Properties of large aspect ratio in tokamak

II. Theoretical & Mathematical Development

2.1 MHD Physics Review in Tokamak

2.1.7 Properties of large aspect ratio in tokamak

In tokamak, the aspect ratio, 𝑅₀/𝑎, is assumed to be large. Introducing a small number, 𝜀, as an inverse large aspect ratio of tokamak, 𝜀 ≡ 𝑎/𝑅 ≪ 1 in tokamak.

The safety factor, 𝑞 is the order of 1 as 𝑞 = 𝑟𝐵_𝜙0

𝑅₀𝐵_𝜃 ~ 𝜀𝐵_𝜙0 𝐵_𝜃 ~ 1.

So, 𝐵_𝜃 is the order of

𝐵_𝜃 ~ 𝜀𝐵_𝜙. (2.36)

For the equilibrium equation, ∇𝑝 = 𝐣×𝐁, each term of Eq. (2.26) is equivalent as d𝑝

d𝑟 ~ 1 𝜇₀{1

𝑟

𝜕

𝜕𝑟(𝑟𝐵_𝜃)} 𝐵_𝜃 (2.37)

and

d𝑝 d𝑟 ~ 1

𝜇₀

𝜕𝐵_𝜙

𝜕𝑟 𝐵_𝜙. (2.38)

From Eq. (2.37), 𝑝 is the order of

𝑝 ~ 𝐵_𝜃²

𝜇₀. (2.39)

So, poloidal beta, 𝛽_𝑝, is the order of 1 as

𝛽_𝑝 ~ 1 (2.40)

and toroidal beta, 𝛽_𝑡, is the order equivalent as

𝛽_𝑡 = 𝐵_𝜃𝑎²

𝐵_𝜙𝑎² 𝛽_𝑝 ~ 𝜀² (2.41)

and beta, 𝛽, is the order equivalant as

𝐵_𝜃𝑎² 𝐵_𝜃𝑎² ₂

(2.42)

From Eq. (2.38), 𝜕𝐵_𝜙/𝜕𝑟 is the order of

𝜕𝐵_𝜙

𝜕𝑟 ~ 𝜇₀ 1 𝐵_𝜙

d𝑝

d𝑟 ~ 𝜇₀ 1 𝐵_𝜙

𝐵_𝜃² 𝜇₀

1 𝑎 ~ 1

𝐵_𝜙(𝜀²𝐵_𝜙²)1 𝑎 ~ 1

𝐵_𝜙(𝜀²𝐵_𝜙²)1

𝑎 ~ 𝜀²𝐵_𝜙 𝑎 .

From Ampere’s law, 𝐣 = 1/𝜇₀(∇×𝐁) each componenets of current, 𝐣 in Eq. (2.28) are equivalent as 𝑗_𝜃 = 1

𝜇₀

𝜕𝐵_𝜙

𝜕𝑟 ~ 𝜀² 𝐵_𝜙

𝜇₀𝑎 ~ 𝜀² 𝐵

𝜇₀𝑎, (2.43)

𝑗_𝜙= 1 𝜇₀{1

𝑟

𝜕

𝜕𝑟(𝑟𝐵_𝜃)} ~ 𝐵_𝜃

𝜇₀𝑎 ~ 𝜀 𝐵_𝜙

𝜇₀𝑎 ~ 𝜀 𝐵

𝜇₀𝑎. (2.44)

So, 𝑗𝜃 is the order of

𝑗_𝜃 ~ 𝜀𝑗_𝜙. (2.45)

Summarizing section 2.1.7, leading order relation is shown in Table 2.2.

Table 2.2 Leading order relations in large aspect ratio tokamak.

Safety factor 𝑞 ~ 1

Magnetic field 𝐵_𝜃 ~ 𝜀𝐵_𝜙

Pressure 𝑝 ~ 𝐵_𝜃²

𝜇₀

Beta, 𝛽

𝛽_𝑝 ~ 1 𝛽_𝑡 ~ 𝜀² 𝛽 ~ 𝜀²

Current density

𝑗_𝜃 ~ 𝜀² 𝐵 𝜇₀𝑎 𝑗_𝜙 ~ 𝜀 𝐵

𝜇₀𝑎

These properties are used to approximate the equations. The MHD equations can be solved more simply by the approximation of large aspect ratio in tokamak.

This concludes the basic physical concept review of tokamak. In this section, the principle of magnetic fields induced by currents was reviewed. Based on the magnetic fields, the concept and the role of the safety factor and beta value were also reviewed. The properties of large aspect ratio in tokamak was also reviewed. Based on this knowledge, MHD instability in the tokamak will be reviewed to study the sawtooth instability.

2.2 Review of MHD instability in tokamak

In this section, the basics of MHD instability will be reviewed.

2.2.1 Basic idea of MHD stability

Whether the equilibrium is stable or unstable depends on whether a small perturbation is growing or is damping [28]. When the perturbation is growing, the plasma is in an unstable condition whereas when the perturbation is damping, the plasma is in a stable condition. Sawtooth oscillation is the result of instability, which shows periodic behavior.

The basic idea about stability is shown in Figure 2.4.

(a) (b)

Figure 2.4 The basic idea of MHD stability. (a) Stable equilibrium, (b) unstable equilibrium.

This can be illustrated by thinking about a ball, as shown in Figure 2.4. The ball tends to move in the direction which gives the potential energy to the outer system. In Figure 2.4 (a), the ball is perturbed and potential energy increases (𝛿𝑊 > 0). As time goes by, the ball will return to its original point. The ball is determined as stable. In Figure 2.4 (b), the ball is perturbed and potential energy deceases (𝛿𝑊 <

0). As time goes by, the ball will roll away.

Stable Equilibrium

𝑥₀ 𝑥₀+ 𝜉

Unstable Equilibrium

𝑥₀ 𝑥₀+ 𝜉

2.2.2 Energy principle

Based on the perturbation introduced in section 2.2.1, the energy principle is introduced here. The energy principle is used to determine stability by the sign of change of potential energy, 𝛿𝑊. By energy conservation, the summation of the potential energy change and the kinetic energy change is zero. The change of kinetic energy from the perturbation is

𝛿𝐾 =1

2∫ 𝛏 ∙ 𝐟₁(𝛏)d𝑉

where 𝐟₁ is the first order perturbation of force density. According to the law of energy conservation, the potential energy comes from the result of the plasma displacement, 𝛏 as

𝛿𝑊 = −1

2∫ 𝛏 ∙ 𝐟₁(𝛏)d𝑉. (2.46) The change of potential energy, 𝛿𝑊 is derived from ideal MHD equations and small displacement, 𝛏 . Using the perturbation theory, but neglecting the second order term and putting the equilibrium velocity to zero, u₀(r) = 0, since the plasma is in static equilibrium. Thus

𝐮(𝐫, 𝑡) = 𝐮₁(𝐫, 𝑡), (2.47)

𝜌(𝐫, 𝑡) = 𝜌₀(𝐫) + 𝜌₁(𝐫, 𝑡), (2.48) 𝑝(𝐫, 𝑡) = 𝑝₀(𝐫) + 𝑝₁(𝐫, 𝑡), (2.49) 𝐁(𝐫, 𝑡) = 𝐁₀(𝐫) + 𝐁₁(𝐫, 𝑡), (2.50)

The mass conservation, separating equilibrium term and first order perturbation term, is given as

𝜕𝜌₀

𝜕𝑡 +𝜕𝜌₁

𝜕𝑡 + (𝐮₁∙ 𝛁)𝜌₀= −𝜌₀𝛁 ∙ 𝐮₁ The equilibrium term for mass conservation is

𝜕𝜌₀

𝜕𝑡 = 0,

For momentum conservation, the plasma is assumed to be in laminar flow, then (𝐮 ∙ 𝛁)𝐮 = 0 and separating equilibrium term and first order perturbation term, the momentum conservation is expressed as

𝜌₀𝜕𝐮₁

𝜕𝑡 = 𝐣₀×𝐁₀+ 𝐣₀×𝐁₁+ 𝐣₁×𝐁₀− 𝛁𝑝₀− 𝛁𝑝₁ The equilibrium term for momentum conservation is

𝛁𝑝₀= 𝐣₀×𝐁₀

and the first order perturbation term of momentum conservation is 𝜌₀𝜕𝐮₁

𝜕𝑡 = 𝐣₀×𝐁₁+ 𝐣₁×𝐁₀− 𝛁𝑝₁. (2.52)

The adiabatic equation, separating equilibrium term and first order perturbation term neglecting second order perturbation term is

𝜕𝑝₀

𝜕𝑡 +𝜕𝑝₁

𝜕𝑡 + (𝐮₁∙ 𝛁)𝑝₀= −𝛾𝑝₀𝛁 ∙ 𝐮₁. The equilibrium term for the adiabatic equation is

𝜕𝑝₀

𝜕𝑡 = 0

and the first order perturbation term of the adiabatic equation is

𝜕𝑝₁

𝜕𝑡 = −𝐮₁∙ 𝛁𝑝 − 𝛾𝑝𝛁 ∙ 𝐮₁. (2.53)

Combining Faraday’s law and Ohm’s law,

𝜕𝐁

𝜕𝑡 = −𝛁×(−𝐮×𝐁) = 𝛁×(𝐮×𝐁).

