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Spectral Properties in Modon Stability Theory
Author(s) -
Swaters Gordon E.
Publication year - 2004
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/j.0022-2526.2004.01506.x
Subject(s) - eigenfunction , eigenvalues and eigenvectors , mathematics , mathematical analysis , barotropic fluid , essential spectrum , vortex , upper and lower bounds , enstrophy , hilbert space , invariant (physics) , mathematical physics , physics , vorticity , quantum mechanics , mechanics , thermodynamics
Spectral properties of an invariant functional, denoted by H, for the linear stability equation associated with the modon , or solitary drift vortex, solutions of the quasi‐geostrophic equivalent barotropic potential vorticity, or Charney–Hasegawa–Mima (CHM), equation are investigated. It is shown that H, which is the only known quadratic invariant in modon stability theory, is identical in form to the second variation of a “Benjamin‐like” variational principle for solitary vortices. However, such a principle does not exist for the modon. The discrete spectrum of the “form operator” in H contains two simple negative eigenvalues and the simple zero eigenvalue. For the leftward‐traveling solution there are only a finite number of positive eigenvalues. For the rightward‐traveling solution, there are a countable infinity of positive eigenvalues. A sharp lower bound on the spectrum, for both the rightward‐ and leftward‐traveling solutions, and a sharp upper bound for the leftward traveling solution, is determined. For the leftward‐traveling solutions, the eigenfunctions span a finite‐dimensional vector space and are orthogonal with respect to an inner product which is valid for all of L 2 . For the rightward‐traveling solutions, the eigenfunctions span an infinite‐dimensional Hilbert space, but are orthogonal with respect to an inner product, which is not valid for all of L 2 .