Trapped Particle Modes Research Articles

Theory of negative energy holes in current carrying plasmas

The theory of hole structures or phase-space vortices in a one-dimensional current-carrying plasma is extended, focusing on the energy of trapped-particle modes in comparison to a homogeneous plasma. It is shown how the energy expression presented in [J.-M. Grießmeier and H. Schamel, Phys. Plasmas 9, 2462 (2002)] is obtained for small amplitude structures. Parameter regimes admitting negative energy solutions are given. It is demonstrated how negative energy structures can be found analytically for the case of a generalized solitary electron hole (where ion trapping is shown to further lower the critical drift velocity), for a generalized solitary ion hole (where the influence of electron trapping increases the critical drift velocity) and for a harmonic (monochromatic) structure. Consequently, a plasma may become nonlinearly unstable well below linear threshold already for infinitesimal wave amplitudes.

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.

Electrostatic wave variety and the origins of BEN

Spiky pulses in broadband electrostatic “noise” (BEN) in the plasma sheet boundary layer (PSBL) observed by Geotail led to a model for BEN as Bernstein–Greene–Kruskal (BGK) modes, which has been explored by a number of simulations. A recent modified model proposed by Grabbe (Phys. Rev. Lett. 84 (2000a) 3614; Rec. Res. Dev. Plasma 1 (2000b) 89) for BEN well into the magnetotail, in which these trapped-particle modes only occur in the top of the spectrum, whereas the bulk of the spectrum below that which propagates obliquely to the magnetic field consists of standard modes driven by unstable electron/ion beams, predicts that such solitary waves are only present in the source region, whereas they are not in BEN that has propagated well outside the source region. Motivated by that prediction and other features of the model, observations are examined from POLAR of BEN obtained poleward of and within the near-Earth extension of the PSBL. The wave data examined for BEN poleward of the near-Earth PSBL (in the plasma mantle) exhibit frequent turbulent waveforms, but little evidence of solitary waves. However, the wave data near the PSBL source region shows a persistent level of turbulence, within which solitary-like waves are embedded, several of which saturate the receiver. The non-solitary portion of the observed wave data, which appears throughout these regions, fits well the theory of standard electron/ion beam instabilities. However, the higher-frequency portion above the plasma frequency, which is confined to the PSBL vicinity, appears nonlinear in character, with a stronger magnetic field apparently playing a vital role in that nonlinearity.

Stability properties of high‐pressure geotail flux tubes

Kinetic theory is used to investigate the stability of ballooning interchange modes in the high‐pressure geotail plasma. A variational form of the stability problem is used to compare new kinetic stability results with MHD, fast MHD, and Kruskal and Oberman [1958] stability results. Two types of drift modes are analyzed: A kinetic ion pressure gradient drift wave with a frequency given by the ion diamagnetic drift frequency ω*pi and a very low frequency mode |ω| ≪ ω*pi, ωDi that is often called a convective cell or the trapped particle mode. In the high‐pressure geotail plasma a general procedure for solving the stability problem in a 1/β expansion for the minimizing δB‖ is carried out to derive an integral differential equation for the kinetically valid displacement field ξψ for a flux tube. The plasma energy released by these modes is estimated in the nonlinear state. The role of these instabilities in substorm dynamics is assessed in the substorm scenarios described by Maynard et al. [1996].

Hole equilibria in Vlasov–Poisson systems: A challenge to wave theories of ideal plasmas

A unified description of weak hole equilibria in collisionless plasmas is given. Two approaches, relying on the potential method rather than on the Bernstein, Greene, Kruskal method and associated with electron and ion holes, respectively, are shown to be equivalent. A traveling wave solution is thereby uniquely characterized by the nonlinear dispersion relation and the “classical” potential V(φ), which determine the phase velocity and the spectral decomposition of the wave structure, respectively. A new energy expression for a hole carrying plasma is found. It is dominated by a trapped particle contribution occurring one order earlier in the expansion scheme than the leading term in conventional schemes based on a truncation of Vlasov’s equation. Linear wave theory— reconsidered by taking the infinitesimal amplitude limit—is found to be deficient, as well. Neither Landau nor van Kampen modes and their general superpositions can adequately describe these trapped particle modes due to an incorrect treatment of resonant particles for phase velocities in the thermal range. It is therefore concluded that wave theories in their present form, dictated by linearity, are not yet properly shaped to describe the dynamics of ideal plasmas (and fluids) correctly.