Separating equilibrium term and first order perturbation term and neglecting second order perturbation term, the equation is

𝜕𝐁₀

𝜕𝑡 +𝜕𝐁₁

𝜕𝑡 = 𝛁×(𝐮₁×𝐁₀).

and the first order perturbation term of the equation is

𝜕𝐁₁

𝜕𝑡 = 𝛁×(𝐮₁×𝐁₀). (2.54)

The first order perturbation of ideal MHD equations are as below.

𝜕𝜌₁

𝜕𝑡 = −𝐮₁∙ 𝛁𝜌₀− 𝜌₀𝛁 ∙ 𝐮₁ (2.51) 𝜌₀𝜕𝐮₁

𝜕𝑡 = 𝐣₀×𝐁₁+ 𝐣₁×𝐁₀− 𝛁𝑝₁ (2.52)

𝜕𝑝₁

𝜕𝑡 = −𝐮₁∙ 𝛁𝑝 − 𝛾𝑝𝛁 ∙ 𝐮₁ (2.53)

𝜕𝐁₁

𝜕𝑡 = 𝛁×(𝐮₁×𝐁₀) (2.54)

The displacement vector, 𝛏 is the integration of velocity perturbation as 𝛏(𝐫, 𝑡) = ∫ 𝐮₁(𝐫, 𝑡)𝑑𝑡^′

𝑡 0

. (2.55)

From now on, 𝛏 is used instead of 𝐮₁. Integrating Eq. (2.13), Eq. (2.15) and Eq. (2.16) using Eq. (2.55), the Eqs. (2.51) – (2.54) are expressed as

𝜌₁(𝐫, 𝑡) = −𝛏(𝐫, 𝑡) ∙ 𝛁𝜌₀− 𝜌₀𝛁 ∙ 𝛏(𝐫, 𝑡) (2.56) 𝜌₀𝜕²𝛏

𝜕𝑡²= 1 𝜇0

(𝛁×𝐁₀)×𝐁₁+ 1 𝜇0

(𝛁×𝐁₁)×𝐁₀− 𝛁𝑝₁ (2.57)

𝑝₁(𝐫, 𝑡) = −𝛏(𝐫, 𝑡) ∙ 𝛁𝑝₀− 𝛾𝑝₀𝛁 ∙ 𝛏(𝐫, 𝑡) (2.58) 𝐁₁(𝐫, 𝑡) = 𝛁×{𝛏(𝐫, 𝑡)×𝐁₀(𝐫)} (2.59) Substituting Eq. (2.58) and Eq. (2.59) into Eq. (2.57),

𝜌₀𝜕²𝛏

𝜕𝑡²= 𝐣₀×{𝛁×(𝛏×𝐁₀)} + 1

𝜇₀[𝛁×{𝛁×(𝛏×𝐁₀)}]×𝐁₀+ 𝛁(𝛏 ∙ 𝛁𝑝₀+ 𝛾𝑝₀𝛁 ∙ 𝛏)

Substituting Eq. (2.60) into Eq. (2.46), the change of potential energy, 𝛿𝑊, is obtained as 𝛿𝑊 = −1

2∫ [𝛏 ∙ (𝐣₀×𝐁₁) + 1

𝜇₀𝛏 ∙ [(𝛁×𝐁₁)×𝐁₀] − 𝛏 ∙ 𝛁(𝛾𝑝₀𝛁 ∙ 𝛏 − 𝛏 ∙ 𝛁𝑝₀)] d𝑉. (2.61)

Using the vector relations Eq. (B.2), Eq. (B.7) and Eq. (B.10) to Eq. (2.61), 𝛿𝑊, is obtained as 𝛿𝑊 = −1

2∫ [𝛁 ∙ {(1

𝜇₀𝐁₀∙ 𝐁₁) 𝛏} + 1

𝜇₀𝐁₁∙ {𝛁×(𝛏×𝐁₀)} + 𝐣₀∙ (𝐁₁×𝛏) + 𝛁 ∙ (𝑝₁𝛏) + 𝑝₁(𝛁 ∙ 𝛏)] d𝑉.

Using divergence theorem, 𝛿𝑊 =1

2∫ (1

𝜇₀𝐁₀∙ 𝐁₁+ 𝑝₁) 𝛏 ∙ d𝐀 +1 2∫ {1

𝜇₀𝐵₁²− 𝐣₀∙ (𝐁₁×𝛏) − 𝑝₁(𝛁 ∙ 𝛏)} d𝑉. (2.62) The potential energy change, 𝛿𝑊, can be distinguished by three different regions: plasma region (0 < 𝑟 < 𝑎), 𝛿𝑊_𝑝, plasma surface region (𝑟 = 𝑎), 𝛿𝑊_𝑆 and vacuum region (𝑎 < 𝑟 ≤ 𝑏), 𝛿𝑊_𝑣. At the vacuum region, there is no displacement. So, the Eq. (2.52) is divided as

𝛿𝑊 =1

2∫ {1

𝜇₀𝐵_1𝑝² − 𝐣0∙ (𝐁1×𝛏) − 𝑝1(𝛁 ∙ 𝛏)} d𝑉

plasma

+1

2∫ (1

𝜇₀𝐁₀∙ 𝐁₁+ 𝑝₁) 𝛏 ∙ d𝐀

surface

+ 1

2𝜇₀∫ 𝐵_1𝑣² d𝑉

vacuum

.

(2.63)

where 𝐁_1𝑣 is perturbed magnetic field at vacuum and 𝐣_1𝑣= 1

𝜇₀(∇×𝐁_1𝑣) = 0. (2.64)

is satisfied. Each term 𝛿𝑊_𝑝, 𝛿𝑊_𝑠 and 𝛿𝑊_𝑣 is

𝛿𝑊_𝑝=1

2∫ {1

𝜇₀𝐵_1𝑝² − 𝐣₀∙ (𝐁₁×𝛏) − 𝑝₁(𝛁 ∙ 𝛏)} d𝑉

plasma

, (2.65)

𝛿𝑊_𝑠 =1

2∫ (1

𝜇₀𝐁₀∙ 𝐁₁+ 𝑝₁) 𝛏 ∙ d𝐀

surface

, (2.66)

𝛿𝑊 = 1

2𝜇₀∫ 𝐵_1𝑣² d𝑉

vacuum

. (2.67)

For steady state, the continuity equation, Eq. (2.1) is

∇ ∙ 𝐮₁= 0. (2.68)

Integrating Eq. (2.68) by 𝑡

∇ ∙ 𝛏 = 0. (2.69)

The volume integral in tokamak is

∫ d𝑉 = ∫ 𝑟d𝑟d𝜃d𝑧 = ∫ 𝑟d𝑟d𝜃𝑅₀d𝜙 = 𝑅₀∫ 𝑟d𝑟d𝜃d𝜙

Substituting Eq. (2.69) into Eq. (2.63), the surface term is removed by taking volume integral, and assuming 𝜉_𝑟 ~ 𝜉_𝜃≫ 𝜉_𝜙 and 𝐵_𝑟1 ~ 𝐵_𝜃1≫ 𝐵_𝜙1,

