WKBJ approach to plasma instabilities
- Supervisor
- Prof. Robert Dewar Phone: 6125 2949
In an inhomogeneous medium the Wentzel-Kramers-Brillouin (WKB) method approximates the solution of a differential or integral equation by using simple travelling waves locally and then taking into account inhomogeneity by allowing the amplitude and wavelength to vary slowly with position. Global eigenvalue problems are solved by constructing standing waves using a superposition of travelling waves that connect properly at turning (reflection) points.
From a mathematical point of view the WKB method is a powerful asymptotic approximation technique. From a physics point of view it gives great insight in all areas of physics involving waves (both classical and quantum).
In one dimension the method can be applied to conservative problems, where the eigenfrequencies of normal modes are real, and to dissipative or unstable problems where the eigenfrequencies are complex. In the latter case, and in problems involving tunnelling, the method is often called the WKBJ or JWKB method to acknowledge the work of Jeffries. The problem can also be applied in higher dimensions using local plane wave solutions, but there are interesting conceptual problems to do with "quantum chaos" in cases where the ray equations are not integrable. The problem of complex eigenvalues also seems hardly to have been touched in more than one dimension.
The specific project is to review the paper by Berk and Pfirsch and to apply it to correct the analysis of the "ballooning Schrödinger equation" (a model equation for a plasma instability in the presence of velocity shear) by Dewar. The aim is to derive solutions that continuously connect between the high and low velocity shear cases. If time permits, the resulting formalism can be applied to deriving a first-principles WKBJ solution to this or another physical problem. It is anticipated that tools such as Mathematica or Maple will be valuable in solving this problem, and Mathematica notebooks used in writing are available.
