Rayleigh–Taylor instability

[2][3][4] Examples include the behavior of water suspended above oil in the gravity of Earth,[3] mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions,[5] supernova explosions in which expanding core gas is accelerated into denser shell gas,[6][7] instabilities in plasma fusion reactors and[8] inertial confinement fusion.

[9] 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.

[3] This occurs deep underwater on the surface of an expanding bubble and in a nuclear explosion.

[10] 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 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.

[2] This process is evident not only in many terrestrial examples, from salt domes to weather inversions, but also in astrophysics and electrohydrodynamics.

For example, RT instability structure is 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.

[11] 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 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.

[2] In the first stage, the perturbation amplitudes are small when compared to their wavelengths, the equations of motion can be linearized, resulting in exponential instability growth.

This eventually develops into a region of turbulent mixing, which is the fourth and final stage in the evolution.

It is generally assumed that the mixing region that finally develops is self-similar and turbulent, provided that the Reynolds number is sufficiently large.

[16] 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.

Then the linear stability analysis based on the inviscid governing equations shows that Thus, if

; that is to say, surface tension stabilises large wavenumbers or small length scales.

and its value is The perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude,

where σ is the surface tension and κ is the curvature of the interface, which in a linear approximation is

To understand the implications of this result in full, it is helpful to consider the case of zero surface tension.

It was recently discovered that the fluid equations governing the linear dynamics of the system admit a parity-time symmetry, and the Kelvin–Helmholtz–Rayleigh–Taylor instability occurs when and only when the parity-time symmetry breaks spontaneously.

[18] The RT instability can be seen as the result of baroclinic torque created by the misalignment of the pressure and density gradients at the perturbed interface, as described by the two-dimensional inviscid vorticity equation,

When in the unstable configuration, for a particular harmonic component of the initial perturbation, the torque on the interface creates vorticity that will tend to increase the misalignment of the gradient vectors.

This concept is depicted in the figure, where it is observed that the two counter-rotating vortices have velocity fields that sum at the peak and trough of the perturbed interface.

In the stable configuration, the vorticity, and thus the induced velocity field, will be in a direction that decreases the misalignment and therefore stabilizes the system.

[16][19] A much simpler explanation of the basic physics of the Rayleigh-Taylor instability was published in 2006.

[20] The analysis in 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.

Hydrodynamics simulation of a single "finger" of the Rayleigh–Taylor instability. [ 1 ] Note the formation of Kelvin–Helmholtz instabilities , in the second and later snapshots shown (starting initially around the level ), as well as the formation of a "mushroom cap" at a later stage in the third and fourth frame in the sequence.
RT instability fingers evident in the Crab Nebula
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.
Base state of the Rayleigh–Taylor instability. Gravity points downwards.
Visualization of an unstable Rayleigh–Taylor instability configuration where baroclinic torque at the interface creates vorticity and induces a velocity field that increases the baroclinic torque. Here ω is vorticity, p is pressure, ρ is density, u is velocity and g is gravity. The thick circular arrows represent the velocity field created by the vortex.