Andrei N. Simakov

A drift ordered short mean free path description for magnetized plasma allowing strong spatial anisotropy

Short mean free path descriptions of magnetized plasmas have existed for almost 50 years so it is surprising to find that further modifications are necessary. The earliest work adopted an ordering in which the flow velocity is assumed to be comparable to the ion thermal speed. Later, less well-known studies extended the short mean free path treatment to the normally more interesting drift ordering in which the pressure times the mean flow velocity is comparable to the diamagnetic heat flow. Such an ordering is required to properly retain the temperature gradient terms in the viscosity that arise from the gyrophase dependent and independent portions of the distribution function. The treatment herein corrects the expressions for the parallel and perpendicular collisional ion viscosities found in these later treatments which use an approximate truncated polynomial expression for the distribution function and neglect the nonlinear piece of the collision operator due to its bilinear form. The modified parallel...

Drift-ordered fluid equations for field-aligned modes in low-β collisional plasma with equilibrium pressure pedestals

Starting from the complete short mean-free path fluid equations describing magnetized plasmas, assuming that plasma pressure is small compared to magnetic pressure, considering field-aligned plasma fluctuations, and adopting an ordering in which the plasma species flow velocities are much smaller than the ion thermal speed, a system of nonlinear equations for plasma density, electron and ion temperatures, parallel ion flow velocity, parallel current, electrostatic potential, perturbed parallel electromagnetic potential, and a perturbed magnetic field is derived. The equations obtained allow sharp equilibrium radial gradients of plasma quantities, and are shown to contain the neoclassical (Pfirsch–Schluter) results for plasma current, parallel ion flow velocity (with the correct temperature gradient terms), and parallel gradients of equilibrium electron and ion temperatures. Special care is taken to ensure the divergence-free character of perturbed magnetic field and total plasma current, as well as local ...

Evaluation of the neoclassical radial electric field in a collisional tokamak

The neoclassical electric field in a tokamak is determined by the conservation of toroidal angular momentum. In the steady state in the absence of momentum sources and sinks it is explicitly evaluated by the condition that radial flux of toroidal angular momentum vanishes. For a collisional or Pfirsch-Schluter short mean-free path ordering with subsonic plasma flows we find that there are two limiting cases of interest. The first is the simpler case of a strongly up-down asymmetric tokamak for which the lowest order gyroviscosity does not vanish and must be balanced by the leading order collisional viscosity in order to determine the radial electric field. The second case is the more complicated case of an up-down symmetric tokamak for which the gyroviscosity must be evaluated to higher order and again balanced by the lowest order collisional viscosity to determine the radial electric field. In general, both the lowest and the next order contributions from the gyroviscosity must be retained.

Drift kinetic equation exact through second order in gyroradius expansion

The drift kinetic equation of Hazeltine [R. D. Hazeltine, Plasma Phys. 15, 77 (1973)] for a magnetized plasma of arbitrary collisionality is widely believed to be exact through the second order in the gyroradius expansion. It is demonstrated that this equation is only exact through the first order. The reason is that when evaluating the second-order gyrophase dependent distribution function, Hazeltine neglected contributions from the first-order gyrophase dependent distribution function, and then used this incomplete expression to derive the equation for the gyrophase independent distribution function. Consequently, the second-order distribution function and the stress tensor derived by this approach are incomplete. By relaxing slightly Hazeltine’s orderings one is able to obtain a drift kinetic equation accurate through the second order in the gyroradius expansion. In addition, the gyroviscous stress tensor for plasmas of arbitrary collisionality is obtained.

Kinetic stability of electrostatic plasma modes in a dipolar magnetic field

An axially symmetric plasma immersed in a poloidal magnetic field with closed lines is considered. Low-frequency electrostatic modes are studied kinetically for an “intermediate collisionality” ordering, in which the particle collision frequency is much smaller than the transit or bounce frequency, but much larger than the mode, magnetic drift, and diamagnetic drift frequencies. This ordering is appropriate for the Levitated Dipole Experiment (LDX) [J. Kesner et al., 17th IAEA Fusion Energy Conference, Yokahama, Japan (IAEA, Vienna, 1999)] and some other closed field line devices. “High-frequency” magnetohydrodynamic-like and “low-frequency” entropy modes are found and stability boundaries are determined. Collisional effects are considered and the corresponding ion gyro-relaxation effects are evaluated. These effects introduce dissipation (or inverse dissipation) and are shown to modify the stability picture considerably, while leaving large stability regions in the d, η parametric space, where η is the r...

