
The Rayleigh–Taylor instability, or RT instability (after
Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh ( ; 12 November 1842 – 30 June 1919), was an English physicist who received the Nobel Prize in Physics in 1904 "for his investigations of the densities of the most important gases and for his discovery ...
and
G. I. Taylor
Sir Geoffrey Ingram Taylor Order of Merit, OM Royal Society of London, FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, who made contributions to fluid dynamics and wave theory.
Early life and education
Tayl ...
), is an
instability
In dynamical systems instability means that some of the outputs or internal states increase with time, without bounds. Not all systems that are not stable are unstable; systems can also be marginally stable or exhibit limit cycle behavior.
...
of an
interface
Interface or interfacing may refer to:
Academic journals
* ''Interface'' (journal), by the Electrochemical Society
* '' Interface, Journal of Applied Linguistics'', now merged with ''ITL International Journal of Applied Linguistics''
* '' Inter ...
between two
fluid
In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...
s of different
densities which occurs when the lighter fluid is pushing the heavier fluid.
[
Drazin (2002) pp. 50–51.] Examples include the behavior of water suspended above oil in the
gravity of Earth
The gravity of Earth, denoted by , is the net force, net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation).
It is a Eucl ...
,
[ ]mushroom cloud
A mushroom cloud is a distinctive mushroom-shaped flammagenitus cloud of debris, smoke, and usually condensed water vapour resulting from a large explosion. The effect is most commonly associated with a nuclear explosion, but any sufficiently e ...
s like those from volcanic eruption
A volcanic eruption occurs when material is expelled from a volcanic vent or fissure. Several types of volcanic eruptions have been distinguished by volcanologists. These are often named after famous volcanoes where that type of behavior h ...
s and atmospheric nuclear explosion
A nuclear explosion is an explosion that occurs as a result of the rapid release of energy from a high-speed nuclear reaction. The driving reaction may be nuclear fission or nuclear fusion or a multi-stage cascading combination of the two, th ...
s, supernova
A supernova (: supernovae or supernovas) is a powerful and luminous explosion of a star. A supernova occurs during the last stellar evolution, evolutionary stages of a massive star, or when a white dwarf is triggered into runaway nuclear fusion ...
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
Miscibility () is the property of two chemical substance, substances to mix in all mixing ratio, proportions (that is, to fully dissolution (chemistry), dissolve in each other at any concentration), forming a homogeneity and heterogeneity, homoge ...
fluid, the denser fluid on top of the less dense one and both subject to the Earth's gravity. The equilibrium
Equilibrium may refer to:
Film and television
* ''Equilibrium'' (film), a 2002 science fiction film
* '' The Story of Three Loves'', also known as ''Equilibrium'', a 1953 romantic anthology film
* "Equilibrium" (''seaQuest 2032'')
* ''Equilibr ...
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
In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
, 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.[ 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 denser fluid.][ This occurs deep underwater on the surface of an expanding bubble and in a nuclear explosion.
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 equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
s, 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
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
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.[
This process is evident not only in many terrestrial examples, from ]salt dome
A salt dome is a type of structural dome formed when salt (or other evaporite minerals) intrudes into overlying rocks in a process known as diapirism. Salt domes can have unique surface and subsurface structures, and they can be discovered us ...
s to weather inversions, but also in astrophysics
Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline, James Keeler, said, astrophysics "seeks to ascertain the ...
and electrohydrodynamics
Electrohydrodynamics (EHD), also known as electro-fluid-dynamics (EFD) or electrokinetics, is the study of the dynamics of electrically charged fluids. Electrohydrodynamics (EHD) is a joint domain of electrodynamics and fluid dynamics mainly foc ...
. For example, RT instability structure is evident in the Crab Nebula
The Crab Nebula (catalogue designations M1, NGC 1952, Taurus A) is a supernova remnant and pulsar wind nebula in the constellation of Taurus (constellation), Taurus. The common name comes from a drawing that somewhat resembled a crab with arm ...
