# Analytical Approximations of Dispersion Relations for Internal Gravity Waves Equation with Shear Flows

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Statement of the Problem

## 3. Basic Properties of Dispersion Curves

## 4. Analytical Approximation of Dispersion Relations

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Miropolsky, Y.Z. Dynamics of Internal Gravity Waves in the Ocea; Shishkina, O., Ed.; Springer: Berlin/Heidelberg, Germany, 2001; p. 406. [Google Scholar]
- Pedlosky, J. Waves in the Ocean and Atmosphere: Introduction to Wave Dynamics; Springer: Berlin/Heidelberg, Germany, 2010; p. 260. [Google Scholar]
- Sutherland, B.R. Internal Gravity Waves; Cambridge University Press: Cambridge, UK, 2010; p. 394. [Google Scholar]
- Morozov, E.G. Oceanic Internal Tides: Observations, Analysis and Modeling; Springer: Berlin/Heidelberg, Germany, 2018; p. 317. [Google Scholar]
- Tarakanov, R.Y.; Marchenko, A.V.; Velarde, M.G. (Eds.) Ocean in Motion; Springer: Berlin/Heidelberg, Germany, 2018; p. 625. [Google Scholar]
- Mei, C.C.; Stiassnie, M.; Yue, D.K.-P. Theory and Applications of Ocean Surface Waves: Advanced Series of Ocean Engineering. V.42; World Scientific Publishing: London, UK, 2017; p. 1500. [Google Scholar]
- Fabrikant, A.L.; Stepanyants, Y.A. Propagation of Waves in Shear Flows; World Scientific Publishing: London, UK, 1998; p. 304. [Google Scholar]
- Bulatov, V.V.; Vladimirov, Y.V. Wave Dynamics of Stratified Mediums; Nauka Publishers: Moscow, Russia, 2012; p. 584. [Google Scholar]
- Bulatov, V.V.; Vladimirov, Y.V. A General Approach to Ocean Wave Dynamics Research: Modeling, Asymptotics, Measurements; OntoPrint Publishers: Moscow, Russia, 2012; p. 587. [Google Scholar]
- Fraternale, F.; Domenicale, L.; Staffilan, G.; Tordella, D. Internal waves in sheared flows: Lower bound of the vorticity growth and propagation discontinuities in the parameter space. Phys. Rev.
**2018**, 97, 063102. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bouruet-Aubertot, P.I.; Thorpe, S.A. Numerical experiments of internal gravity waves an accelerating shear flow. Dyn. Atmos. Oceans
**1999**, 29, 41–63. [Google Scholar] [CrossRef] - Frey, D.I.; Novigatsky, A.N.; Kravchishina, M.D.; Morozov, E.G. Water structure and currents in the Bear Island Trough in July–August 2017. Russ. J. Earth Sci.
**2017**, 17, ES3003. [Google Scholar] [CrossRef][Green Version] - Morozov, E.G.; Paka, V.T.; Bakhanov, V.V. Strong internal tides in the Kara Gates Strait. Geophys. Res. Lett.
**2008**, 35, L16603. [Google Scholar] [CrossRef] - Tarakanov, R.Y.; Morozov, E.G.; Frey, D.I. Hydraulic continuation of the abyssal flow from the Vema Channel in the Southwestern part of the Brazil Basin. J. Geophys. Res. Oceans
**2020**, 125, e2020JC016232. [Google Scholar] [CrossRef] - Khimchenko, E.E.; Frey, D.I.; Morozov, E.G. Tidal internal waves in the Bransfield Strait, Antarctica. Russ. J. Earth Sci.
**2020**, 20, ES2006. [Google Scholar] [CrossRef][Green Version] - Bulatov, V.V.; Vladimirov, Y.V. Dynamics of internal gravity waves in the ocean with shear flows. Russ. J. Earth Sci.
**2020**, 20, ES4004. [Google Scholar] [CrossRef] - Bulatov, V.V.; Vladimirov, Y.V. Internal gravity waves in the ocean with multidirectional shear flows. Atmos. Ocean. Phys.
**2020**, 56, 85–91. [Google Scholar] [CrossRef] - Broutman, D.; Rottman, J. A simplied Fourier method for computing the internal wave field generated by an oscillating source in a horizontally moving, depth-dependent background. Phys. Fluids
**2004**, 16, 3682. [Google Scholar] [CrossRef] - Svirkunov, P.N.; Kalashnik, M.V. Phase patterns of dispersive waves from moving localized sources. Phys. Uspekhi
**2014**, 57, 80–91. [Google Scholar] [CrossRef] - Voelker, G.S.; Myers, P.G.; Walter, M.; Sutherland, B.R. Generation of oceanic internal gravity waves by a cyclonic surface stress disturbance. Dyn. Atmos. Oceans
**2019**, 86, 116–133. [Google Scholar] [CrossRef] - Miles, J.W. On the stability of heterogeneous shear flow. J. Fluid Mech.
**1961**, 10, 495–509. [Google Scholar] [CrossRef][Green Version] - Howard, L.N. Note of the paper of John W. Miles. J. Fluid Mech.
**1961**, 10, 509–512. [Google Scholar] [CrossRef] - Gavrileva, A.A.; Gubarev, Y.G.; Lebedev, M.P. The Miles theorem and the first boundary value problem for the Taylor–Goldstein equation. J. Appl. Ind. Math.
**2019**, 13, 460–471. [Google Scholar] [CrossRef] - Hirota, M.; Morrison, P.J. Stability boundaries and sufficient stability conditions for stably stratified, monotonic shear flows. Phys. Lett. A
**2016**, 380, 1856–1860. [Google Scholar] [CrossRef][Green Version] - Churilov, S. On the stability analysis of sharply stratified shear flows. Ocean Dyn.
**2018**, 68, 867–884. [Google Scholar] [CrossRef] - Carpenter, J.R.; Balmforth, N.J.; Lawrence, G.A. Identifying unstable modes in stratified shear layers. Phys. Fluids
**2010**, 22, 054104. [Google Scholar] [CrossRef][Green Version] - Watson, G.N. A Treatise on the Theory of Bessel Functions (Reprint of the 2nd ed.); Cambridge University Press: Cambridge, UK, 1995. [Google Scholar]
- Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions (Reprint of the 1972 ed.); Dover Publications: New York, NY, USA, 1992. [Google Scholar]
- Stoker, J.J. Water Waves: The Mathematical Theory with Applications; Interscience Publishers: New York, NY, USA, 1957. [Google Scholar]
- Whitham, G.B. Linear and Nonlinear Waves; J. Wiley and Sons: New York, NY, USA, 1974. [Google Scholar]
- Kravtsov, Y.; Orlov, Y. Caustics, Catastrophes and Wave Fields; Springer: Berlin/Heidelberg, Germany, 1999. [Google Scholar]
- Bulatov, V.V.; Vladimirov, Y.V. Asymptotical analysis of internal gravity wave dynamics in stratified medium. Appl. Math. Sci.
**2014**, 8, 217–240. [Google Scholar] [CrossRef]

