assumed that this flux surface will form the new centre of the plasma and thus *ψ*<sup>K</sup> *<sup>h</sup>* (0) = *<sup>ψ</sup>*<sup>0</sup> *<sup>h</sup>*(*r*1). The reconnection then proceeds merging each pair of flux surfaces having the same *ψ<sup>h</sup>* value. In the particular example of Fig. 17, the flux surfaces initially located at *r*in and *r*out will reconnect forming a new flux surface at *r*f. The position of the final surface *r*<sup>f</sup> is given by toroidal flux conservation. Assuming that the toroidal field does not change during the process, the area enclosed by the two initial surfaces should be equal to the area inside the final surface (see Fig. 17 (c) and (d)). This means that

$$r\_{\rm f}^2 = r\_{\rm out}^2 - r\_{\rm in}^2 \tag{55}$$

where we have simplified the problem by considering flux surfaces with circular cross section. The reconnection process ends at *ψ<sup>h</sup>* = 0 so that the flux surfaces located outside *r*<sup>K</sup> remain unaffected.

28 Will-be-set-by-IN-TECH

where (*r*ma, *z*ma) is the position of the magnetic axis and this definition is to be used with *r* ≤ *r*ma. In what follows we will use the minor radius *r*˜ = *r*ma − *r* as the abscissa. In order to

Fig. 17. (a) Initial *q* and poloidal field (*Bz*) profiles. (b) Initial (*ψ*<sup>0</sup>

two reconnecting surfaces before reconnection (c) is equal to the area inside the final

not overload the notation we will drop the tilde. Fig. 17 (a) shows *q* and *Bz* as a function of the minor radius. Note that *Bh* changes its sign at *r*1, where *q* = 1, producing a minimum in

*<sup>h</sup>* as shown in Fig. 17 (b). The Kadomtsev's model model provides a simple way to compute

then proceeds merging each pair of flux surfaces having the same *ψ<sup>h</sup>* value. In the particular example of Fig. 17, the flux surfaces initially located at *r*in and *r*out will reconnect forming a new flux surface at *r*f. The position of the final surface *r*<sup>f</sup> is given by toroidal flux conservation. Assuming that the toroidal field does not change during the process, the area enclosed by the two initial surfaces should be equal to the area inside the final surface (see Fig. 17 (c) and (d)).

where we have simplified the problem by considering flux surfaces with circular cross section. The reconnection process ends at *ψ<sup>h</sup>* = 0 so that the flux surfaces located outside *r*<sup>K</sup> remain

*r* 2 <sup>f</sup> = *r* 2 out − *r* 2

predicted by Kadomtsev's model. *ψ*<sup>K</sup>

the helical flux after reconnection *ψ*<sup>K</sup>

The reconnection begins at the minimum value of *ψ*<sup>0</sup>

surface will form the new centre of the plasma and thus *ψ*<sup>K</sup>

reconnected poloidal field profile.

reconnected surface (d).

This means that

unaffected.

*ψ*0

r <sup>1</sup> r K

r in

(c) Before reconnection

(d) After reconnection <sup>r</sup>

*<sup>h</sup>*) and final (*ψ*<sup>K</sup>

*<sup>h</sup>*(*r*1). It is assumed that this flux

in (55)

*<sup>h</sup>*(*r*1). The reconnection

*<sup>h</sup>* is obtained by assuming that the area enclosed by the

*<sup>h</sup>* (see Fig. 17 (b)) from which one can readily obtain the

*<sup>h</sup>* (0) = *<sup>ψ</sup>*<sup>0</sup>

*<sup>h</sup>*, i.e. at *<sup>ψ</sup>*<sup>0</sup>

r out

r f

1 r K

*<sup>h</sup>* ) helical flux

Several aspects of this model are in close agreement with the evolution of the marginally unstable case shown in Fig. 8. First of all, the fact that the reconnection process is restricted to the core, i.e. the region *q* 1, and does not affect the whole configuration (as assumed by relaxation theory). Secondly, in Fig. 8 we effectively see that the small island formed at *r*<sup>1</sup> (*q* = 1) moves until it occupies the position of the magnetic axis. Thirdly, in our simulation we also observe what is called a complete reconnection process. Note that since *ψ*<sup>K</sup> *<sup>h</sup>* is monotonic this means that *Bh* after reconnection does not change its sign. This means in turn that *q* does not cross 1 (in fact *q* is equal to 1 at *r* = 0). The absence of *q* = 1 surfaces prevents the appearance of magnetic islands just after the reconnection and thus it is said that the Kadomtsev's model predicts complete reconnection. Accordingly, we do not observe any island (other that the magnetic axis) after the reconnection (see Fig. 8) and the resulting *q* profile does not cross 1, as observed in Fig. 18 (a). Despite this agreement in the overall

