**4. Linear MHD instabilities**

6 Will-be-set-by-IN-TECH

<sup>=</sup> *<sup>χ</sup>*� J*eq*

> 1 *X*<sup>2</sup>

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

�·�*<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

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

> *q μ*0 *g*� *<sup>ψ</sup>* + *P*� *ψ χ*� *ψ* J

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)

�

**B** × ∇*χ B*2

*<sup>V</sup>*. The force balance equation (7) can be simply expressed as

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

*<sup>V</sup>*∇*ζ* × ∇*V* + *I*

= *qθ* −

 *θeq* 0

*dθeq*

<sup>J</sup> . (24)

*g*J*eq*

*B*2

∇*ψ* × ∇*θ*. (26)

*<sup>V</sup>*∇*V* × ∇*θ*, (27)

*<sup>V</sup>*. (28)

, (29)

*<sup>X</sup>*<sup>2</sup> , (25)

*<sup>s</sup>* /(4*π*2*B*2), where

*<sup>V</sup>*/*μ*0, and *J*� =

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

> *χ*� *q* 1 <sup>J</sup> <sup>−</sup> *<sup>g</sup>*

One can solve Eq. (23), yielding

where Eq. (24) has been used.

Eq. (21) can be used to specify *ζe*.

can be expressed as

*<sup>V</sup>*/*μ*<sup>0</sup> − *P*�

*V*/*χ*�

−*qg*�

**<sup>J</sup>** <sup>=</sup> <sup>−</sup> <sup>1</sup> *μ*0 *g*�

*ν*(*ψ*, *θ*) =

 *θeq* 0

*dθeq*J*eq*

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>*�

can also express equilibrium current density in covariant representation

**J** = *J* �

total equilibrium current is therefore can be expressed as

*<sup>ψ</sup>*∇*ζ* × ∇*ψ* −

where *I*(*V*) and *J*(*V*) are toroidal and poloidal current fluxes, *I*� = −*g*�

*μ*0*P*� *<sup>V</sup>* = *J* � *Vψ*� *<sup>V</sup>* − *I* � *Vχ*�

**<sup>J</sup>** <sup>=</sup> *dP dχ* 2*λ***B** + 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 ourselves on ideal MHD theory.

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

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 equation (14).

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 (14) gives

$$\mathbf{e}\_1 \cdot \nabla \times \delta \mathbf{B} = -\frac{gP'}{B^2} \mathbf{e}\_1 \cdot \delta \mathbf{B} - g' \mathbf{e}\_1 \cdot \delta \mathbf{B} + \frac{1}{B} \mathbf{e}\_2 \cdot \nabla \left( P' | \nabla \psi | \mathbf{e}\_1 \cdot \mathfrak{F} \right)$$

$$+ \Gamma P \frac{1}{B} \mathbf{e}\_2 \cdot \nabla \left( \nabla \cdot \mathfrak{F} \right) + \frac{\rho\_m \omega^2}{B} \mathbf{e}\_2 \cdot \mathfrak{F}. \tag{31}$$

in Toroidal Plasma Confinement 9

Overview of Magnetohydrodynamics Theory in Toroidal Plasma Confinement 9

et al Glasser et al. (1975). However, the details have been omitted in this paper and direct projection method, alternative to the original vorticity equation approach, is used. Here, we detail the derivation of singular layer equation by vorticity equation approach. These derivation can tell analytical techniques to separate the compressional Alfvén wave from low frequency interchange mode and to minimize field line bending effects due to shear Alfvén mode. The singular equation will be used to derive stability criterion for interchange and

In order to investigate the modes which localize around a particular rational (or singular) magnetic surface *V*0, we specialize the Hamada coordinates to the neighborhood of mode rational surface *V*<sup>0</sup> and introduce the localized Hamada coordinates *x*, *u*, *θ* as usual, where *x* = *V* − *V*<sup>0</sup> and *u* = *mθ* − *nζ*. In this coordinate system the parallel derivative becomes

Using the coordinates (*x*, *u*, *θ*), we find that, in an axisymmetric torus, equilibrium scalars are independent of *u*, and therefore perturbations can be assumed to vary as exp{*ikuu*} with

We consider only singular modes whose wavelength across the magnetic surface *<sup>λ</sup>*<sup>⊥</sup> is much smaller than that on the surface and perpendicular to magnetic field line *λ*∧. This leads us to

where *�* � 1, being a small parameter. Furthermore, we consider only the low-frequency

Since the modes vary on a slow time scale, they are decoupled from compressional Alfvén wave. It can be verified *a posteriori* that we can make following ordering assumptions:

> *<sup>ξ</sup>* <sup>=</sup> *�ξ*(1) <sup>+</sup> ··· , *<sup>μ</sup>* <sup>=</sup> *<sup>μ</sup>*(0) <sup>+</sup> ··· , *<sup>δ</sup>P*(2) <sup>=</sup> *�*2*δP*(2) <sup>+</sup> ··· , *<sup>b</sup>* <sup>=</sup> *�*2*b*(2) <sup>+</sup> ··· , *<sup>v</sup>* <sup>=</sup> *�v*(1) <sup>+</sup> ··· , *<sup>τ</sup>* <sup>=</sup> *�τ*(1) <sup>+</sup> ··· ,

where *<sup>δ</sup>P*(2) <sup>=</sup> <sup>−</sup>Γ*P*∇ · *<sup>ξ</sup>*. These ordering assumptions are the same as those in Ref. Glasser et al. (1975), except that we use *δP*(2) as unknown to replace *ν*. With these ordering assumptions we can proceed to analyze the basic set of linearized MHD equations. As usual, perturbed quantities are separated into constant and oscillatory parts along the field lines:

*dl*/*B*, *<sup>l</sup>* is arc length of magnetic field line, and ˜

*<sup>∂</sup><sup>V</sup>* <sup>∼</sup> *�*<sup>−</sup>1, *<sup>∂</sup>*

*χ*�� − *χ*�

. As in Refs. Johnson & Greene (1967) and Glasser et al. (1975), *ξ* and *δ***B** are

**B** × ∇*V <sup>B</sup>*<sup>2</sup> <sup>+</sup> *<sup>ν</sup>*

**B** × ∇*V <sup>B</sup>*<sup>2</sup> <sup>+</sup> *<sup>τ</sup>*

*<sup>∂</sup><sup>u</sup>* <sup>∼</sup> *<sup>∂</sup>*

*ψ*�� and Ξ = *ψ*�

**B** *B*2 ,

**B** *B*2 . /*m* = *χ*�

*∂θ* <sup>∼</sup> 1, (35)


/*n*.

(*∂*/*∂ζ*)+(Λ*x*/Ξ)(*∂*/*∂u*), where Λ = *ψ*�

*<sup>ξ</sup>* <sup>=</sup> *<sup>ξ</sup>* <sup>∇</sup>*<sup>V</sup>*

*<sup>δ</sup>***<sup>B</sup>** <sup>=</sup> *<sup>b</sup>* <sup>∇</sup>*<sup>V</sup>*

choose following ordering scheme as in Ref. Glasser et al. (1975):

*<sup>x</sup>* <sup>∼</sup> *�*, *<sup>∂</sup>*

*<sup>ξ</sup>* <sup>=</sup> �*ξ*� <sup>≡</sup> *<sup>ξ</sup> dl*/*B*/



peeling modes.

**B** · ∇ = *χ*�

regime

*ξ* = ¯ *ξ* + ˜

*ξ* = *ξ* − �*ξ*�.

*ξ*, where ¯

*ku* = 2*πn*/*χ*�

projected in three directions as follows:

where *ωsi* is parallel ion acoustic frequency.

Similarly, the **e**<sup>1</sup> projection of the MHD equation (14) yields

$$\mathbf{e}\_2 \cdot \nabla \times \delta \mathbf{B} = -\frac{gP'}{B^2} \mathbf{e}\_2 \cdot \delta \mathbf{B} - g' \mathbf{e}\_2 \cdot \delta \mathbf{B} - \frac{P'|\nabla \psi|}{B^2} \mathbf{e}\_\theta \cdot \delta \mathbf{B} - \frac{1}{B} \mathbf{e}\_1 \cdot \nabla \left( P'|\nabla \psi| \mathbf{e}\_1 \cdot \mathfrak{F} \right)$$

$$-\Gamma P \frac{1}{B} \mathbf{e}\_1 \cdot \nabla \left( \nabla \cdot \mathfrak{F} \right) - \frac{\rho\_m \omega^2}{B} \mathbf{e}\_1 \cdot \mathfrak{F}. \tag{32}$$

The **e***<sup>b</sup>* projection of MHD equation (14) can be reduced to, using ∇ · *ξ* as an independent unknown,

$$
\Gamma P \mathbf{B} \cdot \nabla \left(\frac{1}{B^2} \mathbf{B} \cdot \nabla \nabla \cdot \boldsymbol{\mathfrak{f}}\right) + \rho\_m \omega^2 \nabla \cdot \boldsymbol{\mathfrak{f}} = \rho\_m \omega^2 \nabla \cdot \boldsymbol{\mathfrak{f}}\_{\perp}.\tag{33}
$$

Noting that *<sup>δ</sup>***<sup>J</sup>** and *<sup>δ</sup>***<sup>B</sup>** are determined completely by *<sup>ξ</sup>*⊥, one can see that the set of equations (31) - (33) is complete to determine two components of *<sup>ξ</sup>*<sup>⊥</sup> and scalar unknown ∇ · *<sup>ξ</sup>*.

Two perpendicular equations of motion, Eqs. (31) and (32), result from perpendicular projections of MHD equation (10) and therefore contain restoration force due to excitation of compressional Alfvén wave. To suppress compressional Alfvén wave from consideration, one can apply the operator ∇ · (**B**/*B*2) <sup>×</sup> (···) on Eq. (10), yielding

$$\nabla \cdot \frac{\mathbf{B}}{B^2} \times \rho\_m \omega^2 \mathbf{\tilde{f}} = \mathbf{B} \cdot \nabla \frac{\mathbf{B} \cdot \delta \mathbf{J}}{B^2} + \delta \mathbf{B} \cdot \nabla \sigma - \mathbf{J} \cdot \nabla \frac{\mathbf{B} \cdot \delta \mathbf{B}}{B^2} + \nabla \times \frac{\mathbf{B}}{B^2} \cdot \nabla \delta P\_\prime \tag{34}$$

where *<sup>σ</sup>* <sup>=</sup> **<sup>J</sup>** · **<sup>B</sup>**/*B*2. Note that compressional Alfvén wave results from the term *<sup>δ</sup>***<sup>J</sup>** <sup>×</sup> **<sup>B</sup>** <sup>+</sup> **J** × *δ***B** + ∇*δP* → ∇(**B** · *δ***B** + *δP*) in Eq. (10). Therefore the curl operation in deriving Eq. (34) can suppress compressional Alfvén wave. Equation (34) is often referred to as shear Alfvén law or vorticity equation.

Equations (34), (31), and (33) characterize respectively three fundamental MHD waves: shear Alfvén, compressional Alfvén, and parallel acoustic waves. From newly developed gyrokinetic theory Zheng et al. (2007) two perpendicular equations (31) and (32) are fully recovered from gyrokinetic formulation, expect the plasma compressibility effect.

#### **4.2 Singular layer equation: interchange and peeling modes**

Interchange modes are most fundamental phenomena in magnetically confined plasmas. It resembles to the so-called Rayleigh-Taylor instability in conventional fluid theory. Through interchange of plasma flux tubes plasma thermal energy can be released, so that instability develops. Perturbation of magnetic energy from field line bending is minimized for interchange instability. In slab or cylinder configurations such an interchange happens due to the existence of bad curvature region. In toroidal geometry with finite *q* value, however, the curvature directions with respect to plasma pressure gradient are different on high and low field sides of plasma torus. Good and bad curvature regions appear alternately along magnetic field line. Therefore, one needs to consider toroidal average in evaluating the change of plasma and magnetic energies. This makes interchange mode theory in plasma torus become complicated. The interchange mode theory is the first successful toroidal theory in this field. It includes the derivations of the so-called singular later equation and interchange stability criterion, i.e., the so-called Mercier criterion Mercier (1962) Greene & Johnson (1962).

Early derivation of singular layer equation relies on the assumption that the modes are somewhat localized poloidally. This assumption was released in a later paper by Glasser 8 Will-be-set-by-IN-TECH

The **e***<sup>b</sup>* projection of MHD equation (14) can be reduced to, using ∇ · *ξ* as an independent

Noting that *<sup>δ</sup>***<sup>J</sup>** and *<sup>δ</sup>***<sup>B</sup>** are determined completely by *<sup>ξ</sup>*⊥, one can see that the set of equations

Two perpendicular equations of motion, Eqs. (31) and (32), result from perpendicular projections of MHD equation (10) and therefore contain restoration force due to excitation of compressional Alfvén wave. To suppress compressional Alfvén wave from consideration,

where *<sup>σ</sup>* <sup>=</sup> **<sup>J</sup>** · **<sup>B</sup>**/*B*2. Note that compressional Alfvén wave results from the term *<sup>δ</sup>***<sup>J</sup>** <sup>×</sup> **<sup>B</sup>** <sup>+</sup> **J** × *δ***B** + ∇*δP* → ∇(**B** · *δ***B** + *δP*) in Eq. (10). Therefore the curl operation in deriving Eq. (34) can suppress compressional Alfvén wave. Equation (34) is often referred to as shear Alfvén

Equations (34), (31), and (33) characterize respectively three fundamental MHD waves: shear Alfvén, compressional Alfvén, and parallel acoustic waves. From newly developed gyrokinetic theory Zheng et al. (2007) two perpendicular equations (31) and (32) are fully

Interchange modes are most fundamental phenomena in magnetically confined plasmas. It resembles to the so-called Rayleigh-Taylor instability in conventional fluid theory. Through interchange of plasma flux tubes plasma thermal energy can be released, so that instability develops. Perturbation of magnetic energy from field line bending is minimized for interchange instability. In slab or cylinder configurations such an interchange happens due to the existence of bad curvature region. In toroidal geometry with finite *q* value, however, the curvature directions with respect to plasma pressure gradient are different on high and low field sides of plasma torus. Good and bad curvature regions appear alternately along magnetic field line. Therefore, one needs to consider toroidal average in evaluating the change of plasma and magnetic energies. This makes interchange mode theory in plasma torus become complicated. The interchange mode theory is the first successful toroidal theory in this field. It includes the derivations of the so-called singular later equation and interchange stability criterion, i.e., the so-called Mercier criterion Mercier (1962) Greene & Johnson (1962). Early derivation of singular layer equation relies on the assumption that the modes are somewhat localized poloidally. This assumption was released in a later paper by Glasser

recovered from gyrokinetic formulation, expect the plasma compressibility effect.

*<sup>B</sup>*<sup>2</sup> <sup>+</sup> *<sup>δ</sup>***<sup>B</sup>** · ∇*<sup>σ</sup>* <sup>−</sup> **<sup>J</sup>** · ∇ **<sup>B</sup>** · *<sup>δ</sup>***<sup>B</sup>**


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

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

*<sup>B</sup>* **<sup>e</sup>**<sup>1</sup> · *<sup>ξ</sup>*. (32)

<sup>+</sup> *<sup>ρ</sup>mω*2∇ · *<sup>ξ</sup>* <sup>=</sup> *<sup>ρ</sup>mω*2∇ · *<sup>ξ</sup>*⊥. (33)

*<sup>B</sup>*<sup>2</sup> <sup>+</sup> ∇ × **<sup>B</sup>**

*P*�


*<sup>B</sup>*<sup>2</sup> · ∇*δP*, (34)

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

(31) - (33) is complete to determine two components of *<sup>ξ</sup>*<sup>⊥</sup> and scalar unknown ∇ · *<sup>ξ</sup>*.

*<sup>B</sup>* **<sup>e</sup>**<sup>1</sup> · ∇ (∇ · *<sup>ξ</sup>*) <sup>−</sup> *<sup>ρ</sup>mω*<sup>2</sup>

*<sup>B</sup>*<sup>2</sup> **<sup>B</sup>** · ∇∇ · *<sup>ξ</sup>*

one can apply the operator ∇ · (**B**/*B*2) <sup>×</sup> (···) on Eq. (10), yielding

Similarly, the **e**<sup>1</sup> projection of the MHD equation (14) yields

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

1

*<sup>B</sup>*<sup>2</sup> <sup>×</sup> *<sup>ρ</sup>mω*2*<sup>ξ</sup>* <sup>=</sup> **<sup>B</sup>** · ∇ **<sup>B</sup>** · *<sup>δ</sup>***<sup>J</sup>**

**4.2 Singular layer equation: interchange and peeling modes**

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

Γ*P***B** · ∇

**<sup>e</sup>**<sup>2</sup> ·∇× *<sup>δ</sup>***<sup>B</sup>** <sup>=</sup> <sup>−</sup> *gP*�

∇ · **<sup>B</sup>**

law or vorticity equation.

unknown,

et al Glasser et al. (1975). However, the details have been omitted in this paper and direct projection method, alternative to the original vorticity equation approach, is used. Here, we detail the derivation of singular layer equation by vorticity equation approach. These derivation can tell analytical techniques to separate the compressional Alfvén wave from low frequency interchange mode and to minimize field line bending effects due to shear Alfvén mode. The singular equation will be used to derive stability criterion for interchange and peeling modes.

