Luca Guazzotto

Numerical study of tokamak equilibria with arbitrary flow

The effects of toroidal and poloidal flows on the equilibrium of tokamak plasmas are numerically investigated using the code FLOW. The code is used to determine the changes in the profiles induced by large toroidal flows on NSTX-like equilibria [with NSTX being the National Spherical Torus Experiment, M. Ono, S.M. Kaye, Y.-K.M. Peng et al., Nucl. Fusion 40, 557 (2000)] where flows exceeding the sound speed lead to a considerable outward shift of the plasma. The code is also used to study the effects of poloidal flow when the flow velocity profile varies from subsonic to supersonic with respect to the poloidal sound speed. It is found that pressure and density profiles develop a pedestal structure characterized by radial discontinuities at the transonic surface where the poloidal velocity abruptly jumps from subsonic to supersonic values. These results confirm the conclusions of the analytic theory of R. Betti and J. P. Freidberg [Phys. Plasmas 7, 2439 (2000)], derived for a low-β, large aspect ratio tokam...

Magnetohydrodynamics Equilibria with Toroidal and Poloidal Flow

In the present work, the effects of flow on tokamak equilibria are investigated, focusing in particular on the effects of poloidal flows. It is shown that discountinuous transonic equilibria with a pedestal structure can be obtained for relatively low values of the poloidal velocity. Equilibria with poloidal flow of the order of the poloidal Alfven speed are shown to develop inverted Shafranov shift. Since the rotation is damped by the neoclassical poloidal viscosity, a quasi-omnigenous solution for equilibria with large rotation is also derived in order to minimize the flow damping. In this solution, the magnetic field is construed to be a function of the poloidal magnetic flux Ψ up to a small correction by an appropriate choice of the flow profiles. All numerical results are obtained with the code FLOW [L. Guazzotto, R. Betti, J. Manickam, and S. Kaye, Phys. Plasmas 11, 604 (2004)].

A family of analytic equilibrium solutions for the Grad–Shafranov equation

A family of exact solutions to the Grad–Shafranov equation, similar to those described by Atanasiu et al. [C. V. Atanasiu, S. Gunter, K. Lackner, and I. G. Miron, Phys. Plasmas 11, 3510 (2004)], is presented. The solution allows for finite plasma aspect ratio, elongation and triangularity, while only requiring the evaluation of a small number of well-known hypergeometric functions. Plasma current, pressure, and pressure gradients are set to zero at the plasma edge. Realistic equilibria for standard and spherical tokamaks are presented.

A general formulation of magnetohydrodynamic stability including flow and a resistive wall

A general formulation is presented for determining the ideal magnetohydrodynamic stability of an axisymmetric toroidal magnetic configuration including the effects of an arbitrary equilibrium flow velocity and a resistive wall. The system is inherently non-self-adjoint with the eigenvalue appearing in both the equations and the boundary conditions. Even so, after substantial analysis we show that the stability problem can be recast in the form of a standard eigenvalue problem, ωA⋅z=B⋅z, which is highly desirable for numerical computation.

The effect of an anisotropic pressure of thermal particles on resistive wall mode stability

The effect of an anisotropic pressure of thermal particles on resistive wall mode stability in tokamak fusion plasmas is derived through kinetic theory and assessed through calculation with the MISK code [B. Hu et al., Phys. Plasmas 12, 0 57301 (2005)]. The fluid anisotropy is treated as a small perturbation on the plasma equilibrium and modeled with a bi-Maxwellian distribution function. A complete stability treatment without an assumption of high frequency mode rotation leads to anisotropic kinetic terms in the dispersion relation in addition to anisotropy corrections to the fluid terms. With the density and the average pressure kept constant, when thermal particles have a higher temperature perpendicular to the magnetic field than parallel, the fluid pressure-driven ballooning destabilization term is reduced. Additionally, the stabilizing kinetic effects of the trapped thermal ions can be enhanced. Together these two effects can lead to a modest increase in resistive wall mode stability.