𝛿𝑊 =1 2∫ {1

𝜇₀𝐵_1𝑝² − 𝐣₀∙ (𝐁_1𝑝×𝛏)} 𝑟d𝑟d𝜃d𝑧 + 1

2𝜇₀∫ 𝐵_1𝑣² 𝑟d𝑟d𝜃d𝑧 =1

2∫ ∫ ∫ {1

𝜇₀𝐵_1𝑝² − 𝐣₀∙ (𝐁_1𝑝×𝛏)} 𝑟d𝑟

𝑎 0

d𝜃

2𝜋 0

𝑅₀d𝜙

2𝜋 0

+ 1

2𝜇₀∫ ∫ ∫ 𝐵_1𝑣² 𝑟d𝑟

𝑏 𝑎

d𝜃

2𝜋 0

𝑅₀d𝜙

2𝜋 0

= 2𝜋²𝑅₀∫ {1

𝜇₀𝐵_1𝑝² − 𝐣₀∙ (𝐁_1𝑝×𝛏)} 𝑟d𝑟

𝑎 0

+2𝜋²𝑅₀

𝜇₀ ∫ 𝐵_1𝑣² 𝑟d𝑟

𝑏 𝑎

= 2𝜋²𝑅₀∫ {𝐵_1𝑝,𝑟² + 𝐵_1𝑝,𝜃²

𝜇₀ − 𝑗_𝜙∙ (𝐵_𝑟1𝜉_𝜃− 𝐵_𝜃1𝜉_𝑟)} 𝑟d𝑟

𝑎 0

+2𝜋²𝑅₀

𝜇₀ ∫ (𝐵_1𝑣,𝑟² + 𝐵_1𝑣,𝜃² )𝑟d𝑟

𝑏 𝑎

So, the change of potential energy, 𝛿𝑊, is

𝛿𝑊 = 2𝜋²𝑅 ∫ {𝐵_1𝑝,𝑟² + 𝐵_1𝑝,𝜃²

𝜇₀ − 𝑗_𝜙∙ (𝐵_1𝑝,𝑟𝜉_𝜃− 𝐵_1𝑝,𝜃𝜉_𝑟)} 𝑟d𝑟

𝑎 0

+2𝜋²𝑅₀

𝜇₀ ∫ (𝐵_1𝑣,𝑟² + 𝐵_1𝑣,𝜃² )𝑟d𝑟

𝑏 𝑎

.

(2.70)

where the first term is plasma term and the second term is vacuum term.

The plasma term, 𝛿𝑊_𝑝 which is the first term of Eq. (2.70), is first calculated. The displacement, 𝛏, analyzed by Fourier analysis as 𝛏 ~ 𝑒^{𝑖(𝑚𝜃−𝑛𝜙)}, is given as

𝛏 = 𝛏₀𝑒^{𝑖(𝑚𝜃−𝑛𝜙)}= (𝜉_𝑟𝑟̂ + 𝜉_𝜃𝜃̂)𝑒^{𝑖(𝑚𝜃−𝑛𝜙)}. (2.71)

Then, the relation between 𝜉_𝑟 and 𝜉_𝜃 is

𝜉_𝜃= − 𝑖 𝑚

d

d𝑟(𝑟𝜉_𝑟). (2.72)

Substituting Eq. (2.72) into Eq. (2.59) and using Eq. (B.18), 𝐁_1𝑝= [−𝑖𝑚𝐵_𝜙0

𝑅₀ (𝑛 𝑚−1

𝑞) 𝜉_𝑟𝑟̂ +𝐵_𝜙0 𝑅₀ [𝜕

𝜕𝑟{(𝑛 𝑚−1

𝑞) 𝑟𝜉_𝑟}] 𝜃̂] 𝑒^{𝑖(𝑚𝜃−𝑛𝜙)} where the safety factor, 𝑞 is

𝑞 = 𝑟𝐵_𝜙0 𝑅₀𝐵_𝜃.

then, the components of perturbed magnetic field at plasma region are 𝐵_1𝑝,𝑟 = −𝑖𝑚𝐵𝜙0

𝑅₀ (𝑛 𝑚−1

𝑞) 𝜉_𝑟, (2.73)

𝐵_1𝑝,𝜃=𝐵_𝜙0 𝑅₀ [𝜕

𝜕𝑟{(𝑛 𝑚−1

𝑞) 𝑟𝜉_𝑟}]. (2.74)

Substituting Eq. (2.72), Eq. (2.73) and Eq. (2.74) into the first term of Eq. (2.70), the plasma term, 𝛿𝑊_𝑝 is obtained as

𝛿𝑊_𝑝= ∫ [

2𝜋²𝑟²𝐵_𝜙0² 𝜇₀𝑅₀

(𝑛 𝑚 −

1 𝑞)

2

1 + 𝑛²𝑟² 𝑚²𝑅₀²

(d𝜉_𝑟 d𝑟)

𝑎 2 0

+ {2𝜋²𝐵_𝜙0² 𝜇₀𝑅₀² (𝑛

𝑚−1 𝑞)

2

+4𝜋²𝑟𝑝^′ 𝑅₀ (𝑛

𝑚)

2 1

1 + ( 𝑛𝑟 𝑚𝑅₀)²

} 𝜉𝑟2

] 𝑟𝑑𝑟

+2𝜋²𝐵_𝜙0² 𝜇₀𝑅₀

(𝑛² 𝑚²− 1

𝑞²) 1 + ( 𝑛𝑟

𝑚𝑅₀)² 𝑎²𝜉_𝑎².

(2.75)

where 𝜉_𝑎 is the radial displacement, 𝜉_𝑟, at 𝑟 = 𝑎.

Subsequently, the vacuum term, 𝑊_𝑣 which is the second term of Eq. (2.70) is calculated. As in Eq.

(2.64), ∇×𝐁1𝑣= 0, and the flux function, 𝛙 satisfies 𝐁1𝑣 = ∇×𝛙. Then

∇×(∇×𝛙) = ∇(∇ ∙ 𝛙) − ∇²𝛙 = 0 or

∇²𝛙 = 0 or

1 𝑟

𝜕

𝜕𝑟(𝑟𝜕𝜓

𝜕𝑟) + 1 𝑟²

𝜕²𝜓

𝜕𝜃² + 𝜕𝜓

𝜕𝑧²= 0.

Then flux function 𝜓 can be expressed as 𝜓₀𝑒^{𝑖(𝑚𝜃−𝑛𝜙)}, and 𝜓 satisfies Bessel’s equation as 1

𝑟 𝑑 𝑑𝑟(𝑟𝑑𝜓

𝑑𝑟) − (𝑛² 𝑅₀²+𝑚²

𝑟²) 𝜓 = 0, (2.76)

and the magnetic field components in the vacuum are expressed in terms of 𝜓 as 𝐵_1𝑣,𝑟= −1

𝑟

𝜕𝜓

𝜕𝜃 = −𝑖𝑚

𝑟 𝜓, (2.77)

𝐵_1𝑣,𝑟 =𝜕𝜓

𝜕𝑟. (2.78)

The solution of 𝜓 is

𝜓 = 𝐶₁𝐼_𝑚(𝑛

𝑅₀𝑟) + 𝐶₂𝐾_𝑚(𝑛

𝑅₀𝑟). (2.79)

At 𝑟 = 𝑎, 𝐵_1,𝑟 is continuous, so

𝐵_1𝑣,𝑟(𝑎) = 𝐵_1𝑝,𝑟(𝑎) =𝑖𝑚

𝑎 𝜓_𝑎 =𝑖𝑚𝐵_𝜙0 𝑅₀ (𝑛

𝑚− 1 𝑞_𝑎) 𝜉_𝑎 Then 𝜓(𝑎) is

𝜓(𝑎) =𝑎𝐵_𝜙0 𝑅₀ (𝑛

𝑚− 1

𝑞_𝑎) 𝜉_𝑎=𝑎𝐵_𝜙0 𝑅₀ (𝑛

𝑚− 1 𝑞_𝑎) 𝜉_𝑎 At 𝑟 = 𝑏, 𝜓(𝑏) = 0, then the solution of 𝜓(𝑟) is

Substituting Eq. (2.77) and Eq. (2.78) into the second term of Eq. (2.70), the vacuum term is calculated as

𝛿𝑊_𝑣=2𝜋²𝑅₀

𝜇₀ ∫ (𝐵_1𝑣,𝑟² + 𝐵_1𝑣,𝜃² )𝑟d𝑟

𝑏 𝑎

=2𝜋²𝑅₀

𝜇₀ ∫ {(𝑚 𝑟 𝜓)

2

+ (d𝜓 d𝑟)

2

} 𝑟d𝑟

𝑏 𝑎

=2𝜋²𝑅₀

𝜇₀ ∫ (𝑟𝑚²

𝑟² 𝜓²+ 𝑟d𝜓 d𝑟

d𝜓 d𝑟) d𝑟

𝑏 𝑎

=2𝜋²𝑅₀

𝜇₀ ∫ (𝑟𝑚²

𝑟² 𝜓²) d𝑟

𝑏 𝑎

+2𝜋²𝑅₀

𝜇₀ {[𝜓𝑟d𝜓 d𝑟]

