The unified ballooning theory with weak up-down asymmetric mode structure and the numerical studies

TL;DR

This paper develops a unified ballooning theory revealing weak up-down asymmetric mode structures in symmetric equilibria, showing these modes can have higher growth rates and providing a framework for analyzing ITG modes.

Contribution

It introduces a unified ballooning theory accounting for higher order effects causing asymmetry, with analytical solutions validated by numerical methods.

Findings

Asymmetric modes can have higher growth rates than symmetric ones.

Analytical mode structures agree well with numerical solutions.

Provides a reliable framework for quasi-linear computations.

Abstract

A unified ballooning theory, constructed on the basis of two special theories [Y. Z. Zhang, S. M. Mahajan, X. D. Zhang, Phys. Fluids B4, 2729 (1992); Y. Z. Zhang, T. Xie, Nucl. Fusion & Plasma Phys. 33, 193 (2013)], shows that a weak up-down asymmetric mode structure is normally formed in an up-down symmetric equilibrium; the weak up-down asymmetry in mode structure is the manifestation of non-trivial higher order effects beyond the standard ballooning equation. It is shown that the asymmetric mode may have even higher growth rate than symmetric modes. Salient features of the theory are illustrated by investigating a fluid model for the ion temperature gradient (ITG) mode. The two dimensional (2D) analytical form of ITG mode, solved in ballooning representation, is then converted into the radial-poloidal space to provide the natural boundary condition for solving the 2D mathematical local eigenmode problem. We find the analytical expression of mode structure in good agreement with finite difference solution. This sets a reliable framework for quasi-linear computation.