**1. Introduction**

In this chapter we address magnetohydrodynamics (MHD) theory for magnetically confined fusion plasmas. To be specific we focus on toroidal confinement of fusion plasmas, especially tokamak physics.

The biggest challenges mankind ever faces are falling energy sources and food shortages. If controlled nuclear fusion were achieved with net energy yield, the energy source problem would be solved. If natural photosynthesis were reproduced, food shortage concern would be addressed. Though both nuclear fusion and photosynthesis are universal, the difficulties to achieve them are disproportionally great. Instead, those discoveries harmful to nature, though naturally unpopular, are invented relatively easily. This tendency reminds us of a bible verse (Genesis 3:19): "In the sweat of thy face shalt thou eat bread". This verse basically sketches the dependence of efforts (sweat) demanded for scientific discoveries on the usefulness (bread) of the discoveries to mankind (see Fig. 1). The more the discovery is relevant to mankind, the more the sweat is needed for that discovery. This may explain why controlled nuclear fusion is so difficult and its underlying plasma physics is so complicated.

Fig. 1. Schematic interpretation of Genesis 3:19: the dependence of efforts (sweat) demanded for scientific discoveries on the usefulness (bread) of the discoveries to mankind.

in Toroidal Plasma Confinement 3

Overview of Magnetohydrodynamics Theory in Toroidal Plasma Confinement 3

This chapter is arranged as follows: In Sec. 2 the basic set of ideal MHD equations is described; In Sec. 3 MHD equilibrium is discussed; In Sec. 4 analytical or semi-analytical theories for four types of major MHD modes are presented; In Sec. 5 the formulation of global numerical analyses of MHD modes are given; In the last section the results are summarized. Gyrokinetic

The basic set of ideal MHD equations are derived from single fluid and Maxwell's equations.

where *ρ<sup>m</sup>* is mass density, **v** denotes fluid velocity, *P* is plasma pressure, Γ represents the ratio of specific heats, **E** and **B** represents respectively electric and magnetic fields, **J** is current

The MHD equations (1)-(6) can be linearized. For brevity we will use the same symbols for both full and equilibrium quantities. Perturbed quantities will be tagged with *δ*, unless

where *ξ* = **v**/(−*iω*) represents plasma displacement, and the time dependence of perturbed quantities is assumed to be of exponential type exp{−*iωt*}. Inserting Eqs. (11)-(13) into

We have not included toroidal rotation effects in the linearized equations (10)-(13). For most of tokamak experiments rotation is subsonic, i.e., the rotation speed is much smaller than

*dt* <sup>=</sup> −∇*<sup>P</sup>* <sup>+</sup> **<sup>J</sup>** <sup>×</sup> **<sup>B</sup>**, (1) **E** = −**v** × **B**, (2)

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>**<sup>v</sup>** · ∇*<sup>P</sup>* <sup>−</sup> <sup>Γ</sup>*P*∇ · **<sup>v</sup>**, (3)

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>**<sup>v</sup>** · ∇*ρ<sup>m</sup>* <sup>−</sup> *<sup>ρ</sup>m*∇ · **<sup>v</sup>**, (4) *μ*0**J** = ∇ × **B**, (5)

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> ∇ × **<sup>E</sup>**, (6)

**J** × **B** = ∇*P*, (7) ∇ × **B** = *μ*0**J**, (8) ∇ · **B** = 0. (9)

<sup>−</sup> *<sup>ρ</sup>mω*2*<sup>ξ</sup>* <sup>=</sup> *<sup>δ</sup>***<sup>J</sup>** <sup>×</sup> **<sup>B</sup>** <sup>+</sup> **<sup>J</sup>** <sup>×</sup> *<sup>δ</sup>***<sup>B</sup>** − ∇*δP*, (10)

∇ × (∇ × *ξ* × **B**) × **B** + **J** ×∇× *ξ* × **B** + ∇ (*ξ* · ∇*P* + Γ*P*∇ · *ξ*). (14)

*δ***B** = ∇ × *ξ* × **B**, (11) *μ*0*δ***J** = ∇ × *δ***B**, (12) *δP* = −*ξ* · ∇*P* − Γ*P*∇ · *ξ*, (13)

and resistive effects are also discussed in this last section.

*ρ d***v**

*∂P*

*∂ρm*

*∂***B**

density, *μ*<sup>0</sup> is vacuum permeability, and bold faces denote vectors.

