Trapped Ion Modes Research Articles

Fourier optics approach to far forward scattering and related refractive index phenomena in laboratory plasmas

The transmission of a gaussian beam of electromagnetic radiation through a refracting medium such as plasma containing within it a localized sinusoidal disturbance, is described entirely in terms of refraction and diffraction. The lowest order terms of an expansion for the intensity profile of a beam in the front focal plane of a lens situated a focal length beyond a beam waist in the medium are identified with well-known optical effects usually treated independently, including shadowgraph, scattering, and schlieren. The latter is shown to be indistinguishable from spontaneous heterodyning of the scattered radiation with the unperturbed beam, and is sensitive to frequency, wavelength, and the location of the disturbance in the medium. An experiment using a HeNe laser with monochromatic ultrasound in atmospheric air has verified the predictions of the model over a wide range of conditions. As a plasma diagnostic, the method may offer a unique means of measuring otherwise inaccessible very long wavelength instabilities such as trapped ion modes in large Tokamaks like JET.

Control of Thermal Instability of Pure and Catalyzed D-D Fusion Reactor Plasmas

The possibility eliminating the thermal instability of self-sustained pure and catalyzed D-D fusion reactor plasmas is examined, using the two feedback models of (a) fueling rate and (b) power control (control by power injection or subtraction). Calculations are performed on the bases of linear stability analysis and non-linear dynamic simulation. In contrast to D-T fusion plasma, Cat. D plasma can be stabilized by fueling rate control in all the confinement time scalings adopted in the study. Pure D plasma is not completely stabilized by fueling rate control except when the confinement times are governed by trapped-ion mode (TIM) scaling. The plasma, however, can be maintained near equilibrium state even in forms of scaling other than TIM. Among the different types of fueling rate control, the one most suitable would appear to be deuterium control, in which the rate of deuterium injection is increased [decreased] to counteract temperature increase [decreased]. With power control, temperature increase [decrease] is counteracted by subtraction [addition] of power out of [into] the plasma. Compared with fueling rate control, this method exerts a more direct effect, although it involves a more complex procedure.

Inhibition of the trapped ion mode by drift wave fluctuations

The influence of a spectrum of drift wave fluctuations on the trapped ion mode is investigated and shown to inhibit the onset of these modes. The stochastic damping of the trapped ion mode becomes comparable to the linear growth rate when the drift wave spectrum is described by previous theoretical models.

Control of thermal instability of pure and catalyzed D-D fusion reactor plasmas.

The possibility eliminating the thermal instability of self-sustained pure and catalyzed D-D fusion reactor plasmas is examined, using the two feedback models of (a) fueling rate and (b) power control (control by power injection or subtraction). Calculations are performed on the bases of linear stability analysis and non-linear dynamic simulation. In contrast to D-T fusion plasma, Cat. D plasma can be stabilized by fueling rate control in all the confinement time scalings adopted in the study. Pure D plasma is not completely stabilized by fueling rate control except when the confinement times are governed by trapped-ion mode (TIM) scaling. The plasma, however, can be maintained near equilibrium state even in forms of scaling other than TIM. Among the different types of fueling rate control, the one most suitable would appear to be deuterium control, in which the rate of deuterium injection is increased [decreased] to counteract temperature increase [decreased]. With power control, temperature increase [decrease] is counteracted by subtraction [addition] of power out of [into] the plasma. Compared with fueling rate control, this method exerts a more direct effect, although it involves a more complex procedure.

Thermal Stability of Pure and Catalyzed D-D Fusion Reactor Plasmas

The thermal stabilities of the D-D fusion reactor plasmas operating on the pure- deuterium and the catalyzed-deuterium cycles have been analyzed by the first-order perturbation method in order to make clear the steady-state operating conditions. When the constant or trapped-ion mode scaling of the confinement-time is used, the critical temperature of the self-heated plasma lies in the vicinity of the minimum point for the confinement parameter nτ. Bohm scaling gives the lowest value for the critical temperature, while neo-classical scaling does the highest one. With increasing fuel injection energy, the growth rate of the instability at low temperature decreases, whereas it does not always decreases at high temperature. In the D-D fusion plasma, the subsidiary reaction 3He(d, p)4 He makes dominant contribution to the plasma heating and the instability at low temperature. The cyclotron radiation, the dominant power loss mechanism at high temperature, plays a significant role in determining the critical temperatures. Calculations neglecting the cyclotron radiation yield the inadequate, considerably higher values for them. The validity of the above results has been confirmed by the non-linear dynamic simulations.

Ignition Approach by Neutral Beam Injection Heating in Impurity Contaminated Tokamak Reactors

Available heating power by neutral beam injection in a tokamak reactor is evaluated semi-empirically. Using this estimated value, device and plasma parameters to ignite the plasma in impurity contaminated tokamak reactors are investigated. By lowering the plasma density and concurrently by enlarging the plasma minor radius or aspect ratio, the difficulty of NBI heating can be avoided, and the ignition is almost always possible both for trapped ion mode and Alcator scaling laws.