𝑎 𝑏

− ∫ 𝜓 d d𝑟(𝑟d𝜓

d𝑟) d𝑟

𝑏 𝑎

}

=2𝜋²𝑅₀

𝜇₀ ∫ {𝑟𝑚²

𝑟² 𝜓²− 𝜓 d d𝑟(𝑟d𝜓

d𝑟) d𝑟} d𝑟

𝑏 𝑎

+2𝜋²𝑅₀

𝜇₀ [𝜓𝑟d𝜓 d𝑟]

𝑎 𝑏

Then,

𝛿𝑊_𝑣≈2𝜋²𝑅₀

𝜇₀ ∫ 𝜓 (𝑚² 𝑟² 𝜓 −1

𝑟 d d𝑟(𝑟d𝜓

d𝑟) d𝑟) 𝑟d𝑟

𝑏 𝑎

+2𝜋²𝑅₀

𝜇₀ [𝜓𝑟d𝜓 d𝑟]

𝑎 𝑏

. (2.81)

Substituting Eq. (2.76) into Eq. (2.81),

𝛿𝑊_𝑣= −2𝜋²𝑅₀ 𝜇0

𝑎𝜓(𝑎)𝜓^′(𝑎). (2.82)

Substituting Eq. (2.80) into Eq. (2.82), the vacuum term is calculated as

𝛿𝑊_𝑣= −2𝜋²𝑛𝐵_𝜙0² 𝜇₀𝑅₀²

𝐾_𝑚^′ (𝑛𝑎

𝑅₀) 𝐼_𝑚(𝑛𝑏

𝑅₀) − 𝐼_𝑚^′ (𝑛𝑎

𝑅₀) 𝐾_𝑚(𝑛𝑎 𝑅₀) 𝐾_𝑚(𝑛𝑎

𝑅₀) 𝐼_𝑚(𝑛𝑏

𝑅₀) − 𝐼_𝑚(𝑛𝑎

𝑅₀) 𝐾_𝑚(𝑛𝑏 𝑅₀)

(𝑛 𝑚− 1

𝑞_𝑎)

2

𝑎³𝜉_𝑎². (2.83)

The properties of modified Bessel function given as d

d𝑥{𝐼_𝜈(𝑥)} =𝜈

𝑥𝐼_𝜈(𝑥) + 𝐼_𝜈+1(𝑥), (2.84) d

d𝑥{𝐾_𝑛(𝑥)} =𝑛

𝑥𝐾𝑛(𝑥) − 𝐾_𝑛+1(𝑥). (2.85)

Using these properties, 𝛿𝑊_𝑣 in Eq. (2.83) is calculated as 𝛿𝑊_𝑣

= −2𝜋²𝑛𝐵_𝜙0² 𝜇₀𝑅₀²

{𝑚 𝑛

𝑅₀

𝑎 𝐾^𝑚(𝑛𝑎

𝑅₀) − 𝐾_𝑚+1(𝑛𝑎

𝑅₀)} 𝐼_𝑚(𝑛𝑏 𝑅₀) − {𝑚

𝑛 𝑅₀

𝑎 𝐼^𝑚(𝑛𝑎

𝑅₀) + 𝐼_𝑚+1(𝑛𝑎

𝑅₀)} 𝐾_𝑚(𝑛𝑎 𝑅₀) 𝐾_𝑚(𝑛𝑎

𝑅₀) 𝐼_𝑚(𝑛𝑏

𝑅₀) − 𝐼_𝑚(𝑛𝑎

𝑅₀) 𝐾_𝑚(𝑛𝑏 𝑅₀)

(𝑛 𝑚

− 1 𝑞_𝑎)

2

𝑎³𝜉_𝑎² = −2𝜋²𝐵_𝜙0²

𝜇₀𝑅₀ 𝑚 (𝑛 𝑚− 1

𝑞_𝑎)

2

𝑎²𝜉_𝑎²

+2𝜋²𝑛𝐵_𝜙0² 𝜇₀𝑅₀²

𝐾_𝑚+1(𝑛𝑎

𝑅₀) 𝐼_𝑚(𝑛𝑏

𝑅₀) + 𝐼_𝑚+1(𝑛𝑎

𝑅₀) 𝐾_𝑚(𝑛𝑎 𝑅₀) 𝐾_𝑚(𝑛𝑎

𝑅₀) 𝐼_𝑚(𝑛𝑏

𝑅₀) − 𝐼_𝑚(𝑛𝑎

𝑅₀) 𝐾_𝑚(𝑛𝑏 𝑅₀)

(𝑛 𝑚− 1

𝑞_𝑎)

2

𝑎³𝜉_𝑎²

=2𝜋²𝐵_𝜙0² 𝜇₀𝑅₀ 𝑚 (𝑛

𝑚− 1 𝑞_𝑎)

2

{ 𝑛𝑎 𝑚𝑅₀

𝐾_𝑚+1(𝑛𝑎

𝑅₀) 𝐼_𝑚(𝑛𝑏

𝑅₀) + 𝐼_𝑚+1(𝑛𝑎

𝑅₀) 𝐾_𝑚(𝑛𝑎 𝑅₀) 𝐾_𝑚(𝑛𝑎

𝑅₀) 𝐼_𝑚(𝑛𝑏

𝑅₀) − 𝐼_𝑚(𝑛𝑎

𝑅₀) 𝐾_𝑚(𝑛𝑏 𝑅₀)

− 1} 𝑎²𝜉_𝑎²

Then, the vacuum term, 𝛿𝑊_𝑣 is obtained as

𝛿𝑊_𝑣=2𝜋²𝐵_𝜙0² 𝜇₀𝑅₀ 𝑚

{

𝑛𝑎𝐾_𝑚+1(𝑛𝑎 𝑅0) 𝑚𝑅₀𝐾_𝑚(𝑛𝑎

𝑅₀) 1 +

𝐼_𝑚+1(𝑛𝑎 𝑅₀) 𝐼_𝑚(𝑛𝑏

𝑅0)

𝐾_𝑚(𝑛𝑎 𝑅₀) 𝐾_𝑚+1(𝑛𝑎 𝑅₀) 1 −

𝐼_𝑚(𝑛𝑎 𝑅₀) 𝐼_𝑚(𝑛𝑏

𝑅₀)

𝐾_𝑚(𝑛𝑏 𝑅₀) 𝐾_𝑚(𝑛𝑎

𝑅₀)

− 1

} (𝑛

𝑚− 1 𝑞_𝑎)

2

𝑎²𝜉_𝑎².

(2.86)

Finally, summing Eq. (2.75) and Eq. (2.86), the total change of potential energy is derived as [8]

𝛿𝑊 = ∫ (𝑓𝜉^′2+ 𝑔𝜉²)𝑟𝑑𝑟

𝑎 0

+2𝜋²𝐵_𝜙0² 𝜇0𝑅0

𝑎²𝜉_𝑎² {

𝑛² 𝑚²− 1

𝑞_𝑎² 1 + ( 𝑛𝑟

𝑚𝑅₀)²

+ 𝑚𝛬 (𝑛 𝑚− 1

𝑞𝑎

)

2

}

. (2.87) where

𝑓 =2𝜋²𝑟²𝐵_𝜙0² 𝜇₀𝑅₀

(𝑛 𝑚 −

1 𝑞)

2

1 + 𝑛²𝑟² 𝑚²𝑅₀²

=2𝜋²𝑟²𝐵_𝜙0² 𝜇₀𝑅₀ (𝑛

𝑚−1 𝑞)

2{1 + ( 𝑛𝑟

𝑚𝑅₀)²} − ( 𝑛𝑟 𝑚𝑅₀)² 1 + ( 𝑛𝑟

𝑚𝑅₀)²

=2𝜋²𝑟²𝐵_𝜙0² 𝜇0𝑅0

(𝑛 𝑚−1

𝑞)

2

{1 − ( 𝑛𝑟

𝑚𝑅0)² 1 + ( 𝑛𝑟

𝑚𝑅₀)²

} =2𝜋²𝑟²𝐵_𝜙0² 𝜇0𝑅0

(𝑛 𝑚−1

𝑞)

2

{1 − (𝑟 𝑅0

)

2

(𝑛 𝑚)

2

}

and

𝑔 =4𝜋²𝑟𝑝^′ 𝑅₀ (𝑛

𝑚)

2 1

1 + ( 𝑛𝑟 𝑚𝑅₀)²

+2𝜋²𝐵_𝜙0² 𝜇₀𝑅₀² (𝑛

𝑚−1 𝑞)

2

{𝑚²− 1 1 + ( 𝑛𝑟

𝑚𝑅₀)² }

+4𝜋²𝐵_𝜙² 𝑅₀ (𝑛

𝑚)

