**1. Introduction**

26 Will-be-set-by-IN-TECH

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The first attempts to get energy from the controlled fusion of two light atoms nuclei date back to the beginning of the fifties of the last century. The crucial difficulty to achieve this goal is that particles need to have a large amount of thermal energy in order to have a significant chance of overcoming the Coulomb repulsion. At such high temperatures the atoms are fully ionized conforming a plasma. Such a hot plasma can not be in contact with solid walls because it will be rapidly cooled down. Two main methods have been developed to confine plasmas: the magnetic confinement and the inertial confinement. Here we are concerned with the magnetic confinement approach.

Under certain conditions some magnetic configurations studied in the context of plasma confinement become unstable and undergo a process called magnetic (or plasma) relaxation. This process generally causes the system to evolve toward a self-organized state with lower magnetic energy and almost the same magnetic helicity. A key physical mechanism that operates during plasma relaxation is the localized reconnection of magnetic field lines. It was demonstrated that magnetic relaxation can be employed to form and sustain configurations relevant to magnetic confinement research.

The theoretical description of magnetic relaxation is given in terms of a variational principle (Taylor, 1974). Despite the remarkable success of this theory to describe the final self-organized state toward which the plasma evolves, it does not provide any information on the dynamics of the plasma during relaxation. Since the process of relaxation always involves fluctuations that degrade plasma confinement it is very important to understand their dynamics.

The dynamics of the fluctuations induced during the relaxation process can be studied in the context the magnetohydrodynamic (MHD) model. In this Chapter, we will study the dynamics of the relaxation in kink unstable spheromak configurations. To that end we will solve the time-dependent non-linear MHD equations in three spatial dimensions.

The rest of the Chapter is organized as follows. In Section 2 we give a general introduction to magnetic confinement of high temperature plasma which is the main motivation of this study. The physical background of this work is the MHD model which is presented in Section 3. In Section 4 we describe the magnetic relaxation theory and its relationship with


 

**-**

 **-**

Fig. 1. Three examples of toroidal axisymmetric configurations used in magnetic confinement research: the tokamak, the reversed field pinch (RFP) and the spheromak.

the toroidal current that flows through the plasma. Typically, this current is produced by the electric field induced by the temporal variation of the magnetic flux linked by the torus. • RFP (reversed field pinch). It is also an axisymmetric toroidal device whose aspect ratio (ratio of the major radius and the minor radius of the torus) is generally larger than that of the tokamak. The toroidal and poloidal fields have similar strengths. This makes the system much more prone to develop MHD instabilities. The magnetic field generation is analogous to that of the tokamak but using smaller coils for the toroidal field. The toroidal field reverses (changes its sign) near the separatrix opposing the externally applied field as

• Spheromak. It belongs to the family of compact tori. These are toroidal magnetic configurations formed inside a simply connected volume. The lack of elements being linked by the plasma represents a great advantage from a constructive and economical point of view. The two components of the magnetic field have similar strength. This

Dynamics of Magnetic Relaxation in Spheromaks 87

B [Tesla]

B [Tesla]

B [Tesla]

Mag. axis Separatrix

Mag. axis Separatrix

Mag. axis Separatrix


> --



a result of a magnetic relaxation process.

plasma self-organization. The role of the magnetic helicity and magnetic reconnection is also discussed. In Section 5 we present a study of the dynamics of magnetic relaxation in kink unstable spheromak configurations. These configurations are of special interest because they approximate quite well the measurements in spheromaks during sustainment (Knox et al., 1986);(Willet et al., 1999). Previous works have shown the existence of a partial relaxation behavior in marginally unstable configurations (Garcia-Martinez & Farengo, 2009a); (Garcia-Martinez & Farengo, 2009b). In this work we analyze this process in detail and we show, in particular, that this behaviour is connected to the presence of a rational surface near the magnetic axis. The main conclusions are summarized in Section 6.

### **2. Magnetic confinement of high temperature plasmas**

The charged particles which constitute a high temperature plasma are subjected to the Lorentz force. The objective of magnetic confinement is to create magnetic field configurations to constrain the motion of the particles trying to keep them trapped far from the container's wall. In order to accomplish this goal the following four conditions must be fulfilled:


Here we will discuss some general aspects of the first three points. A more detailed discussion on these topics may be found, for instance, in the book of Wesson (2004).

