Toroidal Flow Shear Research Articles

Stabilization of sawteeth in tokamaks with toroidal flows

Sheared toroidal flows with a magnitude of the order of the sound speed are shown to stabilize sawteeth in tokamaks. In the absence of flows, tokamak equilibria in which the central safety factor q is less than unity are unstable to resistive tearing modes (resistive internal kink modes) with toroidal mode number n=1. As the ratio β of the plasma pressure to the magnetic field pressure increases, the growth rate of the n=1 mode rises because of the increasing pressure gradient. However, the addition of a toroidal flow to the equilibrium has a stabilizing effect. As the magnitude of the toroidal flow approaches the sound speed, the n=1 resistive tearing mode can be completely stabilized by the flow, eliminating sawteeth.

Toroidal flow stablization of disruptive high β tokamaks

Disruptive high β tokamak plasmas can be stabilized by the addition of a sheared toroidal flow. Nonlinear simulations demonstrate that confinement in flow-free high β tokamaks is rapidly destroyed by growing fingers of hot plasma that jet out from the center of the discharge to the wall. The added toroidal flow eliminates the growing fingers, maintaining confinement. As β increases further, the toroidal flow becomes less effective at maintaining a stable plasma. But a sound speed toroidal flow increases the critical value of β below which confinement is maintained without disruptions.

Axisymmetric Toroidal Equilibrium with Sheared Toroidal Flows

Problem of the axisymmetric toroidal equilibrium with pure sheared toroidal flow is involved. For standard tokamak equilibrium, general approximate solutions are analytically pursued for arbitrary current profile and non-circular cross-section. Equilibrium properties including the flow-induced density asymmetry are analyzed.

Kinetic simulations of the formation and stability of the field-reversed configuration

The Field-Reversed Configuration (FRC) is a high-beta compact toroidal plasma confined primarily by poloidal fields. In the FRC the external field is reversed on axis by the diamagnetic current carried by thermal plasma particles. A three-dimensional, hybrid, particle-in-cell (zero-inertia fluid electrons, and kinetic ions), code FLAME, previously used to study ion rings [Yu. A. Omelchenko and R. N. Sudan, J. Comp. Phys. 133, 146 (1997)], is applied to investigate FRC formation and tilt instability. Axisymmetric FRC equilibria are obtained by simulating the standard experimental reversed theta-pinch technique. These are used to study the nonlinear tilt mode in the “kinetic” and “fluid-like” cases characterized by “small” (∼3) and “large” (∼12) ratios of the characteristic radial plasma size to the mean ion gyro-radius, respectively. The formation simulations have revealed the presence of a substantial toroidal (azimuthal) magnetic field inside the separatrix, generated due to the stretching of the poloidal field by a sheared toroidal electron flow. This is shown to be an important tilt-stabilizing effect in both cases. On the other hand, the tilt mode stabilization by finite Larmor radius effects has been found relatively insignificant for the chosen equilibria.

Magnetohydrodynamic calculations with a nonmonotonic q profile and equilibrium, sheared toroidal flow

The linear and nonlinear stability of a nonmonotonic q profile is examined using a reduced set of magnetohydrodynamic (MHD) equations with an equilibrium, sheared toroidal flow. The reversed shear profile is shown to be unstable to a rich variety of resistive MHD modes including pressure-driven instabilities and tearing instabilities possessing a tearing/interchange character at low Lundquist number, S, and taking on a double/triple tearing structure at high S. Linear calculations show that the destabilizing effect of toroidal velocity shear on tearing modes is enhanced at finite pressure. In addition, this velocity shear decreases the stabilizing effect of finite pressure seen previously for tearing modes at high S. Nonlinear calculations show the generation of a large, m=1, n=0, Reynolds-stress-driven poloidal flow in the absence of significant flow damping. Calculations in which the poloidal flow was heavily damped show that sub-Alfvénic, sheared toroidal flows have a minimal effect on weakly coupled, localized instabilities.

Flow shear stabilization of a hybrid dissipative trapped electron-ion temperature gradient mode in tokamaks

A simple physical model for toroidal flow shear stabilization of a hybrid dissipative trapped electron-ion temperature gradient mode (namely, the hybrid electron-ion drift mode) is presented. Coupling between the sheared flows and non-adiabatic electrons is proposed as the stabilization mechanism of the hybrid electron-ion drift mode. This implies that the confinement improvement can be achieved not only through the poloidal sheared rotation flow in the edge region of tokamaks for the L-H transition but also through the toroidal sheared rotation in the core region for further confinement improvement associated with the internal transport barrier and negative shear.

