Rayleigh–Taylor instability

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, mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion.The perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude, ( u ′ ( x , z , t ) , w ′ ( x , z , t ) ) . {displaystyle (u'(x,z,t),w'(x,z,t)).,} Because the fluid is assumed incompressible, this velocity field has the streamfunction representation 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, mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion. 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 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. 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. 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, equations can be linearized and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, perturbation amplitude is too large for the non-linear terms to be neglected. 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 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. 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 is a clear example of the magnetically modulated RT instability. 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. 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. 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 , {displaystyle U(x,z)=W(x,z)=0,,} where the gravitational field is g = − g z ^ . {displaystyle { extbf {g}}=-g{hat { extbf {z}}}.,} An interface at z = 0 {displaystyle z=0,} separates the fluids of densities ρ G {displaystyle ho _{G},} in the upper region, and ρ L {displaystyle ho _{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 where γ {displaystyle gamma ,} is the temporal growth rate, α {displaystyle alpha ,} is the spatial wavenumber and A {displaystyle {mathcal {A}},} is the Atwood number. The time evolution of the free interface elevation z = η ( x , t ) , {displaystyle z=eta (x,t),,} initially at η ( x , 0 ) = ℜ { B exp ⁡ ( i α x ) } , {displaystyle eta (x,0)=Re left{B,exp left(ialpha x ight) ight},,} is given by: