Trapped Particle Instability Research Articles

Radial electric fields and global electrostatic microinstabilities in tokamaks and stellarators

The effects of applied radial electric fields on the stability of ion temperature gradient (ITG) modes and trapped particle modes are investigated with a full radius gyrokinetic formulation in both axisymmetric and helically symmetric configurations. The validity of a simplified stabilization criterion often applied to experimental analysis (the shearing rate larger than the linear growth rate) is examined for various profiles of E×B flows using an expression for the shearing rate derived for arbitrary quasisymmetric geometries [T. S. Hahm, Phys. Plasmas 4, 4074 (1997)]. The results show that the criterion holds for profiles with finite shearing rate and for toroidal, helical, and slab-like ITG modes. However, it is not always applicable for trapped particle instabilities: a case is shown for which E×B flows are destabilizing. For profiles with zero shearing rate, another stabilizing mechanism is found for toroidal and helical ITG modes but not for slab-like modes.

Density gradient bifurcation in tokamaks

Certain drift waves in tokamaks have the property that they become unstable when thetemperature gradient exceeds a critical value, which increases with increasing density gradient.If instabilities of this type dominate the transport of heat and particles then it is shownthat the particle flux can be a multivalued function of the density gradient. A bifurcation to areduced transport state can then be driven through the density gradient. The conditions for abifurcation to exist are derived. In order for this bifurcation to be observable, trapped particleinstabilities must be suppressed by some other mechanism.

Optical power evolution in the Boeing 1 kW FEL oscillator

A four-dimensional simulation in x, y, z, and t, including betatron motion of the electrons, is used to study optical power evolution and the trapped-particle instability in the Boeing I kW FEL oscillator. Boeing is designing a visible, high power free electron laser (FEL) oscillator, to produce I kW average power at a wavelength A = 0.63 Rm [1]. The large electron beam current of 1-500 A will pass through an L = 5 m long, N = 220 period undulator, creating strong optical fields inside a resonator cavity. Simulations show this may lead to sideband growth and the onset of the trapped-particle instability. Increasing the outcoupling and tapering the undulator would lessen these effects. Fig. I shows the results from a multi-mode four-dimensional simulation in (x, y, , t) of the Boeing oscillator, for n = 50 passes. The longitudinal coordinate z is scaled to the slippage length NA, and the transverse coordinates (x, y) are scaled to the characteristic optical mode radius (LA/ITr) 2 . The model includes the betatron motion of the

Frequency response to an electron energy shift

In a free electron laser (FEL) oscillator, the optical pulse evolves through mode competition as a result of mirror losses, desynchronism, and amplification by the electron beam. The finite time for this evolution determines the optical response to an electron beam energy change. The ability of the FEL operating frequency to follow modulations in electron energy has been demonstrated experimentally [l-3] and theoretically [4]. Using a self-consistent FEL theory with dimensionless parameters [5], a longitudinal multimode simulation follows the evolution of the optical pulse over many passes of a FEL oscillator. The optical frequency response to an electron energy shift is summarized in Fig. 1. The characteristic response time II, measures the number of passes for the optical frequency centroid to reach l/e of its new steady-state position after an electron energy shift is imposed at steady-state saturation. The parameters in the simulation are chosen so that the optical pulse reaches steady-state saturation fields without the trapped-particle instability and the growth of optical sidebands. The current density j can be related to the weak-field, single-mode gain, G = 0.135 j, and the energy loss at each pass is given by l/Q. The pulse length, cr:, and desynchonism, d. are measured in terms of the slippage distance, NA. For small Q and low desynchronism d, the optical pulse resides predominantly in the region of the electron pulse. Within the bulk of the optical pulse, strong-field saturation limits changes in the optical fields and the frequency remains unchanged. In the back of the optical pulse where the fields are weak and the interaction with the energy-shifted electron beam changes the optical frequency. Desynchronism d transfers these changes toward the front of the pulse at each pass. The centroid of the optical power spectrum shifts linearly with the number of passes during the middle of the response. In Fig. 1, for a given Q and oz = 2 the response time n, falls off as l/d. For large Q and small uz, the bulk of the optical pulse resides in an exponentially decaying profile outside the interaction region. in the short interaction region, changes

