A Sturm-Liouville eigenproblem of the fourth kind: a critical latitude with equatorial trapping. (English) Zbl 1020.34080
Summary: Through both analytical and numerical methods, the authors solve the eigenproblem on the unbounded interval , where is the eigenvalue and as . This models an equatorially trapped Rossby wave in a shear flow in the ocean or atmosphere. It is the usual parabolic cylinder equation with Hermite functions as the eigenfunctions except for the addition of an extra term, which is a simple pole. The pole, which is on the interior of the interval, is interpreted as the limit of . The eigenfunction has a branch point of the form at , where the branch cut is on the upper imaginary axis. The eigenvalue is complex valued with an imaginary part, which the authors show, through matched asymptotics, to be approximately . Because is transcendentally small in the small parameter , it lies ‘beyond all orders’ in the usual Rayleigh-Schrödinger power series in . Nonetheless, the authors develop special numerical algorithms that are effective in computing for as small as .
|34L40||Particular ordinary differential operators|
|34E05||Asymptotic expansions (ODE)|