  This paper studies the instability of two-dimensional magnetohydrodynamic
(MHD) systems on a sphere using analytical methods. The underlying flow
consists of a zonal differential rotation and a toroidal magnetic field is
present. Semicircle rules that prescribe the possible domain of the wave
velocity in the complex plane for general flow and field profiles are derived.
The paper then sets out an analytical study of the `clamshell instability',
which features field lines on the two hemispheres tilting in opposite
directions (Cally 2001, Sol. Phys. vol. 199, pp. 231--249). An asymptotic
solution for the instability problem is derived for the limit of weak shear of
the zonal flow, via the method of matched asymptotic expansions. It is shown
that when the zonal flow is solid body rotation, there exists a neutral mode
that tilts the magnetic field lines, referred to as the `tilting mode'. A weak
shear of the zonal flow excites the critical layer of the tilting mode, which
reverses the tilting direction to form the clamshell pattern and induces the
instability. The asymptotic solution provides insights into properties of the
instability for a range of flow and field profiles. A remarkable feature is
that the magnetic field affects the instability only through its local
behaviour in the critical layer.