**Figure 2.**Phase structure for the first model of shear flows. The phase structure consists of curved triangles embedded in the wave wedges with their vertex lying nearer to the origin.

**Figure 3.**Phase structure for the second model of shear flows. The wave pattern looks like a system of longitudinal and transverse waves.

**Figure 4.**Phase structure for the third model of shear flows. The wave pattern looks like closed (symmetric or asymmetric) ellipsoids.

**Figure 5.**Dispersion curves and its approximations for the first model, lines 1—first mode, lines 2—second mode.

**Figure 6.**Dispersion curves and its approximations for the second model, lines 1—first mode, lines 2—second mode.

**Figure 7.**Dispersion curves and its approximations for the third model, lines 1—first mode, lines 2—second mode.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bulatov, V.; Vladimirov, Y. Analytical Approximations of Dispersion Relations for Internal Gravity Waves Equation with Shear Flows. *Symmetry* **2020**, *12*, 1865.
https://doi.org/10.3390/sym12111865

**AMA Style**

Bulatov V, Vladimirov Y. Analytical Approximations of Dispersion Relations for Internal Gravity Waves Equation with Shear Flows. *Symmetry*. 2020; 12(11):1865.
https://doi.org/10.3390/sym12111865

**Chicago/Turabian Style**

Bulatov, Vitaly, and Yury Vladimirov. 2020. "Analytical Approximations of Dispersion Relations for Internal Gravity Waves Equation with Shear Flows" *Symmetry* 12, no. 11: 1865.
https://doi.org/10.3390/sym12111865