Parabolic traps of Rossby waves in the ocean

Analysis of the Rossby wave dynamics shows when waves interact with shear currents vertical focusing of the modes occurs. Due to the inhomogeneity of the background flow, Rossby waves are captured by the current, and there is a compression of the modes on vertical horizons. For the vertical mode, instead of the classical trigonometric cosine, strongly localized solutions appear in the form of exponentially modulated Hermite polynomials. Qualitatively, the situation can be described as follows: an inhomogeneous background current acts like a kind of parabolic antenna. The wave, falling into this parabolic trap, begins to reflect off the narrowing walls of the paraboloid, while the vertical transparency zone narrows and the wave’s progress towards the center of the paraboloid slows down more and more. In the linear formulation, this process lasts infinitely long, while the distance between adjacent reflection points from the paraboloid mirror gradually decreases. There is a mathematical description of this phenomenon for internal waves. Since there are no fundamental differences between internal waves and Rossby in the vicinity of the focus, the mathematical part of the work for internal waves can also be transformed for Rossby waves.
In this paper, in terms of the Fourier integral, we construct a two-dimensional analytical solution of the reference equation for the vertical focusing of a monochromatic wave in the vicinity of the focus. Using the degenerate hypergeometric function of the complex variable, we show the identity of this solution with the solution of the reference equation obtained in previous studies. We find the asymptotic behavior of the solution in the far zone by the stationary phase method. Using exponentially majored Hermite polynomials, we show the correct two-dimensional crosslinking of the obtained solution, which has in the form of a degenerate hypergeometric function of a complex variable, happens with the WKB solution in the far zone. We show the question of absorption in the focal zone is not unambiguous, and therefore both situations are possible: both the passage and the reflection from the feature.

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