In order to investigate the modes which localize around a particular rational (or singular) magnetic surface *V*0, we specialize the Hamada coordinates to the neighborhood of mode rational surface *V*<sup>0</sup> and introduce the localized Hamada coordinates *x*, *u*, *θ* as usual, where *x* = *V* − *V*<sup>0</sup> and *u* = *mθ* − *nζ*. In this coordinate system the parallel derivative becomes **B** · ∇ = *χ*� (*∂*/*∂ζ*)+(Λ*x*/Ξ)(*∂*/*∂u*), where Λ = *ψ*� *χ*�� − *χ*� *ψ*�� and Ξ = *ψ*� /*m* = *χ*� /*n*.

Using the coordinates (*x*, *u*, *θ*), we find that, in an axisymmetric torus, equilibrium scalars are independent of *u*, and therefore perturbations can be assumed to vary as exp{*ikuu*} with *ku* = 2*πn*/*χ*� . As in Refs. Johnson & Greene (1967) and Glasser et al. (1975), *ξ* and *δ***B** are projected in three directions as follows:

$$\begin{aligned} \mathfrak{F} &= \mathfrak{F}\frac{\nabla V}{|\nabla V|^2} + \mu \frac{\mathbf{B} \times \nabla V}{B^2} + \upsilon \frac{\mathbf{B}}{B^{2'}},\\ \delta \mathbf{B} &= b \frac{\nabla V}{|\nabla V|^2} + \upsilon \frac{\mathbf{B} \times \nabla V}{B^2} + \tau \frac{\mathbf{B}}{B^2}.\end{aligned}$$

We consider only singular modes whose wavelength across the magnetic surface *<sup>λ</sup>*<sup>⊥</sup> is much smaller than that on the surface and perpendicular to magnetic field line *λ*∧. This leads us to choose following ordering scheme as in Ref. Glasser et al. (1975):

$$
\varepsilon \sim \varepsilon, \quad \frac{\partial}{\partial V} \sim \varepsilon^{-1}, \quad \frac{\partial}{\partial u} \sim \frac{\partial}{\partial \theta} \sim 1,\tag{35}
$$

where *�* � 1, being a small parameter. Furthermore, we consider only the low-frequency regime

$$|\omega/\omega\_{si}| \lesssim 1.\tag{36}$$

where *ωsi* is parallel ion acoustic frequency.

Since the modes vary on a slow time scale, they are decoupled from compressional Alfvén wave. It can be verified *a posteriori* that we can make following ordering assumptions:

$$\begin{aligned} \mathfrak{F} &= \mathfrak{e}\mathfrak{z}^{(1)} + \cdots \, \prime \quad \mu = \mu^{(0)} + \cdots \, \prime \quad \delta P^{(2)} = \mathfrak{e}^2 \delta P^{(2)} + \cdots \, \prime \\\ b &= \mathfrak{e}^2 b^{(2)} + \cdots \, \prime \quad \upsilon = \mathfrak{e} \upsilon^{(1)} + \cdots \, \prime \quad \tau = \mathfrak{e} \tau^{(1)} + \cdots \, \prime \end{aligned}$$

where *<sup>δ</sup>P*(2) <sup>=</sup> <sup>−</sup>Γ*P*∇ · *<sup>ξ</sup>*. These ordering assumptions are the same as those in Ref. Glasser et al. (1975), except that we use *δP*(2) as unknown to replace *ν*. With these ordering assumptions we can proceed to analyze the basic set of linearized MHD equations. As usual, perturbed quantities are separated into constant and oscillatory parts along the field lines: *ξ* = ¯ *ξ* + ˜ *ξ*, where ¯ *<sup>ξ</sup>* <sup>=</sup> �*ξ*� <sup>≡</sup> *<sup>ξ</sup> dl*/*B*/ *dl*/*B*, *<sup>l</sup>* is arc length of magnetic field line, and ˜ *ξ* = *ξ* − �*ξ*�.

in Toroidal Plasma Confinement 11

Overview of Magnetohydrodynamics Theory in Toroidal Plasma Confinement 11

We will derive the singular layer equation by averaging this equation. Therefore, it is needed

It is trivial to get *μ*(0) from Eqs. (41) and (42), and *τ*(1) from Eq. (43). The rest can be obtained

*∂x*

*B*2 |∇*V*|<sup>2</sup>

> *∂v*(1) *∂θ* <sup>−</sup> *<sup>χ</sup>*� *P*�

We need also to solve the equation of parallel motion, Eq. (33). Taking into consideration of low frequency assumption in Eq. (36) and the result in Eq. (42), the equation of parallel motion

> *ρmω*<sup>2</sup> *ku*Γ*P*

*<sup>B</sup>*2*<sup>σ</sup>* <sup>−</sup>

Inserting these results into Eq. (45) and averaging over *l*, one obtains the singular layer

(*xξ*(1)

 *∂ξ*(1)

*<sup>∂</sup><sup>x</sup>* <sup>−</sup> <sup>Λ</sup> *<sup>B</sup>*2/|∇*V*<sup>|</sup>

*∂θ* <sup>+</sup> *<sup>χ</sup>*� *<sup>∂</sup>*

*<sup>B</sup>*<sup>2</sup> · *<sup>κ</sup> ∂ξ*(1)

 *∂ξ*(1) *<sup>∂</sup><sup>x</sup>* .

*∂σv*(1)

**B** × ∇*V*

(*∂σ*/*∂θ*), equation (47) can be solved to yield

 ,

<sup>2</sup><sup>2</sup> �*B*2/|∇*V*|2� <sup>+</sup> *<sup>P</sup>*�<sup>2</sup>

 *B*2*σ* �*B*2� *<sup>B</sup>*<sup>2</sup>

 *σB*<sup>2</sup> �*B*2�

*<sup>σ</sup>B*2/|∇*V*<sup>|</sup>

(1) <sup>=</sup> <sup>−</sup><sup>Λ</sup> *<sup>∂</sup>*

Using Eq. (46) to determine integration constant, Eq. (44) can be solved, yielding that

2

<sup>+</sup> *<sup>J</sup>*� *P*�

> = *i*

*ρmω*<sup>2</sup> *ku*Γ*P*

> *∂ξ*(1) *∂x* + 1 <sup>4</sup> <sup>+</sup> *DI*

*b*(2) = (Λ*x*/Ξ)(*∂ξ*(1)/*∂u*). With ¯

). (46)

2

*τ*(1) *<sup>B</sup>*<sup>2</sup> .

*∂*2 *<sup>∂</sup>x*<sup>2</sup> (*xξ*(1)

*<sup>∂</sup><sup>x</sup>* . (47)

*ξ*(1) = 0, (48)

 1 *B*2  , ).

�*B*2/|∇*V*|2�

*∂θ*

*b*(2)

to express unknowns in this equation in terms of *ξ*(1).

as follows. From Eqs. (39) and (40) one can find that ¯

*v*¯

*<sup>B</sup>*2*σ*/|∇*V*<sup>|</sup>

�*B*2/|∇*V*|2�

*∂v*˜(1) *∂u*

obtained one can determine *v*¯(1) from Eq. (38):

 *B*2*σ* |∇*V*|<sup>2</sup> <sup>−</sup>

From Eqs. (40) and (37) one obtains

Noting that **<sup>B</sup>** × ∇*<sup>V</sup>* · *<sup>κ</sup>*/*B*<sup>2</sup> <sup>=</sup> *<sup>χ</sup>*�

<sup>−</sup>*χ*� *<sup>∂</sup>*2*ξ*(2)

*<sup>χ</sup>*�<sup>2</sup> *<sup>∂</sup> ∂θ* <sup>1</sup> *B*2 *∂ ∂θ <sup>δ</sup>P*(2)

*<sup>χ</sup>*� *<sup>∂</sup>*

*∂ ∂x* 

where the total mass parameter *M* = *Mc* + *Mt*,

*DI* <sup>≡</sup> *<sup>E</sup>* <sup>+</sup> *<sup>F</sup>* <sup>+</sup> *<sup>H</sup>* <sup>−</sup> <sup>1</sup>

*<sup>B</sup>*2/|∇*V*<sup>|</sup>

Λ<sup>2</sup>

*<sup>B</sup>*2/|∇*V*<sup>|</sup>

Λ<sup>2</sup>

*E* ≡ 

*F* ≡  *∂θ <sup>δ</sup>P*(2) <sup>=</sup> *<sup>i</sup>*

*<sup>x</sup>*<sup>2</sup> <sup>−</sup> *<sup>M</sup>ω*<sup>2</sup>

4 ,

 *J* � *ψ*�� − *I* � *χ*�� + Λ

 *<sup>σ</sup>*2*B*<sup>2</sup> |∇*V*|<sup>2</sup>

 − 

2

2

*∂θ∂<sup>x</sup>* <sup>=</sup> <sup>1</sup>

Ξ

*∂v*(1) *<sup>∂</sup><sup>x</sup>* <sup>=</sup> <sup>−</sup>

can be reduced to

equation

The condition that *δ***B** be divergence free, as required by Eq. (11), yields

$$\frac{\partial b^{(2)}}{\partial \mathbf{x}} + \frac{1}{\Xi} \frac{\partial}{\partial u} v^{(1)} + \frac{\partial v^{(1)}}{\partial \theta} \frac{\mathbf{B} \times \nabla V \cdot \nabla \theta}{B^2} + v^{(1)} \nabla \cdot \frac{\mathbf{B} \times \nabla V}{B^2} + \chi' \frac{\partial}{\partial \theta} \frac{\tau^{(1)}}{B^2} = 0.$$

It can be reduced to

$$\frac{\partial b^{(2)}}{\partial x} + \frac{1}{\Xi} \frac{\partial}{\partial u} v^{(1)} + \frac{f'}{P'} \frac{\partial v^{(1)}}{\partial \theta} - \frac{\chi'}{P'} \frac{\partial \sigma v^{(1)}}{\partial \theta} + \chi' \frac{\partial}{\partial \theta} \frac{\tau^{(1)}}{B^2} = 0. \tag{37}$$

After surface average it gives

$$\frac{\partial \overline{b}^{(2)}}{\partial x} + \frac{1}{\Xi} \frac{\partial \overline{v}^{(1)}}{\partial u} = 0. \tag{38}$$

The two significant orders of induction equation, Eq. (11), in the ∇*V*–direction are

$$0 = \chi' \frac{\partial \xi^{(1)}}{\partial \theta},\tag{39}$$

$$b^{(2)} = \chi' \frac{\partial \xi^{(2)}}{\partial \theta} + \frac{\Lambda x}{\Xi} \frac{\partial \xi^{(1)}}{\partial u}.\tag{40}$$

The component of Eq. (11) in the ∇*u*–direction, in lowest order, yields

$$
\chi' \frac{\partial \mu^{(0)} }{\partial \theta} = 0. \tag{41}
$$

To satisfy the component of Eq. (11) along the magnetic field line, one must set

$$(\nabla \cdot \mathfrak{F}\_{\perp})^{(0)} + 2\kappa \cdot \mathfrak{F}^{(0)} = \frac{\partial \mathfrak{F}^{(1)}}{\partial \mathfrak{x}} + \frac{1}{\Xi} \frac{\partial \mu^{(0)}}{\partial \mathfrak{u}} = 0. \tag{42}$$

where Eq. (41) and ∇ · (**<sup>B</sup>** × ∇*V*/*B*2) = <sup>2</sup>**<sup>B</sup>** <sup>×</sup> *<sup>κ</sup>* · ∇*V*/*B*<sup>2</sup> have been used, and *<sup>κ</sup>* <sup>=</sup> **<sup>b</sup>** · ∇**<sup>b</sup>** is magnetic field line curvature.

Next, we turn to momentum equation (14). The two components perpendicular to **B** of the momentum equation (14) both lead, in lowest order, to

$$
\pi^{(1)} - P'\tilde{\xi}^{(1)} = 0.\tag{43}
$$

This is consistent to Eq. (42). Since both components yield the same information, we can directly work on the vorticity equation Eq. (34) and obtain

$$
\chi' \frac{\partial}{\partial \theta} \left( \frac{|\nabla V|^2}{B^2} \frac{\partial v^{(1)}}{\partial \mathbf{x}} \right) + \chi' \frac{\partial \sigma}{\partial \theta} \frac{\partial \tilde{\xi}^{(1)}}{\partial \mathbf{x}} = \mathbf{0},
\tag{44}
$$

$$\begin{split} & -\omega^{2} \frac{N\_{i}M\_{i}|\nabla V|^{2}}{B^{2}} \frac{\partial \mu^{(0)}}{\partial x} \\ &= -\chi^{'} \frac{\partial}{\partial \theta} \left( \frac{|\nabla V|^{2}}{B^{2}} \frac{\partial v^{(2)}}{\partial x} - v \frac{\mathbf{B}}{B^{2}} \cdot \nabla \times \frac{\mathbf{B} \times \nabla V}{B^{2}} - \tau \frac{\mathbf{B}}{B^{2}} \cdot \nabla \times \frac{\mathbf{B}}{B^{2}} + \frac{l'}{\chi'} \tau^{(1)} \right) \\ & \qquad - v^{(1)} \left( \frac{l'}{P'} - \frac{\chi'}{P'} \sigma \right) \frac{\partial \sigma}{\partial \theta} - \tau^{(1)} \frac{\chi'}{B^{2}} \frac{\partial \sigma}{\partial \theta} - \Lambda \mathbf{x} \frac{|\nabla V|^{2}}{\Xi B^{2}} \frac{\partial}{\partial u} \frac{\partial v^{(1)}}{\partial x} + \frac{P'}{\Xi B^{2}} \frac{\partial \tau^{(1)}}{\partial u} \\ & \qquad + P' \frac{\nabla V \cdot \nabla (P + B^{2})}{\Xi B^{2} |\nabla V|^{2}} \frac{\partial \xi^{(1)}}{\partial u} - \chi' \frac{\partial \sigma}{\partial \theta} \Theta \frac{\partial \xi^{(1)}}{\partial u} + \frac{\chi'}{P'} \frac{\partial \sigma}{\partial \theta} \frac{\partial}{\partial \mathbf{x}} \left( \delta P^{(2)} - P' \xi^{(2)} \right) . \tag{45} \end{split} \tag{46}$$

We will derive the singular layer equation by averaging this equation. Therefore, it is needed to express unknowns in this equation in terms of *ξ*(1).

