D. G. Swanson

Radio frequency heating in the ion‐cyclotron range of frequencies

Both the theory of the absorption process in the ion‐cyclotron range of frequencies and some of the experiments which slow the promise and problems with radio frequency plasma heating in this range are discussed. It is shown that mode conversion is invariably involved in the process and so an extensive review of mode conversion theory, expecially as it applies to problems with back‐to‐back cutoff‐resonance pairs, is included. This includes a discussion of the tunneling equation with and without absorption effects and with and without energy conservation. The general theory is applied to various ion‐cyclotron harmonics, the two‐ion hybrid resonance, and to a case where a wave converts to a Bernstein mode at the plasma edge. The results are given analytically for a variety of cases without absorption, and empirical formulas are given for the second and third harmonics of the ion‐cyclotron frequency, which include effects of absorption. Various problem areas in the theory are also discussed with some of the ...

Derivation of the mode conversion‐tunneling equation from the Vlasov equation

Instead of using the uniform warm plasma dispersion relation with an inverse Fourier transform to form the mode conversion‐tunneling equation, the equation is developed directly from the Vlasov equation and Maxwell’s equations. The plasma is assumed uniform parallel to B0. The expansion parameters are the Larmor orbit and the scale length, keeping terms to order ρ2L and L−1 = (1/ωc)(dωc/dx). For the special case with ω≃2ωc in a single species plasma, the asymptotic form of the mode conversion‐tunneling equation is unchanged, but localized third and first derivative terms appear even as absorption vanishes.

An algorithm to solve the linear mode conversion problem in a weakly inhomogeneous plasma

In a previous article, the wave equation for electromagnetic and electrostatic waves propagating in a weakly inhomogeneous plasma was studied assuming an ordinary differential equation of the Laplace type. An algorithm was developed, based on the contour integral formalism, that gives a prescription for calculating the connection formulas between the set of fast and slow propagating waves. Physical problems connected to the evanescent solutions were not treated, however, so the complete set of solutions was not obtained. In this paper, the method is extended to include both propagating and nonpropagating waves so that a complete set of solutions is obtained, permitting the construction of the Green function and the inclusion of localized sources or sinks.

Dispersion relations for the lower hybrid frequency range

The hot plasma electrostatic dispersion relation for the lower hybrid frequency range has been cast into a form without any sums using the method of steepest descents. This new form of the dispersion relation with the exact resonance term, which is valid for general complex wavenumber and each term of which is identified according to its role of representing physical waves, is shown to be accurate and to be reducible to an expression obtained by Brambilla [Plasma Phys. 18, 699 (1976)] when some approximations are taken. A very simple dispersion relation is also obtained without singular terms near the high ion cyclotron harmonics that are encountered by lower hybrid waves propagating in inhomogeneous magnetic fields. Finally, the damping rate in space is numerically calculated using the equation derived and compared with the result from the unmagnetized ion dispersion relation.

Emission of ion and electron cyclotron harmonic radiation from mode conversion layers

The asymmetry of cyclotron radiation from a mode conversion layer is presented for harmonics of the ion cyclotron frequency and the second harmonic of the electron cyclotron frequency for weakly relativistic electrons. The same form of Kirchhoff’s law is found for all cases, relating the emission along each branch to the absorption of an incident wave along the corresponding branch. Results show that the fast wave radiation is more strongly asymmetric at the third harmonic than at the second harmonic of the ion cyclotron frequency, while the slow wave radiation ratio is about same. At the second cyclotron harmonic of weakly relativistic electrons, the asymmetry of radiation is found to be small at high temperature. The effect of equilibrium Bernstein wave radiation is also discussed.

Synchrotron radiation and absorption at 3ωce with X‐mode–O‐mode coupling

The mode conversion‐tunneling equation with weakly relativistic absorption at 3ωce has been solved by using the Green function method. The numerical results show that the reflection and conversion coefficients are dramatically reduced when relativistic absorption is included, while the transmission coefficients and the conversion coefficient from the X‐mode on the high field side to the O‐mode on the low field side are independent of absorption. The reduction of these coefficients leads to the reduction of the asymmetry in the absorption between different wave branches. The numerical results also show that the relativistic effect is important even for a relatively low temperature plasma. An important relationship between solutions is expressed by the reciprocity relations which have been proved for any harmonic number. These analyses modify the interpretation of electron cyclotron emission (ECE).

Reciprocity relations and the mode conversion‐absorption equation with an inhomogeneous source term

The fourth‐order mode conversion equation is solved completely via the Green’s function to include an inhomogeneous source term. This Green’s function itself contains all the plasma responsive effects such as mode conversion and absorption, and can be used to describe the spontaneous emission. In the course of the analysis, the reciprocity relations between coupling parameters are proved.

Mode conversion and absorption of lower hybrid waves in inhomogeneous magnetic fields

When a cold lower hybrid wave propagates in an inhomogeneous plasma, it undergoes a total mode conversion to the warm wave which is then coupled with the ion Bernstein wave around ion cyclotron harmonic frequencies. Both types of linear mode conversion problems can be described by fourth‐order differential equations if the two processes are separated from each other. The combination of the two types of mode conversion when the lower hybrid turning point is in the vicinity of a cyclotron harmonic requires a sixth‐order equation with a quadratic coefficient, which is not solvable analytically nor numerically. Another type of sixth‐order equation, which has only linear coefficients and represents the three‐wave problem marginally, is solved analytically without including the absorption term. Finally, the damping and the absorption of the wave are compared for the unmagnetized dispersion relation, the full magnetized dispersion relation, and the mode conversion analysis as the wave propagates in a magneticall...

Mode conversion at electron cyclotron harmonics with finite k

Relativistic plasma mode conversion‐tunneling equations at the second and third electron cyclotron harmonics are derived. A finite k∥ is introduced which keeps the coupling between the O‐mode, the X‐mode and the Bernstein wave in the mode conversion problem for the first time. The solutions for these mode conversion problems without absorption are obtained, and the connection formulas between different wave branches are established. The corresponding transmission, reflection and conversion coefficients are also given. A comparison between the coupled equation and the uncoupled equations is also made.

ABSORPTION AND EMISSION OF PLASMA WAVES IN MODE CONVERSION LAYERS AT GENERAL HARMONICS OF THE ION CYCLOTRON FREQUENCY

The mode conversion‐tunneling equation is solved numerically with the resonant absorption term at general harmonics of the ion cyclotron frequency. The numerical technique is based on the Laplace contour integral and direct integration. Results for various scattering parameters as well as absorbed fractions of incident energy are presented. As in the second and the third harmonic cases, simple expressions for some scattering parameters are found for higher harmonic numbers. Finally, emission ratios are calculated from absorption results and the reciprocity relations.