**2. Basic set of ideal MHD equations**

specified. Equilibrium equations are

The linearized perturbed MHD equations become

Eq. (10), one obtains a single equation for *ξ*:

*μ*0

<sup>−</sup> *<sup>ρ</sup>mω*2*<sup>ξ</sup>* <sup>=</sup> <sup>1</sup>

They are given as follows

When God created universe each day, He always claimed "it was so" and "it was good" (Genesis 1). The hard aspect of fusion plasma physics lies in that we often miss simplicity (it was so) and beauty (it was good) in theoretical formalism. MHD theory seems to be unique in plasma physics. Though many charged particle system, with long mean free path and long range correlation, is intrinsically complicated, MHD theory is relatively simple and, nonetheless, gives rise to rather relevant theoretical predictions for experiments: Tokamaks are designed according to MHD equilibrium theory and nowadays none would expect that a magnetic confinement of fusion plasmas could survive if MHD theory predicted major instabilities. As one will see even with MHD description the theoretical formulation of magnetically confined plasmas in toroidal geometry can still be hard to deal with. Thanks to decades-long efforts many beautiful MHD theoretical formulations for toroidal confinement of fusion plasmas have been laid out. In this chapter we try to give a comprehensive review of these prominent theories. Four key types of modes: interchange/peeling modes Mercier (1962) Greene & Johnson (1962) Glasser et al. (1975) Lortz (1975) Wesson (1978), ballooning modes Connor et al. (1979) Chance et al. (1979), toroidal Alfvén egenmodes (TAEs) Cheng et al. (1985) Rosenbluth et al. (1992) Betti & Freidberg (1992) Zheng & Chen (1998), and kinetically driven modes, such as kinetic ballooning modes (KBMs) Tsai & Chen (1993) Chen (1994) and energetic particle driven modes (EPMs) Zheng et al. (2000), are addressed. Besides, we also describe an advanced numerical method (AEGIS Zheng & Kotschenreuther (2006)) for systematically investigating MHD stability of toroidally confined fusion plasmas. Description of global formulation used for numerical computation can also provide an overall picture of MHD eigen mode structure for toroidal plasmas.

MHD theory is a single fluid description of plasmas. Fluid approach is based on the assumption that particle movements are spatially localized so that a local thermal equilibrium can be established. In the conventional fluid theory particle collision is the ingredient for particle localization. However, for magnetically confined fusion plasmas collision frequency usually is low. One cannot expect particle collisions to play the role for localizing particles spatially. The relevance of partial fluid description of magnetically confined fusion plasmas relies on the presence of strong magnetic field. Charged particles are tied to magnetic field lines due to gyro-motions. Therefore, in the direction perpendicular to magnetic field lines magnetic field can play the role of localization, so that MHD description becomes relevant at least in lowest order. One can expect that perpendicular MHD description needs modification only when finite Larmor radius effects become significant.

In the direction parallel to magnetic field, however, particles can move rather freely. Collision frequency is not strong enough hold charged particles together to establish local thermal equilibrium. One cannot define local thermal parameters, such as fluid density, velocity, temperature, etc. The trapped particle effect, wave-particle resonances, and parallel electric field effects need to be included. Plasma behavior in parallel direction is intrinsically non-fuild and needs kinetic description. Surprisingly, even under this circumstance MHD description still yields valuable and relevant theoretical predictions without major modifications in the concerned low (*ω ωsi*) and intermediate (*ωsi ω ωse*) frequency regimes, where *ω* is mode frequency, *ωsi* and *ωse* represent respectively ion and electron acoustic frequencies. In the low frequency regime coupling of parallel motion results only in an enhanced apparent mass effect; while in the intermediate frequency regime kinetic effects only gives rise to a new phenomenological ratio of special heats in leading order.

This chapter is arranged as follows: In Sec. 2 the basic set of ideal MHD equations is described; In Sec. 3 MHD equilibrium is discussed; In Sec. 4 analytical or semi-analytical theories for four types of major MHD modes are presented; In Sec. 5 the formulation of global numerical analyses of MHD modes are given; In the last section the results are summarized. Gyrokinetic and resistive effects are also discussed in this last section.

#### **2. Basic set of ideal MHD equations**

2 Will-be-set-by-IN-TECH

When God created universe each day, He always claimed "it was so" and "it was good" (Genesis 1). The hard aspect of fusion plasma physics lies in that we often miss simplicity (it was so) and beauty (it was good) in theoretical formalism. MHD theory seems to be unique in plasma physics. Though many charged particle system, with long mean free path and long range correlation, is intrinsically complicated, MHD theory is relatively simple and, nonetheless, gives rise to rather relevant theoretical predictions for experiments: Tokamaks are designed according to MHD equilibrium theory and nowadays none would expect that a magnetic confinement of fusion plasmas could survive if MHD theory predicted major instabilities. As one will see even with MHD description the theoretical formulation of magnetically confined plasmas in toroidal geometry can still be hard to deal with. Thanks to decades-long efforts many beautiful MHD theoretical formulations for toroidal confinement of fusion plasmas have been laid out. In this chapter we try to give a comprehensive review of these prominent theories. Four key types of modes: interchange/peeling modes Mercier (1962) Greene & Johnson (1962) Glasser et al. (1975) Lortz (1975) Wesson (1978), ballooning modes Connor et al. (1979) Chance et al. (1979), toroidal Alfvén egenmodes (TAEs) Cheng et al. (1985) Rosenbluth et al. (1992) Betti & Freidberg (1992) Zheng & Chen (1998), and kinetically driven modes, such as kinetic ballooning modes (KBMs) Tsai & Chen (1993) Chen (1994) and energetic particle driven modes (EPMs) Zheng et al. (2000), are addressed. Besides, we also describe an advanced numerical method (AEGIS Zheng & Kotschenreuther (2006)) for systematically investigating MHD stability of toroidally confined fusion plasmas. Description of global formulation used for numerical computation can also provide an overall picture of

