Y. Z. Zhang

Edge turbulence scaling with shear flow

A formula relating turbulence levels with arbitrary shear flow is derived. When the diffusion coefficient is made a functional of the corresponding turbulence level, it is found that the scaling laws governing turbulence suppression are considerably modified. The results are compared with known formulas in various limiting cases, indicating that turbulence suppression mainly pertains in the moderate shear flow regime. The results also show that a flattened (steep) radial equilibrium gradient tends to enhance (eliminate) turbulence suppression due to the shear flow.

Effect of energetic trapped particles on the ‘ideal’ internal kink mode

The internal kink stability of a tokamak in the presence of energetic particles is studied. It is found that there is a stable window when a finite population of energetic particles is present; also the relationship between the predictions of the fishbone theory of Chen-White-Rosenbluth and the fishbone theory of Coppi-Porcelli is explained. The theory indicates why in some experiments, such as those on PDX and TFTR, fishbone oscillations are likely to occur in conjunction with sawtooth modes, while in other experiments sawtooth suppression in the presence of hot particles is observed.

Internal kink stabilization by high‐energy ions with nonstandard orbits

A generalized energy principle that takes into account the nonstandard, potato‐shaped particle orbits of high‐energy ions in the central region of a tokamak is derived. It is shown that, in the limit of zero orbit width, this energy principle reduces to the one formulated by Van Dam et al. [Phys. Fluids 25, 1349 (1982)]. The modification of hot particle stabilization theory when such orbit effects are important is investigated. In particular, a model distribution function is chosen to describe high‐energy trapped ions produced by ion cyclotron resonant frequency (ICRF) heating applied near the axis of a tokamak. Standard banana orbit theory predicts that, for fixed total stored energy of energetic particles peaked about the magnetic axis, the stabilizing influence on internal kink modes is inversely proportional to the spatial spread of the hot particles. However, this scaling saturates when the spatial spread of the distribution function approaches the width of a typical nonstandard orbit. Hence, ICRF he...

Impurity and neutral effects on the dissipative drift wave in tokamak edge plasmas

Possible destabilizing mechanisms for the linear electrostatic dissipative drift waves (in tokamak edge plasmas) are investigated in slab geometry. The effects of processes such as ionization, charge exchange, radiation, and rippling are examined. In particular, the impurity condensation associated with radiation cooling is evaluated appropriately for the drift wave ordering, which is found to be an important driving mechanism in contrast to the results of earlier studies [R. J. Thayer and P. H. Diamond, Phys. Rev. Lett. 65, 2784 (1990)]. It is also shown that the role of ionization is quite complicated, and depends strongly on the manner in which the equilibrium is achieved. The linear eigenmode equation is studied both analytically and numerically. For the range of parameters relevant to TEXT tokamak [K. W. Gentle, Nucl. Fusion Technol. 1, 479 (1981)], both the charge exchange and the rippling effect are found to be unimportant for instability.

Correlation theory of a two-dimensional plasma turbulence with shear flow

When the ion sound effect is neglected, a wide class of electrostatic plasma turbulence can be modeled by a two‐dimensional equation for the generalized enstrophy Ψ, an inviscid constant of motion along the turbulent orbits. Under the assumption of a Gaussian stochastic electrostatic potential, an averaged Green’s function method is used to rigorously derive equations for the N‐particle correlation functions for a dissipative and sheared flow. This approach is equivalent to the cumulant expansion method [T. H. Dupree, Phys. Fluids 15, 334 (1972); 21, 783 (1978)] used to study the Vlasov–Poisson system. For various cases of interest, appropriate equations are solved to obtain the absolute level as well as the detailed structure of the two‐point correlation function C(r), and its Fourier transform, the enstrophy spectral function I(k). Uniformly valid analytical expressions are derived for the dissipative but shearless case resulting in a ‘‘fluctuation–dissipation’’ theorem relating the total spectral intensity to classical viscosity. These self‐consistent results show a strong logarithmic modification of the mixing length estimates for the turbulence levels. For the extremely important and interesting problem of a sheared flow, the suppression of turbulence is demonstrated by using asymptotic analytical techniques in the inviscid range, and uniformly valid numerical methods for the dissipative system. The current asymptotic methods reproduce the results obtained in the orbit picture [Y. Z. Zhang and S. M. Mahajan, Phys. Fluids B 4, 1385 (1992)], but provide much clearer physical perspective and a better definition of crucial parameters like the decorrelation time. The uniformly valid numerical approach allows the determination of the change in spectral shape and intensity due to the presence of shear. It is found that the suppression is more effective for longer wavelengths as compared to the shorter wavelengths. This and other relevant issues, concerning the role of flows with shear (including its radial variation) in the understanding of the L–H transitions in tokamaks, are discussed.

