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.^[1] ^[2] This is the case with an interstellar cloud and shock system. The equivalent situation occurs when gravity is acting on two fluids of different density – with the dense fluid above a fluid of lesser density – such as water balancing on light oil.^[2]

Consider two completely plane-parallel layers of immiscible fluid, the heavier on top of the light one and both subject to the Earth's gravity. The equilibrium here is unstable to certain perturbations or disturbances. An unstable disturbance will grow and lead to a release of potential energy, as the heavier material moves down under the (effective) gravitational field, and the lighter material is displaced upwards. This was the set-up as studied by Lord Rayleigh.^[2] The important insight by G. I. Taylor was, that he realised this situation is equivalent to the situation when the fluids are accelerated (without gravity), with the lighter fluid accelerating into the heavier fluid.^[2] This can be experienced, for example, by accelerating a glass of water downward faster than the Earth's gravitational acceleration.^[2]

As the instability develops, downward-moving irregularities ('dimples') are quickly magnified into sets of inter-penetrating Rayleigh–Taylor fingers. Therefore the Rayleigh–Taylor instability is sometimes qualified to be a fingering instability.^[3] The upward-moving, lighter material is shaped like mushroom caps.^[4]^[5]

This process is evident not only in many terrestrial examples, from salt domes to weather inversions, but also in astrophysics and electrohydrodynamics. RT fingers are especially obvious 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.^[6]

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.

1 Linear stability analysis
2 Late-time behaviour
3 See also
4 Notes
5 References
- 5.1 Original research papers
- 5.2 Other
6 External links

Linear stability analysis

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.^[7] 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 $\textbf{g}=-g\hat{\textbf{z}}.\,$ 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^[2]

$\exp(\gamma\,t)\;, \qquad\text{with}\quad \gamma={\sqrt{\mathcal{A}g\alpha}} \quad\text{and}\quad \mathcal{A}=\frac{\rho_{\text{heavy}}-\rho_{\text{light}}}{\rho_{\text{heavy}}%2B\rho_{\text{light}}},\,$

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

Details of the linear stability analysis^[8]^[7]

The perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude, $(u'(x,z,t),w'(x,z,t)).\,$ Because the fluid is assumed incompressible, this velocity field has the streamfunction representation

$\textbf{u}'=(u'(x,z,t),w'(x,z,t))=(\psi_z,-\psi_x),\,$

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

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

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

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

The domain of the problem is the following: the fluid with label 'L' lives in the region $-\infty<z\leq 0\,$ , while the fluid with the label 'G' lives in the upper half-plane $0\leq z<\infty\,$ . 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 $w'_i\,$ should satisfy a no-flux condition, so that fluid does not leak out at the boundaries $z=\pm\infty.\,$ Thus, $w_L'=0\,$ on $z=-\infty\,$ , and $w_G'=0\,$ on $z=\infty\,$ . In terms of the streamfunction, this is

$\Psi_L\left(-\infty\right)=0,\qquad \Psi_G\left(\infty\right)=0.\,$

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

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

$\Psi_L\left(\eta\right)=\Psi_G\left(\eta\right).\,$

Expanding about $z=0\,$ gives

$\Psi_L\left(0\right)=\Psi_G\left(0\right)%2B\text{H.O.T.},\,$

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

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

$\frac{\partial\eta}{\partial t}%2Bu'\frac{\partial\eta}{\partial x}=w'\left(\eta\right).\,$

Linearizing, this is simply

$\frac{\partial\eta}{\partial t}=w'\left(0\right),\,$

where the velocity $w'\left(\eta\right)\,$ 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:

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

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

$\kappa=\nabla^2\eta=\eta_{xx}.\,$

Thus,

$p_G\left(z=\eta\right)-p_L\left(z=\eta\right)=\sigma\eta_{xx}.\,$

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

$\left[P_G\left(\eta\right)%2Bp'_G\left(0\right)\right]-\left[P_L\left(\eta\right)%2Bp'_L\left(0\right)\right]=\sigma\eta_{xx}.\,$

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

$P_L=-\rho_L g z%2Bp_0,\qquad P_G=-\rho_G gz %2Bp_0,\,$

this becomes

$p'_G-p'_L=g\eta\left(\rho_G-\rho_L\right)%2B\sigma\eta_{xx},\qquad\text{on }z=0.\,$

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

$\frac{\partial u_i'}{\partial t} = - \frac{1}{\rho_i}\frac{\partial p_i'}{\partial x}\,$ with $i=L,G,\,$

to yield

$p_i'=\rho_i c D\Psi_i,\qquad i=L,G.\,$

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

$c\left(\rho_G D\Psi_G-\rho_L D\Psi_L\right)=g\eta\left(\rho_G-\rho_L\right)%2B\sigma\eta_{xx}.\,$