MHD theory is a single fluid description of plasmas. Fluid approach is based on the assumption that particle movements are spatially localized so that a local thermal equilibrium can be established. In the conventional fluid theory particle collision is the ingredient for particle localization. However, for magnetically confined fusion plasmas collision frequency usually is low. One cannot expect particle collisions to play the role for localizing particles spatially. The relevance of partial fluid description of magnetically confined fusion plasmas relies on the presence of strong magnetic field. Charged particles are tied to magnetic field lines due to gyro-motions. Therefore, in the direction perpendicular to magnetic field lines magnetic field can play the role of localization, so that MHD description becomes relevant at least in lowest order. One can expect that perpendicular MHD description needs modification

In the direction parallel to magnetic field, however, particles can move rather freely. Collision frequency is not strong enough hold charged particles together to establish local thermal equilibrium. One cannot define local thermal parameters, such as fluid density, velocity, temperature, etc. The trapped particle effect, wave-particle resonances, and parallel electric field effects need to be included. Plasma behavior in parallel direction is intrinsically non-fuild and needs kinetic description. Surprisingly, even under this circumstance MHD description still yields valuable and relevant theoretical predictions without major modifications in the concerned low (*ω ωsi*) and intermediate (*ωsi ω ωse*) frequency regimes, where *ω* is mode frequency, *ωsi* and *ωse* represent respectively ion and electron acoustic frequencies. In the low frequency regime coupling of parallel motion results only in an enhanced apparent mass effect; while in the intermediate frequency regime kinetic effects only gives rise to a new

MHD eigen mode structure for toroidal plasmas.

only when finite Larmor radius effects become significant.

phenomenological ratio of special heats in leading order.

The basic set of ideal MHD equations are derived from single fluid and Maxwell's equations. They are given as follows

$$
\rho \frac{d\mathbf{v}}{dt} = -\nabla P + \mathbf{J} \times \mathbf{B} \tag{1}
$$

$$\mathbf{E} = -\mathbf{v} \times \mathbf{B} \,\tag{2}$$

$$\frac{\partial P}{\partial t} = -\mathbf{v} \cdot \nabla P - \Gamma P \nabla \cdot \mathbf{v}\_{\prime} \tag{3}$$

$$\frac{\partial \rho\_m}{\partial t} = -\mathbf{v} \cdot \nabla \rho\_m - \rho\_m \nabla \cdot \mathbf{v}\_\prime \tag{4}$$

$$
\mu\_0 \mathbf{J} = \nabla \times \mathbf{B} \,\tag{5}
$$

$$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times \mathbf{E} \tag{6}$$

where *ρ<sup>m</sup>* is mass density, **v** denotes fluid velocity, *P* is plasma pressure, Γ represents the ratio of specific heats, **E** and **B** represents respectively electric and magnetic fields, **J** is current density, *μ*<sup>0</sup> is vacuum permeability, and bold faces denote vectors.

The MHD equations (1)-(6) can be linearized. For brevity we will use the same symbols for both full and equilibrium quantities. Perturbed quantities will be tagged with *δ*, unless specified. Equilibrium equations are

$$\mathbf{J} \times \mathbf{B} = \nabla P\_{\prime} \tag{7}$$

$$
\nabla \times \mathbf{B} = \mu\_0 \mathbf{J}\_\prime \tag{8}
$$

$$
\nabla \cdot \mathbf{B} = 0.\tag{9}
$$

The linearized perturbed MHD equations become

$$-\rho\_{\rm m}\omega^2 \mathbf{\tilde{f}} = \delta \mathbf{J} \times \mathbf{B} + \mathbf{J} \times \delta \mathbf{B} - \nabla \delta P\_{\prime} \tag{10}$$

$$
\delta \mathbf{B} = \nabla \times \mathbf{f} \times \mathbf{B},
\tag{11}
$$

$$
\mu\_0 \delta \mathbf{J} = \nabla \times \delta \mathbf{B} \,\tag{12}
$$

$$
\delta P = -\mathfrak{F} \cdot \nabla P - \Gamma P \nabla \cdot \mathfrak{F}\_{\prime} \tag{13}
$$

where *ξ* = **v**/(−*iω*) represents plasma displacement, and the time dependence of perturbed quantities is assumed to be of exponential type exp{−*iωt*}. Inserting Eqs. (11)-(13) into Eq. (10), one obtains a single equation for *ξ*:

$$-\rho\_{\mathfrak{M}}\omega^{2}\mathfrak{F} = \frac{1}{\mu\_{0}}\nabla \times (\nabla \times \mathfrak{F} \times \mathbf{B}) \times \mathbf{B} + \mathbf{J} \times \nabla \times \mathfrak{F} \times \mathbf{B} + \nabla \left(\mathfrak{F} \cdot \nabla P + \Gamma P \nabla \cdot \mathfrak{F}\right). \tag{14}$$

