Five themes in Chebyshev spectral methods applied to the regularized Charney eigenproblem: Extra numerical boundary conditions, a boundary-layer-resolving change of coordinate, parameterizing a curve which is singular at an endpoint, extending the tau met

The Charney problem, a second order ordinary differential equation eigenproblem on z∈[0,∞]z∈[0,∞] with complex eigenvalues, is of great historical importance in meteorology and oceanography. Here, it is used as a testbed for several extensions of spectral methods. The first is to parameterize a plane curve which is singular at an endpoint, as very common in applications. The second stretch is to extend the Chebyshev tau method to compute eigenfunctions of the form M(z)+log(z)V(z)M(z)+log(z)V(z) where M(z)M(z) and V(z)V(z) are entire functions and where the approximation interval is a line segment in the complex plane. Third, we offer a special procedure for finding the roots of a function which is not a polynomial, but rather the combination of a polynomial plus a logarithm multiplied by a second polynomial. Lastly, to resolve the very thin boundary layer of the regularized Charney problem, we combine a rational Chebyshev (TLnTLn) pseudospectral method with a change of coordinate which is quadratic at the ground. Remarkably, best results are obtained by applying four boundary conditions even though the Charney problem is a differential equation of only second order.