# Rayleigh–Taylor instability

RT fingers evident in the Crab Nebula

The Rayleigh–Taylor instability, or RT instability (after Lord Rayleigh and G. I. Taylor), is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid.[1][2] Examples include supernova explosions in which expanding core gas is accelerated into denser shell gas,[3][4] instabilities in plasma fusion reactors,[5] and the common terrestrial example of a denser fluid such as water suspended above a lighter fluid such as oil in the Earth's gravitational field.[2]

To model the last example, consider two completely plane-parallel layers of immiscible fluid, the more dense on top of the less dense one and both subject to the Earth's gravity. The equilibrium here is unstable to any perturbations or disturbances of the interface: if a parcel of heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards, the potential energy of the configuration is lower than the initial state. Thus the disturbance will grow and lead to a further release of potential energy, as the more dense material moves down under the (effective) gravitational field, and the less dense material is further displaced upwards. This was the set-up as studied by Lord Rayleigh.[2] The important insight by G. I. Taylor was his realisation that this situation is equivalent to the situation when the fluids are accelerated, with the less dense fluid accelerating into the more dense fluid.[2] This occurs deep underwater on the surface of an expanding bubble and in a nuclear explosion.[6]

As the RT instability develops, the initial perturbations progress from a linear growth phase into a non-linear or "exponential" growth phase, eventually developing "plumes" flowing upwards (in the gravitational buoyancy sense) and "spikes" falling downwards. In general, the density disparity between the fluids determines the structure of the subsequent non-linear RT instability flows (assuming other variables such as surface tension and viscosity are negligible here). The difference in the fluid densities divided by their sum is defined as the Atwood number, A. For A close to 0, RT instability flows take the form of symmetric "fingers" of fluid; for A close to 1, the much lighter fluid "below" the heavier fluid takes the form of larger bubble-like plumes.[1]

This process is evident not only in many terrestrial examples, from salt domes to weather inversions, but also in astrophysics and electrohydrodynamics. RT instability structure is also evident in the Crab Nebula, in which the expanding pulsar wind nebula powered by the Crab pulsar is sweeping up ejected material from the supernova explosion 1000 years ago.[7] The RT instability has also recently been discovered in the Sun's outer atmosphere, or solar corona, when a relatively dense solar prominence overlies a less dense plasma bubble.[8] This latter case is an exceptionally clear example of the magnetically modulated RT instability.[9][10]

Note that the RT instability is not to be confused with the Plateau-Rayleigh instability (also known as Rayleigh instability) of a liquid jet. This instability, sometimes called the hosepipe (or firehose) instability, occurs due to surface tension, which acts to break a cylindrical jet into a stream of droplets having the same volume but lower surface area.

Many people have witnessed the RT instability by looking at a lava lamp, although some might claim this is more accurately described as an example of Rayleigh–Bénard convection due to the active heating of the fluid layer at the bottom of the lamp.

## Linear stability analysis

Base state of the Rayleigh–Taylor instability. Gravity points downwards.

The inviscid two-dimensional Rayleigh–Taylor (RT) instability provides an excellent springboard into the mathematical study of stability because of the exceptionally simple nature of the base state.[11] This is the equilibrium state that exists before any perturbation is added to the system, and is described by the mean velocity field $U(x,z)=W(x,z)=0,\,$ where the gravitational field is $\textbf{g}=-g\hat{\textbf{z}}.\,$ An interface at $z=0\,$ separates the fluids of densities $\rho_G\,$ in the upper region, and $\rho_L\,$ in the lower region. In this section it is shown that when the heavy fluid sits on top, the growth of a small perturbation at the interface is exponential, and takes place at the rate[2]

$\exp(\gamma\,t)\;, \qquad\text{with}\quad \gamma={\sqrt{\mathcal{A}g\alpha}} \quad\text{and}\quad \mathcal{A}=\frac{\rho_{\text{heavy}}-\rho_{\text{light}}}{\rho_{\text{heavy}}+\rho_{\text{light}}},\,$

where $\gamma\,$ is the temporal growth rate, $\alpha\,$ is the spatial wavenumber and $\mathcal{A}\,$ is the Atwood number.