2

(𝑟 𝑅₀)

2

(𝑛² 𝑚²− 1

𝑞²) 1 {1 + ( 𝑛𝑟

𝑚𝑅₀)²}

2

=4𝜋²𝑟𝑝^′ 𝑅₀ (𝑛

𝑚)

2{1 + ( 𝑛𝑟

𝑚𝑅₀)²} − ( 𝑛𝑟 𝑚𝑅₀)² 1 + ( 𝑛𝑟

𝑚𝑅₀)²

+2𝜋²𝐵_𝜙0² 𝜇₀𝑅₀ (𝑛

𝑚−1 𝑞)

2

{𝑚²−

{1 + ( 𝑛𝑟

𝑚𝑅₀)²} − ( 𝑛𝑟 𝑚𝑅₀)² 1 + ( 𝑛𝑟

𝑚𝑅₀)²

}

+4𝜋²𝐵_𝜙0² 𝜇₀𝑅₀ (𝑛

𝑚)

2

(𝑟 𝑅₀)

2

(𝑛² 𝑚²− 1

𝑞²)

{1 + ( 𝑛𝑟

𝑚𝑅₀)²} − ( 𝑛𝑟 𝑚𝑅₀)² {1 + ( 𝑛𝑟

𝑚𝑅₀)²}

2 and

𝛬 =

𝑛𝑎𝐾_𝑚+1(𝑛𝑎 𝑅₀) 𝑚𝑅₀𝐾_𝑚(𝑛𝑎

𝑅₀) 1 +

𝐼𝑚+1(𝑛𝑎 𝑅₀) 𝐼_𝑚(𝑛𝑏

𝑅₀)

𝐾𝑚(𝑛𝑎 𝑅₀) 𝐾_𝑚+1(𝑛𝑎 𝑅₀) 1 −

𝐼_𝑚(𝑛𝑎 𝑅₀) 𝐼_𝑚(𝑛𝑏

𝑅0)

𝐾_𝑚(𝑛𝑏 𝑅₀) 𝐾_𝑚(𝑛𝑎

𝑅₀)

− 1.

The change of potential, 𝛿𝑊 consists of plasma term, 𝛿𝑊_𝑝, and the vacuum term, 𝛿𝑊_𝑣, as

𝛿𝑊_𝑝= ∫ (𝑓𝜉^′2+ 𝑔𝜉²)𝑟𝑑𝑟

𝑎 0

+2𝜋²𝐵_𝜙0² 𝜇₀𝑅₀ 𝑎²𝜉_𝑎²

𝑛² 𝑚²− 1

𝑞² 1 + ( 𝑛𝑟

𝑚𝑅₀)² ,

𝛿𝑊_𝑣 =2𝜋²𝐵_𝜙0²

𝜇₀𝑅₀ 𝑎²𝜉_𝑎²𝑚𝛬 (𝑛 𝑚−1

𝑞 )

2

.

Applying large aspect ratio assumption, 𝑟/𝑅 ≪ 1, functions 𝑓 and 𝑔 are approximated as

𝑓 ≈2𝜋²𝑟²𝐵_𝜙0² 𝜇₀𝑅₀ (𝑛

𝑚−1 𝑞)

2

,

𝑔 ≈2𝜋²𝐵_𝜙0² 𝜇₀𝑅₀ (𝑛

𝑚−1 𝑞)

2

(𝑚²− 1) +2𝜋² 𝑅₀ (𝑛

𝑚)

2

{2𝑟𝑝^′+𝐵_𝜙0² 𝜇₀ (𝑟

𝑅₀)

2

(𝑛 𝑚−1

𝑞) (3𝑛 𝑚 +1

𝑞)}.

The beta values are equivalent to 𝛽 ~ 𝛽𝑡 ~ 𝜀² and 𝛽𝑝 ~ 1. Then, the change of potential can be divided into second order and fourth order terms as

𝛿𝑊₂=2𝜋²𝐵_𝜙0²

𝜇₀𝑅₀ ∫ {(𝑟𝑑𝜉 𝑑𝑟)

2

+ (𝑚²− 1)𝜉²} (𝑛 𝑚−1

𝑞)

2

𝑟𝑑𝑟

𝑎 0

+2𝜋²𝐵_𝜙² 𝜇₀𝑅 𝜉_𝑎²𝑚²

{ 𝑛² 𝑚²− 1

𝑞² 𝑛² 𝑅₀²+𝑚²

𝑎² +𝑎²

𝑚𝛬 (1 𝑞− 𝑛

𝑚 )

2

} ,

𝛿𝑊₄=2𝜋² 𝑅₀ (𝑛

𝑚)

2

∫ {2𝑟𝑝^′+𝐵_𝜙0² 𝜇₀ (𝑛

𝑚−1 𝑞) (3𝑛

𝑚 +1

𝑞)} 𝜉²𝑟𝑑𝑟

𝑎 0

and

𝛿𝑊₂ ~ 2𝜋²𝑅₀𝐵_𝜙0² 𝜇0

𝜉²𝑂 (𝑟 𝑅0

)

2

,

𝛿𝑊₄ ~ 2𝜋²𝑅₀𝐵_𝜙0²

𝜇₀ 𝜉²𝑂 (𝑟 𝑅₀)

4

This is the whole derivation process for change of potential energy, 𝛿𝑊, in tokamak using the

2.3 Theories of sawtooth instability

2.3.1 Internal kink, 𝑚/𝑛 = 1/1 mode

As mentioned in section 1.3, sawtooth oscillation is a type of ‘𝑚 = 1 internal kink mode’ [9]. In the internal mode, plasma surface displacement is zero (𝜉_𝑎= 0) whereas the external kink mode is nonzero (𝜉_𝑎≠ 0). For the external mode, stability is effected by both the integral term and the other term of Eq.

(2.87). So, the external kink mode both has 𝑚 = 1 mode and 𝑚 ≥ 2 mode. For the internal kink mode, with 𝜉_𝑎= 0, Eq. (2.87) becomes

𝛿𝑊 =2𝜋²𝐵_𝜙²

𝜇₀𝑅₀ ∫ {(𝑟𝑑𝜉 𝑑𝑟)

2

+ (𝑚²− 1)𝜉²} (𝑛 𝑚−1

𝑞)

2

𝑟𝑑𝑟

𝑎 0

+2𝜋² 𝑅₀ (𝑛

𝑚)

2

∫ {2𝑟𝑝^′+𝐵_𝜙² 𝜇₀ (𝑛

𝑚−1 𝑞) (3𝑛

𝑚 +1

𝑞)} 𝜉²𝑟𝑑𝑟

𝑎 0

.

(2.88)

with

𝛿𝑊₂=2𝜋²𝐵_𝜙²

𝜇₀𝑅₀ ∫ {(𝑟𝑑𝜉 𝑑𝑟)

2

+ (𝑚²− 1)𝜉²} (𝑛 𝑚−1

𝑞)

2

𝑟𝑑𝑟

𝑎 0

,

𝛿𝑊₄=2𝜋² 𝑅₀ (𝑛

𝑚)

2

∫ {2𝑟𝑝^′+𝐵_𝜙² 𝜇₀ (𝑛

𝑚−1 𝑞) (3𝑛

𝑚 +1

𝑞)} 𝜉²𝑟𝑑𝑟

𝑎 0

.

Instability occurs when 𝛿𝑊 < 0, thus the plasma is likely to be unstable, when 𝛿𝑊₂ is minimized.

Then, the poloidal number, 𝑚, is always 𝑚 = 1 for internal kink mode. So,

𝛿𝑊 =2𝜋²𝐵_𝜙²

𝜇₀𝑅₀ ∫ (𝑟𝑑𝜉 𝑑𝑟)

2

(𝑛 −1 𝑞)

2

𝑟𝑑𝑟

𝑎 0

+2𝜋²𝑛²

𝑅₀ ∫ {2𝑟𝑝^′+𝐵_𝜙²

𝜇₀ (𝑛 −1

𝑞) (3𝑛 +1

𝑞)} 𝜉²𝑟𝑑𝑟

𝑎 0

.

(2.89)

with

𝛿𝑊₂=2𝜋²𝐵_𝜙²

𝜇₀𝑅 ∫ (𝑟𝑑𝜉 𝑑𝑟)

2

(𝑛 −1 𝑞)

2

𝑟𝑑𝑟

𝑎 0

,

𝛿𝑊₄=2𝜋²𝑛²

𝑅₀ ∫ {2𝑟𝑝^′+𝐵_𝜙²

𝜇₀ (𝑛 −1

𝑞) (3𝑛 +1

𝑞)} 𝜉²𝑟𝑑𝑟

𝑎 0

.