Simulations of the Stanford FIREFLY 1 kW free electron laser

A modification of Stanford’s Superconducting Accelerator Free Electron Laser (SCAFEL) has been proposed to increase its average optical power to about 1 kW. The FIREFLY Free Electron Laser (FEL) would be a factor of 100 increase in output power over the most powerful FEL in operation now and would cost about 1 million dollars. Simulations show the evolution of longitudinal modes over many passes and the development of the trapped-particle instability. Power versus desynchronism is studied for

Remote feedback stabilization of tokamak instabilities*

A novel remote suppressor consisting of an injected ion beam has been used for the stabilization of plasma instabilities. A collisionless curvature-driven trapped-particle instability, an E×B flute mode and an ion temperature gradient (ITG) instability have been successfully suppressed down to noise levels using this scheme. Furthermore, the first experimental demonstration of a multimode feedback stabilization with a single sensor–suppressor pair has been achieved. Two modes (an E×B flute and an ITG mode) were simultaneously stabilized with a simple state-feedback-type method where more ‘‘state’’ information was generated from a single-sensor Langmuir probe by appropriate signal processing. The above experiments may be considered as paradigms for controlling several important tokamak instabilities. First, feedback suppression of edge fluctuations in a tokamak with a suitable form of insulated segmented poloidal limiter sections used as Langmuir-probe-like suppressors is proposed. Other feedback control schemes are proposed for the suppression of electrostatic core fluctuations via appropriately phased ion density input from a modulated neutral beam. Most importantly, a scheme to control major disruptions in tokamaks via feedback suppression of kink (and possibly) tearing modes is discussed. This may be accomplished by using a modulated neutral beam suppressor in a feedback loop, which will supply a momentum input of appropriate phase and amplitude. Simple theoretical models predict modest levels of beam energy, current, and power.

Feedback stabilization of plasma instabilities using a modulated ion beam

A new technique for feedback suppression using a modulated ion beam as a remote suppressor has been developed for the first time. A Langmuir probe was used as the sensor. The feedback loop was completed with a phase shifter and amplifiers. This technique was demonstrated by stabilizing the collisionless, curvature-driven trapped particle instability down to a baseline noise level with an appropriate feedback gain and phase. An important feature is that the injected ion beam is nonperturbing to the plasma, constituting only 1% of the plasma density. A simple theory to describe the experimental results is presented and its predictions show generally good agreement with the experiment. To indicate that this concept could also be extended to other plasma instabilities, the rotationally driven E×B mode was also stabilized. Since the basic mechanism of suppression is the cancellation of charge separation inherent in an instability, a modulated electron beam performed equally well. Such remote, nonperturbing particle beam suppressors have great potential for remote suppression of plasma instabilities in a variety of plasma devices.

Compact ion beam source for feedback control of plasma instabilities

A compact ion beam source has been constructed for use in the Columbia Linear Machine for feedback control of plasma instabilities. This source has been used in a feedback configuration to stabilize the collisionless trapped particle instability [Phys. Rev. Lett. 67, 204 (1991)]. The source was based on an E×B hot cathode, magnetron-type design. It utilized the background magnetic field and a radial discharge voltage to obtain the azimuthal E×B drifts. The pressure inside the discharge chamber was maintained in the mTorr range via differential pumping. The source had a stable operating window in gas pressure of over 50% about the chosen operating parameters. Typical source plasma parameters were a plasma potential of 100 V, an electron temperature of 10 eV, and a plasma density of 109–1010 cm−3. The power dissipated was around 30 W. The ion beam energy was typically 100 eV with an energy spread of 20 eV and it could be modulated 100%. In addition, external control of the ion beam energy was provided via the anode bias.

