Non-axisymmetric Ballooning Modes in Thin Accretion Structures

The consideration of non-axisymmetric modes in thin accretion structures has been shown to be relevant to the issue of angular momentum transport [1]. To this end we consider the stability of differentially rotating, $u_{\phi}=R\Omega(R)$, axisymmetric equilibria with cylindrical magnetic flux surfaces to perturbations with large toroidal mode numbers $n_0\gg 1$. Assuming the equilibrium plasma rotation is approximately Keplerian about a central compact object and the rotation speed is smaller than the thermal speed, the equilibrium pressure and density depend on both $R$ and $z$. We seek linear solutions to the ideal MHD equations by expanding in the small parameter $1/n_0$ and satisfying requirements similar to those used for ballooning modes in magnetically confined toroidal plasmas: 1) the wavelength parallel to the magnetic field is relatively small, 2) the mode approximately corotates with the plasma, and 3) the compressibility is small but non-zero. Normal mode solutions are constructed using a radially dependent toroidal mode number which is inversely proportional to the rotation frequency such that $k_{\phi} \sim n_0 \Omega_0/u_{\phi}(R)$ where $\Omega_0$ is a constant. Toroidally periodic perturbations are then constructed with a formalism that is typically used in the theory of ballooning modes in toroidal configurations to enforce periodicity in the poloidal direction in the presence of magnetic shear. [1] {B. Coppi, P. Coppi {\it Ann. Phys.} \textbf{291}, 134 (2001)} *Supported by U.S. D.O.E.