On the Behaviour of a Pair of Complex Eigenmodes Near a Crossing

The behaviour of a pair of eigenmodes from a complex matrix which depends on at least two external parameters is examined near some isolated points of the double degeneracy. The thermal Rossby wave provides an example for illustrating typical modal properties. Some of these properties can appear surprising, such as the rapid variation and the abrupt reconnection of the modes. For a class of stability analyses based on the path-tracing technique in the parameter space, these illustrations may provide a particularly beneficial reference.