We have not included toroidal rotation effects in the linearized equations (10)-(13). For most of tokamak experiments rotation is subsonic, i.e., the rotation speed is much smaller than

in Toroidal Plasma Confinement 5

Overview of Magnetohydrodynamics Theory in Toroidal Plasma Confinement 5

Here and later on we use prime to denote derivative with respect to flux coordinate chosen. This is a nonlinear equation for *χ* for given functions *P*(*χ*) and *g*(*χ*). It generally needs numerical solution. Since it is a two dimensional problem, one needs to introduce a poloidal angle coordinate *θeq* around magnetic axis of plasma torus in addition to radial coordinate *χ*. The solution is usually given in (*χ*, *θeq*) grids for *X*(*χ*, *θeq*), *Z*(*χ*, *θeq*), or inversely, in (*X*, *Z*)

Instead of physical cylinder coordinates (*X*, *Z*, *φ*) or (*χ*, *θeq*, *φ*), magnetic flux coordinates are often used in theoretical analyses, which is characterized by that the magnetic field line is straight in the covariant representation of coordinate system. Note that the coordinate system (*ψ*, *θeq*, *φ*) usually is not a flux coordinate system. In most equilibrium codes *θeq* is just an equal-arc length poloidal coordinate. One of flux coordinate systems is the so-called PEST coordinate system Grimm et al. (1976) (*χ*, *θpest*, *φ*), where *θpest* is generalized poloidal

<sup>∇</sup>*<sup>φ</sup>* × ∇*ψpest* <sup>+</sup> *<sup>q</sup>*∇*ψpest* × ∇*θpest*

<sup>∇</sup>*ψpest* × ∇*θpest* · ∇*<sup>φ</sup>* <sup>=</sup> *<sup>q</sup>χ*�

, *cpest* <sup>=</sup> *<sup>X</sup>*<sup>0</sup>

*dθeq*

*g*J*eq <sup>X</sup>*<sup>2</sup> ,

*ζ* = *φ* + *ν*(*ψ*, *θ*), *θ* = *θpest* + *ν*(*ψ*, *θ*)/*q*, (21)

**B** = *χ*� (∇*ζ* × ∇*ψ* + *q*∇*ψ* × ∇*θ*). (22)

*q* 1

2*π v dτ* 1 *X*2 ,

*X*2

By equating Eqs. (19) and (15) one can find that the Jacobian of PEST coordinates should be

. (19)

*<sup>g</sup>* . (20)

*<sup>v</sup> dτ* denotes volume integration over entire

<sup>J</sup> , (23)

coordinate, such that the equilibrium magnetic field can be represented as

<sup>J</sup>*pest* <sup>≡</sup> <sup>1</sup>

 *χ* 0

as follows. Using Eq. (20), one can determine poloidal angle in PEST coordinate

*<sup>θ</sup>pest* <sup>=</sup> <sup>1</sup> *q θeq* 0

*<sup>d</sup><sup>χ</sup> <sup>q</sup> g*

plasma domain. The PEST poloidal angle *θpest* can be related to physical angle coordinate *θeq*

where J*eq* = 1/∇*χ* × ∇*θeq* · ∇*φ*, which can be computed from equilibrium solution. Here,

Next, we discuss construction of general flux coordinates. The covariant type of representation as in Eq. (19) is not unique. It is preserved under the following coordinate

Here, *θ* and *ζ* are referred to as generalized poloidal and toroidal angles, respectively. PEST coordinates are characterized by its toroidal angle coordinate being axisymmetric toroidal angle. In this general case, by equating Eqs. (22) and (15) in ∇*φ* projection one can find that

> + *g* 1 *<sup>X</sup>*<sup>2</sup> <sup>=</sup> *<sup>χ</sup>*�

1 J*eq*

*∂ν ∂θeq ψ*,*φ*

**B** = *χ*�

In PEST coordinate system the flux coordinate is chosen as

*<sup>ψ</sup>pest* <sup>=</sup> <sup>2</sup>*πX*<sup>0</sup> *cpest*

where *<sup>X</sup>*<sup>0</sup> is major radius at magnetic axis and

the integration is along the path of constant *χ* and *φ*.

transforms

such that

grids for *χ*(*X*, *Z*), *θeq*(*X*, *Z*).

ion thermal speed. In this case the centrifugal and Coriolis forces from plasma rotation is smaller than the effects from particle thermal motion — plasma pressure effect. Therefore, rotation effect can be taken into account simply by introducing the Doppler frequency shift: *ω* → *ω* + *n*Ω*not* in MHD equation (14), where Ω*rot* is toroidal rotation frequency and *n* denotes toroidal mode number Waelbroeck & Chen (1991) Zheng et al. (1999).

### **3. Tokamak MHD equilibrium**

In this subsection we discuss tokamak equilibrium theory. MHD equilibrium has been discussed in many MHD books. Here, we focus mainly on how to construct various flux coordinates from numerical solution of MHD equilibrium equations.

