**6. Summary and discussion**

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

 .

<sup>2</sup> *<sup>δ</sup>*W∞*δ*W−<sup>1</sup>

= *τwγN***d***v*1, (111)

 M

2/(|∇*ψ*||∇*θ*|)

**u**|*ψb*<sup>+</sup> − **u**|*ψb*<sup>−</sup> = −**c***v*1. (112)

*<sup>b</sup>* F1**d***v*<sup>1</sup> = O.

*<sup>b</sup>* F1**d***v*<sup>1</sup> = *τwγN***d***v*1, (113)

*<sup>b</sup>* can be neglected

**<sup>u</sup>**|*ψb*<sup>+</sup> <sup>−</sup> **<sup>u</sup>**|*ψb*<sup>−</sup>

where *τ<sup>w</sup>* = *μ*0*σdb*/*τA*, *d* is the wall thickness, *b* is the average wall minor radius, and

J |∇*ψ*||∇*θ*| − J |∇*ψ* · ∇*θ*|

<sup>2</sup>|∇*ψ*|/|∇*θ*<sup>|</sup>

The dispersion relation for this eigen value problem is given by the determinant equation det |D0(*γN*)| = 0. In general the Nyquist diagram can be used to determine the roots of this dispersion relation. For RWMs, however, the growth rate is much smaller than the

for determining the stability condition. Consequently, one can use the reduced eigen value

with the RWM mode growth rate *γ<sup>N</sup>* on the right hand side of this equation used as the eigen

Now let us discuss the connection of current global theory with localized analytical theories described in Sec. 4. The singular layer equation in Eq. (48) is derived by employing mode localization assumption. Only localized mode coupling is considered. The general eigen mode equation Eq. (96) in plasma region contains all side band couplings. Noting that Q ∝ *x*, one can see from Eqs. (97) and (98) that <sup>F</sup> <sup>∝</sup> *<sup>x</sup>*<sup>2</sup> and <sup>K</sup> <sup>∝</sup> *<sup>x</sup>* at marginal stability *<sup>ω</sup>* <sup>=</sup> 0. We can therefore see the root of Eq. (48) in Eq. (96). If ballooning invariance in Eq. (57) is introduced, the set of matrix Eq. (96) can be transformed to a single ballooning equation. The TAE theory in Sec. 4.4 uses just two Fourier components to construct eigen modes. The general Alfvén gap structure can be determined by det |F| = 0. Note that, if an analytical function is given on a line on complex *ω* plane, the function can be determined in whole domain through analytical continuation by using the Cauchy-Riemann condition. Note also that one can avoid MHD continuum by scanning the dispersion relation with real frequency �*e*{*ω*} for a small positive growth rate �*m*{*ω*}. Using the scan by AEGIS one can in principle find damping roots through analytical continuation. Due to its adaptive shooting scheme AEGIS can be used to compute MHD modes with very small growth rate. It has successfully computed Alfvén continuum damping rate by analytical continuation based on AEGIS code Chen et al.

where *σ* is the wall conductivity. Equation (110) can be reduced to

From Eqs. (109), (111), and (112) we find the eigen mode equations

<sup>D</sup>0(*γN*)**d***v*<sup>1</sup> <sup>≡</sup> *<sup>τ</sup>wγN***d***v*<sup>1</sup> <sup>+</sup> VF−<sup>1</sup>

−VF−<sup>1</sup>

Alfvén frequency. Therefore, the growth rate dependence of *<sup>δ</sup>*W∞*δ*W−<sup>1</sup>

<sup>2</sup> *<sup>δ</sup>*W∞*δ*W−<sup>1</sup>

V 

+*n*<sup>2</sup> J |∇*φ*|

Since **c***v*<sup>2</sup> = 0, we find that Eqs. (104) - (106) yield

problem

**5.4 Discussion**

(2010).

value to determine the stability.