Stability of a plasma confined in a dipole field

Plasma confined in a magnetic dipole field is stabilized by the expansion of the magnetic flux. The stability of low beta electrostatic modes in a magnetic dipole field is examined when the distribution function is Maxwellian to lowest order. It is shown using a Nyquist analysis that for sufficiently gentle density and temperature gradients the configuration would be expected to be stable to both magnetohydrodynamic and collisionless interchange modes. Furthermore, it is shown that when it is stable to the interchange mode it is also stable to ion temperature gradient and collisionless trapped particle modes, as well as modes driven by parallel dynamics such as the “universal” instability.

Stability of electrostatic modes in a levitated dipole

Plasma confined in a magnetic dipole is stabilized by the expansion of the magnetic flux. The stability of low beta electrostatic modes in a magnetic dipole field is examined when the distribution function is to lowest order Maxwellian. It is shown that for sufficiently gentle density and temperature gradients the configuration would be expected to be stable to magnetohydrodynamic interchange, as well as to dissipative trapped ion and collisionless trapped particle modes. These results are applicable to any magnetic configuration for which the curvature drift frequency exceeds the diamagnetic drift frequency.

Turbulent equipartition and up-gradient transport

Turbulent transport in tokamaks is often observed to be directed up-gradient. It is proposed that such up-gradient fluxes represent a tendency to approach turbulent equipartition (TEP). At TEP the phase space fluid is well mixed along surfaces defined by those invariants that are not destroyed by the turbulence. This implies that the distribution function f depends only of these invariants, since, according to Liouville's theorem, f is a Lagrangian invariant of the phase space flow. The nature of the TEP depends entirely on what the invariants are. Assuming that the magnetic moment of the particles is conserved, the general form of the transport equations is found for a two-dimensional electrostatic plasma model. These equations describe the relaxation to TEP, and show that in the presence of turbulence there may be fluxes without gradients in the thermodynamic variables. For tokamaks the two first adiabatic invariants of the single-particle motion are assumed to be conserved in the presence of the turbulence. This leads to the density profile n ~ 1/q of trapped particles, which has a maximum in the centre. The proposed model implies that trapped particle modes dominate turbulent transport in tokamaks. Since negative shear stabilizes these modes, it is predicted that negative shear improves confinement significantly.

Localized tearing modes in the magnetotail driven by curvature effects

The stability of collisionless tearing modes is examined in the presence of curvature drift resonances and the trapped particle effects. A kinetic description for both electrons and ions is employed to investigate the stability of a two‐dimensional equilibrium model. The main features of the study are to treat the ion dynamics properly by incorporating effects associated with particle trajectories in the tail fields and to include the linear coupling of trapped particle modes. Generalized dispersion relations are derived in several parameter regimes by considering two important sublayers of the reconnecting region. For a typical choice of parameters appropriate to the current sheet region, we demonstrate that localized tearing modes driven by ion curvature drift resonance effects are excited in the current sheet region with growth time of the order of a few seconds. Also, we examine nonlocal characteristics of tearing modes driven by curvature effects and show that modes growing in a fraction of a second arise when mode widths are larger than the current sheet width. Further, we show that trapped particle effects, in an interesting frequency regime, significantly enhance the growth rate of the tearing mode. The relevance of this theory for substorm onset phase and other features of the substorms is briefly discussed.

Confinement and stability of VH mode discharges in the DIII-D Tokamak

A regime of improved H mode energy confinement, the VH mode, was obtained following boronization of the DIII-D Tokamak vacuum vessel. The gradual confinement improvement in VH mode is associated with expansion of the H mode pedestal and reduction of thermal diffusivity in a region just inside the pedestal. Disappearance of density fluctuation bursts is associated with a confinement improvement in the latter region. The VH mode is terminated in a rapid global energy loss which is initiated by a mode with toroidal mode number, n approximately 5, consistent with calculated instability to an edge localized ideal kink. The improved confinement in VH mode is consistent with the extension of the region of the E*B velocity shear turbulence suppression zone further in from the plasma boundary. Expansion of the ideal ballooning second stability region and of the region with drift reversal stabilization of trapped particle modes are also considered as possible explanations for the confinement improvement