It is trivial to get *μ*(0) from Eqs. (41) and (42), and *τ*(1) from Eq. (43). The rest can be obtained as follows. From Eqs. (39) and (40) one can find that ¯ *b*(2) = (Λ*x*/Ξ)(*∂ξ*(1)/*∂u*). With ¯ *b*(2) obtained one can determine *v*¯(1) from Eq. (38):

$$
\bar{v}^{(1)} = -\Lambda \frac{\partial}{\partial \mathbf{x}} (\mathbf{x} \boldsymbol{\xi}^{(1)}).\tag{46}
$$

Using Eq. (46) to determine integration constant, Eq. (44) can be solved, yielding that

$$\frac{\partial \boldsymbol{v}^{(1)}}{\partial \mathbf{x}} = -\left(\frac{\boldsymbol{B}^{2} \sigma}{|\nabla V|^{2}} - \frac{\langle \boldsymbol{B}^{2} \sigma / |\nabla V|^{2} \rangle}{\langle \boldsymbol{B}^{2} / |\nabla V|^{2} \rangle} \frac{\boldsymbol{B}^{2}}{|\nabla V|^{2}}\right) \frac{\partial \boldsymbol{\mathfrak{g}}^{(1)}}{\partial \mathbf{x}} - \Lambda \frac{\boldsymbol{B}^{2} / |\nabla V|^{2}}{\langle \boldsymbol{B}^{2} / |\nabla V|^{2} \rangle} \frac{\partial^{2}}{\partial \mathbf{x}^{2}} (\mathbf{x}^{(1)}\_{5}) .$$

From Eqs. (40) and (37) one obtains

10 Will-be-set-by-IN-TECH

*<sup>B</sup>*<sup>2</sup> <sup>+</sup> *<sup>v</sup>*(1)

*∂v*¯(1)

*∂θ* <sup>+</sup>

Λ*x* Ξ

*∂x* + 1 Ξ

where Eq. (41) and ∇ · (**<sup>B</sup>** × ∇*V*/*B*2) = <sup>2</sup>**<sup>B</sup>** <sup>×</sup> *<sup>κ</sup>* · ∇*V*/*B*<sup>2</sup> have been used, and *<sup>κ</sup>* <sup>=</sup> **<sup>b</sup>** · ∇**<sup>b</sup>** is

Next, we turn to momentum equation (14). The two components perpendicular to **B** of the

This is consistent to Eq. (42). Since both components yield the same information, we can

*<sup>B</sup>*<sup>2</sup> ·∇× **<sup>B</sup>** × ∇*<sup>V</sup>*

*∂θ* <sup>Θ</sup> *∂ξ*(1) *∂u*

<sup>+</sup> *<sup>χ</sup>*� *∂σ ∂θ*

*<sup>B</sup>*<sup>2</sup> <sup>−</sup> *<sup>τ</sup>*


<sup>+</sup> *<sup>χ</sup>*� *P*� *∂σ ∂θ ∂ ∂x* 

*∂ξ*(1)

**B**

*∂ ∂u*

*<sup>B</sup>*<sup>2</sup> ·∇× **<sup>B</sup>**

*∂v*(1) *∂x* + *P*� Ξ*B*<sup>2</sup>

*<sup>τ</sup>*(1) <sup>−</sup> *<sup>P</sup>*�

*∂v*(1) *∂x*

**B**

*∂θ* <sup>−</sup> *<sup>τ</sup>*(1) *<sup>χ</sup>*�

*<sup>∂</sup><sup>u</sup>* <sup>−</sup> *<sup>χ</sup>*� *∂σ*

*∂ξ*(1)

*B*2 *∂σ ∂θ* <sup>−</sup> <sup>Λ</sup>*<sup>x</sup>* *∂ξ*(1)

*∂μ*(0)

*∂σv*(1)

∇ · **<sup>B</sup>** × ∇*<sup>V</sup>*

*∂θ*

*∂θ* <sup>+</sup> *<sup>χ</sup>*� *<sup>∂</sup>*

*<sup>B</sup>*<sup>2</sup> <sup>+</sup> *<sup>χ</sup>*� *<sup>∂</sup>*

*τ*(1)

*<sup>∂</sup><sup>u</sup>* <sup>=</sup> 0. (38)

*∂θ* , (39)

*∂θ* <sup>=</sup> 0. (41)

*ξ*(1) = 0. (43)

*<sup>∂</sup><sup>u</sup>* . (40)

*<sup>∂</sup><sup>u</sup>* <sup>=</sup> 0. (42)

*<sup>∂</sup><sup>x</sup>* <sup>=</sup> 0, (44)

*<sup>B</sup>*<sup>2</sup> <sup>+</sup> *<sup>J</sup>*�

*<sup>δ</sup>P*(2) <sup>−</sup> *<sup>P</sup>*�

*<sup>χ</sup>*� *<sup>τ</sup>*(1)

*∂τ*(1) *∂u*

> *ξ*(2)

. (45)

*∂θ*

*τ*(1) *<sup>B</sup>*<sup>2</sup> <sup>=</sup> 0.

*<sup>B</sup>*<sup>2</sup> <sup>=</sup> 0. (37)

**B** × ∇*V* · ∇*θ*

*∂v*(1) *∂θ* <sup>−</sup> *<sup>χ</sup>*� *P*�

The two significant orders of induction equation, Eq. (11), in the ∇*V*–direction are <sup>0</sup> <sup>=</sup> *<sup>χ</sup>*� *∂ξ*(1)

*<sup>b</sup>*(2) <sup>=</sup> *<sup>χ</sup>*� *∂ξ*(2)

To satisfy the component of Eq. (11) along the magnetic field line, one must set

(∇ · *<sup>ξ</sup>*⊥)(0) <sup>+</sup> <sup>2</sup>*<sup>κ</sup>* · *<sup>ξ</sup>*(0) <sup>=</sup> *∂ξ*(1)

momentum equation (14) both lead, in lowest order, to

directly work on the vorticity equation Eq. (34) and obtain

 |∇*V*| 2 *B*2

*∂μ*(0) *∂x*

*∂v*(2) *<sup>∂</sup><sup>x</sup>* <sup>−</sup> *<sup>v</sup>*

*∂σ*

*<sup>χ</sup>*� *<sup>∂</sup> ∂θ*

2

<sup>−</sup>*ω*<sup>2</sup> *NiMi*|∇*V*<sup>|</sup>

 |∇*V*| 2 *B*2

 *J*� *<sup>P</sup>*� <sup>−</sup> *<sup>χ</sup>*� *P*� *σ*

<sup>+</sup>*P*� <sup>∇</sup>*<sup>V</sup>* · ∇(*<sup>P</sup>* <sup>+</sup> *<sup>B</sup>*2) <sup>Ξ</sup>*B*2|∇*V*|<sup>2</sup>

<sup>=</sup> <sup>−</sup>*χ*� *<sup>∂</sup> ∂θ*

<sup>−</sup>*v*(1)

*B*2

*<sup>χ</sup>*� *∂μ*(0)

The component of Eq. (11) in the ∇*u*–direction, in lowest order, yields

The condition that *δ***B** be divergence free, as required by Eq. (11), yields

*<sup>v</sup>*(1) <sup>+</sup> *<sup>J</sup>*� *P*�

> *∂*¯ *b*(2) *∂x* + 1 Ξ

*∂v*(1) *∂θ*

*v*(1) +

*∂b*(2) *∂x* + 1 Ξ *∂ ∂u*

*∂b*(2) *∂x* + 1 Ξ *∂ ∂u*

After surface average it gives

magnetic field line curvature.

It can be reduced to

$$-\chi'\frac{\partial^2 \xi^{(2)}}{\partial \theta \partial \mathbf{x}} = \frac{1}{\Xi} \frac{\partial \vec{v}^{(1)}}{\partial \boldsymbol{u}} + \frac{\boldsymbol{J}'}{P'} \frac{\partial v^{(1)}}{\partial \theta} - \frac{\chi'}{P'} \frac{\partial \sigma v^{(1)}}{\partial \theta} + \chi' \frac{\partial}{\partial \theta} \frac{\boldsymbol{\tau}^{(1)}}{\mathbb{B}^2}.$$

We need also to solve the equation of parallel motion, Eq. (33). Taking into consideration of low frequency assumption in Eq. (36) and the result in Eq. (42), the equation of parallel motion can be reduced to

$$
\chi^{\prime 2} \frac{\partial}{\partial \theta} \left( \frac{1}{B^2} \frac{\partial}{\partial \theta} \delta P^{(2)} \right) = i \frac{\rho\_m \omega^2}{k\_u \Gamma P} \frac{\mathbf{B} \times \nabla V}{B^2} \cdot \kappa \frac{\partial \xi^{(1)}}{\partial \mathbf{x}}.\tag{47}
$$

Noting that **<sup>B</sup>** × ∇*<sup>V</sup>* · *<sup>κ</sup>*/*B*<sup>2</sup> <sup>=</sup> *<sup>χ</sup>*� (*∂σ*/*∂θ*), equation (47) can be solved to yield

$$i\chi'\frac{\partial}{\partial\theta}\delta P^{(2)} = i\frac{\rho\_m\omega^2}{k\_\mu\Gamma P} \left(B^2\sigma - \frac{\langle B^2\sigma\rangle}{\langle B^2\rangle}B^2\right)\frac{\partial\xi^{(1)}}{\partial x}.$$

Inserting these results into Eq. (45) and averaging over *l*, one obtains the singular layer equation

$$\frac{\partial}{\partial \mathbf{x}} \left( \mathbf{x}^2 - M \omega^2 \right) \frac{\partial \tilde{\xi}^{(1)}}{\partial \mathbf{x}} + \left( \frac{1}{4} + D\_I \right) \tilde{\xi}^{(1)} = \mathbf{0},\tag{48}$$

where the total mass parameter *M* = *Mc* + *Mt*,

$$\begin{split} D\_{I} & \equiv E + F + H - \frac{1}{4}, \\ E & \equiv \frac{\langle \mathcal{B}^{2}/|\nabla V|^{2} \rangle}{\Lambda^{2}} \left( f' \psi'' - I' \chi'' + \Lambda \frac{\langle \sigma \mathcal{B}^{2} \rangle}{\langle \mathcal{B}^{2} \rangle} \right), \\ F & \equiv \frac{\langle \mathcal{B}^{2}/|\nabla V|^{2} \rangle}{\Lambda^{2}} \left( \left\langle \frac{\sigma^{2} B^{2}}{|\nabla V|^{2}} \right\rangle - \frac{\langle \sigma \mathcal{B}^{2}/|\nabla V|^{2} \rangle^{2}}{\langle \mathcal{B}^{2}/|\nabla V|^{2} \rangle} + P'^{2} \left\langle \frac{1}{B^{2}} \right\rangle \right), \end{split}$$

in Toroidal Plasma Confinement 13

Overview of Magnetohydrodynamics Theory in Toroidal Plasma Confinement 13

i.e., −*DI* > 0. For the case with Δ < 0 we assume that rational surface resides inside plasma

For the case with Δ > 0 we assume that rational surface resides outside plasma region, so that

both cases, Eqs. (51) and (52), give rise to the same stability criterion for peeling mode: Λ*<sup>P</sup>* < 0.

In this section we review high-n ballooning mode theory. The stability criterion for interchange modes takes into account only average magnetic well effect. As it is well-known tokamak plasmas have good and bad curvature regions, referring to whether pressure gradient and magnetic field line curvature point in same direction or not. Usually bad curvature region lies on low field side of plasma torus; good curvature region on high field side. Although tokamaks are usually designed to have average good curvature, i.e., Mercier stable, the ballooning modes can still develop as soon as the release of plasma thermal energy on bad curvature region is sufficient to counter the magnetic energy resulting from field line bending Connor et al. (1979) Chance et al. (1979). In difference from interchange modes ballooning modes have high toroidal mode number *n*, while interchange modes can be either low and high *n*. Also, ballooning modes allow normal and geodesic wave lengths to be of

same order *<sup>λ</sup>*<sup>⊥</sup> <sup>∼</sup> *<sup>λ</sup>*∧, but both of them are much smaller than parallel wave length *<sup>λ</sup>*�.

**<sup>B</sup>** · *<sup>δ</sup>***<sup>B</sup>** <sup>+</sup> *<sup>δ</sup><sup>P</sup>* <sup>=</sup> <sup>−</sup>(*B*<sup>2</sup> <sup>+</sup> <sup>Γ</sup>*P*)∇ · *<sup>ξ</sup>* <sup>+</sup> **<sup>B</sup>** · ∇ **<sup>B</sup>** · *<sup>ξ</sup>*

**B** · ∇*δϕ B*2

*δϕ* Chance et al. (1979): *<sup>ξ</sup>*<sup>⊥</sup> <sup>=</sup> **<sup>B</sup>** × ∇*δϕ*/*B*2. Equation (34) then becomes,

*<sup>B</sup>*<sup>2</sup>∇⊥

*<sup>B</sup>*<sup>2</sup> · ∇ **<sup>B</sup>** × ∇*<sup>ψ</sup>*

*<sup>B</sup>*<sup>2</sup> **<sup>B</sup>** · ∇∇ · *<sup>ξ</sup>*

momentum equation, Eqs. (31) and (32), give the same result

**<sup>B</sup>** · ∇ <sup>1</sup>

+*P*�

where Eq. (53) has been used.

*<sup>B</sup>*<sup>2</sup> ∇ ·

*<sup>ψ</sup>*∇ × **<sup>B</sup>**

Equation (33), meanwhile, can be reduced to

<sup>Γ</sup>*P***<sup>B</sup>** · ∇ <sup>1</sup>

We first derive ballooning mode equation. In high *n* limit, both components of perpendicular

In lowest order, one has ∇ · *<sup>ξ</sup>*<sup>⊥</sup> <sup>∼</sup> *<sup>ξ</sup>*/*R*. This allows to introduce the so-called stream function

*<sup>B</sup>*<sup>2</sup> · ∇*δϕ*

The key formalism to ballooning mode theory is the so-called ballooning representation Lee & Van Dam (1977) Connor et al. (1979). Here, we outline its physics basis and derivation,

<sup>+</sup> ∇ ·

**<sup>J</sup>**2|∇*V*<sup>|</sup>

<sup>2</sup> + *I*�

<sup>−</sup>*DI* <sup>+</sup> <sup>Δ</sup> <sup>&</sup>gt; 0. (51)

<sup>−</sup>*DI* <sup>−</sup> <sup>Δ</sup> <sup>&</sup>gt; 0. (52)

*<sup>χ</sup>*�� *<sup>B</sup>*2|∇*V*<sup>|</sup>

−<sup>2</sup>

− 2*κ* · *ξ* = 0. (53)

*<sup>B</sup>*<sup>2</sup> · ∇∇ · *<sup>ξ</sup>* <sup>=</sup> 0. (54)

*<sup>B</sup>*<sup>2</sup> · ∇*δϕ*, (55)

. Therefore,

*ψ*� − *J*�

*B*2

*ρω*<sup>2</sup> ∇⊥*δϕ B*2

<sup>+</sup> <sup>Γ</sup>*P*∇ × **<sup>B</sup>**

<sup>+</sup> *<sup>ρ</sup>mω*2∇ · *<sup>ξ</sup>* <sup>=</sup> *<sup>ρ</sup>mω*<sup>2</sup> <sup>2</sup>**<sup>B</sup>** <sup>×</sup> *<sup>κ</sup>*

region, so that *b* > 0. In this case the stability condition becomes

*b* < 0. In this case the stability condition becomes

Note that <sup>−</sup>*DI* <sup>≡</sup> <sup>Δ</sup><sup>2</sup> <sup>−</sup> <sup>Λ</sup>*p*, where <sup>Λ</sup>*<sup>p</sup>* <sup>=</sup> *<sup>S</sup>*−<sup>2</sup>

**4.3 Ballooning modes**

This is more stringent than the Mercier criterion.

$$\begin{split} H & \equiv \frac{\langle \mathcal{B}^2 / |\nabla V|^2 \rangle}{\Lambda} \left( \frac{\langle \sigma \mathcal{B}^2 / |\nabla V|^2 \rangle}{\langle \mathcal{B}^2 / |\nabla V|^2 \rangle} - \frac{\langle \sigma \mathcal{B}^2 \rangle}{\langle \mathcal{B}^2 \rangle} \right), \\ M\_{\mathbb{c}} & \equiv \frac{N\_i M\_i}{k\_\mu^2 \Lambda^2} \left\langle \frac{\mathcal{B}^2}{|\nabla V|^2} \right\rangle \left\langle \frac{|\nabla V|^2}{B^2} \right\rangle, \\ M\_{\mathbb{f}} & \equiv \frac{N\_i M\_i}{k\_\mu^2 \Lambda^2 P'^2} \left\langle \frac{\mathcal{B}^2}{|\nabla V|^2} \right\rangle \left( \left\langle \sigma^2 B^2 \right\rangle - \frac{\langle \sigma B^2 \rangle^2}{\langle B^2 \rangle} \right). \end{split}$$

Here, the mass factor *Mc* results from perpendicular motion and *Mt* from parallel motion due to toroidal coupling. *Mt* is often referred to as apparent mass. In the kinetic description the apparent mass is enhanced by the so-called small parallel ion speed effect. In the large aspect ratio configurations this enhancement factor is of order <sup>√</sup>*R*/*a*, where *<sup>R</sup>* and *<sup>a</sup>* are respectively major and minor radii Mikhailovsky (1974) Zheng & Tessarotto (1994b).

From Eq. (48) one can derive the Mercier criterion, i.e., the stability criterion for localized interchange modes in toroidal geometry. In the marginal stability *ω*<sup>2</sup> = 0, Eq. (48) becomes the Euler differential equation. Its solution is

$$
\mathfrak{F} = \mathfrak{F}\_0 \mathfrak{x}^{-\frac{1}{2} \pm \sqrt{-D\_I}}.\tag{49}
$$

The system stability can be determined by Newcomb's theorem 5 Newcomb (1960): system is unstable, if and only if the solution of Eq. (48) vanishes two or more points. From the solution in Eq. (49) one can see that if −*DI* < 0 *ξ* becomes oscillated. Therefore, interchange mode stability criterion is simply −*DI* > 0.

Interchange modes are internal modes. When internal modes are stable, it is still possible to develop unstable external modes. For external modes one needs to consider the matching condition between plasma and vacuum solutions. As discussed in conventional MHD books, these matching conditions are: (1) the tangential magnetic perturbation (*δ***B***t*) should be continuous; and (2) total magnetic and thermal force (**B** · *δ***B** + *δP*) should balance across plasma-vacuum interface in the case without plasma surface current. It can be proved that for localized modes the vacuum contribution is of order *�*<sup>2</sup> and therefore can be neglected Lortz (1975). Consequently, the boundary condition becomes that total magnetic and thermal forces on the plasma side of the plasma-vacuum interface should vanish. This gives the necessary and sufficient stability condition for peeling modes

$$
\left[\frac{\mathbf{x}^2}{2}\left(\mathfrak{f}^\*\frac{d\mathfrak{f}}{d\mathfrak{x}} + \mathfrak{f}\frac{d\mathfrak{f}^\*}{d\mathfrak{x}}\right) + \left(\Delta + \frac{1}{2}\right)\mathfrak{x}|\mathfrak{f}|^2\right]\_{\mathbf{x}=\mathbf{b}} > 0,\tag{50}
$$

where *b* is the coordinate of plasma-vacuum interface, relative to the rational surface, and

$$
\Delta = \frac{1}{2} + \mathcal{S}^{-1} \left\langle \frac{\mathcal{B}^2 \sigma}{|\nabla V|^2} \right\rangle, \quad \mathcal{S} = \chi' \psi'' - \psi' \chi''.
$$

Note that the stability condition Eq. (50) can be alternatively obtained by the approach of minimization of plasma energy Lortz (1975) Wesson (1978).