V = M

In this chapter we have given an overview of MHD theory in toroidal confinement of fusion plasmas. Four types of fundamental MHD modes in toroidal geometry: interchange, ballooning, TAEs, and KDMs, are discussed. In describing these modes we detail some fundamental analytical treatments of MHD modes in toroidal geometry, such as the average technique for singular layer modes, ballooning representation, mode coupling treatment in TAEs/KDMs theories. Note that analytical approach is often limited for toroidal plasma physics. Global numerical treatment of MHD modes is also reviewed in this chapter, especially the AEGIS code formalism. These theories are reviewed in ideal MHD framework. Here, we briefly discuss kinetic and resistive modifications to ideal MHD, as well as the connection of MHD instabilities to transport.

Let us first discuss kinetic effects. Since strong magnetic field is used to contain plasmas in magnetically confined fusion experiments, MHD theory can be rather good to describe fusion plasmas in the direction perpendicular to magnetic field. This is because strong magnetic field can hold plasmas together in perpendicular movement. Therefore, MHD is a very good model to describe perpendicular physics, if FLR effects are insignificant. However, in parallel direction the Lorentz force vanishes and particle collisions are insufficient to keep particles to move collectively. Consequently, kinetic description in parallel direction is generally necessary. Kinetic effect is especially important when wave-particle resonance effect prevails in the comparable frequency regime *ω* ∼ *ωsi* Zheng & Tessarotto (1994a). In the low frequency regime *ω* � *ωsi*, wave-particle resonances can be so small that kinetic description results only in an enhancement of apparent mass effect. Kinetic effect in this case can be included by introducing enhanced apparent mass. Another non-resonance case is the intermediate frequency regime *ωsi* � *ω* � *ωse*. In this regime kinetic description results in a modification of ratio of special heats. By introducing proper Γ MHD can still be a good approximation. Recovery of perpendicular MHD from gyrokinetics has been studied in details in Ref. Zheng et al. (2007).

Fig. 2. CITM physics picture. The dot-dashed line represents mode rational surface. Perturbed current at rational surface due to interchange modes leads to field line reconnection and formation of magnetic islands.

in Toroidal Plasma Confinement 31

Overview of Magnetohydrodynamics Theory in Toroidal Plasma Confinement 31

Fig. 3. Schematic explanation for why flow shear does not de-correlate turbulence eddies.

consideration.

**7. References**

In conclusion significant progresses have been made for linear ideal MHD theories and numerical codes in the past dacades. However, the kinetic effects on MHD remains considerably open. Although correction of gyrokinetics theory has been made recently Zheng et al. (2007), the applications of the new gyrokinetics theory remain to be worked out. The theories for FLR effects on ballooning modes, KTAEs, energetic particle effects, etc. need to be modified with newly corrected gyrokinetics theory. The extension of toroidal resistive MHD theory Glasser et al. (1975) to take into account the small parallel ion speed effect Zheng & Tessarotto (1996) and current interchange effects Zheng & Furukawa (2010) is under

Berk, H. L., Rosenbluth, M. N. & Shohet, J. L. (1983). Ballooning mode calculations in

Berk, H. L., Van Dam, J. W., Guo, Z. & Lindberg, D. M. (1992). Continuum damping

Bernard, L., Helton, F. & Moore, R. (1981). GATO: an MHD stability code for axisymmetric

Betti, R. & Freidberg, J. P. (1992). Stability of Alfvén gap modes in burning plasmas, *Physics of*

Boozer, A. H. (1982). Establishment of magnetic coordinates for a given magnetic field, *Physics*

Chance, M., Greene, J., Grimm, R., Johnson, J., Manickam, J., Kerner, W., Berger, D.,

Bernard, L., Gruber, R. & Troyon, F. (1978). Comparative numerical studies of ideal

plasmas with internal separatrices, *Comput. Phys. Commun.* 24: 377.

of low-n toroidicity-induced shear Alfvén eigenmodes, *Physics of Fluids B: Plasma*

stellarators, *Physics of Fluids* 26(9): 2616–2620. URL: *http://link.aip.org/link/?PFL/26/2616/1*

URL: *http://link.aip.org/link/?PFB/4/1806/1*

*Fluids B: Plasma Physics* 4(6): 1465–1474. URL: *http://link.aip.org/link/?PFB/4/1465/1*

URL: *http://link.aip.org/link/?PFL/25/520/1*

*Physics* 4(7): 1806–1835.

*of Fluids* 25(3): 520–521.