Tokamak equilibria and stability with arbitrary flows

The topic of plasma flows and their influence on the equilibrium and stability properties of tokamak discharges is briefly reviewed. The stability of high-n ballooning modes in the presence of sheared toroidal rotation is discussed to highlight recent advances in our understanding of the spatial structure and temporal evolution of these modes. The eikonal solution, which is mathematically equivalent to the normal mode solutions at the lowest order in 1/n, may not be physically relevant for a variety of reasons. An analytic formulation of the normal mode problem for high-n ballooning modes in the presence of sheared toroidal flow in the high-β large-aspect-ratio limit is presented. In contrast to earlier studies, the present formulation allows for large plasma shifts as well as accounting for changes in the shape of flux surfaces due to rotation effects.

Transport processes and entropy production in toroidally rotating plasmas with electrostatic turbulence

A new gyrokinetic equation is derived for rotating plasmas with large flow velocities on the order of the ion thermal speed. Neoclassical and anomalous transport of particles, energy, and toroidal momentum are systematically formulated from the ensemble-averaged kinetic equation with the gyrokinetic equation. As a conjugate pair of the thermodynamic force and the transport flux, the shear of the toroidal flow, which is caused by the radial electric field shear, and the toroidal viscosity enter both the neoclassical and anomalous entropy production. The interaction between the fluctuations and the sheared toroidal flow is self-consistently described by the gyrokinetic equation containing the flow shear as the thermodynamic force and by the toroidal momentum balance equation including the anomalous viscosity. Effects of the toroidal flow shear on the toroidal ion temperature gradient driven modes are investigated. Linear and quasilinear analyses of the modes show that the toroidal flow shear decreases the growth rates and reduces the anomalous toroidal viscosity.

Microinstability analysis of DIII-D high-performance discharges

The kinetic stability properties in a number of high performance discharges from the DIII-D tokamak [R. D. Stambaugh for the DIII-D Team, Plasma Physics and Controlled Nuclear Fusion Research, 1994 (International Atomic Energy Agency, Vienna, 1995), Vol. 1, p. 83] have been analyzed utilizing a comprehensive kinetic eigenvalue code. The instability considered is the toroidal drift mode [trapped-electron-ion temperature gradient (ηi) mode]. This code has been interfaced with equilibria specific to DIII-D plasmas. Experimentally measured kinetic profile data, along with motional stark effect data and external magnetic data, was used, and the corresponding magnetohydrodynamic (MHD) equilibria were computed numerically. In particular, a low confinement mode (L-mode) case, a high-li high confinement mode (H-mode) case, a very high confinement mode (VH-mode) case, and a high plasma pressure/poloidal magnetic pressure (βp) case have been analyzed. For the L-mode case, a wide region of instability was found, while for the H-mode and VH-mode and high-βp cases, only relatively narrow regions of instability were found. An assessment of the influence of velocity-shear flow on these instabilities has also been made, as well as of changes in the electron and ion temperature gradients and density gradients. While the experimental values of the sheared toroidal flow velocity are not sufficient to stabilize the instability, an increase by a factor of two to four in the flow velocity could completely stabilize this mode.

Effect of toroidal plasma flow and flow shear on global magnetohydrodynamic MHD modes

The effect of a subsonic toroidal flow on the linear magnetohydrodynamic stability of a tokamak plasma surrounded by an external resistive wall is studied. A complex non-self-adjoint eigenvalue problem for the stability of general kink and tearing modes is formulated, solved numerically, and applied to high β tokamaks. Results indicate that toroidal plasma flow, in conjunction with dissipation in the plasma, can open a window of stability for the position of the external wall. In this window, stable plasma beta values can significantly exceed those predicted by the Troyon scaling law with no wall. Computations utilizing experimental data indicate good agreement with observations.

Effect of sheared toroidal flow on the equilibrium of an axisymmetric plasma torus

Several present day Tokamak experiments have reported large plasma rotations (of the order of sound speed) induced by neutral beam injection during auxiliary heating. The measured rotation profiles indicate that velocity shear can be significant. The authors examine the effect of such sheared flows on the equilibrium of Tokamak plasmas. Using the generalized MHD equilibrium equations for axisymmetric rotation and adopting a variational formulation, the authors carry out a detailed numerical investigation of the form of the magnetic surfaces and other equilibrium quantities as a function of the flow function profile. In certain limits they also obtain analytic expressions for the ellipticity and finite shift of the magnetic axis.