Influence of trapped thermal particles on internal kink modes in high temperature tokamaks

The effects of thermal trapped particles on the n=1 internal kink mode are studied using drift kinetic theory. Strong modifications of the magnetohydrodynamic (MHD) results are found, and marginal stability generally occurs at nonzero rotation frequency. For equal electron and ion temperatures, the trapped particles increase the marginal poloidal beta at q=1 substantially above the MHD value. For unequal electron and ion temperatures, the drift resonance with the hotter species becomes increasingly destabilizing and for sufficiently unequal temperatures, this leads to instability below the ideal-MHD threshold. Treatment of trapped thermal particles requires consideration of the effects of an electrostatic potential. The potential is weakly stabilizing for the internal kink mode. Furthermore, finite beta couples unstable, nearly electrostatic, trapped particle modes to the internal kink mode. At high beta, thermal fluctuations of the trapped particle modes can lead to significant internal kink displacements.

Effects of trapped thermal particles on the n=1 internal kink mode in tokamaks.

The effects of thermal trapped particles on the internal kink mode are studied using drift kinetic theory. For equal electron and ion temperatures, the trapped particles increase the marginal poloidal beta at q=1 substantially. For unequal temperatures, drift resonance with the hotter species is destabilizing and can lead to instability below the ideal magnetohyrodynamics threshold. An electrostatic potential is weakly stabilizing for the internal kink. At high beta, fluctuations of trapped particle modes couple to the internal kink mode and can lead to large displacements of the central region.

Fluid models of phase mixing, Landau damping, and nonlinear gyrokinetic dynamics

Fluidlike models have long been used to develop qualitative understanding of the drift-wave class of instabilities (such as the ion temperature gradient mode and various trapped-particle modes) which are prime candidates for explaining anomalous transport in plasmas. Here, the fluid approach is improved by developing fairly realistic models of kinetic effects, such as Landau damping and gyroradius orbit averaging, which strongly affect both the linear mode properties and the resulting nonlinear turbulence. Central to this work is a simple but effective fluid model [Phys. Rev. Lett. 64, 3019 (1990)] of the collisionless phase mixing responsible for Landau damping (and inverse Landau damping). This model is based on a nonlocal damping term with a damping rate ∼ vt‖k∥‖ in the closure approximation for the nth velocity space moment of the distribution function f, resulting in an n-pole approximation of the plasma dispersion function Z. Alternatively, this closure approximation is linearly exact (and therefore physically realizable) for a particular f0 which is close to Maxwellian. ‘‘Gyrofluid’’ equations (conservation laws for the guiding-center density n, momentum mnu∥, and parallel and perpendicular pressures p∥ and p⊥) are derived by taking moments of the gyrokinetic equation in guiding-center coordinates rather than particle coordinates. This naturally yields nonlinear gyroradius terms and an important gyroaveraging of the shear. The gyroradius effects in the Bessel functions are modeled with robust Padé-like approximations. These new fluid models of phase mixing and Landau damping are being applied by others to a broad range of applications outside of drift-wave turbulence, including strong Langmuir turbulence, laser–plasma interactions, and the α-driven toroidicity-induced Alfvén eigenmode (TAE) instability.

Unified fluid/kinetic description of plasma microinstabilities. Part II: Applications

The unified fluid/kinetic equations developed in Part I [Phys. Fluids B 4, ▪▪▪ (1992)] of this work are used to study plasma-drift-type microinstabilities. A generalized perturbed Ohm’s law is derived (for a sheared slab magnetic field model) that is uniformly valid for arbitrary collisionality ω/ν and adiabaticity ω/k∥vt. Applications to electron drift waves, ion temperature gradient modes (ηi modes) and electron electromagnetic modes (microtearing modes) are addressed. It is shown that the unified equations exhibit their simple and uniformed features in the plasma microinstability analysis. The well-known results of these instability branches in both fluid and adiabatic limits are easily recovered using the new fluid/kinetic equations. Generalization of the two-field Hasegawa–Wakatani turbulent equations [Phys. Rev. Lett. 50, 682 (1983)] to include electron temperature fluctuations and linear Landau damping effects is also discussed. Finally, a new method is presented to facilitate the study of magnetic trapped particle modes using the present kinetic closure procedure. It is found that by including the trapped particle effects in the closure relations, the usual separation of the fluid equations into trapped and untrapped components becomes unnecessary.