One can derive the peeling mode stability criterion by inserting Eq. (49) into Eq. (50) Wesson (1978). In the derivation of peeling stability criterion we assume system to be Mercier stable, i.e., −*DI* > 0. For the case with Δ < 0 we assume that rational surface resides inside plasma region, so that *b* > 0. In this case the stability condition becomes

$$
\sqrt{-D\_I} + \Delta > 0.\tag{51}
$$

For the case with Δ > 0 we assume that rational surface resides outside plasma region, so that *b* < 0. In this case the stability condition becomes

$$
\sqrt{-D\_I} - \Delta > 0.\tag{52}
$$

Note that <sup>−</sup>*DI* <sup>≡</sup> <sup>Δ</sup><sup>2</sup> <sup>−</sup> <sup>Λ</sup>*p*, where <sup>Λ</sup>*<sup>p</sup>* <sup>=</sup> *<sup>S</sup>*−<sup>2</sup> **<sup>J</sup>**2|∇*V*<sup>|</sup> <sup>2</sup> + *I*� *ψ*� − *J*� *<sup>χ</sup>*�� *<sup>B</sup>*2|∇*V*<sup>|</sup> −<sup>2</sup> . Therefore, both cases, Eqs. (51) and (52), give rise to the same stability criterion for peeling mode: Λ*<sup>P</sup>* < 0. This is more stringent than the Mercier criterion.

#### **4.3 Ballooning modes**

12 Will-be-set-by-IN-TECH

*<sup>σ</sup>B*2/|∇*V*<sup>|</sup>


Here, the mass factor *Mc* results from perpendicular motion and *Mt* from parallel motion due to toroidal coupling. *Mt* is often referred to as apparent mass. In the kinetic description the apparent mass is enhanced by the so-called small parallel ion speed effect. In the large aspect ratio configurations this enhancement factor is of order <sup>√</sup>*R*/*a*, where *<sup>R</sup>* and *<sup>a</sup>* are respectively

From Eq. (48) one can derive the Mercier criterion, i.e., the stability criterion for localized interchange modes in toroidal geometry. In the marginal stability *ω*<sup>2</sup> = 0, Eq. (48) becomes

The system stability can be determined by Newcomb's theorem 5 Newcomb (1960): system is unstable, if and only if the solution of Eq. (48) vanishes two or more points. From the solution in Eq. (49) one can see that if −*DI* < 0 *ξ* becomes oscillated. Therefore, interchange mode

Interchange modes are internal modes. When internal modes are stable, it is still possible to develop unstable external modes. For external modes one needs to consider the matching condition between plasma and vacuum solutions. As discussed in conventional MHD books, these matching conditions are: (1) the tangential magnetic perturbation (*δ***B***t*) should be continuous; and (2) total magnetic and thermal force (**B** · *δ***B** + *δP*) should balance across plasma-vacuum interface in the case without plasma surface current. It can be proved that for localized modes the vacuum contribution is of order *�*<sup>2</sup> and therefore can be neglected Lortz (1975). Consequently, the boundary condition becomes that total magnetic and thermal forces on the plasma side of the plasma-vacuum interface should vanish. This gives the necessary

*<sup>ξ</sup>* <sup>=</sup> *<sup>ξ</sup>*0*x*<sup>−</sup> <sup>1</sup>

*dξ*∗ *dx* + Δ + 1 2 *x*|*ξ*| 2 

 *B*2*σ* |∇*V*|<sup>2</sup>

where *b* is the coordinate of plasma-vacuum interface, relative to the rational surface, and

Note that the stability condition Eq. (50) can be alternatively obtained by the approach of

One can derive the peeling mode stability criterion by inserting Eq. (49) into Eq. (50) Wesson (1978). In the derivation of peeling stability criterion we assume system to be Mercier stable,

, *S* = *χ*�

�*B*2/|∇*V*|2� <sup>−</sup>

2 *B*2

> *σ*2*B*<sup>2</sup> − *<sup>σ</sup>B*<sup>2</sup><sup>2</sup> �*B*2�

 ,

2

 *σB*<sup>2</sup> �*B*2�

 ,

> .

<sup>2</sup> <sup>±</sup>√−*DI* . (49)

*x*=*b*

*χ*��.

*ψ*�� − *ψ*�

> 0, (50)

*H* ≡ 

the Euler differential equation. Its solution is

and sufficient stability condition for peeling modes

<sup>Δ</sup> <sup>=</sup> <sup>1</sup>

<sup>2</sup> <sup>+</sup> *<sup>S</sup>*−<sup>1</sup>

minimization of plasma energy Lortz (1975) Wesson (1978).

 *x*2 2 *<sup>ξ</sup>*<sup>∗</sup> *<sup>d</sup><sup>ξ</sup> dx* <sup>+</sup> *<sup>ξ</sup>*

stability criterion is simply −*DI* > 0.

*Mc* <sup>≡</sup> *NiMi k*2 *<sup>u</sup>*Λ<sup>2</sup>

*Mt* <sup>≡</sup> *NiMi k*2 *u*Λ2*P*�<sup>2</sup>

*<sup>B</sup>*2/|∇*V*<sup>|</sup>

Λ

major and minor radii Mikhailovsky (1974) Zheng & Tessarotto (1994b).

2

 *B*<sup>2</sup> |∇*V*|<sup>2</sup>

 *B*<sup>2</sup> |∇*V*|<sup>2</sup>

In this section we review high-n ballooning mode theory. The stability criterion for interchange modes takes into account only average magnetic well effect. As it is well-known tokamak plasmas have good and bad curvature regions, referring to whether pressure gradient and magnetic field line curvature point in same direction or not. Usually bad curvature region lies on low field side of plasma torus; good curvature region on high field side. Although tokamaks are usually designed to have average good curvature, i.e., Mercier stable, the ballooning modes can still develop as soon as the release of plasma thermal energy on bad curvature region is sufficient to counter the magnetic energy resulting from field line bending Connor et al. (1979) Chance et al. (1979). In difference from interchange modes ballooning modes have high toroidal mode number *n*, while interchange modes can be either low and high *n*. Also, ballooning modes allow normal and geodesic wave lengths to be of same order *<sup>λ</sup>*<sup>⊥</sup> <sup>∼</sup> *<sup>λ</sup>*∧, but both of them are much smaller than parallel wave length *<sup>λ</sup>*�.

We first derive ballooning mode equation. In high *n* limit, both components of perpendicular momentum equation, Eqs. (31) and (32), give the same result

$$
\mathbf{B} \cdot \delta \mathbf{B} + \delta \mathbf{P} = -(B^2 + \Gamma P) \nabla \cdot \mathbf{\tilde{g}} + \mathbf{B} \cdot \nabla \left(\frac{\mathbf{B} \cdot \mathbf{\tilde{g}}}{B^2}\right) - 2\mathbf{k} \cdot \mathbf{\tilde{g}} = 0. \tag{53}
$$

In lowest order, one has ∇ · *<sup>ξ</sup>*<sup>⊥</sup> <sup>∼</sup> *<sup>ξ</sup>*/*R*. This allows to introduce the so-called stream function *δϕ* Chance et al. (1979): *<sup>ξ</sup>*<sup>⊥</sup> <sup>=</sup> **<sup>B</sup>** × ∇*δϕ*/*B*2. Equation (34) then becomes,

$$\mathbf{B} \cdot \nabla \frac{1}{B^2} \nabla \cdot \left( B^2 \nabla\_\perp \frac{\mathbf{B} \cdot \nabla \delta \rho}{B^2} \right) + \nabla \cdot \left( \rho \omega^2 \frac{\nabla\_\perp \delta \rho}{B^2} \right)$$

$$+ P\_\Psi' \nabla \times \frac{\mathbf{B}}{B^2} \cdot \nabla \left( \frac{\mathbf{B} \times \nabla \Psi}{B^2} \cdot \nabla \delta \rho \right) + \Gamma P \nabla \times \frac{\mathbf{B}}{B^2} \cdot \nabla \nabla \cdot \mathbf{f} = 0. \tag{54}$$

Equation (33), meanwhile, can be reduced to

$$
\Gamma P \mathbf{B} \cdot \nabla \left(\frac{1}{B^2} \mathbf{B} \cdot \nabla \nabla \cdot \boldsymbol{\mathfrak{f}}\right) + \rho\_m \omega^2 \nabla \cdot \boldsymbol{\mathfrak{f}} = \rho\_m \omega^2 \frac{2 \mathbf{B} \times \boldsymbol{\kappa}}{B^2} \cdot \nabla \delta \boldsymbol{\mathfrak{g}},\tag{55}
$$

where Eq. (53) has been used.

The key formalism to ballooning mode theory is the so-called ballooning representation Lee & Van Dam (1977) Connor et al. (1979). Here, we outline its physics basis and derivation,

in Toroidal Plasma Confinement 15

Overview of Magnetohydrodynamics Theory in Toroidal Plasma Confinement 15

Uniqueness and inversion of ballooning mode representation were proved in Ref. Hazeltine

With ballooning mode representation described, we can proceed to derive ballooning mode equation. It is convenient to use the so-called Celbsch coordinates (*ψ*, *β*, *θ*) to construct

Eq. (62) to Eqs. (54) and (55) and employ the high *n* ordering, one can obtain following coupled

∇ × **<sup>B</sup>**

where *δ*Ξ = *i*∇ · *ξ*/*n*. These two equations are coupled second order differential equations. The derivatives here are along a reference magnetic field line labeled by *ψ* and *β*. The boundary conditions are *δϕ*, *δ*Ξ → 0 at *θ* → ±∞ to guarantee the convergence of the Laplace

In studying ballooning stability at finite beta equilibrium, the so called steep-pressure-gradient equilibrium model is often used Connor et al. (1978) Greene & Chance (1981). In this model, finite beta modification is only taken into account for magnetic shear, while others remain to their low beta values. This model has been proved to be successful for ballooning mode studies. Here, we outline the formulation in Ref. Berk et al. (1983). Noting that *β* = *ζ* − *qθ*, one can see that the magnetic shear effect resides at the quantity ∇*β* in the ballooning mode equations (63) and (64). From Eq. (22) one can prove that

where Λ*<sup>s</sup>* is the so-called shear parameter and can be obtained by applying operator

*<sup>d</sup><sup>θ</sup>* <sup>=</sup> <sup>−</sup>**<sup>B</sup>** × ∇*<sup>χ</sup>* ·∇× (**<sup>B</sup>** × ∇*χ*)

We need to determine finite beta modification to Λ. We assume that *χ* = *χ*<sup>0</sup> + *χ*<sup>1</sup> and *β* = *β*<sup>0</sup> + *β*1, where *χ*<sup>0</sup> and *β*<sup>0</sup> are low beta values and *χ*<sup>1</sup> and *β*<sup>1</sup> represent finite beta modifications.

Noting that in the curl operation on left hand side only the gradient component in ∇*χ*

direction needs to be taken, i.e., ∇× →∇*χ*0*∂*/*∂χ*×, equation (67) can be solved

*∂χ*

2*Pλ*∇*β*<sup>0</sup> + **B**0*P* + ∇*Q* = ∇*χ*<sup>0</sup> × ∇*β*<sup>1</sup> + ∇*χ*<sup>1</sup> × ∇*β*0, (68)

2*λ*∇*χ*<sup>0</sup> × ∇*β*<sup>0</sup> +

∇*β* = Λ*s*∇*χ* +

<sup>+</sup> *<sup>ρ</sup>mω*2*δ*<sup>Ξ</sup> <sup>=</sup> *<sup>ρ</sup>mω*<sup>2</sup> <sup>2</sup>**<sup>B</sup>** <sup>×</sup> *<sup>κ</sup>*

**B** × ∇*χ*

*<sup>B</sup>*<sup>2</sup> · ∇*βδϕ* <sup>+</sup> <sup>Γ</sup>*P*∇ × **<sup>B</sup>**

<sup>2</sup>*δϕ* = 0, (63)

(*∂*/*∂θ*). Applying

*<sup>B</sup>*<sup>2</sup> · ∇*βδ*<sup>Ξ</sup>

*<sup>B</sup>*<sup>2</sup> · ∇*βδϕ*, (64)



**B**<sup>0</sup> × ∇*χ*<sup>0</sup> *B*2

. (67)

ballooning mode equations. In this coordinates ∇→−*in*∇*β* and **B** · ∇ = *χ*�

+ *P*�

et al. (1981).

ballooning mode equations

*<sup>χ</sup>*� *<sup>∂</sup> ∂θ*

+ *ω*2 *ω*2 *A* |∇*β*|

transform in Eq. (59).

**B** × ∇*χ* ·∇× ··· on Eq. (65),

<sup>Γ</sup>*Pχ*� *<sup>∂</sup>*


*∂θ* <sup>1</sup>

<sup>2</sup>*χ*� *<sup>∂</sup> ∂θ δϕ*

*<sup>B</sup>*<sup>2</sup> *<sup>χ</sup>*� *<sup>∂</sup> ∂θ <sup>δ</sup>*<sup>Ξ</sup> 

*<sup>χ</sup>*� *<sup>d</sup>*Λ*<sup>s</sup>*

The linearized Ampere's law can be written as follows:

∇ × (∇*χ*<sup>0</sup> × ∇*β*<sup>1</sup> <sup>+</sup> <sup>∇</sup>*χ*<sup>1</sup> × ∇*β*0) = **<sup>J</sup>** <sup>=</sup> *<sup>∂</sup><sup>P</sup>*

especially to explain the equivalence of two kinds of representations in Refs. Lee & Van Dam (1977) and Connor et al. (1979). In tokamak geometry one can introduce the following Fourier decomposition:

$$\delta\varphi(n\eta,\theta,\zeta) = \sum\_{m=-\infty}^{+\infty} \delta\varphi\_m(n\eta) \exp\{-in\zeta + m\theta\}.\tag{56}$$

For simply to describe ballooning mode representation we have used *nq* as flux surface label. This is allowed for systems with finite magnetic shear, in which the ballooning representation applies. For high *n* modes the distance of mode rational surfaces is of order 1/*n*, which is much smaller than equilibrium scale length. Therefore, in lowest order we can neglect the spatial variance of equilibrium quantities and require mode Fourier harmonics to have the so-called ballooning invariance:

$$
\delta\varphi\_m(nq) = \delta\varphi(nq - m)\_\prime \tag{57}
$$

so that the Fourier decomposistion in Eq. (57) can be expressed as

$$\delta\varphi(n\eta,\theta,\zeta) = \sum\_{m=-\infty}^{+\infty} \delta\varphi(n\eta - m) \exp\{-in\zeta + m\theta\}.\tag{58}$$

We can further introduce the Laplace tranform

$$
\delta\varphi(n\eta) = \frac{1}{2\pi} \int\_{-\infty}^{+\infty} \delta\varphi(\eta) \exp\{in\eta\eta\} d\eta. \tag{59}
$$

Using this transform Eq. (58) can be written as

$$\delta\varphi(n\eta,\theta,\phi) = \frac{1}{2\pi} \exp\{-in\zeta\} \int\_{-\infty}^{+\infty} \delta\varphi(\eta) \sum\_{m} \exp\{im(\theta-\eta)\} d\eta. \tag{60}$$

Noting that

$$\frac{1}{2\pi} \sum\_{m=-\infty}^{+\infty} \exp\{im(\theta - \eta)\} = \sum\_{j=-\infty}^{+\infty} \delta(\eta - \theta - j2\pi)\_j$$

Equation (60) is transformed to

$$\delta\varphi(nq\_{\prime}\theta\_{\prime}\zeta\_{\prime}) = \sum\_{j=-\infty}^{+\infty} \delta\varphi(\theta + j2\pi) \exp\{-in(\zeta - q(\theta + j2\pi))\}.\tag{61}$$

This indicates that we can represent high *n* modes at a reference surface as

$$
\delta\varphi(n\eta,\theta,\zeta) = \delta\varphi(\theta)\exp\{-in\beta\}\tag{62}
$$

without concern of periodicity requirement. Here, *β* ≡ *ζ* − *qθ*. The periodic eigenfunction can always be formed through the summation in Eq. (61). This representation characterizes the most important feature of ballooning modes in a plasma torus that perpendicular wave number is much larger than parallel one: *<sup>k</sup>*<sup>⊥</sup> � *<sup>k</sup>*�. This reduction shows the equivalence of two kinds of representations in Eqs. (58) and (61) Lee & Van Dam (1977) Connor et al. (1979). 14 Will-be-set-by-IN-TECH

especially to explain the equivalence of two kinds of representations in Refs. Lee & Van Dam (1977) and Connor et al. (1979). In tokamak geometry one can introduce the following Fourier

For simply to describe ballooning mode representation we have used *nq* as flux surface label. This is allowed for systems with finite magnetic shear, in which the ballooning representation applies. For high *n* modes the distance of mode rational surfaces is of order 1/*n*, which is much smaller than equilibrium scale length. Therefore, in lowest order we can neglect the spatial variance of equilibrium quantities and require mode Fourier harmonics to have the

*δϕm*(*nq*) exp{−*inζ* + *mθ*}. (56)

*δϕm*(*nq*) = *δϕ*(*nq* − *m*), (57)

*δ*(*η* − *θ* − *j*2*π*),

*δϕ*(*θ* + *j*2*π*) exp{−*in*(*ζ* − *q*(*θ* + *j*2*π*))}. (61)

*δϕ*(*nq*, *θ*, *ζ*) = *δϕ*(*θ*) exp{−*inβ*} (62)

*δϕ*(*nq* − *m*) exp{−*inζ* + *mθ*}. (58)

*δϕ*(*η*) exp{*inqη*}*dη*. (59)

exp{*im*(*θ* − *η*)}*dη*. (60)

+∞ ∑ *m*=−∞

*δϕ*(*nq*, *θ*, *ζ*) =

so that the Fourier decomposistion in Eq. (57) can be expressed as

*δϕ*(*nq*) = <sup>1</sup>

+∞ ∑ *m*=−∞

2*π*

<sup>2</sup>*<sup>π</sup>* exp{−*inζ*}

+∞ ∑ *j*=−∞

This indicates that we can represent high *n* modes at a reference surface as

exp{*im*(*θ* − *η*)} =

 +∞ −∞

> +∞ −∞

without concern of periodicity requirement. Here, *β* ≡ *ζ* − *qθ*. The periodic eigenfunction can always be formed through the summation in Eq. (61). This representation characterizes the most important feature of ballooning modes in a plasma torus that perpendicular wave number is much larger than parallel one: *<sup>k</sup>*<sup>⊥</sup> � *<sup>k</sup>*�. This reduction shows the equivalence of two kinds of representations in Eqs. (58) and (61) Lee & Van Dam (1977) Connor et al. (1979).