Hydrodynamics simulation of a single "finger" of the Rayleigh–Taylor instability[12] Note the formation of Kelvin–Helmholtz instabilities, in the second and later snapshots shown (starting initially around the level $y=0$), as well as the formation of a "mushroom cap" at a later stage in the third and fourth frame in the sequence.

The time evolution of the free interface elevation $z = \eta(x,t),\,$ initially at $\eta(x,0)=\Re\left\{B\,\exp\left(i\alpha x\right)\right\},\,$ is given by:

$\eta=\Re\left\{B\,\exp\left(\sqrt{\mathcal{A}g\alpha}\,t\right)\exp\left(i\alpha x\right)\right\}\,$

which grows exponentially in time. Here B is the amplitude of the initial perturbation, and $\Re\left\{\cdot\right\}\,$ denotes the real part of the complex valued expression between brackets.

In general, the condition for linear instability is that the imaginary part of the "wave speed" c be positive. Finally, restoring the surface tension makes c2 less negative and is therefore stabilizing. Indeed, there is a range of short waves for which the surface tension stabilizes the system and prevents the instability forming.

## Late-time behaviour

The analysis of the previous section breaks down when the amplitude of the perturbation is large. The growth then becomes non-linear as the spikes and bubbles of the instability tangle and roll up into vortices. Then, as in the figure, numerical simulation of the full problem is required to describe the system.

## Notes

1. ^ a b Sharp, D.H. (1984). "An Overview of Rayleigh-Taylor Instability". Physica D 12: 3–18. Bibcode:1984PhyD...12....3S. doi:10.1016/0167-2789(84)90510-4.
2. Drazin (2002) pp. 50–51.
3. ^ Wang, C.-Y. & Chevalier R. A. (2000). "Instabilities and Clumping in Type Ia Supernova Remnants". arXiv:astro-ph/0005105v1.
4. ^ Hillebrandt, W.; Höflich, P. (1992). "Supernova 1987a in the Large Magellanic Cloud". In R. J. Tayler. Stellar Astrophysics. CRC Press. pp. 249–302. ISBN 0-7503-0200-3.. See page 274.
5. ^ Chen, H. B.; Hilko, B.; Panarella, E. (1994). "The Rayleigh–Taylor instability in the spherical pinch". Journal of Fusion Energy 13 (4): 275–280. Bibcode:1994JFuE...13..275C. doi:10.1007/BF02215847.
6. ^ John Pritchett (1971). "EVALUATION OF VARIOUS THEORETICAL MODELS FOR UNDERWATER EXPLOSION". U.S. Government. p. 86. Retrieved October 9, 2012.
7. ^ Hester, J. Jeff (2008). "The Crab Nebula: an Astrophysical Chimera". Annual Review of Astronomy and Astrophysics 46: 127–155. Bibcode:2008ARA&A..46..127H. doi:10.1146/annurev.astro.45.051806.110608.
8. ^ Berger, T. E. et al.; Slater, Gregory; Hurlburt, Neal; Shine, Richard; Tarbell, Theodore; Title, Alan; Lites, Bruce W.; Okamoto, Takenori J. et al. (2010). "Quiescent Prominence Dynamics Observed with the Hinode Solar Optical Telescope. I. Turbulent Upflow Plumes". The Astrophysical Journal 716 (2): 1288–1307. Bibcode:2010ApJ...716.1288B. doi:10.1088/0004-637X/716/2/1288.
9. ^ a b Chandrasekhar, S. (1981). Hydrodynamic and Hydromagnetic Instabilties. Dover. ISBN 0-486-64071-X.. See Chap. X.
10. ^ Hillier, A. et al.; Berger, Thomas; Isobe, Hiroaki; Shibata, Kazunari. "Numerical Simulations of the Magnetic Rayleigh-Taylor Instability in the Kippenhahn-Schl{\"u}ter Prominence Model. I. Formation of Upflows". The Astrophysical Journal 716: 120–133. Bibcode:2012ApJ...746..120H. doi:10.1088/0004-637X/746/2/120.
11. ^ a b Drazin (2002) pp. 48–52.
12. ^ Li, Shengtai and Hui Li. "Parallel AMR Code for Compressible MHD or HD Equations". Los Alamos National Laboratory. Retrieved 2006-09-05.