Unified theory of ballooning instabilities and temperature gradient-driven trapped ion modes

A unified theory of temperature gradient-driven trapped ion modes and ballooning instabilities is developed using kinetic theory in banana regimes. All known results such as electrostatic and purely magnetic trapped particle modes and ideal magnetohydrodynamic ballooning modes (or shear Alfvén waves) are readily derived from the present single general dispersion relation. Several new results from ion–ion collision, finite beta stabilization of ion temperature gradient-driven trapped particle modes, and trapped particle modification of ballooning modes are derived and discussed. The interrelationship between these modes is established.

Axial structure of the curvature driven trapped particle mode in tandem mirrors

The ideal rectangular axial structure of the eigenfunction of the curvature driven trapped particle mode, localized in the central cell of a tandem mirror is critically examined and revised. The search for a physical damping mechanism which can truncate the short wavelength part of the spectrum of the ideal eigenfunction has led to the halo coupling of the trapped particle mode. The radial coupling of the trapped particle mode to the halo plasma can excite acoustic modes which are strongly damped at short wavelengths. This mechanism can easily smooth out the ideal rectangular structure. Approximate results indicate a significant departure from the ideal rectangular structure and a corresponding reduction of the growth rate.

Stabilization of collisionless trapped particle modes in tokamaks

A quadratic form is derived to study the stability of collisionless trapped particle modes in tokamaks. Marginal stability criteria show that stability can be obtained by controlling the electron temperature profile. In a low aspect ratio equilibrium (such as the recently proposed Comet [Comments Plasma Phys. XII, 125 (1989)]), the stability requirement is ηe≂O(1) (ηe=∂ ln Te/∂ ln ne); in a standard large aspect ratio tokamak with a broad density profile, the requirement is LTe/R≂O(1) (L−1Te=−∂ln Te/∂r). Possible application of the second scenario to H‐mode plasmas is noted.

Observation of trapped particle modes in a tandem mirror

An instability with azimuthal mode number m≥3 localized to an axisymmetric end cell of the Tara tandem mirror [Nucl. Fusion 22, 549 (1982)] has been observed, most prominently during strong ion cyclotron resonance heating in the end cell. The instability, which causes enhanced radial losses, becomes either more stable or flutelike when the connection (passing fraction) between the central cell and end cell is increased, depending on whether sufficient stabilization is provided in the central cell. The beta is sufficiently low to rule out the possibility of magnetohydrodynamic ballooning modes. Based on the plasma parameters, the instability appears to be a collisionless curvature-driven trapped particle mode that has been predicted to be unstable in linked minimum- and maximum-B mirror devices.

Effects of collisions on the trapped particle mode in a linear machine

The effects of weak Coulomb and neutral collisions on the collisionless curvature driven trapped particle mode are studied in the Columbia Linear Machine [Phys. Rev. Lett. 57, 1729(1986)] by means of particle-conserving Krook and Fokker–Planck operators. Both types of collisionality are then combined, modeling neutral collisions with the conserving Krook and Coulomb collisions with a Lorentz model. The dispersion relation is then integrated over velocity space. Low Coulomb collisionality yields a small stabilizing correction to the magnetohydrodynamic collisionless mode, which scales as νe using the Krook model and ν1/2eC using a Lorentz pitch-angle operator. In higher collisionality regimes, both models tend to yield similar scalings.

A collisional treatment of the trapped particle mode in multiregion mirror systems

A method of analysis has been developed for treating the trapped particle mode in magnetic mirror systems in which three or more spatially distinct regions are present. The method applies for plasmas in which the electrons are sufficiently collisional that a Krook collision operator can be used. The ions have been treated as trapped within individual regions. The approximations allow the problem to be reduced to coupled algebraic equations, which are solved numerically. The numerical solutions are compared with asymptotic analytic solutions including known results. The results are applied to the multiple mirror experiment at Berkeley [Phys. Fluids 29, 1208 (1986)], which has three mirror cells of equal length but variable density, in which the central cell contains a potential barrier produced by a hot electron distribution.