Theory of free-electron-laser heating and current drive in magnetized plasmas

The introduction of a powerful new microwave source, the free-electron laser, provides new opportunities for novel heating and current-drive schemes to be used in toroidal fusion devices. This high-power, pulsed source has a number of technical advantages for these applications, and its use is predicted to lead to improved current-drive efficiencies and opacities in reactor-grade fusion plasmas in specific cases. The Microwave Tokamak Experiment at the Lawrence Livermore National Laboratory will provide a test for some of these new heating and current-drive schemes. Although the motivation for much of this research has derived from the application of a free-electron laser to the heating of a tokamak plasma at a frequency near the electron cyclotron frequency, the underlying physics, i.e., the highly nonlinear interaction of an intense, pulsed, coherent electromagnetic wave with an electron in a magnetized plasma including relativistic effects, is of general interest. Other relevant applications include ionospheric modification by radio-frequency waves, high-energy electron accelerators, and the propagation of intense, pulsed electromagnetic waves in space and astrophysical plasmas. This review reports recent theoretical progress in the analysis and computer simulation of the absorption and current drive produced by intense pulses, and of the possible complications that may arise, e.g., parametric instabilities, nonlinear self-focusing, trapped-particle sideband instability, and instabilities of the heated plasma.

Feedback-modulated ion beam stabilization of a plasma instability.

A novel remote suppressor consisting of an injected ion beam has been used for the stabilization of a plasma instability. Plasma instabilities can, for the first time, be remotely suppressed by modulating the ion beam with an appropriately phased and amplified feedback signal derived from a sensor probe. A collisionless curvature-driven trapped-particle instability has been successfully suppressed down to noise levels using this scheme. The ion beam causes only a small perturbation to the plasma, on the order of 1%, so that the plasma equilibrium parameters are not significantly affected.

Comparing simulations and experimental observations of the trapped-particle instability in the Stanford FEL

Simulations of the Stanford free electron laser describe the trapped-particle instability leading to sidebands and limit-cycle behavior. Comparisons are made to recent experimental results.

Halo feedback control of trapped particle instabilities

A collisionless, curvature-driven trapped-particle instability has been stabilized by a novel remote-stabilization scheme in the Columbia Linear Machine (CLM). The control signal is applied via a segmented endplate to the halo-plasma region, which is cooler and lower in density than the core plasma. The method can be considered as a generic control scheme for the suppression of some instabilities in a variety of plasma devices, including reactor-type machines.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Study of a collisionless, curvature and rotationally driven, trapped particle instability

The Columbia Linear Machine [Plasma Phys. 24, 185 (1982)] has been redesigned to enter the parameter regime of the collisionless, curvature driven, trapped particle instability. The changes that were made reduced the neutral collisionality by over a factor of 2 and the Coulomb collisionality by an order of magnitude, and increased the curvature drive by nearly an order of magnitude. With the new machine configuration, a strong (ñ/N≳60%), coherent, m=1 fluctuation is observed which is identified as a collisionless, curvature and rotationally driven, trapped particle instability. The mode is localized strongly to the region of trapping. The radial structure of the mode is broad and the mode exhibits magnetohydrodynamiclike body motion of the plasma column. Furthermore, the mode displays the expected parametric dependences of increasing mode amplitude with increasing mirror ratio and rf power and decreasing mirror cell length. A theory whose predictions are in fair agreement with the experiment has been developed for the mode in the presence of a dc radial electric field. Preliminary experiments designed to test the concept of charge overshoot stabilization of trapped particle modes show results consistent with that idea. A measurement of the growth rate of the instability has been made and is in fair agreement with the theoretical prediction.

Nonlinear saturation of collisionless trapped particle instability in tandem mirrors

A collisionless trapped particle instability with magnetohydrodynamiclike growth rate is expected in tandem mirrors. To study its nonlinear state, a two-dimensional polarization drift nonlinearity in a cylindrical geometry is considered. Using three-wave nonresonant mode coupling of radially nonlocal eigenfunctions, two-mode coupling equations for the unstable fundamental and stable first-order radial harmonics are derived. The solution of mode-coupled equations yields the nonlinear saturation level. Mode coupling to the damped higher-order radial harmonics is the principal saturation mechanism of the unstable fundamental radial harmonic.

Adiabatic lapse rate and temperature profile of tokamak plasma.