We first outline the derivation of Grad-Shafranov equation Grad & Rubin (1958) Shafranov (1966). The cylindrical coordinate system (*X*, *Z*, *φ*) is introduced, where *X* is radius from axi-symmetry axis of plasma torus, *Z* denotes vertical coordinate, and *φ* is toroidal axi-symmetric angle. We introduce the vector potential **A** to represent magnetic field **B** = ∇ × **A**. Due to toroidal symmetry *φ* is an ignorable coordinate. Using curl expression in cylinder coordinates and noting that *∂AX*/*∂φ* = *∂AZ*/*∂φ* = 0, one can prove that the vector potential **A** in *X* and *Z* directions (**A***XZ*) can be expressed through the single toroidal component: **A***φ*. Without losing generality one can express **A***<sup>φ</sup>* = −*χ*∇*φ*. Therefore, total equilibrium magnetic field can be expressed as, by adding (*X*, *Z*) components and toroidal component,

$$\mathbf{B} = \nabla \times \mathbf{A}\_{\Phi} + X B\_{\Phi} \nabla \phi = \nabla \phi \times \nabla \chi + \mathbf{g} \nabla \phi,\tag{15}$$

where *B<sup>φ</sup>* is toroidal component of magnetic field and *g* = *XBφ*. From Eq. (15) one can prove that **B** · ∇*χ* = 0 and therefore *χ* = *const*. labels magnetic surfaces. Equation (15) can be used to show that 2*πχ* is poloidal magnetic flux. One can also define the toroidal flux 2*πψT*(*χ*). The safety factor is then defined as *q* = *dψT*/*dχ*, which characterizes the field line winding on a magnetic surface.

Using Ampere's law in Eq. (8) one can express equilibrium current density as follows

$$
\mu\_0 \mathbf{J} = \nabla \mathbf{g} \times \nabla \boldsymbol{\phi} + \mathbf{X}^2 \nabla \cdot \left(\frac{\nabla \chi}{X^2}\right) \nabla \boldsymbol{\phi}.\tag{16}
$$

Here, we have noted that <sup>∇</sup>*<sup>φ</sup>* ·∇× (∇*<sup>φ</sup>* × ∇*χ*) = ∇ · (∇*χ*/*X*2) and <sup>∇</sup>*<sup>θ</sup>* ·∇× (∇*<sup>φ</sup>* × ∇*χ*) = ∇*χ* ·∇× (∇*φ* × ∇*χ*) = 0.

Inserting Eqs. (15) and (16) into force balance equation (7) one obtains

$$
\mu\_0 \nabla P = -\nabla \cdot \left(\frac{\nabla \chi}{X^2}\right) \nabla \chi - \frac{1}{X^2} g \nabla g + \nabla \phi \nabla g \times \nabla \phi \cdot \nabla \chi. \tag{17}
$$

From Eq. (7) one can prove that **B** · ∇*P* = 0. Therefore, one can conclude that plasma pressure is a surface faction, i.e., *P*(*χ*). From Eq. (17) one can further determine that *g* is a surface function as well, through projecting Eq. (17) on ∇*φ*. Therefore, Eq. (17) can be reduced to the so-called Grad-Shafranov equation

$$X^2 \nabla \cdot \left(\frac{\nabla \chi}{X^2}\right) = -\mu\_0 X^2 P'\_{\chi} - g g'\_{\chi}.\tag{18}$$

Here and later on we use prime to denote derivative with respect to flux coordinate chosen. This is a nonlinear equation for *χ* for given functions *P*(*χ*) and *g*(*χ*). It generally needs numerical solution. Since it is a two dimensional problem, one needs to introduce a poloidal angle coordinate *θeq* around magnetic axis of plasma torus in addition to radial coordinate *χ*. The solution is usually given in (*χ*, *θeq*) grids for *X*(*χ*, *θeq*), *Z*(*χ*, *θeq*), or inversely, in (*X*, *Z*) grids for *χ*(*X*, *Z*), *θeq*(*X*, *Z*).

Instead of physical cylinder coordinates (*X*, *Z*, *φ*) or (*χ*, *θeq*, *φ*), magnetic flux coordinates are often used in theoretical analyses, which is characterized by that the magnetic field line is straight in the covariant representation of coordinate system. Note that the coordinate system (*ψ*, *θeq*, *φ*) usually is not a flux coordinate system. In most equilibrium codes *θeq* is just an equal-arc length poloidal coordinate. One of flux coordinate systems is the so-called PEST coordinate system Grimm et al. (1976) (*χ*, *θpest*, *φ*), where *θpest* is generalized poloidal coordinate, such that the equilibrium magnetic field can be represented as

$$\mathbf{B} = \chi' \left( \nabla \phi \times \nabla \psi\_{\rm pest} + q \nabla \psi\_{\rm pest} \times \nabla \theta\_{\rm pest} \right) . \tag{19}$$

By equating Eqs. (19) and (15) one can find that the Jacobian of PEST coordinates should be

$$\mathcal{J}\_{\rm pest} \equiv \frac{1}{\nabla \psi\_{\rm pest} \times \nabla \theta\_{\rm pest} \cdot \nabla \phi} = \frac{q \chi' \mathbf{X}^2}{\mathbf{g}}.\tag{20}$$