Two-dimensional magnetohydrodynamic simulations of poloidal flows in tokamaks and MHD pedestal

Poloidal rotation is routinely observed in present-day tokamak experiments, in particular near the plasma edge and in the high-confinement mode of operation. According to the magnetohydrodynamic (MHD) equilibrium theory [R. Betti and J. P. Freidberg, Phys. Plasmas 7, 2439 (2000)], radial discontinuities form when the poloidal velocity exceeds the poloidal sound speed (or rather, more correctly, the poloidal magneto-slow speed). Two-dimensional compressible magnetohydrodynamic simulations show that the transonic discontinuities develop on a time scale of a plasma poloidal revolution to form an edge density pedestal and a localized velocity shear layer at the pedestal location. While such an MHD pedestal surrounds the entire core, the outboard side of the pedestal is driven by the transonic discontinuity while the inboard side is caused by a poloidal redistribution of the mass. The MHD simulations use a smooth momentum source to drive the poloidal flow. Soon after the flow exceeds the poloidal sound speed, ...

Two-fluid equilibrium with flow: FLOW2

The effects of finite macroscopic velocities on axisymmetric ideal equilibria are examined using the two-fluid (ions and electrons) model. A new equilibrium solver, the code FLOW2, is introduced for the two-fluid model and used to investigate the importance of various flow patterns on the equilibrium of tight aspect ratio (NSTX) and regular tokamak (DIII-D) configurations. Several improvements to the understanding and calculation of two-fluid equilibria are presented, including an analytical and numerical proof of the single-fluid and static limits of the two-fluid model, a discussion of boundary conditions, a user-friendly free-function formulation, and the explicit evaluation of velocity components normal to magnetic surfaces.

Equilibrium beta limits in a dipole configuration

The levitated dipole configuration is an innovative concept for fusion research. One of the main advantages of the dipole configuration is the possibility of stably confining high plasma pressure compared to the magnetic pressure, that is, the possibility of achieving high β (where β is the ratio between plasma pressure and magnetic pressure). The present work investigates the limit on equilibrium β existing in the dipole system. It is found that a limit exists, which is considerably modified by the presence of plasma rotation in the toroidal direction (the long way around the torus). Plasma anisotropy instead does not modify the limit in any appreciable way for the moderate anisotropies considered in this work.

Numerical Calculations Demonstrating Complete Stabilization of the Ideal Magnetohydrodynamic Resistive Wall Mode by Longitudinal Flow

The cylindrical ideal magnetohydrodynamic (MHD) stability problem, including flow and a resistive wall, is cast in the standard mathematical form, ωA⋅x=B⋅x, without discretizing the vacuum regions surrounding the plasma. This is accomplished by means of a finite element expansion for the plasma perturbations, by coupling the plasma surface perturbations to the resistive wall using a Green’s function approach, and by expanding the unknown vector, x, to include the perturbed current in the resistive wall as an additional degree of freedom. The ideal MHD resistive wall mode (RWM) can be stabilized when the plasma has a uniform equilibrium flow such that the RWM frequency resonates with the plasma’s Doppler-shifted sound continuum modes. The resonance induces a singularity in the parallel component of the plasma perturbations, which must be adequately resolved. Complete stabilization within the ideal MHD model (i.e., without parallel damping being added) is achieved as the grid spacing in the region of the re...

A model for transonic plasma flow

A linear, two-dimensional model of a transonic plasma flow in equilibrium is constructed and given an explicit solution in the form of a complex Laplace integral. The solution indicates that the transonic state can be solved as an elliptic boundary value problem, as is done in the numerical code FLOW [Guazzotto et al., Phys. Plasmas 11, 604 (2004)]. Moreover, the presence of a hyperbolic region does not necessarily imply the presence of a discontinuity or any other singularity of the solution.