Next, let us discuss resistivity effects. Resistivity usually is small in magnetically confined fusion plasmas. Due to its smallness resistivity effects are only important in the singular layer region. With ideal MHD singular layer theory detailed in Sec. 4.2 one can rederive resistive singular layer equations given in Ref. Glasser et al. (1975). However, it should be pointed out that, when kinetic enhancement of apparent mass effect is taken into account, the ratio of resistivity and inertia layer widths changes. This leads kinetic description of resistive MHD modes to become substantially different from fluid description Zheng & Tessarotto (1996) Zheng & Tessarotto (1995). Kinetic analysis of low frequency resistive MHD modes becomes necessary.

The driving force for ideal MHD instabilities is related to pressure gradient. Resistivity can instead cause field line reconnection and induce the so-called tearing modes. It is important to note that if current gradient is taken into account pressure driven modes and tearing modes are coupled to each other. The underlying driving mechanism for pressure driven modes is the release of plasma thermal energy from the interchange of magnetic flux tubes. Actually, interchange-type modes exchange not only thermal and magnetic energies between flux tubes, but also current. In a plasma with a current (or resistivity) gradient, such an interchange can create a current sheet at a mode resonance surface and result in the excitation of current interchange tearing modes (CITMs) as shown in Fig. 2 Zheng & Furukawa (2010).

Instabilities of interchange type have been widely used to explain anomalous transport in tokamaks in terms of the formation of turbulent eddies through nonlinear coupling. However, the explanation for experimental observations that the electron energy transport is much larger than what one would expect from diffusive process due to Coulomb collisions is still unsatisfactory. The electron Larmor radius is much smaller than ion one. Nonetheless, the electron thermal transport often is stronger than ion transport. In Ref. Rechester & Rosenbluth (1978), the broken magnetic surfaces due to formation of magnetic island and stochastic field lines are used to explain the enhanced electron transport. But, how magnetic islands are formed in axisymmetric tokamak plasmas has not been given. CITM theory shows that interchange-type instabilities can directly convert to current interchange tearing modes. This helps to clarify the source of electron transport in tokamaks.

Another transport issue we need to discuss is the so-called flow shear de-correlation of turbulences. This concept has been widely used for explaining suppression of plasma turbulences. In fact, this picture is not right for systems with magnetic shear. We use Fig. 3 to explain it (L. J. Zheng and M. Tessarotto, private communication). In Fig. 3, the dashed long arrow represents a magnetic field line on a given magnetic surface *ψ*0, and two solid long arrows denote the magnetic field lines respectively at two time sequences *t*<sup>0</sup> and *t*<sup>0</sup> + Δ*t* on an adjacent magnetic surface *ψ*1. Let us examine the correlation pattern in the local frame moving together with equilibrium velocity of the dashed long arrow on surface *ψ*0. The modes are supposed to locate around the point "*O*" initially at *t* = *t*0. After a time interval Δ*t*, the field line on surface *ψ*<sup>1</sup> moves relatively to the dashed long arrow on the surface *ψ*<sup>0</sup> due to flow shear. From Fig. 3 one can see that the fixed pattern has not been de-correlated by flow shear, instead the pattern just propagates from point "*O*" at time *t* = *t*<sup>0</sup> to point "*O* " at subsequent time *t* = *t*<sup>0</sup> + Δ*t*. This indicates that flow shear does not de-correlate turbulence eddies. Only flow curvature can result in the de-correlation. This resembles to ballooning mode behavior in rotating plasmas with Cooper representation Waelbroeck & Chen (1991).

Fig. 3. Schematic explanation for why flow shear does not de-correlate turbulence eddies.

In conclusion significant progresses have been made for linear ideal MHD theories and numerical codes in the past dacades. However, the kinetic effects on MHD remains considerably open. Although correction of gyrokinetics theory has been made recently Zheng et al. (2007), the applications of the new gyrokinetics theory remain to be worked out. The theories for FLR effects on ballooning modes, KTAEs, energetic particle effects, etc. need to be modified with newly corrected gyrokinetics theory. The extension of toroidal resistive MHD theory Glasser et al. (1975) to take into account the small parallel ion speed effect Zheng & Tessarotto (1996) and current interchange effects Zheng & Furukawa (2010) is under consideration.