Finally, the change of potential energy, 𝛿𝑊, for 𝑚/𝑛 = 1/1 mode is derived as

𝛿𝑊 =2𝜋²𝐵_𝜙²

𝜇₀𝑅₀ ∫ (𝑟𝑑𝜉 𝑑𝑟)

2

(1 −1 𝑞)

2

𝑟𝑑𝑟

𝑎 0

+2𝜋² 𝑅0

∫ {2𝑟𝑝^′+𝐵_𝜙² 𝜇0

(1 −1

𝑞) (3 +1

𝑞)} 𝜉²𝑟𝑑𝑟

𝑎 0

(2.90)

with

𝛿𝑊₂ =2𝜋²𝐵_𝜙²

𝜇₀𝑅 ∫ (𝑟𝑑𝜉 𝑑𝑟)

2

(1 −1 𝑞)

2

𝑟𝑑𝑟

𝑎 0

,

𝛿𝑊₄=2𝜋²

𝑅₀ ∫ {2𝑟𝑝^′−𝐵_𝜃²

𝜇₀ (1 − 𝑞)(1 + 3𝑞)} 𝜉²𝑟𝑑𝑟

𝑎 0

.

or

𝛿𝑊 = 2𝜋²𝑅₀∫ 𝐵_𝜃²

𝜇₀(1 − 𝑞)²(d𝜉 d𝑟)

2

𝑟𝑑𝑟

𝑎 0

+2𝜋²

𝑅₀ ∫ {2𝑟𝑝^′−𝐵_𝜃²

𝜇₀ (1 − 𝑞)(1 + 3𝑞)} 𝜉²𝑟𝑑𝑟

𝑎 0

(2.91)

with

𝛿𝑊₂= 2𝜋²𝑅₀∫ 𝐵_𝜃²

𝜇₀(1 − 𝑞)²(𝑑𝜉 𝑑𝑟)

2

𝑟𝑑𝑟

𝑎 0

,

𝛿𝑊₄=2𝜋²

𝑅₀ ∫ {2𝑟𝑝^′−𝐵_𝜃²

𝜇₀ (1 − 𝑞)(1 + 3𝑞)} 𝜉²𝑟𝑑𝑟

𝑎 0

.

Two models for sawtooth instability are introduced to minimize 𝛿𝑊₂ of Eq. (2.91). Kadomtsev’s model tries to explain instability by taking d𝜉/d𝑟 → 0 inside the 𝑞 = 1 surface [30], whereas Wesson’s model tries to explain instability by taking 𝑞 → 1 inside the 𝑞 = 1 surface.

2.3.2 Kadomtsev’s model

As mentioned in section 2.3.1, Kaomtsev’s model tries to minimize 𝛿𝑊₂ by d𝜉/d𝑟 → 0 inside the 𝑞 = 1 surface. At the magnetic axis (𝑟 = 0), the displacement is 𝜉 = 𝜉₀, and at the edge (𝑟 = 𝑎), 𝜉_𝑎= 0. To minimize 𝛿𝑊₂, d𝜉/d𝑟 = 0 except near 𝑞 = 1 surface (∵ At 𝑞 = 1, d𝜉/d𝑟 cannot be zero).

Based on this picture, the radial displacement distribution, 𝜉(𝑟), is described in Figure 2.5 as

Figure 2.5 The displacement distribution, 𝜉(𝑟), based on Kadomtsev’s model.

The corresponding 𝛿𝑊 is 𝛿𝑊 =2𝜋²𝜉₀²

𝑅² ∫ {2𝑟𝑝^′−𝐵_𝜃²

𝜇₀ (1 − 𝑞)(1 + 3𝑞)} 𝑟𝑑𝑟

𝑟₁ 0

(2.92) and 𝑝^′< 0 (∵ 𝑝 = 𝑛𝑘_𝐵𝑇 and 𝑇 is decreased as 𝑟 increased as shown in Figure 1.3), then it is concluded as 𝛿𝑊 < 0.

𝑟 𝜉

𝑟₁ 𝑟₁− 𝛿 𝜉₀

𝜉 = 𝜉₀ as 𝑟 < 𝑟₁

𝜉 = 0 as 𝑟 > 𝑟1

𝑞 = 1 surface

The 𝑚/𝑛 = 1/1 internal kink mode has a radial displacement of 𝜉 ~ 𝑒^{𝑖(𝜃−𝜙)}.

(a) (b)

Figure 2.6 The 𝑚/𝑛 = 1/1 internal kink within 𝑞 = 1 surface. (a) total view of tokamak (b) poloidal view of tokamak.

The corresponding flow pattern of Figure 2.6 (b) - ② is shown in Figure 2.7 as

Figure 2.7 The flow pattern of plasma in

①

②

③

④ 𝜉 = 𝜉₀cos(𝜃 − 𝜙)

① ②

③ ④

𝑣₁ 𝛿 𝑣₂

𝑣₂

The velocity, 𝑣₁, is the hot core velocity and the velocity, 𝑣₂, is the plasma velocity which comes from the narrow layer. The thickness, 𝛿, is the thickness of the narrow resistive layer.

The helical magnetic field line is

𝐵^∗= 𝐵_𝜃− (𝑟/𝑅)𝐵_𝜙= 𝐵_𝜃(1 − 𝑞) (2.93) and current, 𝑗 comes from the electric field as equivalent to

𝑗 ~ 𝜎𝑣₁𝐵^∗=𝑣₁𝐵^∗

𝜂 . (2.94)

From Ampere’s law,

𝑗~𝐵^∗/𝜇₀𝛿. (2.95)

Combining Eq. (2.94) and Eq. (2.95), 𝑣₁ is expressed as 𝑣1= 𝜂

𝜇₀𝛿. (2.96)

The magnetic pressure is equivalent to plasma pressure by 𝑣₂ as 𝐵^∗2

2𝜇0

~ 1

2𝜌𝑣₂²+1

2𝜌𝑣₂²= 𝜌𝑣₂². Then, the velocity 𝑣₂ is

𝑣₂~ 𝐵^∗

√2𝜌𝜇0

. (2.97)

From continuity equation

𝑣₁𝑟₁~𝑣₂𝛿, then

𝛿 ~ 𝑣₁𝑟₁

𝑣₂ =𝜂𝑟₁√2𝜌𝜇0

𝐵^∗𝜇₀𝛿 . or

where

𝜏_𝑅=𝜇₀

𝜂 𝑟₁², (2.99)

𝜏_𝐴= 𝑟₁ 𝐵^∗/√𝜌𝜇0

. (2.100)

The collapse time is given by

𝜏_𝐾 ~ 𝑟₁

𝑣₁. (2.101)

‘

Substituting (2.95) into (2.98), the collapse time is calculated as 𝜏_𝐾 ~ 𝛿²

(𝜏_𝐴/𝜏_𝑅)^1/2 𝜇₀

𝜂 ~ (𝜏_𝑅/𝜏_𝐴)^1/2(𝜏_𝐴 𝜏_𝑅)𝜇₀

𝜂 𝑟₁² = (𝜏_𝐴/𝜏_𝑅)^1/2𝜏_𝑅= (𝜏_𝐴𝜏_𝑅)^1/2 Finally, the collapse time in Kadomtsev’s model is

𝜏𝐾 ~ (𝜏𝐴𝜏𝑅)^1/2. (2.102)

The whole process of sawtooth oscillation in Kadomtsev’s model is shown in Figure 2.9. Initially, 𝑞 > 1 everywhere as in Figure 2.9 (a). As the temperature at the core increases, 𝑞 = 1 surface appears and, instability occurs as in Figure 2.9 (b). The core starts to move by internal kink, and a new magnetic island appears on the other side, resulting in two apparent islands. One magnetic island by internal kink has 𝑞 < 1 whereas the magnetic island on the other side has 𝑞 > 1 . Then, magnetic reconnection occurs at the narrow resistive layer near 𝑞 = 1 surface as in Figure 2.9 (c). One magnetic island which has 𝑞 < 1 is annihilated by magnetic reconnection whereas the new magnetic island which has 𝑞 > 1 is expanded. After the old magnetic island is annihilated completely, it returns to the initial state as 𝑞 >

1 everywhere as in Figure 2.9 (d). Then the periodic behavior is repeated, starting from the state illustrated in Figure 2.9 (a). This is the theoretical review of Kadomtsev’s model to observe fast crash by magnetic reconnection.