Thermal stability of pure and catalyzed D-D fusion reactor plasmas.

The thermal stabilities of the D-D fusion reactor plasmas operating on the pure- deuterium and the catalyzed-deuterium cycles have been analyzed by the first-order perturbation method in order to make clear the steady-state operating conditions. When the constant or trapped-ion mode scaling of the confinement-time is used, the critical temperature of the self-heated plasma lies in the vicinity of the minimum point for the confinement parameter nτ. Bohm scaling gives the lowest value for the critical temperature, while neo-classical scaling does the highest one. With increasing fuel injection energy, the growth rate of the instability at low temperature decreases, whereas it does not always decreases at high temperature. In the D-D fusion plasma, the subsidiary reaction 3He(d, p)4 He makes dominant contribution to the plasma heating and the instability at low temperature. The cyclotron radiation, the dominant power loss mechanism at high temperature, plays a significant role in determining the critical tem...

Non-linear saturation of the trapped-ion mode by mode coupling in two dimensions

A study of the possible non-linear saturation by mode coupling of the dissipative trappedion mode is presented in which both radial and poloidal variations are considered. The saturation mechanism consists of, the non-linear coupling via convection of energy from linearly unstable modes to stable modes. Stabilization is provided at short poloidal wavelengths by Landau damping from trapped and circulating ions for ηi ≡ d ln(Ti)/d ln(n0) < 2/3, at short radial wavelengths by effects associated with the finite ion banana excursions and at long wavelengths by ion collisions. The authors' analysis applies only to the case ηi < 2/3. A one-dimensional, non-linear partial differential equation for the electrostatic potential derived in earlier work is extended to two dimensions and to third order in amplitude. Included approximately in a systematic way are kinetic effects, e.g. Landau damping and its spatial dependence due to magnetic shear. The stability and accessibility of equilibria are considered for cases far from as well as close to marginal stability. In the first case, three-wave interactions are found to be important when the spectrum of unstable modes is sufficiently narrow. In the latter case, it is found that, for a single unstable mode, a four-wave self-interaction can provide a saturation mechanism. Accessibility of equilibria is determined in both cases. In the case where three-wave interactions are dominant, the stability analysis is inconclusive. However, plasma parameters are found for which equilibria established by means of the four-wave self-interaction are stable to linear perturbation. Cross-field transport is calculated, and the scaling of results is considered for tokamak parameters.

2-D fluid calculations of anomalous trapped ion transport in Tokamaks

The collisional, nonlinear trapped-fluid equations of Kadomtsev and Pogutse (K.P.) are used as a description of the dissipative trapped-ion modes. The equations are solved numerically in two spatial dimensions as an initial-value problem. Numerical results are given for the resulting anomalous diffusion coefficient D at late times, when the dissipative trapped-ion instability saturates. Examples of the time development D(t) are given. The numerical values of D (at late times) are generally larger than according to the K.P. formula, and the scaling of D with the equilibrium parameters resembles Bohm scaling. The dependence of the results on the numerical grid and on the initial conditions was also studied. Several analytical results are presented, some of which have been successfully used for testing the numerical code.

Energy Optimization of a Cycled Tokamak

The accumulation of impurities in a controlled thermonuclear reactor makes steady-state operation unlikely. The energy output during the burn phase will depend on the ion temperatures and densities. A dynamic model of the burn cycle of a tokamak is used to investigate the ion densities and temperatures as a function of time. The total energy output per cycle is investigated as a function of the ion feed rates, plasma current, and the divertor efficiency.The point-kinetics model of the plasma incorporates ion and energy balance equations and explicitly accounts for the impurity ion buildup. The D-D, D-T, and D-3He reactions are all considered in this model. The energy carried off by the neutrons in the D-D and D-T reactions is lost from the plasma. Impurities enter the plasma as a result of wall interactions with escaping ions and neutrons. The trapped-ion mode is used for calculating the confinement times.An equilibrium state vector was obtained using currently projected operating parameters. The total energy density for a burn cycle was found to be a monotonically increasing function of the source rates and the plasma current. The energy density was not substantially increased until the divertor efficiency was greater than ∼60% when the other parameters were held constant.

Quasi-linear and nonlinear theory of dissipative trapped particle instabilities

An examination of particle linear orbits reveals how charged particles nonresonantly exchange energy with trapped particle modes in the presence of collisions. Phase space diffusion coefficients are constructed which describe the changes in the equilibrium particle distributions and which allow computation of the linear growth rates of the dissipative trapped particle instabilities. Wave-induced detrapping and resonance broadening are considered as nonlinear modifications of the particle-wave interaction and found to be ineffective as mechanisms to saturate growth of the dissipative trapped ion mode. Mode coupling with plasma diffusion across the magnetic field is a stronger effect for this and the dissipative trapped electron mode. Final evaluation of plasma diffusion coefficients is complicated by an ignorance of the nonlinear spectral shape.

