Order Of Larmor Radius Research Articles

A mixed Fourier-variational approach to solve differential or integro-differential wave equations for magnetised plasmas

The All ORders Spectral Algorithm (AORSA) wave equation solver by Jaeger (Jaeger et al 2001 Phys. Plasmas 8 1573) solves the integro-differential wave equation relevant for the radio frequency (RF) domain and for fusion-relevant conditions in tokamaks or stellarators, retaining all finite Larmor radius corrections by substituting the continuous Fourier integrals by a sum over a discrete set of modes. Its strength is also its weakness: the simplicity of the method results in significant computational effort, a full matrix needing to be inverted to solve the associated linear system. Based on the notion that modes are gradually more independent if their eigenvalues differ, the present paper proposes a straightforward numerical method to partly alleviate this need, allowing to substitute the full system matrix by a banded one. The adopted method can be applied to a wide variety of equations. A few 1D examples—of relevance for solving the wave equation in the RF domain of frequencies—are provided: the tunneling equation is used to illustrate the potential of the method, and the all-FLR wave equation (retaining all Finite Larmor Radius corrections in the dielectric response) adopted by Jaeger is solved comparing the solutions found to those based on simpler models (a cold plasma and a ‘tepid plasma’ - i.e. a kinetic model truncated at zero order in Larmor radius—description).

Hybrid guiding-centre/full-orbit simulations in non-axisymmetric magnetic geometry exploiting general criterion for guiding-centre accuracy

To identify under what conditions guiding-centre or full-orbit tracing should be used, an estimation of the spatial variation of the magnetic field is proposed, not only taking into account gradient and curvature terms but also parallel currents and the local shearing of field-lines. The criterion is derived for general three-dimensional magnetic equilibria including stellarator plasmas. Details are provided on how to implement it in cylindrical coordinates and in flux coordinates that rely on the geometric toroidal angle. A means of switching between guiding-centre and full-orbit equations at first order in Larmor radius with minimal discrepancy is shown. Techniques are applied to a MAST (mega amp spherical tokamak) helical core equilibrium in which the inner kinked flux-surfaces are tightly compressed against the outer axisymmetric mantle and where the parallel current peaks at the nearly rational surface. This is put in relation with the simpler situation B(x, y, z) = B0[sin(kx)ey + cos(kx)ez], for which full orbits and lowest order drifts are obtained analytically. In the kinked equilibrium, the full orbits of NBI fast ions are solved numerically and shown to follow helical drift surfaces. This result partially explains the off-axis redistribution of neutral beam injection fast particles in the presence of MAST long-lived modes (LLM).

Theory and modelling of second harmonic ICRF coupling and heating in inhomogeneous plasmas

The combined theory of ion cyclotron wave coupling, mode conversion and absorption is developed for bounded inhomogeneous plasmas in slab geometry. The plasma response to the RF waves is described by the linearized Vlasov equation, with an equilibrium distribution function and a poloidal magnetic field profile which are selfconsistent solutions of the equilibrium Vlasov and Maxwell's equations. A general form of the wave equation, valid to arbitrary order in Larmor radius and including first order gradient terms, is derived. Expressions are also derived for the plasma intrinsic impedance. The wave equation, keeping second order terms in Larmor radius, is solved numerically for a plasma bounded by two conducting walls using the INTOR tokamak design parameters. It was found that the effect of the one-dimensional poloidal field is small. Also, a rough estimate of the error due to truncation to second order in Larmor radius was made and was found to be very small except in the high temperature regions. The effects of plasma boundedness and reflections from the mode conversion region on the coupling of waves from the RF source as a function of the plasma parameters and antenna spectra are studied. It is found that, at certain values of the plasma and antenna parameters, eigenmodes are formed, resulting in an increase in the loading impedance. It is verified that the impedance is not symmetric with respect to the spectrum in the poloidal direction, with highly peaked resonances. As the plasma temperature is increased, the resonances broaden and eventually disappear at very high temperatures. This is attributed to increased single pass absorption and decreasing reflection from the mode conversion region.

The theory of mode conversion and wave damping near the ion cyclotron frequency

Using a variational technique, a set of coupled model equations for the mode conversion process near the ion cyclotron frequency is derived. The system is truncated to first order in Larmor radius but includes the effects of explicit gradients and a poloidal field. From the equations a conservation rule is extracted which ensures conservation of total energy and provides an explicit expression for the wave damping in differential form. The equations are integrated numerically for the standard cases of fast waves incident from either the low- or high-field sides of the mode conversion layer. The scaling of the damping processes is discussed and implications for current RF heating experiments on the Princeton Large Torus are drawn.

Trajectory of Charged Particles in Weakly Inhomogeneous Magnetic Fields

The detailed time dependence of the trajectory, and particularly of the acceleration, of a charged particle moving nonrelativistically in a weakly inhomogeneous, slowly varying electromagnetic field is calculated by a method that involves superimposing a conventional perturbation expansion in the weak field on the expansion of Berkowitz and Gardner and of Kruskal, the latter expansion being asymptotically valid in the limit of small Larmor radius and slowly varying fields. Fourier transforms are introduced to facilitate isolation of rapidly varying terms from the most general solutions for the coefficients in the expansion. The calculations are carried to first order in the weak inhomogeneity and (at least formally) to all orders in Larmor radius. The resulting velocity reproduces the familiar transverse drift velocities of first-order orbit theory, emphasizes that in the presence of fields varying slowly in time the polarization and curvature drifts are to be calculated from the time derivative of the fields as seen from a frame of reference fixed with respect to the guiding center of the particle's orbit, and permits calculation of higher-order contributions to the drift velocities, contributions of second order in Larmor radius being explicitly evaluated; the resulting acceleration prepares the way for a study of the effect of weak inhomogeneities on the radiative output of single particles.

Hot Plasma in a Cylindrical Waveguide

An approximate theory for the propagation of waves in a hot plasma in a cylindrical waveguide with an arbitrary axial magnetic field is presented. The boundary value problem is treated by a perturbation expansion in the smallness of the Larmor radius compared to the transverse wavelength. In the limit of zero Larmor radius, only the electromagnetic boundary conditions are important, while acoustic and higher-order boundary conditions appear as corrections. The results to zero order in Larmor radius are found to be formally identical to the cold plasma equations. Mode orthogonality and some limiting cases are also discussed.

Theory of the flute instability in mirror machines

The flute instability of a low β and neutral plasma contained in a mirror machine is investigated. The particle motion in a real mirror field is fully taken into account by using a Hamiltonian method for the solution of the Vlasov equation. An eigenvalue equation is obtained for the potential of the flute-like perturbation assuming an arbitrary radial dependence of plasma density and of the electron and ion energies. Using a variational method, the eigenvalue equation has been discussed in detail for a plasma with no energy gradients. The stability condition has been calculated taking into account terms up to second order in Larmor radius. Finite Larmor radius stabilization is effective for the higher-order modes and cold electron plasmas. The lowest mode, k = 1, is practically not stabilized by finite Larmor radius effects. The discussion of the eigenvalue equation with finite energy gradients has been limited to the case of a hot electron plasma surrounded by a sheath of cold plasma. If the density of the cold plasma is an order of magnitude higher than the density of the hot plasma, the system is stable at all densities. The stabilization is due to the high dielectric constant of the cold plasma.