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

$c^2\left(\rho_G D\Psi_G-\rho_L D\Psi_L\right)=g\Psi\left(\rho_G-\rho_L\right)-\sigma\alpha^2\Psi,\,$

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 $\left(D^2-\alpha^2\right)\Psi_i=0,\,$ with the boundary conditions $\Psi\left(\pm\infty\right)\,$ has the solution

$\Psi_L=A_L e^{\alpha z},\qquad \Psi_G = A_G e^{-\alpha z}.\,$

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

$c^2\left(\rho_G D\Psi_G-\rho_L D\Psi_L\right)=g\Psi\left(\rho_G-\rho_L\right)-\sigma\alpha^2\Psi.\,$

Plugging the solution into this equation gives the relation

$Ac^2\alpha\left(-\rho_G-\rho_L\right)=Ag\left(\rho_G-\rho_L\right)-\sigma\alpha^2A.\,$

The A cancels from both sides and we are left with

$c^2=\frac{g}{\alpha}\frac{\rho_L-\rho_G}{\rho_L%2B\rho_G}%2B\frac{\sigma\alpha}{\rho_L%2B\rho_G}.\,$

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

$c^2=\frac{g}{\alpha}\frac{\rho_L-\rho_G}{\rho_L%2B\rho_G},\qquad \sigma=0,\,$

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

$c=\pm i \sqrt{\frac{g\mathcal{A}}{\alpha}},\qquad \mathcal{A}=\frac{\rho_G-\rho_L}{\rho_G%2B\rho_L},\,$

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

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

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

The time evolution of the free interface elevation $z = \eta(x,t),\,$ initially at $\eta(x,0)=\Re\left\{B\,\exp\left(i\alpha x\right)\right\},\,$ is given by:

$\eta=\Re\left\{B\,\exp\left(\sqrt{\mathcal{A}g\alpha}\,t\right)\exp\left(i\alpha x\right)\right\}\,$

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

Notes

^ Sharp, D.H. (1984). "An Overview of Rayleigh-Taylor Instability". Physica D 12: 3–18. Bibcode 1984PhyD...12....3S. doi:10.1016/0167-2789(84)90510-4
^ ^a ^b ^c ^d ^e ^f Drazin (2002) pp. 50–51.
^ Chen, H. B.; Hilko, B.; Panarella, E. (1994). "The Rayleigh–Taylor instability in the spherical pinch". Journal of Fusion Energy 13 (4): 275–280. doi:10.1007/BF02215847
^ Wang, C.-Y. & Chevalier R. A. (2000). "Instabilities and Clumping in Type Ia Supernova Remnants". arXiv:astro-ph/0005105v1.
^ Hillebrandt, W.; Höflich, P. (1992). "Supernova 1987a in the Large Magellanic Cloud". In R. J. Tayler. Stellar Astrophysics. CRC Press. pp. 249–302. ISBN 0750302003 . See page 274.
^ Hester, J. Jeff (2008). "The Crab Nebula: an Astrophysical Chimera". Annual Review of Astronomy and Astrophysics 46: 127–155. Bibcode 2008ARA&A..46..127H. doi:10.1146/annurev.astro.45.051806.110608
^ ^a ^b Drazin (2002) pp. 48–52.
^ A similar derivation appears in Chandrasekhar (1981), §92, pp. 433–435.
^ Li, Shengtai and Hui Li. "Parallel AMR Code for Compressible MHD or HD Equations". Los Alamos National Laboratory. http://math.lanl.gov/Research/Highlights/amrmhd.shtml. Retrieved 2006-09-05.

References

Original research papers

Rayleigh, Lord (John William Strutt) (1883). "Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density". Proceedings of the London Mathematical Society 14: 170–177. doi:10.1112/plms/s1-14.1.170. (Original paper is available at: https://www.irphe.univ-mrs.fr/~clanet/otherpaperfile/articles/Rayleigh/rayleigh1883.pdf .)
Taylor, Sir Geoffrey Ingram (1950). "The instability of liquid surfaces when accelerated in a direction perpendicular to their planes". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 201 (1065): 192–196. Bibcode 1950RSPSA.201..192T. doi:10.1098/rspa.1950.0052.

Other

Chandrasekhar, Subrahmanyan (1981). Hydrodynamic and Hydromagnetic Stability. Dover Publications. ISBN 978-0486640716.
Drazin, P. G. (2002). Introduction to hydrodynamic stability. Cambridge University Press. ISBN 0 521 00965 0. xvii+238 pages.
Drazin, P. G.; Reid, W. H. (2004). Hydrodynamic stability (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-52541-1. 626 pages.