*δϕ*(*η*)∑ *m*

+∞ ∑ *j*=−∞

*δϕ*(*nq*, *θ*, *ζ*) =

We can further introduce the Laplace tranform

Using this transform Eq. (58) can be written as

*δϕ*(*nq*, *<sup>θ</sup>*, *<sup>φ</sup>*) = <sup>1</sup>

1 2*π*

*δϕ*(*nq*, *θ*, *ζ*) =

Equation (60) is transformed to

+∞ ∑ *m*=−∞

decomposition:

Noting that

so-called ballooning invariance:

Uniqueness and inversion of ballooning mode representation were proved in Ref. Hazeltine et al. (1981).

With ballooning mode representation described, we can proceed to derive ballooning mode equation. It is convenient to use the so-called Celbsch coordinates (*ψ*, *β*, *θ*) to construct ballooning mode equations. In this coordinates ∇→−*in*∇*β* and **B** · ∇ = *χ*� (*∂*/*∂θ*). Applying Eq. (62) to Eqs. (54) and (55) and employ the high *n* ordering, one can obtain following coupled ballooning mode equations

$$\begin{split} \chi^{\prime} \frac{\partial}{\partial \theta} \left( |\nabla \beta|^{2} \chi^{\prime} \frac{\partial}{\partial \theta} \delta \varphi \right) &+ P^{\prime} \nabla \times \frac{\mathbf{B}}{B^{2}} \cdot \nabla \beta \delta \varphi + \Gamma P \nabla \times \frac{\mathbf{B}}{B^{2}} \cdot \nabla \beta \delta \Sigma \\ + \frac{\omega^{2}}{\omega\_{A}^{2}} |\nabla \beta|^{2} \delta \varphi &= 0, \end{split} \tag{63}$$

$$
\stackrel{\smile}{\Gamma P \chi'} \frac{\partial}{\partial \theta} \left( \frac{1}{B^2} \chi' \frac{\partial}{\partial \theta} \delta \Xi \right) + \rho\_m \omega^2 \delta \Xi = \rho\_m \omega^2 \frac{2\mathbf{B} \times \boldsymbol{\kappa}}{B^2} \cdot \nabla \beta \delta \varphi,\tag{64}
$$

where *δ*Ξ = *i*∇ · *ξ*/*n*. These two equations are coupled second order differential equations. The derivatives here are along a reference magnetic field line labeled by *ψ* and *β*. The boundary conditions are *δϕ*, *δ*Ξ → 0 at *θ* → ±∞ to guarantee the convergence of the Laplace transform in Eq. (59).

In studying ballooning stability at finite beta equilibrium, the so called steep-pressure-gradient equilibrium model is often used Connor et al. (1978) Greene & Chance (1981). In this model, finite beta modification is only taken into account for magnetic shear, while others remain to their low beta values. This model has been proved to be successful for ballooning mode studies. Here, we outline the formulation in Ref. Berk et al. (1983). Noting that *β* = *ζ* − *qθ*, one can see that the magnetic shear effect resides at the quantity ∇*β* in the ballooning mode equations (63) and (64). From Eq. (22) one can prove that

$$
\nabla \beta = \Lambda\_s \nabla \chi + \frac{\mathbf{B} \times \nabla \chi}{|\nabla \chi|^2} \,' \,\tag{65}
$$

where Λ*<sup>s</sup>* is the so-called shear parameter and can be obtained by applying operator **B** × ∇*χ* ·∇× ··· on Eq. (65),

$$\chi'\frac{d\Lambda\_s}{d\theta} = -\frac{\mathbf{B}\times\nabla\chi\cdot\nabla\times(\mathbf{B}\times\nabla\chi)}{|\nabla\chi|^4}.\tag{66}$$

We need to determine finite beta modification to Λ. We assume that *χ* = *χ*<sup>0</sup> + *χ*<sup>1</sup> and *β* = *β*<sup>0</sup> + *β*1, where *χ*<sup>0</sup> and *β*<sup>0</sup> are low beta values and *χ*<sup>1</sup> and *β*<sup>1</sup> represent finite beta modifications. The linearized Ampere's law can be written as follows:

$$\nabla \times \left(\nabla \chi\_0 \times \nabla \beta\_1 + \nabla \chi\_1 \times \nabla \beta\_0\right) = \mathbf{J} = \frac{\partial P}{\partial \chi} \left(2\lambda \nabla \chi\_0 \times \nabla \beta\_0 + \frac{\mathbf{B}\_0 \times \nabla \chi\_0}{B^2}\right). \tag{67}$$

Noting that in the curl operation on left hand side only the gradient component in ∇*χ* direction needs to be taken, i.e., ∇× →∇*χ*0*∂*/*∂χ*×, equation (67) can be solved

$$2P\lambda \nabla \beta\_0 + \mathbf{B}\_0 P + \nabla Q = \nabla \chi\_0 \times \nabla \beta\_1 + \nabla \chi\_1 \times \nabla \beta\_{0\prime} \tag{68}$$

in Toroidal Plasma Confinement 17

Overview of Magnetohydrodynamics Theory in Toroidal Plasma Confinement 17

Here, it has been considered that in sound wave scale the slow variable *sθ* can be regarded as

*ω*2 *ω*2 *A*

<sup>√</sup>*r*/*<sup>R</sup>* for large aspect ratio case Mikhailovsky (1974) Zheng & Tessarotto

*<sup>s</sup>*)*δϕ* = 0. (76)

(1 + 2*q*2)*s*

(cos *θ* + Λ*<sup>s</sup>* sin *θ*)*δϕ*. (75)

*<sup>B</sup>*<sup>2</sup> (cos *<sup>θ</sup>* <sup>+</sup> <sup>Λ</sup>*<sup>s</sup>* sin *<sup>θ</sup>*)2*δϕ*

<sup>2</sup>*θ*2*δϕ* = 0. (74)

+ *α*(cos *θ* + Λ*<sup>s</sup>* sin *θ*)*δϕ* +

Here, we see that the sound wave coupling results in the so-called apparent mass effect: i.e., the inertia term is enhanced by a factor (1 + 2*q*2) Greene & Johnson (1962). In the kinetic description the 2*q*<sup>2</sup> term is further boosted by the so-called small particle speed effect to

(1994b). In the marginal stability *ω*<sup>2</sup> = 0 the ballooning stability can be determined by Newcomb's theorem 5 Newcomb (1960): system is unstable, if and only if the solution of Eq. (48) vanishes two or more points. Refs. Connor et al. (1978) and Lortz & Nührenberg

In the intermediate frequency regime the first term in Eq. (72) can be neglected and therefore

<sup>+</sup> *<sup>α</sup>*(cos *<sup>θ</sup>* <sup>+</sup> <sup>Λ</sup>*<sup>s</sup>* sin *<sup>θ</sup>*)*δϕ* <sup>−</sup> <sup>4</sup>Γ*q*2*<sup>P</sup>*

The sound wave coupling term (3rd term) results in the so-called second harmonic TAE in the

In this subsection we review TAE theory. In the last two subsections we see that interchange and ballooning modes are characterized by having only single dominant or resonant mode at resonance surfaces. In particular their resonance surfaces locates at mode rational surface where *m* − *nq* = 0. TAEs are different from them. TAEs involve two mode coupling. In particular, the first TAEs are centered at the surface where *q* = (*m*<sup>0</sup> + 1/2)/*n*. Two neighboring Fourier modes (*m*<sup>0</sup> and *m*<sup>0</sup> + 1) propagate roughly with same speed *vA*/2*Rq* but in opposite directions. They form a standing wave. The toroidal geometry can induce the first frequency gap so that the standing wave becomes an eigen mode, i.e., TAEs Cheng et al. (1985) Rosenbluth et al. (1992). In the second TAE case, although they have same mode resonance surfaces as interchange and ballooning modes, *m*<sup>0</sup> ± 1 mode coupling is involved to form standing 2nd TAEs. The frequency gap for second TAEs in circular cross section case

To explain two mode coupling picture, we consider tokamak model equilibrium with circular cross section, low beta, and large aspect ratio (i.e., 1/*�* = *R*/*a* � 1). There is a review paper on TAEs Vlad et al. (1999). Here, we describe the local dispersion relation for even and odd modes and explain the 2nd TAEs together with the 1st TAEs. The magnetic field in this model

constant. Inserting Eq. (73) into Eq. (71) yields that

(1978). have obtained the stability boundaries for ballooning modes.

*<sup>δ</sup>*<sup>Ξ</sup> <sup>=</sup> <sup>−</sup> <sup>2</sup>

Inserting Eq. (75) into Eq. (71) yields Tang et al. (1980)

is due to plasma compressibility effect Zheng & Chen (1998).

*R*2*B<sup>θ</sup>*

(1 + Λ<sup>2</sup> *s*) *dδϕ dθ*

*d dθ* 

become of order 2*q*2/

*d dθ* 

+ *ω*2 *ω*2 *A*

(1 + Λ<sup>2</sup> *s*) *dδϕ dθ*

(1 + Λ<sup>2</sup>

**4.4 Toroidal Alfvén eigen modes**

circular cross section case Zheng et al. (1999).

one obtains

where ∇*Q* is integration factor. Taking the divergence of Eq. (68) for only *∂*{*P*, *Q*}/*∂χ* large gives

$$2\lambda \frac{\partial P}{\partial \chi} (\nabla \chi\_0 \cdot \nabla \beta\_0) + \frac{\partial^2 Q}{\partial \chi^2} |\nabla \chi\_0|^2 = 0,$$

and therefore

$$
\partial \mathbb{Q} / \partial \chi = -2\lambda P \nabla \chi \cdot \nabla \mathbb{B} / |\nabla \chi|^2. \tag{69}
$$

Now taking the dot product of Eq. (68) with ∇*β*<sup>0</sup> gives

$$2P\lambda|\nabla\beta\_0|^2 + (\nabla\chi\_0 \cdot \nabla\beta\_0)\frac{\partial Q}{\partial\chi} = -\chi'\frac{\partial\beta\_1}{\partial\theta}\lambda$$

We can remove subscript 0 afterward for brevity. Now substituting Eq. (69) for *∂Q*/*∂χ* and noting that *∂β*1/*∂χ* ≡ Λ*s*1, one finds that

$$
\chi' \frac{\partial \Lambda\_{\rm s1}}{\partial \theta} = -2\lambda \frac{\partial P}{\partial \chi} \frac{B^2}{|\nabla \chi|^2}.
$$

Therefore, the shear parameter can be evaluated as follows

$$\chi'\frac{d\Lambda\_s}{d\theta} = -\frac{\mathbf{B}\times\nabla\chi\cdot\nabla\times(\mathbf{B}\times\nabla\chi)}{|\nabla\chi|^4} - 2\lambda\frac{\partial P}{\partial\chi}\frac{\mathcal{B}^2}{|\nabla\chi|^2}.\tag{70}$$

The second term here gives rise to the finite beta modification to shear parameter Λ*<sup>s</sup>* in steep pressure gradient model. The rest parameters here and in ballooning equations (63) and (64) can be evaluated with low beta values.

We now consider tokamak model equilibrium with circular cross section, low beta, and large aspect ratio (i.e., 1/*�* = *R*/*a* � 1). The magnetic field in this model can be expressed as **B** = *Bφ*(*r*)/(1 + *�* cos *θ*)**e***<sup>φ</sup>* + *B<sup>θ</sup>* (*r*)**e***<sup>θ</sup>* . The shear parameter can be expressed as Λ*<sup>s</sup>* = *s*(*θ* − *<sup>θ</sup>k*) <sup>−</sup> *<sup>α</sup>* sin *<sup>θ</sup>*. Here, *<sup>α</sup>* <sup>=</sup> <sup>−</sup>(2*Rq*2/*B*2)(*dP*/*dr*), *<sup>s</sup>* <sup>=</sup> *<sup>d</sup>* ln *<sup>q</sup>*/*<sup>d</sup>* ln *<sup>r</sup>*, and *<sup>θ</sup><sup>k</sup>* is integration constant. Therefore, ballooning equations (63) and (64) can be reduced to

$$\begin{split} &\frac{d}{d\theta} \left( (1+\Lambda\_s^2) \frac{d\delta\varphi}{d\theta} \right) + a(\cos\theta + \Lambda\_s \sin\theta)\delta\varphi + \frac{2\Gamma RrqP}{B}(\cos\theta + \Lambda\_s \sin\theta)\delta\Sigma\\ &+ \frac{\omega^2}{\omega\_A^2} (1+\Lambda\_s^2)\delta\varphi = 0,\end{split} \tag{71}$$

$$\frac{\Gamma P}{R^2 q^2} \frac{\partial^2 \delta \Xi}{\partial \theta^2} + \rho\_m \omega^2 \delta \Xi = -\frac{2 \rho\_m \omega^2}{R^2 B\_\theta} (\cos \theta + \Lambda\_s \sin \theta) \delta \rho. \tag{72}$$

To further analyze this set of equations it is interesting to consider two limits: the low frequency (*ω* � *ωsi*) and intermediate frequency limit (*ωsi* � *ω* � *ωse*). In the low frequency limit the second term on left hand side of Eq. (72) can be neglected and inertia term is only important in the outer region *θ* → ∞. Equation (72) can be solved to yield

$$
\delta\Xi = \frac{2\rho\_m q^2 \omega^2}{\Gamma P B\_\theta} s\theta \sin\theta \delta\varphi. \tag{73}
$$

Here, it has been considered that in sound wave scale the slow variable *sθ* can be regarded as constant. Inserting Eq. (73) into Eq. (71) yields that

$$\frac{d}{d\theta}\left((1+\Lambda\_s^2)\frac{d\delta\varphi}{d\theta}\right) + a(\cos\theta + \Lambda\_s\sin\theta)\delta\varphi + \frac{\omega^2}{\omega\_A^2}(1+2\eta^2)s^2\theta^2\delta\varphi = 0.\tag{74}$$

Here, we see that the sound wave coupling results in the so-called apparent mass effect: i.e., the inertia term is enhanced by a factor (1 + 2*q*2) Greene & Johnson (1962). In the kinetic description the 2*q*<sup>2</sup> term is further boosted by the so-called small particle speed effect to become of order 2*q*2/ <sup>√</sup>*r*/*<sup>R</sup>* for large aspect ratio case Mikhailovsky (1974) Zheng & Tessarotto (1994b). In the marginal stability *ω*<sup>2</sup> = 0 the ballooning stability can be determined by Newcomb's theorem 5 Newcomb (1960): system is unstable, if and only if the solution of Eq. (48) vanishes two or more points. Refs. Connor et al. (1978) and Lortz & Nührenberg (1978). have obtained the stability boundaries for ballooning modes.