In PEST coordinate system the flux coordinate is chosen as

$$
\psi\_{\rm pest} = \frac{2\pi X\_0}{c\_{\rm pest}} \int\_0^\chi d\chi \frac{q}{\mathcal{S}} \qquad c\_{\rm pest} = \frac{X\_0}{2\pi} \int\_\upsilon d\tau \frac{1}{X^2} \lambda
$$

where *<sup>X</sup>*<sup>0</sup> is major radius at magnetic axis and *<sup>v</sup> dτ* denotes volume integration over entire plasma domain. The PEST poloidal angle *θpest* can be related to physical angle coordinate *θeq* as follows. Using Eq. (20), one can determine poloidal angle in PEST coordinate

$$
\theta\_{\text{post}} = \frac{1}{q} \int\_0^{\theta\_{\text{eq}}} d\theta\_{\text{eq}} \frac{\mathbf{g} \mathcal{J}\_{\text{eq}}}{X^2} \lambda
$$

where J*eq* = 1/∇*χ* × ∇*θeq* · ∇*φ*, which can be computed from equilibrium solution. Here, the integration is along the path of constant *χ* and *φ*.

Next, we discuss construction of general flux coordinates. The covariant type of representation as in Eq. (19) is not unique. It is preserved under the following coordinate transforms

$$\mathcal{L} = \phi + \nu(\psi, \theta), \ \theta = \theta\_{\text{post}} + \nu(\psi, \theta) / q,\tag{21}$$

such that

4 Will-be-set-by-IN-TECH

ion thermal speed. In this case the centrifugal and Coriolis forces from plasma rotation is smaller than the effects from particle thermal motion — plasma pressure effect. Therefore, rotation effect can be taken into account simply by introducing the Doppler frequency shift: *ω* → *ω* + *n*Ω*not* in MHD equation (14), where Ω*rot* is toroidal rotation frequency and *n* denotes

In this subsection we discuss tokamak equilibrium theory. MHD equilibrium has been discussed in many MHD books. Here, we focus mainly on how to construct various flux

We first outline the derivation of Grad-Shafranov equation Grad & Rubin (1958) Shafranov (1966). The cylindrical coordinate system (*X*, *Z*, *φ*) is introduced, where *X* is radius from axi-symmetry axis of plasma torus, *Z* denotes vertical coordinate, and *φ* is toroidal axi-symmetric angle. We introduce the vector potential **A** to represent magnetic field **B** = ∇ × **A**. Due to toroidal symmetry *φ* is an ignorable coordinate. Using curl expression in cylinder coordinates and noting that *∂AX*/*∂φ* = *∂AZ*/*∂φ* = 0, one can prove that the vector potential **A** in *X* and *Z* directions (**A***XZ*) can be expressed through the single toroidal component: **A***φ*. Without losing generality one can express **A***<sup>φ</sup>* = −*χ*∇*φ*. Therefore, total equilibrium magnetic field can be expressed as, by adding (*X*, *Z*) components and toroidal

where *B<sup>φ</sup>* is toroidal component of magnetic field and *g* = *XBφ*. From Eq. (15) one can prove that **B** · ∇*χ* = 0 and therefore *χ* = *const*. labels magnetic surfaces. Equation (15) can be used to show that 2*πχ* is poloidal magnetic flux. One can also define the toroidal flux 2*πψT*(*χ*). The safety factor is then defined as *q* = *dψT*/*dχ*, which characterizes the field line winding on

Here, we have noted that <sup>∇</sup>*<sup>φ</sup>* ·∇× (∇*<sup>φ</sup>* × ∇*χ*) = ∇ · (∇*χ*/*X*2) and <sup>∇</sup>*<sup>θ</sup>* ·∇× (∇*<sup>φ</sup>* × ∇*χ*) =

From Eq. (7) one can prove that **B** · ∇*P* = 0. Therefore, one can conclude that plasma pressure is a surface faction, i.e., *P*(*χ*). From Eq. (17) one can further determine that *g* is a surface function as well, through projecting Eq. (17) on ∇*φ*. Therefore, Eq. (17) can be reduced to the

<sup>=</sup> <sup>−</sup>*μ*0*X*2*P*�

*<sup>χ</sup>* − *gg*�

<sup>∇</sup>*<sup>χ</sup>* <sup>−</sup> <sup>1</sup>

Using Ampere's law in Eq. (8) one can express equilibrium current density as follows

*<sup>μ</sup>*0**<sup>J</sup>** <sup>=</sup> <sup>∇</sup>*<sup>g</sup>* × ∇*<sup>φ</sup>* <sup>+</sup> *<sup>X</sup>*2∇ ·

Inserting Eqs. (15) and (16) into force balance equation (7) one obtains

 <sup>∇</sup>*<sup>χ</sup> X*<sup>2</sup> 

*<sup>X</sup>*2∇ ·

 <sup>∇</sup>*<sup>χ</sup> X*<sup>2</sup> 

**B** = ∇ × **A***<sup>φ</sup>* + *XBφ*∇*φ* = ∇*φ* × ∇*χ* + *g*∇*φ*, (15)

 <sup>∇</sup>*<sup>χ</sup> X*<sup>2</sup> 

∇*φ*. (16)

*<sup>χ</sup>*. (18)

*<sup>X</sup>*<sup>2</sup> *<sup>g</sup>*∇*<sup>g</sup>* <sup>+</sup> <sup>∇</sup>*φ*∇*<sup>g</sup>* × ∇*<sup>φ</sup>* · ∇*χ*. (17)

toroidal mode number Waelbroeck & Chen (1991) Zheng et al. (1999).

coordinates from numerical solution of MHD equilibrium equations.