Quasi-linear model for heat flow and diffusion in a micro-unstable tokamak

A formulation of transport theory, including cross-field thermal conductivity, is presented and applied to transport in tokamak due to trapped-particle instabilities,in the quasi-linear stage. Ratios of ion heat conduction, convection, and turbulent heating to electron heat conduction, and ratios of electron heat convection and turbulent heating to electron heat conduction are presented in the quasi-linear phase of trapped-electron and trapped-ion modes, along with general techniques for obtaining these ratios for other modes in a quasi-static micro-unstable plasma.

Observed instabilities in the trapped ion regime in the levitated octupole

Two unstable modes were observed in the average minimum-B region of the levitated octupole operating with a weak toroidal field (safety factor of approximately 0.25) and collisionless ions. One mode propagated in the electron diamagnetic drift direction and the other mode propagated in the ion diamagnetic drift direction. The frequencies of the modes were much greater than the ion detrapping collision frequency and slightly less than the trapped ion bounce frequency. The electron collisionality was intermediate between the collisionless trapped ion mode and the dissipative trapped ion mode regimes. Density fluctuation levels from 5% to 50% and enhanced transport approaching Bohm diffusion were the manifest effects of the instabilities. When the ion collisionality was increased by cooling the ion temperature, the unstable fluctuations were not seen.

Overlap of Bounce Resonances and the Motion of Ions in a Trapped-Ion Mode

Motion near a separatrix (the boundary between closed and open orbits) is studied both theoretically and numerically for parameters appropriate to the dissipative trapped-ion mode. I observe disappearance of an invariant in a stochastic layer surrounding the separatrix, as a result of overlapping bounce resonances. The fraction of ions lying within the stochastic layer is large even for a mode of relatively small amplitude.

Trapped ion mode driven by ion magnetic drift resonance in a fat torus

A study of the trapped ion instability in the real geometry of the large tokamaks leads to the consideration of a new branch of that instability, driven by a resonance with the magnetic drift of the particles, both in collisional and non-collisional regimes.

Non-linear saturation of the dissipative trapped-ion mode by mode coupling

The non-linear saturation of the dissipative trapped-ion mode is analysed. The basic mechanism considered is the process whereby energy in long-wavelength unstable modes is non-linearly coupled via E⃗×B⃗ convection to shortwavelength modes stabilized by Landau damping due to both circulating and trapped ions. In the usual limit of the mode frequency being small relative to the effective electron collision frequency, a one-dimensional non-linear partial differential equation for the potential can be derived, as was first shown by LaQuey, Mahajan, Tang, and Rutherford. The stability and accessibility of the possible equilibria for this equation are examined in detail, both analytically and numerically. The equilibrium emphasized by LaQuey et al. is shown to be unstable. However, a class of non-linear saturated states which are stable to linear perturbations is found. Included in the analysis are the effects of both ion collisions and dispersion due to finite-ion-banana-width effects. Cross-field transport is estimated and the scaling of the results is considered for tokamak parameters (specifically those for the Princeton Large Torus). It is concluded that the anomalous cross-field transport can be much lower than the estimate of Kadomtsev and Pogutse, for relevant parameters near marginal stability for the linear modes.

Effects of impurities on collisionless trapped ion modes

An analysis is given of the effects of impurities on trapped ion modes in the collisionless regime. Applications of the results to advanced toroidal systems indicate that the trapped ion collisionless instability is not eliminated by the impurities, at least when the length scales of the impurity gradients and the main plasma density gradient are in the same direction and of the same order. Stabilizing effects, at not too high concentrations, are obtained for sharp (and, in particular, inverse) impurity gradients.

Effect of plasma confinement and impurity level upon the performance of a D-T burning tokamak experimental power reactor

The effect of uncertainties in plasma confinement and impurity concentration upon the performance of a D-T burning tokamak experiment power reactor is analysed. Steady-state performance parameters are established for ignition and beam-driven modes as a function of nτ and impurity concentration. The deleterious effect of wall-sputtered impurities on the burn cycle dynamics is examined for several potential first-wall materials. The effect of different confinement scaling laws — trapped-ion mode and pseudoclassical — on the burn cycle dynamics is studie.

New macroscopic theory of anomalous diffusion induced by the dissipative trapped-ion instability

For an axisymmetric toroidal plasma of the Tokamak type a new set of dissipative trapped-fluid equations is established. In addition to E*B drifts and collisions of the trapped particles, these equations take full account of the effect of E/sub /// (of the trapped ion modes) on free and trapped particles, and of the effect of grad delta 0( delta 0=equilibrium fraction of trapped particles). From the new equations the linear-mode properties of the dissipative trapped-ion instability and the anomalous diffusion flux of the trapped particles are derived.

Ion cyclotron instability induced by the trapped-ion mode

The conditions and consequence of the onset of ion cyclotron instabilities in tokamaks that can be induced by the finite amplitude level of the trapped-ion mode are studied. It is found that such oscillations are likely to be present but are not likely to alter the diffusion rates originally predicted by Kadomtsev and Pogutse.