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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 weakup-down asymmetric mode structure is normally formed in an up-down symmetricequilibrium; the weak up-down asymmetry in mode structure is the manifestationof non-trivial higher order effects beyond the standard ballooning equation. Itis shown that the asymmetric mode may have even higher growth rate thansymmetric modes. Salient features of the theory are illustrated byinvestigating a fluid model for the ion temperature gradient (ITG) mode. Thetwo dimensional (2D) analytical form of ITG mode, solved in ballooningrepresentation, is then converted into the radial-poloidal space to provide thenatural boundary condition for solving the 2D mathematical local eigenmodeproblem. We find the analytical expression of mode structure in good agreementwith finite difference solution. This sets a reliable framework forquasi-linear computation.

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