Ballooning stability of a point dipole equilibrium

The energy principle is employed to show that the equilibrium confined by the magnetic field of a point dipole is stable to ballooning modes.

A New, Explicitly Collisional Contribution to the Gyroviscosity and the Radial Electric Field in a Collisional Tokamak

An additional contribution to the ion viscosity for a collisional plasma is evaluated and found to be the same order as other temperature gradient terms in the collisional perpendicular viscosity. The new contribution arises because of an explicitly collisional portion of the ion distribution function. The evaluation of the Pfirsch-Schluter radial electric field in a collisional tokamak of arbitrary cross section is extended to retain the new contribution. In a spherical tokamak this new contribution must be retained in determining the radial electric field, while in a conventional tokamak it is small by 1∕q2, where q is the safety factor.

Long mean-free path collisional stability of electromagnetic modes in axisymmetric closed magnetic field configurations

An axially symmetric plasma confined by a poloidal magnetic field with closed field lines is considered. A kinetic analysis of electromagnetic modes is performed for an “intermediate collisionality” ordering in which the particle collision frequency is much smaller than the transit or bounce frequency, but much larger than the mode, magnetic drift, and diamagnetic drift frequencies. A second order integrodifferential ballooning equation for electromagnetic modes is derived, which describes “high-frequency” ideal magnetohydrodynamic (MHD) and “low-frequency” entropy modes. The equation recovers the corresponding ideal MHD ballooning equation for the mode frequency greater than the magnetic drift and diamagnetic drift frequencies, and generalizes the results of an earlier electrostatic treatment of the entropy mode to arbitrary plasma beta. Ion gyrorelaxation collisional modifications to the entropy mode are also evaluated for arbitrary plasma beta and specific results are presented for both a point dipole and Z pinch.An axially symmetric plasma confined by a poloidal magnetic field with closed field lines is considered. A kinetic analysis of electromagnetic modes is performed for an “intermediate collisionality” ordering in which the particle collision frequency is much smaller than the transit or bounce frequency, but much larger than the mode, magnetic drift, and diamagnetic drift frequencies. A second order integrodifferential ballooning equation for electromagnetic modes is derived, which describes “high-frequency” ideal magnetohydrodynamic (MHD) and “low-frequency” entropy modes. The equation recovers the corresponding ideal MHD ballooning equation for the mode frequency greater than the magnetic drift and diamagnetic drift frequencies, and generalizes the results of an earlier electrostatic treatment of the entropy mode to arbitrary plasma beta. Ion gyrorelaxation collisional modifications to the entropy mode are also evaluated for arbitrary plasma beta and specific results are presented for both a point dipole ...

Anisotropic pressure stability of a plasma confined in a dipole magnetic field

The interchange and ballooning stability of general anisotropic pressure plasma equilibria in a dipolar magnetic field are investigated. Starting with the Kruskal–Oberman form of the energy principle and using a Schwarz inequality, a fluid form of the anisotropic pressure energy principle is derived, which, after appropriate minimization, gives an interchange stability condition and an integro-differential ballooning equation. These results are applied to the case of an anisotropic pressure equilibrium having the perpendicular pressure equal to the parallel pressure times a constant and, in particular, to a model point dipole equilibrium. It is found that the model equilibrium is interchange stable for all plasma betas = (plasma pressure/magnetic pressure) and ballooning stable for all betas up to some critical value. The interesting planetary case of “tied” field lines is also considered.

Magnetic topology effects on Alcator C-Mod flows

The effect of magnetic topology on ion and impurity flows in a tokamak is considered by investigating the consequences of (i) the reversal of toroidal and poloidal magnetic fields and currents; (ii) a switch from lower to upper X-point operation; (iii) poloidal magnetic field or plasma current reversal, and (iv) toroidal magnetic field reversal. The general symmetries associated with magnetic topology changes in tokamaks are employed to demonstrate that the flux surface flows inside and outside the separatrix observed in Alcator C-Mod [I. H. Hutchinson et al., Phys. Plasmas 1, 1511 (1994)] can be used to determine the flow features, including neoclassical and turbulent effects and in the presence of charge exchange.