In the intermediate frequency regime the first term in Eq. (72) can be neglected and therefore one obtains

$$
\delta\Sigma = -\frac{2}{R^2 B\_\theta} (\cos\theta + \Lambda\_s \sin\theta) \delta\varphi. \tag{75}
$$

Inserting Eq. (75) into Eq. (71) yields Tang et al. (1980)

$$\begin{split} &\frac{d}{d\theta}\left( (1+\Lambda\_s^2)\frac{d\delta\rho}{d\theta} \right) + a(\cos\theta + \Lambda\_s\sin\theta)\delta\rho - \frac{4\Gamma\eta^2 P}{B^2}(\cos\theta + \Lambda\_s\sin\theta)^2 \delta\rho \\ &+ \frac{\omega^2}{\omega\_A^2}(1+\Lambda\_s^2)\delta\rho = 0. \end{split} \tag{76}$$

The sound wave coupling term (3rd term) results in the so-called second harmonic TAE in the circular cross section case Zheng et al. (1999).

#### **4.4 Toroidal Alfvén eigen modes**

16 Will-be-set-by-IN-TECH

where ∇*Q* is integration factor. Taking the divergence of Eq. (68) for only *∂*{*P*, *Q*}/*∂χ* large

*∂Q*/*∂χ* = −2*λP*∇*χ* · ∇*β*/|∇*χ*|

<sup>2</sup> + (∇*χ*<sup>0</sup> · ∇*β*0)

*∂θ* <sup>=</sup> <sup>−</sup>2*<sup>λ</sup> <sup>∂</sup><sup>P</sup>*

*<sup>χ</sup>*� *<sup>∂</sup>*Λ*s*<sup>1</sup>

*<sup>d</sup><sup>θ</sup>* <sup>=</sup> <sup>−</sup>**<sup>B</sup>** × ∇*<sup>χ</sup>* ·∇× (**<sup>B</sup>** × ∇*χ*)

We can remove subscript 0 afterward for brevity. Now substituting Eq. (69) for *∂Q*/*∂χ* and

The second term here gives rise to the finite beta modification to shear parameter Λ*<sup>s</sup>* in steep pressure gradient model. The rest parameters here and in ballooning equations (63) and (64)

We now consider tokamak model equilibrium with circular cross section, low beta, and large aspect ratio (i.e., 1/*�* = *R*/*a* � 1). The magnetic field in this model can be expressed as **B** = *Bφ*(*r*)/(1 + *�* cos *θ*)**e***<sup>φ</sup>* + *B<sup>θ</sup>* (*r*)**e***<sup>θ</sup>* . The shear parameter can be expressed as Λ*<sup>s</sup>* = *s*(*θ* − *<sup>θ</sup>k*) <sup>−</sup> *<sup>α</sup>* sin *<sup>θ</sup>*. Here, *<sup>α</sup>* <sup>=</sup> <sup>−</sup>(2*Rq*2/*B*2)(*dP*/*dr*), *<sup>s</sup>* <sup>=</sup> *<sup>d</sup>* ln *<sup>q</sup>*/*<sup>d</sup>* ln *<sup>r</sup>*, and *<sup>θ</sup><sup>k</sup>* is integration constant.

+ *α*(cos *θ* + Λ*<sup>s</sup>* sin *θ*)*δϕ* +

*R*2*B<sup>θ</sup>*

*<sup>δ</sup>*<sup>Ξ</sup> <sup>=</sup> <sup>2</sup>*ρmq*2*ω*<sup>2</sup> Γ*PB<sup>θ</sup>*

important in the outer region *θ* → ∞. Equation (72) can be solved to yield

To further analyze this set of equations it is interesting to consider two limits: the low frequency (*ω* � *ωsi*) and intermediate frequency limit (*ωsi* � *ω* � *ωse*). In the low frequency limit the second term on left hand side of Eq. (72) can be neglected and inertia term is only

*∂χ*

*∂χ*<sup>2</sup> |∇*χ*0<sup>|</sup>

*∂Q*

*B*2 |∇*χ*|<sup>2</sup> .


*∂χ* <sup>=</sup> <sup>−</sup>*χ*� *∂β*<sup>1</sup>

*∂θ* ,

*∂χ*

2Γ*RrqP*

*<sup>s</sup>*)*δϕ* = 0, (71)

*B*2

*<sup>B</sup>* (cos *<sup>θ</sup>* <sup>+</sup> <sup>Λ</sup>*<sup>s</sup>* sin *<sup>θ</sup>*)*δ*<sup>Ξ</sup>

(cos *θ* + Λ*<sup>s</sup>* sin *θ*)*δϕ*. (72)

*sθ* sin *θδϕ*. (73)

<sup>2</sup> = 0,

2. (69)


*∂χ* (∇*χ*<sup>0</sup> · ∇*β*0) + *<sup>∂</sup>*2*<sup>Q</sup>*

<sup>2</sup>*<sup>λ</sup> <sup>∂</sup><sup>P</sup>*

Now taking the dot product of Eq. (68) with ∇*β*<sup>0</sup> gives

noting that *∂β*1/*∂χ* ≡ Λ*s*1, one finds that

*<sup>χ</sup>*� *<sup>d</sup>*Λ*<sup>s</sup>*

can be evaluated with low beta values.

(1 + Λ<sup>2</sup> *s*) *dδϕ dθ*

(1 + Λ<sup>2</sup>

*∂*2*δ*Ξ

*d dθ* 

+ *ω*2 *ω*2 *A*

Γ*P R*2*q*<sup>2</sup> 2*Pλ*|∇*β*0|

Therefore, the shear parameter can be evaluated as follows

Therefore, ballooning equations (63) and (64) can be reduced to

*∂θ*<sup>2</sup> <sup>+</sup> *<sup>ρ</sup>mω*2*δ*<sup>Ξ</sup> <sup>=</sup> <sup>−</sup>2*ρmω*<sup>2</sup>

gives

and therefore

In this subsection we review TAE theory. In the last two subsections we see that interchange and ballooning modes are characterized by having only single dominant or resonant mode at resonance surfaces. In particular their resonance surfaces locates at mode rational surface where *m* − *nq* = 0. TAEs are different from them. TAEs involve two mode coupling. In particular, the first TAEs are centered at the surface where *q* = (*m*<sup>0</sup> + 1/2)/*n*. Two neighboring Fourier modes (*m*<sup>0</sup> and *m*<sup>0</sup> + 1) propagate roughly with same speed *vA*/2*Rq* but in opposite directions. They form a standing wave. The toroidal geometry can induce the first frequency gap so that the standing wave becomes an eigen mode, i.e., TAEs Cheng et al. (1985) Rosenbluth et al. (1992). In the second TAE case, although they have same mode resonance surfaces as interchange and ballooning modes, *m*<sup>0</sup> ± 1 mode coupling is involved to form standing 2nd TAEs. The frequency gap for second TAEs in circular cross section case is due to plasma compressibility effect Zheng & Chen (1998).

To explain two mode coupling picture, we consider tokamak model equilibrium with circular cross section, low beta, and large aspect ratio (i.e., 1/*�* = *R*/*a* � 1). There is a review paper on TAEs Vlad et al. (1999). Here, we describe the local dispersion relation for even and odd modes and explain the 2nd TAEs together with the 1st TAEs. The magnetic field in this model

in Toroidal Plasma Confinement 19

Overview of Magnetohydrodynamics Theory in Toroidal Plasma Confinement 19

where D is 2 × 2 matrix and *A*<sup>±</sup> are integration constants. Integration of Eq. (78) across

− � *<sup>δ</sup>q*<sup>+</sup> *δq*−

the outer solutions to the left and right of singular layer. As soon as Δ<sup>±</sup> are computed from outer regions, Eq. (79) can be used to determine the frequency. In general this frequency can

The denominators of integrations in Eq. (79) involve det |D|. The singularity emerges at det |D| = 0. In this case the Landau integration orbit needs to be used, as in the case for particle-wave resonances, and continuum damping occurs Berk et al. (1992). The so-called 1st TAE frequency gap, in which eigen modes can exit without continuum damping, can be

> 1 2*m* + 1

���� � *δω* 2*ω*<sup>0</sup>

1 − 1/(2*m* + 1)

*<sup>n</sup>δ<sup>q</sup>* <sup>−</sup> *�* 4 �

> �<sup>2</sup> − *�*2 16 �

> > 2 .

1 − 1/(2*m* + 1)

�

� � *δω* 2*ω*<sup>0</sup> +

<sup>2</sup>*ω*0(2*m*<sup>0</sup> <sup>+</sup> <sup>1</sup>) <sup>±</sup>

To exclude real *δq* solution for det D = 0, mode frequency must fall in the gap between *δω*±,

Alfvén eigen modes with frequency inside this gap. They are marginally stable and tend to be excited by resonances with energetic particles. Note that the gap width is proportional to *�*. In cylinder limit the gap vanishes. Therefore, existence of TAEs is due to toroidal effects. Also, we note that the dispersion relation, Eq. (79), allows two types of TAEs: even and odd

We have discussed the 1st TEA theory through coupling of neighboring modes. In similar way one can also develop the 2nd TAE theory through coupling of *m* ± 1 modes Zheng & Chen (1998). If FLR effects are taken into consideration, the Alfvén types of singularities can be resolved, so that discrete modes can emerge in the continuum. This types of modes are referred to as kinetic TAEs (i.e., KTAEs). Due to correction of gyrokinetic theory Zheng et al. (2007), several missing FLR effects are recovered. Consequently, KTAE theories by far need to

D11*dδq* det |D| ⎞ ⎠ =

�� *<sup>δ</sup>q*<sup>+</sup> *δq*−

= *n*2*δq*2.

<sup>1</sup> <sup>−</sup> <sup>1</sup> (2*m* + 1)

2

2 �

. The 1st TAEs are

. (80)

*δq*<sup>+</sup>

D12*dδq* det |D| �<sup>2</sup>

*<sup>δ</sup>q*<sup>−</sup> /*A*<sup>±</sup> are determined by

. (79)

singular layer (i.e., from *δq*<sup>−</sup> to *δq*+) one obtains the dispersion relation

⎛ ⎝ *δφ*−| *δq*<sup>+</sup> *δq*− *A*<sup>−</sup>

⎞ ⎠

Here, D*ij* are D matrix elements and two parameters Δ<sup>±</sup> ≡ *δφ*±|

D22*dδq* det |D|

⎛ ⎝ *δφ*+| *δq*<sup>+</sup> *δq*− *A*+

be complex.

Its solution is

� 1 −

be reevaluated.

− � *<sup>δ</sup>q*<sup>+</sup> *δq*−

determined by condition det |D| = 0, i.e.,

�<sup>2</sup> �

1 2*m* + 1 *nδq* + *�* 4

*<sup>n</sup>*2*δ<sup>q</sup>* <sup>=</sup> *δω*

*δω*<sup>±</sup> <sup>=</sup> <sup>±</sup> *�*

One can obtain the gap width Δ*ω* = *δω*<sup>+</sup> − *δω*<sup>−</sup> = *�ω*<sup>0</sup>

types (*ϕ*±), depending on the values of Δ±.

2 *ω*0 �

� *δω* 2*ω*<sup>0</sup> +

� 1 2*m*<sup>0</sup> + 1

i.e., *ω*<sup>−</sup> < *ω* < *ω*+, where

can simply be expressed as **B** = *Bφ*(*r*)/(1 + *�* cos *θ*)**e***<sup>φ</sup>* + *B<sup>θ</sup>* (*r*)**e***<sup>θ</sup>* . The general case will be addressed in Sec. 5 with AEGIS code formalism. Since their frequency is much larger than shear Alfvén mode frequency, the compressional Alfvén modes are decoupled. Therefore, we can use Eqs. (54) and (55) as starting equations for TAE investigation. Noting that Alfvén frequency is much larger than sound wave frequency, the first term in Eq. (55) can be dropped. Adopting the Fourier decomposition in Eq. (56), the sound wave equation (55) becomes

$$i(\nabla \cdot \mathfrak{f})\_m = \frac{1}{BR} \left( \frac{d\varrho\_{m+1}}{dr} - \frac{d\varrho\_{m-1}}{dr} \right) \dots$$

Using this solution for ∇ · *ξ*, Eq. (54) can be reduced to Zheng et al. (1999)

$$\frac{d}{dr}\left[r^3\left(\frac{1}{q}-\frac{n}{m}\right)^2\frac{d}{dr}E\_m\right]-\frac{d}{dr}\left(r^3\frac{R^2\omega^2}{m^2v\_A^2}\frac{d}{dr}E\_m\right)-\epsilon\left(r^3\frac{R^2\omega^2}{m^2v\_A^2}\frac{d^2}{dr^2}E\_{m+1}\right)$$

$$-\epsilon\left(r^3\frac{R^2\omega^2}{m^2v\_A^2}\frac{d^2}{dr^2}E\_{m-1}\right)+\frac{\Gamma Pr^3}{B^2m^2}\frac{d^2E\_{m+2}}{dr^2}+\frac{\Gamma Pr^3}{B^2m^2}\frac{d^2E\_{m-2}}{dr^2}-wE\_m$$

$$-\frac{ar^2}{2mq^2}\frac{dE\_{m+1}}{dr}+\frac{ar^2}{2mq^2}\frac{dE\_{m-1}}{dr}-\frac{ar}{2q^2}E\_{m+1}+\frac{ar}{2q^2}E\_{m-1}=0,\tag{77}$$

where *Em* = *ϕm*/*r*, *v*<sup>2</sup> *<sup>A</sup>* <sup>=</sup> *<sup>B</sup>*<sup>2</sup> <sup>0</sup>/*ρm*, *B*<sup>0</sup> denotes magnetic field at magnetic axis, and *w* represents the rest magnetic well terms.

We first examine singular layer physics. In this layer only terms contains second order derivative in *r* need to be taken into consideration. From the first six terms in Eq. (77) one can see that the 2nd TAEs (coupling of *Em*−<sup>1</sup> and *Em*+1) have structure similarity to the 1st TAEs (coupling of *Em* and *Em*+1). The 1st TAE coupling is due to finite value of aspect ratio; while the 2nd TAE coupling is due to finite beta value. For brevity we focus ourselves to discuss the 1st TAE case. Denoting *ω*<sup>0</sup> = *ωA*/2, *q*<sup>0</sup> = (*m* + 1/2)/*n*, *δω* = *ω* − *ω*0, and *δq* = *q* − *q*0, the singular layer equations describing the coupling of *m* and *m* + 1 modes becomes

$$
\frac{\partial}{\partial \delta q} \left[ \frac{\delta \omega}{2 \omega\_0} - \left( 1 - \frac{1}{2m + 1} \right) n \delta q \right] \frac{\partial}{\partial \delta q} \delta \phi\_m = -\frac{\epsilon}{4} \frac{\partial^2}{\partial \delta q^2} \delta \phi\_{m+1}.
$$

$$
\frac{\partial}{\partial \delta q} \left[ \frac{\delta \omega}{2 \omega\_0} + \left( 1 + \frac{1}{2m + 1} \right) n \delta q \right] \frac{\partial}{\partial \delta q} \delta \phi\_{m+1} = -\frac{\epsilon}{4} \frac{\partial^2}{\partial \delta q^2} \delta \phi\_m.
$$

Introducing even and odd modes: *δφ*<sup>±</sup> = *δφ<sup>m</sup>* ± *δφm*+1, these two equations become

$$
\frac{\partial}{\partial \delta q} \left( \frac{\delta \omega}{2 \omega\_0} + \frac{1}{2m\_0 + 1} n \delta q \right) \frac{\partial}{\partial \delta q} \delta \phi\_+ - \frac{\partial}{\partial \delta q} n \delta q \frac{\partial}{\partial \delta q} \phi\_- = -\frac{\varepsilon}{4} \frac{\partial^2}{\partial \delta q^2} \delta \phi\_+,
$$

$$
\frac{\partial}{\partial \delta q} \left( \frac{\delta \omega}{2 \omega\_0} + \frac{1}{2m\_0 + 1} n \delta q \right) \frac{\partial}{\partial \delta q} \delta \phi\_- - \frac{\partial}{\partial \delta q} n \delta q \frac{\partial}{\partial \delta q} \phi\_+ = \frac{\varepsilon}{4} \frac{\partial^2}{\partial \delta q^2} \delta \phi\_-.
$$

Integrating once one obtains

$$\mathcal{D}\begin{pmatrix}\frac{\partial\delta\phi\_{+}\\\frac{\partial\delta\phi\_{-}}{\partial\delta q}\end{pmatrix} \equiv \begin{pmatrix}\frac{\delta\omega}{2\omega\_{0}} + \frac{1}{2m\_{0}+1}n\delta q + \frac{\varepsilon}{4} & -n\delta q\\-n\delta q & \frac{\delta\omega}{2\omega\_{0}} + \frac{1}{2m\_{0}+1}n\delta q - \frac{\varepsilon}{4}\end{pmatrix} \begin{pmatrix}\frac{\partial\delta\phi\_{+}\\\frac{\partial\delta\phi\_{-}}{\partial\delta q}\end{pmatrix} = \begin{pmatrix}A\_{+}\\A\_{-}\end{pmatrix},\tag{78}$$

where D is 2 × 2 matrix and *A*<sup>±</sup> are integration constants. Integration of Eq. (78) across singular layer (i.e., from *δq*<sup>−</sup> to *δq*+) one obtains the dispersion relation

$$\left(\frac{\delta\Phi\_{+}|\_{\delta q^{-}}^{\delta q^{+}}}{A\_{+}} - \int\_{\delta q^{-}}^{\delta q^{+}} \frac{\mathcal{D}\_{\mathbf{22}}d\delta q}{\det|\mathcal{D}|}\right) \left(\frac{\delta\Phi\_{-}|\_{\delta q^{-}}^{\delta q^{+}}}{A\_{-}} - \int\_{\delta q^{-}}^{\delta q^{+}} \frac{\mathcal{D}\_{11}d\delta q}{\det|\mathcal{D}|}\right) = \left(\int\_{\delta q^{-}}^{\delta q^{+}} \frac{\mathcal{D}\_{12}d\delta q}{\det|\mathcal{D}|}\right)^{2}.\tag{79}$$

Here, D*ij* are D matrix elements and two parameters Δ<sup>±</sup> ≡ *δφ*±| *δq*<sup>+</sup> *<sup>δ</sup>q*<sup>−</sup> /*A*<sup>±</sup> are determined by the outer solutions to the left and right of singular layer. As soon as Δ<sup>±</sup> are computed from outer regions, Eq. (79) can be used to determine the frequency. In general this frequency can be complex.