, in which the expanding pulsar wind nebula
A pulsar wind nebula (PWN, plural PWNe), sometimes called a plerion (derived from the Greek "πλήρης", ''pleres'', meaning "full"), is a type of nebula sometimes found inside the shell of a supernova remnant (SNR), powered by winds generate ...
powered by the Crab pulsar
The Crab Pulsar (PSR B0531+21 or Baade's Star) is a relatively young neutron star. The star is the central star in the Crab Nebula, a remnant of the supernova SN 1054, which was widely observed on Earth in the year 1054. is sweeping up ejected material from the supernova
A supernova (: supernovae or supernovas) is a powerful and luminous explosion of a star. A supernova occurs during the last stellar evolution, evolutionary stages of a massive star, or when a white dwarf is triggered into runaway nuclear fusion ...
explosion 1000 years ago. The RT instability has also recently been discovered in the Sun's outer atmosphere, or solar corona
In astronomy, a corona (: coronas or coronae) is the outermost layer of a star's Stellar atmosphere, atmosphere. It is a hot but relatively luminosity, dim region of Plasma (physics), plasma populated by intermittent coronal structures such as so ...
, when a relatively dense solar prominence
In solar physics, a prominence, sometimes referred to as a filament, is a large Plasma (physics), plasma and magnetic field structure extending outward from the Sun's surface, often in a loop shape. Prominences are anchored to the Sun's surface ...
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
In fluid dynamics, the Plateau–Rayleigh instability, often just called the Rayleigh instability, explains why and how a falling stream of fluid breaks up into smaller packets with the same total volume but less surface area per droplet. It is ...
(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
A lava lamp is a decorative lamp that was invented in 1963 by British entrepreneur Edward Craven Walker, the founder of the lighting company Mathmos.
It consists of a bolus of a special coloured wax mixture inside a glass vessel, the remainde ...
, 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
The evolution of the RTI follows four main stages.[ 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. In the early portion of this stage, a sinusoidal initial perturbation retains its sinusoidal shape. However, after the end of this first stage, when non-linear effects begin to appear, one observes the beginnings of the formation of the ubiquitous mushroom-shaped spikes (fluid structures of heavy fluid growing into light fluid) and bubbles (fluid structures of light fluid growing into heavy fluid). The growth of the mushroom structures continues in the second stage and can be modeled using buoyancy drag models, resulting in a growth rate that is approximately constant in time. At this point, nonlinear terms in the equations of motion can no longer be ignored. The spikes and bubbles then begin to interact with one another in the third stage. Bubble merging takes place, where the nonlinear interaction of mode coupling acts to combine smaller spikes and bubbles to produce larger ones. Also, bubble competition takes places, where spikes and bubbles of smaller wavelength that have become saturated are enveloped by larger ones that have not yet saturated. 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.]
Linear stability analysis
The inviscid
Viscosity is a measure of a fluid's rate-dependent resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup h ...
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.[Drazin (2002) pp. 48–52.] Consider a base state in which there is an interface, located at that separates fluid media with different densities, for and for . The gravitational acceleration is described by the vector . The velocity field and pressure field in this equilibrium state, denoted with an overbar, are given by
:
where the reference location for the pressure is taken to be at . Let this interface be slightly perturbed, so that it assumes the position . Correspondingly, the base state is also slightly perturbed. In the linear theory, we can write
:
where is the real wavenumber in the -direction and is the growth rate of the perturbation. Then the linear stability analysis based on the inviscid governing equations shows that
:
Thus, if , the base state is stable and while if , it is unstable for all wavenumbers. If the interface has a surface tension
Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension (physics), tension is what allows objects with a higher density than water such as razor blades and insects (e.g. Ge ...
, then the dispersion relation becomes
:
which indicates that the instability occurs only for a range of wavenumbers