Figure 2.8 The whole process of sawtooth oscillation in Kadomtsev’s model [31].

(a) 𝑞 > 1 everywhere

(d) 𝑞 > 1 everywhere (b) Internal kink

𝑞 < 1

New magnetic island

Core starts to move Two island systems

𝑞 < 1 𝑞 > 1

(c) Magnetic reconnection

Magnetic reconnection

2.3.3 Wesson’s model

Wesson’s model tries to explain the inconsistencies of Kadomtsev’s model. There are a few discrepancies in Kadomtsev’s model when compared to the experimental observations. One discrepancy is the crash time. The collapse times of several types of tokamak, using Kadomtsev’ model, were calculated as in Table 2.3 with 𝑇 = 1 keV, 𝑛 = 2×10²⁰m⁻³ , and 𝑞₀ ~ 0.9 . The detailed calculation process is in Appendix C.2. According to Table 2.1, Kadomtsev’ collapse time is 𝜏𝐾 ∝ 𝑟₁/√𝐵𝜙, and 𝜏_𝐾 ~ 2×10⁻⁴-3×10⁻³s. But the actual tokamaks have always 𝜏_𝐾 ~ 10⁻⁴ s and does not depend on the tokamak geometry [32].

To explain this inconsistency, Wesson assumed a flat 𝑞-profile inside the 𝑞 = 1 surface. By taking 𝑞 → 1 to Eq. (2.91),

𝛿𝑊 →2𝜋²

𝑅₀ ∫ {2𝑟𝑝^′−𝐵_𝜃²

𝜇₀ (1 − 𝑞)(1 + 3𝑞)} 𝜉²𝑟𝑑𝑟

𝑎 0

(2.103) and the corresponding displacement distribution [18] is illustrated in Figure 2.10.

Kadomtsev’s collapse time calculation is not applicable for the 𝑞 → 1 case, since 𝜏_𝐾 becomes infinite. So, Wesson introduced a new model [18]. In this model, there is no magnetic shear stress inside the 𝑞 = 1 surface, because of the magnetic shear stress, given in Eq. (2.103) [12],

𝑠 =𝑟 𝑞

d𝑞

d𝑟 (2.103)

goes to zero as 𝑞 goes to 1. Due to the absence of magnetic shear, the magnetic surface deforms into a crescent shape, and cold plasma comes into the core as shown in Figure 2.10 [14, 33]. This flow pattern is called ‘hot crescent, cold bubble’. The magnetic reconnection takes place during the ramp phase and not in the fast collapse phase [18].

Table 2.3 Calculated Kadomtsev’s collapse time according to several tokamaks [34-39].

Tokamak

Major radius [m]

Toroidal Magnetic

field [T]

q=1 surface radius [m]

Resistive diffusion time [s]

Alfven time [s]

Kadomtsev's collapse

time [s]

DIII-D 1.67 2.2 0.1

(estimated) 2.26×10⁻¹ 4.91×10⁻⁶ 1.05×10⁻³

JET 2.96 3.45 0.2 9.05×10⁻¹ 5.54×10⁻⁶ 2.24×10⁻³

ADITYA 0.75 1.2 0.03

(estimated) 2.04×10⁻² 4.04×10⁻⁶ 2.87×10⁻⁴

ASDEX 1.65 3.1 0.1

(estimated) 2.26×10⁻¹ 3.44×10⁻⁶ 8.82×10⁻⁴

TCV 0.88 1.43 0.03

(estimated) 2.04×10⁻² 3.98×10⁻⁶ 2.85×10⁻⁴

WEST 2.25 4.5 0.1

(estimated) 2.26×10⁻¹ 3.23×10⁻⁶ 8.55×10⁻⁴

COMPASS 0.56 0.9 ~ 2.1 0.03

(estimated) 2.04×10⁻² 2.41×10⁻⁷ 2.22×10⁻⁴

NSTX-U 0.85 1 0.1

(estimated) 2.26×10⁻¹ 5.49×10⁻⁶ 1.11×10⁻³

TEXTOR 1.75 3 0.09±0.01 1.83×10⁻¹ 3.77×10⁻⁶ 8.31×10⁻⁴

SST-1 1.1 3 0.04

(estimated) 3.62×10⁻² 2.36×10⁻⁷ 2.93×10⁻³

EAST 1.75 3.5 0.05~0.09 1.11×10⁻¹ 3.23×10⁻⁶ 5.98×10⁻⁴

KSTAR 1.8 3.5 0.1 2.26×10⁻¹ 3.32×10⁻⁶ 8.67×10⁻⁴

T-15U 2.43 3.5 0.15

(estimated) 5.09×10⁻¹ 4.49×10⁻⁶ 1.51×10⁻³ 0.2

Figure 2.9 The displacement distribution, 𝜉(𝑟), based on Wesson’s model [18].

Figure 2.10 The flow pattern of plasma in Wesson’s model. The red region is hot region, and the blue

𝑞 = 1 surface

𝑟 𝜉

𝑟₁ 𝜉₀

𝜉 = 0 as 𝑟 > 𝑟₁

2.3.4 Comparison of the two models

This concludes the theoretical review of sawtooth instability. Summarizing section 2.3.2 and section 2.3.3, a comparison of Kadomtsev’s model and Wesson’s model is shown in Table 2.4.

Table 2.4 Comparison between Kadomtsev’s model and Wesson’s model.

Kadomtsev’s model Wesson’s model

Model Fast reconnection model Quasi-interchange model

Safety factor,

𝑞 1 − 𝑞 > 𝜀 1 − 𝑞 ≪ 𝜀

To diminish 𝛿𝑊2 term

d𝜉

d𝑟→ 0 except 𝑞 = 1 surface 𝑞 → 1 inside 𝑞 = 1 surface Approximated

𝛿𝑊

2𝜋²𝜉₀²

𝑅₀ ∫ {2𝑟𝑝^′−^𝐵𝜃2

𝜇₀ (1 − 𝑞)(1 +

𝑎 0

3𝑞)} 𝑟𝑑𝑟

2𝜋²

𝑅₀ ∫ {2𝑟𝑝^′−^𝐵𝜃2

𝜇₀ (1 − 𝑞)(1 +

𝑎 0

3𝑞)} 𝜉²𝑟𝑑𝑟 Flow structure Hot bubble, cold crescent Hot crescent, cold bubble

Magnetic

reconnection Occurs at fast collapse phase Occurs at ramp phase

III. Experimental Methods

The ECEI system is a diagnostic system that visualizes electron temperature fluctuation in 2-D/3-D spaces of tokamak plasmas. The ECEI system operates based on the principle of ECE (Electron Cyclotron Emission). First, the physical principle of ECE will be reviewed. Then, the ECEI (Electron Cyclotron Emission Imaging) system in KSTAR will be reviewed. Finally, the experimental set up for the measurement of sawtooth instability will be described.

3.1 ECE and KSTAR ECEI System

In tokamak, it is impossible to measure the plasma electron temperature with commercial thermometers, because the plasma electron temperature is too high to be measured. Instead of a thermometer, an ECE (electron cyclotron emission) radiometer or ECE imaging system can be used to measure electron temperature by measuring radiation intensity emitted from gyrating electrons.

3.1.1 Physical principle of ECE

The electron is gyrating around a magnetic field in tokamak as shown in Figure 3.1. The cyclotron radiation is emitted from gyrating electrons.

Magnetic field line Trace of electron

In this picture, the equation of motion is 𝑚𝑑𝐯

𝑑𝑡 = 𝑞𝐯×𝐁 (3.1)

and the solution of the equation is simple harmonic motion. The angular velocity of the solution is 𝜔 = 𝑒𝐵

𝑚𝑒

. (3.2)

where 𝑚_𝑒 is the mass of the electron, and 𝐵 is the magnitude of magnetic field [20].

In this gyration motion, the radioactive transfer equation is given as 𝑑𝐼𝜈

𝑑𝑠 = 𝑗_𝜈− 𝛼𝐼_𝑣. (3.3)

where 𝐼_𝜈 is the intensity of radiation, 𝑠 is the radiation traveling distance, 𝑗_𝜈 is the emission coefficient, and 𝛼 is the absorption coefficient [40]. The optical depth, 𝜏, is defined as

𝜏 = ∫ 𝛼𝑑𝑠

𝑠

. (3.4)

If the radiation travel from 0 to 𝜏, the solution for the equation is given as 𝐼_𝜈(𝜏) = 𝐼_𝜈(0)𝑒^−𝜏+𝑗𝜈(𝜏)

𝛼 (1 − 𝑒^−𝜏) =𝑗𝜈(𝜏)

𝛼 + {𝐼_𝜈(0) −𝑗𝜈(𝜏)

𝛼 } 𝑒^−𝜏. (3.5)

When the optical depth is thick enough, the radiation density is approximated as 𝐼_𝜈(𝜏) = 𝑗_𝜈/𝛼 [41].