The denominators of integrations in Eq. (79) involve det |D|. The singularity emerges at det |D| = 0. In this case the Landau integration orbit needs to be used, as in the case for particle-wave resonances, and continuum damping occurs Berk et al. (1992). The so-called 1st TAE frequency gap, in which eigen modes can exit without continuum damping, can be determined by condition det |D| = 0, i.e.,

$$
\left(\frac{\delta\omega}{2\omega\_0} + \frac{1}{2m+1}n\delta q + \frac{\epsilon}{4}\right)\left(\frac{\delta\omega}{2\omega\_0} + \frac{1}{2m+1}n\delta q - \frac{\epsilon}{4}\right) = n^2\delta q^2.
$$

Its solution is

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

can simply be expressed as **B** = *Bφ*(*r*)/(1 + *�* cos *θ*)**e***<sup>φ</sup>* + *B<sup>θ</sup>* (*r*)**e***<sup>θ</sup>* . The general case will be addressed in Sec. 5 with AEGIS code formalism. Since their frequency is much larger than shear Alfvén mode frequency, the compressional Alfvén modes are decoupled. Therefore, we can use Eqs. (54) and (55) as starting equations for TAE investigation. Noting that Alfvén frequency is much larger than sound wave frequency, the first term in Eq. (55) can be dropped. Adopting the Fourier decomposition in Eq. (56), the sound wave equation (55) becomes

� *dϕm*+<sup>1</sup>

*d*<sup>2</sup>*Em*+<sup>2</sup> *dr*<sup>2</sup> <sup>+</sup>

<sup>2</sup>*q*<sup>2</sup> *Em*+<sup>1</sup> <sup>+</sup>

We first examine singular layer physics. In this layer only terms contains second order derivative in *r* need to be taken into consideration. From the first six terms in Eq. (77) one can see that the 2nd TAEs (coupling of *Em*−<sup>1</sup> and *Em*+1) have structure similarity to the 1st TAEs (coupling of *Em* and *Em*+1). The 1st TAE coupling is due to finite value of aspect ratio; while the 2nd TAE coupling is due to finite beta value. For brevity we focus ourselves to discuss the 1st TAE case. Denoting *ω*<sup>0</sup> = *ωA*/2, *q*<sup>0</sup> = (*m* + 1/2)/*n*, *δω* = *ω* − *ω*0, and *δq* = *q* − *q*0, the

singular layer equations describing the coupling of *m* and *m* + 1 modes becomes

� *nδq* � *∂ ∂δq*

� *nδq* � *∂ ∂δq*

Introducing even and odd modes: *δφ*<sup>±</sup> = *δφ<sup>m</sup>* ± *δφm*+1, these two equations become

*δφ*<sup>+</sup> <sup>−</sup> *<sup>∂</sup> ∂δq nδq ∂ ∂δq*

*δφ*<sup>−</sup> <sup>−</sup> *<sup>∂</sup> ∂δq nδq ∂ ∂δq*

<sup>4</sup> −*nδq*

<sup>2</sup>*ω*<sup>0</sup> <sup>+</sup> <sup>1</sup>

<sup>2</sup>*m*0+<sup>1</sup> *<sup>n</sup>δ<sup>q</sup>* <sup>−</sup> *�*

<sup>1</sup> <sup>−</sup> <sup>1</sup> 2*m* + 1

> 1 2*m* + 1

*nδq* � *∂ ∂δq*

*nδq* � *∂ ∂δq*

<sup>2</sup>*m*0+<sup>1</sup> *<sup>n</sup>δ<sup>q</sup>* <sup>+</sup> *�*

<sup>−</sup>*nδ<sup>q</sup> δω*

*dr* <sup>−</sup> *<sup>d</sup>ϕm*−<sup>1</sup> *dr*

> � − *�* � *r* <sup>3</sup> *R*2*ω*<sup>2</sup> *m*2*v*<sup>2</sup> *A*

Γ*Pr*<sup>3</sup> *B*2*m*<sup>2</sup>

*αr*

<sup>0</sup>/*ρm*, *B*<sup>0</sup> denotes magnetic field at magnetic axis, and *w* represents

*δφ<sup>m</sup>* <sup>=</sup> <sup>−</sup> *�*

*δφm*+<sup>1</sup> <sup>=</sup> <sup>−</sup> *�*

4 *∂*2 *∂δq*<sup>2</sup> *δφm*+1,

4 *∂*2 *∂δq*<sup>2</sup> *δφm*.

*<sup>φ</sup>*<sup>−</sup> <sup>=</sup> <sup>−</sup> *�* 4 *∂*2 *∂δq*<sup>2</sup> *δφ*+,

*<sup>φ</sup>*<sup>+</sup> <sup>=</sup> *�* 4 *∂*2 *∂δq*<sup>2</sup> *δφ*−.

> � ⎛ ⎝

*∂δφ*<sup>+</sup> *∂δq ∂δφ*<sup>−</sup> *∂δq*

⎞ ⎠ =

�*A*<sup>+</sup> *A*<sup>−</sup> �

, (78)

4

*d dr Em* � .

*<sup>d</sup>*<sup>2</sup>*Em*−<sup>2</sup>

*dr*<sup>2</sup> <sup>−</sup> *wEm*

*d*2 *dr*<sup>2</sup> *Em*+<sup>1</sup>

<sup>2</sup>*q*<sup>2</sup> *Em*−<sup>1</sup> <sup>=</sup> 0, (77)

�

*<sup>i</sup>*(∇ · *<sup>ξ</sup>*)*<sup>m</sup>* <sup>=</sup> <sup>1</sup>

� − *d dr*

� + Γ*Pr*<sup>3</sup> *B*2*m*<sup>2</sup>

*αr*<sup>2</sup> 2*mq*<sup>2</sup>

�<sup>2</sup> *d dr Em*

*d*2 *dr*<sup>2</sup> *Em*−<sup>1</sup>

*<sup>A</sup>* <sup>=</sup> *<sup>B</sup>*<sup>2</sup>

*dEm*+<sup>1</sup> *dr* <sup>+</sup>

*∂ ∂δq*

*∂ ∂δq*

> � *δω* 2*ω*<sup>0</sup> +

> � *δω* 2*ω*<sup>0</sup> +

> > � *δω* <sup>2</sup>*ω*<sup>0</sup> <sup>+</sup> <sup>1</sup>

*∂ ∂δq*

*∂ ∂δq*

Integrating once one obtains

⎞ ⎠ ≡

*∂δφ*<sup>+</sup> *∂δq ∂δφ*<sup>−</sup> *∂δq*

D ⎛ ⎝ � *δω* 2*ω*<sup>0</sup> − �

> 1 2*m*<sup>0</sup> + 1

> 1 2*m*<sup>0</sup> + 1

� *δω* 2*ω*<sup>0</sup> + � 1 +

*d dr*

−*�* � *r* <sup>3</sup> *R*2*ω*<sup>2</sup> *m*2*v*<sup>2</sup> *A*

where *Em* = *ϕm*/*r*, *v*<sup>2</sup>

<sup>−</sup> *<sup>α</sup>r*<sup>2</sup> 2*mq*<sup>2</sup>

the rest magnetic well terms.

� *r* 3 �1 *<sup>q</sup>* <sup>−</sup> *<sup>n</sup> m*

Using this solution for ∇ · *ξ*, Eq. (54) can be reduced to Zheng et al. (1999)

*dEm*−<sup>1</sup> *dr* <sup>−</sup> *<sup>α</sup><sup>r</sup>*

*BR*

� *r* <sup>3</sup> *R*2*ω*<sup>2</sup> *m*2*v*<sup>2</sup> *A*

$$\left[1-\left(\frac{1}{2m\_0+1}\right)^2\right]n^2\delta q = \frac{\delta\omega}{2\omega\_0(2m\_0+1)} \pm \sqrt{\left(\frac{\delta\omega}{2\omega\_0}\right)^2 - \frac{\epsilon^2}{16}\left[1-\frac{1}{\left(2m+1\right)^2}\right]}.\tag{80}$$

To exclude real *δq* solution for det D = 0, mode frequency must fall in the gap between *δω*±, i.e., *ω*<sup>−</sup> < *ω* < *ω*+, where

$$
\delta\omega\_{\pm} = \pm \frac{\epsilon}{2} \omega\_0 \sqrt{1 - 1/(2m + 1)^2}.
$$

One can obtain the gap width Δ*ω* = *δω*<sup>+</sup> − *δω*<sup>−</sup> = *�ω*<sup>0</sup> � 1 − 1/(2*m* + 1) 2 . The 1st TAEs are Alfvén eigen modes with frequency inside this gap. They are marginally stable and tend to be excited by resonances with energetic particles. Note that the gap width is proportional to *�*. In cylinder limit the gap vanishes. Therefore, existence of TAEs is due to toroidal effects. Also, we note that the dispersion relation, Eq. (79), allows two types of TAEs: even and odd types (*ϕ*±), depending on the values of Δ±.

We have discussed the 1st TEA theory through coupling of neighboring modes. In similar way one can also develop the 2nd TAE theory through coupling of *m* ± 1 modes Zheng & Chen (1998). If FLR effects are taken into consideration, the Alfvén types of singularities can be resolved, so that discrete modes can emerge in the continuum. This types of modes are referred to as kinetic TAEs (i.e., KTAEs). Due to correction of gyrokinetic theory Zheng et al. (2007), several missing FLR effects are recovered. Consequently, KTAE theories by far need to be reevaluated.

in Toroidal Plasma Confinement 21

Overview of Magnetohydrodynamics Theory in Toroidal Plasma Confinement 21

where �*m*{Ω} > 0 for causality. Note here that Eq. (85) is valid for general Ω, so that

Next, we discuss TAE-type KDMs, e.g., EPMs. This type of solutions can be expressed as

*<sup>K</sup>* <sup>≈</sup> <sup>Ω</sup>2(<sup>1</sup> <sup>+</sup> <sup>2</sup>*�*2Ω2), *<sup>A</sup>*<sup>2</sup>

frequency at continuum is allowed, as soon as causality condition is satisfied.

+)(Ω<sup>2</sup> <sup>−</sup> <sup>Ω</sup><sup>2</sup>

−) �1/2 |*θ*| � ⎡

(Ω<sup>2</sup> <sup>−</sup> <sup>Ω</sup><sup>2</sup>

−) �1/2

<sup>+</sup>,<sup>−</sup> <sup>=</sup> 1/4 <sup>±</sup> *�*Ω2. The leading order solution can, therefore, be expressed as

+)(Ω<sup>2</sup> <sup>−</sup> <sup>Ω</sup><sup>2</sup>

both TAEs and KDMs of TAE type (e.g., EPMs). Existence of TAE solution requires mode frequency to fall in the gap: Ω<sup>−</sup> < Ω < Ω+, as shown by the TAE theory in configuration space in Sec. 4.4. For KDMs mode frequency can be in the continuum, i.e., outside the gap as soon as causality condition is satisfied. For TAEs *ζ<sup>T</sup>* contains an O(1) back scattering and, hence, the continuum damping is either suppressed or much reduced. On the other hand, for KDMs *ζ<sup>K</sup>* contains no back scattering from periodic potential in Eq. (83), and, consequently, there is significant mount of continuum damping. Note that in principle both types of solutions can co-exist at |Ω| ≈ 1/2 . However, the TAE solution tends to be more unstable in this case than KDMs, since its continuum damping is much less or absent while

With outer solutions given by Eq. (85) or Eq. (88), one can obtain the corresponding dispersion relation by matching outer and inner solutions. For KBMs Eq. (81) can be used to construct

*ζ<sup>K</sup>* = exp{*iγKθ*}(*A*<sup>0</sup> + *A*<sup>2</sup> cos *θ* + ···). (84)

<sup>≈</sup> <sup>2</sup>*�*Ω2.

*ζ<sup>K</sup>* = exp{*i*Ω|*θ*|}, (85)

� ∼ O(*�*) and *�* � 1,

<sup>Ω</sup><sup>2</sup> <sup>−</sup> <sup>Ω</sup><sup>2</sup> −

+ *δWf* + *δWk* = 0, (89)

Ω<sup>2</sup> <sup>+</sup> − <sup>Ω</sup><sup>2</sup>

�1/2

�1/2

< 0. Equation (88) can describe

, (87)

⎤ ⎦ . (88)

sin(*θ*/2)

<sup>Ω</sup><sup>2</sup> <sup>−</sup> <sup>Ω</sup><sup>2</sup> −

Ω<sup>2</sup> <sup>+</sup> − <sup>Ω</sup><sup>2</sup>

*A*0

*ζ<sup>T</sup>* = exp{*iγTθ*}[*A*<sup>1</sup> cos(*θ*/2) + *B*<sup>1</sup> sin(*θ*/2) + ··· ]. (86)

�Ω<sup>2</sup> <sup>−</sup> 1/4�

, *<sup>B</sup>*<sup>1</sup> *A*1 = �

<sup>⎣</sup>cos(*θ*/2) + �

−) �1/2�

We first discuss KBMs. The KBM-type solution is given by

Inserting Eq. (84) into Eq. (83), one obtains, noting *�* � 1,

*γ*2

Inserting Eq. (86) into Eq. (83), one obtains, for �

(Ω<sup>2</sup> <sup>−</sup> <sup>Ω</sup><sup>2</sup>

+)(Ω<sup>2</sup> <sup>−</sup> <sup>Ω</sup><sup>2</sup>

��

<sup>2</sup> *<sup>ζ</sup>*<sup>∗</sup> *<sup>d</sup><sup>ζ</sup> dθ* � � � �

+∞

−∞

Therefore, at leading order, one has

*γ<sup>T</sup>* = �

*i* �

The causality condition is �*m*

(Ω<sup>2</sup> <sup>−</sup> <sup>Ω</sup><sup>2</sup>

the instability drives are generally comparable.

the following quadratic form in inner region:

Zheng et al. (2000)

and Ω<sup>2</sup>

*<sup>ζ</sup><sup>T</sup>* <sup>=</sup> exp �

#### **4.5 Kinetically driven modes: KBMs, EPMs, etc.**

In this subsection we describe the kinetically driven modes (KDMs), such as KBMs, EPMs, etc. The frequencies of these modes usually reside in continuum spectrum. Therefore, they are generally damped without driving effects. Unlike KTAEs, for which FLR effects are taken into account to resolve singularity, for KDMs strong kinetic effects from wave-particle-resonances are included to overcome continuum damping. That is why they are referred to as kinetically driven modes. Energetic particle drives to marginal stable TAEs can instantly lead unstable TAEs, but the drives to KDMs need to accumulate sufficient energy to overcome continuum damping for unstable KDMs to develop Tsai & Chen (1993) Zheng et al. (2000). In Secs. 4.3 and 4.4 one has seen that there are two types of modes: ballooning and TAEs. Therefore, KDMs also have two types. Those related to ballooning modes are referred to as KBMs, while EPMs are related to TAEs and usually driven by wave-energetic-particle resonances. We employ ballooning representation formalism to discuss them.

