# Waves and Instabilities in Rotating and Stratified Flows

## Abstract

This review intended primarily for Master degree students, presents the different types of classical waves that can occur in astro and geophysical flows. Inertial waves, caused by the rotation of the fluid, will first be introduced as well as their 2D version called Rossby waves. Then it will be shown how a density stratification of the fluid can make internal gravity waves appear. In each case and in the case where both rotation and stratification are present, the dispersion relations of the waves are derived. A differential rotation will then be added on the flow. The classical Rayleigh criterium for the centrifugal instability is recovered in the case of an homogeneous fluid but it will be shown that a new instability, called the strato-rotational instability (SRI), can occur when the fluid is stratified. Some experiments will be described. Finally, we will show how the application of a magnetic field can create Alfven waves in a rotating electrically conducting fluid and in which conditions the magneto-rotational instability (MRI) can grow.

## Keywords

Dispersion Relation Rossby Wave Accretion Disk Alfven Wave Inertial Wave## References

- Alboussière T, Cardin P, Debray F, La Rizza P, Masson JP, Plunian F, Ribeiro A, Schmitt D (2011) Experimental evidence of Alfven wave propagation in a Gallium alloy. Phys Fluids 23:096601CrossRefGoogle Scholar
- Alfven H (1942) Existence of electromagnetic-hydrodynamic waves. Nature 150:405–406CrossRefGoogle Scholar
- Balbus SA, Hawley JF (1998) Instability, turbulence, and enhanced transport in accretion disks. Rev Mod Phys 70:1–53CrossRefGoogle Scholar
- Cébron D, Le Bars M, Leontini J, Maubert P, Le Gal P (2010) A systematic numerical study of the tidal instability in a rotating triaxial ellipsoid. Phys Earth Planet Inter 182:119–128CrossRefGoogle Scholar
- Courtesy of University of Paris-Sud. http://www.fast.u-psud.fr/ppcortet/inertialwave.php
- Courtesy of University of Washington. http://www.ocean.washington.edu/people/faculty/rhines/rossbypbr.html
- Courtesy of Sakai S, Iizawa I, Aramaki E (1997) Atmosphere and Ocean in a Laboratory: Hokusai. http://www.gfd-dennou.org/library/gfdexp/expe/exp/iw/1/res.htm
- Courtesy of University of Oregon (1999). http://zebu.uoregon.edu/1999/ph161/images/rossbyfor.gif
- Eloy C, Le Gal P, Le Dizès S (2003) Elliptic and triangular instabilities in rotating cylinders. J Fluid Mech 476:357–388CrossRefGoogle Scholar
- Kelvin L (1880) Vibrations of a columnar vortex. Phil Mag 10:155–68CrossRefGoogle Scholar
- Lacaze L, Le Gal P, Le Dizès S (2005) Elliptical instability in a rotating spheroid. J Fluid Mech 505:1–22CrossRefGoogle Scholar
- Le Bars M, Le Gal P (2007) Experimental analysis of the stratorotational instability in a cylindrical couette flow. Phys Rev Lett 99:064502CrossRefGoogle Scholar
- Messio L, Morize C, Rabaud M, Moisy F (2008) Experimental observation using particle image velocimetry of inertial waves in a rotating fluid. Exp Fluids 44(4):519–528CrossRefGoogle Scholar
- Molemaker MJ, McWilliams JC, Yavneh I (2001) Instability and equilibration of centrifugally stable stratified Taylor-Couette flow. Phys Rev Lett 86:5270–5273CrossRefGoogle Scholar
- Peacock T, Weidman P (2005) The effect of rotation on conical wave beams in a stratified fluid. Exp Fluids 39(1):32–37CrossRefGoogle Scholar
- Riedinger X, Le Dizès S, Meunier P (2011) Radiative instability of the flow around a rotating cylinder in a stratified fluid. J Fluid Mech 672:130–146CrossRefGoogle Scholar
- Rossby C-G (1939) Relation between variations in the intensity of the zonal circulation of the atmosphere and the displacements of the semi-permanent centers of action. J Mar Res 2(1):38–55Google Scholar
- Shalybkov D, Rüdiger G (2005) Non-axisymmetric instability of density-stratified Taylor-Couette flow. J Phys 14:128–137CrossRefGoogle Scholar
- Sisan DR, Mujica N, Tillotson WA, Huang YM, Dorland W, Hassam AB, Antonsen TM, Lathrop DP (2004) Experimental observation and characterization of the magnetorotational instability. Phys Rev Lett 93:114502CrossRefGoogle Scholar
- Stefani F, Gerbeth G, Gundrum T, Hollerbach R, Priede J, Rüdiger G, Szklarski J (2009) Helical magnetorotational instability in a Taylor-Couette flow with strongly reduced Ekman pumping. Phys Rev E 80:066303CrossRefGoogle Scholar
- Tsurutani BT, Lakhina GS, Pickett JS, Guarnieri FL, Lin N, Goldstein BE (2005) Nonlinear Alfven waves, discontinuities, proton perpendicular acceleration, and magnetic holes/decreases in interplanetary space and the magnetosphere: intermediate shocks? Nonlinear Process Geophys 12:321–326CrossRefGoogle Scholar