Complex coordinate methods for hydrodynamic instabilities and Sturm-Liouville eigenproblems with an interior singularity

Abstract

Calculations of inviscid, linearized waves in fluids are very difficult when a mean wind or current U(y) is included because the differential equation is singular wherever U(y)=c, the phase speed. These "critical latitude," "critical level," or "critical point" singularities are particularly severe for Chebyshev methods since these global expansion algorithms are very sensitive to the analytic properties of the solution. A simple remedy is described: by making a change of coordinates y=f(x) where y is the original variable and x is the new coordinate with f(x) a complex function, one can solve the problem on an arc in the complex plane that makes a wide detour around the singularity. Specific guidelines for choosing f(x) for different problems are given in the text. Results are impressive: for an eigenvalue problem with a pole in the middle of the original real interval (a "Sturm-Liouville problem of the fourth kind"), just six basis functions suffice to calculate the real and imaginary parts of the lowest eigenvalue to within 1.4%. For strong instability, i.e., modes whose phase speeds have large imaginary parts, the complex mapping is unnecessary because the critical latitudes are complex and distant from the real axis. Even so, the mapping is useful for instability problems because it can be used to make calculations for very slowly growing modes to follow the changes in c right up to the "neutral curve" where the imaginary part of c=0. Although especially valuable for spectral algorithms, the same trick can be applied with finite difference methods also. The main disadvantage of the algorithm is that the eigenfunction must be calculated in a second, separate step, but this is usually a minor flaw in comparison to the complex mapping's virtues for coping with singular eigenvalue problems.