#### **2.1 MHD equilibrium**

It is said that a magnetic configuration is in static MHD equilibrium if the Lorentz force cancels out exactly the pressure force

$$
\mathbf{J} \times \mathbf{B} = \nabla p.\tag{1}
$$

This force balance is part of the momentum equation of the MHD model that will be presented in Sec. 3. The magnetic configurations employed for plasma confinement almost always have toroidal topology. In this situation, each magnetic field line describes a toroidal magnetic surface. These toroidal magnetic surfaces are nested around a circle called magnetic axis (see Fig. 1). The separatrix is the outermost closed surface that does not touch the vessel. Three axisymmetric toroidal configuration schemes are shown in Fig. 1. It is a common practice to decompose the magnetic field into its toroidal and poloidal components. If we place a cylindrical coordinate system at the center of the torus, aligning the *z*-axis with the axis of symmetry (of revolution) of the torus, the toroidal direction coincides with the azimuthal direction and the poloidal plane lies in the *r*-*z* plane. In the right column of Fig. 1 we show the profiles of the toroidal and poloidal magnetic fields as a function of the distance between the magnetic axis and the separatrix for each configuration. Let's review the main features of these configurations.

• Tokamak. The toroidal magnetic field is much larger than the poloidal one. This intense toroidal field is imposed by a set of large external coils while the poloidal field comes from 2 Will-be-set-by-IN-TECH

plasma self-organization. The role of the magnetic helicity and magnetic reconnection is also discussed. In Section 5 we present a study of the dynamics of magnetic relaxation in kink unstable spheromak configurations. These configurations are of special interest because they approximate quite well the measurements in spheromaks during sustainment (Knox et al., 1986);(Willet et al., 1999). Previous works have shown the existence of a partial relaxation behavior in marginally unstable configurations (Garcia-Martinez & Farengo, 2009a); (Garcia-Martinez & Farengo, 2009b). In this work we analyze this process in detail and we show, in particular, that this behaviour is connected to the presence of a rational surface

The charged particles which constitute a high temperature plasma are subjected to the Lorentz force. The objective of magnetic confinement is to create magnetic field configurations to constrain the motion of the particles trying to keep them trapped far from the container's

2. The configuration must be stable (or it should be possible to mitigate or control potential

4. The losses due to transport of heat and particles must be low enough to allow the system

Here we will discuss some general aspects of the first three points. A more detailed discussion

It is said that a magnetic configuration is in static MHD equilibrium if the Lorentz force cancels

This force balance is part of the momentum equation of the MHD model that will be presented in Sec. 3. The magnetic configurations employed for plasma confinement almost always have toroidal topology. In this situation, each magnetic field line describes a toroidal magnetic surface. These toroidal magnetic surfaces are nested around a circle called magnetic axis (see Fig. 1). The separatrix is the outermost closed surface that does not touch the vessel. Three axisymmetric toroidal configuration schemes are shown in Fig. 1. It is a common practice to decompose the magnetic field into its toroidal and poloidal components. If we place a cylindrical coordinate system at the center of the torus, aligning the *z*-axis with the axis of symmetry (of revolution) of the torus, the toroidal direction coincides with the azimuthal direction and the poloidal plane lies in the *r*-*z* plane. In the right column of Fig. 1 we show the profiles of the toroidal and poloidal magnetic fields as a function of the distance between the magnetic axis and the separatrix for each configuration. Let's review the main features of

• Tokamak. The toroidal magnetic field is much larger than the poloidal one. This intense toroidal field is imposed by a set of large external coils while the poloidal field comes from

**J** × **B** = ∇*p*. (1)

wall. In order to accomplish this goal the following four conditions must be fulfilled:

1. The configuration must be in magnetohydrodynamic (MHD) equilibrium.

3. Methods to produce, heat and sustain the configuration must be available.

on these topics may be found, for instance, in the book of Wesson (2004).

near the magnetic axis. The main conclusions are summarized in Section 6.

**2. Magnetic confinement of high temperature plasmas**

to have an adequate confinement time.

instabilities).

**2.1 MHD equilibrium**

these configurations.

out exactly the pressure force

Fig. 1. Three examples of toroidal axisymmetric configurations used in magnetic confinement research: the tokamak, the reversed field pinch (RFP) and the spheromak.

the toroidal current that flows through the plasma. Typically, this current is produced by the electric field induced by the temporal variation of the magnetic flux linked by the torus.


magnetically confined plasmas can be classified into two groups: the microinstabilities and the macroinstabilities. The first group is responsible for the turbulence at small scales and it is generally related to finite Larmor radius effects (the gyroradius of charged particles turning around the magnetic field) and asymmetries in the velocity distribution function of the different species that compound the plasma. On the other hand, the macroinstabilities involve fluctuations having a length scale comparable to that of the whole system and can, in their simplest version, be described by the MHD model presented in Section 3. Their appearance generally leads to the termination of the discharge and the destruction of the configuration.

