thumb|right|400px|[[Hydrodynamics simulation of a single "finger" of the Rayleigh–Taylor instability. Note the formation of Kelvin–Helmholtz instabilities, in the second and later snapshots shown (starting initially around the level <math>y=0</math>), as well as the formation of a "mushroom cap" at a later stage in the third and fourth frame in the sequence.]]
thumb|right|300px|RT instability 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. Examples include the behavior of water suspended above oil in the gravity of Earth, supernova explosions in which expanding core gas is accelerated into denser shell gas, merging binary quantum fluids in metastable configuration, instabilities in plasma fusion reactors and inertial confinement fusion.
Concept
Water suspended atop oil is an everyday example of Rayleigh–Taylor instability, and it may be modeled by two completely plane-parallel layers of immiscible fluid, the denser fluid 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 denser 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.
As the RT instability develops, the initial perturbations progress from a linear growth phase into a non-linear growth phase, eventually developing "plumes" flowing upwards (in the gravitational buoyancy sense) and "spikes" falling downwards. In the linear phase, the fluid movement can be closely approximated by linear equations, and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, perturbation amplitude is too large for a linear approximation, and non-linear equations are required to describe fluid motions. 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. 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. This latter case resembles magnetically modulated RT instabilities.
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 total volume but higher 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.
Stages of development and eventual evolution into turbulent mixing
thumb|left|400px|This figure represents the evolution of the Rayleigh–Taylor instability from small wavelength perturbations at the interface (a) which grow into the ubiquitous mushroom shaped spikes (fluid structures of heavy into light fluid) and bubbles (fluid structures of light into heavy fluid) (b) and these fluid structures interact due to bubble merging and competition (c) eventually developing into a mixing region (d). Here ρ2 represents the heavy fluid and ρ1 represents the light fluid. Gravity is acting downward and the system is RT unstable.
The evolution of the RTI follows four main stages.
Linear stability analysis
thumb|right|400px|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 simple nature of the base state. Consider a base state in which there is an interface, located at <math>z=0</math> that separates fluid media with different densities, <math>\rho_1</math> for <math>z<0</math> and <math>\rho_2</math> for <math>z>0</math>. The gravitational acceleration is described by the vector <math>\mathbf g = -g\, \mathbf{e}_z</math>. The velocity field and pressure field in this equilibrium state, denoted with an overbar, are given by
<math display="block">\overline{\mathbf v}=\mathbf{0}, \quad \overline p = \begin{cases}-\rho_1 g z \quad \text{for }z<0,\\ -\rho_2 g z \quad \text{for }z>0,\end{cases}</math>
where the reference location for the pressure is taken to be at <math>z = 0</math>. Let this interface be slightly perturbed, so that it assumes the position <math>z = f(x,t)</math>. Correspondingly, the base state is also slightly perturbed. In the linear theory, we can write
<math display="block">\mathbf{v} = \overline{\mathbf v} + \hat{\mathbf{v(z) e^{ikx+\sigma t}, \quad p = \overline{p} + \hat p(z) e^{ikx+\sigma t}, \quad f =\hat{f} e^{ikx+\sigma t}</math>
where <math>k</math> is the real wavenumber in the <math>x</math>-direction and <math>\sigma</math> is the growth rate of the perturbation. Then the linear stability analysis based on the inviscid governing equations shows that
<math display="block">\sigma^2 = \frac{\rho_2-\rho_1}{\rho_2+\rho_1} gk.</math>
Thus, if <math>\rho_2<\rho_1</math>, the base state is stable and while if <math>\rho_2>\rho_1</math>, it is unstable for all wavenumbers. If the interface has a surface tension <math>\gamma</math>, then the dispersion relation becomes
<math display="block">\sigma^2 = \frac{\rho_2-\rho_1}{\rho_2+\rho_1}gk - \frac{\gamma k^3}{\rho_2+\rho_1},</math>
which indicates that the instability occurs only for a range of wavenumbers <math>0<k<k_c</math> where <math>k_c^2 = (\rho_2-\rho_1)g/\gamma</math>; that is to say, surface tension stabilises large wavenumbers or small length scales. Then the maximum growth rate occurs at the wavenumber <math>k_m = k_c/\sqrt 3</math> and its value is
<math display="block">\sigma_m^2 = \frac{2\gamma}{\rho_2-\rho_1} \left[\frac{\left(\rho_2-\rho_1\right) g}{3\gamma}\right]^{3/2}.</math>