This means all radiation is absorbed. In this case, the plasma can be regarded as a blackbody, so it follows Rayleigh-Jean’s law as

𝐼_𝜈= 2ℎ𝜈³/𝑐²

𝑒^{ℎ𝜈/𝑘𝐵𝑇𝑒}− 1. (3.6)

where ℎ is the Planck constant and 𝑇_𝑒 is the electron temperature [42, 43]. With low frequency approximation, ℎ𝜈 ≪ 𝑇_𝑒, Eq. (3.6) approximated as

𝐼_𝜈 =2ℎ𝜈²𝑘_𝐵𝑇_𝑒

𝑐² . (3.7)

Finally, for thick optical depths, the electron temperature is obtained as 𝑇_𝑒=2𝜋²𝑐²𝐼_𝜔

ℎ𝜔²𝑘_𝐵 =2𝜋²𝑐²𝑚²𝐼_𝜔

ℎ𝑒²𝐵²𝑘_𝐵 . (3.9)

This shows that the electron temperature, 𝑇_𝑒, is proportional to radiation intensity, 𝐼_𝜔, for thick optics. In order to satisfy this characteristic, the plasma should be optically thick.

3.1.2 KSTAR ECEI System

The KSTAR ECEI system is used to visualize MHD instability in 3D by measuring relative electron temperature, 𝛿𝑇_𝑒/〈𝑇_𝑒〉, where 𝛿𝑇_𝑒= 𝑇_𝑒− 〈𝑇_𝑒〉, and 〈𝑇_𝑒〉 is the time averaged temperature.

In KSTAR, there are 3 detector arrays for ECEI measurement. Two of them are placed in H-port and another is placed in G-port as shown in Figure 3.2. The angle difference of G-port and H-port is 22.5° toroidally. Each of the two detector arrays in G-port is placed on HFS (High Field Side, 𝑅 < 𝑅₀) and LFS (Low Field Side, 𝑅 > 𝑅₀ ) as shown in Figure 3.2. The magnetic field, 𝐵 can be approximated by 𝐵𝜙 (∵ 𝐵 = √𝐵_𝜃²+ 𝐵_𝜙² and 𝐵𝜙 ≫ 𝐵𝜃 ), and according to Eq. (2.12), 𝐵𝜙 is proportional to 1/𝑅. Thus, 𝐵 ~ 1/𝑅.

Each detector array has 24 vertical and 8 radial detection channels. In other words, each detector has 192 detection channels, and at H-port, there are 384 channels and at G-port, there are 192 channels, so in total, there are 576 channels on the ECEI system. The vertical and radial resolution is ~1.5 cm per channel and the time resolution is 0.5, 1, 2 𝜇s [44].

Figure 3.2 The schematics of the ECEI system. The first system is in H-port, and the second system is in G-port [45].

3.2 Experimental Set-up

The major and minor radii of KSTAR is shown in Table 3.1. These parameters are constant in every experiment.

Table 3.1 Parameters of KSTAR [32]

Radius

Major radius 𝑅₀= 1.8 m

Minor radius 𝑎 = 0.5 m

The intrinsic rotation experiment # 11264 is used to observe the sawtooth oscillation in 3D with the KSTAR ECEI system. The experimental set-up is shown in Table 3.2. In the next chapter, the experimental results will be shown and discussed by the results. Then, the results will be compared with the theoretical models.

Table 3.2 Experiment set-up for # 11264 Set-up

Measurement duration 𝑡 = 0 ~ 10.5 s

Observed time 𝑡 = 3.5 ~ 4.5 s

Time resolution ∆𝑡 = 2 𝜇s

Mode H-mode

Average density of electron 〈𝑛_𝑒〉 = 1.8×10¹⁹m⁻³ Average density of electron at the core 〈𝑛_𝑒0〉 = 3×10³m⁻³

Average electron temperature 〈𝑇_𝑒0〉 = 1.5 ~ 2.7 keV Average electron temperature at the core 〈𝑇_𝑒0〉 = 2.5 keV

Sampling frequency 𝑓_𝑠= 500 kHz

Harmonic extraordinary mode

Total plasma current 𝐼_𝑝= 600 kA

Total driven current 𝐼_𝑡 = 20 kA

Heating power 0.765 MW

Elongation 𝜅 = 1.8

Safety factor at 𝑟 = 0.95𝑎 𝑞₉₅≈ 5 Radial (𝑅) position /

Vertical size / Zoom factor

HFS, H-port 180 m - 190 m / 40 cm / 1.3 G-port 190 m - 200 m / 50 cm / 1.6 LFS, H-port 205 m - 215 m / 40 cm / 1.3

Vertical position 0.02 ~ 0.04 m

Toroidal magnetic field 𝐵_𝑡 = 3.5 T Radial spatial resolution ∆𝑅 = 1 cm Vertical spatial resolution ∆𝑧 = 2 cm

Triangularity 𝛿 = 0.8

Poloidal beta 𝛽_𝑝= 0.4

Rotation speed 𝑣_𝜙 = 110 km/s

IV. Results & Discussion

4.1 Experimental results

4.1.1 Observation of periodic behavior

The relative temperature fluctuation was observed in 10.5 s intervals. Within this time interval, 3.4 - 4.6 s was selected as shown in Figure 4.1. The results show the time traces of the ratio of electron temperature fluctuation to average temperature, observed through two detection channels from the ECEI system.

(a) (b)

The observed sawtooth period, theoretically calculated Kadomtsev’s collapse time, and the observed collapse time, are shown in Table 4.2.

Table 4.1 Sawtooth period, and collapse times.

Times Symbol Result

Sawtooth period (observed) 𝜏_sawtooth 28.2 ms Kadomtsev’s collapse time

(calculated) 𝜏_𝐾 867 𝜇s

Collapse time (observed) 𝜏_𝑐 150 𝜇s

According to the results, the observed collapse time is very short compared to the sawtooth period, i.e., 𝜏_𝑐/𝜏_sawtooth ~ 10⁻². That is why this phenomenon is called fast collapse.

The whole periodic behavior of at the central region (observed in high field side of H-port) and the outer region (observed in G-port) is shown in Figure 4.2. Also, the corresponding three phases are shown in Table 4.2.

(a)

Figure 4.2 1-D time trace of 𝛿𝑇_𝑒/〈𝑇_𝑒〉 in one period (a) at the central region (b) at the outer region.

Table 4.2 The time lengths of the three phases in Figure 4.2.

Times Time interval (s) Duration Ramp phase 3.52482 ~ 3.54982 25 ms

Precursor phase 3.54982 ~ 3.55287 3.05 ms

Fast collapse phase 3.55287 ~ 3.55302 150 𝜇s

The ECE images of periodic behavior are shown in Figure 4.3. The observation time is from 3.52488 s to 3.55308 s. Initially, as shown in Figure 4.3 (a), the core is relatively cold and it is stable. The temperature of the core increases over time in Figure 4.3 (a) – Figure 4.3 (f). Then, the new magnetic island is growing in Figure 4.3 (g) – Figure 4.3 (k). A new magnetic island was first distinctly observed as shown in Figure 4.3 (i). The old magnetic island and new magnetic island were observed alternately, as plasma was noticed to be rotating. Finally, between Figure 4.3 (g) – Figure 4.3 (k), the crash occurred.

After the crash occurred, the heat of the core spread to the outer region. The plasma returned to its initial conditions as shown in Figure 4.3 (l). This behavior of relative electron temperature fluctuation occurs repeatedly.

The stable ramp phase is shown in Figure 4.3 (b) – Figure 4.3 (f); the precursor phase is shown in Figure 4.3 (g) – Figure 4.3 (j); and the fast collapse phase is shown in Figure 4.3 (k) – Figure 4.3 (l). In the ramp phase, the core is heating. In the precursor phase, the temperature reached critical temperature, and the hot core started to move by internal kink. On the other side of the hot core plasma, a new magnetic island appeared, and a new magnetic island grew. The old magnetic island disappeared suddenly, and the electron temperature fluctuation distribution returned to initial sate, which is the first state of the stable ramp phase. This demonstrated that the experiment resembles Kadomtsev’s model more than Wesson’s model, which was described in section 2.3.2.