Dynamics of Magnetic Relaxation in Spheromaks 89

The usual procedure to study the MHD stability of an equilibrium is based on the analysis of the energy increment *δW* introduced by a small perturbation to the equilibrium (Friedberg, 1987). Using the linearized equations of the MHD model it is possible to compute the growth rate of each perturbation. If all possibles modes decay then the equilibrium is MHD stable. According to the source of energy that feeds the instability, the macroinstabilities can be

• External modes. In this case the energy of the instability comes from the interaction between the plasma and the boundary (the separatrix) or the external magnetic fields. Two typical examples appearing in spheromaks are the shift and the tilt instabilities. The first one consists in the displacement of the configuration as a whole while the second one involves the rigid rotation of the magnetic surfaces. The flux conserver (the chamber of conducting walls inside which the spheromak is formed) plays a crucial role in suppressing these instabilities. For instance, in a cylindrical flux conserver the tilt instability can be avoided if the elongation of the cylinder (ratio between height and radius) is lower than

• Current driven modes. They are activated by non uniform current distributions. The most common example of this kind of instabilities is the kink mode, which may be either an internal (it does not affect the separatrix) or an external mode. In tokamaks this instability is closely related to a phenomenon called sawtooth oscillations that limits in practice the maximum value of toroidal current. In spheromaks and RFP's the kink mode triggers the

• Pressure modes. Pressure gradients combined with an adverse magnetic field line curvature may act as a source of energy to develop instabilites (called ballooning or

In Section 5 we will consider internal kink modes in spheromak configurations. A comprehensive description of the MHD modes relevant to magnetic confinement

Once an MHD equilibrium with good stability properties has been devised it is necessary to find appropriate methods to form and sustain the configuration. The formation methods depend on the configuration under consideration. In fact, a given configuration can be obtained using different formation schemes. In most cases, after the formation process the plasma has a temperature sensibly lower than that required for fusion. Moreover, the resistive

relaxation process that forms and sustains the configuration.

configurations can be found elsewhere (Friedberg, 1987);(Wesson, 2004).

In this Chapter we will deal with this kind of instabilities.

divided in:

1.6.

interchange modes).

**2.3 Formation and sustainment**

configuration is formed as a result of a relaxation process that self-organizes the magnetic field, closely related to that occurring in the RFP.

In all these three systems as well as in other important configurations the magnetic surfaces spanned by the magnetic field lines play a central role in confinement. We examine this in more detail. Let *ψ*(*r*, *z*) be the poloidal flux function defined as

$$
\psi(r, z) = \int\_{S(r, z)} \mathbf{B} \cdot d\mathbf{s} \tag{2}
$$

were *S*(*r*, *z*) is the circle of radius *r* centered at the position *z* of the vertical axis. If the configuration is axisymmetric we can express the poloidal flux function as

$$
\psi(r,z) = 2\pi \int\_0^r B\_z(\chi, z) \, \chi d\chi. \tag{3}
$$

With this definition *ψ* reaches its maximum value at the magnetic axis (*ψ*(*r*ma, *z*ma) = *ψ*ma). The magnetic surfaces, or flux surfaces, can be determined by the equation *ψ*(*r*, *z*) = *C* where *C* is a constant. Note that this useful labeling system for the flux surfaces breaks down when the axisymmetry is lost (due to an instability for example).

The poloidal flux function acts as a stream function for the poloidal field since

$$\mathbf{B}\_p = \nabla \times \left(\frac{\psi(r, z)}{2\pi r} \hat{\theta}\right) \tag{4}$$

where ˆ *θ* is the unit vector pointing in the toroidal direction. Note that the poloidal flux function is closely related to the toroidal component of the magnetic vector potential **A** since Eq. (4) implies that

$$
\psi = 2\pi r A\_{\theta}.\tag{5}
$$

This relationship has important consequences for the confinement of the plasma particles. Due to the axisymmetry, the canonical angular momentum *P<sup>θ</sup>* = *mrv<sup>θ</sup>* + *qrA<sup>θ</sup>* turns out to be a constant of the motion of each particle (*m* and *Ze* being the mass and the charge of the particle, respectively). In terms of the poloidal flux we can see that

$$P\_{\theta} = mrv\_{\theta} + \frac{Ze}{2\pi}\psi\tag{6}$$

is a constant of motion. If the magnetic field is strong enough the term *mrv<sup>θ</sup>* may become very small compared with *Zeψ*/(2*π*). In that case the particles are constrained to move along surfaces of constant *ψ*, i.e. along magnetic surfaces. For this reason, an effective way of confining charged particles can be obtained by creating a set of nested toroidal magnetic surfaces. The rupture of flux surfaces caused by asymmetries in the field generation or instabilities certainly has a detrimental effect on confinement.

