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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 u zz +(1/z-λ-(z-1/ε) 2 )u=0 on the unbounded interval z[-,], where λ is the eigenvalue and u(z)0 as |z|. 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 δ0 of 1/(z-iδ). The eigenfunction has a branch point of the form zlog(z) at z=0, 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 πexp(-1/ε 2 ){1-2εlogε+εlog2+γε}. Because Im(λ) is transcendentally small in the small parameter ε, it lies ‘beyond all orders’ in the usual Rayleigh-Schrödinger power series in ε. Nonetheless, the authors develop special numerical algorithms that are effective in computing Im(λ) for ε as small as 1 100.
MSC:
34L40Particular ordinary differential operators
34B24Sturm-Liouville theory
34E05Asymptotic expansions (ODE)
76B65Rossby waves