Dissipative Trapped-ion Instability Research Articles

Growth rate, saturation, and radial transport measurements for the trapped ion instability

The dissipative trapped ion instability (DTII) is observed as an m=1 mode localized to the mirror cell of the Columbia Linear Machine (CLM) [Plasma Phys. 24, 185 (1982)]. A gated feedback diagnostic allows the direct measurement of the DTII linear growth rate. The DTII growth rate is measured as a function of trapped fraction, keeping all other parameters fixed, and clearly displays the scaling with trapped fraction predicted by the radially local linear dispersion relation, including ion Landau damping. The saturated DTII mode amplitude is measured concurrently with the growth rate. The saturated amplitude displays a nearly linear dependence on the measured growth rate, normalized to the DTII real frequency, and is nearly proportional to the square root of the measured growth rate. A striped particle collector is used to measure the ambipolar radial plasma flux as the trapped fraction is varied. Radial diffusion coefficients are calculated from the measured radial particle fluxes and display a scaling intermediate between the Kadomtsev weak and strong turbulence predictions.

Bounce Time Averaged Wave Potential and the Dissipative Trapped Ion Instability

Effect of banana orbit of the trapped particle on the dissipative trapped ion instability (DTII) is investigated. As the frequency of the DTII is much smaller than the bounce frequency, wave potential is averaged over the bounce period or along the banana orbit of the trapped particle. With this averaged wave potential, dispersion relation is obtained including the effect of the banana orbit. Stabilizing effect is not so large as expected. The growth rate is positive for the wide range of the poloidal mode number m . Previous results, based on the growth rate which makes use of an analogy to the finite Larmor radius effect, overestimate the damping by the effect of the banana orbit incorrectly.

Experimental study of the dissipative trapped ion instability

A new, steady-state linear machine has been constructed which produces a low-density, low-temperature hydrogen plasma in the collisionality regime of the dissipative trapped ion instability. This instability was seen and identified through the observed dependence of the wave amplitude on trapped fraction, axial and radial position, and collisionality and through the scaling of the real frequency with trapped fraction. The observed radial and axial mode structure of the instability is compared with the results of a simple radial model and a calculation of the Landau damping for both trapped and transit particles in a linear geometry.

Numerical Investigation of Saturation Mechanism of the Dissipative Trapped Ion Instability II

Expressions of the anomalous diffusion coefficient due to the dissipative trapped ion instability (DTII) are derived for the cases with and without the effect of magnetic shear. Derivation is made by taking into account of the single mode saturation of the DTII previously obtained numerically. In the absence of the shear effect, the diffusion coefficient is proportional to ν i a 2 (ν i is the effective collision frequency of the trapped ions and a is the minor radius of a torus) and is much larger than the neoclassical ion thermal conductivity. In the presence of the shear effect, the diffusion coefficient is much smaller than Kadomtsev and Pogutse's value and is the same order of magnitude as the neoclassical ion thermal conductivity.

Experimental Study of the Dissipative Trapped Ion Instability

Experiments have been performed in a steady-state linear machine with a low temperature, low density plasma in the collisionality regime of the dissipative trapped ion instability. This instability was seen and identified through the observed dependence of wave amplitude on trapped fraction, axial and radial structure, and collisionality and through the scaling of the real frequency with trapped fraction.

Numerical Investigation of Saturation Mechanism of the Dissipative Trapped Ion Instability

A set of model equations of the dissipative trapped ion instability is derived and solved numerically. Multiple time scale expansion scheme is applied to the non-linear fluid equations of Kadomtsev and Pogutse by taking µ=ω/ν e as a small parameter, where ω is the frequency of the wave and ν e is the effective collision frequency of the electrons. All the modes except for the linearly most unstable one die away due to the mode coupling. Final saturation of the mode is attributed to the modification of background trapped ion number density averaged over the poloidal direction. The coefficient of trapped ion anomalous diffusion, which scales as D ∝ν i a 2 ( ν i is the effective collision frequency of the ions and a is the minor radius of a torus) is also derived.

Production and Observation of the Dissipative Trapped-Ion Instability

A new, steady-state linear mirror machine has been constructed which produces a low-density, low-temperature hydrogen plasma in the collisionality regime of the dissipative trapped-ion instability. This instability was seen and identified through the observed dependence of wave amplitude on trapped fraction, axial and radial position, and collisionality and through the scaling of real frequency with trapped fraction.

Mixing Length Hypothesis in the Dissipative Trapped Ion Instability

The mixing length hypothesis is investigated for the purpose of making the underlying idea of the hypothesis clear in the case of the dissipative trapped ion instability. Three different approaches are employed; 1) analogy to the ordinary turbulence in hydrodynamics, 2) a nonlinear turbulent collision theory, and 3) a quasilinear theory of the density flattening. If the isotropic turbulence is assumed, the mixing length hypothesis can be derived by the first two approaches. The formula D ≃γ/ k ⊥ 2 , which is usually obtained from the mixing length hypothesis, can be derived by the third approach.