The role of the longitudinal invariance in the trapped particle instabilities is considered. The breaking of the invariance leads to the stochastic orbits and the transport. The temperature profile that minimizes the change in the longitudinal invariant on the average is conjectured to minimize the transport rate. The profile corresponds to the adiabatic lapse rate of a one-dimensional gas.

Variational quadratic form for low-frequency electromagnetic perturbations. II. Application to tandem mirrors

In this paper the low-frequency quadratic form governing the linear response of electromagnetic trapped particle modes is applied to a tandem mirror geometry and several problems are treated. It is shown that the long-wavelength magnetohydrodynamic (MHD) wall stabilization mechanism persists even when the line bending energy is suppressed by allowing an electrostatic response. In another problem, the amount of charge uncovering needed to suppress electrostatic trapped particle modes is determined as the beta of the plasma rises to its critical MHD beta value βcr and the required charge uncovering is shown to increase substantially as beta approaches βcr. A third problem treats trapped particle instabilities driven by rotation and field line curvature in diffuse and steep boundary models. The steep boundary model offers the possibility of an enhanced ‘‘robust’’ finite Larmor radius (FLR) stabilization due to an amplification of the finite Larmor radius magnetic compressional term when steep pressure gradients are present. Otherwise, one finds relatively small regions of stability in parameter space when electric field drifts are comparable to diamagnetic drifts. Detailed stability plots are presented for parameters applicable to the Tara experiment [Nucl. Fusion 22, 549 (1982)].

Production and identification of a collisionless, curvature-driven, trapped-particle instability.

The Columbia Linear Machine has been redesigned to enter the parameter regime of the collisionless, curvature-driven, trapped-particle instability. With the new machine configuration, a strong ($\frac{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{n}}{N}\ensuremath{\gtrsim}60%$), coherent mode is observed which bears all the characteristics of this instability. These include strong localization to the mirror cell, broad radial-mode structure, and MHD-like body motion of the plasma column. Furthermore, the mode exhibits all the correct parametric dependences of increasing mode amplitude with increasing mirror ratio and rf power, and decreasing mirror-cell length.

The trapped-particle instability in free electron laser oscillators and amplifiers

The electrons in a high-power free electron laser can become trapped and oscillate in deep potential wells. The oscillating component of the current drives the optical field at frequencies around the fundamental. General features of the trapped-particle instability in free electron laser oscillators and amplifiers are discussed. Dimensionless parameters clarify the trends for a wide range of FEL designs.

Tandem mirror trapped particle instability driven by axial shear in E×B rotation

A complete solution is obtained for the growth rate and real frequency of electrostatic trapped particle modes driven by the axial shear in the equilibrium E×B rotational frequency ωE in the eikonal limit for a simplified square well equilibrium model of a tandem mirror. The instability takes on a flute character for some region of parameter space, principally at low mode number and large passing fraction, with a growth rate γ2=m2∼((ΔωE)2), in agreement with prior calculations, but is nonflute elsewhere, even with zero curvature. The mode is more unstable for larger passing density in contrast to the curvature-driven trapped particle mode which is stable for larger passing density. This feature makes the mode probably more dangerous for more strongly collisional devices. In the flute regime this mode appears identical to the magnetohydrodynamic (MHD) flute version of the shear mode. Depending on the sign of the charge of the passing species and the sign of the change in ωE, this trapped particle mode can isolate itself to the center cell and thus avoid the stabilizing effect of strong anchor curvature and can do so without paying the MHD penalty for bending the field lines, thus requiring large center cell ion gyroradius for stabilization. For some typical tandem equilibria, this worst form of the mode is not possible since the mode tends to isolate to the anchor rather than to the center cell. In this case the shear in the rotation can also help to stabilize the curvature-driven mode.

Trapped particle instability induced by axial shear in E×B velocity

A new nonflute form of the E×B shear-driven trapped particle mode is obtained for a simplified square well equilibrium model of a tandem mirror. The mode is more unstable for a larger passing density in contrast to the curvature-driven trapped particle mode. Under some circumstances this mode can avoid the stabilizing effect of strong anchor curvature by isolating to the center cell and can do so without paying the magnetohydrodynamic (MHD) penalty for bending the field lines, thus requiring a large center cell ion gyroradius for stabilization.