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A Sturm–Liouville Eigenproblem of the Fourth Kind: A Critical Latitude with Equatorial Trapping
Author(s) -
Boyd John P.,
Natarov Andrei
Publication year - 1998
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00100
Subject(s) - eigenfunction , eigenvalues and eigenvectors , mathematical analysis , mathematics , interval (graph theory) , sturm–liouville theory , rossby number , limit (mathematics) , series (stratigraphy) , hermite polynomials , mathematical physics , physics , boundary value problem , turbulence , combinatorics , quantum mechanics , paleontology , biology , thermodynamics
Through both analytical and numerical methods, we 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 δ→0 of 1/( z − i δ). The eigenfunction has a branch point of the form z log( z ) at z =0, where the branch cut is on the upper imaginary axis. The eigenvalue is complex valued with an imaginary part, which we show, through matched asymptotics, to be approximately √ π exp(−1/ε 2 ){1−2ε log ε+ε log 2+γε}. Because T ( λ ) is transcendentally small in the small parameter ε, it lies “beyond all orders” in the usual Rayleigh–Schrödinger power series in ε. Nonetheless, we develop special numerical algorithms that are effective in computing T ( λ ) for ε as small as 1/100.