Delta-sigma modulation

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 that occurs when one of the fluids is accelerated into the other. Examples include supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors, 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.

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. The important insight by G. I. Taylor was, that he realised 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.

Instability will occur even though an idealized, perfectly static interface between the fluid layers would remain static forever. For example, the Earth's atmospheric pressure could easily support a layer of water spread evenly across the flat ceiling of an air filled room. However, small perturbations of the interface between the water and the air always set in. These small perturbations grow due to the energetically favorable displacements described above. It is the RT instability that causes the layer of ceiling water to inevitably end up on the floor.

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.

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 an exceptionally 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 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.

Demonstrating the instability in the kitchen

The Rayleigh–Taylor instability can be demonstrated using common household items. The experiment consists of adding three tablespoons of molasses to a large, heat-resistant glass, and filling up with milk. The glass has to be transparent so that the instability can be seen. The glass is then put in a microwave oven, and heated on maximum power until the instability occurs. This happens before the milk boils. When the molasses heats sufficiently, it becomes less dense than the milk above it, and the instability occurs. Shortly after this, the molasses and milk mix together, forming a light brown liquid.

Linear stability analysis

File:Rti base.png

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. 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 $\mathbf{\text{𝐠}}=-g\widehat{\mathbf{\text{𝐳}}}.$ 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

$\exp (\gamma t),\text{with}\gamma =\sqrt{𝒜g\alpha }\text{and}𝒜=\frac{{\rho }_{\text{heavy}}-{\rho }_{\text{light}}}{{\rho }_{\text{heavy}}+{\rho }_{\text{light}}},$

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

Template:Hidden begin The perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude, $(u^{\prime }(x,z,t),w^{\prime }(x,z,t)).$ Because the fluid is assumed incompressible, this velocity field has the streamfunction representation

${\mathbf{\text{𝐮}}}^{\prime }=(u^{\prime }(x,z,t),w^{\prime }(x,z,t))=({\psi }_z,-{\psi }_x),$

where the subscripts indicate partial derivatives. Moreover, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays irrotational, hence $\mathrm{\nabla }\times {\mathbf{\text{𝐮}}}^{\prime }=0$. In the streamfunction representation, ${\mathrm{\nabla }}^2\psi =0.$ Next, because of the translational invariance of the system in the x-direction, it is possible to make the ansatz

$\psi (x,z,t)=e^{i\alpha (x-ct)}\Psi \left(z\right),$

where $\alpha$ is a spatial wavenumber. Thus, the problem reduces to solving the equation

$(D^2-{\alpha }^2){\Psi }_j=0,D=\frac{d}{dz},j=L,G.$

The domain of the problem is the following: the fluid with label 'L' lives in the region $-\mathrm{\infty }<z\le 0$, while the fluid with the label 'G' lives in the upper half-plane $0\le z<\mathrm{\infty }$. To specify the solution fully, it is necessary to fix conditions at the boundaries and interface. This determines the wave speed c, which in turn determines the stability properties of the system.

The first of these conditions is provided by details at the boundary. The perturbation velocities $w{\text{'}}_i$ should satisfy a no-flux condition, so that fluid does not leak out at the boundaries $z=\pm \mathrm{\infty }.$ Thus, ${w_L}^{\prime }=0$ on $z=-\mathrm{\infty }$, and ${w_G}^{\prime }=0$ on $z=\mathrm{\infty }$. In terms of the streamfunction, this is

${\Psi }_L(-\mathrm{\infty })=0,{\Psi }_G\left(\mathrm{\infty }\right)=0.$

The other three conditions are provided by details at the interface $z=\eta (x,t)$.

Continuity of vertical velocity: At $z=\eta$, the vertical velocities match, $w{\text{'}}_L=w{\text{'}}_G$. Using the streamfunction representation, this gives

${\Psi }_L\left(\eta \right)={\Psi }_G\left(\eta \right).$

Expanding about $z=0$ gives

${\Psi }_L\left(0\right)={\Psi }_G\left(0\right)+\text{H.O.T.},$

where H.O.T. means 'higher-order terms'. This equation is the required interfacial condition.

The free-surface condition: At the free surface $z=\eta (x,t)$, the kinematic condition holds:

$\frac{\partial \eta }{\partial t}+u^{\prime }\frac{\partial \eta }{\partial x}=w^{\prime }\left(\eta \right).$

Linearizing, this is simply

$\frac{\partial \eta }{\partial t}=w^{\prime }\left(0\right),$

where the velocity $w^{\prime }\left(\eta \right)$ is linearized on to the surface $z=0$. Using the normal-mode and streamfunction representations, this condition is $c\eta =\Psi$, the second interfacial condition.

