March 18, 2011

Fig. 1: Simple planar RT instability under
gravitational acceleration. The heavy fluid is supported by
the lighter fluid. The interface between the two will
become unstable under any disturbance. |

The quest for sustained, controlled fusion on earth has faced an enormous amount of engineering and physics related problems problems. The fundamental reactions that occurs in fusion involves two isotopes of hydrogen binding together to form a helium atom, a nuetron and 17.6 MeV of energy. The reaction is simple enough and occurs spontaneously in nature in stars and our sun. In these celestial bodies, the reaction feeds on itself, providing enough energy to spawn more fusion reactions. If the sun weren't contained by its enormous gravitational field and if it were allowed to expand, the energy carrying neutrons would escape and fail to dump energy back into the fusion "power plant." The sun's fusion would come to a halt. The conditions of fusion are necessarily so extreme, that it becomes impossible to contain the core plasma by conventional means. [1]

The United States has had a long sustained effort for attempting to contain a fusion reaction by means of inertia that dates back to 1960, when the laser was first invented. The idea is to implode fusion fuel so quickly that when the fusion begins, the particles are contained from blowing apart the fuel capsule because of the inward inertia of the previous implosion. The implosion counteracts the explosion. Eventually the fuel will blow itself apart but not before fusion has been sustained long enough to yield as much power as was input to make the implosion.

Lasers were thought to be the best way to deliver the energy to implode the fusion capsule and in one the first designs built by Lawrence Livermore National Laboratory (LLNL), the Shiva laser was the first attempt and failure at this technology. Shiva shown its lasers directly on the capsule in what is called, direct drive. It was this feature that introduced fusion scientists to a problem that would forever plague ICF efforts. [1]

Fig. 2: Diagram depicting the RT unstable interface
created in an ICF fuel capsule. The lighter material is
pushing on the heavier one. |

As the fuel capsule compresses, there is an instability which occurs, causing the once spherical geometry to become askew. The Rayleigh-Taylor instability (RTI) causes small bumps or perturbations on the sphere or interface of the capsule to grow, exponentially fast. [2] RTI occurs when a heavier or more dense substance is being pushed or accelerated by a lighter one. [3,4] This will be more rigorously defined later but a diagram for a spherical RTI and a planar RTI is shown in Fig. 1. In the case of ICF, the less dense outer material being burned away by the lasers, squeezes the capsule from all directions, and pushes on the more dense center. Small imperfections on the capsule's surface form instabilities.

In Shiva, lasers are directly shown on the capsule which become non-uniformly heated. These non-uniformities act to exacerbate the RTI even further. In fact, this was the main purpose for the failure of Shiva to produce a sustained fusion burn, or ignition.

Subsequent ICF system have been built and seek to ameliorate the development of RTI by uniformly heating the capsule by using Xrays. Both NOVA and NIF make use of a "hohlraum" (meaning "hollow area" in German), which encloses the spherical fusion capsule and is used to convert laser power to a blanket of xrays which heat the capsule more uniformly.

Rayleigh-Taylor Instability has thus become quite important in the for the application of Inertial Confinement Fusion and has, in fact, other astronomical implications as well. [5] Despite the vast complexity of the application in which this hydro-dynamic instability is present, the fundamental laws governing its behavior are beautifully encapsulated in the classical Navier-Stokes equations.

Where u is velocity of plasma, p is pressure, rho is density etc. The remainder of the terms are beyond the scope of this article. These equations are a quantitative model for conservation of mass, momentum and energy and solutions to these equations are rarely found in closed form. More often than not, computer simulation is needed to glean meaningful insight from more complex problems. However, before super-computers, scientists were able to use approximations to the equations of motion to make progress in understanding RTI. The famous astrophysicist, Subrahmanyan Chandrasekhar was one of the first to solve such a system analytically and found that very small perturbations, would grow as: [2,6]

Fig. 3: Simulation of RTI between two materials at
late time performed with the 3D Miranda code.
(With permission of the Lawrence
Livermore National Laboratory.) |

Fig. 4: Click on image to see 2D simulation showing
the rapid non-linear instability growth and transition to
turbulence. The code that generated this result is
available here. The plot is colored by
density, where white is heavy and black is light. |

The h is the height of the disturbance, α is a measure of the wavelength, and ρ are the densities of the fluid. The A is called the Atwood number, which is bounded between 0 and 1, and g is gravity or the acceleration. From this equation, we can see that for a given gamma, reducing the perturbation size has a diminishing effect of the amount of time the RTI can be held off. That is, if we reduce the initial height from h0 -> h1, we can determine how much time we save, Δ t, before a given height is reached. This time is given by Δ t / γ = log (p1/p2). Therefore, making the targets extremely void of imperfections, can only buy so much time before large perturbation break up the surface. This can be seen in the Figs. 2-3 which visualize a computer simulation of RTI which solves equations 1-4 above. One can see that the smooth interface, rapidly degrades into a turbulent. Although this is a simplified configuration of RTI, analogous dynamics are present in the spherical case.

For the plasma to stay contained long enough, ICF scientists must be aware of the time scales of the RT instability in their capsule design and make sure that the fusion is faster than the hydrodynamic instability. Otherwise, this instability, could yet again thwart efforts to achieve a sustained burn.

© Britton J. Olson. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1] C. Seife, *Sun in a Bottle* (Viking Penguin,
2008).

[2] P. G. Drazin, "The Stability of a Shear Layer in
an Unbounded Heterogeneous Inviscid Fluid," J. Fluid Mech. **4**, 214
(1958).

[3] Lord Rayleigh, "Investigation of the Character of
the Equilibrium of an Incompressible Heavy Fuid of Variable Density,"
Proc. Roy. Math. Soc. **14**, 170 (1883).

[4] G. I. Taylor, "The Instability of Liquid Surfaces
When Accelerated in a Direction Perpendicular to Their Plane," Proc.
Roy. Soc. London A **201**, 192 (1950).

[5] W. H. Cabot and A. W. Cook, "Reynolds Number
Effects on Rayleigh-Taylor Instability With Possible Implications for
Type-1a Supernovae," Nature Phys. **2**, 252 (2006).

[6] S. Chandrasekhar, *Hydrodynamic and
Hydromagnetic Stability* (Oxford U. Press, 1961).