#### **2.2 Stability**

An equilibrium is unstable if it is possible to find a small perturbation that growths when is applied. Otherwise, the equilibrium is stable. The instabilities observed in 4 Will-be-set-by-IN-TECH

In all these three systems as well as in other important configurations the magnetic surfaces spanned by the magnetic field lines play a central role in confinement. We examine this in

> *S*(*r*,*z*)

were *S*(*r*, *z*) is the circle of radius *r* centered at the position *z* of the vertical axis. If the

 *r* 0

With this definition *ψ* reaches its maximum value at the magnetic axis (*ψ*(*r*ma, *z*ma) = *ψ*ma). The magnetic surfaces, or flux surfaces, can be determined by the equation *ψ*(*r*, *z*) = *C* where *C* is a constant. Note that this useful labeling system for the flux surfaces breaks down when

> *ψ*(*r*, *z*) <sup>2</sup>*π<sup>r</sup>* <sup>ˆ</sup> *θ*

*θ* is the unit vector pointing in the toroidal direction. Note that the poloidal flux function is closely related to the toroidal component of the magnetic vector potential **A** since

*Ze*

This relationship has important consequences for the confinement of the plasma particles. Due to the axisymmetry, the canonical angular momentum *P<sup>θ</sup>* = *mrv<sup>θ</sup>* + *qrA<sup>θ</sup>* turns out to be a constant of the motion of each particle (*m* and *Ze* being the mass and the charge of the

*P<sup>θ</sup>* = *mrv<sup>θ</sup>* +

is a constant of motion. If the magnetic field is strong enough the term *mrv<sup>θ</sup>* may become very small compared with *Zeψ*/(2*π*). In that case the particles are constrained to move along surfaces of constant *ψ*, i.e. along magnetic surfaces. For this reason, an effective way of confining charged particles can be obtained by creating a set of nested toroidal magnetic surfaces. The rupture of flux surfaces caused by asymmetries in the field generation or

An equilibrium is unstable if it is possible to find a small perturbation that growths when is applied. Otherwise, the equilibrium is stable. The instabilities observed in

**B** · *d***s** (2)

*Bz*(*χ*, *z*) *χdχ*. (3)

*ψ* = 2*πrA<sup>θ</sup>* . (5)

<sup>2</sup>*<sup>π</sup> <sup>ψ</sup>* (6)

(4)

*ψ*(*r*, *z*) =

configuration is axisymmetric we can express the poloidal flux function as

*ψ*(*r*, *z*) = 2*π*

The poloidal flux function acts as a stream function for the poloidal field since

**B***<sup>p</sup>* = ∇ ×

field, closely related to that occurring in the RFP.

more detail. Let *ψ*(*r*, *z*) be the poloidal flux function defined as

the axisymmetry is lost (due to an instability for example).

particle, respectively). In terms of the poloidal flux we can see that

instabilities certainly has a detrimental effect on confinement.

where ˆ

**2.2 Stability**

Eq. (4) implies that

configuration is formed as a result of a relaxation process that self-organizes the magnetic

magnetically confined plasmas can be classified into two groups: the microinstabilities and the macroinstabilities. The first group is responsible for the turbulence at small scales and it is generally related to finite Larmor radius effects (the gyroradius of charged particles turning around the magnetic field) and asymmetries in the velocity distribution function of the different species that compound the plasma. On the other hand, the macroinstabilities involve fluctuations having a length scale comparable to that of the whole system and can, in their simplest version, be described by the MHD model presented in Section 3. Their appearance generally leads to the termination of the discharge and the destruction of the configuration. In this Chapter we will deal with this kind of instabilities.

The usual procedure to study the MHD stability of an equilibrium is based on the analysis of the energy increment *δW* introduced by a small perturbation to the equilibrium (Friedberg, 1987). Using the linearized equations of the MHD model it is possible to compute the growth rate of each perturbation. If all possibles modes decay then the equilibrium is MHD stable.

According to the source of energy that feeds the instability, the macroinstabilities can be divided in:


In Section 5 we will consider internal kink modes in spheromak configurations. A comprehensive description of the MHD modes relevant to magnetic confinement configurations can be found elsewhere (Friedberg, 1987);(Wesson, 2004).