Effects of Impurity Ions on the Anomalous Diffiusion due to the Dissipative Trapped Ion Instability

Anomalous diffusion coefficient is evaluated by using result of the renormalized perturbation theory for the turbulent collision. Two cases are investigated for light impurity ions, one in the plateau regime and the other in the banana regime. Time evolution of the wave speetrum is solved numerically and the diffusion coefficient is determined, by taking the density gradient of impurity ions in the same direction as that of the bulk plasma. Even with a small concentration of the impurity the diffusion coefficient takes a smaller value for both regime cases.

New scaling laws for trapped-particle anomalous transport in tokamaks

An improved formula for anomalous transport caused by the dissipative trapped-ion instability (without shear effect) is derived numerically and by two independent analytical methods. The new shearless result is also used to derive the anomalous diffusion with shear effect by a general method already published.

Scaling law for trapped-ion anomalous diffusion in tokamaks with shear

A 1/B3 scaling law is derived for anomalous diffusion produced by the dissipative trapped-ion instability in axisymmetric toroidal plasmas with shear. The theory presented by the authors helps to explain why transport of this type has not been detected in the PLT heating experiment, so far. In addition, it is shown that the Kadomtsev-Pogutse diffusion formula, will contradict the trapped-fluid equations from which it was deduced, provided that the necessary boundary conditions are observed.

Effect of circulating ion resonances and impurities on the dissipative trapped-ion instability

Trapped-ion instabilities associated with both electron and ion diamagnetic branches are studied. In particular, the effects of circulating ion Landau resonances and impurities are determined. These effects are studied in the local limit and have been implemented in a computer code used to calculate the relevant mode frequencies.

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.

Nonlinear Saturation of Dissipative Trapped Ion Instability and Anomalous Transport

An expression for the turbulent collision frequency is derived by summing up the most dominant terms from each order in a perturbation expansion in order to obtain the nonlinear saturation level of the dissipative trapped ion instability. Numerical calculation shows that the anomalous diffusion coefficient at the saturated state is in good agreement with the result of Kadomtsev-Pogutse when the effect of ion Landau damping is taken into account.

Residual trapped-ion instabilities in tokamaks

It is demonstrated that in addition to the usual dissipative trapped-ion instability, which is basically an electron diamagnetic drift mode, ion diamagnetic modes of this type can also be generated. Such modes can be driven unstable by ions which have average unfavorable magnetic drifts and can persist as residual instabilities in tokamak systems with flat or reversed gradient profiles.

Effect of toroidal field ripple on particle and energy transport in a tokamak

The azimuthal asymmetry introduced in tokamaks by the use of a finite number of toroidal field coils can lead to anomalously large particle and energy losses. The diffusion and ion heat transport coefficients associated with the ripple-induced drift of particles trapped in the usual banana orbits have been calculated and the results compared with the coefficients for neoclassical banana transport in axisymmetric tokamaks, transport governed by the dissipative trapped-ion instability, and the transport associated with particles trapped between the coil planes by the ripple field itself. The calculations were performed for an Experimental Power Reactor design and show that the losses due to the ripple-induced banana drifts can be a major limit on fusion reactor magnet design.

Nonlinear Stabilization of Dissipative Trapped Ion Instability

An analysis of the anomalous diffusion due to the dissipative trapped ion instability in a toroidal magnetic system is presented. Saturation of the instability is attained both by collisional and by anomalous detrapping of trapped ions. The method of characteristics along the perturbed orbit is used to take into account of the anomalous detrapping effect. The anomalous diffusion coefficient in the saturated state agrees with the result of Kadomtsev and Pogutse.

Trapped ion depletion by anomalous diffusion due to the dissipative trapped ion instability

At high temperatures the Kadomtsev-Pogutse diffusion in Tokamaks can become so large as to cause depletion of trapped ions if these are replaced with free ions by means of collisions rather than being directly recycled or injected. Modified Kadomtsev-Pogutse diffusion formulas are employed in order to estimate this effect in the cases of classical and anomalous collisions. The maximum trapped-ion depletion is estimated from the Penrose stability condition. For anomalous collisions a Bohm-type diffusion is derived. Numerical examples are given for JET-like parameters (JET=Joint European Torus). Depletion is found to reduce diffusion by factors of up to 10 and more.

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.

Stabilization of the dissipative trapped-ion instability by impurities

The influence of impurities on the dissipative trapped-ion instability is investigated considering all possible cases of ion impurities in the banana, plateau or Pfirsch-Schlüter regimes. It is found that impurities lead to very significant stabilizing effects on these modes. In particular, the instability is quenched when the impurities are in the Pfirsch-Schlüter regime, and also for plateau impurities, provided the impurity density gradients are directed as the main plasma density gradient. In the case of impurities in the banana regime, a sufficient condition for complete stabilization of the instability has been found in the case of sufficiently sharp and positive impurity gradients. In the unstable cases, there is a noticeable upward shift and shrinking of the unstable range increasing with impurity concentration. These features have been discussed quantitatively, in terms of impurity parameters, both with reference to present-day experiments and for future experiments with reduced impurity levels.