We start with the ballooning mode equation in intermediate frequency regime, Eq. (76), with energetic particle effects included. Introducing the transformation *ζ* = *ϕp*1/2, Eq. (76) becomes

$$\frac{\partial^2 \zeta}{\partial \theta^2} + \Omega^2 (1 + 2\epsilon \cos \theta) \zeta + \frac{a \cos \theta}{p} \zeta - \frac{(s - a \cos \theta)^2}{p^2} \zeta$$

$$-\frac{4\Gamma g^2}{p^2} + \frac{1}{p^{1/2}} \int \frac{d\epsilon d\mu B}{|v\_{\parallel}|} \omega\_d \delta g\_{\parallel l} = 0,\tag{81}$$

$$i\mathbf{v}\_{\parallel} \cdot \nabla \delta \mathbf{g}\_{h} - i\omega \delta \mathbf{g}\_{h} = i\omega \left(\mu \mathcal{B} + v\_{\parallel}^{2}\right) \frac{\partial F\_{0h}}{\partial \varepsilon} (\kappa\_{r} + \kappa\_{\theta} \Lambda) p^{-1/2} \zeta,\tag{82}$$

where *p* = 1 + Λ<sup>2</sup> *<sup>s</sup>*, *g* = cos *θ* + Λ sin *θ*, *δgh* is perturbed distribution functions for hot ions, *κ<sup>r</sup>* and *κθ* are respectively radial and poloidal components of magnetic field line curvature *κ*, Ω = *ω*/*ωA*, *ω<sup>d</sup>* is magnetic drift frequency, **v** is particle speed, the subscripts ⊥ and � represent respectively perpendicular and parallel components to the equilibrium magnetic field line, *ε* = *v*2/2 is particle energy, *μ* = *v*<sup>2</sup> <sup>⊥</sup>/2*<sup>B</sup>* is magnetic moment, and *<sup>F</sup>*0*<sup>h</sup>* is equilibrium distribution function for hot ions. For simplicity we have neglected the finite Larmor radius effects and only take into account the kinetic effects from energetic ions.

To study KDMs one need to investigate singular layer behavior. In ballooning representation space, singular layer corresponds to *θ* → ∞ limit. Again, we exclude the 2nd TAE from discussion (i.e., assuming Γ = 0). Equation (81) in *θ* → ∞ limit becomes:

$$\frac{\partial^2 \zeta}{\partial \theta^2} + \Omega^2 (1 + 2\varepsilon \cos \theta) \zeta = 0. \tag{83}$$

This is the well-known Mathieu equation. According to Floquet's Theorem, its solution takes following form

$$
\zeta(\theta) = P(\theta) \exp\{i\gamma\theta\},
$$

where *P*(*θ* + 2*π*) = *P*(*θ*). Since modes with longer parallel-to-**B** wavelengths tend to be more unstable, we shall examine solutions corresponding to the two lowest periodicities. The first one is related to KBMs Tsai & Chen (1993) and the second one is related to EPMs Zheng et al. (2000).

We first discuss KBMs. The KBM-type solution is given by

$$\mathcal{L}\_K = \exp\{i\gamma\_\mathcal{K}\theta\}(A\_0 + A\_2\cos\theta + \cdots). \tag{84}$$

Inserting Eq. (84) into Eq. (83), one obtains, noting *�* � 1,

$$
\gamma\_K^2 \approx \Omega^2 (1 + 2\epsilon^2 \Omega^2), \quad \frac{A\_2}{A\_0} \approx 2\epsilon \Omega^2.
$$

Therefore, at leading order, one has

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

In this subsection we describe the kinetically driven modes (KDMs), such as KBMs, EPMs, etc. The frequencies of these modes usually reside in continuum spectrum. Therefore, they are generally damped without driving effects. Unlike KTAEs, for which FLR effects are taken into account to resolve singularity, for KDMs strong kinetic effects from wave-particle-resonances are included to overcome continuum damping. That is why they are referred to as kinetically driven modes. Energetic particle drives to marginal stable TAEs can instantly lead unstable TAEs, but the drives to KDMs need to accumulate sufficient energy to overcome continuum damping for unstable KDMs to develop Tsai & Chen (1993) Zheng et al. (2000). In Secs. 4.3 and 4.4 one has seen that there are two types of modes: ballooning and TAEs. Therefore, KDMs also have two types. Those related to ballooning modes are referred to as KBMs, while EPMs are related to TAEs and usually driven by wave-energetic-particle resonances. We employ

We start with the ballooning mode equation in intermediate frequency regime, Eq. (76), with energetic particle effects included. Introducing the transformation *ζ* = *ϕp*1/2, Eq. (76)

*α* cos *θ*

*<sup>p</sup> <sup>ζ</sup>* <sup>−</sup> (*<sup>s</sup>* <sup>−</sup> *<sup>α</sup>* cos *<sup>θ</sup>*)<sup>2</sup>

*<sup>s</sup>*, *g* = cos *θ* + Λ sin *θ*, *δgh* is perturbed distribution functions for hot ions,

*<sup>p</sup>*<sup>2</sup> *<sup>ζ</sup>*

<sup>|</sup>*v*�| *<sup>ω</sup>dδgh* <sup>=</sup> 0, (81)

<sup>⊥</sup>/2*<sup>B</sup>* is magnetic moment, and *<sup>F</sup>*0*<sup>h</sup>* is equilibrium

*∂θ*<sup>2</sup> <sup>+</sup> <sup>Ω</sup>2(<sup>1</sup> <sup>+</sup> <sup>2</sup>*�* cos *<sup>θ</sup>*)*<sup>ζ</sup>* <sup>=</sup> 0. (83)

*∂ε* (*κ<sup>r</sup>* <sup>+</sup> *κθ*Λ)*p*−1/2*ζ*, (82)

**4.5 Kinetically driven modes: KBMs, EPMs, etc.**

ballooning representation formalism to discuss them.

*∂θ*<sup>2</sup> <sup>+</sup> <sup>Ω</sup>2(<sup>1</sup> <sup>+</sup> <sup>2</sup>*�* cos *<sup>θ</sup>*)*<sup>ζ</sup>* <sup>+</sup>

1 *p*1/2

**<sup>v</sup>**� · ∇*δgh* <sup>−</sup> *<sup>i</sup>ωδgh* <sup>=</sup> *<sup>i</sup><sup>ω</sup>*

*dεdμB*

effects and only take into account the kinetic effects from energetic ions.

discussion (i.e., assuming Γ = 0). Equation (81) in *θ* → ∞ limit becomes:

*∂*2*ζ*

 *μB* + *v*<sup>2</sup> � *∂F*0*<sup>h</sup>*

*κ<sup>r</sup>* and *κθ* are respectively radial and poloidal components of magnetic field line curvature *κ*, Ω = *ω*/*ωA*, *ω<sup>d</sup>* is magnetic drift frequency, **v** is particle speed, the subscripts ⊥ and � represent respectively perpendicular and parallel components to the equilibrium magnetic

distribution function for hot ions. For simplicity we have neglected the finite Larmor radius

To study KDMs one need to investigate singular layer behavior. In ballooning representation space, singular layer corresponds to *θ* → ∞ limit. Again, we exclude the 2nd TAE from

This is the well-known Mathieu equation. According to Floquet's Theorem, its solution takes

*ζ*(*θ*) = *P*(*θ*) exp{*iγθ*}, where *P*(*θ* + 2*π*) = *P*(*θ*). Since modes with longer parallel-to-**B** wavelengths tend to be more unstable, we shall examine solutions corresponding to the two lowest periodicities. The first one is related to KBMs Tsai & Chen (1993) and the second one is related to EPMs Zheng et al.

*∂*2*ζ*

−4Γ*g*<sup>2</sup> *<sup>p</sup>*<sup>2</sup> <sup>+</sup>

field line, *ε* = *v*2/2 is particle energy, *μ* = *v*<sup>2</sup>

becomes

where *p* = 1 + Λ<sup>2</sup>

following form

(2000).

$$\mathcal{Z}\_K = \exp\{i\Omega|\theta|\},\tag{85}$$

where �*m*{Ω} > 0 for causality. Note here that Eq. (85) is valid for general Ω, so that frequency at continuum is allowed, as soon as causality condition is satisfied.

Next, we discuss TAE-type KDMs, e.g., EPMs. This type of solutions can be expressed as Zheng et al. (2000)

$$\mathcal{Z}\_T = \exp\{i\gamma\_T \theta\} [A\_1 \cos(\theta/2) + B\_1 \sin(\theta/2) + \dotsb]. \tag{86}$$

Inserting Eq. (86) into Eq. (83), one obtains, for � �Ω<sup>2</sup> <sup>−</sup> 1/4� � ∼ O(*�*) and *�* � 1,

$$\gamma\_T = \left[ (\Omega^2 - \Omega\_+^2)(\Omega^2 - \Omega\_-^2) \right]^{1/2}, \quad \frac{B\_1}{A\_1} = \left( \frac{\Omega^2 - \Omega\_-^2}{\Omega\_+^2 - \Omega^2} \right)^{1/2},\tag{87}$$

and Ω<sup>2</sup> <sup>+</sup>,<sup>−</sup> <sup>=</sup> 1/4 <sup>±</sup> *�*Ω2. The leading order solution can, therefore, be expressed as

$$\mathcal{L}\_T = \exp\left\{i\left[ (\Omega^2 - \Omega\_+^2)(\Omega^2 - \Omega\_-^2) \right]^{1/2} |\theta| \right\} \left[ \cos(\theta/2) + \left( \frac{\Omega^2 - \Omega\_-^2}{\Omega\_+^2 - \Omega^2} \right)^{1/2} \sin(\theta/2) \right]. \tag{88}$$

The causality condition is �*m* �� (Ω<sup>2</sup> <sup>−</sup> <sup>Ω</sup><sup>2</sup> +)(Ω<sup>2</sup> <sup>−</sup> <sup>Ω</sup><sup>2</sup> −) �1/2� < 0. Equation (88) can describe both TAEs and KDMs of TAE type (e.g., EPMs). Existence of TAE solution requires mode frequency to fall in the gap: Ω<sup>−</sup> < Ω < Ω+, as shown by the TAE theory in configuration space in Sec. 4.4. For KDMs mode frequency can be in the continuum, i.e., outside the gap as soon as causality condition is satisfied. For TAEs *ζ<sup>T</sup>* contains an O(1) back scattering and, hence, the continuum damping is either suppressed or much reduced. On the other hand, for KDMs *ζ<sup>K</sup>* contains no back scattering from periodic potential in Eq. (83), and, consequently, there is significant mount of continuum damping. Note that in principle both types of solutions can co-exist at |Ω| ≈ 1/2 . However, the TAE solution tends to be more unstable in this case than KDMs, since its continuum damping is much less or absent while the instability drives are generally comparable.

With outer solutions given by Eq. (85) or Eq. (88), one can obtain the corresponding dispersion relation by matching outer and inner solutions. For KBMs Eq. (81) can be used to construct the following quadratic form in inner region:

$$2\left.\zeta^\* \frac{d\zeta}{d\theta}\right|\_{-\infty}^{+\infty} + \delta W\_f + \delta W\_k = 0,\tag{89}$$

in Toroidal Plasma Confinement 23

Overview of Magnetohydrodynamics Theory in Toroidal Plasma Confinement 23

In this subsection we describe how to reduce MHD equations for global mode analyses. The starting equation is the single fluid MHD equation (14). This is a vector equation and can be projected onto three directions to get scalar equations. The parallel projection has be derived in Eq. (33). From parallel equation one can solve for ∇ · *ξ*, which is the only unknown needed for two-perpendicular equations to become a complete set of equations. In principle the parallel motion can not be described by MHD model, since particles are not localized along magnetic field line. There are wave-partcle resonance, trapped particle, and so-called small parallel particle speed effects, etc. Nevertheless, from analyses in Sec. 4.3 one can see that in low frequency regime the parallel coupling results only in the so-called apparent mass effect, while in intermediate regime the parallel coupling mainly gives rise to the 2nd TAEs. Note that apparent mass effect can be absorbed by rescaling mode frequency and inclusion of the 2nd TAE effect is straightforward as discussed in Sec. 4.4. For brevity we limit ourselves to treat only two perpendicular components of Eq. (14) with Γ set to zero. AEGIS-K code has been developed to include parallel dynamics using kinetic description Zheng et al. (2010). Using general flux coordnates in Eq. (22), the magnetic field line displacement is decomposed

**5.1 MHD equations and numerical solution method for plasma region**

*ξ* × **B** = *ξs*∇*ψ* + *ξψχ*�

∞ ∑ *m*=−∞

quantities in poloidal and toroidal directions,

<sup>−</sup>*<sup>π</sup> <sup>d</sup>θξ* exp{−*imθ*}/

*<sup>J</sup>*2∇*<sup>θ</sup>* × ∇*<sup>ζ</sup>* · **<sup>B</sup>** <sup>×</sup> [··· × **<sup>B</sup>**]/*B*<sup>2</sup> and (1/*qχ*�

lead to the following set of differential equations in matrices

*<sup>ψ</sup>* + E*ξ<sup>ψ</sup>*

� − 

<sup>B</sup>†*ξ<sup>s</sup>* <sup>+</sup> <sup>D</sup>*ξ*�

A*ξ<sup>s</sup>* + B*ξ*�

*ξ* exp{−*inζ*} =

decomposed as matrices in poloidal Fourier space, for example

<sup>J</sup>*mm*� <sup>=</sup> <sup>1</sup>

2*π*

Since we deal with linear problem, the Fourier transform can be used to decompose perturbed

*ξm* 1 <sup>√</sup>2*<sup>π</sup>*

toroidal Fourier component needs to be considered. As usual, equilibrium quantities can be

In the poloidal Fourier decomposition, the Fourier components are cut off both from lower and upper sides respectively by *m*min and *m*max. Therefore, the total Fourier component under consideration is *M* = *m*max − *m*min + 1. We also use bold face (or alternatively [[··· ]]) to represent Fourier space vectors, and calligraphic capital letters (or alternatively �···�) to represent the corresponding equilibrium matrices (*e.g.*, J for *J*) in poloidal Fourier space. To get scalar equations, we project Eq. (14) respectively onto two directions

introduce the Fourier transformation in Eq. (93) to two projected equations. These procedures

<sup>C</sup>†*ξ<sup>s</sup>* <sup>+</sup> <sup>E</sup>†*ξ*�

*<sup>ψ</sup>* + H*ξ<sup>ψ</sup>*

*<sup>ψ</sup>* + C*ξ<sup>ψ</sup>* = 0. (95)

*dθ J*(*θ*)*e*

*i*(*m*� <sup>−</sup>*m*)*<sup>θ</sup>* .

 *π* −*π* (∇*ζ* − *q*∇*θ*). (92)

exp{*i*(*mθ* − *nζ*)}, (93)

)*J*2∇*<sup>ζ</sup>* × ∇*<sup>ψ</sup>* · **<sup>B</sup>** <sup>×</sup> [··· × **<sup>B</sup>**]/*B*2, and then

= 0, (94)

<sup>√</sup>2*π*. With the toroidal symmetry assumed, only a single

as follows

with *<sup>ξ</sup><sup>m</sup>* = *<sup>π</sup>*

where

$$\begin{split} \delta \mathcal{W}\_{f} &= \int\_{-\infty}^{+\infty} d\theta \left\{ \left| \frac{\partial \zeta}{\partial \theta} \right|^{2} - \left[ \frac{a \cos \theta}{p} - \frac{(s - a \cos \theta)^{2}}{p^{2}} \right] |\zeta|^{2} \right\} \,\mathrm{d}\theta \\ \delta \mathcal{W}\_{k} &= \int\_{-\infty}^{+\infty} d\theta \zeta^{\*} \frac{1}{p^{1/2}} \int \frac{d \varepsilon d\mu B}{|\upsilon\_{\parallel}|} \omega\_{d} \delta \mathcal{g}\_{\text{fl}} \,\mathrm{d}\theta \end{split}$$

Here, the superscript ∗ represents complex conjugate. Matching inner (Eq. (89)) and outer (Eq. (85)) solutions one obtains the dispersion relation Tsai & Chen (1993)

$$-i\Omega + \delta \mathcal{W}\_f + \delta \mathcal{W}\_K = 0.\tag{90}$$

Here, we note that kinetic effects from core plasma should also be taken into account in outer region. As proved in Ref. Zheng & Tessarotto (1994a) this results in the so-called apparent mass effect and leads Ω in the first term of Eq. (90) to become complicated function of actual mode frequency.

Similarly, for KDMs of TAE type, for example EPMs, one need to consider even and odd modes. For even modes the dispersion relation is given by Refs. Zheng et al. (2000) and Tsai & Chen (1993)

$$-i\left(\frac{\Omega\_-^2 - \Omega^2}{\Omega\_+^2 - \Omega^2}\right)^2 + \delta T\_f + \delta T\_K = 0,\tag{91}$$

where *δTf* represents MHD fluid contribution and *δTK* is energetic-particle contribution to the quadratic form in inner region.

The dispersion relations in Eqs. (90) and (91) extend respectively MHD ballooning modes in diamagnetic gap and TAEs in Alfvén gap to respective continua. Kinetic drives are the causes to make causality conditions satisfied.
