D. P. Kostomarov

Stability regions of dissipative trapped-ion instability

Conditions for the stability of the dissipative trapped ion mode in axisymmetric toroidal confinement systems are investigated, and the ranges of temperature and magnetic field for which the plasma is expected to be unstable are calculated numerically for several proposed Tokamaks. The collisional damping is studied as a boundary layer problem in the velocity space of the trapped ions. The Fokker-Planck equation is solved by means of a variational form with the ordering νiR/r ω 2π/τi, where R/r is the aspect ratio, νiR/r is the effective collision frequency, and τi is the bounce time. The relative damping rate is found to be proportional to (νiR/rω) 1/2 [In (rω/νiR)1/2]−3/2 rather than νiR/rω. This damping term is compared with the electron driving term, which is proportional to rω/Rνe, and with the Landau damping caused by resonant untrapped ions. The presence of a temperature gradient is found to be destabilizing. The Landau term provides relatively strong damping when dT/dr=0 but changes sign when d ln T/d In n > 2/3. In the latter case it is concluded that ion collisions are insufficient to stabilize the mode under typical operating conditions of the proposed machines. Finally, if d ln T/d In n > 1.75 the ion collisional damping itself changes to growth, and the mode cannot be stabilized.

Analyses of substantially different plasma current densities and safety factors reconstructed from magnetic diagnostics data

The problem of plasma current density and safety factor reconstruction using magnetic field measurements is considered. In the traditional formulation, the problem is strongly ill-posed. In particular, substantially different current densities and safety factors can be equally well attributed to the same set of measurements, given their experimental errors. In other words, the problem can be strongly unstable with respect to the input data. Different constraints are used in practice to make the problem more stable. This paper presents an accurate mathematical formulation of the inverse problem and its variants. A numerical algorithm is provided, which permits us to study the stability with respect to variations in the input data, to find all substantially different solutions, or to prove their absence, and to determine the confidence intervals of the reconstructions. The proposed method also allows establishing the maximum error for a given diagnostic (additional constraint), below which the diagnostic efficiently extracts one solution among several substantially different ones. Examples of very different current density and safety factor reconstructions for measurements with finite accuracy are presented for the original formulation of the inverse problem. Cases of MAST, JET and ITER-like plasmas are considered. It is shown that including the motional Stark effect (MSE) measurements as a constraint, provided the accuracy of MSE measurements is sufficient, allows identifying one solution among several very different ones, obtained without such a constraint. The maximum MSE diagnostics error for efficient identification of this solution is estimated for JET. The approach of this paper can be used for a wide range of ill-posed problems in physics and can help in selecting additional conditions, which can identify the most likely solution among several.

Cyclotron instability of a plasma

In this paper we study the cyclotron instability of a plasma in a magnetic field when the ion distribution function is anisotropic in velocity space. For perturbations in the form of plane waves with various wave vectors k, we determine the values of plasma parameters at which instability appears (that is, the region of instability). Minimum values are found.for the parameter τ = T⊥i/T||i characterizing the anisotropy of ion temperatures – that is, the values at which the plasma begins to be unstable. It is shown that the largest region of instability corresponds to. the long waves propagated almost perpendicularly to the external magnetic field. It is established that two conditions must be fulfilled for cyclotron instability to develop in a plasma with cold electrons: T⊥i/T||i > 2, ω0e > ωi/2 (where ω0e is the plasma frequency of the electrons and ωi the cyclotron frequency of the ions). Heating the electrons improves the stability of the plasma—specifically, by raising the critical value of the parameter T⊥i/T||i from which instability appears. The growth rates of unstable solutions are calculated, and from the formulas for these it is evident that cyclotron instability has a resonance character. The waves that oscillate most strongly are those for which ω0e kz/ωi (kz2 + k⊥2)½ ≈ (n being a whole number).

Calculation of the diffusion of light impurities in tokamaks

Using numerical methods, the authors investigate plasma diffusion in tokamaks when impurity ions of light elements are present. The model considered makes use of neoclassical expressions for diffusion fluxes and takes ionization and recombination processes into account. The question of the boundary conditions for the initial system of equations is discussed. The new boundary conditions for the proton and impurity ion densities are zero partial ion-ion fluxes at the boundary of the plasma column. The computational results show that the presence of impurity ions in the plasma column of a tokamak – even in relatively small amounts – strongly influences the spatial distribution of the protons. The entire diffusion process can be broken down into a number of stages which differ one from the other with regard to the amount of impurities in the system. If the proportion of impurities ξim does not exceed 0.01 of the electron density, the highly ionized ions collect at the centre of the plasma. If ξim > 0.01, the proton profile becomes flat within a short time and the impurities become more evenly distributed over the plasma cross-section.