**3. Tokamak MHD equilibrium**

component,

a magnetic surface.

∇*χ* ·∇× (∇*φ* × ∇*χ*) = 0.

*μ*0∇*P* = −∇ ·

so-called Grad-Shafranov equation

$$\mathbf{B} = \chi' \left( \nabla \zeta \times \nabla \psi + q \nabla \psi \times \nabla \theta \right). \tag{22}$$

Here, *θ* and *ζ* are referred to as generalized poloidal and toroidal angles, respectively. PEST coordinates are characterized by its toroidal angle coordinate being axisymmetric toroidal angle. In this general case, by equating Eqs. (22) and (15) in ∇*φ* projection one can find that

$$\frac{\partial \nu}{\partial \theta\_{eq}}\Big|\_{\psi,\Phi} \frac{1}{\mathcal{I}\_{eq}} + g\frac{1}{X^2} = \chi' q \frac{1}{\mathcal{I}'} \tag{23}$$

in Toroidal Plasma Confinement 7

Overview of Magnetohydrodynamics Theory in Toroidal Plasma Confinement 7

where the second term is diamagnetic current and the first term denotes the Pfirsch-Schlüter

∇ × **<sup>B</sup>**

where *λ*<sup>0</sup> is the integration constant and can be determined by Ohm's law in the parallel

In this subsection we overview the linear MHD stability theories in toroidal geometry. We will detail major analytical techniques developed in this field in the past decades, such as interchange, ballooning, TAE, and EPM/KBM theories. Due to space limitation, we focus

There are three fundamental waves in magnetic confined plasmas. The compressional Alfvén mode characterizes the oscillation due to compression and restoration of magnetic field. It mainly propagates in the derection perpendicular to magnetic field. Since plasmas are frozen in magnetic field, such a magnetic field compression also induces plasma compression. Note that the ratio of plasma pressure to magnetic pressure (referred to as plasma beta *β*) usually is low. The compression and restoration forces mainly result from magnetic field energy. The shear Alfvén wave describes the oscillation due to magnetic field line bending and restoration. It mainly propagates along the magnetic field lines. Since long wave length is allowed for shear Alfvén wave, shear Alfvén wave frequency (or restoration force) is usually lower than that of compressional Alfvén wave. Therefore, shear Alfvén wave is often coupled to plasma instabilities. Another fundamental wave in magnetic confined plasmas is parallel acoustic wave (sound wave). Since plasma can move freely along magnetic field lines without being affected by Lorentz's force. Parallel acoustic wave can prevail in plasmas. The various types of electrostatic drift waves are related to it. Due to low beta assumption, the frequency of ion sound wave is lower than that of shear Alfvén wave by oder *β*. The behaviors of these three waves in simplified geometry have been widely studied in many MHD books. Here, we focus on toroidal geometry theories. MHD equation (14) in toroidal geometry can be hard to deal with. One usually needs to separate the time scales for three fundamental waves to reduce the problem. This scale separation is realized through proper projections and reduction of MHD

There are three projections for MHD equation, Eq. (14). We introduce three unit vectors: **e***<sup>b</sup>* = **B**/*B*, **e**<sup>1</sup> = ∇*ψ*/|∇*ψ*|, and **e**<sup>2</sup> = **e***<sup>b</sup>* × **e**<sup>1</sup> for projections. The **e**<sup>2</sup> projection of the MHD equation

*<sup>B</sup>* **<sup>e</sup>**<sup>2</sup> · ∇ (∇ · *<sup>ξ</sup>*) <sup>+</sup> *<sup>ρ</sup>mω*<sup>2</sup>

**e**<sup>1</sup> · *δ***B** +

1 *<sup>B</sup>* **<sup>e</sup>**<sup>2</sup> · ∇

*P*�


*<sup>B</sup>* **<sup>e</sup>**<sup>2</sup> · *<sup>ξ</sup>*. (31)

*<sup>B</sup>*<sup>2</sup> **<sup>e</sup>**<sup>1</sup> · *<sup>δ</sup>***<sup>B</sup>** <sup>−</sup> *<sup>g</sup>*�

<sup>+</sup>Γ*<sup>P</sup>* <sup>1</sup>

*<sup>B</sup>*<sup>2</sup> · ∇*χd<sup>θ</sup>* <sup>+</sup> *<sup>λ</sup>*0, (30)

current. We can determine the Pfirsch-Schlüter current from ∇ · **J** = 0,

**4.1 Decomposition of linearized MHD equations, three basic MHD waves**

 *θ* 0

*<sup>λ</sup>* <sup>=</sup> <sup>−</sup> <sup>1</sup> 2*χ*

direction.

equation (14).