Multiple-gap theory of toroidal Alfven waves with kinetic effects

The stability of kinetic toroidal Alfven waves with multigap coupling is analyzed by using the two‐dimensional ballooning transform. An alternate convergence scheme, based on the smallness of the inverse aspect ratio, is devised. The resulting wave functions are oscillatory and do not balloon in contrast to the wave functions of conventional ballooning theory. It is shown that the single‐gap theory is a special, weak shear (s→0) limit of the formalism. Analytical and numerical results for the two fundamental branches, the ideal toroidal Alfven eigenmode (TAE), and the kinetic toroidal Alfven eigenmode (KTAE), are presented and discussed.

Two‐dimensional aspects of toroidal drift waves in the ballooning representation

By systematically doing the higher‐order theory, the predictions of the conventional ballooning theory (CBT) are examined for nonideal systems. For the complex solvability condition to be satisfied, radial variation of the lowest‐order mode amplitude needs to be invoked. It turns out, however, that even this procedure with its concomitant modifications of eigenvalues and eigenstructures, is not sufficient to justify the predictions of many CBT solutions; only a small set of the CBT solutions could be put on firm footing. To demonstrate this work’s general conclusions, theoretical and numerical results are presented for a system of fluid drift waves with nonadiabatic electron response.

M = 1 KINK MODE FOR LAYER WIDTHS COMPARABLE TO THE ION LARMOR RADIUS

A kink‐tearing eigenmode equation is derived for a slab layer geometry in the limit me/mi≪βi≪L2n/L2s (me/mi ≡mass ratio, βi≡ ion beta value, Ln ≡gradient scale length, Ls ≡shear length) and when the electron collision frequency is comparable to the eigenfrequency. It is essential for consistency to retain arbitrary ion Larmor radius effects, which are described with the use of the Pade approximation. The asymptotic solution of the inhomogeneous eigenvalue problem is obtained using simple approximations to the eigenfunction. A dispersion relation duplicates previously derived results when a Lorentzian conductivity model is used for electrons, while a new dispersion relation is obtained if electrons are described by kinetic dynamics. The dispersion relation is analytically and numerically investigated. The numerical results are compared to a more complicated and presumably more ‘‘rigorous’’ asymptotic expression. It is shown that this asymptotic expression requires quite a small value for (me/mi)βi for accu...

Continuum damping of ideal toroidal Alfven eigenmodes

A perturbation theory based on the two‐dimensional (2‐D) ballooning transform is systematically developed for ideal toroidal Alfven eigenmodes (TAEs). A formula, similar to the Fermi golden rule for decaying systems in quantum mechanics, is derived for the continuum damping rate of the TAE; the decay (damping) rate is expressed explicitly in terms of the coupling of the TAE to the continuum spectrum. Numerical results are compared with previous calculations. It is found that in some narrow intervals of the parameter me, the damping rate varies very rapidly. These regions correspond precisely to the root missing intervals of the numerical solution by Rosenbluth et al. [Phys. Fluids B 4, 2189 (1992)].

Renormalized perturbation theory: Vlasov–Poisson system, weak turbulence limit, and gyrokinetics

The self‐consistency of the renormalized perturbation theory of Zhang and Mahajan [Phys. Rev. 132, 1759 (1985)] is demonstrated by applying it to the Vlasov–Poisson system and showing that the theory has the correct weak turbulence limit. Energy conservation is proved to arbitrary high order for the electrostatic drift waves. The theory is applied to derive renormalized equations for a low‐beta gyrokinetic system. Comparison of this theory with other current theories is presented.