Kinetic convective ripple transport of ions in tokamaks

Convective transport of fast ions in the toroidal field ripples of a tokamak was investigated experimentally and theoretically. Comprehensive numerical computations of this effect were performed on the basis of the previously developed theory. Simultaneously, detailed experimental studies were made of the energetic ions in the T-4 and T-10 tokamaks. The experiments demonstrate that the ion distribution function is substantially different from the Maxwellian one, being strongly enriched with fast particles at the plasma column periphery. The local trapped-ion distribution is, in addition, asymmetric throughout the plasma cross-section. A detailed comparison with the numerical results shows that the observed effects can be explained in terms of the kinetic convective transport theory. Kinetic convection is shown to contribute significantly to energy transport. With increasing ion temperature, convective ripple transport may become dominant. Therefore, the variation of rippling may become an effective means of controlling the temperature and ignition regimes of a thermonuclear reactor. The results obtained demonstrate that transport processes in modern tokamaks have to be described in terms of kinetic theory.

Automating Computations in the Virtual Tokamak Software System

We describe the concept, functional capabilities, graphic user interface (GUI), and operating technique of a specialized system for distributing the computational burden encountered in solving typical problems of controlled thermonuclear fusion. The system is employed in the Virtual Tokamak simulation modeling complex to automate the distribution of computing on a network of computers, making it possible to dramatically improve the productivity of a researcher’s work. The system is useful in various applications that require massive multivariate calculations using one or more application codes, and for supporting websites that provide computing services using locally stored science-intensive application software.

CYCLOTRON INSTABILITY OF A PLASMA IN MAGNETIC MIRROR MACHINES.

The authors consider cyclotron instability in a homogeneous plasma with anisotropic non-monotonic ion distribution in velocity space. In the plane of the parameters and τ = ⊥i/||i (where ω0e is the plasma frequency of the electrons, ωi is the Larmor frequency of the ions, ⊥i and ||i are the transverse and longitudinal energy of the ions respectively) regions are constructed by numerical methods which correspond to the unstable states of a plasma at cyclotron resonances of different orders. The authors study the dependence of these regions on electron temperature, the dimensions of the trap, and the width of the ion transverse-velocity distribution peak. The stability of a plasma in machines of the Ogra-2 type is discussed. It is shown that the results of recent experiments on the Phoenix device aimed at suppressing cyclotron instability cannot be explained by broadening of the ion distribution peak. To stabilize the oscillations in this machine it is sufficient to increase the temperature of the electrons to 30 – 50 eV, which is apparently what happens in the experiment.

Probing a plasma by electromagnetic waves

The authors examine the problem of determining the spatial density distribution of a magnetized plasma containing cold electrons by using the measured phase difference between the incident and reflected waves in a given frequency range. With the methods of geometrical optics the problem yields Volterras integral equations of the first kind. If the external magnetic field is homogeneous the equations are linear; otherwise they are nonlinear. The equations are studied by numerical methods. The authors also consider the absorption of electromagnetic waves in relation to the problem of probing a plasma.

Modification of the canonical profile transport model on the basis of new DIII-D experiments

Recent DIII-D experiments have shown that the stiffness of the ion temperature profile κiPC in the region 0.4 < ρ < 0.7 increases by one order of magnitude with increasing radius. At ρ < 0.4, the stiffness is low and the ion temperature profile is “soft.” The stiffness of the temperature profile also increases with decreasing the toroidal rotation velocity. The approximation of the experimental stiffness profiles allows one to modify the canonical profile transport model. The heat conductivity κi0 in the plasma core is determined by minimizing the r.m.s. deviations of the calculated ion temperature from the measured one. This procedure also makes it possible to determine how κi0 depends on the central ion temperature.

Toroidal current control in the problem of plasma equilibrium evolution

The results of application of automatic control methods to the problem of the given total current sustainment in toroidal plasma are presented. Plasma current control is performed mainly by solenoid with the use of electromagnetic inductance effect. The mathematical problem is formulated, the methods and developed software are described, and the solutions of test and real problems are given. The comparison of experimental and numerical data allows drawing a conclusion on the nature of plasma conductivity in spherical tokamaks.