Pressure relation across the interface: For the case with surface tension, the pressure difference over the interface at $z=\eta$ is given by the Young–Laplace equation:

$p_G(z=\eta )-p_L(z=\eta )=\sigma \kappa ,$

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

$\kappa ={\mathrm{\nabla }}^2\eta ={\eta }_{xx}.$

Thus,

$p_G(z=\eta )-p_L(z=\eta )=\sigma {\eta }_{xx}.$

However, this condition refers to the total pressure (base+perturbed), thus

$[P_G\left(\eta \right)+p{\text{'}}_G\left(0\right)]-[P_L\left(\eta \right)+p{\text{'}}_L\left(0\right)]=\sigma {\eta }_{xx}.$

(As usual, The perturbed quantities can be linearized onto the surface z=0.) Using hydrostatic balance, in the form

$P_L=-{\rho }_{L}gz+p_0,P_G=-{\rho }_{G}gz+p_0,$

this becomes

$p{\text{'}}_G-p{\text{'}}_L=g\eta ({\rho }_G-{\rho }_L)+\sigma {\eta }_{xx},\text{on }z=0.$

The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised Euler equations for the perturbations,

$\frac{\partial {u_i}^{\prime }}{\partial t}=-\frac{1}{{\rho }_i}\frac{\partial {p_i}^{\prime }}{\partial x}$ Template:Pad with $i=L,G,$

to yield

${p_i}^{\prime }={\rho }_{i}cD{\Psi }_i,i=L,G.$

Putting this last equation and the jump condition on $p{\text{'}}_G-p{\text{'}}_L$ together,

$c({\rho }_{G}D{\Psi }_G-{\rho }_{L}D{\Psi }_L)=g\eta ({\rho }_G-{\rho }_L)+\sigma {\eta }_{xx}.$

Substituting the second interfacial condition $c\eta =\Psi$ and using the normal-mode representation, this relation becomes

$c^2({\rho }_{G}D{\Psi }_G-{\rho }_{L}D{\Psi }_L)=g\Psi ({\rho }_G-{\rho }_L)-\sigma {\alpha }^2\Psi ,$

where there is no need to label $\Psi$ (only its derivatives) because ${\Psi }_L={\Psi }_G$ at $z=0.$

Solution

Now that the model of stratified flow has been set up, the solution is at hand. The streamfunction equation $(D^2-{\alpha }^2){\Psi }_i=0,$ with the boundary conditions $\Psi (\pm \mathrm{\infty })$ has the solution

${\Psi }_L=A_L{e}^{\alpha z},{\Psi }_G=A_G{e}^{-\alpha z}.$

The first interfacial condition states that ${\Psi }_L={\Psi }_G$ at $z=0$, which forces $A_L=A_G=A.$ The third interfacial condition states that

$c^2({\rho }_{G}D{\Psi }_G-{\rho }_{L}D{\Psi }_L)=g\Psi ({\rho }_G-{\rho }_L)-\sigma {\alpha }^2\Psi .$

Plugging the solution into this equation gives the relation

$A{c}^2\alpha (-{\rho }_G-{\rho }_L)=Ag({\rho }_G-{\rho }_L)-\sigma {\alpha }^{2}A.$

The A cancels from both sides and we are left with

$c^2=\frac{g}{\alpha }\frac{{\rho }_L-{\rho }_G}{{\rho }_L+{\rho }_G}+\frac{\sigma \alpha }{{\rho }_L+{\rho }_G}.$

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

$c^2=\frac{g}{\alpha }\frac{{\rho }_L-{\rho }_G}{{\rho }_L+{\rho }_G},\sigma =0,$

and clearly

• If ${\rho }_G<{\rho }_L$, $c^2>0$ and c is real. This happens when the

lighter fluid sits on top;

• If ${\rho }_G>{\rho }_L$, $c^2<0$ and c is purely imaginary. This happens

when the heavier fluid sits on top.

Now, when the heavier fluid sits on top, $c^2<0$, and

$c=\pm i\sqrt{\frac{g𝒜}{\alpha }},𝒜=\frac{{\rho }_G-{\rho }_L}{{\rho }_G+{\rho }_L},$

where $𝒜$ is the Atwood number. By taking the positive solution, we see that the solution has the form

$\Psi (x,z,t)=A{e}^{-\alpha |z|}\exp \left[i\alpha (x-ct)\right]=A\exp \left(\alpha \sqrt{\frac{g\widetilde{𝒜}}{\alpha }}t\right)\exp (i\alpha x-\alpha |z|)$

and this is associated to the interface position η by: $c\eta =\Psi .$ Now define $B=A/c.$ Template:Hidden end

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 $y=0$), 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 $\eta (x,0)=\mathrm{\Re }\{B\exp \left(i\alpha x\right)\},$ is given by:

$\eta =\mathrm{\Re }\{B\exp \left(\sqrt{𝒜g\alpha }t\right)\exp \left(i\alpha x\right)\}$

which grows exponentially in time. Here B is the amplitude of the initial perturbation, and $\mathrm{\Re }\{\cdot \}$ 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.

See also

• Richtmyer–Meshkov instability
• Kelvin–Helmholtz instability
• Mushroom cloud
• Plateau–Rayleigh instability
• Salt fingering
• Hydrodynamic stability
• Kármán vortex street
• Fluid thread breakup

Notes

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References

Original research papers

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Other

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 xvii+238 pages.
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 626 pages.

External links

• Java demonstration of the RT instability in fluids
• Actual images and videos of RT fingers
• Experiments on Rayleigh-Taylor experiments at the University of Arizona
• plasma Rayleigh-Taylor instability experiment at California Institute of Technology