Abstract
Essential Spectrum of Internal Waves in the Ocean and Atmosphere: the Relation between the Frequency of the Induced Vibrations and Non-Uniqueness of the Limit Amplitude
Highlights
Theorem 1: Let the exterior mass force be a periodic function with frequency w ≥ 0 : F (x, t) = f (x)e−iwt
Let us recall that, for a linear self-ad joint operator M acting in a Hilbert space, the essential spectrum is defined as the set of points of the continuous spectrum, limit points of the point spectrum, and eigenvalues of infinite multiplicity [3]
Theorem 2: There exists a finite constant, which is uniquely determined by the matrix, such that for the problem of normal vibrations, the essential spectrum of the operator is the interval of the imaginary axis : σ ess (M ) = [−iB,iB]
Summary
For various models of three-dimensional fluid which describe the flows in the Atmosphere and the Ocean, we find a relation between the essential spectrum of normal vibrations of internal waves and non-uniqueness of the limit amplitude of vibrations induced by external mass forces.
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