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
Department of Mathematics, Los Andes University, Colombia, South America
Submission: November 12, 2017; Published: February 19, 2018
*Corresponding author: Andrei Giniatoulline, Associate Professor, Department of Mathematics, Los Andes University, Bogota D.C., Colombia, South America, Tel: 5713394949, Fax: 5713324340, Email: [email protected]
How to cite this article: Andrei G. 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. Civil Eng Res J. 2018; 3(3): 555612. DOI: 10.19080/CERJ.2018.03.555612
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.
Keywords: Water Resources, Applied Ocean Research, Physical and Computational Models,Structural Dynamics, Internal Waves, Spectral Theory, Essential Spectrum, Limit Amplitude, Uniqueness of Mathematical Solutions.
For the Cauchy problem, in  there was proved the following result for the rotational motion.
Theorem 1: Let the exterior mass force be a periodic function with frequency w ≥ 0 :
Then, the solution v(x,t) of the Cauchy problem is also periodic and the stabilization property is valid :
is the solution ofthe stationary system with the external force .
We observe that for the case of the stratified fluid, the Theorem 1 was proved in .
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 . Therefore, t
he spectral points outside of the essential spectrum may only be eigenvalues of finite multiplicity.
For various types of the operator A, in [4,5] the following result was proved.
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 :
Now, we would like to compare the relation between the uniqueness of the above limit amplitude of the induced vibrations of frequency , and the structure and localization of the essential spectrum of the normal vibrations.
The following result was proved in  for the rotational model, and later in  it was proved for the model of stratified fluid.
Theorem 3: If the frequency w of the exterior mass forces does not belong to the essential spectrum of the normal vibrations, then the limit amplitude is unique. On the other hand, if, however, the frequency w belongs to the essential spectrum , then the limit amplitude loses the uniqueness. From the physical point of view, it means that the resonance effect takes place.
We also would like to observe that, particularly, the explicit example of non-uniqueness of the solution may be found in  for the construction of the Weyl sequence  in the proof of the result for the essential spectrum.
The remarkable relation between the essential spectrum of normal vibrations and non-uniqueness of the limit amplitude may explain the effect of resonance of internal waves from a new mathematical point of view. May we also express a conjecture that similar results are valid for more general types of the operator A.