Fig. 18. (a) Initial (solid) and reconnected, i.e. at *t* = 100, (dashed) *q* and toroidal field profiles. (b) Several helical fluxes profiles: initial (*ψ*<sup>0</sup> *<sup>h</sup>*), final (*ψ*<sup>f</sup> *<sup>h</sup>*), predicted by Kadomtsev (*ψ*<sup>K</sup> *h* ), predicted by the modified model (*ψ*<sup>m</sup> *<sup>h</sup>* ) and predicted by the modified model with correction due to resistive decay (*ψ<sup>η</sup> <sup>h</sup>* ). The modified model proposed involves the reconnection of the surface at *r*<sup>0</sup> (c) with the surface at *r*<sup>f</sup> (d). The shaded regions have the same area.

behavior we will show that the results of the *α* = −0.4 case are better described by introducing a modification to the Kadomtsev's model. In Fig. 18 (b) the initial helical flux *ψ*<sup>0</sup> *<sup>h</sup>* and the final *ψ*K *<sup>h</sup>* predicted by Kadomtsev are compared with the actual final helical flux *<sup>ψ</sup>*<sup>f</sup> *<sup>h</sup>* (the red curve) obtained at *t* = 100 for the *α* = −0.4 case. Note that the agreement is not good.

A better approximation can be obtained by looking at Fig. 8 more carefully and noting that the reconnection process takes place *inside* the *q* = 1 surface. Since little or no effect is observed outside *r*<sup>1</sup> we propose a modified procedure for the construction of the reconnected helical flux *ψ*<sup>m</sup> *<sup>h</sup>* . Again, the flux surface at *<sup>r</sup>*<sup>1</sup> is reconnected with the magnetic axis so *<sup>ψ</sup>*<sup>m</sup> *<sup>h</sup>* (0) = *<sup>ψ</sup>*<sup>0</sup> *<sup>h</sup>*(*r*1).

the relaxation theory as formulated by Taylor (1974) is applicable to highly unstable plasmas but it becomes useless to study the operation of configurations near an instability threshold. The kink instability produces the helical deformation of the flux surfaces near the magnetic axis. This drives the reconnection of the inner flux surfaces with the outer ones. This process has been studied in detail. The reconnection layer has been identified as well as the new structure resulting from the reconnection of the two flux tubes. Taking the low (high) *λ* value of the inner (outer) tube on its inner (outer) side, these crescent shaped structures average the *λ* value inside the *q* = 1 surface. Even when the flux surfaces remain regular during this evolution, the process involves the full reconnection of all the magnetic tubes inside the *q* = 1 surface. This is of course undesired from the point of view of confinement and could partially explain the poor performance of spheromak operation (compared to tokamaks and RFP's). However further studies are required on this topic regarding the coupled dynamics between the kink and the external driving of the system. This could be done by applying appropriate boundary conditions to model the injection of helicity from a source (Garcia-Martinez &

Dynamics of Magnetic Relaxation in Spheromaks 115

Finally, models for the reconnection process driven by the kink mode were discussed. The Kadomtsev's model was presented and showed to give a poor description of the actual simulation results. A modification to this model that greatly improves the agreement with simulations was proposed. A method to incorporate the correction due to the resistive decay

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**7. References**

of the configuration was described.

Cambridge/New York.

*Phys.: Conf. Ser.*, 166: 012010.

Then, the flux surface initially placed at *r*0, see Fig. 18 (c), reconnects and ends at *r*<sup>f</sup> , Fig. 18 (d), in such a way that

$$r\_{\rm f}^2 = r\_1^2 - r\_0^2 \tag{56}$$

which expresses the conservation of the area of the shaded regions of Fig. 18 (c) and (d). With the initial helical flux *ψ*<sup>0</sup> *<sup>h</sup>* and Eq. (56) it is possible to compute the reconnected helical flux predicted by the modified model *ψ*<sup>m</sup> *<sup>h</sup>* . This is shown by the green curve of Fig. 18 (b). While this prediction is much closer than the Kadomtsev's model to the actual final state there is still a significant difference. In what follows we will show that this difference is due to resistive dissipation.

Relations (55) and (56) express the toroidal flux conservation assuming that it does not decay, i.e. the toroidal fluxes inside *r*<sup>K</sup> and *r*<sup>1</sup> do not change. However, as can be observed in Fig. 18 (a), the toroidal magnetic field is visibly reduced due to resistivity. One way to take into account this resistive decay is to change the reference radius with which we make the construction of *ψ*<sup>m</sup> *<sup>h</sup>* given by Eq. (56). In particular, we define *r<sup>η</sup>* as the radius of the circle that contains at *t* = 0 the same amount of toroidal flux that is contained inside *r*<sup>1</sup> at *t* = 100. If we now compute the reconnected helical flux using Eq. (56) but changing *r*<sup>1</sup> by *r<sup>η</sup>* we obtain *ψη <sup>h</sup>* , shown by the dashed line of Fig. 18 (b). The agreement with the actual final helical flux is very good and this suggests that the modified model indeed captures the basic physics of the reconnection process.