**<sup>e</sup>**<sup>1</sup> ·∇× *<sup>δ</sup>***<sup>B</sup>** <sup>=</sup> <sup>−</sup> *gP*�

(14) gives

**4. Linear MHD instabilities**

ourselves on ideal MHD theory.

where J = 1/∇*ψ* × ∇*θ* · ∇*ζ*. Using J*eq* and J definitions, one can prove that

$$\left.\frac{\partial\theta}{\partial\theta\_{eq}}\right|\_{\psi,\phi} = \frac{\chi'\mathcal{I}\_{eq}}{\mathcal{I}}.\tag{24}$$

One can solve Eq. (23), yielding

$$\nu(\psi, \theta) = \int\_0^{\theta\_{\text{eq}}} d\theta\_{\text{eq}} \mathcal{J}\_{\text{eq}} \left( \chi' q \frac{1}{\mathcal{J}} - \mathcal{g} \frac{1}{X^2} \right) = q\theta - \int\_0^{\theta\_{\text{eq}}} d\theta\_{\text{eq}} \frac{\mathcal{g} \mathcal{J}\_{\text{eq}}}{X^2} \,\tag{25}$$

where Eq. (24) has been used.

Equations (21)-(25) can be used to construct various types of flux coordinate systems. There are two classes of them: One is by specifying Jacobian (e.g., Hamada coordinates Hamada (1962) and Boozer coordinates Boozer (1982)) and the other by directly choosing generalized poloidal angle (e.g., equal arc-length coordinate). In the Hamada coordinates the volume inside a magnetic surface is used to label magnetic surfaces, i.e., *ψ* = *V*, and Jacobian J*<sup>h</sup>* = 1/∇*V* · ∇*θ<sup>h</sup>* × ∇*ζ<sup>h</sup>* is set to be unity. With Jacobian specified, Eq. (24) can be used to solve for *θ<sup>h</sup>* at given (*V*,*φ*). With *ν* determined by Eq. (25) the definition Eq. (21) can be used to specify *<sup>ζ</sup>h*. In the Boozer coordinates Jacobian is chosen to be <sup>J</sup>*<sup>B</sup>* <sup>=</sup> *<sup>V</sup>*� *B*2 *<sup>s</sup>* /(4*π*2*B*2), where �·�*<sup>s</sup>* represents surface average. The procedure for specifying Boozer poloidal and toroidal coordinates *θ<sup>B</sup>* and *ζ<sup>B</sup>* is similar to that for Hamada coordinates. In the equal-arc-length coordinates poloidal angle is directly specified as equal-arc-length coordinate *θe*. In this case, Jacobian J*<sup>e</sup>* can be computed through Eq. (24). With *ν* determined by Eq. (25) the definition Eq. (21) can be used to specify *ζe*.

We can also express current density vector in covariant representation with generalized flux coordinates. Using Ampere's law in Eq. (8) for determining **J** · ∇*θ* and Eq. (7) for **J** · ∇*ζ*, one can also express equilibrium current density in covariant representation

$$\mathcal{J} = -\frac{1}{\mu\_0} g\_{\psi}^{\prime} \nabla \zeta \times \nabla \psi - \left(\frac{q}{\mu\_0} g\_{\psi}^{\prime} + \frac{P\_{\psi}^{\prime}}{\chi\_{\psi}^{\prime}} \mathcal{I}\right) \nabla \psi \times \nabla \theta. \tag{26}$$

This general coordinate expression for **J** can be alternatively obtained from PEST representation in Eq. (16) and Grad-Shafranov equation (18) through coordinate transform. Equation (26) is significantly simplified in the Hamada coordinates. Due to J = 1, Eq. (26) can be expressed as

$$\mathbf{J} = f\_V' \nabla \zeta \times \nabla V + I\_V' \nabla V \times \nabla \theta\_\prime \tag{27}$$

where *I*(*V*) and *J*(*V*) are toroidal and poloidal current fluxes, *I*� = −*g*� *<sup>V</sup>*/*μ*0, and *J*� = −*qg*� *<sup>V</sup>*/*μ*<sup>0</sup> − *P*� *V*/*χ*� *<sup>V</sup>*. The force balance equation (7) can be simply expressed as

$$
\mu\_0 P\_V' = f\_V' \psi\_V' - I\_V' \chi\_V'. \tag{28}
$$

It is also interesting to discuss diamagnetic current and Pfirsch-Schlüter current in plasma torus. Due to the existence of plasma pressure there is diamagnetic current in tokamak system. The diamagnetic current alone is not divergence-free and is always accompanied by a return current in the parallel direction, i.e., the so-called Pfirsch-Schlüter current. The total equilibrium current is therefore can be expressed as

$$\mathbf{J} = \frac{dP}{d\chi} \left( 2\lambda \mathbf{B} + \frac{\mathbf{B} \times \nabla \chi}{B^2} \right) ,\tag{29}$$

where the second term is diamagnetic current and the first term denotes the Pfirsch-Schlüter current. We can determine the Pfirsch-Schlüter current from ∇ · **J** = 0,

$$\lambda = -\frac{1}{2\chi} \int\_0^{\theta} \nabla \times \frac{\mathbf{B}}{B^2} \cdot \nabla \chi d\theta + \lambda\_0. \tag{30}$$

where *λ*<sup>0</sup> is the integration constant and can be determined by Ohm's law in the parallel direction.
