## Vortex Layer on the β-Plane in the Miles – Ribner Formulation. Pole on the Real Axis

*V. G. Gnevyshev ^{1}, T. V. Belonenko^{2, ✉}*

^{1} Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow, Russian Federation

^{2} Saint Petersburg State University, St. Petersburg, Russian Federation

^{✉} e-mail: btvlisab@yandex.ru

### Abstract

*Purpose*. The problem of a non-zonal vortex layer on the β-plane in the Miles – Ribner formulation is considered. It is known that in the absence of the β-effect, the vortex layer has no neutral eigenmodes, and the available two ones (varicose and sinusoidal) are unstable. Initially, generalization of the problem to the β-plane concerned only the zonal case. The problem for a non-zonal vortex layer is examined for the first time in the paper. It is known that in the WKB approximation for the linear wave disturbances (regardless of whether a zonal or non-zonal background flow is considered), there is an adiabatic invariant in the form of the law of the enstrophy (vorticity) conservation. For the zonal vortex layer, the enstrophy conservation law also holds, and no vorticity exchange occurs between the waves and the flow in the zonal case. The non-zonal vortex layer has qualitatively different features; particularly, it does not retain enstrophy. Thus, as a result, there appears a new class of solutions which can be interpreted as pure radiation of the Rossby waves by a non-zonal flow. Generalizing the vortex layer problem on the β-plane to the non-zonal case constitutes the basic aim of the present study.

*Methods and Results*. A new class of linear stationary wave solutions, namely the Rossby waves, is found. It is shown a non-zonal flow can be directed in one way, whereas the stationary wave disturbances can move in the opposite (contrary) direction. The coexistence of such solutions for the shear non-zonal flow and stationary wave disturbances takes place due to the influence of the external force and mathematically comes from a non-self-adjoining character of the linear operator for a non-zonal background flow.

*Conclusions*. There exists a new class of solutions that can be interpreted as pure radiation of the Rossby waves by a non-zonal flow. There is no such solution for a zonal flow. It is just non-zoning that gives the effect of pure radiation and corresponds to the classical definition of radiation. This approach makes it possible to eliminate inconsistency in terminology, when instabilities are mistakenly called radiation, and radiation – pure radiation.

### Keywords

Rossby waves, vortex layer, non-zonal flow, Miles – Ribner problem

### Acknowledgements

The work was supported by the RFBR grant No. 20-05-00066. The research by V.G. Gnevyshev was also supported by the IO RAS, ST-ASS-0128-2021-0003.

### Original russian text

Original Russian Text © V. G. Gnevyshev, T. V. Belonenko, 2021, published in MORSKOY GIDROFIZICHESKIY ZHURNAL, Vol. 37, Iss. 5, pp. 525-537 (2021)

### For citation

Gnevyshev, V.G. and Belonenko, T.V., 2021. Vortex Layer on the β-Plane in the Miles – Ribner Formulation. Pole on the Real Axis. *Physical Oceanography*, 28(5), pp. 486-498. doi:10.22449/1573-160X-2021-5-486-498

### DOI

10.22449/1573-160X-2021-5-486-498

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