On the construction of a local curvilinear coordinate system conforming to the native curved geometry of the plasma focus sheath

TL;DR

This paper details constructing a specialized local curvilinear coordinate system aligned with the plasma focus sheath's geometry, facilitating analysis of physical phenomena with natural symmetry considerations.

Contribution

It introduces a detailed method for creating a local orthogonal coordinate system tailored to the plasma focus sheath's shape, aiding in physical problem formulation.

Findings

Coordinate system aligns with sheath geometry

Simplifies analysis of current density and velocity

Provides a reference for future research

Abstract

This paper describes in detail the construction of a local right-handed, orthogonal curvilinear coordinate system whose axes are along the local tangent, the local azimuth and the local normal of an analytically defined 3D surface of rotation, whose shape mimics the shape of a plasma focus current sheath. Expressions for various differential operators are derived in a tutorial format for the benefit of young researchers and non-specialists. Physical problems expressed in this coordinate system would benefit from the natural symmetry properties of the plasma focus sheath. For example, the normal component of current density is zero and the velocity has mainly the normal component. This paper is meant to serve as a readily available reference in the hope that it would be found useful.