Effect of sheared toroidal flow on the equilibrium of an axisymmetric plasma torus

Several present day Tokamak experiments have reported large plasma rotations (of the order of sound speed) induced by neutral beam injection during auxiliary heating. The measured rotation profiles indicate that velocity shear can be significant. The authors examine the effect of such sheared flows on the equilibrium of Tokamak plasmas. Using the generalized MHD equilibrium equations for axisymmetric rotation and adopting a variational formulation, the authors carry out a detailed numerical investigation of the form of the magnetic surfaces and other equilibrium quantities as a function of the flow function profile. In certain limits they also obtain analytic expressions for the ellipticity and finite shift of the magnetic axis.

Ballooning instabilities in tokamaks with sheared toroidal flows

The stability of ballooning modes in the presence of sheared toroidal flows is investigated. The eigenmodes are shown to be related by a Fourier transformation to the nonexponentially growing Floquet solutions found by Cooper [Plasma Phys. Controlled Fusion 30, 1805 (1988)]. It is further shown that the problem cannot be reduced further than to a two-dimensional partial differential equation. Next, the generalized ballooning equation is solved analytically for a circular tokamak equilibrium with sonic flows, but with a small rotation shear compared to the sound speed. With this ordering, the centrifugal forces are comparable to the pressure gradient forces driving the instability, but coupling of the mode with the sound wave is avoided. A new stability criterion is derived that explicitly demonstrates that flow shear is stabilizing at constant centrifugal force gradient.

Ballooning stability of axisymmetric plasmas with sheared equilibrium flows

A WKB theory is formulated for large-n ballooning modes in axisymmetric, toroidal plasmas with sheared equilibrium flows. The validity of the standard ballooning representation is severely restricted in the presence of sheared toroidal flow, despite the fact that to leading order in (1/n), where n is the azimuthal number, the eigenmode equation contains only derivatives along a field line. Necessary and sufficient conditions for stability are obtained in a high-beta ordering for rigid toroidal rotation as well as field-aligned flows.

Ballooning instabilities in tokamaks with sheared toroidal flows

The ballooning-mode eikonal representation is applied to the linearized incompressible magnetohydrodynamic (MHD) equations in axisymmetric systems with toroidal mass flows to obtain a set of initial value partial differential equations in which the time t and the poloidal angle theta are the independent variables. To derive these equations, the eikonal function S is assumed to satisfy the usual condition B.gradS=0 to guarantee that the modes vary slowly along the magnetic field. In addition, to resolve the V.grad operator acting on perturbed quantities, the eikonal must also satisfy the condition dS/dt=0. this induces a Doppler shift in S. This description of the instability, however, is incompatible with normal mode solutions of the MHD equations because the wave vector gradS becomes time dependent when the velocity shear is finite. Nevertheless, the author is able to investigate the effects of the sheared toroidal flows on localized ballooning instabilities because the initial value formulation of the problem developed does not constrain the solutions to evolve as exp (i omega t). Fixed boundary MHD equilibria with isothermal toroidal flows that model the JET device are generated numerically with a variational inverse moments code. As the initial value evaluations are evolved in time, periodic burst of ballooning activity are observed which are correlated with the formation of a ballooning structure at the outside edge of the torus that becomes displaced by 2 pi in the extended poloidal angle domain from one burst to the next. The velocity shear has a stabilizing influence on plasma ballooning.

Spontaneous development of toroidal magnetic field during formation of field-reversed theta pinch

The formation of an FRC (Field-Reversed Configuration) is considered by using a hybrid simulation model that treats the electrons as a zero-inertia thermal fluid and the ions either kinetically or as a simple fluid. The unexpected spontaneous generation of a substantial toroidal magnetic field Bθ near magnetic reconnection points requires explanation. Although ion kinetic effects are responsible for some of the internal reconnection, simulations with a fluid ion representation also show similar self-generation of the toroidal field. Further investigation reveals that the origin of the toroidal field can be found in some of the terms that arise from retention of the Hall term – an automatic feature of the two-component model. Conceptually, the phenomena can be understood by noting that, if the magnetic field is ‘frozen into the electron fluid’, a sheared toroidal electron flow will stretch the poloidal field to make the toroidal field.